Sihem, Maha, Wilna Sushi Camilla

Mme Sevahnee

Félix

Mélanie, Thibault, Ronan, Djanella, Christophe, Selma, Iris, Abdoul lakhad, Bass, Nico, Sarah Mariano Paulo

Viviane Roger

Katia

Stef

Parrain ! Leïla

Audrey

Babacar, Ahmadou, Moctar, Bill, Séverin Mr David

Charlotte Diaby

J'espère

Pape, Michel Christophe

Jean Louis

Jean Claude

Serge Rolland

Phillipe Marie-Claude Ahmed

Mati

Allyx

Thomas

Mme Olivier

Madame Allouache

Aouizerate

Corinne Luc

Madame Mr Criton

Ho-Coui-Youn J'espère

Mireille Dominique

Camille Hadrien

Craig

Rami

̃= Τ ̃1 + R

Mouhamet Diallo
email: mouhametdiallo@gmail.com

Antoine Primerose

Laurent Linguet

2 R 1 * R 2 Τ ̃1 + R 2 R 1 * R 2 R 1 * R 2 Τ ̃1 D

Keywords: stress, Mr steele, osef, espoir, Global horizontal irradiance, Global Forecast System (GFS), Integrated Forecast System (IFS), French Guiana, La Réunion, Singapore Global horizontal irradiance, radiative transfer parameterization, WRF, tropical zone, AROME

À cet effet je tiens à remercier mes directeurs de thèse : Mr Linguet, Mme Seyler. Je les remercie tout d'abord de m'avoir choisi pour mener à bout ce projet de recherche. Je leur exprime ma reconnaissance et les remercie ensuite pour le concours qu'ils ont apportés pour que cette thèse arrive à terme.

Pour ne froisser personne pas faire de jaloux je tiens à préciser que je cite chaque individu par date de connaissance et en cas de doute par ordre alphabétique! Donc si vous vous retrouvez plus bas dans la liste, vous n'aviez qu'à me rencontrer plus tôt  Je remercie ensuite mes collègues et amis doctorants de l'ird du cnrs : Marco, Tommy, Youven, Marjorie, Justine,

Résumé de la thése

La Guyane est un territoire d'outre-mer, situé en zone intertropicale (ZIT). Cette zone est le lieu de phénomènes de convections intenses. De ce fait, l'énergie solaire incidente au sol est très variable ce qui constitue un frein à son exploitation à grande échelle.

La question de recherche étudiée dans ce manuscrit est: comment peut-on améliorer les estimations et prédictions de rayonnement au sol en ZIT de façon à augmenter le taux de pénétration dans le réseau électrique de cette énergie renouvelable intermittente? Afin de répondre à cette question, nous avons utilisé deux outils. Le code Héliosat-II (HII) et le modéle de prévisions météorologiques Weather and research forecast (WRF). Nous avons utilisé ces outils de manière à améliorer les estimations et prévisions de rayonnement global au sol (IGH) dans la ZIT.

La première partie de ce manuscrit présente le contexte de la thèse. La seconde présente une modification d'H-II permettant d'améliorer les estimations d'IGH par une modélisation explicite de l'absorption de nuages. Ces estimations améliorées donnent ainsi des outils décisionnels permettant de situer au mieux une centrale solaire en fonction du potentiel solaire du site et des systèmes services avoisinants. La seconde partie traite dans un premier précision des prévisions des modèles globaux IFS et GFS (i.e integrated forecast system, global forecast system GFS) en ZIT. Ces produits téléchargés sont validés par comparaison avec des mesures in situ de trois pays situés dans la ZIT et caractérisés par des climats tropicaux. Cette étude permet de combler un vide dans l'étude des prévisions d'IGH des modèles globaux en ZIT. Nous proposons ensuite une méthode générique permettant de calibrer le modèle WRF en ZIT. Cette méthodologie vise à limiter le nombre de simulations à effectuer en sélectionnant et en faisant varier uniquement les paramètres ayant le plus d'influence sur le rayonnement au sol en ZIT. Pour valider cette méthodologie nous avons comparé les prévisions d'IGH du modelé WRF calibré avec celle du modelé AROME ainsi qu'avec des mesures in situ en Guyane. La quatrième partie présente l'utilisation d'une méthode hybride ensembliste variationnelle d'assimilation de donnée permettant d'améliorer les prévisions de rayonnements en ZIT. Cette méthode initialement utilisée pour améliorer la description de phénomènes convectifs extrêmes tels que prévision de la trajectoire des cyclones est pour la première fois appliquée pour améliorer les prévisions d'IGH. Cette méthodologie appliquée à la ZIT fournie alors des prévisions améliorées d'IGH permettant ainsi une gestion améliorée de centrale solaire.

Les travaux effectués dans la seconde section ont donnés lieu à une publication dans solar energy journal. Les travaux effectués dans la troisième section ont été soumis, mais non publiés à la date de [START_REF] Liou | An introduction to atmospheric radiation[END_REF] .............. Fig. II.5 schematic representation of scattering angle for an incoming ray (In) in the direction 𝛺(𝜇, 𝜙)that is deviated in the direction 𝛺′(𝜇′, 𝜙′) [START_REF] Liou | An introduction to atmospheric radiation[END_REF] The upward transmission and reflection of the i th layer are 𝛵𝑖 * and 𝑅𝑖 , respectively whereas the downward transmission and reflection are 𝛵𝑖 and 𝑅𝑖 * . 𝐷 and 𝑈 are the the combined total transmission and reflection functions between layers 1 and 2. Source [START_REF] Liou | An introduction to atmospheric radiation[END_REF] ........................... Fig. II.11 solar radiation flux at the atmosphere inspired after (Noia et al., 1993a) ............................. Fig. II.12 atmospheric model of [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] adapted from (Noia et al., 1993b) [START_REF] Barker | Hybrid Variational-Ensemble Data Assimilation[END_REF][START_REF] Kutty | Impact of Representing Model Error in a Hybrid Ensemble-Variational Data Assimilation System for Track Forecast of Tropical Cyclones over the Bay of Bengal[END_REF] 𝛽 a,λ : absorption coefficient 𝛽 e,λ : extinction coefficient 𝛽 s,λ : scattering coefficient 𝐶:cloud amount 𝑑 0 : innovation for the analysis at 𝑡 0 𝛿𝑥 ⃗(0): analysis increment 𝜀 : squared ratio between the mean earth-sun distance and the instaneous earth-sun 𝜀 𝑎 ⃗⃗⃗⃗⃗⃗: analysis error 𝜀 𝑏 ⃗⃗⃗⃗: background error 𝜀 𝑒 : the emissivity 𝜀 𝑚 ⃗⃗⃗⃗⃗⃗: model errors 𝜀 0 : observation error 𝐸 λ : irradiance 𝐻: observation operator 𝐻 ̃(𝑡 0 ): the linearization of the observation observatory at the analysis time 𝑡 0 𝐼𝑜 : Solar constant K: Kalman gain K c : clear sky index K t : clearness index 𝐿: radiance 𝑀(𝑥 ⃗): model operator 𝜇 cosines of solar zenith angle 𝜔 𝜆 : single scattering albedo 𝑃: analysis error covariance matrix 𝑄: model error covariance matrix 𝑅: observation error covariance matrix 𝜌 𝑐 : cloud albedo 𝜌 𝑔 : ground albedo 𝜌 𝑠,𝑑 : scattering coefficient for the upward diffuse radiation due to a Rayleigh atmosphere 𝜌 𝑠,𝑏 : scattering coefficient for the downward beam radiation due to a Rayleigh atmosphere Τ ̅ (𝜇 0 ): broadband atmospheric transmission function Τ: the transmissivity 𝑇: the temperature 𝜏 λ :optical depth 𝑥 ⃗: NWP model state vector 𝑥 𝑏 ⃗⃗⃗⃗⃗ : background state 𝑦 𝑛 : observation of the state variable over a given time interval n CHAPTER I : Thesis framework and research issue

List of figures

I. Introduction I.1 Technical challenges of solar energy for electricity production

The sun is a star with a 6.96 10 5 km radius and a surface temperature of 5800K [START_REF] Liou | An introduction to atmospheric radiation[END_REF]. According to Boltzmann law, it emits 3.94 10 26 W.s toward space. Earth is located on average 1.5 10 8 km from the sun; having a 6378.137 km radius [START_REF] Liou | An introduction to atmospheric radiation[END_REF], it receives 1.8 10 17 W.s. This energy is approximately 4000 times of the World's total primary energy consumption in 2016 (BP Statistical Review, 2017;Enerdata, 2017). First studies dealing with the conversion of solar energy into electricity date back to the mid-twenties (Albert [START_REF] Einstein | The photoelectric effect[END_REF]Bell labs, 1954;[START_REF] Fairley | Solar-Cell Squabble[END_REF]. However, at that time, the efficiency and cost of solar cells were not competitive with oil thermal energy. The interest in solar energy was renewed with the launch of the first satellites, as it was a reliable and renewable option for space applications [START_REF] House | History of satellite missions and measurements of the Earth Radiation Budget (1957-1984)[END_REF]. The combination of increasing energy needs [START_REF] Birol | World energy outlook 2010[END_REF], increasing concern for global warming [START_REF] Pachauri | Changements climatiques 2014: rapport de synthèse : contribution des Groupes de travail I, II et III au cinquième Rapport d'évaluation du Groupe d'experts intergouvernemental sur l'évolution du climat[END_REF], fossil fuel depletion [START_REF] Shafiee | When will fossil fuel reserves be diminished?[END_REF] and cost decrease [START_REF] Irena | renewable cost IRENA_Power_to_Change_2016[END_REF], has established solar energy as a viable solution to meet the needs. According to IEA (2017) the global average levelized cost of electricity (LCOE) from utility-scale solar photovoltaic (PV) projects declined by 70% from 2010 to 2016. Nevertheless, there are high discrepancies in the LCOE between regions. The LCOE in 2016 for solar PV (IRENA, 2018) was ~ 0.1 USD per kWh in Europe whereas in Africa it was ~0.17 USD per kWh. The LCOE in Africa for 2016 was higher than the fossil fuel cost range (i.e. [0.05:0.15] USD per kWh).

Regions with higher solar potential are located in the intertropical zone (ITZ) [START_REF] Löf | World distribution of solar radiation[END_REF][START_REF] Müller | Surface Solar Radiation Data Set -Heliosat (SARAH) -Edition[END_REF][START_REF] Trieb | Global Potential of Concentrating Solar Power[END_REF]. Nevertheless, the solar irradiance in the ITZ is highly variable due to the rapid cloud transformations that occur in this area [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF][START_REF] Rossow | Tropical Precipitation Extremes[END_REF][START_REF] Wheeler | Convectively Coupled Equatorial Waves: Analysis of Clouds and Temperature in the Wavenumber-Frequency Domain[END_REF]. Consequently, the solar energy available at the ground in this area is highly variable. This variability causes economical and technical challenges to fully exploit this resource.

I.2 Thesis framework and goals

French Guiana is a French territory located in the ITZ. This territory has fragile supply and demand equilibrium because it has to produce its energy locally. From 2009 to 2014 French Guiana's population increased on average of 2.4% each year (INSEE: Rémi Charrier, 2017). The 2030 demography projection Tab. I.1 shows that this trend will be magnified. Based on these projections, EDF the historical energy provider in France assessed that there will be an increase of electricity consumption that need to be addressed (Ministry of the environment, 2016a).

Tab. I.1Projection of population evolution and electricity needs. The electricity production is based on the BPEOD reference scenario. The multiannual energy plan-programmation pluriannuelle de l'énergie (PPE) (Ministry of the environment, 2016a) aims to meet the electricity needs using renewable energies (REs). Previous studies [START_REF] Fillol | Spatiotemporal indicators of solar energy potential in the Guiana Shield using GOES images[END_REF] showed that French Guiana has a high solar potential. The global horizontal irradiance (GHI) ranges between 1650 and 2000 kWh/kWc/year; the direct normal irradiance (DNI) ranges between 1850 and 2250 kWh/kWc/year. Therefore, solar energy will be a driving force to meet the growing demand in electricity. In 2014, solar photovoltaic has a 6% (48 GWh) share in French Guiana electrical mix with an installed capacity of 34 MW (Ministry of the environment, 2016a). The goal set for French Guiana is to increase this installed capacity by 111.7% by 2023. As an intermittent RE source, the integration of high loads of solar photovoltaic output into the electricity grid introduces instabilities (Clean Energy Council, 2017;IER, 2013;Zoulias: CRES, 2016). This issue led the government to set in 2018 a 35% penetration threshold for intermittent RE sources. In 2012, a penetration rate of 22% was already reached in French Guiana [START_REF] Dambreville | Prévision du rayonnement solaire global par télédétection pour la gestion de la production d'énergie photovoltaïque[END_REF]. Increasing the intermittent REs installed capacity may, on the one hand, provide enough electricity but on the other may destabilize the grid. Various solutions were suggested to increase the penetration of intermittent REs and ensure, at the same time, grid safety. These solutions can be classified into five categories (Zoulias: CRES, 2016):

-use thermal fossil fuel power plant -update the electricity grid -use energy storage systems -use geographical information systems -forecast the incoming solar energy

The first solution deals with the use of fossil fuel power plants as backup production systems. As the electricity production from REs depend on the weather conditions, it cannot meet user's needs consistently. Fossil fuel power plants have low inertia, they can increase or decrease (i.e. ramp) the electricity production quickly to meet user's needs and ensure grid stability (Ministry of the environment, 2016a). In 2014, thermal fossil fuel represents in French Guiana an installed capacity of 127.4 MW and this capacity is to increase up to 160 MW by 2030 (Ministry of the environment, 2016a).

The second solution deals with extending the grid, improving transport lines and incorporating grid elements with "smart" functionalities in order to balance supply and demand. Extending the grid allows importing and exporting power with the neighboring countries. This extension helps in maintaining the equilibrium between supply and demand, which in turn helps in stabilizing the grid. Inverters are grid devices that feed solar power into the grid, converting the power from direct current to alternating current [START_REF] Táczi | System Effects of Intermittent Renewable Generators (Wind, Solar)-Balancing[END_REF]. These devices cut off the solar plant output to the grid if they detect an abnormal state. As solar energy becomes more and more important if all grid inverter cut off all the solar plant output, it can result in system instabilities. Therefore, each inverter of the grid should vary on its own, adapting the cut-off frequency to grid state [START_REF] Irena | renewable cost IRENA_Power_to_Change_2016[END_REF]. To best of the author knowledge this technology has not yet been implemented in French Guiana. Fig. I.1 shows the 2014 French Guiana's electricity grid; it is 414 km long and is not connected to neighboring countries. Grid lines extend from Saint Laurent to Cayenne. Feasibility studies of grid extension to neighboring countries are planned for 2018 (Ministry of the environment, 2016a).

The third solution is to increase the penetration of intermittent renewable energy into the electricity grid and ensure grid safety by using energy storage systems [START_REF] Irena | renewable cost IRENA_Power_to_Change_2016[END_REF]. Energy storage systems allow storing electricity from variable renewable generation when the production exceeds the demand [START_REF] Irena | Renewable Energy Integration in Power Grids[END_REF]. The energy previously stored can then be supplied upon demand, when renewable energy production is not available or insufficient. Energy storage systems increase system flexibility and provide supply security. In 2014, the available energy storage systems in French Guiana are distributed amongst the 2 PV plant of 5 MW installed capacity. By 2023, out of the 53 MW intermittent REs installed capacity, 45 MW will be backed up by energy storage systems. However, energy storage systems require high investments. The cost of a PV facility with backup production systems is estimated to be 3100 k€/MW whereas the cost of a PV facility without backup is estimated to be 2900 k€/MW (Ministry of the environment, 2016a).

The fourth solution deal with the use of Geographical Information Systems (Zoulias: CRES, 2016). This tool is used to identify the most suitable locations to install REs power plants with respect to both energy potential and surrounding facilities that favor grid stability.

The fifth solution is to forecast the solar energy available to solar power plants in order to increase the penetration of solar energy into French Guiana's electricity grid.

In this thesis we consider the fourth and fifth solutions applied in the ITZ.

II. Manuscript outline

This thesis dissertation aims to answer the following scientific issue: how could the solar irradiance be assessed and forecasted in French Guiana to increase the penetration rate of this intermittent renewable energy into the electricity grid? Weather conditions in French Guiana are driven by the physical phenomena in the ITZ; consequently, methods applied in French Guiana can be extended to all the ITZ.

To answer this scientific issue, we use two tools: Heliosat-II method (H-II) and weather and research forecast (WRF) numerical weather prediction model (NWP). We used these tools in order to produce improved GHI estimates in French Guiana and the ITZ.

The first chapter introduces the thesis and the research issue. The second chapter presents a modification to H-II; with this modification H-II can account for cloud absorption. The GHI estimates from modified H-II provide tools for decision making in French Guiana. These tools allow identifying the most suitable locations to install solar facilities in French Guiana with respect to both solar potential and surrounding facilities favoring grid stability. In the third chapter we first study the accuracy of the GHI forecasts from integrated forecast system (IFS) and global forecast system (GFS) NWP models in French Guiana, La Reunion and Singapore. We validate the accuracy of these downloaded products by comparison with ground measurements from these three territories located in the ITZ that have tropical climate. This study aims to fill the gaps in the accuracy of global NWP model in the ITZ. Secondly, we propose a methodology to calibrate WRF to produce improved GHI forecasts in the ITZ. The goal is to restrain and select the minimum number of simulations to run, to obtain improved GHI forecasts compared to a non-calibrated NWP model. This methodology to calibrate WRF is validated in French Guiana by comparison with the GHI forecasts of AROME and ground measurements. The fourth chapter deals with the use of an hybrid 3D variational (3D-Var) ensemble transform Kalman filter (ENTKF) to further improve the GHI forecasts of WRF calibrated for French Guiana. This methodology originally used in the tracking of extreme convection events such as cyclones is applied for the first time for GHI forecasts. Its application in French Guiana allows us to obtain an improved GHI forecast which makes monitoring the electricity production from solar facilities easier. In the fifth chapter, we summarize and conclude thesis.

The research undergone in the second chapter is published in solar energy journal whereas the research on IFS, GFS, WRF and AROME from chapter three were submitted to journal for publication but were not published yet at the time of thesis dissertation submission.

Note:

In this manuscript we aimed to remain consistent with notations and abbreviations. Consequently, notations and abbreviations are explained only when they are first introduced. For later mention to these notations and abbreviations, the reader is invited to refer to the nomenclature and definition sections.

CHAPTER II : Improving ground irradiance assessment in the intertropical zone

I. Modeling and measuring the solar irradiance at the ground

The solar irradiance at the ground that may be converted into electricity is a function of the irradiance at the outer edge of the atmosphere and the atmosphere constituents as seen in Fig. II.1. To describe this quantity we first define relevant radiometric and geometric variables. Second we describe how the outer edge of the atmosphere solar irradiance is computed. Third we discuss the measurement devices for the global component of the solar radiation and quality control procedures. Last we discuss the modeling of the solar radiation at the ground and the metrics used to validate the accuracy of a model.

Fig. II.1 Earth radiation budget credit American meteorological society (AMS).

I.1 Concepts and definitions

I.1.a Solar constant I o

The solar constant 𝐼 0 is the total amount of solar energy reaching the top of the atmosphere at the mean distance between the sun and the earth across a surface of unit area normal to the solar beam. The solar constant is computed from the energy 𝐹 0 emitted from the sun using an energy conservation principle [START_REF] Liou | An introduction to atmospheric radiation[END_REF]:

I 0 = F 0 ( r s r o ) 2 , II.1
where 𝑟 𝑠 and 𝑟 𝑜 are the sun radius and earth-sun mean distance respectively; these variables are expressed in m. 𝐹 0 is the energy emitted from the sun; its value is given by the Boltzmann law for black bodies. 𝐹 0 and 𝐼 0 are expressed in W/m². The ratio 𝜀 between the mean earth-sun distance and the instantaneous earth-sun distance is the sun correction factor, its approximation can be found in [START_REF] Mather | Computer processing of remotely-sensed images: an introduction[END_REF].

I.1.b Solar zenith angle 𝜽 𝟎 , 𝝁 𝟎

The solar zenith angle is the angle between the local zenith and the center of the sun disc. It is a function of the hour angle h, the sun declination 𝛿 and the local latitude ϕ and writes: The azimuth ∅ is the angle between a reference direction (here the north) and a line from the observer to a point of interest (here the sun) projected on the same plane as the reference direction orthogonal to the zenith.

I.1.d Radiance 𝑳, radiation

The monochromatic radiance 𝐿 λ or luminance is the amount of radiant energy 𝑑𝐸 λ emitted by an element of area 𝑑𝐴 in a specified wavelength interval [𝜆, 𝜆 + 𝑑𝜆] , in a direction 𝑑𝛺 and during a 𝑑𝑡 time span. The radiance 𝐿 λ is an instantaneous value, it is expressed in W/m 3 .sr.

L λ = dE λ cosθ dΩ dλ dt dA

II.3

Radiation is the accumulation of radiance over a time span. If the radiance is monochromatic, it is expressed in J/m 3 .sr. The total or polychromatic radiance (radiation) is the integration over all wavelengths of the monochromatic radiance (radiation); it is expressed in W/m 2 .sr (J/m 2 .sr).

I.1.e Irradiance 𝑬, Irradiation

The monochromatic irradiance or emittance is the radiant flux received from all directions by an element of area 𝑑𝐴. The radiance 𝐸 λ is an instantaneous value; it is expressed in W/m 3 .

E λ = ∫ L λ Ω cosθ dΩ II.4
Irradiation is the accumulation of irradiance over time. If the irradiance is monochromatic it is expressed in J/m 3 . The total or polychromatic irradiance (irradiation) is the integration over all wavelengths of the monochromatic irradiance (irradiation); it is expressed in W/m 2 (J/m 2 ).

I.1.f Top of atmosphere irradiance, G TOA

The top of atmosphere irradiance (𝐺 𝑇𝑂𝐴 ) is the solar flux density at the top of the atmosphere when the instantaneous and mean earth-sun are 𝑟 𝑠 and 𝑟 𝑜 respectively, and the sun is located by the solar zenith angle 𝜃 0 in radians.

G TOA = I 0 μ 0 II.5

G TOA and I 0 are expressed in W/m 2 .

I.1.g Global horizontal irradiance, Global horizontal irradiation

The global horizontal irradiance (GHI) is the irradiance 𝐸 λ received on a horizontal surface located on the ground. [START_REF] Liou | An introduction to atmospheric radiation[END_REF] The global horizontal irradiance is composed by two components: the irradiance received directly in a straight line from the sun on a normal surface (DNI) and the irradiance received on the ground after multiple scattering in the atmosphere (DHI).

Fig. II.4 schematic representation of angles involved in radiometric quantities

𝐺𝐻𝐼 = 𝑐𝑜𝑠(𝜃 0 ) . 𝐷𝑁𝐼 + 𝐷𝐻𝐼 II.6 The global horizontal irradiation is the accumulation of GHI over a time span. The GHI and global horizontal irradiation are expressed in W/m 2 and J/m 2 respectively.

I.1.h Clear sky model 𝑮𝒄

A clear sky model 𝐺𝑐 approximates the maximal irradiance value available at the ground under cloudless sky conditions; it accounts for the depletion caused by aerosols and water vapor. 𝐺𝑐 always depends on the solar zenith angle; it also requires a varying number of other inputs data that describe the local atmospheric conditions. According to Gueymard (2012a) these input parameters are typically less than or equal to eight.

I.1.i Clearness index 𝑲 𝒕 and Clear sky index 𝑲 𝒄

The clearness index 𝐾 𝑡 is the ratio of the GHI to the irradiance outside the atmosphere. The clear sky index 𝐾 𝑐 instead divides the GHI by the clear sky atmosphere irradiance. Both 𝐾 𝑐 and 𝐾 𝑡 serve as proxies for the atmosphere transmission factor; these dimensionless numbers are expressed as follows:

K t = GHI G TOA 𝑎𝑛𝑑 II.7 𝐾 𝑐 = 𝐺𝐻𝐼 𝐺𝑐 . II.8
Clear sky conditions are defined for K c > 0.65 or K t > 0.7; cloudy sky conditions are defined for 0.4 < K c < 0.65 or 0.2 < K t < 0.7 and overcast conditions for K c < 0.4 or K t < 0.2 [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF][START_REF] Yousif | Comparison of solar radiation in Marsaxlokk, Malta and Valladolid, Spain[END_REF]. There is no consensus whether to use clear sky index or clearness index [START_REF] Smith | Cloud cover effect of clear-sky index distributions and differences between human and automatic cloud observations[END_REF]. The clear sky index allows the removal of diurnal and seasonal signals from a given set of radiation data [START_REF] Langella | Solar Radiation Forecasting, Accounting for Daily Variability[END_REF], besides it is less dependent to the air mass than the clearness index. Nevertheless, the clear-sky index subject to many errors due to the clear sky model input parameters.

I.1.j Albedo, reflectance and reflectivity

The monochromatic bidirectional reflectance is defined as the ratio of the radiant flux reflected by a surface to the incoming radiant flux [START_REF] Schaepman-Strub | Reflectance quantities in optical remote sensing-definitions and case studies[END_REF]. It is a dimensionless number and is expressed mathematically as follows:

ρ(T, λ, θ i , ϕ i , θ r , ϕ r ) = dL r (T, λ, θ r , ϕ r ) dL i (T, λ, θ i , ϕ i ) .

II.9

Where 𝜃 𝑖 and 𝜙 𝑖 give the direction of the incoming radiance 𝑑𝐿 𝑖 ; 𝜃 𝑟 and 𝜙 𝑟 the direction of the reflected radiance 𝑑𝐿 𝑟 . T is the medium temperature. According to the International Commission on Illumination CIE (2011), reflectivity is distinguished from reflectance by the fact that reflectivity is a value that applies to thick reflecting objects.

The albedo is the ratio of the radiant flux reflected from a unit surface area into the whole hemisphere to the incident radiant flux of hemispherical angular extent. Consequently, in Eq.II.9, the albedo is defined when 𝜃 𝑟 = ϕ r = 2𝜋 [START_REF] Schaepman-Strub | Reflectance quantities in optical remote sensing-definitions and case studies[END_REF].

When the term surface reflectance is used for remote sensing it implies that narrow waveband or single wavelength is considered. Surface albedo, on the other hand, is used as a substitute for the integrated hemispherical albedo, of several wave-lengths [START_REF] Duguay | Estimating Surface Reflectance and Albedo from Landsat-5 Thematic Mapper over Rugged Terrain[END_REF].

I.1.k Absorptance, absorption coefficient, absorptivity

The absorptance is defined as the ratio of the radiant flux absorbed by a surface to the incoming radiant flux. It is a dimensionless number and is expressed mathematically as follows:

α(T, λ, θ i , ϕ i ) = dL a (T, λ, θ i , ϕ i ) dL i (T, λ, θ i , ϕ i ) . II.10
According to [START_REF] Boulet | On the Influence of the Sample Absorptivity when Studying the Thermal Degradation of Materials[END_REF] absorptance and absorption coefficients are inter changeable; yet absorptance refers to volumetric properties whereas absorptivity is used for surface properties. However, as showed by the literature review of [START_REF] Hu | Absorbance, absorption coefficient, and apparent quantum yield: A comment on common ambiguity in the use of these optical concepts[END_REF] there are still ambiguities in the usage of these optical terms.

I.1.l The emissivity

The emissivity is defined as the ratio of the ratio of the radiance of a specific object or surface to that of a standard black body. It is a dimensionless number and is expressed mathematically as follows:

ε e (T, λ, θ, ϕ) = dL(T, λ, θ, ϕ) dL 0 (T, λ) . II.11

I.1.m Scattering

Scattering (diffuse reflection) similarly to reflection redirects an incoming radiation stream. Contrary to the reflection, there is a total absorption and emission of the incoming particle or photon. Scattering phenomenon happens when the surface is rough relative to wavelengths whereas reflection happens when the surface is smooth. To characterize the angular distribution of the scattered particle or photon the phase function 𝑝 is used. It describes photons with initial direction Ω′ ⃗⃗⃗⃗ (𝜇 ′ , 𝜙 ′ ) that follow the direction Ω ⃗⃗⃗ (𝜇, 𝜙) after single or multiple scattering such that Eq. II.12 is verified.

∫ p (λ, Ω ⃗⃗⃗ , Ω′ ⃗⃗⃗⃗ , t) 4π dΩ ′ = 1.

II.12

Fig. II.5 schematic representation of scattering angle for an incoming ray (In) in the direction

𝛺 ⃗⃗ (𝜇, 𝜙)that is deviated in the direction 𝛺 ′ ⃗⃗⃗⃗⃗ (𝜇 ′ , 𝜙 ′ ) [START_REF] Liou | An introduction to atmospheric radiation[END_REF] For spherical geometry, the scattering angle is related to the incoming and outgoing directions following Eq.II.13 as shown Fig.

II.5 cosΘ = μμ ′ + (1 -μ 2 ) 1 2 (1 -μ ′2 ) 1 2 cos(ϕ ′ -ϕ).
II.13

I.1.n The transmittance and transmissivity

The transmittance or transmissivity Τ is the ratio of the radiant flux directly transmitted after passing through a medium (atmosphere) to the amount of radiant flux that would have passed the same distance through a vacuum. It is the amount of light that remains after the absorption and scattering by the media. Τ is related to the absorptance and reflectance as follows:

Τ + ρ + α = 1. II.14

I.2 Modeling the extraterrestrial irradiance

The top of atmosphere irradiance (𝐺 𝑇𝑂𝐴 ) is a function of astronomical geometric parameters [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas -with respect to the heliosat method[END_REF], it is defined Eq.II. 5. Fig. II.6 shows the spectral distribution of the top of atmosphere irradiance.

Fig. II.6 Solar radiation spectrum credit geosciencebigpicture

The accumulation over time of the top of atmosphere irradiance is the top of atmosphere irradiation. Its expression can be found below:

G TOA,t = ∫ G TOA dt sunset sunrise .

II.15

Firstly, substituting the solar zenith angle expression Eq.II.2 and II.5 into Eq.II.15; secondly, expressing the time in function of the hour angle and earth angular velocity; lastly, assuming that the variation of the earth-sun distance in one day can be neglected yields [START_REF] Liou | An introduction to atmospheric radiation[END_REF]:

G TOA,t ≅ Io ε π (sinφ sinδ H + cosφ cosδ sin H),

II.16

where H represents a half-day expressed in radians. 𝐺 𝑇𝑂𝐴,𝑡 is expressed in J/m².

I.3 Instruments for measuring the GHI and quality control procedures

In manuscript only the global horizontal (GHI) component of the solar irradiance is studied. We do not consider the direct normal (DNI) and the diffuse horizontal parts (DHI) because the French weather services in French Guiana do not measure them.

I.3.a Instruments for measuring the GHI

The GHI is measured by two types of radiometers: thermopile and photodiode (e.g photoelectric or silicon) pyranometers [START_REF] Driesse | Investigation of pyranometer and photodiode calibrations under different conditions[END_REF][START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF][START_REF] Myers | Solar radiation: practical modeling for renewable energy applications[END_REF][START_REF] Sengupta | Best Practices Handbook for the Collection and Use of Solar Resource Data for Solar Energy Applications[END_REF]. They differ from each other by the type of sensor they use to infer the amount solar irradiance received.

Thermopile pyranometers measure indirectly the irradiance by quantifying the temperature difference between two surfaces: one absorbing the other non-absorbing. These surfaces are in contact with two metals with different heat capacities called thermocouple. When the absorbing surface of the thermopile is heated by the incident solar irradiance it creates a thermal flux upon the thermocouple junction. Therefore, this junction produces a voltage proportional to the difference in temperature [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF][START_REF] Myers | Solar radiation: practical modeling for renewable energy applications[END_REF]. A commonly used combination is the T type thermocouple, consisting of junctions of copper and constantan; it produces about 40 microvolts per degree centigrade.

Photodiode pyranometers measure indirectly the irradiance by quantifying the photocurrent generated in response to an incident irradiance flux. Photodiode pyranometers are less expensive and have a lower response time than thermopile pyranometers. The response time of thermopile pyranometers ranges between 1 and 240s [START_REF] Sen | Solar Energy Fundamentals and Modeling Techniques[END_REF] whereas the response time of photodiode pyranometers is approximately 10µs [START_REF] Patil | PHOTODIODE BASED PYRANOMETER[END_REF]. Nevertheless, photodiode pyranometers are less accurate than most thermopile pyranometers [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF]. A large part of photodiode pyranometers inaccuracies are related to their narrow spectral responses and or related to calibration issues [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. Thermopile pyranometers are sensitive to the whole shortwave spectrum whereas photodiode pyranometers have a narrow spectral sensitivity range: 350-1000 nm [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF]. The most common solid-state detector used on solar photodiodes is crystalline silicon [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF][START_REF] Myers | Solar radiation: practical modeling for renewable energy applications[END_REF].

According to [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF], [START_REF] Younes | Quality control of solar radiation data: Present status and proposed new approaches[END_REF], [START_REF] Myers | Solar radiation: practical modeling for renewable energy applications[END_REF] and [START_REF] Sengupta | Best Practices Handbook for the Collection and Use of Solar Resource Data for Solar Energy Applications[END_REF] the measurement uncertainty of pyranometers are associated with: the calibration of the instrumentation, the data acquisition equipment, and the data processing step. The uncertainties related to the calibration process include the pyranometer spectral response, temperature response, non-linearity, aging and longwave radiation errors. The uncertainties related to the data acquisition include the error associated with the measurement of the sensor signals and environmental influences such as the cosine, azimuth and zenith response. The Uncertainties due to data processing results for instance from the average of sub-hourly irradiance measurements to hourly measurements.

Fig. II.7 thermopile (left) and photodiode (right) pyranometer. Source Wikipedia, Licor

Based on these criteria, the World Meteorological Organization (WMO) classifies pyranometers into three categories based on their accuracies: secondary standard, first class and second class. Tab. II.1 shows the characteristic of each category. [START_REF] Sen | Solar Energy Fundamentals and Modeling Techniques[END_REF]

Tab. II.1 WMO classification of pyranometers, extracted from

Characteristics

Secondary standard Non linearity (% of full scale) ±0.5 ±2 ±5 Spectral sensitivity (% deviation from mean absorbance 0.3 -3μm )

±2

±5 ±10

Response time (99% response) <25s <60s <240s

In French Guiana, French weather services use a First class pyranometers. They are thermopile of brand Kipp & zonen type CM6B.

I.3.b Quality control procedures

Before comparing global NWP GHI forecasts to ground measurements, one should quality control (QC, ore quality check) the ground data used as reference. QC procedures aim to detect faulty measurements related to calibration of the instrumentation, data acquisition equipment, and or data processing. QC procedures can be classified into four categories [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF]:

-QC based on redundancy -QC based on limits within range -step check QC -and consistency QC.

QC based on redundancy use two instruments for measuring the same meteorological variable. These measurements from two different instruments are then compared and should be equal within the coupled uncertainty of both instruments. However, this approach is expensive; measurement sites do not often have several instruments that measure simultaneously the GHI. QC based on limits within a range makes sure that GHI measurements comply to physical limits and statistical knowledge. Statistical knowledge allows accounting for rare observation cases [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF].

Step checks QC aims to detect unrealistic jumps in values from two consecutive GHI measurements or stagnation within a given time interval. Consistency QC consists in measuring simultaneously at the same location the three components of the solar radiation, i.e. GHI, DNI, DHI (Eq.II.6) and verifying their inner consistencies. To achieve maximum benefit the WMO combining these QC procedure starting by the QC based on limits within a range then step and consistency check [START_REF] Wmo | WMO Guide to the Global Observing System[END_REF].

In French Guiana the GHI is measured by French weather services (Météo-France) every 15 min, then accumulated to hourly values. These hourly values are then stored and released to the users. As subhourly measurements are not available in 2016, this QC procedure is not applicable. Besides, only the GHI is measured and only one pyranometer is available in each measurement sites; therefore, the redundancy and consistency QC cannot be applied.

Tab. II.2 shows the different types of limits within range QC procedures found in the literature. Lower and higher limit values apply to daylight time: between sunrise and sunset. This table shows that [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF] and Helioclim [START_REF] Younes | Quality control of solar radiation data: Present status and proposed new approaches[END_REF] are the most constrained QC. They share the same lowest GHI value; however, their highest observable values differ. The lowest value was set up according to the analysis of collected data and clearness index value from the European Solar Radiation Atlas for solar altitude greater than 2° [START_REF] Geiger | A web service for controlling the quality of measurements of global solar irradiation[END_REF]. Helioclim [START_REF] Younes | Quality control of solar radiation data: Present status and proposed new approaches[END_REF] maximum observable value involves the use of a clear sky model [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas -with respect to the heliosat method[END_REF]. [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF] higher limit is defined as the minimum between Baseline Surface Radiation Network (BSRN) [START_REF] Long | BSRN recommended QC tests V2[END_REF] and [START_REF] Muneer | Quality control of solar radiation and sunshine measurements -lessons learnt from processing worldwide databases[END_REF] based limits. It has the advantage of accounting for rare observations using a constant.

Tab. II.2 literature review of limits within range QC procedure

QC

Minimum range

Maximum range [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF] 0.03 𝐺𝐻𝐼 𝑇𝑂𝐴 min (1.2 𝐼 0 , 1.5 𝐼 0 cos(𝜃 𝑧 ) BSRN [START_REF] Long | BSRN recommended QC tests V2[END_REF] -4 1.5 𝐼 0 cos(𝜃 𝑧 ) 1.2 + 100 WMO (WMO, 2007) 0 1600 Younes [START_REF] Younes | Quality control of solar radiation data: Present status and proposed new approaches[END_REF] none 𝐺𝑐

Most QC procedures shown Tab. II.2 have restrictions on the minimum solar elevation angle they should be used for. [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF] and Helioclim [START_REF] Younes | Quality control of solar radiation data: Present status and proposed new approaches[END_REF] QC are valid for solar altitude greater than 2° [START_REF] Geiger | A web service for controlling the quality of measurements of global solar irradiation[END_REF]. CIE [START_REF] Younes | Quality control of solar radiation data: Present status and proposed new approaches[END_REF], [START_REF] Muneer | Quality control of solar radiation and sunshine measurements -lessons learnt from processing worldwide databases[END_REF] QC are valid only for solar elevation greater than 4°. Younes [START_REF] Younes | Quality control of solar radiation data: Present status and proposed new approaches[END_REF] QC is the most restrictive, it assumes that the solar elevation is greater than 7°. Restrictions on the solar elevation aim to limit the cosine effect error at sunrise and sunset. The more acute the angle of the sun (at sunrise and sunset), the greater the error associated with the cosine effect is. Besides, at low sunrise and sunset angle the change of GHI value might be too small to be detectable change by pyranometers [START_REF] Sen | Solar Energy Fundamentals and Modeling Techniques[END_REF].

When disproportional number of observations are rejected by the QC it may either be an indication of strongly biased data that should not be considered as outliers or an indication that the QC is not adequate and should be further studied and possibly modified [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF].

I.4 Modeling the solar irradiance depletion due to the atmospheric constituent: the radiative transfer equation

The following equation is the radiative transfer equation (RTE); it describes the extinction and generation process of radiance through the atmosphere path [START_REF] Liou | An introduction to atmospheric radiation[END_REF][START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF].

𝑐

𝜕𝐿 𝜆 (𝑟 ⃗, 𝛺 ⃗⃗ , 𝑡) 𝜕𝑡 + (𝛺 ⃗⃗ . 𝛻 ⃗⃗ )𝐿 𝜆 (𝑟 ⃗, 𝛺 ⃗⃗ , 𝑡) = -𝛽 𝑒,𝜆 𝐿 𝜆 (𝑟 ⃗, 𝛺 ⃗⃗ , 𝑡) + 𝛽 𝑒,𝜆 𝐽 𝜆 (𝑟 ⃗, 𝛺 ⃗⃗ , 𝑡), II.17

where 𝐿 λ (𝑟 ⃗, Ω, t) is the monochromatic radiance in the position 𝑟 ⃗ from direction Ω ⃗⃗⃗ at a time t.

The first term of the left-hand side (lhs) describes the radiance time evolution within a domain whereas the second term of the lhs describes the radiance exchange through the domain boundaries. The first term of the right hand side (rhs) describes the extinction causes by the absorption and scattering (Eq.II.18) whereas the second term of the rhs describes the generation of radiance by source terms (Eq.II.19). The extinction coefficient β e,λ and radiance source term J λ are expressed as follows:

𝛽 𝑒,𝜆 = (𝛽 𝑠,𝜆 + 𝛽 𝑎,𝜆 ) 𝑎𝑛𝑑 II.18

J λ (r ⃗, Ω ⃗⃗⃗ , t) = β s,λ 4π β e,λ ∫ L λ (r ⃗, Ω, t) p(λ, Ω ⃗⃗⃗ , Ω ′ ⃗⃗⃗⃗⃗ )d Ω ′ ⃗⃗⃗⃗⃗ Ω ′ + β a,λ β e,λ B λ (r ⃗, Ω ⃗⃗⃗ , t).

II.19

Where 𝛽 s,λ and 𝛽 a,λ are the scattering and absorption coefficients respectively. The source term J λ includes photons scattered inward the domain and photons emitted by the medium. In Eq.II.19, 𝐵 𝜆 (𝑟 ⃗, Ω, t) is the radiant flux emitted by the atmosphere, mostly in the thermal infrared [START_REF] Liou | An introduction to atmospheric radiation[END_REF]; 𝑝 (𝜆, Ω ⃗⃗⃗ , , Ω′ ⃗⃗⃗⃗ ) is the phase function defined Eq.II.12.

The radiance stationarity approximation is often made to simplify the RTE [START_REF] Liou | An introduction to atmospheric radiation[END_REF][START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF]. Applying these assumptions to Eq.II.17 yields:

(Ω ⃗⃗⃗ . ∇ ⃗ ⃗⃗ )L λ (r ⃗, Ω ⃗⃗⃗ ) β e,λ = -L λ (r ⃗, Ω ⃗⃗⃗ ) + ω λ 4π ∫ L λ (r ⃗, Ω ⃗⃗⃗ ) p(λ, Ω ⃗⃗⃗ , Ω ′ ⃗⃗⃗⃗⃗ )d Ω ′ ⃗⃗⃗⃗⃗ Ω ′ + (1 -ω λ )B λ (r ⃗, Ω ⃗⃗⃗ ), II.20 where, 𝜔 𝜆 = 𝛽 s,λ 𝛽 e,λ
is the single scattering albedo. The scattering term may be written as a sum of single scattering and multiple scattering terms.

ω λ 4π ∫ L λ (r ⃗, Ω ⃗⃗⃗ ) p (λ, Ω ⃗⃗⃗ , Ω′ ⃗⃗⃗⃗ ) d Ω′ ⃗⃗⃗⃗⃗ Ω ′ = ω λ 4π Fo p (λ, Ω ⃗⃗⃗ 0 , Ω′ ⃗⃗⃗⃗ ) e -τ λ + ω λ 4π ∫ L λ (r ⃗, Ω′ ⃗⃗⃗⃗ ) p (λ, Ω ⃗⃗⃗ , Ω′ ⃗⃗⃗⃗ ) d Ω′ ⃗⃗⃗⃗⃗ Ω ′ ,

II.21

where Ω ⃗⃗⃗ 0 shown Fig. II.9 gives the direction of incoming TOA radiation 𝐹𝑜 (Fig. II.9; Eq.II.1).For computations speed purposes the atmosphere is considered as plane-parallel in localized portions [START_REF] Liou | An introduction to atmospheric radiation[END_REF]. The plane-parallel atmosphere hypothesis considers the atmosphere as a sum of vertical atmospheric layers. Each layer is characterized by homogeneous properties and bordered by the bottom and top infinite plates called boundaries.

Fig. II.8 Plane Parallel atmosphere

Applying these assumptions to Eq.II.20 and assuming the angular symmetry 𝜃 of the radiance yields [START_REF] Fouquart | Intercomparing shortwave radiation codes for climate studies[END_REF][START_REF] Liou | An introduction to atmospheric radiation[END_REF][START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF]:

μ dL λ (τ, μ, ϕ) dτ = L λ (τ, μ, ϕ) - ω λ(τ) 4π F 0 p(τ, μ 0 , ϕ 0 , μ ′ , ϕ ′ )e -τ λ μ - ω λ(τ) 4π ∫ ∫ L λ (τ, μ ′ , ϕ ′ ) 1 -1 p(τ, μ, ϕ, μ ′ , ϕ ′ ) dμ ′ dϕ ′ 2π 0 + (1 -ω λ (τ))B λ (τ, μ), II.22
where 𝜇, 𝜇 ′ and 𝜇 0 angles are defined Fig. II.9. τ is the optical depth, it is expressed as follows:

τ(z) = ∫ -β e,λ z 2 z 1
dz.

II.23

Eq.II.22 describes the effect of the optical constituents of the atmosphere (cloud water droplets, cloud ice crystals, water vapor, ozone, aerosols, carbon dioxide and other minor trace gases) on the transfer of radiance for each atmospheric layer 𝑑𝜏.

A common approximation to Eq.II.22 is to consider that the azimuthal dependence of the radiance can be neglected; then we can define the azimuthally average radiance and phase function as follows [START_REF] Liou | An introduction to atmospheric radiation[END_REF]:

L λ (τ, μ) = 1 2π ∫ L λ (τ, μ, ϕ)dϕ 2π 0 𝑎𝑛𝑑 II.24 p(τ, μ, μ ′ ) = 1 2π ∫ p(τ, μ, ϕ, μ ′ , ϕ ′ )dϕ ′ . 2π 0

II.25

For simplicity Eq. II.22 can be split in two uncoupled equations, one that describes the beam component of the radiation 𝐿 𝜆,𝑏(𝜏,𝜇) and the other the diffuse component 𝐿 𝜆,𝑑(𝜏,𝜇) . They are written as follows [START_REF] Liou | An introduction to atmospheric radiation[END_REF]:

μ dL λ,b (τ, μ) dτ = L λ,b (τ, μ) 𝑎𝑛𝑑 II.26 μ dL λ,d (τ, μ) dτ = L λ,d (τ, μ) - ω λ(τ) 2 F 0 p(τ, μ 0 , μ)e -τ λ μ - ω λ(τ) 2 ∫ L λ,d (τ, μ)p 1 -1 (τ, μ, μ ′ ) dμ ′ + (1 -ω λ (τ))B λ (τ, μ).

II.27

The beam component equation can be solved analytically; its solution is written as follows:

L λ,b (τ) = μ 0 I 0 e -τ/μ 0 , II.28
where 𝐿 0 is the TOA radiance and 𝜇 0 is the azimuth angle of the incoming solar radiation.

A further simplification made to solve the diffuse component of RTE Eq.II.27 is to separate the contributions from shortwave and the contribution from longwave radiations. The overlap between these ranges is relatively small which allows treating the two types of radiative transfer and their source functions separately. [START_REF] Stephens | The parameterization of radiation for numerical weather prediction and climate models[END_REF] indicates that Rayleigh scatter is dominant only for the shorter wavelengths, while liquid water absorption in clouds occurs only for the longer wavelengths. The transfer of shortwave radiation is not as complex as longwave radiation, since the problem of the simultaneous absorption and emission of radiation from layer to layer does not occur [START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF]. These hypotheses simplify the complexity of the radiative transfer problem [START_REF] Liou | An introduction to atmospheric radiation[END_REF]; for shortwave the thermal emission 𝐵 𝜆 can be neglected since scattering process is dominant [START_REF] Stephens | The parameterization of radiation for numerical weather prediction and climate models[END_REF]. The following subsection introduces the different method to solve the RTE. Since the shortwave radiation is of interest for solar PV because it generates significant PV power [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF], only the method dealing with downward shortwave radiation modeling is discussed in this manuscript. [START_REF] Liou | An introduction to atmospheric radiation[END_REF]

I.5 Input datasets for radiative transfer models

To solve the RTE Eq.II.26, II.27; a knowledge on the optical state of the atmosphere is required. Tab. II.3 shows the input datasets commonly required by radiative transfer codes (ECMWF, 2016a;[START_REF] Kneizys | The MODTRAN 2/3 Report and LOWTRAN 7 MODEL[END_REF][START_REF] Mayer | Technical note: The libRadtran software package for radiative transfer calculations -description and examples of use[END_REF][START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF][START_REF] Ricchiazzi | SBDART: A Research and Teaching Software Tool for Plane-Parallel Radiative Transfer in the Earth's Atmosphere[END_REF][START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF][START_REF] Vermote | Second simulation of the satellite signal in the solar spectrum, 6S: An overview[END_REF].

Tab. II.3 input dataset for radiative transfer code Source (ECMWF, 2016a;[START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF][START_REF] Stephens | The parameterization of radiation for numerical weather prediction and climate models[END_REF] Variable Source Vertical profile of temperature [START_REF] Nabat | of the monthly tropospheric aerosol optical depth distribution over the Mediterranean region from a comparative evaluation and blending of remote sensing and model products[END_REF][START_REF] Zubler | Intercomparison of aerosol climatologies for use in a regional climate model over Europe: INTERCOMPARISON OF AEROSOL CLIMATOLOGIES[END_REF] -Climatology from chemistry transport models [START_REF] Tegen | Contribution of different aerosol species to the global aerosol extinction optical thickness: Estimates from model results[END_REF] Cloud properties: content (liquid, ice, snow, water), optical thickness (𝜏) -Climatological value from satellites observations [START_REF] Rossow | Advances in Understanding Clouds from ISCCP[END_REF] -or diagnosed from variable forecasted by NWP models [START_REF] Bengtsson | The HARMONIE-AROME Model Configuration in the ALADIN-HIRLAM NWP System[END_REF][START_REF] Slingo | A GCM parameterization for the shortwave radiative properties of water clouds[END_REF][START_REF] Stephens | The Relevance of the Microphysical and Radiative Properties of Cirrus Clouds to Climate and Climatic Feedback[END_REF] Effective radius of cloud particles -Climatological value from satellites observations [START_REF] Rossow | Advances in Understanding Clouds from ISCCP[END_REF] -or diagnosed from variable forecasted by NWP models [START_REF] Bengtsson | The HARMONIE-AROME Model Configuration in the ALADIN-HIRLAM NWP System[END_REF]ECMWF, 2016a;[START_REF] Troccoli | Skill of Direct Solar Radiation Predicted by the ECMWF Global Atmospheric Model over Australia[END_REF]) Single scattering albedo (wo)

-approximated as a function the effective radius of particles sizes [START_REF] Stephens | The parameterization of radiation for numerical weather prediction and climate models[END_REF]) Asymmetry parameter (g) -approximated as a function the effective radius of particles sizes [START_REF] Hansen | Light scattering in planetary atmospheres[END_REF]) Solar zenith angle -Computed from trigonometric formulas [START_REF] Liou | An introduction to atmospheric radiation[END_REF]

Altitude

-From digital elevation model (USGS, 2018)

Climatological datasets shown Tab. II.3 are primarily inferred from satellite observations, atmospheric sounding and ground measurements that are combined using regressions [START_REF] Anderson | km)[END_REF][START_REF] Cionni | Ozone database in support of CMIP5 simulations: results and corresponding radiative forcing[END_REF]. According to [START_REF] Cionni | Ozone database in support of CMIP5 simulations: results and corresponding radiative forcing[END_REF] this procedure introduces biases. For unobserved or sparsely observed variable multi-year simulations of global numerical weather prediction model may be used [START_REF] Chin | Atmospheric sulfur cycle simulated in the global model GOCART: Model description and global properties[END_REF][START_REF] Cionni | Ozone database in support of CMIP5 simulations: results and corresponding radiative forcing[END_REF].

These input datasets have different coverage, uneven and sometimes coarse resolutions. We found in the literature [START_REF] Anderson | km)[END_REF] that vertical profiles of temperature, pressure and water vapor mixing ratio from climatology have a typical resolution of 1 km. NWP model have higher spatial and temporal resolution in; for Integrated Forecast System (IFS) the spatial resolution ranges between 20 and 6000 m (ECMWF, 2018). Nevertheless, the forecast evaluation of IFS [START_REF] Richardson | Evaluation of ECMWF forecasts, including 2012-2013 upgrades[END_REF][START_REF] Haiden | Evaluation of ECMWF forecasts, including 2013-2014 upgrades[END_REF][START_REF] Haiden | Evaluation of ECMWF forecasts, including 2014-2015 upgrades[END_REF][START_REF] Haiden | Evaluation of ECMWF forecasts, including the 2016 resolution[END_REF] shows that its accuracy varies with the time of the year and the considered location. Besides the aerosol dataset currently used in a large number of operational center (ECMWF, 2016a;[START_REF] Noaa | The Global Forecast System (GFS) -Global Spectral Model (GSM)[END_REF][START_REF] Termonia | The ALADIN System and its canonical model configurations AROME CY41T1 and ALARO CY40T1[END_REF] has a 5° by 4° spatial resolution and monthly temporal resolution. According to [START_REF] Nabat | of the monthly tropospheric aerosol optical depth distribution over the Mediterranean region from a comparative evaluation and blending of remote sensing and model products[END_REF] the coarse spatial resolution of this dataset misrepresents the regional aerosol loads; this misrepresentation of aerosol loads lead to inaccurate irradiance estimates [START_REF] Janjic | WRF-Solar: An augmented NWP model for solar power prediction. Model description and clear sky assessment[END_REF][START_REF] Ruiz-Arias | Surface clear-sky shortwave radiative closure intercomparisons in the Weather Research and Forecasting model: SW CLOSURE INTERCOMPARISONS IN WRF[END_REF]Zhong et al., 2016). Consequently, to solve the RTE, lots of input datasets are required; these datasets despite being available worldwide have a coarse resolution and or rely on interpolation methods which misrepresent the regional variability and cause inaccurate assessment of the GHI.

I.6 Accurate Methods to solve the radiative transfer equation for shortwaves

Accurate methods to solve the RTE Eq.II.27 we found in the are the discrete ordinate (disort) and adding methods. However, these methods are not the preferred choice to assess the GHI operationally because they are computationally expensive and require a lot of input data. They are used in some NWP model to assess the GHI; however, the RTE is not solved for every model time step and or is solved in coarser grid than the NWP model resolution (ECMWF, 2016a;[START_REF] Ruiz-Arias | Surface clear-sky shortwave radiative closure intercomparisons in the Weather Research and Forecasting model: SW CLOSURE INTERCOMPARISONS IN WRF[END_REF].

Disort method uses Eq.II.27 as a starting point; the integral over all the directions 𝜇 is replaced by a summation over a finite number of quadrature points N [START_REF] Liou | An introduction to atmospheric radiation[END_REF]. N is typically equal to 16 (Yang et al., 2016). Using Gaussian quadrature to approximate Eq. II.27 yields:

μ i dL λ,d (τ, μ i ) dτ = L λ,d (τ, μ i ) - ω λ(τ) 2 F 0 p(τ, μ 0 , μ i )e -τ λ μ 0 - ω λ(τ) 2 ∑ L λ,d (τ, μ i )a j p(τ, μ i , μ j ) N j=-N , II.29
where 𝑎 𝑗 are the Gaussian quadrature points.

The second level of approximation in the disort method consists in expanding the radiance and phase function with Fourier cosine series. In case both are independent of azimuth Eq.II.24-II.25, they are written as follows:

L λ,d (τ, μ i ) = ∑ L λ,d m (τ, μ i ) 𝑎𝑛𝑑 N m=0 II.30 p(τ, μ i , μ j ) = ∑ N m=0
∑ ω l m p l m (τ, μ i )p l m (τ, μ j ) N l=m .

II.31

Substituting Eq.II.30 and Eq.II.31 into Eq.II.29 yields a system of 2N coupled equations with nonconstant coefficients. These equations describe the upwelling (Eq.II.32) and downwelling (Eq.II.33) radiance:

μ i dL λ,d m (τ, μ i ) dτ = L λ,d m (τ, μ i ) - ω λ(τ) 2 F 0 ∑ N m=0 ∑ ω l m p l m (τ, μ i )p l m (τ, μ 0 ) N l=m e -τ λ μ 0 - ω λ(τ) 2 ∑ L λ,d m (τ, μ i )a j ∑ N m=0 ∑ ω l m p l m (τ, μ i )p l m (τ, μ j ) 𝑎𝑛𝑑 N l=m N j=-N II.32 -μ i dL λ,d m (τ, -μ i ) dτ = L λ,d m (τ, -μ i ) - ω λ(τ) 2 F 0 ∑ N m=0 ∑ ω l m p l m (τ, -μ i )p l m (τ, μ 0 ) N l=m e -τ λ μ 0 - ω λ(τ) 2 ∑ L λ,d m (τ, -μ i )a j ∑ N m=0 ∑ ω l m p l m (τ, -μ i )p l m (τ, μ j ). N l=m N j=-N II.33
Eq.II.32 and II.33 assume that the medium consists of several adjacent homogeneous layers in which the single-scattering albedo and phase function are taken to be constant within each layer but allowed to vary from layer to layer. For large numbers of N, this system of equation is solved using numerical methods [START_REF] Campbell | A numerical method for discrete ordinate and moment equations in radiative transfer[END_REF][START_REF] Stamnes | Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media[END_REF][START_REF] Stamnes | A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres[END_REF].

Adding is an accurate method that allows calculating the shortwave radiative flux across multiple vertical levels. The underlying idea of the adding method is that knowing the reflection and transmission of two individual layers, one can deduce the reflection and transmission of the combined layer by calculating the successive reflections and transmissions between these two layers [START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF]. Accounting for multiple reflections of the light beam in the two layers, as shown in Fig. II.10, the combined reflection and transmission functions are given by: The upward transmission and reflection of the i th layer are 𝛵 ̃𝑖 * and 𝑅 𝑖 , respectively whereas the downward transmission and reflection are 𝛵 ̃𝑖 and 𝑅 𝑖 * . 𝐷 ̃ and 𝑈 are the the combined total transmission and reflection functions between layers 1 and 2. Source [START_REF] Liou | An introduction to atmospheric radiation[END_REF] In Eq.II.34 the infinite series can be replaced by a single inverse function such that:

{ R 12 = R 1 + Τ ̃1 * R 1 Τ ̃1 + Τ ̃1 * R 2 R 1 * R 2 Τ ̃1 + Τ ̃1 * R 2 R 1
{ ∑(𝑅 1 * 𝑅 2 ) 𝑛 = 1 1 -𝑅 1 * 𝑅 2 = 1 + 𝑆, 𝑤𝑖𝑡ℎ 𝑛≥0 𝑆 = 𝑅 2 𝑅 1 *
(1 -𝑅 2 𝑅 1 * ) .

II.35

As seen in Eq.II.26-II.27 the diffuse and beam component of the radiance can be treated separately. We can similarly separate the diffuse and direct components of the total transmission as follows:

Τ ̃i = Τ i + e -τ i μ ′ , II.36
where 𝜇 ′ = 𝜇 0 when transmission is associated with the incident solar beam and 𝜇 ′ = 𝜇 when it is associated with the emergent light beam in the direction 𝜇.

On the basis of Eq.II.34, II.35 and II.36, a set of iterative equations for the computation of diffuse transmission and reflection involving the two layers may be written as follows [START_REF] Liou | An introduction to atmospheric radiation[END_REF]:

{ Q = R 1 * R 2 S = Q(1 -Q) -1 D = Τ 1 + SΤ 1 + Se -τ 1 μ 0 U = R 2 D + R 2 e -τ 1 μ 0 R 12 = R 1 + Ue -τ 1 μ 0 + Τ 1 * U Τ 12 = De -τ 2 μ 0 + Τ 2 e -τ 1 μ 0 + Τ 2 D

II.37

In Eq.II.37 the product of any two parameters implies that integration over the solid angle (Fig. II.4) is to be performed so as to take into account all the possible multiple scattering contributions. Therefore, the product of A and B where they can be any of the parameters R,Τ, U, and D refers to the following equation:

A 1 B 2 = 2 ∫ A(τ 1 , μ, μ ′ )B(τ 2 , μ, μ 0 )μ ′ dμ ′ 1 0
.

II.38

In practice, one may begin with the computations for initial layers with small optical depths Δ𝜏 ≈ 10 -8 so that the single scattering approximation is sufficiently accurate. Using the single scattering approximation one can find an analytical solution to Eq.II.22; this solution is written as follows:

L λ (τ, μ) = L(τ 1 , μ)e - (τ 1 -τ) μ + ω 4 F 0 p(μ, -μ 0 ) ∫ e (- (τ ′ -τ) μ + τ ′ μ 0
) dτ ′ μ and τ 1 τ II.39

L λ (τ, -μ) = L(0, -μ)e - -τ μ + ω 4 F 0 p(μ, -μ 0 ) ∫ e (- (τ ′ -τ) μ + τ ′ μ 0 ) dτ ′ μ . τ 1 τ

II.40

Where 𝐿(𝜏, 𝜇)is the upward (reflected) and downward (transmitted) radiance for a finite atmosphere bounded on two sides at 𝜏 = 0 and 𝜏 = 𝜏 1 . Eq.II.37 are then used to compute the reflection and transmission functions for an optical depth of 2Δ𝜏. This procedure may be repeated until the desirable optical depth is reached [START_REF] Liou | An introduction to atmospheric radiation[END_REF].

Adding and disort methods that are described above assume a plane parallel homogeneous atmosphere; therefore, they cannot describe accurately the horizontal variations of optical properties induced by clouds [START_REF] Barker | The Monte Carlo Independent Column Approximation: an assessment using several global atmospheric models[END_REF][START_REF] Oreopoulos | Radiative impacts of cloud heterogeneity and overlap in an atmospheric General Circulation Model[END_REF][START_REF] Räisänen | The Monte Carlo Independent Column Approximation's Conditional Random Noise: Impact on Simulated Climate[END_REF]. According to [START_REF] Oreopoulos | Radiative impacts of cloud heterogeneity and overlap in an atmospheric General Circulation Model[END_REF] neglecting the horizontal variation in cloud optical depth leads to sizable bias on the estimated solar flux. To solve this issue, the fraction of cloud in the layer is modeled using method such as the maximum overlap, random overlap and Monte Carlo independent assumption column. They allow addressing columns whose layers are only partially filled by clouds which, in turn, overlapped vertically [START_REF] Barker | The Monte Carlo Independent Column Approximation: an assessment using several global atmospheric models[END_REF].

To compute the broadband radiation calculations the radiance is first written in a flux form as follow [START_REF] Pincus | Paths to accuracy for radiation parameterizations in atmospheric models: PATHS TO PARAMETERIZATION ACCURACY[END_REF]:

F λ (τ) = ∫ L λ (τ, μ) μ μ dμ. II.41
Eq.II.26-II.27 are then discretized using a quadrature formula written as follows: (Gueymard, 2012a). RTP are physical, semi-physical or statistical based methods. These methods aim to find the GHI value with a limited accuracy loss and faster than accurate methods. RTP are favored for operational purposes.

F(τ) = ∑ b g G g F λ,

I.7.a Statistical RTP

Statistical RTPs are non-spectral model. They either use regression between irradiance and meteorological parameters; regression between irradiance and the digital count of satellite images or use historical measurements of irradiance to assess ground irradiance [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF]Noia et al., 1993aNoia et al., , 1993b)). [START_REF] Angstrom | Solar and terrestrial radiation. Report to the international commission for solar research on actinometric investigations of solar and atmospheric radiation[END_REF] pioneered solar radiation statistical modeling; he used a linear relation to relate GHI and sunshine hours [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF][START_REF] Sen | Solar Energy Fundamentals and Modeling Techniques[END_REF]:

GHI G TOA = a + b S S 0 . II.43
Where S is the monthly average of daily bright sunshine hours; which is the total hours when the sunlight is higher than a specified threshold value; a and b are Angstrom coefficients. The ratio 𝑆 𝑆 0 is related to the cloud amount n as demonstrated by [START_REF] Transvalor | Estimating Solar Irradiation on Horizontal Surfaces[END_REF].

S S 0 = 1 -n. II.44
Angstrom coefficients were derived using yearly GHI and sunshine duration measurements in Stockholm. A major difficulty of [START_REF] Angstrom | Solar and terrestrial radiation. Report to the international commission for solar research on actinometric investigations of solar and atmospheric radiation[END_REF] relationship arises from the unicity of its coefficients: these coefficients depend on physical and spatial properties of the atmosphere at the region of interest [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF]. [START_REF] Akinoǧlu | A review of sunshine-based models used to estimate monthly average global solar radiation[END_REF] tabulated Angstrom coefficient for 100 locations and noticed a wide spread of values. Later, Ögelman et al. (1984) suggested the addition of a nonlinear term to [START_REF] Angstrom | Solar and terrestrial radiation. Report to the international commission for solar research on actinometric investigations of solar and atmospheric radiation[END_REF] formulation. This model was validated in Turkey by [START_REF] Akinoǧlu | Construction of a quadratic model using modified \AAngstrom coefficients to estimate global solar radiation[END_REF]; results found showed an improvement of the GHI estimates compared to [START_REF] Angstrom | Solar and terrestrial radiation. Report to the international commission for solar research on actinometric investigations of solar and atmospheric radiation[END_REF] original model.

However, regression analyses are limited in their accuracies and the number of variables they can process. Studies showed that GHI can be correlated to several parameters [START_REF] Elizondo | Development of a neural network model to predict daily solar radiation[END_REF]. Therefore, artificial neural networks were used to correlate GHI, sunshine and or meteorological variable because of their ability to deal with linear and nonlinear correlations between several variables [START_REF] Zarzalejo | Artificial intelligence techniques applied to hourly global irradiance estimation from satellite-derived cloud index[END_REF]. One of the pioneer of this research area is [START_REF] Elizondo | Development of a neural network model to predict daily solar radiation[END_REF], he correlated the GHI to: daily precipitation, maximum and minimum daily temperature, day length, daily total. Several other parameters were also suggested as predictor: the location coordinate [START_REF] Al-Alawi | An ANN-based approach for predicting global radiation in locations with no direct measurement instrumentation[END_REF][START_REF] Reddy | Solar resource estimation using artificial neural networks and comparison with other correlation models[END_REF][START_REF] Sözen | Estimation of solar potential in Turkey by artificial neural networks using meteorological and geographical data[END_REF], the pressure [START_REF] Al-Alawi | An ANN-based approach for predicting global radiation in locations with no direct measurement instrumentation[END_REF][START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF], the relative humidity [START_REF] Al-Alawi | An ANN-based approach for predicting global radiation in locations with no direct measurement instrumentation[END_REF][START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF][START_REF] Reddy | Solar resource estimation using artificial neural networks and comparison with other correlation models[END_REF], the wind speed [START_REF] Al-Alawi | An ANN-based approach for predicting global radiation in locations with no direct measurement instrumentation[END_REF][START_REF] Reddy | Solar resource estimation using artificial neural networks and comparison with other correlation models[END_REF], the clear sky radiation [START_REF] Elizondo | Development of a neural network model to predict daily solar radiation[END_REF], solar geometry angle [START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF][START_REF] Zarzalejo | Artificial intelligence techniques applied to hourly global irradiance estimation from satellite-derived cloud index[END_REF], the clearness index [START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF], the relative air mass [START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF], the dew point temperature [START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF] and the precipitable water [START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF]. According to [START_REF] Mao | An optimal model output calibration algorithm suitable for objective temperature forecasting[END_REF], the predictor choice is related to the geographical location. A major difficulty arises from the choice of relevant predictors. Having redundant predictors causes over learning which prevent the model to be applied generally to other datasets than the training dataset [START_REF] Elizondo | Development of a neural network model to predict daily solar radiation[END_REF]. [START_REF] López | Selection of input parameters to model direct solar irradiance by using artificial neural networks[END_REF] suggested that the selection process should be done using an automatic relevance determination method (ARD).

The launch of the first satellite promoted the use of statistical model using satellite images as input [START_REF] House | History of satellite missions and measurements of the Earth Radiation Budget (1957-1984)[END_REF][START_REF] Levanon | Determination of the sea surface slope distribution and wind velocity using sun glitter viewed from a synchronous satellite[END_REF]. Statistical satellite methods establish a regression between the digital counts measured by the satellites radiometers and simultaneous ground irradiance values measured at the ground (Noia et al., 1993a). [START_REF] Tarpley | Estimating incident solar radiation at the surface from geostationary satellite Data[END_REF]; [START_REF] Cano | A method for the determination of the global solar radiation from meteorological satellites data[END_REF]; [START_REF] Hay | A satellite-based methodology for determining solar irradiance at the ocean surface during GATE[END_REF] pioneered the statistical modeling of solar irradiance using satellites images. Their methods were regularly enhanced, but their core principles remain unchanged. These methods may be generalized as follows:

GHI G c = a + bn + cn 2 . a, b, c ∈ ℜ II.45
Where n is called the cloud index; it is a measure effective cloud cover obtained using satellite measurements. 𝑎, 𝑏 and 𝑐 are either fixed parameters or parameters varying with n; they are determined with sets of satellite images and ground-based radiation measurements covering the same period.

For example, [START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF] improved the model of Cano (1985) by adding a calibration relationship between the atmosphere emerging radiance and the numerical count measured by satellite sensors. [START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF] also improved the model of Cano (1985) by adding a physical parameterization to describe the cloud and ground albedo, and adding several parameters such as the Linke turbidity factor and air mass to account for atmospheric extinction. The updated calibration step for the satellite digital count allowed processing images taken by different sensors (Rigollier et al., 2002a). Besides, to improve GHI estimates the cloud and ground albedo formulations that were previously defined empirically were updated so that their new formulations are expressed using physical laws.

I.7.b Semi-physical RTP

Semi-physical RTPs attempt to model the absorption and scattering physical processes occurring in the atmosphere using physical principle; however, with parameters that are defined empirically.

Semi-physical RTPs are non-spectral models, they treat the atmosphere as plane parallel and account for scattering [START_REF] Transvalor | Estimating Solar Irradiation on Horizontal Surfaces[END_REF]. They generally approximate the downward shortwave radiative transfers as follows [START_REF] Transvalor | Estimating Solar Irradiation on Horizontal Surfaces[END_REF][START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF]Noia et al., 1993b):

GHI = G TOA T ̅ (𝜇 0 )f(ρ g ). II.46
𝑇 ̅ is the atmospheric transmission function averaged over the entire solar spectrum; it is defined as:

𝑇 ̅ (𝜇 0 ) = 1 𝐺 𝑇𝑂𝐴 ∫ 𝐺 𝑇𝑂𝐴,λ 𝑇 λ ( ∞ 0 𝜇 0 )𝑑λ, II.47
where 𝐺 𝑇𝑂𝐴,λ and 𝑇 λ are the monochromatic top of atmosphere irradiance and transmission function, respectively. To compute the GHI, 𝑇 ̅ is obtained using an energy conservation equation in an earthatmosphere column. This energy conservation following Fig. II.11 writes [START_REF] Hay | A satellite-based methodology for determining solar irradiance at the ocean surface during GATE[END_REF]:

IE ↑ -IE ↓ = E a + E g ↓ (1 -ρ g ), II.48
where 𝐼𝐸 ↑ and 𝐼𝐸 ↓ are: the solar stream reflected to the space and the extraterrestrial solar irradiance (G TOA ) respectively. 𝐸 𝑎 is the solar stream absorbed by the atmosphere; 𝐸 𝑔 ↓ is solar irradiance at the earth's surface (GHI). 𝑓(𝜌 𝑔 ) describes the multiple reflections between the ground and the atmosphere; it is obtained using empirical relationships that correlate measurements made by the satellite in the satellite band to the broadband irradiance. (Noia et al., 1993a) The bibliographic review we carried out highlighted the following model: [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF][START_REF] Janjai | A method for estimating direct normal solar irradiation from satellite data for a tropical environment[END_REF][START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF]. These models use Eq.II.47 to define 𝑇 ̅ ; they, however, differ in the atmospheric processes that are accounted for. [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] model the atmosphere for the downward path of the solar radiation as a scattering layer on top of an absorbing layer whereas in the upward path these layers are inverted. These layers model the absorption due to the water vapor and the scattering from a Rayleigh atmosphere. A pure Rayleigh atmosphere contains only the permanent atmospheric gases that scatter radiation by Rayleigh scattering. It excludes the effects of water vapor, clouds, and aerosols (AMS, 2012). Clouds add an extra layer that both absorb and reflect the incoming radiation as shown Fig. II.12. Gautier et al. (1980) model does not account for the ozone absorption and multiple scattering. Besides, it assumes isotropic reflection, isotropic scattering and a single homogeneous layer of clouds [START_REF] Diak | Improvements to a Simple Physical Model for Estimating Insolation from GOES Data[END_REF].

Under clear sky conditions the GHI is expressed as follows [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF]Noia et al., 1993b):

{ GHI = G TOA T ̅ (μ 0 )(1 + ρ s,d (θ)ρ g ) 𝑎𝑛𝑑 T ̅ (μ 0 ) = (1 -ρ s,b )(1 -α w ↓ ).

II.49

Where 𝜌 𝑠,𝑑 and 𝜌 𝑠,𝑏 are the broadband scattering coefficients for the upward diffuse radiation and the broadband downward beam radiation coefficient due to a Rayleigh atmosphere. 𝜌 𝑔 is the broadband surface albedo; 𝛼 𝑤 ↓ is the broadband water vapor absorption coefficient for the downward path of the solar radiation. 𝜌 𝑑 ,𝜌 𝑏 and 𝛼 𝑤 ↓ values were extracted from [START_REF] Coulson | Characteristics of the radiation emerging from the top of a Rayleigh atmosphere-II[END_REF] study. 𝜌 𝑔 was computed using a conservation equation for the upwelling radiance. This equation related the ground albedo to the solar radiation measured by Geostationary Operational Environmental Satellite (GOES) in the visible band for cloud-free days. [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] assumed that there was no need for a conversion relation between satellite visible band measurements into a broadband equivalent [START_REF] Diak | Improvements to a Simple Physical Model for Estimating Insolation from GOES Data[END_REF]. The satellite images are also used to detect the occurrence of clouds by applying an empirical brightness threshold (Noia et al., 1993b).

Under cloudy sky conditions the atmospheric transmission function:

{ GHI = G TOA T ̅ (μ 0 ) 𝑎𝑛𝑑 T ̅ (μ 0 ) = (1 -ρ s,b )(1 -α w,t ↓ )(1 -α c )(1 -ρ c )(1 -α w,b ↓ ) .

II.50

Where 𝛼 𝑤,𝑡 ↓ and 𝛼 𝑤,𝑏 ↓ are water vapor absorption coefficient above and below the cloud level respectively. [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] estimated empirically that for low and middle clouds there is an average of 30% of water vapor above the cloud level (i.e. 𝛼 𝑤,𝑡 ↓ = 0.3 𝛼 ↓ ). Besides 𝛼 𝑐 was modeled empirically as linear function of 𝜌 𝑐 which was obtained through satellite measurements.

Janjai (2010) atmosphere model is more complex; it consists of two layers. The first layer where ozone absorption occurs is from the TOA to the top of the cloud layer. The second layer is from the top of the cloud layer to the earth's surface, containing air molecules, water vapor, aerosols and clouds. Contrary to the model of [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF], [START_REF] Janjai | A method for estimating direct normal solar irradiation from satellite data for a tropical environment[END_REF] includes the effect of ozone and aerosols and assumes that the optical state of the atmosphere on the downwelling path is identical to the upwelling path (Fig. II.13). Therefore, the GHI is expressed with the following equation:

{ GHI = G TOA T ̅ (μ 0 ) [(1 -ρ g ) + ρ g (α w + α aer )]𝑎𝑛𝑑
T ̅ (μ 0 ) = τ 0 (1 -ρ a -α w -α aer ).

II.51

Where 𝜌 𝑔 was determined empirically from several cloud free satellite images. 𝛼 𝑤 is the broadband water vapor absorption computed using [START_REF] Lacis | A Parameterization for the Absorption of Solar Radiation in the Earth's Atmosphere[END_REF] approximation for water vapor; 𝛼 𝑎𝑒𝑟 the broadband aerosol absorption is computed using ground measurement of the visibility and aerosol single scattering albedo; 𝜏 0 the broadband ozone absorption transmission coefficient is computed from [START_REF] Lacis | A Parameterization for the Absorption of Solar Radiation in the Earth's Atmosphere[END_REF] approximation for ozone assuming that the ozone layer is purely absorbing. 𝜌 𝑎 is the broadband earth-atmosphere reflectance obtained empirically using satellite band earth-atmosphere reflectance. The information about clouds are included in 𝜌 𝑎 .

Fig. II

.12 atmospheric model of [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] adapted from (Noia et al., 1993b) The atmosphere model of [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF] has a more detailed description of the atmospheric processes than the model of [START_REF] Janjai | A method for estimating direct normal solar irradiation from satellite data for a tropical environment[END_REF]. Similarly to the model of Janjai (2010) the water vapor absorption, ozone absorption, aerosol absorption and scattering are included. The atmosphere model of [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF] also includes the absorption of carbon dioxide and oxygen; and the multiple scattering between the ground and an atmosphere that is clear or cloudy. Under clear sky the monochromatic irradiance 𝐺𝐻𝐼 c,λ is expressed as follows [START_REF] Frouin | A simple analytical formula to compute clear sky total and photosynthetically available solar irradiance at the ocean surface[END_REF]:

{ dGHI c,λ = G TOA,λ T λ (μ 0 ) e -(1-ρ g,λ (θ)s λ ) dλ and T λ (μ 0 ) = T g,λ (μ 0 )e ( -τ μ 0 +T d,λ (μ 0 ))
.

II.52

Where 𝑇 g,λ and 𝑇 d,λ are the transmittance due to absorbing gases and due to diffuse sky. 𝜌 𝑔,λ and 𝑠 λ are the ground reflectance and spherical albedo of the atmosphere. Under cloudy sky conditions the GHI writes:

GHI = GHI c (1 -ρ c -α c ) 1 -ρ c ρ g , II.53
where 𝐺𝐻𝐼 c is the value of irradiance under clear sky for the shortwave frequency range. This model assumes that in the spectral range of interest, solar radiation is: scattered by air molecules and aerosols, absorbed by ozone, water vapor, carbon dioxide, oxygen, and aerosols.

Fig. II.13 atmospheric model of Janjai, (2010)

Besides, reflection and absorption by clouds is supposed to occur in one layer. 𝜌 𝑔 is determined empirically from a time series of satellite images. The minimum brightness value for each satellite pixel during the daytime defines a threshold that is used to classify the satellite pixel as clear or cloudy. Under clear sky conditions, the satellite sensors measure the reflection from the ground. 𝑠 λ , 𝑇 d,λ are approximated using the relationship suggested by [START_REF] Tanre | Atmospheric modeling for space measurements of ground reflectances, including bidirectional properties[END_REF]. Similarly 𝑇 g,λ is approximated using a relationship suggested in [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF]. 𝛼 𝑐 was modeled as linear function of 𝜌 𝑐 which is obtained through satellite measurements in the same way than [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF].

I.7.c Physical RTP

Physical RTPs study the physical processes occurring in the atmosphere and influencing solar radiation; namely absorption and scattering. They make approximations to Eq.II.22 that simplify its resolution. The two stream and Eddington approximation are the most commonly used methods to assess the GHI [START_REF] Anderson | The New GFDL Global Atmosphere and Land Model AM2-LM2: Evaluation with Prescribed SST Simulations[END_REF][START_REF] Brunner | Comparative analysis of meteorological performance of coupled chemistry-meteorology models in the context of AQMEII phase 2[END_REF][START_REF] Mathiesen | Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States[END_REF][START_REF] Morcrette | Recent advances in radiation transfer parameterizations[END_REF][START_REF] Ritter | A Comprehensive Radiation Scheme for Numerical Weather Prediction Models with Potential Applications in Climate Simulations[END_REF][START_REF] Slingo | A GCM parameterization for the shortwave radiative properties of water clouds[END_REF][START_REF] Stuhlmann | An Improvement of the IGMK Model to Derive Total and Diffuse Solar Radiation at the Surface from Satellite Data[END_REF][START_REF] Troccoli | Skill of Direct Solar Radiation Predicted by the ECMWF Global Atmospheric Model over Australia[END_REF][START_REF] Zhong | Dissecting surface clear sky irradiance bias in numerical weather prediction: Application and corrections to the New Goddard Shortwave Scheme[END_REF].

Two-stream approximation is similar to the disort method; it assumes that there are only upward and downward radiance in the directions 𝜇 1 and -𝜇 1 . Consequently, 𝑁 = 1 in Eq.II.29, which gives 𝑗 = -1 for the downward stream and 𝑗 = 1 for the upward stream. The upward and downward directions are given by the Gauss quadrature formula. The values of 𝜇 𝑖,𝑗 , 𝑝 𝑙 and 𝜔 𝑙 Eq.II.30 -II.31 for the two-stream approximation are shown Tab. II.5.

Tab. II.5 Gaussian Points, Gaussian

Weights and Legendre polynomials values for two-stream approximation. g is the asymmetry factor defined Eq. II.52. Inspired after [START_REF] Liou | An introduction to atmospheric radiation[END_REF] n

0 𝑃 0 = 1 𝜔 0 = 1 1 𝜇 1 = 0.5773503 𝑎 1 = 1 𝑃 1 = 𝑥 𝜔 1 = 3𝑔
Consequently, we obtain two equations from Eq.II.29: one equations that describe the downward stream and the other describing the upward stream. After some mathematical manipulations Eq.II.29 yields [START_REF] Liou | An introduction to atmospheric radiation[END_REF]:

μ 1 dM(τ, μ 1 ) dτ = (1 -ω λ (τ)g)N(τ, μ 1 ) -(S --S + )e e -τ λ μ 0 𝑎𝑛𝑑 II.54 μ 1 dN(τ, μ 1 ) dτ = (1 -ω λ (τ))M(τ, μ 1 ) -(S -+ S + )e e -τ λ μ 0 . II.55
Where,

{ M = L(τ, μ 1 ) + L(τ, -μ 1 ), N = L(τ, μ 1 ) -L(τ, -μ 1 ), g = ω 1 3 = 1 2 ∫ p(cosΘ)cosΘ dcosΘ, 1 -1 b = 1 -g 2 𝑎𝑛𝑑 S ± = F 0 ω λ (τ) 4π (1 ± 3gμ 1 μ 0 ).

II.56

g is called the asymmetry factor, it is a function of 𝜔 1 , the 1 𝑡ℎ moment of the phase function p. The two stream resolution of the RTE is used mostly in the radiative transfer code of NWP model: McRad [START_REF] Mathiesen | Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States[END_REF][START_REF] Morcrette | Recent advances in radiation transfer parameterizations[END_REF][START_REF] Troccoli | Skill of Direct Solar Radiation Predicted by the ECMWF Global Atmospheric Model over Australia[END_REF], New Goddard [START_REF] Zhong | Dissecting surface clear sky irradiance bias in numerical weather prediction: Application and corrections to the New Goddard Shortwave Scheme[END_REF] and COSMO [START_REF] Brunner | Comparative analysis of meteorological performance of coupled chemistry-meteorology models in the context of AQMEII phase 2[END_REF][START_REF] Ritter | A Comprehensive Radiation Scheme for Numerical Weather Prediction Models with Potential Applications in Climate Simulations[END_REF]. It was also used in a satellite based method by [START_REF] Stuhlmann | An Improvement of the IGMK Model to Derive Total and Diffuse Solar Radiation at the Surface from Satellite Data[END_REF]. [START_REF] Stuhlmann | An Improvement of the IGMK Model to Derive Total and Diffuse Solar Radiation at the Surface from Satellite Data[END_REF] used the Two stream method to relate the cloud index n Eq.II.45 measured in the satellite narrow band to a broadband cloud transmittance.

In Eddington's approximation the phase function is approximated similarly as the twostream but the radiance is expanded using Legendre polynomial and Eq.II.27 is not discretized using Gaussian quadrature.

L λ,d (τ, μ i ) = ∑ L lλ,d (τ, ) p l (μ i ) N=1 l=0 i = N, -N with N ∈ ℜ II.57
Substituting Eq.II.57 into in Eq.II.27 and rearranging it yields [START_REF] Liou | An introduction to atmospheric radiation[END_REF]:

dL 1λ,d (τ, μ) dτ = 3(1 -ω λ (τ))L 0 (τ, μ) - 3ω 4π F 0 e e -τ λ μ 0 𝑎𝑛𝑑 II.58 dL 0λ,d (τ, μ) dτ = (1 -ω λ (τ)g)L 1 (τ, μ) + 3ω 4π
F 0 e e -τ λ μ 0 .

II.59

The Eddington resolution of the RTE is used in GFDL-SW [START_REF] Anderson | The New GFDL Global Atmosphere and Land Model AM2-LM2: Evaluation with Prescribed SST Simulations[END_REF][START_REF] Mathiesen | Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States[END_REF] and [START_REF] Slingo | A GCM parameterization for the shortwave radiative properties of water clouds[END_REF].

The two-stream and Eddington methods are good approximations for optically thick layers, but they produce inaccurate results for thin layers and when significant absorption is involved [START_REF] Liou | An introduction to atmospheric radiation[END_REF]. These inaccuracies are related to simplicity of the phase function approximation Eq.II.31; according to [START_REF] Liou | An introduction to atmospheric radiation[END_REF] a two term Legendre polynomial expansion is not adequate to describe the strong forward scattering of aerosols and cloud particles. To solve this issue Delta-Function adjustment was introduced; this procedure lead to the 𝛿-Eddington and the 𝛿-two-stream. Consequently, the optical depth, single-scattering albedo, and phase function were scaled such tat:

{ f = ω 2 5 , τ ′ = (1 -ωf)τ, ω ′ = (1 -f)ω 1 -ωf 𝑎𝑛𝑑 p ′ (τ, μ, μ ′ ) = 1 + 3g ′ μμ ′ .

II.60

Where f is the fraction of the energy scattered in the forward direction; it is expressed a function of the phase function second moment. 𝜏 ′ , 𝜔 ′ ,𝑝 ′ are the scaled optical depth, single scattering albedo and phase function respectively [START_REF] Liou | An introduction to atmospheric radiation[END_REF]. Under clear sky both the 𝛿-Eddington and 𝛿-twostreams have relative errors of less than 1.5%. However, under cloudy skies relative errors might be as high as 10%, which indicates that higher order approximation scheme is necessary in order to obtain the accurate solar cloud absorption in weather and climate models (F. Zhang et al., 2013).

Method such as the Heliosat-4 [START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF][START_REF] Qu | Fast radiative transfer parameterisation for assessing the surface solar irradiance: The Heliosat-4 method[END_REF] aim to find a tradeoff between the accuracy of method such as disort, adding and the computational efficiency of two-stream and Eddington. In the Heliosat-4 [START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF][START_REF] Qu | Fast radiative transfer parameterisation for assessing the surface solar irradiance: The Heliosat-4 method[END_REF] the RTE was resolved for a selected number of atmosphere characterized by fixed optical properties. These computations define nodes that can be used to interpolate the GHI for other types of atmosphere characterized by other optical properties.

I.8 Validation metrics

Various metrics have been proposed and used to quantify the accuracy of solar irradiance forecasts. They can be broadly divided into five categories (J. Zhang et al., 2013): statistical metrics, variability estimation metrics, uncertainty quantification and propagation metrics, ramping characterization metrics and economic and reliability metrics. These metrics describe different characteristics of the solar irradiance obtained from a model. These metrics are insufficient on their own and must be combined to establish a complete, coherent comparison for benchmarking [START_REF] Beyer | D 1.1.3 Report on Benchmarking of Radiation Products 160[END_REF]. Determining which metrics are most appropriate depends on the user: system operators are interested in metrics that accurately reflect the costs of forecast errors whereas researchers are interested in metrics that give relative performance of different forecast models under different conditions [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. In this section we describe conventional statistical metrics, temporal variability and ramping metric that are commonly used to assess models performances.

Statistical metrics commonly used [START_REF] Jimenez | WRF-Solar: Description and Clear-Sky Assessment of an Augmented NWP Model for Solar Power Prediction[END_REF][START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF][START_REF] Perez | Validation of short and medium term operational solar radiation forecasts in the US[END_REF][START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF][START_REF] Zhong | Dissecting surface clear sky irradiance bias in numerical weather prediction: Application and corrections to the New Goddard Shortwave Scheme[END_REF] are the correlation coefficient (R), the root mean square error (RMSE), the mean average error (MAE), the mean bias error (MBE) and the Kolmogorov Smirnov integral (KSI). The RMSE and MAE are indicators of the model spread [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF], the RMSE allows to put more weight on large forecast errors whereas the MAE weighs linearly all deviations to the observations (Lorenz et al., 2009b). According to [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF] the RMSE is most important model validation metric in the context of renewable power forecasting where small errors are more tolerable and large forecast errors have a disproportionately high impact in the grid management. Nevertheless, according to [START_REF] Chai | Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature[END_REF] there is no definite consensus on the most appropriate metric between MAE and RMSE to assess a model performance. The use of RMSE is recommended when the error distribution is expected to be Gaussian [START_REF] Chai | Root mean square error (RMSE) or mean absolute error (MAE)? -Arguments against avoiding RMSE in the literature[END_REF]. The MBE describes systematic deviations of the model GHI estimates [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF]; it indicates the model average bias or deviation. This metric contrary to the RMSE and MAE indicate if the model under or overestimate the GHI. Contrary to the MAE and RMSE a deviation to the observation does not automatically increase the MBE value; low MBE may be caused by highly over-estimated values of GHI that are balanced by highly underestimated values of GHI or vice versa. KSI and R measure correlation between datasets; the KSI instead of comparing raw datasets, it compares their cumulative distribution function (CDF). The evaluation of CDF is helpful for applications where decisions are related to threshold values [START_REF] Espinar | Analysis of different comparison parameters applied to solar radiation data from satellite and German radiometric stations[END_REF][START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF]. The KSI value is advantageous because it shows how dataset values are statistically distributed. It not only describes the distribution of the points around a unit line, similarly to MBE, but also allows the user to distinguish the behaviors of stations with similar RMSE values [START_REF] Espinar | Analysis of different comparison parameters applied to solar radiation data from satellite and German radiometric stations[END_REF]. Statistical metrics discussed above are expressed as follows:

𝑀A𝐸 = ∑ |𝑦 𝑖 -𝑥 𝑖 | 𝑖 𝑁 ∑ 𝑥 𝑖 𝑖 , II.61 MBE = ∑ y i -x i i N ∑ x i i , II.62 RMSE = √ 1 n ∑ (y i -x i ) 2 i N ∑ x i i , II.63 R = ∑ (x i -x ̅)(y i -y ̅) i √∑ (x i -x ̅) i √∑ (y i -y ̅) i = 1 - σ 2 (y i -x i ) σ 2 (x i ) 𝑎𝑛𝑑 II.64 KSI = ∫ |CDF comp (x j ) - xmax xmin CDF ref (x j )|dx.

II.65

Where 𝑥 𝑗 ∈ [𝑥 𝑚𝑖𝑛 + (𝑛 -1)𝑝, 𝑥 𝑚𝑖𝑛 + 𝑛𝑝) with 𝑝 = (𝑥 𝑚𝑎𝑥 -𝑥 𝑚𝑖𝑛 ) 𝑚 ⁄ ; n is the discretisation subdivision n ∈ [1,.., m]; m is the total number of discretisation subdivisions, taken as ≥ 100 for all integrations; and x max and x min are the extreme values of GHI; y i is the model estimate and x i is the ground measurement value. 𝑥̅ and 𝑦 ̅ are the model-estimated and ground-measured means, respectively; N is the total number of data points. RMSE and MBE are expressed in W/m². 𝜎 is the standard deviation error and 𝜎 2 the variance.

These statistical metric are related to each other as follows (Lorenz et al., 2009a):

RMSE 2 = MBE 2 + σ 2 .

II.66

Temporal variability and ramping characterization metrics intend to address one of the biggest concerns associated with integrating a large amount of solar power into the grid [START_REF] Bright | A synthetic, spatially decorrelating solar irradiance generator and application to a LV grid model with high PV penetration[END_REF]. They give indication on the ability of a model to describe sudden and large variations of GHI values; therefore, sudden and large variations of electricity output caused by clouds s and extreme weather events. Variability metrics include the mean variability of the clearness index: 𝑀 𝐾 𝑐 Eq.II.67, and the mean squared variability of the clearness index : V Eq.II.68 [START_REF] Marquez | Proposed Metric for Evaluation of Solar Forecasting Models[END_REF]. M K c and 𝑉 are expressed as follows:

𝑉 = √ 1 𝑁 ∑(𝐾 𝑐 (𝑘 + ∆𝑡) -𝐾 𝑐 (𝑘)) 2 𝑁 𝑘=1
.

II.68

M K c = 1 N ∑ K c (k) N k=1 𝑎𝑛𝑑 II.67 𝑉 = √ 1 𝑁 ∑(𝐾 𝑐 (𝑘 + ∆𝑡) -𝐾 𝑐 (𝑘)) 2 𝑁 𝑘=1
.

II.68

Where N is the number of sample is the consider time interval. The GHI ramp rate is defined as the change in irradiance divided by the time over which it occurs; it is expressed as follows [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF]:

RR(∆t) = x i+∆t -x i ∆t . II.69
The ramp rate may also be used to describe the overall ability of a model to reproduce the variability of ground measurement datasets using the CDF as the show by [START_REF] Bright | A synthetic, spatially decorrelating solar irradiance generator and application to a LV grid model with high PV penetration[END_REF].

Consequently, we used the metric described here in this section to describe the ability of a model to reproduce specific characteristics of the GHI measured at the ground.

II. Improving the GHI estimate from satellite method in the inter-tropical zone

The first section presented the different methods used to solve the complex RTE. This section gives the advantage and drawback of statistical, semi-physical and physical parameterizations and how they can be combined to improve the GHI estimates in French Guiana and the ITZ using the Heliosat-2 method.

II.1 Comparison between statistical, semi-physical and physical parameterizations of the radiative transfer equation

The main advantage of statistical parameterization methods is their simplicity, they rely on linear or nonlinear relationship between the GHI and other measured variables (Noia et al., 1993a). Nevertheless, a major drawback of statistical parameterization methods arises from the unicity of the relationship coefficients. These coefficients depend on the physical and spatial properties of the atmosphere at the region of interest [START_REF] Badescu | Modeling solar radiation at the earth's surface: recent advances[END_REF][START_REF] Raphael | An Assessment of Models which use Satellite Data to Estimate Solar Irradiance at the Earth's Surface[END_REF]. Statistical methods such as the Heliosat-2 was validated in several type of climates ranging from oceanic, Mediterranean, desert to semi-arid climate regions [START_REF] Dagestad | Mean bias deviation of the Heliosat algorithm for varying cloud properties and sun-ground-satellite geometry[END_REF][START_REF] Lefèvre | Using reduced data sets ISCCP-B2 from the Meteosat satellites to assess surface solar irradiance[END_REF][START_REF] Dürr | Deriving surface global irradiance over the Alpine region from METEOSAT Second Generation data by supplementing the HELIOSAT method[END_REF][START_REF] Moradi | Evaluation of the Heliosat-II method using daily irradiation data for four stations in Iran[END_REF][START_REF] Wahab | Assessing surface solar irradiance in Northern Africa desert climate and its long-term variations from Meteosat images[END_REF][START_REF] Jumaily | Evaluation of Meteosat-8 measurements using daily global solar radiation for two stations in Iraq[END_REF][START_REF] Blanc | The HelioClim project: Surface solar irradiance data for climate applications[END_REF]. These studies showed a varying level of accuracy and sizable biases under cloudy skies. Inaccuracies under cloudy sky conditions [START_REF] Girodo | Influence of three-dimensional cloud effects on satellite derived solar irradiance estimation-First approaches to improve the Heliosat method[END_REF][START_REF] Suárez | Brightness-dependent Tarpley model for global solar radiation estimation using GOES satellite images: Application to Uruguay[END_REF][START_REF] Polo | Sensitivity of satellite-based methods for deriving solar radiation to different choice of aerosol input and models[END_REF][START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF] were explained by the inability to model properly the optical properties of clouds such as cloud optical depth which influence cloud reflectance, absorptance and transmittance [START_REF] Welch | Solar radiation and clouds[END_REF].

Semi-physical methods presented rely on physical principles. They are broadband and consider that for shortwave the most important radiative processes occurring within the atmosphere are: scattering and absorption by molecules, clouds, and aerosols. These methods use satellites images to provide information on the optical properties of the cloud and for some case other processes [START_REF] Janjai | A method for estimating direct normal solar irradiation from satellite data for a tropical environment[END_REF]. Nevertheless, semi-physical methods require an extensive knowledge of the optical properties of the atmosphere that might not be available. In that case they are obtained through empirical relationships [START_REF] Hay | A satellite-based methodology for determining solar irradiance at the ocean surface during GATE[END_REF][START_REF] Lacis | A Parameterization for the Absorption of Solar Radiation in the Earth's Atmosphere[END_REF]Noia et al., 1993b;[START_REF] Stephens | The parameterization of radiation for numerical weather prediction and climate models[END_REF]. When these data are available they are climatological and do not vary with the atmospheric state [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF][START_REF] Stephens | The parameterization of radiation for numerical weather prediction and climate models[END_REF][START_REF] Stephens | The Relevance of the Microphysical and Radiative Properties of Cirrus Clouds to Climate and Climatic Feedback[END_REF]. Nevertheless, satellites semi-physical method offer the advantage of: (1) being more computationally efficient, (2) having a wide geographical coverage, (3) having a high spatial and temporal resolution [START_REF] Hay | A satellite-based methodology for determining solar irradiance at the ocean surface during GATE[END_REF].

Physical parameterization methods similarly to semi-physical methods consider the most important radiative processes occurring within the atmosphere; besides, they include a spectral dependency. Physical methods have a more detailed description of the atmosphere than semi-physical methods; therefore, they require a lot more input datasets that might not be accurate or might be available sparsely. According to [START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF] inaccurate aerosol optical depths and ozone amounts inputs lead to high bias in GHI estimates. One advantage of physical parameterization methods is their potential to assess more accurately the GHI; however, with high computational cost [START_REF] Pincus | Paths to accuracy for radiation parameterizations in atmospheric models: PATHS TO PARAMETERIZATION ACCURACY[END_REF]. According to [START_REF] Fouquart | Intercomparing shortwave radiation codes for climate studies[END_REF] [START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF] there are two main causes of the uncertainty in the shortwave calculations from physical parameterization methods. The first is related to the approximation made to compute vapor absorption and the lower spectral resolution used to reduce the computation time.

Recently, the Heliosat-2 statistical method was used to produce GHI estimates in French Guiana with GOES images [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]. On cloudy sky days, the root mean square error (RMSE) of hourly GHI estimates attained values of up to 78%. These unsatisfactory results can be attributed to the significant cloud cover seen in intertropical zones, as clouds are known to be an important source of bias [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF][START_REF] Pereira | Effects of burning of biomass on satellite estimations of solar irradiation in Brazil[END_REF][START_REF] Janjai | A method for estimating direct normal solar irradiation from satellite data for a tropical environment[END_REF]. To improve the GHI estimates from this statistical method we considered coupling it with semi-physical methods that account for cloud properties. This approach was chosen because: (1) satellite images can provide information on the optical properties of the cloud; according to [START_REF] Stuhlmann | An Improvement of the IGMK Model to Derive Total and Diffuse Solar Radiation at the Surface from Satellite Data[END_REF] the cloud fraction observed by satellite is used as a proxy of cloud transmittance. ( 2) Semi-physical methods are less computationally expensive than Physical parameterization methods and require less input parameters.

The article below describes how the statistical heliosat-2 method was coupled with a radiative transfer parameterization under cloudy sky conditions.

II.2 Improving the Heliosat-2 Method for Surface Solar Irradiation Estimation Under Cloudy Sky Areas

(Preprint version of: https://doi.org/10.1016/j.solener.2018.05.032 )

AUTHORS:

Diallo Mouhamet a , Albarelo Tommy a , Antoine Primerose a , Linguet Laurent a a Université de la Guyane -UMR BP 792,97 337,Cayenne,France Corresponding author: laurent.linguet@univ-guyane.fr

Abstract

The purpose of this study was to improve global horizontal irradiation (GHI) estimates under cloudy sky conditions using an optimised version of the Heliosat-2 method calibrated with Geostationary Operational Environmental Satellite images. The optimised version was coupled with a radiative transfer parameterisation 1 (RTP) to better account for local cloud properties. The key element of this parameterisation is the cloud absorption coefficient, which was the only element to be computed. The obtained estimates were compared against GHI measurements from six meteorological stations in French Guiana over four years. We used root mean square error (RMSE), mean bias error (MBE), correlation coefficient, and the Kolmogorov Smirnov test integral to assess GHI estimate accuracy. The estimates were also compared with those obtained with the original optimised Heliosat-2 method. Our results show an improvement in the GHI estimates under cloudy, clear, and all sky conditions. Under cloudy skies, the RMSE and MBE of our method ranged from 43% to 63% and -2% to -22%, respectively, whereas those of the original optimised Heliosat-2 method ranged from 66% to 87% and -48% to -65%, respectively. Another effect of the new method was the improvement in clear sky GHI estimates: the RMSE and MBE ranged from 16% to 24% and -20% to -8%, respectively, while those of the optimised Heliosat-2 ranged from 19% to 28% and -23% to -10%, respectively. The improvement in GHI estimates under cloudy sky conditions led to better GHI estimates under all sky conditions: the RMSE and MBE ranged from 19% to 26% and -7% to -2%, respectively, while the corresponding values for the original optimised Heliosat-2 ranged from 22% to 32% and -15% to -8%, respectively. The suggested cloudy sky RTP offers the advantages of simple implementation and good computation speed. This method requires only one parameter; users need to configure a cloud absorption coefficient suitable for their local conditions. We conclude that coupling the optimised Heliosat-2 method with an RTP improves GHI estimates in French Guiana and reduces discrepancies between satellite-derived irradiation and ground measurements in areas with high cloudiness.

Keywords: Heliosat-2, GOES, cloudy sky, global horizontal irradiance, radiative transfer parameterization

Introduction

Accurate knowledge of global horizontal irradiation (GHI) is vital in several fields, including climatology, agriculture and energy. For instance, GHI is used in climatology to assess ocean heat flux balance and in agronomy to monitor the growth rate of crops. GHI mapping can be inferred through surface radiation network measurements using pyranometers [START_REF] Perez | Using satellite-derived insolation data for the site/time specific simulation of solar energy systems[END_REF]. However, the high manufacturing cost, maintenance costs, and an uneven network density of pyranometers limit their use [START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF][START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF]. Several studies (e.g., [START_REF] Perez | Using satellite-derived insolation data for the site/time specific simulation of solar energy systems[END_REF][START_REF] Zelenka | Effective accuracy of satellite-derived hourly irradiances[END_REF] have shown that when distance between the studied site and the pyranometers is greater than 30 km, solar radiation obtained through interpolation methods is of lower quality than satellite-derived solar radiation. Satellite-derived GHI methods were developed with the first satellite launches [START_REF] Levanon | Determination of the sea surface slope distribution and wind velocity using sun glitter viewed from a synchronous satellite[END_REF][START_REF] House | History of satellite missions and measurements of the Earth Radiation Budget (1957-1984)[END_REF] and were either classified as statistical [START_REF] Tarpley | Estimating incident solar radiation at the surface from geostationary satellite Data[END_REF]Cano et al., 1985;[START_REF] Möser | Incident Solar Radiation over Europe Estimated from METEOSAT Data[END_REF]Noia et al., 1993aNoia et al., , 1993b) ) or physical [START_REF] Ellis | Zonal average earth radiation budget measurements from satellites for climate studies[END_REF][START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF].

Heliosat-2 is one of the most widely validated methods developed for estimating solar irradiation from satellite images. One of the assets this method is that it is an open source method and is modular enough to be employed worldwide. It has been validated in oceanic, Mediterranean, desert and semi-arid climate regions [START_REF] Dagestad | Mean bias deviation of the Heliosat algorithm for varying cloud properties and sun-ground-satellite geometry[END_REF][START_REF] Lefèvre | Using reduced data sets ISCCP-B2 from the Meteosat satellites to assess surface solar irradiance[END_REF][START_REF] Dürr | Deriving surface global irradiance over the Alpine region from METEOSAT Second Generation data by supplementing the HELIOSAT method[END_REF][START_REF] Moradi | Evaluation of the Heliosat-II method using daily irradiation data for four stations in Iran[END_REF][START_REF] Wahab | Assessing surface solar irradiance in Northern Africa desert climate and its long-term variations from Meteosat images[END_REF][START_REF] Jumaily | Evaluation of Meteosat-8 measurements using daily global solar radiation for two stations in Iraq[END_REF][START_REF] Blanc | The HelioClim project: Surface solar irradiance data for climate applications[END_REF]. Furthermore, it has been used in tropical areas over which the Intertropical Convergence Zone (ITCZ) transits: Mozambique [START_REF] Wald | The performances of the HelioClim databases in Mozambique[END_REF], Zimbabwe [START_REF] Blanc | The HelioClim project: Surface solar irradiance data for climate applications[END_REF]French Guiana (Marie-Joseph et al., 2013). However, GHI estimates from the stations in these tropical areas show greater biases than those from stations in oceanic, Mediterranean and semi-arid climate regions [START_REF] Blanc | The HelioClim project: Surface solar irradiance data for climate applications[END_REF]. Therefore, improvement of GHI estimates from Heliosat-2 in intertropical zones remains an open topic of discussion.

Recently, Heliosat-2 was used to produce GHI estimates in French Guiana [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF] with Geostationary Operational Environmental Satellite (GOES) images. On cloudy sky days, the root mean square error (RMSE) of hourly GHI estimates attained values of up to 78%. These unsatisfactory results can be attributed to the significant cloud cover seen in intertropical zones, as clouds are known to be an important source of bias [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF][START_REF] Pereira | Effects of burning of biomass on satellite estimations of solar irradiation in Brazil[END_REF][START_REF] Janjai | A method for estimating direct normal solar irradiation from satellite data for a tropical environment[END_REF]. Overall, satellite-derived GHI methods exhibit positive biases under cloudy sky conditions [START_REF] Diak | An operational system for mapping insolation from goes satellite data[END_REF][START_REF] Girodo | Influence of three-dimensional cloud effects on satellite derived solar irradiance estimation-First approaches to improve the Heliosat method[END_REF][START_REF] Suárez | Brightness-dependent Tarpley model for global solar radiation estimation using GOES satellite images: Application to Uruguay[END_REF][START_REF] Polo | Sensitivity of satellite-based methods for deriving solar radiation to different choice of aerosol input and models[END_REF][START_REF] Tarpley | Estimating incident solar radiation at the surface from geostationary satellite Data[END_REF], reflecting the inability of the satellite sensor to observe the lowest layer of clouds in the visible spectrum [START_REF] Heinle | Automatic cloud classification of whole sky images[END_REF]. In the Heliosat-2 method, cloud attenuation is modelled empirically using a polynomial function of a cloud index that tends to one when the sky is overcast and zero when the sky is clear. However, the cloud index does not model cloud properties such as cloud optical depth, which influences cloud reflectance, absorbance and transmittance [START_REF] Welch | Solar radiation and clouds[END_REF]. Therefore, Heliosat-2 GHI estimates under cloudy sky conditions need to be improved to provide accurate GHI estimates in tropical areas. Several corrections have been proposed to improve the Heliosat-2 method irrespective of whether the sky is clear or cloudy; however, to the best of our knowledge, no methods have been proposed to correct the GHI estimates obtained using the Heliosat-2 method in tropical zones with a high occurrence of cloudy skies. The objective of this study was to improve GHI estimates from the Heliosat-2 method and GOES images for areas with high cloudiness.

Two main approaches have been used to assess GHI under cloudy sky conditions. The first involves radiative transfer models (RTMs) that simultaneously compute absorbed, scattered and emitted electromagnetic radiation through the Earth's atmosphere. This approach provides the most accurate results; however, it is time consuming and requires many input parameters [START_REF] Stephens | Parameterization of atmospheric radiative transfer. Part I: Validity of simple models[END_REF]. [START_REF] Oumbe | new method for estimating solar energy resource[END_REF] developed the Heliosat-4 method that relies on an approximation of the RTM equation using a product of clear sky irradiance and a parameter representing cloud extinction. This approximation allows faster GHI computation when abaci, previously computed by an RTM, are combined with interpolation functions. Under cloudy skies, the tested cases exhibited an overall overestimation of GHI; the greatest positive biases were found when cloud coverage was between 2% and 10%, where biases were sometimes greater than 60 W/m 2 . High RMSE values were encountered for desert stations where cloud fractions were between 20% and 30% and RMSE was up to 220 W/m 2 .

The second approach involves radiative transfer parameterisations (RTPs), which are simpler spectral models based on parameterisations of transmittance and absorption functions for basic atmospheric constituents such as Rayleigh scattering and ozone, water vapour and aerosol transmittances [START_REF] Khalil | Parameterization models for solar radiation and solar technology applications[END_REF]. These parameterisations provide faster results with accuracy similar to the more timeconsuming RTMs [START_REF] Fouquart | Intercomparing shortwave radiation codes for climate studies[END_REF]. [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] developed a method based on an RTP to estimate GHI. Overall, the parameterisation showed potential; however, the results under cloudy sky conditions were not optimal. This first RTP was updated by [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF] to better account for cloud effects; they found that under all sky conditions, the first version's monthly RMSE was 9%, while in the improved version it decreased to 4%. [START_REF] Janjai | Development of a method for generating operational solar radiation maps from satellite data for a tropical environment[END_REF]2011;2013) also developed a method based on an RTP to estimate GHI in a tropical climate region under all sky conditions. Contrary to [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF], they did not develop a separate RTP for clear and cloudy sky conditions; the monthly RMSEs were 6.8% for Vietnam [START_REF] Janjai | Development of a method for generating operational solar radiation maps from satellite data for a tropical environment[END_REF], 6.3% for Cambodia [START_REF] Janjai | Estimation of solar radiation over Cambodia from long-term satellite data[END_REF] and 9.6% for Myanmar [START_REF] Janjai | Satellite-derived solar resource maps for Myanmar[END_REF]. [START_REF] Geiger | Near real-time provision of downwelling shortwave radiation estimates derived from satellite observations[END_REF] parameterised GHI as top-ofatmosphere irradiance lowered by an atmospheric transmission factor. Under cloudy sky conditions, the atmospheric transmission factor was inferred from the [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] cloudy sky RTM. [START_REF] Geiger | Near real-time provision of downwelling shortwave radiation estimates derived from satellite observations[END_REF] noted the monthly means of surface solar estimated biases to be within ±5% in most cases with clear sky conditions and within ±15% under cloudy sky conditions. The standard deviation was noted to be between 13 and 111 W/m² under clear sky conditions and between 21 and 191 W/m² under cloudy sky conditions. [START_REF] Bisht | Estimation of net radiation from the MODIS data under all sky conditions: Southern Great Plains case study[END_REF] also used an RTM under clear and cloudy sky conditions. To improve GHI estimates under cloudy sky conditions, they chose a cloudy sky RTM suggested by [START_REF] Slingo | A GCM parameterization for the shortwave radiative properties of water clouds[END_REF]; this parameterisation describes the effect of clouds based on their fraction and optical thickness, both provided by MODIS cloud data products. Under clear sky conditions, the daily averaged bias was about 17.82 W/m² and the daily averaged RMSE was about 42.05 W/m²; under cloudy sky conditions, the values were 25.64 W/m² and 66.52 W/m², respectively.

With increasing interest in the development of solar technologies in tropical areas, the merits of Heliosat-2 in these areas are questioned. The solar energy potential of tropical areas can be up to three times higher than that of extra-tropical countries [START_REF] Fillol | Spatiotemporal indicators of solar energy potential in the Guiana Shield using GOES images[END_REF]; however, variability in solar energy is also higher in these areas due to the dynamic cloud cover and the hot and humid weather [START_REF] Laing | Introduction to tropical meteorology, Educational material from The COMET Program[END_REF][START_REF] Galvin | An Introduction to the Meteorology and Climate of the Tropics[END_REF]. Designing and sizing systems using solar energy input (such as solar water heaters, photovoltaic cells or solar thermal concentrators) require solar data estimates with high accuracy even under tropical zones with a high occurrence of cloudy skies.

We propose an improvement to the Heliosat-2 method so that it can be widely used to obtain GHI estimates of tropical climate regions with as high standards as those of other climate regions. To achieve this goal, we used an RTP model [START_REF] Polo | Preliminary survey on siteadaptation techniques for satellite-derived and reanalysis solar radiation datasets[END_REF] and focused on adjusting the atmospheric input data and cloud absorption to better match the local cloud regime. Our new method (RTP_OPT_H2) achieves the following: (1) no significant increase in computation time and (2) no deterioration of clear sky GHI estimation accuracy. This paper is structured as follows. Section 2 describes both the ground and satellite data used in the study. Section 3 introduces the Heliosat-1 and Heliosat-2 methods with GOES images and describes how RTP_OPT_H2 is implemented to improve GHI estimates under cloudy sky conditions. Section 4 compares the results obtained for cloudy, clear and all sky conditions using both the optimised Heliosat-2 method (OPT_H2) and RTP_OPT_H2. Finally, Section 5 summarises the paper.

Data

Ground measurements

We exploited hourly GHI data from six stations operated by the French national weather services (Météo-France) over four years (Table 1). CM6B is a first-class pyranometer, and CMP11 fulfils the accuracy requirements of a secondary standard pyranometer defined in [START_REF] Wmo | WMO Guide to the Global Observing System[END_REF]. Preventive maintenance is performed every two months, and the pyranometers are replaced every two years. Prior to final installation, each pyranometer was calibrated at the national radiometry centre in Météo-France; once installed, the coefficients of the new pyranometer were entered into the data acquisition system of the station [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF].

Hourly GHI measurements were run through a quality control process based on extreme values, following the method of [START_REF] Geiger | A web service for controlling the quality of measurements of global solar irradiation[END_REF] and the Solar Radiation Data (SoDa 2 ) website. For extreme values, hourly GHI (Gh, in Wh/m 2 ) was considered valid when it adhered to the following condition:

0.03 𝐺ℎ 𝑇𝑂𝐴 < 𝐺ℎ < 𝑚𝑖𝑛 ((1.2 𝐼 0 ), ((1.5 𝐼 0 cos(𝜃 0 ) 1.2 ) + 100)),

where Gh TOA is the top-of-atmosphere hourly surface solar insolation, 𝜃 0 is the sun zenith angle and I 0 is the solar constant (1367 W/m 2 ).

Satellite data

The satellite data used in this study were sourced from the Comprehensive Large-Array Stewardship System (CLASS), provided by the National Oceanic and Atmospheric Administration (NOAA). These data were collected by GOES 13, which was launched in May 2006 [START_REF] Hillger | Observing Systems: The GOES-13 Science Test: A Synopsis[END_REF]. GOES 13 has five band imagers; in this study, images from the visible band (0.55-0.75 μm) were considered. Images from GOES 13 were taken every 30 min with a 1 × 1 km spatial resolution. The downloaded dataset had 30-min time steps in a series of 16-bit images from January 2010 to December 2013.

Climate

The climate in French Guiana is regulated by the ITCZ, which is the meeting point of the northeast and southeast trade winds [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]. The ITCZ is characterised by the rapid ascent of hot air associated with the development of large and homogeneous cumulonimbus clouds (Marie-Joseph , 2013). It is mainly an oceanic phenomenon [START_REF] Vasquez | The Intertropical Convergence Zone[END_REF]; however, its latitude changes over the years, resulting in high humidity clouds that bring continuous rain over continental areas.

The ITCZ moves over French Guiana twice a year, although variations in the timings of the start and end of the seasons show that there are annual variations in the ITCZ movement [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]. In general, the ITCZ moves northward to ~7° N from May to July and southward to ~15° S from November to January [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]. This shift creates a seasonal cycle with four periods [START_REF] Bovolo | Fine-scale regional climate patterns in the Guianas, tropical South America, based on observations and reanalysis data[END_REF]: (1) July to November, when the ITCZ lies to the north, producing a dry season characterised by sky that is mostly clear, although weak precipitations may occur [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]. During this period cirrus and cumulus fractus clouds are predominant; (2) November to January, a short rainy season when the ITCZ moves south of French Guiana [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]; (3) February to March, a transition period before the ITCZ begins its northward motion, characterised by a short dry season [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]; and (4) April to May, when the ITCZ moves northward and a rainy season occurs. During this time, cumulonimbus clouds are predominant.

3 Methods

The Heliosat method

For details of the principles governing the Heliosat method and the development of the optimised Heliosat-2 method (hereafter, OPT_H2) that exploits GOES images over French Guiana, readers are directed to [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF] and [START_REF] Fillol | Spatiotemporal indicators of solar energy potential in the Guiana Shield using GOES images[END_REF]. In summary, the Heliosat method involves the construction of a 'cloud index', n, resulting from a comparison of what is observed by the sensor with what should be observed over that pixel if the sky were clear, which is related to the 'clearness' of the atmosphere (Rigollier et al., 2002b):

𝑛 𝑡 (𝑖, 𝑗) = 𝜌 𝑡 (𝑖,𝑗)-𝜌 𝑔 𝑡 (𝑖,𝑗) 𝜌 𝑐 𝑡 (𝑖,𝑗)-𝜌 𝑔 𝑡 (𝑖,𝑗) , (2)
where 𝑛 (𝑖, 𝑗) 𝑡 is the cloud index (unitless); 𝜌(𝑖, 𝑗) 𝑡 is the apparent albedo, observed by a spaceborne sensor for time t (unitless); 𝜌 𝑔 𝑡 (𝑖, 𝑗) is the apparent albedo of the ground under clear sky conditions (unitless); and 𝜌 𝑐 𝑡 (𝑖, 𝑗) is the apparent albedo of the brightest clouds (unitless).

As per the Heliosat-1 method core hypothesis (Cano et al., 1985), GHI is a linear function of n and is expressed as follows:

𝐺𝐻𝐼 = 𝑛 𝐺 𝑏 + (1 -𝑛)𝐺 𝑐 , (3)
where GHI is the global horizontal irradiance (Wh/m 2 ), n is the cloud index (unitless), G c is the clear sky RTP (Wh/m 2 ) and G b is the cloudy sky RTP (Wh/m 2 ). GHI is also related to top-of-atmosphere irradiance as follows:

𝐺𝐻𝐼 = 𝐾 𝑡 𝐺 𝑇𝑂𝐴 , (4)
where K t is the clearness index or atmospheric transmission factor and G TOA is the top-of-atmosphere irradiance. Using equations ( 3) and ( 4), Cano et al., (1985) found that K t is linearly dependent on the cloud index equation ( 5):

𝐾 𝑡 = 𝑎 𝑛 + 𝑏. (5)
The coefficients a and b have to be determined empirically with sets of satellite images and groundbased radiation measurements covering the same period [START_REF] Beyer | Modifications of the Heliosat procedure for irradiance estimates from satellite images[END_REF]. [START_REF] Beyer | Modifications of the Heliosat procedure for irradiance estimates from satellite images[END_REF] adapted the GHI formula to the following:

𝐺𝐻𝐼 = 𝐾 𝑐 𝐺 𝑐 , (6)
where K c is the clear sky index or an atmospheric transmission factor and G c is the clear sky RTP. Beyer et al. suggested the use of a clear sky RTP to better account for the irradiation dependence on the solar zenith angle and atmospheric aerosol and water vapour content.

Using the linear relationship ( 5) between the atmospheric transmission factor K t and the cloud index used by Cano et al., (1985), [START_REF] Beyer | Modifications of the Heliosat procedure for irradiance estimates from satellite images[END_REF] also suggested a linear relationship between the atmospheric transmission factor for clear sky, K c , and the cloud index. Using sets of satellite images and ground-based radiation measurements covering the same period, [START_REF] Beyer | Modifications of the Heliosat procedure for irradiance estimates from satellite images[END_REF] found that K c is expressed as follows:

𝐾 𝑐 = 1 -𝑛, (7)
where K c tends to one for clear sky conditions and tends to zero for cloudy sky conditions. Later, (Rigollier et al., 2002c) showed that for overcast skies (n > 0.8), a linear relationship is inappropriate and underestimates K c ; therefore, they suggested a quadratic equation:

0.8 < 𝑛 ≤ 1.1 𝐾 𝑐 = 2.0667 -3.6667𝑛 + 1.6667𝑛 ² . ( 8
)
The Heliosat-2 method was developed by [START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF], considering previous works (Cano et al., 1985;[START_REF] Beyer | Modifications of the Heliosat procedure for irradiance estimates from satellite images[END_REF]Rigollier et al., 2002a;2002b) and setting the maximal and minimal values of the clear sky index to 1.2 and 0.05 [START_REF] Rigollier | Towards operational mapping of solar radiation from Meteosat images[END_REF], respectively (6). The clear sky index K C was defined as follows.

𝑛 ≤ -0.2 𝐾 𝑐 = 1.2, -0.2 < 𝑛 ≤ 0.8 𝐾 𝑐 = 1 -𝑛, 0.8 < 𝑛 ≤ 1.1 𝐾 𝑐 = 2.0667 -3.6667𝑛 + 1.6667𝑛 ² , 1.1 < 𝑛 𝐾 𝑐 = 0.05. (9)
The European Solar Radiation Atlas clear sky model in the Heliosat-2 method [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas -with respect to the heliosat method[END_REF] uses the Linke turbidity factor (TL) as an input. TL is a parameter that describes the attenuation of solar radiation by the atmosphere under clear skies [START_REF] Remund | Worldwide Linke turbidity information[END_REF]. TL varies for each site and each month and is calculated using monthly values from the SoDa database. The methodology used to obtain this database of TL values is described in [START_REF] Remund | Worldwide Linke turbidity information[END_REF].

The OPT_H2 was developed by [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF] to consider GOES images as input instead of METEOSAT images. The main change involved the modification of the calibration step to process data from GOES. A sensitivity analysis was also conducted with different values of cloud albedo (maximum value, Rigollier value, quantile 95) and TL. However, in this version, overcast sky conditions are associated with significant errors in GHI estimates. Since the studied area is subject to strong climatic instability linked to the regular presence of the ITCZ, the accuracy of GHI estimates when the sky is cloudy degrades quickly; thus, there is a need for a GHI calculation formula based only on a clear sky RTP. In the Heliosat-2 code [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF][START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF], two types of solar irradiation attenuation are accounted for: attenuation by aerosols and attenuation by clouds.

TL models the attenuation caused by aerosol absorption and scattering, assesses the optical depth of aerosols and incorporates water vapour and NO 2 optical depths [START_REF] Gueymard | Clear-sky irradiance predictions for solar resource mapping and large-scale applications: Improved validation methodology and detailed performance analysis of 18 broadband radiative models[END_REF]. The value of this parameter increases as scattering increases [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas -with respect to the heliosat method[END_REF].

Cloud attenuation is modelled empirically using a clear sky index (K c ) function of cloud index (n) [START_REF] Dürr | Deriving surface global irradiance over the Alpine region from METEOSAT Second Generation data by supplementing the HELIOSAT method[END_REF]. When the sky is overcast, n tends to one and K c tends to zero, which means that clouds fully attenuate the incoming radiation. However, using parameters that account for both aerosol and cloud attenuation have drawbacks. One of which is that TL is subjected to daily variations [START_REF] Gueymard | Turbidity determination from broadband irradiance measurements: a detailed multicoefficient approach[END_REF] that are not accounted for in the monthly climatological values provided by the SoDa database used in the Heliosat-2 code. Another drawback is that K c does not model cloud properties such as cloud optical depth, which influences cloud reflectance, absorbance and transmittance [START_REF] Welch | Solar radiation and clouds[END_REF].

Optimised Heliosat-2 method coupled with cloudy sky RTP

The limited modelling of cloudy skies by the Heliosat-2 method generates inaccuracies in GHI estimation under cloudy sky conditions in tropical areas, where skies are regularly disturbed by cloud formations with significant variability throughout the year. Therefore, a GHI calculation formula that considers a cloudy sky parameterisation is required. We modified the Heliosat-2 method by using equation (3) instead of (6) to estimate the GHI values, where G b is a cloudy sky model that represents the extinction of irradiance due to clouds. Equation (3) was used by Cano et al., (1985) in the form

𝐺𝐻𝐼 𝐺 𝑇𝑂𝐴 = 𝑛. 𝐺 𝑏 𝐺 𝑇𝑂𝐴 + (1 -𝑛) 𝐺 𝑐 𝐺 𝑇𝑂𝐴 = 𝑛 (. 𝐺 𝑏 𝐺 𝑇𝑂𝐴 - 𝐺 𝑐 𝐺 𝑇𝑂𝐴 ) + 𝐺 𝑐 𝐺 𝑇𝑂𝐴 (10)
to empirically determine the regression coefficients between GHI and n (4-5). However, Cano et al., (1985) did not explain or propose a cloudy sky model G b .

To consider the significant presence of clouds in the atmosphere of the studied area, we integrated a spatiotemporally dependent cloudy sky RTP into the GHI estimate formula of Cano et al., (1985). For this, we used equation (3) proposed by Cano et al., (1985), in which G b now represents a cloudy sky RTP and 1-n is replaced by the clear sky coefficient K c [START_REF] Dürr | Deriving surface global irradiance over the Alpine region from METEOSAT Second Generation data by supplementing the HELIOSAT method[END_REF]. The GHI equation transforms into the following:

𝐺𝐻𝐼 = 𝑛 𝐺 𝑏 + 𝐾 𝑐 𝐺 𝑐 . (11)
A bibliographic review highlighted four cloudy sky RTPs used in [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF], [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF], [START_REF] Slingo | A GCM parameterization for the shortwave radiative properties of water clouds[END_REF] and [START_REF] Bisht | Estimation of net radiation from the MODIS data under all sky conditions: Southern Great Plains case study[END_REF]. The following criteria were used when choosing the appropriate model: (1) the RTP should not significantly increase the computation time, and (2) it should not worsen the accuracy of clear sky GHI estimates.

Considering these constraints, we kept the cloudy sky RTP proposed by [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF].

It is written as follows:

𝐺 𝑏 = 𝐺 𝑐 (1-𝜌 𝑒𝑓𝑓 -𝛼 𝑐 ) (1-𝜌 𝑒𝑓𝑓 𝜌 𝑔 ) , (12)
where G b is the cloudy sky RTP (Wh/m 2 ), G c is the clear sky RTP (Wh/m 2 ), ρ eff is the effective cloud albedo (unitless), α c is the cloud absorption (unitless) and ρ g is the apparent ground albedo (unitless).

For the effective cloud albedo, we used ρ eff , as defined by [START_REF] Lefèvre | Using reduced data sets ISCCP-B2 from the Meteosat satellites to assess surface solar irradiance[END_REF]:

eff  𝑡 (𝑖, 𝑗) = 0.85 -0.13 [1 -𝑒𝑥𝑝(-4 𝑐𝑜𝑠(𝜃 0 )) 5 ], (13)
where 𝜃 0 is the solar zenith angle.

The apparent ground albedo ρ g was used by [START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF] in the Heliosat-2 model and is defined as the minimisation of the following equation, since it is assumed that the presence of clouds increases the albedo.

𝜌 𝑔 𝑡 (𝑖, 𝑗) = 𝑚𝑖𝑛[ 𝜌 𝑡 (𝑖,𝑗)-𝜌 𝑎𝑡𝑚 𝑡 (𝜃 𝑠 ,𝜃 𝑣 ,𝛹)] 𝑇(𝜃 𝑠 )𝑇(𝜃 𝑣 ) . ( 14
)
For further details, the reader is encouraged to refer to [START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF].

The cloud absorption α c of [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF] is linearly dependent on the effective cloud albedo:

𝛼 𝑐 = 𝑎 𝜌 𝑒𝑓𝑓 . ( 15
)
The proportionality coefficient a can be determined empirically, and [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF] suggested a value of 0.07 for cirrus-type clouds. This RTP considers plane parallel clouds [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF] as well as reflection and absorption by clouds, which are assumed to occur in one layer [START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF].

To compute the cloudy sky RTP, a cloud absorption index that describes the local cloud regime was required. As suggested by [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF], the tendency of the satellite-derived GHI to underestimate or overestimate the ground-measured surface solar irradiation may be linked to a misrepresentation of cloud absorption. According to [START_REF] Diak | Improvements to a Simple Physical Model for Estimating Insolation from GOES Data[END_REF] and [START_REF] Welch | Solar radiation and clouds[END_REF], the proportionality coefficient a (15) between cloud absorption and cloud albedo ranges from 0.04 to 0.17.

To find a (15) the author minimised the bias between the parameterised cloud absorption (15) and a cloud absorption computed from a RTM [START_REF] Liou | On the absorption, reflection and transmission of solar radiation in Cloudy atmospheres[END_REF] for solar zenith angle between 0° and 75°. First, the cloud type that mainly represents the local cloud regime was chosen. Second, the selected cloud type's absorption per solar zenith angle 𝛼 𝑐𝑡 was extracted from [START_REF] Liou | On the absorption, reflection and transmission of solar radiation in Cloudy atmospheres[END_REF] drawings, where the cloud absorption for five different types of clouds are available. Third, the effective cloud albedo 𝜌 𝑒𝑓𝑓 was computed following ( 14). Finally, a was chosen to minimise the bias between the cloud absorption defined in equation ( 15) and the cloud absorption 𝛼 𝑐𝑡 obtained previously in [START_REF] Liou | On the absorption, reflection and transmission of solar radiation in Cloudy atmospheres[END_REF] for solar zenith angle 𝜃 𝑠 between 0° and 75°. To fit these two cloud absorptions, a (15) was varied between 0.04 and 0.17, and the authors solved the following equation:

𝑚𝑖𝑛 𝑗∈𝑅 ( 1 𝑁 ∑ 𝛼 𝑐𝑡 (𝜃𝑠 𝑖 ) -𝑎 𝑗 𝜌 𝑒𝑓𝑓 (𝜃𝑠 𝑖 )) 𝑁 𝑖=1 , ( 16
)
where a j is the value that can be obtained from the proportionality coefficient a and discrete values of solar zenith angle 𝜃𝑠 𝑖 between 0° and 75°, chosen according to [START_REF] Liou | On the absorption, reflection and transmission of solar radiation in Cloudy atmospheres[END_REF] cloud type absorption per solar zenith curves. Equation ( 16) is minimised when 𝑎 𝑗 = 𝑎.

The optimised Heliosat-2 method used in this study (RTP_OPT_H2) was designed for 16-bit GOES images instead of the 8-bit GOES images used by [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]. Monthly TL values for each site were obtained from the SoDa online database. GHI estimates using the RTP_OPT_H2 method (10, 11) were computed as follows: the clear sky RTP G c , effective cloud albedo ρ eff and ground albedo ρ g used to compute the cloudy sky RTP as well as G b were considered as outputs of the optimised Heliosat-2 code. These parameters were then used to compute the cloud index and GHI estimates. Within the Heliosat-2 code, G b , K c and G c were computed simultaneously and combined to produce the GHI estimates.

Results and discussion

The accuracies of the GHI estimates from RTP_OPT_H2 were benchmarked against those of GHI estimates from the optimised Heliosat-2 and of GHI from in situ measurements.

The GHI datasets were compared using four criteria: mean bias error (MBE), the root mean square (RMSE, the Pearson correlation coefficient (R), and the Kolmogorov Smirnov test integral (KSI). These accuracy criteria were defined as follows:

𝐵𝐼𝐴𝑆 = ∑ y i -x i i 𝑁 ∑ x i i , ( 17
) 𝑅𝑀𝑆𝐸 = √ 1 𝑛 ∑ (y i -x i ) 2 i 𝑁 ∑ x i i , and (18)
𝑅 = ∑ (𝑥 𝑖 -𝑥̅ )(𝑦 𝑖 -𝑦 ̅) 𝑖 √∑ (𝑥 𝑖 -𝑥̅ ) 𝑖 √∑ (𝑦 𝑖 -𝑦 ̅) 𝑖 . ( 19
)
where y i is the model estimate and x i is the ground measurement value; 𝑥̅ and 𝑦 ̅ are the modelestimated and ground-measured means, respectively; and N is the total number of data points. RMSE and MBE are expressed in Wh/m².

According to [START_REF] Espinar | Analysis of different comparison parameters applied to solar radiation data from satellite and German radiometric stations[END_REF], RMSE, MBE and R describe different characteristics of the behaviour of a dataset but are often insufficient to establish a complete, coherent comparison for benchmarking. While RMSE describes how points are clustered around a regression line, MBE describes the distribution of the points around a unit line [START_REF] Espinar | Analysis of different comparison parameters applied to solar radiation data from satellite and German radiometric stations[END_REF]. Therefore, the KSI can be used in such cases as a measure of similarity between two datasets described by their cumulative distribution function (CDF) [START_REF] Espinar | Analysis of different comparison parameters applied to solar radiation data from satellite and German radiometric stations[END_REF]; the lower the KSI value, the closer the CDFs of the two datasets. The KSI value is advantageous because it shows how dataset values are statistically distributed. It not only describes the distribution of the points around a unit line, similar to MBE, but also allows the user to distinguish the behaviours of stations with similar RMSE values [START_REF] Espinar | Analysis of different comparison parameters applied to solar radiation data from satellite and German radiometric stations[END_REF]. KSI is defined as follows:

𝐾𝑆𝐼 = ∫ |𝐶𝐷𝐹 𝑐𝑜𝑚𝑝 (𝑥 𝑗 ) - 𝑥𝑚𝑎𝑥 𝑥𝑚𝑖𝑛 𝐶𝐷𝐹 𝑟𝑒𝑓 (𝑥 𝑗 )|𝑑𝑥, (20)
where 𝑥 𝑗 ∈ [𝑥 𝑚𝑖𝑛 + (𝑛 -1)𝑝, 𝑥 𝑚𝑖𝑛 + 𝑛𝑝) with 𝑝 = (𝑥 𝑚𝑎𝑥 -𝑥 𝑚𝑖𝑛 ) 𝑚 ⁄ ; n is the discretisation subdivision n∈ [1,.., m]; m is the total number of discretisation subdivisions, taken as ≥ 100 for all integrations; and x max and x min are the extreme values of GHI.

Clear and cloudy skies were split with respect to the clearness index K t values:

𝐾 𝑡 = 𝐺𝐻𝐼 𝐺 𝑇𝑂𝐴 . ( 21
)
Although it would have been also possible to separate cloudy skies from clear skies using the clear sky index (see eq. 6), we used the clearness index (K t ) instead. This is because the calculation of the clear sky index implies the use of a clear sky model, and we did not want to add uncertainties related to this model to our calculation. While we are also aware that K t is more related to the turbidity in the atmosphere, the calculation of this index only requires GHI and the top-of-atmosphere GHI (GHI TOA ), as opposed to that of the clear sky index. Furthermore, GHI used to calculate K t is obtained from validated and quality-checked ground measurements. K t has already been used in studies in tropical climate areas, with highly variable sky conditions during the day:

- [START_REF] Soubdhan | Classification of daily solar radiation distributions using a mixture of Dirichlet distributions[END_REF] studied the daily distribution of K t in Guadeloupe, French West Indies. They found 4 classes of days: clear sky days, intermittent clear sky days, cloudy sky days and intermittent cloudy sky days. The classes were separated according to the daily K t value, a sunshine threshold, a cloudy level and a dynamic level.

-Marie-Joseph et al. ( 2013) studied the applicability of the Heliosat-2 method to French Guiana in the north-eastern part of South America. Cloudy skies were assumed when K t < 0.2 and clear skies were assumed when K t > 0.7. We calculated the daily K t value from the daily averaged GHI measured at the ground stations, as well as the standard deviation of the hourly K t for each day; only data for days with a standard deviation of less than 15% were retained. Because of this constraint, the K t values ranged between 0.7 and 0.1; K t for clear sky days was almost equal to 0.7, while that for cloudy sky days was less than 0.2 [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF].

Over all the years and stations, the studied dataset includes 70480 hourly all sky values, 4052 hourly clear sky values and 3700 hourly cloudy sky values.

Quality of GHI estimates from the OPT_H2 method

To compare RTP_OPT_H2 GHI estimates against OPT_H2 GHI estimates, we first estimated the accuracy of the OPT_H2 GHI estimates using ground measurement data for clear, cloudy and all sky conditions (Table 2). [START_REF] Janjai | Development of a method for generating operational solar radiation maps from satellite data for a tropical environment[END_REF]. However, available data are monthly climatological values. According to [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas -with respect to the heliosat method[END_REF] a change of 1 in TL leads to a relative change of approximately 10-15% in clear-sky GHI estimates.

Fig. 1. GHI time series of two days where all stations experienced on average a clear sky day. ( 1) is for Julian day 259 (September 16 th ) on year 2010, ( 2) is for Julian day 277 (October 4 th ) on year 2013.

For all sky conditions, MBE and RMSE ranged from -15% to -8% and from 22% to 32%, respectively, while the correlation coefficient R ranged from 0.89 to 0.94. Under cloudy sky conditions, MBE increased by more than three times, with values ranging from -48% to -64%. RMSE also increased in this case, usually by almost four times, with values ranging from 66% to 84%. The correlation coefficient, a measure of the similarity between the irradiance estimated by the model and that measured by the pyranometers, was close to maximum under clear sky conditions. For each year and station, the correlation coefficient was better than 0.93 under clear sky conditions. Under cloudy sky conditions, there was a significant decrease in the correlation coefficient values: the highest value was 0.78 and the lowest value was 0.62. As seen in Table 2 and Figure 2, under cloudy sky conditions, there was a large discrepancy between GHI estimates from the OPT_H2 model and the measured values. To address this discrepancy, we coupled the optimised model with an RTP, as described in Section 4.2.

Quality of GHI estimates from the OPT_H2 method coupled with an RTP

Using the methodology described in Section 3.2, the cloud absorption to effective cloud albedo proportionality coefficient, a, was computed to describe the effect of cumulonimbus clouds over French Guiana [START_REF] Linguet | On the applicability of the Heliosat-2 method to assess surface solar irradiance in the Intertropical Convergence Zone, French Guiana[END_REF]. Equation ( 15) is minimised when 𝑎 𝑗 = a= 0.165. This value was substituted in equation ( 14) to yield the following:

α c = 0.165 ρ eff . ( 22
)
This value of α c was integrated in the cloudy sky RTP, and GHI estimates were generated with this parameterisation for the years 2010-2013 (Table 3).

GHI estimates under cloudy sky conditions

Table 3 shows the impact of coupling a cloudy sky RTP with OPT_H2 under cloudy sky conditions. Figure 3 shows the relationship between GHI from ground measurements and from OPT_H2.

Comparing Figures 2 and3, one can see that the gap between the identity line and the linear regression line between the ground measurements and OPT_H2 has decreased, and in some cases, they intersect.

There was an overall decrease in both MBE and RMSE, and the correlation coefficient improved for most stations. The RTP_OPT_H2 MBE ranged from -2% to -22% over the entire study period, while OPT_H2 MBE ranged from -48% to -65%. A noticeable decrease in the RMSE was also observed; OPT_H2 RMSE ranged from 66% to 87%, while RTP_OPT_H2 RMSE ranged from 43% to 63%.

Figure 4 shows a clear reduction in the gap between the ground-measured values and RTP_OPT_H2 values under cloudy sky conditions for all stations. The clearness index CDF (Fig. 4) show that for these days considered as cloudy on average, at least 50% of the hourly K t values are lower than 0.2 for each station. K t values higher than 0.4 represent less than 1%. The accuracy of the GHI estimates for this cloud absorption coefficient parameterisation ( 22) varies according to the location of the station, as seen in Table 3. One can classify stations into three groups with respect to the magnitude of the improvement: (1) Saint-Georges and Rochambeau, (2) Kourou and Ile Royale and (3) St-Laurent and Maripasoula. This classification is based on rainfall amount in the regions: the first group has a yearly mean rainfall between 3700 and 3500 mm, the second group has a yearly mean rainfall of approximately 3000 mm and the third group has a yearly mean rainfall between 2500 and 2650 mm according to Météo-France [START_REF] Héritier | Le climat guyanais ; petit atlas climatique de la Guyane française[END_REF]. [START_REF] Hong | Precipitation estimation from remotely sensed imagery using an artificial neural network cloud classification system[END_REF] and [START_REF] Richards | on the relationship between satellite observed cloud cover and precipitation[END_REF] noticed significant correlation between cloud groups and rainfall; therefore, the regions within each group may also have a similar cloud regime. These values represent approximately the 85 th percentile on average for all stations. The highest improvement was noted for irradiance of approximately 42 to 56 Wh/m² (Fig. 4), which, depending on the station, represents approximately 20% to 30% of the cloudy sky samples. However, the correction effect was less obvious for irradiance values of less than 42 Wh/m² or more than 333 Wh/m² (Fig. 4), except at Maripasoula, where noticeable improvements were found for values of up to 472 Wh/m².

The results for percentiles of values higher than 333 Wh/m² (Fig. 4) are explained by the occurrences of clear sky periods on days considered cloudy. The results for percentiles of values lower than 42

Wh/m² correspond to periods near sunrise or sunset (high  s ), where [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF] found discrepancies between their cloudy sky model (used in this study) and ground measurements. One of the objectives for RTP_OPT_H2 development was to not degrade the clear sky surface solar radiation estimates. Table 3 shows significant improvements in RMSE, MBE and correlation coefficients under clear sky conditions. MBE of the GHI estimates from OPT_H2 ranged from -23% to -10%, while that from RTP_OPT_H2 ranged from -8% to -20%. RMSE followed the same trend, ranging from 19% to 28% for OPT_H2 and 16% to 24% for RTP_OPT_H2. Overall, there was a minor increase (0.01) in the correlation coefficient. To further analyse the impact of RTP_OPT_H2 under clear sky conditions, we considered the CDF of the GHI estimated by RTP_OPT_H2, OPT_H2 and from ground measurements (Fig. 5). Figure 5 shows that, although RTP_OPT_H2 improved GHI estimation from each site similarly regardless of geographical location under clear sky conditions, this improvement was less noticeable than that observed under cloudy sky conditions (Fig. 4). The clearness index CDF (Fig. 5) show that for these days considered as clear on average, at least 50% of the hourly K t values are higher than 0.65 for each station. K t values lower than 0.5 represent less than 1%.

Quality of GHI estimates under all sky conditions

The computation results showed an overall improvement in GHI estimates under all sky conditions (Table 3). MBE of the GHI estimates from RTP_OPT_H2 ranged from -2% to -7% over the entire study period, while that from OPT_H2 ranged from -15% to -8%. RMSE also showed a decrease; RMSE of the GHI estimates from OPT_H2 ranged from 22% to 32%, while that from RTP_OPT_H2 ranged from 19% to 26%. Under all sky conditions, there was a noticeable improvement in the correlation coefficient; lower and higher values for RTP_OPT_H2 were 0.91 and 0.95, respectively, whereas those for OPT_H2 were 0.89 and 0.94, respectively. The clearness index CDF (Fig. 6) show that under all sky conditions there is a high occurrence of clouds, at least 50% of the hourly K t values are lower than 0.5 for each station. The overall improvement under all sky conditions may be explained by improved cloud feature characterisation.

Previously, cloud optical depth was not modelled in OPT_H2 by K c but is now modelled in RTP_OPT_H2 using cloud absorption α c as a proxy for cloud optical depth [START_REF] Gautier | Surface solar radiation flux and cloud radiative forcing for the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP): A satellite, surface observations, and radiative transfer model study[END_REF]. [START_REF] Welch | Solar radiation and clouds[END_REF] showed that when cloud optical depth decreases, cloud transmittance increases, and vice versa. Using the cloudy sky parameterisation allowed tweaking of the cloud optical depth to match the ITCZ regional cloud properties. To further analyse RTP_OPT_H2's effect under all sky conditions, the CDF of the GHI from RTP_OPT_H2, OPT_H2 and ground measurements was considered (Fig. 6). Figure 6 shows a clear reduction in the discrepancy between ground-measured and RTP_OPT_H2 GHI values under all sky conditions for all stations, similar to the improvements noted for clear and cloudy sky conditions. Table 4 shows KSI for both OPT_H2 and RTP_OPT_H2 computed for the six stations under clear, cloudy and all sky conditions using CDF. One can see that the magnitude of the improvement clearly distinguishes the three groups described in Section 4.2.1. The first group (Saint Georges and Rochambeau) showed the greatest improvement, followed by the second and third groups. There was also an improvement under clear sky conditions, albeit lower than that under cloudy sky conditions. This is explained by the occurrence of clouds during days that are, on average, clear, as defined in equation ( 21).

Conclusions

In this article, we have attempted to improve GHI estimates under cloudy sky conditions encountered in tropical areas. We coupled a cloudy sky RTP that included cloud features in an optimised version of the Heliosat-2 method calibrated with GOES images. Using this method improved GHI estimation accuracy in the ITCZ, a zone characterised by high cloudiness.

The novelty of this modification to the Heliosat-2 method is its design, which (i) is based on a cloudy sky RTP with only one parameter-the cloud absorption and (ii) does not increase computation time.

The bias improvement of GHI estimates under cloudy sky conditions lies between 25% and 39%.

RTP_OPT_H2 had a greater effect on bias; for the entire dataset, under cloudy sky conditions, the bias changed from a mean of -60% for OPT_H2 to a mean of -16% for RTP_OPT_H2. This improvement under cloudy sky conditions has a direct effect on that under all sky conditions; the overall MBE and RMSE changed from mean values (based on all years and stations combined) of -11% and 28%, respectively, for OPT_H2 to -4% and 23%, respectively, for RTP_OPT_H2. This improvement is explained by the improved cloud feature characterisation resulting from the cloudy sky parameterisation integration in the GHI formula. KSI was also computed for all stations under clear, cloudy and all sky conditions.

RTP_OPT_H2 was found to be highly sensitive to cloud absorption parameterisation. We have shown that the magnitude of RTP_OPT_H2 bias improvement varied between three groups: (1) Saint-Georges and Rochambeau, (2) Kourou and Ile Royale and (3) St-Laurent and Maripasoula. This phenomenon can be explained by the similar cloud regimes amongst these groups. Experiments showed that better results are obtained if cloud absorption is tweaked for each group of stations. Nevertheless, the cloud absorption value for cumulonimbus clouds was found to be good enough to describe the cloud regimes in French Guiana. Thus, we have shown that, despite having dynamic cloud cover, a single cloud albedo and cloud absorption proportionality coefficient can be used at different sites to provide good results.

The scope for future research involves creating a monthly map of cloud absorption, first, for French Guiana, and then, for other tropical areas using several satellite spectral channels and/or ground measurements to identify cloud types [START_REF] Tapakis | Equipment and methodologies for cloud detection and classification: A review[END_REF]. Once the cloud type is known, the cloud absorption can be computed using Liou, (1976). To conclude, our Heliosat-2 method with RTP coupling can be applied universally and at various locations, provided the correct cloud absorption parameterisation, describing the cloud regime at the said location, is used.

III. Summary and conclusion of Chapter II

In Chapter II we presented first the definition of radiometric quantities. Using these definitions we discussed methods used to model the GHI at the ground and how they can be validated against quality control ground measurements. These methods either solve accurately or approximate the RTE. Approximate methods answer a major concern that is to find a good compromise between the accuracy of the GHI estimates and the computation cost. Approximate method can either be classified as physical, semi-physical or statistical parameterizations. A key component for modelling the GHI in the ITZ is the ability of a model to describe cloud coverage and cloud content. Physical parameterizations account for cloud properties; however, they do not account for cloud coverage as statistical satellite methods. For this reason we presented in this chapter a new method that couple a statistical satellite methods (ie, Heliosat-II) and a physical parameterization for cloudy sky conditions. This methodology (RTP_OPT_H2) was validated against ground measurements and the native Heliosat-II (OPT_H2) calibrated with GOES images. We found that the bias improvement of GHI estimates under cloudy sky conditions lies between 25% and 39%. We also found that the improvement under cloudy sky conditions has a direct effect on the GHI estimates under all sky conditions The overall MBE and RMSE changed from mean values (based on all years and stations combined) of -11% and 28%, respectively, for OPT_H2 to -4% and 23%, respectively, for RTP_OPT_H2. Consequently, we have shown that our method improves the GHI estimates under cloudy and under all sky condition in French Guiana. The scope for future research includes extend this methodology to other location in the intertropical zone and provide accurate GHI estimate to improve solar facilities design and siting in in the ITZ.

CHAPTER III : Assessing the accuracy of numerical weather prediction model in the intertropical zone

I.

Forecasting the solar irradiance at the ground I.1 Recommendation for solar energy forecast method depending on the horizon Knowledge of the global horizontal irradiance (GHI), its temporal and spatial distribution is crucial for many purposes, such as electricity production [START_REF] Clack | Modeling Solar Irradiance and Solar PV Power Output to Create a Resource Assessment Using Linear Multiple Multivariate Regression[END_REF], crop growth monitoring [START_REF] Campillo | Solar radiation effect on crop production[END_REF], and energy efficient building design [START_REF] Oldewurtel | Use of model predictive control and weather forecasts for energy efficient building climate control[END_REF]. Several countries, including Australia (Clean Energy Council, 2017), Costa Rica [START_REF] Nandwani | Uses of solar energy in Costa Rica[END_REF], France (Ministry of the environment, 2016b), Germany [START_REF] Lehr | Renewable energy and employment in Germany[END_REF], the United States (O'Connor and Cleveland, 2015), and Sweden (Centeno [START_REF] Centeno López | energy2016_EvaCentenoLopez[END_REF] have begun an energy transition. A primary goal of this energy transition is to increase the share of electricity produced from solar and or wind energy and decrease the share of electricity produced from fuel and or nuclear energy. However, the integration into the grid of electricity generated by intermittent renewable energies present economic and technical challenges (IER, 2013). To ensure grid stability and safety: accurate forecasts of solar irradiance and weather conditions up to 48 hours ahead are required [START_REF] Dambreville | Very short term forecasting of the Global Horizontal Irradiance using a spatio-temporal autoregressive model[END_REF]. Many models have been developed to assess the GHI at the ground depending upon the forecast horizon [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF][START_REF] Glassley | CALIFORNIA RENEWABLE ENERGY FORECASTING[END_REF][START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF][START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF][START_REF] Pelland | Photovoltaic and Solar Forecasting[END_REF]. These methods can be divided into three [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]Ruiz-Arias and Goenka, 2017) For intra hour GHI forecasts the use of sky imagers is suggested [START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. Sky imagers forecast the GHI based on a cloud advection approach. First cloud motion vectors are generated by cross-correlating two consecutive sky images [START_REF] Chow | Intrahour forecasting with a total sky imager at the UC San Diego solar energy testbed[END_REF]. Second the cloud images are then propagated forward in time using the motion vector previously defined. The second step results in a forecast of the cloud position [START_REF] Quesada-Ruiz | Cloud-tracking methodology for intra-hour DNI forecasting[END_REF]. Therefore, using a clear sky model and knowing the location of clouds, a GHI forecast can be obtained. The main challenge of these imagers forecasts are: the cloud detection method [START_REF] Fu | Predicting solar irradiance with all-sky image features via regression[END_REF][START_REF] Yang | An automated cloud detection method based on the green channel of total-sky visible images[END_REF] and the computation of the 3D-cloud map extrapolated from sky imagers images. The main advantage of sky imagers forecasts is that they have higher spatial and temporal resolution than numerical weather prediction and satellite-based forecasts [START_REF] Fu | Predicting solar irradiance with all-sky image features via regression[END_REF]. The temporal resolution is typically inferior or equal to 1 min whereas the spatial resolution is determined by the sky imager field of view [START_REF] Chow | Intrahour forecasting with a total sky imager at the UC San Diego solar energy testbed[END_REF]Marquez and Coimbra, 2013;[START_REF] Quesada-Ruiz | Cloud-tracking methodology for intra-hour DNI forecasting[END_REF][START_REF] Schmidt | Evaluating the spatio-temporal performance of sky imager based solar irradiance analysis and forecasts[END_REF]. Using sky imagers to forecast the GHI Schmidt et al., (2015) found for a forecast horizon of 10 min a RMSE ranging between 70 and 250 W/m². for a 5 minute forecast horizon, [START_REF] Quesada-Ruiz | Cloud-tracking methodology for intra-hour DNI forecasting[END_REF] found that overall the RMSE was 105.26 W/m². For a 15-minute forecast horizon (Marquez and Coimbra, 2013) found a RMSE ranging between 299 and 401 W/m². Marquez et al. (2013) showed that intra hours forecasting using stochastic learning technic is possible, for a 30 min forecast horizon he found out a 1.78 W/m². The main disadvantage of this method is their reliance on training datasets [START_REF] Mathiesen | Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States[END_REF].

For intraday, between 1h and 6h forecast horizon, the use of satellite method is suggested [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF][START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. Nevertheless, stochastic learning technics can be used forecast the GHI up to 2h ahead [START_REF] Pedro | Assessment of forecasting techniques for solar power production with no exogenous inputs[END_REF]. Satellite methods are also based on a cloud motion forecasts approach. They allow a greater forecast range because of their spatial resolution that is coarser than those of sky imagers [START_REF] Diagne | Post-processing of solar irradiance forecasts from WRF model at Reunion Island[END_REF]. Satellite advection forecast extends up to 6h at a resolution of 1 km² ( C. W. [START_REF] Chow | Intrahour forecasting with a total sky imager at the UC San Diego solar energy testbed[END_REF]. The main challenges for satellitebased advection based forecasts are the cloud detection algorithm and cloud height determination [START_REF] Qu | La nouvelle méthode Heliosat-4 pour l'évaluation du rayonnement solaire au sol[END_REF]. [START_REF] Perez | Validation of short and medium term operational solar radiation forecasts in the US[END_REF] found that the RMSE of 6h GHI forecasts using a satellite method ranges between -31% and 38%.

Beyond 6h forecast horizon, cloud advection which is observed through satellite images become less important than cloud development and dissipation. NWP models describe the atmospheric state using conservation equations; therefore they account for dissipation phenomenon [START_REF] Hamill | A Short-Term Cloud Forecast Scheme Using Cross Correlations[END_REF]. Consequently, for forecast horizon larger than 6h, the use of numerical weather prediction (NWP) model is suggested [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF][START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. In this manuscript we aim to forecast the GHI up to 48 h ahead; therefore, we are interested in using NWP to assess the GHI. In the following section we present the different type of NWP models and how they describe the atmospheric state.

I.2 Introduction to Numerical weather prediction models

A numerical weather prediction model is a set of non-linear equations that are discretized and resolved using numerical algorithms [START_REF] Coiffier | Fundamentals of Numerical Weather Prediction[END_REF]. These equations model the physical process that occur in the atmosphere and its interaction with the ocean and land surface (Fig. III.2). These physical processes account for: the transfer of momentum and temperature by advection, convection, smallscale turbulent motions (diffusion) as well as the selective absorption and emission of radiation and the release of latent heat accompanying condensation [START_REF] Schlesinger | Physically-Based Modelling and Simulation of Climate and Climatic Change[END_REF]. According to Jacobson, (2005) Molecular diffusion occur over distances much smaller than 2 mm whereas advection motion such as thunderstorm occur over distances of 2-2000 km. This wide disparity of length scale introduces modeling and computation feasibility issues. To solve this issue NWP models were divided into two categories:

-global NWP models -mesoscale models Global models, on one hand forecast the state of the entire earth atmosphere; they have a typical resolution of 16-50 km. Mesoscale models, on the other forecast the state of a portion of earth atmosphere. The spatial resolution of mesoscale models is defined by the user with respect to the computing power; it usually ranges between 5 and 20 km [START_REF] Diagne | Post-processing of solar irradiance forecasts from WRF model at Reunion Island[END_REF]. Mesoscale models take global model forecasts as initials and boundary conditions inputs so they can account for large scale phenomena; they do not include an ocean model. Yet the resolution of current NWP models is still too coarse to resolve the micro-scale physics associated with cloud formations [START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. Because of NWP model coarse grid resolution, sub-grid scale (i.e unresolved) physical process must be approximated (hereafter, parameterized). Consequently, parameterization are used so that the physical effect of smaller scale processes are accounted for on larger scale [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. Physical parameterization scheme can be divided in five categories [START_REF] Dudhia | A history of mesoscale model development[END_REF][START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF][START_REF] Schlesinger | Physically-Based Modelling and Simulation of Climate and Climatic Change[END_REF][START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF]:

-microphysics (or stratiform-microphysic) -cumulus (or convective) -radiation (or radiative transfer) -surface (land surface-atmosphere) -planetary boundary layer (or turbulent transport)

Microphysics scheme governs the formation of hydrometeors for stratiform clouds, and their growth to precipitation-size raindrops and snow particles. Cumulus scheme intend to represent vertical fluxes and latent heat due to unresolved upward or downward moving air current. Radiation scheme compute absorption by gases as well as scattering and absorption by particles. Surface scheme model the exchange of mass and energy between land surface and the atmosphere with respect to the land-surface properties. The planetary boundary layer scheme represents the small scale turbulence that transports heat and moisture in the vertical, thereby creating the conditions needed for saturation and cloud formation. These parameterizations are not independent, they interact directly as shown Fig.

. [START_REF] Dudhia | A history of mesoscale model development[END_REF] The advantage of global models is that there GHI forecasts are available globally and freely for most cases; they do not require any computing facilities. There drawback is that these forecasts are available on grid resolutions greater than 10 km and with frequency greater or equal to 1-hour. Tab. III.1 shows the spatial and temporal resolution of GHI forecasts from global NWP for that are available for download. Among these models, the uncertainties of IFS and GFS GHI forecasts at the ground are the most well-documented [START_REF] Mathiesen | Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States[END_REF][START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF][START_REF] Remund | Comparison of solar radiation forecasts for the USA[END_REF][START_REF] Schroedter-Homscheidt | Verification of ECMWF and ECMWF/MACC's global and direct irradiance forecasts with respect to solar electricity production forecasts[END_REF][START_REF] Troccoli | Skill of Direct Solar Radiation Predicted by the ECMWF Global Atmospheric Model over Australia[END_REF]. The aforementioned studies gathered data from a total of 43 stations. A large number of these validation stations represent Mediterranean, oceanic, continental, and arid climates, and are located in the United States, Canada. Nevertheless, these models have not been evaluated yet in the ITZ. Therefore, the accuracy the accuracy of IFS and GFS in French Guiana and in tropical climates is unknown.

Fig. III.3 Interaction between physical parameterization. Extracted from

Mesoscale model have finer resolution than global because they are ran for smaller scale domains.

They have the potential to produce more realistic forecasts than lower resolution models by resolving smaller scale processes [START_REF] Lorenz | Comparison of global horizontal irradiance forecasts based on numerical weather prediction models with different spatio-temporal resolutions: Comparsion of irradiance forecasts based on NWP models[END_REF]. Nevertheless, for a mesoscale model to be used it must be carefully calibrated, using physical parameterization that describes the region climate patterns [START_REF] Ruiz-Arias | Evaluation of two MM5-PBL parameterizations for solar radiation and temperature estimation in the South-Eastern area of the Iberian Peninsula[END_REF]. Most studies dealing with mesoscale model calibration concentrate on locations in the USA, Spain and Germany with continental climate [START_REF] Lara-Fanego | Evaluation of the WRF model solar irradiance forecasts in Andalusia (southern Spain)[END_REF][START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF][START_REF] Prabha | Evaluation of solar irradiance at the surface-inferences from in situ and satellite observations and a mesoscale model[END_REF][START_REF] Ruiz-Arias | Surface clear-sky shortwave radiative closure intercomparisons in the Weather Research and Forecasting model: SW CLOSURE INTERCOMPARISONS IN WRF[END_REF][START_REF] Zempila | Evaluation of Wrf shortwave radiation parametrization in predicting global horizontal irradiance in Greece[END_REF]. These countries are amongst the world leaders in installed photovoltaic power capacity [START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF].

Tab. III.1 spatial and temporal resolution of GHI forecast products from global model operated by

forecast center for the first 48 hour forecasts. The temporal and spatial resolution of these products refers to those of datasets available for download. Source (BAM, 2018;CMC, 2016;DWD, 2018;ECMWF, 2016b;Figueroa et al., 2016;[START_REF] Noaa | The Global Forecast System (GFS) -Global Spectral Model (GSM)[END_REF][START_REF] Walters | The Met Office Unified Model Global Atmosphere 6.0/6.1 and JULES Global Land 6.0/6.1 configurations[END_REF] -determine the prevailing physical process in the geographic area of interest -define the horizontal resolution so that physical process are resolved within the model grid -define the vertical resolution so it describe accurately vertical structures such as the boundary layers gradients -define the map projection -validate the model accuracy for different season -study the accuracy of the model with respect to the location of the computation domain and domain size.

-Perform tests to determine the sensitivity of the model accuracy to the vertical and horizontal grid increments Tab. III.2 shows a comparison of commonly used mesoscale model [START_REF] Dudhia | A history of mesoscale model development[END_REF]WMO, 2014). This comparison focuses on the numerical aspect and radiation parameterization options. Tab. III.2 shows that weather and research forecast model (WRF) [START_REF] Skamarock | A Description of the Advanced Research WRF Version 3[END_REF] has the numerical methods with higher order of accuracy. According to [START_REF] Wang | High-order CFD methods: current status and perspective: HIGH-ORDER CFD METHODS[END_REF] high order numerical method achieve better accuracy than low order numerical method for the same grid resolution. WRF [START_REF] Skamarock | A Description of the Advanced Research WRF Version 3[END_REF] model is the most documented, besides it has physical parameterization that allows for feedbacks between aerosol, ozone, cloud and radiation which is not the case for all mesoscale NWP.

Tab. III.2 Comparison of commonly used mesoscale model

Source: for numerics ( [START_REF] Burridge | A split semi-implict reformulation of the Bushby-Timpson 10-level model[END_REF][START_REF] Doms | A Description of the Nonhydrostatic Regional COSMO-Model 164[END_REF][START_REF] Käellen | HIRLAM DOCUMENTATION[END_REF][START_REF] Majewski | [END_REF][START_REF] Ncar | ARW Version 3.8 Modeling System User's Guide January[END_REF][START_REF] Termonia | The ALADIN System and its canonical model configurations AROME CY41T1 and ALARO CY40T1[END_REF][START_REF] Tudor | Forecasting Weather in Croatia Using ALADIN Numerical Weather Prediction Model[END_REF]Undén and co-authors, 2002); for Radiation [START_REF] Doms | A Description of the Nonhydrostatic Regional COSMO-Model 164[END_REF][START_REF] Jimenez | WRF-Solar: Description and Clear-Sky Assessment of an Augmented NWP Model for Solar Power Prediction[END_REF][START_REF] Termonia | The ALADIN System and its canonical model configurations AROME CY41T1 and ALARO CY40T1[END_REF][START_REF] Walters | The Met Office Unified Model Global Atmosphere 6.0/6.1 and JULES Global Land 6.0/6.1 configurations[END_REF][START_REF] Zubler | Intercomparison of aerosol climatologies for use in a regional climate model over Europe: INTERCOMPARISON OF AEROSOL CLIMATOLOGIES[END_REF]

II. Studying the GHI forecast accuracy of global NWP models in the intertropical zone

Assessing GFS and IFS global weather prediction and numerical model forecast accuracy in the intertropical zone and for tropical climates 3

Mouhamet Diallo a,* , Hadrien Verbois b, c, d , Fabrice Chane-Ming e , Antoine Primerose a , Wilfred Walsh c , and Laurent Linguet a the forecast skill (FS) were used to study: (1) the seasonal and yearly accuracy of IFS and GFS GHI forecasts (2) the bias distribution of both models as a function of the clear sky index and the clear sky index variability (3) the effect of temporally averaging IFS native hourly GHI outputs (IFS-1h) to 3hour GHI outputs, and (4) the accuracy of both model forecasts relative to other climates. To compare GFS 3-hour forecast outputs to IFS-1h forecast outputs, we considered two approaches. (i)

We averaged temporally IFS-1h forecasts following GFS cycle to obtain 3-hour forecasts; these 3-hour output forecasts were named IFS-3h-p. (ii)Starting from 00h UTC we kept every 3 hour forecasts of hourly IFS ; these 3-hour output forecasts were named IFS-3h. Results show that under all sky conditions the GHI forecasts of IFS-3h and IFS-3h-p outperform those of 3-hour GFS in all territories. We found that averaging temporally the GHI forecasts using GFS cycle increase the MAE value; under all sky conditions the relative improvement of the MAE of IFS-3h over the MAE of IFS-3h-p ranges between 21% and 46% depending on the territory. The magnitude of the improvement of the MAE of IFS-3h over the MAE of GFS-3h was found to be 48%, 57%, 37% in French Guiana, Reunion Island, and Singapore, respectively. GFS and IFS behave differently with respect to the sky conditions. IFS-3h and IFS-3h-p underestimate the GHI under clear sky conditions and overestimate the GHI under cloudy and overcast sky conditions. Under clear sky conditions GFS underestimates the GHI only in Reunion Island. Under cloudy and overcast sky conditions GFS overestimate the GHI except for Reunion Island where the GHI under cloudy sky is underestimated. We found that except for GFS in Reunion Island, the GHI forecasts of IFS-3h, IFS-3h-p and GFS are more accurate under clear sky conditions than under overcast and cloudy sky conditions; this result is similar to result found in previous study assessing IFS and GFS accuracies in extra tropical climate. However, we also find that the clear sky index variability has a key role on model accuracies which was not addressed previously. The FS computed for GHI 48 hours ahead in Mediterranean, oceanic, continental, and arid climates is superior, for both GFS and IFS, than in tropical regions. This study provides forecast accuracy metrics that aim to respond to the need for reliable solar power forecasts driven by increasing interest in the development of solar technologies in tropical areas.

Introduction

The integration into the grid of electricity generated by intermittent renewable energies presents economic and technical challenges (IER, 2013). To deal with the uncontrollable nature of these resources, they are forecasted at different time scales so that the global production can be balanced to match users' consumption [START_REF] Vallance | Towards a standardized procedure to assess solar forecast accuracy: A new ramp and time alignment metric[END_REF].

Many models have been developed to assess solar radiation at the ground depending upon the forecast horizon [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF][START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF][START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. For forecast horizons greater than six hours, it is widely accepted that numerical weather prediction (NWP) models are needed [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF][START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. These models forecast either the state of the entirety of Earth's atmosphere, or some part of it. The following global NWP models were developed to forecast the state of the entire Earth atmosphere: the Global Forecast System (GFS) (Environmental Modeling Center, 2003), the Integrated Forecast System (IFS) (ECMWF, 2016b), the Unified Model (UM) [START_REF] Staniforth | Unified model documentation paper N°15[END_REF], the Global Environmental Multiscale (GEM) [START_REF] Côté | The Operational CMC-MRB Global Environmental Multiscale (GEM) Model. Part I: Design Considerations and Formulation[END_REF], and the GME [START_REF] Majewski | The operational global icosahedral-hexagonal gridpoint model GME: Description and high-resolution tests[END_REF]. Among these models, IFS [START_REF] Richardson | Evaluation of ECMWF forecasts, including 2012-2013 upgrades[END_REF][START_REF] Haiden | Evaluation of ECMWF forecasts, including 2013-2014 upgrades[END_REF][START_REF] Haiden | Evaluation of ECMWF forecasts, including 2014-2015 upgrades[END_REF][START_REF] Haiden | Evaluation of ECMWF forecasts, including the 2016 resolution[END_REF] and GFS forecast evaluations are the most well-documented [START_REF] Fanglin | Review of GFS Forecast Skills in 2013[END_REF][START_REF] Fanglin | Review of GFS Forecast Skills in 2014[END_REF][START_REF] Fanglin | Review of GFS Forecast Skills in[END_REF].

Many studies have evaluated GFS and IFS solar irradiance forecast uncertainty at the ground by comparison with validation stations [START_REF] Mathiesen | Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States[END_REF][START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF][START_REF] Remund | Comparison of solar radiation forecasts for the USA[END_REF][START_REF] Schroedter-Homscheidt | Verification of ECMWF and ECMWF/MACC's global and direct irradiance forecasts with respect to solar electricity production forecasts[END_REF][START_REF] Troccoli | Skill of Direct Solar Radiation Predicted by the ECMWF Global Atmospheric Model over Australia[END_REF]. The aforementioned studies gathered data from a total of 43 stations. A large number of these validation stations represent Mediterranean, oceanic, continental, and arid climates, and are located in the United States, Canada, and Europe as these regions are among the world leaders for installed photovoltaic power capacity [START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. Therefore, although many studies have evaluated GFS and IFS irradiance estimates at the ground, their performances under several specific climate zones still remain to be evaluated. The present work focuses on assessing the GHI forecast accuracy of GFS and IFS in the intertropical zone for tropical climates. This paper is structured as follows: section 2 describes both the ground and NWP forecast data used in the study; section 3 addresses GFS and IFS monthly and annual accuracy metrics; in section 4, we present our conclusions.

Datasets and Methods

In this section, we first describe the downloaded GHI data and the operative setup used in GFS and IFS models to produce these forecasts. Second, we describe the GHI measurements and the quality check procedure followed prior to using the data to validate IFS and GFS forecasts. Lastly, we describe the climates of French Guiana, Reunion Island, and Singapore as well as the microclimate at each measurement station.

Numerical Weather Prediction models

GFS (GSM v13.0.2) and IFS (cycle 41R2) forecasts used in this study were downloaded online respectively from the National Center for Atmospheric Research data archive ( [dataset] NCEP-NWS-NOAA-USDC, 2015) and the European Center for Medium-Range Weather Forecasts (ECMWF) Meteorological Archival and Retrieval System (MARS) ([dataset] Maass, 2017). The downloaded datasets began from January 1st and ended December 31st of 2016. Both model GHI forecasts originated at 00:00 UTC, and were run for 48 hours. The primary IFS and GFS setups used to produce GHI datasets are presented below. GFS was initialized using a hybrid 3D variational ensemble method [START_REF] Buehner | Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction[END_REF] with a 6-hour data assimilation window. GFS horizontal and vertical computational resolutions were T1534 (~13 km) and 64 levels. GHI forecasts were distributed with a lower resolution; the dataset exploited in this study was 0.25° (~28 km), 46 levels of spatial resolution, and a 3-hour temporal resolution. GFS GHI forecasts (W/m²) alternated between 3-and 6-hour averages [START_REF] Noaa | The Global Forecast System (GFS) -Global Spectral Model (GSM)[END_REF]. GHI values (W/m²) at 03:00, 09:00, 15:00, and 18:00 UTC were the means of 3-hour irradiance values ending at forecast time. GHI values (W/m²) at 06:00, 12:00, 18:00, and 24:00 UTC were the means of 6-hour irradiance values ending at forecast time. IFS was initialized using a 4D variational method, with a 12-hour data assimilation window (ECMWF, 2016c). The horizontal and vertical computational resolution were TCo1279 (~9 km) and 137 levels, respectively. IFS GHI forecasts were also distributed with a lower resolution; the dataset exploited in this study was 0.125°, 125 levels of spatial resolution, and a 1-hour temporal resolution. GHI values were accumulated from the forecast initialization [START_REF] Hogan | Radiation Quantities in the ECMWF model and MARS[END_REF]. Both IFS and GFS used rapid radiative transfer model-G (RRTMG), [START_REF] Mlawer | Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave[END_REF] to parameterize radiative transfer for longwave and shortwave radiations; the scheme was called hourly by the dynamic core of each model. Under cloudless skies, irradiance forecast accuracy is driven by aerosols [START_REF] Janjic | WRF-Solar: An augmented NWP model for solar power prediction. Model description and clear sky assessment[END_REF]. GFS models the effects of aerosols on the troposphere and stratosphere separately (DTC, 2016). In the troposphere, aerosol data are climatological values with a 5° horizontal resolution [START_REF] Chin | Atmospheric sulfur cycle simulated in the global model GOCART: Model description and global properties[END_REF][START_REF] Hess | Optical properties of aerosols and clouds: The software package OPAC[END_REF]. In the stratosphere, climatological aerosol values derived from [START_REF] Sato | Stratospheric aerosol optical depths, 1850-1990[END_REF] were used, and they varied with latitude, and exhibited an uneven resolution greater than or equal to 30° depending upon the zonal band. GFS aerosols were divided into 11 types: insoluble, water soluble, soot, sea salt (2 types), minerals (3 types), transported minerals, sulfate, and stratospheric volcanic aerosols. IFS models aerosols using climatological values [START_REF] Tegen | Contribution of different aerosol species to the global aerosol extinction optical thickness: Estimates from model results[END_REF]; the stratosphere and troposphere are not treated separately as in GFS. The aerosol data used were of six types: organic carbon, soot, sulfate, sea salt, minerals, and stratospheric volcanic aerosols. Aerosol data had 5x4 longitudinal grids, latitudinal spatial resolution, height variation (59 levels), and monthly temporal variations.

Under cloudy skies, irradiance forecast accuracy is driven by cloud cover and content [START_REF] Welch | Solar radiation and clouds[END_REF]. The planetary boundary layer (i.e., turbulent transport) parameterization scheme and the cloud microphysics scheme have a strong influence on the distribution and physical properties of the simulated cloud fields (Cintineo et al., 2014;[START_REF] Otkin | Comparison of WRF Model-Simulated and MODIS-Derived Cloud Data[END_REF]Xie et al., 2012). The microphysics scheme governs the formation of hydrometeors that scatter and absorb radiation and describes their growth to precipitation-size raindrops and snow particles. In the NWP model, microphysics schemes are developed separately for stratiform and cumulus clouds [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. In this study, GFS modeled stratiform microphysics after the scheme of Zhao et al. (1997), which uses prognostic equations for cloud vapor, liquid water, and cloud ice. The precipitation types included rain, freezing rain, and snow. Liquid and ice phases do not coexist in this model. Deep cumulus clouds were parametrized after the bulk mass flux scheme of Arakawa and Schubert (1993). The shallow cumulus scheme used was modified from Arakawa and Schubert (1973) as described by [START_REF] Han | Revision of Convection and Vertical Diffusion Schemes in the NCEP Global Forecast System[END_REF]. Both deep and shallow cumulus schemes use prognostic equations to compute the cloud vapor and liquid water. IFS stratiform microphysics were evaluated after the methods of [START_REF] Forbes | A New Prognastic Bulk Microphysics Scheme for the IFS[END_REF], and this scheme forecasts cloud liquid water, cloud ice, rain, snow and fraction. The precipitation types included rain, freezing rain, and snow. Deep and shallow cumulus microphysics were parametrized by a bulk mass flux scheme originally described by [START_REF] Tiedtke | A comprehensive mass flux scheme for cumulus parameterization in large-scale models[END_REF]. They both use a prognostic equation to compute the cloud vapor and liquid water.

The turbulent transport scheme used parameterizes the sub-grid-scale vertical transfer of heat, moisture, and momentum between the surface and atmosphere. Therefore, it interacts with cumulus microphysics to produce the cloud field (Xie et al., 2012). Vertical turbulent transport was treated differently in the surface layer and above. In the surface layer, the turbulence fluxes were based on the Monin Obukhov similarity theory both for IFS and GFS [START_REF] Dtc | GFS Operational Physics Documentation[END_REF]ECMWF, 2016a). Above the surface layer, IFS used a weakly unstable boundary layer with a K-diffusion turbulence closure (ECMWF, 2016a), whereas GFS used an eddy diffusivity counter-gradient parameterization [START_REF] Hong | Precipitation estimation from remotely sensed imagery using an artificial neural network cloud classification system[END_REF]. For unstable boundary layers, both GFS and IFS used eddy diffusivity mass flux schemes, as described by [START_REF] Han | Implementation in the NCEP GFS of a Hybrid Eddy-Diffusivity Mass-Flux (EDMF) Boundary Layer Parameterization with Dissipative Heating and Modified Stable Boundary Layer Mixing[END_REF] and [START_REF] Köhler | Unified treatment of dry convective and stratocumulustopped boundary layers in the ECMWF model[END_REF], respectively.

Cloud radiation feedback was taken into account by GFS (GSM v13.0.2) and IFS (cycle 41R2) using the values for liquid, ice, and snow water contents from the microphysics scheme and a cloud fraction parameterization [START_REF] Dtc | GFS Operational Physics Documentation[END_REF]ECMWF, 2016a). GFS and IFS assume that each grid box is either entirely filled with clouds or entirely clear for any given time step. A cloud fraction parameterization scheme assigns the fractional volume of a grid box that is occupied by clouds [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. IFS uses a prognostic equation to compute cloud fraction [START_REF] Tiedtke | Representation of clouds in large scale models[END_REF], whereas GFS diagnoses cloud fraction from liquid-water content [START_REF] Xu | A multimodel data assimilation framework via the ensemble Kalman filter[END_REF]. Both GFS and IFS assume maximum cloud overlap.

Measurement stations

To assess GFS and IFS forecast accuracy for tropical climates, we selected GHI measurements from three tropical locations: (1) French Guiana, located in South America between the latitudes of 2° and 6° N, and longitudes of 51 ° and 55° W, (2) Reunion Island, located in the Southwest Indian Ocean (21° S, 55° E), and (3) Singapore, located in Southeast (1.37° N, 103.87° E). Our study exploits hourly GHI data from twelve stations described in Table 1; these data extend from January 1st to December 31st of 2016. First the GHI value before sunrise and after sunset were removed; second the hourly GHI data between sunrise and sunset were quality checked after the methods of [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF] shown in Equation ( 1): 0.03 𝐺𝐻𝐼 𝑇𝑂𝐴 < 𝐺𝐻𝐼 < min(1.2 𝐼 0 , 1.5 𝐼 0 cos(𝜃 𝑧 ) 1.2 + 100),

where 𝐺𝐻𝐼 𝑇𝑂𝐴 (W/m²) is the top of atmospheric irradiance, 𝜃 𝑧 is the sun zenith angle (°), and 𝐼 0 is the solar constant (1367 W/m²). The top of atmospheric irradiance is defined as Equation ( 2):

𝐺𝐻𝐼 𝑇𝑂𝐴 = 𝐼 0 𝜀 cos(𝜃 𝑧 ), (2)
where 𝜀 is the solar correction [START_REF] Mather | Computer processing of remotely-sensed images: an introduction[END_REF].

French Guiana and Reunion Island stations were equipped with type CM6B Kipp and Zonen pyranometers, whereas Singapore stations were equipped with Delta-T SPN1 pyranometers. French weather services in French Guiana and Reunion Island carry out preventive maintenance every two months, and pyranometers are replaced every two years. Singapore pyranometers are operated by the Solar Energy Research Institute of Singapore (SERIS) and are calibrated at the National Metrology Center of Singapore every two years. The GHI values resulting from the quality check procedure (Eq. 1) were then compared to the coincident IFS and GFS time stamps for a 48h forecast horizon. Figure 1 shows the monthly means GHI value for each station with hourly and 3 hourly frequency.

Climate

In tropical regions, seasons are based on the precipitation amount [START_REF] Hess | McKnight's physical geography: a landscape appreciation[END_REF]. Under the Köppen climate classification, a dry season month has an average monthly precipitation amount of less than 60 mm, whereas a rainy season month has precipitation amounts greater than 60 mm. Based on the Köppen climate classification system, tropical climates are divided into three categories: tropical rainforest, tropical monsoon, and tropical savanna [START_REF] Hess | McKnight's physical geography: a landscape appreciation[END_REF]. Tropical rainforest climates have annual precipitation amounts between 150 and 250 cm, and precipitation amounts greater than 60 mm every month. Tropical monsoon and savanna climates have at least one month with precipitation amounts of less than 60 mm. Tropical monsoon climates have annual precipitation amounts between 250 and 500 cm, whereas the annual precipitation of tropical savanna climates is between 90 and 180 cm [START_REF] Hess | McKnight's physical geography: a landscape appreciation[END_REF]. Table 1 shows the climates for stations of validation.

In the following subsections, we describe the climate of each territory, and the microclimates at each of the measurement sites.

French Guiana

The climate of French Guiana is regulated by the Intertropical-Convergence Zone (ITCZ), which is the meeting point of the northeast and southeast trade winds [START_REF] Bovolo | Fine-scale regional climate patterns in the Guianas, tropical South America, based on observations and reanalysis data[END_REF]. The ITCZ moves over French Guiana twice a year, this shift from north to south creates a seasonal cycle with four periods (Philippe Héritier, 2011): (1) From July to November, the ITCZ lies north of French Guiana, and this period represents the dry season during which time the sky is mostly clear, although weak precipitation may occur [START_REF] Albarelo | Optimizing the Heliosat-II Method for Surface Solar Irradiation Estimation with GOES Images[END_REF]. During this period cirrus and cumulus fractus clouds are predominant. ( 2) From November to January, there is a short rainy season when the ITCZ moves southward over French Guiana. (3) From February to March, there is a transition period between the southward and northward motions of the ITCZ, and depending on the year, this month behaves as a rainy or dry. (4) From April to May, the ITCZ moves northward, and a second rainy season begins. During this period, cumulonimbus clouds predominate. Rochambeau Station is located 13 km of the Atlantic Ocean. Maripasoula Station is located inland, ~230 km from the coast. Kourou Station is located, ~2 km from the Atlantic Coast. Saint Georges Station is located and inland, ~50 km from the Atlantic Ocean (Philippe Héritier, 2011).

Reunion Island

Reunion Island is a volcanic island with a complex orography driving several microclimates, but its overall climate is classified as tropical. There are two main seasons, the rainy season (January to March) and dry season (May to November). April and December are transitional months with rainy or dry periods. From November to April, clouds are observed at low and high altitude, below 3.5 km, and above 6.5 km, respectively. High altitude cirrus clouds represent ~13% of the cloud coverage [START_REF] Cadet | A sub-tropical cirrus clouds climatology from Reunion Island[END_REF]. From May to October, there is a high occurrence of low altitude clouds. This cloud cover is three times greater than that observed from November to April, and clouds are denser, and observed at altitudes below 3.5 km [START_REF] Badosa | Scales of spatial and temporal variation of solar irradiance on Reunion tropical island[END_REF][START_REF] Badosa | Reliability of day-ahead solar irradiance forecasts on Reunion Island depending on synoptic wind and humidity conditions[END_REF]. Ligne-Paradis Station is located on the south of the island in the volcanic foothills, ~6 km from the Indian Ocean. Trop. rainforest ~100 m from the Indian Ocean. Gillot Aéroport is located in the north, ~300 m from the ocean. Pierrefonds-Aéroport is located in the southwest, ~500 m from the ocean, and has a tropical savanna climate.

Singapore

Singapore has a tropical rainfall climate characterized by high temperatures and humidity throughout the year. There is no distinct wet or dry season in Singapore; the main seasonal events are the Northeast (NE) and the Southwest (SW) monsoons, separated by two inter-monsoon periods. The NE monsoon lasts from December to early March and is characterized by prevailing north and northeastern winds and is divided into two phases. The wet phase lasts from December to January and sees rapid development of monsoon surges in the afternoon and early evening. The dry phase, from January to early March, is relatively drier. The SW monsoon extends from June to September with prevailing winds from the south and southwest. This period witnesses occasional Sumatra squall lines (SSL) that bring organized lines of thunderstorms to Singapore.

During the two inter-monsoon periods, there is little prevailing wind, and thunderstorms typically develop in the afternoon and the evening. The first inter-monsoon period (late March to May) is usually hotter and relatively drier than the second one (October to November). The four meteorological stations operated by SERIS are located in all cardinal directions of the island. Depending on their locations, each station is located less than 6 km from Singapore or the Johor Strait. Due to the small size of Singapore, the stations are relatively close and all experience the same tropical rainfall climate. However, there are small variations in the hourly GHI received by each station over the year, as illustrated in Figure 1.

Results and discussion

In this section we consider the GHI forecasts at the single grid points closest to each station; we do not average spatially the GHI forecasts of IFS and GFS. We present first the metrics used to assess the accuracy of both IFS and GFS; second we study their yearly accuracies and third their monthly accuracies. Lastly, we compare the accuracy of IFS and GFS in the different tropical climate to their respective accuracy in extra-tropical climates.

Error metrics

IFS and GFS GHI forecasts are compared with in situ measurements. Their performance is evaluated in terms of mean bias error (MBE), mean average error (MAE), root mean squared error (RMSE) and forecast skill (FS), defined as by Equations 3-7, respectively:

𝑀𝐵𝐸 % = ∑ 𝐺𝐻𝐼 𝑁𝑊𝑃 (𝑡)-𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡 (𝑡) 𝑁 ∑ 𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡 (𝑡)
;

(3)

𝑀𝐴𝐸 % = ∑ |𝐺𝐻𝐼 𝑁𝑊𝑃 (𝑡)-𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 (𝑡)| 𝑖 𝑁 ∑ 𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡 (𝑡) ; (4)
𝑅𝑀𝑆𝐸 % = √ 1 𝑛 ∑ (𝐺𝐻𝐼 𝑁𝑊𝑃 (𝑡)-𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 (𝑡)) 2 𝑖 𝑁 ∑ 𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡 , (5)
where 𝐺𝐻𝐼 𝑁𝑊𝑃 refers to the NWP model GHI estimate, 𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 to in-situ observed measurements and N is the total number of data points. RMSE is arguably the most important metric in renewable energy forecasting [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF] because it puts more weight on large forecast errors that have a higher impact on electricity grid management. MAE provides a measurement of global error, and it is suitable for applications where forecast errors are proportional to the induced error cost [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF]. The MBE shows a model tendency to underestimate or to overestimate the GHI. 𝑀𝐵𝐸 % , 𝑀𝐴𝐸 % and 𝑅𝑀𝑆𝐸 % are unitless, they are expressed as percentages (%). Absolute MBE, MAE and RMSE in W/m² are also used; they are obtained by multiplying their relative value by the mean 𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 value over the considered time span.

Persistence forecasts (Eq. 6) were implemented using the 24-hour previous irradiance values. Persistence forecasts serve as references to compute the forecast skills (FS) of the models [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF]:

𝐺𝐻𝐼(𝑡 + 24ℎ) = 𝐺𝐻𝐼(𝑡) (6) 𝐹𝑆 = (𝑅𝑀𝑆𝐸 𝑝𝑒𝑟𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑒 )²-(𝑅𝑀𝑆𝐸 𝑁𝑊𝑃 )² (𝑅𝑀𝑆𝐸 𝑝𝑒𝑟𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑒 )² . ( 7
)
FS allows comparison between different forecast results from different locations, it is unitless. FS uses the absolute RMSE (in W/m²) computed from the GHI estimate of a NWP model. FS = 1 corresponds to a perfect model, while negative scores indicate performance worse than persistence [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF].

The clear sky index 𝐾 𝑐 is used as a proxy for sky conditions, it is unitless. 𝐾 𝑐 > 0.65 indicates a clear sky, 0.4 < K c < 0.65 a cloudy sky, and 𝐾 𝑐 < 0.4 overcast conditions [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF]. 𝐾 𝑐 is defined as follows (Eq. 7):

𝐾 𝑐 = 𝐺𝐻𝐼 𝐺 𝑐 , ( 8
)
where 𝐺 𝑐 is the ESRA clear sky model as defined by [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas -with respect to the heliosat method[END_REF]. Link turbidity, the clear sky attenuation factor input to 𝐺 𝑐 , is defined for each site and each month using monthly climatological values provided by the SoDa database ( [dataset] Transvalor, 2014).

As rapid cloud transformations occur in tropical climates [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF], the mean value of K c , 𝑀 𝐾 𝑐 (Eq. 9), and the mean absolute variability of K c , V (Eq. 10) [START_REF] Marquez | Proposed Metric for Evaluation of Solar Forecasting Models[END_REF] are introduced to study the ability of IFS and GFS to reproduce the GHI variability in tropical climates; both MK c and 𝑉 are unitless:

𝑀 𝐾 𝑐 = 1 𝑁 ∑ 𝐾 𝑐 (𝑡) 𝑡+𝑁 𝑡 , ( 9
) 𝑉 = √ 1 𝑁 ∑ (𝐾 𝑐 (𝑡 + 𝑑𝑡) -𝐾 𝑐 (𝑡)) 2 𝑡+𝑁 𝑡 . ( 10
)
Where dt is the time step of ground measurements, i.e. hourly. For a model with hourly GHI outputs N=1 whereas for a model with a three hour GHI outputs N=2. To illustrate how MK c and 𝑉 were computed, we give the following example for a one hour and a three-hour output model. At 12 h UTC, and for a one hour output model, MK c and 𝑉 are computed using the K c value at 11h UTC and 12 h UTC (N=1). At 12 h UTC, and for a three hour output model, MK c and 𝑉 are computed using the K c value at 10h, 11h UTC and 12 h UTC (N=2).

Assessing the accuracy of hourly IFS forecasts

In this section we study the accuracy of hourly IFS forecasts; we name these forecasts IFS-1h. Table 2 gives the MAE and MBE for each territory under clear, cloudy, overcast and all sky conditions; figure 2 gives IFS-1h MAE distribution with respect to the mean clearness index (𝑀 𝐾 𝑐 )and mean clearness index variability (V).

Table 2 shows that the MAE of IFS-1h under clear sky condition is lower than the MAE of IFS-1h under cloudy and overcast sky conditions. Under clear sky conditions the MAE values range between 100 W/m² and 122 W/m² whereas under overcast and cloudy sky conditions the MAE values range between 127 W/m² and 225 W/m². Consequently IFS-1h forecasts are more accurate in clear sky conditions than in cloudy and overcast sky conditions. Nevertheless, Figure 2 shows that the accuracy of IFS forecasts does not decrease systematically as the sky conditions become clearer; IFS-1h is sensitive to the mean clear sky index and mean clear sky index variability. Table 2 also shows that under clear sky conditions, the MBE are negative for each territory. Oppositely, under cloudy and overcast sky conditions the MBE are positive. Consequently IFS-1h underestimates the GHI value under clear sky conditions whereas under cloudy and overcast sky conditions IFS-1h overestimate the GHI.

Table 2: IFS-1hour absolute MAE and MBE by territory under clear, cloudy, and overcast sky conditions. We computed 𝑀 𝐾 𝑐 (Eq. 9) using N=1, and dt=1hour. Months and stations were merged by territory. The classification of sky conditions was based on ground measurements. Figure 2: IFS-1h MAE distribution with respect to 𝑀 𝐾 𝑐 (Eq. 9) and V (Eq. 10). We computed 𝑀 𝐾 𝑐 (Eq. 9) and V (Eq. 10) using N=1, and dt=1hour. Months and stations were merged by territory. The classification of sky conditions was based on ground measurements

Comparing IFS and GFS yearly irradiance forecast accuracy in French Guiana, Reunion Island and Singapore

To compare IFS-1h outputs to GFS 3 hour outputs (GFS-3h), there are three alternatives. Firstly, the GHI forecasts of GFS can be interpolated to hourly outputs using 𝐾 𝑐 [START_REF] Mathiesen | Evaluation of numerical weather prediction for intra-day solar forecasting in the continental United States[END_REF]. [START_REF] Verzijlbergh | Improved model output statistics of numerical weather prediction based irradiance forecasts for solar power applications[END_REF] found in his study by comparing hourly 𝐾 𝑐 values computed from ground measurement to hourly interpolated 𝐾 𝑐 values obtained using the perfect 3h average forecasts (also computed from ground measurements) that the RMSE was as high as 21%. Consequently, to compare IFS to GFS we did not consider this alternative. Secondly, IFS-1h GHI forecasts can be averaged temporally to obtain three hours forecasts outputs following GFS cycle; we named these forecasts IFS-3h-p. The GHI values of IFS-3h-p (W/m²) at 03:00, 09:00, 15:00, and 18:00 UTC are the means of 3-hour irradiance values ending at forecast time whereas the GHI values of IFS-3h-p (W/m²) at 06:00, 12:00, 18:00, and 24:00 UTC are the means of 6-hour irradiance values ending at forecast time. Lastly, from 00h UTC to 24h UTC we keep only every 3-hour forecasts of hourly IFS; we named these forecasts IFS-3h.

Figure 3 gives the MAE and MBE of IFS-3h-p, GFS-3h and IFS-3h for each territory under clear, cloudy, overcast and all sky conditions. Table 3 gives the Relative improvement of the MAE of IFS-3h over the MAE of IFS-3h-p and GFS-3h for each territory under clear, cloudy, overcast and all sky conditions. Figure 4 gives IFS-3h-p, GFS-3h and IFS-3h MAE distribution with respect to 𝑀 𝐾 𝑐 and V.

Figure 3 shows that for all territories, the MBE of IFS-3h and IFS-3h-p are negative under clear sky conditions whereas under cloudy and overcast sky conditions the MBE of both are positive. Consequently, IFS-3h and IFS-3h-p under estimate the GHI under clear sky conditions whereas under cloudy and overcast sky conditions both overestimate consistently the GHI. Oppositely, the MBE of GFS-3h is consistently positive in Singapore and French Guiana. In Reunion Island, the MBE is positive under overcast sky conditions and negative under clear and cloudy sky conditions. Consequently, GFS-3h overestimates the GHI consistently in Singapore and French Guiana; in Reunion Island GFS-3h overestimates the GHI under overcast sky conditions whereas it underestimates the GHI under clear and cloudy sky conditions.

Figure 4 shows that the MAE of IFS-3h-p, GFS-3h and IFS-3h are highly correlated to 𝑀 𝐾 𝑐 and V. This figure shows that except for GFS in Reunion Island, each model has a tendency to perform better under clear sky conditions. To conclude on the accuracy of each model with respect to the sky condition, we gather each GHI sample according to sky condition type that is reported by the ground measurements (Figure 3).

The MAE values (Fig. 3) show that for each territory IFS-3h and IFS-3h-p are more accurate in clear sky conditions and least accurate in cloudy and overcast sky conditions (Fig. 3). Oppositely, we find that in Reunion Island, GFS-3h MAE of clear sky condition forecasts is higher than the MAE of cloudy and overcast sky conditions forecasts. Under clear sky conditions, the MAE of GFS-3h is 319 W/m² whereas under cloudy and overcast sky conditions it is 249 W/m² and 242 W/m², respectively. Under all sky conditions and for each territory, the MAE of IFS-3h-p is lower than the MAE of GFS-3h (Fig. 3). Figure 3 also shows that for each territory and under all sky conditions the MAE of IFS-3h is lower than the MAE of IFS-3h-p. Table 3 shows that under all sky conditions and for each territory, the relative improvement of the MAE of IFS-3h over IFS-3h-p ranges between 21% and 46%. This result contrasts with [START_REF] Lorenz | Comparison of global horizontal irradiance forecasts based on numerical weather prediction models with different spatio-temporal resolutions: Comparsion of irradiance forecasts based on NWP models[END_REF] who found that averaging temporally improve the accuracy of NWP GHI outputs. Therefore, we conclude that ( 1) averaging temporally IFS-1h following GFS cycle to obtain IFS-3h-p worsen the MAE in tropical climate and ( 2) that both IFS-3h and IFS-3h-p are more accurate than GFS.

IFS-3h GFS-3h IFS-3h-p

Figure 4: IFS and GFS bias distributions with respect to the mean K c values (Mkc) of the previous three hours, and the mean of the K c Variability (V). We computed 𝑀 𝐾 𝑐 (Eq. 9) and V (Eq. 10) using N=2, and dt=1hour. Months and stations were merged by territory. The classification of sky conditions was based on ground measurements

Analyzing the sources of error of IFS and GFS GHI forecasts

We have shown in section 3.3 that the accuracy of IFS-3h-p, IFS-3h and GFS-3h forecasts was related to the K c value. Consequently, to explain the sources of IFS and GFS GHI forecasts errors in tropical climates, we analyze and compare the K c values computed using ground measurement and the K c value computed using IFS-3h-p, IFS-3h and GFS-3h GHI forecasts. Figure 5 shows a comparison between the sky conditions measured at the ground i.e clear, cloudy or overcast to the sky conditions forecasted by the NWP model (i.e IFS-3h-p, IFS-3h or GFS-3h). This comparison uses the cdf of K c values computed using ground measurements and the cdf K c values forecasted by the NWP GHI forecasts.

Figure 5: cdf of K C computed using ground measurements and IFS-3h-p, GFS-3h and IFS-3h forecasts under clear, cloudy and overcast sky conditions. Months and stations were merged. The classification of sky conditions was based on ground measurements

IFS-3h

IFS-3h-p

GFS-3h

Ground

Analyzing the sources of error under clear sky condition

Figure 5 shows that under measured clear sky conditions, for IFS-3h, there are 77%, 87% and 78% of the forecasted K C values that are higher than 0.65 in French Guiana, Reunion Island and Singapore, respectively. Consequently, when the sky condition is reported as clear by the ground measurements, it is reported as clear by IFS-3h between 77% and 87% of the time.

For GFS-3h, under measured clear sky conditions, Figure 5 shows there are 69%, 32% and 80% of the forecasted K C values that are higher than 0.65 in French Guiana, Reunion Island and Singapore, respectively. Consequently, when the sky condition is reported as clear by the ground measurements, it is reported by GFS-3h as clear between 32% and 80%.

For IFS-3h-p, under measured clear sky conditions, Figure 5 shows there are 56%, 57% and 40% of the forecasted K C values that are higher than 0.65 in French Guiana, Reunion Island and Singapore, respectively. Consequently, when the sky condition is reported as clear by the ground measurements, it is reported by IFS-3h-p as clear between 40% and 57% of the time.

Therefore, IFS-3h-p, GFS-3h and IFS-3h are unable to forecast realistic cloud cover and or amount for sky conditions reported as clear by the ground measurements. Consequently, this inability contributes to the source of errors under clear sky. To quantify the source of errors due to sky conditions that are reported as clear by the ground measurements, but forecasted otherwise by IFS-3h-p, GFS-3h and IFS-3h; firstly, we computed first the MAE when IFS-3h-p forecast and the ground measurements assess simultaneously that the sky conditions are clear. Secondly, we computed the MAE when GFS-3h forecasts and the ground measurements assess simultaneously that the sky conditions are clear. Thirdly, we computed the MAE when IFS-3h forecast and the ground measurements assess simultaneously that the sky conditions are clear.

Firstly, we found that IFS-3h-p has a MAE of 123 W/m², 133 W/m² and 134 W/m² in French Guiana, Reunion Island and Singapore respectively. These values represent 78%, 58% and 61%, respectively of clear sky MAE values that were shown figure 3. Consequently, the source of errors due to sky conditions that are reported as clear by the ground measurements, but forecasted otherwise by IFS-3h-p represent between 22% and 39%.

Secondly, we found that GFS-3h has a MAE of 139 W/m², 131 W/m² and 172 W/m² in French Guiana, Reunion Island and Singapore respectively. These values represent 81%, 41% and 77%, respectively of clear sky MAE values that were shown figure 3. Consequently, the source of errors due to sky conditions that are reported as clear by the ground measurements, but forecasted otherwise by GFS-3h represent between 19% and 59%, Thirdly, we found that IFS-3h has a MAE of 69 W/m², 80 W/m² and 96 W/m² in French Guiana, Reunion Island and Singapore respectively. These values represent 64%, 68% and 63%, respectively of clear sky MAE values that were shown figure 3. Consequently, the source of errors due to sky conditions that are reported as clear by the ground measurements, but forecasted otherwise by IFS-3h represent between 32% and 37%, Therefore, the sky conditions that are reported as clear by the ground measurements, but forecasted otherwise by IFS-3h-p, GFS-3h and IFS-3h lead to sizable MAE values; for GFS-3h in Reunion Island they are as high as 59% of the overall MAE.

According to [START_REF] Zhong | Dissecting surface clear sky irradiance bias in numerical weather prediction: Application and corrections to the New Goddard Shortwave Scheme[END_REF] : "often, but not always clear sky model biases are related to aerosol modeling issues"; for [START_REF] Janjic | WRF-Solar: An augmented NWP model for solar power prediction. Model description and clear sky assessment[END_REF] aerosols represent the largest source of uncertainty in the GHI forecasts under clear sky conditions. Consequently, we could explained the MAE when IFS-3h-p or GFS-3h or IFS-h forecasts and the ground measurements assess simultaneously that the sky conditions are clear by a misrepresentation of the aerosols.

To conclude IFS-3h-p, IFS-3h, and GFS-3h sources of error under clear sky conditions may include: (1) sky conditions reported as clear by the ground measurements that are inaccurately forecasted as cloudy or overcast and ( 2) an inaccurate representation of the aerosols.

Analyzing the sources of error under cloudy and overcast sky conditions

Figure 5 shows that for all territories considered, when the sky condition is reported as cloudy or overcast by the ground measurements, it is inaccurately forecasted by IFS-3h-p, GFS-3h and IFS-3h.

In French Guiana, under measured cloudy sky conditions and for IFS-3h-p, figure 5 shows that 26% of forecasted K C values are lower than 0.4 whereas 32% of the forecasted K C are higher than 0.65. Therefore, the match between sky conditions type that is measured at the ground and sky conditions type that is forecasted by IFS-3h-p is 6%. In French Guiana, under measured overcast sky conditions, figure 5 shows that 23% of forecasted K C values are higher than 0.4. Therefore, the match between sky conditions type that is measured at the ground and sky conditions type that is forecasted by IFS-3h-p is 23%.

Using the same procedure described previously, for each territory and each NWP model: we find that under cloudy and overcast sky condition, the match between sky conditions measured at the ground and sky conditions forecasted by any NWP model (eg. IFS-3h-p, GFS-3h and IFS-3h) is lower than 50%. Consequently, these results may be explained by the inability of the NWP to either forecast accurately the cloud position and or extent. Similar conclusions were drawn by [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF], [START_REF] Zempila | Evaluation of Wrf shortwave radiation parametrization in predicting global horizontal irradiance in Greece[END_REF] and [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]; they explained high biases under cloudy and overcast skies were caused by the inability of NWP models to accurately predict cloud position and cloud extent. [START_REF] Lin | Simulation of low clouds from the CAM and the regional WRF with multiple nested resolutions[END_REF] demonstrated that increasing the horizontal resolution of NWP models improves the accuracy of the simulated low cloud field; they concluded that clouds can be realistically simulated with horizontal resolutions higher than 4 km, but not with coarser resolutions. Consequently, we could explained the forecast accuracy of IFS-3h, IFS-3h-p and GFS-3h under cloudy and overcast sky conditions by their coarse resolution; this coarse resolution causes a misrepresentation of the sky conditions which lead to high MAE value.

IFS and GFS seasonal forecast accuracy assessment

In this section, seasonal forecast accuracies of both 3-hour IFS and GFS are assessed using the metrics described in section 3.1. To obtain the percent RMSE and MAE relative values, absolute value in W/m² were normalized using the monthly mean values shown in Figure 1, and following Equations 4 and 5.

French Guiana Sites

Figure 6 gives on the one hand IFS and GFS monthly RMSE and MAE, figure 7 gives on the other the monthly mean K c value.

Figure 6 shows that both GFS and IFS GHI forecasts follow seasonal trends for French Guiana (i.e., there is a clear distinction between dry and rainy seasons). RMSE and MAE during the dry season (i.e., July to November) are on average lower than those found in rainy season (i.e., from November to January and from April to May). The different forecast accuracies found may be explained by different clearness index value that varies according to the seasons (Fig. 7).

During the dry season, Kourou station exhibits the lowest average RMSE and MAE. Figure 7 shows that during this season, Kourou station exhibits the highest monthly mean K c values. During the rainy seasons, Maripasoula exhibits lowest average RMSE and MAE. Figure 7 shows that during these seasons, Maripasoula have on average the highest monthly mean K c values, respectively. These results suggest that there is a correlation between the monthly mean forecast accuracy and the monthly mean K c value. Consequently IFS-3h and GFS-3h GHI forecasts are less accurate in cloudier sky conditions; the accuracy of IFS-3h and GFS-3h GHI forecasts are also influenced by the variability of the sky conditions.

Reunion Island sites

Figure 8 gives on the one hand IFS and GFS monthly RMSE and MAE, figure 9 gives on the other the monthly mean K c value.

When comparing figure 9 to Figure 7, it can be seen that, unlike in French Guiana, there is no distinct wet or dry season on Reunion Island. Figure 9 shows that the difference between monthly mean Kc values during the rainy and dry season is less visible. According to the French Weather Service, 2016 ranks 13th amongst the driest years since 1960. For Station PI, during the dry season, the lowest GFS RMSE and MAE (60% and 49%, respectively) are found in August; the lowest IFS RMSE and MAE (24% and 14%, respectively), they are found in June. For this station, the highest monthly mean K c is found in June. Consequently, the monthly RMSE and MAE of the GHI forecasts of GFS are not as strongly correlated with monthly mean K c as they are in French Guiana. . From January to August, station SC has the highest monthly mean K c value, as seen in figure 11. During this time span, this station exhibits overall the lowest RMSE and MAE. Consequently, IFS-3h and GFS-3h GHI forecasts are correlated to the monthly mean K c value and they are less accurate in cloudier sky conditions. To compare the forecast accuracy for the different tropical climates we used IFS-1h forecasts because we showed that this model was able to reproduce the seasonal variation of GHI contrary to GFS-3h. Table 4 shows that overall the highest RMEs are found in tropical rainforest climate followed by tropical monsoon and tropical savanna climate. Nevertheless, there are discrepancies in the GHI forecast accuracies between stations sharing the same tropical climate type (Fig. 12). We have shown in section 3.3 that the forecast accuracy of IFS and GFS were highly correlated with the MK c and V. Consequently, to explain the results shown table 4 we analyze the cdf of MK c and V for each climate type (Fig. 13).

Figure 13 presents the cdf of M c and V for stations with tropical monsoon, savanna and rainforest climate.We find, based on the analysis of the cdf MK c shown figure 13, that stations with tropical rainforest stations have the highest occurrence of cloudy and overcast sky conditions sample followed by stations with tropical monsoon and savanna climate. For stations with tropical rainforest climate 50% of the sample has MK c values lower than 0.70. For stations with tropical monsoon and tropical savanna climate 50% of the sample has K c values lower than 0.75 and 0.85, respectively. Similarly the analysis of the cdf of V, shows that stations with tropical rainforest stations have the highest hourly mean variability followed by stations with tropical monsoon and savanna climate. computed 𝑀 𝐾 𝑐 (Eq. 9) and V (Eq. 10) using N=1, and dt=1hour

To compare the accuracy of the different tropical climates, the FS was also used. The FS allows the comparison of the forecast accuracy of a NWP model to that of a persistence model (Eq. 6) at different locations [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF]. A negative FS value indicates that the NWP underperformed in comparison to the persistence model. A FS value close to 0 indicates that the NWP model forecast accuracy is similar to the minor persistence model forecast accuracy. A FS value close to 1 indicates that the NWP model outperforms the persistence model. Table 4 shows, by comparison with the persistence model, that it is more difficult for IFS-1h to forecast the GHI 48 hours in advance for tropical monsoon, savanna and rainforest in decreasing order.

Comparing GFS and IFS forecast accuracy in tropical climates to their forecast accuracy in extratropical climates

Table 5 gives the yearly FS and RMSE for different climate zones; we used this table to compare the GHI forecasts accuracy of IFS and GFS found in tropical climates to those found in extratropical climates.. The FS metric was not computed in all previous studies; when it was not available, it was computed if the absolute 24-hour persistence and 48-hour forecast RMSE were given by the authors (e.g., [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF][START_REF] Schroedter-Homscheidt | Verification of ECMWF and ECMWF/MACC's global and direct irradiance forecasts with respect to solar electricity production forecasts[END_REF][START_REF] Troccoli | Skill of Direct Solar Radiation Predicted by the ECMWF Global Atmospheric Model over Australia[END_REF]. FS and RMSE ranges were given for stations using the lowest and highest FS values, and the lowest and

IFS-3h FS

GFS-3h FS

IFS-1f FS

IFS-3h RMSE

GFS-3h RMSE

highest RMSE values, respectively. GFS-based models NDFD and BLUE FORECAST are detailed in [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF]. Every NWP was run for 48 hours, and was initialized at 00:00 UTC. The FS and RMSE values that were computed for tropical climates are shown Figure 12.

Table 5 shows that the 48h forecasts of GFS-3h have higher RMSE in tropical climates than in extratropical climate. The RMSEs of IFS-3h in tropical climate are in agreement with those found in extratropical climates. In Mediterranean, arid, semi-arid, continental, and humid continental stations, all GFS and IFS FS scores are positive. Among all tropical climate stations considered in Figure 12, GFS has a positive FS for only 1 station out of 12 (i.e., SQ, where GFS GHI forecast accuracy is similar to the accuracy of persistence models, and the FS score is 0.04). Conversely, IFS-3h FS is always positive; however, FS values found in tropical climates are lower than values found in other climates. Consequently, forecasting the GHI in tropical climate is more difficult for IFS and GFS. The lower GFS FS results might be explained by the coarser spatial resolution of GFS. This resolution may not be able to reproduce the higher variability of solar energy due to the dynamic cloud cover and hot and humid weather of tropical climates [START_REF] Laing | Introduction to tropical meteorology, Educational material from The COMET Program[END_REF][START_REF] Galvin | An Introduction to the Meteorology and Climate of the Tropics[END_REF]. Recently, [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF] and [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF] downscaled 0.5° GFS model forecasts using Weather and Research Forecast [START_REF] Skamarock | A description of the advanced research WRF version 2[END_REF] limited-area NWP to forecast GHI on Reunion Island and in Singapore. Both studies forecasted the GHI hourly, on a 3 km resolution grid size. The yearly FS were computed from the RMSE values found by [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF] and [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF]; these FS were -0.177, 0.07, and 0 for the tropical savanna station (Saint Pierre) and the tropical rainforest stations (station 305 and 500), respectively. These results are worse than those found in this study in tropical savannas (i.e., 0.32, 0.44), and tropical rainforest climates (i.e., 0.25, 0.50) using IFS-1h GHI forecasts, despite having a higher spatial resolution. Therefore, using IFS global NWP for GHI forecast is relevant in tropical climates. Using IFS and GFS global NWP forecasts, one might avoid calibrating a limited area NWP model, instead using physical parameterization that describes regional climate patterns. This process requires high computational resources, and does not guarantee better forecast results [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF].

Conclusions

The goal of this study was to assess the accuracy of GFS and IFS global NWP model GHI forecasts for tropical climates. As the solar energy potential of tropical countries can be up to three times greater than that of extra-tropical countries, yet more variable due to the dynamic cloud cover [START_REF] Fillol | Spatiotemporal indicators of solar energy potential in the Guiana Shield using GOES images[END_REF], predicting the yield of solar energy systems and their input into the electricity grid up to 48 hours in advance is crucial. Therefore, knowledge about the accuracy of the GHI forecasts from NWP models is needed.

To characterize GFS and IFS forecast accuracy for tropical climates, three countries were selected for study. Our results reveal that: (1) Higher RMSEs are found in tropical rainforest climate, followed by tropical monsoon, and tropical savanna climates. This result was explained by the higher occurrence of cloudy and overcast sky conditions sample found in stations with tropical rainforest climate followed by stations with tropical monsoon and savanna climates. ( 2) GFS and IFS behave differently with respect to the sky conditions. IFS underestimates the GHI under clear sky conditions and overestimate the GHI under cloudy and overcast sky conditions. Under clear sky conditions GFS underestimates the GHI only in Reunion Island. Under cloudy and overcast sky conditions GFS overestimate the GHI except for Reunion Island where the GHI under cloudy sky is underestimated. The inaccuracies under clear sky conditions were explained by the combination of two phenomena: the misrepresentation of aerosols and the inability of model forecast inaccurately clouds under measured clear sky. This inability to forecast realistic clouds cover also translates in cloudy and overcast sky conditions. (3) By comparing the MAE of IFS-3h and the MAE of IFS-3h-p we found that using the average cycle of GFS to obtain 3-hour worsen the model forecasts. ( 4) IFS and GFS model biases under clear sky conditions were lower than their biases under cloudy and overcast sky conditions. However, the biases of both models do not decrease systematically as the sky became clearer; the sky index variability also plays an important role in both model biases. ( 5) IFS-3h, IFS-3h-p outperforms the 3-hour GFS model in all countries. IFS-3h outperforms GFS-3h to a higher extent. Under all sky conditions, with all months and stations merged by territory, the magnitude of improvement for IFS-3h over GFS-3h were found to be 48%, 57%, 37% in French Guiana, Reunion Island, and Singapore, respectively. This study fills the gap of global NWP models GHI forecast accuracy in tropical climates by studying two of the most widely used NWPs, IFS and GFS, for various types of tropical climates. Comparing the FS score for tropical climate stations to those of Mediterranean, arid, semi-arid, continental, and humid continental stations, we found that forecasting solar irradiance is more difficult in tropical climates. By comparing the FS found in this study with recent studies that calibrated limited-area NWP to forecast GHI on Reunion Island and in Singapore, we showed that using IFS global NWP for GHI forecast is relevant in tropical climates.

Nevertheless, GHI forecast accuracy in tropical climate regions is too poor to be exploited directly; they must be post-processed. Our future research will involve using post-processing techniques [START_REF] Verbois | Solar irradiance forecasting in the tropics using numerical weather prediction and statistical learning[END_REF] to further improve GHI forecast accuracy. The current study may be useful in the following ways: (1) as a benchmark tool to assess the forecast accuracy of a limited area NWP model GHI forecast, and (2) as a reference for post-processing or data assimilation methods to characterize the improvement obtained when forecasting GHI in tropical climates.

Introduction

The world current energetic model is unsustainable; it relies primarily on fossil fuel powered energy [START_REF] Shafiee | When will fossil fuel reserves be diminished?[END_REF]. According to BP Statistical Review, (2017) oil make up the third of the 2016 primary energy consumption and coal share is approximately 28.1%. However, oil resources are expected to be depleted by 2040, the remaining fossil fuel resources by 2212 [START_REF] Shafiee | When will fossil fuel reserves be diminished?[END_REF]. The combination of increasing energy needs [START_REF] Birol | World energy outlook 2010[END_REF], increasing concern for global warming [START_REF] Pachauri | Changements climatiques 2014: rapport de synthèse : contribution des Groupes de travail I, II et III au cinquième Rapport d'évaluation du Groupe d'experts intergouvernemental sur l'évolution du climat[END_REF] fossil fuel depletion [START_REF] Shafiee | When will fossil fuel reserves be diminished?[END_REF] and cost decrease (IRENA, 2016) established the solar energy as a solution to meet the needs. Regions with higher solar potential are located in the intertropical zone (ITZ) [START_REF] Löf | World distribution of solar radiation[END_REF][START_REF] Müller | Surface Solar Radiation Data Set -Heliosat (SARAH) -Edition[END_REF][START_REF] Trieb | Global Potential of Concentrating Solar Power[END_REF]. Nevertheless, the solar irradiance in the ITZ is highly variable due to the rapid cloud transformations that occur in this area [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF][START_REF] Rossow | Tropical Precipitation Extremes[END_REF][START_REF] Wheeler | Convectively Coupled Equatorial Waves: Analysis of Clouds and Temperature in the Wavenumber-Frequency Domain[END_REF]. To deal with the uncontrollable nature of the solar energy, the available resource must be forecasted at different time scales so that the global production can be balanced to match users' consumption [START_REF] Vallance | Towards a standardized procedure to assess solar forecast accuracy: A new ramp and time alignment metric[END_REF]. It is commonly accepted that for forecast horizon larger than 6h, numerical weather prediction (NWP) models are needed [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF][START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. They are two types of NWP model, global and mesoscale. Global model forecast the state of the entire earth atmosphere; they have a typical resolution of 16-50 km, with a forecast horizon up to 15 days ahead [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF]. Global NWP models have a coarse resolution, to account for local effect mesoscale models were introduced. They have a typical resolution of 5-20 km [START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF] and forecast the state of the atmosphere on a regional domain. They take global NWP model forecasts as initials and boundary conditions inputs. NWP models with high spatial resolution have the potential to produce more realistic cloud forecasts than lower resolution models (Lorenz et al., 2016) by resolving explicitly smaller scale processes. Yet the resolution of current NWP model is still too coarse to resolve the micro-scale physics associated with cloud formations [START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF][START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF]. Because of NWP model coarse grid resolution, sub-grid scale physical process must be approximated (hereafter, parameterized) so that effects of smaller scale processes are accounted for on larger scale [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. For a mesoscale model to be used it must be carefully calibrated, using physical parameterization that describes the region climate patterns [START_REF] Ruiz-Arias | Evaluation of two MM5-PBL parameterizations for solar radiation and temperature estimation in the South-Eastern area of the Iberian Peninsula[END_REF]. The parameterization schemes express the effect of sub grid, subscale processes on the variables resolved by the NWP spatial resolution [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF]. Most studies dealing with mesoscale models calibration concentrate on locations in the USA, Spain and Germany with continental climate [START_REF] Lara-Fanego | Evaluation of the WRF model solar irradiance forecasts in Andalusia (southern Spain)[END_REF][START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF][START_REF] Prabha | Evaluation of solar irradiance at the surface-inferences from in situ and satellite observations and a mesoscale model[END_REF][START_REF] Ruiz-Arias | Surface clear-sky shortwave radiative closure intercomparisons in the Weather Research and Forecasting model: SW CLOSURE INTERCOMPARISONS IN WRF[END_REF][START_REF] Zempila | Evaluation of Wrf shortwave radiation parametrization in predicting global horizontal irradiance in Greece[END_REF]. These countries are amongst the world leaders in installed photovoltaic power capacity [START_REF] Inman | Solar forecasting methods for renewable energy integration[END_REF]. Tropical regions have been less studied [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF][START_REF] Diagne | Post-processing of solar irradiance forecasts from WRF model at Reunion Island[END_REF] and are more challenging due to the dynamic weather. Recently [START_REF] Diallo | Assessing GFS and IFS global numerical weather prediction model forecast accuracy under tropical climate[END_REF] studied the accuracy of GFS and IFS global horizontal irradiance (GHI) forecast in the ITZ using measurement forecast from French Guiana, Reunion Island and Singapore. They showed that forecasting the GHI is more difficult in tropical climate than in extra tropical.

In this study is we propose a methodology to calibrate the Weather and Research Forecast (WRF) mesoscale model to produce improved GHI forecast in the intertropical zone. This methodology uses an ensemble of 34 members with high-resolution model simulations. It aims to restrain and select the minimum number of simulations to run, to obtain improved GHI forecasts in the ITZ compared to a non-calibrated model. We validated this methodology against GHI measurements during 2016 from six meteorological stations and AROME "outre-mer" (AROME-OM) GHI forecasts in French Guiana. This paper is structured as follows: Section 2 describes the dataset used to validate WRF GHI forecast. Section 3 describes the methodology followed to build the ensemble members (EM). Section 4 presents the results of this study. Finally section 5 summarizes the paper and concludes.

Data

Ground Dataset

Our study exploits 2016 hourly data of GHI from six stations of the French national weather services. The characteristics of these stations are described table 1. Due computational capacity constraints associated with the computation of the 34 EM, three months representative of French Guiana seasons were selected. September represents the dry season, May the rainy season and March to describe the latency period between the ITCZ southward to northward motion over French Guiana. Before comparing the NWP GHI forecast to the ground data, ground measurements were quality checked following [START_REF] Espinar | Project ENDORSE-Excerpt of the report on the harmonization and qualification of meteorological data: Procedures for quality check of meteorological data[END_REF] range check formula as follows:

0.03𝐺𝐻𝐼 𝑇𝑂𝐴 < 𝐺𝐻𝐼 < 𝑚𝑖𝑛(1.2𝐼 0 , 1.5𝐼 0 𝑐𝑜𝑠(𝜃 𝑧 ) 1.2 + 100), (1)
where 𝐺𝐻𝐼 𝑇𝑂𝐴 (W/m²) is the top of atmosphere irradiance, 𝜃 𝑧 the sun zenith angle (°),𝐼 0 the solar constant (1367 W/m²). The top of atmosphere irradiance is defined as follows:

𝐺𝐻𝐼 𝑇𝑂𝐴 = 𝐼 0 𝜀𝑐𝑜𝑠(𝜃 𝑧 ), (2)
where 𝜀 is the sun correction [START_REF] Mather | Computer processing of remotely-sensed images: an introduction[END_REF]. Only valid hourly GHI measurements were used to assess the accuracy of WRF and AROME; when the hourly GHI measurements passed the quality check procedure it was then compared to the coincident NWP time stamp to compute the accuracy metrics. [START_REF] Seity | The AROME-France Convective-Scale Operational Model[END_REF]. It was developed to better characterize convection processes and used operationally since December 2008. In this study the authors used AROME "Outre mer" (AROME-OM). AROME-OM produces meteorological forecast in French West Indies while AROME-France which produces meteorological forecasts in mainland France. Unlike AROME France it doesn't have its own data assimilation system, it is initialized using ARPEGE forecast as lateral boundary conditions and IFS forecast as initial conditions. AROME-OM GHI forecasts used in this study were downloaded online from French weather services catalogue servers (Meteo France, 2017). All forecasts originated at 00:00 UTC and were run for 36h. The main setups used to produce AROME-OM GHI forecasts are shown table 2. -Turbulent kinetic energy scheme [START_REF] Cuxart | A turbulence scheme allowing for mesoscale and large-eddy simulations[END_REF], associated with a mass flux scheme for shallow convection [START_REF] Bechtold | A mass-flux convection scheme for regional and global models[END_REF] -Call frequency: 1 min Land surface scheme (LSM)

-ISBA scheme [START_REF] Noilhan | A Simple Parameterization of Land Surface Processes for Meteorological Models[END_REF]) Cumulus scheme (Cu) -Eddy diffusivity max flux approach (Pergaud et al., 2009) -Call frequency: 1 min

Methodology

According to Warner, (2010) calibrating a NWP model involves the following steps: (1) determine the prevailing physical process in the geographic area of interest; (2) define the horizontal resolution so that physical processes are resolved within the model grid;

(3) define the vertical resolution so it describes accurately vertical structures such as the boundary layers gradients; (4) validate the model accuracy for different seasons. Consequently, the following subsections explain the calibration process for WRF in the ITZ.

Physical parameterization

IFS and GFS NWP models GHI forecasts are inaccurate in the ITZ; their accuracies are highly sensitive to the cloudiness and its variability [START_REF] Diallo | Assessing GFS and IFS global numerical weather prediction model forecast accuracy under tropical climate[END_REF]. Results obtained under measured clear sky conditions were explained by a misrepresentation of aerosols and by clouds wrongly forecasted by the NWP. Under cloudy sky, they were explained by the inability of the NWP model to predict accurately cloud position, extent and content due to the NWP model coarse resolution. From this investigation on the origin of IFS and GFS inaccuracies, we concluded that to improve NWP GHI forecast in the ITZ, sub-grid scale phenomena such as clouds, aerosols and their interaction with radiation should be better characterized.

Parameterization schemes allow modeling the effect of a sub-grid scale physical process; however, their accuracies highly depends on the weather regime and parameterization schemes interactions in the NWP model [START_REF] Cossu | Influence of microphysical schemes on atmospheric water in the Weather Research and Forecasting model[END_REF][START_REF] Dudhia | A history of mesoscale model development[END_REF]. Therefore, in order to find the set of parameterization that forecast the most accurately the GHI in the ITZ the authors built and computed an ensemble of several 48h GHI forecasts using WRF V3.8. 1 (NCAR, 2017;[START_REF] Skamarock | A Description of the Advanced Research WRF Version 3[END_REF].

The 34 EM used to calibrate WRF 48h GHI forecast, varied the MPs scheme, LSMs scheme, and PBLs schemes. The ITZ is an area with highly variable dynamic in which we encounter significant amounts of convective clouds [START_REF] Galvin | An Introduction to the Meteorology and Climate of the Tropics[END_REF][START_REF] Laing | Introduction to tropical meteorology, Educational material from The COMET Program[END_REF]. The parameterization scheme we chose to vary have a strong influence on the distribution and physical properties of the simulated cloud field (Cintineo et al., 2014;[START_REF] Otkin | Comparison of WRF Model-Simulated and MODIS-Derived Cloud Data[END_REF]Xie et al., 2012). The EM also includes set of parameterizations previously used in Singapore and Reunion tropical climate by [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF] and [START_REF] Diagne | Post-processing of solar irradiance forecasts from WRF model at Reunion Island[END_REF] respectively; they are shown table 3. The MPs scheme, LSMs scheme, and PBLs schemes varied in this study (members 1 to 32) were chosen with respect to the following constraint:

Chosen Microphysics schemes are either double or triple moment. For triple moment microphysics schemes, the mixing ratio of each hydrometeor, their number concentrations and reflectivities are independent whereas for double moment schemes only mixing ratio of each hydrometeor and number concentrations are independent. Triple and double moment schemes are more time consuming than single moment scheme but more computationally efficient than Bin scheme. They allow removing internal assumptions regarding reflectivity and number concentrations respectively which increases the flexibility of these schemes to adapt to the availability of cloud condensation or ice nuclei [START_REF] Dudhia | A history of mesoscale model development[END_REF]. According to [START_REF] Chosson | Adapting Two-Moment Microphysics Schemes across Model Resolutions: Subgrid Cloud and Precipitation Fraction and Microphysical Sub-Time Step[END_REF] using microphysics scheme with higher number of moments improve the representation of microphysical processes which may improve the radiative transfer computations. Yet as higher moment schemes are more realistic, they may expose other NWP model deficiencies and might not lead to better results [START_REF] Morrison | An overview of cloud and precipitation microphysics and its parameterization in models[END_REF]. Consequently, the following third and second moment microphysics schemes were selected: Thompson aerosol-aware [START_REF] Thompson | A Study of Aerosol Impacts on Clouds and Precipitation Development in a Large Winter Cyclone[END_REF], [START_REF] Milbrandt | A Multimoment Bulk Microphysics Parameterization. Part II: A Proposed Three-Moment Closure and Scheme Description[END_REF], Morrison (Morrison et al., 2009), WDM6 (Lim and[START_REF] Liang | Development of an Effective Double-Moment Cloud Microphysics Scheme with Prognostic Cloud Condensation Nuclei (CCN) for Weather and Climate Models[END_REF].

Chosen PBL schemes equally represent local and non-local schemes. Local and non-local schemes differ in the determination of which model layers influence atmospheric condition at a given model level [START_REF] Cohen | A Review of Planetary Boundary Layer Parameterization Schemes and Their Sensitivity in Simulating Southeastern U.S. Cold Season Severe Weather Environments[END_REF]. According to [START_REF] Cohen | A Review of Planetary Boundary Layer Parameterization Schemes and Their Sensitivity in Simulating Southeastern U.S. Cold Season Severe Weather Environments[END_REF] non-local schemes represent more accurately deep PBL circulations, though similar accuracy could be obtained using higher order of closure. Therefore, both local and non-local PBL with various orders of closure from 1 to 2 was accounted for. The major weakness of non-local scheme is their high sensitivity to diagnosed quantities that are difficult to define accurately in NWP models [START_REF] Milovac | Investigation of PBL schemes combining the WRF model simulations with scanning water vapor differential absorption lidar measurements: WRF Sensitivity to PBL Schemes and LSMs[END_REF]. Consequently, the following local and non-local PBL schemes were selected: the non-local first order YSU [START_REF] Hong | A New Vertical Diffusion Package with an Explicit Treatment of Entrainment Processes[END_REF], the local second order MYNN3 [START_REF] Nakanishi | Development of an Improved Turbulence Closure Model for the Atmospheric Boundary Layer[END_REF] scheme, the local 1.5 order QNSE [START_REF] Sukoriansky | Application of a New Spectral Theory of Stably Stratified Turbulence to the Atmospheric Boundary Layer over Sea Ice[END_REF] scheme, and the mixed local, non-local first order Shin-Hong [START_REF] Shin | Representation of the Subgrid-Scale Turbulent Transport in Convective Boundary Layers at Gray-Zone Resolutions[END_REF][START_REF] Xu | The Performance of a Scale-Aware Nonlocal PBL Scheme for the Subkilometer Simulation of a Deep CBL over the Taklimakan Desert[END_REF] scheme.

Chosen land surface scheme should at least represent vegetation, and soil moisture processes and computationally efficient. According to [START_REF] Liang | Development of an Effective Double-Moment Cloud Microphysics Scheme with Prognostic Cloud Condensation Nuclei (CCN) for Weather and Climate Models[END_REF] LSMs that have more comprehensive treatments of physical processes of land reduce the uncertainties of regional climate model. However, [START_REF] Kotsopoulos | A Comparative Study of Two Land Surface Schemes in WRF-ARW Model[END_REF] showed that more complex LSM may expose other NWP model deficiencies similarly to microphysics scheme [START_REF] Morrison | An overview of cloud and precipitation microphysics and its parameterization in models[END_REF]. Land surface schemes such as Noah MP [START_REF] Niu | The community Noah land surface model with multiparameterization options (Noah-MP): 1. Model description and evaluation with localscale measurements[END_REF], [START_REF] Pleim | Development of a Land Surface Model. Part II: Data Assimilation[END_REF][START_REF] Xiu | Development of a Land Surface Model. Part I: Application in a Mesoscale Meteorological Model[END_REF] and CLM4 [START_REF] Dai | Technical Description of version 4.0 of the Community Land Model[END_REF] despite modeling more complex of land physical process were not used in this study because they were not computationally efficient; therefore, they could not be used for operational forecasting [START_REF] Kotsopoulos | A Comparative Study of Two Land Surface Schemes in WRF-ARW Model[END_REF][START_REF] Ncar | ARW Version 3.8 Modeling System User's Guide January[END_REF]. Consequently, the following LSMs schemes were selected: Noah [START_REF] Tewari | Implementation and verification of the unified NOAH land surface model in the WRF model[END_REF], RUC [START_REF] Benjamin | Mesoscale Weather Prediction with the RUC Hybrid Isentropic-Terrain-Following Coordinate Model[END_REF]. These LSMs communicate with the PBL schemes through the surface layer scheme. The surface layer scheme of each member were selected in accordance with (NCAR, 2017) recommendations, the one affiliated with the PBL was always preferred.

Due to computational constraints, the Cumulus scheme and radiation scheme may be fixed (members 1-32). RRTMG [START_REF] Iacono | Radiative forcing by long-lived greenhouse gases: Calculations with the AER radiative transfer models[END_REF] radiation schemes were chosen for both long wave and shortwave radiations. Previous work [START_REF] Ruiz-Arias | Surface clear-sky shortwave radiative closure intercomparisons in the Weather Research and Forecasting model: SW CLOSURE INTERCOMPARISONS IN WRF[END_REF][START_REF] Zempila | Evaluation of Wrf shortwave radiation parametrization in predicting global horizontal irradiance in Greece[END_REF] showed that this scheme outperform the other commonly used radiation scheme independently of the other parameterization. This scheme is one of the rare WRF radiative schemes to allow aerosols, ozone radiation feedback in WRF [START_REF] Jimenez | WRF-Solar: Description and Clear-Sky Assessment of an Augmented NWP Model for Solar Power Prediction[END_REF]. According to [START_REF] Jimenez | WRF-Solar: Description and Clear-Sky Assessment of an Augmented NWP Model for Solar Power Prediction[END_REF][START_REF] Ruiz-Arias | Surface clear-sky shortwave radiative closure intercomparisons in the Weather Research and Forecasting model: SW CLOSURE INTERCOMPARISONS IN WRF[END_REF] the inclusion of aerosols in the calculations shows that RRTMG outperform Dudhia's scheme [START_REF] Dudhia | Numerical Study of Convection Observed during the Winter Monsoon Experiment Using a Mesoscale Two-Dimensional Model[END_REF] and is able to provide an excellent agreement with GHI observation. Grell 3D [START_REF] Grell | A generalized approach to parameterizing convection combining ensemble and data assimilation techniques: PARAMETERIZING CONVECTION COMBINING ENSEMBLE AND DATA ASSIMILATION TECHNIQUES[END_REF] cumulus scheme was chosen to model unresolved convective column physics. Previous work in the ITZ [START_REF] Crétat | Uncertainties in simulating regional climate of Southern Africa: sensitivity to physical parameterizations using WRF[END_REF][START_REF] Raghavan | Regional climate simulations over Vietnam using the WRF model[END_REF] showed that this scheme has the ability to reproduce Inter Tropical Convergence Zone patterns. Besides Grell 3D [START_REF] Grell | A generalized approach to parameterizing convection combining ensemble and data assimilation techniques: PARAMETERIZING CONVECTION COMBINING ENSEMBLE AND DATA ASSIMILATION TECHNIQUES[END_REF] scheme accounts for the shallow convection which improves sub grid-scale clouds, shortwave irradiance feedback (NCAR, 2017).

Dynamical core setup

To initialize WRF initial and boundary conditions from a global NWP model [START_REF] Côté | The Operational CMC-MRB Global Environmental Multiscale (GEM) Model. Part I: Design Considerations and Formulation[END_REF][START_REF] Déqué | The ARPEGE/IFS atmosphere model: a contribution to the French community climate modelling[END_REF]ECMWF, 2016b;Environmental Modeling Center, 2003;[START_REF] Majewski | The operational global icosahedral-hexagonal gridpoint model GME: Description and high-resolution tests[END_REF]. Among the existing global NWP model the usage of Global Forecast System (Environmental Modeling Center, 2003) is the most straightforward; there is no need for users pre-processing steps to ingest GFS forecasts into WRF. Besides GFS forecasts are available free of charge and archives dating back to 1997 are available (NCAR-RDA, 2017). Consequently, we chose in this study GFS forecast for initial and boundary conditions. To choose the initialization, the time needed for WRF to produce a balanced state from GFS coarser initial and boundary condition must be considered. According to [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF][START_REF] Diagne | Post-processing of solar irradiance forecasts from WRF model at Reunion Island[END_REF] the spin up time in the ITZ ranges between 6 and 12 hours. The GFS model is initialized every 6 hour starting at 00h; consequently, we chose the initialization at 00 UTC to allow for a spin up of 12 hours before the sunrise in French Guiana4 .

To choose the spatial discretization of WRF the spatial resolution of the initial and boundary conditions (icbc) must be accounted for. GFS (GSM v13.0.2) forecasts used in this study ( [dataset] NCEP-NWS-NOAA-USDC, 2015) have a spatial resolution of approximately 24 km. According to [START_REF] Lin | Simulation of low clouds from the CAM and the regional WRF with multiple nested resolutions[END_REF] findings, clouds can be realistically simulated with horizontal resolutions higher than 4 km, but not with coarser resolutions. Therefore, there is a dimension mismatch between the resolution needed to simulate realistic cloud and the resolution of the icbcs. To solve this mismatch issue we used three ways nested domain; D01 with a 27 km horizontal resolution, D02 with a 9 km horizontal resolution and D03 with a 3 km horizontal resolution. To allow for each nest to produce accurate sink sand sources we have approximately 1 3 of D01 surrounding each side of D02 and similarly 1 3 of D02 surrounding each side of D01 [START_REF] Gill | WRF tutorial: nesting in WRF[END_REF]. GHI forecasts resulting from the inner domain with a 3 km resolution centered on French Guiana are exploited (Fig. 1). We chose this resolution to find a balance between the computational power required and the ability of the model to produce realistic clouds. Previous studies in the ITZ used a vertical discretization ranging between 35 and 100 levels [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF][START_REF] Diagne | Review of solar irradiance forecasting methods and a proposition for small-scale insular grids[END_REF][START_REF] Lima | Forecast for surface solar irradiance at the Brazilian Northeastern region using NWP model and artificial neural networks[END_REF][START_REF] Verbois | Solar irradiance forecasting in the tropics using numerical weather prediction and statistical learning[END_REF] for solar energy forecasts. According to [START_REF] Tselioudis | Evaluation of midlatitude cloud properties in a weather and a climate model: Dependence on dynamic regime and spatial resolution[END_REF] the cloud coverage and cloud properties are more accurately forecasted with increased vertical resolution. Consequently, in our study all domains have 100 levels vertical resolution.

Based on the horizontal and vertical resolution the time step of the WRF dynamic core was set to 50s to achieve convergence. All parameterization schemes are called every model time-step except for the radiation schemes called every 3 min. .

(5)

In the metrics mention above 𝐺𝐻𝐼(𝑡) 𝑁𝑊𝑃 is the GHI forecasted by the NWP whereas 𝐺𝐻𝐼(𝑡) 𝑔𝑟𝑜𝑢𝑛𝑑 is the ground measured GHI. They are both expressed in W/m².

To assess the accuracy of the NWP models with respect to the sky condition: the clearness index (𝐾 𝑐 ), clearness index mean value (𝑀 𝐾 𝑐 ) and clearness index mean variability (𝑉) over a time span was used. The sky condition were considered as clear when 𝐾 𝑐 > 0.65, cloudy when 0.4 < 𝐾 𝑐 < 0.65 and overcast when 𝐾 𝑐 < 0.4 [START_REF] Aryaputera | Day-Ahead Solar Irradiance Forecasting in a Tropical Environment[END_REF].

𝐾 𝑐 (𝑡) = 𝐺𝐻𝐼 𝑔𝑟𝑜𝑢𝑛𝑑 (𝑡) 𝐺 𝑐 (𝑡)
, ( 6)

𝑀 𝐾 𝑐 = 1 ∆𝑡 ∑ 𝐾 𝑐 (𝑡)
𝑡+∆𝑡 𝑡 and ( 7)

𝑉 = √ 1 ∆𝑡 ∑ (𝐾 𝑐 (𝑡 + ∆𝑡) -𝐾 𝑐 (𝑡)) 2 𝑁 𝑘=1 . ( 8
)
where 𝐺 𝑐 is ESRA clear sky model as defined in [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas -with respect to the heliosat method[END_REF]. Link turbidity, the clear sky attenuation factor inputted to 𝐺 𝑐 is defined for each site and each month using monthly climatological values provided by the SoDa database. In equation 7-8, dt is the time step of ground measurements, i.e. hourly. For a model with hourly GHI outputs N=1. To illustrate how MK c and 𝑉 were computed, we give the following: at 12 h UTC, MK c and 𝑉 are computed using the K c values at 11h UTC and 12 h UTC.

Using previously described metrics WRF and AROME GHI forecasts were validated against, quality checked in situ measurements.

Intercomparing the accuracies of the ensemble members

According to [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF] the most important metric in renewable energy forecast is the RMSE because give more weight to the large forecasts errors that has higher impact on the electrical grid management. Therefore, to select the greatest ensemble member we used the RMSE as validation metric.

Figure 2 gives the RMSE of each member for all the month and stations merged under clear cloudy and overcast sky conditions. It shows that there are high discrepancies between the most and least accurate members. Under clear, cloudy, overcast and all sky conditions these discrepancies are 87 W/m², 157 W/m², 201W/m² and 96 W/m² respectively. They are observed between members 16 and 23; members 3 and 34; members 7 and 34; members 9 and 34 respectively. These results show that:

(1) varying the PBL scheme, microphysics scheme and land surface model has a strong influence on the forecasted GHI. ( 2) Using parameterization previously defined in similar climate is not suitable.

The member with the worst accuracy under all sky condition is 34, it was previously used by [START_REF] Diagne | Post-processing of solar irradiance forecasts from WRF model at Reunion Island[END_REF] to forecast the GHI in Reunion Island tropical climate.

The best compromise between clear cloudy and overcast sky GHI forecast accuracy is obtained using member 9 (WRF-M9)6 as show figure 2. This results may be explained by two phenomena: (1) Member 9 uses Thompson aerosol aware microphysics scheme [START_REF] Thompson | A Study of Aerosol Impacts on Clouds and Precipitation Development in a Large Winter Cyclone[END_REF]; this scheme exploits an aerosol climatology derived from eight years (2001)(2002)(2003)(2004)(2005)(2006)(2007) global model simulations instead of using Tegen [START_REF] Tegen | Contribution of different aerosol species to the global aerosol extinction optical thickness: Estimates from model results[END_REF] aerosol datasets used in the other microphysics scheme. [START_REF] Thompson | A Study of Aerosol Impacts on Clouds and Precipitation Development in a Large Winter Cyclone[END_REF]) aerosol dataset has a higher spatial resolution: 0.5-degree longitude by 1.25-degree latitude spacing whereas [START_REF] Tegen | Contribution of different aerosol species to the global aerosol extinction optical thickness: Estimates from model results[END_REF]) aerosol dataset has 5 by 4 longitudes, latitude spatial resolution. (2) This scheme is the only microphysics scheme up to WRF V3.8 that allows cloud radiation feedback; the effective cloud water, ice and snow radii from Thompson are fed into RRTMG [START_REF] Janjic | WRF-Solar: An augmented NWP model for solar power prediction. Model description and clear sky assessment[END_REF]. Consequently, WRF-M9 might improve aerosols and their interaction with radiation characterization which result in a better forecast under all sky conditions.

Comparing the GHI forecast of WRF Member against AROME French West Indies

In this section the GHI forecasts of WRF using Member 9 (WRF-M9) is compared against AROME French West Indies described section 2.2. We consider the GHI forecasts at the single grid points closest to each station and do not average spatially the GHI forecasts of IFS and GFS. First we study in section 4.3.1 the accuracy of WRF-M9 and AROME-OM when all stations and months are merged. Second we study in section 4.3.2 their accuracies for each station and each month.

Influence of the sky conditions

To analyze the influence of the sky conditions on the accuracy of WRF-M9 and AROME-OM: we give Figure 3, the MAE and MBE of each model under clear, cloudy and overcast sky conditions. Lastly, Figure 4 gives AROME-OM and WRF-M9 the MAE as a function of the clearness index mean value (𝑀 𝐾 𝑐 ) and clearness index mean variability (𝑉) over an hour time span.

Figure 3 shows that under all sky conditions WRF-M9 GHI forecast outperforms AROME-OM GHI forecast; the MAE of WRF-M9 is 139 W/m² whereas the MAE of AROME-OM is 206 W/m². To explain the behavior of WRF-M9 and AROME-OM under all sky, we study their individual performance under clear cloudy and overcast sky conditions.

Under measured clear sky conditions Figure 3 shows that the MAE and MBE of WRF-M9 and AROME-OM are the lowest. This result is in agreement with [START_REF] Diallo | Assessing GFS and IFS global numerical weather prediction model forecast accuracy under tropical climate[END_REF] findings in French Guiana. According to [START_REF] Janjic | WRF-Solar: An augmented NWP model for solar power prediction. Model description and clear sky assessment[END_REF] under clear skies, the forecast accuracy of the GHI is driven by aerosols. To analyze the effect of the different aerosol dataset used by WRF-M9 and AROME-OM on the GHI forecasts, we computed the MAE for samples that are reported as clear both by the NWP model and ground measurements. We found for all months and stations merged by territory that aerosols cause a MAE (MBE) of 80 W/m² (40 W/m²), for WRF-M9. For AROME-OM aerosols cause a MAE (MBE) of 167 W/m² (112 W/m²). From the positive MBE of WRF-M9 and AROME-OM we deduce that both WRF-M9 and AROME-OM overestimate of the aerosol optical depth [START_REF] Jimenez | WRF-Solar: Description and Clear-Sky Assessment of an Augmented NWP Model for Solar Power Prediction[END_REF]; however, WRF-M9 to a lesser extent. WRF-M9 uses [START_REF] Thompson | A Study of Aerosol Impacts on Clouds and Precipitation Development in a Large Winter Cyclone[END_REF] aerosol datasets which has a higher spatial than the one used in AROME-OM [START_REF] Tegen | Contribution of different aerosol species to the global aerosol extinction optical thickness: Estimates from model results[END_REF] which is also available in WRF. Consequently, we may believe that WRF-M9 aerosol dataset represent more accurately the aerosol load and distribution in French Guiana which in turn lead to better forecasts under clear sky condition.

Despite the ability of WRF-M9 and AROME-OM detect the occurrence of cloudy and overcast sky conditions being weak; they are in agreement with value reported in [START_REF] Diallo | Assessing GFS and IFS global numerical weather prediction model forecast accuracy under tropical climate[END_REF]. According to [START_REF] Lin | Simulation of low clouds from the CAM and the regional WRF with multiple nested resolutions[END_REF] findings, clouds can be realistically simulated with horizontal resolutions higher than 4 km, but not with coarser resolutions. Consequently, both models have the potential to simulate realistic clouds; nevertheless, there are still high MAEs under cloudy and overcast sky conditions. The MAEs of WRF-M9 under cloudy and overcast sky conditions are 172 W/m² and 230 whereas the MAEs of AROME-OM are 185 W/m² and 231 W/m² respectively (Fig. 3). According to [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF]; [START_REF] Yang | Preprocessing WRF initial conditions for coastal stratocumulus forecasting[END_REF] the model initialization is critical for NWP forecast accuracy; initial conditions derived from large-scale models will inherit the error of the parent model. We have showed in [START_REF] Diallo | Assessing GFS and IFS global numerical weather prediction model forecast accuracy under tropical climate[END_REF] that both IFS and GFS are inaccurate in cloudy and overcast sky conditions which may translate in AROME-OM and WRF-M9 forecasts respectively. As WRF-M9 and AROME-OM has quasi-similar horizontal (e.g 3 km vs 2.5 km) and vertical resolution (e.g 100 levels vs 90 levels); we may explained WRF-M9 improvements over AROME-OM (Tab.4) under cloudy and overcast sky conditions by its more frequent call of the radiation physics. WRF-M9 hourly GHI forecast is the result of 20 accumulated irradiance values divided by the accumulation time (eg, 1 hour) whereas AROME-OM hourly GHI forecast is the result of 4 accumulated irradiance values divided by the accumulation time (also 1 hour).

figure 4 shows that the MAE does not decrease systematically as the sky conditions become clearer. Figure 4 also show that the variability plays a key role on each model accuracy. For highly variable sky conditions 𝑉 ≥ 0.4, WRF-M9 outperforms AROME-OM (Fig. 4). We explain this result by the higher frequency of the radiation physics of WRF-M9. Monthly GHI of SG in March,May,September and all months merged are 316,312,450 and 359 W/m² respectively;for RO they are 307,342,456 and 370 W/m² respectively;for KR they are 335,362,484 and 393 W/m² respectively;for IR they are 352,358,468 and 390 W/m² respectively. For SL they are 392,324,422 and 381 W/m² respectively;for MP they are 348,386,454 and 397 W/m² respectively WRF-M9

AROME-OM

The different forecast accuracies found figure 5 may be explained by different cloud cover and cloud cover variability. The Lowest MAE are found during September for all stations (Fig. 5); Oppositely higher RMSE are found during May in all stations (Fig. 5) except for Rochambeau and Maripasoula. We explain results found in SG, KR, IR and SL by the higher variability of sky conditions found during May compare to March; figure 4 shows that both model underperform for highly variable sky conditions. For RO and MP May as higher occurrence of cloudy sky than March; we believe that it outweigh the high variable sky conditions and gives better RMSE.

Under all sky conditions and all months merged the MAE of WRF-M9 ranges between 29% and 33% whereas the MAE of AROME-OM ranges between 40% and 68% (Fig. 5).

Conclusions

The goal of this study was to propose a methodology to calibrate WRF so that it produced improved GHI forecasts in the ITZ compared to non-calibrated mesoscale model. This methodology was validated against ground measurements and AROME-OM GHI forecasts in French Guiana during 2016. French Guiana is a French territory located in the ITZ; its photovoltaic installed capacity is projected to increase by 112% by 2023. Because there are higher solar potential in the ITZ predicting the solar input to the solar facilities which will lead the electricity injected to the grid is vital. Therefore, knowledge of the forecast accuracy of NWP models is needed. We found in this study that using set of physical parameterization previously used in other similar climate is not suitable. It can worsen the all sky conditions RMSE by 96 W/m² compared to the calibrated model. The set of parameterization derived from our calibration (WRF-M9) outperform AROME-GHI forecast in clear cloudy and overcast sky conditions. Improvements under clear sky conditions were explained by the finer and more recent aerosol dataset used by WRF-M9 and the ability of WRF to reproduce more accurately the sky conditions. Improvements under cloudy and overcast sky conditions are explained by the higher call frequency of the radiation physics. The seasonal and accuracy study showed that MAE were lower in the dry season represented by September and higher in the rainy season represented by May. These different forecast accuracies was explained by different cloud cover and cloud cover variability found during these months. Under all sky conditions and month merged, the improvement of WRF-M9 MAE over AROME-OM ranges between [35:52]% depending on the stations.

To the best of the author's knowledge several studied showed that the MPs scheme, LSMs scheme, and PBLs schemes, as these schemes have a strong influence on the distribution and physical properties of the simulated cloud field (Cintineo et al., 2014;[START_REF] Otkin | Comparison of WRF Model-Simulated and MODIS-Derived Cloud Data[END_REF]Xie et al., 2012); however, no other studies analyzed the effect of varying these physics on the irradiance forecast in the ITZ. This study is the first dealing with the calibration of mesoscale models in French Guiana; results found that this calibration of WRF has similar standards as those of other climate regions.

IV. Summary and conclusion of chapter III

In Chapter III we presented first methods recommended for forecasting the GHI depending on the horizon of interest. Using this knowledge, we presented methods that forecast the GHI using NWP models. First we downloaded and studied the accuracy of IFS and GFS forecast products in the ITZ for tropical climate areas. That study filled the gap on the knowledge of global NWP models performance in the ITZ and for tropical climates. We studied two of the most widely used NWP models for various types of tropical climates located in French Guiana, Reunion Island and Singapore.

Results found showed by comparison to the persistence model that forecasting the GHI in tropical climate is more difficult than forecasting the GHI in extra-tropical climates. Besides, under all sky conditions the GHI forecasts of IFS outperform those GFS in all territories; nevertheless, both models have high MAE values under clear, cloudy and overcast sky conditions. High MAE under measured clear sky conditions were explained by a misrepresentation of aerosols and clouds wrongly forecasted by the NWP whereas high MAE under cloudy and overcast sky conditions were explained by the coarse resolution of IFS and GFS that render them unable to produce realistic clouds. From this investigation on the origin of IFS and GFS inaccuracies, we concluded that to improve NWP GHI forecasts in the ITZ, sub-grid scale phenomena such as clouds, aerosols and their interaction with radiation should be better characterized. Consequently, the authors built and computed an ensemble of several 48h GHI forecasts using WRF. The aim of the second study was to suggest a generic method to calibrate mesoscale models to obtain improved GHI forecasts in the ITZ. The ensemble members used to calibrate WRF 48h GHI forecast varied the MPs scheme, LSMs scheme, and PBLs schemes. The ITZ is an area with highly variable dynamic in which we encounter significant amounts of convective clouds [START_REF] Galvin | An Introduction to the Meteorology and Climate of the Tropics[END_REF][START_REF] Laing | Introduction to tropical meteorology, Educational material from The COMET Program[END_REF]. The parameterization scheme we chose to vary have a strong influence on the distribution and physical properties of the simulated cloud fields (Cintineo et al., 2014;[START_REF] Otkin | Comparison of WRF Model-Simulated and MODIS-Derived Cloud Data[END_REF]Xie et al., 2012). Results found showed that under all sky conditions, the improvement of WRF MAE over AROME-OM ranges between [35:52]% depending on the stations. We also found that MAE values have similar standards as those of extra-tropical regions despite forecast the GHI in tropical climate being more difficult.

The scope for future research includes extending this methodology to other locations in the intertropical zone and using data assimilation methods to improve the initialization of WRF forecasts [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF][START_REF] Yang | Preprocessing WRF initial conditions for coastal stratocumulus forecasting[END_REF].

CHAPTER IV : Improving the irradiance forecast of numerical weather prediction model using data assimilation

I. Improving the solar irradiance forecast under tropical climate for long range forecast horizon

Solar radiation forecasts and especially GHI forecasts are needed for integrating into the grid and at a large scale the electricity production from PV plants. Increasing the installed capacity of PV power requires that solar irradiance forecasts be more and more accurate in terms of spatial and temporal resolutions [START_REF] Thorey | Ensemble forecast of solar radiation using TIGGE weather forecasts and HelioClim database[END_REF]. However, NWP models are still highly biased in cloudy sky conditions; these conditions occur more frequently in the ITZ. According to [START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF], errors related to irradiance forecasts can have a wide variety of sources. These sources include: poor model initialization, excessively coarse vertical grid spacing, and inaccurate assumptions in the physical parameterizations. For this reason several methods were developed to increase the accuracy of NWP models irradiance forecasts. These methods could be classified into three categories:

-Model output statistics (MOS) -Multi model ensemble (MME) -Data assimilation (DA) MOS is the most frequent mean to improve solar irradiance forecasts; it is a post-processing method, which consists in the determination of a relationship between the GHI and other ground measured variables in order to refine the output of NWP models [START_REF] Glahn | MOS) The use of model output statistic in objective weather forecasting[END_REF]. This method was successfully used by [START_REF] Diagne | Post-processing of solar irradiance forecasts from WRF model at Reunion Island[END_REF], [START_REF] Lauret | A Neural Network Post-processing Approach to Improving NWP Solar Radiation Forecasts[END_REF], [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF], and Verbois et al. (2018). They found RMSE improvement of WRF GHI forecasts was high as 40%, 18%, 19%, and 33% respectively using a Kalman filter, a multivariate fourth-order regression and combination of a principal component analysis with a stepwise variable selection. However, these methods were exclusively based on statistical properties of the time series and do not use the physical properties of solar irradiance; therefore, they are site specific. Beside MOS need a training period require at least one year of archive. During the training and validation period the NWP setup and physics must not be changed [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF]. MME combines several forecasts from one or several NWP models (e.g. an ensemble) in order to produce a single forecast hopefully more skillful than any individual model of the ensemble [START_REF] Mallet | Ozone ensemble forecast with machine learning algorithms[END_REF]. This methodology was used by [START_REF] Thorey | Ensemble forecast of solar radiation using TIGGE weather forecasts and HelioClim database[END_REF] who combined linearly GHI forecasts from six NWP models. [START_REF] Thorey | Ensemble forecast of solar radiation using TIGGE weather forecasts and HelioClim database[END_REF] found that for a 42-hour forecast the RMSE of the worst NWP is improved by as much as -24%. The advantage of multi-model ensemble methods is that they rely on several NWP; different models capture the same physical phenomenon differently which tend to reduce the uncertainty based on each individual model (Xue and Zhang, 2014). Nevertheless, multi-model ensemble has several limitations. The combination of the ensemble forecasts is determined to minimize its discrepancy with the observations; since the observations are not perfect, this approach is not entirely satisfactory. The second limitation is that the weights are computed only at the locations and for the variables that are observed. Computing weights for other locations and other do not guaranty similar accuracy [START_REF] Mallet | Ensemble forecast of analyses: Coupling data assimilation and sequential aggregation: ENSEMBLE FORECAST OF ANALYSES[END_REF]. Besides, similar to MOS methods, MME methods need an extensive training period and the NWP models must be frozen during the training and validation periods.

Data assimilation (DA) methods merge the physical information provided by numerical models and the information brought by the observations, in order to improve the forecast. These methods date back to the early seventies [START_REF] Daley | Atmospheric data analysis[END_REF] and have been tested for several purposes such as improving storm forecasts [START_REF] Fierro | Evaluation of a Cloud-Scale Lightning Data Assimilation Technique and a 3DVAR Method for the Analysis and Short-Term Forecast of the 29 June 2012 Derecho Event[END_REF], improve wind forecasts [START_REF] Portabella | A probabilistic approach for SeaWinds data assimilation: SEAWINDS DATA ASSIMILATION[END_REF], improve cloud forecasts [START_REF] Benjamin | Cloud/hydrometeor initialization in the 20-km RUC using GOES and radar data[END_REF], improving aerosol concentration forecasts. However, a literature study we carried out show that only a few publications using DA to improve solar irradiance forecasts were available [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF][START_REF] Sahu | Assimilating in-situ observations over Southern California for improved solar forecasting[END_REF][START_REF] White | Improving cloud prediction in WRF through the use of GOES satellite assimilation[END_REF][START_REF] Yang | Preprocessing WRF initial conditions for coastal stratocumulus forecasting[END_REF]. Nevertheless, these studies showed promising results. Clear advantages of DA over MME are that (1) DA methods take into account observational errors which make a difference when significant instrumental errors are involved, (2) DA methods are not site specific: DA improvements apply to the NWP forecast region (3) they are autonomous, they may be used with different NWP models and (4) they do not require a training period. [START_REF] Bourgin | Investigating the interactions between data assimilation and post-processing in hydrological ensemble forecasting[END_REF] investigated the interactions between data assimilation and post-processing in hydrological ensemble forecasting; he found that DA method outperform MOS especially for forecast horizon lower or equal to 12h. For these reasons we considered DA methods to improve solar irradiance forecasts in this thesis.

II. Data assimilation methods

DA is a technique in which an accurate image of the true state of the atmosphere at a given time (i.e analysis) is accumulated into a NWP model state to improve its forecasts [START_REF] Bouttier | Data Assimilation Concepts and Methods[END_REF][START_REF] Holm | Lecture notes on assimilation algorithms[END_REF]. This accurate image of the true state of the atmosphere is called the analysis. This analysis is constrained by the physical laws of the NWP model set of equations. DA methods can be divided into four categories (Bannister, 2008a;[START_REF] Daley | Atmospheric data analysis[END_REF][START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF][START_REF] Lakshmivarahan | Nudging Methods: A Critical Overview[END_REF]:

-interpolation method (e.g., function fitting) -empirical methods (e.g. successive correction method) -variational method (e.g., 4D-var, 3D-var) -stochastic method (e.g., Kalman filter and ensemble Kalman filters) -nudging methods

In this manuscript we focus only on real time assimilation algorithms systems because they only consider observations from the past [START_REF] Bouttier | Data Assimilation Concepts and Methods[END_REF]. Therefore, they are relevant in the operational management of solar energy systems. Consequently, interpolation and empirical methods are not discussed as they are outdated and no longer used in current NWP (ECMWF, 2016b;Environmental Modeling Center, 2003;[START_REF] Majewski | The operational global icosahedral-hexagonal gridpoint model GME: Description and high-resolution tests[END_REF][START_REF] Staniforth | Unified model documentation paper N°15[END_REF]. We present in the main body of the manuscript the general framework of stochastic, variational and nudging DA algorithms. Further details on the different DA methods, are given Appendix A.

II.1 Stochastic method and variational methods

Stochastic DA methods include commonly used in commonly used in atmospheric science [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] include ensemble square root filters (EnSRF) and ensemble Kalman filters (ENKF); while variational DA algorithms include 3D-Var and 4D-Var (Appendix A). [START_REF] Bennett | Inverse Modeling of the Ocean and Atmosphere[END_REF][START_REF] Holm | Lecture notes on assimilation algorithms[END_REF] and [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] showed that stochastic and variational DA algorithms can be derived from one common source. Consequently, The DA problem from the stochastic and variational framework is expressed as follows:

x ⃗⃗(t + 1) = M ⃗⃗⃗⃗ (x ⃗⃗(t)) + ε m ⃗⃗⃗⃗⃗⃗, IV.1

x ⃗⃗(t

0 ) = x b ⃗⃗⃗⃗⃗(t 0 ) + ε b ⃗⃗⃗⃗ 𝑎𝑛𝑑 IV.2 y ⃗⃗(t n ) = ℋ ⃗⃗⃗ (x ⃗⃗(t n )) + ε 0,n ⃗⃗⃗⃗⃗⃗⃗. IV.3
Where 𝑥 ⃗ ∈ ℜ 𝑛 is the NWP model state vector; 𝑥 𝑏 ⃗⃗⃗⃗⃗ ∈ ℜ 𝑛 the background state; 𝑦 ⃗(𝑡 𝑛 ) ∈ ℜ 𝑝 the observation of the state variable over a given time interval n; 𝑀 ⃗⃗⃗ (𝑥 ⃗) ∈ ℜ 𝑛 the model operator;

ℋ ⃗⃗⃗ ∈ ℜ 𝑛⟶𝑝 the observation operator which computes the model equivalent of an observation 𝑦 ⃗ made a given time and location; 𝜀 𝑚 ⃗⃗⃗⃗⃗⃗⃗ ∈ ℜ 𝑛 ,𝜀 𝑏 ⃗⃗⃗⃗ ∈ ℜ 𝑛 and 𝜀 0 ⃗⃗⃗⃗ ∈ ℜ 𝑝 approximate the model errors, the initial condition error (i.e., background error), and observation error respectively. 𝜀 0,𝑛 ⃗⃗⃗⃗⃗⃗⃗ is used a simplification for 𝜀 0 ⃗⃗⃗⃗(𝑡 𝑛 ).

To model the uncertainty related to the model errors, background errors, and observation errors, probability distribution functions (pdfs) are used. Under the assumption that the background observation and model error pdfs are Gaussian, the probability to find the model state at x knowing the observation error is:

P(x ⃗⃗|y ⃗⃗) = 1 (2π) 3 2| B||R||Q| e -1 2 [ ε b ⃗⃗⃗⃗⃗ T B -1 ε b ⃗⃗⃗⃗⃗+ ε m ⃗⃗⃗⃗⃗⃗⃗ T Q -1 ε m ⃗⃗⃗⃗⃗⃗⃗+∑ ε 0,n ⃗⃗⃗⃗⃗⃗⃗⃗⃗ T R -1 N n=0 ε 0,n ⃗⃗⃗⃗⃗⃗⃗⃗⃗] ,
IV.4

where 𝐵 ∈ ℜ 𝑛.𝑛 ,𝑄 ∈ ℜ 𝑛.𝑛 and 𝑅 ∈ ℜ 𝑝.𝑝 are the model error covariance matrix, background error covariance matrix and observation error covariance matrix respectively. They are defined as follows:

𝐵 =< (𝜀 𝑏 ⃗⃗⃗⃗-< 𝜀 𝑏 ⃗⃗⃗⃗ >) (𝜀 𝑏 ⃗⃗⃗⃗-< 𝜀 𝑏 ⃗⃗⃗⃗ >) 𝑇 >, IV.5 R =< (ε 0 ⃗⃗⃗⃗-< ε 0 ⃗⃗⃗⃗ >) (ε 0 ⃗⃗⃗⃗-< ε 0 ⃗⃗⃗⃗ >) T > 𝑎𝑛𝑑 IV.6 𝑄 =< (𝜀 𝑚 ⃗⃗⃗⃗⃗⃗-< 𝜀 𝑚 ⃗⃗⃗⃗⃗⃗ >) (𝜀 𝑚 ⃗⃗⃗⃗⃗⃗-< 𝜀 𝑚 ⃗⃗⃗⃗⃗⃗ >) 𝑇 >. IV.7
Similarly we define the analysis error 𝜀 𝑎 ⃗⃗⃗⃗ ∈ ℜ 𝑛 which measures the departure of the analysis to the true state and the analysis covariance matrix 𝑃 ∈ ℜ 𝑛.𝑛 :

𝑃 =< (𝜀 𝑎 ⃗⃗⃗⃗-< 𝜀 𝑎 ⃗⃗⃗⃗ >) (𝜀 𝑎 ⃗⃗⃗⃗-< 𝜀 𝑎 ⃗⃗⃗⃗ >) 𝑇 >. IV.8
In Eq.IV.5 to IV.8 < > and 𝑇 represent the expectation and transposition operators, respectively.

The most important elements of DA methods are the covariance matrices B, Q and R. They determine to what extent the background fields will be corrected to match the observations [START_REF] Bouttier | Data Assimilation Concepts and Methods[END_REF]) and how the model error influence the analysis.

𝑄 describes the correlations between errors in the model between several locations and for several time instances. Model errors can be caused by physical processes not described by the model equations, or by inaccurate physical parameterizations [START_REF] Holm | Lecture notes on assimilation algorithms[END_REF]. In most assimilation algorithm the model is assumed to have no error; therefore, 𝑄 = 0. 𝐵 spreads the information vertically and horizontally with proper weights to observation 𝑦 𝑛 and the background 𝑥 𝑏 ⃗⃗⃗⃗⃗(𝑡 0 ). Thus, if |𝐵| is very large compared to observation errors, the analysis is closer to observation, otherwise the analysis is closer to the background [START_REF] Descombes | Generalized background error covariance matrix model (GEN_BE v2.0)[END_REF][START_REF] Liu | Algorithm (2): Background Error Modeling and Estimation[END_REF][START_REF] Rizvi | WRFDA Background Error (Modeling and Estimation)[END_REF]. B can be either be approximated as static or dynamic. For the static case, B is approximated using long term simulations (i.e climatological); For the dynamic case B is approximated using a set of different numerical simulation (i.e an ensemble) that varies with the time. To approximate 𝐵 several approaches may be used: the analysis of innovations method, the differences of varying length forecasts (NMC method), the forecast time lags method (CQ) and the ensemble method (Bannister, 2008a).

𝑅 includes the effects of measurement errors, errors in the design of the observation operators and representativeness errors [START_REF] Bouttier | Data Assimilation Concepts and Methods[END_REF][START_REF] Holm | Lecture notes on assimilation algorithms[END_REF][START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] . R is specified according to the knowledge of instrumental characteristics; in WRF [START_REF] Skamarock | A Description of the Advanced Research WRF Version 3[END_REF] for instance a standard table of observation error values provided by the US Air Force Weather Agency is used [START_REF] Kavulich | Observations Lecture 1: Conventional observations and Observation Preprocessing for WRFDA[END_REF]. This table contains error values broken down by observed variable, observation type, and pressure level.

The most likely state of the atmosphere (analysis) maximizes the joint probability Eq.IV.9. This maximum is attained when 𝑥 ⃗ = 𝑥 𝑎 ⃗⃗⃗⃗⃗ , the analysis state vector. Consequently, the following cost function is minimized [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF]:

J(x ⃗⃗(t 0 )) = 1 2 [ ε b ⃗⃗⃗⃗⃗⃗ T B -1 ε b ⃗⃗⃗⃗ + ε m ⃗⃗⃗⃗⃗⃗ T Q -1 ε m ⃗⃗⃗⃗⃗⃗ + ∑ ε 0,n ⃗⃗⃗⃗⃗⃗⃗ T R -1 N n=1 ε 0,n ⃗⃗⃗⃗⃗⃗⃗] = J b + J m + J 0 .
IV.9

Where 𝐽 𝑏 , 𝐽 𝑚 and 𝐽 0 are the background, observation and model cost functions respectively. The cost function 𝐽 aims to find an optimal estimate for the initial state of the system 𝑥 ⃗(𝑡 0 ) given a prior estimate 𝑥 𝑏 ⃗⃗⃗⃗⃗(𝑡 0 ) of the initial state and observations at several points 𝑡 𝑛 ; 𝑛 = 0, … , 𝑁 distributed over a time window[𝑡 0 : 𝑡 𝑁 ].

Variational DA algorithms minimize Eq.IV.9 to obtain an analysis whereas stochastic DA solve directly the analysis equation (Eq.IV.9) directly by inversion (Appendix A). In operational NWP centers, to reduce computational costs, a sequence of linear approximations to the nonlinear minimization problem Eq.IV.9 using successive small increments δx ⃗⃗, given an estimated analysis 𝑥 ⃗(𝑡 0 ). δ𝑥 ⃗(t 0 ) = 𝑥 ⃗(t 0 ) -𝑥 ⃗ 𝑏 (t 0 )

IV.10

In practice the assimilation cycle has two step, the update and prediction step. If observations 𝑦(𝑡 𝑛 ) are available, they are used in the update step to compute the analysis state vector 𝑥 𝑎 ⃗⃗⃗⃗⃗ and its error covariance matrix 𝑃 is then propagated into time during the prediction step.

II.2 Nudging methods

The nudging DA (NDA) algorithm is originally an empirical method that consists in adding to the prognostic equations a term that relaxes (nudges) the solution towards the observations. Current nudging DA assimilation method are no longer empirical, they derive the nudging coefficient from variational or ensemble methods. They are two types of nudging: the observation nudging and the analysis nudging [START_REF] Auroux | A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm[END_REF][START_REF] Lakshmivarahan | Nudging Methods: A Critical Overview[END_REF][START_REF] Lei | A hybrid nudging-ensemble Kalman filter approach to data assimilation[END_REF][START_REF] Lei | Nudging, Ensemble, and Nudging Ensembles for Data Assimilation in the Presence of Model Error[END_REF]. In the case of analysis nudging, the model state is nudged toward gridded analysis based on the analysis error whereas in observation nudging, the model predictions are nudged to match better with observations at individual locations both on the surface and above (Xiangshang Li et al., 2016) based on the model error. The DA problem from the observation nudging framework is expressed as follows [START_REF] Lakshmivarahan | Nudging Methods: A Critical Overview[END_REF]:

{ x ⃗⃗(t + 1) = M ⃗⃗⃗⃗ (x ⃗⃗(t)) + G 0 (t)ε 0 ⃗⃗⃗⃗(t) 𝑎𝑛𝑑 x ⃗⃗(t 0 ) = x ⃗⃗ 0 .

IV.11

Whereas from analysis nudging, it is express as:

{

x ⃗⃗(t + 1) = M ⃗⃗⃗⃗ (x ⃗⃗(t)) + G a (t) ε a ⃗⃗⃗⃗(t) 𝑎𝑛𝑑 x ⃗⃗(t 0 ) = x ⃗⃗ 0 .

IV.12

Where 𝜀 0 ⃗⃗⃗⃗, 𝐺 0 , 𝜀⃗ 𝑎 , 𝐺 𝑎 are: the observation error, observation nudging coefficient matrix, analysis error and analysis nudging coefficient matrix respectively.

A variational variant of nudging method named back and forth nudging method was developed recently by Auroux andBlum (2008), (2005). The back and forth nudging algorithm consists in first solving the forward nudging equation (Eq.IV.11) and then the backward nudging equation. The initial condition of the backward integration is the final state obtained after integration of the forward nudging equation. At the end of this process, one obtains an estimate of the initial state of the system (Appendix A). The main difficulty in the NDA scheme resides in the estimation of the nudging coefficient G. They could be either computed empirically, statistically or using a variational approach [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF][START_REF] Zou | An Optimal Nudging Data Assimilation Scheme Using Parameter Estimation[END_REF]). The latter approach gives the optimal nudging method [START_REF] Lakshmivarahan | Nudging Methods: A Critical Overview[END_REF].

III. Comparison between DA algorithms

A literature study we carried out showed that among DA methods commonly use in atmospheric sciences, only 3D-Var and nudging were used to improve solar irradiance forecasts. [START_REF] Sahu | Assimilating in-situ observations over Southern California for improved solar forecasting[END_REF] recently used 3D-var data assimilation to improve irradiance forecast over southern California. [START_REF] Sahu | Assimilating in-situ observations over Southern California for improved solar forecasting[END_REF] ran WRF for two consecutive days and noticed that 3-DVAR improved the MAE by 65%.

Most available studies used nudging type method [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF][START_REF] White | Improving cloud prediction in WRF through the use of GOES satellite assimilation[END_REF][START_REF] Yang | Preprocessing WRF initial conditions for coastal stratocumulus forecasting[END_REF]. [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF] and [START_REF] Yang | Preprocessing WRF initial conditions for coastal stratocumulus forecasting[END_REF] used conjointly WRF [START_REF] Skamarock | A Description of the Advanced Research WRF Version 3[END_REF] NWP model with GOES cloud cover images to increase or decrease artificially the water vapor mixing ratio to produce or dissipate clouds simulated in the NWP models so that they match GOES imagery. [START_REF] Yang | Preprocessing WRF initial conditions for coastal stratocumulus forecasting[END_REF] found that their nudging method improved by 47% the solar irradiance forecast of WRF; yet under clear sky conditions, the model was negatively biased [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF]; indicating that for some forecasts clouds were incorrectly dissipated. [START_REF] White | Improving cloud prediction in WRF through the use of GOES satellite assimilation[END_REF] also used conjointly WRF NWP model with GOES cloud cover images but nudged the horizontal wind fields positively to produce cloud and negatively to dissipate clouds. This methodology was validated against two stations and for two months; it was found that the correlation coefficient was found to be greater for the cloud assimilation simulation, 0.71 compared to 0.65 and 0.67 compared to 0.47 at both locations. The main disadvantage of these methods is: these methods introduce fake sink/source term which creates stability issues [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF]. Beside on their current stage of development, they rely on site specific parameters. [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF] used a mixing ratio optically similar to the marine stratocumulus clouds observed in coastal California.

For these reasons a bibliographic review highlighting the advantage and drawback of each DA method commonly used in atmospheric science is presented in the next section. This bibliographic review was used to select the DA method to improve the solar irradiance forecast in the ITZ. -4Dvar treats observations at the correct time [START_REF] Bannister | Variational data assimilation Background and method[END_REF] improve analysis -4Dvar evolves B is implicitly [START_REF] Bannister | Variational data assimilation Background and method[END_REF]Schwartz et al., 2014) -4D-Var computationally expensive compared  depending on the computational facilities cannot be used operationally [START_REF] Lorenc | Why does 4D-Var beat 3D-Var? Q[END_REF] -4DVAR analysis is consistent with the model [START_REF] Fowler | Comparison between a multivariate nudging method and the ensemble Kalman filter for sea-ice data assimilation[END_REF][START_REF] Lorenc | The potential of the ensemble Kalman filter for NWP-a comparison with 4D-Var[END_REF] appropriate B -4DVAR relies on the validity of: TL and perfect model [START_REF] Fowler | Comparison between a multivariate nudging method and the ensemble Kalman filter for sea-ice data assimilation[END_REF][START_REF] Lorenc | The potential of the ensemble Kalman filter for NWP-a comparison with 4D-Var[END_REF] restrict assimilation window -B not evolved explicitly [START_REF] Fowler | Comparison between a multivariate nudging method and the ensemble Kalman filter for sea-ice data assimilation[END_REF]Schwartz et al., 2014) -4DVAR assumes no model error  give the same credence to older observations at the beginning of the interval as to newer observations at the end of the interval [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] Ensemble filters -ENKF is more computationally demanding [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] -ENKF treat explicitly the model error  improve the forecast [START_REF] Mitchell | Ensemble Size, Balance, and Model-Error Representation in an Ensemble Kalman Filter[END_REF] -ENKF better spread information in space and across variables [START_REF] Fowler | Comparison between a multivariate nudging method and the ensemble Kalman filter for sea-ice data assimilation[END_REF] -Ensembles are propagated with full nonlinear numerical (Janjic, n.d.) Model no need to linearize -ENKF is more computationally demanding [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] -B is flow dependent [START_REF] Fowler | Comparison between a multivariate nudging method and the ensemble Kalman filter for sea-ice data assimilation[END_REF][START_REF] Lorenc | The potential of the ensemble Kalman filter for NWP-a comparison with 4D-Var[END_REF] -No need to linearize the model [START_REF] Fowler | Comparison between a multivariate nudging method and the ensemble Kalman filter for sea-ice data assimilation[END_REF][START_REF] Lorenc | The potential of the ensemble Kalman filter for NWP-a comparison with 4D-Var[END_REF]. However, due to computational cost we can only afford small ensemble sizes thus offering minimal if any benefit for strongly nonlinear problems (Vetra-Carvalho, 2018)

-Sensitive to ensemble size [START_REF] Fowler | Comparison between a multivariate nudging method and the ensemble Kalman filter for sea-ice data assimilation[END_REF] filter divergence -ENKF is more computationally demanding [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] As shown Tab. IV.1 each DA method as its own set of advantages and drawbacks. Main drawbacks of variational method are that they do not natively take into account the model error and that they do not evolve explicitly B within the assimilation window. ENKF main drawback deal with the tradeoff between ensemble size, computational affordability and accuracy. For these reasons hybrid method were developed such as ETKF-3DVar [START_REF] Wang | A Hybrid ETKF-3DVAR Data Assimilation Scheme for the WRF Model. Part I: Observing System Simulation Experiment[END_REF], ENKF-3DVAR [START_REF] Gao | The Development of a Hybrid EnKF-3DVAR Algorithm for Storm-Scale Data Assimilation[END_REF], ENSRF-3DVAR (Schwartz et al., 2014), ETKF-4DVar [START_REF] Clayton | Operational implementation of a hybrid ensemble/4D-Var global data assimilation system at the Met Office[END_REF], ENKF with nudging [START_REF] Lei | Nudging, Ensemble, and Nudging Ensembles for Data Assimilation in the Presence of Model Error[END_REF], to benefit from the strengths of each individual DA methods. Most methods that couple several DA methods involve variational and ENKF methods; they are called EnVar [START_REF] Bannister | A review of operational methods of variational and ensemble-variational data assimilation: Ensemble-variational Data Assimilation[END_REF]. [START_REF] Bannister | A review of operational methods of variational and ensemble-variational data assimilation: Ensemble-variational Data Assimilation[END_REF] classified in his study EnVar DA methods in three categories:

-pure EnVar system that uses a tangent linear model (TLM) similarly to variational methods (e.g En4dvar,En3dvar) -pure EnVar system that does not use a TLM similarly to ENKF (4DEnvar, 3DEnvar) -hybrid EnVar system whose B is computed with the contribution of a static part similarly to variational methods and a contribution from a flow dependent part similar to ENKF methods According to [START_REF] Bannister | A review of operational methods of variational and ensemble-variational data assimilation: Ensemble-variational Data Assimilation[END_REF] hybrid methods tend to be better and more robust for smaller ensembles size than pure variational method, ensemble, or EnVar methods. Bannister (2017) also found that pure EnVar system that uses a TLM outperforms pure EnVar system that does not use a TLM. However, an EnVar system that uses a TLM requires more significant computational resources.

IV. Data assimilation for solar energy forecast purposes in the intertropical zone

The ITZ is an area with highly variable dynamic in which we encounter significant amounts of convective clouds. Consequently, the solar energy available at the ground is highly variable. Recently [START_REF] Schwartz | Improving Large-Domain Convection-Allowing Forecasts with High-Resolution Analyses and Ensemble Data Assimilation[END_REF]; and Schwartz and Liu (2014) showed that using continuously cycling an hybrid 3DEnVar DA systems improve the characterization of convective phenomena. As clouds convective phenomena have a sizeable effect on accuracy of NWP model GHI forecasts; therefore, hybrid 3DEnVar DA method was selected to improve WRF irradiance forecasts in the ITZ.

IV.1 Description of the Hybrid 3D Ensemble variational

The hybrid 3DEnVar DA algorithm is available in WRF-DA; it aims to minimize a cost function similar to that of Eq.IV.9; However, in the hybrid 3DEnVar the analysis increment (δ𝑥 ⃗) is partitioned into a weighted linear combination of the ensemble and static contributions; it is expressed as follows [START_REF] Li | Implementation of a dynamic equation constraint based on the steady state momentum equations within the WRF hybrid ensemble-3DVar data assimilation system and test with radar T-TREC wind assimilation for tropical Cyclone Chanthu (2010): HYBRID DATA ASSIMILATION WITH CONSTRAINT[END_REF]Schwartz et al., 2014):

δ𝑥 ⃗(t 0 ) = δx ⃗⃗ s (t 0 ) + 1 √N -1 ∑ a i N i=1 (δx (𝑖) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (t 0 )-< δx (𝑖) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (t 0 ) >) IV.13
Where 𝛿𝑥 ⃗ 𝑠 is the increment associated with the static BECs, and 𝑎 𝑖 the ith member's extended control variable [START_REF] Lorenc | The potential of the ensemble Kalman filter for NWP-a comparison with 4D-Var[END_REF] that give proper weight to the i th member's departure from the ensemble mean. Therefore, the Hybrid 3DEnVar minimization problem is written as follows [START_REF] Li | Implementation of a dynamic equation constraint based on the steady state momentum equations within the WRF hybrid ensemble-3DVar data assimilation system and test with radar T-TREC wind assimilation for tropical Cyclone Chanthu (2010): HYBRID DATA ASSIMILATION WITH CONSTRAINT[END_REF]Schwartz et al., 2014):

J(δx ⃗⃗(t 0 )) = β 1 2 δx ⃗⃗ s (t 0 ) B -1 δx ⃗⃗ s (t 0 ) + β 2 2 a ⃗⃗ T A -1 a ⃗⃗ + 1 2 (d 0 -Hδ𝑥 ⃗(t 0 )) T R -1 (d 0 -Hδ𝑥 ⃗(t 0 )), IV.14
where 𝑎 ⃗ is an array containing the extended control variable for each ensemble member. A is the ensemble covariance localization matrix; it controls the spatial correlation of 𝑎 ⃗ ; 𝛽 1 and 𝛽 2 are user defined parameters that determine how much weight is given to the ensemble and static BECs. 𝛽 1 and 𝛽 2 are constrained such that:

1 = 1 β 1 + 1 β 2 .
IV.15 In this figure we can see that the coupling between the ETKF and the 3DVAR algorithm. First the ensemble perturbation supplies an estimate of the flow dependent forecast error A to the 3DVAR analysis. Second the ensemble member analysis is recentered about the hybrid analysis. The high-resolution hybrid analysis is then used as initial and boundary conditions for a WRF forecast instead of the GFS coarse resolution analysis. In our Studies we performed the hybrid analysis for the three domains of WRF showed previously chapter III section III.

Fig. IV.3 Flowchart describing the hybrid ETKF-3DVAR

DA system in WRF model; inspired after [START_REF] Barker | Hybrid Variational-Ensemble Data Assimilation[END_REF][START_REF] Kutty | Impact of Representing Model Error in a Hybrid Ensemble-Variational Data Assimilation System for Track Forecast of Tropical Cyclones over the Bay of Bengal[END_REF]

IV.2 Building the ensemble members

As stated in earlier chapter III section III, we chose the initialization at 00 UTC to allow for a spin up of 12 hours before the sunrise in French Guiana. Consequently, each member should be valid at 00h

UTC and allows for 12 hours spin-up at least. To build the ensemble valid for the day d at 00h UTC; first we perturbed GFS initial and boundary condition valid for the day d-1 at 00h UTC using randomcv utility provided in WRFDA (NCAR, 2017). Second we use this perturbed set of initial and boundary conditions to run WRF for a 24-hour forecast, which is valid for day d at 00h UTC. These forecasts were computed using WRF-M9 setup for the same three way nested domain shown chapter III section III. We repeated this cycle 40 times to obtain a 40 ensemble member. This ensemble size was based on Vetra-Carvalho (2013) finding that ETKF is able predicts the initial cloud growth for ensemble containing at least 30 members.

IV.3 Modeling of the background error covariances A and B

B plays a key role in Var, ENKF, pure and hybrid EnVar DA algorithms. It weights the importance of the background state by spreading information from observation points and imposing balance between analysis variables 𝑥 𝑎 ⃗⃗⃗⃗⃗(𝑡) [START_REF] Wang | Inhomogeneous Background Error Modeling for WRF-Var Using the NMC Method[END_REF]. Current NWP models have a typical dimension of 10 8 which gives B matrix with 10 16 elements [START_REF] Descombes | Generalized background error covariance matrix model (GEN_BE v2.0)[END_REF]. This number of elements is too important to be calculated explicitly and stored. Therefore, B needs to be preconditioned such that it becomes numerically manageable [START_REF] Wang | Inhomogeneous Background Error Modeling for WRF-Var Using the NMC Method[END_REF]. To precondition B, first a control variable transform (CVT) is applied [START_REF] Descombes | Generalized background error covariance matrix model (GEN_BE v2.0)[END_REF][START_REF] Wang | Inhomogeneous Background Error Modeling for WRF-Var Using the NMC Method[END_REF]. Instead of using the state variable 𝑥 ⃗ of the NWP model, other variables (control variables) that are likely to be mutually uncorrelated are selected. Second B is decomposed in a series of sub-matrices expressed in the control variables space. Each matrix corresponds to an elemental transform that can be individually modeled. These two steps aim to renders B diagonal, they are expressed mathematically as follows [START_REF] Descombes | Generalized background error covariance matrix model (GEN_BE v2.0)[END_REF][START_REF] Liu | Algorithm (2): Background Error Modeling and Estimation[END_REF][START_REF] Wang | Inhomogeneous Background Error Modeling for WRF-Var Using the NMC Method[END_REF]:

{ δx ⃗⃗ = Uu ⃗⃗, B = UU T 𝑎𝑛𝑑 U = U p U v U h .

IV.16

where 𝑢 ⃗⃗ is the control variable array; U the transformation matrix; 𝑈 𝑝 , 𝑈 𝑣 and 𝑈 ℎ are the physical transform operator, vertical transform matrix, and the horizontal transform matrix respectively. They aim to remove the remaining correlation (e.g, balanced part) between the control variable using linear regressions, define the vertical correlation between control variables using recursive filters and horizontal correlations via empirical orthogonal decomposition of vertical covariance. 𝑆 is a diagonal matrix composed of the standard deviations of the background error.

According to [START_REF] Dhanya | Impact of variational assimilation using multivariate background error covariances on the simulation of monsoon depressions over India[END_REF]; Xin Li et al. (2016); [START_REF] Wang | Inhomogeneous Background Error Modeling for WRF-Var Using the NMC Method[END_REF] the formulation and choice of control variables have an impact on the analysis as well as model forecasts. Bannister (2008b) and [START_REF] Gustafsson | Survey of data assimilation methods for convective-scale numerical weather prediction at operational centres[END_REF] listed in their study the control variable used in NWP forecast centers. In WRF V3.8 (NCAR, 2017) four choices of control variable are available (cv3, cv5, cv6, cv7). In cv3 the control variables are in physical space; i.e. the state vector variable are used. It is assumed that there is no spatial and multivariate correlation [START_REF] Liu | Algorithm (2): Background Error Modeling and Estimation[END_REF]. This option uses stream function, unbalanced velocity potential, unbalanced temperature, mixing ratio and unbalanced logarithm surface pressure. Cv3 provides a generic B that has a global coverage, however, it needs to be tuned to get a an improved analysis [START_REF] Guo | Impact of WRF-Var (3DVar) Background Error Statistics on Typhoon analysis and Forecast[END_REF][START_REF] Radi | Tuning of WRF 3D-Var data assimilation system over Middle-East and Arabian Peninsula 10[END_REF]. In cv5, cv6, cv7 the control variable are instead in eigen-vector space (NCAR, 2017). cv5 and cv6 use quasi-similar control variables; cv5 uses stream function, unbalanced velocity potential, unbalanced temperature, pseudo-relative humidity and unbalanced surface pressure. Cv6 uses an unbalanced pseudo relative humidity instead of using the pseudo-relative humidity. A major difference between cv5 and cv6 is that cv5 correlation used between the velocity potential and temperature or the velocity potential and surface pressure [START_REF] Chen | Balance characteristics of multivariate background error covariances and their impact on analyses and forecasts in tropical and Arctic regions[END_REF]. Therefore, neither temperature nor surface pressure observations can directly influence the divergent part of the wind. Similarly the moisture observations will not have any impact on other variables like wind, temperature, and surface pressure. To overcome these limitations, cv6 includes linear relationship between temperature and surface pressure with unbalanced velocity potential as well as the inclusion of moisture correlations with all other analysis variables [START_REF] Chen | Balance characteristics of multivariate background error covariances and their impact on analyses and forecasts in tropical and Arctic regions[END_REF]. [START_REF] Chen | Balance characteristics of multivariate background error covariances and their impact on analyses and forecasts in tropical and Arctic regions[END_REF] showed that in tropical regions the inclusion of these extra correlations improved the forecast accuracy. According to [START_REF] Dhanya | Impact of variational assimilation using multivariate background error covariances on the simulation of monsoon depressions over India[END_REF] Moisture is a highly variable component in the atmosphere especially in tropical regions where convection is a major driving factor for weather systems, better formulation of moisture fields in the analysis system can improve the analysis and consequently the model predictions. Cv7 uses a different set of control variables than cv5 and cv6; it uses horizontal wind components as momentum control variable temperature, pseudo-relative humidity and surface pressure. For this cv7 the transformation matrix U p Eq.IV.16 becomes an identity matrix [START_REF] Sun | Comparison of the Impacts of Momentum Control Variables on High-Resolution Variational Data Assimilation and Precipitation Forecasting[END_REF]. Recent studies [START_REF] Li | Evaluation of two momentum control variable schemes and their impact on the variational assimilation of radarwind data: Case study of a squall line[END_REF]Sun et al., 2016;[START_REF] Xie | Selection of Momentum Variables for a Three-Dimensional Variational Analysis[END_REF] have shown that using stream function and velocity potential instead of horizontal wind components tends to produce non-physical wind increments with opposite direction to the observed wind in the neighborhood around the observation point. [START_REF] White | Improving cloud prediction in WRF through the use of GOES satellite assimilation[END_REF] showed in this study that nudging horizontal velocity has the ability to improve GHI forecasts. Therefore, a CVT that describes accurately wind increments is considered because in our study there are a significant number of wind velocity observations. Consequently, for the static background B, the NMC method [START_REF] Parrish | The National Meteorological Center's Spectral Statistical-Interpolation Analysis System[END_REF] Eq.IV.17 is performed using the horizontal wind component as momentum control variable temperature, pseudo-relative humidity and surface pressure control variables. For the flow dependent background we use the ensemble method Eq.IV.18.

B NMC = x ⃗⃗ 24 -x ⃗⃗ 12 𝑎𝑛𝑑 IV.17

B ENS = 1 N -1 ∑(x ⃗⃗ i -〈x ⃗⃗ i 〉 N i=1 )(x ⃗⃗ i -〈x ⃗⃗ i 〉) T .

IV.18

Where 𝑥 ⃗ 24 and 𝑥 ⃗ 12 are forecasts which are valid at the same time for different initialization times; N and 𝑥 ⃗ 𝑖 are the ensemble size and an ensemble member respectively.

IV.4 Experiments

Due to computational constraints we selected a limited number of experiments for a limited time span to validate 3DEnVar GHI forecasts. In CHAPTER IIIIII we have calibrated and validated WRF GHI forecasts in French for March, May and September. To choose which month to use we set the following constraint. The GHI forecasts obtained using 3DEnVar should not deteriorate significantly the GHI forecast under clear sky conditions and should improve the GHI forecasts under cloudy and overcast sky conditions. To meet this constraint we validated the GHI forecasts 3DEnVar over September because this month has the highest number of clear sky conditions samples. Fig. IV.4 shows the cdf of 𝐾 𝑐 values when the GHI measurements of all stations are merged from September 1 st to September 29 th . In September 5% of 𝐾 𝑐 value are lower than 0.4; approximately 12% range between 0.4 and 0.65 and 83% of the sample size have 𝐾 𝑐 values higher than 0.65. In the 3DEnVar DA algorithm two parameters can be adjusted: the ensemble weight 𝛽 2 Eq.IV.14 and the correlation length scale. The correlation length scale was defined to 500 km, the default value.

The ensemble weight was defined so that there is first a 50% and second a 75% contribution from the flow dependent background i.e. 𝛽 2 = 2 and 𝛽 2 = 4 3 respectively. The experiments performed are described in this study are described as follows:

(i)

The control experiment; WRF-M9 described chapter III. In this experiment there is no data assimilation; WRF is initialized using GFS forecasts at 00h UTC and ran for 48h. WRF forecasts are forced every 3h by GFS. Once a 48h forecast is obtained, a new one is launched so that the last 48h is valid for September 29 th at 00h UTC.

(ii) 3DEnVar DA with no cycling (WRF-M9-3DEnVar). In this experiment 𝛽 2 = 2 and WRF is initialized at 00h UTC using the analysis produce by 3DEnVar and ran for 48h. From +3h to +45h WRF forecasts are forced using GFS. Once a 48h forecast is obtained, a new one is launched so that the last 48h is valid for September 29 th at 00h UTC.

(iii) 3DEnVar DA with a 6h cycling (WRF-M9-3DEnVar-c). In this experiment 𝛽 2 = 2; the hybrid data assimilation is performed once at 00h UTC and WRF forecasts are cycled every 6 hours. WRF is run for 6h and the forecast obtained at t+6h is used a background for another 6h forecast. This forecast cycle is repeated so at the end we obtain a 48h 12% respectively. Nevertheless, the greater results of WRF-M9-3DEnVar-c compared to WRF-M9 meet the constraint initially set: not deteriorate significantly the GHI forecast under clear sky conditions and should improve the GHI forecasts under cloudy and overcast sky conditions. Consequently these results show that WRF-M9-3DEnVar-c is able to improve the forecast of cloudy and overcast sky conditions during September a rainy season month. Tab. IV.2 also shows that the hybrid DA impact the GHI forecasts under clear sky conditions. The GHI forecasts increase by as much as 4 W/m² (+5%) whereas the MBE decreases as much as 15 W/m² (-88%). [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF] showed in this study that IFS is the most accurate global model in extra-tropical climate. This model even outperformed GHI forecasts from optimized mesoscale model including WRF. We have shown in Chapter III section that similarly to extra-tropical climate IFS is more accurate than GFS in tropical climate. Consequently, in this section we compare our calibrated WRF model with data assimilation (e.g., WRF-M9-3DEnVar-c) to IFS forecasts in French Guiana. According to [START_REF] Lorenz | Comparison of global horizontal irradiance forecasts based on numerical weather prediction models with different spatio-temporal resolutions: Comparsion of irradiance forecasts based on NWP models[END_REF] spatial average improve the forecast metrics. Consequently, we averaged WRF GHI forecasts over a 12² km area and compared it against IFS hourly forecast available over a 14² km grid.

IV.4.b Comparison of WRF-M9-3DEnVar-c against WRF-M9 and IFS forecast in September

We have shown in chapter III sections II and III that IFS and WRF accuracy varied with K c and V values. Consequently, we represented Fig. IV.6 WRF-M9, WRF-M9-3DEnVar-c and IFS MAE as a function of K c and V. We can see Fig. IV.6 and Fig. IV.7 that averaging spatially the GHI forecasts of WRF-M9 and WRF-M9-3DEnVar-c increase the MAE improvement of WRF-M9-3DEnVar-c over WRF-M9 obtained previously for a single grid cell. Under all sky conditions and for a single grid cell the MAE improvement of WRF-M9-3DEnVar-c over WRF-M9 was 2%. For an average over a 12² km area the MAE improvement of WRF-M9-3DEnVar-c over WRF-M9 is 6%. Under cloudy and overcast sky conditions the improvement cause by the spatial average is 9% and 7% respectively.

When comparing the GHI forecasts of WRF-M9-3DEnVar-c to those of IFS we find that under clear sky conditions WRF-M9-3DEnVar-c outperforms the GHI forecasts of IFS-1h. The relative improvement of WRF-M9-3DEnVar-c MAE over IFS MAE is 10% (Fig. IV.7). However, under cloudy and overcast sky conditions it is the opposite; the relative worsening of WRF-M9-3DEnVar-c MAE over IFS MAE is -37% and -45% respectively. Nevertheless, due to the higher occurrence of clear sky conditions during September the accuracies of WRF-M9-3DEnVar-c and IFS are close. Under all the relative worsening of WRF-M9-3DEnVar-c MAE over the MAE of IFS is -5%. Tab. IV.4 shows a comparison of IFS and postprocessed WRF for various types of extra-tropical climate. For arid and humid continental climates the GHI forecasts of WRF are worse than the GHI forecasts of IFS; the relative worsening of WRF over IFS is -25% and -30 %respectively. In Mediterranean WRF with MOS has a similar accuracy than IFS. These results showed that the hybrid data we used performed successfully comparatively to MOS methods.

Fig. IV.6 WRF-M9, WRF-M9-3DEnVar, and IFS MAE as a function of K c and V

IV.5 Conclusions

In this section we have suggested a method to improve the GHI forecasts of mesoscale models in French Guiana and in the ITZ. This method is based on an hybrid ensemble variational ensemble data assimilation: 3DEnVar; it was validated in French Guiana. Due to computational constraints we selected a limited number of experiments and over a limited time span to validate 3DEnVar GHI forecasts. The validation period was chosen so that it guarantees the ability of the method to not deteriorate significantly the GHI forecast under clear sky conditions and to improve the GHI forecasts under cloudy and overcast sky conditions. We found that : (i) our method meet the constraint initially set: In September during the dry season and under clear sky conditions WRF-M9 and WRF-M9-3DEnVar have similar MAE values; however, under cloudy sky and overcast conditions the MAE of WRF-M9 is improved by 6% and 7% respectively. (ii) the balance between the flow dependent background and static background highly influence the forecast accuracy. Due to lack of computational resources we did not optimized this balance. Nevertheless, the fact that the forecasts under cloudy and overcasts sky conditions are improved whereas the GHI forecasts under clear sky are not deteriorated shows that this method is promising. (iii) Cycling WRF GHI forecasts improve the GHI forecasts. The GHI forecasts of WRF-M9-3DEnVar were also compared to the GHI of IFS. We found that under clear sky conditions WRF-M9-3DEnVar-c outperforms the GHI forecasts of IFS-1h.

The relative improvement of WRF-M9-3DEnVar-c MAE over IFS MAE is 10%. However, under cloudy and overcast sky conditions it is the opposite; the relative worsening of WRF-M9-3DEnVar-c MAE over IFS MAE is -37% and -45% respectively. Nevertheless, due to the higher occurrence of clear sky conditions during September the accuracies of WRF-M9-3DEnVar-c and IFS are close; the relative improvement of WRF-M9-3DEnVar-c MAE over the MAE of WRF-M9 is -5%. Our future research will involve optimizing the balance between the flow dependent background and static background and validating the method in the rainy season.

CHAPTER V : Thesis summary and conclusions

The contribution of this thesis to the scientific community was: to use existing and well-known methods (i.e Heliosat-II (H-II) and weather and research forecast (WRF)) and adapt them so that they produce improved Global Horizontal Irradiance (GHI) estimates in French Guiana. Weather conditions in French Guiana are driven by the physical phenomena in the Intertropical zone (ITZ); consequently, methods developed in this manuscript for French Guiana aimed to be extended to all the ITZ.

Our first contribution to the scientific community was to modify the H-II method so it can account for cloud absorption. The rationale behind this choice came from a literature review of the physical properties of the ITZ. The ITZ is an area with highly variable cloud cover dynamic. In this area we encounter significant amounts of convective clouds. Consequently, to improve the GHI forecasts in the ITZ the effect of clouds must be modeled accurately. Native H-II method accounts for cloud attenuation empirically by using a clear sky index (K c ) that is a function of a cloud index derived from satellite images. A drawback of using K c is that it does not model cloud properties such as cloud optical depth, which influences cloud reflectance, absorbance and transmittance. Therefore, to consider the significant presence of clouds we integrated a spatiotemporally dependent cloudy sky RTP into the GHI estimate formula of native H-II. This cloudy sky model integrates the cloud albedo and the cloud absorption which is a proxy for the cloud optical depth. This study showed that the GHI forecasts of the H-II method can be improved by accounting for the cloud absorption. Our method does not increase computation time and only require one parameter that must be computed beforehand using cloud absorption abacus available in the literature. The scope for future research involves validating this methodology to other locations; for this purpose a monthly map of cloud absorption must be computed, first, for French Guiana, and then, for other tropical areas.

Our second contribution to the scientific community was to fill the gap on the knowledge about the forecasts accuracy of GHI products from global NWP models in the ITZ. We compared IFS and GFS against GHI measurements from three territories located in the ITZ and with tropical climates. That study provided benchmark tools to assess the forecast accuracy of a limited area NWP model GHI forecast and for assessing the quality of post-processing methods. It also identified the source of inaccuracies of the GHI forecasts from NWP in the ITZ. This study showed the inability of model forecast realistic clouds cover; this inability impact cloudy, overcast and clear sky conditions. Under sky conditions reported as clear by the ground measurements, IFS and GFS forecasts cloudy and overcast sky conditions a significant amount of time. We compared results in tropical climates to results in extra tropical climates (Mediterranean, oceanic, continental, and arid climates) and found by comparison with the persistence that is more difficult to forecasts the GHI in tropical climates. The scope for future research involves studying for other locations and other climate in the ITZ. The scope for future research involves validating this methodology to other locations in the ITZ.

Our third contribution to the scientific community was to suggest based on the analysis of IFS and GFS a generic methodology to WRF mesoscale model in the ITZ. The aim of this methodology was to restrain and select the minimum number of simulations to run, to obtain improved GHI forecasts in the ITZ compared to a non-calibrated model. Consequently, we built 34 Ensemble Members (EM) by varying the microphysics, land surface model, and planetary boundary layer parameterization schemes. These parameterizations were varied because they have a strong influence on the distribution and physical properties of the simulated cloud field. This methodology was validated against GHI forecasts from AROME mesoscale model and ground measurements in French Guiana. This study showed that the calibration process of a mesoscale model must be carefully considered and that using physical parameterization previously selected in similar climate is not suitable.

Our last contribution to the scientific community was to use an hybrid ensemble variational data assimilation method to improve the GHI forecasts of WRF in the ITZ. The rationale behind this choice was to find a method that does not require an extensive training dataset; that is applicable for the whole computation domain and that does not require that the setup of the NWP model to be frozen. Ensemble variational data assimilation methods have proved their abilities to improve the characterization of severe convection events but have not been used yet for solar energy purposes. This method used the greatest ensemble member previously computed on the calibration process of WRF first to compute the static background and to build the initial ensemble needed for the assimilation method. Results found showed that this method improves the GHI forecast under cloudy and overcast sky conditions without lowering significantly the accuracy under clear sky conditions. The accuracy of the GHI forecasts depend highly on the balance between the flow dependent and static background which needs to be optimized. Nevertheless, this method is computationally demanding: it requires computing the background and the initial ensemble. The scope for future research is to find the adequate balance between the static and the flow dependent background covariance and to modify our method so it can be used operationally. In order the computational time the following solution may be considered: (i) instead of computing the initial ensemble, we may use ensemble forecasts available from forecasts center at coarser resolution and or (ii) use and optimize background covariance also provided globally for coarser resolutions. Using these two suggestions may one hand reduce the computing time but on the other reduce the accuracy of the GHI forecasts.

Conclusions et apport de la thése

La contribution de ce travail de thèse à la communauté scientifique a été : d'utiliser des outils connus et réputés et de les adapter pour qu'ils fournissent des estimations (avec H-II) et des prévisions (avec WRF) améliorées en Guyane. Les conditions météorologiques en Guyane sont influencées par les phénomènes physiques se déroulant dans la zone intertropicale; en conséquence, les méthodes appliquées en Guyane française ont pour vocation à être étendues à toute la zone intertropicale.

Notre première contribution à la communauté scientifique a été de modifier H-II afin qu'il puisse tenir compte de l'absorption des nuages. Ce choix a été motivé par une étude des propriétés physiques de l'ITZ. Cette zone est caractérisée par une dynamique nuageuse importante; en conséquence les nuages et leurs propriétés physiques doivent être modélisés pour obtenir de meilleurs estimés de GHI. La méthode H-II originelle modèle l'atténuation des nuages en utilisant une relation empirique fonction d'un index nuageux déduit à partir d'images satellites. Un inconvénient de cette méthode est que les propriétés physiques des nuages tels leurs épaisseurs optiques ne sont pas modélisées. L'épaisseur optique influence à la fois l'absorption, la réflexion et la transmission des nuages. En conséquence, afin de tenir compte de la présence significative de nuage, nous intégrons un modèle de transfert radiatif en ciel couvert dans la formulation de GHI de la méthode H-II originelle. Ce modèle de ciel couvert intègre l'albédo des nuages qui est un proxy de l'épaisseur optique. Cette étude a montré que la prise en compte de l'absorption des nuages permet une amélioration des estimés de GHI. Notre méthode n'augmente pas de manière significative le temps de calcul et ne nécessite qu'un seul paramètre supplémentaire ; celui-ci est calculé à priori à partir d'abaque d'absorption des nuages disponible dans la littérature. Les recherches futures consisteraient à valider cette méthode dans d'autre pays de la zone intertropicale.

Notre deuxième contribution à la communauté scientifique d'étudier les performances des modèles IFS et GFS en zone intertropicale. Nous avons comparé les prévisions d'IFS et de GFS avec des mesures au sol provenant de trois territoires dans l'ITZ de climat tropical. Cette étude a permis des outils comparatifs permettant de valider la calibration du modèle WRF et la validation de méthode de post-processing. De plus ils ont permis d'identifier les sources d'erreurs des prévisions de GHI en zone intertropicale. Cette étude a montré que les modèles IFS et GFS étaient incapables de simuler une couverture nuageuse réaliste ; cette inhabilité se répercute aussi bien en ciel clair que couvert. Nous avons comparé les résultats obtenus en climat tropical aux résultats obtenus en climat extratropical (Méditerranéen, océanique, continental, et aride); ceci nous a permis de montrer par comparaison avec le modèle de persistance que la prévision est climat extratropical est plus aisé. Les recherches futures consisteraient à valider cette méthode dans d'autre pays de la zone intertropicale.

Notre troisième contribution à la communauté scientifique a été de suggéré une méthode générique permettant de calibrer le modèle mésoéchelle WRF dans l'ITZ. La méthodologie développée pour calibrer WRF se base sur l'identification des sources d'erreurs des prévisions de GHI en zone intertropicale. L'objectif de cette étude a été de choisir restreindre et choisir le nombre minimum de simulations à effectuer afin que le modèle WRF produise des prévisions de GHI de meilleures qualités qu'un modèle numérique non calibré. En conséquence nous avons construit un ensemble de simulation comprenant 34 membres. Cet ensemble fut construit en faisant varier le schéma de microphysique, le schéma de surface et le schéma de couche limite. Ces éléments ont été variés car une étude de la littérature a montré qu'ils avaient le plus d'influence sur les propriétés physiques de la couche nuageuse simulée. Cette méthode a été validée avec des mesures au sol ainsi que des prévisions de rayonnement du modèle AROME. Cette méthode a montré l'intérêt du processus de calibration avant l'utilisation d'un modèle de prévisions météorologiques mésoéchelle.

Where 𝛿𝑥 ⃗(0) ∈ ℜ 𝑛 is the analysis increment; 𝐻 ̃(𝑡 0 ) ∈ ℜ 𝑛⟶𝑝 the linearization of the observation operator at the analysis time 𝑡 0 and 𝑑 0 the innovation for the analysis at𝑡 0 . Therefore the Linearize model 𝑀 ̃ prediction step follows Eq.0.3: δx ⃗⃗⃗⃗⃗ (t + 1) = M ̃(t + 1, t)δx a ⃗⃗⃗⃗⃗⃗⃗ (t), 0.3

Where 𝛿𝑥 ⃗⃗⃗⃗⃗ ∈ ℜ 𝑛 is the forecast model increment and 𝑀 ̃∈ ℜ 𝑛.𝑛 the linearized model between t+1 and t.

I.2 4D-Var

4D-Var is an extension of the 3D-Var which allows for observations distributed within a time interval [𝑡 1 : 𝑡 𝑁 ]. It attempts to find an approximate solution of the minimum of Eq.IV.9, under the following approximations [START_REF] Holm | Lecture notes on assimilation algorithms[END_REF]:

-no model error -linearity of M ⃗⃗⃗⃗ and ℋ ⃗⃗⃗ -unbiased errors -errors are uncorrelated Under the above assumptions Eq.IV.9 becomes:

J = 1 2 [ ε b ⃗⃗⃗⃗ T B -1 ε b ⃗⃗⃗⃗ + ∑ ε 0,n ⃗⃗⃗⃗⃗⃗⃗ T R -1 N n=1 ε 0,n ⃗⃗⃗⃗⃗⃗⃗] = J b + J 0 .
0.4 4D-Var contrary to 3D-Var does not assimilate observation at a particular point within the assimilation window [START_REF] Rihan | Four-dimensional variational data assimilation for Doppler radar wind data[END_REF], instead the assimilation window is divided in several sub interval [START_REF] Laroche | Impact of the Different Components of 4DVAR on the Global Forecast System of the Meteorological Service of Canada[END_REF] to assimilate observation at their correct time. Therefore, 4D-Var has the ability to implicitly evolve the initial 𝐵 over the length of the assimilation window with the tangent linear dynamics: B(t) ≈ MB(t = t 0 )M T , 0.5

where 𝐵(𝑡 = 𝑡 0 ) is the background at the beginning of the assimilation windows and M the linearized forecast model value between the beginning and the end of the assimilation window. However, B does not propagate error information from one assimilation cycle to the next; it revert to the stationary value at the beginning of each assimilation window [START_REF] Miyoshi | Accounting for Flow-dependence in the Background Error Variance within the JMA Global Four-dimensional Variational Data Assimilation System[END_REF][START_REF] Raynaud | An extended specification of flow-dependent background error variances in the Météo-France global 4D-Var system[END_REF]Schwartz et al., 2014;[START_REF] Thepaut | Dynamical structure functions in a four-dimensional variational assimilation: A case study 27[END_REF]. Only the information about the state 𝑥 ⃗ is propagated from one cycle to the next following. Similarly to 3D-Var is exploited in operational NWP centers in the incremental form; it is expressed as follows [START_REF] Haben | Conditioning of the 3DVAR Data Assimilation Problem[END_REF][START_REF] Lawless | An investigation of incremental 4D-Var using nontangent linear models[END_REF]

0.6

The information about the state 𝑥 ⃗ is propagated from one assimilation cycle similar to 3-DVar Eq. 0.3.

II.

Stochastic method II.1 Kalman filter

In the Kalman filter (KF) DA, the analysis equation is solved directly by inversion.

∇J(x ⃗⃗ → x a ⃗⃗⃗⃗) = 0, 0.7

The KF DA assumes the following:

-the analysis is performed at each time step of the model, it uses only the observations available during that time step.

-linearity of M ⃗⃗⃗⃗ and ℋ ⃗⃗⃗ -unbiased errors -errors are uncorrelated Under the assumptions listed above [START_REF] Holm | Lecture notes on assimilation algorithms[END_REF], Eq.0.7 yields:

x a ⃗⃗⃗⃗(t) = x b ⃗⃗⃗⃗⃗(t) + K(t) (y(t) -H(x ⃗⃗ b (t))) with 0.8 K(t) = BH T (HBH T + R) -1 and 0.9

P(t) = (I -KH)B. 0.10
Where K is the Kalman gain. Contrary to 3DVar and 4DVar B is propagated from one cycle to the next alongside the information about the state 𝑥 ⃗. Therefore, the prediction step follows the following equations:

x ⃗⃗(t + 1) = Mx a ⃗⃗⃗⃗(t) 𝑎𝑛𝑑 0.11 𝐵(𝑡 + 1) = 𝑀𝑃(𝑡)𝑀 𝑇 + 𝑄. 0.12 However, KF is very expensive to evaluate directly for atmospheric science due to size of K e.g 𝑂(10 7 ) [START_REF] Ghil | Data Assimilation in Meteorology and Oceanography[END_REF]. Therefore, several approximations were made to achieve a computationally efficient DA algorithm. One simplification of KF is the ensemble Kalman filter (ENKF) is presented in the following section.

II.2 stochastic and deterministic ensemble Kalman filter

ENKF is an approximate version of the KF where the distribution of possible states is represented by an ensemble. This ensemble is propagated forward through time and updated when new data become available [START_REF] Katzfuss | Understanding the Ensemble Kalman Filter[END_REF]. The ensemble representation is a form of dimension reduction, instead of specifying B, the EnKF uses an ensemble of possible forecasts that contains valuable flow-dependent information about the background-error statistics [START_REF] Bannister | A review of operational methods of variational and ensemble-variational data assimilation: Ensemble-variational Data Assimilation[END_REF]. This dimension reduction leads to computational feasibility even for very high-dimensional systems. ENKF DA algorithms are either classified as stochastics (SENKF) or deterministic (DENKF). They differ on the methodology used to get the updated ensemble; they share the same prediction step but have different update state. SENKF directly use the Kalman gain together with random perturbations whereas DENKF uses a non-random transformation on the forecast ensemble [START_REF] Lei | Comparison of Ensemble Kalman Filters under Non-Gaussianity[END_REF].

Stochastic ENKF methods were first introduced by [START_REF] Evensen | Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics[END_REF], since then many variations on the original algorithm have been developed [START_REF] Evensen | The Ensemble Kalman Filter: theoretical formulation and practical implementation[END_REF][START_REF] Houtekamer | Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation[END_REF][START_REF] Lei | Comparison of Ensemble Kalman Filters under Non-Gaussianity[END_REF]. Nevertheless, the update step from the SENKF framework can be expressed generally as follows:

x a ⃗⃗⃗⃗⃗⃗⃗ are the analysis and state vector of the ensemble member i. 𝑦 ̃ is the perturbed observations and 𝑅 𝑒 the observation error covariance matrix consistent with𝑦 ̃. Eq.0.13-0.14 differ from Eq. 0.8-0.9 because of the addition of a perturbation to the observation. [START_REF] Burgers | Analysis Scheme in the Ensemble Kalman Filter[END_REF] showed that the observations must be treated as random variables at the analysis steps in order for P, the variance of the ensemble to be consistent with Eq.0.10 after the update step. [START_REF] Burgers | Analysis Scheme in the Ensemble Kalman Filter[END_REF] found that for the SENKF that P is expressed as follows: P(t) = (I -KH)B + 𝒪 (N -1 2 ), 0.15

where N is the ensemble size and 𝒪(𝑁 -1/2 ) a negligible term that decreases as the ensemble size increase. Therefore, 𝑦 ̃ and 𝑅 𝑒 were defined as follows:

{ y ̃= y ⃗⃗ + ε ⃗ 𝑎𝑛𝑑 R e =< ε ⃗ε ⃗ T >.

0.16

Where 𝑦 ⃗ and 𝜀⃗ are the observation as defined in the KF and 𝜀 the observation perturbation. 𝑅 𝑒 assumes that errors are unbiased. In SENKF K is not computed directly as in KF but using ensemblebased approximations [START_REF] Houtekamer | Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation[END_REF]: 0.17

{ BH T = 1 N -1 ∑ (x b (
Similarly to KF, B is evolved explicitly propagated from one cycle to the next alongside the information about the state 𝑥 ⃗. First the state of the member number is evolved:

x (i) ⃗⃗⃗⃗⃗⃗⃗ (t + 1) = M (x a (i) ⃗⃗⃗⃗⃗⃗⃗⃗⃗ (t)) + ε m ⃗⃗⃗⃗⃗ .

0.18

Then when all members are evolved, the ensemble mean and its covariance are reconstructed as follows:

{ < x (i) ⃗⃗⃗⃗⃗⃗⃗ >= 1 N ∑ x (i) ⃗⃗⃗⃗⃗⃗⃗ N i=1 𝑎𝑛𝑑 B = 1 N -1
∑(x (i) ⃗⃗⃗⃗⃗⃗⃗ -< x (i) ⃗⃗⃗⃗⃗⃗⃗ >)(x (i) ⃗⃗⃗⃗⃗⃗⃗ -< x (i) ⃗⃗⃗⃗⃗⃗⃗ >) T = XX T . N i=1

0.19

Where 𝑋 is an array that contains the ensemble perturbations. In the SENKF if the ensemble number is not sufficient due to the computational resources, perturbing the observation with random perturbations will cause sampling errors, which makes the filter suboptimal [START_REF] Sakov | A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters[END_REF]. Statistics will not be necessarily representative of the state with estimated variances that are systematically too small [START_REF] Bannister | A review of operational methods of variational and ensemble-variational data assimilation: Ensemble-variational Data Assimilation[END_REF]. For this reason DENKF methods such as the ensemble square root filter (EnSRF), the ensemble adjustment Kalman filter (EAKF), and the ensemble transform Kalman filter (ETKF) were proposed [START_REF] Houtekamer | Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation[END_REF]. The idea of DENKF is to create an updated ensemble with covariance consistent with Eq.0.10 using a transformation matrix𝑇 𝑀 . Instead of updating each ensemble member separately as in Eq.0.13, the ESRF generates the new ensemble simultaneously as follows [START_REF] Sakov | A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters[END_REF][START_REF] Zhang | Ensemble filter methods with perturbed observations applied to nonlinear problems[END_REF]):

Fig. I. 1 French

 1 Fig. I.1 French Guiana electricity grid (Ministry of the environment, 2016a) ....................................... Fig. II.1 Earth radiation budget credit American meteorological society (AMS). .................................. Fig. II.2 schematic representation of the solar zenith angle 𝜃0, sun declination δ, the hour angle h (Liou, 2002) ............................................................................................................................................ Fig. II.3 schematic representation of the azimuth angle ∅ (Sidek et al., 2014) .................................... Fig. II.4 schematic representation of angles involved in radiometric quantities (Liou, 2002) .............. Fig. II.5 schematic representation of scattering angle for an incoming ray (In) in the direction 𝛺(𝜇, 𝜙)that is deviated in the direction 𝛺′(𝜇′, 𝜙′) (Liou, 2002) ........................................................... Fig. II.6 Solar radiation spectrum credit geosciencebigpicture ............................................................. Fig. II.7 thermopile (left) and photodiode (right) pyranometer. Source Wikipedia, Licor .................... Fig. II.8 Plane Parallel atmosphere ........................................................................................................ Fig. II.9 Transfer of solar radiation in plane-parallel layers, illustrating attenuation by extinction, a; multiple scattering, b; and single scattering of the unscattered solar flux, c. Source (Liou, 2002) ...... Fig. II.10 Configuration of the adding method for two layers of optical depth 𝜏1 and 𝜏2respectively. The upward transmission and reflection of the i th layer are 𝛵𝑖 * and 𝑅𝑖 , respectively whereas the downward transmission and reflection are 𝛵𝑖 and 𝑅𝑖 * . 𝐷 and 𝑈 are the the combined total transmission and reflection functions between layers 1 and 2. Source (Liou, 2002) ........................... Fig. II.11 solar radiation flux at the atmosphere inspired after (Noia et al., 1993a) ............................. Fig. II.12 atmospheric model of[START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] adapted from(Noia et al., 1993b) ..................... Fig. II.13 atmospheric model of Janjai, (2010) ...................................................................................... Fig. III.1 Conceptual diagram of forecast skill as a function of forecast lead time for different forecast methods. Extracted from (Kleissl, 2013; Ruiz-Arias and Goenka, 2017) ............................................... Fig. III.2 Physical process accounted for in NWP models extracted from (Helmert, 2016) .................. Fig. III.3 Interaction between physical parameterization. Extracted from (Dudhia, 2014) ................... Fig. VII.1 Measurements instrument used in DA system credit WMO ............................................... Fig. VII.2 Observation network in ECMWF DA system ........................................................................ Fig. VII.3 Flowchart describing the hybrid ETKF-3DVAR DA system in WRF model; inspired after[START_REF] Barker | Hybrid Variational-Ensemble Data Assimilation[END_REF][START_REF] Kutty | Impact of Representing Model Error in a Hybrid Ensemble-Variational Data Assimilation System for Track Forecast of Tropical Cyclones over the Bay of Bengal[END_REF] .................................................................................... Fig. VII.4 Cdf of Kc values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.5 Cdf of V values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.6 WRF-M9, WRF-M9-3DEnVar, and IFS MAE as a function of K c and V .................................. Fig. VII.7 MAE (W/m²) and MBE (W/m²) of WRF-M9, WRF-M9-3DEnVar-c and IFS ...........................

  Fig. I.1 French Guiana electricity grid (Ministry of the environment, 2016a) ....................................... Fig. II.1 Earth radiation budget credit American meteorological society (AMS). .................................. Fig. II.2 schematic representation of the solar zenith angle 𝜃0, sun declination δ, the hour angle h (Liou, 2002) ............................................................................................................................................ Fig. II.3 schematic representation of the azimuth angle ∅ (Sidek et al., 2014) .................................... Fig. II.4 schematic representation of angles involved in radiometric quantities (Liou, 2002) .............. Fig. II.5 schematic representation of scattering angle for an incoming ray (In) in the direction 𝛺(𝜇, 𝜙)that is deviated in the direction 𝛺′(𝜇′, 𝜙′) (Liou, 2002) ........................................................... Fig. II.6 Solar radiation spectrum credit geosciencebigpicture ............................................................. Fig. II.7 thermopile (left) and photodiode (right) pyranometer. Source Wikipedia, Licor .................... Fig. II.8 Plane Parallel atmosphere ........................................................................................................ Fig. II.9 Transfer of solar radiation in plane-parallel layers, illustrating attenuation by extinction, a; multiple scattering, b; and single scattering of the unscattered solar flux, c. Source (Liou, 2002) ...... Fig. II.10 Configuration of the adding method for two layers of optical depth 𝜏1 and 𝜏2respectively. The upward transmission and reflection of the i th layer are 𝛵𝑖 * and 𝑅𝑖 , respectively whereas the downward transmission and reflection are 𝛵𝑖 and 𝑅𝑖 * . 𝐷 and 𝑈 are the the combined total transmission and reflection functions between layers 1 and 2. Source (Liou, 2002) ........................... Fig. II.11 solar radiation flux at the atmosphere inspired after (Noia et al., 1993a) ............................. Fig. II.12 atmospheric model of[START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] adapted from(Noia et al., 1993b) ..................... Fig. II.13 atmospheric model of Janjai, (2010) ...................................................................................... Fig. III.1 Conceptual diagram of forecast skill as a function of forecast lead time for different forecast methods. Extracted from (Kleissl, 2013; Ruiz-Arias and Goenka, 2017) ............................................... Fig. III.2 Physical process accounted for in NWP models extracted from (Helmert, 2016) .................. Fig. III.3 Interaction between physical parameterization. Extracted from (Dudhia, 2014) ................... Fig. VII.1 Measurements instrument used in DA system credit WMO ............................................... Fig. VII.2 Observation network in ECMWF DA system ........................................................................ Fig. VII.3 Flowchart describing the hybrid ETKF-3DVAR DA system in WRF model; inspired after[START_REF] Barker | Hybrid Variational-Ensemble Data Assimilation[END_REF][START_REF] Kutty | Impact of Representing Model Error in a Hybrid Ensemble-Variational Data Assimilation System for Track Forecast of Tropical Cyclones over the Bay of Bengal[END_REF] .................................................................................... Fig. VII.4 Cdf of Kc values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.5 Cdf of V values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.6 WRF-M9, WRF-M9-3DEnVar, and IFS MAE as a function of K c and V .................................. Fig. VII.7 MAE (W/m²) and MBE (W/m²) of WRF-M9, WRF-M9-3DEnVar-c and IFS ...........................

  Fig. I.1 French Guiana electricity grid (Ministry of the environment, 2016a) ....................................... Fig. II.1 Earth radiation budget credit American meteorological society (AMS). .................................. Fig. II.2 schematic representation of the solar zenith angle 𝜃0, sun declination δ, the hour angle h (Liou, 2002) ............................................................................................................................................ Fig. II.3 schematic representation of the azimuth angle ∅ (Sidek et al., 2014) .................................... Fig. II.4 schematic representation of angles involved in radiometric quantities (Liou, 2002) .............. Fig. II.5 schematic representation of scattering angle for an incoming ray (In) in the direction 𝛺(𝜇, 𝜙)that is deviated in the direction 𝛺′(𝜇′, 𝜙′) (Liou, 2002) ........................................................... Fig. II.6 Solar radiation spectrum credit geosciencebigpicture ............................................................. Fig. II.7 thermopile (left) and photodiode (right) pyranometer. Source Wikipedia, Licor .................... Fig. II.8 Plane Parallel atmosphere ........................................................................................................ Fig. II.9 Transfer of solar radiation in plane-parallel layers, illustrating attenuation by extinction, a; multiple scattering, b; and single scattering of the unscattered solar flux, c. Source (Liou, 2002) ...... Fig. II.10 Configuration of the adding method for two layers of optical depth 𝜏1 and 𝜏2respectively. The upward transmission and reflection of the i th layer are 𝛵𝑖 * and 𝑅𝑖 , respectively whereas the downward transmission and reflection are 𝛵𝑖 and 𝑅𝑖 * . 𝐷 and 𝑈 are the the combined total transmission and reflection functions between layers 1 and 2. Source (Liou, 2002) ........................... Fig. II.11 solar radiation flux at the atmosphere inspired after (Noia et al., 1993a) ............................. Fig. II.12 atmospheric model of[START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] adapted from(Noia et al., 1993b) ..................... Fig. II.13 atmospheric model of Janjai, (2010) ...................................................................................... Fig. III.1 Conceptual diagram of forecast skill as a function of forecast lead time for different forecast methods. Extracted from (Kleissl, 2013; Ruiz-Arias and Goenka, 2017) ............................................... Fig. III.2 Physical process accounted for in NWP models extracted from (Helmert, 2016) .................. Fig. III.3 Interaction between physical parameterization. Extracted from (Dudhia, 2014) ................... Fig. VII.1 Measurements instrument used in DA system credit WMO ............................................... Fig. VII.2 Observation network in ECMWF DA system ........................................................................ Fig. VII.3 Flowchart describing the hybrid ETKF-3DVAR DA system in WRF model; inspired after[START_REF] Barker | Hybrid Variational-Ensemble Data Assimilation[END_REF][START_REF] Kutty | Impact of Representing Model Error in a Hybrid Ensemble-Variational Data Assimilation System for Track Forecast of Tropical Cyclones over the Bay of Bengal[END_REF] .................................................................................... Fig. VII.4 Cdf of Kc values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.5 Cdf of V values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.6 WRF-M9, WRF-M9-3DEnVar, and IFS MAE as a function of K c and V .................................. Fig. VII.7 MAE (W/m²) and MBE (W/m²) of WRF-M9, WRF-M9-3DEnVar-c and IFS ...........................

  Fig. I.1 French Guiana electricity grid (Ministry of the environment, 2016a) ....................................... Fig. II.1 Earth radiation budget credit American meteorological society (AMS). .................................. Fig. II.2 schematic representation of the solar zenith angle 𝜃0, sun declination δ, the hour angle h (Liou, 2002) ............................................................................................................................................ Fig. II.3 schematic representation of the azimuth angle ∅ (Sidek et al., 2014) .................................... Fig. II.4 schematic representation of angles involved in radiometric quantities (Liou, 2002) .............. Fig. II.5 schematic representation of scattering angle for an incoming ray (In) in the direction 𝛺(𝜇, 𝜙)that is deviated in the direction 𝛺′(𝜇′, 𝜙′) (Liou, 2002) ........................................................... Fig. II.6 Solar radiation spectrum credit geosciencebigpicture ............................................................. Fig. II.7 thermopile (left) and photodiode (right) pyranometer. Source Wikipedia, Licor .................... Fig. II.8 Plane Parallel atmosphere ........................................................................................................ Fig. II.9 Transfer of solar radiation in plane-parallel layers, illustrating attenuation by extinction, a; multiple scattering, b; and single scattering of the unscattered solar flux, c. Source (Liou, 2002) ...... Fig. II.10 Configuration of the adding method for two layers of optical depth 𝜏1 and 𝜏2respectively. The upward transmission and reflection of the i th layer are 𝛵𝑖 * and 𝑅𝑖 , respectively whereas the downward transmission and reflection are 𝛵𝑖 and 𝑅𝑖 * . 𝐷 and 𝑈 are the the combined total transmission and reflection functions between layers 1 and 2. Source (Liou, 2002) ........................... Fig. II.11 solar radiation flux at the atmosphere inspired after (Noia et al., 1993a) ............................. Fig. II.12 atmospheric model of[START_REF] Gautier | A simple physical model to estimate incident solar radiation at the surface from GOES satellite Data[END_REF] adapted from(Noia et al., 1993b) ..................... Fig. II.13 atmospheric model of Janjai, (2010) ...................................................................................... Fig. III.1 Conceptual diagram of forecast skill as a function of forecast lead time for different forecast methods. Extracted from (Kleissl, 2013; Ruiz-Arias and Goenka, 2017) ............................................... Fig. III.2 Physical process accounted for in NWP models extracted from (Helmert, 2016) .................. Fig. III.3 Interaction between physical parameterization. Extracted from (Dudhia, 2014) ................... Fig. VII.1 Measurements instrument used in DA system credit WMO ............................................... Fig. VII.2 Observation network in ECMWF DA system ........................................................................ Fig. VII.3 Flowchart describing the hybrid ETKF-3DVAR DA system in WRF model; inspired after[START_REF] Barker | Hybrid Variational-Ensemble Data Assimilation[END_REF][START_REF] Kutty | Impact of Representing Model Error in a Hybrid Ensemble-Variational Data Assimilation System for Track Forecast of Tropical Cyclones over the Bay of Bengal[END_REF] .................................................................................... Fig. VII.4 Cdf of Kc values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.5 Cdf of V values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale ............................................................ Fig. VII.6 WRF-M9, WRF-M9-3DEnVar, and IFS MAE as a function of K c and V .................................. Fig. VII.7 MAE (W/m²) and MBE (W/m²) of WRF-M9, WRF-M9-3DEnVar-c and IFS ...........................

Fig. I. 1 French

 1 Fig. I.1 French Guiana electricity grid (Ministry of the environment, 2016a)

μ 0 = 2 Fig

 02 Fig. II.2 schematic representation of the solar zenith angle 𝜃 0 , sun declination δ, the hour angle h (Liou, 2002) I.1.c Azimuth ∅

Fig

  Fig. II.3 schematic representation of the azimuth angle ∅ ([START_REF] Sidek | GPS based portable dual-axis solar tracking system using astronomical equation[END_REF]

Fig

  Fig. II.9 Transfer of solar radiation in plane-parallel layers, illustrating attenuation by extinction, a; multiple scattering, b; and single scattering of the unscattered solar flux, c. Source[START_REF] Liou | An introduction to atmospheric radiation[END_REF]

Fig

  Fig. II.10 Configuration of the adding method for two layers of optical depth 𝜏 1 and 𝜏 2 respectively. The upward transmission and reflection of the i th layer are 𝛵 ̃𝑖 * and 𝑅 𝑖 , respectively whereas the downward transmission and reflection are 𝛵 ̃𝑖 and 𝑅 𝑖 *. 𝐷 ̃ and 𝑈 are the the combined total transmission and reflection functions between layers 1 and 2. Source[START_REF] Liou | An introduction to atmospheric radiation[END_REF]

Fig

  Fig. II.11 solar radiation flux at the atmosphere inspired after(Noia et al., 1993a)

Fig. 2 .

 2 Fig. 2. Measured GHI versus OPT_H2 GHI estimates under cloudy conditions from 2010 to 2013 for all stations. The full line is the identity line and the dashed line is the linear regression between the

Fig. 3 .

 3 Fig. 3. All stations' GHI ground-measured data versus RTP_OPT_H2 estimates under cloudy sky conditions from 2010 to 2013. The full line is the identity line and the dashed line is the linear regression between the ground measurements and RTP_OPT_H2

Figure 4

 4 Figure 4 also shows that modelling cloud absorption with RTP_OPT_H2 improved the Heliosat-2 cloudy sky GHI CDF for irradiance values between 42 and 278 Wh/m². These values are obtained my multiplying the normalized value Fig.4by the maximum GHI value measured under cloudy sky condition. These values represent approximately the 85 th percentile on average for all stations. The highest improvement was noted for irradiance of approximately 42 to 56 Wh/m² (Fig.4), which, depending on the station, represents approximately 20% to 30% of the cloudy sky samples. However, the correction effect was less obvious for irradiance values of less than 42 Wh/m² or more than 333 Wh/m² (Fig.4), except at Maripasoula, where noticeable improvements were found for values of up to 472 Wh/m².

  [START_REF] Rigollier | The method Heliosat-2 for deriving shortwave solar radiation from satellite images[END_REF] also found that Heliosat-2 performs poorly for high  s (values greater than or equal to 75°). Differences observed between stations in the intermediate percentiles can be explained by varying cloudiness conditions during the day. An analysis of satellite images containing the ground stations showed that cloud index values can vary rapidly for days considered cloudy; there were few days where the cloud index was constantly high.

Fig. 4 .

 4 Fig. 4. CDF of GHI estimates from RTP_OPT_H2 and OPT_H2 compared with CDF of measured GHI from 2010 to 2013 under cloudy sky conditions. The clearness index CDF (computed using ground measurements) under cloudy sky conditions is shown alongside the GHI cdfs for each station. The normalized GHI CDFs are obtained by dividing the GHI values by the maximum ground GHI value measured under cloudy sky condition. These normalization values are: for Saint Georges 847 Wh/m², Rochambeau 661 Wh/m², Kourou 697 Wh/m², Iles Royale 822 Wh/m², Saint Laurent 852 Wh/m², Maripasoula 966 Wh/m² 4.4 Quality of GHI estimates under clear sky conditions

Fig. 5 .

 5 Fig. 5. CDF of GHI estimates from RTP_OPT_H2 and OPT_H2 compared with CDF of measured GHI from 2010 to 2013 under clear sky condition. The clearness index CDF (computed using ground measurements) under clear sky conditions is shown alongside the GHI cdfs for each station. The normalized GHI CDFs are obtained by dividing the GHI values by the maximum ground GHI value measured under clear sky condition. These normalization values are: for Saint Georges 1058 Wh/m², Rochambeau 1075 Wh/m², Kourou 1019 Wh/m², Iles Royale 1055 Wh/m², Saint Laurent 1083 Wh/m², Maripasoula 1038 Wh/m²

Fig. 6 .

 6 Fig. 6. CDF of GHI estimates from RTP_OPT_H2 and OPT_H2 compared with the CDF of measured GHI from 2010 to 2013 under all sky conditions. The clearness index CDF (computed using ground measurements) under all sky conditions is shown alongside the GHI cdfs for each station. The normalized GHI CDFs are obtained by dividing the GHI values by the maximum ground GHI value measured under all sky condition. These normalization values are: for Saint Georges 1108 Wh/m², Rochambeau 1172 Wh/m², Kourou 1063 Wh/m², Iles Royale 1108 Wh/m², Saint Laurent 1122 Wh/m², Maripasoula 1066 Wh/m²

  categories: ground based, satellite based and NWP based Fig. III.1.

Fig. III. 1

 1 Fig. III.1 Conceptual diagram of forecast skill as a function of forecast lead time for different forecast methods. Extracted from[START_REF] Kleissl | Solar Energy Forecasting and Resource Assessment[END_REF] Ruiz-Arias and Goenka, 2017)

Fig. III. 2

 2 Fig. III.2 Physical process accounted for in NWP models extracted from(Helmert, 2016)

Figure 1 :

 1 Figure 1: Monthly mean of global horizontal irradiance (GHI) values at 1 and 3-hour intervals. Values were computed using hourly quality checked GHI values. The abbreviations of the station name and location are shown Table1.

Figure 3 :

 3 Figure 3: MAE and MBE of GHI forecast from IFS-3h, IFS-3hp and GFS-3h in French Guiana, Reunion Island and Singapore. Months and stations were merged by territory. The classification of sky conditions was based on ground measurements.

Figure 6 : 1 Figure 7 :

 617 Figure 6: GFS-3h and IFS-1h monthly relative MAE and RMSE (%) for French Guiana stations. The monthly mean value used to normalized the MAE and RMSE are shown Figure 1

Figure 8 : 1 Figure 9 :

 819 Figure 8: GFS-3h and IFS-1h monthly relative MAE and RMSE (%) for Reunion island stations. The monthly mean value used to normalized the MAE and RMSE are shown Figure 1

Figure 10 :

 10 Figure 10: GFS-3h and IFS-1h monthly relative MAE and RMSE (%) for Singapore stations. The monthly mean value used to normalized the MAE and RMSE are shown Figure 1

Figure 11 :

 11 Figure 11: Monthly K c value and associated standard deviations for Singapore stations. The Monthly K c value were computed using hourly quality checked GHI measurement between sunrise and sunset 3.6 Comparing GFS and IFS forecast accuracy for tropical climates and comparing their accuracy in tropical climate to extra tropical climates 3.6.1 Comparing GFS and IFS forecast accuracy for the different tropical climates

Figure 12 :

 12 Figure 12: Yearly IFS and GFS FS and relative RMSE for all stations. The yearly GHI values used to normalize the RMSE were respectively: SG: 387 W/m², KR: 443W/m², GI: 482 W/m²; LP: 474 W/m², PO: 451 W/m², PI: 507 W/m²; ST: 359 W/m², SQ: 390 W/m², SC: 373 W/m², SH: 364 W/m², RO: 394 W/m², MP: 425 W/m²

Figure 1 2 𝑖

 12 Figure 1: WRF computation domain 5

Figure 2 :

 2 Figure 2: Normalized RMSE for each ensemble members all months and stations merged. The absolute RMSE (W/m²) of each member are normalized with the RMSE value of the less accurate member for the considered sky conditions. The normalization value under all, clear, cloudy and overcast sky conditions are 272, 208, 315, and 437 W/m² respectively. They are obtained for member34, 23, 34, 34

Figure 4 :

 4 Figure 4: WRF and AROME-OM bias distributions with respect to the mean of the previous hours K c (MK c ) values and the mean of the previous hour K c variability (V). Months and stations were merged

Figure 5

 5 Figure 5 shows WRF-M9 and AROME-OM monthly MBE and MAE. Both AROME-OM and WRF-M9 GHI forecasts follow seasonal trends for French Guiana. There is a clear distinction between the MAE value dry season represented by September and the MAE value in the rainy seasons represented by May. March is a transition period between the southward and northward motions of the Inter Tropical Convergence Zone, and depending on the year, this month behaves as a rainy or dry. Consequently, except for Maripasoula the MAE in March is overall higher than the MAE value in the dry season but lower than the MAE values in the rainy season.

Figure 5 :

 5 Figure 5: WRF and AROME-OM relative MBE and MAE for rainy season (May), dry season (September) and March transition month. The MAE and MBE were normalized using monthly mean values to obtain % values. Monthly GHI of SG in March, May, September and all months merged are 316, 312, 450 and 359 W/m² respectively; for RO they are307, 342, 456 and 370 W/m² respectively; for KR they are 335, 362, 484 and 393 W/m² respectively; for IR they are 352, 358, 468 and 390 W/m² respectively. For SL they are 392, 324, 422 and 381 W/m² respectively; for MP they are 348, 386, 454 and 397 W/m² respectively

  Fig. IV.1 and IV.2 shows the type of observation used to build this accurate image, and their distribution around the globe.

Fig. IV. 1

 1 Fig. IV.1 Measurements instrument used in DA system credit WMO

Fig

  Fig. IV.3 describes the hybrid 3DEnVar algorithm.In this figure we can see that the coupling between the ETKF and the 3DVAR algorithm. First the ensemble perturbation supplies an estimate of the flow dependent forecast error A to the 3DVAR analysis. Second the ensemble member analysis is recentered about the hybrid analysis. The high-resolution hybrid analysis is then used as initial and boundary conditions for a WRF forecast instead of the GFS coarse resolution analysis. In our Studies we performed the hybrid analysis for the three domains of WRF showed previously chapter III section III.

Fig. IV. 4

 4 Fig. IV.4 Cdf of Kc values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale

Fig. IV. 5

 5 Fig. IV.5 Cdf of V values in March, May, September. The GHI forecasts from Rochambeau, Saint Georges, Maripasoula, Saint Laurent, Kourou and Ile Royale

  The population evolution is based on the INSEE projection as reported in the PPE(Ministry of the environment, 2016a)

		2015	2020	2025	2030
	Population number	~280 000	~330 000	~297 000	~380 000
	Electricity needs GWh	879	1026	1158	1280

Approximate methods to solve the radiative transfer equation for shortwaves

	Tab. II.4 spectral interval choose in RRTM, extracted from (ECMWF, 2016a)
	Spectral intervals cm -1 Number of g-points	Gases included
			Troposphere	Stratosphere
	800-2600	12	H 2 O	CO 2
	2600-3250	6	H 2 O, CH 4
	3250-4000	12	H 2 O, CO 2	H 2 O, CO 2
	4000-4650	8	H 2 O, CH 4	CH 4
	4650-5150	8	H 2 O, CO 2	CO 2
	5150-6150	10	H 2 O, CH 4	H 2 O, CH 4
	6150-7700	10	H 2 O, CO 2	H 2 O, CO 2
	7700-8050	2	H 2 O, O 2	O 2
	8050-12850	10	H 2 O
	12850-16000	8	H 2 O, O 2	O 2
	16000-22650	6	H 2 O
	22650-29000	6
	29000-38000	8	O 3	O 3
	38000-50000	6	O 2, O 3	O 2, O 3
	I.7
	Radiative transfer parametrizations (RTPs) are simplified version of the sophisticated, rigorous
	radiative transfer models Eq.II.22
				II.42
			g (τ),
	where 𝐹 𝜆,𝑔 is the individual flux in the spectral interval defined by g and 𝑏 𝑔 the quadrature points for
	the corresponding spectral interval. Tab. II.3 shows the spectral interval used in Rapid radiative
	transfer model RRTM (ECMWF, 2016a).

Table 1 :

 1 Ground meteorological stations in French GuianaThree stations are located on the Atlantic Coast, namely, Rochambeau, Ile Royale and Kourou. Rochambeau is located 13 km from the Atlantic Ocean and Ile Royale is on a 0.6-km² island, 7 km offshore from the coast. The other three are located inland; Saint-George, Saint-Laurent and Maripasoula are located between 30 and 230 km from the coast. These stations are equipped with Kipp and Zonen pyranometers of type CM6B and CMP11; both types are fitted with a ventilation fan.

	Station	Latitude (°)	Longitude (°)	Height (m)	Period
	Rochambeau	4.81	-52.37	4	2010-2013
	Saint Georges 3.88	-51.80	6	2010-2013

Table 2 :

 2 Results of the comparison between hourly GHI estimates from OPT_H2 and measured GHI under cloudy, clear and all sky conditions from 2010 to 2013 at all stations. Relative RMSE and MBE are given in brackets.For clear, cloudy and all sky conditions, biases were negative, indicating that OPT_H2 underestimated the GHI. Under clear skies, MBE ranged from -23% to -10% for the study period, and RMSE ranged from 19% to 28%. High biases found in clear skies are explained by two phenomena: The first phenomenon is the occurrence of clouds on days that are on average clear. Fig1show time series of GHI for two days on each station. One can notice for instance that clouds occur over Saint Georges (Julian day 259) around 2 pm UTC because of the high discrepancies between the clear sky modelled GHI and the ground measured GHI. The bias between OPT_H2 and the ground measurement is approximately 50 Wh/m². The second phenomenon that explains high biases found in clear skies is

	Station	MBE	RMSE	R
		Wh/m² (%)	Wh/m² (%)
		cloudy sky conditions
	Saint
	Georges	-97.1 (-61)	133.6 (84)	0.62
	Rochambeau	-113.5 (-65)	150.8 (86)	0.62
	Kourou	-101.9 (-62)	147.0 (81)	0.66
	Ile Royale	-109.0 (-62)	146.9 (84)	0.64
	Saint
	Laurent	-130.1 (-64)	159.8 (82)	0.72
	Maripasoula	-123.1 (-49)	162.3 (66)	0.78
		clear sky conditions
	Saint
	Georges	-85.4 (-15)	123.3 (22)	0.94
	Rochambeau	-103.5 (-19)	136.1 (24)	0.94
	Kourou	-96.0 (-16)	118.6 (20)	0.96
	Ile Royale	-109.4 (-18)	140.1 (22)	0.96
	Saint
	Laurent	-133.6 (-23)	159.4 (28)	0.95
	Maripasoula	-63.3 (-11)	113.3 (19)	0.94
		all sky conditions
	Saint
	Georges	-44.5 (-11)	130.3 (31)	0.89
	Rochambeau	-55.5 (-13)	133.1 (30)	0.90
	Kourou	-48.2 (-10)	120.6 (26)	0.92
	Ile Royale	-40.6 (-8)	112.0 (23)	0.94
	Saint
	Laurent	-63.8 (-15)	134.3 (31)	0.90
	Maripasoula	-37.3 (-8)	125.5 (27)	0.90

TL. TL describes the optical thickness of the atmosphere due to both the absorption by the water vapour and the absorption and scattering by the aerosols. Tropical climate is characterised by hot and humid weather conditions; therefore, the water vapour modelling heavily influence the GHI

Table 3 :

 3 Results of the comparison between hourly GHI estimates from RTP_OPT_H2 and measured GHI under clear, cloudy and all sky conditions from 2010 to 2013. Relative RMSE and MBE are given in brackets.

	Station	MBE	RMSE	R
		Wh/m² (%)	Wh/m² (%)
		cloudy sky conditions
	Saint
	Georges	-4.1 (-2)	99.1 (62)	0.62
	Rochambeau -19.4 (-11)	108.0 (62)	0.60
	Kourou	-22.6 (-12)	103.2 (57)	0.66
	Ile Royale	-20.1 (-11.14)	102.2 (58)	0.67
	Saint
	Laurent	-42.9 (-21)	109.0 (54)	0.74
	Maripasoula -49.3 (-20)	108.1 (43)	0.84
		clear sky conditions
	Saint
	Georges	-70.0 (-12)	106.4 (18)	0.94
	Rochambeau -86.8 (-15)	117.1 (20)	0.95
	Kourou	-86.8 (-14)	105.7 (17)	0.97
	Ile Royale	-100.0 (-16)	126.8 (20)	0.97
	Saint
	Laurent	-116.2 (-20)	139.7 (24)	0.96
	Maripasoula	-51.7 (-9)	101.3 (17)	0.94
		all sky conditions
	Saint
	Georges	-9.4 (-2)	107.8 (26)	0.91
	Rochambeau	-21.7 (-5)	108.1 (24)	0.92
	Kourou	-20.7 (-4)	99.7 (21)	0.94
	Ile Royale	-17.5 (-4)	94.2 (19)	0.95
	Saint
	Laurent	-30.4 (-7)	109.0 (25)	0.92

Table 4 :

 4 KSI between CDF of GHI from RTP_OPT_H2, OPT_H2 and ground measurements under clear, cloudy and all sky conditions from 2010 to 2013. These values are obtained my multiplying the normalized value Fig.4-6 by the maximum GHI value measured in the station under the considered sky condition

	Station	KSI between OPT_H2 GHI estimates	KSI between RTP_OPT_H2 GHI
		and ground measurements		estimates and ground measurements
		2010-2013 (Wh/m²)		2010-2013 (Wh/m²)
		Cloudy sky Clear sky	All sky	Cloudy sky Clear sky	All sky
	Saint Georges	63.7	36.1	18.1	15.9	29.1	4.7
	Rochambeau	57.3	42.5	22.7	15.2	34.7	8.6
	Kourou	58.8	40.9	18.1	11.6	35.8	7.6
	Ile Royale	67.5	43.3	15.3	13.3	38.8	7.8
	Saint Laurent	76.2	58.7	26.4	20.7	50.6	12.5
	Maripasoula	72.7	25.8	13.6	28.8	21.3	5.4

  [START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF] studied the accuracy of global and mesoscale model for several location in Canada, Europe and the USA. They found that despite their higher resolution the accuracy of hourly GHI forecasts of mesoscale models are similar or worse than those of global models. According[START_REF] Kotsopoulos | A Comparative Study of Two Land Surface Schemes in WRF-ARW Model[END_REF][START_REF] Morrison | An overview of cloud and precipitation microphysics and its parameterization in models[END_REF] describing more accurately physical phenomena may expose other NWP model deficiencies. For this reason we decided first study the accuracy of global NWP model in the ITZ and for tropical climates. Second, to propose a methodology to calibrate WRF in the ITZ so that it has improved GHI forecasted compared to non-calibrated mesoscale models. This methodology was validated in French Guiana using ground measurements and AROME[START_REF] Seity | The AROME-France Convective-Scale Operational Model[END_REF] mesoscale model.

					2011)
		WRF		ALADIN	COSMO		HIRLAM	Unified	HRM
		v3.8		cycle 43	v5.1		V5	model UM	v2.5
								v6.0
	Numerics temporal	3 rd order Runge	2 nd order	3 rd order		1 st order	2 nd order	Explicit two
	integration	Kutta		semi-implicit	Runge		Eulerian semi	semi-implicit	step Lax-
	order of accuracy			semi-Lagrangian	Kutta		implicit	semi-Lagrangian	Wendroff 2 nd order
	Horizontal advection	5th order default (6	th order	Semi-Lagrangian.	5th order default (6	th	Eulerian or semi-	Semi-Lagrangian.	Adjusted Lax-
	order of	available)		Order not	order		Lagrangian.	Order not	Wendroff.
	accuracy			mentionned	available)		Order not	found	Order not
							mentionned		mentioned
	Vertical advection	3 rd order default (6	th order	Semi-Lagrangian.	2 nd order Crank-		Eulerian or semi-	Semi-Lagrangian.	Adjusted Lax-
	order of	available)		Order not	Nicholson	Lagrangian.	Order not	Wendroff.
	accuracy			mentioned			Order not	mentioned	Order not
							mentioned		mentioned
	Horizontal	6 th order		4 th order	4 th Order		6th order	2 nd order	4 th order
	Diffusion			numerical	linear		semi-		linear
				diffusion			Lagragian		diffusion
	Radiation Ozone	CAM climatology:	UGAMP	Not		Not	SPARC-II	Not
	dataset	2.82 ° latitudinal,	climatology:	mentionne	mentionned	monthly	documented
	resolution	monthly		2.5°*2.5°	d			variable
	available			monthly				longitude
								latitude
	Climatologic	-Tegen		-Tegen	-Tegen		-Tegen	CLASSIC	Not
	Aerosol	climatology:	climatology	climatolog	climatology	aerosol	documented
	dataset	Monthly, 5°*4°		y			climatology.
	available	-Thompson and		-Tanré			resolution
		Eidhammer		climatolog		not found
		climatology :		y: ~11°,
		Monthly, 0.5°*		constant in
		1.25°			time
	Cloud	yes		yes	yes		Not	yes	Not
	radiation						mentioned		documented
	feedback
	Aerosol	yes		yes	yes		yes	yes	Not
	radiation								documented
	feedback
	Ozone	yes		Yes	yes		yes	yes	Not
	radiation								documented
	feedback
	cloud	yes		Yes	Not		Not	yes	Not
	aerosol				mentioned	mentioned		documented
	feedback

Table

Table 1 :

 1 1. Validation station description.

	Stations	Latitude (°) Longitude (°) Height (m) Climate
	French Guiana
	Saint Georges (SG)	3.890	-51.804	6	Trop. monsoon
	Rochambeau (RO)	4.822	-52.365	4	Trop. rainforest
	Kourou (KR)	5.209	-52.748	12	Trop. monsoon
	Maripasoula (MP)	3.640	-54.028	106	Trop. rainforest
	Reunion Island
	Ligne-Paradis (LP)	-21.318	55.485	156	Trop. savanna
	Le Port (PO)	-20.946	55.281	9	Trop. savanna
	Gillot-Aéroport (GI)	-20.891	55.528	8	Trop. monsoon
	Pierrefonds-Aéroport (PI)	-21.320	55.425	21	Trop. savanna

  Le Port Station is located in the northwest,

	Singapore
	Seris 403 (ST)	1.443	103.784	45	Trop. rainforest
	Seris 404 (SQ)	1.249	103.841	36	Trop. rainforest
	Seris 405 (SC)	1.352	103.965	45	Trop. rainforest
	Seris 408 (SH)	1.355	103.692	57

Table 3 :

 3

		Clear	Cloudy	Overcast	All
		Vs	Vs	Vs	Vs	Vs	Vs	Vs	Vs
		IFS-3h-p	GFS-3h	IFS-3h-p	GFS-3h	IFS-3h-p	GFS-3h	IFS-3h-p	GFS-3h
	French Guiana	32%	37%	22%	54%	20%	61%	27%	48%
	Reunion Island	49%	63%	43%	34%	33%	4%	46%	57%
	Singapore	31%	32%	14%	39%	7%	43%	21%	37%

Relative improvement of the MAE of IFS-3h over the MAE of IFS-3h-p and GFS-3h under clear, cloudy, and overcast sky conditions. The classification of sky the conditions for each 3-hour sample is based on 𝑀 𝐾 𝑐 (Eq. 9) which is computed using ground measurements with N=2, and dt=1hour. Months and stations were merged by territory.

Territory

Relative improvement (%) of the MAE of IFS-3h (W/m²) over the MAE of IFS-3h-p (W/m²) and GFS-3h (W/m²)

Table 4 :

 4 Table 4 present the RMSE and FS for the different tropical climates. This table merged the yearly GHI values per tropical climate type; it allows comparing the forecast accuracy in tropical monsoon climate compared to tropical savanna and rainforest climates. Figure12presents the RMSE and FS per station. The yearly GHI values per stations were merged in order to analyze the variability amongst the same climate type. This approach allows us to obtain a range of RMSE value instead of a single value. RMSE of FS of IFS-1h GHI forecasts for tropical monsoon, savanna and rainforest climates.

	The yearly mean value used to normalized the RMSE are: 434 W/m² for stations with tropical
	monsoon climate; 468 W/m² for stations with tropical savanna climate, 392 W/m² for station with
		tropical rainforest climate
		RMSE	FS
	Tropical monsoon	35%	0.39
	Tropical savanna	33%	0.40
	Tropical rainforest	46%	0.42

Table 5 :

 5 Yearly FS and RMSE for different climates. The native temporal resolution of IFS and GFS models used by[START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF]are underlined.

	Climate	Stations	GFS/ GFS based	IFS yearly RMSE,
			yearly RMSE, MAE, FS	MAE, FS range
			range
	Mediterranean	Huelva, Cordoba, Granada (Perez et al.,	N.A	3-hour
		2013)	RMSE: N.A	RMSE: [21:25]% FS: [0.45:0.63]
			FS: N.A
	Arid	Desert Rock (Perez et al., 2013)	3-hour	3-hour
			RMSE: ~27%	RMSE: ~22%
			FS: ~0.13	FS:~0.42
	Semi-arid	Boulder (Perez et al., 2013)	3-hour	3-hour
			RMSE: ~45%	RMSE: ~39%
			FS:~0.22	FS:0.41 ~
	Continental	Bondville, Sioux Falls, Fort Peck, Furstenzell,	3-hour	3-hour
		Stuttgart, Wurzburg, Linz, and Wien (Perez et al.,	RMSE: [39:49]%	RMSE: [34:52]%
		2013)	FS: [0.22:0.57]	FS:[0.41:0.61]
	Humid	Goodwin Creek, Penn State, Egbert, Bratt's,	3-hour	3-hour
	continental	Varennes (Perez et al., 2013)	RMSE: [40:49]%	RMSE: [34:41]%
			FS: [0.32:0.43]	FS: [0.54:0.60]
	Tropical monsoon Saint Georges (SG), Kourou (KR), Gillot-	3-hour	3-hour
		aéroport(GI)	RMSE: [59:78]%	RMSE: [31:40]%
			FS: [-2.32.:-1.41]	FS: [0.26:0.46]
				hourly
				RMSE: [32:40]%
				FS: [0.28:0.49]
	Tropical Savanna	Le port (PO), Pierrefonds aéroport(PI), Ligne-	3-hour	3-hour
		Paradis(LP)	RMSE: [61:69]%	RMSE: [28:37]%
			FS: [-2.24:-1.52]	FS: [0.31:0.44]
				hourly
				RMSE: [29:38]%
				FS: [0.32:0.44]
	Tropical	Rochambeau (RO), Maripasoula (MP),	3-hour	3-hour
	rainforest	Station three (ST), Station four (SQ), Station Five	RMSE: [53:73]%	RMSE: [33:52]%
		(SC), Station eight (SH)	FS: [-1.36:0.04]	FS: [0.26:0.49]
				hourly
				RMSE: [34:52]%

Table 1 :

 1 French Guiana GHI ground measurement stations

	Stations	Latitude (°) Longitude (°) Height (m) Instruments
	Rochambeau	4.822	-52.365	4	Kipp & Zonen CM6B
	Saint Georges	3.890	-51.804	6	Kipp & Zonen CM6B
	Maripasoula	3.640	-54.028	106	Kipp & Zonen CM6B
	Saint-Laurent	5.485	-54.031	5	Kipp & Zonen CM6B
	Kourou	5.209	-52.748	12	Kipp & Zonen CM6B
	Iles Royales	5.283	-52.583	48	Kipp & Zonen CM6B
	2.2 AROME irradiance forecast

AROME France was developed by French weather services

Table 2 :

 2 AROME-OM operational setup[START_REF] Termonia | The ALADIN System and its canonical model configurations AROME CY41T1 and ALARO CY40T1[END_REF][START_REF] Yessad | BASICS ABOUT ARPEGE/IFS[END_REF]

	Dynamical Core

Table 3 :

 3 WRF 34 ensemble members' physical parameterization

		Member number
	Physics	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
	PBL
	YSU
	MYNN
	QNSE
	SHIN-HONG
	MP
	Thompson
	Milbrandt
	Morisson
	WDM6
	WSM6
	WSM3
	LSM
	Noah
	RUC
	CU
	Grell-3D
	Grell-Devenyi
	Kain Frisch
	RAD LW
	RRTMG
	RAD SW
	RRTMG
	Dudhia

Table 4 :

 4 Relative improvement (%) of the MAE of WRF-M9 (W/m²) over the MAE of AROME-OM (W/m²). Months and stations were merged.

		Relative improvement (%) of the MAE of WRF-M9 (W/m²) over the MAE of
	Territory		AROME-OM (W/m²)
		Clear	Cloudy	Overcast	All
	French Guiana	44%	26%	8%	33%
	4.3.

2 Seasonal forecast accuracy under all sky conditions

  The improvement of WRF-M9 MAE over AROME-OM ranges between [35:52]% depending on the stations. WRF-M9 MAE values has similar standards as those of other climate regions despite forecast the GHI in tropical climate being more difficult[START_REF] Diallo | Assessing GFS and IFS global numerical weather prediction model forecast accuracy under tropical climate[END_REF].[START_REF] Perez | Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe[END_REF] showed that WRF 48 h GHI forecasts in the USA, Central Europe, Spain and Canada ranged between [17:39] %, [32:40] %, [16:14] %, [30:32] %.

  Tab. IV.1 comparison of advantage and disadvantage of existing DA algorithms used in NWP models. These DA algorithm are detailed Appendix A

	4D-Var	-4DVar requires the development of
	DA methods	Nudging linear and adjoint models (Bannister, 2018)  more difficult to code and	3D-Var	4D-Var	Ensemble filters (ENKF,ENSRF)
	Nudging	maintain	-Nudging does not need to linearize the	-Nudging does not need to linearize the	-Nudging does not require a to compute
			model (Asch et al., 2016)	model (Asch et al., 2016)	B (Asch et al., 2016)  less computational
		-4DVar can process a wider range of observations (Skamarock et al., 2008)	-Nudging does not require a to compute B (Asch et al., 2016)  less computational burden	easier to code -Nudging does not need the development of an adjoint model (Asch et al., 2016) 	burden networks with heterogeneous reporting -Nudging is more adapted to observing
					times (Lei and Hacker, 2015)
			-Observations are treated at the correct	-Nudging does not require a covariance
			time (Lorenc and Rawlins, 2005) improve	matrices B (Asch et al., 2016)  less
			the forecast	computational burden
			-can model implicitly the model error	-can model implicitly the model error
			(Lakshmivarahan and Lewis, 2013)	(Lakshmivarahan and Lewis, 2013)
			improve the forecast (Howes et al., 2017)	improve the forecast (Howes et al.,
				2017)
	3D-Var	-3DVar can process a wider range of		-3DVAR does not requires the	-Perfect model assumption wrong
		observations  In nudging only		development of linear and adjoint	analysis when model error is high relative
		variables in the prognostic equations		models(i.e TLM) (Bannister, 2018)	to observation and background errors
		can be nudged (Asch et al., 2016)		computationally cheaper but worst	(Holm, 2008)
				forecast (Lorenc and Rawlins, 2005)
		-Perfect model assumption wrong
		analysis when model error is high
		relative to observation and
		background errors (Holm, 2008)

  (t 0 ) -(x ⃗⃗ b (t 0 ) -x ⃗⃗(t 0 ))] T B -1 [δx ⃗⃗(t 0 ) -(x ⃗⃗ b (t 0 ) -x ⃗⃗(t 0 ))] + R -1 [H ̃δx ⃗⃗(t 0 ) -d ⃗⃗ ] ,with H ̃= [H ̃(t 0 ), H ̃(t 1 )M ̃(t 1 , t 0 ) T , … , H ̃(t n )M ̃(t n , t 0 ) T ] T 𝑎𝑛𝑑

	J ̃(δx ⃗⃗(t 0 )) =	1 2	[δx ⃗⃗+	1 2	[H ̃δx ⃗⃗(t 0 ) -d ⃗⃗ ]
	{				d ⃗⃗ = [d 0	T , d 1	T , … , d N	T ].
	:

T

RTP: radiative transfer parameterisation

Abbreviations: FS, Forecast Skill; GEM, Global Environmental Multiscale; GFS, Global Forecast System; GHI, Global Horizontal Irradiance; IFS, Integrated Forecast System; ITCZ, Intertropical-Convergence Zone; MAE, mean average error; NWP, numerical weather prediction; RMSE, root mean squared error; RRTMG, rapid radiative transfer model-G; UM, Unified Model.

RMSE, MAE GFS 3h RMSE, MAE IFS 1h

Reference point : WRF spin up

Reference point : WRF computation domain

Reference point : WRF-M9

Remerciements

Je commencerai les remerciements par deux citations qui en résument le contenu. Celui qui ne sait pas d'où il vient ne peut savoir où il va ; tout seul on va plus vite ensemble on va plus loin…

Acknowledgments

The authors are grateful to Météo-France for supplying ground data and to NOAA for supplying satellite data via CLASS.

Acknowledgments

The authors are grateful to the National Environment Association (NEA), Solar Energy Research Institute of Singapore (SERIS), and Météo-France (French Guiana and Reunion Island) for supplying ground data. The authors also thank the European Center for Medium-Range Weather Forecasts (ECMWF) and the United States National Oceanic and Atmospheric Administration (NOAA) for providing IFS and GFS forecasts.

Acknowledgments

The authors are grateful to the Météo-France (French Guiana and Reunion Island) for supplying ground data and AROME forecasts. The authors also thank the European Center for Medium-Range Weather Forecasts (ECMWF) for providing access to their computing facilities to run WRF.

Funding

This research did receive grant from the following public agencies: "le Centre national d'étude spatiales (CNES)" and "la collectivité territorial de Guyane (CTG)"

Funding

This research did receive grant from the following public agencies: "le Centre national d'étude spatiales (CNES)" and "la collectivité territorial de Guyane (CTG)"

List of tables

Tab. I.1Projection of population evolution and electricity needs. The electricity production is based on the BPEOD reference scenario. The population evolution is based on the INSEE projection as reported in the PPE (Ministry of the environment, 2016a) .................................................................. Tab. II.1 WMO classification of pyranometers, extracted from [START_REF] Sen | Solar Energy Fundamentals and Modeling Techniques[END_REF] (ECMWF, 2016a;[START_REF] Stensrud | Parameterization schemes: keys to understanding numerical weather prediction models[END_REF][START_REF] Stephens | The parameterization of radiation for numerical weather prediction and climate models[END_REF] g is the asymmetry factor defined Eq. II.52. Inspired after [START_REF] Liou | An introduction to atmospheric radiation[END_REF] ..................... Tab. III.1 spatial and temporal resolution of GHI forecast products from global model operated by forecast center for the first 48 hour forecasts. The temporal and spatial resolution of these products refers to those of datasets available for download. Source (BAM, 2018;CMC, 2016;DWD, 2018;ECMWF, 2016b;Figueroa et al., 2016;[START_REF] Noaa | The Global Forecast System (GFS) -Global Spectral Model (GSM)[END_REF][START_REF] Walters | The Met Office Unified Model Global Atmosphere 6.0/6.1 and JULES Global Land 6.0/6.1 configurations[END_REF] [START_REF] Burridge | A split semi-implict reformulation of the Bushby-Timpson 10-level model[END_REF][START_REF] Doms | A Description of the Nonhydrostatic Regional COSMO-Model 164[END_REF][START_REF] Käellen | HIRLAM DOCUMENTATION[END_REF][START_REF] Majewski | [END_REF][START_REF] Ncar | ARW Version 3.8 Modeling System User's Guide January[END_REF][START_REF] Termonia | The ALADIN System and its canonical model configurations AROME CY41T1 and ALARO CY40T1[END_REF][START_REF] Tudor | Forecasting Weather in Croatia Using ALADIN Numerical Weather Prediction Model[END_REF]Undén and co-authors, 2002); for Radiation [START_REF] Doms | A Description of the Nonhydrostatic Regional COSMO-Model 164[END_REF][START_REF] Jimenez | WRF-Solar: Description and Clear-Sky Assessment of an Augmented NWP Model for Solar Power Prediction[END_REF][START_REF] Termonia | The ALADIN System and its canonical model configurations AROME CY41T1 and ALARO CY40T1[END_REF][START_REF] Walters | The Met Office Unified Model Global Atmosphere 6.0/6.1 and JULES Global Land 6.0/6.1 configurations[END_REF][START_REF] Zubler | Intercomparison of aerosol climatologies for use in a regional climate model over Europe: INTERCOMPARISON OF AEROSOL CLIMATOLOGIES[END_REF]

Declaration of Interest

The authors declare that there are no competing interests.

Nevertheless, these GHI forecasts must be improved as PV managers have financial incentive to produce accurate forecasts of electricity production. Our future research will involve using data assimilation methods to improve the initialization of WRF forecasts [START_REF] Mathiesen | A high-resolution, cloud-assimilating numerical weather prediction model for solar irradiance forecasting[END_REF][START_REF] Yang | Preprocessing WRF initial conditions for coastal stratocumulus forecasting[END_REF].

Declaration of Interest

The authors declare that there are no competing interests. forecasts. Once a 48h forecast is obtained, a new one is launched so that the last 48h is valid for September 29 th at 00h UTC.

(iv) 3DEnVar DA with a 6h cycling (WRF-M9-3DEnVar-c2). In this experiment 𝛽 2 = 4 3 .

The observations we assimilated were conventional observation downloaded online from NCEP servers (NCEP, 2018), they are Global Upper Air and Surface Weather Observations in PREPBUFR format.

IV.4.a Studying the impact of the hybrid DA on the GHI forecasts

Tab. IV. Tab. IV.2 shows that under overcast sky conditions each experiment (ii, iii, iv) involving the use of the hybrid DA improves the GHI forecasts. The greatest improvement of the GHI forecasts under cloudy and overcast sky conditions is obtained using WRF-M9-3DEnVar-c; the relative improvement of the MAE and MBE of WRF-M9-3DEnVar-c compared to WRF-M9 are 6% and 7% respectively. We can also see Tab. IV.2 that cycling WRF forecasts as initial and boundary conditions improve the accuracy. Nevertheless, in September, increasing 𝛽 2 outweigh this gain; under cloudy sky conditions, the MAE of WRF-M9-3DEnVar-c2 GHI forecasts are worse than those of WRF-M9.

Under all sky conditions the lowest MAE is obtained using WRF-M9-3DEnVar-c (Tab. IV.2). The improvement of WRF-M9-3DEnVar-c MAE and MBE over the MAE and MBE of WRF-M9 are 2% and 22% respectively. Under clear sky conditions WRF-M9-3DEnVar-c and WRF-M9 have similar MAE values (Tab. IV.2); consequently, we explain this improvement by the better GHI forecasts under cloudy and overcast sky conditions. The magnitude of the improvement under all sky conditions (e.g. 2%) is explained by the fewer occurrence of cloudy and overcast conditions in September; 5% and

Appendix A

We present in this section how the stochastic and variational DA algorithms that are commonly used in atmospheric science can be derived from one common source Eq.IV.9. Each DA methods has two step; when observations 𝑦(𝑡 𝑛 ) are available, they are used in the update step to compute the analysis state vector 𝑥 𝑎 ⃗⃗⃗⃗⃗ and its error covariance matrix 𝑃 then propagated into time during the prediction step.

I.

Variational method

I.1 3D-Var

In practice, the 3D-Var method attempts to find an approximate solution of the minimum of Eq.IV.9 under the following hypothesis [START_REF] Holm | Lecture notes on assimilation algorithms[END_REF]:

-no model error 𝑄 = 0 -all observations in the time window [𝑡 0 -ℎ: 𝑡 0 + ℎ] are treated as if they were at 𝑡 0 , the analysis time which is close to their average time.

-linearity of M ⃗⃗⃗⃗ and ℋ ⃗⃗⃗ -unbiased errors < 𝜀 𝑏 ⃗⃗⃗⃗ >=< 𝜀 0 ⃗⃗⃗⃗ >=< 𝜀 𝑚 ⃗⃗⃗⃗⃗⃗ >= 0 -errors are uncorrelated < 𝜀 𝑏 ⃗⃗⃗⃗ 𝜀 0 ⃗⃗⃗⃗ >=< 𝜀 0 ⃗⃗⃗⃗ 𝜀 𝑚 ⃗⃗⃗⃗⃗⃗ >=< 𝜀 𝑚 ⃗⃗⃗⃗⃗⃗ 𝜀 0 ⃗⃗⃗⃗ >= 0

Under the above assumptions Eq.IV.9 yields [START_REF] Rihan | Four-dimensional variational data assimilation for Doppler radar wind data[END_REF]:

B is stationary, it is computed beforehand using the one of the algorithm described Eq.IV.17 [START_REF] Bouttier | Data Assimilation Concepts and Methods[END_REF]. As one hypothesis of the 3D-VAR is linearity, the forward model M ⃗⃗⃗⃗ should be linearized to evolve the uncertainty in the state from one observation time to the next.

In operational NWP centers, to reduce computational costs, a sequence of linear approximations to the nonlinear minimization problem Eq.0.1 using successive small increments δx ⃗⃗. Given an estimated analysis 𝑥 ⃗(𝑡 0 ) ; the nonlinear cost function J is then linearized about the corresponding model trajectory 𝑥 ⃗(𝑡 𝑖 ), 𝑖 = 1. . 𝑛 satisfying the nonlinear forecast model M ⃗⃗⃗⃗ . This incremental variational3D-Var DA scheme is written as follows [START_REF] Haben | Conditioning of the 3DVAR Data Assimilation Problem[END_REF][START_REF] Lawless | An investigation of incremental 4D-Var using nontangent linear models[END_REF]: Where 𝑌 𝑏 and T are the perturbation and transformation matrices. 𝑌 𝑏 is expressed as follows:

𝑇 𝑀 is not unique, its different values lead to the EnSRF, EAKF and ETKF. To the best of the author knowledge no previous studies using ENKF to improve the solar irradiance forecast was found in the literature similarly to 4D-var.

III. Back and forth nudging

A variational variant of nudging method named back and forth nudging method was developed recently by Auroux andBlum (2008), (2005). The back and forth nudging algorithm consists in first solving the forward nudging equation (Eq.IV.11) and then the backward nudging equation. The initial condition of the backward integration is the final state obtained after integration of the forward nudging equation. At the end of this process, one obtains an estimate of the initial state of the system. These forward and backward integrations are then cycled until convergence of the algorithm. Therefore, the DA problem from the back and forth nudging (BFN) framework is expressed as follows:

0.24

where k is the iteration step; 𝑥 ⃗𝑘 and 𝑥 ⃗ 𝑘 are the state vector in the backward and forward NDA respectively; 𝜀 0,𝑘 ⃗⃗⃗⃗⃗⃗⃗⃗ and 𝜀 0,𝑘 ⃗⃗⃗⃗⃗⃗⃗ and are he observation error in the backward and forward NDA respectively. [START_REF] Auroux | A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm[END_REF] tested his algorithm by comparison with the 4D-var using the Lorenz, Burgers and quasi-geostrophic model non-linear systems. Result found showed that BFN algorithm is better than the variational method for the same number of iterations (and hence for the same computing time). It converges in a small number of iterations. However, the BFN was not tested using more complex models such as primitive equations with various types of observations