Thierry Ranchin

Lucien Wald

Didier Mayer

SOLAR AND WIND RESOURCES ESTIMATION: AN OVERVIEW

The renewable energies are of tremendous importance for the fight against greenhouse gas effects. They are also of paramount importance for the rural electrification and to open access to energy for all. Before making investment in this domain, there is a need for knowing their potential. This can only be achieved through the evaluation of natural resources of solar and wind energies. This paper presents some of the means and tools for estimation.

1.

Introduction

Renewable energies are a challenge for durable development around the World. The use of solar and wind sources are growing to several hundred thousand of megawatts with hope of reducing the greenhouse gas effects. But data on resources are difficult to acquire and compile in some countries, and even their availability is problematic. This lack of data and knowledge have impacts on:

policy and national planning favouring the development of traditional techniques of electricity production, security of supply of energy, large scale investment that can not be done without confidence on the potential of production.

Other problems are linked such as those related to the management of distributed generation of electricity (connection to the network reliability, safety, etc.) or to reliability of resources.

Hence, the evaluation of resources is a central point for a thoughtful and durable approach of energy development.

In this paper, we propose a small review of tools for measuring solar and wind resources. The usual measuring tools for solar radiation estimation are described and the benefits of using satellite data and modelling are shown through the application of the Heliosat method. The same approach is proposed for the evaluation of wind resources.

The principal models used in Europe are presented and the benefits of prediction evoked. Some elements of conclusions are proposed

Tools for Solar resources estimation

Solar energy applications make use of Solar radiation. Solar radiation describes the radiation emitted by the Sun and reaching the ground after diffusion and absorption by the atmosphere. The spectrum of Solar radiation is composed by visible (0.4 0.7 µm), ultraviolet (0.3 0.4 µm), and infrared (0.7 5 µm) wavelengths. Global Solar radiation is the solar radiation received on a plane surface. It is composed of direct and diffuse components. Global Solar radiation is the flux of solar radiation falling on a receiving surface from a solid angle of 2π sr.

TRADITIONAL MEANS FOR SOLAR RESOURCES ESTIMATION

Measuring instruments are designed for the whole range of wavelengths of solar radiation. But they react also to other wavelengths such as far infrared radiation emitted naturally by the atmosphere (atmospheric radiation) and by all natural bodies (ground radiation). These wavelengths, called terrestrial radiation, cannot be clearly separated from solar radiation.

Various instruments are used for the measurements of solar radiation:

pyranometers measure global, diffuse and reflected solar radiation, pyrheliometers measure direct solar radiation at normal incidence, pyrgeometers measure longwave terrestrial radiation, pyrradiometers measure the total radiation, solar and terrestrial.

Pyranometers are the most used instruments in meteorological stations for determination of the energy reaching the ground from the Sun. For allowing comparisons, all the measuring devices in stations should be calibrated. This task is far from being simple and should be regularly done.

Solar radiation is observed by the means of networks of meteorological stations. Various measurements can be made: cloud coverage, sunshine duration, global irradiation, the direct or diffuse irradiation, the spectral distribution of the irradiance etc. For irradiation measurements, the relative RMSE is approximately 3 to 5 percent if the station is well maintained. Costs for installation and maintenance of such networks are very high; consequently, there are a few measuring stations in the world. In the Western Europe and Northern America, where the situation is best, the average distance between stations is approximately 100 km. In Eastern Europe, the average distance is approximately 500 km and is approximately 1000 km in Africa. The situation is worse over the oceans. In the present situation, the irradiation over a particular site in the world is performed by interpolation / extrapolation of the measurements available and closest to this site. [START_REF] Anonymous | Spatial interpolation of daily meteorological data[END_REF] performed a study of the errors resulting from interpolation techniques for the assessment of global daily irradiation for agricultural purposes (agricultural policy for Europe). Using eight neighbouring stations, the root mean square error (RMSE) was 20 per cent in relative value. [START_REF] Supit | Global radiation. Agriculture series, EUR 15745[END_REF] reported relative RMSE of 15 25 per cent, using several techniques and five stations. Both authors and others stress the importance of the geographical and climate parameters upon the results and the importance of the adequacy between the spatial and temporal scales of the optical state of the atmosphere, the density of the network and the quantities to interpolate. The most complete study is that of [START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF]. Various techniques were defined and applied to six very dense networks in Europe and North America. For daily irradiation, relative RMSE of 15 per cent in July up to 40 per cent in January was found. [START_REF] Hulme | A 196190 climatology for Europe for climate change modelling and impact applications[END_REF], [START_REF] Beyer | Merging groundmeasurements and satellitederived data for the construction of global radiation map[END_REF] and [START_REF] Beyer | Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA)[END_REF] computed climatological means (respectively 30 and 10 years) of monthly means of respectively sunshine duration and daily irradiation using sophisticated interpolation methods calling upon satellitederived estimates and 800 and 600 stations in Europe.

The relative RMSE ranges from 5 per cent in July up to 11 per cent in January.

Several authors [START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF][START_REF] Hay | The mesoscale distribution of solar radiation at the Earth's surface and the ability of satellites to resolve it[END_REF][START_REF] Hay | An assessment of the mesoscale variability of solar radiation at the Earth's surface[END_REF][START_REF] Hay | Evaluating the solar resource: a review of problems resulting from temporal, spatial and angular variations[END_REF]WMO, 1981;[START_REF] Perez | Comparing satellite remote sensing and ground network measurements for the production of site/time specific irradiance data[END_REF] studied the extrapolation and interpolation errors as a function of the distance to the measuring stations. The errors are very similar for both cases, though the interpolation errors are lower. The larger the distance, the larger the RMSE.

For global daily irradiation, the relative RMSE is 10 per cent for a distance of 16 km, 20 per cent for 64 km and 3040 per cent for 256 km. For the global hourly irradiation, the errors are worse: 15 per cent for 4 km, 20 per cent for 16 km, 25 per cent for 64 km and 45 per cent for 256 km.

Mapping the solar radiation by interpolation / extrapolation of measurements is possible but leads to large errors, except if the network is dense. It should be noted that a deficient station might strongly affect the final map. Finally, interpolation / extrapolation techniques cannot reproduce radiation features that are smaller than twice the average distance between stations.

ESTIMATION BY MEANS OF SATELLITE

Several authors have shown the potentialities of the images of the Earth taken by the meteorological satellites for the mapping of the global irradiation impinging on a horizontal surface at the ground level. One advantage is that the errors in the estimates are constant throughout the mapped area, except in areas with highly variable relief or for very large areas, where illumination and observation angles may change dramatically. Consequently, an estimate of the quality may be provided for every site. [START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF][START_REF] Zelenka | Effective accuracy of satellite derived hourly irradiances[END_REF] and [START_REF] Perez | Comparing satellite remote sensing and ground network measurements for the production of site/time specific irradiance data[END_REF] demonstrate that for the best methods, the irradiation assessed by satellite is better than that estimated from interpolation technique as soon as the distance to the stations is greater than 34 km for the hourly irradiation and 50 km for the daily irradiation.

Many methods were developed especially in the years '80s. Among them was the method called Heliosat, developed by Armines / ENSMP under the auspices of the European Commission [START_REF] Cano | Etude de l'ennuagement par analyse de séquences d'images de satellite. Application à l'évaluation du rayonnement solaire global au sol[END_REF][START_REF] Cano | A method for the determination of the global solar radiation from meteorological satellite data[END_REF][START_REF] Michaudregas | Mise en oeuvre et validation d'une méthode opérationnelle et automatique pour l'évaluation d'atlas solaires en Europe à l'aide de mesures satellitaires Meteosat[END_REF][START_REF] Diabaté | Détermination du rayonnement solaire à l'aide d'images satellitaires[END_REF]. This method was among the most accurate methods as shown by several authors [START_REF] Grüter | Solar Radiation Data from Satellite Images[END_REF][START_REF] Raschke | Solar Radiation Atlas of Africa[END_REF]. It was simple enough to widely disseminate in the world (Diabaté et al. 1988a(Diabaté et al. b, 1989;;[START_REF] Wald | A technical note on a lowcost highquality system for the acquisition and digital processing of images of WEFAX type provided by meteorological geostationary satellites[END_REF]. It became popular and several modifications were proposed [START_REF] Moussu | A method for the mapping of the apparent ground brightness using visible images from geostationary satellites[END_REF][START_REF] Obrecht | Météorologie solaire et images satellitaires : cartographie du rayonnement solaire, détermination de l'albédo des sols et évaluation de l'ennuagement[END_REF][START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF][START_REF] Zelenka | Effective accuracy of satellite derived hourly irradiances[END_REF][START_REF] Beyer | Merging groundmeasurements and satellitederived data for the construction of global radiation map[END_REF][START_REF] Fontoynont | SatelLight, Processing of Meteosat data for the production of high quality daylight and solar radiation data available on a World Wide Web Internet server[END_REF][START_REF] Iehlé | Meteosat: A valuable tool for agrometeorology[END_REF]Ineichen, Perez 1999).

The principle of the method HeliosatI is the construction of a "cloud index" resulting from a comparison of what is observed by the sensor to what should be observed over that pixel if the sky were clear, which is related to the "clearness" of the atmosphere. At each pixel (i, j) of the current image at time t, a normalised count CN* t (i,j) is computed as:

CN* t (i,j) = (CN t (i,j) CN0 t ) / [I 0 ε(t) (sin γ S (i,j)) (sin γ S (i,j)) 0,15 ]
where • CN t (i,j) is the numerical count observed by the sensor at time t for this pixel (i, j), CN0 t being what can be called the sensor zero (the numerical counts are not necessarily calibrated),

• (sin γ S ) 0,15 is the clear sky transmittance (model of Perrin de Brichambaut, Vauge 1982).

This normalised count CN* t (i,j) is related to the apparent albedo ρ t (i,j) observed by the spaceborne sensor. In preparation to the determination of the cloud index n t (i,j), a reference map of the normalised count for clear sky is constructed [START_REF] Moussu | A method for the mapping of the apparent ground brightness using visible images from geostationary satellites[END_REF].

It is also called the normalised count for ground, CN* g (i,j). Given a timeseries of images, it is evaluated at each pixel in a recursive fashion by minimising the variance between the digital counts and those resulting from the clear sky model, the cloudy cases being eliminated at each step [START_REF] Cano | A method for the determination of the global solar radiation from meteorological satellite data[END_REF]). The cloud index is defined as a function of CN* g (i,j), CN* t (i,j), and the typical normalised count of the brightest clouds tops CN* cloud :

n t (i,j) = [CN* t (i,j) CN* g (i,j) ] / [CN* cloud CN* g (i,j)) ]
The computation of CN* cloud is performed using the inverse of the algorithm used for determining the reference albedo map and retaining only the cloudy areas. The histogram of this "only cloud" image provides an estimation of CN* cloud . The clearness index KT h (i,j) may be defined for the hour h centred on t as:

KT h (i,j) = G h (i,j) / G 0h (i,j)
where G 0h (i,j) is approximated by I 0 ε(t) sin γ S , where γ S is for the middle t of the hour.

Care should be taken of the sunset and sunrise. Several previous studies did show a linear relationship between the cloud index and the clearness index, where the parameters A and B are positive and have been determined once for ever (Diabaté et al. 1988;[START_REF] Diabaté | Détermination du rayonnement solaire à l'aide d'images satellitaires[END_REF]:

KT h (i,j) = A n t (i,j) + B
This relationship between KT h (i,j) and n t (i,j) leads to the computation of the global hourly irradiation G h (i,j).

The global daily irradiation G d (i,j) is computed from the set of hourly irradiations available for that day. The larger the number of images used per day, the lower the level of error. A model has been proposed by [START_REF] Diabaté | Détermination du rayonnement solaire à l'aide d'images satellitaires[END_REF], using an analytical law fitted onto climatological hourly values. However most users adopted the following model. Let denote the horizontal daily irradiation outside the atmosphere by G 0d (i,j) and the daily clearness index by KT d (i,j). G d (i,j) is then computed from the N assessments of the hourly irradiation G h (i,j) made during the day:

G d (i,j) = KT d (i,j) G 0d (i,j) = G 0d (i,j) ) , ( 1 j i KT w h N h h ∑ = where w h = ∑ = N h h h j i G j i G 1 0 0 ) , ( ) , ( It comes G d (i,j) = G 0d (i,j) ∑ ∑ = = N h h N h h j i G j i G 1 0 1 ) , ( ) , (
The method HeliosatI is currently used by several institutes in Europe and elsewhere with geostationary satellites like Meteosat (Europe), GOES (USA) or GMS (Japan). A scientific network ensures collaboration between these institutes and improvements in the method. There is also a software, the SunUNIX version of it made at Armines / Ecole des mines de Paris being in public domain and available on an Internet server (http://wwwcenerg.cma.fr/tele). A PCbased package including a satellite data receiver has also been developed and marketed by a French company.

There are several empirical parameters in the method HeliosatI, especially in the computation of the apparent albedoes of the ground and clouds and in the followup of the changes of the ground albedo. The clearsky model is also sitedependent. In this model, the optical state of the clear atmosphere is expressed by the sole exponent. As a basis, a value of 0.15 has been selected but other values may perform better. The parameters c and d in the relationship between KT h (i,j) and n t (i,j) may be adjusted by comparison with measurements made at meteorological stations. These parameters were well tuned during the construction of the method or of its varieties and this explains the good results attained by the authors when performing a comparison with ground observations. For example, according to [START_REF] Diabaté | Détermination du rayonnement solaire à l'aide d'images satellitaires[END_REF], Diabaté et al. (1988) or [START_REF] Grüter | Solar Radiation Data from Satellite Images[END_REF], the typical relative error (RMSE) is about • 718 percent for the assessment of the hourly irradiation,

• 10 percent or better for the monthly mean of the hourly irradiation,

• 1015 percent for the assessment of the daily irradiation,

• 10 percent or better for the monthly mean of daily irradiation. [START_REF] Rigollier | Vers un accès à une climatologie du rayonnement solaire : estimation de l'irradiation globale à partir d'images satellitales[END_REF] performed a survey of the literature and reported the accuracy found by several authors when comparing irradiations measured by ground stations and irradiations derived from satellite data by various methods. She retained only the comparisons performed by authors, who were not the inventors of a method. Her work thus demonstrates the actual accuracy that can be attained by a naive implementation of a method.

EXAMPLE OF APPLICATION OF THE HELIOSAT METHOD

The Heliosat method was firstly applied to the Meteosat data.

Tools for Wind resources estimation

In this part are presented the way of obtaining wind resources. Wind resources start from the general estimation of the mean energy from wind over a large area and go till the mean annual production of a wind machine. In our paper we focus on the evaluation of wind parameters which are prerequired for wind resources estimation.

MEANS FOR WIND PARAMETERS ESTIMATION

CupAnemometers are certainly the mostused tools in the World for estimation of wind parameters. But these tools are liable to mechanical component deterioration. This leads to errors in wind speed are systematically biased to under representation of wind speed. Meteorological systematic tracking is considered a national baseline responsibility.

Hence meteorological stations are also a basis of information on wind parameters. Some experience were also made with very interesting results on the use of anemometers with kites. Anemometers give access to a pointmeasurement and not to spatial information on the wind parameters.

But others tools were developed to try to catch a good representation of wind fields such as:

-USA1 (Ultra Sonic Anemometer). This system is an anemometer measuring turbulent fluctuations of horizontal and vertical wind on a path of 10 cm.

-Wind profilers. During the last 20 years, socalled Doppler radars were systematically developed to probe the atmosphere and derive the wind profile (i.e., the speed and direction as function of height) from echoes of the transmitted radio waves produced by turbulence in the clear air. A wind profiler is the operational application of a radar originally developed by scientists for measuring the echo intensity and the wind profile up to about 30 km with height resolutions from 100 to 1500 m. Sequences of high power pulses are radiated in the vertical and in oblique directions. By analysing the received echoes, the radial velocity and the turbulence intensity can be computed. Observations from at least three directions are necessary to determine direction and speed of the wind. Implementation Conditions", Tashkent, Uzbekistan, 2224 May, 2002.

-SODAR (SOund Detection And Ranging). This system allows to obtain a 3D profile of the wind parameters in the first hundreds of meters of the atmosphere.

The SODAR processes the echo of an acoustic pulse which is directed into the atmosphere. The frequency shift of the echo varies according to the wind speed. This is the wellknown DOPPLER effect. The echo intensity varies according to thermal turbulence and structure.

-LIDAR (LIght Detection And Ranging). The transverse wind component is measured by projecting a series of beams which are evenly spaced in the transverse direction. The technique provides instantaneous velocities as well as mean velocities. Thus some turbulence quantities ( e.g., turbulent intensities, Reynolds stresses, and higher moments or statistics) can be derived. In addition, particulate related quantities can also be measured to obtain such quantities as cloud height and optical depth / reflectivity or boundary layer height and relative particulate loading with altitude. The vertical wind velocity is found from a correlation analysis. This tool is most often operated from an airborne platform.

The calibration of all tools for wind parameters measurement is mandatory to obtain possible comparison or combination of all the measures. But all these tools does not give a complete view of the wind configuration over a large area. Hence the only solutions is to use modelling tools for wind resources estimation.

MODELLING TOOLS FOR WIND RESOURCES ESTIMATION

Meteorological models such as those used by the ECMWF (European Centre for MediumRange Weather Forecasts) allow to access to wind parameters statistics. In the European energetic domain, the most used model for wind resources estimation is certainly WAsP [START_REF] Troen | European Wind Atlas[END_REF][START_REF] Bowen | Exploring the limits of WAsP: the Wind Atlas Analysis and Application Program[END_REF] from the Risoe National Laboratory (Denmark). It is based on the physical principles of flows in the atmospheric boundary layer and try to take into account the effect of the surface roughness, of obstacles and of terrain variations. In fact it is a combination of a set of models for each effect. Figure 5 presents the wind atlas methodology of WAsP. -Analysis of raw data. This option enables an analysis of any timeseries of wind measurements.

-Generation of wind atlas data. Analysed wind data can be converted into a wind atlas data set. In a wind atlas data set the wind observations have been "cleaned" with respect to site specific conditions and reduced to standard conditions.

- -Estimation of wind power potential. The total energy content of the mean wind (wind resource) is calculated by WAsP. Furthermore, an estimate of the actual, yearly mean power production of a wind turbine can be obtained by providing WAsP with the power curve of the turbine in question.

The main other family of models is based on Computational Fluid Mechanics (CFD).

Examples of such models are Aria Local and Aria Wind from Aria Technologies(http://www.aria.fr). They are based on an implementation of the Navier Stokes equations. NavierStokes models require considerable computational effort. It has been a serious drawback for their wide use. Even so, they constitute the most complete approximation and so they could be useful at least for checking and calibrating simpler models and to obtain good solutions when managing wind fields over complex terrains.

Aria Wind for example implements a linearised version of the NavierStokes equation and allows an accurate representation of 3D wind fields over complex terrain. It has two objectives:

-At regional scale: To provide a cartography of wind potential and define the best sites for wind energy production.

-At local scale: To define the pertinent implantation for the more convenient type of turbines. Precise calculation of each wind turbine production (kWh/year).

PREDICTION OF WIND RESOURCES

Accurate forecasting of the wind resource up to two days ahead is recognised as a major contribution for reliable largescale wind power integration. Especially, in a liberalised electricity market, prediction tools enhance the position of wind energy compared to other forms of dispatchable generation. Nowadays, several tools have been developed for wind power forecasting (i.e. Zephyr, Predictor, Previento, WPPT, MoreCare, Sipreolico and others). The stateofthe art tools are usually developed under the frame of specific applications. They are based either on physical (detailed terrain representation, roughness etc) or statistical modelling (i.e. blackbox type of models based only on measurements). Physical modelling benefits from advances in the area of wind resource assessment.

In this important topic, a large European project, called ANEMOS (http://anemos.cma.fr) aims to develop advanced forecasting models that will substantially outperform current methods. Emphasis is given to situations like complex terrain, extreme weather conditions, as well as to offshore prediction for which no specific tools currently exist. The prediction models will be implemented in a software platform and installed for online operation at onshore and offshore wind farms by the endusers participating in the project. The project will try to demonstrate the economic and technical benefits from accurate wind prediction at different levels: national, regional or at single wind farm level and for time horizons ranging from minutes up to several days ahead.

Conclusion

The means and tools for estimation of solar and wind resources are numerous and potentially sufficient for assessing the potential of a specific area for solar or wind production of energy. Of course they do not provide an exact view of the situation, but they can help to plan new implantations of solar and wind farms. Connected with a tool allowing a geographical analysis of the energy potential of a country, such as geographical information system (GIS), this estimation will provide basis for a decision support system (DSS) if it includes the network description, the energy demand and some societal issues [START_REF] Lamache | Les outils modernes de la géomatique au service de l'ERD[END_REF][START_REF] Vandenbergh | Energie Solaire et production d'électricité : un outil d'analyse technologique, économique et géographiques du potentiel des filières photovoltaïques et thermiques. Thèse de Doctorat "Energétique[END_REF]) . Such a kind a tool exists and will help in a near future to propose economically and environmentally consistent solutions to provide energy to population.

Figure 1

 1 Figure 1 Meteosat original image in the visible channel acquired 1rst August 1992 at 11h30 GMT

Figure 2 .

 2 Figure 2. Clearness index computed from Figure 1 by the Heliosat method.

Figure 3 .

 3 Figure 3. Global hourly irradiation computed by the Heliosat methodFrom best available measured solar data complemented with other meteorological data necessary for solar engineering, digital maps for the European continents are produced. It comprises a hardcopy atlas and a CD for PC, including the ESRA data base with the mapping in digital form, algorithms to calculate derived parameters and input files for engineering issues. It is complemented by a users guidebook(ESRA 2000a(ESRA , 2000b)).

Figure 4 .

 4 Figure 4. Map of the climatological average of the monthly mean of the daily sum of the global radiation received on horizontal plane for May.

Figure 4

 4 Figure 4 is an example of the tools proposed by ESRA. From this kind of maps potential energetic resources can be derived. A set of Web services are designed to provide information on and tools for exploiting Solar radiation database (see e.g. http://www.helioclim.net, http://soda.jrc.it, http://www.satellight.com).

Figure 5 .

 5 Figure 5. Wind atlas methodology of WAsP WAsP consists of four main blocks of computation:

  Wind climate estimation. Using a wind atlas data set calculated by WAsP or one obtained from another source eg. the European Wind Atlas the program can estimate the wind climate at a particular point by performing the inverse calculation as is used to generate a wind atlas. Implementation Conditions", Tashkent, Uzbekistan, 2224 May, 2002.


O El-Fallah

K Kellay
email: kellay@cmi.univ-mrs.fr

M Shabankhah

H Youssfi
email: youssfi@cmi.univ-mrs.fr

LEVEL SETS AND COMPOSITION OPERATORS ON THE DIRICHLET SPACE

Keywords: 2000 Mathematics Subject Classification. 47B38, 30H05, 30C85, 47A15 Dirichlet space, composition operators, capacity

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal.

Introduction

In this note we consider composition operators in the Dirichlet space of the unit disc. A comprehensive study of composition operators in function spaces and their spectral behavior could be found in [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF][START_REF] Shapiro | Composition operators and classical function theory[END_REF][START_REF] Zhu | Operator theory in function spaces[END_REF]. See also [START_REF] Gallardo-Gutiérrez | Exceptional sets and Hilbert-Schmidt composition operators[END_REF][START_REF] Gallardo-Gutiérrez | Hilbert-Schmidt composition operators on Dirichlet spaces[END_REF][START_REF] Gallardo-Gutiérrez | Hausdorff measures, capacities and compact composition operators[END_REF][START_REF] Tjani | Compact composition operators on Besov spaces[END_REF][START_REF] Wirths | Global integral criteria for composition operators[END_REF][START_REF] Zorboska | Composition operators on weighted Dirichlet spaces[END_REF] for a treatment of some of the questions addressed in this paper.

Let D be the unit disc in the complex plane and let T = ∂D be its boundary. We denote by D the classical Dirichlet space. This is the space of all analytic functions f on D such that

D(f ) := D |f ′ (z)| 2 dA(z) < ∞,
where dA(z) = dxdy/π stands for the normalized area measure in D. We call D(f ) the Dirichlet integral of f . The space D is endowed with the norm

f 2 D := |f (0)| 2 + D(f ). It is standard that a function f (z) = ∞ n=0 f (n) z n , holomorphic on D, belongs to D if and only if n≥0 | f (n)| 2 (1 + n) < ∞,
and that this series defines an equivalent norm on D.

Since the Dirichlet space is contained in the Hardy space H 2 (D), every function f ∈ D has non-tangential limits f * almost everywhere on T. In this case, however, more can be said. Indeed, Beurling [START_REF] Carleson | Selected Problems on Exceptional Sets[END_REF] showed that if f ∈ D then f * (ζ) = lim r→1 f (rζ) exists for ζ ∈ T outside of a set of logarithmic capacity zero.

Let ϕ be a holomorphic self-map of D. The composition operator C ϕ on D is defined by

C ϕ (f ) = f • ϕ, f ∈ D.
We are interested herein in describing the spectral properties of the composition operator C ϕ , such as compactness and Hilbert-Schmidt class membership, in terms of the size of the level set of ϕ. For s ∈ (0, 1), the level set E ϕ (s) of ϕ is given by

E ϕ (s) = {ζ ∈ T: |ϕ(ζ)| ≥ s}.
We give new characterizations of Hilbert-Schmidt class membership in the case of the Dirichlet space. We also establish the sharpness of these results.

A general criterion

For α > -1, dA α will denote the finite measure on D given by

dA α (z) := (1 + α)(1 -|z| 2 ) α dA(z).
For p ≥ 1 and α > -1, the weighted Bergman space A p α consists of the holomorphic functions f on D for which

f p,α := D |f (z)| p dA α (z) 1/p < ∞.
We denote by D p α the space consisting of analytic functions f on D such that f p

D p α := |f (0)| p + f ′ p p,α < ∞.
Appropriate choices of the parameter α give, with equivalent norm, all the standard holomorphic function spaces. Indeed, The Hardy space H 2 can be identified with D 2 1 . The classical Besov space is precisely

D p p-2 , and if p < α + 1, D p α = A p α-2 .
Finally, the classical Dirichlet space D is identical to D 2 0 . We recall that, by the reproducing formula, one has

f (z) = D f (w) (1 -wz) 2+α dA α (w), z ∈ D, (1)
for every f ∈ A p α (see [START_REF] Zhu | Operator theory in function spaces[END_REF]).

Lemma 2.1. Let p ≥ 1 and let σ > -1. Then, there exists a constant C depending only on p and σ such that

|f (z)| p ≤ C D |f (λ)| p |1 -λz| 2+σ dA σ (λ),
for every f ∈ A p σ and z ∈ D. Proof. By the above reproducing formula,

f (z) 1 -zw = D f (λ) 1 -λw dA σ (λ) (1 -λz) 2+σ , z, w ∈ D,
for every f ∈ A p σ . By Hölder's inequality, with q = p/(p -1),

|f (z)| p |1 -zw| p ≤ D |f (λ)| p dA σ (λ) |1 -λz| 2+σ × D dA σ (λ) |1 -λw| q |1 -λz| (2+σ)p p q .
Taking w = z, and using the standard estimate ([16, Lemma 3.10])

D dA c (λ) |1 -zλ| 2+c+d ≍ 1 (1 -|z| 2 ) d , if d > 0 , c > -1, (2)
we get the desired conclusion.

For λ ∈ D, consider the test function

F λ,β (z) = 1 (1 -λz) 1+β , z ∈ D. If β ≥ 0 is chosen such that δ := δ(p, α, β) = 2 + β -(2 + α)/p > 0, by (2)
, we have

F λ,β p D p α ≍ (1 -|λ| 2 ) -pδ .
The following theorem unifies and generalizes the previously known results of MacCluer [3, Theorem 3.12], Tjani [START_REF] Tjani | Compact composition operators on Besov spaces[END_REF]Theorem 3.5] and Wriths-Xiao [13, Theorem 3.2] on Hardy, Besov and weighted Dirichlet spaces, respectively.

As mentioned before, the proof we provide here is short and simple.

Theorem 2.2. Let p > 1. Suppose ϕ ∈ D p α satisfies ϕ(D) ⊂ D. Fix β ≥ 0 such that δ := δ(p, α, β) = 2 + β -(2 + α)/p > 0. Then (a) C ϕ is bounded on D p α ⇐⇒ sup λ∈D (1 -|λ| 2 ) δ F λ,β • ϕ D p α < ∞; (b) C ϕ is compact on D p α ⇐⇒ lim |λ|→1 (1 -|λ| 2 ) δ F λ,β • ϕ D p α = 0.
Proof. To prove (a), we observe that if C ϕ is bounded , then

F λ,β • ϕ D p α = O((1 -|λ| 2 ) -δ
). For the converse we may assume, without loss of generality, that ϕ fixes the origin. It follows from Lemma 2.1 that, for f ∈ D p α ,

D |ϕ ′ (z)| p |f ′ (ϕ(z))| p dA α (z) ≤ C D |ϕ ′ (z)| p D |f ′ (λ)| p |1 -λϕ(z)| (2+β)p dA 2p+βp-2 (λ) dA α (z) = C D |f ′ (λ)| p (1 -|λ| 2 ) 2p+βp-2-α D |ϕ ′ (z)| p |1 -λϕ(z)| (2+β)p dA α (z) dA α (λ) = C D |f ′ (λ)| p (1 -|λ| 2 ) pδ (F λ,β • ϕ) ′ p p,α dA α (λ).
Therefore part (a) follows.

(b) Without loss of generality we assume that ϕ(0

) = 0. Note that C ϕ is compact on D p α if and only if for every bounded sequence (f n ) n ⊂ D p α such that f n → 0 uniformly on compact subsets of D, we have C ϕ (f n ) D p α → 0, as n → ∞. Suppose that C ϕ is compact. Since (1 -|λ| 2 ) δ F λ,β → 0 uniformly on compact subsets of the unit disc, as |λ| → 1, we see that C ϕ (F λ,β ) D p α = o((1 -|λ| 2 ) -δ ). Conversely, assume that lim |λ|→1 (1 -|λ| 2 ) δ F λ,β • ϕ D p α = 0. Let (f n ) n be a bounded sequence of D p
α such that f n → 0 uniformly on compact sets. Since f ′ n → 0 uniformly on compact sets, it follows from the proof of part (a) and the hypothesis that, for r close enough to 1,

C ϕ (f n ) p D p α -|f n (0)| p ≤ rD |f ′ n (λ)| p (1 -|λ| 2 ) pδ (F λ,β • ϕ) p p,α dA α (λ) + D\rD |f ′ n (λ)| p (1 -|λ| 2 ) pδ (F λ,β • ϕ) ′ p p,α dA α (λ) → 0, n → ∞
which finishes the proof.

The following result is an immediate consequence of Theorem 2.2.

Corollary 2.3. Let ϕ : D → D such that ϕ ∈ D. (a) If sup n≥1 D(ϕ n ) < ∞, then C ϕ is bounded; (b) If lim n→∞ D(ϕ n ) = 0, then C ϕ is compact.
Proof. We consider the test function F λ,0 with β = α = 0 and p = 2. Both (a) and (b) follow from the following inequality:

D(F λ,0 • ϕ) ≤ 2 ( 1 -|λ| 2 ) 2 D |ϕ ′ (z)| 2 (1 -|λ| 2 ϕ(z)| 2 ) 4 dA(z) ≤ c (1 -|λ| 2 ) 2 n≥0 (n + 1) 3 |λ| 2n D |ϕ ′ (z)| 2 | ϕ n (z)| 2 dA(z) = c (1 -|λ| 2 ) 2 n≥0 (1 + n) |λ| 2n D(ϕ n+1 ) ≤ c lim sup n→∞ D(ϕ n+1 ).
Remarks 2.4.

1. The compactness criterion for C ϕ in the Bloch space is equivalent to ϕ n B → 0 as was shown in [START_REF] Wulan | Compact composition operators on BMOA and the Bloch space[END_REF] (see also [START_REF] Montes-Rodriguez | The essential norm of a composition operator on Bloch spaces[END_REF][START_REF] Tjani | Compact composition operators on Besov spaces[END_REF]). In the case of the Hardy space H 2 , however, we know that if C ϕ is compact on H 2 then ϕ n H 2 → 0 but the converse does not hold [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF]. Note that as before in the proof of Corollary 2.3 (β = 0, α = 1 and p = 2) if

ϕ n H 2 = o(1/ √ n), then C ϕ is compact on H 2 .
2. The characterization of compact composition operators on the Dirichlet space in terms of Carleson measures can be found in [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF][START_REF] Tjani | Compact composition operators on Besov spaces[END_REF][START_REF] Zorboska | Composition operators on weighted Dirichlet spaces[END_REF]. A positive Borel measure µ given on D satisfying

D |f (z)| 2 dµ(z) ≤ f 2 2,0 , f ∈ A 2 0 ,
is called a Carleson measure for A 2 0 , i.e., the identity map i 0 : A 2 0 → L 2 (µ) is a bounded operator. Such a measure has the following equivalent properties (see [START_REF] Zhu | Operator theory in function spaces[END_REF]Theorem 7.4]). A positive Borel measure µ is a Carleson measure for A 2 0 if and only if Let ϕ : D → D be analytic and denote by n ϕ (z) the multiplicity of ϕ at z. By the change of variable formula [START_REF] Shapiro | Composition operators and classical function theory[END_REF],

sup λ∈D (1 -|λ| 2 ) 2 D dµ(z) |1 -λz| 4 < ∞,
F λ,0 p 2,0 = (1 -|λ| 2 ) 2 D n ϕ (z) dA(z) |1 -λz| 4 .
Therefore, as a consequence of Theorem 2.

       C ϕ is bounded in D ⇐⇒ sup I⊂T 1 |I| 2 S(I) n ϕ (z) dA(z) < ∞; C ϕ is compact in D ⇐⇒ lim |I|→0 1 |I| 2 S(I) n ϕ (z) dA(z) = 0.

Hilbert-Schmidt membership

In the case of the Hardy space H 2 , one can completely describe the membership of C ϕ in the Hilbert-Schmidt class in terms of the size of the level sets of the inducing map ϕ. Indeed, C ϕ is Hilbert-Schmidt in H 2 if and only if

n≥0 ϕ n 2 H 2 = T |dζ| 1 -|ϕ(ζ)| 2 < ∞.
Given an arbitrary measurable function f on T, consider the associated distribution function m f defined by

m f (λ) = |{ζ ∈ T : |f (ζ)| > λ|}, λ > 0.
It then follows that C ϕ is in the Hilbert-Schmidt class of H 2 if and only if

T |dζ| 1 -|ϕ(ζ)| 2 = ∞ 1 m (1-|ϕ| 2 ) -1 (λ) dλ ≍ 1 0 |E ϕ (s)| (1 -s) 2 ds < ∞.
It was shown by Gallardo-González [START_REF] Gallardo-Gutiérrez | Hausdorff measures, capacities and compact composition operators[END_REF]Theorem] that there is a mapping ϕ taking D to itself such that C ϕ is compact in H 2 , and that the level set E ϕ (1) has Hausdorff measure equal to one. Recall that the Hausdorff dimension of E

d(E) = inf{α : Λ α (E) = 0}
where Λ α (E) is the α-Hausdorff measure of E given by ) and E has Hausdorff dimension one. (such examples can be given by generalized Cantor sets [START_REF] Carleson | Selected Problems on Exceptional Sets[END_REF]). Let ω(t) = (log(e/t)) -2 , and consider the outer function given by

Λ α (E) = lim ǫ→0 inf i |∆ i | α : E ⊂ i ∆ i , |∆ i | < ǫ .
|f ω,E (ζ)| = e -w(d(ζ,E)) ,
a.e on T.

Since ω satisfies the Dini condition

0 ω(t) t dt < ∞, it follows that f ω,E ∈ A(D) := Hol(D) ∩ C(D)
, disc algebra (see [START_REF] Garnett | Bounded analytic functions[END_REF] p.105-106) and so E f ω,E (1) = E. On the other hand

T |dζ| 1 -|f ω,E (ζ)| 2 ≍ T |dζ| ω(d(ζ, E)) ≍ 0 |E t | ω ′ (t) ω(t) 2 dt,
(see [4, Proposition A.1 ] for the last equality). Since the last integral converges, C ϕ is a Hilbert-Schmidt operator in H 2 .

We have the following more precise result.

Theorem 3.1. Let E be a closed subset of T with Lebesgue measure zero. There exists a mapping ϕ : D → D, ϕ ∈ A(D) such that C ϕ is a Hilbert-Schmidt operator on H 2 and that E ϕ (1) = E.

Proof. The proof is based a well known construction of peak functions in the disc algebras. Let T \ E = ∪ n≥1 (e ian , e ibn ). For t ∈ (a n , b n ), we define

g(e it ) = τ n (b n -a n ) 1/2 ((b n -a n ) 2 -(2t -(b n + a n )) 2 ) 1/4 ,
where (τ n ) n ⊂ (0, ∞) will be chosen later, and g(e it ) := +∞ if e it ∈ E. Note that

2π 0 g(e it ) 2 dt = π ∞ n=1 τ 2 n (b n -a n ). Since ∞ n=1 (b n -a n ) = 2π, there exists a sequence (τ n ) n such that lim n→+∞ τ n = +∞ and ∞ n=1 τ 2 n (b n -a n ) < ∞.
Let U denote the harmonic extension of g on the unit disc given by

U(re iθ ) = 1 2π 2π 0 1 -r 2 |e it -re iθ | 2 g(e it ) dt = n∈Z g(n) r |n| e inθ .
Since τ n → ∞, one can easily verify that lim t→θ g(e it ) = +∞, for e iθ ∈ E.

Hence, lim

r→1 -U(re iθ ) = +∞, for e iθ ∈ E. Let V be the harmonic conjugate of U, with V (0) = 0. It is given by

V (re iθ ) = n =0 n |n| g(n) r |n| e inθ .
Now, since g is a C 1 function on T \ E, we see that the holomor-

phic function f = U + iV is continuous on D \ E. Knowing that lim r→1 -U(re it ) = +∞, for e it ∈ E, we get that ϕ = f f + 1 ∈ A(D), disc algebra, and E ϕ (1) = E. Finally 1 2π 2π 0 dt 1 -|ϕ(e it )| 2 = 1 2π 2π 0 (U(e it ) + 1) 2 + V 2 (e it ) (U(e it ) + 1) 2 -U 2 (e it ) dt ≤ 1 2π 2π 0 (U(e it ) + 1) 2 + V 2 (e it )dt ≤ 1 + 2 n∈Z | g(n)| 2 , which shows that C ϕ is Hilbert-Schmidt because g ∈ L 2 (T).
Let E be a closed subset of the unit circle T. Fix a non-negative

function w ∈ C 1 (0, π] such that T w(d(ζ, E)) |dζ| < ∞,
where d denotes the arclength distance. Now, let f w,E be the outer function given by

|f * w,E (ζ)| = e -w(d(ζ,E)) , a.e. on T. (3)
The following lemma gives an estimate for the Dirichlet integral of f w,E in terms of w and the distance function on E. The proof is based on Carleson's formula, and can be achieved by slightly modifying the arguments used in [5, Theorem 4.1].

Lemma 3.2. Assume that the function ω is nondecreasing and ω(t γ ) is concave for all γ > 2. Then

D(f w,E ) ≍ T ω ′ (d(ζ, E)) 2 e -2w(d(ζ,E)) d(ζ, E) |dζ|.
Since the sequence {z n / √ n + 1} ∞ n=0 is an orthonormal basis of D, the operator C ϕ is Hilbert-Schmidt on the Dirichlet space if and only if 1

π D |ϕ ′ (z)| 2 (1 -|ϕ(z)| 2 ) 2 dA(z) = n≥1 D(ϕ n ) n < ∞.
Theorem 3.3. Assume that the function ω is nondecreasing and ω(t γ ) is concave for some γ > 2.

Then C f w,E is in the Hilbert-Schmidt class in D if and only if T ω ′ (d(ζ, E)) 2 w(d(ζ, E)) 2 d(ζ, E) |dζ| < ∞.
Proof. We first note that f n w,E = f nw,E . Therefore, by Lemma 3.2, we have d(ζ,E)), the result fllows.

D |f ′ w,E (z)| 2 (1 -|f w,E (z)| 2 ) 2 dA(z) = ∞ n=1 D(f nw,E ) n ≍ T ω ′ (d(ζ, E)) 2 d(ζ, E) ∞ n=1 ne -2nw(d(ζ,E)) |dζ| ≍ T ω ′ (d(ζ, E)) 2 [1 -e -2w(d(ζ,E)) ] 2 d(ζ, E) |dζ|. Since 1 -e -2w(d(ζ,E)) ≍ w(
Given a (Borel) probability measure µ on T, we define its α-energy, 0 ≤ α < 1, by

I α (µ) = ∞ n=1 | µ(n)| 2 n 1-α .
For a closed set E ⊂ T, its α-capacity cap α (E) is defined by cap α (E) := 1/ inf{I α (µ) : µ is a probability measure on E}.

If α = 0, we simply note cap(E) and this means the logarithmic capacity of E.

The weak-type inequality for capacity [START_REF] Carleson | Selected Problems on Exceptional Sets[END_REF] states that, for f ∈ D and

t ≥ 4 f 2 D , cap({ζ : |f (ζ)| ≥ t}) ≤ 16 f 2 D t 2 .
As a result of this inequality, we see that if lim inf ϕ n D = 0, then cap(E ϕ (1)) = 0. Indeed, since E ϕ (1) = E ϕ n (1), the weak capacity inequality implies that

cap(E ϕ (1)) = cap(E ϕ n (1)) ≤ 16 ϕ n 2 D .
Now let n → ∞. Hence, in particular, if the operator C ϕ is in the Hilbert-Schmidt class in D, then cap(E ϕ (1)) = 0. This result was first obtained by Gallardo-González [START_REF] Gallardo-Gutiérrez | Exceptional sets and Hilbert-Schmidt composition operators[END_REF][START_REF] Gallardo-Gutiérrez | Hilbert-Schmidt composition operators on Dirichlet spaces[END_REF] using a completely different method. Theorems 3.4 and 3.6 give quantitative versions of this result.

Theorem 3.4. If C ϕ is a Hilbert-Schmidt operator in D, then 1 0 cap(E ϕ (s)) 1 -s log 1 1 -s ds < ∞. (4)
Proof. Fix λ ∈ T and let

ϕ λ (ζ) = log Re 1 + λϕ(ζ) 1 -λϕ(ζ) , ζ ∈ T. Since D |ϕ ′ (z)| 2 (1 -|ϕ(z) 2 |) 2 dA(z) < ∞,
it follows that ϕ λ ∈ D(T), see [START_REF] Gallardo-Gutiérrez | Exceptional sets and Hilbert-Schmidt composition operators[END_REF], where

D(T) := {f ∈ L 2 (T) : f 2 D(T) = n∈Z | f (n)| 2 (1 + |n|) < ∞}. Setting ∆ λ := {ζ ∈ T : |1 -λϕ(ζ)| ≥ 1}, we see that |ϕ λ (ζ)| ≍ log 1 1 -|ϕ(ζ)| 2 , ∀ζ ∈ ∆ λ .
Applying the strong capacity inequality [14, Theorem 2.2] to ϕ λ , we get

∞ > ϕ λ 2 D(T) ≥ c ∞ cap {ζ ∈ T : |ϕ λ (ζ)| > s} ds 2 = c ∞ cap ζ ∈ T : log 1 -|ϕ(ζ)| 2 |1 -λϕ(ζ)| 2 > s ds 2 ≥ c ∞ cap ζ ∈ T ∩ ∆ λ : log 1 -|ϕ(ζ)| 2 |1 -λϕ(ζ)| 2 > s ds 2 ≥ c ∞ cap ζ ∈ T ∩ ∆ λ : log 1 1 -|ϕ(ζ)| 2 > 4s ds 2 ≥ c 1 1 cap ζ ∈ T ∩ ∆ λ : |ϕ(ζ)| > u d log 1 1 -u 2 .
Since T = ∆ 1 ∪ ∆ -1 , the subadditivity of the capacity implies that

∞ > ϕ 1 2 D(T) + ϕ -1 2
D(T) ≥ c 2 1 cap ζ ∈ T : |ϕ(ζ)| > u d log 1 1 -u 2 ,
and hence the theorem follows.

Remarks 3.5.

Since {z

n /(1 + n) 1-α 2 } ∞ n=0 is an orthonormal basis in D α , α ∈ (0, 1), C ϕ is a Hilbert-Schmidt operator in D α if and only if ∞ n=1 D α (ϕ n ) n 1-α ≍ D |ϕ ′ (z)| 2 (1 -|ϕ(z)| 2 ) 2+α dA α (z) < ∞.
Therefore, for fixed λ ∈ T, the function

ϕ λ (ζ) = Re 1 + λϕ(ζ) 1 -λϕ(ζ) -α/2 , (ζ ∈ T),
belongs to the weighted harmonic Dirichlet space

D α (T) := {f ∈ L 2 (T) : f 2 Dα(T) = n∈Z | f (n)| 2 (1 + |n|) 1-α < ∞},
(see [START_REF] Gallardo-Gutiérrez | Hilbert-Schmidt composition operators on Dirichlet spaces[END_REF]) . Applying again the strong capacity inequality [14, Theorem 2.2] for D α to ϕ λ , we get as before

( 1 -|λ| 2 ) 2 D

 12 or, equivalently, sup I⊂T µ(S(I))/|I| 2 < ∞, for any subarc I ⊂ T with arclengh |I|, and S(I) is the Carleson box. The measure µ is called vanishing (or compact) Carleson measure for A 2 0 if the identity map i α : A 2 0 → L 2 (µ) is a compact operator. This happens if and only if lim |λ|→1 dµ(z) |1 -λz| 4 = 0 ⇐⇒ lim |I|→0 µ(S(I))/|I| 2 = 0.

  Given E ⊂ T and t > 0, let us write E t = {ζ : d(ζ, E) ≤ t} where d denotes the arclength distance and |E t | denotes the Lebesgue measure of E. Let E be a closed subset of T with |E t | = O((log(e/t)) -3

1 0Theorem 3 . 6 .

 136 cap α (E ϕ (s)) (1 -s) 1+α ds < ∞.The following theorem is the analogue of Proposition 3.1 for the Dirichlet space. It shows that condition (4) is optimal.Let h : [1, +∞[→ [1, +∞[ be a function such that lim x→∞ h(x) = +∞. Let E be a closed subset of T such that cap(E) = 0.Then there is ϕ ∈ A(D) ∩ D, ϕ(D) ⊂ D such that :(1) E ϕ (1) = E; (2) C ϕ is in the Hilbert-Schmidt class in D; Let k(x) = h(e x ), there exists a continuous decreasing functionψ such that +∞ ψ(x) dx 2 < ∞ and +∞ ψ(x) k(x) dx 2 = ∞. Set η(t) = ψ -1 (cap(E t )). We have 0 cap(E t ) |dη 2 (t)| ≍ 0 ψ(η(t)) |dη 2 (t)| ≍ +∞ ψ(x) dx 2 < ∞,and,0 cap(E t ) h(e η(t) )) |dη 2 (t)| ≍ 0 ψ(η(t)) k(η(t)) |dη 2 (t)| ≍ +∞ ψ(x) k(x) dx 2 = ∞. Since 0 cap(E t ) |dη 2 (t)| < ∞,by [4, Theorem 5], there exists a function f ∈ D such that Ref (ζ) ≥ η(d(ζ, E)) and |Imf (ζ)| < π/4, q.e. on T. By harmonicity, |Imf (z)| < π/4 , |z| < 1, Now take ϕ = exp(-e -f ).By a simple modification in the construction of f as in[START_REF] Brown | Some examples of cyclic vectors in Dirichlet space[END_REF], we can suppose that ϕ ∈ A(D). Hence E ϕ (1) = E andD |ϕ ′ (z)| 2 (1 -|ϕ(z)| 2 ) 2 ≍ D |f ′ (z)| 2 e -2 Ref (z) e -2 e -Ref (z) cos(Imf (z)) e -2 Ref (z) cos 2 (Imf (z)) dA(z) ≤ D |f ′ (z)| 2 exp(-√ 2 e -Ref (z) ) dA(z) ≤ c D |f ′ (z)| 2 dA(z) < ∞.Hence C ϕ is in the Hilbert-Schmidt class. Finally, since E ϕ (s) ⊇ {ζ ∈ T : η(d(ζ, E)) ≥ log(1/1 -s)}, we get 0 cap(E ϕ (s))h(1/1-s) d log(1/1-s) 2 ≥ 0 cap(E t )h(e η(t) ) |dη 2 (t)| = +∞.

  2, C ϕ is bounded in D if and only if n ϕ (z)dA(z) is a Carleson measure for A 2 0 and C ϕ is compact in D if and only if n ϕ (z)dA(z) is a vanishing Carleson measure for A 2 0 . More explicitly, we have

Département de Mathématiques, Université Mohamed V, B.P. 1014 Rabat, Morocco E-mail address: elfallah@fsr.ac.ma

CMI, LATP, Université de Provence,

39, rue F. Joliot-Curie, 13453 Marseille, France

1. Research partially supported by a grant from Egide Volubilis. 2. Research partially supported by grants from Egide Volubilis and ANR Dynop.


Marion Schroedter Homscheidt

Jethro Betcke

Hanne Breitkreuz

Annette Hammer

Detlev Heinnemann

Sybille Petrack

Thomas Holzer-Popp

Lucien Wald

Marion Schroedterhomscheidt
email: marion.schroedterhomscheidt@dlr.de

Detlev Heinemann

Sibylle Petrack

Thomas Holzerpopp

Energy-specific solar radiation data from MSG: The Heliosat-3 project

INTRODUCTION

The Heliosat method [START_REF] Cano | A method for the determination of the global solar radiation from meteorological satellite data[END_REF][START_REF] Beyer | Modifications of the Heliosat procedure for irradiance estimates from satellite images[END_REF] was developed for Meteosat First Generation to calculate surface solar irradiance from measurements in the visible channel. It is supposed that the sunlight reflected by clouds is inversely related to the sunlight transmitted by them. The scheme takes the variability of the cloud field and its impact on hourly global and diffuse irradiance into account. Together with a sophisticated model of clearsky irradiance [START_REF] Müller | Rethinking satellitebased solar irradiance modeling The SOLIS clearsky module[END_REF] this method gives reliable results for Europe for most of the year.

Potential for improvement is expected in winter time especially in snow covered areas, during morning and evening hours due to threedimensional effects of clouds to be described with cloud physical parameters and in the clear sky case using more detailed input information on aerosols and water vapor. Therefore, a new scheme called Heliosat3 was developed in the MSGRAO and EU FP5 programs (Fig. 1).

Figure 1. Heliosat3 scheme [4]

SNOW COVER

The old Heliosat method fails in wintertime, when snow is covering the surface under clear skies. Due to their high reflectances these cases are falsely classified as overcast, which leads to a remarkable error in the global irradiance values. Fig. 2 gives an example of an existing photovoltaic (PV) system showing high energy yield on 11 th February 2006 while the satellitebased estimate is lower due to an error in snowcloud discrimination. Even the opposite effect is observed as e.g. in Fig. 3. In this case the PV system delivers an energy yield close to zero due to snow cover on the modules while the satellitebased energy production estimate reaches 1500 W as usual for a sunny situation in wintertime. In case the snow cover is known, the assumption of a total hardware failure can be denied and a false alarm to the system maintenance staff can be avoided.

Figure 3. Hourly effective energy production (solid line) and reference yields (yellow) of a PV plant (Source Enecolo SPYCE service).

The MSG channels give the opportunity to discriminate snow from clouds and to reduce this error source. A multitemporal snow cover detection algorithm was developed [START_REF] Heinicke | Titel, Annette????[END_REF][START_REF] De Ruyter De Wildt | Snow mapping using multitemporal Meteosat8 data[END_REF]. Snow maps are derived in 3x3 km² nadir SEVIRI pixel resolution and superimposed to cloud index maps derived from HRV reflectances (1x1 km²) by an adapted Heliosat method. Results are given in Fig. 4. While some falsely detected snow pixels lead to an overestimation of global irradiance, the remarkable underestimation of a method without snow detection vanishes. GEOMETRICAL CORRECTION USING CLOUD HEIGHT MSG has another opportunity to derive a better quality in the derived surface irradiance, namely the higher resolution in space and time compared to MFG. When making full use of this high resolution data, a three dimensional consideration is optimal. A cloud in the sky has its largest effect on the irradiance measured at the ground when it is affecting the direct beam. For a given pixel the clouds between pixel and sun must be analysed instead of the clouds between the pixel and the observer (MSG). For a geometric correction of this effect cloud top height is needed and can be provided by MSG SEVIRI. In a first approach a standard cloud of 1 km thickness and a seasonal variation of cloud top height was assumed. A significant improvement can be noticed in the winter months due to snow detection. During the summer months a slight improvement in standard error can also be noticed which is attributed to the increased spatial resolution of the HRV channel.

NEED FOR AEROSOL INFORMATION

It has been stated e.g. by [START_REF] Gueymard | Direct solar transmittance and irradiance predictions with broadband models. Part I: detailed theoretical performance assessment[END_REF] that accurate aerosol information is crucial for irradiance calculations in the clear sky case. Climatologies or model output data sets with a typical grid size of 3 to 5° are currently available. Fig. 8 shows the RMSE in aerosol optical thickness at 550 nm to be expected for hourly values if such climatologies are used. This analysis was performed using AERONET ground measurements [START_REF] Holben | AERONET: A federated instrument network and data archive for aerosol characterization[END_REF] for 30 European stations in 2003. Data from the year 2004 or 2005 reveal similar results.

From that analysis it can not be expected that new global data sets with a spatial resolution of several degrees can improve the situation remarkably. This is in line with the findings of the solar energy community that the use of different existing aerosol climatologies or recently made available model data sets did not increase the accuracy. On the other hand, the new ENVISAT and the upcoming METOP satellites with the SCIAMACHY and GOME2 instruments and a pixel size around 0.5° have the potential to deliver aerosol information in a spatial resolution more suitable for this user community. Yet, to account for the temporal and spatial variability of aerosols, an optimum retrieval scheme would deliver products in the same temporal and spatial resolution as MSGSEVIRI. AEROSOL STUDY BASED ON MSG Therefore, the SYNAER method was applied for SEVIRI. The SYNAER method (SYNergetic AErosol Retrieval, [HolzerPopp]) delivers boundary layer aerosol optical depth (AOD) and type over both land and ocean. After cloud detection, AOD is calculated over automatically selected and characterized dark pixels and surface albedo correction for a set of 40 different predefined boundary layer aerosol mixtures is performed. A least square fit of a simulated spectrum based on the corrected surface albedo to the MSG SEVIRI channels delivers the AOD value. Unfortunately, MSG SEVIRI has no blue channels, which means that the spectrum is not covered complete. Therefore, accurate aerosol retrieval cannot be expected. Nevertheless, in the HELIOSAT3 project a case study was made to quantify the dependence on aerosol type for an example case. Fig. 9 gives the SYNAER results for MSGSEVIRI on 1 st August 2004 assuming fixed aerosol types of pure sulfate and of a polluted continental aerosol. Large differences in quantitative values can be observed due to the missing aerosol type information as result of the missing spectral resolution of SEVIRI in the blue spectral region. This test trial reveals clearly, that aerosol type information has to be provided as auxiliary data set to a SEVIRIbased aerosol retrieval scheme. It is proposed to provide such information from polar satellites as e.g. ENVISAT and in future METOP as they have opportunities to separate major aerosol types. Therefore, the next section reports on recent results in retrieving aerosol type information from ENVISAT AATSR and SCIAMACHY using SYNAER in the PROMOTE project.

AEROSOLS FROM ENVISAT

For ENVISAT, the high spatial resolution of the AATSR instrument (Advanced Along Track Scanning Radiometer) permits accurate cloud detection, dark pixel selection and surface albedo correction. After spatial integration to the larger pixels of the spectrometer SCIAMACHY (SCanning Imaging Absorption spectroMeter for Atmospheric CartograpHY) these parameters are used to simulate SCIAMACHY spectra for the same set of different aerosol mixtures. SYNAER has been implemented for operational processing at the German Remote Sensing Data Center within the ESA GSE PROMOTE (Protocol Monitoring for the GMES Service Element; see also http://www.gsepromote.org) and delivers daily near real time observations (within the same day) and an evolving archive of historic data. The seasonal behavior visible in the SYNAER data is shown in Fig. 9 for the soot component. Here, three major features can be seen: From July to September the tropical African biomass burning moves South and East together with its plume over the Atlantic (which is slightly visible). The Amazonian biomass burning season mainly extends from August to October. Over large parts of Europe, no data are available in December (due to low sun, high cloudiness and snow cover). Highest European soot concentrations are detected, when observations are becoming available still in the heating season (April). Additionally, wildfires in August (hot summer 2003) contribute to elevated soot over the Iberian peninsula.

CONCLUSIONS

Starting from the fact that large inconsistencies in existing irradiance data bases available for solar energy engineering, the Heliosat3 RAO project aimed at using the enhanced temporal, spatial and spectral resolution of the Meteosat Second Generation SEVIRI instrument for an improved derivation of global and direct irradiance. It was shown, that cloud top height is needed to account for basic threedimensional effects of clouds in radiative transfer. In future, SEVIRI derived cloud top heights will be included in a modified cloud index scheme as part of the new Heliosat3 method. Additionally, snow detection allows avoiding errors due to misdetection of clouds as snow and vice versa. It helps to decide on a possible hardware failure in automatic failure detection routines for photovoltaic system monitoring. A parameterization scheme (SOLIS, [START_REF] Müller | Rethinking satellitebased solar irradiance modeling The SOLIS clearsky module[END_REF]) based on radiative transfer calculations was developed which allows to take actual information on aerosol optical depth, angstrom coefficient and water vapor concentration into account. There is an urgent need for improved aerosol optical depth information in a better spatial and temporal resolution than available nowadays from global models and climatologies. Therefore, SYNAER aerosol retrievals were analyzed both for a test trial with MSG SEVIRI and for ENVISAT. The trial for SEVIRI shows the clear need for aerosol type information as input for any aerosol optical depth retrieval scheme using SEVIRI. ENVISAT based SYNAER results show first successful results on the discrimination of water soluble and soot results, while the large aerosol particles from mineral and sea salt sources are still a subject of further work. Followon activities will be performed in the virtual Institute of Energy Meteorology (vIEM), the ESA GSE PROMOTE and on the market development for solar energy services based on Earth observation within the ESA EOMD project ENVISOLAR.

Figure 2 .

 2 Figure 2. Hourly effective energy production (solid line) and reference yields (yellow) of a PV plant (Source Enecolo SPYCE service).

Figure 4 .

 4 Figure 4. Hourly global irradiance with (blue) and without snow detection (red) for 20 German stations in February -March 2005

Fig. 5

 5 gives validation results for global irradiances measured at the ground station Vaulxen Velin (France) for the period March -August 2004 and November 2004 -February 2005. Fig. 6 presents the same analysis but for Freiburg (Germany). A remarkable improvement in monthly root mean square errors (RMSE) can be noticed. The next step will be the use of MSG based cloud top height information. global irradiance ground measured global irradiance satellite [W/m2]

Figure 5 .

 5 Figure 5. Improvement in monthly RMSE of hourly global irradiance for VaulxenVelin (France) due to a geometrical correction for cloud height.

Figure 6 .

 6 Figure 6. Improvement in monthly RMSE of hourly global irradiance for VaulxenVelin (France) due to a geometrical correction for cloud height.

Figure 7 .

 7 Figure 7. Improvement in monthly bias and standard error of hourly global irradiance for 20 DWD ground stations in 2005.

Figure 8 .

 8 Figure 8. Variogram based on 30 European AERONET ground measurements of AOT at 550 nm in 2003.

Figure 9 .

 9 Figure 9. Aerosol optical depth at 550nm derived from MSG SEVIRI on 1st August 2004 assuming fixed aerosol types.

Figure 9 :

 9 Figure 9: Seasonal variation of soot: monthly mean maps of soot aerosol optical depths for the months April, July, August, September, October 2003, and