Keywords: Radar, Sentinel-1, surfaces agricoles, rugosité du sol, modèles de rétrodiffusion radar II Radar, Sentinel-1, agricultural areas, soil roughness, radar backscattering models III

I

RESUME

La télédétection spatiale est d'une importance primordiale pour la cartographie et la surveillance des problèmes environnementaux. Son intérêt réside dans la capacité des capteurs satellitaires spatiaux à fournir des informations globales et permanentes de la planète, aux échelles locale à globale. La télédétection radar a montré son grand potentiel ces dernières années dans la caractérisation des états de surface du sol. L'état de la surface du sol, et en particulier l'humidité et la rugosité, exerce une influence fondamentale sur la répartition de la pluie entre infiltration, rétention superficielle et ruissellement. Il a un rôle essentiel dans les processus hydrologiques de surface et ceux associés à l'érosion et aux processus d'évapotranspiration. La caractérisation et la prise en compte de ces conditions de surface constituent actuellement un enjeu important pour la modélisation à base physique des processus hydrologiques ou pour le couplage surfaceatmosphère. Dans ce cadre et depuis plusieurs années, plusieurs études scientifiques ont montré le potentiel des données micro-ondes actives dans l'estimation de l'état hydrique du sol et de sa rugosité de surface.

Les nouveaux systèmes radar (SAR ʺSynthetic Aperture Radarʺ) ont permis d'ouvrir de nouvelles perspectives pour l'observation de la terre grâce à l'amélioration de la résolution spatiale (métrique sur TerraSAR-X et COSMO-SkyMed) et temporelle (TerraSAR-X, COSMO-SkyMed, Sentinel-1). La disponibilité depuis peu des nouveaux capteurs radar bande C Sentinel-1 (Sentinel-1A et Sentinel-1B) rend indispensable l'évaluation des données Sentinel-1 pour la caractérisation des états de surface du sol et en particulier la rugosité de surface.

I.1 Context

The soil is not just the surface we walk on, build on, nor the land we cultivate; it is the source of our life and a wealth that we must preserve. From the 1950s, the intensification of agricultural land accelerated the processes of soil degradation, thus affecting their biophysical-chemical properties. The first component directly affected by intensification is soil fertility in particular (organic matter and soil structural stability).

Generally, several soil processes could be identified based on specific physical conditions and physical, chemical or biological activities. The soil processes are classified mainly into seven categories [START_REF] Bockheim | The role of soil-forming processes in the definition of taxa in Soil Taxonomy and the World Soil Reference Base[END_REF]: (i) the translocation which is related to physical movements and always in the downward direction, (ii) the Organic Changes which occur mainly on the surface and follow a specific sequence, (iii) the Podzolization which happens in cool, humid climates where the bacterial activity is low, (iv) the gleying which takes place under water-logged and anaerobic conditions, (v) the salinization, (vi) the recarbonatation, (vii) eventually, the desilication that is common in hot-wet tropical and equatorial climates.

The study and modeling of the continental surfaces functioning and their interactions with the atmosphere are essential research subjects to understand the climatic system of the earth.

These surfaces constantly exchange the amount of movement, energy, water and chemical constituents, such as carbon, nitrogen, etc… As for continental hydrology, the processes involved in the water cycle and quantifying the exchanges of matter and energy shall be understood. Surface states essentially control the distribution of rainfall between soil storage, underground infiltration, runoff and evapotranspiration. Hydrodynamic characteristics, roughness and vegetation cover are major determinants of hydrological processes (soil water interception and return to the atmosphere), and those associated with erosion [START_REF] Ambroise | La dynamique du cycle de l'eau dans un bassin versant: processus, facteurs, modèles[END_REF][START_REF] Auzet | Surface characterisation for soil erosion forecasting[END_REF].

Runoff and erosion in agricultural soils are major problem for territorial managers. For several years, several regions in Europe have been facing an increase in natural disasters: floods, loss of fertile land, soil degradation and water quality. The soil conditions (i.e. moisture and surface roughness) have an essential role in surface hydrological processes.

Runoff occurs when the amount of rain exceeds the infiltration capacity of the soil (Le [START_REF] Bissonnais | Experimental study and modelling of soil surface crusting processes[END_REF][START_REF] Brun | Mapping saturated areas with a helicopter-borne C band scatterometer[END_REF][START_REF] François | Influence des systèmes de culture sur les risques d'érosion par ruissellement concentré. I. — Ana-lyse des conditions de déclenchement de l'érosion[END_REF][START_REF] Zobeck | Tillage and rainfall effects on random roughness: a review[END_REF].

Soils with silty texture are particularly sensitive to runoff because they are subjected to crusting phenomenon as a result of episodes (Le [START_REF] Bissonnais | Crusting, runoff, and erosion response to soil water content and successive rainfalls[END_REF]. The interactions between meteorological conditions, agricultural practices and soil texture cause significant and fast changes in the hydraulic properties of the soil surfaces. The deterioration of soil infiltrability and surface storage capacity is strongly influenced by the phenomena of crusting and degradation of roughness and the genesis conditions of runoff, causing erosive problems [START_REF] Govers | Soil roughness and overland flow[END_REF][START_REF] King | Rôle des sols et des pratiques culturales dans l'infiltration et l'écoulement des eaux. Exemple du ruissellement et de l'érosion sur les plateaux limoneux du nord de l'Europe[END_REF].

Soil moisture is a key parameter in the different processes involved in the hydrological cycle (water cycle). Knowledge of moisture is necessary to assess water resources and to carry out water balances. Information on the spatial distribution of soil moisture optimizes water reassignment during droughts and provides support for flood forecasting and management.

From an agronomic point of view, soil moisture is a crucial variable for crop development.

Thus, assessing it allows better monitoring and management of irrigation, leading to a more precise farming.

Another characteristic of the soil to be considered is the surface roughness. It is a physical parameter that characterizes the surface state. For agricultural soils, roughness defines the microrelief of the soil surface on the clods scale and is due to small accidents in the field (natural, cultivation techniques or both). Knowing the state of surface roughness is necessary for understanding the different processes. It is a key parameter in the estimation of water storage capacity within the soil horizons as well as modeling runoff. This latter phenomenon has a great influence on the erosion processes [START_REF] Roose | Méthodes de mesure des états de surface du sol, de la rugosité et des autres caractéristiques qui peuvent aider au diagnostic de terrain des risques de ruissellement et d'érosion, en particulier sur les versants cultivés des montagnes[END_REF] and determines the floods following a rainy event.

The evolution in time and space of the physical, hydraulic and geometric properties of soil surfaces is an information that can be integrated into hydrological models for forecasting the water balance and the processes of runoff and erosion [START_REF] Auzet | Surface characterisation for soil erosion forecasting[END_REF][START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF][START_REF] Boiffin | La dégradation structurale des couches superficielles du sol sous l'action des pluies[END_REF][START_REF] Casenave | Les états de surface de la zone sahélienne: influence sur l'infiltration[END_REF][START_REF] King | Remote-sensing data as an alternative input for the 'STREAM'runoff model[END_REF][START_REF] Ludwig | Hydrological structure and erosion damage caused by concentrated flow in cultivated catchments[END_REF][START_REF] Quesney | Estimation of watershed soil moisture index from ERS/SAR data[END_REF][START_REF] Weisse | Assimilation of soil moisture into hydrological models for flood forecasting: comparison of a conceptual[END_REF]. Moreover, soil and water resources' management are key issues, not only from the environmental point of view, but also from a socioeconomic perspective [START_REF] Condrea | Environmental issues from an economic perspective[END_REF].

There are many in situ experimental methods to measure surface soil moisture [START_REF] Gardner | Water content. Methods Soil Anal. Part 1-Physical Mineral[END_REF][START_REF] Topp | Electromagnetic determination of soil water content: Measurements in coaxial transmission lines[END_REF]. These methods allow precise soil moisture estimates only at the local scale. Since several years, the scientific community has demonstrated the potential of spatial observation for estimating soil parameters. Spatial remote sensing allows repetitive measurements and provides access to spatial information at scales that can be very fine.

In this context, remote sensing is of paramount importance for mapping and monitoring environmental problems. Its interest lies in the ability of space-based satellite sensors to provide global and permanent information about the planet.

I.2 State of art I.2.1 Remote sensing data for soil characterization

Active microwave remote sensing is specifically well suitable in agricultural fields concerning the characterization of soil surface conditions. Synthetic Aperture Radar (SAR) sensors allow all-weather measurements, independently of weather conditions (cloud cover, day/night…).

They use microwave frequencies with wavelengths between 1 mm to 1 m. Theses microwave frequencies are very sensitive to the geometric and dielectric properties which are themselves dependent on surface parameters (roughness, soil moisture, soil composition and vegetation cover). SAR signal also depends on different instrumental parameters (polarization, incidence angle, and radar wavelength).

Studies using radar remote sensing started at the end of the 70s with in-situ or airborne scatterometers [START_REF] Ulaby | Microwave backscatter dependence on surface roughness, soil moisture, and soil texture: Part I-bare soil[END_REF]. The developments of these studies became more important in the 1990s with satellite and airborne SAR (ERS-1/2, JERS, SIR-C, RADARSAT-1 …). Most studies were carried out in C-band (wavelength ~6 cm), L-band (wavelength ~22 cm), and more recently in X-band (wavelength ~3 cm). Firstly, the satellite SAR sensors that were accessible to the scientific community had an instrumental configuration of mono-polarization and a single incidence angle (ERS-1/2, JERS). The second generation of radar sensors with new instrumental configurations (RADARSAT, ASAR/ENVISAT, PALSAR/ALOS, TerraSAR-X, COSMO-SkyMed, Sentinel-1) allowed the scientific community to gather images in multi-polarization and sometimes polarimetric mode (scattering matrix). These SAR second generation provide images in high spatial resolution (about 1 m for TerraSAR-X and COSMO-SkyMed) and high temporal resolution (up to one image by day). In addition, the launch of the Sentinel-1 C-band SAR, which is based on a constellation of two satellites (A and B units) makes it possible to obtain SAR data for global areas at high spatial and temporal resolutions (spatial resolution of 10 m and time revisit of 6 days over Europe) with free and open access Sentinel satellites. These new SAR sensors with C-band are suitable for hydrological and agronomic applications [START_REF] Alexakis | Soil Moisture Content Estimation Based on Sentinel-1 and Auxiliary Earth Observation Products[END_REF][START_REF] Aubert | Analysis of TerraSAR-X data sensitivity to bare soil moisture, roughness, composition and soil crust[END_REF]Baghdadi et al., 2002aBaghdadi et al., , 2011aBaghdadi et al., , 2012a;;[START_REF] Hajnsek | Potential of estimating soil moisture under vegetation cover by means of PolSAR[END_REF][START_REF] Holah | Potential of ASAR/ENVISAT for the characterization of soil surface parameters over bare agricultural fields[END_REF][START_REF] Paloscia | A comparison of algorithms for retrieving soil moisture from ENVISAT/ASAR images[END_REF][START_REF] Srivastava | Use of multiincidence angle RADARSAT-1 SAR data to incorporate the effect of surface roughness in soil moisture estimation[END_REF][START_REF] Srivastava | Large-area soil moisture estimation using multi-incidence-angle RADARSAT-1 SAR data[END_REF]Zribi et al., 2005a). Some low resolution spatial sensors are also suitable for meteorological and climatic applications on a global scale such as, AMSR-E, AMSR2 (microwave radiometers), SMOS, SMAP (L-band microwave radiometers), and ASCAT/METOP (C-band scatterometer). They provide users of soil moisture products with a temporal frequency on few days with a spatial resolution around 25-40 km [START_REF] Champagne | Satellite surface soil moisture from SMOS and Aquarius: Assessment for applications in agricultural landscapes[END_REF][START_REF] Chan | Assessment of the SMAP passive soil moisture product[END_REF][START_REF] Entekhabi | The soil moisture active passive (SMAP) mission[END_REF][START_REF] Jackson | Validation of Soil Moisture and Ocean Salinity (SMOS) soil moisture over watershed networks in the US[END_REF][START_REF] Mohanty | Soil moisture remote sensing: State-of-the-science[END_REF][START_REF] Wigneron | Modelling the passive microwave signature from land surfaces: A review of recent results and application to the Lband SMOS & SMAP soil moisture retrieval algorithms[END_REF].

For bare agricultural soils or soils with little vegetation, the radar signal is dependent on the two parameters of the surface: the dielectric constant related to the soil moisture and the surface roughness. Several radar backscatter models have been developed in recent years with the aim to model the backscattering of natural surfaces and to reverse the radar response to find the different parameters of the soil surface. These models depend on the sensor characteristics (incidence, frequency, polarization...) and those of the target (soil moisture and surface roughness). From a perspective of anticipation or coherence of information on various natures, it is essential to rely on models capable for estimating soil parameters. Numerous radar backscattering models have been developed in order to estimate soil parameters (i.e. soil moisture and surface roughness) [START_REF] Zribi | A new empirical model to retrieve soil moisture and roughness from C-band radar data[END_REF][START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF], 2006a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF], 2015, 2016a;[START_REF] Chen | Emission of rough surfaces calculated by the integral equation method with comparison to three-dimensional moment method simulations[END_REF][START_REF] Dubois | Measuring soil moisture with imaging radars[END_REF][START_REF] Fung | Backscattering from a randomly rough dielectric surface[END_REF][START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF][START_REF] Oh | An empirical model and an inversion technique for radar scattering from bare soil surfaces[END_REF][START_REF] Oh | An inversion algorithm for retrieving soil moisture and surface roughness from polarimetric radar observation[END_REF][START_REF] Oh | Semi-empirical model of the ensembleaveraged differential Mueller matrix for microwave backscattering from bare soil surfaces[END_REF]. However, ground measurements of different soil parameters are necessary to calibrate these models in order to have accurate estimations.

I.2.2 Potential of radar data for monitoring soil conditions

Synthetic Aperture Radar (SAR) data have been used for a long time to estimate and map soil moisture. Several radar backscattering models were developed in order to estimate soil parameters (i.e. soil moisture and surface roughness). The availability of high spatial and temporal resolutions SAR Sentinel-1 data and these models make the intere st to estimate soil parameters accurately. In the case of bare soils (or soils with little vegetation), the estimation of soil moisture and surface roughness was performed by inverting the measured SAR backscatter through SAR backscattering models (both empirical and physical). Unlike physical models, empirical or semi-empirical models need to be calibrated each time the study area changes by using site-specific in situ measurements and SAR observations. The most commonly semi-empirical models are the models of Oh [START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF][START_REF] Oh | An empirical model and an inversion technique for radar scattering from bare soil surfaces[END_REF][START_REF] Oh | An inversion algorithm for retrieving soil moisture and surface roughness from polarimetric radar observation[END_REF][START_REF] Oh | Semi-empirical model of the ensembleaveraged differential Mueller matrix for microwave backscattering from bare soil surfaces[END_REF] , Dubois [START_REF] Dubois | Measuring soil moisture with imaging radars[END_REF] and Baghdadi (Baghdadi et al., 2016a); while, the most popular physical models are Integral Equation Model (IEM) [START_REF] Fung | Backscattering from a randomly rough dielectric surface[END_REF], IEM calibrated by Baghdadi, called in this thesis "IEM_B" [START_REF] Baghdadi | An empirical calibration of the integral equation model based on SAR data, soil moisture and surface roughness measurement over bare soils[END_REF][START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF](Baghdadi et al., , 2006a(Baghdadi et al., , 2011a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF], and Advanced Integral Equation Model (AIEM) [START_REF] Chen | Emission of rough surfaces calculated by the integral equation method with comparison to three-dimensional moment method simulations[END_REF].

Several studies have been carried out to evaluate and compare the robustness of the backscattering models such as, Oh, Dubois and IEM (original IEM, IEM_B and AIEM). [START_REF] Zribi | Backscattering behavior and simulation comparison over bare soils using SIR-C/X-SAR and ERASME 1994 data over Orgeval[END_REF] evaluated the Oh model and IEM using L-, C-and X-bands SAR data and in situ measurements. Results showed that the IEM provides accurate simulations (RMSE about 2.0 dB) only over smooth surfaces. In addition, for rough surfaces and medium incidence angle, Oh model simulations retrieve backscattering values very close to the measured ones, while showing poor correlation with measured backscattering coefficients over smooth areas. Baghdadi and Zribi (2006) evaluated the backscattering models IEM, Oh and Dubois by using large C-band SAR data and in situ measurements. Results showed that these models frequently tend to over-estimate or under-estimate the radar signal (in the order of 3.0 dB) and the errors on model simulation depend on height surface roughness, Hrms, soil moisture, mv, and/or incidence angle. [START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF] evaluated the potential of IEM, Oh and Dubois models by using TerraSAR-X images acquired over France and Tunisia and experimental datasets of in situ measurements (mv ranged between 5 vol. % and 41 vol. % and Hrms between 0.42 cm and 4.55 cm). In this case, the semi-empirical Oh model correctly simulated the backscattering (showing over or under-estimation of the backscatter <1 dB, and RMSE <3 dB), while Dubois model showed a poor correlation between real data and simulations, with RMSE between 2.2 and 4.4 dB and over or under-estimation of the backscatter of about 3.4 dB. In addition, the IEM correctly simulates the backscattering at Xband for Hrms < 1.5 cm by using the exponential correlation function and for Hrms > 1.5 cm by using the Gaussian correlation function. [START_REF] Panciera | Evaluation of IEM, Dubois, and Oh radar backscatter models using airborne L-band SAR[END_REF] compared the performances of the IEM, Dubois and Oh models by using fully polarized L-band airborne data (incidence angles between 24° and 38°) and in situ measurements (mv between 5 vol. % and 39 vol. % and Hrms between 1 cm and 7.6 cm) acquired over the study area in southeastern Australia.

At HH polarization, the three models simulated the backscattering with almost similar accuracy, showing a mean error between the simulated and the observed backscattering coefficients of about 1.6 dB in absolute value (standard deviation "std" about ±2.5 dB). At VV polarization, the Oh model resulted to be more accurate than IEM and Dubois models: the mean errors between the simulated and observed backscattering were equal to 4.5 dB (std = ±2.0 dB), 1.7 dB (std = ±2.3 dB), and -0.4 dB (std = ±2.4 dB) for IEM, Dubois, and Oh model, respectively. Moreover, several studies confirmed that the use of the calibrated correlation length, as proposed by [START_REF] Baghdadi | Potential of ERS and RADARSAT data for surface roughness monitoring over bare agricultural fields: application to catchments in Northern France[END_REF][START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF]Baghdadi et al. ( , 2006Baghdadi et al. ( , 2011a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF] is able to improve the performance of the IEM at both HH and VV polarizations [START_REF] Dong | Validation of the AIEM through correlation length parameterization at field scale using radar imagery in a semi-arid environment[END_REF][START_REF] Mcnairn | Estimating surface soil moisture using Radarsat-2[END_REF][START_REF] Panciera | Evaluation of IEM, Dubois, and Oh radar backscatter models using airborne L-band SAR[END_REF]. [START_REF] Dong | Validation of the AIEM through correlation length parameterization at field scale using radar imagery in a semi-arid environment[END_REF] used the calibrated correlation length in the AIEM to simulate SAR data in C-band. Results showing that the RMSE reduced from 3.1 to 1.7 dB at HH and VV polarizations and from 31.0 dB to 5.1 dB at HV polarization. [START_REF] Panciera | Evaluation of IEM, Dubois, and Oh radar backscatter models using airborne L-band SAR[END_REF] showed that the use of calibrated correlation length decreases the errors on IEM simulation with a bias equal to about -0.3 dB (standard deviation about ±1.1 dB) at both HH and VV polarizations.

Several studies were done in order to investigate the SAR data for monitoring roughness states over bare agricultural soils. Baghdadi et al., (2002a) examined the potential of the first generation of SAR data (ERS-2 and RADARSAT-1) to estimate surface roughness over bare agricultural soils. Results showed that the use of high incidence angles about 45° are more appropriate to differentiate numerous roughness classes (smooth, medium and rough) over bare agricultural soils. Moreover, Baghdadi et al. (2012a) used neural networks (NNs)techniques to estimate soil moisture mv and surface roughness Hrms from C-band polarimetric RADARSAT-2 data. Results indicated that the accuracy on the soil roughness estimates was about 0.5 cm using polarimetric data. The estimation is better for Hrms-values lower than 2 cm than for Hrms-values higher than 2 cm. For higher Hrms, the neural networks under-estimate the surface roughness.

I.3 Plan of the thesis

The general objective of this work is to explore and evaluate the potential of the Sentinel-1 radar sensor to estimate surface roughness over bare agricultural soil. The originality is that few studies have been done to estimate soil roughness from SAR data and never before have been estimated from Sentinel-1 data. The recent launch of Sentinel-1 C-band SAR providing full earth coverage at high spatial and temporal resolutions with free and open access satellite justifies this work.

Thus, the first part of this thesis is to analyze the quality of the most popular radar backscattering models in order to find the model that best fit the SAR measurements. Integral Equation Model "IEM", Integral Equation Model calibrated by Baghdadi "IEM_B", Oh, Dubois, and Advanced Integral Equation Model "AIEM" will be evaluated using a wide dataset of SAR data and experimental soil measurements. The results will show the performance of each model in order to identify the most robust backscattering model that will be used later in the inversion procedure for estimating the soil roughness.

After evaluating the existing radar backscattering models (semi-empirical, empirical and physical) using a wide reference dataset of SAR (Synthetic Aperture Radar) data and experimental soil measurements, a new radar backscattering model will be proposed. The objective is to develop an empirical radar backscattering model. Never before a backscattering model has been built and validated on such a large dataset: wide range of incidence angles (18°-57°), dataset in L, C and X bands, dataset well distributed geographically for regions with different climate conditions (humid, semi-arid and arid sites) and involving many SAR sensors.

The last research part of this thesis consists of proposing a method to invert the radar signal using neural networks technique. The objective is to evaluate the potential of Sentinel-1 data for estimating soil roughness. The best model found in the first part of this thesis and the new empirical model developed in the second part will be used to train and validate neural networks. Finally, the neural networks will be validated using a real dataset composed of Sentinel-1 images and in-situ measurements, collected in Tunisia and France. This thesis is composed of several chapters. This first chapter is a general introduction that describes the importance of soil parameters and the potential of remote sensing techniques for their monitoring. The second chapter introduces the concept of radar remote sensing technique and describes the interaction of electromagnetic waves with agricultural soils. The soil surface parameters (roughness and moisture content) as well as the different methods for estimating these parameters are described. Next, the most popular radar backscattering models (empirical, semi-empirical and analytical) will be described and evaluated in chapter 3. A new semi-empirical backscattered model will be proposed in chapter 4. Finally in chapter 5, neural networks trained on dataset simulated from radar backscattering models (the IEM modified by Baghdadi and the new proposed model) will be used to estimate the soil roughness from Sentinel-1 radar images. Conclusions and perspectives are presented in chapter 6.

II.1 Introduction

This chapter is an introduction to radar remote sensing, its principle measurement and its instrumental characteristics. Some reminders are introduced in section 2 on electromagnetic waves and their interactions with surfaces in agricultural areas. Section 3 of this chapter describes the descriptive parameters of soil (moisture and roughness). Section 4 shows the sensitivity of radar signal to soil moisture and surface roughness. In this thesis, our study focuses on soils in bare agricultural areas. Thus, to conclude this chapter, the electromagnetic backscattering models are introduced, which simulate the radar signal by linking it to the geophysical parameters of the soil surface, mainly moisture and roughness.

II.2 Radar remote sensing

Radar is an active sensor for Earth observation, operates in the microwave frequencies of the electromagnetic spectrum (300 MHz to 30 GHz). The principle of radar, for the observation of surfaces, consists of emitting an electromagnetic wave in a selected configuration (frequency, polarization, incidence angle). This wave propagates in space to the observed surface and part of the emitted energy is returned in the direction of observation. It is then said that the energy is backscattered. It is a function of both the characteristics of the system and the nature of the surface (electrical properties and surface state) (Baghdadi and Zribi, 2016).

In the general introduction, we have already mentioned the advantages of the radar. It is used day and night and is almost insensitive to the weather conditions. For the observation of the Earth, most radars are synthetic aperture radars (SARs) which provide images with high and very high spatial resolution (from 1m for TerraSAR-X and CosmoSky-Med to 10 m for Sentinel-1 for example). These new SAR images have made it possible to retrieve surface soil parameters with a high spatial resolution.

II.2.1 Instrumental Parameters

The main instrumental parameters of radar remote sensing described in this subsection are frequency, polarization and incidence angle.

II.2.1.1 Radar frequency

Radar systems operate on the microwave domain. The frequency of the transmitted signal is the number of waves passing through a given point during the interval of one second. It is measured in Hertz. The frequency ranges in the microwave domain (from 0.3 to 300GHz)

Wavelength is a measure of the physical distance between peaks of a sine wave propagated in space. Most radar signals have wavelengths measured in centimeters or millimeters.

Wavelength and frequency have inverse relationship = 𝑐/𝜆 : the higher the frequency, the shorter the wavelength. The wavelengths range in the microwave domain between 1m and 1mm.

For the Earth observation, the frequency bands particularly used in radar imagery are summarized in the table II.1. TerraSAR-X,

Band

Cosmo-SkyMed

Table II.1. The main frequency band used in radar imagery and examples of space sensors (past, present or future).

Using radar images in L, C, and X bands to perform studies for the characterization of the soil surface moisture in agricultural areas, in-situ measurements of soil moisture are taken at a depth between 0 and 10 cm. This measurement depth is related to the penetration depth of the radar wave (p) that is generally equal to few centimeters in C and X bands. In L-band, this depth could reach a few dozen cm for very dry soils. Moreover, the penetration depth of the radar signal in C-band decreases from 5 to 1 cm for a clay soil when the soil moisture increases from 10 to 30 vol. % (HH polarization and 15º incidence angle) [START_REF] Bruckler | Near surface soil moisture estimation from microwave measurements[END_REF]. Indeed, the thickness of this surface layer depends on the radar wavelength () (more penetration with greater wavelengths) and the dielectric constant of soil (water content and soil composition) [START_REF] Ulaby | Microwave backscatter dependence on surface roughness, soil moisture, and soil texture: Part I-bare soil[END_REF]:

𝛿 𝑝 ≅ λ√𝜀 ′ 2𝜋√𝜀 ′′ (2.1)
Where ε' is the real part of the dielectric constant and ε" its imaginary part.

The dielectric constant is a physical quantity also known as complex permittivity. The amount of water present in a soil affects its electrical properties and consequently the radar signal.

The microwave dielectric constant of soil is related to soil moisture content and to a lesser extent soil texture [START_REF] Ulaby | Microwave backscatter dependence on surface roughness, soil moisture, and soil texture: Part I-bare soil[END_REF]. This constant is a complex number expressed in the form =-jʺ. The real part ε' affects the moisture content more, while the imaginary part ε" essentially depends on the electrical conductivity of the soil solution.

II.2.1.2 Polarization

The polarization is a property of the electromagnetic wave that describes the orientation of the electric field in the plane perpendicular to the direction of propagation. For radars existing in the Earth observation field, this is a linear polarization in two directions (horizontal and vertical). For example, Sentinel-1 has selectable single polarizations (VV or HH) for the Wave mode and selectable dual polarizations (two polarizations: VV+VH or HH+HV) for all other modes.

II.2.1.3 Incidence angle

The incidence angle (θ) is the angle between the incident propagation direction and the normal surface, in the plane of propagation. The first generation of satellite radar had a fixed sight (for ERS-1/2, θ is centered at 23°). Other sensors have been equipped with variablefocus antennas that can acquire images with incidence angles ranging from 20° to 46° for Sentinel-1.

The launch of multi-incidence SAR satellites (ASAR, RADARSAT-1/2, TerraSAR-X, COSMO-SkyMed, Sentinel-1) have allowed to estimate under very limited conditions the soil moisture and surface roughness simultaneously, even though multi-incidence acquisitions actually represent two acquisitions on two different dates. However, it is assumed that the soil parameters have not changed between the two acquisition dates. For this configuration, the radar images are acquired at two different incidence angles, generally one image with a weak incidence (weak~20°) and another image with a strong incidence (strong~40°) (Baghdadi and Zribi, 2016).

II.2.2 Radar backscattering coefficient

The backscattering coefficient (σ°) is the usual radar term for the measurement of the backscattering of a target by the radar. It expresses the ratio between the power transmitted by the antenna to the ground and that returned by the target for a given configuration system (polarization, frequency, angle of incidence). It defines the ability of an illuminated surface to reflect incident energy towards the antenna. It is usually expressed in decibels (dB), on a logarithmic scale:

σ o dB = 10 . log10 (σ°linear) (2.2)

II.3 Description of soil parameters

II.3.1 Soil moisture

Soil moisture is defined as the water contained in the soil. Obtained from rainfall, snowmelt, irrigation, or from the tube of liquid of groundwater. Soil moisture content is an important variable of climatological, hydrological and environmental systems.

The moisture content in the surface layers of the soil is an important parameter for many applications in hydrology, agriculture and meteorology. Soil moisture is one of the few directly observable hydrological variables that play an important role in the water and energy budgets necessary for climate studies. Estimation of soil properties as soil moisture is an important variable for many water management and agricultural applications [START_REF] Verhoest | On the soil roughness parameterization problem in soil moisture retrieval of bare surfaces from synthetic aperture radar[END_REF]. Moreover soil moisture information could also be used to predict natural disasters such as flooding and for environment changing such as soil erosions [START_REF] Lakhankar | Effect of sub-pixel variability and land-cover on soil moisture retrieval from RADARSAT-1 data[END_REF].

Soil moisture measurements in situ are expensive and take time. Also these measurements might have some problems during the sampling process that make all the measurement incorrect. There are several methods to measure soil moisture, such as the gravimetric method and Time-Domain Reflectrometry (TDR).

II.3.2 In situ measurements

II.3.2.1 The gravimetric method

The gravimetric method consists in first measuring the moisture content of a soil sample taken from a cylinder. The wet content Wp (% or g.g -1 ) is calculated using the wet weight (Ph: soil weight after sampling) and the dry weight (Ps). The dry weight (Ps) is obtained by drying the sample from the sampled soil with a temperature of 105 °C for 24 h.

This method determines the wet weight of a soil sample by comparing the wet mass to its dry mass, according to the following equation:

𝑊𝑝(𝑣𝑜𝑙. %) = 100 * [ 𝑃h -Ps 𝑃h ]
(2.3) With:

Ph: soil moisture mass

Ps: dry soil mass

Wp: water content in weight %

The moisture volume mv (% or cm 3 /cm 3 ) is deduced from the wet weight Wp (% or g.g -1 ) by multiplying it by the apparent density of the soil:

𝑚𝑣(%) = Da .Wp (%)
(2.4)

Da: apparent density = the dry soil mass / the cylinder volume

II.3.2.2 The TDR (Time Domain Reflectometry)

This instrument measures the propagation speed of a microwave signal by waveguides pressed into the soil. This speed is a function of the soil dielectric permittivity, related to the soil water content. For surface measurements, this method is fast with equipment that allows extensive measurements of soil moisture. This instrument can be placed deeply in a horizontal position for automated measurements. It is better to precede a previous calibration of the device by doing a comparison with measurements obtained by the gravimetric method (calibration for each soil type). Moreover, the TDR measurements are not valid for frozen soil where an important drop in the moistures registered by the probe can be observed.

The gravimetric method is considered the most accurate but it requires a lot of effort and time

II.3.3 Surface roughness

Roughness is a parameter that describes the soil microtopography. When the transmitted radar signal interacts with a rough surface, the energy of the wave is reflected in all directions and in particular in the radar's direction. The description of the surface roughness of bare agricultural soils is based on three parameters: the standard deviation height (Hrms), the correlation length (L) and the autocorrelation function [START_REF] Ogilvy | Theory of wave scattering from random rough surfaces[END_REF][START_REF] Ulaby | Microwave remote sensing: Active and passive. Volume 2-Radar remote sensing and surface scattering and emission theory[END_REF].

There are several techniques that could be used for soil roughness measurement: pin profilometer (Figure II.2), laser profilometer (Figure II.3), and 3D photogrammetry. The pin profilometer 1D can be represented by a function f(x) = z, where x is on the horizontal axis and z is the profile height with respect to this axis. Moreover, the use of the laser or 3D photogrammetry allows for the most accurate rendering of soil roughness (high spatial resolution) with a precise estimation of the roughness parameters, Hrms and L (Davidson et al., 2000;[START_REF] Mattia | Backscattering properties of multi-scale rough surfaces[END_REF].

However, the pin profilometer is the very widely used due to cost reasons. Most pin profilometers are 1 or 2 m long with a sampling interval of 0.5, 1, or 2 cm. The measures are often taken in both directions, parallel and perpendicular to the row direction, in order to consider the directional effect of soil tilling (several profiles in each direction). The autocorrelation functions, calculated using different roughness profiles of a reference plot, are averaged, and the roughness parameters Hrms and L estimated (Baghdadi and Zribi, 2016). The standard deviation height (Hrms) expresses the vertical variation of the soil roughness.

The Hrms values depend on the agricultural operations and the rain or snow ground effects.

The parameter is defined as follows:

𝐻𝑟𝑚𝑠 2 = 〈(𝑧(𝑥) -〈𝑧〉) 2 〉 (2.5)
Where z (x) is the measured altitude on the x axis, and <z> is the mean height.

The Hrms parameter is not sufficient to characterize the soil surface. It does not take into account the relation which may exist between different surface points.

In order to take into account the relationship that may exist between the altitudes of two surface points separated by a distance u, we define the autocorrelation function (ρ(u)) of the surface and calculating the correlation length L:

𝜌(𝑢) = [ 〈{𝑧(𝑥 + 𝑢) -〈𝑧〉}. {𝑧(𝑥) -〈𝑧〉}〉 𝐻𝑟𝑚𝑠 2 ] (2.6)
When u = 0, the distance between the altitudes of two surface points is zero, the correlation between these two points is maximum and the autocorrelation function is one: ρ(u) = 1. When u increases, the points move away and become less correlated and the function ρ(u) decreases.

Finally, there is a distance for which the points are considered to be uncorrelated. This distance is the correlation length (L). It is defined as the distance (from the origin profile) in which the autocorrelation function equals e -1 (Figure II.4). When the roughness is low and the ground is smooth, the autocorrelation function has an exponential shape. Conversely, for high roughness, the autocorrelation function has a Gaussian shape [START_REF] Ulaby | Microwave remote sensing: Active and passive. Volume 2-Radar remote sensing and surface scattering and emission theory[END_REF] (Equation 2.7). [START_REF] Zribi | Développement de nouvelles méthodes de modélisation de la rugosité pour la rétrodiffusion hyperfréquence de la surface du sol[END_REF] introduced the fractal dimension to the description of the autocorrelation function's shape for bare soils in agricultural fields. For one-dimensional roughness profiles, the autocorrelation functions are defined as follows: Several studies (e.g. [START_REF] Lievens | Error in radar-derived soil moisture due to roughness parameterization: An analysis based on synthetical surface profiles[END_REF][START_REF] Oh | Condition for precise measurement of soil surface roughness[END_REF] showed that the precision of insitu measurements root mean surface height (Hrms) and correlation length (L) are very sensitive to the length and the horizontal resolution x of the roughness profiles [START_REF] Lievens | Error in radar-derived soil moisture due to roughness parameterization: An analysis based on synthetical surface profiles[END_REF][START_REF] Oh | Condition for precise measurement of soil surface roughness[END_REF]). An underestimation is observed of Hrms and L using small profiles. Moreover, the estimate correlation length (L) increase using roughness profiles with large sampling intervals. For agricultural soils, with about ten profiles measuring 1 m, Hrms can be measured with a precision of 10%, while the precision of the estimation of L is around 15 to 20% (a more important error for strong L values). Measuring L and Hrms with an error better than 5%, the sampling interval x must be smaller than 0.2L and 0.5L respectively [START_REF] Oh | Condition for precise measurement of soil surface roughness[END_REF]. The Hrms values are generally seen between 0.3 cm (very smooth fields that have just been sown) and 4 cm (fields that have just been plowed). The lengths of correlation L measured on the agricultural plots are predominantly between 3 and 25 cm.

fractal : e ² Gaussian : ² l exponentia : ² ) ( L x 2                          

Hrms

II.4 Radar signal sensitivity to soil parameters

II.4.1 Sensitivity of radar signal to soil roughness

Many studies (Aubert et al., 2011a;Baghdadi and Zribi, 2016;Baghdadi et al., 2008a;Gorrab et al., 2015a) showed that the backscattering radar signal for bare soil increases with the rms surface height (Hrms) according to the logarithmic or exponential law. Then after certain    L Hrms Hrms Zg  thresholds, the backscattering radar signal becomes constant (Figure II.5). This threshold after which the signal becomes constant depends on the wavelength and the radar's incidence angle. According to several studies, the radar signal rapidly saturates with the soils roughness (Hrms) when the wavelength and or the incidence angle are weak. This saturation of the radar signal occurs when kHrms is higher than 1 (where k is the radar wave number = 2π/λ) (Figure II.5). Moreover, this saturation corresponds to Hrms values of 4 cm in L-band, 1 cm in Cband and around 0.5 cm in X-band (Baghdadi and Zribi, 2016). (Baghdadi and Zribi, 2016). radar signal stabilizes and starting to decrease with the increasing of the soil moisture. So that, the estimation of soil moisture is difficult after 35 vol. % (e.g. [START_REF] Baghdadi | Operational mapping of soil moisture using synthetic aperture radar data: application to the Touch basin (France)[END_REF]Holah et al., 2005). (Baghdadi and Zribi, 2016).

Numerous studies (Aubert et al., 2011a;Baghdadi and Zribi, 2016;[START_REF] Baghdadi | Operational mapping of soil moisture using synthetic aperture radar data: application to the Touch basin (France)[END_REF][START_REF] Choker | Evaluation of the oh, dubois and iem backscatter models using a large dataset of sar data and experimental soil measurements[END_REF] show the radar signal's sensitivity to soil moisture as a function of different radar parameters (incidence angle, polarization, wavelength). Over bare soil, the optimal radar signal configuration to get the better sensitivity to soil moisture consists of Xband (in comparison to L and C bands), HH polarization and a low incidence angle [START_REF] Anguela | Analysis of local variation of soil surface parameters with TerraSAR-X radar data over bare agricultural fields[END_REF][START_REF] Aubert | Analysis of TerraSAR-X data sensitivity to bare soil moisture, roughness, composition and soil crust[END_REF]Baghdadi and Zribi, 2016;[START_REF] Beaudoin | SAR observations and modeling of the C-band backscatter variability due to multiscale geometry and soil moisture[END_REF][START_REF] Toan | Active microwave signatures of soil and crops-Significant results of three years of experiments[END_REF]; Soil moisture (vol. %) HH ; 50 to 52 [START_REF] Ulaby | Microwave remote sensing: Active and passive. Volume 2-Radar remote sensing and surface scattering and emission theory[END_REF][START_REF] Weisse | Assimilation of soil moisture into hydrological models for flood forecasting: comparison of a conceptual[END_REF]. These radar's incidence angles are ranging from 15° to 35° [START_REF] Beaudoin | SAR observations and modeling of the C-band backscatter variability due to multiscale geometry and soil moisture[END_REF][START_REF] Lievens | Error in radar-derived soil moisture due to roughness parameterization: An analysis based on synthetical surface profiles[END_REF][START_REF] Ulaby | Microwave remote sensing: Active and passive. Volume 2-Radar remote sensing and surface scattering and emission theory[END_REF].

The sensitivity of radar signal to soil moisture, in C-band it is approximately between 0.15 dB/vol.% and 0.3 dB/vol.% [START_REF] Baghdadi | Analysis of TerraSAR-X data and their sensitivity to soil surface parameters over bare agricultural fields[END_REF][START_REF] Hégarat-Mascle | Soil moisture estimation from ERS/SAR data: Toward an operational methodology[END_REF][START_REF] Quesney | Estimation of watershed soil moisture index from ERS/SAR data[END_REF][START_REF] Srivastava | Use of multiincidence angle RADARSAT-1 SAR data to incorporate the effect of surface roughness in soil moisture estimation[END_REF]. For the effect of wavelength, [START_REF] Aubert | Analysis of TerraSAR-X data sensitivity to bare soil moisture, roughness, composition and soil crust[END_REF] showed that the sensitivity of the radar signal toward the soil moisture is twice high in X-band than in C-band (about 0.35 dB/vol. % in X-band and 0.15 dB/Vol. % in C-band). [START_REF] Narvekar | Soil moisture retrieval using L-band radar observations[END_REF] showed that the sensitivity to soil moisture in L-band is approximately the same as in C-band. Moreover, the sensitivity for all frequency decreases as the incidence angle increases (Baghdadi and Zribi, 2016;Baghdadi et al., 2008a) (Figure II.6).

II.5 SAR data processing

Before processing the SAR images, the data are radiometrically calibrated, which allows the backscattering coefficient (°) to be extracted from the signal intensity of each pixel. This calibration enables to carry out multi-temporal analysis of different images (using either the same, or different sensors, but the same radar frequency, incidence angle and polarization).

The pixel-by-pixel interpretation of SAR images are extremely difficult because of the presence of speckle noise. It is due to the coherent interference of waves reflected from many elementary scatterers. Due to these reasons, soil surface characteristics are always estimated over homogeneous sectors including several pixels, or at the scale of single fields (which helps to reduce the speckle effect). The mean backscattering coefficients are calculated from calibrated SAR images, by averaging the linear intensity values of all pixels within the field (or sub-field). The reduction in speckle noise and the improvement in the quality of backscattering estimations are thus highly dependent on the size of homogeneous units used [START_REF] Joughin | Maximum likelihood estimation of K distribution parameters for SAR data[END_REF][START_REF] Lee | Speckle filtering of synthetic aperture radar images: A review[END_REF].

II.6 Radar backscattering modeling and evaluation

The radar backscattered models have been the subject of many studies based on theoretical or experimental research. In general, there are several classes of models: empirical, semiempirical and physical models. These models will be briefly discussed in chapter three.

II.6.1 Case of bare soil

II.6.1.1 Modeling of radar backscattering on bare soils

The modeling of the radar backscattered signal was developed in order to link and analyze the radar signal's sensitivity to the physical parameters of the soil surface (roughness and water content in particular) as a function of SAR configurations (mainly radar wavelength, incidence angle and polarization) (Baghdadi and Zribi, 2016;[START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF]Baghdadi et al., , 2006a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF]Baghdadi et al., , 2016a;;[START_REF] Beckmann | The scattering of electromagnetic waves from rough surfaces[END_REF][START_REF] Fung | Backscattering from a randomly rough dielectric surface[END_REF][START_REF] Rice | Reflection of electromagnetic waves from slightly rough surfaces[END_REF][START_REF] Ulaby | Microwave remote sensing: Active and passive. Volume 2-Radar remote sensing and surface scattering and emission theory[END_REF].

The empirical models require calibration using in situ measurements and SAR observations acquired [START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF](Baghdadi et al., , 2006a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF](Baghdadi et al., , 2016a;;[START_REF] Dubois | Measuring soil moisture with imaging radars[END_REF][START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF][START_REF] Oh | An empirical model and an inversion technique for radar scattering from bare soil surfaces[END_REF][START_REF] Oh | An inversion algorithm for retrieving soil moisture and surface roughness from polarimetric radar observation[END_REF][START_REF] Oh | Semi-empirical model of the ensembleaveraged differential Mueller matrix for microwave backscattering from bare soil surfaces[END_REF]. In addition, the range of validity of the empirical models is limited to the range of variations in the data used for model calibration.

In addition, the physical models are based on laws of the resolution of Maxwell's equations, with physical approximations limiting their areas of validity. The disadvantages of these models are the complexity of implementations and require many parameters in simulations.

The development of these models have been the goal of several studies such as [START_REF] Chen | Emission of rough surfaces calculated by the integral equation method with comparison to three-dimensional moment method simulations[END_REF][START_REF] Fung | Microwave Scattering and Emission Models and their Applications[END_REF][START_REF] Fung | Backscattering from a randomly rough dielectric surface[END_REF][START_REF] Ulaby | Microwave remote sensing: Active and passive. Volume 2-Radar remote sensing and surface scattering and emission theory[END_REF]. In the IEM model [START_REF] Fung | Backscattering from a randomly rough dielectric surface[END_REF], the discrepancy between SAR simulations and SAR measurements is mainly related to the description of surface roughness which is an important input to SAR backscattering models [START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Mattia | A comparison between soil roughness statistics used in surface scattering models derived from mechanical and laser profilers[END_REF][START_REF] Verhoest | On the soil roughness parameterization problem in soil moisture retrieval of bare surfaces from synthetic aperture radar[END_REF]. The surface roughness is described by three parameters: the standard deviation of the height (Hrms), the correlation length (L) and the shape of the correlation function [START_REF] Fung | Backscattering from a randomly rough dielectric surface[END_REF]. The correlation length is usually measured with an uncertainty which introduces an error on simulated backscattering coefficients [START_REF] Baghdadi | Relationship between profile length and roughness variables for natural surfaces[END_REF][START_REF] Davidson | On the characterization of agricultural soil roughness for radar remote sensing studies[END_REF][START_REF] Le Toan | Improved observation and modelling of bare soil surfaces for soil moisture retrieval[END_REF][START_REF] Lievens | Effective roughness modelling as a tool for soil moisture retrieval from C-and L-band SAR[END_REF]. A few studies proposed a semi-empirical calibration of SAR backscattering models in order to reduce the uncertainty on SAR simulations [START_REF] Baghdadi | An empirical calibration of the integral equation model based on SAR data, soil moisture and surface roughness measurement over bare soils[END_REF][START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF](Baghdadi et al., , 2006a(Baghdadi et al., , 2011a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF][START_REF] Rahman | A derivation of roughness correlation length for parameterizing radar backscatter models[END_REF] . In [START_REF] Baghdadi | An empirical calibration of the integral equation model based on SAR data, soil moisture and surface roughness measurement over bare soils[END_REF][START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF]Baghdadi et al. ( , 2006Baghdadi et al. ( , 2011a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF] the method consisted of replacing the measured L by a fitting parameter, so-called Lopt, which was found to be related to Hrms (Lopt increases with Hrms). Lopt is a function of Hrms (linear, exponential, or power calibration) which depends on SAR parameters (incidence angle, polarization and frequency). This calibration reduces IEM's input soil parameters (Hrms and mv instead of Hrms, L and mv). [START_REF] Rahman | A derivation of roughness correlation length for parameterizing radar backscatter models[END_REF] proposed a method for deriving L through the IEM. In this method, the radar signal is modeled as a function of only Hrms and L, and the contribution of soil moisture on backscattering coefficients is ignored (dry soil). Thus, L could be estimated by inverting the IEM.

II.6.1.2 Estimation of soil parameters using radar backscattering on bare soils For bare soil, many studies have shown the potential of radar data to retrieve soil parameters (moisture and roughness) [START_REF] Aubert | Analysis of TerraSAR-X data sensitivity to bare soil moisture, roughness, composition and soil crust[END_REF]Baghdadi and Zribi, 2006;Baghdadi et al., 2002a[START_REF] Baghdadi | Operational mapping of soil moisture using synthetic aperture radar data: application to the Touch basin (France)[END_REF]Baghdadi et al., , 2008aBaghdadi et al., , 2012a;;[START_REF] Hégarat-Mascle | Soil moisture estimation from ERS/SAR data: Toward an operational methodology[END_REF][START_REF] Zribi | Evaluation of a rough soil surface description with ASAR-ENVISAT radar data[END_REF]. The SAR signal increases with increasing soil moisture for values between 0 and 35% (Aubert et al., 2011a;[START_REF] Baghdadi | Operational mapping of soil moisture using synthetic aperture radar data: application to the Touch basin (France)[END_REF]Gorrab et al., 2015a;[START_REF] Holah | Potential of ASAR/ENVISAT for the characterization of soil surface parameters over bare agricultural fields[END_REF]. Beyond this threshold, the backscattering coefficients tend to saturate and then decreases with increasing soil moisture [START_REF] Holah | Potential of ASAR/ENVISAT for the characterization of soil surface parameters over bare agricultural fields[END_REF]. Most bare soil moisture estimation studies have used SAR data in X and C bands and the results show a precision on the estimation of soil moisture between 3 and 6 vol.% [START_REF] Aubert | Analysis of TerraSAR-X data sensitivity to bare soil moisture, roughness, composition and soil crust[END_REF][START_REF] Baghdadi | Estimation of soil parameters over bare agriculture areas from C-band polarimetric SAR data using neural networks[END_REF][START_REF] Baghdadi | Coupling SAR C-band and optical data for soil moisture and leaf area index retrieval over irrigated grasslands[END_REF][START_REF] El Hajj | Irrigated grassland monitoring using a time series of terraSAR-X and COSMO-skyMed X-Band SAR Data[END_REF][START_REF] Paloscia | Soil moisture mapping using Sentinel-1 images: Algorithm and preliminary validation[END_REF][START_REF] Srivastava | Use of multiincidence angle RADARSAT-1 SAR data to incorporate the effect of surface roughness in soil moisture estimation[END_REF][START_REF] Srivastava | Large-area soil moisture estimation using multi-incidence-angle RADARSAT-1 SAR data[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF]. Moreover, in C-band, the accuracy of the soil moisture estimates depends on the effect of surface roughness and of the sensor incidence angle. On the other hand, in X-band, the effect of roughness on the accuracy of the soil moisture estimation is negligible and the quality of estimates is slightly better with low incidence angle (RMSE < 1 vol.%) (Aubert et al., 2011a[START_REF] Aubert | Toward an operational bare soil moisture mapping using TerraSAR-X data acquired over agricultural areas[END_REF][START_REF] Galarneau | Utilization of Radarsat in integrated catchment management[END_REF][START_REF] Hégarat-Mascle | Estimation of Watershed Soil Moisture Index from ERS/SAR Data[END_REF][START_REF] Quesney | Estimation of watershed soil moisture index from ERS/SAR data[END_REF]. Thus, the accuracy of the moisture estimates in X-band is twice as well as that obtained in C-band data (3 vol.% in the X-band compared with 6 vol.% in the C-band) (Baghdadi and Zribi, 2016). Baghdadi et al. (2002a) analyzed the potential of synthetic aperture radar (SAR) data for monitoring roughness states over bare agricultural fields using one ERS image (23°) and two RADARSAT images (39° and 47°). The relationships between the backscattering coefficient, incidence angle, soil surface roughness and row direction have been examined in order to determine the best SAR configuration for such monitoring. The result showed a strong dependence of incidence angle on the discrimination between radar return over areas of different surface roughness. The influence of soil roughness on radar return is more sensitive at a high incidence angle (47°), over the influence of other soil parameters, making it possible to differ and map various surface roughness classes (smooth, medium and rough) over agricultural fields.

In addition, Baghdadi et al. (2012a) developed an approach to estimate soil surface parameters from C-band polarimetric SAR data in the case of bare agricultural soils. An inversion technique based on multi-layer perceptron (MLP) neural networks was introduced. The neural networks were trained and validated on a noisy simulated dataset generated from the Integral Equation Model (IEM) on a wide range of surface roughness and soil moisture. The performances of neural networks in retrieving soil moisture and surface roughness over bare soils were tested using or not a priori knowledge on soil moisture. The inversion approach was then validated using RADARSAT-2 images in polarimetric mode. The polarimetric parameter α1 (alpha angle that corresponds to the first eigenvector of coherency matrix) was used to discriminate two soil moisture classes (very wet soils, and dry to wet soils) and the anisotropy parameter A to separate two soil roughness's (smooth with kHrms<1.0 and rough with kHrms≥1.0). The inversion errors obtained with the RADARSAT-2 images on the mv estimates is about 6.5 vol.% with a priori information on mv compared with 9.8 vol.% without a priori information on mv. The use of polarimetric parameters slightly improves the soil moisture estimates in comparison to the case without a priori information on mv (8.3 vol.% as compared to 9.8 vol.%). This is due to the weak dynamics of the polarimetric parameters (alpha angle, entropy, anisotropy) with the soil parameters for the C-band. Results show also that the estimation of soil surface roughness (Hrms) is possible with accuracy around 0.5 cm (RMSE). The estimation is better for Hrms lower than 2 cm. For higher Hrms, the NNs underestimate the surface roughness.

II.6.2 Case of soil with vegetation cover

The presence of vegetation makes the inversion of the radar signal much more complicated because the vegetation not only attenuates the backscattered signal but also produces its own contribution. Thus, in the presence of vegetation, the total backscattered radar signal is the result of contributions from soil attenuated by the vegetation and the vegetation contribution.

The possibility of estimating soil moisture in the presence of vegetation from SAR images has been widely studied using C-band radar data [START_REF] De Roo | A semi-empirical backscattering model at L-band and C-band for a soybean canopy with soil moisture inversion[END_REF][START_REF] Gherboudj | Soil moisture retrieval over agricultural fields from multi-polarized and multi-angular RADARSAT-2 SAR data[END_REF][START_REF] He | A synergistic methodology for soil moisture estimation in an Alpine prairie using radar and optical satellite data[END_REF][START_REF] Notarnicola | Use of radar and optical remotely sensed data for soil moisture retrieval over vegetated areas[END_REF][START_REF] Prevot | Estimating surface soil moisture and leaf area index of a wheat canopy using a dual-frequency (C and X bands) scatterometer[END_REF][START_REF] Sikdar | A modified empirical model for soil moisture estimation in vegetated areas using SAR data[END_REF][START_REF] Wang | Estimation of surface soil moisture and roughness from multi-angular ASAR imagery in the Watershed Allied Telemetry Experimental Research (WATER)[END_REF][START_REF] Yu | A new semi-empirical model for soil moisture content retrieval by ASAR and TM data in vegetation-covered areas[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF]. Conversely, very few studies have been conducted using X-band radar data [START_REF] El Hajj | Irrigated grassland monitoring using a time series of terraSAR-X and COSMO-skyMed X-Band SAR Data[END_REF][START_REF] Kseneman | Soil-moisture estimation from TerraSAR-X data using neural networks[END_REF][START_REF] Kweon | COSMO SkyMed AO projects-soil moisture detection for vegetation fields based on a modified water-cloud model using COSMO-SkyMed SAR data[END_REF] or L-band [START_REF] Paloscia | Combining L and X band SAR data for estimating biomass and soil moisture of agricultural fields[END_REF][START_REF] Wigneron | A simple algorithm to retrieve soil moisture and vegetation biomass using passive microwave measurements over crop fields[END_REF].

The most used models to estimate soil moisture in the presence of vegetation are the Michigan Microwave Canopy Scattering model ʺMIMICSʺ [START_REF] Ulaby | Michigan microwave canopy scattering model[END_REF] and the Water Cloud Model ʺWCMʺ [START_REF] Attema | Vegetation modeled as a water cloud[END_REF]. MIMICS is based on the theoretical principle of the first order radiative transfer model to simulate radar backscatter from soil and vegetation parameters. In MIMICS vegetation is described in detail (stem diameter and leaf slope, among others), which makes it possible to better quantify the contribution of vegetation to the total backscattered radar signal and thus a better estimation of soil moisture. However, MIMICS is complicated to use, and requires a large number of input parameters. For this reason, most studies have used the WC model because it is relatively simple and requires only few input variables [START_REF] De Roo | A semi-empirical backscattering model at L-band and C-band for a soybean canopy with soil moisture inversion[END_REF][START_REF] Gherboudj | Soil moisture retrieval over agricultural fields from multi-polarized and multi-angular RADARSAT-2 SAR data[END_REF][START_REF] Kweon | COSMO SkyMed AO projects-soil moisture detection for vegetation fields based on a modified water-cloud model using COSMO-SkyMed SAR data[END_REF][START_REF] Prevot | Estimating surface soil moisture and leaf area index of a wheat canopy using a dual-frequency (C and X bands) scatterometer[END_REF][START_REF] Sikdar | A modified empirical model for soil moisture estimation in vegetated areas using SAR data[END_REF][START_REF] Wang | Estimation of surface soil moisture and roughness from multi-angular ASAR imagery in the Watershed Allied Telemetry Experimental Research (WATER)[END_REF][START_REF] Yang | Estimation of soil moisture from multipolarized SAR data over wheat coverage areas[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF].

In the WCM, the total backscattered radar signal is modeled as the sum of soil attenuated by vegetation and vegetation contribution. The contribution of vegetation (direct backscatter and attenuation) is calculated using mainly a single biophysical parameter (leaf area index, vegetation water content, NDVI, biomass, or vegetation height) representing the effect of vegetation. The soil contribution is calculated as a function of soil moisture and roughness using a radar backscattering model. Optical data are often required to reverse the radar signal using the WCM. Indeed, the optical data are complementary to the radar data, and their interest lies in their sensitivity to the physical vegetation parameters. In the WCM, the biophysical parameters allow to evaluate the vegetation contribution to the radar signal, and to then reverse the soil contribution in order to estimate soil moisture [START_REF] Baghdadi | Coupling SAR C-band and optical data for soil moisture and leaf area index retrieval over irrigated grasslands[END_REF][START_REF] El Hajj | Irrigated grassland monitoring using a time series of terraSAR-X and COSMO-skyMed X-Band SAR Data[END_REF][START_REF] Fieuzal | Combined use of optical and radar satellite data for the monitoring of irrigation and soil moisture of wheat crops[END_REF][START_REF] He | A synergistic methodology for soil moisture estimation in an Alpine prairie using radar and optical satellite data[END_REF][START_REF] Hosseini | Soil moisture estimation based on integration of optical and SAR images[END_REF][START_REF] Notarnicola | Use of radar and optical remotely sensed data for soil moisture retrieval over vegetated areas[END_REF][START_REF] Prakash | A fusion approach to retrieve soil moisture with SAR and optical data[END_REF].

The Water Cloud model (WCM) defines the backscattered radar signal in a linear scale (σ 0 tot) which is the sum of the contribution from the vegetation (σ 0 veg), the soil (σ 0 soil) attenuated by the vegetation (T 2 σ 0 soil), and multiple soil-vegetation scatterings that are often neglected: σ 0 tot= σ 0 veg + T 2 σ 0 soil (2.9) The soil contribution σ 0 soil that depends on the soil moisture and surface roughness with SAR instrumental parameters can be simulated using physical backscattering model (IEM) or empirical models (Oh, Dubois, Baghdadi) [START_REF] Baghdadi | Coupling SAR C-band and optical data for soil moisture and leaf area index retrieval over irrigated grasslands[END_REF][START_REF] Dubois | Measuring soil moisture with imaging radars[END_REF][START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF][START_REF] Oh | An empirical model and an inversion technique for radar scattering from bare soil surfaces[END_REF][START_REF] Oh | An inversion algorithm for retrieving soil moisture and surface roughness from polarimetric radar observation[END_REF][START_REF] Oh | Semi-empirical model of the ensembleaveraged differential Mueller matrix for microwave backscattering from bare soil surfaces[END_REF].

σ 0 veg = A.V1.cos θ (1-T 2 ) (2.
Several studies have used the WCM to estimate soil moisture in presence of vegetation. The results showed a soil moisture precision between 2 and 8 vol.% [START_REF] De Roo | A semi-empirical backscattering model at L-band and C-band for a soybean canopy with soil moisture inversion[END_REF][START_REF] El Hajj | Irrigated grassland monitoring using a time series of terraSAR-X and COSMO-skyMed X-Band SAR Data[END_REF][START_REF] Gherboudj | Soil moisture retrieval over agricultural fields from multi-polarized and multi-angular RADARSAT-2 SAR data[END_REF][START_REF] He | A synergistic methodology for soil moisture estimation in an Alpine prairie using radar and optical satellite data[END_REF][START_REF] Notarnicola | Use of radar and optical remotely sensed data for soil moisture retrieval over vegetated areas[END_REF][START_REF] Prevot | Estimating surface soil moisture and leaf area index of a wheat canopy using a dual-frequency (C and X bands) scatterometer[END_REF][START_REF] Sikdar | A modified empirical model for soil moisture estimation in vegetated areas using SAR data[END_REF][START_REF] Wang | Estimation of surface soil moisture and roughness from multi-angular ASAR imagery in the Watershed Allied Telemetry Experimental Research (WATER)[END_REF][START_REF] Yang | Estimation of soil moisture from multipolarized SAR data over wheat coverage areas[END_REF][START_REF] Yu | A new semi-empirical model for soil moisture content retrieval by ASAR and TM data in vegetation-covered areas[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF]. [START_REF] Prevot | Estimating surface soil moisture and leaf area index of a wheat canopy using a dual-frequency (C and X bands) scatterometer[END_REF] combined radar data in C and X bands to estimate the soil moisture of wheat plots (LAI of wheat between 0.1 and 8 m 2 /m 2 ). The accuracy on the soil moisture estimation was 6.5 vol. %. Similar precision was obtained by [START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF] using ASAR (C band) images acquired on wheat plots (LAI between 0.01 and 3.7 m 2 /m 2 ). [START_REF] Kweon | COSMO SkyMed AO projects-soil moisture detection for vegetation fields based on a modified water-cloud model using COSMO-SkyMed SAR data[END_REF] estimated soil moisture from soybean plots using X-band SAR data with an accuracy of 3 vol. % (VWC "water content" and LAI "leaf index" at 1.8 kg/m 2 and 4.5 m 2 /m 2 , respectively). [START_REF] He | A synergistic methodology for soil moisture estimation in an Alpine prairie using radar and optical satellite data[END_REF] combined radar data (C-HH and C-VV) and optical data to estimate soil moisture. The results showed an accuracy of about 3 vol.%. [START_REF] Gherboudj | Soil moisture retrieval over agricultural fields from multi-polarized and multi-angular RADARSAT-2 SAR data[END_REF] combined "WC" and "Oh" models to estimate soil moisture with vegetation (wheat, peas, lentils, fallows, pasture and canola) from C-band radar data. Soil moisture was estimated with an accuracy of 6 vol.% for vegetation heights between 11 and 97 cm. Finally, De Roo et al. (2001) used the MIMICS model in conjunction with the "Oh model" to estimate soil moisture in soybean plots (with VWC between 0.02 and 0.97 kg/m 2 ) from the C and L polarimetric data. Accuracy of soil moisture estimates was about 2 vol.%.

II.7 Conclusion

Through the diversity of the radar backscattering models over bare soils, my first work is to evaluate the most commonly backscatter models using a wide dataset of SAR data and in situ measurements acquired over numerous agricultural sites in France, Italy, Germany, Belgium, Luxembourg, Canada and Tunisia. Thus, this study could be of a great importance for scientific community since it helps understanding backscatter models performance for a wide range of soil surface conditions, acquired for several study areas through the world by numerous SAR sensors. Never before have all these backscatter models been evaluated together in the same literature with such a wide dataset. This step is very important to find the model that produces good agreement between the radar data and the simulations in order to be used in the inverse mode later in this thesis.

III. Chapter 3: Evaluation of radar backscattering models

III.1 Introduction

The aim of this chapter is to evaluate the most popular backscattering SAR models (Oh, Dubois, IEM, IEM_B, and AIEM) by using a wide range of SAR data and in situ measurements. With the arrival of Sentinel-1A and -1B satellites that provide free high resolution SAR data with 6 days revisit time, several research teams work actually on developing methods for mapping soil moisture using these Sentinel-1 data. Most of methods for soil moisture mapping are based on backscatter models for soil moisture estimations. The objective of this part is to evaluate the most commonly backscatter models using a wide dataset of SAR data and in situ measurements acquired over numerous agricultural sites in France, Italy, Germany, Belgium, Luxembourg, Canada and Tunisia. Thus, this study could be of a great importance for scientific community since it help on understand backscatter models performance for wide range of soil surface conditions, acquired for several study areas through the world by numerous SAR sensors. Never before have been evaluated all these backscatter models together in the same literature with such a wide dataset. In addition, this study is the first that evaluates the backscatter models using L-, C-and X-bands together. A description of the study areas and different datasets used in this study is provided in Section 2.

Section 3 the models are described. The results are shown in Section 4. Finally, Section 5 presents the conclusion.

III.2 Dataset

III.2.1 Study Areas

A wide range of datasets composed of AIRSAR, SIR-C, JERS-1, PALSAR-1, ESAR, ERS, RADARSAT, ASAR and TerraSAR-X acquisitions over numerous agricultural sites in France, Italy, Germany, Belgium, Luxembourg, Canada and Tunisia (Table III.1), have been used in this research work. In addition, in situ measurements of soil moisture and surface roughness were carried out simultaneously to SAR acquisitions over bare soil surfaces.

III.2.2 Satellite Data

A large number of L-, C-and X-band images (approximately 1.25 GHz,5.3 GHz and 9.6 GHz,respectively) were acquired between 1994 and 2014 with different incidence angles (between 18° and 57°) and in HH, VV and HV polarizations (Table III.1). The spatial resolution of SAR images is between 1 m and 30 m (Table III.1). Images were first radiometrically calibrated to enable the extraction of the backscattering coefficients ( 0  ).

Then, the mean backscattering coefficients were computed from calibrated SAR images by linearly averaging the 0  values of all pixels within the plot.

III.2.3 Field Data

Field measurements of soil moisture and surface roughness have been collected from bare plots selected over the study areas. Each plot is a homogeneous surface (similar soil type, moisture content and surface roughness) of around one hectare or more. In situ measurements of soil moisture (mv, in vol. %) were carried out for a soil layer of 5 cm or 10 cm in each reference plot by using both the gravimetric method or a calibrated TDR (time domain reflectometry) probe. For each bare soil reference field the average soil moisture (mv) of all samples was calculated. The soil moisture ranged between 2 vol. % and 47 vol. %.

Roughness measurements were carried out by using laser or needle profilometers (mainly 1 m and 2 m long, and with 1 cm and 2 cm sampling intervals); while for some in situ measurement campaigns, a meshboard technique was used. Several roughness profiles along and across the direction of tillage were acquired in each reference field. The standard deviation of surface heights (Hrms) and the correlation length (L) were calculated by using the mean of all experimental correlation functions. In our dataset, Hrms ranged from 0.2 cm to 9.6 cm and the L from 1.2 cm to 38.5 cm. The reference plots in the datasets were chosen with low density of stones.

A total of 2442 experimental data of soil moisture content and surface roughness were available, together with the corresponding values of backscattering coefficient, of which 1262 at HH polarization, 790 at VV polarization, and 390 at HV polarization (Table III.1).

Site SAR Sensor Spatial Resolution

Freq Year

Number of Data

Orgeval (Fr) [START_REF] Zribi | Backscattering behavior and simulation comparison over bare soils using SIR-C/X-SAR and ERASME 1994 data over Orgeval[END_REF] (Baghdadi et al., 2008a[START_REF] Baghdadi | Sensitivity of main polarimetric parameters of multifrequency polarimetric SAR data to soil moisture and surface roughness over bare agricultural soils[END_REF][START_REF] Zribi | Backscattering behavior and simulation comparison over bare soils using SIR-C/X-SAR and ERASME 1994 data over Orgeval[END_REF] Villamblain (Fr) (Baghdadi et al., 2006a;[START_REF] Holah | Potential of ASAR/ENVISAT for the characterization of soil surface parameters over bare agricultural fields[END_REF]) Villamblain (Fr) (Baghdadi et al., 2011c[START_REF] Baghdadi | Sensitivity of main polarimetric parameters of multifrequency polarimetric SAR data to soil moisture and surface roughness over bare agricultural soils[END_REF]) Touch (Fr) [START_REF] Baghdadi | Operational mapping of soil moisture using synthetic aperture radar data: application to the Touch basin (France)[END_REF][START_REF] Holah | Potential of ASAR/ENVISAT for the characterization of soil surface parameters over bare agricultural fields[END_REF] ERS-2; ASAR 30 m × 30 m C 2004; 2006; 2007 Mauzac (Fr) (Baghdadi et al., 2011c)

ASAR

TerraSAR-X 1 m × 1 m X 2009
Garons (Fr) (Baghdadi et al., 2011c)

TerraSAR-X 1 m × 1 m X 2009
Kairouan (Tu) [START_REF] Zribi | Influence of radar frequency on the relationship between bare surface soil moisture vertical profile and radar backscatter[END_REF] Kairouan (Tu) (Baghdadi et al., 2011c;[START_REF] Gorrab | Retrieval of both soil moisture and texture using TerraSAR-X images[END_REF][START_REF] Zribi | Influence of radar frequency on the relationship between bare surface soil moisture vertical profile and radar backscatter[END_REF]) Yzerons (Fr) [START_REF] Aubert | Toward an operational bare soil moisture mapping using TerraSAR-X data acquired over agricultural areas[END_REF] TerraSAR-X 1 m × 1 m X 2009

ASAR

Versailles (Fr) (Baghdadi et al., 2011c) TerraSAR-X 1 m × 1 m X 2010

Seysses (Fr) (Baghdadi et al., TerraSAR-X 1 m × 1 m X 2010

Table III.1. Description of the dataset used in this study. "Fr": France, "It": Italy, "Ge": Germany, "Be": Belgium, "Lu": Luxembourg, "Ca": Canada, "Tu": Tunisia. The radiometric accuracy of SAR data is about 1 dB.

III.2.4 Soil texture

Twenty one agricultural study sites were used. The texture compositions (Silt; Clay; Sand) are described below (Table III.2).

2011c) Chateauguay (Ca) [START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF] RADARSAT 30 m × 30 m C 1999 Brochet (Ca) [START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF] RADARSAT 30 m × 30 m C 1999 Alpilles (Fr) [START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF] Montespertoli (It) [START_REF] Baronti | SAR polarimetric features of agricultural areas[END_REF][START_REF] Panciera | Evaluation of IEM, Dubois, and Oh radar backscatter models using airborne L-band SAR[END_REF] Montespertoli (It) [START_REF] Macelloni | The SIR-C/X-SAR experiment on Montespertoli: sensitivity to hydrological parameters[END_REF] Montespertoli (It) [START_REF] Paloscia | The potential of C-and L-band SAR in estimating vegetation biomass: the ERS-1 and JERS-1 experiments[END_REF]

                Hrms k HH (3.1)
where θ is expressed in radians and λ in cm, and 0 HH  and 0 VV  are expressed in linear units.

III.3.2 The Semi-Empirical Oh Model

Oh ( 2004) and Oh et al. (1992b[START_REF] Oh | An inversion algorithm for retrieving soil moisture and surface roughness from polarimetric radar observation[END_REF][START_REF] Oh | Semi-empirical model of the ensembleaveraged differential Mueller matrix for microwave backscattering from bare soil surfaces[END_REF] The initial version of the Oh model [START_REF] Oh | An empirical model and an inversion technique for radar scattering from bare soil surfaces[END_REF] is defined as:

0 2 1 0 3Γ 0 1 . 90 k Hrms HH VV p e                      (3.2)   0 0 0 0.23 Γ 1 k Hrms HV VV q e       (3.3)
where:

      2 0 1 1 r r
(3.4) [START_REF] Oh | An inversion algorithm for retrieving soil moisture and surface roughness from polarimetric radar observation[END_REF] proposed a new expression for q to incorporate the effect of the incidence angle:

   0 0 1.4 1.6Γ 0.9 0 0 0.25 Γ 0.1 sin θ 1 k Hrms HV VV q e          
(3.5) [START_REF] Oh | Semi-empirical model of the ensembleaveraged differential Mueller matrix for microwave backscattering from bare soil surfaces[END_REF] again modified the expressions for p and q, and the following expression for the cross-polarized backscatter coefficient was proposed:

0.65 1.4 0.35 0 0.4( ) 0 σ θ 1 . σ 90 mv k Hrms HH VV p e            (3.6)   0.8 1.2 0 0.9 ( ) 0 σ 0.1 sin1.3θ 1 σ k Hrms HV VV Hrms q e L            (3.7)   1.8 0 0.7 2.2 0.32 ( ) σ 0.11 cos θ 1 k Hrms HV mv e    (3.8)
Oh and Kay (1998) demonstrated that the measurement of the correlation length is not accurate and that the ratio q is not sensitive to the roughness parameter (defined as Hrms/L).

Thus, [START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF] proposed a new equation for q that ignores the correlation length (L):

    0.9 0 1.4 1.3( ) 0 σ 0.095 0.13 sin1.5θ 1 σ k Hrms HV VV q e      (3.9)
The Oh model [START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF]  in the inversion process of SAR data, while the ratio p/q, as well as 0 HV  , is used in the case where SAR data are available in the both HH and HV polarizations.

III.3.3 The Physical Integral Equation Model (IEM)

The Integral Equation IEM is a physical model [START_REF] Fung | Microwave Scattering and Emission Models and their Applications[END_REF], where the soil is characterized by the dielectric constant ( r  ), the standard deviation of surface height (Hrms), the form of the correlation function, and the correlation length (L). The IEM also takes into account the sensor parameters such as the incidence angle (θ), the polarization (pq with p,q = H or V), and the radar wave number (k = 2π/λ where λ is the wavelength). The IEM has a validity domain that covers the range of roughness values that are commonly encountered for agricultural surfaces:

kHrms ≤ 3       2 ( cos θ) 0.46 exp 0.92 1 sin θ 0.25 k Hrms k L k L    (3.10)
Over bare soils in agricultural areas, the backscattering coefficient of the surface contribution is expressed at HH and VV polarizations as:

2 2 2 2 2 2 2 2 2 2 2 2 2 0 4 cos θ ( ) 1 2 2 2 * 3 cos θ ( ) 1 2 2 2 2 2 cos θ ( ) 1 ² (4 cos θ) (2 sin θ, 0) 2 ! ² (4 cos θ) Re( ) (2 sin θ, 0) 2 ! ² ( cos θ) (2 sin θ, 0) 8 ! n k Hrms n pp pp n n k Hrms n pp pp n n k Hrms n pp n k k Hrms f e W k n k k Hrms f F e W k n k k Hrms F e W k n                 (3.11)
At cross polarization, the backscattering coefficient is as follows:

2 2 2 2 2 2 0 2 cos θ 1 1 2 * ( ) ( ) ² ( cos θ) 16 ! ! ( , ) ( , ) ( , ) ( sin θ, ) ( sin θ,) n m k Hrms hv n m n m hv hv hv k k Hrms e nm F u v F u v F u v W u k v W u k v du dv                     (3.12)
where:

2 cos θ h hh R f   ; 2 cos θ v vv R f  (3.13) 2 2 cosθ sin θ : Fresnel coefficient at horizontal polarization cosθ sin θ r r r h r r r R            (3.14) 2 2 cosθ sin θ : Fresnel coefficient at vertical polarization cosθ sin θ r r r v r r r R            (3.15)   2 2 sin θ 1 2 4 1 1 cos θ hh h h r F R R                    (3.16)     2 2 2 2 2 cos θ sin θ 1 2 1 1 1 1 cos θ sin θ r vv v v r r r F R R                               (3.17) 2 2 2 2 2 2 2 2 2 2 (1 ) 2 6 (1 ) 8 ( , ) cos θ r r hv r R R R uv R F u v k k u v k u v                            (3.18) 2 h v R R R   (3.19) r  : dielectric constant,
obtained on the basis of volumetric water content (mv). In our study, Hallikainen empirical model is used [START_REF] Hallikainen | Microwave dielectric behavior of wet soil-part 1: empirical models and experimental observations[END_REF].

dy dx e y x b a W by ax i n n     ) ( ) ( ) , ( 2

III.3.4 IEM Modified by Baghdadi (IEM_B)

Several studies reported important discrepancies between backscattering coefficients simulated by IEM and those measured by SAR sensors (Baghdadi et al., 2002a;[START_REF] Boisvert | Effect of surface soil moisture gradients on modelling radar backscattering from bare fields[END_REF][START_REF] Gorrab | Retrieval of both soil moisture and texture using TerraSAR-X images[END_REF][START_REF] Panciera | Evaluation of IEM, Dubois, and Oh radar backscatter models using airborne L-band SAR[END_REF][START_REF] Rakotoarivony | Radar backscattering over agricultural bare soils[END_REF][START_REF] Remond | Image SAR: potentialités d'extraction d'un paramètre physique du ruissellement, la rugosité (modélisation et expérimentation)[END_REF][START_REF] Zribi | Backscattering behavior and simulation comparison over bare soils using SIR-C/X-SAR and ERASME 1994 data over Orgeval[END_REF]. [START_REF] Baghdadi | An empirical calibration of the integral equation model based on SAR data, soil moisture and surface roughness measurement over bare soils[END_REF] showed that the discrepancy between the observed and IEM simulated backscattering coefficients is mainly due to the correlation length parameter which is difficult to measure with a good accuracy. To reduce such incongruities between simulated and measured backscattering values, Baghdadi et al. (2006Baghdadi et al. ( , 2011aBaghdadi et al. ( , 2011c[START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF] proposed a semi-empirical calibration of the IEM backscattering, which consists of replacing the in situ measured correlation length by a fitting parameter (Lopt). Lopt depends on surface roughness conditions and SAR configurations (incidence angle, polarization and radar wavelength). This calibration has been performed by using large experimental datasets and SAR configurations (incidence angles from 23° to 57°, and HH, HV, and VV polarizations), and it has been carried separately at X-band in (Baghdadi et al., 2011c), C-band in (Baghdadi et al., 2006a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF] and L-band in [START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF]. The proposed calibration reduces the IEM's input soil parameters from three to two (Hrms and mv only, instead of Hrms, L and mv).

Lopt is computed at L-, C-, and X-bands using a Gaussian correlation function and it is described as follows:

In X-band: { 𝐿𝑜𝑝𝑡(𝐻𝑟𝑚𝑠, θ, 𝐻𝐻) = 18.102𝑒 -1.891θ 𝐻𝑟𝑚𝑠 0.7644𝑒 0.2005 θ 𝐿𝑜𝑝𝑡(𝐻𝑟𝑚𝑠, θ, 𝑉𝑉) = 18.075𝑒 -2.1715θ 𝐻𝑟𝑚𝑠 1.2594𝑒 -0.8308θ (3.21) In C-band: { 𝐿𝑜𝑝𝑡(𝐻𝑟𝑚𝑠, θ, 𝐻𝐻) = 0.162 + 3.006 (sin 1.23θ) -1.494 𝐻𝑟𝑚𝑠 𝐿𝑜𝑝𝑡(𝐻𝑟𝑚𝑠, θ, 𝐻𝑉) = 0.9157 + 1.2289 (sin 0.1543 θ) -0.3139 𝐻𝑟𝑚𝑠 𝐿𝑜𝑝𝑡(𝐻𝑟𝑚𝑠, θ, 𝑉𝑉) = 1.281 + 0.134 (sin 0.19 θ) -1.59 𝐻𝑟𝑚𝑠 (3.22) In L-band: { 𝐿𝑜𝑝 (𝐻𝑟𝑚𝑠, θ, 𝐻𝐻) = 2.6590 θ -1.4493 + 3.0484 𝐻𝑟𝑚𝑠 θ -0.8044 𝐿𝑜𝑝𝑡(𝐻𝑟𝑚𝑠, θ, 𝑉𝑉) = 5.8735 θ -1.0814 + 1.3015 𝐻𝑟𝑚𝑠 θ -1.4498 (3.23)
where θ is in radians; Lopt and Hrms are in centimeters. Several studies showed that the use of the fitting parameter Lopt allows more correct estimations of the radar backscattering coefficient [START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF][START_REF] Dong | Validation of the AIEM through correlation length parameterization at field scale using radar imagery in a semi-arid environment[END_REF]Gorrab et al., 2015a).

III.3.5 The Advanced Integral Equation Model

The Advanced Integral Equation Model (AIEM) [START_REF] Chen | Emission of rough surfaces calculated by the integral equation method with comparison to three-dimensional moment method simulations[END_REF] is the updated version of the Integral Equation Model (IEM) [START_REF] Fung | Microwave Scattering and Emission Models and their Applications[END_REF]. In a comparison with the IEM, two improvements have been integrated into the AIEM: (1) the complete expressions for the Kirchhoff field coefficient and the complementary field coefficient based on the removal of the simplification assumption of the Green's function have been included in the AIEM [START_REF] Chen | Emission of rough surfaces calculated by the integral equation method with comparison to three-dimensional moment method simulations[END_REF] and (2) a continuous Fresnel reflection coefficient is obtained using a transition model [START_REF] Wu | A transition model for the reflection coefficient in surface scattering[END_REF]. This update allows a more precise calculation of the simple scattering for a surface with a wide range of dielectric constant ( r  ), large standard deviation of heights Hrms, and various remote sensing configurations. The AIEM simulates the radar backscattering coefficients basing on the same parameters as the IEM.

III.4 Results and Discussion

This section shows the evaluation results of the five radar backscattering models Dubois, Oh, IEM, IEM_B and AIEM using large datasets, characterized by various radar wavelength (L, C and X), wide range of incidence angles and large geographical distribution in regions with different climate conditions (humid, semi-arid and arid sites). The size of reference plots is at least of 2 ha. For each plot, SAR data was simulated through backscatter models using in situ measurements (mv, Hrms and L) averaged within that plot. Then, the simulated SAR signal were compared with the backscattering coefficients computed from calibrated SAR images by linearly averaging the 0  values of all pixels within the plot

III.4.1 Evaluation of the Dubois Model

The evaluation of Dubois model was carried out for different scenarios using all data, per radar wavelength, and by range of soil moisture, kHrms, and incidence angle.

Using all data, the Dubois model slightly over-estimates the radar signal by about 1.0 dB in HH polarization and slightly under-estimates the radar signal by about 0.7 dB in VV polarization (Table III.3, Figures III.1 and III.2). RMSE is about 4.0 dB and 2.9 dB at HH and VV polarization, respectively (Table III.3). The analysis of the error according to each radar frequency band separately (L, C and X) shows an over-estimation in HH polarization, which is almost the same at L-, C-and X-bands (between 0.9 dB and 1.1 dB). In VV polarization, the Dubois model under-estimates the radar signal by about 1.8 dB and 0.4 dB for X and C bands, respectively. For L band, the Dubois model fits correctly the radar signal in VV because the difference between real data and simulations is about 0.2 dB. The RMSE in HH is the same as at X-and C-bands, and is about 4.1 dB and decreases to 3.0 dB at L-band. In VV polarization, the RMSE increases with the radar frequency (2.5 dB at L-band, 2.8 dB at Cband and 3.1 dB at X-band). 2.9 2.5 2.8 3.1 3.0 2.5 2.8 3.0 2.9 2.9

Table III.3 The analysis of the error of the Dubois model according to the validity domain was studied by range of surface roughness (kHrms), soil moisture (mv) and incidence angle (Table III.3). The Dubois model underestimates the radar signal for kHrms < 2.5 (validity domain of the Dubois model) by about 0.4 dB and 1.2 dB in HH and VV polarizations, respectively. In the case of kHrms < 2.5, the RMSE is about 3.6 and 3.0 dB for HH and VV polarizations, respectively. In addition, the Dubois model overestimates the radar signal for kHrms > 2.5 by about 2.9 dB in HH polarization with RMSE about 4.6 dB. In VV polarization, the Dubois model fits correctly the radar signal in the case of kHrms > 2.5 with a difference between real and simulated data of about 0.2 dB and a RMSE of 2.5 dB (Table III.3).

Moreover, the evaluation of the Dubois model was carried out by range of soil moisture (mv).

Results show an overestimation in HH pol. by about 2.6 dB and a slightly underestimation in VV by about 0.5 dB with mv-values lower than 20 vol.% (RMSE = 4.6 and 2.8 dB at HH and VV, respectively) (Table III.3). Besides, the Dubois model correctly simulates the backscattering coefficient in HH pol. with a difference between real data and simulations about 0.3 dB and underestimates the radar signal in VV by about 1.0 dB with mv-values greater than 20 vol. %. In the case of mv-values greater than 20 vol. %, the RMSE is about 3.4 dB and 3.0 dB for HH and VV polarization respectively. Finally, the performance of Dubois model was studied according to ranges of incidence angle (Table III.3

III.4.2 Evaluation of the Oh Model

The Oh model versions developed in 1992Oh model versions developed in , 1994Oh model versions developed in , 2002Oh model versions developed in and 2004 were applied to our datasets.

The evaluation of the different Oh model versions was carried out firstly using all data, successively for each radar wavelength (L, C and X bands), and finally by range of soil moisture, kHrms and incidence angle (Table III .4,.

Using the entire dataset, results showed that the different versions of Oh model correctly simulate the backscattering at both HH and VV polarizations with difference between real data and simulations varying between -0.9 and +0.4 dB at HH pol. and between (-1.3 dB and +0.4 dB) in VV pol. The RMSE values are approximately the same for all models and in both HH and VV polarizations, i.e., between 2.4 dB and 2.8 dB. The Oh 1992 model simulates slightly better the backscattering than the other versions (Table III.4). For HV polarization, the Oh 2002 model simulates correctly the backscattering with a difference between real and simulated data of about +0.7 dB, with RMSE equal to 2.9 dB. Table III.4. Comparison between real data and Oh models for all data and different ranges of kHrms and soil moisture (mv). Bias = real data -simulations.

Model

In L-band, the different versions of the Oh model underestimate the backscattering at both HH and VV polarizations. This underestimation varies between 1.3 dB and 2.5 dB in HH polarization and between 0.7 dB and 2.1 dB in VV polarization (Table III.4). The RMSE is slightly higher in HH than in VV polarization (between 2.8 dB and 3.7 dB in HH and between 2.6 dB and 3.4 dB in VV). The Oh 1994 version better simulates the backscattering than other versions of Oh model, with an underestimation of the backscattering between 1.3 dB and 0.7 dB and RMSE of 2.8 and 2.6 dB for HH and VV polarizations, respectively. At HV polarization, the Oh model underestimates the backscattering by about 1.5 dB with RMSE equal to 3.1 dB.

In C-band, the Oh 1992 model correctly simulates the backscattering in both HH and VV polarizations with differences between real and simulated data of 0.1 dB and 0.4 dB at HH and VV polarizations, respectively (Table III.4). Besides, the RMSE is of 2.4 dB at HH and 2.3 dB at VV pol. Moreover, the other Oh versions overestimate the backscattering in both HH and VV polarizations (between 0.9 dB and 1.5 dB) with similar RMSE between 2.6 dB and 2.8 dB. At HV polarization, the Oh 2002 model slightly underestimates the backscattering by about 1.0 dB with a RMSE of 2.7 dB.

The analysis of results obtained in X-band shows that Oh model versions simulate the radar signal with difference between real data and simulations between 0.0 and -1.2 dB in HH and between +0.4 and -2.1 dB in VV (Table III .4,. The RMSE is between 2.3 and 2.8 dB in HH and between 2.0 and 2.7 dB in VV polarization. For HV polarization, the Oh model over-estimates the backscattering by about 0.9 dB with RMSE of 3.8 dB.

The analysis of the error was studied by selecting two ranges of surface roughness (kHrms < 2.0 and kHrms > 2.0) (Table III.4). This range is different from the general validity domain of the Oh model (0.13 ≤ kHrms ≤ 6.98) because it covers the entire dataset except only a few points. For kHrms < 2.0, the 1994, 2002 and 2004 Oh models simulate correctly the backscattering at both HH and VV polarizations with differences between real data and simulations between -0.5 and +0.6 dB and RMSE between 2.4 dB and 2.7 dB. The Oh 1992 model underestimates the backscattering by 1.3 dB and 1.0 dB at HH and VV polarizations, respectively (RMSE is 2.9 for HH pol. and 2.7 dB for VV pol.). For kHrms > 2.0, the 1992 and 2002 Oh versions simulate correctly backscattering at both HH and VV polarizations with difference between real and simulated data between -0.5 dB and -1.0 dB with RMSE between 2.3 and 2.6 dB. The 1994 Oh model over-estimates the backscattering at both HH and VV polarizations by about 1.7 dB and 2.1 dB, respectively (RMSE = 2.9 dB). The last version of the Oh model [START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF] underestimates the backscattering in HH polarization by about 1.5 dB (RMSE = 2.6 dB) and over-estimates it in VV polarization by about 2.0 dB (RMSE= 2.8 dB). At HV polarization, for kHrms < 2, the Oh 2002 model underestimates the backscattering in HV by 1.8 dB (RMSE = 2.5 dB). In addition, Oh model correctly fits the backscattering for kHrms > 2.0, with a difference between the real and simulated data of about -0.7 dB and RMSE of 2.5 dB.

Finally, the performance of the Oh model was studied according to its validity domain by selecting two intervals of soil moisture (mv < 29.1 and mv > 29.1 vol. %: validity domain of Oh model). For mv < 29.1 vol. %, the 1992 and 2002 Oh versions correctly simulate the backscattering coefficient at both HH and VV polarizations with a difference between real and simulated data varying between -0.3 dB and -0.7 dB. In addition, the 1994 and 2004 Oh models overestimate the backscattering at both HH and VV polarizations (Table III.4) with RMSE between 2.6 dB and 2.9 dB. In conclusion, for mv < 29.1 vol. %, the 1992 Oh model provides the best simulations. For mv > 29.1 vol. %, the 1994, 2002 and 2004 Oh models correctly simulate the backscattering with a difference between real and simulated data between -0.8 dB and +0.5 dB, while the 1992 Oh model underestimates the backscattering by about 1.9 dB and 1.5 dB at HH and VV polarizations, respectively (RMSE = 3.1 dB for HH and 2.7 dB for VV). The RMSE values are approximately the same in the Oh 1994, 2002 and 2004 versions, and range between 2.2 dB and 2.6 dB. At HV polarization, the Oh model correctly simulates the backscattering for both range of mv-values, with RMSE of 3.0 dB for mv < 29.1 vol. % and RMSE of 2.6 dB for mv > 29.1 vol. %.

The validity domain of Oh model according to the incidence angle (10° ≤ θ ≤ 70°) covers the entire dataset. Moreover, our results showed that the performance of the Oh model is not dependent on the incidence angle.

In conclusion, the Oh models simulate correctly the backscattering. Results showed that Oh

III.4.3 Evaluation of the IEM

The IEM was tested on our dataset using both a Gaussian correlation function (GCF) and an exponential correlation function (ECF). The evaluation of the IEM was carried out firstly using the entire dataset, later on for each radar wavelength (L-, C-and X-bands) and finally according to the validity domain of the IEM (Equation 3.10).

Using all data, the IEM simulates the backscattering in HH polarization with an RMSE of 10.5 dB and 5.6 dB for GCF and ECF, respectively (Table III.5). At VV polarization, the RMSE is 9.2 dB for GCF and 6.5 dB for ECF. At HV polarization, the RMSE is higher than 30.0 dB for both GCF and ECF. Some points show a large discrepancy between the real data and the IEM simulations performed using both ECF and GCF (Figures III.12-III.17 [START_REF] Altese | Retrieving soil moisture over bare soil from ERS 1 synthetic aperture radar data: Sensitivity analysis based on a theoretical surface scattering model and field data[END_REF], [START_REF] Zribi | Backscattering behavior and simulation comparison over bare soils using SIR-C/X-SAR and ERASME 1994 data over Orgeval[END_REF]Zribi et al. ( , 2005a) ) and [START_REF] Callens | Parameterization of tillage-induced single-scale soil roughness from 4-m profiles[END_REF] showed that in agricultural areas, the ECF usually provides better agreement to real data than the GCF.

The results obtained in L-band show that the IEM simulates the backscattering in HH pol.

using both GCF and ECF with differences between real data and model simulations ranges between -0.9 dB and +0.6 dB, with an RMSE of 3.6 dB for GCF and 2.9 dB for ECF (Table III.5). At VV polarization, the IEM overestimates the backscattering by about 2.5 dB and 1.3 dB for GCF and ECF, respectively (RMSE of 5.0 dB for GCF and 3.5 dB for ECF). At HV polarization, the IEM simulates the backscattering using GCF with RMSE of 14.5 dB using GCF, and lower RMSE (6.8 dB) using ECF.

According to the results observed in C-band, the IEM simulates the backscattering using GCF with RMSE of 11.2 dB and 8.6 dB for HH and VV polarizations, respectively (Table III.5).

The RMSE is lower with ECF than GCF about 4.1 dB for HH and 4.9 dB for VV polarizations. At HV polarization, the RMSE is higher than 25.0 dB using both GCF and ECF.

The results obtained in X-band show that the IEM simulates the backscattering with higher The analysis of the error was also studied according to the validity domain of the IEM (Equation (3.10)). Inside the validity domain, the RMSE is larger than 11.5 dB for both HH and VV polarizations using GCF. Better results were obtained using ECF, where the IEM correctly simulates the backscattering at both HH and VV polarizations with differences between real and simulated data between -1.2 dB and -0.9 dB with RMSE of 3.2 dB at HH and 3.7 dB at VV polarizations, using data concerning the IEM validity domain. Outside the IEM validity domain, the IEM simulates the backscattering with RMSE of 6.7 dB for HH and 3.1 dB for VV using GCF; whereas RMSE is 7.8 dB for HH and 9.4 dB for VV polarization using ECF. At HV polarizations, model simulations show large differences from real data for both GCF and ECF for points inside or outside the validity domain of the IEM (in this case, RMSE is larger than 20 dB). Errors observed on IEM simulations were also studied as a function of the difference between Lopt and the measured correlation length (L). Results show that the IEM using GCF gives poor simulations mainly when the measured correlation length was over-estimated (L > Lopt). In this case, the IEM strongly under-estimates the SAR backscatter. In addition, the performance of the IEM was also analyzed using ECF according to the difference between Lopt and L. Results show the same performance of the IEM whatever the difference between Lopt and L. As a conclusion, we could say that the IEM better simulates the backscattering in L-band than in C-and X-bands. Moreover, the results show a better fitting with real data using ECF instead than GCF, which agrees with the validity domain of the IEM.

III.4.4 Evaluation of IEM Modified by Baghdadi (IEM_B)

The IEM_B was also tested on our dataset. This model version was run using GCF (Figures

III.4.5 Evaluation of the Advanced Integral Equation Model (AIEM)

The AIEM was tested on our dataset at HH and VV polarizations using both GCF and ECF.

For all data, the AIEM simulates the backscattering at HH and VV polarizations using GCF with RMSE larger than 10 dB (Table III discrepancies between the real data and the AIEM simulations using GCF. Due to the high sensitivity to surface roughness of the AIEM using GCF, these points correspond mainly to surface with kHrms < 3, L > 4 cm and θ > 35°.

The performance of the AIEM was also evaluated for each SAR wavelength. Results show that in L-band the AIEM simulates the backscattering with RMSE of about 5.0 dB at both HH and VV polarizations using the GCF. In C and X-bands, the AIEM using GCF simulates the backscattering with RMSE higher than in L-band (RMSE > 11 dB). Moreover, AIEM better simulates better the backscattering in using GCF than ECF for all wavelength (RMSE about 4 dB).

In conclusions, the AIEM is able to better simulate better the backscattering than the original IEM only using the ECF with better results in X-band than in C-and L-bands.

III.5 Conclusions

Physical (IEM, IEM_B and AIEM) and semi-empirical (Oh and Dubois) backscattering models were tested using a wide dataset composed by large intervals of surface conditions (mv between 2 vol. % and 47 vol. %, Hrms between 0.2 cm and 9.6 cm and kHrms from 0.2 and 13.4), the dataset was acquired over bare soils in various agricultural study sites (France, Italy, Germany, Belgium, Luxembourg, Canada and Tunisia) characterized by large variety of climatological conditions and using SAR sensors in L-, C-and X-bands with incidence angle between 18° and 57°.

Results (Figure III.25) show that the IEM modified by Baghdadi (IEM_B used the empirical correlation length instead of measured correlation length) provides the most accurate SAR simulations (bias lower than 1.0 dB and RMSE lower than 2.0 dB) with slightly better performance in X-band (RMSE = 1.8 dB) than in L-and C-bands (RMSE between 1.9 and 2.3 dB). At HV polarization, the IEM_B was only run at C-band. Results show that the RMSE strongly decreases from values higher than 25.1 dB, using the original IEM, to 3.1 dB, using IEM_B. In contrast, high RMSE were found using both IEM and AIEM using Gaussian

IV.1 Introduction

Soil moisture content plays an important role in meteorology, hydrology, agronomy, agriculture, and risk assessment. This soil parameter can be estimated using synthetic aperture radar (SAR). Today, it is possible to obtain SAR data for global areas at high spatial and temporal resolutions with free and open access Sentinel-1 satellites (6 days with the two Sentinel-1 satellites, at 10 m spatial resolution).

The retrieval of soil moisture content and surface roughness requires the use of radar backscatter models capable of correctly modeling the radar signal for a wide range of soil parameter values. This estimation from imaging radar data implies the use of backscattering electromagnetic models, which can be physical, semi-empirical or empirical.

This estimation from imaging radar data implicates the use of backscattering electromagnetic models (Physical, empirical or semi-empirical. The physical models such as Integral Equation Model (IEM), Small Perturbation Model (SPM), Geometrical Optic Model (GOM) and Physical Optic Model (POM) that based on physical approximations corresponding to a range of surface conditions (soil moisture and surface roughness) provide site-independent relationships but have limited validity depending upon the soil roughness. Moreover, the semi-empirical or empirical models are often valid only for specific soil conditions and needs calibration on other soil conditions. Users preferred the empirical models because of their facility in implementation and inversion [START_REF] Chai | Modeling and mapping soil moisture of plateau pasture using RADARSAT-2 imagery[END_REF][START_REF] Gherboudj | Soil moisture retrieval over agricultural fields from multi-polarized and multi-angular RADARSAT-2 SAR data[END_REF][START_REF] Kirimi | Influence of Vegetation Cover on the Oh Soil Moisture Retrieval Model: A Case Study of the Malinda Wetland, Tanzania[END_REF][START_REF] Hégarat-Mascle | Soil moisture estimation from ERS/SAR data: Toward an operational methodology[END_REF][START_REF] Rao | Modified Dubois model for estimating soil moisture with dual polarized SAR data[END_REF][START_REF] Zribi | A new empirical model to retrieve soil moisture and roughness from C-band radar data[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF].

Popular semi-empirical models developed over bare soils as Oh model [START_REF] Oh | Quantitative retrieval of soil moisture content and surface roughness from multipolarized radar observations of bare soil surfaces[END_REF][START_REF] Oh | An empirical model and an inversion technique for radar scattering from bare soil surfaces[END_REF][START_REF] Oh | An inversion algorithm for retrieving soil moisture and surface roughness from polarimetric radar observation[END_REF][START_REF] Oh | Semi-empirical model of the ensembleaveraged differential Mueller matrix for microwave backscattering from bare soil surfaces[END_REF] and Dubois model [START_REF] Dubois | Measuring soil moisture with imaging radars[END_REF]. The Oh model uses the ratios of the measured backscatter coefficients HH/VV and HV/VV to estimate volumetric soil moisture (mv) and surface roughness (Hrms), while the Dubois model links the backscatter coefficients in HH and VV polarizations to the soil's dielectric constant and surface roughness. Numerous studies evaluated several semi-empirical models, but these models showed conflict in the results obtained. Some studies show good agreement between measured backscatter coefficients and those predicted by the models, while others have found great discrepancies between them (Baghdadi and Zribi, 2006;Baghdadi et al., 2011c;[START_REF] Hégarat-Mascle | Soil moisture estimation from ERS/SAR data: Toward an operational methodology[END_REF][START_REF] Wang | Adaptation of Oh Model for soil parameters retrieval using multi-angular RADARSAT-2 datasets[END_REF][START_REF] Wang | A comparison of soil moisture retrieval models using SIR-C measurements over the Little Washita River watershed[END_REF][START_REF] Zribi | A new empirical model to retrieve soil moisture and roughness from C-band radar data[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF]. The discrepancy between simulations and measurements often reaches several decibels, making soil parameter estimates not useful.

The goal of this chapter is to produce a new robust, empirical, radar backscattering model. A description of the dataset is presented in section 2, section 3 describes and analyzes the potential and the limitations of the Dubois model in radar signal simulations over bare soils.

In section 4, the new model is described and its performance is evaluated for different available SAR data (L-, C-and X-bands, incidence angles between 20° and 45°). Conclusions are presented in section 5.

IV.2 Dataset description

A wide experimental dataset was used, consisting of SAR images and ground measurements of soil moisture content and roughness collected over bare soils at several agricultural study sites (Chapter III, Table III.1). SAR images were acquired by various airborne and spaceborne sensors (AIRSAR, SIR-C, JERS-1, PALSAR-1, ESAR, ERS, RADARSAT, ASAR, TerraSAR-X). The radar data were available in L-, C-and X-bands (approximately 1.25 GHz, 5.3 GHz and 9.6 GHz, respectively); with incidence angles between 18° and 57°; and in HH,

VV and HV polarizations. For several reference plots, the mean backscatter coefficients have been obtained from radiometrically and geometrically calibrated SAR images by averaging backscatter coefficient values for each plot for all pixels within the plot.

A total of 1569 experimental data acquisitions with radar signal, soil moisture content and roughness were available for HH polarization, 930 for VV polarization, and 605 for HV polarization. This dataset is approximately the same as descripted in chapter III, section III.2.

IV.3 Validation and analysis of the Dubois model

IV.3.1 Description of Dubois model

A complete description of Dubois model is done in chapter III, section III.3.1.

IV.3.2 Comparison between simulated and real data

The Dubois model overestimates the radar signal by 0.7 dB in HH polarization and underestimates the radar signal by 0.9 dB in VV polarization for all data combined (Table IV.1). The results show that the overestimation in HH is of the same order for L-, C-and Xbands (between 0.6 dB and 0.8 dB). For the L-band, a slight overestimation of approximately 0.2 dB of SAR data is observed in VV polarization. Also in VV polarization, Dubois model based simulations underestimate the SAR data in C-and X-bands by approximately 0.7 dB and 2.0 dB, respectively.

The rms error (RMSE) is approximately 3.8 dB and 2.8 dB in HH and VV, respectively (Table IV.1). Analysis of the RMSE according to the radar frequency band (L, C and X separately) shows in HH an increase of the RMSE with the radar frequency (2.9 dB in Lband, 3.7 dB in C-band, and 4.1 dB in X-band). In VV polarization, the quality of Dubois simulations (RMSE) is similar for L-and C-bands but is less accurate in X-band (2.3 dB in Lband, 2.6 dB in C-band, and 3.2 dB in X-band). Finally, the discrepancy between SAR and the model is larger in HH for incidence angles lower than 30° (outside of the Dubois validity domain) than for incidence angles higher than 30° (Table IV.1). The Dubois model strongly overestimates the radar signal in HH for incidence angles lower than 30° but agrees closely with the measured data for incidence angles higher than 30° (Figures IV.1d, IV.2d; Table IV.1). In VV polarization, the Dubois model slightly overestimates the radar signal for incidence angles lower than 30° and underestimates the signal for incidence angles higher than 30° by +1.5 dB (Figures IV.1d and, IV.2d; Table IV.1).

In conclusion, the Dubois model simulates VV better than it does HH (RMSE=2.8 and 3.8 dB, respectively). The disagreements observed between the Dubois model and measured data are not limited to data that are outside the optimal application domain of the Dubois model. The new model is based on the Dubois model and uses the dependency observed between the SAR signal and soil parameters according to results obtained in various studies. For bare soils, the backscattering coefficient depends on soil parameters (roughness and moisture) and SAR instrumental parameters (incidence angle, polarization and wavelength). For bare soils, the radar signal in pq polarization (p and q = H or V, with HV=VH) can be expressed as the product of three components:

𝜎 𝑝𝑞 °= 𝑓 𝑝𝑞 (𝜃) 𝑔 𝑝𝑞 (𝑚𝑣, 𝜃) Γ 𝑝𝑞 (𝑘𝐻𝑟𝑚𝑠, 𝜃) (4.1)
The radar backscatter coefficient is related to the incidence angle () by the relation 𝑓 𝑝𝑞 (𝜃) = 𝛼(𝑐𝑜𝑠 𝜃) 𝛽 [START_REF] Baghdadi | Evaluation of C-band SAR data for wetlands mapping[END_REF][START_REF] Beauchemin | Modelling forest stands with MIMICS: implications for calibration[END_REF][START_REF] Ulaby | Microwave remote sensing: Active and passive. Volume 2-Radar remote sensing and surface scattering and emission theory[END_REF]. This relationship describes the decrease of ° with the incidence angle (decrease higher for low angles than for high angles).

The second term represents the relationship between the radar backscatter coefficient and soil moisture (mv). The results obtained in several investigations show that, for bare soils, the radar signal (°) in decibels linearly increases with soil moisture (mv) when mv is in the range between approximately 5 and 35 vol.% [START_REF] Baghdadi | Soil moisture estimation using multiincidence and multi-polarization ASAR data[END_REF](Baghdadi et al., , 2008a;;[START_REF] Hégarat-Mascle | Soil moisture estimation from ERS/SAR data: Toward an operational methodology[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF]. In linear scale 𝑔 𝑝𝑞 (𝑚𝑣, 𝜃) can be written as δ 10 𝛾 𝑚𝑣 . The sensitivity of the radar signal to the soil moisture  depends on . Higher sensitivity is observed for low than for high incidence angles (Aubert et al., 2011a;[START_REF] Baghdadi | Operational performance of current synthetic aperture radar sensors in mapping soil surface characteristics in agricultural environments: application to hydrological and erosion modelling[END_REF].

To include this dependence on incidence angle, the soil moisture value is multiplied with the term 𝑐𝑜𝑡𝑎𝑛(𝜃). Thus, 𝑔 𝑝𝑞 (𝑚𝑣, 𝜃) can be written as δ 10 𝛾 𝑐𝑜𝑡𝑎𝑛(𝜃) 𝑚𝑣 .

The last term Γ 𝑝𝑞 (𝑘𝐻𝑟𝑚𝑠, 𝜃) represents the behaviour of ° with soil roughness. An exponential or logarithmic function is often used to express the radar signal (in dB) in terms of surface roughness [START_REF] Baghdadi | Soil moisture estimation using multiincidence and multi-polarization ASAR data[END_REF][START_REF] Sahebi | A comparison of multi-polarization and multiangular approaches for estimating bare soil surface roughness from spaceborne radar data[END_REF][START_REF] Srivastava | Use of multiincidence angle RADARSAT-1 SAR data to incorporate the effect of surface roughness in soil moisture estimation[END_REF][START_REF] Zribi | A new empirical model to retrieve soil moisture and roughness from C-band radar data[END_REF]. For a logarithmic behaviour of °(dB) with k Hrms, Γ 𝑝𝑞 in linear scale can be written as 𝜇(𝑘𝐻𝑟𝑚𝑠) 𝜉 . Baghdadi et al. (2002a) showed that at high incidence angles, radar return is highly sensitive to surface roughness and shows much larger dynamics than at a low incidence angle. In addition, the term 𝑠𝑖𝑛(𝜃) is intended to include this dependence with the incidence angle: Γ 𝑝𝑞= 𝜇(𝑘𝐻𝑟𝑚𝑠) 𝜉 𝑠𝑖𝑛 (𝜃) .

Finally, the relationship between the radar backscattering coefficient (°) and the soil parameters (soil moisture and surface roughness) for bare soil surfaces can be written by equation (4.2):

𝜎 𝑝𝑞 °= 𝛿(𝑐𝑜𝑠 𝜃) 𝛽 10 𝛾 𝑐𝑜𝑡𝑎𝑛(𝜃) 𝑚𝑣 (𝑘𝐻𝑟𝑚𝑠) 𝜉 𝑠𝑖𝑛 (𝜃) (4.

2)

The coefficients , , , and  are then estimated for each radar polarization using the method of least squares by minimizing the sum of squares of the differences between the measured The fitting of various coefficients parameter in the equation (4.2) was done using all dataset (fitting errors are about 2 dB for all polarizations). This fitting allows writing ° as a function of the rms surface height (Hrms) and incidence angle (), by equations (4.3), (4.4) and (4.5):

𝜎 𝐻𝐻 °= 10 -1.287 (𝑐𝑜𝑠 𝜃) 1.227 10 0.009 𝑐𝑜𝑡𝑎𝑛(𝜃) 𝑚𝑣 (𝑘𝐻𝑟𝑚𝑠) 0.86 𝑠𝑖𝑛 (𝜃) , (4.3)

𝜎 𝑉𝑉 °= 10 -1.138 (𝑐𝑜𝑠 𝜃) 1.528 10 0.008 𝑐𝑜𝑡𝑎𝑛(𝜃) 𝑚𝑣 (𝑘𝐻𝑟𝑚𝑠) 0.71 𝑠𝑖𝑛 (𝜃) (4.4)

𝜎 𝐻𝑉 °= 10 -2.325 (𝑐𝑜𝑠 𝜃) -0.01 10 0.011 𝑐𝑜𝑡𝑎𝑛(𝜃) 𝑚𝑣 (𝑘𝐻𝑟𝑚𝑠) 0.44 𝑠𝑖𝑛 (𝜃) , (4.5)

where  is expressed in radians and mv is in vol.%. Equations (4.3), (4.4), and (4.5) show that the sensitivity () of the radar signal to the soil moisture in decibel scale is 0.25 dB/vol.% in HH polarization, 0.22 dB/vol.% in VV polarization and 0.30 dB/vol.% in HV polarization for an incidence angle of 20°. This sensitivity decreases to 0.09 dB/vol.% in HH, 0.08 dB/vol.% in VV and 0.11 dB/vol.% for an incidence angle of 45°. As for the signal's sensitivity to soil roughness, it is of the same order of magnitude in HH and VV and twice as large than that of the HV signal. The few data in HV polarization may explain the higher sensitivity observed in HV than in HH and VV. Indeed, several studies showed very near sensitivity of radar signal to soil moisture in HV, HH and VV polarizations.

The availability of a backscatter model for the cross polarization component is required because most spaceborne SAR acquisitions are made with one co-polarization and one crosspolarization in case of dual-polarization mode.

IV.4.2 Comparison between Dubois model and new model

In comparison between Dubois model (Equation 3.1) and the new model (Equation 4.2), several terms have been changed, simplified or removed:

 In the new model, the first term that describes the relationship between the radar backscatter coefficient and the incidence angle () which describes the decrease of ° with the incidence angle was simplified using the function 𝑓 𝑝𝑞 (𝜃) = 𝛼(𝑐𝑜𝑠 𝜃) 𝛽 . In Dubois model, the relation between the radar backscatter signal and the incidence angle () is more complex (Equation 4.6).

 The second term which describes the relationship between the radar backscatter coefficient and soil moisture (mv) was modified. First, the dielectric constant was changed into the volumetric soil moisture. Second, the dependence between the sensitivity of the radar signal to the incidence angle () described by tan() in Dubois model was corrected into cotan() in the new model. Indeed, higher sensitivity is observed for low than for high incidence angles. To include this dependence on incidence angle, the soil moisture value is multiplied with the term 𝑐𝑜𝑡𝑎𝑛(𝜃). Thus, 𝑔 𝑝𝑞 (𝑚𝑣, 𝜃) can be written as δ 10 𝛾 𝑐𝑜𝑡𝑎𝑛(𝜃) 𝑚𝑣 .

 In the new model, the term Γ 𝑝𝑞 (𝑘𝐻𝑟𝑚𝑠, 𝜃) represents the behaviour of ° with soil roughness. For a logarithmic behaviour of °(dB) with k Hrms, Γ 𝑝𝑞 in linear scale can be written as 𝜇(𝑘𝐻𝑟𝑚𝑠) 𝜉 . At high incidence angles, radar return is highly sensitive to surface roughness and shows much larger dynamics than at a low incidence angle. Thus, the term 𝑠𝑖𝑛(𝜃) is intended to include this dependence with the incidence angle:

Γ 𝑝𝑞= 𝜇(𝑘𝐻𝑟𝑚𝑠) 𝜉 𝑠𝑖𝑛 (𝜃) .

 Finally the dependence of the radar signal on the radar wavelength described in Dubois model by () 𝑐 (Equation 4.6) does not seem validated using our in situ dataset. Thus, this term has been removed in the new model.

IV.4.3 Results and discussion

IV.4.3.1 Performance of the new model

Results show that the new model provides more accurate results. The biases and the RMSE decrease for both HH and VV polarizations. The RMSE decreases from 3.8 dB to 2.0 dB for HH and from 2.8 dB to 1.9 dB for VV (Table IV Analysis of the new model's performance for each radar wavelength separately (L-, C-and Xbands) shows that the most significant improvement is observed in X-band with an RMSE that decreases from 4.1 dB to 1.9 dB in HH and from 3.2 dB to 1.8 dB in VV. In L-band, the performance of the new model is not better than that of the Dubois model because the RMSE decreases slightly with the new model of 3.0 dB to 2.3 dB in HH and remains similar in VV (RMSE = 2.3 dB with the Dubois model and 2.7 dB with the new model). The improvement is also important for the C-band with an RMSE that decreases from 3.7 dB to 1.9 dB in HH and from 2.6 dB to 1.9 dB in VV. With respect to bias, the results show that it decreases with the new model for all radar wavelengths. In addition, the new model does not show bias according the range of soil moisture, surface roughness, and radar incidence angle.

IV.4.3.2 Behaviour of the new model

The physical behaviour of the new radar backscatter model was studied in function of incidence angle (), soil moisture (mv) and surface roughness (kHrms).

Figure IV.6 shows that the radar signal is strongly sensitive to surface roughness, especially for small values of kHrms. In addition, this sensitivity increases with the incidence angle.

Concerning the influence of polarization, the new model shows, as do many theories and experimental studies, that a given soil roughness leads to slightly higher signal dynamics with the soil moisture in HH than in VV polarization (Figure IV.6). The radar signal ° increases with kHrms. This increase is higher for either low kHrms values or high -values than it is for either high kHrms values or low -values. For =45°, ° increases approximately 8 dB in HH and 6.5 dB in VV when kHrms increases from 0.1 to 2 compared with only 3 dB when kHrms increases from 2 to 6 (for both HH and VV). This dynamic of ° is only half for =25° in comparison to that for =45°. In HV, the dynamic of ° to kHrms is half that observed for HH and VV.

The behaviour of ° according to soil moisture shows a larger increase of ° with mv for low incidence angles than for high incidence angles. Based on the same equation as that used for HH and VV, a radar signal in HV polarization was also proposed. Finally, the new empirical model proposed in the present study would allow more accurate soil moisture estimates using the new Sentinel-1A and -1B SAR data.

V.1 Introduction

Soil surface characteristics (mainly soil moisture and surface roughness) play a key role in different hydrological processes (floods, runoff, evapotranspiration, infiltration, soil erosion, and imbalances in the water and carbon cycles). Surface roughness has a role in trapping water at the surface and reducing flow velocity, which increases infiltration and in turn reduces downstream runoff. The roughness scales observed by a radar sensor have a strong dependence on the frequency and radar incidence [START_REF] Ogilvy | Theory of wave scattering from random rough surfaces[END_REF].

Radar data were used since a long time for estimating and mapping the surface soil parameters (mainly soil moisture and roughness) of bare soils. Soil moisture and surface roughness can be estimated from SAR images by using physical or statistical models (Baghdadi and Zribi, 2016;[START_REF] Baghdadi | Retrieving surface roughness and soil moisture from synthetic aperture radar (SAR) data using neural networks[END_REF]Baghdadi et al., , 2012a;;[START_REF] Merzouki | Mapping soil moisture using RADARSAT-2 data and local autocorrelation statistics[END_REF][START_REF] Rahman | Mapping surface roughness and soil moisture using multi-angle radar imagery without ancillary data[END_REF]. The best known physical model is the Integral Equation Model (IEM) [START_REF] Fung | Microwave Scattering and Emission Models and their Applications[END_REF][START_REF] Fung | Backscattering from a randomly rough dielectric surface[END_REF]. This model simulates the radar backscattering coefficients from SAR and soil parameters (radar wavelength, polarization, incidence angle, surface roughness and soil moisture 'dielectric constant'). The validity domain of IEM in C-band covers the range of roughness values that are commonly encountered for bare agricultural surfaces (k Hrms ≤ 3,

where Hrms is the root mean square surface height and k the radar wave number ≈ 1.12 cm-1 for a frequency in C-band of 5.4 GHz as Sentinel-1 SAR). Most Hrms values of agricultural bare soils range from 0.5 to 4 cm (Baghdadi et al., 2012a).

The discrepancies observed between the IEM and the SAR data had encouraged [START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF]Baghdadi et al., ( , 2006aBaghdadi et al., ( , 2011aBaghdadi et al., ( , 2015) ) to propose an empirical calibration of IEM model. Moreover, Baghdadi et al. (2016a) proposed an new empirical model based on Dubois model that make the estimation of soil moisture and surface roughness possible in an easy way. Actually, physical, empirical and semi-empirical models were developed to invert the radar signal in order to monitor the soil parameters (moisture and roughness). 2012) developed an approach to estimate soil moisture and surface roughness from C-band polarimetric RADARSAT-2 data based on neural networks (NNs). Results showed that the accuracy on the soil roughness estimates was about 0.5 cm using polarimetric data. The estimation is better for Hrms-values lower than 2 cm than for Hrms-values higher than 2 cm. For higher Hrms, the NNs under-estimate the surface roughness. Moreover, [START_REF] Zribi | A new empirical model to inverse soil moisture and roughness using two radar configurations[END_REF] proposed an approach based on the use of two SAR images acquired at two different incidence angles, one image with a weak incidence (~20°) and one image with a strong incidence (~40°) for estimating both soil moisture and surface roughness. The surface roughness defined by Zs=Hrms/L (L is the correlation length) is estimated with an RMSE of 0.08 cm for Zs-values between 0.075 and 0.75 cm.

The aim of this part is to develop an approach to estimate the soil surface roughness from Cband Sentinel-1 SAR data in the case of bare agricultural soils. This approach is an inversion technique based on Multi-Layer Perceptron (MLP) neural networks. The training of the neural networks is performed using synthetic dataset simulated by the Integral equation model calibrated by Baghdadi [START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF](Baghdadi et al., , 2006a[START_REF] Baghdadi | Semiempirical calibration of the integral equation model for SAR data in C-band and cross polarization using radar images and field measurements[END_REF][START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF] and the new proposed model by Baghdadi modified Dubois (Baghdadi et al., 2016a) on a wide range of surface roughness and soil moisture. The inversion approach was then validated in using Sentinel-1 datasets (one in France and one in Tunisia) composed on Sentinel 1 images and in-situ measurements. This work is done in order to evaluate the potential of Sentinel-1 SAR sensors for retrieving soil roughness. Section 2 presents a review of datasets. A presentation of the methodology developed in order to estimate the soil roughness is done is section 3. The results and discussions are presented in section 4, and finally, the main conclusion is presented in section 5.

V.2 Dataset

V.2.1 Synthetic dataset

The Integral Equation Model calibrated by [START_REF] Baghdadi | Semiempirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements[END_REF]Baghdadi et al., ( , 2006aBaghdadi et al., ( , 2011a[START_REF] Baghdadi | Semi-empirical calibration of the integral equation model for co-polarized L-band backscattering[END_REF] 'IEM_B' and the modified Dubois Model (Baghdadi et al., 2016a) A synthetic dataset combining a wide range of soil parameters (soil roughness "Hrms" and soil moisture "mv") and corresponding backscattering coefficients was generated from the calibrated IEM and the modified Dubois in order to evaluate the performance of the NN technique. 18 soil roughness values (Hrms between 0.35 and 3.8 cm with a step of 0.2 cm), 20 soil moisture values (mv between 2 vol.%. and 40 vol.%.with a step of 2 vol.%), 25 radar incidence angles ( between 20°and 45° with a step of 1°) are considered. In order to make the IEM simulations more realistic, the SAR measurement error which includes both calibration errors and measurements precision errors is added to the simulated backscattering coefficients. Realistic values of measurements errors are 0.75 for VV and 1dB for VH [START_REF] Schwerdt | Independent System Calibration of Sentinel-1B. Remote Sens[END_REF].To better simulate an experimental dataset, the synthetic dataset is then obtained by adding a zero mean Gaussian random noise with a standard deviation of 0.75 and 1 dB to the simulated backscattering coefficients VV and VH (in dB scale), respectively. In order to obtain a statistically significant dataset, 250 noise samples are generated for each couple of mv and Hrms. A total of 1350000 elements (C-band VV and VH) are also obtained to produce the synthetic dataset.

The noisy synthetic datasets are then divided into two equal datasets one of which is used for training the NNs, the remaining is being used for the validation of the NNs.

V.2.2 Real dataset

An experimental dataset is used in this study, consisting of Sentinel-1 images as well as ground measurements of soil moisture and surface roughness collected over two agricultural study sites: one in France and one in Tunisia (Figure V.1, Table V.1). Sentinel-1 images (Cband, radar wavelength about 6 cm) were acquired with incidence angles between 37° and 41°, and in VV and HV polarizations.

V.2.2.1 Study sites

The French study site is the Versailles plain. It is located west of Paris and covers about 221 km² (48°46ʹ -48°56ʹ N; 1°50ʹ-2°07ʹ E, Figure V.1) [START_REF] Vaudour | Mapping tillage operations over a periurban region using combined SPOT4 and ASAR/ENVISAT images[END_REF]. This agricultural peri-urban site is characterized by a semi-oceanic climate with an average rainfall of 570 mm/year and an average annual temperature of 11.3°C (INRA meteorological station of Thiverval-Grignon, 1986-2016). Rainfed annual crop systems cover 99 km² and develop over two embedded plateaus, the gentle slopes at their edges and the valleys at their bottom. The main crop rotations in the area involve winter wheat, winter rapeseed, winter and spring barley and maize on occasion [START_REF] Vaudour | An overview of the recent approaches to terroir functional modelling, footprinting and zoning[END_REF]. Conventional tillage practices are used:

ploughing in November-December, followed by chisel in March then seedbed preparation for spring cereals (spring barley in March, maize in April). The main cultivated soils according to the FAO classification (World Reference Base (WRB) [START_REF] Vaudour | Mapping tillage operations over a periurban region using combined SPOT4 and ASAR/ENVISAT images[END_REF] is close to 1600 mm. The landscape is mainly flat, and the vegetation is dominated by agricultural production (cereals, olive groves, fruit trees, market gardens and bare soils). Soil texture measurements showed a clay percentage between 2.4% and 53.1% and sand percentage between 4.4% and 84.3% (Gorrab et al., 2015a). The soil roughness was assumed isotropic. So, the row direction is not considered.

V.2.2.2 . SAR Satellite images

Four Sentinel-1 images were acquired in March and April 2017 (Table V.1) over the French study site. In addition, 7 Sentinel-1 images acquired over the Kairouan plain between 2015 and 2017 are used in this study. All Sentinel-1 images acquired with a spatial resolution of 10 m and in VV and VH polarizations are radiometrically calibrated in order to convert the digital number to radar backscattering coefficients.

Site

V.2.2.3 In situ measurements

Simultaneously with the Sentinel-1 acquisitions, in situ measurements of soil moisture and surface roughness were collected on several reference bare plots of a few hectares. Soil moisture was determined gravimetrically at each reference plot of the French site in using soil samples collected between 0 and 8 cm depth (one measure by plot). For the Tunisian site, between 20 and 30 volumetric soil moisture measurements (mv) were performed in the first top 5 cm using calibrated TDR (Time Domain Reflectometry) probes. The mean volumetric soil moisture was then calculated for each reference plot and each date. The soil moisture on the reference plots ranged between 11.5 et 25.1 vol.% for the French site and between 4.6 and 41.7 vol.% for the Tunisia site.

The soil roughness measurements made in Tunisia on the reference plots use 1 m long pin profiler with a resolution of 2 cm. Ten roughness profiles (five parallel and five perpendicular to the tillage row direction) were made in each reference field using a 1 m long needleprofilometer and a sampling interval of 2 cm. From these roughness profiles, the root mean square surface height (Hrms) were then calculated for each reference plot using the mean of all autocorrelation functions acquired for each reference plot. All data bases are described in (Bousbih et al., 2017). For the French site, soil roughness was estimated with a fully automatic photogrammetric method [START_REF] Gilliot | Soil surface roughness measurement: A new fully automatic photogrammetric approach applied to agricultural bare fields[END_REF]. The rms surface height ranged between 0.56 cm and 4.55 cm for the reference plots in the Tunisian site and between 0.41 cm and 2.90 cm for the reference plots in the French site.

Finally, each element of our real dataset corresponds to in situ measurements (mv and Hrms)

and Sentinel-1 information (mean of radar backscattered coefficients in VV and VH, and radar incidence angle). The mean of radar backscattered coefficients was calculated by averaging for each reference plot the values of all pixels within the reference plot.

V.3 Methodology for estimating soil moisture

V.3.1 Neural Networks

In this study, surface roughness was estimated by means of multi-layer perceptron (MLP) neural networks. The Levenberg-Marquardt optimization algorithm [START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF]

V.3.2 Methodological overview

An approach based on neural networks is chosen to estimate the soil roughness from Sentinel-1 images (SAR data) over bare agricultural soils at very high spatial resolution ʺVHSRʺ (plot scale or on a finer scale). Two networks are applied one after the other, the first to estimate soil moisture and the second to estimate soil roughness. Three SAR configurations corresponding to the standard acquisition mode of Sentinel-1 with image acquisitions in both VV and VH polarizations are tested: VV alone, VH alone, VV and VH together. In order to improve the soil parameters estimates, a priori knowledge about soil moisture mv is introduced. Baghdadi et al. (2012a) showed that the use of a priori knowledge on the soil moisture (dry to slightly wet or very wet information) improves the soil moisture estimates.

The priori information on mv is provided in using meteorological data (precipitations, temperature) and terrain knowledge. Indeed, it is easily to define from the weather forecasts (precipitation and temperature) if the soil is either dry to slightly wet (no precipitation for many days before SAR acquisition) or very wet (heavy rainfall preceding SAR acquisition).

The integration of a priori information constrains the range of possible soil moisture parameter values estimated and thus leads to a better estimation of the soil moisture.

Three neural networks are developed for the estimation of mv, with and without a priori information on the soil moisture state:

 Case 1: No a priori information is available on the soil moisture state. In this case mv will be estimated between 2 and 40 vol.%.

 Case 2: A priori information is available on mv. The soil is supposed to be dry to the soil surface roughness. The validation of these NNs will be made using the soil moisture estimated without and with a priori information on mv.

V.4 Results and discussion

The different neural networks are tested for the evaluation of the precision on soil roughness estimates using synthetic (built from IEM model and Baghdadi model) and real datasets. First the performance of the inversion algorithm was analyzed according to Hrms and incidence angle "" (Figure V.3) in the case without a priori on mv is used. For VV, results

show that the bias (estimated mv -measured mv) and the RMSE are strongly dependent on Hrms (Figures V.3a and V.3b). The RMSE on mv in the case of inversion without a priori information on mv increases from 4.40 vol.% for Hrms=0.5 cm to 7.0 vol.% for Hrms = 3.8 cm for mv between 2 and 25 vol.% (dry to slightly wet soils). In very wet soil conditions, the RMSE on mv decreases from 11.5 vol.% for Hrms=0.5 cm to 4.0 vol.% for Hrms = 3.8 cm.

The high RMSE values of in the case of dry to slightly wet conditions and high Hrms-values are due to an overestimation of mv (bias increases from -3.0 to +5.0 vol.% for Hrms between 0.5 and 3.8 cm). Similarly, the high RMSE values in the case of very wet conditions and low

Hrms-values are due to an underestimation of mv (bias decreases from -11.0 to -1.5 vol.% for Hrms between 0.5 and 3.8 cm). In addition, results show that the RMSE on mv slightly depends on  in the case of no a priori information on mv (Figures V.3c and V.3d). The RMSE on mv is between 4.2 vol.% (for =20°) and 5.0 vol.% (for  between 25° and 45°) for dry to slightly wet soil conditions and between 6.0 and 7.0 vol.% for very wet soils. The overestimation of mv in dry to slightly conditions is approximately +2.5 vol.% for  between 20° and 45°. For very wet soil conditions, the underestimation of mv is approximately -4.0 vol.% for  between 20° and 45°.

In the case of a priori information on mv with dry to slightly wet soil conditions, the RMSE on mv estimates varies between 2.2 and 5.0 vol.% for all mv and Hrms values of the validation synthetic dataset (case of dry to slightly wet conditions). The bias reduction varies between -3.0 vol. % (low Hrms) and +3.0 vol.% (high Hrms). In addition, RMSE and bias on mv estimates are slightly dependent on the incidence angle.

In the case of a priori information on mv with very wet soil conditions, the RMSE on mv retrieved from the synthetic dataset in VV polarization using IEM. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

V.4.1.1.1.2 Use of VH polarization alone

In the case of VH polarization and mv between 2 and 40 vol.%, the RMSE on the mv estimates is of 5.27 vol.% for mv between 2 and 25 vol.% and 8.27 vol.% for mv between 25 and 40 vol.%. An overestimation of 2.94 vol.% on mv is observed for mv between 2 and 25 vol.%, and an underestimation of -4.64 vol.% is obtained for mv between 25 and 40 vol.%.

Moreover, the RMSE for all the range of mv between 2 and 40% is 6.63 vol.% (Figure V.4a).

The RMSE on mv estimates decreases from 5.27 vol.% without a priori information on mv to 4.16 vol.% in the case of a priori information on mv for dry to slightly wet soil conditions.

Also, the difference between estimated and measured mv is well reduced (approximately by a factor of 2) from 2.94 vol.% to 1.23 vol.% (Figure V.4b). Results also show that the use of a priori information on mv in the case of very wet soil conditions improves slightly the mv estimates. The RMSE on mv estimates decreases from 8.27 vol.% without a priori information on mv to 5.06 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also well reduced from -4.64 vol.% to -2.26 vol.% (Figure V.4c).

First the performance of the inversion algorithm was analyzed according to Hrms and incidence angle "" in the case without a priori on mv is used. Hrms-values are due to an overestimation of mv (bias increases from -6.0 to 6.0 vol.% for Hrms between 0.5 and 3.8 cm). Moreover, results show that the RMSE on mv slightly depends on  in the case of no a priori information on mv (Figures V.5c and V.5d). The RMSE slightly decreases from 5.70 vol.% for =20° to 4.8 vol.% for  = 45° in the case of dry to slightly wet soil conditions. It also slightly decreases from 8.5 vol.% for =20° to 7.5 vol.%(for  = 45° for very wet soils. The overestimation of mv in dry to slightly conditions is approximately +3.0 vol.% for  between 20° and 45°. For very wet soil conditions, the underestimation of mv is approximately about 5.0 vol.% for  between 20° and 45°.

In the case of a priori information on mv with dry to slightly wet soil conditions, the quality of the estimation is also well improved when the accuracy on mv estimates is analyzed according without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

V.4.1.1.1.3 Use of VV and VH polarizations together

In the case of VV and VH polarizations together and mv between 2 and 40 vol.%, the RMSE on the mv estimates is of 4.33 vol.% for mv between 2 and 25 vol.% and 6.27 vol.% for mv With a priori information on mv with dry to slightly soil conditions, the RMSE on mv estimates decreases from 4.33 vol.% without a priori information on mv to 3.26 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also reduced from +2.00 vol.% to +0.91 vol.% (Figure V.6b).

The use of a priori information on mv in the case of very wet soil conditions improves the estimation of mv. The RMSE on mv estimates decreases from 6.27 vol.% without a priori information on mv to 4.85 vol.% in the case of a priori information on mv. Also, the difference between estimated and measured mv is reduced from -3.23 vol.% to -2.02 vol.% (Figure

V.6c).
First the performance of the inversion algorithm was analyzed according to Hrms and incidence angle "" in the case without a priori on mv is used ( With a priori information on mv with dry to slightly soil conditions, the estimation quality is also well improved when the accuracy on mv estimates is analyzed according to Hrms and  (Figures V.7). The RMSE on mv estimates varies between 2.0 and 5.0 vol.% for all mv and Hrms values of the validation synthetic dataset (case of dry to slightly wet conditions). The bias reduction varies between -4.0 vol. % (low Hrms) and +4.0 vol.% (high Hrms). Finally, RMSE and bias on mv estimates are slightly dependent on the incidence angle. The RMSE is about 3.00 vol. % for incidence angle between 20° and 45° and the bias is about -2.50 vol. % for incidence angle between 20° and 45°. retrieved from the synthetic dataset in VV and VH polarizations together using IEM. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

With a priori information on mv with very wet soil conditions, the RMSE on mv estimates varies between 3.0 and 7.0 vol.% for all mv and Hrms values of the validation synthetic dataset in the case of very wet conditions. The highest RMSE-values correspond approximately to low Hrms-values. The bias is also well reduced mainly for low Hrms-values from -6.0 vol.% for Hrms-values of 0.5 cm to +1.0 vol.% for Hrms-values of 3.8 cm.

According to the incidence angle, the RMSE and the bias shows relatively close values. The RMSE is about 4.9 vol.% for incidence angle between 20° and 45° and the bias is about -2.0 vol.% for incidence angle between 20° and 45° (Figures V.7c and V.7d).

V.4.1.1.2 Using Baghdadi model

Three radar configurations will be tested in order to analyze the accuracy on mv estimates:

VV alone, VH alone, VV and VH together.

V.4.1.1.2.1 Use of VV polarization alone

In the case of no a priori information on mv, the RMSE on the mv estimates is of 7.03 vol.% for mv between 2 and 25 vol.% and 8.43 vol.% for mv between 25 and 40 vol.%. An overestimation of +3.39 vol.% on mv is observed for mv between 2 and 25 vol.%, and an underestimation of -5.09 vol.% is obtained for mv between 25 and 40 vol.%. For the entire range of mv, between 2 and 40%, the RMSE on mv is of 7.62 vol.% (Figure V.8a).

Results show that the introducing of a priori information on mv improves the mv estimates.

The RMSE on mv estimates decreases from 7.03 vol.% without a priori information on mv to 5.88 vol.% in the case of a priori information on mv for dry to slightly wet soil conditions. In addition, the difference between estimated and measured mv is also reduced from +3.39 vol.% to +1.64 vol.% (Figure V.8b).The use of a priori information on mv in the case of very wet soil conditions improves the mv estimates. The RMSE on mv estimates decreases from 8.43 vol.% without a priori information on mv to 4.79 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also well reduced from -5.09 vol.% to -1.52 vol.% (Figure V.8c). The performance of the algorithm was also analyzed according to Hrms and the incidence angle "" (Figure V.9). Results show that the bias (estimated mv -measured mv) and the RMSE are strongly dependent on Hrms and . According to Hrms, the RMSE on mv in the case of inversion without a priori information on mv increases from 6.0 vol.% for Hrms=0.5 cm to 10.0 vol.% for Hrms = 3.8 cm in the case of dry to slightly wet soils. In very wet soil conditions, the RMSE on mv decreases from 14.5 vol.% for Hrms=0.5 cm to 4.0 vol.% for

(a) (b) (c) (d)
Figure V.9. Accuracy on the mv estimates (RMSE and bias "=estimatedmeasured") retrieved from the synthetic dataset in VV polarization using Baghdadi model. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

V.4.1.1.2.2 Use of VH polarization alone

In using VH alone, the RMSE on mv is of 6.05 vol.% in the case of no a priori information on mv (Figure V.10a). For mv between 2 and 25 vol.%, the RMSE on mv is of 5.91 vol.%. It is of 6.25 vol.% for mv between 25 and 40 vol.%. An overestimation of +2.08 vol.% on mv is observed for mv between 2 and 25 vol.%, and an underestimation of -3.15 vol.% is obtained for mv between 25 and 40 vol.%.

Results show that the introduction of a priori information on mv improves the mv estimates.

The RMSE on mv estimates decreases from 5.91 vol.% without a priori information on mv to 5.15 vol.% in the case of a priori information on mv for dry to slightly wet soils. For very wet soils, the RMSE on mv estimates decreases from 6.27 vol.% without a priori information on mv to 4.34 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also reduced from +2. retrieved from the synthetic dataset in VH polarization using Baghdadi model. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

V.4.1.1.2.3 Use of VV and VH polarizations together

In using VV and VH polarizations together without a priori information on mv, the RMSE on the mv estimates is of 5.68 vol.% for mv between 2 and 25 vol.% and 6.14 vol.% for mv between 25 and 40 vol.%. An overestimation of +1.98 vol.% on mv is observed for mv between 2 and 25 vol.%, and an underestimation of -2.94 vol.% is obtained for mv between 25 and 40 vol.%. For mv between 2 and 40 vol.%, the RMSE on mv is of 5.87 vol.% (Figure

V.12a).

123

The introduction of a priori information on mv in the case of dry to slightly wet soil conditions improves the mv estimates. The RMSE decreases from 5.68 vol.% without a priori information on mv to 4.97 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also reduced from +1. the case of no a priori information on mv (Figures V.13c and V.13d). The RMSE increases from 3.0 vol.% for =20° to 8.0 vol.% for = 45° for dry to slightly wet soil conditions and increases from 3.0 for =20° to 9.0 vol.% for = 45° for very wet soils.

In the case of a priori information on mv, the RMSE on mv shows slight dependence with Hrms and the incidence angle (Figures V.13a and V.13c).

V.4.1.1.3 Conclusion

Using the two models (IEM and Baghdadi), the use of a priori information on mv strongly improves the estimation of mv. With IEM, better results are obtained with VV polarization.

The use of VV and VH together provides similar performances than those obtained with VV alone. For the range of surface roughness the most encountered in agricultural environments with Hrms between 1 and 2 cm, results show that the RMSE on mv in VV polarization varies between 3.0 and 6.0 vol.%. The difference between estimated and real mv varies between -1.0 and +1.0 vol.% in the case of dry to slightly wet soils. An underestimation of mv from -4.5 to -2.5 vol.% in the case of very wet soils is observed for Hrms between 1 and 2 cm.

The use of Baghdadi model shows slightly better results when VV and VH are used together (RMSE about 5.9 vol.% with VV alone, 5.2 vol.% with VH alone and 4.9 vol.% with VV and VH together). For surface roughness between 1 and 2 cm, the RMSE on mv varies between 3.9 and 5.5 vol.%. The difference between estimated and real mv varies between -0.5 and +2.0 vol.% in the case of dry to slightly wet soils. An underestimation of mv from -4.0 to -1.3 vol.% is observed in the case of very wet soils are observed for Hrms between 1 and 2 cm.

In a comparison between the two models with Hrms ranged between 1 and 2 cm, Baghdadi model shows slightly better results than IEM model mainly with VH polarization alone and with VV and VH polarizations together. Using the IEM model, results show slightly better results for Hrms between 1 and 2 cm with VV than with VH alone or with VV and VH polarizations together.

V.4.1.2 Estimation of Hrms

The estimation of the soil roughness (Hrms) is carried out after a first step which consisted to estimate mv. Indeed, the neural network (NN) which should estimate Hrms needs an estimate of mv. The two conditions on the soil moisture state are considered in input to the NN: no a priori information on mv, a priori information on mv (dry to slightly wet soil conditions or very wet soils). As input to the network, these are the mv estimated by the previous networks built to estimate mv which are used. In addition to these cases corresponding to operational conditions for estimating soil roughness, the configuration where the input mv to the network corresponds to exact mv without estimation error (those that are in the validation dataset) is also tested.

According to the results obtained on the estimation of the soil moisture mv, results showed that the IEM model shows better performance in using VV polarization in comparison to VH alone or to the use of VV and VH together. Baghdadi model provides better results in using VV and VH polarizations together in comparison to VV alone or to the use of VH alone.

Therefore, in the case of surface roughness (Hrms) estimation, only the results obtained from the IEM model corresponding to VV alone and results obtained from Baghdadi model corresponding to VV and VH polarizations together are presented briefly. Other results corresponding to VH alone, VV and VH polarizations together for data generated from the IEM model and corresponding to the use of VV alone and VH alone for data generated from Baghdadi model are presented in Annex 1 and Annex 2, respectively.

V.4.1.2.1 Using IEM model

Figure 1V.8 shows the results for estimating the soil roughness using the synthetic dataset generated from the IEM model with VV polarization alone. Better estimates of Hrms are obtained when the mv used at the input of the NN corresponds to the exact mv (RMSE=0.72 cm). The results obtained using the mv estimated without and with a priori information on mv are with a higher RMSE, respectively 1.01 cm and 0.94 cm. This shows that the use of mv estimates with an accuracy of about 5 vol.% is not sufficient to accurately estimate the soil roughness in C-band and VV polarization. In addition, figure V.14 shows an overestimation of Hrms for low Hrms-values and an underestimation of Hrms for Hrms higher than 2 cm. The results obtained using the mv estimated without or with a priori information on mv show higher RMSE with respectively 0.84 cm and 0.78 cm. This shows that the use of mv estimates with an accuracy of about 5 vol.% is not sufficient to accurately estimate the soil roughness in

V.4.2.2 Estimation of surface roughness (Hrms)

The results of the estimation of Hrms are done in using two inversion configurations:

-At the input of the network for the estimation of Hrms, the mv used corresponds to mv estimated at plot scale (using the mean radar signal calculated by averaging for each reference plot the values of all pixels within the reference plot).

-At the input of the network for the estimation of Hrms, the mv used corresponds to mv estimated at the scale of the study site (using the mean radar signal calculated by averaging the values of all bare soil pixels within the study site). This second approach assumes that the soil moisture is of the same order for all bare agricultural plots located in the area under consideration (no irrigation activities and similar soil composition). The use of mv estimated at the scale of the study site in the estimation technique of Hrms could be relevant only when the study site is not irrigated. Indeed, if the SAR images are acquired during the dry season with irrigation activities on the study site, the use of an average soil moisture estimated at the scale of the study site (average mv calculated on all the bare soil agricultural plots) will lead to a strong overestimation of Hrms since the mv used for irrigated plots in the neural network for estimating Hrms will most probably be underestimated.

For these two configurations a priori information on mv is used in the network for estimating mv.

V.4.2.2.1 Using the IEM model

In the case of VV polarization alone, results show that the accuracy on the estimates of Hrms is similar in using the mv estimated at the study site scale (a few tens of km²) and in using the mv estimated at the plot scale. The RMSE is of 0.98 cm when the mv used correspond to mv estimated at the scale of the study site and of 0.81 cm when the mv is estimated at the plot scale (Figure V.24). Figure V.25 shows the result using VH alone. The accuracy on the estimates of Hrms is mostly the same in using the mv estimated at the study site scale (RMSE = 0.82cm) and in using the mv estimated at the plot scale (RMSE=0.74 cm). In the case of VV and VH used together, results show that the accuracy on the estimates of Hrms is better in using the mv estimated at the plot scale (RMSE = 0.81 cm ) than in using the mv estimated at

V.4.2.2.2 Using Baghdadi model

With VV polarization alone, the accuracy on the estimates of Hrms is mostly the same in using the mv estimated at the study site scale and in using the mv estimated at the plot scale with RMSE of 1.04 cm and 0.95 cm, respectively (Figure V.27). Figure V.28 shows the result according to VH polarization alone. The precision on the estimates of Hrms is similar in using the mv estimated at the study site scale is approximately the same as in using the mv estimated at the plot scale (RMSE of 0.92 cm and of 0.84 cm, respectively). Figure V.29 shows that the estimation of Hrms using VV and VH together is carried out with a precision about 1.0 cm for the two inversion configurations.

V.4.2.2.3 Discussion

Results obtained in using the NN built with a priori information on mv show estimates of Hrms with an RMSE higher than 0.7 cm. This accuracy on Hrms obtained shows that the use of mv estimates with an accuracy of about 6 vol.% is not sufficient to accurately estimate the soil roughness in C-band. From the real dataset, results show that the accuracy on Hrms estimates in using the mv estimated at the study site scale is similar to that in using the mv estimated at the plot scale. The use of mv estimated at the scale of the study site is possible only when the study site is not irrigated. In addition, results are similar using the neural networks trained with data simulated from IEM model and Baghdadi model.

This first study on the potential of Sentinel-1 data for estimating the soil roughness shows that the development of an automatic and generalizable inversion procedure of the C-band radar signal does not allow a pertinent estimation of the soil roughness. The accuracy on soil roughness estimates obtained in this study cannot satisfy the requirements of operational users of soil roughness products (in particular to modelers) because the need is at least three roughness classes: smooth (sowing), medium (small plowing) and rough (large plowing).

Only methods based on the use of experimental relationships, which are often difficult to apply to sites other than those for which they were developed and are generally valid only for specific soil conditions, allows the mapping of three roughness classes (Baghdadi et al., 2002a). Indeed, different experimental studies have revealed that the sensitivity of the radar signal to surface roughness (i.e. the slope of the regression lines) can be highly variable from one site to another. In addition, the experimental relationships between the radar signal and Hrms are established for a given incidence angle and a range of soil moisture. The soil composition could be also different from one site to another. All these reasons explain why the experimental relationships are not generalizable.

V.4.3 Estimation of Hrms and mv both at very high spatial resolution ʺVHSRʺ

In this second approach, soil roughness Hrms estimates are analyzed when the output of the neural network is both soil moisture and surface roughness at the same time. In this configuration, both VV and VH polarizations are used as input of the neural networks. The transfer function that are used in the Neural Networks for this configuration is Logsig.

Three neural networks are analyzed corresponding to the three cases of soil moisture conditions with and without a priori information on the soil moisture state:

 Case 1: No a priori information on the soil moisture state is available. In this case mv will be estimated between 2 and 40 vol.%.  Case 2: A priori information is available on mv. The soil is supposed to be dry to slightly wet according to expertise based mainly on meteorological data (precipitations, temperature). Soil moisture values are assumed to range from 2 to 25 vol.%.

 Case 3: A priori information is available on mv. The soil is supposed to be very wet according to expertise based on meteorological data. mv-values are assumed to vary between 25 and 40 vol.%.

The three NNs use the backscattering coefficients in VV and VH polarizations and the incidence angle as input. The NN outputs are the soil moisture mv and the surface roughness

Hrms. An overlapping of 10 vol.% on mv was used on the training datasets of the two networks in the case of a priori information on the soil moisture mv. So that, in the case of dry to slightly wet soils, the mv-values used for the training is ranged from 2 to 30 vol.%. In the case of very wet soils, the mv-values used for the training is ranged from 20 to 40 vol.%.

The different neural networks are tested for the evaluation of the precision on soil roughness and moisture estimates using synthetic and real datasets.

First, we will discuss the performance of networks for the estimation of mv. Then, the performance of the same networks for estimating Hrms is analyzed.

V. 4.3.1.1 Estimation of mv V.4.3.1.1.1 Using the IEM model

In the case of no a priori information on mv, the RMSE on the mv estimates is of 4.55 vol.% for mv between 2 and 25 vol.% and 6.37 vol.% for mv between 25 and 40 vol.%, with an overestimation of +2.14 vol.% and an underestimation of -3.22 vol.% on mv respectively for each mv range. For the entire range of mv, between 2 and 40%, the RMSE on mv is of 5.35 vol.% (Figure V.30a).

In the case where the NNs were trained using a priori information on mv with dry to slightly wet soil condition, the RMSE on mv estimates decreases from 4.55 vol.% without a priori information on mv to 3.40 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also reduced from +2.14 vol.% to +1.01 vol.% (Figure V.30b).

In addition, the use of a priori information on mv in the case of very wet soil conditions also improves the mv estimates. The RMSE on mv estimates decreases from 6.37 vol.% without a priori information on mv to 4.89 vol.% in the case of a priori information on mv. Also, the difference between estimated and measured mv is reduced from -3.23 vol.% to -1.96 vol.%

(Figure V.30c).
The performance of the inversion algorithm was analyzed according to Hrms and incidence angle "" (Figure V.31). Results show that the bias (estimated mv -measured mv) and the RMSE are strongly dependent on Hrms. The RMSE on mv in the case of inversion without a priori information on mv increases from 4.10 vol.% for Hrms=0.5 cm to 6.50 vol.% for Hrms = 3.8 cm for mv between 2 and 25 vol.% (bias increases from -8.8 to +1.2 vol.% for Hrms between 0.5 and 3.8 cm). In very wet soil conditions, the RMSE on mv decreases from 11.00 vol.% for Hrms=0.5 cm to 4.1 vol.% for Hrms = 3.8 cm (bias increases from -1.2 to 5.1 vol.% for Hrms between 0.5 and 3.8 cm). Moreover, results show that the RMSE on mv slightly depends on The RMSE on mv estimates is about 5 vol.% for  between 20° and 45°

for dry to slightly wet soil conditions ( overestimation of mv of +2.0 vol.%) and about 6 vol.% for  between 20° and 45° for very wet soils in the case of no a priori information on mv ( underestimation of mv of about -3.0 vol.%) (Figures V.31c and V.31d). In the case of a priori information on mv, the RMSE on mv estimates varies between 4.0 and 4.5 vol.% for all mv and Hrms values of the validation synthetic dataset for dry to slightly wet soil conditions (Figures V.31). With the use of a priori information on mv, the bias reduction varies between -1.0 vol. % (low Hrms) and +4.0 vol.% (high Hrms). In addition, RMSE and bias on mv estimates are slightly dependent on the incidence angle. The RMSE is about 3 vol. The analysis according to Hrms and incidence angle "" show that the bias (estimated mvmeasured mv) and the RMSE are strongly dependent on Hrms and incidence angle "" (Figure V.33). In the case of a priori information on mv, the RMSE on mv estimates varies between 5.1 and 6.0 vol.% for all mv and Hrms values considered in this study for dry to slightly wet soil conditions. In comparison to the case where no a prior information on mv is used, the use of a priori information shows that the bias reduction varies between -2.5 vol. % (low Hrms) and +4.0 vol.% (high Hrms). Finally, RMSE and bias on mv estimates are slightly dependent on the incidence angle "". The RMSE increases from 3.0 vol. % (for =20°) to 6.1 vol. % (for =45°). The bias on mv is approximately about +1.0 vol. % for "" between 20° and 45°.

In addition, Figure V.33 shows that the RMSE on mv estimates is well reduced in the case of a priori information on mv for very wet soil conditions (between 4.0 and 6.0 vol.%). The highest RMSE values correspond to low Hrms-values. The underestimation of mv is well reduced mainly for low Hrms-values from -8.80 vol.% without a priori information on mv to -4.5 vol.% with a prior information on mv for very wet soils. In addition, the analysis of the RMSE on mv estimates shows that the RMSE is well reduced mainly for high incidence angles (=45°) from 9.00 vol.% without a priori information on mv to 5.00 vol.% with a priori information on mv. V .4.3.1.1.3 Discussion The comparison between results obtained with IEM and Baghdadi model shows similar performances in the estimation of soil moisture.

Using IEM and Baghdadi model, the use of a priori information on mv strongly improves the estimation of mv. For Hrms between 1 and 2 cm (the range of surface roughness the most encountered in agricultural environments) results show that the RMSE on mv varies between 3.0 and 6.0 vol.% using the IEM model and between 3.9 and 5.5 vol.% using Baghdadi model. The difference between estimated and real mv in the case of dry to slightly wet soils varies between -1.0 and +1.0 vol.% using IEM model and between -0.5 and +2.0 vol.% using Baghdadi model. For very wet soils and Hrms between 1 and 2 cm, an underestimation of mv is observed using both IEM and Baghdadi model, from -5.0 to -2.5 vol.% and from -4.0 to -1.3 vol.%, respectively. The RMSE is 0.96 cm in the case of no a priori information on mv. Similar RMSE is obtained for dry to slightly wet conditions about 0.98 cm when a priori information on mv is used.

Lower RMSE is obtained with a priori information on mv in the case of very wet soils (RMSE=0.65 cm).

In addition Figure V.34 shows an overestimation of Hrms for Hrms lower than 2 cm and an underestimation of Hrms for Hrms higher than 2 cm.

V.5 Conclusions

The objective of this study was to investigate the potential of Sentinel-1 C-band SAR in several polarizations (VV alone, VH alone, VV and VH together) for estimating the soil roughness over bare agricultural areas using the neural networks technique (NNs). Neural networks were trained with radar backscattering coefficients generated from two models: the Integral Equation Model 'IEM' and the new semi-empirical model developed recently by Baghdadi et al. (2016). An additional simulated dataset and a real dataset composed of Sentinel-1 images and in situ measurements were then used to analyze the performance of the inversion technique for estimating the surface roughness (Hrms).

Two inversion configurations were proposed. The first based on estimation of soil roughness at very high spatial resolution ʺVHSRʺ (plot scale or on a finer scale). Two networks were applied one after the other, the first to estimate the soil moisture (mv) and the second using the soil moisture estimates for estimating the soil roughness. Three SAR configurations are tested: VV alone, VH alone, VV and VH together. In order to improve the soil parameters estimates, a priori knowledge about soil moisture mv (dry to slightly wet or very wet soil conditions) is introduced. Three neural networks are developed for the estimation of mv, with and without a priori information on the soil moisture state. Next, the soil roughness is estimated at a fine spatial scale (plot or sub-plot scale) using the soil moisture estimated by the first network. The second inversion configuration concerns the estimation of both soil roughness (Hrms) and soil moisture (mv) at very high spatial resolution ʺVHSRʺ. Both VV and VH polarizations together are used as inputs of these neural networks. Three neural networks are developed, with and without a priori information on the soil moisture state.

Using the first inversion configuration and using the two radar backscattering models, best results are obtained using the VV polarization alone for the IEM Model and the VV and VH polarizations together for Baghdadi model. The soil moisture could be estimated with an RMSE better than 6 vol.% when a priori information on mv is used in the neural network for the two models. The second neural network uses this estimation of mv in order to estimate the soil roughness at the plot scale. Results obtained show estimates of Hrms with an RMSE of 0.94 cm using the IEM model (VV polarization alone) and 0.78 cm using Baghdadi model (VV and VH polarizations together). This accuracy on Hrms obtained in using the NN built with a priori information on mv shows that the use of mv estimates with an accuracy of about 6 vol.% is not sufficient to accurately estimate the soil roughness in C-band. Results obtained from the real dataset show that the accuracy on Hrms estimates in using the mv estimated at the study site scale is better to that in using the mv estimated at the plot scale, with an RMSE on Hrms about 0.81 cm (RMSE=0.98 cm using mv estimated at plot scale) in using the IEM model (VV polarization alone) and an RMSE on Hrms about 1.03 cm (RMSE=1.05 cm using mv estimated at plot scale) in using Baghdadi model (VV and VH polarizations together).

For the second inversion configuration (VV and VH polarizations together), the use of simulated dataset from the Integral Equation Model 'IEM' or the new semi-empirical Baghdadi model show approximately similar results and close to the results obtained in the first inversion configuration. Using the real dataset, the soil moisture mv could be estimated with an RMSE better than 6.0 vol.% and 6.6 vol.% for the IEM and Baghdadi models, respectively when a priori information on mv is used in the neural networks. The RMSEs on Hrms are of 0.84 cm without a priori information on mv and 0.75 cm with a priori information on mv in using the IEM model. Using Baghdadi model, the RMSEs on Hrms are about 1.01 cm without a priori information on mv and 0.97 cm with a priori information on mv. The use

VI.1 General conclusion

The state of the soil surface and in particular through roughness and moisture exerts a fundamental influence on the distribution of rain between infiltration, surface retention and runoff. In addition, it has a key role in surface hydrological processes. The characterization and consideration of these surface conditions is currently an important issue for the physicalbased modeling of the processes of infiltration, runoff and erosion. The main objective of this thesis was to evaluate the potential of the new Sentinel-1 SAR for the mapping of surface roughness in bare agricultural areas.

In order to achieve this objective, several steps were followed. The first step was to evaluate the potential of the five most popular radar backscattering models (Oh, Dubois, IEM, AIEM and IEM modified by Baghdadi "IEM_B") using a wide dataset composed of AIRSAR, SIR-C, JERS-1, PALSAR-1, ESAR, ERS, RADARSAT, ASAR, TerraSAR-X, CosmoSky-Med, Sentinel-1 acquisitions over numerous agricultural sites in France, Italy, Germany, Belgium, Luxembourg, Canada and Tunisia. In addition, in situ measurements of soil moisture and surface roughness were carried out simultaneously to SAR acquisitions over bare soil surfaces (soil moisture "mv" ranged between 2 vol. % and 47 vol. %, surface roughness "Hrms" between 0.2 cm and 9.6 cm which corresponds to kHrms from 0.2 and 13.4). The SAR sensors used are in L-, C-and X-bands with incidence angle between 18° and 57°. Results

showed that the IEM modified by Baghdadi "IEM_B" using a fitting parameter instead of measured correlation length provides the most accurate SAR simulations with bias (real data -model simulations) lower than 1.0 dB and Root Mean Square Error "RMSE" lower than 2.0 dB. The IEM_B model showed slightly better performance in X-band (RMSE = 1.8 dB) than in L-and C-bands (RMSE between 1.9 and 2.3 dB). The IEM and AIEM models showed better simulations of measured backscattering coefficients using exponential correlation function (RMSE of 5.6 dB for HH and 6.5 dB for VV using the IEM model; RMSE of 4.4 dB for HH and 3.8 dB for VV using the AIEM model) than in using Gaussian correlation function (RMSE about 10 dB for both HH and VV and in using both IEM and AIEM models).

The Oh models showed good results in simulations with slightly better performance of the Oh 1992 version with bias less than 1.0 dB and RMSE of 2.6 dB and 2.4 dB respectively for HH and VV. The Oh model showed better agreement in simulations than Dubois model which simulates the backscattering in HH polarization with RMSE of 4.0 dB, and slightly better simulations for VV polarization with RMSE of 2.9 dB. The neural networks were trained using synthetic dataset simulated by the IEM model calibrated by Baghdadi "IEM_B" and the new semi-empirical model developed in the previous step (Baghdadi model). For this purpose, the inversion approach was then validated in using both synthetic and real dataset. The synthetic datasets consist of a wide range of surface roughness "Hrms" and soil moisture "mv" simulated from IEM-B and Baghdadi models. The real dataset used in the validation of the inversion approach consists of C-band Sentinel-1 images (one in France and one in Tunisia) and in-situ measurements. This work was done in order to evaluate the potential of Sentinel-1 SAR sensors for retrieving soil roughness in several polarizations (VV alone, VH alone, VV and VH together) using the neural networks technique (NNs).

In order to achieve this last step and to estimate the soil roughness, two inversion configurations were proposed. The first configuration is composed of two consecutive neural networks (NNs) that are applied consecutively, the first to estimate the soil moisture (mv) and the second using the soil moisture estimates for estimating the soil roughness (Hrms). Three cases were tested: VV alone, VH alone, VV and VH together. The second inversion configuration uses a single NN to estimate both soil roughness (Hrms) and soil moisture (mv).

Both VV and VH polarizations are used as inputs of the neural network. A priori knowledge about soil moisture mv (dry to slightly wet or very wet soil conditions) was also introduced in the two inversion configurations in order to improve the soil parameters estimates (soil moisture and surface roughness).

The use of the first inversion configuration showed better estimation of mv and Hrms in using VV alone and the synthetic dataset simulated from the IEM-B model. When Baghdadi model was used to simulate the synthetic dataset, better results were obtained in using VV and VH together. Results showed that the soil moisture (mv) could be estimated at the plot scale with an RMSE better than 6 vol.% using a priori information on mv. Then, this estimated soil moisture "mv" was used in order to estimate the soil roughness at the plot scale. Results

showed an estimation of Hrms with an RMSE of 0.94 cm using the IEM-B model with VV polarization alone and 0.78 cm using Baghdadi model with both VV and VH. This result is not sufficient to accurately estimate the soil roughness in C-band. Using the real dataset, results showed that the RMSE on Hrms is of 0.98 cm in using the IEM model (VV alone) and of 1.05 cm in using Baghdadi model (VV and VH together).

The use of the second inversion configuration (VV and VH together for estimating both mv and Hrms by the same neural network) showed close performances in using synthetic datasets simulated from the two backscattering models. Moreover, the performances are similar with the two inversion configurations. Using the real dataset, the soil moisture mv could be estimated with an RMSE about 6.0 vol.% and 6.6 vol.% respectively for the IEM-B and Baghdadi models, when a priori information on mv is used in the neural networks. Using a priori information on mv and using the IEM_B model, the RMSEs on Hrms are of 0.84 cm without a priori information on mv and 0.75 cm with a priori information on mv. Using Baghdadi model, the RMSEs on Hrms are 1.01 cm without a priori information on mv and 0.97 cm with a priori information on mv.

In general, at least three roughness classes (smooth, medium and rough) are needed in order to fulfill the operational users of soil roughness products. This first study on the potential of Sentinel-1 data for estimating the soil roughness didn't satisfy these requirements. Results

show that the inversion procedure of the C-band radar signal does not permit an accurate estimation of the soil roughness. Thus, the three roughness classes are not achievable with an incidence of about 40° (nominal incidence of Sentinel-1). Indeed, the incidence angles used in this study, between 37° and 41° using VV, VH or VV and VH together, are not optimal for the characterization of the soil roughness. [START_REF] Fung | Microwave Scattering and Emission Models and their Applications[END_REF] has shown that HH polarization is more sensitive to soil roughness than VV for high incidence angles. Moreover, Baghdadi et al., (2002a) indicated that the high incidence angles (about 45°) are more suitable to discriminate various roughness classes (smooth, medium and rough) over bare agricultural fields. So that, the results obtained by the Sentinel-1 SAR sensor for the estimation of surface roughness are logical with such medium incidence angles. Moreover, the use of two polarizations does not improve the estimation of the soil roughness.

VI.2 Perspectives

In general, the work of this thesis focused on several research sectors (evaluation, modeling and estimation of soil parameters). This thesis showed the potential of radar images (i.e.

Sentinel-1 radar sensor) to estimate soil surface parameters. The results obtained could be used as a guide to support research in several fields.

Several research fields emerge from this work. These axes relate, on the one hand, to the potential of SAR sensors to estimate soil parameters (soil moisture and surface roughness) and, on the other hand, the best use the SAR's instrumental parameters (i.e. wavelengths) and configurations (i.e. incidence angles) in order to attain better accurate estimations.

The results obtained in this thesis lead to indicate that the C-band Sentinel-1 SAR data with incidence angle about 40° are not suitable to retrieve surface roughness. So, It is essential to further develop multi-sensor methods combining radar data acquired from two radar wavelengths (L and C bands, or C and X bands). Indeed, the three frequencies L, C and X have complementary capabilities for estimating soil moisture and surface roughness. As the radar signal in X-band is more sensitive to soil moisture than the radar signal in C-band (accuracy on soil moisture estimates in X-band is twice greater than the once obtained in Cband, [START_REF] Aubert | Analysis of TerraSAR-X data sensitivity to bare soil moisture, roughness, composition and soil crust[END_REF]), it might be relevant to combine SAR data in C and X bands to estimate both soil parameters (i.e. soil moisture and surface roughness). The X-band data will be used to estimate soil moisture and the C-band data will be used to estimate the surface roughness. The use of the higher radar wavelengths (L-band) may also be relevant due to the high potential of higher wavelengths for the estimation of soil roughness. In this context, different spatial missions with L band SAR systems are in preparation, as NISAR (NASA and ISRO mission). European Space Agency also discusses the possibility of adding a new L band SAR to the constellations of Sentinels. The arriving of these missions will open a serious opportunity to develop studies based on the combination of multi-sensor acquired data, particularly with Sentinel-1 & Sentinel-2 missions. The multi-sensor, and particularly multifrequency algorithms will be essential to reach the operational algorithms which are able to separate effects of different surface parameters on radar signals. This is particularly the case of covered vegetation surface, for which, retrieving of roughness is still very complicated with one frequency configuration. The final objective is to reach in next years the assimilation of roughness maps, as for soil moisture in different surface process models, as runoff and erosion models. Cette première étude sur le potentiel des données Sentinel-1 (incidence de l'ordre de 40°) pour l'estimation de la rugosité du sol montre que le développement d'une procédure d'inversion automatique et généralisable du signal radar en bande C ne permet pas une estimation pertinente de la rugosité du sol. Les estimations de la rugosité du sol obtenues dans cette étude ne peuvent pas satisfaire les exigences des utilisateurs de la rugosité du sol (en

Frequency

  Figure II.1. (a) The Thetaprobe instrument. (b) Thetaprobe calibration curve, black circles represent moisture measurements.

Figure II. 2 .

 2 Figure II.2. Roughness profile made by a needle profile-meter in the parallel direction to the furrows (1D profile from a pin profilometer: 1 m long and a sampling interval of 2 cm) (Baghdadi and Zribi, 2016).

Figure II. 3 .

 3 Figure II.3. Examples of roughness profile: 3D profile from a laser scanner (Baghdadi and Zribi, 2016).

   = -2D+4, where D is the fractal dimension. When the fractal dimension varies, the shape of the autocorrelation function changes: it goes from an exponential function from for D=1.5 to a Gaussian shape for D=1. The experimental measurements show a fractal dimension between 1.25 and 1.45, where an autocorrelation function power  between 1.1 and 1.5.

Figure II. 4 .

 4 Figure II.4. Example of a correlation function, L is the correlation length.

Figure

  Figure II.5. Sensitivity of the radar signal to soil roughness. SAR sensors in C and L bands were used. The signal is represented as a function of kHrms(Baghdadi and Zribi, 2016).

  of the radar signal to soil moisture The radar signal approximately follows a logarithmic law with soil moisture. Moreover, this logarithmic function represented approximately as a linear function for soil moisture between 10 and 35 vol. % (Figure II.6). When the soil moisture increases than about 35 vol. %, the

Figure II. 6 .

 6 Figure II.6. Sensitivity of the radar signal in C and X bands to soil moisture. The Hrms values vary between 0.5 and 1.5 cm. (a): C-HH(20°-24°), (b): C-HH(43°-45°) (c): X-HH(25°-28°), (d): X-HH(50°-52° )(Baghdadi and Zribi, 2016).

  developed between 1992 and 2004 several versions of a semi empirical backscattering model. Basing on theoretical models, scatterometer measurements and airborne SAR observations, the Oh model is built over a wide variety of bare soil surfaces. The Oh model relates the co-polarized ratio p (incident angle (θ), wave number (k), standard deviation of surface height (Hrms), correlation length (L), and soil moisture (mv) or dielectric constant ( r  ).

  is optimized for bare soils in the following validity domain: 0.13 ≤ kHrms ≤ 6.98, 4 ≤ mv (vol. %) ≤ 29.1, and 10° ≤ θ ≤ 70°.The estimation of soil moisture and surface roughness from Oh model requires two backscattering coefficients at least, with one co-polarized coefficient (

  transform of the nth power of the surface correlation )

  low surface roughness values and Gaussian for high surface roughness values.

.Figure III. 1 .Figure III. 2 .

 12 Figure III.1. Comparison between backscattering coefficient values obtained from SAR images and those estimated from the Dubois model at HH polarization. (a) Dubois model simulations vs. SAR data; (b) difference between SAR signal and the Dubois model vs. soil roughness (kHrms); (c) difference between SAR signal and the Dubois model vs. soil moisture (mv); (d) difference between SAR signal and Dubois model vs. incidence angle.

  ). For θ < 30° (outside the validity domain of the Dubois model), the Dubois model overestimates the radar signal by -4.2 dB in HH polarization (RMSE = 5.5 dB) and slightly underestimates the radar signal in VV polarization (real data -simulations = -0.6 dB) with a RMSE of 2.9 dB. At θ > 30°, the Dubois model correctly simulates the backscattering coefficient in HH pol. with a difference between real data and model of 0.3 dB at HH polarization and underestimates the backscattering at VV pol. by about 1.5 dB (RMSE = 3.2 dB and 2.9 dB for HH and VV polarizations, respectively).

Figure III. 3 .Figure III. 4 .FigureFigureFigure

 34 Figure III.3. Comparison between backscattering coefficients derived from SAR images and those estimated from the Oh 1992 model at HH polarization, (a) Oh model simulations vs. SAR data; (b) difference between SAR signal and Oh model results vs. soil roughness (kHrms); (c) difference between SAR signal and Oh model results vs. soil moisture (mv); (d) difference between SAR signal and Oh model results vs. incidence angle.

FigureFigureFigureFigureFigure

  Figure III.12. Comparison between backscattering coefficients derived from SAR images and those estimated from IEM at HH polarization using GCF. (a) IEM simulations vs. SAR data; (b) difference between SAR signal and IEM vs. soil roughness (kHrms); (c) difference between SAR signal and IEM vs. soil moisture (mv); (d) difference between SAR signal and IEM vs. incidence angle.

FigureFigureFigure

  Figure III.18. Comparison between backscattering coefficients derived from SAR images and those estimated from IEM_B at HH polarization using GCF. (a) IEM_B simulations vs. SAR data; (b) difference between SAR signal and IEM_B vs. soil roughness (kHrms); (c) difference between SAR signal and IEM_B vs. soil moisture (mv); (d) difference between SAR signal and IEM_B vs. incidence angle.

FigureFigureFigureFigureFigure

  Figure III.21. Comparison between backscattering coefficients derived from SAR images and those estimated from AIEM at HH polarization using GCF. (a) AIEM simulations vs. SAR data; (b) difference between SAR signal and AIEM vs. soil roughness (kHrms); (c) difference between SAR signal and AIEM vs. soil moisture (mv); (d) difference between SAR signal and AIEM vs. incidence angle.

  correlation function (RMSE higher than 9.2 dB) for both HH and VV polarizations because of the high sensitivity of the Gaussian correlation function to roughness parameters, mainly for kHrms < 3 and L > 4 cm. Moreover, results show better simulations of measured backscattering coefficients for both IEM and AIEM using exponential correlation function (RMSE > 5.6 dB for IEM and RMSE > 3.8 dB for AIEM) at HH and VV polarizations. At HV polarization, IEM results show very high errors (RMSE larger than 30.0 dB using both Gaussian correlation function and exponential correlation function). The AIEM better simulates the backscattering than the original IEM only using the exponential correlation function with slightly better results in X-band than in C-and L-bands. In contrast, the IEM simulates better the backscattering in L-band than C-and X-bands (

  The simple formulations of Dubois model that directly relates the radar signal to soil (dielectric constant and soil roughness) and SAR parameters (incidence angle, polarization and radar wavelength) lead us to select the formulations of Dubois model as basic of the new model. The formulations of Oh model was not used because only the co-polarized ratio p and cross-polarized ratio q are available.First, the performance of the Dubois model is analyzed using a large dataset acquired at several worldwide study sites by numerous SAR sensors. The dataset consists of SAR data (multi-angular and multi-frequency) and measurements of soil moisture and surface roughness over bare soils. Then, the different terms of Dubois equations that describe the dependence between the SAR signal and both sensor and soil parameters have been validated or modified to improve the modelling of the radar signal. Ultimately, a new semi-empirical backscattering model has been developed for radar scattering in the HH, VV, and HV polarization from bare soil surfaces.

Figure IV. 1 .Figure IV. 2 .

 12 Figure IV.1. For HH polarization, (a) comparison between radar backscattering coefficients calculated from SAR images and estimated from the Dubois model, (b) difference between the SAR signal and the Dubois model relative to soil roughness (kHrms), (c) difference between the SAR signal and the Dubois model relative to soil moisture (mv), (d) difference between the SAR signal and the Dubois model relative to incidence angle. The best regression model is ploted in gray.

  and modelled radar signal. The error in the modelling of radar backscatter coefficients by the new backscattering model was assessed for each polarization using a 5-fold cross-validation to validate the predictive performance of the new model. To do the 5-fold cross-validation, the dataset was first randomly divided into 5 equal size subsets. Next, 4 of the subsets are used to train the new model and one was retained to validate its predictive performance. The cross-validation process was then repeated 5 times, with each of the 5 sub-datasets used exactly once as the validation data. The final validation result combines the 5 validation results. The advantage of this method over repeated random sub-sampling is that all observations are used for both training and validation, and each observation is used for validation exactly once.

Figure IV. 3 .Figure

 3 Figure IV.3. (a) Comparison between ° modelled in the new model and ° measured (for all SAR bands) for HH polarization, (b) difference between SAR and the new model as a function of surface roughness (kHrms), (c) difference between SAR and the new model as a function of soil moisture (mv), (d) difference between SAR and the new model as a function of incidence angle. The best regression model is ploted in gray.

Figure IV. 6 .

 6 Figure IV.6. Behavior of the new model as a function of incidence angle, surface roughness (k Hrms) and soil moisture (mv) in HH, VV and HV polarizations.

  is to propose a new empirical model for radar backscatter from bare soil surfaces. The new model is based on the formulation made in the Dubois model where the radar signal in HH and VV polarizations is described according to radar wavelength, incidence angle, soil moisture and roughness. This new model is based on the formulation made in the Dubois model. A large dataset was used, composed of ground measurements and SAR images over bare agricultural soils. Results show that the new model provides improved results in comparison to the Dubois model (in the case of HH and VV). Biases and RMSE have decreased for both HH and VV polarizations. In addition, the high over-or under-estimations observed with the Dubois model for some ranges of soil moisture, surface roughness and radar incidence angle were clearly eliminated with the new model. Analysis of the new model's performance for each radar wavelength separately (L, C and X) shows that in the L-band, the performance of the new model was similar to that of the Dubois model (may be due to the few data used in Lband). The model shows significant improvement in C-and X-bands (RMSE approximately 1.9 dB with the new model and between 2.6 and 4.1 dB with the Dubois model).

  Baghdadi et al., (2002a) investigated the potential of the first generation of SAR data (ERS-2 and RADARSAT-1) for monitoring roughness states over bare agricultural fields. Results indicate that high incidence angles (about 45°) are more suitable to discriminate various roughness classes (smooth, medium and rough) over bare agricultural fields. An algorithm based on an experimental exponential relationship between the radar backscattering coefficient and the surface roughness (root mean square surface height, Hrms) independently of the soil moisture was used. Next,Baghdadi et al. (

  Figure V.1. Location of the two study sites, (a): location of Versailles in France. (b): location of Kairouan in Tunisia.

  was used to train the Neural Networks. The Neural Networks (NN) architecture is created from three layers: input, hidden, and output. The NNs have two dimensional input vectors when using one polarization (VV or VH) which are the backscattered signal and the incidence angle. Using two polarizations (VV and VH), the NNs have three dimensional input vectors which are the two backscattered signals (VV and VH) and the incidence angle. In order to estimate only mv or Hrms, the output vector contains only the soil moisture (mv) or the soil surface roughness (Hrms). When the estimation concern both Hmrs and mv, the two dimensional output vector contains both soil moisture and surface roughness. The numbers of neurons associated with the hidden layer was determined by training the Neural Networks using different numbers of neurons. 20 hidden neurons provided accurate estimates of reference parameters(Baghdadi et al., 2012a;[START_REF] Chai | Use of soil moisture variability in artificial neural network retrieval of soil moisture[END_REF]. To develop a neural network, it is necessary to train the network with training synthetic dataset composed of input and output vectors. Training is accomplished to minimize the mean square error between the predicted Neural Networks outputs and the reference values. All transfer functions were tested in the Neural Networks which give different results. Best results are shown by the Purlin and Tansig transfer functions for the estimation of soil moisture and Logsig for the estimation of the surface roughness.

  slightly wet according to expertise based mainly on meteorological data (precipitations, temperature). Soil moisture values are assumed to range from 2 to 25 vol.%.  Case 3: A priori information is available on mv. The soil is supposed to be very wet according to expertise based on meteorological data. mv-values are assumed to vary between 25 and 40 vol.%. The three NNs use the backscattering coefficient in each SAR configuration (VV polarization alone, VH polarization alone, VV and VH polarizations together) and the incidence angle as input. The output is only the soil moisture mv. An overlapping of 10 vol.% on mv was used on the training datasets of the two networks in the cases of a priori information on the soil moisture mv. So that, in the case of dry to slightly wet soils, the mv-values used for the training ranged from 2 to 30 vol.%. In the case of very wet soils, the mv-values used for the training ranged from 20 to 40 vol.%. Next, the soil roughness could be estimated at a fine spatial scale (plot or sub-plot scale) using the soil moisture estimated by the first network. The standard acquisition mode of Sentinel-1 corresponds to acquisitions in both VV and VH polarizations. The Neural Networks used to estimate the soil roughness use the backscattering coefficient in VV alone, VH alone, VV and VH together and the incidence angle and the estimated soil moisture as input. The output is

Figure V. 2 .

 2 Figure V.2. Box plots of mv estimates retrieved from the synthetic dataset generated using IEM. Neural networks were trained and validated using VV polarization alone. (a): no a priori information on mv; (b): with a prior information on mv and dry to slightly wet soil conditions; (c): with a prior information on mv and very wet soil condition.

  estimates varies between 4.3 and 7.0 vol.% for all mv and Hrms values of the validation synthetic dataset in the case of very wet conditions. The highest RMSE-values correspond to low Hrms-values. The bias is also well reduced mainly for low Hrms-values from -6.0 vol.% to -1.0 vol.% for Hrms-values of 3.8 cm. The analysis of the RMSE and the bias shows relatively close values according to the incidence angle. The RMSE is about 5.00 vol.% for incidence angle between 20° and 45° and the bias is about -2.5 vol.% for incidence angle between20° and 45° (Figures V.3c and V.3d).

Figure V. 3 .

 3 Figure V.3.Accuracy on the mv estimates (RMSE and bias "=estimatedmeasured") retrieved from the synthetic dataset in VV polarization using IEM. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

Figure V. 4 .

 4 Figure V.4. Box plots of mv estimates retrieved from the synthetic dataset generated using IEM. Neural networks were trained and validated using VH polarization alone. (a): no a priori information on mv; (b): with a prior information on mv and dry to slightly wet soil conditions; (c): with a prior information on mv and very wet soil condition.

  For VH polarization alone(Figure V.5), the performance analysis of the inversion algorithm shows that the bias (estimated mv -measured mv) and the RMSE on mv are strongly dependent on Hrms (Figures V.5a and V.5b). The RMSE on mv in the case of inversion without a priori information on mv increases from 6.60 vol.% for Hrms=0.5 cm to 7.0 vol.% for Hrms = 3.8 cm for mv between 2 and 25 vol.% (dry to slightly wet soils). In very wet soil conditions, the RMSE on mv decreases from 19.0 vol.% for Hrms=0.5 cm to 4.0 vol.% for Hrms = 3.8 cm. The high RMSE values of in the case of very wet conditions and low Hrms-values are due to an underestimation of mv (bias increases from -19.0 to 0.0 vol.% for Hrms between 0.5 and 3.8 cm). Similarly, the high RMSE values in the case of dry to slightly wet conditions and low

to

  Hrms and  (Figure V.5). The RMSE on mv estimates varies between 3.0 and 7.5 vol.% for all mv and Hrms values of the validation synthetic dataset (case of dry to slightly wet conditions). The bias reduction varies between -6.0 vol. % for low Hrms-values and +4.0 vol.% for high Hrms-values. In addition, RMSE and bias on mv estimates are slightly dependent on the incidence angle.In the case of a priori information on mv with very wet soil conditions, the RMSE on mv estimates varies between 3.0 and 7.0 vol.% for all mv and Hrms values of the validation synthetic dataset (case of very wet conditions). The highest RMSE-values correspond to low Hrms-values. The bias is also well reduced mainly for low Hrms-values from -6.0 vol.% for Hrms-values of 0.5 cm to -1.0 vol.% for Hrms-values of 3.8 cm. The analysis of the RMSE and the bias shows relatively close values according to the incidence angle. The RMSE is about 5.00 vol.% for incidence angle between 20° and 45° and the bias is about -2.5 vol.% for incidence angle between 20° and 45° (Figures V.5c and V.5d).

Figure V. 5 .

 5 Figure V.5. Accuracy on the mv estimates (RMSE and Bias "=estimatedmeasured") retrieved from the synthetic dataset in VH polarization using IEM. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

  between 25 and 40 vol.%. An overestimation of +2.vol.% on mv is observed for mv between 2 and 25 vol.%, and an underestimation of -3.23 vol.% is obtained for mv between 25 and 40 vol.%. For the entire range of mv, between 2 and 40%, the RMSE on mv is of 5.19 vol.% (Figure V.6a).

Figure V. 6 .

 6 Figure V.6. Box plots of mv estimates retrieved from the synthetic dataset generated using IEM. Neural networks were trained and validated using VV and VH polarization together. (a): no a priori information on mv; (b): with a prior information on mv and dry to slightly wet soil conditions; (c): with a prior information on mv and very wet soil condition.

  Figure V.7). Results show that the RMSE on mv in the case of inversion without a priori information on mv increases from 4.80 vol.% for Hrms=0.5 cm to 7.0 vol.% for Hrms = 3.8 cm in dry to slightly wet soil conditions (mv between 2 and 25 vol.%) (Figures V.7a and V.7b). In very wet soil conditions, the RMSE on mv decreases from 12.50 vol.% for Hrms=0.5 cm to 4.0 vol.% for Hrms = 3.8 cm. The high RMSE values in the case of very wet conditions and low Hrms-values are due to an underestimation of mv (bias increases from -5.0 to +5.0 vol.% for Hrms between 0.5 and 3.8 cm). Similarly, the high RMSE values in the case of dry to slightly wet conditions and low Hrms-values are due to an overestimation of mv (bias increases from -12.0 to 0.0 vol.% for Hrms between 0.5 and 3.8 cm). Moreover, results show that the RMSE on mv slightly depends on  in the case of no a priori information on mv in the inversion process in the case of VV and VH polarizations together (Figures V.7c and V.7d). The RMSE is approximately about 6.0 vol.% for  between 20° and 45° for dry to slightly wet soil conditions and about 4.50 vol.% for  between 20° and 45° for very wet soils. The overestimation of mv in dry to slightly conditions is approximately +2.0 vol.% for  between 20° and 45°. For very wet soil conditions, the underestimation of mv is approximately about -3.0 vol.% for  between 20° and 45°.

Figure V. 7 .

 7 Figure V.7.Accuracy on the mv estimates (RMSE and Bias "=estimatedmeasured") retrieved from the synthetic dataset in VV and VH polarizations together using IEM. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

Figure V. 8 .

 8 Figure V.8. Box plots of mv estimates retrieved from the synthetic dataset generated using Baghdadi model. Neural networks were trained and validated using VV polarization alone. (a): no a priori information on mv; (b): with a prior information on mv and dry to slightly wet soil conditions (mv between 2 and 25 vol.%); (c): with a prior information on mv and very wet soil condition (mv between 25 and 40 vol.%).

  Figure V.10. Box plots of mv estimates retrieved from the synthetic dataset generated using Baghdadi model. Neural networks were trained and validated using VH polarization alone. (a): no a priori information on mv; (b): with a prior information on mv and dry to slightly wet soil conditions (mv between 2 and 25 vol.%); (c): with a prior information on mv and very wet soil condition (mv between 25 and 40 vol.%).

  Figure V.11. Accuracy on the mv estimates (RMSE and bias "=estimatedmeasured") retrieved from the synthetic dataset in VH polarization using Baghdadi model. Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

  98 vol.% to +1.11 vol.% (Figure V.12c). The RMSE on mv estimates in the case of very wet soil conditions decreases from 6.14 vol.% without a priori information on mv to 4.24 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also well reduced from -2.94 vol.% to -1.52 vol.% (Figure V.12c). Box plots of mv estimates retrieved from the synthetic dataset generated using Baghdadi model. Neural networks were trained and validated using VV and VH polarization together. (a): no a priori information on mv; (b): with a prior information on mv and dry to slightly wet soil conditions (mv between 2 and 25 vol.%); (c): with a prior information on mv and very wet soil condition (mv between 25 and 40 vol.%).

  Figure V.14. Box plots of Hrms (cm) retrieved from the synthetic dataset generated from the IEM model in using VV polarization. (a) the input mv to the NN corresponds to exact mv (those that are in the validation dataset without estimation error), (b) the input mv to the NN corresponds to mv estimated by the NN built for estimating mv without a priori information on mv, (c) the input mv to the NN corresponds to mv estimated by the NN built for estimating mv with a priori information on mv.

C

  Figure V.16c).

  Figure V.19. Retrieved mv versus in situ measurements in using the IEM model. (a): using VH without a priori information on mv; (b): using VH with a priori information on mv. Each point corresponds to one reference plot.

  Figure V.20. Retrieved mv versus in situ measurements in using the IEM model. (a): using VV and VH together without a priori information on mv; (b): using VV and VH together with a priori information on mv. Each point corresponds to one reference plot.

  Figure V.21. Retrieved mv versus in situ measurements in using Baghdadi model. (a): using VV without a priori information on mv; (b): using VV with a priori information on mv. Each point corresponds to one reference plot.

Figure V. 22

 22 Figure V.22 shows the results obtained for the estimation of mv in the case of VH polarization alone. Better results are obtained when a priori information on mv is used (RMSE=6.45 vol.%with a priori on mv and RMSE=7.97 vol.% without a priori on mv). Figure V.23 shows the results obtained for the estimation of mv in the case of VV and VH polarizations together. An RMSE of 6.67 vol.% is obtained with the introduction of a priori information on mv and an RMSE of 8.25 vol.% in the case without a priori information on mv.

  Figure V.22. Retrieved mv versus in situ measurements in using Baghdadi model. (a): using VH without a priori information on mv; (b): using VH with a priori information on mv. Each point corresponds to one reference plot.

  Figure V.23. Retrieved mv versus in situ measurements in using Baghdadi model. (a): using VV and VH together without a priori information on mv; (b): using VV and VH together with a priori information on mv. Each point corresponds to one reference plot.

  24. Retrieved Hrms versus measured measurements in VV polarization alone using the NN trained with synthetic data simulated from IEM model. (a): the mv used at the input of the network corresponds to mv estimated at plot scale; (b): the mv used at the input of the network corresponds to mv estimated at the scale of the study site. (a) (b) Figure V.25. Retrieved Hrms versus measured measurements in VH polarization alone using the NN trained with synthetic data simulated from IEM model. (a): the mv used at the input of the network corresponds to mv estimated at plot scale; (b): the mv used at the input of the network corresponds to mv estimated at the scale of the study site. (a) (b) Figure V.26. Retrieved Hrms versus measured measurements in VV and VH polarizations together using the NN trained with synthetic data simulated from IEM model. (a): the mv used at the input of the network corresponds to mv estimated at plot scale; (b): the mv used at the input of the network corresponds to mv estimated at the scale of the study site.

  27. Retrieved Hrms versus measured measurements in VV polarization alone using the NN trained with synthetic data simulated from Baghdadi model. (a): the mv used at the input of the network corresponds to mv estimated at plot scale; (b): the mv used at the input of the network corresponds to mv estimated at the scale of the study site. (a) (b) Figure V.28. Retrieved Hrms versus measured measurements in VH polarization alone using the NN trained with synthetic data simulated from Baghdadi model. (a): the mv used at the input of the network corresponds to mv estimated at plot scale; (b): the mv used at the input of the network corresponds to mv estimated at the scale of the study site. (a) (b) Figure V.29. Retrieved Hrms versus measured measurements in VV and VH polarizations together using the NN trained with synthetic data simulated from Baghdadi model. (a): the mv used at the input of the network corresponds to mv estimated at plot scale; (b): the mv used at the input of the network corresponds to mv estimated at the scale of the study site.

  Figure V.30. Box plots of mv estimates retrieved from the synthetic dataset generated using IEM. Neural networks were trained and validated using VV and VH polarizations. The NNs outputs are Hrms and mv together. (a): no a priori information on mv; (b): with a prior information on mv and dry to slightly wet soil condition; (c): with a prior information on mv and very wet soil condition.

  soil conditions, the RMSE on mv estimates varies in the case of a priori information on mv between 4.1 and 7.0 vol.% for all mv and Hrms values of the validation synthetic dataset. The highest RMSE-values correspond approximately to low Hrms-values. The bias is also well reduced mainly for low Hrms-values (-4.0 vol.% for Hrms-values of 0.5 cm).

  Figure V.31. Accuracy on the mv estimates (RMSE and bias "=estimatedmeasured") retrieved from the synthetic dataset generated from IEM. VV and VH are the inputs of the NNs. The NNs outputs are Hrms and mv together. Three NNs are tested: without a priori information on mv, with a priori information on mv with dry to slightly wet soil conditions, with a priori information on mv with very wet conditions.

  Figure V.33. Accuracy on the mv estimates (RMSE and bias "=estimatedmeasured") retrieved from the synthetic dataset generated from Baghdadi model. VV and VH are the inputs of the NNs. The NNs outputs are Hrms and mv together. Three NNs are tested: without a priori information on mv, with a priori information on mv with dry to slightly wet soil conditions, with a priori information on mv with very wet conditions.

V. 4

 4 Figure V.34 shows the results for estimating the soil roughness using the synthetic dataset generated from the IEM model with VV and VH polarizations in input to neural networks.

  Figure V.34. Box plots of Hrms (cm) retrieved from the synthetic dataset generated from the IEM model in using VV and VH polarizations together. (a): without a priori information on mv, (b): with a priori information on mv and dry to slightly wet soils, (c): with a priori information on mv and very wet soils.

  Figure V.35. Box plots of Hrms (cm) retrieved from the synthetic dataset generated from Baghdadi model in using VV and VH polarizations together. (a): without a priori information on mv, (b): with a priori information on mv and dry to slightly wet soils, (c): with a priori information on mv and very wet soils.

  The second step was to produce a new semi-empirical backscattering model(Baghdadi model) for bare soils based on the formulation of Dubois model. The different terms of Dubois model equations that describe the dependence between the SAR signal and both sensor and soil parameters have been validated or modified to improve the modeling of the radar signal. The new radar backscattering model was developed for HH, VV and HV polarizations. Analysis of this new model showed very good performances in simulating the radar signal. Results showed that this new proposed model improved the performances of simulations (Biases and RMSEs were well decreased) in comparison to the Dubois model for both cases HH and VV polarizations. The radar signal in HV polarization which was not modeled in the Dubois model was added in the new model. The high over-and underestimations for some ranges of soil moisture, surface roughness and radar incidence angle observed with Dubois model were obviously eliminated with the new backscattering model. The performances of this new model in the L-band were similar to the Dubois model's simulations. For HH and VV, better results were observed with this new model in C-and Xbands (RMSE approximately about 1.9 dB), while the Dubois model simulated the radar backscattering signal with RMSE between 2.6 dB and 4.1 dB in C-and X-bands, respectively. Moreover, the difference between the new model simulations in HV polarization and the real SAR data show an RMSE of 2.1 dB. The last step was to develop an inversion procedure based on neural networks to estimate the soil surface roughness from C-band Sentinel-1 SAR data in the case of bare agricultural soils.

  Figure A1.1. Box plots of Hrms (cm) retrieved from the synthetic dataset generated from the IEM model in using VH polarization. (a) the input mv to the network corresponds to real mv, (b) the input mv to the network corresponds to mv estimated by the NN built for estimating mv without a priori information on mv, (c) the input mv to the network corresponds to mv estimated by the NN built for estimating mv with a priori information on mv.

4 5

 45 Figure A2.1. Box plots of Hrms (cm) retrieved from the synthetic dataset generated from Baghdadi model in using VV polarization alone. (a) the input mv to the network corresponds to real mv (those that are in the validation dataset), (b) the input mv to the network corresponds to mv estimated by the NN built for estimating mv without a priori information on mv, (c) the input mv to the network corresponds to mv estimated by the NN built for estimating mv with a priori information on mv.

) SIR-C 30 m × 30 m L 1994 HH: 1262 measurements 66 in L-band 766 in C-band 430 in X-band VV: 790 measurements 159 in L-band 411 in C-band 220 in X-band HV: 390 measurements 13 in L-band 313 in C-band 64 in X-band Orgeval (Fr)

) AIRSAR SIR-C JERS-1 30 m × 30 m L L; C L 1991 1994 1994 Site Texture Composition (Silt; Clay; Sand) Orgeval (Fr) (78%; 17%; 5%) Pays de Caux (Fr) (70%, 13%, 17%) Villamblain (Fr) (60%,30%, 10% ) Thau (Fr) (53%, 35%, 12%) Touch (Fr) (55%, 21%, 24%)

Table III.2. Description of the Texture Composition dataset

	(Silt; Clay; Sand) used in this
	study. "Fr": France, "It": Italy, "Ge": Germany, "Be": Belgium, "Lu": Luxembourg, "Ca":
	Canada, "Tu": Tunisia.

III.3 Description of the Backscattering Models III.3.1 The Semi-Empirical Dubois Model

	Dubois et al. (1995) proposed a semi-empirical model for simulating the backscattering
	coefficients in HH and VV polarizations (	0 HH  and 0 VV  ) over bare soils. The expression of
	0  and 0  depends on the incident angle (θ), the soil dielectric constant (ε, which is a
	HH	VV
	function of the soil moisture content), the soil roughness defined by the standard deviation of

Mauzac (Fr) (48%, 16%, 36%) Garons (Fr) (54%, 40%, 6% ) Kairouan (Tu) (11%, 32%, 57% ) Yzerons (Fr) (13%, 20%, 67% ) Versailles (Fr) (58%, 24%, 18%) Seysses (Fr) (50%, 16%, 34%) Chateauguay (Ca) (43%, 37%, 20%) Brochet (Ca) (43%, 37%, 20%) Alpilles (Fr) (54%, 40%, 6%) Sardaigne (It) (23%, 30%, 47%) Matera (It) (59%; 14%; 27%) Alzette (Lu) (50%; 30%; 20%) Dijle (Be) (84%; 12%; 4%) Zwalm (Be) (72%; 13%; 15%) Demmin (Ge) (25%; 7%; 68%) Montespertoli (It) (40%; 20%; 40%)

	surface height (Hrms), and the radar wavelength (λ = 2π/k where k is the wave number). The
	model optimized for bare soils according to the validity domain defined by kHrms ≤ 2.5, mv ≤
	35 vol. %, and θ ≥ 30° is expressed as:
		0 VV		10		. 2	35	   	sin cos	3 3	 	   	10	. 0	046 	tan		 k	Hrms	sin		 7 . 0 1 . 1 
	0	10	2	75 .		sin 1 5 5 . cos		10	. 0	028	tan		sin	 7 . 0 4 . 1 

  RMSE than L-and C-bands, the RMSE in HH pol. being about 10.6 dB for GCF and 8.3 dB for ECF. At VV polarization, the RMSE is 11.3 dB for GCF and 9.4 dB for ECF. At HV polarization, the IEM simulates the backscattering with high RMSE which is larger than 54.0 dB using both GCF and ECF.

	Model	Pol.	Statistics	All Data	L-Band	C-Band	X-Band	Inside the Validity Domain	Outside the Validity Domain
		HH	Bias (dB) RMSE (dB)	+0.8 10.5	-0.9 3.6	+0.7 11.2	+1.5 10.6	+2.6 12.4	-1.8 6.7
	IEM using GCF	HV	Bias (dB) RMSE (dB)	+17.2 +5.2 +11.8 +46.3 38.4 14.5 26.7 74.0	+18.0 28.5	+14.1 50.1
		VV	Bias (dB) RMSE (dB)	+0.4 9.2	-2.5 5.0	+0.7 8.6	+3.5 11.3	+1.2 11.5	-0.9 3.1
		HH	Bias (dB) RMSE (dB)	+0.8 5.6	+0.6 2.9	-1.0 4.1	+4.2 8.3	-1.2 3.2	+3.8 7.8
	IEM using ECF	HV	Bias (dB) RMSE (dB)	-15.8 +1.2 -19.9 31.4 6.8 25.1	0.0 54.4	-15.8 20.1	-17.1 44.3
		VV	Bias (dB) RMSE (dB)	+2.2 6.5	-1.3 3.5	+0.5 4.9	+6.7 9.4	-0.9 3.7	+7.1 9.4
		HH	Bias (dB) RMSE (dB)	-0.3 2.0	-0.1 2.3	-0.6 2.1	+0.3 1.8
	IEM_B with Lopt using GCF	HV	Bias (dB) RMSE (dB)			-1.3 3.1
		VV	Bias (dB) RMSE (dB)	+0.1 1.9	+0.2 2.3	0 1.9	+0.3 1.8
	AIEM using	HH	Bias (dB) RMSE (dB)	+2.3 12.2	-3.2 5.4	+2.9 13.4	+3.1 11.7
	GCF	VV	Bias (dB) RMSE (dB)	0.0 10.8	-4.1 5.9	+0.5 11.4	+0.5 11.0
	AIEM using	HH	Bias (dB) RMSE (dB)	-2.3 4.4	-3.0 4.4	-3.6 4.6	+0.2 4.2
	ECF	VV	Bias (dB) RMSE (dB)	-1.8 3.8	-2.4 4.4	-2.3 3.8	-0.7 3.7

Table III.5. Comparison between real data and IEM versions (original IEM model, IEM_B and AIEM) using both GCF and ECF. (1) all data; (2) for different SAR wavelength; (3) according to the validity domain of IEM. Bias = real data -model simulations.

  . In comparison to the original IEM, results show that the RMSE was significantly lower. Using the entire dataset, the IEM_B correctly simulates the backscattering at both HH and VV polarizations showing low differences between real data and model simulations (-0.3 dB for HH and +0.1 dB for VV) with approximately similar RMSE of about 2.0 dB (TableIII.5). Moreover, the evaluation of the IEM_B was tested separately for each

SAR band. Results show that the IEM_B correctly simulates the backscattering in comparison to the original model for all bands and in both HH and VV polarizations with a difference between real data and model simulations lower than 1.0 dB and with approximately similar RMSE between 1.8 and 2.3 dB (Table

III

.5). At HV polarization, the IEM_B slightly over-

  angles higher than35°. Moreover, Figures III.23 and III.24 show high underestimation of the radar signal (using ECF) in both HH and VV polarizations for points with high surface roughness (kHrms > 6), low mv-values (mv < 5 vol. %, and with low incidence angles (θ < 20°). FiguresIII.21 and III.22 show that some points show high

	.5, Figures III.21 and III.22). Moreover, results show
	better agreements of the AIEM with real data using ECF (Figures III.23 and III.24). Indeed,

the AIEM tends to overestimates the backscattering by about 2.3 dB at HH and 1.8 dB at VV (RMSE is 4.4 dB for HH and 3.8 dB for VV). Using the ECF, Figures III.23 and III.24 show high overestimations of the backscattering for low values of surface roughness (kHrms < 4) and for incidence

  Table III.5). version in HH and VV polarizations is better in C-and X-bands (bias between -1.2 and +0.4 dB with RMSE < 2.5 dB) than in L-band (bias > +2.0 with RMSE > 3.0 dB).

	It should be mentioned that the use of different in situ sampling methods and SAR acquisition
	techniques may also contribute to the modelling errors. Indeed, the datasets comprises both
	airborne and space-borne acquisitions, which may cause scaling effects. In addition, in situ
	data have been collected using different techniques, both regarding soil moisture (gravimetric
	and TDR, sometimes at different sampling depths) and roughness (different profile length and
	sampling intervals, and post-processing methods).
	This study evaluated the robustness of the most used backscattering models by means of
	Using the empirical models, all the Oh model versions show good agreements (RMSE < 3.0
	dB) with measured backscattering with slightly better performance of the Oh 1992 version
	(bias less than 1.0 dB and RMSE less than 2.6 dB) at both HH and VV polarizations. The Oh
	model provides better results than Dubois model which simulates the backscattering in HH

with RMSE of 4.0 dB, and slightly better simulations for VV with RMSE of 2.9 dB. At HV polarization, the Oh 2002 version correctly simulates the backscattering with difference between real and simulated data of about +0.7 dB and RMSE of 2.9 dB. The performance of the Oh 1992 statistical indices (Bias and RMSE). These statistical indices should guide in choosing the appropriate model for backscattering coefficients simulation. As it has been shown in the present study, the IEM modified by Baghdadi (IEM_B) was the most accurate model among the others. Thus, it is preferred to use the IEM_B in the inversion procedure of SAR backscattering coefficient in order to more accurately estimate soil moisture and roughness parameters.

  In addition, the agreement between Dubois model simulations and SAR data is analyzed according to soil roughness, moisture content and incidence angle (Figures IV.1 and IV.2). The results indicate a slight underestimation of the radar signal by the Dubois model in the case of kHrms lower than 2.5 (Dubois validation domain) for both HH and VV polarizations (Figures IV.1b, IV.2b; Table IV.1). For surfaces with a roughness kHrms greater than 2.5, an overestimation of the radar signal is obtained with the Dubois model in HH while the model works correctly in VV (Figures IV.1b, IV.2b; Table IV.1). Higher under-and overestimations are observed in HH than they are in VV (reach approximately 10 dB in HH).Analysis of the error as a function of soil moisture (mv) shows that for both VV-polarized data, whatever the mv-values, and HH-polarized data with mv-values higher than 20 vol.%, the observed bias between real and simulated data is small (Figures IV.1c and IV.2c; TableIV.1). However, a strong overestimation of the radar signal is observed by the Dubois model in HH for mv-values lower than 20 vol.% (-2.0 dB, TableIV.1).

		Dubois for HH	Dubois for VV
		Bias (dB)	RMSE (dB)	Bias (dB)	RMSE (dB)
	For all data	-0.7	3.8	+0.9	2.8
	L-band	-0.8	2.9	-0.2	2.3
	C-band	-0.6	3.7	+0.7	2.6
	X-band	-0.7	4.1	+2.0	3.2
	kHrms < 2.5	+0.4	3.4	+1.3	2.9
	kHrms > 2.5	-2.7	4.5	-0.1	2.5
	mv < 20 vol.%	-2.0	4.3	+0.9	2.8
	mv > 20 vol. %	+0.5	3.2	+0.9	2.8
	 < 30°	-4.1	5.4	-0.6	2.9
	 > 30°	+0.6	3.0	+1.5	2.7

Table IV.1. Comparison between the Dubois model and real data for all data and by range of kHrms, soil moisture (mv) and incidence angle (). Bias = real datamodel.

  Table IV.2. Comparison between the results obtained with the Dubois model and those obtained with the new model. Bias = realmodel.

		Dubois for HH and VV	New model
		Bias (dB)	RMSE (dB)	Bias (dB)	RMSE (dB)
	HH for all data	-0.7	3.8	0.4	2.0
	VV for all data	+0.9	2.8	0.0	1.9
	HV for all data	-	-	0.0	2.1
	HH, L-band	-0.8	2.9	-0.1	2.3
	HH, C-band	-0.6	3.7	+0.3	1.9
	HH, X-band	-0.7	4.1	0.7	1.9
	VV, L-band	-0.2	2.3	-0.1	2.7
	VV, C-band	+0.7	2.6	+0.1	1.9
	VV, X-band	+2.0	3.2	-0.4	1.8
	HV, L-band	-	-	-1.3	1.6
	HV, C-band	-	-	+0.2	2.2
	HV, X-band	-	-	-1.3	1.9

.2). In addition, the high over-or underestimations of radar backscattering coefficients observed with the Dubois model according to soil moisture, surface roughness and radar incidence angle are clearly eliminated with the new model (Figures IV.3 and IV.4).

  The standard mode of Sentinel-1 corresponds only to acquisitions in both VV and VH polarizations. For this reason, only VV and VH polarizations will be used in this chapter.

are used to generate the reference datasets for the inversion of SAR data by the neural networks (NN) technique. The IEM modified by Baghdadi and the new semi-empirical modified Dubois model are able to reproduce the radar signal at VV, HH and VH from SAR parameters (incidence angle and radar wavelength) and soil surface characteristics (soil moisture and surface roughness).

SAR sensor Incidence angle (°) Dates (dd/mm/yyyy) Number of data

	French site Sentinel-1	~37°	15/03/2017 ; 27/03/2017	24 measurements
				02/04/2017 ; 08/04/2017
	Tunisian	Sentinel-1	~39° to 41° 18/12/2015 ; 04/02/2016	85 measurements
	site			03/04/2016 ; 04/04/2016
				23/12/2016 ; 05/01/2017
				09/02/2017
	Table V.1. Description of the real dataset used in this study for validating the inversion
	approach.

Le travail de thèse se structure en trois parties. La première partie consiste à évaluer les modèles de rétrodiffusion de radar les plus utilisés (IEM, Oh, Dubois and AIEM) en utilisant un large ensemble de données de SAR et des mesures expérimentales des paramètres du sol. Cette évaluation permet de trouver le modèle de rétrodiffusion le plus robuste qui simule le mieux le signal radar afin de l'utiliser par la suite dans la procédure d'inversion du signal radar pour estimer la rugosité du sol. Le deuxième axe de recherche de cette thèse consiste à proposer un modèle de rétrodiffusion radar semi-empirique pour les polarisations HH, HV et VV. Ce nouveau modèle sera construit à l'aide d'une grande base de données réelle. Ce nouveau modèle sera également utilisé dans la procédure d'inversion du signal radar pour estimer la rugosité du sol. Le dernier axe de cette thèse consiste à construire une méthode d'inversion du signal radar en utilisant les réseaux de neurones afin d'évaluer le potentiel des données Sentinel-1 pour l'estimation de la rugosité des sols en milieux agricoles. Ces réseaux de neurones seront entraînés à l'aide d'un ensemble de données synthétiques élaborées à partir des modèles de rétrodiffusion radar choisis (IEM calibré par Baghdadi et du nouveau modèle proposé) et validés en utilisant deux ensembles de données: un ensemble de données synthétiques et une base de données réelle (images Sentinel-1 et mesures in situ d'humidité et de rugosité du sol). La base de données réelle a été collectée en Tunisie (Kairouan) et en France (Versailles).

II. Chapter 2: Generalities

IV. Chapter 4: A New Empirical Model for Radar

Scattering from Bare Soil Surfaces V. Chapter 5: Estimation of soil roughness using neural networks from sentinel-1 SAR data

V.4.1 Synthetic dataset

In this first approach, the estimation of soil roughness (Hrms) requires the use of an estimate of the soil moisture (mv). First, we will discuss the performance of networks developed for the estimation of mv. Then, network built for estimating Hrms is analyzed.

V.4.1.1 Estimation of mv V.4.1.1.1 Using the IEM model

In order to estimate the soil moisture mv, three radar configurations will be tested: VV alone, VH alone, VV and VH together.

V.4.1.1.1.1 Use of VV polarization alone

First the results are discussed in using the synthetic dataset simulated from the Integral Equation Model (IEM). In the case of VV polarization and mv between 2 and 40 vol.%, the RMSE on the mv estimates is of 4.89 vol.% for mv between 2 and 25 vol.% and 6.64 vol.% for mv between 25 and 40 vol.%. An overestimation of +2.40 vol.% on mv is observed for mv between 2 and 25 vol.%, and an underestimation of -3.84 vol.% is obtained for mv between 25 and 40 vol.%. For the entire range of mv, between 2 and 40%, the RMSE on mv is of 5.66 vol.% (Figure V.2a).

In the case where the NNs were trained using a priori information on mv with dry to slightly wet soil conditions (training with mv between 2 and 30 vol.% and validation using mv between 2 and 25 vol.%), results show that the introduction of a priori information on mv improves the mv estimates. The RMSE on mv estimates decreases from 4.89 vol.% without a priori information on mv to 3.58 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also reduced from 2.40 vol.% to 1.06 vol.% (Figure V.2b).

In addition, the use of a priori information on mv in the case of very wet soil conditions also improves the mv estimates. The RMSE on mv estimates decreases from 6.64 vol.% without a priori information on mv to 5.04 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also reduced from -3.84 vol.% to -2.29 vol.% (Figure V.2c).

Hrms = 3.8 cm. The high RMSE values of in the case of dry to slightly wet conditions and high Hrms-values are due to an overestimation of mv (bias increases from -2.5 to +8.0 vol.%

for Hrms between 0.5 and 3.8 cm). Similarly, the high RMSE values in the case of very wet conditions and low Hrms-values are due to an underestimation of mv (bias about -13.0 for Hrms=0.5cm).

According to the incidence angle "", results show that the RMSE on mv is strongly dependent on  in the case of no a priori information on mv (Figures V.9c and V.9d). The RMSE increases from 4.0 vol.% (for =20°) to 9.0 vol.% (for = 45°) for dry to slightly wet soils and increases from 4.0 (for =20°) to 11.0 vol.% (for =45°) for very wet soil conditions. In the case of dry to slightly wet conditions with high incidence angle values, the high RMSE values are due to an overestimation of mv (bias increases from +1.0 to +5.5 vol.% for  between 20° and 45°). Similarly, the high RMSE values in the case of very wet conditions and high incidence angle values are due to an underestimation of mv (bias decreases from -1.5 to -9.0 vol.% for  between 20° and 45°).

The RMSE on mv estimates varies slightly with Hrms in the case of a priori information on mv for dry to slightly wet soils (between 6.0 and 7.1 vol.%) (Figure V.9).In addition, RMSE and bias on mv estimates are also dependent on the incidence angle "". The RMSE increases from 4.0 vol. % for =20° to 7.0 vol. % for =45°. The overestimation of mv increases from +1.0 to +2.5 vol.% for  between 20° and 45°.

Figure V.9 shows that the RMSE on mv estimates is well reduced in the case of a priori information on mv for very wet soil conditions (it varies between 4.0 and 7.0 vol.%). The highest RMSE-values correspond to low Hrms-values. The underestimation of mv is well reduced mainly for low Hrms-values from -13.0 vol.% without a priori information on mv to -6.0 vol.% with a prior information on mv (case of very wet conditions). In addition, the analysis of the RMSE on mv estimates shows that the RMSE is well reduced mainly for high incidence angles (=45°) from 11.0 vol.% without a priori information on mv to 5.1 vol.% with a priori information on mv. Moreover, the underestimation on mv is well decreased from -9.0 vol.% .% without a priori information on mv to -2.5 vol.% with a priori information on mv (case of very wet conditions and =45°).

The analysis of the accuracy on mv estimates according to Hrms shows similar results than in the case of VV alone. The RMSE on mv in the case of inversion without a priori information on mv increases from 5.0 vol.% for Hrms=0.5 cm to 7.0 vol.% for Hrms = 3.8 cm for dry to slightly wet soils and decreases from 10.0 vol.% for Hrms=0.5 cm to 4.0 vol.% for Hrms = 3.8 cm for very wet soil conditions (Figure V.13). Three NNs are tested: without a priori information on mv (case 1), with a priori information on mv with dry to slightly wet soil conditions (case 2), with a priori information on mv with very wet conditions (case 3).

Figure V.13b shows that the bias on mv estimates increases when Hrms increases. According to the incidence angle "", results show that the RMSE on mv is strongly dependent on  in

V.4.2 Real dataset

The NNs built for estimating mv and Hrms are then analyzed using the real Sentinel-1 dataset.

V.4. priori on mv and RMSE=7.46 vol.% without a priori on mv).

V.4.3.1.1.2 Using Baghdadi model

The results show that the introducing of a priori information on mv in the case of dry to slightly wet soil conditions improves the mv estimates. The RMSE on mv estimates decreases from 5.67 vol.% without a priori information on mv to 4.97 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also reduced from +1.96 vol.% to +1.1 vol.% (Figure V.32). The RMSE on mv estimates in the case of very wet soil conditions decreases from 6.14 vol.% without a priori information on mv to 4.24 vol.% in the case of a priori information on mv. In addition, the difference between estimated and measured mv is also well reduced from -2.9 vol.% to -1.

V.4.3.2 Real dataset

The NNs built for estimating mv and Hrms are then analyzed using the real Sentinel-1 dataset.

VV and VH are the inputs of NNs. Results show that the introduction of a priori information on mv provides better accuracy on the mv estimates than the case without a priori information on mv (RMSE=5.83 vol.% with a priori information on mv and RMSE=7.25 vol.% without a priori on mv).

The analysis of the difference between the estimated and measured mv shows that the strong underestimates of the mv corresponds to low Hrms-values (Hrms<2 cm) and the strong overestimates corresponds to high Hrms-values (Hrms>2 cm). of mv estimated at the scale of the study site is possible only when the study site is not irrigated.

This first study on the potential of Sentinel-1 data for estimating the soil roughness shows that the development of an automatic and generalizable inversion procedure of the C-band radar signal does not allow a pertinent estimation of the soil roughness. The accuracy on soil roughness estimates obtained in this study cannot satisfy the requirements of operational users of soil roughness products (in particular to modelers) because the need is at least three roughness classes: smooth (sowing), medium (small plowing) and rough (large plowing).

Only methods based on the use of experimental relationships, which are often difficult to apply to sites other than those for which they were developed and are generally valid only for specific soil conditions, allows the mapping of three roughness classes (Baghdadi et al., 2002a). Indeed, different experimental studies have revealed that the sensitivity of the radar signal to surface roughness (i.e. the slope of the regression lines) can be highly variable from one site to another. In addition, the experimental relationships between the radar signal and

Hrms are established for a given incidence angle and a range of soil moisture. The soil composition could be also different from one site to another. All these reasons explain why the experimental relationships are not generalizable.

Annex 2: Results on soil roughness estimates using synthetic dataset generated from Baghdadi model

List of figures and tables

List of figures

Chapter 2-Generalities Chapter 5-Estimation of soil roughness using neural networks from sentinel-1 SAR data Toutes ces raisons expliquent pourquoi les relations expérimentales ne sont pas généralisables.

Chapitre 6

Les travaux de cette thèse ont porté sur plusieurs secteurs de recherche (évaluation,