New empirical backscattering models for estimating bare soil surface parameters

ABSTRACT Various models have been proposed to estimate the degree of backscatter in Synthetic Aperture Radar (SAR) images. However, it is still necessary to calibrate these models based on the characteristics of different study areas and to propose new models to achieve the highest possible accuracy in estimating the backscattering coefficient ( ) SAR. In this study, three empirical models, including Champion, Sahebi and Zribi/Dechambre, were initially calibrated for two SAR datasets (i.e. The Airborne Synthetic Aperture Radar (AIRSAR) and Canadian Space Agency radar satellite (RADARSAT-1)) acquired over two bare soil study areas with various soil characteristics. The Zribi/Dechambre model was then modified by revising the roughness parameter to obtain higher accuracy in estimating over a larger range of incidence angles (θ). A new empirical model was also proposed by combining the four parameters of Soil Moisture (SM), standard deviation of surface height -root mean square- (rms), correlation length (l), and θ. To this end, the most appropriate form of the regression model was investigated and used for each of these parameters to obtain the highest correlation between the in-situ data and values. A comparison of the empirical models showed that the modified Zribi/Dechambre had the highest accuracy in predicting values with the Root Mean Square Errors (RMSE) of 1.20 dB and 1.59 dB over Oklahoma and Quebec, respectively. Furthermore, coefficients values of the new proposed model remained stable in the two datasets unlike the other investigated models. In this study, the effects of l on the accuracy of the new proposed model were also assessed. It was concluded that l had a considerable impact on the accuracy of the proposed model and including this parameter can improve the accuracy by up to 1 dB.

resolution, multi-dimensional, and multi-mode datasets by new SAR systems has posed many challenges in the processing and interpretation of the corresponding datasets (Sun, Shimada, and Xu. 2017). However, SAR images contain valuable information about their targets, making them unique compared to other remote sensing datasets. For example, SAR data are great resources for estimating soil parameter because SAR signals are highly sensitive to the soil dielectric constant and surface roughness (Ulaby, Dubois, and Van Zyl 1996;Sahebi, Bonn, and Bénié 2004;Azimi et al. 2020). SAR can also collect data in almost any atmospheric condition, which makes it a suitable option in areas with frequent cloud cover and bad weather conditions (Amani et al. 2018;Zakharov et al. 2020). Moreover, SAR microwave signals can penetrate not only clouds, but also vegetation canopies and soil surfaces and, therefore, acquire information from areas invisible in other satellite imagery, such as optical satellite data . Open-access SAR data, such as Sentinel-1 imagery, has facilitated frequent and high-resolution soil moisture retrieval (El Hajj et al. 2019;Hachani et al. 2019;Ezzahar et al. 2020;Zhang et al. 2020). Consequently, SAR has been utilized in different environmental applications, including Earth surface monitoring, land cover classification, geohazard events monitoring, climate change analysis, and drought monitoring (Berardino et al. 2002;Tofani et al. 2013;Amani et al. 2017;Li et al. 2020).
SAR systems generally collect the backscattering coefficient (σ 0 ) from different targets. The fraction σ 0 describes the amount of average backscattered energy compared to the energy of the incident field. This parameter depends on the SAR configuration (e.g. incidence angle (θ) and wavelength (λ)) and the target properties (e.g., Soil Moisture (SM) content and soil surface roughness) (Champion 1996). Linear correlation of radar signal and SM was first investigated as a function of co-polarized C-band configuration by Ulaby and Batlivala (1976). Later, considering the depth of soil, stronger sensitivity of radar signal to SM was observed using a linear relationship (Attema and Ulaby 1978) and empirical or physical radar backscattering models (Oh, Sarabandi, and Ulaby 2004;Fung, Li, and Chen 1992;Dubois, Van Zyl, and Engman 1995;Baghdadi, Aubert, and Zribi 2011), which present an increase of radar signal with SM. Different models have been developed to formulate the relationship between σ 0 and land surface parameters, which are generally called backscattering models. Over the past 40 years, many backscattering models, from simple linear to complex theoretical models, have been proposed based on radar data availability and different in-situ datasets (e.g., Attema and Ulaby 1978;McNairn et al. 1996;Baghdadi et al. 2012;MirMazloumi and Sahebi 2016;Mirsoleimani et al. 2019;Ezzahar et al. 2020). In case of the effects soil roughness and θ on the radar signal, although low θ minimizes the effect of soil roughness (Verhoest et al. 2008), roughness estimation is more accurate for θ >30 � (Bousbih et al. 2017). Furthermore, low to medium values of θ (i.e., 20 � to35 � ) have been suggested an optimal range to estimate SM (Holah et al. 2005). Since relationship between σ 0 and surface roughness and moisture are not linear (Zribi et al. 2013), several studies have proposed logarithmic (Sahebi, Bonn, and Bénié 2004) and exponential terms (Zribi and Dechambre 2003;Baghdadi, Holah, and Zribi 2006;Baghdadi et al. 2007) to highlight the impacts of surface roughness on σ 0 (Petropoulos, Ireland, and Barrett 2015).
Backscattering models are generally divided into three categories: empirical, semiempirical, and theoretical or physical (MirMazloumi and Sahebi 2016). Theoretical backscattering models, such as the Integral Equation Model (IEM) (Fung, Li, and Chen 1992), Small Perturbation Model (SPM) (Ulaby, Moore, and Fung 1982;Hajnsek, Pottier, and Cloude 2003;Iodice, Natale, and Riccio 2011), Geometrical Optic Model (GOM) (Ulaby, Moore, and Fung 1982), and Physical Optic Model (POM) (Ulaby, Moore, and Fung 1986) are relatively complex and there are several limitations to obtain high accuracy in estimating soil parameters from these models. However, physical models have been applied to provide site-independent relationships (Baghdadi et al. 2004), indicating their sensitivity to dielectric constant and roughness (Baghdadi et al. 2012), as well as their independency to specific site calibration (Mirsoleimani et al. 2019). For instance, it is challenging to apply the IEM model to estimate SM content over a large area because surface roughness is not usually well described by the statistical representations used in this model (Zribi et al. 2000;Davidson et al. 2003;Callens, Verhoest, and Davidson 2006). Moreover, additional information, such as topography and soil type are required to optimize the performance of IEM (Zribi and Dechambre 2003;Notarnicola and Solorza 2014). Furthermore, the validity domain of SPM, GOM, and POM are restricted to a limited surface roughness range. These models have been generally employed for smooth or rough surfaces. Although Hajnsek, Pottier, and Cloude (2003) and Iodice, Natale, and Riccio (2011) have proposed second-order polarimetric scattering models to expand the validity range of SPM, they were only designed for cross-polarization and were not suitable for short wavelength C-band (Hajnsek, Papathanassiou, and Cloude 2001;Barrett, Dwyer, and Whelan 2009). Therefore, the estimation of soil surface parameters using theoretical models requires very detailed knowledge of surface roughness, which is only achievable through intensive roughness measurement campaigns. Semiempirical models, such as Oh (2004) and Dubois (Dubois, Van Zyl, and Engman 1995), on the other hand, require a large amount of field data to accurately model the nonlinear response of SAR backscattering to SM and surface roughness parameters (Zribi, Gorrab, and Baghdadi 2014;Petropoulos, Ireland, and Barrett 2015). Moreover, there are limitations to invert the physical part of the semi-empirical models to a specific geographic area with similar roughness characteristics (Baghdadi, Holah, and Zribi 2006;Paloscia et al. 2008).
Compared to theoretical and semi-empirical models, empirical models are often favoured by users because they are straightforward to implement and can easily invert σ 0 to soil surface parameters (Zribi and Dechambre 2003;Gherboudj et al. 2011;Rao et al. 2013;Chai et al. 2015;Kirimi et al. 2016). The simplest empirical model was developed to represent the relationship between SM and radar response (Attema and Ulaby 1978). In fact, many of the first empirical models were generally proposed based on the linear relationship between σ 0 and SM when SM content values were between about 10% and 35% and under the assumption that roughness does not change between successive radar measurements (Wang and Qu 2009;Zribi et al. 2013). An exponential term was then included in the linear relationship to consider the effects of roughness on σ 0 and, thus, to make empirical models more applicable (Zribi and Dechambre 2003;Baghdadi, Holah, and Zribi 2006;Baghdadi, Aubert, and Zribi 2011;Zribi et al. 2013). During that time, several other ideas were also proposed to improve the efficiency of empirical models. For examples, Baghdadi, Aubert, and Zribi (2011) and Baup et al. (2011) suggested that considering the difference between one image acquired during the wet season and a reference image acquired during the dry season could further eliminate the effects of surface roughness.
Although many empirical backscattering models have been developed, their applicability and accuracy should be investigated in different study areas. This is because the coefficients of empirical models are usually calculated using specific in-situ datasets over a study area. Accordingly, these models may need to be refined or new empirical models developed to obtain reasonable accuracies in other regions. Thus, in this study, three empirical backscattering models, including those proposed by Champion (1996), Sahebi, Bonn, and Gwyn (2003), and Zribi and Dechambre (2003) were first investigated in study areas in Oklahoma, USA and in Quebec, Canada. Then, the coefficients of these models were calibrated to obtain better accuracies based on characteristics specific to the study areas. Furthermore, a modified version of the Zribi/Dechambre model (Zribi and Dechambre 2003) and a new empirical backscattering model that is a function of SM, θ, standard deviation of surface height -root mean square -(rms), and correlation length (l) were proposed to obtain the most accurate estimation of σ 0 over the study areas.

Study areas
The datasets from three bare soil regions were used in this study to consider soil types with various characteristics in the backscattering models. The first study area (Figure 1 (a)) was the Little Washita Experimental Watershed (LWREW) with an area of 611 km 2 , located in Oklahoma in the Southern Great Plains of the United States (34° 57′ N, 98° 01′ W). In this study area, summers are typically long, hot, and relatively dry and winters are generally short, temperate, and dry but are usually very cold for a few weeks. The topography of the LWREW is moderately rolling with a maximum relief less than 200 m. Except for a few rocky and steep hills near the Cement region, the upland topography is gently to moderately rolling. Furthermore, soils include a wide range of textures. Land use in this watershed is also dominated by rangeland and pasture (63%) with large areas of winter wheat and other croplands concentrated in the floodplain and western portions of the watershed (Jackson et al. 2002).
The second study area includes the Chateauguay River (45° 19′ N, 73°46′ W) and Pike River (45° 08′ N, 72° 54′ W) watersheds (i.e., the CRPRW study area). This study area is located on the south shore of the St. Lawrence River, southeast of Montréal, Quebec, Canada (Figure 1(b)). This region contains many agricultural fields on a relatively flat and relief plateau with a homogeneous soil texture composed of about 36% clay, 42% silt and 22% sand (Sahebi, Bonn, and Gwyn 2003).

In-situ datasets
In-situ datasets collected over the study areas were used to evaluate the accuracy of the empirical models (see Table 1 for the amount of field data). The field data from LWREW were collected by the Soil Moisture Experiments (SMEX03) administration and campaign members. SMEX03 stations are equipped with different types of sensors to measure parameters such as the dielectric constant, volumetric and gravimetric SM contents, soil temperature, bulk density, and soil surface roughness (Jackson et al. 2002). In this study, only the volumetric SM at the depth of 0 to 6 cm and soil surface roughness collected over 13 sites were considered. A ThetaProbe™ sensor was used to measure SM content. The rms height (0.2 to 2.6 cm) and l (1.9 to 7.5 cm) were also calculated through analysing the digitized photographs using the method described in Jackson and Cosh (2006).  The in-situ data collected over CRPRW study area, also included SM and surface roughness data over 27 sites. A ThetaProbe™ soil moisture sensor, which measured the apparent dielectric constant, was used to derive the SM content at a depth of 0 to 5 cm (dictated by the length of the probe's needle). Using the equation presented in the ThetaProbe SM user manual, the direct outputs (DC voltage in mV) were converted to soil water content ðm 3 m À 3 ) and dielectric constant. Moreover, six 2 m long surface profiles with 1 cm sampling intervals (three parallel and three perpendicular to the soil furrows) were used at each site to calculate the height of the rms (1.5 to 5.4 cm). The method to extract and model roughness parameters is described in detail in Beaulieu, Leclerc, and Moisan (1995).

Remote sensing datasets
The information about the satellite images used in this study is provided in Table 2. Two types of SAR datasets were used in assessing the backscattering models to obtain a more comprehensive conclusion about the accuracy and limitations of each backscattering model. These images were acquired at the same time as the in-situ data measurements. Four the Airborne Synthetic Aperture Radar (AIRSAR) and two Canadian Space Agency radar satellite (RADARSAT-1) images were used over the LWREW and CRPRW study areas, respectively.

Satellite image processing
The AIRSAR data were initially in the Stokes matrix format and so the synthesized and decompressed procedures were applied to derive σ 0 . By doing so, the radar data were available in C-band and HH (Horizontal Horizontal) polarization Jackson et al. (2009). A resampling process was then carried out to convert the 9.26 m spatial resolution to a 6.70 m. Finally, Digital Ortho-photo Quadrangles (DOQs) with a 1 m pixel size were used for geo-referencing. To this end, 40 points distributed throughout the scene were employed, by which a Root Mean Square Errors (RMSE) of half a pixel size (3.3 m) was obtained. The RADARSAT-1 datasets included here were mainly pre-processed by Shepherd (1997) and Wickel, Jackson, and Wood (2001). A Landsat Thematic Mapper (TM) image, acquired on 25 July 1997, was applied to geo-register the RADARSAT-1 scenes using several ground control points. Then, the RADARSAT-1 raw values were converted to σ 0 according to the methodology described in Srivastava (1999). Finally, an average σ 0 value was assigned to each sample with an area of 20 m × 30 m (Sahebi, Bonn, and Bénié 2004).

Empirical backscattering models
One of the basic empirical models used to characterize the relationship between σ 0 and soil surface parameters was presented by Attema and Ulaby (1978) as follows: in which σ 0 (dB) and A (dB) are the total backscattering coefficient value and backscattering coefficient of dry soil, respectively; m v is the volumetric SM; and B is the coefficient that indicates the sensitivity of the radar signal to SM. Subsequently, several advanced empirical models have been developed based on this basic relationship (Zribi et al. 2013).
In the following sections, three of these models, as well as two new models developed in this study, are discussed in more detail.

The Champion model
The Champion model (Equation (2)) is a nonlinear model, developed by Champion (1996), to include θ in the basic backscattering model (see Equation (1)).
where C 1 , C 2 , C 3 (dB) refer to the dependence of σ 0 on θ. C 1 is σ 0 of dry soil when θ ¼ 90 � ; C 2 is the total dynamics of soil response over θ ¼ 0 � to60 � ; and C 3 is the shape factor of the cosine function. The main limitation of this model is that it neglects the effects of other surface parameters, such as soil surface roughness, on the backscattering values.

The Sahebi model
Because the Champion model does not consider roughness parameters, Sahebi, Bonn, and Bénié (2004) included the effects of soil surface roughness on σ 0 by adding rms in their empirical model (Equation (3)).
where A 1 , A 2 , A 3 , A 4 , and D are coefficients determined using training datasets. In Sahebi, Bonn, and Gwyn (2003), the constants of A 1 , A 2 , A 3 , A 4 , and D were calculated for a configuration of C-band and HH polarization using the nonlinear least squares method.

The Zribi and Dechambre model
One of the limitations of empirical models such as the Sahebi model is that they do not consider l, although several studies have demonstrated the effects of this parameter on σ 0 (Baghdadi, Holah, and Zribi 2006;Verhoest et al. 2008;Choker et al. 2017). The l parameter describes the horizontal distance over which the surface profile is autocorrelated. The effect of l on surface analysis have been investigated in several studies. For example, Davidson et al. (2003), Verhoest et al. (2008), and Zribi, Gorrab, and Baghdadi (2014) proposed l and rms as a single term or a function of surface roughness. Moreover, Baghdadi, Holah, and Zribi (2006), Choker et al. (2017), and Mirsoleimani et al. (2019) fitted an optimum parameter in which l was a function of θ, rms, and polarization type. Finally, Álvarez-Mozos, Gonzalez-Audícana, and Casalí (2007) and MirMazloumi and Sahebi (2016) considered l separately from rms. Zribi and Dechambre (2003) developed an empirical model (the Zribi/Dechambre model, Equation (4)) to consider SM, rms height, and l. The authors proposed the use of Z rms cm ð Þ= rms 2 ð Þ=l in their empirical model because it was challenging to separately estimate the effects of the rms height and l using empirical backscattering models over rough surfaces.

The modified Zribi and Dechambre model
The Zribi/Dechambre model used only backscatters with θ ¼ 25 � ; 39 � and, thus, their model did not result in reasonable accuracies for θ ¼ 20 � to25 � and θ ¼ 45 � to70 � (Zribi and Dechambre 2003). In the current study, this model was modified by (1) considering SAR data acquired with θ ¼ 20 � to64 � and (2) adding minus power to Z rms in the original equation (named the modified Zribi/Dechambre model) as provided in Equation (5). In this study, this was added because it was observed that exp À Z rms ð Þ had highest correlation with σ 0 when compared to ln Z rms ð Þ.

New proposed empirical backscattering model
In this study, a new empirical backscattering model (Equation (6)) was developed to (1) consider all parameters that can affect σ 0 (i.e., SM, θ, rms and l) (MirMazloumi and Sahebi 2016), and (2) consider each of these parameters as a separate term within a single model. Moreover, the proposed model was assumed to be comprehensive because it was developed based on datasets collected from areas with various land cover characteristics. Compared to other empirical models, another new aspect of the proposed model is that it utilizes rms in the power of the exponential. This was because it was observed that using rms in the power of exponential increased the correlation with σ 0 compared to other forms of statistical models, such as logarithmic or linear.
In which the series of A coefficients were calculated for band C and HH polarization using the least square method. Each term of the equation was obtained by the method proposed by Attema and Ulaby (1978), in which achieving the best function for each parameter was attempted. The correlation assessment was performed among all four parameters used in the proposed model (i.e., m v , θ, rms and l) and σ 0 using different forms of regression equations, including linear, polynomial (second-order), geometrical, and logarithmic. Subsequently, the model that produced the highest correlation coefficient was selected for each term. For example, both sinusoidal and cosine functions were tested for θ ¼ 20 � to64 � . The data were correlated to σ 0 in the two datasets using the sinθ function. However, σ 0 was increased by increasing θ using sinusoidal function, which was not in agreement with the fact that σ 0 from a rough surface decreases with increasingθ (Champion 1996;Sahebi, Bonn, and Gwyn 2003;Verhoest et al. 2008;Bousbih et al. 2017).
Furthermore, the cosine function provided more accurate results along with other terms in the model. Thus, more experiments over various datasets needs to be performed to confirm the use of sinusoidal function in empirical models. Therefore, cosine function was used in the proposed model (i.e., cosθ ð Þ A 2 : second term in Equation (6)). Moreover, it has been extensively discussed that linear functions are the best models to relate SM to σ 0 (Champion 1996;Sahebi, Bonn, and Bénié 2004;Zribi and Dechambre 2003;Zribi, Gorrab, and Baghdadi 2014;Petropoulos, Ireland, and Barrett 2015) and, thus, a linear model was used in the fourth term of the proposed model (A 4 m v ). The relationship between surface roughness parameters (i.e., both rms and l) and σ 0 was the critical point due to wide range of θ and roughness, which were used in this study. The low values of θ (e.g., near to 10 � ) decreases the influence of soil roughness. The decreasing rate of σ 0 can be also observed after 30 � (Oh, Sarabandi, and Ulaby 2004). Although surface roughness is a confounding factor on bare soil analysis, it has been reported that the backscattering from a rough surface increases with roughness (Toure et al. 1991;Oh, Sarabandi, and Ulaby 2004;Verhoest et al. 2008). Thus, finding a straightforward correlation between rms ð Þ=l and σ 0 for the datasets were complicated. To find the most appropriate mathematical function for soil surface parameters, sinusoidal and cosine functions were initially tested, although this did not result in a high correlation with. However, logarithmic functions resulted in a high correlation coefficient. For instance, an exponential function resulted the best fitting for rms (see exp A 3 rms ð Þ ð Þ in Equation (6)). It is also worth noting that the coefficient (A 3 ) was used in the power of the exponential function to increase the correlation and, thus, to improve the prediction accuracy of the model.

Model development and accuracy assessment
All the in-situ data (see subsection In-situ datasets) were randomly divided into training (75%) and test (25%) datasets. The training data were used to develop the empirical models and, in fact, to calculate the coefficients of each model using least square method. In total, 39 and 29 samples of LWREW and CRPRW were, respectively, selected to train the models. The test data (13 and 10 samples of LWREW and CRPRW, respectively) were also employed for accuracy assessment through calculating the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and unbiased RMSE (ubRMSE) metrics (or the error standard deviation).
ubRMSE ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 N RMSE ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 N where N is the number of points with both estimated backscattering coefficient from models (σ 0 est ) and that of extracted from and image (σ 0 im ). The overbar represents the average of the variables. The ubRMSE measures the RMSE excluding the bias (b). It is worth noting that RMSE is critically undermined if there are biases in either the average or the amplitude of fluctuations in the estimation (Taylor 2001;Entekhabi et al. 2010;Liu, Yang, and Yue 2018). Thus, the mean-bias can easily be removed by defining the unbiased RMSE, which implies RMSE � b j j and emphasizes on the deficiency of the RMSE in the presence of mean-bias.
The development of models and accuracy assessment included the following: (1) The accuracies of the Champion, Sahebi, and Zribi/Dechambre models with the corresponding original coefficients were assessed using the test data.
(2) The coefficients of the Champion, Sahebi, and Zribi/Dechambre models were recalculated using the training data and least square method and the accuracies of these refined models were then evaluated using the test data. (3) The coefficients of the modified Zribi/Dechambre model (Equation (5)) were calculated using the training data and least square method and, then, its accuracy was assessed using the test data. (4) The new proposed model (Equation (6)) was developed by calculating its coefficients using the training datasets. Then, its accuracy was evaluated using the test data.

Results and discussion
The procedure described in subsection Empirical backscattering models was applied to develop each empirical model, discussed by followed, and to obtain the level of accuracy. Consequently, all empirical backscattering models, including their coefficients and accuracy parameters, are provided in Tables 3-7. Additionally, the distribution of the estimated σ 0 from each model versus the σ 0 obtained from the AIRSAR and RADARSAT-1 datasets for the test data are illustrated in Figure 2. As clear from Figure 2(a,b), the σ 0 values obtained from the calibrated Champion model are closer to the 1:1 line compared to those obtained from the original model. In fact, the original Champion model significantly underestimated and overestimated σ 0 values over LWREW and CRPRW, respectively. This can be also observed in Table 3, in which the MAE, RMSE and ubRMSE errors for the calibrated Champion model is considerably lower than those of the original model. Additionally, considerable differences were observed for the values of C 2 and D in the original and calibrated Champion model (see Table 3). This may be due to the local characteristics of each study area (Champion 1996). As explained in Champion (1996), the difference could be due to the influence of k � rms ð Þ and k � l on C 2 (k is the wavenumber), which expresses roughness values, and their values are about 0.7 and 3.0 in the original Champion model. It was reported that there is a negative relationship between k � rms ð Þ and C 2 , wherein as the latter decreases as the former increases (Champion 1996). Figure 2(c,d) illustrates the σ 0 derived from the original and calibrated Sahebi model (see Table 4 for the coefficient and the accuracies for the model with original and calibrated coefficients). As is clear, the accuracy of the calibrated model was higher than that of the original model. For instance, the RMSE values were 1.80 dB and 2.48 dB in the LWREW study area for the calibrated and original Sahebi models, respectively. The most important reason is that the calibrated model was developed based on the datasets collected in this study. Another reason might be due to the large difference between θ in these two cases. The original coefficients were calculated using the data from the Chateauguay site acquired with 20° to 47° angles; however, θ was between 47° and 64°     in this study. Overall, the accuracy increased in both datasets using the calibrated model (Table 4); however, better improvement was observed in the CRPRW study region, in which θ was smaller than that of the LWREW study area. Therefore, it was concluded that the Sahebi model more accurately estimates σ 0 when θ < 47 � . Finally, the Sahebi model did not underestimate to the same degree as the Champion model when σ 0 was between −15 dB and −10 dB. Figure 2(e-h) and Tables 5-6 provide the results of both the original and calibrated Zribi/Dechambre models, as well as the modified Zribi/Dechambre model proposed in this study. Comparing the accuracy parameters provided in Tables 5 and 6, it was observed that the modified Zribi/Dechambre model had the highest accuracy in estimating σ 0 , followed by the calibrated and original Zribi/Dechambre models, respectively. Over LWREW, a slight overestimation of approximately 0.5 dB was observed when σ 0 was between −18 dB and −15, while a slight underestimation was observed for greater σ 0 values except when σ 0 = −10 dB. Furthermore, Figure 2(e-h) shows the sensitivity of the original Zribi/Dechambre model to small θ values, as the original model underestimated all σ 0 values in RADARSAT-1 data acquired over CRPRW. It was also observed that the distribution of the estimated σ 0 values obtained from the modified Zribi/Dechambre model was more accurate than those of the original and calibrated Zribi/Dechambre models. Figure 2(i,j) and Table 7 provide the results of the backscattering model proposed in this study. As clearly illustrated in Figure 2, the estimated σ 0 values from the proposed model were closer to the centreline compared to Champion, Sahebi, and Zribi/ Dechambre models, indicating the higher accuracy of the proposed model. This was also observed by comparing all the results provided in Table 3 to 5 and 7, where the proposed model had the lowest MAE, RMSE and ubRMSE values. For instance, the RMSE errors decreased by between 0.10 dB and 1.50 dB for the LWREW and between 0.15 dB to 1.00 dB for the CRPRW study areas compared to the Champion, Sahebi, and Zribi/Dechambre empirical models. As another example, the high overestimation and underestimation of σ 0 values when using the Champion and Sahebi models were improved by the new proposed model (compare Figure 2 (a-d,i,j)). However, the new backscattering model slightly overestimated and underestimated σ 0 by up to 1 dB in some cases.
The modified Zribi/Dechambre model provided the smallest metrics in Table 3 to 7, where estimated values were the closest results to the centreline in Figure 2. The MAE, RMSE, and ubRMSE of this model were, respectively, 1.33, 1.20, and 0.58 over the LWREW study area; and they were, respectively, 2.30, 1.59, and 1.65 over the CRPRW study area. According to the results in Tables 3-6, significant differences were observed for four models (the Champion, Sahebi, modified and original Zribi/Dechambre) in the two datasets. For instance, there were 40 and 36 values differences in D in the modified and Zribi/Dechambre models, respectively. Additionally, considerable differences were obtained in the Champion and Sahebi models. However, in the new proposed model, the model coefficients values, which were calibrated using field data, remained stable in the two datasets. The average variations of datasets coefficients were approximately 7, which was lower two, three, three, and five times compared to other models. Therefore, differences reflected the necessity of the calibration of the four models, which is not always practical.
Another experiment carried out in this study was investigating the importance of l in improving the accuracy of estimating σ 0 , and the results are demonstrated in Table 7 and Figure 3. Specifically,lwas removed from the proposed model and the accuracy of the model was recalculated and then compared with that of the original proposed model. According to Table 7, the accuracy of the proposed model decreased whenlwas removed. For instance, RMSE increased from 1.20 dB to 1.82 dB in the LWREW study area and it increased from 1.83 dB to 1.97 dB in the CRPRW study region whenlwas removed. Most studies discussed in the Zribi and Dechambre model section, exploited surface responses using exponential correlation function, while the inverse function of exponential (logarithm) performed efficiently over the datasets of this study. This showed the importance of considering l as an independent parameter to highlight its effect. It could be concluded that since removing l decreased the estimation capability, the effects of l might be relatively limited by describing l as a function of other independent parameters, which have their own limitations. Therefore, l could affect the backscattering models by choosing a proper function and there are four important parameters that can affect σ 0 values in a SAR images (i.e., SM, θ, rms and l), all of which should be considered in backscattering models to achieve the highest possible accuracies.

Conclusion
SAR systems acquire valuable datasets from which soil surface parameters may be derived using backscattering models. The objectives of this study were to (1) calibrate several empirical backscattering models, including the Champion, Sahebi and Zribi/Dechambre models, and (2) propose two new models, including the modified Zribi/Dechambre and a novel model proposed in this study, to accurately estimate σ 0 values. Two SAR datasets, acquired by AIRSAR and RADARSAT-1, over two study areas with different soil characteristics were used to evaluate the accuracy of each model. Both the original and calibrated Champion, Sahebi and Zribi/Dechambre models were investigated in this study, and it was concluded that the calibrated models more accurately estimated σ 0 over the study areas. Moreover, it was concluded that the modified Zribi/Dechambre model developed in this study had the highest accuracy in estimating σ 0 values compared to all models evaluated in this study. However, the new proposed empirical model had higher stability in estimating the coefficients over two datasets, i.e. the least differences of the estimated coefficients for two datasets indicated that the new proposed model remained stable in the two datasets. Thus, the new proposed model can be potentially utilized over other datasets. Finally, it was concluded that all important soil surface parameters and SAR configuration parameters that can affect σ 0 , including l, should be considered in any empirical model developed to obtain the highest possible accuracy. This is why multiple previous studies have implemented the correlation length using exponential function relating to other parameters (e.g., rms, θ, polarization) based on IEM model modifications (Baghdadi et al. 2004;Baghdadi, Holah, and Zribi 2006;Baghdadi et al. 2012;Choker et al. 2017;El Hajj et al. 2019;Mirsoleimani et al. 2019). Although the effect of vertical soil roughness has been examined exponentially, a logarithmic function was proposed in this study to exploit the horizontal impact. It was tried to produce an accurate empirical model by (1) calibrating the coefficients of the original models through fitting/investigating different forms of statistical models (e.g., exponential, linear, and logarithmic), and (2) considering the well-known relationships between soil surface parameters and σ 0 (e.g. backscattering from a rough surface increases with roughness and backscattering from a rough surface decreases with increasing θ). However, considering both facts together was challenging and resulted in a limitation in improving the accuracy of the proposed model. Moreover, finding the most optimal coefficients for the proposed model was challenging considering the wide range of parameters of the datasets, which were used in this study. Finally, the unknown effects related to the nature of the study area and radar noise range might disturb empirical estimations in this study. Since all the experiments in this study were performed using SAR data with C-band and HH polarizations, it is suggested that future studies calibrate previous empirical models and evaluate the accuracy of the proposed models for SAR data with other bands and polarizations types (i.e., L-, P-, and X-band/VV (Vertical Vertical) and VH (Vertical Horizontal) polarizations).