**3.1. Electromagnetic modeling**

The development of physical or theoretical models simulating direct backscatter coefficients in terms of soil attributes as dielectric constant and the surface roughness for an area of known characteristics is one of the most common approaches for SM estimation (Barrett et al., 2009). Electromagnetic models allow a direct relationship between the surface parameters and the backscattered radiation and are useful for understanding the sensitivity of the radar response to changes in these biophysical variables.

Despite its complexity, only theoretical models can produce a meaningful understanding of the interaction between electromagnetic waves and the Earth's surface. However, exact solutions of the equations that govern the rough surface scattering are not yet available and various approach methods have been developed with different ranges of validity [10]. The standard backscatter models are known as Kirchhoff Approximation (KA), which includes the Geometrical Optics Model (GOM), the Physical Optics Model (POM) and the Small Perturba‐ tion Model (SPM). These models can be applied to specific conditions of roughness in relation to the sensor wavelength. For example, GOM is considered for very rough surfaces, POM middle roughness surfaces and SPM smooth surfaces.

The Integral Equation Model (IEM), based on the radiative transfer model, has been developed by Fung and Chen in 1992 [27]. The model unifies the KA and the SPM model, a condition that makes it applicable to a wide range of roughness conditions. The IEM requires, as inputs, sensor parameters such as polarization, frequency and incidence angle, and target parameters such as the real part of the dielectric constant, the RMS height, *s*, and the correlation length, *l* [27].

For the IEM model, the like polarized backscattering coefficients for surfaces are expressed by this formula:

$$
\sigma\_{pp}^{0} = \frac{k^2}{2} \exp(-2\mathbf{k}\_z^2 \mathbf{s}^2) \sum\_{k=1}^{n} \left| I\_{pp}^n \right| \frac{\mathcal{W}^{(n)}(-2\mathbf{k}\_x, 0)}{n!},\tag{1}
$$

where *k* is the wave number, *θ* is the incidence angle, *kz*=*kcosθ, kx*=*ksenθ* and *pp* refers to the horizontal (HH) or vertical (VV) polarization state and *s* is the standard deviation of terrain heights. The term *I pp <sup>n</sup>* depends on these parameters, *k*, *s* and on *RH*, *RV,* the Fresnel reflection coefficients in horizontal and vertical polarizations. The Fresnel coefficients are strictly related to the incidence angle and the dielectric constant. The symbol *W (-2kx,0)* is the Fourier transform of the nth power of the surface correlation coefficient. For this analysis, an exponential correlation function has been adopted that seems to better describe the properties of natural surfaces [27].

tions are considered as flat, diffuse or non-informative. The rational to use such types of distributions is to let the inference being not affected by external information and based

The proposed Bayesian approach is driven by both experimental data and theoretical electro‐ magnetic models. The theoretical electromagnetic model has the main aim to simulate the

In order to have a better understanding of the proposed methodology, described in section

The development of physical or theoretical models simulating direct backscatter coefficients in terms of soil attributes as dielectric constant and the surface roughness for an area of known characteristics is one of the most common approaches for SM estimation (Barrett et al., 2009). Electromagnetic models allow a direct relationship between the surface parameters and the backscattered radiation and are useful for understanding the sensitivity of the radar response

Despite its complexity, only theoretical models can produce a meaningful understanding of the interaction between electromagnetic waves and the Earth's surface. However, exact solutions of the equations that govern the rough surface scattering are not yet available and various approach methods have been developed with different ranges of validity [10]. The standard backscatter models are known as Kirchhoff Approximation (KA), which includes the Geometrical Optics Model (GOM), the Physical Optics Model (POM) and the Small Perturba‐ tion Model (SPM). These models can be applied to specific conditions of roughness in relation to the sensor wavelength. For example, GOM is considered for very rough surfaces, POM

The Integral Equation Model (IEM), based on the radiative transfer model, has been developed by Fung and Chen in 1992 [27]. The model unifies the KA and the SPM model, a condition that makes it applicable to a wide range of roughness conditions. The IEM requires, as inputs, sensor parameters such as polarization, frequency and incidence angle, and target parameters such as the real part of the dielectric constant, the RMS height, *s*,

For the IEM model, the like polarized backscattering coefficients for surfaces are expressed by

*(n) n x*

*n!*

¥ - - <sup>å</sup> (1)

1 ( 2k ,0) exp( 2k ) , <sup>2</sup>

where *k* is the wave number, *θ* is the incidence angle, *kz*=*kcosθ, kx*=*ksenθ* and *pp* refers to the horizontal (HH) or vertical (VV) polarization state and *s* is the standard deviation of terrain

*k=*

sensor response by considering the characteristics of the soil and vegetation surface.

3.2, a brief description of the electromagnetic models is presented in the next section.

exclusively on data [34].

48 Dynamic Programming and Bayesian Inference, Concepts and Applications

**3.1. Electromagnetic modeling**

to changes in these biophysical variables.

and the correlation length, *l* [27].

this formula:

middle roughness surfaces and SPM smooth surfaces.

<sup>2</sup> <sup>0</sup> 2 2

*pp z pp*

*<sup>k</sup> <sup>W</sup> σ= s I*

For vegetated soils, the simple approach, based on the so-called Water Cloud Model (WCM), developed by [35] has been considered in this analysis. In this radiative transfer model, the vegetation canopy as a uniform cloud whose spherical droplets are held in place structurally by dry matter. The WCM represents the power backscattered by the whole canopy *σ<sup>0</sup>* as the incoherent sum of the contribution of the vegetation σ<sup>0</sup> veg and the contribution of the under‐ lying soil *σ<sup>0</sup> soil*, which is attenuated by the vegetation layer through the vegetation transmis‐ sivity parameters *τ<sup>2</sup>* . For a given incidence angle the backscatter coefficient is represented by the following expression:

$$
\sigma^0 = \sigma^0\_{\text{reg}} + \pi^2 \sigma^0\_{\text{soil}}.\tag{2}
$$

If the terms related to vegetation and incidence angle are explicitly written in more detailed way, the backscattering coefficients become:

$$\sigma^0 = \mathbf{A} \cdot \text{VWC}^{\mathbb{E}} \cos \theta \cdot (\mathbf{1} - \exp(-\mathbf{2} \cdot \mathbf{B} \cdot \text{VWC} / \cos \theta)) + \sigma^0\_{\text{soil}} \cdot \exp(-\mathbf{2} \cdot \mathbf{B} \cdot \text{VWC} / \cos \theta), \tag{3}$$

where *VWC* is the vegetation water content (kg/m2 ), *θ* the incidence angle, *σ<sup>0</sup> soil* represents the backscattering coefficient of bare soil that in this case calculated by using the IEM model, *τ<sup>2</sup>* is the two-way vegetation transmissivity with *τ<sup>2</sup> =exp(-2B VWC/ cosθ).* The parameters *A*, *B* and *E* depend on the canopy type and require an initial calibration phase where they have to be found in dependence of the canopy type and with the use of ground data.

In this work the model simulation enters directly in the inversion procedure. For the Bayesian approach, the simulated data are generated in order to compare them to the measured data and to create the noise probability density function (PDF) as detailed in the section devoted to this approach. For this reason, it is needed to perform a preliminary validation of the proposed model as their simulation enters directly the inversion procedure.

Calibration constant values of the WCM, namely *A*, *B* and *E* were taken initially from literature to take into account the effect of vegetation on the SAR signal [36]. Subsequently through a Maximum Likelihood approach they were determined to fit the data used in this work from both test sites. The application of calibration equations considers two different kind of vegetation, with respect to the sensor response: very dense vegetation (as corn and sunflower) and less dense vegetation (soybean and grass). This step includes the NDVI calculation from some SPOT and LANDSAT optical images for the Argentinean and SMEX'02 test site respec‐ tively acquired close in time to the SAR image. Then the NDVI values were transformed in VWC through empirical approach defined by Jackson et al, 2004 [37].

For bare soil, these unknown parameters are the real part of the dielectric constant (*ε*), the standard deviation of the height (*s*) and the correlation length (*l*), the latter two describing the morphology of the surface. For vegetated fields, the Bayesian inversion was performed under two different approaches. In one case the Water Cloud Model (WCM) [35] is used to simulate the backscattering coefficients from vegetation. In the second case, the PDF parameters are properly modified to take into account vegetation effect through empirical relation with vegetation [38]. In both cases, the Vegetation Water Content (VWC) is added as unknown parameter, and it is derived from optical images. In this way, the approach exploits a synergy

Integration of Remotely Sensed Images and Electromagnetic Models into a Bayesian Approach for…

At the beginning, the conditional probability is assumed as normal distribution. In the training phase, the conditional PDF is evaluated using measured data (*fim*) and simulated values from the IEM model (*fith*). The distribution assumption is then verified with a chi-squared statistics.

are calculated from the statistics of the ration between measured and simulated backscattering

. *im*

*ith*

Subsequently a joint PDF is obtained as a convolution of single independent PDFs. The joint PDF is a posterior probability derived from prior probability on roughness and soil moisture values and to the conditional probability which relates the variations of backscattering coefficients to variations of soil moisture and roughness. The relationship can be expressed as

( ) ( )

*prior i post i i i*

*p S p S dS*

*p Sp S*

where the term at the denominator is a normalization factor with integration over all variables

**•** For bare soil: dielectric constant (*ε*), the standard deviation of the height (*s*) and the corre‐

**•** For vegetated soil: dielectric constant (*ε*), the standard deviation of the height (*s*) and the

**•** Backscattering coefficients at L-band HH and VV pol for the Argentinean test sites;

<sup>0</sup> , *prior i post i i*

0

refer to the input values derived from remote sensing data, which in the

s

s

*i*

*<sup>f</sup> <sup>N</sup>*

( ) ( ) ( )

<sup>=</sup>

0

*S i*

*i i*

correlation length (*l*), vegetation water content (*VWC*).

s

*p S*

(eq. 4) and the related PDF parameters (mean and standard deviation)

*<sup>f</sup>* <sup>=</sup> (4)

http://dx.doi.org/10.5772/57562

51

ò (5)

between SAR and optical images.

The noise function Nl

follows:

coefficients as follows [11, 39]:

*Si.* The variables *Si* can be:

*i*

lation length (*l*);

The variables *σ<sup>0</sup>*

presented approach are:

The free parameters are illustrated in table 3, where also the RMSE of the difference between measured and simulated backscattering coefficients are reported. Figure 7 depicts the com‐ parison between simulated and measured backscattering coefficients.


**Table 3.** Calibration parameters for simulation of L band backscattering coefficients with the Water Cloud Model with distinction between soybean and corn types.

## **3.2. Bayesian approach for SM estimation**

The objective of this research is to examine the capability and accuracy of a Bayesian approach to retrieve surface soil moisture under different assumptions for prior information on rough‐ ness and vegetation conditions in view of an operational use of the algorithm.

In the Bayesian approach, the scope is to infer biophysical parameters (e.g. soil moisture), from a set of backscattering responses measured by the sensor. The proposed algorithm is based on experimental data and theoretical models. The problem of having a few amounts of experi‐ mental data to build a reliable PDF has been overcome by the use of the simulated data from theoretical models. The Integral Equation Model (IEM) [27] was selected because it has the advantage of being applicable to a wide range of roughness scales. The general condition of validity of the model is ks < 3, where k is the wave number (≈ 0.2732 cm−1 for 1.3 GHz).

**Figure 7.** Comparison of measured backscattering coefficients (dots) with simulation from Water Cloud Model (surfa‐ ces) after the proper calibration of the free parameters. Two simulated surfaces are reported, one for corn (the red one) and the other soybean (the blue one).

For bare soil, these unknown parameters are the real part of the dielectric constant (*ε*), the standard deviation of the height (*s*) and the correlation length (*l*), the latter two describing the morphology of the surface. For vegetated fields, the Bayesian inversion was performed under two different approaches. In one case the Water Cloud Model (WCM) [35] is used to simulate the backscattering coefficients from vegetation. In the second case, the PDF parameters are properly modified to take into account vegetation effect through empirical relation with vegetation [38]. In both cases, the Vegetation Water Content (VWC) is added as unknown parameter, and it is derived from optical images. In this way, the approach exploits a synergy between SAR and optical images.

tively acquired close in time to the SAR image. Then the NDVI values were transformed in

The free parameters are illustrated in table 3, where also the RMSE of the difference between measured and simulated backscattering coefficients are reported. Figure 7 depicts the com‐

> **Model A B E RMSE** Soya 0.00119 0.03 0.634 1.7 dB Corn 0.2 0.003 2.2 2.6 dB

**Table 3.** Calibration parameters for simulation of L band backscattering coefficients with the Water Cloud Model with

The objective of this research is to examine the capability and accuracy of a Bayesian approach to retrieve surface soil moisture under different assumptions for prior information on rough‐

In the Bayesian approach, the scope is to infer biophysical parameters (e.g. soil moisture), from a set of backscattering responses measured by the sensor. The proposed algorithm is based on experimental data and theoretical models. The problem of having a few amounts of experi‐ mental data to build a reliable PDF has been overcome by the use of the simulated data from theoretical models. The Integral Equation Model (IEM) [27] was selected because it has the advantage of being applicable to a wide range of roughness scales. The general condition of validity of the model is ks < 3, where k is the wave number (≈ 0.2732 cm−1 for 1.3 GHz).

**Figure 7.** Comparison of measured backscattering coefficients (dots) with simulation from Water Cloud Model (surfa‐ ces) after the proper calibration of the free parameters. Two simulated surfaces are reported, one for corn (the red

ness and vegetation conditions in view of an operational use of the algorithm.

VWC through empirical approach defined by Jackson et al, 2004 [37].

50 Dynamic Programming and Bayesian Inference, Concepts and Applications

parison between simulated and measured backscattering coefficients.

distinction between soybean and corn types.

one) and the other soybean (the blue one).

**3.2. Bayesian approach for SM estimation**

At the beginning, the conditional probability is assumed as normal distribution. In the training phase, the conditional PDF is evaluated using measured data (*fim*) and simulated values from the IEM model (*fith*). The distribution assumption is then verified with a chi-squared statistics. The noise function Nl (eq. 4) and the related PDF parameters (mean and standard deviation) are calculated from the statistics of the ration between measured and simulated backscattering coefficients as follows [11, 39]:

$$N\_i = \frac{f\_{im}}{f\_{ith}}.\tag{4}$$

Subsequently a joint PDF is obtained as a convolution of single independent PDFs. The joint PDF is a posterior probability derived from prior probability on roughness and soil moisture values and to the conditional probability which relates the variations of backscattering coefficients to variations of soil moisture and roughness. The relationship can be expressed as follows:

$$p\left(S\_i \middle| \sigma\_i^0\right) = \frac{p\_{prior}\left(S\_i\right)p\_{post}\left(\sigma\_i^0 \middle| S\_i\right)}{\int p\_{prior}\left(S\_i\right)p\_{post}\left(\sigma\_i^0 \middle| S\_i\right)dS\_i},\tag{5}$$

where the term at the denominator is a normalization factor with integration over all variables *Si.* The variables *Si* can be:


The variables *σ<sup>0</sup> i* refer to the input values derived from remote sensing data, which in the presented approach are:

**•** Backscattering coefficients at L-band HH and VV pol for the Argentinean test sites;

**•** Backscattering coefficients at C-band HH and VV pol, L-band HH and VV pol and the combination of C and L band at HH pol for SMEX'02 test site.

**4. Results and discussion**

**4.1. Argentinean test site**

generated.

cm < *s* < 1.4 cm; *l*=5.0 cm.

cm < *s* < 1.4 cm; *l*=15.0 cm.

*s*=0.5 cm; *l*=5.0 cm.

The main aim of the work is to verify the sensitivity of the algorithm to prior conditions of roughness and vegetation in order to optimize the accuracy of the results. Based on this concept several retrievals were performed for different conditions of surface roughness, with specific algorithms for each coverage type in the study area. In the following the results on the

Integration of Remotely Sensed Images and Electromagnetic Models into a Bayesian Approach for…

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53

Over the Argentinean test site, the algorithm (fig.8) is divided in two main parts: one to be used in plots with bare soil or covered with sparse vegetation and another for vegetated soils. In both cases, two versions of the algorithm were developed: a simplified one working on a vector of mean values for each plot where the aim is to analyze the backscatter coefficient behavior using random values within ranges of *s* and *l*, and another one to work on the whole image, on pixel basis, to investigate the SM spatial distributions. Working with average values of backscattering coefficients has two objectives: to understand the effect on the SM estimates when the signal noise in the single plot is strongly reduced and to lower the computation

An extensive analysis was conducted in order to understand the behavior of variables such as surface roughness and vegetation presence in the final SM estimation through the variability

**•** Case 1: Pixel based algorithm for bare soil with fixed roughness. Three runs were executed: *s*=0.3 cm, *l*=5.0 cm; *s*=0.5 cm, *l*=5.0 cm and *s*=0.9 cm, *l*=5.0 cm. Then a mean value map is

**•** Case 2: Pixel based algorithm for bare soil with an integration over a roughness range: 0.6

**•** Case 3: Pixel based algorithm for bare soil with an integration over a roughness range: 0.6

**•** Case 4: Pixel based algorithm for bare soil with an integration over a roughness range: 1.0 cm< *s* < 1.5 cm; *l*=5.0 cm. In this case a very small integration range was considered.

**•** Case 5: Algorithm applied to backscattering coefficients averaged at plot level with a

**•** Case 6: VWC is calculated using a SPOT image. Fixed roughness and correlation length.

**•** Case 7: VWC is calculated using a SPOT image and a random function is implemented for *s* and *l* calculation, considering expected mean and standard deviation values for each parameter: mean value of *s*=0.7 cm and standard deviation value of 0.5 cm, mean value of *l*=5.0 cm and standard deviation of 5.0 cm. The random function is built as a noise function

random function. Values range: 0.5 cm < *s* < 1.2 cm; 5.0 cm < *l* < 10.0 cm.

Argentinean and SMEX'02 test sites are presented and discussed.

burden when applying a random function for *s* and *l* variables.

of the prior information. The different cases analyzed are listed below:

Based on the field data, the integration ranges for Bayesian inference were selected with different values as is illustrated in the following part. The main aim of using different intervals was to test the sensitivity of the methods to prior information, Through these integrations, to each pixel a value of dielectric constant is associated, starting from the corresponding back‐ scattering coefficient values. Finally, with the formula proposed by [40] the dielectric constant values have been transformed to estimated values of soil moisture. The flowchart in Fig.8 outlines the main steps of the algorithm, including training and test phase.

**Figure 8.** Flowchart of the Bayesian soil moisture approach applied to the Argentinean test site.

As above mentioned, another version of the Bayesian algorithm was developed to take into account the effect of vegetation into the PDF. The flowchart of the algorithm is the same as shown in fig.8, but instead of Water Cloud Model there is an adaptation of the PDF mean to an empirical function related to VWC as detailed described in Notarnicola et al., 2007 [38]. The algorithm was developed to work with C, L and combination of C and L band. In this work, this specific algorithm is applied to SMEX'02 data.
