**3. Description of the methodology for SM estimation**

The retrieval algorithm for SM is based on a Bayesian approach. Bayesian data analysis determines methods to make inference from data by using probabilities models for quantities.

The main characteristic of Bayesian methods is the explicit use of probability for quantifying uncertainty in inference based on statistical data analysis.

The process of Bayesian data analysis consists of three main steps:


**Figure 4.** Detail of the sampled plots during SARAT campaign acquisitions. N and S indicate North and South test

**Figure 5.** Soybeans, wheat (winter development), corn and sunflower.

46 Dynamic Programming and Bayesian Inference, Concepts and Applications

**Figure 6.** Bare plot with induced low (left) and high roughness (right).

fields.

Prior distributions can express our knowledge and uncertainty about the target variable. In this case the target variable could be thought as a random realization from the prior distribu‐ tion.

The application of Bayesian approach implies passing from a prior distribution to a posterior distribution. Based on this concept, a relationship is expected between these two distributions [29, 30, 33]. A general feature of Bayesian inference is that the posterior distribution is centered at a point which represents a compromise between the prior information and the data. This compromise is strongly controlled by the data as the sample size increase.

A prior distribution may not have a population basis and for this reason it is desirable to have a prior which plays a minor role in the posterior distributions. These prior distribu‐ 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 exclusively on data [34].

heights. The term *I pp*

lying soil *σ<sup>0</sup>*

sivity parameters *τ<sup>2</sup>*

the following expression:

properties of natural surfaces [27].

incoherent sum of the contribution of the vegetation σ<sup>0</sup>

way, the backscattering coefficients become:

where *VWC* is the vegetation water content (kg/m2

the two-way vegetation transmissivity with *τ<sup>2</sup>*

*soil σ = A VWC* × × - -× ×

qq

found in dependence of the canopy type and with the use of ground data.

model as their simulation enters directly the inversion procedure.

*<sup>n</sup>* depends on these parameters, *k*, *s* and on *RH*, *RV,* the Fresnel reflection

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

as the

49

veg and the contribution of the under‐

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

 q

*soil* represents the

is

(2)

*=exp(-2B VWC/ cosθ).* The parameters *A*, *B* and

*B VWC σ B VWC* + × -× × (3)

), *θ* the incidence angle, *σ<sup>0</sup>*

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

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>*

> 0 0 20 . *veg soil σ =σ σ* <sup>+</sup>t

0 0 cos (1 exp( 2 / cos )) exp( 2 / cos ), *<sup>E</sup>*

backscattering coefficient of bare soil that in this case calculated by using the IEM model, *τ<sup>2</sup>*

*E* depend on the canopy type and require an initial calibration phase where they have to be

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

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‐

If the terms related to vegetation and incidence angle are explicitly written in more detailed

*soil*, which is attenuated by the vegetation layer through the vegetation transmis‐

. For a given incidence angle the backscatter coefficient is represented by

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

In order to have a better understanding of the proposed methodology, described in section 3.2, a brief description of the electromagnetic models is presented in the next section.
