INVERSION OF THREE LAYERS MULTI-SCALE SPM MODEL BASED ON NEURAL NETWORK TECHNIQUE FOR THE RETRIEVAL OF SOIL MULTI-SCALE ROUGHNESS AND MOISTURE PARAMETERS

In this paper, a multi-layered multi-scale backscattering model for a lossy medium and a neural network inversion procedure has been presented. We have used a bi-dimensional multi-scale (2D MLS) roughness description where the surface is considered as a superposition of a finite number of one-dimensional Gaussian processes each one having a spatial scale using the wavelet transform and the Mallat algorithm to describe natural surface roughness. An adapted three layers 2D MLS small perturbations (SPM) model has been used to describe radar backscattering response of semiarid sub-surfaces. The total reflection coefficients of the natural soil are computed using the multilayer model, and volumetric scattering is approximated by the internal reflections between layers. The original multi-scale SPM model includes only the surface scattering of the natural bare soil, while the multilayer soil modified 2D MLS SPM model includes both the surface scattering and the volumetric scattering within the soil. This multi-layered model has been used to calculate the total surface reflection coefficients of a natural soil surface for both horizontal and vertical co-polarizations. A parametric analysis presents the dependence of the backscattering coefficient on multi scale roughness and soil. The overall objective of this work is to retrieve soil surfaces parameters namely roughness and soil moisture related to the dielectric constant by inverting the radar backscattered signal from natural soil surfaces. To perform the inversion of the modified three layers 2D MLS SPM model, we used a multilayer neural network (NN) architecture trained by a back-propagation learning rule.


INTRODUCTION
Over the last two decades, microwave remote sensing has become an efficient tool for indirectly estimating soil moisture and soil properties in the top few centimeters of soils at different spatial and temporal scales.Soil moisture affects the partitioning of rainfall into infiltration and runoff and modulates soilatmosphere feedback interactions and it also affects groundwater recharge, crop growth.
In that context, modeling radar backscattering through natural surfaces has become an important theme of research and active remote sensing and has shown its utility for many applications in hydrology, geology, astrophysics, etc The characterization of soil surface roughness is a key requirement for the correct analysis of radar backscattering behavior.Many previous works have been devoted to the analysis of the backscattering characteristics of bare soils and several backscattering models (theoretical, semi-empirical and empirical) were developed ( [1] [2] [6] [9]).They used the classical statistical description of natural surfaces and characterized roughness by statistical parameters namely correlation length and standard deviation.
However, the weakness of the classical description of natural surfaces is the large spatial variability which affects the correlation function and makes classical roughness parameters very variable.Several works have proposed various approaches for the improvement of roughness descriptions ([3] [4] [5] [7] [11]) and have suggested that natural surfaces are better described as self-affine random processes (1/f processes) than as stationary processes.In previous works, we have analyzed radar backscattering on multi-scale bi-dimensional surfaces [3][7] [10] which description does not depend on classical roughness parameters standard deviation and correlation length but on new parameters related to multi-scale surfaces properties.
Extracting soil moisture and roughness parameters of natural surfaces from this data has been problematic for many reasons and many researchers have encountered many problems like the lack of information about the characteristics of natural surface roughness.In addition, the relation-ship between the backscattering coefficients is non-linear and the problem of retrieving parameters is frequently ill-posed and it may be impossible to separate the contributions from different mechanisms making the retrieval of several parameters simultaneously necessary.
The objective of this paper is to develop and test an inversion algorithm for soil moisture and multi-scale roughness parameters retrieval from radar backscattering coefficients simulated by the modified SPM model using a neural network inversion procedure based on a multilayer neural network (NN) architecture trained by a back propagation learning rule.This paper is organized into five sections.The first section describes the two dimensional multi-scale description of natural rough surfaces.Section 2 presents the multi-layers MLS SPM model.The third section discusses the influence of multi-scale roughness and the dielectric constant related to soil moisture on the backscattering simulations using our three layers multi-scale bi-dimensional small perturbation model (SPM).In the next section the neural network based inversion procedure, the results and their accuracy are presented.Finally, our conclusions are presented in the last section.

A MULTISCALE DESCRIPTION OF NATURAL SOILS ROUGHNESS
In this section, we present the multi-scale surface model used in the SPM model.
Natural roughness is described as a multi-scale process having a 1/f spectrum with a finite range of spatial scales going from a few millimeters b (ܾ ఒ ଵ ሻ) to several meters (Bresolution cell) [3] [7].We have considered the surface as a superposition of a finite number of one-dimensional Gaussian processes each one having a spatial scale [1] characterized by: Whereܼ is a collection of gaussian random independent variables with varianceߛ ଶ 2 ି , x a normalized distance with respect to an arbitrary length L= 2 b b and Ѱ a collection of orthonormal wavelet (4th Daubechies).The roughness multiscale parameter ν is related to the fractal dimension (ν=5-2D for mono-dimensional Euclidean surfaces and ν=7-2D for bidimensional surfaces [7]) and γ is related to the standard deviation and the number of spatial scales is equal to P. In a previous work [4][5], to describe more adequately natural surfaced, we have used the separable dyadic multi-resolution analysis introduced by Mallat [8] to extend the wavelet theory from one-dimensional to two-dimensional case.
Using the bi-dimensional wavelet transform, we have obtained respectively the vertical wavelet component, the horizontal wavelet component (3) and the diagonal wavelet component (4) of the heightܼ , (where i=Vertical, Horizontal or Diagonal.

ሺ4ሻ
Their autocorrelation function (ACF) is given by: And the standard deviation can be written as:

MLS SPM Model
In this study, we modeled radar backscattering over a three layer model [10] by taking into account volume scattering.Each layer is described as a multi-scale bi-dimensional surface using our multi-scale description.
In this study the small perturbation model SPM is used for the simulation of backscattering coefficients.SPM input parameters are the dielectric constant (deduced from the surface volumetric moisture content), the fractal parameter and the standard deviation of surface height.A multi-scale correlation function was therefore used in thisstudy.To remain within the domain of validity of the SPM used surfaces with ks< 0.3 (k: wave number, s: rms height).
Whereϴ is the incident angle and qp I is given by given by Fung [6] and Where W (n) is the Fourier transform of the nth power of the multi-scale autocorrelation function given by Mattia in [7] with n=1 for the SPM model [3][4][5] [7].Surfaces are characterized by the dielectric constant related to soil moisture, the albedo, the optical depth and surface roughness.Previous works used classical statistical parameters namely correlation length and standard deviation in the expression of the autocorrelation function W. The principal aim of this study is to use the multiscale surface description in the backscattering coefficient.

Multilayer modified SPM model
In this section we present the multilayer reflection model given by Fung [6] and Song [10] using our multi-scale 2D description of surface roughness.
The natural soil is composed by a dense media composed by multiple species of particles [10] and water, of a discrete dielectric soil componentε.We considered the half-space below the ground surface (z<0) as a three-layer medium (figure 5), where D is the radar penetration depth.This multilayer soil model includes three uniform layers [10]: -The medium 1 with thickness ݀ ଵ and permittivityߝ ௦௪ represents the mixture of soil particles and liquid water contents; -The medium 2 with thickness d ଶ and permittivity ε represents the air in soil; The medium 3 represents the soil layer below the radar penetration depth D (d ଵା ݀ ଶ ), with permittivity ε ୰ .It is semiinfinite and has no thickness.

Figure 5. Multilayer reflection model soil [10]
We consider the reflection of an electromagnetic wave from each layer.The incident wave is from layer1, and as layer3 is semi-infinite there is no electromagnetic wave reflection from the bottom of layer3.The incidence and reflected radar signal between medium 0 (air) and medium 1 can be expressed as: (12) We have to take into account these equalities since medium 2 is air: The total surface reflection coefficient of the multilayer soil can be expressed from the incident and reflected radar signal at airmedium 1 interface as where ܴ is the surface scattering of the soil (the specular surface reflection term), ܴ ܶ ଵ ܶ ଵ ‫ܣ‬ ଶ ሺܶ ଵ ܶ ଵ 1ሻ is the internal reflections between layers (the equivalent volumetric scattering term), with Aൌ ݁ ೖ భ ౙ౩ ഇ , ߠ ௧ the refraction angle at the air medium 1 interface, R ୟ the specular reflection coefficient of air at airmedium,ܶ the transmission coefficient from medium m to medium n (n=0, 1, 2, 3), ‫ܭ‬ the extinction coefficient of the medium 1 ([7], [10]) ݀ ଵ the thickness of the medium 1.The total reflection coefficients of the natural soil are computed using the multilayer model, and volumetric scattering is approximated by the internal reflections between layers.The surface reflection terms in the modified SPM model are replaced by the total reflection coefficients from the multilayer soil surface.The original multi-scale SPM model includes only the surface scattering of the natural bare soil, while the multilayer soil modified 2D MLS SPM model includes both the surface scattering and the volumetric scattering within the soil.This multilayered model has been used to calculate the total surface reflection coefficients of a natural soil surface for both horizontal and vertical co-polarizations.Each layer is described as a multi-scale bi dimensional surface using our multi-scale description ([4] [5] [10]) and the modified SPM.

Sensitivity to multi-scale roughness parameters
We have considered the VV and HH polarizations and studied the sensitivity of radar backscattering and angular trends for different multi-scale roughness and for different dielectric constants of each layer.
We have simulated the angular trends of the three layers multiscale backscattering coefficient from 20 to 80 degrees for different roughness parameters.As a first step, we fixed the parameter related to the Root Mean Square at 0.0031cm in VV and HH polarizations for five spatial scales to find out the effect of fractal dimension on the radar backscattered signal (Figure 6 and figure 7).As surfaces with ν between 1.5 and 2.3 are considered as smooth, we set, as a second step, this parameter at 2.1 in VV polarization and 1.9 in HH polarization for five spatial scales (Figure 8 and figure 9).When γ, the parameter related to the fractal dimension, increases the backscattering coefficient increases.The backscattered signal in VV polarization is higher than the backscattered signal in HH polarization.For all the simulations, the backscattering coefficient decreases with the incidence angle.

Sensitivity to Soil Moisture
Soil moisture is related to the complex dielectric constant ε.In Figure 10, figure 11, figure 12 and figure 13, we have represented radar backscattering as angular trends for different values of the complex permittivity of the second layer in the two polarizations VV and HH.The backscattering coefficient decreases as ߝ ᇱ 1 increases, whereas it increases where ߝ ᇱ 1 increases also.Indeed when the layers are dry corresponding to a lower humidity and as a consequence a lower dielectric constant, the penetration of the signal is more important and the backscattered signal is lower.As the dielectric constant increases, the surfaces and subsurface become wetter and the backscattered signal increases because the penetration is lower.

Inversion procedure
We present in this section, an algorithm to retrieve multi-scale roughness parameters and soil moisture parameter.In this study, the direct problem is represented by the SPM model.Thus, a sensitivity analysis of the SPM model has been performed and presented in the section 5.3 to examine the dependence of the output of the scattering model to the inputs parameters.When the outputs of the scattering model became saturated or insensitive to a parameter, the parameter inversion range was narrowed.
The method consists of inverting the SPM direct model using multilayer perceptron architecture [4] and [6].The inversion consists in retrieving roughness and soil moisture parameters γ1, γ2, ν1, ν2, ε1 et ε2 by using as input parameters the radar backscattering coefficients σHH,, σVV and the incident angle To illustrate the inversion techniques we propose a methodology given by figure 14.

Inversion Algorithm Results
To illustrate the inversion techniques described in the previous section, we apply them to the data simulated by the SPM.Before using the NN for the inversion, we have to calculate the mean rms error of the network.It converges well to a value smaller than 0.05 after 6000 iterations so that the NN is ready for the inversion procedure.
In figure 18 we present the sum squared network error for 35390 epochs.The inversion has given quite satisfactory results as the original values were retrieved with an error of 2.75%.After this sensitivity study we performed the inversion using a neural network technique witch leaded to quite satisfactory results with a mean error of 2.75 %.Future work will be dedicated to the study of radar backscattering on n layered media.

The
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-7/W3, 2015 36th International Symposium on Remote Sensing of Environment, 11-15 May 2015, Berlin, Germany θ varied from 30 to 60 degrees.The NN is trained by learning rules using the back propagation method.Simulated data sets based on the SPM surface scattering model are used to train the neural network.

Figure 14 . The proposed methodology 5 . 2
Figure 14.The proposed methodology5.2Neural Network TrainingThe first step in the inversion procedure is the generation of a set of training patterns.In this study, a total of 35390 training patterns were generated by using each of the signal models σ of the SPM backscattering coefficient.The parameters of interest σ used to generate the training patterns were randomly selected from within the range of parameters given by the sensitivity analysis.

Figure
Figure 15.The retrieving soil moisture parameter ε ε ε ε 1 after the inversion by the NN function of its original value

Figure 18 .
Figure 18.The sum squared network error