Statistical Characterization of Bare Soil Surface Microrelief

Because the soil surface occurs at the boundary between the atmosphere and the pedosphere, it plays an important role for geomorphologic processes. Roughness of soil surface is a key parameter to understand soil properties and physical processes related to substrate movement, water infiltration or runoff, and soil erosion. It has been noted by many authors that most of the soil surface and water interaction processes have characteristic lengths in millimeter scales. Soil irregularities at small scale, such as aggregates, clods and interrill depressions, influence water outflow and infiltration rate. They undergo rapid changes caused by farming imple‐ ments, followed by a slow evolution due to rainfall events. Another objective of soil surface roughness study is investigating the effects of different tillage implements on soil physical properties (friability, compaction, fragmentation and water content) to obtain an optimal crop emergence. Seedbed preparation focuses on the creation of fine aggregates and the size distribution of aggregates and clods produced by tillage operations is frequently measured. Active microwave remote sensing allows potential monitoring of soil surface roughness or moisture retrieving at field scale using space-based Synthetic Aperture Radars (SAR) with high spatial resolution (metric or decametric). The scattering of microwaves depends on several surface characteristics as well as on imagery configuration. The SAR signal is very sensitive to soil surface irregularities and structures (clod arrangement, furrows) and moisture content in the first few centimeters of soil (depending on the radar wavelength). In order to link the remote sensing observations to scattering physical models as well as for modelling purpose, key features of the soil microtopography should be characterized. However, this characteri‐ zation is not fully understood and some dispersion of roughness parameters can be observed in the same field according to the methodology used. It seems also, that when describing surface roughness as a whole, some information related to structured elements of the micro‐ topography is lost

Model with two roughness levels for diffraction gratings: the generalized Rayleigh expansion

International audienceA model with two roughness levels for the diffraction of a plane wave by a metallic grating with periodic imperfections is presented. The grating surface is the sum of a reference profile and a perturbation profile. First, the diffraction by the reference grating is treated. At this stage the Chandezon method is used. This method leads to the resolution of eigenvalue systems. Each eigensolution defines an elementary wave function that characterizes a propagating or an evanescent wave. Second, the periodic errors are taken into account and a Rayleigh hypothesis is expressed: Everywhere in space the diffracted fields can be written as a linear combination of reference wave functions. The boundary conditions on the perturbed grating allow the diffraction amplitudes to be determined and therefore lead to the energetic magnitudes (efficiencies). The domain of analytical validity of this hypothesis is not defined. In fact, this method is considered to be an approximation. The proposed numerical study leads to some utilization rules. With a plane as the reference surface, the electromagnetic fields are given by classical Rayleigh expansions. Here the reference profile is a grating, hence the term generalized Rayleigh expansion

Multilook Intensity Ratio Distribution for 3-D Layered Structures with Slightly Rough Interfaces

International audienceWe derive the statistics for the multilook intensity ratio for a multilayered medium bounded by randomly rough surfaces. Calculations are carried out in the context of the first-order small perturbation method and assume slightly rough surfaces of infinite extent and centered Gaussian height distributions. We show that the probability distributions for the co-polar and cross-polarized intensity ratios for n-look data are functions of three parameters and that the mean exists for n > 1 and the variance for n > 2 The obtained theoretical expressions are verified by comparison with Monte-Carlo results

Scattering by anisotropic rough layered 2D interfaces

International audienceWe propose a statistical study of the scattering of an incident plane wave by a stack of two two-dimensional rough interfaces. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. The electromagnetic fields are represented by Rayleigh expansions, and a perturbation method is used for solving the boundary value problem and determining the firstorder scattering amplitudes. For slightly rough interfaces with a finite extension, we show that the modulus of the co- and crosspolarized scattering amplitudes follows a Hoyt law and that the phase is not uniformly distributed. For interfaces with an infinite extension, the modulus follows a Rayleigh law and the phase is uniformly distributed. We show that these results are true for correlated or uncorrelated interfaces and for isotropic or anisotropic interfaces

Statistics of the Stokes parameters for multilook signals scattered by layered structures with slightly rough random interfaces

International audienceThe polarization state of waves scattered from multi-layered structures with randomly rough boundaries is given by the Stokes vector and its four components. Polarimetric radars use multilook processing to reduce statistical fluctuations by spatially averaging the scattered signals. In this paper, we obtain and express in closed-form the probability density function, the cumulative density function, the expected value and the variance for each Stokes parameter in the multilook case. The slightly rough boundaries are realizations of second-order stationary centered Gaussian bi-dimensional spatial random processes and the layered structure is illuminated by a monochromatic plane wave. The zenithal and azimuthal components of the far scattered electric field are obtained from the first-order small perturbation method. The theory leads to a multivariate Gaussian model. Finally, we obtain a universal function for the probability distribution for the last three Stokes parameters. Numerical simulations are performed and coincide with the theoretical derivations. For a ground snow system represented by three interfaces air / snow cover / frozen soil / unfrozen soil, we obtain the marginal probabilities and compare the theoretical results with Monte-Carlo simulations. We show that the marginal probability distributions for multilook data are very different from those for single-look returns. Some features of coherent phenomena are observed since multilook processing reduced statistical fluctuations

Loi de probabilité du rapport des intensités en approche multivisée pour des milieux stratifiés à interfaces faiblement rugueuses

International audienceNous déterminons l'expression analytique de la loi de probabilité du rapport des intensités diffusées par un milieu stratifié présentant deux interfaces faiblement rugueuses éclairé par une onde plane électromagnétique. Les formules obtenues sont valides pour des intensités en polarisations directes et croisées et en approche simple visée et en multivisées. Les calculs sont menés dans le cadre de la méthode des petites perturbations à l'ordre 1 et suppose que les scènes éclairées sont d'extensions infinies et que les interfaces présentent des distributions des hauteurs gaussiennes. Nous montrons que pour les signatures électromagnétiques basées sur plus de deux visées, la loi de probabilité a une moyenne finie et une variance finie. Les lois de probabilité théoriques sont validées par comparaison avec des histogrammes issus de tirages de Monte-Carlo. Mots clés-Rapport d'intensités, amplitude de diffraction, simple visée, multivisée, densité de probabilité, milieu stratifié, surface aléatoire, méthode des petites perturbations

Multilook phase difference distribution for slightly rough boundary layered ground

International audienceThe authors derive the distribution of the phase difference between two multilook scattering signals for a multilayer stack with randomly rough surfaces under a plane wave excitation. First, for infinite slightly rough surfaces described by Gaussian centred stochastic processes, the authors show that the underlying complex scattering signals follow a Gaussian joint distribution. Also, it is demonstrated that this property is within the scope of the first-order perturbation theory. Secondly, the authors use this joint probability law to derive the closed-form expression for the probability density function of the phase difference. The theoretical formula is verified by comparison with Monte-Carlo simulations. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited

Scattering from a slightly rough chiral surface: analysis with the small perturbation method and the small slope approximation

International audienceWe study the scattering of a linearly polarized electromagnetic plane wave by a two-dimensional random slightly rough surface separating the vacuum from a chiral medium. We implement the first-order small perturbation method (SPM) and the first-order small slope approximation (SSA) and determine the analytical expressions of the coherent and incoherent intensities. The effects of chirality on the polarization of the wave scattered within the vacuum are analyzed. The coherent intensity has a cross-polarized component as well as the incoherent intensity in the incidence plane. We show that there are configurations for which a total polarization coupling occurs with the co-polarized incoherent intensity equal to zer

Soil surface roughness modelling with the bidirectional autocorrelation function

International audienceSurface roughness is a major part of soil surface condition. It results from tillage operations and weathering. Surface roughness parameterisation is still a scientific lock and the object of many studies. An efficient parametrisation of soil surface roughness by modelling the bidirectional autocorrelation function estimated from 2.5D digital elevation models of soil surfaces is introduced. It not only provides geostatistical parameters that can be related to other soil surface characteristics, but let us emphasise that it reproduces the autocorrelation function with very good accuracy. The autocorrelation function is often modelled by a function of three parameters, the height variance, a single correlation length, and a roughness exponent. We added two parameters in order to take into account the anisotropy of soil surfaces and to align the coordinate system in the direction of the maximum correlation length. We propose the way to estimate roughness parameters and show its robustness for soil surfaces using laboratory tests with repeated rainfall events. One soil surface evolves from isotropy to anisotropy, and the other undergoes a reduction of initial anisotropy. The improvement brought by a second correlation length is thus highlighted. Under rainfall impact, the variation of the correlation lengths is more marked than that of the usual roughness parameter that is the root mean squared of the heights. Both parameters are complementary, capturing horizontal or vertical variation respectively. The evolution of the roughness exponent showed a slight increasing trend, which can be related to surface smoothing

Statistics of the Stokes parameters for the signal scattered by layered structures with an arbitrary number of slightly rough interfaces

International audienceHorizontally stratified structures are commonly used to represent naturally occurring structures, such as soils. The electromagnetic signature of such a medium illuminated by a radar and the polarization state of the scattered wave are fully determined by the knowledge of the four Stokes parameters. In this paper, we determine the statistics of the four Stokes parameters for the signal scattered by layered structures with an arbitrary number of slightly rough interfaces. The rough interfaces are realizations of second-order stationary centered Gaussian stochastic processes and the layered structure is illuminated by an elliptically polarized monochromatic wave. The zenithal and azimuthal components of the far scattered electric field are derived from the first-order small perturbation method. The derivation leads to a multivariate Gaussian model for the underlying complex scattered amplitudes and we establish the closed-form expressions of the probability density function, the cumulative density function and the first- and second-order moments for the four Stokes parameters. For an observation direction outside the incidence plane, we establish the condition on the incidence wave parameters for which the zenithal and azimuthal components are uncorrelated. For an air / snow cover / frozen soil / unfrozen soil structure, we analyze the marginal probabilities and validate the theory by comparison with Monte-Carlo simulations. More generally, when the two complex components of the field scattered by the illuminated zone are Gaussian random variables, these statistics offer possibilities for in-depth investigating the polarization of scattering processes from random media