Small Slope Approximation Applied to the Electromagnetic Scattering From a Rough Soil Surface With Furrows

The scattering of an electromagnetic wave by agricultural tilled soils is studied by using the first-order small-slope approximation. The rough surface is the superposition of two levels of roughness. A two-dimensional Gaussian random process describes the small-scale roughness, and a two-dimensional narrow-band Gaussian random process defined around a certain spatial frequency describes the large-scale component. The incoherent intensity obtained by the small-slope approximation is expressed as a combination of three terms. The first term is the intensity of small-scale roughness multiplied by an attenuation factor, the second is that of large-scale roughness multiplied by an attenuation factor and the third is the coupling intensity between the two roughness levels. When the surface is illuminated perpendicularly to the furrows and the conditions are right, the Floquet modes can be observed in the signature of the large-scale roughness. These effects are not observable for an impinging wave parallel to the furrows. Knowing that the correlation lengths of the large-scale roughness are much larger than those of the small-scale component, we derive an approximate expression for the coupling intensity which then depends only on the small-scale roughness spectrum and the root-mean-square height of the large-scale spectrum. Resulting expressions are studied for a cloddy sowing soil and a ploughed soil in monostatic configurations, in L- and C-bands. We also analyze the influence of the anisotropy of the large-scale roughness component autocorrelation function on the electromagnetic signature.