Application of perturbation techniques to the problem of low‐frequency electromagnetic wave scattering from an air‐ground interface

Abstract

Abstract. The two-dimensional problem. of horizontally polarized wave scattering from. an airground interface is considered. The diffraction problem is formulated by means of the extinction theorem., leading to a system of two simultaneous surface integral equations. Three perturbation techniques, namely the small-perturbation method, the phase perturbation method, and the smallslope approximation, have been applied to this system.. Iterative relations for the terms of arbitrary order are obtained for each perturbation series. For the sinusoidal surface a benchmark solution, which remains stable near the points of Wood's anomalies, is developed. It is used to establish the area of validity of each perturbation technique by means of numerical comparison.

Numerical Simulations of the Scattered Field Near a Statistically Rough Air-Ground Interface

“…For a single realization of the random surface from each ensemble, a comparison with the method of moments [10] results has been made. The accuracy of the simulation has been estimated based on the relative error, which is defined as the normalized rms difference between the reflected field as found by the perturbation technique and the MoM solution (19) where the superscript denotes the corresponding values found by the benchmark.…”

“…For the selected power spectrum of the surface roughness (see next section) and the surface root-mean-square (rms) height , (which results in a surface rms slope of about 0.14 radians), the perturbative solution (13) and (14) is roughly 500 times faster than the method of moment solution developed in [10].…”

“…The implementation of different numerical methods is still limited by the amount of required CPU time and computer memory. We apply to this system the small-slope approximation [13]- [16] in a similar manner as in [10]. We represent both unknown functions in the form (13) where , are functions of order .…”

“…We do not discuss the convergence of (14) here, assuming that it holds in some appropriate sense. Using the procedure described in [10], one can derive a pair of iterative relations for and (14) Here we used the notations…”

“…Direct numerical simulation based on integral equation methods requires considerable computational resources (for more details, see, e.g., [6]). If one takes into account that the surface roughness has a typical height below 10 cm, that the surface slope is not very steep and that the GPR systems typically use frequencies below 1 GHz [1], one can see that it is possible to use a relatively simple perturbative approach like the small-slope approximation [7] or similar techniques [8]- [10] to solve the forward scattering problem. Such an approach can speed up Monte Carlo simulations tremendously in comparison with a direct numerical solution based on the integral equation methods.…”