A Study of the Kirchhoff Approximation for the Coherent Reflection Coefficient of a 1-D Perfectly Conducting Rough Surface

Applications involving the measurement of specular reflections from the Earth’s surface have recently renewed interest in the coherent reflection coefficient of a rough surface, particularly in the limit where the reflection coefficient becomes small. In this letter, a statistical model is first presented that characterizes the number of surface realizations required to obtain a specified precision in estimating the coherent reflection coefficient; the model shows that the number of realizations increases significantly as the coherence of the reflected signal decreases. A study of rough surface reflection coefficient amplitudes is also reported. The Kirchhoff Approximation (KA) is frequently used to describe the coherent reflection coefficient amplitude as a function of the angle of incidence and the surface root-mean-square (rms) height relative to the electromagnetic wavelength. KA predicted horizontally polarized reflection coefficients are compared with those obtained from Monte Carlo method of moments (MOM) simulations using 1-D perfectly conducting surfaces described as stationary Gaussian random processes with a Gaussian correlation function. The comparisons show reasonable agreement between the KA and MOM results for a variety of observing scenarios and reflection coefficient amplitudes, and an improved level of agreement if a correction to the KA prediction derived from the small slope approximation (SSA) theory of surface scattering is included.