Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions

The Wagner (1966) and Smith (1967) classical monostatic one-dimensional (1-D) shadowing functions assume that the joint probability density of heights and slopes is uncorrelated, thus inducing an overestimation of the shadowing function. The goal of this article is to quantify this assumption. More recently, Ricciardi and Sato (1983, 1986) proved that the shadowing function is given rigorously by Rice´s infinite series of integrals. We observe that the approach proposed by Wagner retains only the first term of this series, whereas the Smith formulation uses the Wagner model by introducing a normalization function. In this article, we first calculate the shadowing function based on the work of Ricciardi and Sato for an uncorrelated process. We see that the uncorrelated results do not have any physical sense. Next, the Wagner and Smith formulations are modified in order to introduce the correlation. Correlated and uncorrelated results are compared with the reference solution, which is determined by generating a surface for a Gaussian autocorrelation function. So, we show that the correlation improves the results for values μ⩽2σ, where μ represents the slope of incident ray and σ the slopes variance of the surface. Finally, our results are compared to those given by Kapp and Brown (1994), determined from the first three terms of Rice´s series, but the shadowing function used is not averaged over the slopes