Backscattering from Multi-Scale and Exponentially Correlated Surfaces

Most natural surfaces have been reported to have an exponential-like correlation function and generally contain more than one scale of roughness. In this paper we want to show that a multiscale Gaussian-distributed surface with a differentiable correlation function also possesses an exponential-like correlation function except near the origin. As a result, angular backscattering from such a surface in the low frequency region behaves the same as if it is from an exponentially correlated surface. In the high frequency region, scattering from the two differently correlated surfaces should be quite different as expected. Furthermore, we show that the differentiable correlation function for a multi-scale, Gaussian-distributed surface can be used to interpret backscattering from a known randomly rough surface that appears to possess an exponential correlation over a wide range of frequencies. These findings indicate that for natural surfaces the exponential correlation function may be a valid approximation to the real correlation in the low frequency region, although it is not the correct function in the high frequency region. It also offers an explanation as to why many natural surfaces appear to be exponentially correlated.