The surface correlation function p(ξ) determines the degree of correlation between two lateral separated locations of one surface. Thereby, ξ is defined as displacement between two locations, (x, y) and (x', y') on the surface and given by (1).
With increasing separation between two locations on the surface p(ξ) decreases, and at a certain distance, the surface correlation length l, the heights at the two locations are considered statistically uncorrelated.
The surface scattering of electromagnetic waves can be simulated with various models. Depending on the observed roughness scale multiple surface scattering models are valid for specific roughness conditions. For example, one of the first surface scattering models for slightly rough surfaces, the small perturbation model (SPM), deals with roughness scales that are small relative to the wavelength and hence has validity conditions for ks < 0.3, kl < 3, and m < 0.3. Since then, various surface scattering models for computing the scattering and emission behavior of natural surfaces in the microwave region have been proposed, such as the Kirchhoff scattering model (KH), the geometric optics model (GO), the physical optics model (PO), or the integral equation model (IEM), to name the most common used in literature. For simulations of EM scattering at soil surfaces, assumptions of the functional forms of p(ξ) have to be made. The two most common forms for mathematically describing the surface correlation of natural surfaces are the exponential pE(ξ) and the Gaussian pG(ξ) correlation functions, defined by (2) and (3).
For some mathematically sophisticated surface scattering models, an x-Power correlation function p(x-Power)(ξ) can be assumed (4), with x as value between 1 and 2.
In literature, rather smooth surfaces are characterized by an exponential surface correlation function, while rather rough surfaces are characterized by a Gaussian surface correlation function.