Correlation of the Fields Scattered by a Fractal Surface at Two Closely Spaced Receivers

Natural surfaces show scale invariance statistical properties over a wide range of scales, and exhibit power-law spectra over a wide range of spatial frequencies: these properties are well modelled by fractional Brownian motion (fBm) two-dimensional processes. We here present a closed form expression of the correlation coefficient of the fields scattered by a fBm surface and measured at two closely spaced positions. The obtained formulation shows that the correlation coefficient in the near-specular scattering case depends on a parameter that is related to the rms surface slope measured at the electromagnetic wavelength scale: when this parameter decreases, the correlation coefficient smoothly increases from the value obtained by the roughness-independent expression already available in literature to a value close to unity.