A structure function is a way to represent a surface in the spatial frequency domain. It is similar to the Fourier transform of a surface. It plots the rms of differences of randomly sampled pairs of points on a surface. The pairs are a fixed distance apart and have random orientation. The height difference between the pair of points is tabulated for many locations and the rms of the population tabulated. This is repeated for a range of different pair separations.
Structure functions are commonly used in atmospheric turbulence models. A Kolmogorov atmospheric model characterizes turbulence using a 2/3 power law based on a coherence length. Phase MOSAIC allows the structure function of a surface to be plotted and compared to an atmospheric structure function. This allows a measured surface to be compared to modeled atmospheric turbulence to predict optical performance.
Additional effects modeled using structure functions include scattering effects at high spatial frequency and tilt compensation (pointing error) at low spatial frequencies.