Auto-Correlation Function

The auto correlation function [1] is useful for observing directly the lateral extent of the structures on the surface by studying the decay in the function near zero shift.  It is given by





In digitized form, the unbiased auto-correlation [2] function may be given by



where

: the shift distance
_x: the x shift distance (=l_x)
_y: the y shift distance (=l_y)
_x and _y: the lateral point spacing of the digitized data points
z(x,y): surface topography
L_x and L_y: size of a definition area
N_x and N_y: number data points in x and y direction of the definition area


References

1. K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Lou, T. Mathia and H. Zahouani, The Development of Methods for The Characterisation of Roughness in Three Dimensions, Report EUR 15178 EN. EC Brussels, 1993

2. G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day, San Francisco, 1968, pp. 171