Auto-Correlation

The autocorrelation 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

C(tau) = integral from 0 to L of (z(x)*z(x+tau)*dx / integral from 0 to L of square(z(x))*dx


where

Tau is the shift distance
z(x) is surface profile
L is evaluation length

or in digitized form for unbiased auto-correlation

Cj=C(tau)=1/(N-j)*sum from i=1 to N-j of z(i)*z(i+j)/sum from i=1 to N of square(z(i))

Reference


T. V. Vorburger, J. Raja,  Surface Finish Metrology Tutorial, NISTIR 89­4088 (National Institute of Standards and Technology, Gaithersburg, MD, 1990)