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**3D S Parameters - Spatial Parameters**

ACF (Autocorrelation Function)

The development of the spatial parameters involves the use of the mathematical technique of the autocorrelation function (ACF). This section will review the basic concepts behind the ACF necessary to understand the various spatial parameters.

The ACF is found by taking a duplicate surface (Z(x-Dx,y-Dy)) of the measured surface ((Z (x, y)) and mathematically multiplying the two surfaces together, with a relative lateral displacement (Dx,Dy) between the two surfaces. Once multiplied together, the resulting function is integrated and normalized to Sq, to yield a measure of the degree of overlap between the two functions. If the shifted version of the surface is identical to the original surface then the ACF is 1.00. If the shifted surface is such that all peaks align with corresponding valleys then the ACF will approach –1.00.

Thus the ACF is a measure of how similar the texture is at a given distance from the original location. If the ACF stays near 1.00 for a given amount of shift, then the texture is similar along that direction. If the ACF falls rapidly to zero along a given direction, then the surface is different and thus “uncorrelated” with the original measurement location.

For the turned surface above, the ACF in the X direction falls to zero quickly as the peaks of the shifted surface align with the mean plane. The ACF along X becomes negative as the peaks of the surface align with the valleys of the shifted surface. Shifting along the Y direction, the surface is near identical to the original, resulting in the ACF in the Y direction remaining near 1.00.