A 2D convolution often applied to potential-field maps, whereby each point on the map grid is replaced by a weighted sum of the values at other grid points.
where ψx,y are the output values, x,y the input values, and fα,β is the weighting scheme called the template. Simple residual maps are made by subtracting an average of values around the point from the value at the point. The values for different distances may be weighted and sometimes a bias is included (i.e., ΣΣfα,β≠0), so that the residual does not change sign locally.
By weighting the points to give horizontal derivatives and using Laplace's equation, second-vertical-derivative maps may be made. Other weighting schemes can be used for field continuation, wavelength filtering, etc. See grid residual.