# Dictionary:Map convolution

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 ψ_{x,y} are the output values, _{x,y} the input values, and *f*_{α,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., ΣΣ*f*_{α,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*.