# Dictionary:Convolution

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(kon, v∂ loo’ sh∂n) Change in waveshape as a result of passing through a linear filter.

1. A mathematical operation between two functions, g(t) and f(t), often symbolized by an asterisk:

$g(t)\ast f(t)=\int _{\tau }g(\tau )f(t-\tau )d\tau$ .

Convolution is not restricted to one dimension. For example, in two dimensions,

$g(x,y)^{\ast }f(x,y)=\int _{\alpha }\int _{\beta }g(\alpha ,\beta )f(x-\alpha ,y-\beta )d\alpha d\beta$ .

2. Linear filtering. If a waveform g(t) is passed into a linear filter with the impulse response f(t), then the output is given by the convolution of g with f. In discrete form where the input is a series of impulses of varying size, each will generate an f(t) of proportional amplitude and the output will be the superposition of these. This can be expressed as

$g_{t}\ast f_{t}=\sum _{k=0}^{L}g_{k}f_{t-k}$ .

This expresses that the output of a linear filter at the instant t is a weighted linear combination of the inputs. L is the convolution operator length and (L+1) is the number of points in the operator. (A simple computer program is shown in Figure F-14.) The frequency-domain operation equivalent to time-domain convolution consists of multiplying frequency-amplitude curves and adding frequency-phase curves. Convolution is sometimes done by (a) replacing each spike of the input with a proportionately scaled version of the impulse response and superposition forms the output; (b) folding where the impulse response of the filter is reversed in time and slid past the input, the output for each position of the impulse response being the sum of the products of input and folded impulse response for corresponding points; (c) multiplying z-transforms of the input and of the impulse response to give the z-transform of the output; or (d) multiplying Fourier or Laplace transforms to give the Fourier or Laplace transform of the output. See Sheriff and Geldart (1995, 279-81 and 540-2). Well logs may be thought of as the convolution of the response of the earth adjacent to the borehole with the logging sonde impulse response.

3. Convolution in two dimensions is used with gravity, magnetic, and other data to produce grid residual, second derivative, continuation maps, etc. see Fuller (1967).