**1. n. []**

A mathematical operation on two functions that is the most general representation of the process of linear (invariant) filtering. Convolution can be applied to any two functions of time or space (or other variables) to yield a third function, the output of the convolution. Although the mathematical definition is symmetric with respect to the two input functions, it is common in signal processing to say that one of the functions is a filter acting on the other function. The response of many physical systems can be represented mathematically by a convolution. For example, a convolution can be used to model the filtering of seismic energy by the various rock layers in the Earth; deconvolution is used extensively in seismic processing to counteract that filtering.

The mathematical form of the convolution of two functions, a filter *f*(*t*) and a time-series *x*(*t*), is

*y*(*t*) = ∫ *f*(*t*−τ)*x*(τ)dτ,

where *y*(*t*) is the output of the convolution.

In the frequency domain, convolution is simply the product of the Fourier transforms (FT) of the two functions:

*Y*(ω) = *F*(ω)**X*(ω),

where

*X*(ω) = FT of the time series *x*(*t*)

*F*(ω) = FT of the filter *f*(*t*)

*Y*(ω) = FT of the output *y*(*t*)

ω = angular frequency.

**See related terms:**
embedded wavelet,
synthetic seismogram,
wavelet

**2. n. []**

A mathematical operation that uses downhole flow-rate measurements to transform bottomhole pressure measurements distorted by variable rates to an interpretable transient. Convolution also can use surface rates to transform wellhead pressures to an interpretable form. Convolution assumes a particular model for the pressure-transient response, usually infinite-acting radial flow. This operation is similar to what is done to account for the flow history in rigorous pressure-transient analysis.

**See related terms:**
transient rate and pressure test analysis