# Inverse filtering

If a filter operator *f*(*t*) were defined such that convolution of *f*(*t*) with the known seismogram *x*(*t*) yields an estimate of the earth’s impulse response *e*(*t*), then

**(**)

By substituting equation (**4**) into equation (**3a**), we get

**(**)

**(**)

When *x*(*t*) is eliminated from both sides of the equation, the following expression results:

**(**)

where *δ*(*t*) represents the Kronecker delta function:

**(**)

By solving equation (**6**) for the filter operator *f*(*t*), we obtain

**(**)

Therefore, the filter operator *f*(*t*) needed to compute the earth’s impulse response from the recorded seismogram turns out to be the mathematical inverse of the seismic wavelet *w*(*t*). Equation (**8**) implies that the inverse filter converts the basic wavelet to a spike at *t* = 0. Likewise, the inverse filter converts the seismogram to a series of spikes that defines the earth’s impulse response. Therefore, inverse filtering is a method of deconvolution, provided the source waveform is known (*deterministic deconvolution*). The procedure for inverse filtering is described in Figure 2.2-1.

## See also

- The inverse of the source wavelet
- Least-squares inverse filtering
- Minimum phase
- Exercises
- Mathematical foundation of deconvolution