# 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

 ${\displaystyle {e(t)=f(t)\ast x(t)}.}$ (4)

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

 ${\displaystyle {x(t)=w(t)\ast e(t)}.}$ (3a)
 ${\displaystyle {x(t)=w(t)\ast f(t)\ast x(t)}.}$ (5)

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

 ${\displaystyle {\delta (t)=w(t)\ast f(t)},}$ (6)

where δ(t) represents the Kronecker delta function:

 ${\displaystyle \delta (t)=\left\{{\begin{array}{ll}1,&t=0,\\0,&{\mbox{otherwise}}.\end{array}}\right.}$ (7)

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

 ${\displaystyle f(t)=\delta (t)\ast {\frac {1}{w(t)}}.}$ (8)

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.

Figure 2.2-1  A flowchart for inverse filtering.