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
$ {e(t)=f(t)\ast x(t)}. $ ()
By substituting equation (4) into equation (3a), we get
$ {x(t)=w(t)\ast e(t)}. $ ()
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {x(t)=w(t)\ast f(t)\ast x(t)}. ()
When x(t) is eliminated from both sides of the equation, the following expression results:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\delta (t)=w(t)\ast f(t)}, ()
where δ(t) represents the Kronecker delta function:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta(t) = \left\{ \begin{array}{ll} 1, & t = 0,\\ 0, & \mbox{otherwise}. \end{array} \right. ()
By solving equation (6) for the filter operator f(t), we obtain
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(t)=\delta (t)\ast \frac{1}{w(t)}. ()
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
External links
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