Deconvolution or spiking filters are frequently employed to sharpen the character of seismograms and hence improve resolution of the earth's layering. In media having significant attenuation or scattering, the shape of an input waveform will change as it propagates. On a reflection seismogram this results in a change with time in the character and frequency content of the seismic signal. Since a statistical description of this situation would be time-dependent, the seismogram should be treated as a nonstationary random process. A spiking filter to deconvolve such a seismogram must be time-variable. Optimum filters for nonstationary inputs can be designed using the least-mean-square error criterion. For a zero-delay filter to give satisfactory results, the signal waveform must at all times be minimum phase. This will be true if the input waveform and the law governing the change of waveshape with time are both minimum phase. If these conditions are not met, a filter which has a time delay should be used. For a linear attenuation mechanism with frequency-independent Q, there is some justification for expecting a minimum-phase law. In the absence of a priori knowledge of the input waveform and the attenuation structure of the layered earth, it is necessary to estimate the time-dependent autocorrelation function of the recorded seismic trace.

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