Abstract

A layered-earth seismic model is subdivided into two subsystems. The upper subsystem can have any sequence of reflection coefficients but the lower subsystem has a sequence of reflection coefficients which are small in magnitude and have the characteristics of random white noise. It is shown that if an arbitrary wavelet is the input to the lower lithologic section, the same wavelet convolved with the white sequence of reflection coefficients will be the reflected output. That is, a white sedimentary system passes a wavelet in reflection as a linear time-invariant filter with impulse response given by the reflection coefficients. Thus, the small white lithologic section acts as an ideal reflecting window, producing perfect primary reflections with no multiple reflections and no transmission losses. The upper subsystem produces a minimum-delay multiple-reflection waveform. The seismic wavelet is the convolution of the source wavelet, the absorption effect, this multiple-reflection waveform, and the instrument effect. Therefore, the seismic trace within the time gate corresponding to the lower subsystem is given by the convolution of the seismic wavelet with the white reflection coefficients of the lower subsystem. The linear time-invariant seismic model used in predictive deconvolution has been derived. Furthermore, it is shown that any layered subsystem which has small reflection coefficients acts as a linear time-invariant filter. This explains why time-invariant deconvolution filters can be used within various time gates on a seismic trace which at first appearance might look like a continually time-varying phenomenon.

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