MULTICHANNEL Z‐TRANSFORMS AND MINIMUM‐DELAY
Authors: Robinson Enders A. 
Published: October 2002

In the standard deterministic model of water reverberation generation, the reverberation pulse‐train resulting from a deep reflection is minimum‐delay. Even in the more complex physical situations encountered in the field, there is evidence that in many cases the reverberation pulse‐train waveforms are minimum‐delay, or at least approximately so. The reason for this minimum‐delay property is that a pulse‐train waveform results from multiple reflections and transmissions within the layered earth; because reflection coefficients are less than unity in magnitude, the concentration of energy in a pulse‐train must appear at its beginning rather than its end; this early concentration of energy is the condition that pulse‐train waveform be minimum‐delay. Each deep reflection horizon contributes a minimum‐delay reverberation pulse‐train waveform to a seismic trace. If we let a spike series represent the deep horizons in the sense that the timing of a spike represents the direct arrival time of a reflection and the amplitude of the spike represents the strength of the reflection, then the seismic trace may be considered as the convolution of the spike series with the reverberation pulse‐train waveform. Because the reverberation pulse‐train waveform is minimum‐delay, and because at least approximately the deep horizon spike series represents a statistically uncorrelated series, the two conditions required for the application of the method of predictive deconvolution (Robinson, 1954, 1957) are met, and hence this method can be used as a practical digital data processing method to eliminate water reverberations on field seismic traces. The concept of minimum‐delay therefore is an important link in chaining together the deterministic approach and the statistical approach to seismic record analysis in the single‐channel case. The concept of minimum‐delay can be extended to the multichannel case. The theory of multichannel digital filters can be regarded as the matrix‐valued counterpart of single‐channel digital filter theory. A reflection seismogram consists of many traces; these traces are interrelated. A multichannel filter operates simultaneously on all these traces, and thus it can take advantage of the seismogram structure between traces as well as along a single trace. An important objective of seismogram analysis is to increase the resolution of overlapping waveforms by deconvolution. This goal can be accomplished through the use of inverse multichannel digital filters. Without proper design, an inverse multichannel filter can have the undesirable property that its impulse response function is unstable. In the case when there is the same number of input channels as output channels, then each of the coefficients of a multichannel digital filter may be regarded as a square matrix, and the z‐transform of the filter coefficients is a matrix‐valued polynomial in z. The determinant of its matrix‐valued z‐transform plays a central role in the classification of the delay properties of such a multichannel filter. This determinant is a scalar‐valued polynomial in z. If the coefficients of this polynomial represents a single‐channel minimum‐delay filter, then the original multichannel filter is also minimum‐delay; if they represent a single‐channel maximum‐delay filter, then the multichannel filter is also maximum‐delay; if they represent a single‐channel mixed‐delay filter, then the multichannel filter is also mixed‐delay. A minimum‐delay multichannel digital filter has an inverse which is a stable memory function. On the other hand, a maximum‐delay multichannel digital filter has an inverse that consists of a stable anticipation function. A mixed‐delay multichannel digital filter has a stable inverse, this inverse being made up of a memory component and an anticipation component.