The Spectral Function of a Layered System and the Determination of the Waveforms at Depth*

Abstract

Consider the mathematical model of a horizontally layered system subject to an initial downgoing source pulse in the upper layer and to the condition that no upgoing waveforms enter the layered system from below the deepest interface. The downgoing waveform (as measured from its first arrival) in each layer is necessarily minimum‐phase. The net downgoing energy in any layer, defined as the difference of the energy spectrum of the downgoing wave minus the energy spectrum of the upgoing wave, is itself in the form of an energy spectrum, that is, it is non‐negative for all frequencies. The z‐transform of the autocorrelation function corresponding to the net downgoing energy spectrum is called the net downgoing spectral function for the layer in question. The net downgoing spectral functions of any two layers A and B are related as follows: the product of the net downgoing spectral function of layer A times the overall transmission coefficient from A to B equals the product of the net downgoing spectral function of layer B times the overall transmission coefficient from B to A. The net downgoing spectral function for the upper layer is called simply the spectral function of the system. In the case of a marine seismogram, the autocorrelation function corresponding to the spectral function can be used to recursively generate prediction error operators of successively increasing lengths, and at the same time the reflection coefficients at successively increasing depths. This recursive method is mathematically equivalent to that used in solving the normal equations in the case of Toeplitz forms. The upgoing wave‐form in any given layer multiplied by the direct transmission coefficient from that layer to the surface is equal to the convolution of the corresponding prediction error operator with the surface seismogram. The downgoing waveform in this given layer multiplied by the direct transmission coefficient from that layer to the surface is equal to the convolution of the corresponding hindsight error operator (i.e., the time reverse of the prediction error operator) with the surface seismogram.

A Time‐domain Approach to the Normal‐incidence Inverse Problem*

“…We do not set the right-hand side of equation (40) to zero, because, as we shall see, p is a common factor to all P ( E ) and is also a factor in fl in equation (20); hence p cancels out on both sides of (20). Turning next to P ( E ) in equation (27), we see that Robinson (1967) and Robinson and Treitel (1977). Is these works do = 1, ro = _+ 1, and z = T/2 (T is usually chosen as 1 time unit).…”

“…, 2 K z . Before we substitute equations (40) (40), (41) and (42) into equation (20) we obtain the following solvable modified normal equation Equation (44) is precisely the normal equation obtained by Robinson (1967) and Robinson and Treitel (1977). Is these works do = 1, ro = _+ 1, and z = T/2 (T is usually chosen as 1 time unit).…”

An Algorithm for Fast Time‐domain Computation of One‐dimensional Synthetic Vertical Seismic Profiles*

“…Since the early sixties many articles have been published on different aspects of solving the one-dimensional wave equation to obtain synthetic seismograms and lately, also SVSPs. Baranov and Kunetz (1960), Wuenschel (1960), Trorey (1962), Sherwood and Trorey (1965), Darby and Neidell(1966), Claerbout (1968), Robinson and Treitel (1977), Wyatt (1981) and Choate (1982), are all working with a time domain solution and a Goupillaud layered model. Further, the following authors utilize arbitrary horizontally layered models : Berryman, Goupillaud and Waters (1958), Nielsen (1978) and Ganley (1981): they perform computations in the Fourier domain and therefore may include all kinds of absorption models.…”

Use of the Seismic Dynamic Deconvolution Algorithm in Direct Resistivity Interpretation*

“…The major contributions to the dynamic deconvolution of synthetic seismograms have been reviewed by Robinson and Treitel (1977). This review background is supplemented by Hemon (1966) and recent contributions by Bardan (1977), Koehler and Taner (1977), and Robinson and Treitel(l978).…”