Comment on ‘Aspects of 1D seismic modelling using the Goupillaud principle’ by Evert Slob and Anton Ziolkowski1

The paper by Slob and Ziolkowski (1993) is apparently a comment on my paper (Szaraniec 1984) on odd-depth structure. In fact the basic understanding of a seismogram is in question. The fundamental equation for an odd-depth model and its subsequent deconvolution is correct with no additional geological constraints. This is the essence of my reply which is contained in the following points.

• 1 The discussion by Slob and Ziolkowski suffers from incoherence. On page 142 the Goupillaud (1961) paper is quoted: “… we must use a sampling rate at least double that… minimum interval…”. In the following analysis of such a postulated model Slob and Ziolkowski say that “… two constants are used in the model: Δt as sampling rate and 2Δt as two-way traveltime”. By reversing the Goupillaud postulation all the subsequent criticism becomes unreliable for the real Goupillaud postulation as well as the odd-depth model.
• 2 Slob and Ziolkowski take into consideration what they call the total impulse response. This is over and above the demands of the fundamental property of an odd-depth model. Following a similar approach I take truncated data in the form of a source function, S(z), convolved with a synthetic seismogram (earth impulse response), R?(z), the free surface being included. The problem of data modelling is a crucial one and will be discussed in more detail below. By my reasoning, however, the function may be considered as a mathematical construction introduced purely to work out the fundamental property. In this connection there is no question of this construction having a physical meaning. It is implicit that in terms of system theory, K(z) stands for what is known as input impedance.
• 3 Our understandings of data are divergent but Slob and Ziolkowski state erroneously that: “Szaraniec (1984) gives (21) as the total impulse response…”. This point was not made. This inappropriate statement is repeated and echoed throughout the paper making the discussion by Slob and Ziolkowski, as well as the corrections proposed in their Appendix A, ineffective. Thus, my equation (2) is quoted in the form which is in terms of the reflection response G^sc and holds true at least in mathematical terms. No wonder that “this identity is not valid for the total impulse response” (sic), which is denoted as G(z). None the less a substitution of G for G^sc is made in Appendix A, equation (A3). The equation numbers in my paper and in Appendix A are irrelevant, but (A3) is substituted for (32) (both numbers of equations from the authors’ paper). Afterwards, the mathematical incorrectness of the resulting equation is proved (which was already evident) and the final result (A16) is quite obviously different from my equation (2). However, the substitution in question is not my invention.
• 4 With regard to the problem of data modelling, I consider a bi-directional ID seismic source located just below the earth's surface. The downgoing unit impulse response is accompanied by a reflected upgoing unit impulse and the earth response is now doubled. The total impulse response for this model is thus given by where (—r₀) =— 1 stands for the surface reflection coefficient in an upward direction. Thus that is to say, the total response to a unit excitation is identical with the input impedance as it must be in system theory. The one-directional 1D seismic source model is in question. There must be a reaction to every action. When only the downgoing unit impulse of energy is considered, what about the compensation?
• 5 In more realistic modelling, an early part of a total seismogram is unknown (absent) and the seismogram is seen in segments or through the windows. That is why in the usual approach, especially in dynamic deconvolution problems, synthetic data in the presence of the free surface are considered as an equivalent of the global reflection coefficient. It is implicit that model arises from a truncated total seismogram represented as a source function convolved with a truncated global reflection coefficient.

Validation or invalidation of the truncation procedure for a numerically specified model may be attempted in the frame of the odd-depth assumption. My equations (22) and (23) have been designed for investigating the absence or presence of truncated energy. The odd-depth formalism allows the possibility of reconstructing an earlier part of a seismogram (Szaraniec 1984), that is to say, a numerical recovery of unknown moments which are unlikely designed by Slob and Ziolkowski for the data.