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ASPECTS OF 1D SEISMIC MODELLING USING THE GOUPILLAUD PRINCIPLE 1
Author(s) -
SLOB EVERT,
ZIOLKOWSK ANTON
Publication year - 1993
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1111/j.1365-2478.1993.tb00862.x
Subject(s) - seismogram , synthetic seismogram , reflection (computer programming) , wavelet , geology , impulse response , convolution (computer science) , geophone , mathematical analysis , function (biology) , mathematics , geometry , geophysics , seismology , computer science , artificial intelligence , machine learning , artificial neural network , programming language , evolutionary biology , biology
A bstract A reflection response function for a 1D discretized earth model can be obtained using ray‐theory and Z‐transforms with the Goupillaud model. This is usually done by taking the source function as a plane wave impinging normally on the layered earth. Two important problems have been tackled with this basic idea. The first, extraction of the source wavelet, and the second, a description of the free‐surface related problems. In the Goupillaud model, the one‐way traveltime in each layer is taken to be the same time interval At, which is also the time unit for the Z‐transform. The two‐way traveltime in any layer is 2Δ t , corresponding to a multiplication by Z 2 . The reflection impulse response therefore contains only even powers of Z . The convolution of the reflection response with the wavelet yields a seismogram whose Z ‐transform contains both odd and even powers of Z . However, even though the seismogram contains more coefficients than unknowns, the wavelet cannot be extracted, because the coefficients are not independent: later coefficients are functions of earlier ones, which does not make sense physically. To overcome this physical problem for the reflection seismogram, the two‐way traveltime through the layer should be Δ t. It is then impossible to extract the wavelet, as there are fewer coefficients in the seismogram than unknowns. Szaraniec has proposed a modification to the Goupillaud model, known as the odd‐depth model, that includes the free surface and a top layer whose two‐way traveltime Δ t is half the two‐way traveltime 2Δ t of all the other layers. Using what Szaraniec calls the fundamental identity of the odd‐depth model, it is possible to extract the source wavelet from the seismogram. We show that this fundamental identity holds only if reflection coefficients of deeper interfaces are functions of the reflection coefficients of shallower interfaces; that is, for extremely improbable geologies. Neither of these approaches offers a solution to the deconvolution problem. It is better to obtain the source signature from measurements in the field. Only Szaraniec's model offers the possibility of tackling the problem of the free surface but because of an inherent flaw in the model, it fails to address the problem.