ISSN:
1365-2478
Source:
Blackwell Publishing Journal Backfiles 1879-2005
Topics:
Geosciences
,
Physics
Notes:
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 2Δt, corresponding to a multiplication by Z2. 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 Δt. 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 Δt is half the two-way traveltime 2Δt 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1111/j.1365-2478.1993.tb00862.x
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