Geophysical Prospecting zyxwvuts 41, zyxwvuts 135-148,1993 zyxwvu ASPECTS OF 1D SEISMIC MODELLING USING THE GOUPILLAUD PRINCIPLE’ EVERT SLOB’ and ANTON ZIOLKOWSK13 ABSTRACT SLOB, E. and ZIOLKOWSKI, A. 1993. Aspects of 1D seismic modelling using the Goupillaud model. Geophysical Prospecting 41, 135-148. 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 2At, corresponding to a multiplication by Zz. 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 zyx : 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 At. It is then impossible to extract the wavelet, as there are fewer coefficients in the seismo- gram 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 At is half the two-way traveltime 2At of all the other layers. Using what Szaraniec calls the funda- Received May 1991, revision accepted August 1992. * Faculty of Electrotechnical Engineering, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands. Dept of Geology and Geophysics, University of Edinburgh, Grant Inst., West Mains Road, Edinburgh EH9 3LA, U.K. zyxwvu 135