Seismic representation theorem coupling: synthetic P-SV mode sum seismograms for non-homogeneous paths

Abstract

The hybrid technique developed and demonstrated in this paper will be useful in studies of the propagation of any phase that can be represented in terms of a P-SV mode sum seismogram, across regional transition zones or other heterogeneities. These heterogeneities may exist in regions which form subsegments of a longer, mostly plane-layered, path. Examples of structures of interest through which such waves can be propagated using these techniques include regions of crustal thickening or thinning such as continent ocean transitions or basins, anomalous bodies of any shape located in the path, and sudden transitions from one layered structure to another. Examples of the types of phases than may be propagated through these structures include Rayleigh waves, Lg, Pn, Rg, Sn, and Pg. The hybrid method integrates three different types of numerical methods. The combination of these methods into a single hybrid technique allows consideration of problems which, due to computational and theoretical constraints, cannot be solved by any one of the individual component methods. The first two methods which are part of the hybrid technique are the finite element (FE) or finite difference (FD) method, and the modal propagator matrix (PM) method. Existing implementations of these methods have been modified to be compatible parts of the hybrid method. The third method which forms an integral part of the hybrid technique is the representation theorem (RT) integration method. It is developed, explained and illustrated in this paper. The hybrid method is a generalization of the hybrid method of Regan & Harkrider (1989a, b) for generation of hybrid SH mode-sum seismograms. It is used to produce synthetics for propagation along a long plane-layered path which contains a short segment or segments of heterogeneous structures. For simplicity consider one such short heterogeneous path segment between the source and the receiver. The hybrid method consists of a series of three procedures which propagate the wavefield across the heterogeneous segment of the path. This series of operations may be repeated if other heterogeneous segment occur between the source and the receiver. The first operation of the series uses the PM method to propagate the wavefield from the point or line source, located in a layered half-space, to the beginning of the heterogeneous segment of the path. The second procedure in the series couples the wavefield into the FE or FD grid and propagates it through the heterogeneous segment of the path using the FE or FD method. The third and final procedure in the series couples the wavefield back into and through another layered medium using the RT integration coupling method. The uncertainties inherent in the hybrid method are well understood. A discussion based on the orthogonality relations for Rayleigh waves is used as a basis for two methods, a method for the determination of the accuracy of the hybrid results, and a method which allows the decomposition of the wavefield into seismograms composed of single overtones. Relations expressing the expected accuracy of the seismograms on a mode by mode basis are derived. They can be used to choose the parameters needed for the hybrid calculation such that a desired level of accuracy is attained in the final hybrid seismograms. Seismograms generated using this new hybrid method with parameters chosen to produce seismograms accurate at the 2 per cent level are illustrated. Both mode-sum seismograms including the fundamental mode and the first 12 overtone branches and single-overtone mode-sum seismograms are generated. They show that the hybrid method is highly accurate and that the analysis of uncertainty gives a correct understanding of the sources of uncertainty. All hybrid seismograms agree with the appropriate theoretical comparison seismograms with less than the pre-determined 2 per cent misfit. In addition, the overtone by overtone hybrid seismograms show that the sources of this misfit are well modelled by the mode by mode uncertainty estimates derived in the analysis of uncertainties. Green's function filtering allows the separation of the wavefield into its component overtones. Such separation will considerably increase the usefulness of the hybrid technique (additional sources can be studied without repeating the FD step). It will also allow results of future studies using the hybrid method to be compared directly with studies of reflection and transmission coefficients for transition regions already in the literature. Using the example structure results of calculations are presented that show that single-overtone seismograms extracted from multiple-overtone mode sum seismograms recorded during the FD calculations are visually indistinguishable from single-overtone seismograms calculated using only that overtone at all steps. The rms amplitude discrepancies are less than the 2 per cent tolerance used to choose the parameters of the hybrid method. Clearly, the hybrid method can produce accurate seismograms, with well-understood uncertainties whose magnitudes are easily predictable. Thus, thus the hybrid method should be a valuable tool in many areas of waveform modelling.