Fast evaluation of complete synthetic SH seismograms based on asymptotic mode theory

Abstract

In this thesis we have developed an asymptotic mode theory with the following features. 1) Complete synthetic SH seismograms can be evaluated for both realistic models of Earth and crust. 2) The method is of practical value and can be used even on small computers wi th reasonable computation times on a routine basis. The bulk of computation time is for the eigenveccu, w, which need not be recalculated for a different source or station. This is the greatest advantage of modal summation over all other methods. In many cases such as treated in section 7.1 for relatively shallow sampling of structure it becomes practical to calculate seismograms interactively. 3) In chapter V section 5.2 - 5.3 we have indicated a method to speed up the dispersion calculations considerably for a large range of slownesses. However, such a method would require more memory in the present programcode then was available on our minicomputer. 4) Typi cal for mode programs is that its in'" and out put data are si mpl e. Users do not need a profound knowledge of asymptotic mode theory. whereas ray theoretical methods and their generalized versions usually demand skillful users in order to make an appropiate choice of the ray sum. 5) In this thesis we have also indicated a method in chapter IV section 4.3 to improve the accuracy of eigenfrequencies and eigenfunctions in gradient zones, where asymptotic theory fails. For efficiency, these corrections should be restri cted to regions wi thin a few wavelengths around the turning point, if gradients there are strong. 6) It is relati vely simple to incorporate the effect of attenuation, while for example, the generalized ray theory accounts for attenuation in an adhoc manner. 7) At least one of the integrals in the double inverse transformation to the space time domain is done analytically by summation of residues, whereas in the reflectivity method both inverse transforms are evaluated numerically with a computational cost proportional to the product of the frequency band and the distance. Both the general i zed ray method (C hapman, 1976; Burdi ck & Grcut t , 1981 ) and the reflectivity method (Kennett,1983) can be extended to vertically inhomogeneous layers. Yet mode theory seems to us a f as ter and more reliable computational technique. For the future the following extensions of the current mode pogram seems relevant to us. Firstly, the implementation of the synthesis of P~SV motion in a 32 bi ts computing machine is feasi ble, whereas it is impractical in the 16 bits Hewlett Packard 1000 of our geophysical department. The method of first order corrections can be extended to pclSV case and the same applies to the evaluation of energy integrals with slightly modified expressions for the integrands. Secondl y, if low vel oci ty zones are of maj or importance in the appli cation of the present method, rather awkward sol utions are generated in these regions. Parabolic cylinder functions are the most appropriate solutions in this case, but their evaluation would require a table of these functions and first order corrections would yield rather complicated algorithms. Another strategy which seems more promising to us with minor modifications in the present structure of the program flow is a linear approximation to the potential function of the wave equation instead of a transformation to the Airy equation. The advantages are that (i) the Airy functions are still the basic solutions for an interval whose size depends on the behaviour of the vertical slowness. (ii) although near the change of sign of the derivative of the vertical slowness finer subintervals are needed, the accuracy can be controll ed by decreasi ng these intervals. (i i i) analytical solutions to the energy integrals remain the same. Although only shown by a way of illustrations: the comparison with NARS data seems to imply that current realistic Earth models are too simple and our ignorance about the struct ure of the upper mantl e isstill very large. Therefore, the usefulness of the present method, althoug valid for lateral homegenous media, is still very promising for seismologic research.