COMPLETE SYNTHETIC SEISMOGRAMS FOR A SPHERICALLY SYMMETRICAL EARTH BYA NUMERICAL COMPUTATION OF THE GREENS-FUNCTION IN THE FREQUENCY-DOMAIN

We present a new method to calculate complete synthetic seismograms fo r a spherically symmetric earth model which uses neither eigenfrequenc ies and eigenfunctions nor an earth-flattening transformation. The res ponse of the earth to a moment tenser point source is evaluated in the frequency domain for both spheroidal and toroidal motion by numerical integration of the appropriate system of ordinary differential equati ons with source term and summation over vector spherical harmonics. At tenuation is included by using complex elastic moduli. Owing to the di screte sampling of the response in the frequency domain, the numerical effort is proportional to the length of the desired time series for a fixed maximum frequency. This makes the method much more efficient th an normal-mode calculations for higher frequency applications, where o ften seismogram lengths of 20 to 40 min are sufficient. Since the angu lar degree of the spherical harmonics provide a natural discretization in the wavenumber domain, spatial aliasing is unimportant. Time alias ing is suppressed by evaluating the response at complex frequencies wi th constant imaginary part. We have compared synthetic seismograms obt ained by the new method with normal-mode seismograms up to a frequency of 20mHz and achieve excellent agreement for all three components. Th e accuracy of the method is further corroborated by comparisons with r eal data up to a frequency of 200mHz. We tested the numerical scheme u p to frequencies of 1Hz and harmonic degrees of 12000 and did not find any numerical instabilities. Incidentally, the approach sheds some li ght on how normal modes make up body waves.