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

We present a new method to calculate complete synthetic seismograms for a spherically symmetric earth model which uses neither eigenfrequencies and eigenfunctions nor an earth-flattening transformation. The response 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 equations with source term and summation over vector spherical harmonics. Attenuation is included by using complex elastic moduli. Owing to the discrete 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 than normal-mode calculations for higher frequency applications, where often seismogram lengths of 20 to 40 min are sufficient. Since the angular degree of the spherical harmonics provide a natural discretization in the wavenumber domain, spatial aliasing is unimportant. Time aliasing is suppressed by evaluating the response at complex frequencies with constant imaginary part. We have compared synthetic seismograms obtained by the new method with normal-mode seismograms up to a frequency of 20mHz and achieve excellent agreement for all three components. The accuracy of the method is further corroborated by comparisons with real data up to a frequency of 200mHz. We tested the numerical scheme up to frequencies of 1Hz and harmonic degrees of 12000 and did not find any numerical instabilities. Incidentally, the approach sheds some light on how normal modes make up body waves.