MATHEMATICS OF SURFACE WAVES AND THEIR USE FOR DETECTING AND IMAGING LATERAL HETEROGENEITIES IN THE EARTH'S CRUST AND UPPER MANTLE *)

The rapidly growing computer capabilities suggest to think about refining the theory which stands behind wave propagation in seismology. In this connection the equations of motion governing surface wave propagation are investigated from a fundamental mathematical point of view and the present state of art is sketched. While Love-wave motion is governed by an operator of the type Sturm-Liouville the Rayleigh-wave operator is in its original form more complex and leads to generalized eigenvalue problems which are beyond the theory usually employed in functional analysis. Great progress in this field was achieved recently, but nevertheless there are a lot of mathematically unsolved problems as e. g. the coupling between Rayleigh and Love modes and the completeness of Rayleigh motion within an open waveguide. These problems are important for the expansion of wave motion into surface wave modes. But just this procedure is desirable for the consideration of lateral heterogeneities which play an important role in all Earth sciences. On the other hand, we are able to demonstrate that reflected and scattered surface waves, which are produced by such inhomogeneities, can be successfully used for tomographic purposes by applying approximate techniques. Because the reflection coefficients of surface waves are very small and there is a great ambiguity in locating discontinuities, seismograms of many earthquakes from different regions and of many stations (German Regional Seismic Network) were used. Prominent discontinuities in Central Europe were investigated in such a way. The theory was extended on the spherical Earth and theoretical seismograms in presence of an ocean-continent transition were calculated. The application of the method proposed is nowadays yet far behind its full potential.