Kirchhoff-Helmholtz Synthetic Seismograms

In this paper, author systematically and succinctly introduced the basic theory and the basisequations of Kirchhoff-Helmholtz (KH) integral, and applied the method to the seismologicalproblems of reflected wave, converted wave and refracted wave. The KH integral is one of theboundary integral methods, it is derived in terms of wave equation and transforms body integral tothe boundary integral. When computing the reflected wave and transmitted wave, every point overthe surface of the boundary can be regarded as a point source which contributes to the reflectedwave or transmitted wave and the wave responses are the summation of the contribution ofevery point over the boundary.First, we computed the response of reflected wave on a single horizontal interface using themethod of KH integral. The numberical results are good agreement with those of reflectivirymethod and finite-difference method. Furthermore, the response of reflected wave on a curvedinterface are calculated, comparison with the results from the finite-difference method shows thatKH method can work quickly on curved or irregular interface, its computing speeds is superior tofinite-difference method. In addition, the reflected waves from the free surface are obtained by themethod of KH integral. From computing the six kinds of phases: PP, PS, pPP, pPS, pSP, pSS, myresult indicates that both of the amplitude and the wave form for the model of layered medium arebasically similar to the reflectivity method.From above basis, the KH integral theory is also applied to compute the conversion waveresponses. With the derivation of the conversion wave of KH integral equation, we simulatted thenear seism conversion wave response and the teleseismic conversion wave response respectively.The results compared with that of the reflectivity method and the dynamic ray tracing givessatisfactory result.Finally, we try to calculate the theoretical seismogram in the multi-layered medium and thesituation that the material parameters are linear below the surface. The result is consistent with thatfrom the reflectivity method.Our analysis indicates that KH integral method is a preferable synthetic seismogram method in a laterally inhomogeneous elastic medium. The accuracy of this approximation is excelled othermethods and it has the high computing efficiency also.