Generalized Multipolar Ray Theory for Surface and Shallow Sources

Abstract

The study of surface and shallow sources has lately become of importance, especially in connection with nuclear-explosion seismology (Ben-Menahem & Vered 1975; Ben-Menahem 1975). An ad-hoc study of surface sources was deemed necessary in view of the observation that, in general, the radiation pattern of a surface source differs substantially from the radiation pattern of the same source buried in an infinite space (White 1960). Consequently, an exact generalized multipolar ray theory (GMRT) is developed for surface sources. This theory holds but approximately for buried sources. However, for shallow sources the approximation is quite acceptable, while making the tedious evaluation of surface reflections redundant. The same problem of computing theoretical seismograms for shallow sources was recently investigated, in an approximate way, by Langston & Helmberger (1975). Radiation patterns of surface sources may be derived in different ways: ab initio solution of a surface source problem (e.g. Miller & Pursey 1954; Cherry 1962) or finding the radiation field of a buried source and taking the limit as the burial depth tends to zero (e.g. Pekeris 1955). We use a version of the second technique. The direct and reflected displacement fields for buried multipolar sources were given in detail by Ben-Menahem & Vered (1973). We form the total field by adding the direct and reflected fields. The resultant expression may be expanded in a Taylor series in a non-dimensional quantity which eventually may be shown to be kh where k is the wave number and h the burial depth of the source. Asymptotic ray expressions may be obtained by approximately evaluating the exact field entities using the stationary phase method and the initial motion approximation. The two main advantages of our method are that: (1) it applies equally easily to all multipolar sources, thus avoiding the need to re-evaluate the radiation pattern anew for every source, and (2) it may be straightforwardly applied to multilayered media. The first term in the above mentioned expansion yields the radiation pattern for surface sources. The second term in the series may be considered as a correction to be added to the first term, for small non-vanishing h, and so on. In this note we give (Table 1) the results for h = 0 only, which apply either exactly to surface sources or approximately to shallow sources such that the wavelength