Scattering of elastic waves by a spherical inclusion–II. Limitations of asymptotic solutions

SUMMARY Starting with the exact solution for the scattering of a plane P wave by a homogeneous spherical inclusion, various types of approximate solutions are developed and discussed. The standard Rayleigh and Mie approximations are extended to the case of inclusions having arbitrary contrasts in material properties. For the low-contrast case, solutions are developed which are valid over a wide frequency range. Several aspects of these solutions are discussed, including the importance of near-field terms and the relative strength of the scattered P and S fields. The various types of approximate solutions are compared with each other and with the exact solution by calculating and displaying their normalized scattering cross-sections. Scattering of seismic waves plays a significant role in seismic wave propagation in the earth. Such phenomena as coda-wave excitation, attenuation, precursors, and irregular arrival times are often assumed to be caused by scattering from inhomogeneities in the crust and upper mantle of the earth. The scattering mechanism also plays a role in some of the algorithms for migration and diffraction tomography which are used in exploration seismology. Exact solutions for scattering problems are known for only a few types of obstacles, and even for these cases the calculations are not easily implemented because of their complicated mathematical representations. The basic approach to the scattering problem which has been developed in such disciplines as optics, acoustics and quantum mechanics makes extensive use of asymptotic solutions to explain the main physical features of the scattered fields. In these disciplines, where the critical parameters of the scattering obstacle and the observation system are usually well known and controllable, experiments can be arranged so that the conditions for the asymptotic solutions are strongly satisfied, and thus these solutions give a satisfactory representation of the observational results. However, in the case of scattered seismic waves the situation is more complicated. Generally there are a wide variety of shapes, locations, boundaries and material properties in the local heterogeneity of the earth’s crust and upper mantle which represent the scattering obstacles for the seismic problem. Additional complications arise from the fact that these local heterogeneities are measured with respect to a surrounding material which is itself typically inhomogeneous, but on a larger distance scale. In most cases the parameters of the scattering obstacles are either unknown or poorly known. Furthermore, parameters of the observation system, such as the locations of the sources and receivers and the wavelengths of the incident waves, are typically difficult to control. Therefore, a common situation in seismic problems is that it is unknown whether the conditions necessary for the application of asymptotic solutions are actually satisfied. Establishing proper conditions for the use of different asymptotic approaches in the treatment of scattered seismic waves is not a simple problem because of a number of reasons: