Multiparameter two-dimensional inversion of scattered teleseismic body waves 1. Theory for oblique incidence

This is the first paper in a three-part series that examines formal inversion of the telescismic P wave coda for discontinuous variations in elastic properties beneath dense, three-component, seismic arrays. In this paper, we develop the theoretical framework for a migration method that draws upon the tenets of inverse scattering theory and is amenable to practical implementation. The forward problem is formulated for two-dimensional (2-D) heterogeneity in observance of formal sampling requirements and currently accessible instrumentation. A ray theoretic Green's function, corresponding to a line source with axial component of forcing, is employed within the 2-D Born approximation to accommodate planar, incident wave fields at arbitrary back azimuths. Both the forward scattered response generated by the upgoing incident wave field and the backscattered response created by its reflection at the free surface are included within the formulation. In accordance with the high-frequency and single-scattering approximations employed in the forward problem the inverse problem is cast as a generalized Radon transform, The resulting back projection operator is well suited to the teleseismic context in several respects. It is tolerant of irregularities in array geometry and source distribution and allows a full complement of global seismicity to be utilized through its accommodation of oblique incidence. By permitting both independent and simultaneous treatment of different scattering modes (reflections, transmissions, conversions) the inversion formula facilitates a direct appraisal of individual mode contributions to the recovery of structure. In particular, it becomes evident that incorporation of backscattered modes leads to (1) a better localization of structure than possible using forward scattered energy and (2) the imposition of complementary constraints on elastic properties.