Mathematical Physics Characterization and ‘Source-Receiver ’ Continuation of Seismic Reflection Data

Abstract: In reflection seismology one places sources and receivers on the Earth’s surface. The source generates elastic waves in the subsurface, that are reflected where the medium properties, stiffness and density, vary discontinuously. In the field, often, there are obstructions to collect seismic data for all source-receiver pairs desirable or needed for data processing and application of inverse scattering methods. Typically, data are measured on the Earth’s surface; the location of the receiver relative to the source can be coordinated by offset and azimuth. We employ the term data continuation to describe the act of computing data that have not been collected in the field. Seismic data are commonly modeled by a scattering operator developed in a high-frequency, single scattering approximation. We initially focus on the determination of the range of the forward scattering operator that models the singular part of the data in the mentioned approximation. This encompasses the analysis of the properties of, and the construction of, a minimal elliptic projector that projects a space of distributions on the data acqui-sition manifold to the range of the mentioned scattering operator. This projector can be directly used for the purpose of seismic data continuation, and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator. We illustrate the data continuation by a numerical example. 1