2001/12/17 by Christiaan C. Stolk, Maarten V. de Hoop, Stolk, Christiaan C. +1
Earth and Planetary Sciences · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Seismic Imaging and Inversion Techniques #Seismic Waves and Analysis
paper · pdf · doi:10.48550/arxiv.math/0112172
openalex publication_date 2001/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Seismic data are commonly modeled by a high-frequency single scattering approximation. This amounts to a linearization in the medium coefficient about a smooth background. The discontinuities are contained in the medium perturbation. The wave solutions in the background medium admit a geometrical optics representation. Here we describe the wave propagation in the background medium by a one-way wave equation. Based on this we derive the double-square-root equation, which is a first order pseudodifferential equation, that describes the continuation of seismic data in depth. We consider the modeling operator, its adjoint and reconstruction based on this equation. If the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal, the singular part of the data is described by the solution to an inhomogeneous double-square-root equation. We derive a microlocal reconstruction equation. The main result is a characterization of the angle transform that generates the common image point gathers, and a proof that this transform contains no artifacts. Finally, pseudodifferential annihilators based on the double-square-root equation are constructed. The double-square-root equation approach is used in seismic data processing.