Abstract
SUMMARY We carry out high-frequency analyses of Claerbout's double-square-root equation and its (numerical) solution procedures in heterogeneous media. We show that the double-square-root equation generates the adjoint of the single-scattering modelling operator upon substituting the leading term of the generalized Bremmer series for the background Green function. This adjoint operator yields the process of ‘wave-equation’ imaging. We finally decompose the wave-equation imaging process into common image point gathers in accordance with the characteristic strips in the wavefront set of the data.
Highlights
Directional wavefield decomposition is a tool for analysing and computing the propagation of waves in configurations with a certain global directionality, such as waveguiding structures
The method consists of three main steps: (1) decomposing the field into two constituents and propagating them upward or downward along a principal or preferred direction, (2) computing the interaction of the counterpropagating constituents and (3) recomposing the constituents into observables at the positions of interest. (The preferred direction is ‘vertical’, whereas the directions orthogonal to the preferred direction are referred to as the ‘lateral’ directions.) The method allows one to ‘trace’ wave constituents in a global coordinate system in a given medium, and distinguish constituents that have been scattered along the up/down direction a different number of times
Computational aspects of the Bremmer series can be found in Van Stralen et al (1998) and Le Rousseau & De Hoop (2001a)
Summary
Directional wavefield decomposition is a tool for analysing and computing the propagation of waves in configurations with a certain global directionality, such as waveguiding structures. If the medium of the configuration were laterally homogeneous, the directional decomposition would become an algebraic operation in the lateral Fourier or wavenumber domain (see, e.g., Kennett 1985) In such a medium, the phase-shift method (Gazdag & Sguazzero 1984) is amongst the fastest (and accurate) one-way algorithms. The measurements of the sources and the receivers are propagated into the subsurface in a presumed background medium back to the time when the scattering took place. This continuation is governed by the double-square-root equation. Throughout we will make use of microlocal techniques, the fundamental concepts of which are introduced in Appendix A
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