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# Mixed-domain migration theory

Downward continuation is the process in which we recursively extrapolate in depth the wavefield recorded at the surface. Mathematically, this operation amounts to a phase shift applied to the wavefield Claerbout (1985)

where Uz is the wavefield at depth z, and is the extrapolated wavefield at depth . The downward continuation operator at depth z is
 (1)
with the vertical wavenumber, kz, given by the one-way wave equation, also known as the single square root (SSR) equation

where is the temporal frequency, s is the laterally variable slowness of the medium, and is the horizontal wavenumber.

Since downward continuation by phase shift can be applied for slowness models that only vary with depth, we need to split the operator into two parts: a constant slowness continuation operator applied in the domain, which accounts for the propagation in depth, and a screen operator applied in the domain, which accounts for the wavefield perturbations due to the lateral slowness variations. In essence, we approximate the vertical wavenumber kz with its constant slowness counterpart kzo, corrected by a term describing the spatial variability of the slowness function (Figure ).

 screen Figure 1 A sketch of mixed-domain migration. The wavefield at depth z is downward continued to depth trough a variable-slowness screen.

Next: Fourier finite-difference Up: Sava: Mixed-domain operators Previous: Introduction
Stanford Exploration Project
9/5/2000