Wave-equation migration in generalized coordinate systems

Abstract

Wave-equation migration using one-way wavefield extrapolation operators is commonly used in industry to generate images of complex geologic structure from 3D seismic data. By design, most conventional wave-equation approaches restrict propagation to downward continuation, where wavefields are recursively extrapolated to depth on Cartesian meshes. In practice, this approach is limited in high-angle accuracy and is restricted to down-going waves, which precludes the use of some steep dip and all turning wave components important for imaging targets in such areas as steep salt body flanks.This thesis discusses a strategy for improving wavefield extrapolation based on extending wavefield propagation to generalized coordinate system geometries that are more conformal to the wavefield propagation direction and permit imaging with turning waves. Wavefield propagation in non-Cartesian coordinates requires properly specifying the Laplacian operator in the governing Helmholtz equation. By employing differential geometry theory, I demonstrate how generalized a Riemannian wavefield extrapolation (RWE) procedure can be developed for any 3D non-orthogonal coordinate system, including those constructed by smoothing ray-based coordinate meshes formed from a suite of traced rays. I present 2D and 3D generalized RWE propagation examples illustrating the improved steep-dip propagation afforded by the coordinate transformation.One consequence of using non-Cartesian coordinates, though, is that the corresponding 3D extrapolation operators have up to 10 non-stationary coefficients, which can lead to imposing (and limiting) computer memory constraints for realistic 3D v applications. To circumvent this difficulty, I apply the generalized RWE theory to analytic coordinate systems, rather than numerically generated meshes. Analytic coordinates offer the advantage of having straightforward analytic dispersion relationships and easy-to-implement extrapolation operators that add little computational overhead. In particular, I demonstrate that the dispersion relationship for 2D elliptical geometry introduces only an effective velocity model stretch, permitting the use of existing high-order Cartesian extrapolators. The results of elliptical coordinate shotprofile migration tests demonstrate the improvements in steep dip reflector imaging facilitated by the coordinate system transformation approach.I extend the analytic coordinate system approach to 3D geometries using tilted elliptical-cylindrical (TEC) meshes. I demonstrate that propagation in a TEC coordinate system is equivalent to wavefield extrapolation in elliptically anisotropic media, which is easily handled by existing industry practice. TEC coordinates also allow steep dip propagation in both the inline and cross-line directions by virtue of the associated elliptical and tilting Cartesian geometries. Observing that a TEC coordinate system conforms closely to the shape of a line-source impulse response, I develop an TEC-coordinate, inline-delay-source migration strategy tha...

Inline Delayed-shot Migration in Tilted Elliptical-cylindrical Coordinates

“…where A 2 =a 2 sinh 2  3 +sin 2  2 . An expression for the extrapolation wavenumber, k 3 , for the TEC coordinate system can be obtained by using the procedure described in Shragge (2009), who formulated an expression for the extrapolation wavenumber in any analytic coordinate system defined by metric coefficient g jk (see Equations B-1 and B-2 therein). Inserting metric g jk into these equations yields the following extrapolation wavenumber:…”

“…However, the TEC coordinate dispersion relationship closely resembles that of elliptically anisotropic media (Tsvankin, 1996), now routinely handled with finite-difference approaches [e.g., Zhang et al (2001); Shan and Biondi (2004)]. For the following example, we advance TEC coordinate wavefields at each extrapolation step using an implicit finite-difference splitting method that alternatively advances the wavefield in the ξ 1 and ξ 2 directions (Shragge, 2009).…”