Initially, RWE was designed for dynamic applications where wavefields are extrapolated on ray-based coordinate systems oriented in the wave propagation direction. This approach generally generates high-quality Green's functions; however, numerical instability (i.e. zero divisions) occurs wherever the ray coordinate system triplicates. Sava and Fomel (2005) addressed this issue by iteratively smoothing the velocity model until coordinate system triplications vanish. This solution, though, is somewhat less than ideal because it counters the original purpose of RWE: coordinate systems conformal to propagation directions.
A second more geometric RWE application is performing wavefield extrapolation to and from surfaces of irregular geometry. Shragge and Sava (2004) formulate a wave-equation migration from topography strategy that poses wavefield extrapolation directly in locally orthogonal meshes conformal to the acquisition surface. Although successful in areas with longer wavelength and lower amplitude topography, imaging results degraded in situations involving more rugged acquisition topography. However, a more general observation is the genetic link between degraded image quality and the grid compression/extension demanded by semi-orthogonality.
A solution to these two problems is to extend RWE to include non-orthogonal coordinate systems. This generalized RWE (GRWE) framework removes the semi-orthogonal constraint to allow propagation in non-orthogonal Riemannian spaces. Non-orthogonality introduces two additional terms in the GRWE dispersion relationship and makes existing coefficient terms slightly more involved. The non-stationary coefficients in the resulting extrapolation wavenumber can be handled with an extended split-step Fourier approach Stoffa et al. (1990). Importantly, this solution affords greater flexibility in coordinate system design while facilitating more rapid mesh generation. Furthermore, greater emphasis can be placed on optimizing grid quality by controlling grid clustering and generating smoother coefficient fields Shragge (2006b).
This paper develops the 3D wave-equation dispersion relationship for performing GRWE. I first discuss generalized Riemannian geometry and show how the acoustic wave-equation can be formulated in a non-orthogonal Riemannian space. Subsequently, I develop an expression for a one-way wavefield extrapolation wavenumber and present the corresponding split-step Fourier approximation. I then present two analytic 2-D non-orthogonal coordinate systems to help validate the developed extrapolation wavenumber expressions. The paper concludes with a more realistic example of GRWE generated Green's function estimates through a slice of the SEG-EAGE salt model.