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# Riemannian Wavefield Extrapolation

Performing wavefield extrapolation on topographic computational meshes computed through conformal mapping requires parameterizing the acoustic wave-equation by a set of variables that describe the coordinate system. In 2-D, we denote these variables the extrapolation direction, , (analogous to depth in Cartesian wavefield extrapolation), and the direction orthogonal, (analogous to horizontal offset in Cartesian wavefield extrapolation). Variables and are related to the topographic coordinate system point set through .Figure  presents a sketch of the topographic coordinate system geometry.

 topocoord Figure 4 Cartoon illustrating the topography coordinate system. Variable is the extrapolation direction and parameter may be considered a topographic front''. Variable is the coordinate across the extrapolation step at a constant step, and parameter may be considered a topographic ray''.

The 2-D acoustic wave-equation for wavefield, , at frequency, , governing propagation in topographic coordinates is Sava and Fomel (2004),
 (3)
where s is the slowness of the medium, a distance scaling factor in the extrapolation direction , and J a Jacobian of transformation of coordinate (analogous to a geometrical ray spreading factor). Parameters and J are defined by,
 (4)
where x and z are the coordinates of the underlying Cartesian basis. Note that parameters and J are solely components of the coordinate system and are independent of the extrapolated wavefield values.

Analogous to wavefield continuation on a Cartesian mesh, a dispersion relation must be specified that forms the basis for all derived extrapolation operators in a topographic coordinate system. The relation being sought is the wavenumber along the extrapolation direction, . Following Sava and Fomel (2004), the partial derivative operators in (3) are expanded out to generate a second-order partial differential equation with non-zero cross derivatives. Fourier-domain wavenumbers are then substituted for the partial differential operators acting on wavefield, , and the quadratic formula is applied to yield the expression for ,
 (5)

One relatively straightforward way to apply wavenumber in an extrapolation scheme is to develop the topographic coordinate system equivalent to a phase-screen extrapolation operator Sava (2004). In the following example, we treat solely the kinematic, one-way propagation of recorded wavefields. This asymptotic approximation leads us to drop the first order partial differential terms in (5),
 (6)
where and . The expansion of about reference parameters a0 and b0 is,
 (7)
where subscript denotes reference. Partial derivatives with respect to parameters a and b are,
 (8)
where the square root function in the denominator has been expanded using a Padé approximation. The choice of numerical constants and c2=0 yields a 15 finite-difference term. Thus, the phase-screen approximation for extrapolation wavenumber, , is,
 (9)
This expression can be generalized to include multiple reference media through a phase-shift plus interpolation (PSPI) approach Gazdag and Sguazzero (1984) over the two parameters; however, this extension is not treated here. The approximation for wavenumber, , given in (9) is used in a conventional wavefield extrapolation scheme that extends the recorded wavefield away from the acquisition surface to the required subsurface locations. This involves solving a one-way wave-equation which, in discrete extrapolation steps of , requires a recursive computation of the following:
 (10)

Our prestack migration example is computed using a shot profile migration code. This involves extrapolating the source and receiver wavefields, and , independently using,
 (11)
and applying an imaging condition at each extrapolation level to generate image, ,
 (12)
where the line over the receiver wavefield indicates complex conjugate. Image is then mapped to a Cartesian coordinate system using sinc-based interpolation operators in the neighborhood of each mapped point to generate the final image, .

Next: Numerical examples Up: Shragge and Sava: Migration Previous: Conformal Mapping
Stanford Exploration Project
10/23/2004