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## Ray-coordinate wavefield extrapolation

Wavefield extrapolation in ray-coordinates requires casting the acoustic wave-equation not in the usual Cartesian representation, but rather in a system parameterized by phase-ray variables. In 2-D, these variables consist of , the one-way travel time from a source/receiver point along the direction of a ray, and ,the direction across the rayfront at a constant time step. A cartoon illustrating ray-coordinate geometry is presented in Figure 1.

 raycoord Figure 1 Cartoon illustrating the phase-ray coordinate system and its relation to the Cartesian basis. Variable is the direction along a single ray, and parameter is an isochron or rayfront. Variable is the coordinate across the rayfront at a constant step, and parameter is a ray. Grey lines illustrate the mapping between ray point and Cartesian point (xo,zo). Angle is a rotation angle between the ray and the z-axis (assumed to be positive downward).

Note that the dimensions of time and space coordinates and are seconds and meters, respectively.

The 2-D acoustic wave-equation for wavefield, , at frequency, , in ray-coordinates is Sava and Fomel (2003),
 (6)
where v is the velocity function, and J is the ray-coordinate Jacobian or geometrical ray spreading factor given by,
 (7)
Importantly, parameter J is solely a component of ray-coordinates and is independent of wavefield extrapolated on the coordinate system.

Analogous to wave-equation extrapolation in Cartesian coordinates, a dispersion relation must be specified that forms the basis for all derived ray-coordinate extrapolation operators. The relation being sought is the wavenumber along the ray direction, .Following Sava and Fomel (2003), the partial derivative operators in equation (6) 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 ,
 (8)
Note, again, that quantity depends solely on the coordinate system and is independent of the wavefield being propagated.

One relatively straightforward manner to apply wavenumber in an extrapolation scheme is to develop the ray-coordinate equivalent of Claerbout's classic 15 equation Claerbout (1985). This involves a second-order Taylor series expansion of the radical in equation (8), and the identification of Fourier dual parameters and with their space domain derivative counterparts and . The ray-coordinate formula corresponding to the 15equation is,
 (9)
where may be considered as the effective (non-stationary) frequency,
 (10)
Equation (9) may be solved in 2-D using fully implicit finite difference methods (e.g. Crank-Nicolson) and fast tridiagonal solvers. After wavefield solution, , has been computed at all rayfield locations, the result is mapped to Cartesian coordinates using sinc-based interpolation operators in a neighborhood about each mapped point.

Next: The chicken and the Up: Theory Previous: Phase-rayfields
Stanford Exploration Project
5/23/2004