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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 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),
where v is the velocity function, and J is the ray-coordinate
Jacobian or geometrical ray spreading factor given by,
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 ,
Note, again, that quantity depends solely on the
coordinate system and is independent of the wavefield being
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,
where may be considered as the effective (non-stationary)
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.
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