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- Alkhalifah, T., 2003, Tau migration and velocity analysis: Theory and synthetic examples: Geophysics, 68, 1331-1339.
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- Guggenheimer, H., 1977, Differential Geometry: Dover Publications, Inc., New York.
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- Liseikin, V., 2004, A Computational Differential Geometry Approach to Grid Generation: Springer-Verlag, Berlin.
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- Rüger, A. and D. Hale, 2006, Meshing for velocity modeling and ray tracing in complex velocity fields: Geophysics, 71, U1-U11.
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- Sava, P. and S. Fomel, 2001, 3-D traveltime computation using Huygens wavefront tracing: Geophysics, 66, 883-889.
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- Sava, P. and S. Fomel, 2005, Riemannian wavefield extrapolation: Geophysics, 70, T45-T56.
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- Shragge, J. and P. Sava, 2005, Wave-equation migration from topography: 75st Ann. Internat. Mtg., SEG Technical Program Expanded Abstracts, 1842-1845.
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- Shragge, J., 2006, Generalized riemannian wavefield extrapolation: SEP-124.
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A
This appendix details a numerical scheme for solving the differential
gridding equations discussed in Liseikin (2004). The set of
parabolic equations to solve are,
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(19) |
| (20) |
| (21) |
| (22) |
| (23) |
| (24) |
where,
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(25) |
| (26) |
Computational domain is the unit square divided into N
intervals equally spaced in the directions. The
first transformation interrelates the known
coordinate values on boundaries of domains S2 and ,
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(27) |
The interior points of S2 are generated using blending functions,
, where is linear
function defined by,
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(28) |
Blended coordinates are generated on S2 with,
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| (29) |
where,
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(30) |
Equations 1924 can be solved using
finite difference approximations that march forward in time. To
simplify notation coordinates s1 and s2 are redefined as u=s1
and v=s2. The
finite difference solution is split into a two-stage process along the
different coordinate axes. The first stage calculates solutions
and for a step in the u
direction at time using the values u0 and
v0. The second stage calculates solutions u0+1 and v0+1
for a step in the v direction at time t=0+1 using intermediate
values and .
Explicitly, the four equations comprising the finite difference scheme
for unij and vnij, on uniform grid
are,
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| (31) |
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| (32) |
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| (33) |
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| (34) |
where,
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| (35) |
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| (36) |
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| (37) |
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| (38) |
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| (39) |
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| (40) |
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| (41) |
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| (42) |