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- Alkhalifah, T., 2003, Tau migration and velocity analysis: Theory and synthetic examples: Geophysics, 68, 1331-1339.
- Guggenheimer, H., 1977, Differential Geometry: Dover Publications, Inc., New York.
- Liseikin, V., 2004, A Computational Differential Geometry Approach to Grid Generation: Springer-Verlag, Berlin.
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- Sava, P. and S. Fomel, 2001, 3-D traveltime computation using Huygens wavefront tracing: Geophysics, 66, 883-889.
- Sava, P. and S. Fomel, 2005, Riemannian wavefield extrapolation: Geophysics, 70, T45-T56.
- 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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This appendix details a numerical scheme for solving the differential
gridding equations discussed in Liseikin (2004). The set of
parabolic equations to solve are,
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 ,
The interior points of S2 are generated using blending functions,
, where is linear
function defined by,
Blended coordinates are generated on S2 with,
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