Next: Algorithm for Computation
Up: Shragge: Differential gridding methods
Previous: Acknowledgements
 Alkhalifah, T., 2003, Tau migration and velocity analysis: Theory and synthetic examples: Geophysics, 68, 13311339.

 Guggenheimer, H., 1977, Differential Geometry: Dover Publications, Inc., New York.

 Liseikin, V., 2004, A Computational Differential Geometry Approach to Grid Generation: SpringerVerlag, Berlin.

 Rüger, A. and D. Hale, 2006, Meshing for velocity modeling and ray tracing in complex velocity fields: Geophysics, 71, U1U11.

 Sava, P. and S. Fomel, 2001, 3D traveltime computation using Huygens wavefront tracing: Geophysics, 66, 883889.

 Sava, P. and S. Fomel, 2005, Riemannian wavefield extrapolation: Geophysics, 70, T45T56.

 Shragge, J. and P. Sava, 2005, Waveequation migration from topography: 75st Ann. Internat. Mtg., SEG Technical Program Expanded Abstracts, 18421845.

 Shragge, J., 2006, Generalized riemannian wavefield extrapolation: SEP124.

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,
 
(19) 
 (20) 
 (21) 
 (22) 
 (23) 
 (24) 
where,
 
(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 S^{2} and ,
 
(27) 
The interior points of S^{2} are generated using blending functions,
, where is linear
function defined by,
 
(28) 
Blended coordinates are generated on S^{2} with,
 

 (29) 
where,
 
(30) 
Equations 1924 can be solved using
finite difference approximations that march forward in time. To
simplify notation coordinates s^{1} and s^{2} are redefined as u=s^{1}
and v=s^{2}. The
finite difference solution is split into a twostage process along the
different coordinate axes. The first stage calculates solutions
and for a step in the u
direction at time using the values u^{0} and
v^{0}. The second stage calculates solutions u^{0+1} and v^{0+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 u^{n}_{ij} and v^{n}_{ij}, on uniform grid
are,
 

 (31) 
 
 
 (32) 
 
 
 (33) 
 
 
 (34) 
where,
 

 (35) 
 
 
 
 (36) 
 
 
 
 (37) 
 
 (38) 
 
 (39) 
 
 (40) 
 
 (41) 
 
 
 (42) 