next up previous print clean
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, 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.

Rüger, A. and D. Hale, 2006, Meshing for velocity modeling and ray tracing in complex velocity fields: Geophysics, 71, U1-U11.

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.

Shragge, J., 2006, Generalized riemannian wavefield extrapolation: SEP-124.

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,
\frac{ \partial s^1}{ \partial t} = D^{\xi}[s^1] + D^{\xi}[f^k]...
 ...1,\xi^2), & t=0 \ 
s^2(\xi^1,\xi^2,0) = s^2_0(\xi^1,\xi^2), & t=0\end{eqnarray} (19)
D^{\xi}[v] = & g_{22}^{\xi} \frac{\partial^2 v}{\partial \xi^1
 ...}(\mathbf{\xi})] }{\partial \xi^j}, 
\quad i,j,k=1,2, \quad m=1,l.\end{eqnarray} (25)
Computational domain $\Xi^2$ is the unit square divided into N intervals equally spaced in the $\{ \xi^1,\xi^2 \}$ directions. The first transformation $\Xi^2 \rightarrow S^2$ interrelates the known coordinate values on boundaries of domains S2 and $\Xi^2$,
\Phi (\xi^j) = \left[\phi^1 (\xi^j),\phi^2 (\xi^j) \right], \quad j=1,2. \end{displaymath} (27)
The interior points of S2 are generated using blending functions, $\alpha^i_{ij}(s), \; 0 \le s \le 1$, where $\alpha^u_{ij}$ is linear function defined by,  
\alpha^i_{0j} (s) = 1-s, \quad \alpha^i_{1j} = s.\end{displaymath} (28)
Blended coordinates $\{ s^1,s^2 \}$ are generated on S2 with,
s^{1}(\xi^{1},\xi^{2},0) = F^{1}(\xi^{1},\xi^{2})+(1-\xi^{2})
\xi^{2} \left[\phi^{i}(\xi^{1},1)-F^{2}(\xi^{1},1) \right],\end{eqnarray}
F^{i}(\xi^{1},\xi^{2}) = (1-\xi^{1})
\phi^{i}(0,\xi^{2})+\xi^{1}\phi^{i} (1,\xi^{2}). \end{displaymath} (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 $u^{0+\frac{1}{2}}$ and $v^{0+\frac{1}{2}}$ for a step in the u direction at time $t=0+\frac{1}{2}$ 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 $u^{0+\frac{1}{2}}$ and $v^{0+\frac{1}{2}}$.

Explicitly, the four equations comprising the finite difference scheme for unij and vnij, $0 \le i,j \le N, \; 0 \le n$ on uniform grid $(ih,jh,n\tau)$ are,
u^{n+1/2}_{ij}- u^n_{ij}= &\frac{\tau}{h^2}
 ...{ij}(v^{n+1}-v^n) & \nonumber \ \; & 1 \le i,j \le N-1, n \ge 0 &\end{eqnarray}
g_{11} \left(\mathbf{s}^n_{ij}\right)= & 
\left(\frac{ u^n_{i+1...
 ...right]& \nonumber \  
\;& k=1,l \quad \; \quad 1\le i,j \le N-1 &\end{eqnarray}