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Generating a 2-D coordinate system through differential methods requires
solving for coordinates
within domain X2.
Incorporating l monitor functions for grid regularization expands
the dimensionality of the mapping to,
| ![\begin{displaymath}
\mathbf{x}(\mathbf{s}) \; : \; S^2 \rightarrow X^{2+l},\;\;\...
...hbf{s}) = \{ s^1,s^2,f^1(\mathbf{s}),..., f^l (\mathbf{s}) \}. \end{displaymath}](img32.gif) |
(9) |
Coordinate system
is related to an underlying Cartesian
grid, which is chosen to be a unit square defined by
. Transformation
is assumed to be
piece-wise smooth and known on the boundary of
such that:
.
Within this framework, the 2-D gridding equations become,
| ![\begin{eqnarray}
D^{\xi}[s^j] = - D^{\xi} [ f^k ] \frac{\partial f^k}{\partial s...
...l \Xi^2} =
\mathbf{\phi}(\mathbf{\xi}), \quad j=1,2, \quad k=1,l,\end{eqnarray}](img36.gif) |
(10) |
where Laplacian operator
and metric tensor gij
are written explicitly Liseikin (2004),
| ![\begin{eqnarray}
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}](img38.gif) |
(11) |
| (12) |
One convenient way to solve the set of elliptical
equations 10 is by transforming them to a set of
parabolic equations (i.e. include time-dependence) that have a common
steady-state solution. Thus, equations 10 are
reformulated to include time-dependence -
-
leading to the six governing equations,
| ![\begin{eqnarray}
\frac{ \partial s^1}{ \partial t} = D[s^1] + D[ f^k] \frac{\par...
...1,\xi^2), & t=0 \
s^2(\xi^1,\xi^2,0) = s^2_0(\xi^1,\xi^2), & t=0\end{eqnarray}](img40.gif) |
(13) |
| (14) |
| (15) |
| (16) |
| (17) |
| (18) |
Solutions
and
satisfying
equations 13-18 will converge to the
solutions
of equations 10 as
. Hence, the answer to within tolerance factor
occurs at some Tn. Details of an iterative scheme and an
algorithm for computation to solve equations 13 are
provided in Appendix A.
Next: Numerical Examples
Up: Shragge: Differential gridding methods
Previous: Regularization through Monitor Functions
Stanford Exploration Project
4/5/2006