Next: Example 3  GRWE
Up: Shragge: GRWE
Previous: Example 1  2D
A second instructive example is a stretched polar coordinate system
(see figure ). A polar ellipsoidal coordinate
system is specified by,
 
(29) 
Parameter is a smooth function controlling coordinate
system ellipticity and has curvature parameters and .The metric tensor g_{ij} is,
 
(30) 
with determinant . The associated metric and
weighted associated metric tensors are given by,
 
(31) 
2Dex2
Figure 3 Polar ellipsoidal coordinate
system example.

 
Tensors g^{ij} and m^{ij} specify a wavenumber appropriate for
extrapolating wavefields on a 2D nonorthogonal mesh (see
equation 41). However, because the coordinate system is
spatially variant, we must also compute the n_{i} fields:
and n_{3}=0. Inserting these values
yields the following extrapolation wavenumber ,
 
(32) 
The kinematic version of equation 32 is,
 
(33) 
while the orthogonal polar case (i.e. a=1) recovers the following,
 
(34) 
Figure shows an ellipsoidal polar coordinate
system defined by . The upper
left panel shows a velocity function overlain by
a coordinate system mesh. The upper right panel presents velocity
model as mapped into the GRWE domain. The data used in this test
consisted of 4 flat planewaves. Given this experimental setup,
propagating flat planewaves should not bend in the Cartesian domain
because of the v=v(z) velocity model, even though there is velocity
variation across each extrapolation step in the GRWE domain. Hence,
the impulses have curvature in the GRWE domain (lower right panel).
The lower left panel shows the GRWE domain imaging results mapped back
to a Cartesian domain. Consistent with theory, the flat planewaves
are imaged as flat reflectors. Note that the edge effects are again
caused by coordinate system and planewave truncation.
Polar
Figure 4 Ellipsoidal polar coordinate
system test example. Upper left: velocity
function overlain by a polar ellipsoidal coordinate system defined
by parameter . Upper right: velocity
model in the GRWE domain. Bottom right: Imaged reflectors in GRWE
domain. Bottom left: the GRWE domain image mapped to a Cartesian
mesh.
Next: Example 3  GRWE
Up: Shragge: GRWE
Previous: Example 1  2D
Stanford Exploration Project
4/5/2006