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The reflection coefficient of the multiple-generating layer can be estimated
directly, by comparing the amplitude of the primary reflection with its first
multiple, after the latter has undergone Snell resampling (normalized to account
for the amplitude increase due to compression of the offset axis) and
differential geometric spreading correction. The following procedure is utilized
to obtain a smooth estimate the reflection coefficient as a function of midpoint.
- Pick zero-offset traveltime to reflector for all midpoints: .
- Loop over midpoints (y'):
- Apply NMO to data. Extract small time window around .
Output is .
- Apply NMO for first-order multiple Brown (2003b),
normalized Snell resampling, and differential geometrical spreading
to data. Extract time small window around . Output is
.
- Optional: Align with
on a trace-by-trace basis using (for example) the crosscorrelation
technique of Rickett and Lumley (2001). Output is
.
- Optional: Compute residual weight which reflects ``quality''
of the data at this midpoint. Output is .
- Minimize the following quadratic functional for unknown seabed reflection
coefficient vector :
| |
(6) |
Minimization of equation (6) is the equivalent to
solving a regularized least-squares problem. To minimize the first term,
we adjust to force the scaled primary to match the multiple.
To minimize the second (regularization) term, we force to
vary slowly across midpoint. The scalar term balances data
fitting with model smoothness.
Figure shows the stack of the raw Mississippi Canyon 2-D
test dataset, acquired and distributed by WesternGeco. The LSJIMP method is
tested on this data by Brown (2003a). In addition to the
seabed peglegs, peglegs from the top of salt and strong reflector R1 are included
in the inversion. Rather than including each of these surfaces separately, the R1
and top salt events are assumed to arise from a single reflector; from midpoint 0
m to roughly 6000 m, it is R1, while from 6000 m to 20000 m it is the top of salt.
Figure illustrates the reflection coefficient estimation
procedure applied to 750 midpoints of the Mississippi Canyon data. The geology
is quite complex in some areas, and the data looks decidedly incoherent. For
this reason, the residual weight w(y) is quite important. I eyeballed Figure
and heuristically picked the residual weights shown therein.
The seabed reflection coefficient is nearly constant (-0.12) across the entire
spread. The rugosity of the top of salt severely limited the sections of data
considered ``good'', but since salt is generally somewhat homogeneous in material
properties, we can confidently assume some degree of spatial similarity.
gulf.stackraw
Figure 3 Stacked Mississippi Canyon 2-D dataset (750
midpoints), annotated with important horizons and multiples. Labeled events:
R1 - strong reflection; TSR - top of salt; BSR - bottom of salt; WBM - first
seabed multiple; R1PL - seabed pegleg of R1; R1PM - R1 pure surface multiple;
TSPL - seabed pegleg of TSR; BSPL - seabed pegleg of BSR; TSPM - TSR pure
surface multiple.
rc.gulf
Figure 4
a) Stack of time window around seabed reflection.
b) Stack of time window around first seabed multiple.
c) Seabed residual weight (either 0 or 1).
d) Estimated seabed reflection coefficient.
e) Stack of time window around top of salt reflection.
f) Stack of time window around first top of salt multiple.
g) Top of salt residual weight (either 0 or 1).
h) Estimated top of salt reflection coefficient.
Next: Applying the Seabed Reflection
Up: Brown: Pegleg amplitudes
Previous: Differential Geometrical Spreading for
Stanford Exploration Project
7/8/2003