In this thesis, I assume that the reflection coefficient varies in space, but
does not vary with reflection angle. Assume that and
are, respectively, small windows in time, offset, and midpoint
of dimension
, around a primary reflection and its first
pure multiple after normalized Snell Resampling and differential geometric
spreading correction. The reflection coefficients,
, are chosen to
minimize the following quadratic functional:
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Recall from Figure that for a multiple and primary recorded
at the same midpoint, there exists a shift in the target reflection point,
, described by equation (
) for a first-order pegleg.
In my LSJIMP implementation, variations in reflection strength of the target
reflector are ignored, but not those in the multiple generator. A first
justification of this assumption is convenience: the strength of the target
reflection is, after all, unknown. Secondly, since the target reflection points
of all legs of a pegleg are symmetric about the midpoint, the average of the
reflection stregths is the same as the primary's if the true reflection strength
is locally linear with midpoint.
decreases with target depth, so for
deep targets the local linearity assumption is likely to hold to first-order
accuracy. Thirdly, ignoring target reflector variation implies that the model
space of the LSJIMP inversion consists of one midpoint location only, which
reduces memory usage and permits coarse-grained computer parallelization over
midpoints. Therefore, when applying the reflection coefficient, we apply the
coefficient at the assumed reflection point for the particular multiple being
imaged. Second order multiples would be scaled by reflection coefficients from
two locations, and so on.