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An *l*_{2} solution for the reflection angles can be estimated directly
from the reflection data *D* by substituting the solution (6)
into the normal equation . However, an
efficient ad hoc solution is available based on physical intuition.
Consider a fixed point in the subsurface. As we migrate a constant
offset section into , the angle between the source
and receiver rays ranges from at the
midpoint position , to
near the specular midpoint, and back to
at . Analogously, the differential
reflection coefficient varies from (diffraction) to
(specular reflection), to (diffraction) again, over the same midpoint integration range.
Hence, it is apparent that will
attain a maximal peak amplitude at the specular midpoint, whereupon
and .This physical argument suggests performing a first moment weighted estimate
of as follows:

| |
(7) |

where
| |
(8) |

and
| |
(9) |

where is another damping parameter, and is a function
which can be arbitrarily chosen to optimize the estimate. In practice,
choosing *f* to be a low power of the cosine function works well, such that
. I have found works well
by experience.
It should be noted that the estimate (7) is very similar to the
result of Bleistein (1987) for , except that the slightly
different WKBJ weighting and the absolute value signs may add a certain
robustness advantage, especially at subsurface points where
is small or zero.

The two estimates and can
be mapped uniquely to the desired output , which
completes the least-squares angle-dependent reflectivity estimation.

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** Previous:** Reflectivity estimation
Stanford Exploration Project

11/17/1997