An *l _{2}* solution for the reflection angles can be estimated directly
from the reflection data

However, in the meantime 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 , 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:

(17) |

(18) |

(19) |

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 use .It should be noted that the estimate (17) 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
tends to be small or zero.

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

11/17/1997