An l2 solution for the reflection angles can be estimated directly from the reflection data D by substituting the solution (16) into the normal equation . Unfortunately, I have not yet completed the required analysis. A difficulty arises from the fact that the kernel functions of (5) are nonlinearly related to .
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:
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.