Reflection angles

An l₂ solution for the reflection angles $\Theta$ can be estimated directly from the reflection data D by substituting the solution (16) into the normal equation $\partial E / \partial \Theta = 0$. 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 $\Theta$.

However, in the meantime an efficient ad hoc solution is available based on physical intuition. Consider a fixed point ${\bf x}$ in the subsurface. As we migrate a constant offset section into ${\bf x}$, the angle $\theta_{sr}$ between the source and receiver rays ranges from $\theta_{sr}\approx 0$ at ${\bf x}_m= {\bf x}_{min}$, to $\theta_{sr} \approx 2\Theta$ near the specular midpoint, and back to $\theta_{sr}\approx 0$ at ${\bf x}_m={\bf x}_{max}$. Analogously, the differential reflection coefficient $R_{_{P\!P}}$ varies from $R_{_{P\!P}}\approx 0$ (diffraction) to $R_{_{P\!P}}\approx \grave{P}\!\acute{P}(\Theta)$ (specular reflection), to $R_{_{P\!P}}\approx 0$ (diffraction) again, over the same midpoint integration range. Hence, it is apparent that $R_{_{P\!P}}$ will attain a maximal peak amplitude at the specular midpoint, whereupon $R_{_{P\!P}}= \grave{P}\!\acute{P}(\Theta)$ and $\theta_{sr}= 2\Theta$. This physical argument suggests performing a first moment weighted estimate of $\Theta$ as follows:

 

$$f(2\Theta({\bf x};{\bf x}_h)) \approx \frac{{\bf g}_2({\bf x};{\bf x}_h)} { \H_2({\bf x};{\bf x}_h)}$$ (17)

where

$${\bf g}_2({\bf x};{\bf x}_h) = \int_w \int_{{\bf x}_m} \ve... ...\, A_s A_r D \vert f(\theta_{sr}) e^{-iw\tau} \,d{\bf x}_m\,dw$$ (18)

and

$$\H_2({\bf x};{\bf x}_h) = \int_w \int_{{\bf x}_m} \vert k\... ...{-iw\tau} \,d{\bf x}_m\,dw + \lambda^2({\bf x};{\bf x}_h) \;,$$ (19)

where $\lambda^2$ is another damping parameter, and $f(\Theta)$ 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 $f \approx \cos^{\nu}\theta_{sr}$. I use $\nu=2$.It should be noted that the estimate (17) is very similar to the result of Bleistein (1987) for $\nu=1$, except that the slightly different WKBJ weighting and the absolute value signs may add a certain robustness advantage, especially at subsurface points ${\bf x}$ where $\vert\grave{P}\!\acute{P}\vert$ tends to be small or zero.

Finally, the two estimates $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ and $\Theta({\bf x};{\bf x}_h)$ can be mapped uniquely to the desired output $\grave{P}\!\acute{P}({\bf x};\Theta)$, which completes the least-squares angle-dependent reflectivity estimation process.