Reflectivity estimation

Under the stationary phase approximation, the $\grave{P}\!\acute{P}$ equation is in the standard linear form: ${\bf d}= {\bf F}{\bf m}$. This has a well-known damped least-squares Gauss-Newton solution: ${\bf m}\approx -\H^{-1}{\bf g}= ({\bf F}'{\bf F}+ \epsilon^2)^{-1} {\bf F}'{\bf d}$, where ${\bf g}$ and $\H$ are the gradient and Hessian of E respectively, and ${\bf F}'$ is the adjoint operator. In this case, the gradient can be derived as

 

$${\bf g}({\bf x};{\bf x}_h) = - {\bf F}'{\bf d}= \int_w \int_{{\bf x}_m} ik\, A_s A_r D e^{-iw\tau} \,d{\bf x}_m\,dw \;.$$ (4)

where $k=w/\alpha$ is the spatial wavenumber. An approximate diagonal Hessian can be derived as  

$$\H = {\bf F}'{\bf F}+ \epsilon^2 \approx \int_w \int_{{\bf x... ...s A^2_r \, d{\bf x}_m\, dw + \epsilon^2({\bf x};{\bf x}_h) \;,$$ (5)

resulting in the l₂ reflectivity solution  

$$\grave{P}\!\acute{P}({\bf x};{\bf x}_h) \approx - \frac{{\bf g}({\bf x};{\bf x}_h)} {\H({\bf x};{\bf x}_h)} \;.$$ (6)

The approximate diagonal Hessian solution (6) costs little more to implement than a standard prestack migration, but requires twice the memory in order to store the gradient and Hessian images separately. In particular, (6) involves two separate but simultaneous images to be evaluated, the gradient (numerator) being a weighted Kirchhoff prestack depth migration, and the approximate Hessian (denominator) being an accumulation of the squared migration weights.