Regularization of the angle domain

In essence, the angle-gather method, introduced in this paper, amounts to a stretch of the offset angle according to equation (4). The stretch takes every point on the offset wavenumber axis and repositions it on the angle axis, most likely not on its regular grid. We therefore need to interpolate the unevenly sampled axis to the regular one. In other words, we need to solve a simple linear interpolation problem

$$\bold L\bold m\approx \d$$

where the model ($\bold m$) is represented by the evenly-spaced values on the angle axis, the data ($\d$) is represented by the unevenly-spaced values on the angle axis, and ($\bold L$) represents a 1-D linear interpolation operator. Since parts of the model space will not be covered because of the uneven distribution of the data, we need to regularize the interpolation process and solve a system such as

$$\begin{eqnarray} \bold L\bold m\approx \d \\ \epsilon\bold A\bold m\approx 0\end{eqnarray}$$ (5) (6)

where ($\bold A$) represents a 1-D roughener operator. Consequently, the least-squares solution to the system (5) is  

$$\bold m= \left (\bold L^T\bold L+ \epsilon^2\bold A^T\bold A\right )^{-1} \bold L^T\d.$$ (7)

In the special case of the angle-domain stretch, the inverted term on the right side of equation (7) is a tridiagonal matrix. Given the sparseness of the stretched data, the least-squares tridiagonal matrix corresponding to the operator $\bold L$ has zeros present along the diagonals, which results in instability during inversion. However, the regularization term fills the gaps; therefore, the inversion of the matrix in equation (7) is well-behaved.

Since the matrix $\bold L^T\bold L+ \epsilon^2\bold A^T\bold A$ is tridiagonal, we can invert it using a fast tridiagonal solver Golub and Van Loan (1989); Consequently, we obtain smoothly interpolated values for the ADCIGs. A similar approach could also be used for other problems, for example in Stolt migration Vaillant and Fomel (1999), residual migration Sava (1999a,b), or in velocity continuation Fomel (1998).

The main benefit of solving the least-squares problem this way is that we can obtain a very inexpensive regularized solution, with important benefits not only in data visualization, but also in other problems such as wave-equation migration velocity analysis Biondi and Sava (1999); Sava and Biondi (2000) and imaging Prucha et al. (1999b).