Regularization of the angle-domain

The angle gather transformation introduced in this chapter amounts to a stretch of the offset to reflection angle according to adcig. The stretch takes every point on the offset wavenumber axis and repositions it on the angle axis, most likely not on a regular grid. Therefore, we need to interpolate the unevenly sampled axis to the regular one, i.e. we need to solve a simple linear interpolation problem  

$$\Lop \mod \approx \dat \;,$$ (27)

where the model ($\mod$) is represented by the evenly-spaced values on the angle axis, the data ($\dat$) is represented by the unevenly-spaced values on the angle axis, and $\Lop$ represents a 1D linear interpolation operator. Both $\mod$ and $\dat$ in interp are Fourier-domain quantities. Since parts of the model space are not covered because of the uneven distribution of the data, we need to regularize the interpolation process and solve a system such as  
0

where ($\Rop$) represents a 1D regularization operator (1D gradient, for example). The least-squares solution to the system () takes the usual form  

$$\mod = \lp \Lop^T\Lop + \epsilon^2\Rop^T\Rop \rp^{-1} \Lop^T\dat.$$ (28)

In the special case of the angle-domain stretch, the inverted factor on the right side of LSsol is a tridiagonal matrix and can be easily inverted using a tridiagonal solver. However, given the sparseness of the stretched data, the least-squares tridiagonal matrix corresponding to the operator $\Lop$ has zeros present along the diagonals, which results in instability during inversion and artifacts in the angle gathers. Regularization fills those gaps and the inversion of the matrix in LSsol is well-behaved.

Finally, I emphasize that regularization is not applied in the space domain, but in the Fourier domain and so this method does not smooth reflection events spatially. Consequently, the amplitude response in ADCIGs is not altered, although, as noted earlier, there are other reasons why direct AVA interpretation is not straightforward.