Diffraction imaging and time-migration velocity analysis using oriented velocity continuation

Slope decomposition

In order to perform the initial slope decomposition (Step 4 in the algorithm above), we adopt the method of Ghosh and Fomel (2012). The idea of slope decomposition was discussed previously by Ottolini (1983) and implemented using the local-slant stack transform (Ventosa et al., 2012). The slope-decomposition algorithm suggested by Ghosh and Fomel (2012) is based on the time-frequency decomposition of Liu and Fomel (2013). Namely, at each frequency $\omega$, we apply regularized non-stationary regression (Fomel, 2009a) to transform from space $x$ to space-slope $x$-$q$ domain. The non-stationary regression amounts to finding complex coefficients $A_n(\omega,x)$ in the decomposition

$$D(\omega,x) = \sum\limits_{n=1}^{N_p} D_n(\omega,x)\;,$$ (14)

where $D(\omega,x)$ is the image slice, and $D_n(\omega,x)$ is its slope component corresponding to slope $q_n$:

$$D_n(\omega,x) = A_n(\omega,x) e^{i \omega x q_n}\;.$$ (15)

Equation (14) is the discrete analog of equation (6). Similarly to the time-frequency decomposition proposed by Liu and Fomel (2013), shaping regularization is used to control the variability of $A_n$ coefficients and to accelerate the algorithm.

Diffraction imaging and time-migration velocity analysis using oriented velocity continuation