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 ,
we apply regularized non-stationary regression (Fomel, 2009a) to
transform from space to space-slope - domain. The
non-stationary regression amounts to finding complex coefficients
in the decomposition
(14)
where is the image slice, and is its slope component corresponding to slope :
(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
coefficients and to accelerate the algorithm.
Diffraction imaging and time-migration velocity analysis using oriented velocity continuation