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![]() | Wave-equation time migration | ![]() |
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Conversion from time to depth coordinates and from to
is a nontrivial inverse problem. The problem involves not
only a coordinate transformation (Hatton et al., 1981; Larner et al., 1981) but also a
correction for the geometrical spreading of image rays. As shown
by Cameron et al. (2009), the problem can be reduced to solving an initial-value
(Cauchy) problem for an elliptic PDE (partial differential equation),
which is a classic example of a mathematically ill-posed problem.
To arrive at this formulation, let us transform the system of
equations 4-6 into the image-ray coordinate
system. The system of equations for inverse functions takes the form (Li and Fomel, 2015)
From equation 12, it follows that
Li and Fomel (2015) develop
robust time-to-depth conversion, which uses
equations 4-6 in the Cartesian coordinate
system and formulates time-to-depth conversion as a regularized
least-squares optimization problem. Using linearization with respect to velocity perturbations Sripanich and Fomel (2018) reformulate equations 17-18 for fast time-to-depth conversion appropriate for handling weak lateral variations. Weak lateral variation assumption is important because in case of strong lateral variations, there is no longer a one-to-one mapping between image-ray coordinates and Cartesian coordinates, and the coordinate transformation will also have a zero determinant (at the caustics of the image-ray field). Using Sripanich and Fomel (2018), the squared Dix velocity converted to depth is given as
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![]() | Wave-equation time migration | ![]() |
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