RTM imaging conditions

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RTM imaging conditions

Since the one-step wave extrapolation kernel operates in the complex domain, it requires a definition of data and reflectivity with complex values. The analytical data follows the definition of the complex wavefield in equations 7 and 8. It implies that the input data need to be Hilbert-transformed along the time axis and supplied as the imaginary part before the migration process, creating an analytical signal (Taner et al., 1979). We adopt the following complex-valued cross-correlation imaging condition (Claerbout, 1985):

$\displaystyle I_c(\mathbf{x}) = \sum\limits_s \sum\limits_t\bar{S}_s(\mathbf{x},t) R_s(\mathbf{x},t) \; ,$

(29)

where the lower case

denotes shots and

denotes time samples. The real part of the complex image $I_c(\mathbf{x})$ is extracted and used as the final image.

Extended imaging conditions (Sava and Vasconcelos, 2011), including space-shift (Sava and Fomel, 2003; Rickett and Sava, 2002) and time-shift (Sava and Fomel, 2006) imaging conditions, can provide additional information for migration velocity analysis. The complex-valued space-shift and time-shift imaging condition for lowrank one-step RTM takes the form

$\displaystyle I_e(\mathbf{x},\mathbf{\lambda},\tau) = \sum\limits_s \sum\limits... ..._s(\mathbf{x}-\mathbf{\lambda},t-\tau) R_s(\mathbf{x}+\mathbf{\lambda},t+\tau)$

(30)

and can be easily implemented in the time-space domain.

Next: Examples Up: Theory Previous: Lowrank approximation

2016-11-16