Velocity continuation describes vertical and lateral shifts of time-migrated events under the change of migration velocity.
The continuous process in a zero-offset isotropic case is described by equation (Claerbout, 1986; Fomel, 2003b):
|
(1) |
Applying double-Fourier transform after time warpping
, the solution for equation 1 can be expressed as (Fomel, 2003a):
|
(2) |
where is migration velocity, is the Fourier dual of and is wavenumber.
The input zero-offset stack
is transformed to a constant velocity time migrated image
.
Burnett and Fomel (2011) provide an extension to the 3-D anisotropic case.
The path-integral formulation creates velocity independent images (Landa et al., 2006) in time domain; the formulation of velocity-weighted path-integral of VC images is:
|
(3) |
where is the velocity-weighting function used to fine-tune velocity constraints formed by and .
Efficient workflow of the integral above, proposed by Merzlikin and Fomel (2015), is integrating velocity analytically in the double-Fourier domain.
For example, an unweighted integral takes the form:
which turns path-integral into an analytical filter in double-Fourier domain.
Velocity-weighting function with analytical forms can be included also in this integral.
The efficient path-integral time migration workflow for passive seismic data imaging can be summarized as:
- Apply different time shifts to passive seismic data to cancel onsets; all operations afterwards are done for each constant slice.
- Apply
time warpping to -- domain data.
- Double-FFT transform from -- domain to -- domain.
- Apply equation 4 to build the filter in the double-Fourier domain.
- Apply the filter in last step to -- data.
- Inverse FFT and inverse time warpping into -- image.
Because the path-integral filtering is implemented in -- domain independently without data communication, all computations can be performed efficiently and in parallel.
2024-07-04