1. The FOCI method versus other wavefield extrapolation methods
Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
The FOCI method versus other wavefield extrapolation methods
Saleh Al-Saleh, Gary Margrave, and Hugh Geiger
November 19, 2004
Saleh Al-Saleh

2. Motivation
To compare the forward operator and conjugate inverse (FOCI) method for calculating wavefield extrapolators with
the Hale (1991) method
the weighted least square (WLSQ) method (Thorbecke et al., 2004)

3. Outline Brief review of the theory of Hale’s extrapolator
Brief review of the theory of WLSQ’s extrapolator
Comparisons of the three extrapolators:
Amplitude spectra
Phase errors
Impulse responses
and prestack depth migrations of the Marmousi dataset using
Hale’s, WLSQ’s, and FOCI’s extrapolators

4. Wavefield extrapolation methods:
Are more powerful in handling strong lateral velocity variations than ray theory based methods
Have two major problems:
Computationally expensive
Instability of the extrapolation operator

5. Wavefield extrapolation methods
where

6. Hale’s extrapolator (Hale, 1991)
Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
Basis function
We start with phase-shift operator in the f-kx domain,
We want to design an operator in the w-x domain that has N-coefficients, Hale’s extrapolator can be designed by in the f-k domain by superposition of M weighted basis functions where the criterion for M is that it has to be less than (N+1)/2. This is Hale’s choice of the basis function in the f-k domain. By matching the even derivatives of the basis functions and the phase-shift operator, we obtain a system of linear equations that can be solved to obtain cm. So the only criterion is that M is less than (N+1)/2. The problem with that is that there is no direct formula for choosing M value to ensure stability and it has to be done subjectively.
N operator length
M number of basis functions
Saleh Al-Saleh

7. Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
Hale’s extrapolator
Also, choosing a constant M value to all frequencies will not be stable for all of them. So each range of frequencies should have a distinct M value to ensure stability for all frequencies.
Saleh Al-Saleh

8. WLSQ’s extrapolator (Thorbecke et al., 2004)
Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
WLSQ’s extrapolator (Thorbecke et al., 2004)
On the other hand, the theory of WLSQ extrapolator is simpler. This is the discrete Fourier transform of an operator that has a finite length. This can be also represented by a matrix multiplication. Then we can solve for the operator in the w-x domain by weighted least square approach. Where this ^ is the weight function.
where,
Saleh Al-Saleh

9. WLSQ’s extrapolator v=2000 m/s and frequency=50 Hz
Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
WLSQ’s extrapolator v=2000 m/s and frequency=50 Hz
dx=10m, dz=2m, and N=25
dx=10m, dz=2m, and N=19
dx=10m, dz=10m, and N=19
dx=10m, dz=10m, and N=101
This is a zoomed look at the amplitude spectrum of WLSQ extrapolator in the f-k domain. These are the parameters. Note that it has a controlled instability. So after 300 depth steps in a homogenous medium the maximum amplitude will be around 1.05 which very good. When changing the operator size from 25 points to 19 points, it becomes less stable. Also, using a 10 m spacing with 19 coefficient, it becomes less stable. It generally more stable with long operators such as 101. Our tests show that the stability of WLSQ is dependant on the size of depth step and operator length.
Saleh Al-Saleh

10. Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
Amplitude Spectra of Hale’s, WLSQ’s, and FOCI’s extrapolators v=2000 m/s and frequency=50 Hz
dx=10m, dz=2m, and N=19
dx=10m, dz=10m, and N=31
This is a comparison of the three extrapolators. Note how Hale’s extrapolator is more stable than WLSQ and FOCI but it starts decaying before the evanescent boundary. That is why Hale’s extrapolator cannot migrate high angles of propagation. For the set of parameters, our tests show that FOCI’s extrapolator is more stable than WLSQ’s extrapolator. When changing the operator length to 31 points and the size of the depth step to 10 m, Hale’s is also more stable than WLSQ and FOCI but note that FOCI is more stable than WLSQ’s extrapolator. This shows that the stability of FOCI is less dependant on the parameters than WLSQ’s.
Saleh Al-Saleh

11. Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
Phase error of Hale’s, WLSQ’s, and FOCI’s extrapolators v=2000 m/s and frequency=50 Hz
dx=10m, dz=10m, and N=31
This slide shows the phase errors of the three extrapolators. The phase error is the phase of the extrapolator minus the phase of the exact phase-shift operator. This is the evanescent boundary. We do not care to what is happening in the evanescent region because the extrapolator will have amplitudes less than unity for that region. In the wave-like region, the three extrapolator have relatively small phase error. In the previous slide, the amplitude of Hale’s extrapolator started to decay before the evanescent boundary. So it will have a small phase error until its amplitude starts decaying. The phases of WLSQ and FOCI are similar. This slide shows that the three extrapolators have small phase error until their amplitudes start decaying.
Saleh Al-Saleh

12. Impulse responses N=31 velocity=2000 m/s
Poststack and prestack depth migrations using Hale's extrapolator
Impulse responses N=31 velocity=2000 m/s
12/8/2018
Phase-shift
Hale
WLSQ
FOCI
These are impulse responses of the three extrapolators compared the phase-shift migration. As expected from Hale’s extrapolator, it could not migrate the high angles of propagation. The WLSQ and FOCI results are comparable. This comparison shows also that both the WLSQ and FOCI could migrate high angles of propagations.
Saleh Al-Saleh

13. Marmousi Prestack Depth Migrations
Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
Marmousi Prestack Depth Migrations
The Marmousi dataset will be used to evaluate the different extrapolators. the Marmousi has strong lateral velocity variations and steeply dipping events.
Saleh Al-Saleh

14. Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
Hale’s and FOCI’s extrapolators dx=25 m dz=25 m operator length= 19 points
The first test is between Hale’s and FOCI with spatial resampling for dx=dz=25 meter and operator length of 19 points. The WLSQ result is not shown because we could not stable extrapolation operators with these parameters. The objective of this comparison is to show that both Hale’s and FOCI is more stable for a wider range of parameters.
Saleh Al-Saleh

15. Hale’s extrapolator run time=3.5 hours
Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
Hale’s extrapolator run time=3.5 hours
Our tests show that the WLSQ extrapolator is not stable for depth step=25 m. This test is to show that FOCI is more stable than WLSQ for a
Saleh Al-Saleh

16. FOCI’s extrapolator run time=2.0 hours

17. Poststack and prestack depth migrations using Hale's extrapolator
12/8/2018
WLSQ’s and FOCI’s extrapolators dx=12.5 m dz=12.5 m operator length= 51 points
The Hale’s result is not shown here because we think that there is an error in our implementation of Hale’s extrapolator for long operators.
Saleh Al-Saleh

18. WLSQ’s extrapolator Run time=16 hours

19. FOCI’s extrapolator Run time=12 hours

21. Conclusions
FOCI results are comparable with Hale’s and WLSQ’s results.
FOCI is computationally more efficient than the other methods due to spatial resampling.
Spatial resampling can not be easily implemented in the other methods.
This new method is a promising technique for seismic imaging.

22. Acknowledgments We would like to thank:
The sponsors of the CREWES project.
The sponsors of the POTSI project.
NSERC, MITACS, and PIMS.

23. WLSQ’s extrapolator run time=23.7 hours

24. FOCI’s extrapolator run time=15.8 hours