1. Prestack Depth Migration in a Viscoacoustic Medium
Prestack Depth Migration in a Viscoelastic Medium
Prestack Depth Migration in a Viscoacoustic Medium
Hello, everyone. It is my great honors to give a talk here. My topic is prestack depth migration in a viscoelastic medium.
Jianjun Cui
Jianjun Cui

2. Prestack Depth Migration in a Viscoelastic Medium
Outline
Introduction
Theory and methodology
Numerical example
Conclusions
Acknowledgments
My talk includes five parts, introduction, theory and methodology,and then in numerical example I will show a synthetic example, Conclusion, finally, Ackownledgment.
Jianjun Cui

3. Prestack Depth Migration in a Viscoelastic Medium
Introduction
Prestack Depth Migration in a Viscoelastic Medium
Elasticity is a good model for mechanical wave propagation through the earth. According to classical theory, the deformation should obey Hooke’s Law. The equation for a pressure wave can be expressed as
when we study the propagation of the seismic wave underground, we usually suppose that the earth is a elastomer, means it is elastic. Elasticity is a good model for mechanical wave propagation through the earth, and the problem become simple. According to the classic theory, the deformation obey the Hooke’s law. We usually use this equation to describe the seismic wave propagation. It is pressure equation.
where
Jianjun Cui

4. Prestack Depth Migration in a Viscoelastic Medium
Introduction
ρi,νi …
Pulse
ρj,νj
ρk,νk
ρl,νl
Layer model
According to the previous wave equation, if we send a pulse like this into this layer model. At every layer, there should be a reflection. And if we have a receiver on the ground, we should get a result like this. That is great, the resolution is wonderful. Because in the seismic exploration, the source could generate a pulse signal. But, in fact, what we have gotten is not like this.
Anticipated Result
Jianjun Cui

5. Prestack Depth Migration in a Viscoelastic Medium
Introduction
In fact, we get this kind of results. This a real seismic recorder, what I want to say is that:
No real materials are perfectly elastic. Wave energy is gradually converted into heat.
The propagation of seismic waves in real medium is in many respects different from propagation in an ideal solid.
Real media will cause dissipation of seismic energy, and decreasing the amplitude and modifying the frequency content of the propagating wavelet.
Attenuation of propagating waveforms is,quite significant, if neglected, it could be a source of erroneous results in forward modelling, inversion, and imaging.
There is,no justification for neglecting the absorption and dispersion of seismic energy.
if the quality factor Q is satisfactorily approximated. We can extract more detailed information about the subsurface from seismic data or to construct images with better resolution,
In this paper, we will try to compensate for some of the loss energy in migration.
Blackfoot oilfield shot gather
Jianjun Cui

6. Theory and methodology
Prestack Depth Migration in a Viscoelastic Medium
Theory and methodology
In viscoacoustic media, the viscoacoustic wave equation can be expressed as
Attenuation function
As what I said just now, that the earth is not a elastomer.
Wave energy is gradually converted into heat.
About this question, Stokes had studied in 19th century andhad set up the viscoelastic wave equation, which has considered the energy dissipation of internal friction. But because of the application is limited and that wave equation could not describe the wave propagation , there were seldom people pay attention to that.
Till last century, Ricker modified Stoke’s equation, which can describe the wave propagation properly.
This is the modified Stokes’s wave equation, and this part is the attenuation function. If we get ride of this part , it is an acoustic wave equation, we call it transfer frequency. And this part is direct ratio to the attenuation Q.
We will study this equation in Frequency-Wavenumber domain.
where
Jianjun Cui

7. Theory and methodology
Prestack Depth Migration in a Viscoelastic Medium
Theory and methodology
Viscoacoustic wave extrapolation in the frequency-wavenumber domain
After a series of transition, we gain this wavefield extrapolating formula. Ok, lets analyze this formula.
where
Jianjun Cui

8. Theory and methodology
Prestack Depth Migration in a Viscoelastic Medium
Theory and methodology
For the downgoing wave:
the extrapolator can be used to compensate for the energy-loss that the wave has experienced during propagation from source to reflecting interface.
For a downgoing wave, we have ….
the extrapolator can be used to compensate for the energy-loss that the wave has experienced during propagation from source to the reflecting interface
Jianjun Cui

9. Theory and methodology
Prestack Depth Migration in a Viscoelastic Medium
Theory and methodology
The upgoing wave extrapolator is:
The extrapolator will boost the amplitude, thereby compensating for the amplitude loss on the way up from the reflection point.
For Upgoing wave has been lost amplitude on its way, the extrapolation will give
The extrapolator will boost the amplitude, thereby compensating for the lost amplitude on the way up from the reflection point.
Jianjun Cui

10. Theory and methodology
Prestack Depth Migration in a Viscoelastic Medium
Theory and methodology
Migration with the previous equations is limited to homogeneous media with constant velocity function. This problem can be solved by using:
There is a problem with the previous equation,
Such solutions is limited to homogeneous media with constant-velocity function. We tried to solve the problem this way.
Jianjun Cui

11. Theory and methodology
Prestack Depth Migration in a Viscoelastic Medium
Theory and methodology
SOURCE RECEIVER
Reflector
Raypath of a reflected signal
Prestack depth migrations are commonly obtained by using Claerbout’s imaging principle (Claerbout, 1971). The upcoming wavefield is correlated with the down-going wavefield at each depth level. Then, the results is the depth image underground.
Imaging theory
Jianjun Cui

12. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
distance (m)
250
750
Depth (m)
V=1500m/s.
Trace space is 30m.
Depth of points is 250m and 750m, respectively.
Sampling interval is 1ms.
Ok, here we show some numerical example. First I want to show an example about the seismic wave change in the viscoacoustic medium. We design an geological model like this, there are two exploding points in the model. and other parameter are shown here. We create elastic and viscoacoustic zero offset profile.
Acoustic geological model for forward modeling
Jianjun Cui

13. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
This is acoustic zero offset modelling result, Attenuation is zero. Other parameters are same to the model. We all know that in the acoustic medium, the frequency, amplitude and phase of the wave are not changed at different trance, as well as the different depth
An acoustic forward modelling example (no attenuation)
Jianjun Cui

14. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
Prestack Depth Migration in a Viscoelastic Medium
This is the zero offset viscoacoustic forward mordelling result. The transition frequency is It is very clear that the energy of the seismic wave is absorbed when it is propagating in the viscoacoustic media. Look at the amplitude and frequency are changed.
the longer propagation, the absorption the more serious.
Forward modeling result (with attenuation)
Transition frequency is 2000
Jianjun Cui

15. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
This figure is partially enlarged from last figure, in order that the absorption can be observed easily. Note with exploding point being far away from the receiver, amplitude of wave is decreased, and frequency become lower, and the phase is also influenced. It is very clear.
Partially enlarged of viscoelastic forward modelling result
Jianjun Cui

16. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
V=1500m/s
V=1800m/s
V=2000m/s
V=2300m/s
V=2500m/s
200
600
850
1700
Depth (m)
distance (m)
Direction of acquisition
Transition frequency is 20000
Ok, here we show an example of viscoacoustic prestack depth migration. This is the geological model, we try to model a complex geological situation, so there are decline layers, faults and horizontal layer. The transition frequency of this model is
Viscoacoustic model for prestack depth migration
Jianjun Cui

17. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
Offset
600
1200
1800
Times (ms)
This is the shot gather created form the geological model, look the absorption with depth.
Shot gather
Jianjun Cui

18. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
Offset
Depth (m)
Here, we show the result using elastic prestack depth migration.
Note the diffuse image of the reflectors due to the dispersion of the wavelet as it propagates down into the subsurface and up again.
Also the locations of some of the reflectors are incorrect.
This is a consequence of the fact that both an amplitudes and arrival times are changed in a viscoacoustic medium with absorption
Acoustic prestack depth migration result
Jianjun Cui

19. Prestack Depth Migration in a Viscoelastic Medium
Numerical example
Offset
Depth (m)
This is the result of viscoacoustic prestack depth migration. Note the improved quality of the image; at the points of the faults and reflectors are imaged clearly. Because some of the lost energy are compensated when the wave field extrapolated. (downgoing wave and upgoing wave)
Viscoacoustic prestack depth migration result
Jianjun Cui

20. Prestack Depth Migration in a Viscoelastic Medium
Conclusions
Real media are attenuative and cause amplitude loss and phase changes.
The viscoacoustic wave equation can be used to describe wave propagation.
An f-k viscoacoustic migration scheme has been outlined here.
The results show considerable promise
Ok, through this research we can get these conclusion:
Real media are attenuative and cause dissipation of seismic energy, and decreasing the amplitude and modifying the frequency content of the propagating wavelet.
viscoacoustic wave eqauation can be used to describe wave propagation.
An f-k viscoacoustic migration scheme has been outlined here.
The results show considerable promise
Jianjun Cui

21. Prestack Depth Migration in a Viscoelastic Medium
Acknowledgements
CREWES
Robert R. Stewart, Gary Margrave, Hanxin Lu, Kevin Hall, Chuck Ursenbach, Rolf Maier, Zhihong Cao and … …
CREWES sponsors
CSU (Central South University)
Jianjun Cui