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

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

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

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

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

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

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

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

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

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

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

Prestack Depth Migration in a Viscoelastic Medium Numerical example 0 1000 2000 3000 distance (m) 4000 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

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

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 2000. 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

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

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) 0 1000 2000 3000 4000 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 20000. Viscoacoustic model for prestack depth migration Jianjun Cui

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

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

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

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

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