Image-based Q tomography using reverse time Q migrationYi Shen, Stanford University, Tieyuan Zhu, The University of Texas at Austin

SUMMARY

This paper first developed a technique to tomographically es-timate a Q model from the migrated image using two-waywavefield continuation method. When compared with the pre-vious work with a one-way downward-continuation method,this technique exhibits better capability of handling the steepstructure, e.g. the salt flank. Numerical results on a complexmodel with a salt body demonstrate the effectiveness of thistwo-way method on resolving the overturned wave propaga-tion introduced by the steep structures.

INTRODUCTION

Attenuation, parametrized by a seismic quality factor, Q, hasconsiderable impacts on surface seismic reflection data. Thisimpact degrades the image quality by lowering the amplitudeof the events, distorting the phase of wavelet, attenuating thehigh frequency of the image, and dispersing the velocity. Thoseeffects make the reservoir interpretation and identification dif-ficult. Therefore, it is important to estimate the properties ofthis attenuation parameter, and to use it for compensating theattenuated image.

Shen et al. (2013) and Shen et al. (2014) presented a newmethod, wave-equation migration Q analysis (WEMQA) toproduce a reliable Q model. This method uses one-way wave-field continuation to first migrate the viscoacoustic data with Qcompensation for measuring the Q effects; and then, to back-project these effects to update the current Q model. This methodhandles the structure well with gentle horizontal variation. How-ever, a one-way method is not able to represent the actualwave propagation at the steep structure, because propagatingthe wavefield along only one direction of the depth axis cannotproperly deal with the overturned events. In some situations,overturned events provide extremely useful information e.g.,the reservoir properties beside the salt flank. While at least inprinciple, two-way wavefield continuation has the capabilityof resolving these overturned wave propagations. Therefore,we apply two-way wavefield continuation migration with Qcompensation (i.e reverse time migration with Q compensa-tion) to produce an image for Q effects measurements (Zhuet al., 2014), and update the Q model using our designed two-way wave-equation tomographic operator to better handle thesteep structures.

In this paper, we first describe the theory of two-way migrationwith Q compensation and the two-way wave-equation tomo-graphic operator. Then, we apply WEMQA with the two-waymethod to a complex model with a salt body to demonstratethe effectiveness of the two-way method on steep structures.

THEORY

We incorporate reverse time migration with Q compensation(Zhu et al., 2014) and two-way wave-equation tomographicoperator into WEMQA (Shen et al., 2013, 2014) to estimate Qmodel and compensate Q effects through migration for betterhandling the steep structures.

Reverse time migration with Q compensation

Zhu and Harris (2014) first introduced the time-domain viscoa-coustic wave equation based on the constant Q model (Kjar-tansson, 1979), which is shown as follows,

„ηL+ τH d

dt− v−2 ∂ 2

∂ t2

«P(t) = f (t), (1)

where t is time, P is the propagated wavefield, and f is thesource wavefield; v is the acoustic velocity at the referencefrequency ω0 ; the two fractional Laplacian operator are L =`−∇2´γ+1 and H =

`−∇2´γ+1/2, where the variable γ is de-

fined as γ = 1/π tan−1 (1/Q). The absorption and dispersionoperator coefficients are given by η = −v2γ ω−2γ

0 cosπγ andτ = −v2γ−1ω−2γ

0 sinπγ . The term of (ηLP(t)−∇2P(t)) in-dicates the dispersion effects, and the second term in the leftside of Equation 1 indicates the absorption effects (Zhu et al.,2014). A pseudo-spectral method is applied for the numericalimplementation.

According to the character of attenuation, higher frequency hasmore loss than the lower frequency of the propagating wave.Compensation through Q migration (Zhu et al., 2014), the in-verse procedure of the wave forward propagation, amplifiesmore on the higher frequency. The risk is that the amplificationmay gain the high frequency noise and make these frequenciesdominate the image. Therefore, a low-pass filter is added tohelp reduce the high-frequency noise through migration:

„∇2 +N(ηL−∇2)+NτH d

dt− v−2 ∂ 2

∂ t2

«P(t) = f (t), (2)

where N is the low-pass filter in the spatial frequency domain.

Two-way wave-equation tomographic operator

Wave-equation Q tomography is a nonlinear inversion processthat aims to find the Q model that minimizes the residual fieldin the image space, ∆I (x,h). This residual image can be ap-proximated by a linearized operator being performed on themodel perturbation ∆Q:

∆I (x,h) =X

y

∂ I (x,h)∂Q(y)

˛̨Q0 ∆Q(y) (3)

where x and y are the coordinates of the image and model Q,respectively. And h is the subsurface offset, Q0 is the back-

ground quality factor. The adjoint of this tomographic oper-ator projects the image perturbation back into the Q modelspace. The back-projected changes in the model space are usedas gradient directions to conduct a line-search in optimizationschemes. This back-projection can be expressed as follows:

∆Q(y) =X

y

„∂ I (x,h)∂Q(y)

˛̨Q0

«∗∆I (x,h) (4)

The derivative of I (x,h) with respect to Q is shown as follows,which is the sum of the attenuation-induced perturbed sourcewavefield timed by the background receiver wavefield and theattenuation-induced perturbed receiver wavefield timed by thebackground source wavefield:

∂ I (x,h)∂Q(y)

˛̨Q0

=X

xs,xr

„∂G(x−h,xs;Q0)

∆Q(y)

«∗G∗ (x+h,xr;Q0)d (xr,xs)

+X

xs,xr

G∗ (x−h,xs;Q0)„

∂G(x+h,xr;Q0)∆Q(y)

«∗d (xr,xs)

(5)where xs and xr are the source and receiver coordinates re-spectively, G is the Green’s function, d is the surface data. Theattenuation-induced perturbed wavefield can be linearized us-ing Taylor expansion as follows,

∂G(x,xs;Q0)∂Q(y)

=−∂η (y;Q)L(y;Q)∂Q(y)

˛̨Q0 G(y,xs;Q0)G(x,y;Q0)

−∂τ (y;Q)H(y;Q)∂γ

˛̨Q0

ddt

G(y,xs;Q0)G(x,y;Q0)

∂G(x,xr;Q0)∂Q(y)

=−∂η (y;Q)L(y;Q)∂Q(y)

˛̨Q0 G(y,xr;Q0)G(x,y;Q0)

−∂τ (y;Q)H(y;Q)∂γ

˛̨Q0

ddt

G(y,xr;Q0)G(x,y;Q0)

(6)

and

∂η (y;Q)L(y;Q)∂Q(y)

˛̨Q0

=−(2η lnv−2η lnω0−πvτ)L+ηL ln

`−∇2´

π`Q2

0 +1´

∂τ (y;Q)H(y;Q)∂γ

˛̨Q0

=−`2τ lnv−2τ lnω0 +πv−1η

´H+ τH ln

`−∇2´

π`Q2

0 +1´

(7)

NUMERICAL RESULTS

The numerical example test employs a dataset generated bySchlumberger (Cavalca et al., 2013) using a 2D viscoacoustic

version of the 2004 BP benchmark model (Billette and Brandsberg-Dahl, 2005). An attenuation model has been added by Schlum-berger to the original 2004 BP models. This true Q modelis not released, but its location and value has been shown bySchlumberger (Cavalca et al., 2013). The attenuation modelis a space- and depth-variant absorption model made of sev-eral Q heterogeneities and a nonattenuative background (1/Q= 0.0002). To test the effectiveness of our method on the steepstructure, we only focus our work on the salt region, whichhas a large attenuative zone (1/Q = 0.02) near the left of thesalt flank. The velocity model of this region is shown in Fig-ure 1. The viscoacoustic surface seismic data generated bySchlumberger used a finite difference modeling code based onstandard linear solid theory. In my example, 248 shots are usedwith 100 m spacing, and the offsets range from -15,000 m to15,000 m. Receivers are distributed on both sides of each shotat an increment of 25 m. The cut-off frequency that we usedto filter out the noise in the reverse-time migration is 30 Hz,which is 60% of the maximum useful frequency. The workflowused in this numerical example has been presented in Shenet al. (2013, 2014).

Figure 2 and Figure 3 are the attenuated images from the vis-coacoustic data using one-way and two-way Q migration witha nonattenuating model, respectively. The results show thatthe events beside the salt flank are attenuated, in terms of theamplitude dimming, incoherency of the events, and stretchingof the wavelets. Both migrations image the structure well withgentle horizontal variation. However, Figure 2 shows a poorerimage around the salt flank when compared with Figure 3. Thesalt flank in Figure 2 is not focused. The regions beside thesalt have discontinued events and are contaminated with high-frequency noises. Figure 3 shows a sharper and clearer saltflank. The image around the salt is cleaner with less high-frequency noises, and the events become more coherent.

The next step is to invert for the Q model besides the salt flankfrom these attenuated images using wave-equation migrationQ analysis developed by Shen et al. (2013, 2014). The initialmodel for the inversion is without attenuation. We analyze theattenuation effects by calculating the slope of the logarithm ofthe spectral ratio between the windowed events of each traceand the events in the reference window in the same depth. Thewindow size is 1500 m, and 100 sliding windows are used foreach trace. The reference trace is the one at 24,000 m. Figure 5and Figure 6 are the inverted Q model using one-way and two-way WEMQA, respectively. Figure 5 fails in resolving the Qmodel in that area, because one-way propagation is not ableto accurately reflect the steep structure. Figure 6 shows thata two-way method retrieves well the Q model in the reservoirregion beside the salt; especially at the upper part of the saltflank, which matches well the true Q model. However, it stillfails updating the Q model at the lower part of the salt flank,because of the high-frequency image artifacts at that area. Fig-ure 4 is the image using two-way migration with Q compen-sation. The results show that the image at the lower regionis not well compensated because of the weak updating of theQ model there. Therefore, a filter, e.g., a curvelet-transform-based filter, is needed to mitigate the image noise. But, theevents at the rest of the part beside the salt flank are compen-

sated. The amplitudes are partially recovered, the events be-come more coherent, and the phase of the events get corrected.

Figure 1: BP velocity model.

Figure 2: Attenuated images from the viscoacoustic data usingone-way Q migration with a nonattenuating model. The saltflank is not focused. The regions beside the salt have discontin-ued events and are contaminated with high-frequency noises.

CONCLUSION

This paper presented a new wave equation migration Q anal-ysis based on two-way wavefield continuation method. Whencompared with the previous work with a one-way downward-continuation method, the two-way based method has bettercapability of handling the steep structure, e.g. the salt flank.Numerical tests on a complex model with a salt body demon-strate the effectiveness of this two-way method on resolvingthe overturned wave propagation caused by the steep struc-tures.

ACKNOWLEDGMENTS

The authors thank Biondo Biondi and Robert Clapp for theiradvice and suggestions, Schlumberger and BP for providing

Figure 3: Attenuated images from the viscoacoustic data usingtwo-way reverse-time Q migration with nonattenuating model.The salt flank is sharp and well-focused. The image around thesalt is cleaner with less high-frequency noises, and the eventsbecomes more coherent, when compared with Figure 2.

Figure 4: Compensated image using two-way reverse-time Qmigration. The results show that the image at the lower regionis not well compensated because of the weak updating of theQ model there. But the events at the rest of the part beside thesalt flank are compensated.

Figure 5: The inverted Q model using one-way wave-equationmigration Q analysis. The result fails in resolving the Q modelin that area, because one-way propagation is not able to accu-rately reflect the steep structure.

Figure 6: The inverted Q model using two-way wave-equationmigration Q analysis. The result retrieves well the Q modelin the reservoir region beside the salt; especially at the upperpart of the salt flank. However, it still fails in updating the Qmodel at the lower part of the salt flank, because of the high-frequency image artifacts at that area.

the 2D synthetic dataset, Yaxun Tang and Ali Almonin for helpwith programming, and Stanford Exploration Project sponsorsfor their financial support. The second author wish to thankthe Jackson Distinguished Postdoctoral Fellowship at the Uni-versity of Texas at Austin.