least-squares migration via a gradient projection method ...kazemino/papers/lsgp-deblending.pdf ·...

30 May – 2 June 2016 | Reed Messe Wien

78th EAGE Conference & Exhibition 2016 Vienna, Austria, 30 May – 2 June 2016

Th P4 07Least-squares Migration via a Gradient ProjectionMethod - Application to Seismic Data DeblendingJ. Cheng (University of Alberta), N. Kazemi* (University of Alberta) & M.Sacchi (University of Alberta)

SUMMARYWe propose a gradient projection method that is applicable to least-squares migration for separation ofsimultaneous source seismic data. Using shot-profile split-step migration and de-migration operators, wenotice that, in shot-index image domain, the simultaneous source interferences appear random whereas thedesired signal is coherent. The latter is used as a coherency constraint for least-squares migration. Weincorporate a projection operator, which is the Singular Spectrum Analysis (SSA) filter in shot-indexdomain, into gradient projection method to solve for a volume of artifacts-reduced shot-index gathers thathonors the observed data. The method effectively suppresses simultaneous source crosstalk and improvesthe quality of shot-index image gathers. The outputs of our method can be a crosstalk-free migrated imageand the deblended data set that can be used in conventional processing workflows.



Introduction

Seismic migration is a key step in seismic data processing to acquire the structural images of the sub-surface. It entails moving the energy in seismic records to the corresponding reflection points in thesubsurface. One problem associated with seismic migration is the generation of migration artifacts andthe distortion of the subsurface image due to insufficient illumination, the complexity of the subsurface,and poor acquisition geometry. Least-squares migration has been proposed to improve the quality ofimages and to reduce the migration artifacts. Instead of directly using the migration operator, in least-squares migration, the solution is acquired by iteratively seeking a model that best fits the observed data.During the process, various constraints have been introduced in order to suppress the migration artifacts(Nemeth et al., 1999; Kuehl and Sacchi, 2003; Dai and Schuster, 2012).

Another strategy to resolve the distortion issue is to acquire large amount of information in seismicdata acquisition. For instance, simultaneous source acquisition techniques have been gaining popularityas a low-cost strategy to increase seismic survey density (Beasley et al., 1998; Berkhout, 2008; Abmaet al., 2010). In simultaneous source acquisition, seismic shots are fired in an overlapping fashionwith small randomized time delay. As the non-overlapping criterion in conventional seismic acquisitionhas been eliminated, simultaneous source acquisition is extremely helpful in saving time and costs ofseismic data acquisition. However, severe interferences are introduced since the firing interval is smallerthan the recording time. One strategy of processing simultaneous source seismic data is to introducean additional processing step, which is referred to as simultaneous source separation or deblending, toconvert the data to the conventional seismic data processing flow (Abma et al., 2010; Mahdad et al., 2011;van Borselen et al., 2012). Another strategy is to apply direct imaging on the blended gathers. Romeroet al. (1999) introduced randomized phase-encoding among sources to suppress the interferences inmigration. Different source encoding schemes are studied by Dai and Schuster (2009) and Godwin andSava (2013). Particularly, source blending has been a strategy to improve the computational efficiency insome large scale problems such as reverse-time migration (Romero et al., 1999), least-squares migration(Dai and Schuster, 2009; Xue et al., 2014) and Full waveform inversion (Krebs et al., 2009; Anagaw andSacchi, 2014).

In this article, we propose a gradient projection algorithm for least-squares migration and apply themethod to deblend simultaneous source seismic data. The cost function is the data misfit in the blendeddomain. Since the desired signal is coherent in shot-index domain, we introduce the singular spec-trum analysis (SSA) filter (Sacchi, 2009) as a projection operator to the gradient descent iterations forsuppressing the migration artifacts. We tested the efficacy of the algorithm via a synthetic example.The method leads to a framework in which different projections can be adopted for constraining theleast-squares migration.

Theory

We remind the readers that a single shot gather acquired in the conventional seismic acquisition can besimulated via the following equation

di = Limi (1)

where mi indicates a vectorized model of length N corresponding to the shot i, Li is a matrix of size M×N and represents the shot-profile split-step forward modeling, or de-migration operator that transformsthe partial image to seismic data di. In simultaneous source acquisition, several seismic sources fire atclose time intervals. The responses are then recorded by the same set of receivers. We express the dataacquired by simultaneous sources as follows

b =Ns

∑i

ΓiLimi = Am, (2)

where m = [m1,m2, ...,mNs ]T and Γi is a time-shift operator that maps di to the blended data b after

summation. Ns denotes the total number of sources that are blended. The migrated image mi can be



expressed as followsmi = L∗i Γ

∗i b , or m = A∗b . (3)

L∗i represents the shot-profile split-step migration operator that maps the ith shot record to a partial imageof the subsurface. We adopt the pseudo-deblending operator as the adjoint operator Γ∗i for deblending.It implies the process of shifting the time delays back and splitting the blended observation to the ithshot (Berkhout, 2008). The operator A∗ cannot remove the crosstalk artifacts (Figure 1a). In shot-indexdomain, as is shown by Figure 1b, the desired signal is coherent whereas the source interferences areperturbed by the randomized time delay. Therefore, a coherence constraint can be adopted to suppressthe artifacts.

To find an optimal m that honors the blended observation b is equivalent to finding the minimum of thecost function

J(m) = ||b − Am||22 . (4)

Equation 4 is the unconstrained cost function for the least-squares problem. As the pseudo-deblendingoperator cannot eliminate the source crosstalk, a more efficient method is to introduce the previouslydescribed coherency constraint to suppress simultaneous source crosstalk. We introduce a projectionoperator PC that projects m to a set (C : m = PC [m]) that is corresponding to the desired coherentsignal

J(m) = ||b − Am||2 s.t. m ∈ C . (5)

The solution to Equaiton 5 can be acquired via the gradient projection algorithm as follows

mk+1 = PC [mk−λGk], (6)

where λ is the step size and Gk denotes the gradient for Equation 4 given by

Gk = A∗(Amk − b). (7)

The gradient descent iterations are adopted in order to search for an optimal m that best fits the blendedobservation. In each iteration, the current solution is projected to a set that m is coherent along theshot-index gathers. The projection may not necessarily improve the convergence rate of the algorithm.However, when the step size λ is sufficiently small, the convergence of the gradient projection methodis guaranteed (Bertsekas, 2009). In this article, we choose the projection operator as the reduced-rankfilter for incoherent noise attenuation in shot-index domain. The method can be found in the literatureas Cazdow filtering or Singular Spectrum Analysis (SSA). SSA entails forming Hankel matrices fromkz− x domain data, performing rank reduction and then recovering the data via anti-diagonal averagingof the reduced-rank Hankel form. The details associated to the implementation of SSA for seismic noiseattenuation, seismic data reconstruction, and debelnding can be found in Oropeza and Sacchi (2011)and Cheng and Sacchi (2015). It can be shown that SSA can preserve linear events and filter incoherentevents (Sacchi, 2009; Trickett and Burroughs, 2009). In addition, it is easy to show that for a data setcomposed of the superposition of k dips, the Hankel matrix of the data is a rank k matrix. If the dataare contaminated with noise, the rank of the associated Hankel matrix will increase and therefore, rankreduction is an effective way of noise attenuation. It is important to mention that a variety of projectionoperators can be adopted to replace the SSA reduced-rank filtering. The projection operators include butnot limited to sort and stacking, median filters, f-x prediction filters and sparse transforms.

Example

We use a 2D synthetic exmaple to test the efficacy of the proposed algorithm. The sources are fired from1km to 5km according to the self-simultaneous source acquisition design. With this acquisition design,50% of the conventional acquisition time is saved. The locations are then binned to a regular grid with20m spatial interval. The receivers are deployed in the bottom of the first layer to mimic the oceanbottom nodes. The spatial interval for receivers is 20m as well. Then we adopt the proposed method todeblend the simultaneous source data. In this example, we choose the rank for the SSA reduced-rankfilter as k = 10 and the step size λ equals 2.5. Figure 1 shows the migrated partial images after 15



iterations. The crosstalk artifacts are effectively suppressed in shot-index domain. The deblended databecome comparable with the true unblended data set from conventional seismic acquisition.

0

1

2

z (k

m)

0 1 2 3 4 5x (km)

(a)

0

1

2

z (k

m)

50 100 150 200shot number

(b)

0

1

2

z (k

m)

0 1 2 3 4 5x (km)

(c)

0

1

2z

(km

)

50 100 150 200shot nunmber

(d)

Figure 1 (a) Partial image for the pseudo-deblended shot gather. (b) Center shot-index gather. (c) Partial image after 15 iterations of gradient projection. (d) Center shot-index gather after 15 iterations of gradient projection.

0

t (s)

0 1 2 3 4 5receiver position (km)

(a)

0

t (s)


(b)

0

t (s)


(c)

0

t (s)


(d)

Figure 2 The debelnding result via LSM: (a) An unblended shot gather. (b) The pseudo-deblended shot gather. (c) Deblending results after 15 iterations of gradient projection. (d) The difference between (a) and (c). The amplitude difference is because the algorithm has yet reached the convergence in 15 iterations.



Conclusions

We have presented a gradient projection algorithm to solve least-squares migration where constraints tosuppress simultaneous source crosstalk artifacts are implemented by a projection operator. The methodadopts the gradient descent method and iteratively search for a solution that fits the observed data. Wetest the method via a synthetic example where we adopt least-squares migration to improve the qualityof common image gathers and to separate the responses from simultaneously fired sources. The methodis effective in suppressing the artifacts introduced by simultaneous shooting.

Acknowledgements

The authors thank the sponsors of the Signal Analysis and Imaging Group (SAIG) at the Department ofPhysics of University of Alberta for supporting this research.

References

Abma, R., Manning, T., Tanis, M., Yu, J. and Foster, M. [2010] High quality separation of simultaneoussources by sparse inversion. 72nd Annual International Conference and Exhibition, EAGE, ExpandedAbstracts.

Anagaw, A.Y. and Sacchi, M.D. [2014] Comparison of multifrequency selection strategies forsimultaneous-source full-waveform inversion. Geophysics, 79(5), R165–R181.

Beasley, C.J., Chambers, R.E. and Jiang, Z. [1998] A new look at simultaneous sources. 68th AnnualInternational Meeting, SEG, Expanded Abstracts, 133–135.

Berkhout, A.J. [2008] Changing the mindset in seismic data acquisition. The Leading Edge, 27, 924–938.

Bertsekas, D.P. [2009] Convex optimization theory. Athena Scientific.van Borselen, R., Baardman, R., Martin, T., Goswami, B. and Fromyr, E. [2012] An inversion approach

to separating sources in marine simultaneous shooting acquisition - application to a Gulf of Mexicodata set. Geophysical Prospecting, 60(4), 640–647.

Cheng, J. and Sacchi, M.D. [2015] Separation and reconstruction of simultaneous source data via itera-tive rank reduction. Geophysics, 80(4), V57–V66.

Dai, W. and Schuster, J. [2009] Least-squares migration of simultaneous source data with a deblurringfilter. 79th Annual International Meeting, SEG, Expanded Abstracts, 2990–2994.

Dai, W. and Schuster, J. [2012] Multisource least-squares reverse time migration. Geophysical Prospect-ing, 60(4), 681–695.

Godwin, J. and Sava, P. [2013] A comparison of shot-encoding schemes for wave-equation migration.Geophysical Prospecting, 61(s1), 391–408.

Krebs, J.R., Anderson, J.E., Hinkley, D., Neelamani, R., Lee, S., Baumstein, A. and Lacasse, M.D.[2009] Fast full-wavefield seismic inversion using encoded sources. Geophysics, 74(6), WCC177–WCC188.

Kuehl, H. and Sacchi, M.D. [2003] Least-squares wave-equation migration for AVP/AVA inversion.Geophysics, 68(1), 262–273.

Mahdad, A., Doulgeris, P. and Blacquiere, G. [2011] Separation of blended data by iterative estimationand subtraction of blending interference noise. Geophysics, 76, 9–17.

Nemeth, T., Wu, C. and Schuster, G.T. [1999] Least-squares migration of incomplete reflection data.Geophysics, 64(1), 208–221.

Oropeza, V. and Sacchi, M. [2011] Simultaneous seismic data denoising and reconstruction via multi-channel singular spectrum analysis. Geophysics, 76(3), V25–V32.

Romero, L.A., Ghiglia, D.C., Ober, C.C. and Morton, S.A. [1999] Phase encoding of shot records inprestack migration. Geophysics, 65(2), 426–436.

Sacchi, M.D. [2009] F-X singular spectrum analysis. CSPG CSEG CWLS Convention, 392–395.Trickett, S.R. and Burroughs, L. [2009] Prestack rank-reducing noise suppression. 79th Annual Interna-

tional Meeting, SEG, Expanded Abstracts Annual International Meeting, SEG, Expanded Abstracts.Xue, Z., Chen, Y., Fomel, S. and Sun, J. [2014] Imaging incomplete data and simultaneous-source data

using least-squares reverse-time migration with shaping regularization. 84th Annual InternationalMeeting, SEG, Expanded Abstracts, 3991–3996.

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