Processing seismic data in the presence of residual staticsAaron Stanton*, Nasser Kazemi, and Mauricio D. Sacchi, Department of Physics, University of Alberta

SUMMARY

Many common processing steps are degraded by static shiftsin the data. The effects of static shifts are analogous to noise ormissing samples in the data, and therefore can be treated usingconstraints on sparsity or simplicity. In this paper we show thatrandom static shifts decrease sparsity in the Fourier and Radontransforms, as well as increase the rank of seismic data. Wealso show that the concepts of sparsity promotion and rank re-duction can be used to solve for static shifts as well as to carryout conventional processes in the presence of statics. The firstalgorithm presented is a modification to the reinsertion step ofProjection Onto Convex Sets (POCS) and Tensor Completion(TCOM) that allows for the compensation of residual staticsduring 5D denoising and interpolation of seismic data. Themethod allows preserving residual statics during denoising, orcorrection of residual statics in the case of simultaneous de-noising and interpolation. An example is shown for a 5D re-construction of synthetic data with added noise, missing tracesand random static shifts, as well as for a 2D stacked sectionwith missing traces and static shifts. While standard recon-struction struggles in the presence of even small static shifts,reconstruction with simultaneous estimation of statics is ableto accurately reconstruct the data. The second algorithm pre-sented is a Statics Preserving Sparse Radon transform (SPSR).This algorithm includes statics in the Radon bases functions,allowing for a sparse representation of statics-contaminateddata in the Radon domain.

INTRODUCTION

Many processing steps can fail in the presence of even smallstatic shifts (≤±10ms). Creating robust methods that can han-dle residual static shifts can benefit early stages of processingworkflows, such as denoising, interpolation, and multiple at-tenuation. This paper is divided into two parts. In the firstpart a modification to the reinsertion step of Tensor Comple-tion (TCOM) and Projection Onto Convex Sets (POCS) basedreconstruction is provided. This step inserts the estimated datainto the original data and can be used for both interpolationand denoising. We propose a modification to the reinsertionthat allows for the compensation of residual static shifts dur-ing this step. In the case of POCS the property being exploitedis sparsity in the frequency-wavenumber domain, while in thecase of TCOM the property exploited is rank in the frequency-space domain.

Residual statics are typically attributed to near surface lateralvelocity and topographical variations (Ronen and Claerbout,1984). Because these effects are considered to be surface con-sistent their correction involves a single static shift for eachtrace. A common practice for residual static estimation is touse the cross-correlation of unstacked data within a CMP gathercompared with the stacked trace, taking the maximum lags for

each trace as an estimate of the residual statics. A problemwith this method is that it is sensitive to the velocity used toNMO correct the input gathers (Eriksen and Willen, 1990).Traonmilin and Gulunay (2011) tackle the problem by simulta-neously estimating the statics during projection filtering for thepurpose of denoising seismic data. Recently Gholami (2013)showed that sparsity maximization can be used as a criterionfor short period statics correction. Building off the work ofStanton and Sacchi (2012) we propose two algorithms that al-low for statics to be estimated during 5D trace interpolationand denoising as well as for demultiple using the sparse Radontransform in the presence of statics.

THEORY

5D ReconstructionOur method begins with the observation that statics appearto have the same character as noise or missing traces in theFourier domain. Figure 1 shows a 2D synthetic gather (a) withadded noise (b), missing traces (c), static shifts (d), and theirrespective f-k amplitude spectra (e)-(h). The destruction of thesignal observed in the f-k amplitude spectra for each of thesethree cases are remarkably similar.

The assumption of any Fourier reconstruction method is thatthe desired noise-free, fully-sampled signal can be sparselyrepresented in the Fourier domain. In this paper we show thatthis same sparsity relation can be used for the removal of staticshifts within the data.

In the case of 5D Projection Onto Convex Sets (POCS) recon-struction a Fourier estimate of the data is found by iterativelythresholding the amplitude spectrum of the data (Abma andKabir, 2006). For a given temporal frequency, ω , the data inthe ω−mx−my−hx−hy domain at the kth iteration of POCSare given by

Dk = α1Dobs +(1−α1S)F−1D T FDDk−1,k = 1, ...,N, (1)

where mx, my, hx, hy are midpoints and offsets in the x and ydirections respectively, Dobs are the original data with missingtraces, and FD and F−1

D are the forward and reverse 4D Fouriertransforms in the spatial dimensions respectively. In this nota-tion Dk(ω,kmx ,kmy ,khx ,khy) = FDDk(ω,mx,my,hx,hy), and Tis an iteration dependent threshold operator that is designed us-ing the amplitudes of the input data (Gao and Sacchi, 2011). Sis the sampling operator and is equal to one for points with ex-isting traces and zero for points with unrecorded observations.The scaling factor α1 ≤ 1 can be used to simultaneously de-noise the data. A choice of α1 = 1 reinserts the noisy originaldata at each iteration, whereas a lower value of α1 will denoisethe volume by averaging the original and reconstructed data.

The modification we propose to allow for statics to be com-pensated for during the reconstruction is to derive a static shiftbetween the thresholded data and the data from the previous

Processing seismic data in the presence of residual statics

iteration. The data in the ω−mx−my−hx−hy domain at thekth iteration of POCS with statics compensation are given by

Dk =α1Dobse−iω(1−α2)τk+(1−α1S)F−1

D T FDDk−1,k= 1, ...,N,(2)

where τk(mx,my,hx,hy) are the estimated static shifts at the kth

iteration and are constant for all frequencies, ω . The scalingfactor α2 ≤ 1 can be used to control the level of static cor-rection. A choice of α2 = 1 can be used for data with littleto no residual statics, whereas a value of α2 = 0 will removestatics more aggressively by fully applying the estimated staticshifts at a given iteration. The time shifts τk(mx,my,hx,hy) arethe lags given by the maximum values of the cross correlationof the static corrected input data from the previous iteration,Dobse−iω(1−α2)τ

k−1(where τk−1 =

∑k−1n=1 τn), with the thresh-

olded data from the current iteration, F−1D T FDDk−1. This al-

lows for the iterative application of noise attenuation, missingtrace interpolation, and static correction.

The reinsertion step of POCS is identical to that used during5D Tensor Completion (TCOM) (Stanton et al., 2012). Thisimplies that we can replace the Fourier estimate of the data,F−1

D T FDDk−1 , at a given iteration, k, with a rank-reducedversion of the data R(Dk−1). This formulation has the advan-tage that it can deal with the reconstruction of curved events(Kreimer and Sacchi, 2011). In the case of noise attenuationgiven data that is fully spatially sampled one may wish to de-noise the data while preserving residual statics. The total resid-ual static corrections applied during the denoising are con-tained in τN(mx,my,hx,hy) allowing for them to be removedfrom the data. In the case of interpolation it is preferable toleave the static corrections applied to avoid static shifts be-tween interpolated and original traces. The fact that the algo-rithm produces a tensor of time-shifts τ(mx,my,hx,hy) couldoffer an advantage for other processing steps. Converting thistensor to shot and receiver coordinates, τ(sx,sy,gx,gy), thetime shifts could then be used for further processing or to gaina better understanding of the near surface.

Sparse Radon TransformThe Sparse Radon Transform can be written as the linear op-eration. Data, d can be generated from a model m under theaction of the Radon operator L as follows:

d = Lm+n, (3)

where d is data, m is the Radon model and n is the noise con-tent. Thorson and Claerbout (1985) showed that given the data,equation (3) can be solved via damped least squares approach.In other words, the objective function is

minimize ||m||2 s.t. ||d−Lm||2 < ε (4)

where ε is some estimate of noise level in the data. To increasethe resolution of the Radon model, one can adopt l2 norm fordata misfit and l1 norm for the model

m̂ = argminm

12‖d−Lm‖2 + τ ‖m‖1, (5)

where m is desired sparse model and τ is a regularization pa-rameter that balances the importance of the misfit functional

and regularization term. The cost function of equation (5) iscomplete if data has no statics. As discussed earlier statics in-troduce artifacts and smears the Radon model. Consideringstatics changes equation (5) to

argminm,Q

J = argminm,Q

12‖Qd−Lm‖2 + τ ‖m‖1, (6)

where Q is shifting operator that corrects statics in data. Equa-tion (6) can also be written as

argminm,Q

J = argminm,Q

12‖d−QT Lm‖2 + τ ‖m‖1, (7)

where QT is the adjoint operator of Q and puts statics back into the Radon predicted data. The only advantage in using equa-tion (7) instead of equation (6) is that equation (7) preservesstatics in the predicted data. The cost function of equation (7)can be minimized by alternatively solving the following sub-problems:

m- step m̂= argminm

12‖d−Q̂T Lm‖2+τ ‖m‖1, (8)

Q- step Q̂ = argminQ

12‖d−QT Lm̂‖2. (9)

By considering the initial shifting operator as the identity ma-trix, the m- step can be solved using Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) (Beck 2009). The Q- stephas a closed form solution but it could be considered a fullmatrix and without physical meaning. So, instead of solvingdirectly the Q- step we update the shifting operator by simplycross correlating the d and Lm̂ vectors. Note that this is also avalid solution by defining the Q operator space as a combina-tion of some shifting basis functions.

EXAMPLES

To test the reconstruction algorithm we apply the algorithm toa 5D synthetic dataset with dimension 100x12x12x12x12. Thedata has hyperbolic moveout in all four of the spatial directionsas seen in figure 2. This figure shows one central bin locationout of a total of 144 bins that comprise the complete data. Thecomplete noise free data is shown in figure 2 (a). Before recon-struction random noise was added to the data giving a signal tonoise ratio of 2. Random traces were then decimated from thedata leaving 50% of the original traces. Random static shiftsbetween ±10ms were then applied to the data producing thedata seen in figure 2 (b). Standard 5D POCS reconstructionwas applied to the data resulting in the data shown in figure 2(c). The static shifts cause a very low quality reconstructionthat smears the signal. 5D POCS Reconstruction with staticcompensation gives a much higher quality result as seen in fig-ure 2 (d). Figure 2 shows reconstruction results for a stackedinline of data that has been corrupted with random static shifts(a). The result after simultaneous statics computation and re-construction (c) is of higher quality compared to the result afterstandard reconstruction (b). To test the Radon demultiple al-gorithm we show an NMO corrected CMP gather from a Gulf

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Figure 2: a) A portion of noise-free, static-free, fully sampled 5-D synthetic data. b) Data after adding random noise (SNR =2), random ±10ms static shifts, and randomly removing traces (50%). c) Data after standard 5D reconstruction d) Data aftersimultaneous 5D reconstruction and statics computation.

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Figure 4: Marine CMP gather with random +/- 10ms static shifts, (a) input data, (b) sparse Radon Transform with no staticscomputation, (c) sparse Radon transform with statics computation.

of Mexico towed-streamer dataset in figure 4 (a). The gather iscorrupted with random +/-10ms static shifts. The sparse Radontransform is shown in (b), while the sparse Radon transformwith simultaneous statics computation is shown in (c). Thesparsity of the Radon panel is improved through the use of thenew algorithm. Figure 5 shows the estimated data using theconventional sparse Radon transform (a), compared with us-ing SPSR. Notice the dimming of amplitudes in the estimateproduced using the conventional sparse Radon transform. TheSPSR result preserves the statics on the data, allowing for mul-tiple suppression of statics-contaminated data.

CONCLUSION

We presented methods to process seismic data in the presenceof residual statics. The methods are able to preserve the staticshifts in the case of noise and multiple removal, or to compen-sate for the shifts in the case of simultaneous denoising andtrace interpolation. The methods make use of sparsity or sim-plicity of the static-free seismic data. A topic of future researchis to incorporate surface consistency into the methodology tocharacterize anomalies in the near surface.

ACKNOWLEDGEMENTS

The authors would like to thank the sponsors of the SignalAnalysis and Imaging Group (SAIG) at the University of Al-berta, and the USGS for use of example seismic data.

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Figure 5: Marine CMP gather with random +/- 10ms staticshifts, (a) data estimated using conventional sparse Radontransform, (b) data estimated using sparse Radon transformwith statics computation.

Processing seismic data in the presence of residual statics

REFERENCES

Abma, R., and N. Kabir, 2006, 3D interpolation of irregulardata with a POCS algorithm: Geophysics, 71, E91–E97.

Eriksen, E. A., and D. E. Willen, 1990, Interaction of stackingvelocity errors and residual static corrections: SEG Techni-cal Program Expanded Abstracts, 9, 1730–1733.

Gao, J., and M. Sacchi, 2011, Convergence improvement andnoise attenuation considerations for POCS reconstruction:Presented at the 73rd EAGE conference and exhibition.

Gholami, A., 2013, Residual statics estimation by sparsitymaximization: Geophysics, 78, V11–V19.

Kreimer, N., and M. D. Sacchi, 2011, A tensor higher-ordersingular value decomposition (hosvd) for pre-stack simul-taneous noise-reduction and interpolation: SEG TechnicalProgram Expanded Abstracts, 30, 3069–3074.

Ronen, J., and J. F. Claerbout, 1984, Surface-consistent resid-ual statics estimation by stack optimization: SEG TechnicalProgram Expanded Abstracts, 3, 420–422.

Stanton, A., N. Kreimer, D. Bonar, M. Naghizadeh, and M.Sacchi, 2012, A comparison of 5d reconstruction methods:SEG Technical Program Expanded Abstracts, 1–5.

Stanton, A., and M. D. Sacchi, 2012, 5D reconstruction in thepresence of residual statics: Presented at the CSEG AnnualMeeting.

Traonmilin, Y., and N. Gulunay, 2011, Statics preserving pro-jection filtering: SEG Technical Program Expanded Ab-stracts, 30, 3638–3642.