Regularizing seismic inverse problems by model reparameterization using plane-wave construction

Transcription

1 GEOPHYSICS, VOL. 71, NO. 5 SEPTEMBER-OCTOBER 2006 ; P. A43 A47, 6 FIGS / Regularizing seismic inverse problems by model reparameterization using plane-wave construction Sergey Fomel 1 and Antoine Guitton 2 ABSTRACT We define plane-wave construction PWC, an operator for generating data aligned along predefined locally variable slopes, as the inverse of plane-wave destruction, an operator used for measuring the slopes. PWC can be applied for efficient regularization of seismic estimation problems. Using simple examples, we demonstrate how PWC enhances the coherency of seismic images, improves velocity estimation methods, and separates primaries and multiples with a pattern-based approach. INTRODUCTION Model reparameterization Harlan, 1995 is a form of inverse problem regularization particularly attractive for large-scale, underdetermined estimation problems typical in geophysics Fomel and Claerbout, Reparameterization constrains the estimated model by using a preconditioning operator to force a particular behavior. In seismic application, the desirable behavior is often associated with locally planar structures. To estimate slopes of locally planar events, Fomel 2002 developed a method of plane-wave destruction PWD following earlier ideas of Claerbout PWD predicts each seismic trace from its neighbor by following local plane-wave slopes, then estimates the slopes by subtracting the prediction and minimizing the error. Highorder, accurate destruction filters are used for estimating local slopes of seismic events and can be applied to problems such as fault detection, data regularization, and noise attenuation. Estimating local slopes can also replace traditional velocity analysis and enable velocity-independent time-domain seismic imaging Fomel, 2005b. In this paper, we introduce plane-wave construction PWC, the inverse of the PWD operator. We show that PWC is an efficient regularizer for speeding up iterative optimization that involves models with local plane-wave structure. In that sense, one can view PWC as a higher-order generalization of steering filters, which steer model estimations toward local dips Clapp et al., 1998a, We illustrate PWC applications using existing data sets from the field. The immediate effect of PWC is to smooth the data along dominant event slopes. In application to coherency enhancement in seismic images, iterative least-squares inversion with PWC reparameterization extracts portions of the image aligned with the dominant slopes. Using PWC with iterative reweighting preserves fault geometry during coherency enhancement. When applied to the classic velocity estimation problem, PWC forces consistency between the velocity structure and reflector geometry. In the multiple suppression problem, PWC reparameterization can separate primary and multiple events based on differences in their local slopes. PWC requires sequential seismic traces. It may not be immediately applicable to 3D problems, but transformation from plane-wave construction to plane-wave shaping Fomel, 2005a addresses this problem Fomel and Guitton, PLANE-WAVE CONSTRUCTION DEFINED The PWC operator is the mathematical inverse of the PWD operator. Let us represent a seismic section s as a collection of traces: s = s 1 s 2... s N T. A PWD operator Fomel, 2002 predicts each trace from its neighbor and subtracts the prediction from the original trace. In the linear operator notation, we can write the PWD operator as r = Ds, where r is the destruction residual and D is the destruction operator, defined as 1 Manuscript received by the Editor January 12, 2006; revised manuscript received March 1, 2006; published online September 11, The University of Texas ataustin, JohnA. and Katherine G. Jackson School of Geosciences, Bureau of Economic Geology, University Station, Box X,Austin, Texas sergey.fomel@beg.utexas.edu. 2 3DGeo Development Inc., 4633 Old Ironsides Drive, Suite 401, Santa Clara, California antoine@3dgeo.com Society of Exploration Geophysicists. All rights reserved. A43

2 A44 Fomel and Guitton P 1,2 I 0 0 D = I N P 0 P 2,3 I 0, P N 1,N I where I stands for the identity operator, P is the prediction operator defined by Fomel 2002, and P i,j describes the prediction of trace j from trace i. We predict a trace by shifting the original trace along the dominant event slope. We then estimate the dominant slope by minimizing the prediction error which is the output of D using regularized least-squares optimization. Regularization constrains the estimated slopes to vary smoothly inside the data space. The PWC operator C is the inverse of D: P 1,2 I 0 0 C 0 P 2,3 I P N 1,N I 1 P 1,2 P N 1,N P 2,3 P N 1,N P N 1,N I P 1,2 I 0 0 P 1,2 P 2,3 P 2,3 I 0 I P 1,2 I 0 0 P 1,3 P 2,3 I 0, 3 P 1,N P 2,N P N 1,N I For efficiency, we apply PWC as a recursive triangular inversion. We compute the output of recursively as c = Cs = c 1 c 2... s N T 4 c 1 = s 1, c k = s k + P k 1,k c k 1 k = 2, 3,..., N. The PWC operator c is a smoothing operator along local plane waves. Reparameterization by the PWC operator C provides effective regularization and can help accelerate the convergence of iterative optimization in inverse problems Harlan, 1995; Fomel and Claerbout, When we use PWC reparameterization in leastsquares inversion of the forward modeling operator d = Lm = LCp, PWC leads to the formal inversion mˆ = Cpˆ = CC T LCC T L T + 2 I N 1 d. 7 Here, m is the model, pˆ is the reparameterized model, d is the observed data, is the regularization parameter, and mˆ is the regularized model estimate. The CC T operator functions as the model covariance operator. In large-scale problems, we can compute the inversion in equation 7 by an iterative conjugate-gradient algorithm. Fomel and Claerbout 2003 suggest a different approach to recursive filter reparameterization, based on the helix transformation Claerbout, The total cost of plane-wave construction is proportional to the data size and to the cost of an elementary prediction, P k 1,k, which operates at the speed of tridiagonal inversion for a matrix of the tracelength size Fomel, This operation is comfortably efficient in practical applications. APPLICATIONS Following are three practical applications of PWC reparameterization in solving inverse problems. 5 6 Coherency enhancement The simplest kind of regularized inversion involves I from equation 6 as the identity operator I N. The corresponding application of PWC reparameterization results in a deconvolution that enhances both the continuity and structural consistency of seismic images. A coherency enhancement operator is h = Hs = CWC T CWC T + 2 I N 1 s. 8 Figure 1. a Seismic image after prestack time migration. b Local dips estimated with plane-wave destruction. We add a diagonal weighting operator W to prevent plane-wave smoothing across structural discontinuities, and we define W as a diagonalized magnitude of p from an unweighted inversion in equation 7. When iterated, this method corresponds to iteratively reweighted least squares with a simulated L 1 norm for vector p Trad et al., We apply coherency enhancement to a timemigrated seismic image from a Gulf of Mexico data set Claerbout, 2005, shown in Figure 1. We

3 Plane-wave construction A45 estimate local event slopes Figure 1a to define the PWC operator. Figure 2 shows the output of coherency enhancement using equation 8 and the corresponding noise component removed from the data. Coherency enhancement highlights locally continuous reflectors while preserving the geometry of faults. Hockers and Fehmers 2002 describe a similar effect, which they called Van Gogh filtering for the brush-stroke appearance of the resulting seismic image. Figure 2. a Seismic image after coherency enhancement. b Difference between the enhanced image and the original image. Figure 3. Seismic image from Figure 2 overlaid on the interval velocity model estimated by a Dix inversion reparameterized by planewave construction. Figure 4. a Migration velocity used for prestack time migration. b Migration velocity predicted by regularized Dix inversion. Velocity estimation To demonstrate an application of PWC to the seismic velocity estimation problem, we used the same data set shown in Figure 1. We then chose a simple Dix inversion formula Dix, 1955 applied for interval velocity estimation. Formulating Dix inversion as a regularized estimation problem rendered the forward operator L in equations 6 and 7 into a simple integration Clapp et al., 1998b; Valenciano et al., PWC reparameterization, using the dip field estimated from the seismic image, forces the estimated velocity to follow the geological structure. Figure 3 shows the resultant interval velocity model. Compare this model with Figure 4, which shows the contrast between the input and predicted rms migration velocity. The estimated model not only explains the observed data but also follows a geological structure consistent with the seismic image. Multiple suppression Separation of primary and multiple reflections is one of the most important tasks in seismic data processing. Slope-based prediction, such as PWC reparameterization, can be a useful method for separating multiples from primaries. A characteristic that distinguishes surface-related multiple events is variation among slopes as a result of differing apparent velocities. The advantage of using a slope-based prediction is that, for estimating dominant slopes of multiple events, we can utilize models of the multiples that may have incorrect amplitudes and wavelets as long as they correctly predict event geometry Guitton, Figure 5 shows an example of a multiple-infested common-midpoint CMP gather from the Mobil amplitude-variation-with-offset AVO data set Keys and Foster, 1998, its prediction with the surface-related multiple elimination SRME method Verschuur et al., 1992, and two dominant slopes estimated from the data and corresponding to primary and multiple reflections. Even though SRME does not provide correct amplitudes and wavelets in predicting the multiple events, it can guide the slope estimation method toward extracting the dominant slopes of these events.

4 A46 Fomel and Guitton Using the PWC operator, the model of the data is Nemeth et al., 2000; Guitton, 2002 comparison of velocity semblance scans before and after multiple suppression. Using PWC removes a large portion of the multiple energy from the data. d = C p p + C n n = C p C n p n, 9 CONCLUSIONS where C p is plane-wave construction along primary slopes and C n is plane-wave construction along multiple slopes. In accordance with equation 7, the least-squares estimate of the multiples is mˆ = C n nˆ = C n C n T C p C p T + C n C n T + 2 I N 1 d. 10 Figure 6 shows the estimated primary and multiple events and The PWC operator generates models aligned with predefined locally variable dips. It is the inverse of plane-wave destruction, an operator used for measuring local dips. It is applicable as a model regularizer in seismic estimation problems such as coherency enhancement, velocity estimation, and multiple suppression. As an efficient operator for smoothing data along dominant event slopes, PWC can be incorporated for regularization of any inversion problems that operate with locally planar data. We anticipate more applications of the proposed method that go beyond applications discussed here. ACKNOWLEDGMENTS Sergey Fomel thanks Norsk Hydro for partially supporting this research. REFERENCES Figure 5. a CMP gather from the Mobil AVO data set b multiple model from surfacerelated multiple elimination prediction, c estimated dominant slope of the primary reflections. d Estimated dominant slope of the multiple reflections. Figure 6. a Estimated primary reflections data with multiples removed, b estimated multiple reflections, c velocity scan of the original gather, d velocity scan of the gather after multiple suppression. Claerbout, J. F., 1992, Earth soundings analysis: Processing versus inversion: Blackwell Scientific Publications, Inc., 1998, Multidimensional recursive filters via a helix: Geophysics, 63, , 2005, Basic Earth imaging: Stanford Exploration Project, bei/toc_html/index.html. Clapp, R. G., B. L. Biondi, and J. F. Claerbout, 2004, Incorporating geologic information into reflection tomography: Geophysics, 69, Clapp, R. G., B. L. Biondi, S. B. Fomel, and J. F. Claerbout, 1998a, Regularizing velocity estimation using geologic dip information: 68th Annual International Meeting, SEG, ExpandedAbstracts, Clapp, R. G., P. Sava, and J. F. Claerbout, 1998b, Interval velocity estimation with a null-space, in SEP-97, , Stanford Exploration Project. Dix, C. H., 1955, Seismic velocities from surface measurements: Geophysics, 20, Fomel, S. B., 2002, Applications of plane-wave destruction filters: Geophysics, 67, , 2005a, Shaping regularization in geophysical estimation problems: 75th Annual International Meeting, SEG, ExpandedAbstracts, , 2005b, Velocity-independent time-domain seismic imaging using local event slopes: 75th Annual International Meeting, SEG, Expanded Abstracts, Fomel, S. B., and J. F. Claerbout, 2003, Multidimensional recursive filter preconditioning in geophysical estimation problems: Geophysics, 68, Fomel, S. B., and A. Guitton, 2005, Model preconditioning by plane-wave construction in geophysical estimation problems: 75th Annual International Meeting, SEG, ExpandedAbstracts, Guitton, A., 2002, Coherent noise attenuation using inverse problems and prediction-error filters: First Break, 20, , 2005, Multiple attenuation in complex geology with a pattern-based approach: Geophysics, 70, V97 V107. Harlan, W. S., 1995, Regularization by model reparameterization: Hockers, C., and G. Fehmers, 2002, Fast structural interpretation with structure-oriented filtering: The Leading Edge, 21,

5 Plane-wave construction A47 Keys, R. G., and D. J. Foster, eds., 1998, Comparison of seismic inversion methods on a single real data set: SEG. Nemeth, T., H. Sun, and G. T. Schuster, 2000, Separation of signal and coherent noise by migration filtering: Geophysics, 65, Trad, D., T. Ulrych, and M. Sacchi, 2003, Latest views of the sparse Radon transform: Geophysics, 68, Valenciano, A. A., M. Brown, A. Guitton, and M. D. Sacchi, 2004, Interval velocity estimation using edge-preserving regularization: 74th Annual International Meeting, SEG, ExpandedAbstracts, Verschuur, D. J., A. J. Berkhout, and C. P. A. Wapenaar, 1992, Adaptive surface-related multiple elimination: Geophysics, 57,