Least-squares joint imaging of primaries and pegleg multiples: 2-D field data test

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1 Stanford Exploration Project, Report 113, July 8, 2003, pages Least-squares joint imaging of primaries and pegleg multiples: 2-D field data test Morgan Brown 1 ABSTRACT In this paper I present an improved least-squares joint imaging algorithm for primary reflections and pegleg multiples (LSJIMP) which utilizes improved amplitude modeling and imaging operators presented in two companion papers in this report (Brown, 2003b,a). This algorithm is applied to the entire Mississippi Canyon 2-D multiples test dataset and demonstrates a good capability to separate pegleg multiples from the data. INTRODUCTION Multiple reflections are generally treated as noise in the reflection seismic experiment, and strategies for their removal occupy a prominent place in the exploration seismic literature. Tantalizingly, though, multiples often penetrate deeply into the earth, and thus have the potential to aid the imaging of the prospect zone. Algorithms to structurally image multiples have been in the exploration geophysics literature for over a decade (Reiter et al., 1991; Berkhout and Verschuur, 1994; Yu and Schuster, 2001; Guitton, 2002; Shan, 2003), though none successfully addresses perhaps the most important question: How can any usable information from the multiples be rigorously extracted and exploited? In a previous work (Brown, 2002), I presented a least-squares joint imaging technique which tentatively answers this question. In two companion papers in this report (Brown, 2003b,a), I derived operators which, to reasonable accuracy, model the physical (kinematic and amplitude) connection between pegleg multiples and their corresponding primary events. These operators transform a pegleg multiple into a copy of its primary. My regularized least-squares joint imaging scheme, LSJIMP, (Least-squares Joint Imaging of Multiples and Primaries) then exploits this redundancy to separate the pegleg multiples from the data, and at the same time, spreads additional information provided by the multiples into the primary image. I apply LSJIMP to the Mississippi Canyon 2-D multiples test data, and it demonstrates good noise removal and signal preservation characteristics. LSJIMP s inherent efficiency, combined with the use of Message Passing Interface (MPI) parallelization ensure that its computational performance compares favorably with other advanced multiple suppression techniques. 1 morgan@sep.stanford.edu 17

2 18 Brown SEP 113 LEAST-SQUARES JOINT IMAGING OF MULTIPLES AND PRIMARIES To the uninitiated, LSJIMP may seem at best a perplexing idea; at worst, an egregious display of circular reasoning. Signal/noise separation and imaging have traditionally occupied mutually exclusive domains. Usually multiple suppression is performed as a prerequisite to imaging. While some recent approaches do the multiple suppression after imaging (Sava and Guitton, 2003), the two steps are still largely independent. With LSJIMP, the multiple separation and imaging steps are innately intertwined. Figure 1 motivates the fundamental difference between LSJIMP and conventional multiple suppression and imaging. The NMO for primaries panel is the usual domain of multiple suppression/imaging algorithms. There, the signal is the primary reflections and the noise is the multiples. LSJIMP expands the dimensionality of the problem. In the other two panels, the signal is not the primaries, but instead the flattened multiples. The key observation is that the signal in each panel is consistent with the signal in the other panels. Nearly every existing multiple suppression technique exploits differences between multiples and primaries in the left panel only. LSJIMP does that, but the novelty of the method is that it also exploits similarities between signal across panels to improve the separation. Noise is sometimes signal, and vice versa, depending on which image one views. Why not use all the information at our disposal? Figure 1: Solid lines in each panel are signal ; dashed lines are noise. Left: A CMP gather after NMO. The seabed (WB) produces pegleg multiples WBM and RPL. Reflector R also produces pegleg RM. Center: After NMO for seabed peglegs, WBM and RPL are aligned with WB and R. Right: After NMO for R s peglegs, RM is aligned with R and RPL. morgan3-schem-lsjimp [NR] t x NMO for primaries WB R WBM RPL RM R WBM RPL RM NMO for WB peglegs NMO for R peglegs RPL RM Nemeth Forward Model Nemeth et al. (1999) introduced an under-determined least-squares formulation to jointly image compressional waves and various (non-multiple) coherent noise modes. Guitton et al. (2001) extended this technique to multiple suppression, using a prior noise model and prediction-error filters to model the noise and signal. We can model the recorded data, here a CMP gather, as the superposition of the primary reflections and p orders of pegleg multiples from a single interface (e.g., the seabed). The

3 SEP 113 LSJIMP field data test 19 i th order pegleg has i + 1 legs, or independent arrivals (Brown, 2003a). If we denote the recorded data as d, the primaries as d 0, and the k th leg of the i th order pegleg as d i,k, the model of the data takes the following form. d = d 0 + p i=1 k=0 i d i,k, (1) The main goal of LSJIMP is to use the multiples as a constraint on the primaries. Thus the model space of the LSJIMP process will be a collection of images, one corresponding to the primaries, and one for each leg of each order of pegleg. When the process is finished, the events in each image should have the same timing and AVO signature as the primaries; they should be copies of the primaries. The forward modeling equation is the physical mapping which transforms these copies of the primaries into the multiples recorded in the data. Brown (2003b) derived HEMNO (Heterogeneous Earth Multiple NMO Operator), an operator which independently images each leg of a pegleg (flattens and shifts to the zero-offset traveltime of the primary). Brown (2003a) derived a series of linear operators which modify the amplitude of peglegs to account the effects of the multiple leg of the raypath (reflection and transmission). Taken together, these operators transform a single-cmp image that resembles the primaries, into data that resembles a pegleg multiple. Let us rewrite equation (1): d = N 0 m 0 + p i=1 k=0 i R i N i,k S i G i m i,k (2) m 0 is the primary image, flattened by the adjoint of the NMO operator N 0. m i,k is image of the k th leg of the i th -order pegleg, flattened by N i,k, the adjoint of the HEMNO equation. To all i th -order peglegs, G i applies a differential geometric spreading correction, S i applies Snell resampling, and R i applies the appropriate reflection coefficient. The motivation and implementation of G i, S i, and R i are discussed in detail by Brown (2003a). For clarity, we can rewrite equation (2) in matrix notation: [ N0 (R 1 N 1,0 S 1 G 1 ) (R 1 N 1,1 S 1 G 1 ) (R p N p,0 S p G p ) (R p N p,p S p G p ) ] m 0 m 1,0 m 1,1. m p,0. m p,p and define the data residual as the difference between modeled and recorded data: = L n m, (3) r d = d L n m. (4) Viewed as a standard least-squares inversion problem, where the model is adjusted to minimize the L 2 norm of the data residual, equation (4) is under-determined. The addition of model regularization operators, defined in later sections, forces the problem to be over-determined.

4 20 Brown SEP 113 Consistency of the Data and the Crosstalk Problem Our hope is that, after solving equation (3) for the unknown model, m 0 contains only primaries (and other non-pegleg events) and the m i,k contain only peglegs. Unfortunately, simple least-squares minimization of the data residual (4) proves insufficient to properly segregate the various modes. Nemeth et al. (1999) pins the problem on operator overlap, or coincident operator range. In this application, operator overlap is most troublesome at near offsets, where primaries and multiples are both flat. If (for instance) m 0 contains residual first-order pegleg multiple energy, equation (1) will map this energy back into data space, at the position of a first-order multiple at near offsets. The residual multiple energy is called crosstalk (Claerbout, 1992). Luckily, Nemeth et al. show that that properly designed model regularization operators can at least partially mitigate the crosstalk problem. REGULARIZATION OF THE LEAST-SQUARES PROBLEM In this section, three discriminants between crosstalk and signal are exploited to devise model regularization operators which guide the final model toward the one with the peglegs best separated from the data. 1. Inconsistency with multiple order - After imaging, corresponding crosstalk events on two model panels have different residual moveout, e.g. residual first-order multiples on m 0 and residual second-order multiples on m 1,k. The moveout difference is negligible at near offsets, but larger than the Fresnel Zone (half a wavelength) at long offsets. Conversely, actual signal events are flat on all the m i,k. Conclusion: For fixed (t, x), the difference between one model panel and another will be relatively small where there is signal, but large where there is crosstalk noise, especially at far offsets. 2. Curvature with offset - After imaging, signal events are flat, while crosstalk events have at least some residual curvature, especially at far offsets and in regions with a strong velocity gradient. Conclusion: Provided that the AVO response of the signal changes slowly with offset, the difference (in x) between adjacent samples of any m i,k will be relatively small where there is signal, but large where there is crosstalk noise. 3. Predictability of pre-seabed multiple events - The third discriminant exploits the inherent predictability of crosstalk to suppress it. Between the seabed reflection and the onset of its first multiple, the recorded data contains only primaries (inter-bed multiples and locally-converted shear waves are generally weak); these strong events spawn the (usually) most troublesome crosstalk events. Fortunately, the pre-seabed-mutiple primaries can be used to directly construct a prior model of the crosstalk noise, valid even at near offsets. Conclusion: Given an accurate kinematic model of crosstalk noise, a corresponding set of model-space weights used in a model regularization term penalizes crosstalk events but not signal.

5 SEP 113 LSJIMP field data test 21 Model Regularization 1: Differencing across multiple order We can write the model residual corresponding to the model regularization operator which differences between adjacent m i at a fixed (t, x). r [1] m (τ, x, j) = m j (τ, x) m j+1 (τ, x) where j = [0, p(p + 3)/2]. (5) p is the maximum order of multiple included in equation (2). Here we have modified the notation a bit and written m j rather than m i,k. This is because the difference (5) is blind to the order or leg of the pegleg corresponding to m j ; it is simply a straight difference across all the model panels. By design, signal (non-crosstalk) events on the m i,k in equation (3) are assumed physically invariant for all i and k everything is a copy of the primary. Again, this is the crucial fact underlying LSJIMP. Minimally, multiples provide a redundant constraint on the amplitude of the primaries; where no data is recorded (missing traces, near offsets), they provide additional information about the primaries. Equation (5) is a systematic way in which to exploit the multiples redundancy, and to integrate any additional information that they might provide. Model Regularization 2: Differencing across offset We can similarly write the model residual corresponding to the model regularization operator which applies a difference with offset to each m i,k at fixed t. r [2] m (τ, x,i) = m i,k(τ, x) m i,k (τ, x + x). (6) A similar approach is used by Prucha and Biondi (2002) to regularize prestack depth migration in the angle domain. Model Regularization 3: Crosstalk-boosting weighting NMO (for primaries or multiples) flattens events of a single order, but leaves events of other orders (crosstalk) remaining with residual curvature. To compute (for instance) the crosstalk model for first-order pegleg multiples on the primary model panel, m 0, the following steps are followed. Apply NMO for primaries to data: c 0 = N T 0 d. Zero c 0 in the range τ τ 2τ, where τ (y) is the zero-offset traveltime to the multiple-generating layer. Simulate the kinematics of the data s total first-order pegleg by applying inverse HEMNO: c 1 = (N 0,1 + N 1,1 )c 0. Apply NMO for primaries to simulate the kinematics of the first-order pegleg crosstalk event in the primary model panel, m 0 : c = N T 0 c 1.

6 22 Brown SEP 113 Normalize c to [0,1] and clip if desired. Figure 2 illustrates the application of the crosstalk weight for first and second-order peglegs on m 0 applied to a NMOed synthetic CMP gather. Notice how the multiples are picked cleanly out of the data, while strong primaries are left largely intact. Denoting the crosstalk Figure 2: Synthetic CMP gather with and without crosstalk weights for k = 0 and j = 1,2,3 applied. morgan3-crosstalk.hask [CR] weights for each m i,k as a vector w i,k, we can write the model residual corresponding to the third model regularization operator: r [3] m (τ, x,i,k) = w i,k(τ, x) m i,k (τ, x). (7) Although the crosstalk weights will likely overlap some primaries, the primaries flatness ensures that regularization operators (5) and (6) spread redundant information about the primaries from other m i,k and other offsets to compensate for any losses. Combined Data and Model Residuals To solve equation (3) for the optimal set of m i,k, we minimize a quadratic objective function, Q(m), which consists of the sum of the weighted l 2 norms of a data residual [equation (4)] and of three model residuals [equations (5), (6), and (7)]. minq(m) = r d 2 + ɛ 2 1 r[1] m 2 + ɛ 2 2 r[2] m 2 + ɛ 2 3 r[3] m 2 (8) ɛ 1,ɛ 2, and ɛ 3 are scalars which balance the relative weight of the three model residuals with the data residual. In practice I suggest setting ɛ 1 = 2.0, ɛ 2 = 1.0, and ɛ 3 = 2.0. I use the conjugate gradient method to minimize Q(m); the method is well-suited for large-scale least-squares optimization problems like this one. EXTENSION TO NON-SEABED PEGLEGS Reflectors other than the seabed also produce pegleg multiples, and the strength of these events sometimes rivals, and even surpasses, the strength of the seabed peglegs. The Mississippi

7 SEP 113 LSJIMP field data test 23 Canyon dataset, with its shallow tabular salt body, falls into this category. Figure 3 shows the stack of this data, with important reflectors and multiples labeled. Computationally, the extension of this joint imaging method to include non-seabed peglegs is straightforward. We need only add an index for the multiple generating layer (m) to equation (2), where we model n sur f multiple-generating layers in the data. d = N 0 m 0 + p i n sur f i=1 k=0 m=1 R i,m N i,k,m S i,m G i,m m i,k,m (9) Physically, however, modeling non-seabed peglegs requires some additional thinking. Notice that all modeling operators in equation (9) have an extra index. Brown (2003a) discusses how to extend the HEMNO operator (N i,k,m ) to non-seabed peglegs, while Brown (2003b) discusses how to extend R i,m, S i,m, and G i,m. The regularization operators all remain the same, though non-seabed peglegs add additional crosstalk energy that must be modeled when the crosstalk weights are generated. RESULTS In 1997, WesternGeco distributed a 2-D test dataset, acquired in the Mississippi Canyon region of the Gulf of Mexico, for the testing of multiple suppression algorithms. The data contain a variety of strong surface-related multiples, and enough geologic complexity to render onedimensional methods ineffectual. Figure 3 shows the stack of the raw data. The geologic setting of the area is a fairly well-behaved deepwater sedimentary basin, interrupted between 4000 and 16000m by a shallow tabular salt body. In addition to the seabed peglegs, peglegs from the top of salt and strong reflector R1 are included in the inversion. Rather than including each of these surfaces separately, the R1 and top salt events are assumed to arise from a single reflector; from midpoint 0 m to roughly 6000 m, it is R1, while from 6000 m to m it is the top of salt. Only first order multiples are included in the inversion. Thus in equation (9), n sur f = 2 and p = 1. I tested LSJIMP on 750 CMP gathers of the Mississippi Canyon dataset. Figure 4 illustrates the stack of the LSJIMP result. From the difference panel, note that important peglegs (TSPL and WBM) are almost entirely removed. Primaries (like PR) are not damaged. R1PM, and to a lesser extent TSPM, are removed effectively. Salt rugosity contributes negatively to the separation, by forming diffractions which are not modeled by HEMNO, and by violating HEMNO s small reflector dip assumption. Much deep multiple energy remains. Some of it is likely internal salt multiples. Additionally, time imaging operators (like HEMNO) are notoriously poor at imaging subsalt events. Figures 5 and 6 show the LSJIMP results at two midpoint locations (0 m and m, respectively). From the top rows of these figures, notice that most important peglegs are separated from m 0, while the primaries are preserved. The estimated seabed and R1/top of salt peglegs seem to match the events in the data, both in term of kinematics and amplitudes.

8 24 Brown SEP 113 Figure 3: Stacked Mississippi Canyon 2-D dataset (750 midpoints), annotated with important horizons and multiples. Labeled events: R1 - strong reflection; TSR - top of salt; BSR - bottom of salt; WBM - first seabed multiple; R1PL - seabed pegleg of R1; R1PM - R1 pure surface multiple; TSPL - seabed pegleg of TSR; BSPL - seabed pegleg of BSR; TSPM - TSR pure surface multiple. morgan3-gulf.stackraw [CR] Figures 7 and 8 illustrate the superior performance of the HEMNO operator versus 1-D NMO when used in LSJIMP. The strong top of salt seabed pegleg in Figure 7, dipping over this midpoint range, is clearly better removed when HEMNO is used. Figure 8 illustrates a more insidious problem: because the 1-D NMO operator did not focus a pegleg at the correct time, it caused a spurious event to be manufactured at a slightly smaller time. CONCLUSIONS I introduced many refinements to my earlier least-squares joint imaging for multiples and primaries scheme (Brown, 2002), and denoted it LSJIMP for short. Two companion papers in this report (Brown, 2003b,a) derived operators which transform pegleg multiples into copies

9 SEP 113 LSJIMP field data test 25 Figure 4: Stack comparison of Mississippi Canyon data before and after LSJIMP. All panels windowed in time from 3.5 to 5.5 seconds. Top: Raw data stack. Center: Stack of estimated primary image, m 0. Bottom: Stack of the subtracted multiples. Labeled events: PR - underlying primary; WBM - first seabed multiple; R1PL - seabed pegleg of reflector R1; R1PM - R1 pure surface multiple; TSPL - seabed pegleg of TSR (top of salt); BSPL - seabed pegleg of BSR (bottom of salt); TSPM - TSR pure surface multiple. BSTSPL - TSR pegleg of BSR. morgan3-stackcomp.gulf [CR,M]

10 26 Brown SEP 113 Figure 5: Mississippi Canyon CMP 1 (y = 0 m). All panels NMO ed with stacking velocity and windowed in time from 3.5 to 5.5 seconds. Top row (L to R): Raw data; Estimated primaries (m 0 ); Estimated non-primaries (difference). Center row (L to R): Raw data; Estimated total first order seabed multiple ( 1 k=0 R 1,1N 1,k,1 S 1,1 G 1,1 m 1,k,1 ); difference. Bottom row (L to R): Raw data; Estimated total first order salt (R1 at this location) multiple ( 1 k=0 R 1,2N 1,k,2 S 1,2 G 1,2 m 1,k,2 ); difference. morgan3-comp1.lsrow.gulf.0 [CR,M]

11 SEP 113 LSJIMP field data test 27 Figure 6: Mississippi Canyon CMP 540 (y = 14400m). All panels NMO ed with stacking velocity and windowed in time from 3.5 to 5.5 seconds. Top row (L to R): Raw data; Estimated primaries (m 0 ); Estimated non-primaries (difference). Center row (L to R): Raw data; Estimated total first order seabed multiple ( 1 k=0 R 1,1N 1,k,1 S 1,1 G 1,1 m 1,k,1 ); difference. Bottom row (L to R): Raw data; Estimated total first order salt multiple ( 1 k=0 R 1,2N 1,k,2 S 1,2 G 1,2 m 1,k,2 ); difference. morgan3-comp1.lsrow.gulf.540 [CR]

12 28 Brown SEP 113 Figure 7: LSJIMP stack comparison of HEMNO versus 1-D NMO operator. All panels windowed from 4.4 to 4.8 seconds in time; to meters in midpoint. From L to R: Raw data stack; Stack of estimated m 0 using HEMNO; Stack of estimated m 0 using 1-D NMO operator; HEMNO difference; 1-D NMO difference. Seabed pegleg from top of salt reflection is outlined in all panels. morgan3-stackcomp-dipcomp.1.gulf [CR] Figure 8: LSJIMP stack comparison of HEMNO versus 1-D NMO operator. All panels windowed from 4.8 to 5.2 seconds in time; 9200 to meters in midpoint. From L to R: Raw data stack; Stack of estimated m 0 using HEMNO; Stack of estimated m 0 using 1-D NMO operator; HEMNO difference; 1-D NMO difference. The ovals highlight a nonexistent event removed from data due to 1-D NMO s inferior performance over nonflat structure. morgan3-stackcomp-dipcomp.2.gulf [CR] of their primary. These operators enable LSJIMP to separate peglegs and primaries by exploiting the mutual consistency of events after imaging. LSJIMP produces good multiple separation results on the 2-D Mississippi Canyon multiples test dataset. Encouragingly, this approach looks well-suited both in terms of physics and computation to be applied to 3-D data under the industry s current acquisition constraints (poor crossline shot coverage), which inhibit the multiple prediction ability of methods like Verschuur et al. (1992), which work well in 2-D. ACKNOWLEDGEMENT WesternGeco acquired and released the Mississippi Canyon dataset for public use. REFERENCES Berkhout, A. J., and Verschuur, D. J., 1994, Multiple technology: Part 2, migration of multiple reflections: 64th Ann. Internat. Mtg, Soc. Expl. Geophys., Expanded Abstracts,

13 SEP 113 LSJIMP field data test 29 Brown, M., 2002, Least-squares joint imaging of primaries and multiples: 72nd Ann. Internat. Mtg., Soc. of Expl. Geophys., Expanded Abstracts, Brown, M., 2003a, Amplitude modeling of pegleg multiple reflections: SEP 113, Brown, M., 2003b, Prestack time imaging operator for 2-D and 3-D pegleg multiples over nonflat geology: SEP 113, Claerbout, J. F., 1992, Earth Soundings Analysis: Processing Versus Inversion: Blackwell Scientific Publications. Guitton, A., Brown, M., Rickett, J., and Clapp, R., 2001, Multiple attenuation using a t-x pattern-based subtraction method: 71st Ann. Internat. Mtg, Soc. Expl. Geophys., Expanded Abstracts, Guitton, A., 2002, Shot-profile migration of multiple reflections: 72nd Ann. Internat. Mtg., Soc. of Expl. Geophys., Expanded Abstracts, Nemeth, T., Wu, C., and Schuster, G. T., 1999, Least-squares migration of incomplete reflection data: Geophysics, 64, no. 1, Prucha, M. L., and Biondi, B. L., 2002, Subsalt event regularization with steering filters: 72nd Ann. Internat. Mtg., Soc. of Expl. Geophys., Expanded Abstracts, Reiter, E. C., Toksoz, M. N., Keho, T. H., and Purdy, G. M., 1991, Imaging with deep-water multiples: Geophysics, 56, no. 07, Sava, P., and Guitton, A., 2003, Multiple attenuation in the image space: SEP 113, Shan, G., 2003, Source-receiver migration of multiple reflections: SEP 113, Verschuur, D. J., Berkhout, A. J., and Wapenaar, C. P. A., 1992, Adaptive surface-related multiple elimination: Geophysics, 57, no. 09, Yu, J., and Schuster, G., 2001, Crosscorrelogram migration of IVSPWD data: 71st Ann. Internat. Mtg, Soc. Expl. Geophys., Expanded Abstracts,