Stanford Exploration Project, Report 113, July 8, 2003, pages 97–106
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Stanford Exploration Project, Report 113, July 8, 2003, pages 97–106
Short Note Amplitude modeling of pegleg multiple reflections Morgan Brown1
INTRODUCTION My Least-Squares Joint Imaging of Multiples and Primaries (LSJIMP) algorithm (Brown, 2003a) separates pegleg multiples and primaries. LSJIMP computes separate images of the peglegs and primaries, and then uses the mutual consistency of the images to discriminate against unwanted noise types in each image. The images must be consistent in two respects. Kinematically, the events must be correctly positioned in time and flat with offset. This is accomplished by an improved normal moveout operator (HEMNO) introduced in a companion paper in this report (Brown, 2003b). This paper addresses the second aspect of image consistency: amplitudes. LSJIMP requires that the amplitudes of the pegleg images be correct not in absolute terms, but relative to the primary image. Correct modeling of the relative amplitudes of pegleg multiples requires more work than a scaling by a reflection coefficient. For one, the reflection coefficient of the multiple-generating layer generally varies with position. Multiples also have longer raypaths through the earth, and thus suffer greater geometric spreading and anelastic attenuation losses than do primaries with the same offset. Furthermore, multiples and primaries have different reflection angles for a fixed offset, so the multiples and primaries will exhibit different amplitude-versus-offset (AVO) behavior. In this paper, I present three innovations which transform pegleg multiples into events which are directly comparable to their corresponding primary reflections.
SNELL RESAMPLING REMOVES AVO/ATTENUATION DIFFERENCES If we are modeling seabed pegleg multiples, Figure 1 illustrates the fact that (ignoring the seabed reflection) in a v(z) medium, there exists a single offset x p such that a pegleg with offset x and primary with offset x p are physically invariant with respect to AVO behavior and anelastic attenuation (water is assumed perfectly elastic). Ottolini (1982) introduced the concept of “Snell Traces” – a resampling of multi-offset reflection data along curves of constant time dip, or “stepout”. I adopt a similar line of reasoning to infer x p as a function of x. Since 1 email:
[email protected]
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x xp Figure 1: A primary and pegleg multiple with the same emergence angle (θ) and midpoint (y). Note different offsets (x and x p ) and a shift (1y) in reflection point. morgan2-schem-snell [NR]
y
θ θ
τ∗
τ
∆y
the pegleg multiple and primary in Figure 1 have the same emergence angle, θ, the stepout of the two events is the same at x and x p . This fact is the basis of the derivation in Appendix A, which obtains the following result for x p as a function of x. x 2τ 2 V 4 , where = (τ + jτ ∗ )2 Ve4f f + x 2 (Ve2f f − V 2 ) jτ ∗ v ∗ (τ ∗ ) + τ V (τ ) Ve f f (τ ) = jτ ∗ + τ x p2
(1) (2)
jτ ∗ is the two-way traveltime of a j th -order pegleg multiple in the top layer. Equation (1) defines a time-variable compression of the offset axis. In constant velocity, Ve f f = V , and equation (1) reduces to the radial trace resampling used by Taner (1980) for long-period deconvolution of peglegs. Figure 2 demonstrates the Snell resampling on the first- and second-order pegleg multiples of a synthetic dataset. Graphically (Figure 1), we infer that the shift in midpoint, 1y, of the reflection points of the primary and pegleg is (3) 1y = x − x p /2.
As a function of time, 1y decreases asymptotically to zero from a maximum of x/4 at the seabed. The deeper the reflector, the smaller 1y becomes.
Oftentimes, non-seabed pegleg multiples (e.g. top of salt) are strong enough to merit modeling. In a v(z) earth, the results derived in this section are equally valid, with one exception: attenuation. In this case, the effects of attenuation are encoded – in a possibly non-linear way – in the effective reflection coefficient that we estimate in a subsequent section.
DIFFERENTIAL GEOMETRICAL SPREADING FOR PEGLEG MULTIPLES The Snell resampling transformation derived in the previous section renders pegleg multiples and primaries invariant with respect to AVO and attenuation. However, because of their longer raypaths, multiples suffer greater geometric spreading losses than the corresponding primary. Following previous authors (Ursin, 1990; Lu et al., 1999), I write offset-dependent geometric
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Figure 2: Snell resampling demonstration. A synthetic dataset with flattened primaries and resampled first- and second-order peglegs. The transformation warps the vertical black lines horizontally, according to equation (1). Notice that the raw data has five unrecorded near offset traces and two dead traces at medium offsets. Snell resampling spreads information from the multiples into these no-data zones. The multiples provide a direct, additional constraint on the amplitude of the primaries where no data is recorded. morgan2-snell.hask [CR,M]
spreading corrections for a primary (g prim ) and its pegleg multiples (gmult ), respectively:
∗
s
∗
s
g prim = v t prim (x) = gmult = v tmult (x) =
(τ v ∗)2 +
[(τ +
xv ∗ V
2
jτ ∗ )v ∗ ]2 +
(4) xv ∗ Ve f f
2
.
(5)
Ve f f is defined in equation (2). After scaling by the ratio gmult /g prim and Snell resampling, a pegleg multiple should be an exact copy of its associated primary, to within a scaling by the appropriate reflection coefficient.
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ESTIMATION OF SEABED REFLECTION COEFFICIENT The reflection coefficient of the multiple-generating layer can be estimated directly, by comparing the amplitude of the primary reflection with its first multiple, after the latter has undergone Snell resampling (normalized to account for the amplitude increase due to compression of the offset axis) and differential geometric spreading correction. The following procedure is utilized to obtain a smooth estimate the reflection coefficient as a function of midpoint. • Pick zero-offset traveltime to reflector for all midpoints: τ ∗ (y). • Loop over midpoints (y 0 ): – Apply NMO to data. Extract small time window around τ ∗ (y 0 ). Output is p0 (t, x, y 0). – Apply NMO for first-order multiple (Brown, 2003b), normalized Snell resampling, and differential geometrical spreading to data. Extract time small window around 2τ ∗ (y 0 ). Output is m(t, x, y 0). – Optional: Align p0 (t, x, y 0) with m(t, x, y 0) on a trace-by-trace basis using (for example) the crosscorrelation technique of Rickett and Lumley (2001). Output is p(t, x, y 0). – Optional: Compute residual weight which reflects “quality” of the data at this midpoint. Output is w(y 0 ). • Minimize the following quadratic functional for unknown seabed reflection coefficient vector r(y): # " ny ny nx nt X X XX (r(k) − r(k − 1))2 . (6) w(k)2 (r(k) p(i , j, k) − m(i , j, k))2 + 2 k=1 j =1 i=1
k=2
Minimization of equation (6) is the equivalent to solving a regularized least-squares problem. To minimize the first term, we adjust r(y) to force the scaled primary to match the multiple. To minimize the second (regularization) term, we force r(y) to vary slowly across midpoint. The scalar term balances data fitting with model smoothness.
Figure 3 shows the stack of the raw Mississippi Canyon 2-D test dataset, acquired and distributed by WesternGeco. The LSJIMP method is tested on this data by Brown (2003a). 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 20000 m it is the top of salt. Figure 4 illustrates the reflection coefficient estimation procedure applied to 750 midpoints of the Mississippi Canyon data. The geology is quite complex in some areas, and the data looks decidedly incoherent. For this reason, the residual weight w(y) is quite important. I eyeballed Figure 4 and heuristically picked the residual weights shown therein. The seabed
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reflection coefficient is nearly constant (-0.12) across the entire spread. The rugosity of the top of salt severely limited the sections of data considered “good”, but since salt is generally somewhat homogeneous in material properties, we can confidently assume some degree of spatial similarity.
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. morgan2-gulf.stackraw [CR]
APPLYING THE SEABED REFLECTION COEFFICIENT IN PRACTICE Figure 5 illustrates that a first-order pegleg multiple consists of two unique arrivals, each with the same traveltime in a v(z) medium. If the material properties of the water bottom and target reflector do not vary across midpoint, then the two arrivals also have the same strength. In this
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Figure 4: a) Stack of time window around seabed reflection. b) Stack of time window around first seabed multiple. c) Seabed residual weight (either 0 or 1). d) Estimated seabed reflection coefficient. e) Stack of time window around top of salt reflection. f) Stack of time window around first top of salt multiple. g) Top of salt residual weight (either 0 or 1). h) Estimated top of salt reflection coefficient. morgan2-rc.gulf [CR]
idealized case, the expression for total pegleg multiple amplitude derived by Backus (1959), rtot = r p · ( j + 1)r j ,
(7)
holds, where j is the order of the multiple, r is the seabed reflection coefficient, and r p is the target reflector reflection coefficient. In real data, sedimentary bedding layers always exhibit at least some “texture”, or variation of reflection strength with midpoint (Claerbout, 1985). If r(y) and r p (y) are space-variable reflection coefficients of the seabed and subsea reflector, respectively, then the total strength of the first-order pegleg is rtot = r(ym,1 ) · r p (y p,1 ) + r(ym,2 ) · r p (y p,2 ).
(8)
In practice, a tractable compromise between the simplistic Backus model of pegleg amplitude [equation (7)] and the complicated model of equation (8) is justified. Let us assume that the
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total reflection strength of a first-order pegleg can be modeled as rtot = r(ym,1 ) + r(ym,2 ) r p (y).
(9)
In other words, we ignore variations in reflection strength of the target reflector, but not the seabed. Notice from Figure 5 that the two pegleg “splits” impinge upon the seafloor at y p,1 and y p,2 , and that the primary reflection occurs at y. Therefore, if r p varies linearly in the neighborhood of y, then the average of r p (y p,1 ) and r p (y p,2 ) is r p (y), and we can safely ignore the variation in reflection strength of the target. Equation (3) shows that 1y p is always less or equal to than |ym,1 − ym,2 |, making local linearity in reflection strength more likely.
Figure 5: In a 1-D earth, both pegleg multiple events shown here arrive at the same time. At fixed offset, the multiple legs of the two events impinge on the multiple-generating layer at ym,1 and ym,2 , and on the target reflector at y p,1 and y p,2 . 1y p , which we ignore in equation (9), goes asmptotically to zero as τ increases. morgan2-schem-pegleg [NR]
ym,1
ym,2 y
τ∗
τ
yp,2
yp,1
∆ yp
EXTENSION TO 3-D All operators derived in this paper extend fairly readily to 3-D. In equation (1), x p becomes a vector quantity ( xEp = [x p,1 , x p,2 ]T ), though the derivation is the same. In equations (4) and q (5) for the differential geometric spreading correction, x becomes x12 + x22 , and τ ∗ becomes a function of two midpoint axes. The reflection coefficient estimation scheme extends easily to 3-D, provided that HEMNO correctly aligns the primary and multiple.
CONCLUSIONS In this short note I derived three linear operators that transform a pegleg multiple into an event that is directly comparable to its primary. Snell Resampling normalizes multiples to primaries with respect to AVO response and attenuation. Differential geometric spreading accounts for the longer raypaths of multiples through the earth. Finally, I introduce a flow for computing and applying the reflection coefficient of the multiple-generating layer. These operators extend naturally to three dimensions.
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ACKNOWLEDGMENT WesternGeco acquired and released the Mississippi Canyon dataset in 1997.
APPENDIX A Derivation of Snell Resampling Operator In the following appendix, I derive the Snell resampling operation, equation (1). The graphical basis for the derivation is Figure 1. Since the pegleg multiple and primary in the figure have the same emergence angle, θ, the stepout of the two events is the same at x and x p . First we compute the stepout of the primary event (standard NMO equation): t p2 = τ +
x p2
V2 dt 2x p d p = 2 t p2 = 2t p d xp d xp V xp dt p = . d xp tp V 2
(A-1) (A-2) (A-3)
Brown (2002) derived an extension to the conventional NMO equation which flattens peglegs to the zero-offset traveltime of the reflector of interest. tm =
s
(τ + jτ ∗ )2 +
x2 Ve2f f
(A-4)
Using equations (A-4) and (2), we can similarly compute the stepout of the corresponding j th -order pegleg multiple: dtm x = . (A-5) dx tm Ve2f f Finally, we compute x p as a function of x. We square equations (A-3) and (A-5), set them equal, then substitute equation (A-4) for tm and t p . x p2
x2 . t p2 V 4 tm2 Ve4f f x 2 V 4 τ 2 + x p2 V 2 = x p2 Ve4f f (τ + jτ ∗ )2 + x 2 Ve2f f =
x p2 =
x 2τ 2 V 4 (τ + jτ ∗ )2 Ve4f f + x 2 (Ve2f f − V 2 )
(A-6) (A-7) (A-8)
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REFERENCES Backus, M. M., 1959, Water reverberations their nature and elimination: Geophysics, 24, no. 02, 233–261. Brown, M., 2002, Least-squares joint imaging of primaries and multiples: 72nd Ann. Internat. Mtg., Soc. of Expl. Geophys., Expanded Abstracts, 890–893. Brown, M., 2003a, Least-squares joint imaging of primaries and pegleg multiples: 2-D field data test: SEP–113, 17–30. Brown, M., 2003b, Prestack time imaging operator for 2-D and 3-D pegleg multiples over nonflat geology: SEP–113, 85–96. Claerbout, J. F., 1985, Imaging the Earth’s Interior: Blackwell Scientific Publications. Lu, G., Ursin, B., and Lutro, J., 1999, Model-based removal of water-layer multiple reflections: Geophysics, 64, no. 6, 1816–1827. Ottolini, R., 1982, Migration of reflection seismic data in angle-midpoint coordinates: Ph.D. thesis, Stanford University. Rickett, J., and Lumley, D. E., 2001, Cross-equalization data processing for time-lapse seismic reservoir monitoring: A case study from the Gulf of Mexico: Geophysics, 66, no. 4, 1015– 1025. Taner, M. T., 1980, Long-period sea-floor multiples and their suppression: Geophys. Prosp., 28, no. 01, 30–48. Ursin, B., 1990, Offset-dependent geometrical spreading in a layered medium (short note): Geophysics, 55, no. 04, 492–496.
