CRS stack and redatuming for rugged surface topography: a synthetic data example Yonghai Zhang * , Ru-Shan Wu, Institute of Geophysics and Planetary Physics, University of California, Santa Cruz, CA95064 Summary Imaging complex structures in the earth is a challenging task in seismic imaging. In addition to the complexities in the earth, very often the seismic data are acquired along a rugged surface topography. This makes the conventional imaging techniques unusable without any modification. In contrast to the conventional imaging methods, which require a velocity model, the Common-Reflection-Surface (CRS) stack is a model-independent, data-oriented imag- ing technique. The handling of the surface topography in the CRS stack is a natural extension of the newly estab- lished CRS stacking method. The extended CRS stacking technique is powerful in imaging complex structures in the presence of a rugged surface topography. In this work we apply the extended CRS stack to a synthetic data set. The data set was created at Amoco in 1997. Introduction From its introduction in the 1990s, the CRS stack (Mann et al., 1999) has been shown to be an efficient alter- native to the conventional stacking methods, such as NMO/DMO. The CRS stack is based on the fundamen- tal seismic ray theory. This provides an easy-to-handle formalism. An important feature of the CRS stack is its independence from a velocity model, which saves a lot of effort in velocity model building and model updating. The CRS stacking method is preferable to conventional stacking methods because it provides improved imaging quality, an enhanced signal to noise ratio, and a better continuity of the reflection events. Moreover, the wave field attributes determined through the CRS stack may be useful for a large variety of applications. These include the inversion of the attributes for a macro-velocity model (Duveneck and Hubral, 2002) and the residual static cor- rections by means of the CRS attributes (Koglin and Ewig, 2003). The presence of the surface topography prevents the stack section from being easily interpreted and being further applied, such as in depth migration. An algorithm to redatum the stack section to an a priori chosen datum level is thus required. The wave field attributes deter- mined through the CRS stack find their simple utility in the redatuming of the stack section. The resulting section after the redatuming is a section with a perfectly plane acquisition level (Zhang et al., 2004; Zhang, 2003). In this paper we present the results of the application of the CRS stack to a synthetic data set. The data set was created at Amoco in 1997. Together with the sim- ulated zero-offset (ZO) section, we show the redatumed CRS stack section. Methodology The implementation of the 2D CRS stack for a rugged topography is based on a three-parameter moveout formula for the paraxial rays in the vicinity of a ZO normal ray. Both the receivers and the shots of the reflections are located on an irregular surface topography (Zhang and Hubral, 2002; Zhang et al., 2004). The three parameters are the emergence angle of the ZO normal ray, and the wavefront curvatures of each of two up-coming hypothetical waves. This algorithm is an extension of the newly established CRS stack over a plane surface. The implementation of the CRS stack makes use of the semblance for the determination of the wave field attributes, which is also referred to as coherence analysis. The coherence is a measure of the relative similarities between the sample amplitudes. In the data volume with the dimensions midpoint, half-offset, and traveltime, each combination of the three parameters defines a surface in the vicinity of the point pertaining to a ZO normal ray. This is the blue surface in Figure 1. We also display the actual reflection in this data volume with the red surface. Varying the wave field attributes in the moveout formula changes the contour of the blue surface. If we assume that the blue surface which best matches the actual reflection yields the highest coherence, the wave field attributes can then be determined to be the triplet that yields the highest coherence. The blue surface in the data volume defined by the triplet is taken to be the stacking operator. The CRS stack delivers the ZO CRS stack section to- gether with the associated coherence section. This co- herence section helps evaluate the imaging quality of the CRS stack and the accuracy of the wave field attributes. The CRS attributes carry all the kinematic information of the data and can be used in further analysis, such as the redatuming of the stack section to a plane surface, the determination of the velocity model, and the residual correction analysis. The redatuming makes use of the wave field attributes determined through the CRS stack and does not require a large additional effort. Synthetic data example The CRS stack was applied to a synthetic data set. The data set was created at Amoco in 1997 for testing the static corrections. In Figure 2 we show the model used for creating the data. Please note the variation in the
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CRS stack and redatuming for rugged surface topography: a synthetic data example
Yonghai Zhang ∗, Ru-Shan Wu, Institute of Geophysics and Planetary Physics, University of California,Santa Cruz, CA95064
Summary
Imaging complex structures in the earth is a challengingtask in seismic imaging. In addition to the complexitiesin the earth, very often the seismic data are acquiredalong a rugged surface topography. This makes theconventional imaging techniques unusable without anymodification.
In contrast to the conventional imaging methods, whichrequire a velocity model, the Common-Reflection-Surface(CRS) stack is a model-independent, data-oriented imag-ing technique. The handling of the surface topography inthe CRS stack is a natural extension of the newly estab-lished CRS stacking method. The extended CRS stackingtechnique is powerful in imaging complex structures in thepresence of a rugged surface topography. In this work weapply the extended CRS stack to a synthetic data set.The data set was created at Amoco in 1997.
Introduction
From its introduction in the 1990s, the CRS stack (Mannet al., 1999) has been shown to be an efficient alter-native to the conventional stacking methods, such asNMO/DMO. The CRS stack is based on the fundamen-tal seismic ray theory. This provides an easy-to-handleformalism. An important feature of the CRS stack is itsindependence from a velocity model, which saves a lot ofeffort in velocity model building and model updating.
The CRS stacking method is preferable to conventionalstacking methods because it provides improved imagingquality, an enhanced signal to noise ratio, and a bettercontinuity of the reflection events. Moreover, the wavefield attributes determined through the CRS stack maybe useful for a large variety of applications. These includethe inversion of the attributes for a macro-velocity model(Duveneck and Hubral, 2002) and the residual static cor-rections by means of the CRS attributes (Koglin andEwig, 2003).
The presence of the surface topography prevents the stacksection from being easily interpreted and being furtherapplied, such as in depth migration. An algorithm toredatum the stack section to an a priori chosen datumlevel is thus required. The wave field attributes deter-mined through the CRS stack find their simple utility inthe redatuming of the stack section. The resulting sectionafter the redatuming is a section with a perfectly planeacquisition level (Zhang et al., 2004; Zhang, 2003).
In this paper we present the results of the applicationof the CRS stack to a synthetic data set. The data set
was created at Amoco in 1997. Together with the sim-ulated zero-offset (ZO) section, we show the redatumedCRS stack section.
Methodology
The implementation of the 2D CRS stack for a ruggedtopography is based on a three-parameter moveoutformula for the paraxial rays in the vicinity of a ZOnormal ray. Both the receivers and the shots of thereflections are located on an irregular surface topography(Zhang and Hubral, 2002; Zhang et al., 2004). Thethree parameters are the emergence angle of the ZOnormal ray, and the wavefront curvatures of each of twoup-coming hypothetical waves. This algorithm is anextension of the newly established CRS stack over a planesurface. The implementation of the CRS stack makes useof the semblance for the determination of the wave fieldattributes, which is also referred to as coherence analysis.The coherence is a measure of the relative similaritiesbetween the sample amplitudes. In the data volume withthe dimensions midpoint, half-offset, and traveltime, eachcombination of the three parameters defines a surfacein the vicinity of the point pertaining to a ZO normalray. This is the blue surface in Figure 1. We also displaythe actual reflection in this data volume with the redsurface. Varying the wave field attributes in the moveoutformula changes the contour of the blue surface. If weassume that the blue surface which best matches theactual reflection yields the highest coherence, the wavefield attributes can then be determined to be the tripletthat yields the highest coherence. The blue surface inthe data volume defined by the triplet is taken to be thestacking operator.
The CRS stack delivers the ZO CRS stack section to-gether with the associated coherence section. This co-herence section helps evaluate the imaging quality of theCRS stack and the accuracy of the wave field attributes.
The CRS attributes carry all the kinematic informationof the data and can be used in further analysis, such asthe redatuming of the stack section to a plane surface,the determination of the velocity model, and the residualcorrection analysis. The redatuming makes use of thewave field attributes determined through the CRS stackand does not require a large additional effort.
Synthetic data example
The CRS stack was applied to a synthetic data set. Thedata set was created at Amoco in 1997 for testing thestatic corrections. In Figure 2 we show the model usedfor creating the data. Please note the variation in the
CRS stack and redatuming
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Fig. 1: Stacking operator for the CRS stack.
velocity of the top layer. The acquisition parametersused in creating the data set are tabulated in Table 1.
We applied the CRS stack to the data set. As one cannotice, the velocity in the top layer varies laterally. Wecan roughly divide the model into two parts at the pointaround 20 km. The left part possesses a relatively highertop layer velocity around 4.8 km/s, while the right parthas a velocity of 3.2 km/s. The reason for the partitionlies in that, currently in CRS we assume a homogeneousvelocity in the top layer. Such a partition provides a goodapproximation that satisfies this assumption for the CRSstack. Of course a better approximation should result bytaking the gradient of the velocity into account. The CRSstack can also be followed by a residual correction anal-ysis using the wave field attributes determined through
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the CRS stack (Koglin and Ewig, 2003). Such a residualanalysis should enhance the imaging quality, especiallyin the region with very rugged topography and/or strongvariation in velocity of the top layer.
The presence of the top surface topography makes themoveout curve in the CDP gathers quite complicated andthey cease to be hyperbolic. We notice that, as the shotsare quite sparsely distributed (shot spacing 100 m), thenumber of coverage folds in the CDP gather is quite small.In Figure 3 we show a CDP gather, in which the deviationof the moveout curve from a hyperbolic shape can be eas-ily seen. All these facts make the conventional stackingmethods such as NMO/DMO inapplicable without addi-tional modification.
However, the CRS stack makes use of not only the tracesin the CDP gather but also as many traces as possible inthe vicinity of the ZO normal ray. The problem with thestacking folds can be compensated for with the help of thecontributions from different CDP gathers. In Figure 4 weshow the CRS stack section. In spite of all the difficulties,such as the variation in velocity, the lack of a high foldnumber, and the ruggedness of the topography, the CRSstack provides very good imaging quality.
Number of shots 252Shot interval 100 mNumber of receivers 301Receiver interval 25 mMaximum CDP fold 38CDP bin interval 12.5 mOffset range -3762 . . . 3738 mDepth of shots 630. . . 1620 mDepth of receivers 630. . . 1630 mRecording time 4.096 sSampling interval 8 msDominant frequency 18 HzMaximum frequency 40 Hz
Table 1: Acquisition parameters of the experiment.
We find that, even when we use a different near-surfacevelocity in the CRS stack for the two parts, the reflectionevents at the connection boundary in the two CRS stacksections have identical traveltimes. This implies that, ifwe combine the two sections into one section, the reflec-tion events are continuous. This is due to the fact thatthe CRS stack is data-oriented. The CRS stack will yielda non-zero amplitude only at the position where there isa reflection. Otherwise, it gives zero output. This conti-nuity is intrinsic in data and cannot be influenced by thestacking velocity. However, a deviation in the amplitudecan be observed. This difference in amplitude occurs be-cause the near surface velocity in this region is neither4.8 km/s nor 3.2 km/s. Therefore, some residual staticsexist in the stacking operators. This residual correctioncauses the amplitude of the reflection events in both stacksections to no longer be identical. This deviation in am-plitude can be compensated for by means of the residualstatic corrections.
CRS stack and redatuming
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Fig. 3: One CDP gather in the data. Note the negative move-out of the traces in the vicinity of the ZO normal ray.
For the redatuming of the CRS stack section we needto choose a level, to which the stack section should beredatumed. The process makes use of the emergence an-gle of the ZO normal ray, which was determined throughthe CRS stack. As indicated in Table 1, the shots aredistributed between depths of 630 m and 1620 m and thereceivers are distributed between 630 m and 1630 m. Thisis an unusually large variation in depth. A compromisechoice of the redatuming level is the one of depth 1130 m.This value is simply the mean value of the depths of thereceivers. The results of the redatuming are demonstratedin Figure 5.
In practice, the redatuming algorithm assumes a homo-geneous velocity. This velocity is set equal to the near-surface velocity used in the CRS stack. As the redatum-ing is a “time migration” of the events, the positioning ofthe redatumed reflections depends on the near surface ve-locity. Therefore, the reflection events at the connectionboundary in the two sections will no longer be continuousafter redatuming. This slight deviation can be removedby taking the gradient in the near surface velocity intoaccount.
Conclusions
The data-driven, model-independent CRS stack was ap-plied to a synthetic data set. The data set containsrugged statics and strong velocity variations. Thesefeatures make the conventional imaging methods in-applicable without additional modification. However,the CRS stack is shown to be very robust even in thiscomplex situation and to be a good alternative to theconventional imaging methods.
Besides the ZO CRS stack section, the stacking processdelivers some useful wave field attributes, i. e., the emer-
gence angle of the ZO normal ray, and the wavefront cur-vatures of two up-coming hypothetical waves. These threewave field attributes are useful for velocity model build-ing, residual correction analysis, and other problems. Inthis paper we demonstrated a simple method for redatum-ing the stack section to an arbitrarily chosen elevationusing the emergence angle of the ZO normal ray. Thisaddition to the CRS stack does not require a large effort.
Acknowledgments
We thank the sponsors of the Wavelet Transform onPropagation and Imaging (WTOPI) research consortiumfor the financial support.
The author Yonghai Zhang is grateful to Amoco for thepermission to publish the results on the synthetic data setused in this paper.
References
Duveneck, E., and Hubral, P., 2002, Tomographic velocitymodel inversion using kinematic wavefield attributes:72nd Annual Meeting and Exposition, Soc. Expl. Geo-phys., Expanded Abstracts, 862–865.
Koglin, I., and Ewig, E., 2003, Residual static correctionby means of crs attributes: 73rd Annual Meeting andExposition, Soc. Expl. Geophys., Expanded Abstracts,SP 1.4.
Mann, J., Jager, R., Muller, T., Hocht, G., and Hubral,P., 1999, Common-reflection-surface stack - a real dataexample: Journal of Applied Geophysics, 42, no. 3,4,301–318.
Zhang, Y., and Hubral, P., 2002, 2D ZO CRS stack andredatuming for a complex top surface topography: 72thAnnual Meeting and Exposition, Soc. Expl. Geophys.,Expanded Abstracts, 2051–2053.
Zhang, Y., Mann, J., Wu, R., and Hubral, P., 2004, Han-dling of top surface topography by means of the zo crsstack: CPS/SEG International Geophysical Conferenceand Exposition, Beijing, Expanded Abstract.
Zhang, Y., 2003, Common-Reflection-Surface Stack andthe Handling of Top Surface Topography: Ph.D. thesis,University of Karlsruhe, Germany, http://www.logos-verlag.de.
CRS stack and redatuming
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Fig. 4: The CRS stack section
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Fig. 5: Redatumed CRS stack section
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