CRS as a tool for true amplitude imaging

R. S. Portugal, R. Biloti, L. T. Santos and M. Tygel

State University of Campinas, Brazil

Abstract

We present a method to obtain a true-amplitudemigration and amplitude-versus-angle (AVA) at se-lected points using the attributes generated by theCommon Reflection Surface (CRS) Stack. Our ap-proach combines the CRS stack/inversion processapplied to multicoverage data, together with theuse of a kinematic Kirchhoff migration, to achievetrue-amplitudes (TA) at assigned depth points ofthe migrated images. The proposed method con-sists of the following steps: (i) apply the CRS pro-cess to the given multicoverage data; the obtainedCRS attributes are next used to produce a simplemacro-velocity depth model; (ii) perform an un-weighted Kirchhoff migration for imaging purposesonly; for selected points on target reflectors in themigrated image, we use the macro-velocity modelto determine, by ray tracing, common-reflection-point (CRP) gathers that belong to the input data;for these rays, we compute the incident angles andthe geometrical spreadings; (iii) go back to CRPgathers and compensate the amplitudes for geomet-rical spreading. The results permit to constructAVA curves on the assigned CRPs. In summary,our method is designed to aggregate amplitude in-formation on selected points of a reflector, after apurely kinematic image (migration) has been ob-tained. The method is tested on a synthetic inho-mogeneous layered model with good results.

Introduction

One of the main objectives of processing seismic re-flection data for hydrocarbon prospecting is to ob-tain meaningful images of the geological structures,in particular reservoir structures in the subsurface.The geological structures to be imaged are definedby seismic reflectors.

Kinematical images, in which only the lo-cation and orientation of the reflectors (with noregard to amplitudes) are considered, can beachieved, for example, by efficient Kirchhoff migra-tion procedures using simple weights or no weightsat all. Kirchhoff migration requires a given macro-velocity model. Moreover, special methods existto combine the migration outputs to update themodel, so as to refine and improve the image. The

final result is, in many cases, a fairly adequate(kinematical) image of the structures of interest.

The problems that concern us in this paperis how to aggregate dynamical information (ampli-tudes) to the obtained image. In fact, the ampli-tudes are needed essentially on selected points atkey interfaces, where the determination of angle-dependent reflection coefficients is the most desir-able information.

According to zero-order ray theory, the ampli-tude of a primary-reflection event can be describedby

U = A Rc

L , (1)

where Rc is the angle-dependent reflection coeffi-cient of the primary reflection ray and θ is the inci-dence angle of that ray with respect to the interfacenormal. The reflection coefficient is the quantity ofinterest to be estimated from the data. The quan-tity L is the angle-dependent geometrical-spreadingfactor of the reflection ray. It accounts for the am-plitude variations due to focusing and defocusing ofthe energy carried by the ray along its ray path. Allfactors which affects amplitudes other than the ge-ometrical spreading are combined and representedby overall quantity A in equation (1). The geomet-rical spreading, L, is generally singled out as oneof the major sources of amplitude distortion in theobserved data. That is the reason why the termtrue-amplitude is typically attached to a primary-reflection amplitude that has been corrected for ge-ometrical spreading.

In the case of depth migration, the term true-amplitude (TA) migration refers to the case inwhich the migration output equals the observedamplitudes automatically corrected for geometri-cal spreading (see, e.g., Hubral et al., 1996). FullTA algorithms are significantly more expensive andtime-consuming than their kinematic unweightedcounterparts. As another complication, the ac-curacy requirements on the macro-velocity depthmodel are higher for the application of TA migra-tion than for purely kinematic migration. The flex-ibility of using migration outputs to update the ve-locity model is lost when such a heavy migrationalgorithm is applied. As a last, and probably thebest, argument against the application of a full TAmigration algorithm to an overall region is that,in fact, the amplitude information is required only

1

Portugal et al.

on some specific target points or reflectors. Awayfrom these points, the obtained amplitudes are notuseful.

In this paper, we propose a method to ag-gregate true amplitudes (i.e., observed primary-reflection amplitudes after geometrical-spreadingcorrection) at selected CRPs of interest, after animage of the subsurface has been obtained. Thisimage can be, for example, the result of one or sev-eral kinematic migrations.

Strategy

The kernel of the method is summarized by thefluxogram shown in Figure 1. Our strategy ismainly divided in three steps: CRS attribute ex-traction & macro-model inversion, kinematic imag-ing through unweighted Kirchhoff migrations andsubsequent geometrical spreading corrections in theinput data.

KinematicMigration

MulticoverageData

True amplitude onMigrated section

CRS Stack

CRS macrovelocity inversion

Geometricalspreading tables

CRS attribute sectionCoherence section

ZO simulated section

Layered macrovelocity model

Figure 1: Fluxogram.

CRS stack. The 2-D common-reflection-surface(CRS) stack (see, e.g., Muller 1999), applied tomulticoverage data on a seismic line, is designedto produce a stacked section (an approximation ofa zero-offset section), together with three auxiliarsections of CRS attributes and a coherence section.

For each fixed central point (e.g., a CMP loca-tion of the original data), on which the output traceis to be computed, the CRS uses a multiparametrictraveltime formula to stack all data that correspondto arbitrary source and receiver locations in thevicinity of that point. In this sense, it differs sig-nificantly from the conventional NMO/DMO stack(that employs only reflections from CMP gathers)to achieve much more redundancy with a conse-quent improvement of signal-to-noise ratio. Thethree CRS attributes assigned to each point of the

stacked section are the parameters of the traveltimemoveout formula. These are the emergence angleof the normal reflection ray and the wavefront cur-vatures of the NIP- and N-waves that arrive at thatpoint. For the definition of the NIP-wave and theN-wave, the reader is referred to the original paperof Hubral (1983).

The CRS attributes are extracted upon theuse of coherency analysis strategy directly appliedto the data. The determination of more efficientand accurate parameter extraction methods is atopic of active research (Birgin et al., 1999).

CRS macro-velocity model inversion. Thephilosophy of the CRS Stack method is to use asmuch data as possible during the stacking process.Therefore, the most relevant events are better de-fined on the stacked section and available for fur-ther inversion.

The input data for CRS velocity inversion arethe CRS attributes that refer to those selected tar-get reflections. Also the near-surface velocity fieldneeds to be known. In fact, this is already a re-quirement for the application of the CRS method.

The classical layer-stripping velocity inversionalgorithm of Hubral and Krey (1980) can be recastin terms use of the CRS, inverts iteratively on thedepth the homogeneous layer velocities and the in-terface positions. The interfaces are constructed ascubic splines, which are suitable for further blockyray tracing algorithms.

Kinematic image. As soon as the homogeneouslayered velocity model is provided by the CRS in-version, it is smoothed in order to perform a kine-matic migration. The traveltime tables are gen-erated on the fly by the wavefront constructionmethod, therefore each seismic trace can be mi-grated independently from each other. To enhancethe signal-to-noise ratio, the final image is built bystacking all common-offset migrated section, thissection is called stacked migrated section.

Geometrical spreading correction. On thestacked migrated section, it is possible to choosedepth points on a target reflector. For each onechosen point, using the approximated homogeneouslayered model, we compute, by standard dynamicray-tracing, the traveltimes, the incident angles andthe geometrical spreading factors. These quantitieshelp to extract a common-reflection-point (CRP)gather from the original data. For each trace, us-ing the computed reflection traveltime, we can pick

2

CRS as a tool for true amplitude imaging

the amplitude, which is multiplied by the corre-sponding geometrical spreading factor. Applyingthis successively for all traces in the CRP gather,we obtain the desired AVO/AVA curves.

Synthetic example

The synthetic model, depicted in Figure 2, is com-posed by four layers separated by smooth inter-faces. The first and fourth layers are homogeneouswith constant compressional velocity of 2.0 km/sand 2.7 km/s, respectively. The second and the

2

2.2

2.4

2.6

Dep

th (

km)

Distance (km)0 4 8

0

1

2

3

4

1.2

1.3

1.4

1.5

Dep

th (

km)

Distance (km)0 4 8

0

1

2

3

4

Figure 2: Velocity model for synthetic data. Top:compressional velocity. Bottom: shear velocity.

third layers are inhomogeneous, and their veloci-ties are composed as a linear combination of thevelocity just below the upper interface of the layerand the velocity just above the lower interface thatbounds the layer. For those layers, the compres-sional velocity varies from 2.2 km/s to 2.4 km/sand from 2.5 km/s to 2.55 km/s, respectively. Theshear velocity in each point of the model is the com-pressional velocity divided by

√3, and the density

is unitary in the whole model. The multicoveragedata is composed by 501 common-source experi-ments, where the sources are 20 m spaced. EachCS section has 151 receivers 20 m spaced. The ra-tio signal-to-noise in the data is 7:1. Figure 3 showsa typical common-offset section.

Each common-offset section was migrated sep-arately, and then stacked to generate a stacked mi-grated section, depicted in Figure 4. The veloc-ity model employed on the migration process (Fig-ure 5) was obtained from the CRS attributes by

0

1

2

3

4

Tim

e (s

)

0 2 4 6 8 10Midpoint coordinate (km)

Figure 3: A typical common-offset section for theoffset 1500 m.

the inversion process briefly describe above. Notethat the inverted model is as accurate as possible,since only homogeneous layers could be inverted.

Using this kinematic image we have chosen apoint locate on the second interface to analyze theamplitude variation (AVO and AVA curves). Fig-ures 6 and 7 show the CRP section and the AVOand AVA curves for the selected point, respectively.

1000

1500

2000

2500

3000

3500

Dep

th (

m)

3000 4000 5000 6000 7000Distance (m)

Figure 4: Stacked migrated section.

Conclusions

We have presented a method which provides acomplete process to obtain AVA curves for chosenpoints on target interfaces. It is mainly composedby three steps: (i) construction of a layered macro-velocity model by using CRS attributes (obtainedfrom the multicoverage data); (ii) kinematic migra-tion of data using that macro-velocity model; (iii) a

3

Portugal et al.

posteriori correction of amplitude of chosen pointson the migrated section (using traveltime, reflec-tion angle and geometrical spreading computed onthe approximated model). The numerical resultsare encouraging, concerning accurancy and com-putational effort. As a next step, further tests inreal data will be carried out.

2

2.2

2.4

2.6

Dep

th (

km)

Distance (km)0 4 8

0

1

2

3

4

Figure 5: Compressional velocity model obtainedby the CRS inversion algorithm.

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Tim

e (s

)

0 500 1000 1500 2000Offset (m)

Figure 6: CRP section for the selected point on thesecond interface. The red tube confines the regionwhere the picking process was carried out. Thisregion was found out by the traveltime estimationthat came out of the modelling process.

Acknowledgments

This work was partially supported by FAPESP(Grants 97/12125-8 and 97/12318-0) and by thesponsors of the WIT Consortium.

0 10 20

Reflection angle (degree)

0

0.5

1.0

1.5

Nor

mal

ized

R(θ

)

AVA Curve

0 1000 2000

Offset (m)

0

0.5

1.0

1.5

Nor

mal

ized

R(h

)

AVO Curve

Figure 7: AVO and AVA curves for the chosen pointon the second interface. The solid red line is the ex-pected normalized reflection coefficient. The bluecrosses are the picked amplitude correct for geo-metrical spreading, computed on the approximatedmodel, and normalized.

References

Birgin, E. G., Biloti, R., Tygel, M., and Santos,L. T., 1999, Restricted optimization: a clue toa fast and accurate implementation of the com-mon reflection surface method: Journal of Ap-plied Geophysics, 42, 143–155.

Hubral, P., and Krey, T., 1980, Interval velocitiesfrom seismic reflection time measurements: Soc.of Expl. Geophys.

Hubral, P., Schleicher, J., and Tygel, M., 1996, Aunified approach to 3-d seismic reflection imag-ing, part i: Basic concepts: Geophysics, 61, no.03, 742–758.

Hubral, P., 1983, Computing true amplitude reflec-tions in a laterally inhomogeneous earth: Geo-physics, 48, no. 08, 1051–1062.

Muller, T., 1999, Common reflection surfacestack method – seismic imaging without explicitknowledge of the velocity model: Ph.D. the-sis, Geophysical Institute, Karlsruhe University,Germany.

4