Gaussian-beam-based Seismic Illumination Analysis

8/12/2019 Gaussian-beam-based Seismic Illumination Analysis

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Gaussian-beam-based seismic illumination analysisPeng Wen*, Jiang Xianyi, and Zhou Shuang, BGP, CNPC

Summary

Seismic wave illumination analysis provides the seismicwave energy and other information at the scattering pointon a target layer, so it can be used to analyze imaging

shadow and acquisition footprint, optimize acquisitiongeometry design and improve the imaging quality of pre-stack depth migration. However, the seismic illumination

analysis based on either one-way or two-way waveequation is time-consuming. Therefore, this analysis is notwidely used in practice. In order to improve thecomputational efficiency of seismic illumination withoutlosing wave field characteristics, a new illuminationanalysis method which employs Gaussian beams to

calculate seismic wave field is presented in this paper.Numerical examples indicate that the Gaussian-beam-basedseismic illumination is a highly efficient tool to evaluatethe effects of the designed seismic geometry.

Introduction

Seismic illumination analysis can be applied to seismic

acquisition and seismic imaging. It has been proven thatseismic wave illumination analysis is an importantapproach to the target-oriented seismic survey design. Bycalculating the illumination generated from differentshooting patterns and geometries, the optimumacquisitiongeometry can be identified.

One of the key steps in seismic illumination analysis isgenerating seismic wave field through seismic modeling.Seismic illumination analysis can therefore be classifiedinto two groups, ray-based method (Schneider and Winbow,

1999; Bear et al., 2000) and wave-equation-based method(Wu and Chen, 2002, 2003; Xie et al., 2003, 2006).

The method of two-point ray tracing is characterized by its

high efficiency. However it can only simulate the kinematiccharacteristics of seismic wave field. In addition, althoughthe ray-based illumination analysis can handle bothirregular acquisition geometry and laterally varyingvelocity models, it can not generate accurate results in the

caustic and shadow zones (Hoffmann, 2001).

By contrast, the feature of wave-equation simulation is highprecision, but it is time-consuming. At present, the wave-equation-based method is still too expensive for

illumination analysis. As a result, it is difficult to use thewave-equation-based illumination to optimize seismicacquisition design in practice.

The Gaussian beam method is developed from paraxialasymptotic ray theory. It has been proven that this method

is a very stable asymptotic approach for the computation ofhigh-frequency wave fields in smoothly varyinginhomogeneous media. One of the advantages of thismethod is that the individual Gaussian beams have nosingularities even in regions where the ray method fails,such as the caustic region, critical region, etc. Another

advantage is that the Gaussian beam algorithm does notrequire two-point (source-to-receiver) ray tracing. Thenumerical modeling based on Gaussian beam algorithmrequires approximately the same computer running time as

ray method and the memory requirements of these twomethods are also comparable. By using complex travel time

and amplitude in the dynamic ray tracing, Gaussian beamapproach can provide the dynamic characteristics ofseismic wave field (erven, 1985; Nowack, 2003). Underthe condition of computational capabilities of the presentcomputers, the Gaussian beam method can meet the need ofthe seismic acquisition design in terms of seismicillumination analysis.

In this paper, we present an illumination analysis methodusing Gaussian beams to calculate wave field energy. Asthe illumination of one source-receiver pair acts as the

primary role in the seismic wave illumination analysis, we

focus on the illumination of one source-receiver pair, whichis related to source, receiver and reflection points. Bysummation of the illumination of the user-defined source-

receiver pairs, different illumination analysis results can beobtained. For example, by summation of the illumination of

the source-receiver pairs of the same offset and azimuthrange, the common offset-azimuth illumination can begenerated, including the source side, receiver side and boththe source and receiver side illumination. We use ahorizontal layer model to demonstrate the illuminationanalysis of one source-receiver pair and a complex three-

dimensional model to demonstrate the agreement betweenthe forward stacked section and the illumination result.

Theory and Method

Three steps are involved in the seismic illuminationanalysis based on Gaussian beams. The first step is the

calculation of the seismic wave energy based on theseismic wave field generated from the Gaussian beamalgorithm. The second step is to calculate the illuminationof each source-receiver pair. And the last step is to compute

all kinds of user-defined illumination for a given seismicacquisition geometry.

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Gaussian beam based illumination

Step 1: Calculation of the seismic wave energy based on

Gaussian beam algorithm

As seismic wave field can be expressed as the weighted

summation of some Gaussian beams, the synthetic seismicdata at a point on the target layer can be obtained byweighted summation of the Gaussian beams in the Fresnelzone of this point. (erven, 1985).

Given the starting, ending and increment values of the

incident angle and azimuth angle, the ray tracing of thedirect wave from the source to the target layer is performedin the azimuth angle range of 0 to 360 degrees and theincident angle range of 0 to 90 degrees. These raysconstitute the central ray set. Next the Gaussian beams

corresponding to the target layer can be figured out bydynamic ray tracing of each central ray using Runge-Kuttaalgorithm.

Suppose that a geophone is located at the grid point i

),,( iii zyx ),3,2,1( L=i on the target layer, the seismic record

of this geophone can be obtained by summation of the

Gaussian beams in the Gaussian beams subset. TheGaussian beams subset consists of the Gaussian beams inwhich the distance from the grid point ito the central ray isless than the half width of the corresponding Gaussian

beam. Finally, the seismic energy or the source side

illuminationiU at each grid point ican be calculated.

Step 2: seismic Illumination of a source-receiver pair

Illumination analysis of a source-receiver pair consists ofsource side and receiver side illumination and relates to

source, receiver and reflection points. If the reflector or thetarget layer is a smooth formation boundary, the seismicwave propagation from the source S will obey the Snell law,that is, down-going to the reflection point B with incidentangle and then up-going to the point D with reflectionangle . Now given the receiver position R, we want tocalculate the seismic energy at this point as shown in figure1. In this figure, BM is the normal direction of the reflector

at reflection point B. ( SBM ) is the incident angle of thedown-going ray. BD is the up-going ray which obeys theSnell law or the Snell up-going ray, and ( DBR ) is theincluded angle between the Snell up-going ray BD and the

paraxial asymptotic ray BR.

Suppose that the reflection coefficient at the reflectionpoint B is )(BK , the down-going wave field energy

emitted from the source S at reflection point B isSU , and

the seismic energy after wave propagation from the

reflection point B to the receiver location R isSBRU , then

SBRU can be formulated as:

2

)1(cos =

eKUFU BSSBR (1)

The function )(F represents the seismic wave energy

variation during seismic wave propagation from B through

the overlying layers to B. SBRU can be used as theillumination of the source-receiver pair at thereflector B in the target layer. In fact, it is difficult to

construct the function )(F analytically. According to the

principle of reciprocity which says that the sameseismogram should be recorded if the locations of thesource and geophone are exchanged, the seismic wave

energy recorded at point R and excited at B is equivalent tothe one recorded at B and excited at R if the source energy

is identical. LetRU be the seismic wave energy received at

reflection point B and excited at point R with impulsivesource, we get the following relationship:

2)1(cos

eKUUU BRSSBR (2)

Let SBRI be the illumination of the source-receiver pair at the reflector B or the source-receiver pair

illumination, thenSBRI can be defined as

2)1(cos

= eKUUI BRSSBR (3)

After calculating the illumination of the source-receiver

pair at each grid point on the target layer according to theequation (3), we can get the illumination distribution of thesource-receiver pair on the whole target layer.

Step 3: Illumination analysis of a seismic geometry

Based on source-receiver pair illumination of all thereflection points on the target layer, we can derive other

illumination attributes of the target layer from the source-receiver pair illumination. For example, by summation ofall the source-receiver pair illumination in a gather, we can

get the common shot illumination and the common receiverillumination, and by summation of all the source-receiver

pair illumination in an offset and/or azimuth range, we can

get the common offset and/or azimuth illumination of

Figure 1: Schematic diagram of the illumination analysis of one

source-receiver pair

B

SM

R

D



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reflection points on the target layer. In short, given a

seismic geometry and a target layer, we can performillumination analysis to optimize the seismic geometrydesign using various illumination attributes as shown in

figure 4.

Examples

Figure 2 shows the results of the source side, receiver sideand source-receiver pair illumination corresponding to a

three-dimensional model with one horizontal layer. Asshown in figure 2(b) and (c), the patterns of both the sourceside and receiver side illumination are circles. Thestrongest illumination is at the centre of the circle and thestrength of the illumination decreases gradually as the

radius increases. Furthermore, the source-receiver pairillumination exhibits an elliptical pattern as shown in figure2(d). The major axis of the ellipse links the centers of thesource side and receiver side illumination circles. And in

the centre part of the ellipse, the illumination is strongerthan the other regions. In addition, at the CMP location ofthe source-receiver pair, the strongest illumination isobserved, which agrees with the seismic reflection principle.

Figure 3 shows a narrow swath with orthogonal geometry.

This geometrys illumination attributes on the geologymodel of figure 2(a) are displayed in figure 4.

Figure 5 shows a complex three-dimensional geology

model and a stack section. The geological model consists ofsome steep-dipping faults. A geometry as shown in figure 3is used to perform seismic modeling based on the Gaussian

beam forward method. After generating all the synthetic

shot gathers, the full three-dimensional seismic datastacking processing for the synthetic data is performed. Thebottom picture in figure 5 shows the stack section of theCMP inline No.5 located at the center of the geometry.

Figure 6 and 7 show the comparisons between the stack

sections of the synthetic data and the results of theGaussian-beam-based illumination analysis for two giventarget layers in the complex three-dimensional model. Thered lines in the stack sections indicate the faults while the

red lines in the illumination results indicate the location ofCMP inline No.5. The pink lines with double arrows inthese figures show the locations of the corresponding points.It is clear that amplitude variation of these events in the

stack section generally agree with variation of the source-

receiver illumination.

Conclusions

1) The Gaussian-beam-based illumination analysis is more

efficient than the illumination analysis based on theone-way or two-way wave equation.

2) The source-receiver pair illumination and the otherillumination attributes of a geometry introduced in this

paper are practicable in the seismic survey design.3) Synthetic data examples show that the illumination

results agree with the corresponding seismic section forthe complex geological model.

Acknowledgements

We would like to thank Dr. Ke Benxi for revising thispaper and the management at BGP for allowing us topublish this work.

Figure 3: The orthogonal geometry with a swath of 4 lines, 125

shots and 400 channels. Group interval is 50m and line spacing

is 100m.

( a ) ( b )

( c ) ( d )

Figure 2: A three-dimensional model of 10000m*5000m*5000m

with single horizontal layer buried in depth of 4000m, given the

P-wave velocity of the overlying formation as 3000m/s, Vs as

1700m/s, density as 1.5g/cm3, those of the underlying formationare 4500m/s, 2600m/s and 1.5g/cm3, source S locates at point

(3500, 2500, 0) and receiver R at point (6500,2500,0). Scan from

0~360 degree horizontally and 0~90 degree perpendicularly,

angular interval is 1 degree. (a) is the model, (b) and (c) are theenergy distribution of the source and the receiver. (d) is the

illumination analysis result of the source-receiver pair.



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Fault

Fault

InLine

No.5

Figure 6: The top figure is the horizontal stack section of inline

No.5 of layer No.5. The bottom figure is the i llumination resultof layer No.5, in which each block represents the illumination

result of the slices of the layer shaped by the faults.

( a ) ( b )

( c ) ( d )

( e ) ( f )

Color : illumination values

Figure 4: Illumination attributes of a narrow swath with

orthogonal geometry. (a) and (b) are the number of source-

receiver pairs (NSRP) corresponding to different offsets and

azimuths. (c) is the illumination result of 0~1250m offset range.(d) is the illumination result of 1250~3000m offset range. (e) is

the illumination result of 230~290 degree azimuth range. (f) is

the illumination result of 140~200 degree azimuth range.

InLineNo.5

Figure 7: The top figure is the horizontal stack section of inline

No.5 of layer No.6. The bottom figure is the i llumination result

of layer No.6, in which each block represents the illumination

result of the slices of the layer shaped by the faults.

Figure 5: The top figure is a three-dimensional geology model

with six layers. The bottom figure is the horizontal stack section

of inline No.5 which locates in the center of the swath at station

(North: 2525m, East: 1362m to 8612m).



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Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2011SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for

each paper will achieve a high degree of linking to cited sources that appear on the Web.

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Cerven, V., 1985, Gaussian beam synthetic seismograms: Journal of Geophysics, 58, 4472.

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Gaussian-beam-based Seismic Illumination Analysis