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GEOPHYSICS, VOL. 68, NO. 3 (MAY-JUNE 2003); P. 849–862, 20 FIGS., 2 TABLES.10.1190/1.1581037

AVO polarization and hodograms: AVO strength and polarization product

Patrice Nsoga Mahob∗ and John P. Castagna‡

ABSTRACT

An alternative approach to identifying amplitude-variation-with offset (AVO) anomalies is to consider theAVO polarization in the AVO intercept–AVO gradient(A-B) plane. This method does not require deviations orseparations from a background trend exhibited in tra-ditional crossplots such as intercept-gradient (A-B) ornear trace–far trace (N-F). A benefit of the hodogramor polarization method is that the wavelet is taken intoconsideration. Crossplotted intercept and gradient arepolarized along a “background trend” for nonanoma-lous events and at angles different from the “backgroundtrend” for anomalous events. This allows recognition ofanomalous behavior otherwise buried in a background.

Attributes resulting from this methodology include(1) the polarization angle, (2) the polarization angledifference, (3) the AVO strength, (4) the polarizationproduct, and (5) the linear-correlation coefficient. Thesedifferent attributes can then be used to enhance AVOinterpretation. Synthetic modeling for a succession ofgas and brine layers encased in shale units indicates thatthe method can potentially be an effective hydrocarbonindicator. Application of the method to a real seismicdataset shows polarization anomalies associated withhydrocarbons.

INTRODUCTION

Amplitude-variation-with-offset (AVO) attributes havebeen used in a variety of applications in exploration andproduction. They can be applied effectively in discriminat-ing hydrocarbon-filled reservoirs (Ostrander, 1984). More re-cently, to improve prospect evaluations in new areas suchas deep offshore environments, AVO attributes are beingused as an analysis tool for quantitative prospect ranking(e.g., Adamick et al., 1994, Cardamone et al., 1998). AVO

Manuscript received by the Editor October 26, 2001; revised manuscript received November 5, 2002.∗Formerly University of Oklahoma, Norman, Oklahoma; presently BP Exploration Inc., 900 East Benson Boulevard, Anchorage, Alaska 99519.E-mail: [email protected].‡University of Oklahoma, School of Geology and Geophysics, 100 East Boyd Street, Room 810, Norman, Oklahoma 73019. E-mail: [email protected]© 2003 Society of Exploration Geophysicists. All rights reserved.

attributes are considered valuable for evaluating anomalousseismic amplitude responses on large 3D datasets (e.g., Bartonand Gullette, 1996). Evolving applications of AVO attributesare the detection and characterization of fractured reservoirs(e.g., Lefeuvre, 1994; Rueger and Tsvankin, 1995; Ramos,1996) and in lithology determination (e.g., Nada and Shrallow,1994).

Crossplotting AVO attributes helps in establishing trendsagainst which anomalous amplitude behavior can be seen(e.g., Smith and Gidlow, 1987; Foster et al., 1997; Castagnaet al., 1998; Sams, 1998; Ross, 2000). Successful use of an AVOcrossplot requires a deviation of anomalous events (presumedhydrocarbon-saturated reservoirs) from a well-defined “back-ground” trend. However, when there is no deviation from thebackground trend, the AVO crossplot cannot be used as anAVO indicator. Rather, determining preferred orientations ofthe sample points in the intercept-gradient (A-B) plane is analternative approach (Keho, 2000; Keho et al., 2001; NsogaMahob and Castagna, 2002). This approach does not requiredeviations from a background trend and takes into consid-eration the wavelet as it is convolved with the reflectivityseries.

At any given interface, sample points resulting from a re-flection have a preferred orientation and can be spread acrossthe four quadrants in the A-B plane (intercept-gradient space).The angle defining any preferred orientation in the intercept-gradient space is called the polarization angle. Nonanomalousevents related to shales and brine sands can exhibit a well-defined orientation (or background angle).

In this paper, we investigate the following polarization at-tributes: the polarization angle, the polarization angle differ-ence, the AVO strength (distance from the origin in A-B plane),the product of strength and polarization angle difference, andthe linear-correlation coefficient. In this study, we investigatedpolarization attributes for a synthetic model of a succession ofgas and brine layers encased in shale units and then for a realdata case where we compared conventional AVO attributeswith the polarization attributes.

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850 Nsoga Mahob and Castagna

ALGORITHM

Concepts

For a time window about a single reflection from a giveninterface, the AVO intercept and gradient have a preferredorientation in the A-B plane (Figure 1). The angle defining thepreferred orientation in the intercept-gradient space is calledthe polarization angle. Nonanomalous reflections related toshales and brine sands may exhibit a small range of orientationswith a dominant background angle. Hence, reflections with an-gles different from the background angle can be considered asanomalous. Therefore, the angle of polarization can potentiallybe used in identifying AVO anomalies of any Rutherford andWilliam (1989) class (I, II, III), or Castagna and Swan (1997)class (IV).

One of the main benefits of this approach is the enhancementof seismic anomalies that either exhibit small anomalies or areembedded in the background trend (Figure 2) using traditionalcrossplot-derived AVO indicators. For example, an event cor-responding to a gas sand whose points are plotted close to thebackground trend on the A-B crossplot will not show a largeseparation (distance from the trend). However, such an eventmay show up as a large anomaly based on polarization angleand related attributes.

Window-size analysis.—The size of the time window is veryimportant in computing the polarization attributes. There isan optimum size that gives a good temporal resolution of seis-mic events. For a given preferred orientation or polarizationdirection, the magnitude of the attribute will have the max-imum value. The polarization attributes resulting from verysmall windows are noisy, whereas attributes from very largewindows do not represent temporally the seismic reflections(Nsoga Mahob, 2001). The optimum window size will greatly

FIG. 1. Schematic diagram of a reflection at a boundary. The reflectivity series is convolved with a wavelet, andthe resulting AVO intercept (A) and gradient (B) traces are crossplotted. Note that the points are spread acrossall the quarants.

depend on the data set in hand. However, a rule of thumb ispresented below.

Equations governing the attributes

Some equations and principles related to AVO hodogramsare determined by analogy with three-component vertical seis-mic profile (VSP) data analyses (DiSiena et al., 1981; Esmersoy,1984).

Polarization angle.—The polarization characteristics of aseismic event change in time. Therefore, the angle of polar-ization is characterized by the preferred orientation within atime window for a given time sample point (Figure 3). The po-larization angle can be determined by eigenvector analysis assuggested by Keho (2000) for AVO hodograms and Esmersoy

FIG. 2. AVO intercept and gradient crossplot showing ananomalous trend that is embedded in a background trend.


AVO Strength and Polarization Product 851

(1984) for polarization analysis of three-component VSP. Theformulation and derivation of the polarization vector compo-nents are described in Appendix A.

The polarization angle, φ, is determined for a sliding timewindow. The size of the time window should be from one-halfto a wave period (Keho, 2000). For any given window, the angle

FIG. 3. Angle of polarization definition. The angle φ is mea-sured counterclockwise from the point and the horizontal axis.At any point M on the hodogram, a unit polarization vectorP (PX , PY) can be computed. The polarization direction is thepreferred orientation in the time window [−N, N].

FIG. 4. Definition of the AVO strength attribute, L , as is relatedto the minimum and maximum values of the AVO intercept (A)and gradient (B).

of polarization at a time sample is

φ = tan−1(

Py

Px

), (1)

where Px and Py are the components of the eigenvector (seeAppendix A).

The values of the polarization angle range from −90◦ to+90◦.

Polarization angle difference.—The polarization angle dif-ference, 1φ, is the difference between the polarization angleand the “background” angle or trend angle:

1φ = φ − φtrend, (2)

FIG. 5. Linear-correlation coefficient attribute (r ) and samplepoints scattering. (1) Small values of r are related to high scat-tering of data points within the analysis window. (2) High valuesof r correspond to small scattering of data points.



where φ is the polarization angle and φtrend is the backgroundpolarization angle.

The background polarization angle or trend angle is com-puted from a larger time window that can be several hundredmilliseconds long. We should note that as the VP/VS ratio de-creases with depth (or two-way time) and as wave propagationeffects accumulate or the signal-to-noise ratio varies, the back-ground angle could change (Castagna et al., 1998).

The polarization angle difference attribute should visuallymagnify any polarization angle anomaly, thus enhancing visualdetection of the seismic amplitude anomaly.

AVO strength.—The AVO strength is the measure of thedistance of the hodogram points from the origin within thetime window of the analysis. It is one way to measure the AVOhodogram magnitude, as mentioned in Keho et al. (2001). Thesample points from the intercept (A) and the gradient (B)traces, on the plot can be considered as a cloud of points ofa certain length (Figure 4). The strength, L , is defined as

L = Lmin + Lmax, (3)

with

Lmin =√

A2min + B2

min (4)

and

Lmax =√

A2max + B2

max, (5)

FIG. 6. Extracted AVO intercept (A) and gradient (B) traces along with the synthetic gather. The first eventaround 1450 ms is the top of B Sand reflection.

where Amin is the minimum signed value within the time win-dow of the analysis of A and Bmin is the corresponding B atAmin, and Amax is the maximum signed value within the timewindow of the analysis of A and Bmax is the corresponding Bat Amax.

FIG. 7. AVO hodogram of the event for the top of the B Sand,D Sand, G1 Sand, and I Sand. The B Sand and I Sand are gassands; the D Sand and G1 Sand are brine sands. The apertureof the data is less or equal to 32◦.



Polarization product.—The product of AVO strength andpolarization angle difference, also called the polarization prod-uct, is a measure of the magnitude of the AVO effect alongthe trace. Large seismic amplitude anomalies will exhibit largevalues, whereas small values will be related to nonanoma-lous events. This attribute, L1φ, can be used to identify AVOanomalies of magnitude above noise level.

FIG. 8. Display of intercept trace, gradient trace, polarization angle, AVO strength, and square of linear-correlation coefficient for the model.

FIG. 9. Display of the synthetic gather, the product of AVO strength and polarization angle difference, and thelinear-correlation coefficient for the model.

Linear-correlation coefficient.—The linear-correlation coef-ficient, r , of the polarization analysis is the measure of how well-defined the polarization spread is (Figure 5). This attribute canexhibit various effects resulting from seismic processing suchas residual normal moveout (NMO), NMO stretching, and/ormigration artifacts (Dong, 1996, 1998; and Ross, 2000). Thelinear-correlation coefficient, r , is defined as (Rawlings et al.,



1998)

r 2 = (Cov(At+i , Bt+i ))2

Var(At+i )× Var(Bt+i ), (6)

where Cov and Var are the covariance and the variance,respectively.

In our application, we use r 2 which ranges from 0, when thereis very high scattering of hodogram points, to 1, when there isno scattering of hodogram points about the polarization trendwithin the analysis window.

MODELING EXAMPLE

Model parameters

Consider a flat-layered model made of a succession of gas-and water-saturated sand units encased in shale or silt units(Table 1) based on well log data from the Northwest Shelf ofAustralia. The B Sand, C Sand, I Sand, and M Sand are gassands. The D Sand, G1 Sand, and L Sand are water-saturatedsands. The G Sand is a tight gas sand. The model elasticparameters are presented in Table 1.

FIG. 10. Stacked seismic line with known gas- and brine-sandintervals. The gas-sand zones are indicated in dark gray;brine-sand intervals are shown in light gray.

Table 1. Elastic parameters of the flat-layered model.

Layer Name Thickness (ft) VP (m/s) VS (m/s) ρ(g/cm3) Poisson’s ratio

1 Barrow Group 1000 3640 2000 2.45 0.332 B (gas) 300 3530 2390 2.27 0.103 shale/silt 200 3610 2040 2.42 0.304 C (gas) 200 3625 2235 2.35 0.105 shale/silt 200 3450 1900 2.30 0.316 D (brine) 200 3915 2540 2.40 0.257 shale/silt 200 3615 2025 2.49 0.308 G (tight gas) 250 3985 2435 2.45 0.209 shale/silt 175 3755 2020 2.47 0.3010 G1 (brine) 250 3830 2425 2.37 0.2011 shale/silt 200 3740 2125 2.45 0.3012 I (gas) 300 3550 2415 2.33 0.1013 silt 200 3960 2080 2.45 0.3014 L (brine) 200 4140 2555 2.43 0.2015 shale/silt 200 3995 2140 2.40 0.2816 M (gas) 300 3830 2540 2.33 0.1017 shale/silt 350 4320 2460 2.50 0.30

A synthetic common-depth-point (CDP) gather was gener-ated using the full elastic wave algorithm of the AVO mod-eling module of the AVO Hampson-Russell software. A zero-phase Ricker wavelet with a dominant frequency of 40 Hz and a

FIG. 11. Display of a NMO-corrected CDP gather close to thewell used in the study. The interval of interest is between 2900and 3300 ms.



length of 200 ms was used. The range of offsets modeled variesfrom 0 to 16405 ft (5000 m).

Polarization attribute generation

A gradient analysis was performed to extract the intercept(A) and the gradient (B) traces using a maximum incidenceangle of 32◦. The resulting intercept and gradient traces aredepicted in Figure 6.

Representative hodograms of events corresponding to thetop of B Sand, D Sand, G1 Sand, and I sand are shown inFigure 7. A time window size of 20 ms, corresponding to about80% of the period of the seismic data was used to compute thepolarization attributes displayed in Figure 8. A constant back-ground angle of −20◦ was used to compute the polarizationangle difference.

From the synthetic results (Figure 9), note that the porous gassands correspond to large product of strength and polarizationangle difference (L1φ), whereas brine sands do not. The tightgas sand is represented by a very small value of the polarizationproduct.

REAL CASE EXAMPLE

Polarization attributes are computed using real seismic datafrom the Northwest Shelf of Australia to investigate the

FIG. 12. Display of the AVO product (A×B) attribute along the seismic line. The red boxes and arrows indicate the gas-sandintervals and the light blue boxes and arrows show the brine-sand zones.

methodology. Known hydrocarbon intervals are compared tothe derived attributes.

Seismic data

One prestack time-migrated (PSTM) 2D line, extracted froma 3D survey, is used for this study (Figure 10). The dominantfrequency of the seismic data is roughly 30 Hz, and the ap-proximate tuning thickness is about 30 m (or 18 ms two-waytime) in the reservoir section of interest. Some of the knowngas and brine intervals are highlighted on the seismic sections(Figure 10). A CDP gather close to the well of interest is de-picted in Figure 11.

Conventional AVO attribute generation

To perform the gradient extraction, the smoothed correctedsonic curve at a nearby well location was used for the velocityfunction. The following constraints are set during the analy-sis: range of incidence angles= 8–32◦, range of offsets= 280–3160 m. The resulting AVO product (A×B) and scaledPoisson’s ratio change (0.5A+ 0.5 B) (Verm and Hilterman,1995) sections for the line are depicted in Figures 12 and 13,respectively. The known hydrocarbon and brine zones are high-lighted and colorcoded. Overall, note that porous gas intervalscorrespond to larger AVO products and scaled Poisson’s ratios



than the brine-sand intervals do. However, brine sands also ex-hibit large values. A crossplot of AVO intercepts and gradientsand the corresponding seismic section is shown in Figure 14.There is a clear separation from the defined background trendof the seismic reflections related to the porous gas sand at2910 ms, but the other known gas sand at roughly 3200 msis not exhibited as anomalous on both the crossplot and theseismic section.

Polarization attribute computation

The extracted intercept (A) and gradient (B) traces for theseismic lines were used to compute the polarization attributes:(1) polarization angle (φ), (2) polarization angle difference(1φ), (3) AVO strength (L), (4) polarization product (prod-uct of strength and polarization angle difference (L1φ), and(5) square of linear-correlation coefficient (r 2). A 24-ms slid-ing window was chosen for the computation. Since the domi-nant frequency of the seismic data is approximately 30 Hz, thetime window for the computation is roughly 0.727T (whereT is the seismic wave period of the data), a value within thesuggested range (Keho, 2000). A constant background polar-ization angle of −20◦ was used for the entire trace to calcu-late the polarization angle difference. The background valueis determined after examining the polarization angles alonga series of traces (12) of the seismic line, particularly out-side the zone of interest (Figure 15, Table 2). The five at-

FIG. 13. Display of the scaled Poisson’s ratio change (0.5A× 0.5 B) attribute along the seismic line. The red boxes and arrowsindicate the gas-sand intervals and the light blue boxes and arrows show the brine-sand zones.

tributes at the well location are depicted in Figures 16–20,respectively.

RESULTS

Figures 12–13 and 15–19 indicate that the porous gas sandscan be better identified on the polarization product sectionthan on the conventional AVO attributes, where gas and brinesands can exhibit the same signature. The lateral extent of the

Table 2. Average background angle values for the seismicline. Two time windows are considered: 2500–2700 ms and2600–2800 ms. The angles obtained from the other seismic linesare similar to the values presented in the table.

Window 1 Window 2Trace (2500–2700 ms) (2600–2800 ms)

431 −12.60 −22.20432 −13.40 −12.60433 −12.60 −13.40434 −22.20 −15.80435 −22.20 −15.80436 −19.00 −12.60437 −12.60 −16.60438 −17.40 −15.80439 −15.80 −22.20440 −16.60 −12.60441 −18.20 −17.40Average −16.60 −16.09



sand bodies seem well defined on the polarization attributes,whereas the delineation is not clear on the AVO product sec-tion or on the scaled Poisson’s ratio attribute.

Known hydrocarbon and brine intervals for the case-studyseismic line exhibit different signatures on the polarization at-tributes. Overall, gas-sand zones are indicated by a large po-larization product (positive), whereas brine-sand zones exhibitsmaller (or negative) values. This is validated by the high val-ues of the square of linear-correlation coefficient (≥0.60) ingas intervals, but intervals of large polarization product corre-sponding to brine sands have small values of r 2 (≤0.2), mean-

FIG. 14. (a) Crossplot of AVO intercepts and gradients. The purple indicates the top of gas-sand reflections,yellow illustrates the base of gas-sand reflection, and blue represents the background. (b) Overlay of intercepttraces and color zones generated from the crossplot in (a).

ing that there is a high scattering of time sample points aboutthe polarization trend within these analysis windows. A largepolarization product with large r 2 identifies every productivegas zone. The single large polarization product associated withbrine had a low r 2.

CONCLUSIONS

It has been shown from the synthetic results that porous gassands correspond to large polarization product (L1φ), whereasbrine sands do not. This is in agreement with the real data result.



FIG. 15. Background polarization angle determination for seismic line 2. Two time windows are considered for the analysis:2500–2700 ms and 2600–2800 ms. The background angle calculated at a trace from each window is the arithmetical average ofthe polarization angles within the window. The trace average values for each are presented in Table 2. After examination of thevalues, a rounded value of −20◦ is chosen for the attribute computation.

The square of the linear-correlation coefficient (r 2) providesan indication of the reliability of the result. For a good en-hancement of AVO interpretation, the polarization attributesshould be used in conjunction with the correlation coefficient.The polarization product and the linear-correlation coefficientseem to be the most useful attributes for the synthetic and realdata investigated.

The study results from the polarization methodology suggestthat:

1) Polarization attributes should be considered as an alter-native approach to identifying AVO anomalies.

2) Polarization attributes can enhance AVO interpretation.3) Polarization attributes can potentially be used as a re-

connaissance tool to identify possible hydrocarbon (gas)intervals.

AVO polarization attributes are potentially useful hydro-carbon indicators. For a synthetic model and real seismicdata example, large polarization products combined with highlinear-correlation coefficients were found to correlate with thepresence of hydrocarbons. However, this technique will notwork properly if the signal-to-noise ratio of the data is verypoor. In addition, a seismic event, corresponding to a very thingas sand, hidden within the sidelobe of a large background

event, would not be detectable. This method might fail whenanalyzing very low frequency data.

AVO strength and polarization product attributes enhanceand highlight amplitude anomalies related to gas sands andbrines better than conventional AVO attributes.

ACKNOWLEDGMENTS

The authors express sincere thanks to Chevron OverseasPetroleum Inc. for making the data available to us. We thankBP for financial support. We also thank Dr. Bill Lamb for use-ful discussions and advice in programming the polarization at-tributes. Thanks to Hampson-Russell Software Services for useof its AVO software. The authors show their sincere appreci-ation to Herbert Swan and the other reviewers for their con-structive comments.

REFERENCES

Adamick, J., Hall, D., Skoyles, D., DeWildt, J., and Erickson, J., 1994,AVO as an exploration tool; Gulf of Mexico case studies and ex-amples: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., ExpandedAbstracts, 1107–1111.

Barton, J., and Gullette, K., 1996, Reconnaissance amplitude versusoffset techniques in the Niger Delta (abs.): AAPG Bulletin, 80,1272.



FIG. 16. Display of the polarization angle (φ) attribute along the seismic line. The black arrows indicate known gas-sand zones; thewhite arrows show brine-sand intervals.

Cardamone, M., Marini, I., and Bertelli, L., 1998, New 3D visualizationand analysis tools improve prospect evaluation in a deep offshoreenvironment (abs.): AAPG Bulletin, 82, 1898.

Castagna, J. P., and Swan, H. W., 1997, Principles of AVO crossplotting:The Leading Edge, 16, 337–342.

Castagna, J. P., Swan, H. W., and Foster, D. J., 1998, Framework forAVO gradient and intercept interpretation: Geophysics, 63, 948–956.

DiSiena, J. P., Gaiser, J. E., and Corrigan, D., 1981, Horizontal com-ponents and shear wave analysis of three-component VSP data, inToksoz, M. N., and Stewart, R. R., Eds, Handbook of geophysicalexploration, vol. 14B: Advanced concepts: Geophysical Press, 177–188.

Dong, W., 1996, Fluid line distortion due to migration stretch: 66th Ann.Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1345–1348.

——— 1998, AVO detectability against tuning and stretching artifacts:Geophysics, 64, 494–503.

Esmersoy, C., 1984, Polarization analysis, rotation and velocity estima-tion in three-component VSP, in Toksoz, M. N., and Stewart, R. R.,Eds, Handbook of geophysical exploration, vol. 14B: Advanced con-cepts: Geophysical Press, 236–255.

Foster, D. J., Keys, R. G., and Reilly, J. M., 1997, Another perspectiveon AVO crossplotting: The Leading Edge, 16, 1233–1237.

Keho, T. H., 2000, The AVO hodogram: Using polarization to identifyanomalies: 70th Ann. Internat. Mtg., Soc. Expl. Geophys., ExpandedAbstracts, 118–121.

Keho, T. H., Lemanski, S., Ripple, R., and Tambunan, B. R., 2001, TheAVO hodogram: The Leading Edge, 20, 1214–1224.

Lefeuvre, F., 1994, Fractured related anisotropy detection and analysis:“and if P-waves were enough?”: 64th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 942–945.

Nada, H., and Shrallow, J., 1994, Evaluating geophysical lithologydetermination method in the central offshore Nile Delta, Egypt:64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts,1112–1113.

Nsoga Mahob, Z. P., 2001, AVO polarization attributes and hodograms:Ph.D. diss., Univ. of Oklahoma.

Nsoga Mahob, Z. P., and Castagna, J. P., 2002, AVO hodograms andpolarization attributes: The Leading Edge, 21, 18–27.

Ostrander, W. J., 1984, Plane-wave reflection coefficients for gas sandsat nonnormal angles of incidence: Geophysics, 49, 1637–1648.

Ramos, A. C. B., 1996, Three-dimensional AVO analysis andanisotropic modeling applied to fracture characterization in coalbedmethane reservoirs, Cedar Hill field, San Juan Basin, New Mexico:Ph.D. diss., Colorado School of Mines.

Rawlings, J. O., Pantula, S. G., and Dickey, D. A., 1998, Applied regres-sion analysis: A research tool, 2nd ed.: Springer-Verlag New York,Inc.

Ross, C. P., 2000, Effective AVO crossplot modeling: A tutorial: Geo-physics, 65, 700–711.

Rueger, A., and Tsvankin, I., 1995, Azimuthal variation of AVO re-sponse for fractured reservoirs: 65th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 1103–1106.

Rutherford, S. R., and Williams, R. H., 1989, Amplitude-versus-offsetvariations in gas sands: Geophysics, 54, 680–688.

Sams, M., 1998, Yet another perspective on AVO crossplotting: TheLeading Edge, 17, 911–917.

Smith, G. C., and Gidlow, P. M., 1987, Weighted stacking for rockproperty estimation and detection of gas: Geophys. Prosp., 35, 993–1014.

Verm, R., and Hilterman, F., 1995, Lithology color-coded seismic sec-tions, The calibration of AVO crossplotting to rock properties: TheLeading Edge, 14, 847–853.



FIG. 17. Display of the polarization angle difference (1φ) attribute along the seismic line. The black arrows indicate known gas-sandzones; the white arrows show brine-sand intervals.

FIG. 18. Display of the AVO strength (L) attribute along the seismic line. The black arrows indicate known gas-sand zones; thewhite arrows show brine-sand intervals.



FIG. 19. Display of the polarization product (L ×1φ) attribute along the seismic line. The black arrows indicate known gas-sandzones; the white arrows show brine-sand intervals.

FIG. 20. Display of the square of linear-correlation coefficient (r 2) attribute along the seismic line. The black arrows indicate knowngas-sand zones; the white arrows show brine-sand intervals.



APPENDIX APOLARIZATION-VECTOR COMPONENTS DERIVATION

The formulation of the polarization vector is derived fromthe correlation matrix Rm that is used to compute the eigen-vectors (Esmersoy, 1984):

Rm = 12N + 1

N∑i=−N

r (i )r T (i ), (A-1)

where N is half of the length of the time window (in samplepoints) and r (i ) represents the observed data in the time win-dow of interest. The subscript m is the center sample point ofthe time window [-N, N], which is rectangular.

The matrix Rm from equation (1) can be expanded in theA-B plane as follows:

Rm = 12N + 1

N∑

i=−N

A2t+i

N∑i=−N

At+i Bt+i

N∑i=−N

At+i Bt+i

N∑i=−N

B2t+i

,(A-2)

where At+i is the AVO intercept value at time sample t , andBt+i is the AVO gradient value at time sample t . Rm is a 2× 2symmetric matrix, and its eigenanalysis can be done efficiently.

The eigenvalues are obtained by solving the equation

|Rm − λI| = 0, (A-3)

where λ represents the eigenvalues or characteristic roots andI is the unity matrix:

I =(

1 0

0 1

). (A-4)

The eigenvectors corresponding to the characteristic roots (λ)calculated from equation (A-3). The two vectors are orthogo-nal. Nsoga Mahob (2001) showed that the components of thevector corresponding to the larger eigenvalue are(

Px

Py

)=

√2

[1+ D]12

∑i

At+i Bt+i√√√√4

(∑i

At+i Bt+i

)2

+(∑

i

A2t+i −

∑i

B2t+i

)2

√2

2[1+ D]

12

,

(A-5)

where

D =

(∑i

A2t+i −

∑i

B2t+i

)√√√√4

(∑i

At+i Bt+i

)2

+(∑

i

A2t+i −

∑i

B2t+i

)2

(A-6)

with i =−N, . . . , N. Px and Py are normalized so that a unitpolarization vector is considered. That is,

‖P‖ = 1. (A-7)


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