1.1) geophysics96 vpvs corr analysis

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GEOPHYSICS, VOL. 61, NO. 4 (JULY-AUGUST 1996); P. 1137-1149, 11 FIGS, 1 TABLE.

Multicomponent V P /V s correlation analysis

James E. Gaiser*

ABSTRACT

An important step in the simultaneous interpretationor inversion of multicomponent data sets is to quantita-tively estimate the ratio of P-wave velocity to S-wave

velocity (V p /V s). In this endeavor, I have developed cor-relation techniques to determine long-wavelength compo-nents of C that can lead to more accurate measure-ments of rock properties and processing parameters.

P-wave reflections are correlated with converted P- toS-wave reflections (or S-wave reflections) from the samelocation to determine which events are related to thesame subsurface impedance contrasts. Shear waves aretransformed (compressed) to P-wave time via averageV p /V s conjugate operators before correlation. Aided byconventional P-wave velocity information and petro-physical relationships, this technique provides optimalV p /V s estimates in a similar manner that semblanceanalyses provide stacking velocities. These estimates can

be used to transform the entire S-wave trace to P-wavetime for short-wavelength amplitude inversion. Also, atarget-oriented correlation analysis quantitatively deter-mines interval V p /V s at a specific horizon or group of horizons.

Data from vertical seismic profile (VSP) stackedtraces are used to evaluate these techniques. Long-wavelength average and interval V p /V s estimates ob-tained from the correlation analyses agree closely withV p /V s results determined from VSP direct-arrival trav-eltimes.

INTRODUCTION

The ratio of compressional-wave (P-wave) to shear-wave(S-wave) velocity (Vp/Vs ) and Poissons ratio are importantquantities for interpreting lithology and fluid properties fromseismic data. For marine data a practical method to obtainS-wave information is to analyze P-wave amplitude variations

with offset (AVO). For land data, heterogeneous near-surfacegeology often complicates AVO analyses and so it is oftennecessary to collect S-wave data with the P-wave data. In thissituation, the problem is relating S-wave reflections quantita-tively with P-wave reflections from the same horizons. Usingmulticomponent land data, such quantities as sand-shale ratios(McCormack et al., 1984), carbonate porosity (Robertson,1987), limestone-dolomite content (Pardus et al., 1990), andeven anisotropy (Justice et al., 1987) have been inferred fromV p /V s estimates. However, these measurements often rely onvisually correlating reflections before traveltimes are com-puted.

Attempts have been made to objectively correlate multicom-ponent data to compute V p /V s over the entire seismic section.Garotta (1985) developed a semiautomatic correlation methodthat is based on preliminary visual correlations of the P-waveand S-wave sections. A time scale-factor guess of the averageV p /vs ratio is applied to the S-wave data for cross-correlationwithin small windows with the P-wave data. These guesses are

then corrected using the time shifts on the peak values of thecross-correlations to obtain the interval V p /V s value betweenspecified windows. This technique suffers from instability de-pending on the size of the intervals. Closely spaced windowsresult in poor estimates of V p /V s and must be heavilysmoothed. The problem is a result of estimating short-wave-length V p /V s values from long-wavelength information.

Behle and Dohr (1985) describe a combined velocity analysisbetween P-waves and converted P to S V-waves (PS V-waves)for correlating reflections. Using measured P-wave values of vertical times and normal-moveout velocities, PS V-wave timesand moveout velocities are cast in terms of V p /V s and theircorresponding coherence is computed. This results in a PSV-wavetime versus V p /V s coherence plot that is interpreted for

peaks along specified windows; however, analyses applied toshot records yield sparse and ambiguous peaks.

McCormack (1990) has patented a method that establishesthe correlation between radially polarized S-wave (S V-wave)and transversely polarized S-wave (SH-wave) stacked data tocompute S-wave velocity ratios for fracture detection. This

Presented at the 63rd Annual Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor October 24, 1994; revisedmanuscript received October 30, 1995.*Formerly ARCO Oil and Gas Co.; presently Western Geophysical Co., Research and Development, 7229 S. Alton Way, Englewood, Colorado80112-2202. 1996 Society of Exploration Geophysicists. All rights reserved.

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technique can also be applied to P-wave and S-wave data tocompute Vp/Vs at each time sample. The method does notexplicitly transform S-wave seismic traces but requires anamplitude scale factor such that high-frequency interval trav-eltimes match interpreted estimates of low-frequency intervaltraveltimes. Also, it cannot be applied to converted waves.

In this paper, I introduce an alternative, general approach

that correlates multicomponent data to simplify the determi-nation of just the long-wavelength component of V p /V s. It canbe applied to any pair of P-wave, S-wave, or PSV-wave data,either globally over an entire data set or locally as a target-oriented analysis. Results from the V p /V s correlation analysesdefine operators that can be used to align the wavefields forimproved interpretation or short-wavelength amplitude inver-sion. The specific objective of this study is to examine thesetechniques using vertical seismic profile (VSP) stacked data.Vp/Vs estimates are evaluated by comparison with V p /V scalculations from VSP direct-arrival traveltimes.

MULTICOMPONENT INTERPRETATION

Throughout this paper, S-wave refers to PSI/-wave orSH -wave reflections unless otherwise specified. The focus of this study is on PSV-wave data because these are moreeconomical to acquire and generally provide higher signal-to-noise ratio (S/N) information than S-wave source data. Disad-vantages of PSV-wave data are low-amplitude reflection coef-ficients near zero offset and asymmetric raypaths that requireunconventional processing techniques.

Figure 1 shows a portion of the P-wave, PSV-wave, andSH -wave surface-seismic data centered at the well locationwhere the VSP data were acquired. For this study, horizontalstratigraphy is desired to avoid complications of lateral varia-

tions; however, a small amount of dip, roughly parallel to theseismic line direction, influences V P /V s estimates from theVSP correlation analyses that must be corrected.

Confronted with the simultaneous interpretation or correla-tion of the three components in Figure 1, there are severalrequirements that must be satisfied, aside from source charac-teristics, receiver coupling, and processing (Garotta, 1985).One of these is that the wavefields sample the same volume of rock in a common-depth point (CDP) sense. In terms of theVSP data, the near offset P-wave and W-wave VSPs havereflection points at the borehole, but the PSI/-wave VSP,acquired with a 4000 ft (1219 m) source offset, has reflectionpoints that occur some distance from the borehole in theup-dip direction.

Equally important as lateral sampling, the wavefields mustprovide the same resolution of stratigraphic horizons in depth.Hence, P-wave and S-wave wavelengths must match in orderto sample the reflectivity sequence in an equivalent manner.This means that P-wave frequencies should be filtered to abouttwice that of S-wave frequencies. If the frequency content of P-waves and S-waves are the same, S-waves will have abouthalf the wavelength and twice the resolution as P-waves.

Another requirement or assumption for correlating thewavefields is that the P-wave reflectivity is similar to S-wavereflectivity in a long-wavelength sense. There is justification forthis assumption based on the linear relationship betweenP-wave and S-wave velocities for saturated mudrocks (Cast-agna et al., 1985). These empirical data indicate that as S-wavevelocities increase from zero (water) to that of quartz, there isa corresponding linear increase in P-wave velocities. Specificrelationships also exist for other lithologies (Pickett, 1963). It isreasonable to expect that a particular sequence of these

FIG . 1. Multicomponent surface-seismic data are the (a) P-wave, (b) PSV-wave, and (c) SH-wave, centered at the well locationwhere the VSP data were acquired. Different time scales are used to enhance the visual correlation of similar reflections betweenthe three wavefields.


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Multicomponent V P/Vs Correlation Analysis 1139

materials would produce P-wave and S-wave velocity profileswith similar long-wavelength characteristics. Furthermore, anycorrelation technique based on the similarity between P-waveand S-wave data would tend to be biased in favor of such arelationship.

It is important to recall from AVO studies (Ostrander, 1984;Smith and Gidlow, 1987; Rutherford and Williams, 1989;Swan, 1991) that hydrocarbon indicators in exploration seis-

mology are related to cases where the P-wave and S-wavevelocity may not follow Castagnas mudrock trend. The pres-ence of gas can lower P-wave velocity but have little effect onthe S-wave velocity. Correlating P-waves with S-waves, toobtain the long-wavelength V p /V s structure, is biased awayfrom such hydrocarbon indicators, which will have to bedetermined by short-wavelength inversion of reflection ampli-tudes. Thus, it is assumed that short-wavelength impedancevariations provide hydrocarbon indicator information. Themajor focus of this paper is to present the techniques andresults of correlating multicomponent data to obtain long-wavelength V p /V s structure. Inversion for short-wavelengthcomponents of V P /V s is briefly examined in the discussion.

V p

/V s CORRELATION ANALYSIS

An important part of the V P /V S analysis is the use of operators to time scale or transform S-wave reflections toP-wave time before correlation. This section discusses themethodology of the V P /V s analysis and its sensitivity.

Conjugate operators

Many of the calculations done in geophysical modeling andinversion involve conjugate operators (Claerbout, 1992). Sev-eral examples and their conjugate are convolution and cross-correlation, stretching and squeezing, diffraction modeling andimaging by migration, and many others. As applied to theV P /V s analysis, the conjugate of the operator that stretches

P-waves to S-wave time will compress the S-waves to P-wavetime.

Suppose the exact operator to transform S-wave reflec-tions to P-wave two-way time is known (i.e., the averageV P /V s as a function of P-wave two-way time, isknown for every sample). Then, the transformed S-wave trace,b, is given by

(1)where b is the reflectivity of the S-wave trace. The matrix hasa form characterized by

where dots indicate zeros and ones are impulses. Index irepresents P-wave samples and j represents S-wave samples. Itis important to note that nonzero do not occur along adiagonal but have an index ratio i/j that is inversely propor-tional to as a function of P-wave two-way time.

Claerbout (1992) points out that it is desirable to work withoperators whose transpose is their inverse or as near aspossible to their inverse. Although data can be lost in conju-gate transformations as opposed to inverse transforms, theyare in general safer in terms of noise amplification and areoften exact. The operation of the transpose matrix on toobtain the desired inverse gives

(3)where represents pseudodata because high-frequency infor-mation from the original b j has been lost. To demonstrate this,attempting an inverse of equation (1) results in

(4)

Clearly in this form, the system is not invertible becausenumerous zeros along the diagonal in make itsingular. The data lost are those S-wave samples correspond-ing to the columns in that contain only zeros wheresignificant squeezing has occurred. In contrast, it is interestingto note that stretching P-waves to S-wave time, results in

which is invertible.In practice, in equation (2) may consist of interpolation

operators (Sicking, 1980) instead of impulses to reduce high-frequency loss. For linear interpolation, each operator wouldhave two coefficients. Also, if the assumption stated above issatisfied (S-waves have the same wavelength as P-waves), thenno data will be lost within the wavelengths of interest. TheS-wave bandwidth will be transformed or mapped to theP-wave bandwidth; low-frequency S-waves are compressed tohigher frequencies by the transformation.

It should be pointed out that the entire matrix is notconstructed because it contains mostly zeros. Instead, the ithP-wave interpolation operator is applied at the appropriate jthS-wave sample. This eliminates unnecessary multiplications.

Multicomponent correlation

The objective of the correlation analysis is to quantitativelyrelate common P-wave and S-wave reflections with little or noa priori information. It is assumed that the two wavefields havehad appropriate waveform corrections, have been filtered tomatch wavelengths, have been corrected to the same datum,and sample the same subsurface material. The P-wave data aredefined as a reference, and the S-wave data are transformed toP-wave time before correlation. A single transformation takesthe form of equation

(5)

where is defined by a constant (average V P /V s ratiofrom the surface at time zero to the end of record). In terms of equation (2) interpolation operators occur along a fixeddiagonal with slope i/j that depends on and the particularwavefields being analyzed. The relationships for i/j are

(6)

for an SKwave (SV-wave) transformed to P-wave time,

( 7 )


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Multicomponent V P /V s Correlation Analysis 1141

seismology. Note that an analysis between P-wave data andeither SH-wave or PSV -wave data provides about the samesensitivity for the full range of although there is a smalldifference. Below about = 2.5, P-waves and SH-wavesprovide better resolution and above this point P-waves andPSV-waves have better resolution. In the next section, thissensitivity is demonstrated using stacked traces from themulticomponent VSP example.

VSP ANALYSIS

VSPs are excellent data to evaluate this multicomponentcorrelation analysis because of their high S/N and broadbandwidth. From the first arrival times, average V p /V s can becomputed for calibrating the analyses and can provide gooddepth control. Reflections are nearly multiple free because of deconvolution with the downgoing wavefield. Also, the re-flected wavefield is essentially migrated for near source offsetVSPs; reflection points occur near the borehole and so thedifferent wavefields sample the same material. For the con-verted wave VSP, this is not the case.

The field data in this study were acquired in a cased well

located in south Texas. The source consisted of two ARIS units(side by side) and were operated in four orthogonal directions.A three-component sonde with a gyroscope was included formeasuring geophone orientation. A 500 ft (152 m) offsetP-wave, SH-wave, and SV-wave VSP were recorded from adepth of 9000 ft (2743 m) to 1000 ft (305 m) at a 50 ft (15 m)interval. Also, the sources were positioned at 4000 ft (1219 m)offset to acquire a converted P- to SV -wave VSP over the samedepth range (up dip in Figure 1).

Standard processing was performed for multicomponentVSP data. The four directions were summed to obtain P-wavedata and opposite directions were differenced to obtain S-wavedata. Rotation of the three-component seismograms was ac-complished using the gyroscope data. Four component S-wave

rotation (Alford, 1986) was not performed as there appearedto be no azimuthal anisotropy. Upgoing and downgoing waveswere separated with an f-k filter and the reflected wavefieldwas deconvolved using the transmitted wavefield as signatures.These processed data were aligned to two-way time andsummed to provide the VSP stacked traces. The convertedwave data were processed in a similar fashion except thatalignment and summing of the data involved ray tracing usingP-wave and S-wave velocities and a near offset VSP-CDPstack, (Geis et al., 1990). Care was taken to maintain a

Table 1. Values corresponding to equations (ll), (12), and(13) are a relative measure of the sensitivity in resolving

average V P /V s for a fixed time of j = 1. Sensitivitydecreases as increases, and the sensitivity for correlationbetween PSV-waves and H-waves is about half compared toanalyses that use the P-wave.

WavefieldsCorrelated

Average V P / V s

1.5 2.0 2.5 3.0 4.0 5.0- - - - - -

P-wave and SH-wave .444 .250 .160 .111 .063 .040P-wave and PS V-wave .320 .222 .163 .125 .080 .056PSI/-wave and SH- .222 .125 .080 .056 .031 .020

wave

consistent polarity such that positive amplitude reflections repre-sented a positive impedance contrast for the three wave types.

An important aspect of this analysis is that frequencies of theP-wave and transformed S-wave must be the same for optimalcorrelation. This means the wavelengths of the three wave-fields must be the same. Figure 3 shows the VSP stacked tracesfor the P-wave, PSV-wave, and SH-wave. Based on band-passfilter tests, the P-wave consists of 4-40 Hz, the PSI/-wave

4-26 Hz, and the SH-wave 4-16 Hz.

Constant analysis

Figures 4, 5 and 6 show the correlation analyses for theP-wave with PSV-wave, the P-wave with SH-wave, and the

FIG. 3. The (a) P-wave, (b) PSI/-wave, and (c) SH-wave VSPstacked traces used in the correlation analyses for this studyhave been band-pass filtered such that their wavelengths areequivalent. Different time scales are used to enhance the visualcorrelation of similar reflections between the three wavefields.


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FIG . 4. Correlation analysis obtained between the P-wave and PSV -wave is displayed. The PSV -wave is transformed to P-wavereference time using constant average V P /V s conjugate operators and then correlated with the P-wave. Average V P /V Sranges from 2.3 to 2.9 at an increment of 0.005, and correlations are performed every 20 ms over a 50 ms window. A high correlationtrend extending from 0.6 s at = 2.9 to about 2.1 s at = 2.4 represents the average relationship between these twowavefields.

FIG , 5. Correlation analysis obtained between the P -wave and SH -wave is displayed. The W-wave is trareference time using constant average conjugate operators and then correlated with the P-wranges from 2.3 to 2.9 at an increment of 0.005, and correlations are performed every 20 ms over a 50 ms windotrend extending from 0.6 s at = 2.9 to about 2.2 s at = 2.4 represents the average V P /V s relationshwavefields.

nsformed to P-vave. Average V P W. A high correlaip between these

vave

tiontWO


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Multicomponent V P/Vs Correlation Analysis 1143

PSV-wave with SH-wave, respectively. The time axis is two-way vertical time and the horizontal axis is average V P /V s.Color contours indicate the cross-correlation coefficient, rang-ing from -0.2 (dark purple) to 1.0 (white) as shown in thescale.

The results in Figure 4 are obtained over an average V P /V srange of 2.3 to 2.9 at an increment of = 0.005. Timerepresents two-way P-wave reflection time. The PSV-wave

trace is transformed via equations (5) and (7) to for each and then correlated with the P-wave trace (a i) using

equation (9). Correlations are performed using parameterssimilar to high-resolution velocity analyses with an incrementof k = 20 ms and a window length of = 50 ms. Each traceof the analysis is computed for a fixed as a function of time. The fact that this restriction is not necessary and also notpreferred will be discussed later.

A high correlation trend extends from 0.6 s at = 2.9 toabout 2.1 s at = 2.4 that represents the average V P /V srelationship between these two wavefields. Clearly this trend isnot the maxima for a given time or and many spuriousmaxima occur elsewhere. The trend can be enhanced byincreasing the correlation window width, but at the expense of resolution. Note that for a fixed bandwidth, the width of the

ave trend decreases with increasing time. This is in generalagreement with the sensitivity shown in Table 1. To resolve thetrend at later times (2.0 s), a fine increment of = 0.005is necessary; however, this is too fine for earlier times.

An interesting feature is that the trend contains distinct gapswhere the correlation coefficient drops significantly because of differences between the P-wave and PSV-wave reflectivity.Since the VSPs were of good quality and the same wave-lengths, it is likely that the gaps result from differences betweenthe P-wave and S-wave velocity structure. To demonstrate thisfor near vertical incidence and small contrasts in elastic

FIG. 6. Coreferenceranges fro

properties, P-wave reflections can be approximated to firstorder in velocity and density contrasts and (Aki andRichards, 1980) by the familiar relationship,

where is average

density, and is average P-wave velocity.Although converted wave reflection amplitudes go throughzero at vertical incidence, their behavior to first order invelocity and density contrasts is proportional to sin Ap-proximating the reflection coefficient of the sin term forconverted waves in the limit as the incident P-wave anglegoes to zero gives,

(15)

where is the S-wave velocity contrast, is average S-wavevelocity, and is the average V p /V s ratio across the contrast.One can interpret the continuous portions of the trend inFigure 4 a result of similar P-wave and PSV-wave reflectivity;the contrasts and in equations (14) and (15) have thesame polarity and similar magnitudes. In the gaps where thecorrelation coefficient is low, similarity between andmay be poor; polarity may be opposite or one of the contrastsmay be zero. These are precisely zones of interest that could berelated to changes in rock properties or to the presence of hydrocarbons. For the most part, the P-wave and S-waveimpedance contrasts appear to be similar. At least for thisexample, it confirms the mudrock relationship (Castagna et al.,1985) for long wavelengths.

The results between the P-wave and SH-wave in Figure 5are computed in a similar manner as the PS V-wave, exceptusing equation (6). A coherent trend extends from 0.6 s at

rrelation analysis obtained between the PSI/-wave and SH-wave is displayed. The SH-wave is transtime using constant average V p /V s conjugate operators and then correlated with the PSV-m 2.4 to 3.0 at an increment of 0.005, and correlations are performed every 20 ms over a 50 ms wind

formed to PSV- -wavewave. Average V p/Vsow, A high correl ation

trend extending from 0.9 s at = 3.0 to about 3.8 s at = 2.4 represents the-average V P /V s relationship between these twowavefields.


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FIG . 8. Long-wavelength average V p /V s from the three analyses are compared to the average V p /V s determined from VSPtraveltimes of the near offset P -wave and SH -wave direct arrivals. The correlation results between the P-wave and SH-wave agreewith the VSP traveltime data, but results using the PSV -wave do not agree.

FIG . 9. Same as Figure 8 except that the PSI/-wave analyses have been time corrected for dip observed at the borehole. Convertedwave reflections are offset from the well in the up-dip direction in Figure 1 and do not sample the same material as the P-wave andSH-wave VSPs.


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Multicomponent V P /V s Correlation Analysis 1149

strong similarities between the PSV -wave and SH -wave VSPs eltime V p /vs indicate good accuracy. Case studies on surfaceas observed in the correlation analysis in Figure 6. Therefore, seismic data need to be performed to evaluate these methodsit is likely an inversion for the three elastic parameters would in resolving lateral variations in V p /vs associated with hydro-be underdetermined, particularly around a P-wave time of 2 s. carbon deposits.

It is of interest to eliminate this singularity at = 2 becauseit occurs in the region of most importance to explorationseismology. One possible solution is to use other S-waveinformation, such as the slope term, B, from AVO analyses

(e.g., from Aki and Richards, 1980) given by,

ACKNOWLEDGMENTS

(25)

Using equation (25) with P -waves and PSV -waves provides adifferent set of equations to invert for elastic parameters. It canbe shown that a singularity will still exist at = 2. In fact, anycombination of P -wave, B, PSV -wave, and S -wave results inthe singularity. Reparameterization to other elastic variables isalso to no avail. The problem originates from zero-offsetapproximations in defining B and Including higher-orderterms in amplitude reflection equations, (i.e., relying on fartheroffset information) results in shifting the singularity to slightly

higher values of However, estimating these terms isless reliable and subject to poor accuracy.

I thank Dennis Corrigan, John Castagna, and Mike McCor-mack for their suggestions and valuable discussions, and EliasAta for providing technical support. Scott MacKay, SubhashisMallick, and Wendell Wiggins gave many useful comments forwhich I am grateful. I also thank Robert Tathum and DonLawton for their thorough and helpful reviews. Finally, I wishto thank ARCO Oil and Gas Company for permission topublish this work.

REFERENCES

Aki, K., and Richards, P., 1980, Quantitative seismology, 1: W. H.Freeman and Co.

Alford, R. M., 1986, Shear data in the presence of azimuthal anisot-ropy: Diley, Texas: 56th Ann. Internat. Mtg., Soc. Expl. Geophys.,Expanded Abstracts, 476-479.

It appears that only two parameters can be reliably esti-mated, which is not bad in itself. A reparameterization to say,contrasts in P-wave impedance and S-wave impedance, orcontrasts in P-wave impedance and V P /V s ratio is possible. Inthese cases, density is removed from the problem and incor-porated into the remaining parameters. A contrast in tofirst order is the sum of the contrasts in P-wave and S-waveimpedance.

Behle, A., and Dohr, G., 1985, Converted waves in explorationseismics, in Dohr, G., Ed., Seismic shear waves, Part B: Applica-

tions: Geophys. Press, 15B, 178-223.Castagna, J. P., Batzle, M. L., and Eastwood, R. L., 1985, Relationshipsbetween compressional- and shear-wave velocities in elastic silicaterocks: Geophysics, 50, 571-581.

Claerbout, Jon. F., 1992, Earth soundings analysis: Processing versusinversion: Blackwell Scientific Publ.

Garotta, R., 1985, Observation of shear waves and correlation with P events, in Dohr, G., Ed., Seismic shear waves, Part B: Applications:Geophys. Press, 15B, l-86.

For land and ocean bottom cable applications, the mostfeasible approach economically for multicomponent data, is tocollect P-wave and converted wave data using three-compo-nent receivers and conventional P-wave sources. These two

wavefields could provide a reliable two parameter inversion inthe absence of strong azimuthal anisotropy. At present, col-lecting S-wave source data are prohibitively expensive.

Geis, W. T., Stewart, R. R., Jones, M. J., and Katopodis, P. E., 1990,Processing, correlating and interpreting converted shear waves fromborehole data in southern Alberta: Geophysics, 55, 660-669.

Justice, M. G., McCormack, M. D., and Lee, S. S., 1987, Anisotropy inthe Morrow formation of southeast New Mexico, in Danbom, S. H.,and Domenico, S. N., Eds., Shear-wave exploration: Geophysicaldevelopment Series, 1: Society Expl. Geophys. 154-164.

McCormack, M. D., 1990, Detection of subsurface anisotropy: U.S.Patent No. 4947381.

CONCLUSIONS

The analyses presented in this study demonstrate a robusttechnique for determining long-wavelength relation-ships between P-waves and converted waves or S-waves.Correlation analyses are parameterized by average V P /V susing conjugate operators. Aided by conventional P-wavevelocity information and empirical rock-physics relationships,this technique provides optimal estimates in a similarmanner that semblance analyses provide stacking velocities.Average as a function of two-way time can be measuredand used to transform (align) converted wave or S-wave datawith P-wave data for multicomponent interpretation or short-wavelength amplitude inversion. Comparisons with VSP trav-

McCormack, M. D., Dunbar, J. A., and Sharp, W. W., 1984, A case

study of stratigraphic interpretation using shear and compressionalseismic data: Geophysics, 49, 509 -520.Ostrander, W. J., 1984, Plane-wave reflection coefficients for gas sands

at nonnormal angles of incidence: Geophysics, 49, 1637-1648.Pardus, Y. C., Conner, J., Schuler, N. R., and Tatham, R. H., 1990,

and lithology in carbonate rocks: A case study in the Scipiotrend in southern Michigan: 60th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 169-172.

Pickett, G. R., 1963, Acoustic character logs and their applications information evaluation: J. Petr. Tech., 15, 650-667.

Robertson, J. D., 1987, Carbonate porosity from S/P traveltime ratios:Geophysics, 52, 1346 -1354.

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

Sicking, C. J., 1980, Sampling requirements for reflection seismogramsin geophysical data acquisition: Ph.D. Thesis, Univ. of Texas atAustin.

Smith, G. C., and Gidlow, P. M., 1987, Weighted stacking for rock propertyestimation and detection of gas: Geophys. Prosp., 35,

Swan, H. W., 1991, Amplitude-versus-offset measurement errors in afinely layered medium: Geophysics, 56, 41-49.

Toldi, J. L., 1989, Velocity analysis without picking: Geophysics, 54,191-199.

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