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Automatic velocity analysis of crosswell se ism ic data

Guoping Li and Robert R . S tew art

ABSTRACT

A new method for velocity analysis of crosswell seism ic data is discussed in thispaper. Based on semblance analysis, this method derives velocities from crosswell directarrivals in an autom atic m anner and so avoids tim e-consum ing hand picking of traveltim es.T o develop the m ethod, an isotropic continuous elastic m edium w ith a linear velocity-depthrelationship is assumed. Two steps are involved. In the first step , traveltimes for directarrivals are calculated for different velocity guesses, using theoretical eq uations w e derivefor a linear velocity function. T heoretically calculated traveltim e trajectories of directarrivals have been found to exhibit a quasi-hyperbolic pattern, one characteristic appearingon real crosswell data. A numerical study shows that they agree very well w ith thosem easured from synthetic data. In the second step, coherency (sem blance) analysis is donefor amplitudes of direct arrivals within a time window along each travehime trajectorycalculated for different velocity guesses. The velocity with the largest semblance value isthen picked and used as final inversion output.

Examples of inverting velocity information from direct arrivals in synthetic,physical modeling, and field crosswell seismic data gathers, by using the new velocityanalysis method, are given. R esults prove this method to be a potential velocity inversiontech niq ue w ith efficiency and reliability .

INTRODUCTION

Crosswell seismic data are acquired by using two (or more) drilling wells: shootingthe seismic source in one well, and recording the propagating waves in the other well offsetby a certain distance. In crosswell surveys, seism ic waves usually are generated andrecorded below the highly attenuative near-surface zone and travel a relatively shortdistance between the wells, so high-fidelity data can be obtained. Therefore, compared toother seism ic inform ation sources (surface seism ic or VSP), high resolution crosswellseism ic data should provide more reliable inform ation regarding subsurface physicalparam eters such as velocity.For many years, attempts have been made to obtain velocity information fromcrosswell seismic data using tomographic inversion methods (B ois et al., 1972; Ivansson,1985; Peterson et al., 1985; Bregman et al., 1989; Lines and LaFehr, 1989; A bdalla et al.,1990; Lines and Tan, 1990; S tewart, 1990). However, it has been recognized that mostcurrent tomographic techniques involve time-consum ing hand picking of traveltim es.W hen hundreds of crosswell data gathers, for exam ple, need to be analysed, hand-pickingof traveltimes is not efficient.

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In this paper, we will discuss a different crosswell velocity analysis method whichmak es use of the basic concept of coherency-based velocity analysis used for conventionalsurface seism ic data. we assum e that a linear velocity variation with depth is good enoughto describe the intervening m edium betw een w ells. T raveltim es are calculated theoreticallyfor crosswell direct arrivals for each velocity guess. W ithin a fixed time window along thecalculated travehim e trajectory, seism ic traces are scanned to find the best coherency(largest sem blance value). T he procedure is repeated until all possible velocity guesses aretried. A s a result, the largest semblance value corresponds to the best-fit velocity. S incesemblance analysis can be done automatically, the whole velocity analysis is done in anautom atic m anner, thus avoiding the process of hand-picking lraveltim es w hich is requiredby traditional tomographic methods. In the following sections, we will discuss thisautomatic velocity analysis method in detail. Some exam ples of application to variouscrossw ell seism ic datasets w ill be given.

THEORY

Linear gradient m edium

In seismic exploration, it is significant to make a proper assumption about theseism ic wave velocity distribution in subsurface solid materials. A lthough adopted attimes, the assumption of homogeneous isotropic media is a rough approximation to theactual earth. However, the concept of gradient m edium , where velocity changes accordingto some sim ple m athematical function of distance from a reference plane, has been foundm ore useful (Helbig, 1990). In such m edia, rays are curved everywhere, but are sim ilarfor all ray param eters. M oreover, w avefronts are of sim ilar shapes for all traveltim es.Particularly, a linear-gradient medium has been investigated by many geophysicists

and found to be a very good approximation to the real solid earth. Slotnick (1959) wrotethat the velocity of seismic wave propagation in Tertiary basins can be closelyapproximated by expressing it as linear function of depth. He gave some examples ofareas, including the Gulf of M exico, San Joaquin Valley of C alifornia, and a Venezuelabasin, where 'one can safely assume a linear velocity relationship with depth'.Northwards, Jaln (1987) finds, after inspecting sonic logs from western Canada, that mostlogs in the western Canadian basin justify a linear increase in velocity with depth down tothe Paleozoic unconform ity. The values of the velocity gradient he obtains from theCretaceous section range from 0.25 to 1.0 ft/sec/ft.

A commonly-used expression for the linear velocity relation with depth is

v(z)=v0+_z, (1)where Vo is the initial velocity (ft/sec or rn/sec), r is the velocity gradient (ft/sec/ft, orm/sec/m, or 1/second), Z is depth (ft or m). The value of r indicates an increase (when xis positive ) or decrease (when t is negative) in velocity per unit of depth.

Som etim es the linear velocity function is expressed in the alternative formV(Z)Vo(1NZ). (2)

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Here 7/is called velocity gradient factor, and its dimension is (feet) 1 [or (meters)t].C om paring equations (1) and (2), w e have_c= VOrl. (3)

In this pap er, relation (I) is used.Traveltime equations for direct arrivals

In Appendix I, we have discussed some fundamental characteristics of propagationof crosswell direct waves in the isotropic elastic medium where the velocity function hasthe form of equation (1). In this kind of m edia, seism ic waves propagate along circularraypaths whose radii and centers are closely related to velocity parameters.A lso in A ppendix I, we have derived expressions for direct arrival traveltimes,namely, equation (A-I-4b) and equation (A-I-1 lb). In fact, they are equivalent except forthe difference in sign. So we can write them into a combined form since traveltime isalways positive:

= 1 in (Vo + t_Z)(_/1 - P2(Vo+ KZs)2+ 111_ (Vo+r_Zs)(41_p2(Vo+r.Z)2 . (4)where t - tmveltime for direct arrivals;Z - depth of the receiver,Zs - depth of the source; andp - ray parameter.

To calculate traveltimes, we need to know the unknown parameter p in equation(4). W e have known that the paths of direct seism ic waves, traveling in a linear velocitygradient medium between the recording well and the source well, are circular arcs.Therefore they m ust satisfy the equation for a circle. M athem atically, two points (a pair ofsource and receiver positions) cannot determine a circle uniquely and sufficiently. But it isknown from Appendix I that the vertical coordinate of the center of a cricular raypath is aconstant determined by given velocity parameters. Then for some circular ray connectingthe source point (Xs, Zs ) and the receiver point (XR, ZR), we have

(Xs- Xc)2 + (Zs- Zc}2 = R2 , (5a)

(XR-XC)2 + (ZR-ZC)2 = R2 , (5b)where Xc, Zc -centerfthecircularrc;ndR - redius of the circular arc.S olv in g (5 ) g iv es

Xc-XR2-xs2+(ZR-C)-(Zs-C)2(XR- Xs) ' (6a)

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a n d

R = 4(Xs- Xc}2 + (Zs- Zc}2 (6b)Therefore the ray parameter p for this particular circular locus is given by

p= !RK (7)With the parameter p determined, we are now able to calculate the taveltime from the sourceto th e re ce iver.

The discussion in Appendix I tells us that if the initial emission angle of themyceo < 900 we need to take into account the effect of diving waves on lraveltimes. Onemethod (as used by Slotnick, 1959) is to find the deepest my penetration point and do thetraveltime integration from the source position to this point and then from this point to thereceiver position. The deepest penetration point can be readily found by setting X = Xc ineq uation (A -I-5 ), w hich g ives

Zm=- 1 V0=R_V0 .rp _ K (8)Now, the traveltime can be calculated via

t= dz _ dz_v0_/_v0+_zl, _/_lVo+rCZ_-_ RI( ! , (V0+K:Z RK P,vo+,(q,.ivo+z,.i,1)1 In _ RK ]_ _vo+_z_,(ql.(Vo_q1)

+ 1 in (Vo+ lcZmax)(q 1- ('VOR_ZR-)Z+ 1)_ ,vo+z.,(ql_(%_l) (9,S ubstituting equation (8), the above equation is reduced to

=1111 R_(_/1- (-VR_S-; + 1) RK(_/1- (_R-)2 + I)+llnr (Vo+_Zs) ) (Vo+KZR) (10)When ZR = Zs, we have

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t=2 In _ R_: /_c (V0+ rZs) (11)Slotnick (1959) and Grant and West (1965) obtained the same result for the case ofZR = Zs = 0.

The method to determine when equation (4) or (10) should be used to calculatetraveltimes is to know the depth at which the ray would leave the source at a 90 angle.This depth can be easily found. In crosswell seismic surveys, the separation distancebetween the wells, X, is usually fixed, and X = XR - Xs (in the vertical borehole case).For ao = 90 , P = 1/(V0 + _Zs). Then from equation (A-I-5) or (A-I-12), we have2, (12)which g iv es one solution

Z .//V0+rZs__x 2 V01 ='Vl _ 1 -_- (13)Now, if the receiver depth ZR is less than or equal to Z1, then equation (4) shouldbe used. IfzI

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The automatic velocity analysis method comprises two steps. In the first step,direct arrival traveltim es are calculated at all receiver (or source) positions in a givencrossw ell g eom etry, using equations (4) and (10), or eq uation (14). A ll possib le velocityguesses are tried, resulting in a set of traveltim e curves (trajectories). T hen in the secondstep, seismic traces in a crossweU data gather are scanned within a time window along eachtraveltim e trajectory calculated in the fkrst step. B ased on this, sem blance analysis is thenconducted.

Semblance analysis

A useful measure of coherency of signals is their energy. According to Yilmaz(1987) and K rebes (1989), the output energy is defined as

t(i)+At ( N )2Eout = t_t(i_).At ,i=_1 fi, t ,- (15)and the input energy as

t( t )t=t(i).At \i=l (16)where

i -the i th seismic trace;t(i) - the traveltime corresponding to the ithtrace;fi,t - the amplitude of the ithtrace at time t(i) within window [-At, At]; andN - the total number of traces involved.

Therefore, the semblance value can be found fromSo= Eout 0

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CROSSWELLGEOM'- I MY

ALLaATH_____. SINGLE DATAGATHERMANYF.LOCmESA.Dal_u)r,E_S GUESSVELOCITY

CALCULATETRAVELT IME TRAJECTORY

STA CK A MPLITUDES ON-- T RA JE CT OR Y, S EM B LA NC E

P IC K V EL OC ITYWITHLARGEST SEMBLANCE

_L[ EACT -1NTERVAL VELOCn'YFIG. 1. Processing procedure for automatic velocity analysisof crossw ell seism ic data.

En ter cro sswell g eometry; Input a crosswell data gather (common source or common receiver); Scan through a range of t0andV0 values;- C alcu late trave lfim e, t(i), fo r ev ery trace ;

- S tack the trace am plitudes along the calculated travelfim e trajectory, w ithina given time window [-A t, A t];- C alculate sem blance value;

Pick the velocity w hich corresponds to the largest sem blance value; D o this whole procedure for next gather from a different depth aperture, and find anew velocity function; until all g athers are processed.

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Extract the f'mal interval velocity distribution by fitting all velocity functionsobtained from different depth apertures.

TESTING

T he autom atic crossw eU velocity analysis m ethod has been tested with different datasets. In this paper, we give some examples of applying this method to synthetic data,physical m odeling data, and a field crossw ell data gather.

Synthetic data

Synthetic erosswell seismic data used to test our method were generated with theUNISEIS ray tracing program. The geologic model we used is shown in Figure 2. It iscomposed of 71 horizontal layers w ith equal thickness (20 m) in each layer. Velocitydistribution in this m ultiple layered m odel obeys a linear velocity-depth relationship asfollowsV = 2000 + 0.8Z (m/s), (18)

but in individual layers, velocities are constant and take the value calculated using (18) inthe m iddle of each layer.T o record crossw ell seism ic d ata, sou rce and receivers w ere p osition ed respectivelyon the two sides of the model, which were separated 500 m. Receiver depths ranged from0 m to 1200 m . Common shot gathers were collected when the source was 'excited' atdifferent depths. Shot gathers contain 121 seism ic traces each. Only rays for directarrivals were traced for our purpose. During ray tracing, the source was fired at depths of

0 m , 260 m , 500 m , 760 m , and 1000 m, separatively. To generate crosswell sections,zero-phase R icker wavelets were used. The wavelets were 60 ms long and had a centerfrequency of 40 Hz. Shown in Figures (3) and (4) are two shot gathers corresponding tothe source depths of 0 m and 500 m, respectively. Direct arrivals are displayed clearly inboth figure s.

T o test precision of the traveltim e equations we derived, theoretical traveltim es w erecalculated for one common shot gathers (source depth is 500 m ), using equations (4) and(10). Their com parison with traveltim es m easured on the synthetic section (Figure 4) isgiven in Figure 5. It can be seen from Figure 4 or Figure 5(a) that the traveltime curve hasa quasi-hyperbolic shape, w hich is observable on real crossw ell data. Figure 5(b) revealsthat the difference between the theoretical and the traced traveltim es is very sm all (themaximum absolute differential time is about 3 ms), indicating that our traveltime equationsp rovide sufficient p recision. It im plies that if correct velocity is used, traveltim e trajectoryof real crosswell data can be approximated with high precision. This provides the basis forau toma tic v elo city an aly sis.

T o run the autom atic velocity analysis program , crosswell parameters for a shotgather must be provided. A wide range of initial velocities and of velocity gradients isscanned for the best selection w hich agrees w ith the largest sem blance value.For the synthetic data in hand, we scanned for the best velocity from 1900 m/s to2100 m/s (at interval of 10 m/s) and for the best gradient from 0.5 m/s/m to 1.1 m/s/m (at

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interval of 0.05 m/s/m). A time window equal to the length of the Ricker wavelet used wasopened for semblance analysis. T he results of semblance analysis were finally contouredfor the ease of picking the largest values. Figure 6 show s the result of semblance analysisfor the shot gather for source depth of 0 m. The largest semblance value appears at thepoint where the initial velocity is 2000 m/s and velocity gradient is 0.8 rn/s/m. Theinversion m ethod has reconstructed the velocity function used to generate the geologicmodel. The same results are obtained from semblance analysis of shot gatherscorresponding to source depths of 500 m (Figure 7), and 760 m (Figure 8). Thus, velocitydistribution in a layered model can be confidently recovered by applying the automaticvelocity analysis m ethod to crossw ell data.

However, small errors in velocity analysis are also noted. Figure 9 shows thesem blance analysis result for the gather with the source 260 m deep. Largest semblancevalue is obtained when initial velocity is 2010 m/s and gradient is 0.8 m/s/m. Figure 10shows the semblance analysis result for the data gather at the source depth of 1000 m . A tthe point at which the initial velocity is 1990 m/s and the gradient is 0.825 m/s/m, thereexists the largest sem blance value. T hus, the final inversion results from analysis of thesetwo gathers have a shift, but very sm all, from the actual velocity function.

Physical modeling data

Physical modeling data were from an ultrasonic borehole seism ic modelingexperiment accomplished at the U niversity of Calgary (S tewart and C headle, 1989). 40crosswell shot gathers, each having 40 traces, were collected in a geometry whereultrasonic source and receiver transducers were deployed along the two sides of a targetm odel (a Teflon cylinder 3.81 cm) located in water tank. Source spacing and receiverspacing were 50 m , and the well separation was 600 m, in a scaled distance. Two shotrecords with the source shot at 0 m and 1000 m are shown in Figure 11. Note that thedirect arrivals at several traces in the middle of Figure 11 (b) are pulled dow n because of thevelocity of the T eflon m odel lower than that of the surrounding w ater.

The automatic velocity analysis method was applied to three gathers for sourcedepths of 0 m, 450 m, and 1000 m, respectively, in order to see whether or not thebackground velocity (that is, water velocity which is around 1490 m /s) can be invertedfrom direct arrivals. The guessed initial velocities ranged from 1000 m/s to 1800 m/s atinterval of 20 m/s. The velocity gradient was guessed between 0.00001 m/s/m and 0.01m/s/m. The length of the wavelet, which was used as width of the time window forvelocity analysis, was 30 m s.

Figuers 12-14 show the results of sem blance analysis for the gathers from differentsource positions. In Figure 12, the best velocity should be picked from where the initialvelocity is 1520 m/s and gradient is 0.0095 m/s/m. In Figure 13, the best inverted velocityis formed by the initial velocity of 1520 m/s and the gradient of 0.0085 m/s/m. Figure 14shows that the largest semblance value corresponding to the initial velocity of 1520 m/s andvelocity gradient of 0.0065 m/s/m gives the best selection for velocity. From the aboveresults, the initial velocity inverted is consistently 1520 m /s, w ith a 2.0% difference invalue from the real velocity of 1490 m/s. The gradient value obtained from the velocityanalysis change from 0.0065 m/s/m to 0.0095 m/s/m, causing velocity variation by 13 ~ 19rn/s within the depth aperture of 2000 m. Thus, the velocity changes caused by the velocitygradient is neglegible. In short, the velocity inverted from the physical m odeling crossw elldata using our m ethod is very close to the actual m odel velocity.

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Field data

The real crosswell seismic data were acquired in Humble, Texas (courtesy ofTexaco Inc.). In Figure 15, a common-receiver gather is shown, which is composed of113 traces, representing a depth aperture of 300 ft (91.5 m) to 2540 ft (774.4 m). Thereceiver was at the depth of 1500 ft (457.3 m ). W ell-to-well separation was 815 ft (248.5m ). Seism ic traces were recorded at sam ple interval of 0.25 m s. P-wave direct arrivals,denoted with the letter D, has a quasi-hyperbolic traveltime trajectory, as mentioned before.The hyperbolic trajectory is not symmetric partly because of variations of velocity in thesubsurface, and partly because of crosswell recording geometry. It can be seen in Figure15 that the data contain very strong tube wave energy, which dominates the record.

T he autom atic velocity analysis program searches for the optim al velocity in thatparticular depth aperture by scanning velocity in a range of 4900 ft/sec to 16500 ft/sec(interval 200 ft/sec) and gradient in a range of 0.045 ft/sec/ft to 2.0 ft/sec/ft (interval 0.1ft/sec/ft). Time window of 14 ms wide (approximately wavelet width) was used to scanseism ic traces around direct arrivals. Figure 16 is the result of velocity analysis. V elocityis picked at the initial velocity of 6100 ft/sec and gradient of 0.145 ft/sec. From the newvelocity function, velocity changes from 6143.5 ft/sec to 6468.3 ft/sec in an aperture from300 ft to 2540 ft. Although unfortunately we currently do not have other velocityinformation in this area to confirm the inverted velocity function, velocities we have shownap pear to b e reaso nab le.DISCUSSION

The inversion problem may not be unique. For example, in Figures 6, 9 , and 10,there are several local highs within a narrow belt in the semblance map. These local highshave values very close to the one that we picked, making interpretation difficult. Thisproblem arises because com binations of sm aller initial velocity and larger gradient or oflarger initial velocity and smaller g radient may generate the same effect as does thecombination of the correct initial velocity and gradient. Therefore, precaution must betaken in interpreting the inversion results. Fortunately, in our case, the semblance valuewhich gave the inverted velocity is after all larger than these highs around it.

Problems may also be caused when the available crosswell data are noisy. Noisehas an obvious effect on the results of semblance analysis because in addition to signals,noise within the given time window is also involved in calculation of semblance values [seeequations (15) and (16)]. T hus, application of band-pass filters to crosswell seism ic data,prior to velocity analysis, is recommended. It is also found that inversion result w ill begood if the width of time window is close to the wavelet length. It w ill be useful to apply awavelet shaping process to real data to make wavelets consistent from trace to trace.

In this paper, we did not discuss the case of negative velocity gradient, a case thatmay exist in some areas. B ut as we can see, it is not difficult to generalize our discussion.The basic idea we have developed here still applies, except for some modifications intraveltime equations we derived previously. M athem atically, it is not difficult either togeneralize our discussion to the case of deviated wells. This velocity analysis m ethod canalso be applied to S-wave crosswell data.

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CONCLUSIONS

It is possible now to automatically derive velocity from crosswell seismic data, in asim ple but efficient w ay w ith the velocity analysis m ethod w e have discussed in this paper.Without having to pick traveltimes by hand, this method estimates a velocity distributionfrom crosswell direct arrivals by conducting an automatic semblance analysis for seismictraces around traveltimes of direct arrivals theoretically calculated, assuming a linearvelocity-depth relationship. This method has been tested on a num ber of crosswell datagathers. Inversion results are satisfactory.

FUTURE WORK

The automatic crosswell velocity analysis method discussed here has shown anexceptionally encouraging perspective. B ut as this m ethod is still in the early stage, moreresearch is required to im prove it.It is not unusual that in some areas, the subsurface velocity is distributed withseveral velocity g radients w ithin different depths. S uch m ultiple gradients in the sam e areawould cause the direct arrivlas to behave differently than w hat we have dealt with before.In this situation, direct arrival events may no longer be described by the smooth quasi-hyperbolic trajectory which is obtained from our traveltime equations. Therefore, in suchcases, multi-scan m ay be necessary. That is, the entire depth aperture of interest m ay needto be divided into a number of sub-apertures, w ithin each of which, velocity analysis isconducted using our method. Inversion results from these sub-apertures are finallycomb in ed to ge th er.B esides, it is worth investigating other velocity-depth functions, which may be

more accurate and more suitable to describe subsurface velocity patterns than the linear onewe have used in this paper.S ince random or coherent noise has a strong influence on the result of semblanceanalysis, we m ay not always be able to obtain reliable velocity inversion results. Thus,som e noise-resistant coherency m easures m ay need to be considered.

ACKNOWLEDGEMENTS

We are very grateful for technical help and valuable suggestions of M. Harrison, H.Bland, D . Easley, D r. D . Lawton, T . Howell, E. Gallant, S . M iller, J. B aerg and Y. Song,during this research. T he generous donation of real crosswell seism ic data by Texaco Inc.(USA) is sincerely acknowledged. W e also wish to thank all sponsors of the CREW ESProject for their continuous support. This research was supported by the CREW ESProject.

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REFERENCES

Abdalla, A.A., Stewart, R.R., and Henley, D.C., 1990, Reflection processing of crosswell seismic data:Midale field, Saskatchewan: CREWES Project Research Report, The University of Calgary, Vol.2, 225-259.

Baerg, J.R., 1985, Analysis of cecrnst seismic refraction and wide angle reflection experiments fromsouthern Saskatchewan and Manitoba: M.Sc. thesis, The University of Western Ontario.Baerg, J.R., 1991, Traveltime in media with linear velocity-depth functions: this volume.Bois, P., La Porte, M., Lavergne, M., and Thomas, G., 1972, Well-to-well seismic measurements:

Geophysics, 37, 471-480.Bregman, N.D., Bailey, R.C., and Chapman, C.H., 1989, Crosshole seismic tomography: Geophysics,54, 200-215.Grant, F.S., and West, G.F., 1965, Interpretation theory in applied geophysics: McGraw-Hill Book

Company, New York.Helbig, K., 1990, Rays and wavefront charts in gradient media: Geophysical Prospecting, 38, 189-220.Ivansson S., 1985, A study of methods for tomographic velocity estimation in the presence of low-velocityzones: Geophysics, 50, 969-988.Jain, S., 1987, Amplitude-vs-offset analysis: A review with reference to application in western Canada: J.

Can. Soc. Expl. Geophys. 23, No. 1.Krebes, E.S., 1989, Geophysical dataprocessing: Course Notes. The University of Calgary.Lines, L., and Tan, H., 1990, Cross-borehole analysis of velocity and density: Expanded Abstracts, 1990

SEG meeting, San Francisco.Lines, L.R., and I..aFehr,E.D., 1989, Tomographic modeling ofa cross-borehole data set: Geophysics, 54,1249-1257.Peterson, J.E., Paulsson, B.N.P., and McEvilly, T.V., 1985, Applications of algebraic reconstruction

techniques to crosshole seismic data: Geophysics, 50, 1566-1580.Slomick, M.M., 1959, Lessons in seismic computing, Ed. by Geyer R.A., Society of ExplorationGeophysicists.Stewart, R.R., 1990, Exploration seismic temography: Fundamentals:Continning Education Course Notes,

Society of Exploration Geophysicists.Stewart, R.R., and Chcadle, S.P., 1989, Ultrasonic modeling of borehole seismic surveys: CREWESProject Research Report, The University of Calgary, Vol. 1,206 - 224.

Telford, W.M., Geldart, L.P., Sheriff, R.E., and Keys, D.A., 1976, Applied Geophysics: CambridgeUniversity Press.Yilmaz, O., 1987, Seismic data processing: Society of Exploration Geophysicists.

APPENDIX I

WAVE PROPAGATION IN A LINEAR GRADIENT MEDIUM

In this section, we will examine some fundamental characteristics of seismic wavepropagation in a linear velocity-gradient medium. Because we concern ourselves mainlywith applications to crosswell seismic data, we choose to consider the crosswell surveyinggeometry. In addition, for our purpose, we consider direct arrivals only. Therefore,relevant expressions for raypaths and travehimes of direct waves will be derived.

Let us first establish a Cartesian coordinate system such that the X-axis is on the flatsurface of the earth, the Z-axis is along the symmetrical axis of a vertical borehole, and theorigin of the coordinate system is at the wellhead. Suppose that the source S is at (0, Zs),where Zs >-0, and an arbitrary point R is at (X, Z). A seismic wave leaves the source S at

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Athe angle Xoo the vertical axis, and travels to R along a curved raypath SR. The geometryis shown in Figure A -I-1.

01 V=Vo+ tcZ _XSource lS(O,Zs) _._o

Z

FIG. A -I-1. Geometry showing a seismic ray leaving the source andpropagating in a continuous m edium .

The energy generated by the seismic source will radiate outwards in all directions.In crosswell surveying geometries, angles of the rays emitted from the source rangebetween 0 and 180 . W hen the emission angle (7-0is 90 , the ray leaves the sourcehorizontally, and then gradually turns upward. However, when the angle is less than 90 ,diving waves would be expected to occur. Therefore, it is necessary to discuss two cases:1) o,090.

Case I: Ray emission angle not larger than 90

This is the case in which the seismic wave, upon leaving the source, travelsdownward into the medium below the source, or in a special situation when the angle ofemission is 90 , it leaves the source horizontally and then travels upward. Let us look at aninfinitesimal segment of the ray, dl (see Figure A-I-l). It makes an angle o_with thevertical axis. W e assume that all positive angles are measured counterclockwise from thevertical axis. This segment has a vertical component of dz and a horizontal component ofdx. The following relations can be found:

dx = tanot, dl= dz , dt = dldz 41- (sincz)2 V " (A-I-I)

The velocity V and the depth-dependent angle _zare related by Snell's law:sinctP=V' (A-I-2)

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which is the law governing wave propagation along a least-time path. Here, the rayparameter p is a constant which depends upon the direction in which the ray left the source,that is, upon the angle a0. By integrating equation (A-I-l) and substituting equation (A-I-2), we get two integral equations for the total horizontal distance X, and total Iraveltime t:

f_ pVdz= a/1 - p2V2 (A-I-3a)

t = V'f_- p2V2 " (A-I-3b)C learly, equation (A -l-3a) describes a fam ily of curved raypaths, characterized by thecorresponding values of the ray parameter p. Since p affects the raypaths, the traveltimesgiven by (A -I-3b) differ from one path to another. The assumption of linear velocitygradient, that is, equation (1) in the text body, leads to solutions of the above integrals:

X = 1 (41- pZ(Vo+ gZs) 2- _/1- p2(Vo+ gZ)2 ) ,_cp (A-I-4a)

t = 1 In (Vo + KZ)(41- p2(Vo+ KZs)2 + 1)1< (Vo+ rZs)(a/1 - p2(V0+ KZ)2 + 1) " (A-I-4b)

The solutions (A-I-4a) and (A-I-4b) can also be expressed in a different approach (Telfordet al., 1976; Baerg, 1985).The raypath given by equation (A-I-4a) is a circle in the X-Z plane; this can beshown by rearranging terms in equation (A-I-4a):

x 41-p vo+ s-p (A-I-5)The center of the circular raypath is at C(Xc, Zc), where

Xc =dl- p2(Vo + _Zs)2 ,lop (A-I-6a)Zc=. vo , (A-I-6b)

and the radius R is

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4 4 1

R=| rp (A-I-7)T herefore, in a linear velocity-gradient m edium , seismic w aves travel along circularraypaths, characterized by equation (A-I-5). Figure A-I-2 shows a seismic ray leaving the

source at the angle tt0and traveling along a circular path. The center, C, of the circular raylies above the earth's surface a distance V0/t.

c (xc,zc)

Z

FIG. A-I-2. Circular raypath leaving the source S at the angle eto .

We find, in equation (A-I-6b), that the vertical coordinate of the center, Zc, isindependent of ao. The value of Zc is determined by a given velocity function alone andthus is a constant. This means that the centers of all circular rays lie on the same horizontalline. This line is located where the velocity would be zero if the velocity function wereextrapolated up to an elevation where Z = -V0/r (Telford et al., 1976).

Furthermore, since parameters V0 and r, and ray parameter p, are given positive,the centers of those rays are all located within the (+X, -Z) quadrant of the coordinatesystem . Equation (A -I-7) indicates that the radius of the circular ray depends upon the rayparameterp. Figure A-I-3 shows schematically some of the circular raypaths whose radiiare different .

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442

CentersCl C2 C3 C4 C5-- -0--0... --0-0-- -- --

_XSource

1

FIG . A -I-3. C ircular raypaths with different radii and their centers.

From equation (A -I-6a), w e can see that the horizontal coordinates of the centers ofthe circular rays are determined by non-negative values of Xc. In particular, when theemission angle ao=90 , the raypath is such a circle whose center is at (0, Zc). This can beshown by substitu ting

sina0 = sin90 o = 1P = Vo +rdZs Vo +_Zs Vo +lcZs ' (A-I-8)into equation (A-I-6a). This situation is shown in Figure A-I-4. We see that at this time,no effect of diving w aves (or turning w aves, G rant and W est, 1965) occurs.

From Figure A-I-4, it can be predicted that when % > 90 , the centers of rays willbe moved into the (-X, -Z) quadrant of the coordinate system. This will be discussed in thenex t sec ti on .

Case II: Ray emission angle larger than 90

In this case, the wave travels upward along a curved raypath as shown in Figure A-1-5. Let us discuss this situation and derive form ulae for the raypath and traveltime. T hederivation can be accomplished in a similar approach that we have used before. But note

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44 3

ZS_ _ SurfaceP, XCircular Arc so = 900

Z

FIG. A-I-4. When the emission angle of the rayis 90 o, the center ofthe circular raypath is on the vertical axis.

that the ray's angle a, in this case, is always greater than 90. From Figure A-I-5, we seethat0=c_90 , (A-I-9)

therefore,dx = cotan0 = cotan(ct - 90) = - tanct = - sinet ,dz 4 1 - (sinot)2 (A-I- 10a)

dt = dl = 4(dx) 2 + (dz)zV V (A-I-10b)Applying Snell's law (A-I-2) and the linear velocity function (Eq.1), and integratingeq uations (A -I-1 0a) and (A -I-1 0b ) result in the follow ing relations:

X = _ (41- p_{Vo+ 1Z)2- 41- p2{Vo+ KZs)2 ) (A-I-I la)t =lln(Vo+ 1Zs)(41p2(Vo r_Z)+1)

_ {Vo+KZ)(_/1 -p_{Vo+_Zs} z + l) " (A-I-lib)R earranging the term s in equation (A -I-1 la), we have

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44 4

0 _XR(X, Z)

Source

I

Z

FIG. A-I-5. Geometry showing a seismic ray leaving the sourceand traveling upward in a continuous medium.

Centers [

o T

SourceZ

FIG. A-I-6. Circular raypaths and their centers.

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44 5

[X+ _/1-p2(V+r'Zs)2 _+(z+V)2= 1lp K K2p2 (A-I-12)Equation (A-I-12) tells us that the resulting raypath is a circular arc, whose center andr ad ius a re , r espect ively,

Centerat:C(.a/1-P2( V+_Zs)2 ,.V0) ,pand

Radius: R = _1 _lcpAs can be seen, when the velocity gradient r > 0, the center of the circular path isalways located within the coordinate quadrant (-X, -Z). The vertical coordinate of thecenter is independent of the ray parameter p while the radius varies with it. Again, thecenters of all possible rays lie on the same horizontal line above the X-axis (the earth'ssurface) a certain distance. FigureA-I-6 shows some raypaths and their center positions.In summary, seismic waves propagate along circular raypaths in a medium of avelocity increasing linearly with depth. The radii of these circular rays are inverselyproportional to the product of the ray parameterp and the given velocity gradient r. All thecenters of these rays lie on the same line which is above the surface of the earth a distancedetermined by the two parameters of the given velocity function. When the emission angle

ao, atwhich seismic rays leave the source position, is greater than 90, rays travel upwardand the centers of circles are located within the (-X, -Z) zone. When _x0

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446

SO0 meters _-o

20 0

400I , , . , ,G J

e-- 4_Q.a 1000

1 2 0 0

1 40 0

FIG . 2. Layered model used to generate synthetic crosswell seism ic data.Each layer has the same thickness. Velocities in all layers satisfya linear velocity-depth relationship: V = 2000 + 0.8 Z (m/s).

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Depth (me te r)

1.0

FIG . 3. Synthetic crossw ell shot gather for source depth = 0 m . (O nly direct arrival iss hown h er e.)

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Dept h (met er )

1.0

FIG . 4. Synthetic crossw ell shot gather for source depth = 500 m . (O nly direct arrival iss hown h er e.)

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44 9

340

320

300

2802 6 02 4 02 2 O200 .... I .... I .... I .... I .... I ....0 200 400 600 800 1000 1200

Depth (m)(a) Theo retic al tra veltime curv e

1 0864

_ 2"_ 0N -2oI[i, -4

-6-8

-I0 , I I I I0 200 400 600 800 1000 1200Depth (m)

(b) Differentialtraveltime

FIG. 5. Theoretically calculatedtraveltimes (a) and comparison withobserved traveltimes from synthetic crosswell shot gather (b)(source depth = 500 m ).

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CONTOUR MAP FOR SEMBLANCE VALUES

208 02050

2020

_ 1990;> 1960;> 19301900

0

_ - V elocity G rad ient ( x 1 0 -3mNm )

FIG . 6. R esult of sem blance analysis for synthetic crossw eU common-shot gather (sourcedepth = 0m). The maximum semblance value appears where V = 2000 m/s and= 0.8 m/s/m.

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CONTOUR MAP FOR SEMBLANCE VALUES

2O8O20 5O2020

"_ 19901960>19511

d_

- Velocity Gradient ( x 10.3m/s/m)

FIG . 7 . R esult of sem blance analysis for synthetic crossw ell common-shot g ather (sourcedepth = 500m). The m aximum semblance value appears where V = 2000 m/s and= 0.8 m/s/m.

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CO NT OU R MAP FOR SEM BLA NCE VALU ES

2O8O205O2020

Iggog60;>1930IgO0 o o o o o o o o o o o o Q_ ,_ _ o _ o _ o _ o Og) _, tD r,. I_. CO _ (n 01 0 0 _'- _'L#

- Velocity Gradient ( x 10 .3 m/s/m)

FIG. 8. Result of semblanceanalysis for synthetic crosswellcommon-shot gather (sourcedepth = 760m). The maximum semblance value appears where V = 2000 m/s and= 0.8 m/s/m.

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CONTOUR MAP FOR SEMBLANCE VALUES

205O2020

> t960 Io o o o o o o o o o o o o

_ _ (C

I- Velocity Gradient ( x 10 -3 m/s/m)

FIG. 9. Result of semblance analysis for synthetic crosswell common-shot gather (sourcedepth = 260m). The maximum semblance value appears where V = 2010 m/s andK = 0.8 m/s/m.

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CONTOUR MAPFORSEMBLANCEVALUES

2O8O

2020

1990960> .

19301900

1,_-'Velocity Gradient ( x 10 "3 m/s/m)

FIG . 1 0. R esult of sem blance analy sis for synthetic crossw ell common-shot gather(source depth = 1000m ). The m axim um sem blance value appears w here V= 1990m/s and K = 0.825 rrds/m.

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45 5

Trace Trace0. 0

0. l

0. 2

0.30. 4

0. 5

0.b

0. 7

0. 0

._ 1.0o 1.!

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

(a) (b)

FIG. 11. Shot records from an ultrasonic seismic modeling experiment in a water tank(S tew art and C headle, 1989). (a) Source depth = 0 m ; and (b) Source depth= 1000 m.

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45 6

CONTOUR MAP FOR SEMBLANCE VALUES1800. 0

1700.0

1600.0

"-" 1500.0 _ . - "_

"_ 1400.0 _ _ ---_.._ __-1300. 0

1200. 0

11000

1000.0 .r I I I I I I _ I I i f I i I I I I I I I I I I 1 I I I I I I I I. J_4---LI_4-'-_fi ,.o oo c5 _J _o 05 c5

C'_ rtl

Velocitygradient 1(xl0 -5m/s/m)FIG. 12. Result of semblance analysis for physical modelingcrosswell common-shotgather (source depth = 0 m ). T he largest sem blance value appears w hereV = 1520m/s and i= 0.0095 rn/s/m.

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45 7

CONTOUR MAP FOR SEMBLANCE VALUES1800. 0

1700.0 ___

1600.0

1500.0 --

_, 1400.0 - ---

_- 1300.0

1200. 0

1100.0

:Z1000.0 I I I I I I I I I I P q I I I I I I I P P I r I I I I I I I I I I 1 1 1 1 1

0 0 0 _ _ _ _

Velocity gradient _ (xlO "5m/#m)

FIG . 13. R esult of sem blance analysis for physical m odeling crossw ell common-shotgather (source depth = 450 m ). The largest sem blance value appears whereV = 1520 rn/s and _c= 0.0085 m/s/m.

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45 8

CONTOUR MAP FOR SEMBLANCE VALUES

1760,0 1

1680,0

1600.0_>' i52o.o

1440.0

F

1280.0

t200.O

Velocity gradient 1

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45 9

depth (fl)

FIG. 15. Common-receiver gather of crosswell seismic data acquired in Humble, Texas.Source depths range from 300 ft to 2540 ft at interval of 20 ft. The geophoneis located at 1500 ft. Offset between wells is 815 ft. D represents P-wavedirect arrivals. (Courtesy by Texaco Inc.)

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460

CONTOUR MAP FOR SEMBLANCE VALUES

157.0

o

_ ,_.0 o _ _210 0N _"_" 109,0 __' 97.0 o

> as.o 073.0 _ _

0

61.0

4 90

Velocity gradient _ (xlO "2ft]sec/ft)

FIG. 16. Result of semblance analysis for real crosswell seismic data shown in Figure 15.Initial velocity of 6100 ft/sec and velocity gradient of 0.145 ft/sec/ft are picked atthe point w ith the largest sem blance value.