GEOPHYSICS, VOL. 50, NO.6 (JUNE 1985); P. 903-923, 18 FIGS., 2 TABLES.

Tomographic determination of velocity and depth in laterallyvarying media

T. N. Bishop*, K. P. Bubet, R. T. Cutler§, R. T. Langan**,P. L. Love**, J. R. Resnick**, R. T. Shuey**, D. A. Spindler**,and H. W. WyldH

ABSTRACT INTRODUCTION

Estimation of reflector depth and seismic velocityfrom seismic reflection data can be formulated as a gen­eral inverse problem. The method used to solve thisproblem is similar to tomographic techniques in med­ical diagnosis and we refer to it as seismic reflectiontomography.

Seismic tomography is formulated as an iterativeGauss-Newton algorithm that produces a velocity­depth model which minimizes the difference betweentraveltimes generated by tracing rays through the modeland traveltimes measured from the data. The input tothe process consists of traveltimes measured from select­ed events on unstacked seismic data and a first-guessvelocity-depth model. Usually this first-guess model hasvelocities which are laterally constant and is usuallybased on nearby well information and/or an analysis ofthe stacked section. The final model generated by thetomographic method yields traveltimes from ray tracingwhich differ from the measured values in recorded databy approximately 5 ms root-mean-square.

The indeterminancy of the inversion and the associ­ated nonuniqueness of the output model are both ana­lyzed theoretically and tested numerically. It is foundthat certain aspects of the velocity field are poorly de­termined or undetermined.

This technique is applied to an example using realdata where the presence of permafrost causes a near­surface lateral change in velocity. The permafrost is suc­cessfully imaged in the model output from tomography.In addition, depth estimates at the intersection of twolines differ by a significantly smaller amount than thecorresponding estimates derived from conventional pro­cessing.

Estimation of velocity and depth is often an important stepin prospect evaluation in areas where lithology and structureundergo significant lateral change. Depth estimation is usuallyaccomplished by converting zero-offset traveltimes, interpretedfrom a stacked section, to depth using a velocity field obtainedfrom a normal-movement (NMO) analysis. In areas with com­plex lateral changes, a depth migration technique may be neces­sary to obtain the correct depth estimate (Lamer et aI., 1981).Both of these methods require an accurate representation of theroot-mean-square (rms) velocity field. However, the stackingvelocities used for such analyses can deviate significantly fromrms velocities because analysis of stacking velocities assumesthat the medium is laterally invariant and that traveltime tra­jectories for reflection events in CDP gathers are hyperbolic.

Media vary laterally due to either reflector dip or curvature,or due to lateral velocity variations, or both. A large portion ofthe effect of reflector dip or curvature on the stacking velocitycan be removed approximately by first migrating common­offset panels with a first-guess velocity function, and then recal­culating the stacking velocity in the migrated common-depth­point (CDP) gathers (Doherty and Claerbout, 1976). The influ­ence of lateral variations in velocity on the stacking velocitycannot be corrected this way. For lateral variations in velocitywhose wavelength is longer than a cable length (the maximumspread offset), the smoothed stacking velocity is usually a goodrepresentation of the rms velocity. Velocity variations on thescale of a cable length or shorter can produce large differencesbetween the stacking velocity and the rms velocity. These differ­ences are sometimes referred to as stacking-velocity anomalies(Pollet, 1974).

A conventional residual-statics approach can often correctfor near-surface variations in velocity on a scale shorter than acable length. However, the spatial resolution of statics analysesdecreases rapidly beyond one cable length (Wiggins et al.,

Presented at the 54th Annual International SEG Meeting December 5, 1984 in Atlanta. Manuscript received by the Editor August 13, 1984;revised manuscript received November 20,1984.*Gulf Oil Exploration and Production Co., Bakersfield District Office, P. O. Box 1392, Bakersfield, CA 93302:j:Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90024.§Gulf Oil Exploration and Production Co., New Orleans District Office, P. O. Box 61590, New Orleans, LA 70161.**Gulf Research and Development Co., Exploration Research Division, P. O. Box 37048, Houston, TX 77236.HLoomis Laboratory of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801.© 1985 Society of Exploration Geophysicists. All rights reserved.

903

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

904 Bishop et al.

_-,"-__~Zp.m

III

I Wk-I.lIII I

j-- ---- - j- ----- -j-- - - - --iI I I II I I I

: Wk,I_1 : Wk,l ~ Wk,l+l :I I I II I Iw __ - ----t--- - ---t-- - ----

II

Wk+I,! :I

surface are denoted by Zl(X), Zz(x), ... , Zp(x), ... , Zn,(x). Theslowness in the region is modeled by a function w(x, z), whichrepresents the reciprocal of seismic velocity at points (x, z) inthe subsurface.

For computation, it is convenient to characterize the slow­ness and reflector depth functions by a finite set of parameters.Those functions are then restricted to lie in a certain finite­dimensional space of functions. The intent is to make thedimension of that space sufficiently high that functions whichare not in the space, but may more accurately describe the realEarth, can be well approximated by functions which do lie inthe space. However, the dimension must not be so high that theproblem of inverting the available data is hopelessly indetermi­nate.

Consider first the slowness function w(x, z). Divide the regionof interest into a matrix of rectangular boxes having n" columnsand nz rows (Figure 1). The horizontal dimension of a box istypically taken as four times the COP spacing, with the verticaldimension roughly twice that. The value of w(x, z) at the centerof the box in the kth row and tth column is denoted Wk(.

Elsewhere in the box the slowness is assumed to vary in such afashion that the gradient of velocity within that box is constant.The value of slowness anywhere within the box can be obtainedfrom an interpolation formula which depends on Wkt and onWk-1,t, Wk+l,t, W k• t - 1' W k• t + 1, that is, on the values of slow­ness at the centers of adjacent boxes. This procedure is moti­vated by the ray-tracing scheme. Rays are traced from source toreceiver by a shooting method which uses the entrance angle ateach box and the velocity gradient to compute the radius ofcurvature of an arc and the arclength in the box. This method isdescribed in greater detail in Appendix A.

The reflector depth functions Zp(x) are parameterized by thedepths at which the reflector intersects vertical boundaries ofsuccessive columns of boxes. For reflector p these depths are

1976). Another method was proposed in Lynn and Claerbout(1982) which inverts the observed stacking velocity and zero­offset traveltime, as a function of common midpoint, to obtainan approximate true-vertical rms velocity function. Thismethod is limited to velocity variations longer and shallowerthan a cable length.

Here we present a method which uses traveltimes to selectedevents in unstacked seismic reflection data to image velocityanomalies above and between these events and to infer theirdepths. The depths are determined even in the presence ofreflector dip or curvature and lateral changes in velocity. Whenthus determined, the depths are those that would be found fromdepth migration using the correct rms velocity field. With somelimitations (to be discussed), the method is capable of imagingboth short- and long-wavelength variations in velocity.

Conceptually, this new method is closely related to the itera­tive inversion method used in Hawley et al. (1981) for simulta­neously determining hypocenters of earthquakes and three­dimensional (3-D) velocity variations. Their method is a gener­alization of the one-step general inverse method proposed inAki and Lee (1976). The technique presented here differs fromtransmission problems (earthquake to receiver, see Andersonand Dziewonski, 1984) in that rays start at the surface, reflectoff interfaces whose depths are to be determined, and return tothe surface. Another major difference is the much largernumber of model parameters in the reflection seismology prob­lem due to the increased density of data. This difference makesthis inversion method very computer-intensive. This method isalso similar to work done by Kjartansson (1980). While hiswork also used reflected rays, it was limited to straight raypathsand fixed reflector depths.

All these methods are closely related to x-ray computerizedtomography (CT) used in medical diagnosis. In CT the mea­sured data can be modeled as line integrals through a dis­cretized density field along a straight raypath. An efficientconvolutional method exists for inverting these measurementsto obtain the spatial distribution of the density. Recent geo­physical applications of similar tomographic techniques haveincluded imaging between boreholes for site characterization(Dines and Lytle, 1979) and mapping oflarge-scale changes inthe density of the oceans (Cornuelle, 1982). The general inversemethods are computationally similar to CT, but they usetraveltimes to image slowness as a discretized field. The tomog­raphy problem for seismic reflection data is complicated by theunknown depth to the reflectors, ray-bending effects which arenonlinear, and raypath coverage which is irregular and haslimited view-angle coverage.

This paper discusses the following: the method of reflectiontomography as a two-dimensional (2-D) general inverse prob­lem; the inversion algorithm, illustrated using a synthetic ex­ample; the uniqueness of solutions to the inverse problem; andfinally an example of velocity and depth determination for aseismic line where a large near-surface change in lateral velocityis caused by permafrost.

PROBLEM FORMULATION

__--.....,..-_Z=-'P+I. m

First we establish notation, and then formulate the problemin a precise fashion. Let x be horizontal distance along theearth's surface and let z be depth. In the region of interest, thereare nr reflectors whose depths as a function of position on the

FIG. 1. Discretization of velocities and depths used in themodel earth.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Tomographic Determination of Velocity 905

where a is a real constant and W(k) a weighting matrix whichwill be discussed below. With L\p chosen, p(k+ 1) is then simply

If ~(p) is the Jacobian matrix of t(p), that is, the matrix whoseijth element is otdopj' then equation (2) implies

The vector L\p is chosen so that, to first order, the right-handside of equation (4) is zero. Since typically ~'(k)~(k) is either notinvertible or is, at best, very poorly conditioned, there are manyvectors L\p which can play such a role. The algorithm choosesL\p such that

(3)

(2)

(5)

(6)

(4)

V<I>(p) = o.

p(k+ 1) = p(k) + L\p.

~'(p) [t(p) - td ] = O.

~'(k)[t(p(k+1») _ td

] = ~'(k)~(k)L\p _ ~'(k)r(k)

+ 0(11 L\p 11 2).

The vector p* is a solution regardless of how a or W(k) arechosen, although both a and W(k) can affect the rate of conver­gence. However, if many solutions of equation (3) exist, thevalues of a and W(k), as well as the choice of the initial modelp(O), determine which solution is obtained. Presumably all threeterms may also influence whether or not convergence occurs. Ifthe algorithm does converge, p* satisfies the necessary con­dition [equation (3)] for a local minimum, but there is no apriori guarantee that it represents a global minimum.

In numerical tests it is observed that the residual decreasesfor several (between three and five) Gauss-Newton steps; afterabout five steps, changes in its value are sufficiently slow thatfurther iteration seems pointless. A useful measure of the algo­rithm's effectiveness is the quantity

Suppose the algorithm outlined in equations (5) and (6) con­verges. Call the vector to which p(k) converges p*. Then p* mustbe a solution of equation (3). This follows since p(k) -> p* impliesL\p -> O. So from equation (5),

~'(p*)r(p*) = lim ~'(k)r(k) = O.k~oo

The algorithm actually attempts to find a solution of equation(3). To understand how the procedure works, suppose p(k) isknown and it is desired to find p(k+ 1). Let L\p = p(k+ 1) _ p(k),~(k) = ~(p(k»), and r(k) = r(p(k»). Consider the quantity~'(k)t(p(k+ 1»). If t(p(k+ 1») is expanded in a Taylor series about p(k),we have

<I>(p) = II r(p) 11 2 = [td - t(p)]'[td - t(p)]. (1)

The description of an algorithm to minimize <I>(p) follows.

denoted Zpm' 1 ~ p ~ nr , 0 ~ m ~ nx • Away from box bound­aries, Zp (x) is regarded as a cubic spline function which interpo­lates the points specified by the Zpm'

Let M be the total number of parameters which characterizethis model of the earth. From the above discussion, it is clearthat

M = nxnz + nr(nx + 1).

It will prove useful to define an M-dimensional vector p havingthe property that every parameter of the model appears as acomponent of p. This can be done by reindexing the parame­ters: the first nx nz components of p are the Wkt , and the lastnr(nx + 1) components are the Zpm'

In addition to the vector p, which is the primary descriptor ofthe model, some notation is needed to describe the data set ofreflection tomography. The data consist of two-way traveltimemeasurements obtained from seismic recordings made on thesurface. Let there be ns shots with a maximum of ng geophonestation locations per shot. The observed traveltime for a raywhich emerges from source 11, travels to reflector p, and returnsto receiver v is represented as I;.pv, 1 ~ Il ~ ns , 1 ~ P ~ nr ,

1 ~ v ~ ng • (In practice, every reflector will not necessarilyyield reflections on every geophone trace of sufficient amplitudeto facilitate the picking of event traveltimes; for such reflectors,the geophone index will remain below ng in some shot gathers.)The total number of rays is denoted N. As was the case with theparameters, it is convenient to renumber the data so the travel­time for each ray appears as an entry in a single N-componentvector. This vector of traveltime measurements is called td .

For a particular choice of the parameter vector p, rays can betraced through the model to yield a collection of traveltimeswhich correspond to the same shot-receiver pairs that are rep­resented in the data. Indexing these traveltimes in a manneranalogous to that oftraveltimes from the seismic line yields t(p),an N-component vector function ofp.

The goal of reflection tomography is to find a model forwhich the model traveltimes closely match the real traveltimes.In more mathematical language, the problem can be posed asfollows. Let the residual vector r(p) be defined by r(p) = td

- t(p). Double bars (II • III denote the Euclidean norm of avector and prime denotes the transpose. The objective of to­mography is to choose p such that the quantity II r(p) II isminimized. The solution of this problem is identical to that ofminimizing the mathematically more convenient function

(7)

INVERSION ALGORITHM

To look for a parameter vector which minimizes <I>(p), wehave used an algorithm which has been used previously ingeophysical applications (Hawley et aI., 1981). The procedure isan iterative one which starts with an estimate p(k) of the solutionand then generates a new estimate p(k+ 1) based on a linearizedapproximation to t(p) at p(k). In the mathematics literature, thisapproach is known as the Gauss-Newton method (Marquardt,1963; Gill et aI., 1981).

A necessary condition for a local minimum of <I>(p) is that atthe minimum,

which represents the rms difference between the traveltimespredicted by the model and those found in the data. Although Emay be as large as 50 ms initially, by the third or fourthiteration it is often reduced to 5 ms, close to the sampling rateof standard seismic data.

In the scheme outlined above, the Jacobian plays a dominantrole. A single row of the matrix ~(k) consists of the derivatives ofthe traveltime for a particular ray with respect to each of theparameters in the model. The first nx nz of these terms areslowness derivatives, and the rest are depth derivatives. Thus,the matrix ~(k) can be partitioned into two matrices ~~) andA(k)_z

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

906 Bishop et 81.

oW(k)

w

where W~) = (trace ~~k)~~)IN)1/2 and W~k) = (trace~~(k)~~k)IN)1/2. This choice serves to ensure that the unitlessparameter a will have a value near 1.0 at the point at which itbegins to affect significantly the solution of equation (5).

Despite the relative sparsity of ~(k), the large values of NandM typical of seismic data (N ~ 4 X 104

, M ~ 2 X 103) make

the determination of .1p in equation (5) a challenging task. Thenumerical technique employed for inferrring .1p consists of aGauss-Seidel algorithm with successive overrelaxation (Varga,1962). A considerable effort has been expended in optimizing

the code for use with an array processor attached to aUNIVAC 1100/84 mainframe.

To illustrate some of the ideas described, we present a com­putational example of reflection tomography. The exampleconsists of a simulation designed to demonstrate the effects of anear-surface velocity anomaly. The simulated earth is shown inFigure 2. The simulated region contains flat reflectors andlaterally homogeneous velocity, except in the uppermost layer,where the velocity changes laterally from 7 272 ftls to 8 890 ft/s.The parameters of this simulated earth are denoted by Ptarget.

Each box here is 880 ft long and 1 000 ft deep, with the totalnumber of parameters M = 2 203. (Also nx = 110, n, = 3, andnz = 17.) The simulated data are obtained by tracingN = 27 300 rays through this model. A single-shot gather ofrays striking one of the reflectors is shown in Figure 3.

Five Gauss-Newton inversion steps are employed in thisdemonstration. The initial guess p(O), known as the "gray"model, is similar to Ptarget except that p(O) contains no lateralvelocity anomaly. Figure 4 displays the difference between thetarget slowness structure and the gray-model slowness struc­ture. Also shown are the reflector depths, which are identical forboth Ptarget and p(O). The difference between the slowness distri­butions ofp(5) and p(O) appears in Figure 5. Figure 6 exhibits thedepth of the first reflector on an enlarged scale after one, three,and five Gauss-Newton steps, respectively.

If this procedure worked perfectly, Figures 4 and 5 would beidentical, and the depth corresponding to p(5) in Figure 6 wouldsimply be constant at 5 000 ft. It is clear that in the solutionthere is some coupling between the velocity anomaly and thedepth. Nonetheless, it is significant that the depth has beencorrectly determined to within 40 ft. That the depth is deter­mined relatively accurately is shown by analysis of what is"well-determined" and what is "poorly determined" in thesolution of equation (5).

W(k)z

W(k)z

W(k)w

o

where~~)isN X nxnz,and~~k)isN X n,(nx + 1).How are the entries in ~(k) calculated? First, since one ray

passes through relatively few boxes, many of the entries in arow of ~(k) are zero, and consequently ~(k) is quite sparse.Moreover, an argument based on Fermat's principle of leasttime (see Appendix B) shows that the derivative with respect toslowness in a particular box is approximately the pathlength inthat box. The depth derivative can also be obtained in closedform involving terms readily available from the ray tracing (seeAppendix B).

The weighting matrix W(k), which serves to stabilize thecomputation of .1p in equation (5), has been discussed by othersin a variety of contexts (Franklin, 1970; Wiggins, 1972). In thepresent application it has the form

COP 31.1 111.1 191.1 271.1 351.1

OOOOOOOOOOOOOOOOOOODOOOOOOOOOOOOOOOOOOOOOOOOOOO~OOOOOO00

21l1lll.

4111.I-lUlU 61l1lll •...z...:J: Bill.l-ll..lUc

11111

12111

Hili

000000000000000000000000000000000000000000000000000000OOOOOQOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOODOC

000000000000000000000000000000000000000000000000000000OOOOOQOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO~OOOOOOODOC

00000000000000000000000:000000000000000000000000000000000000000000000000000000000000000000000000000000000000oooe

~oooooooooooooooooooooooooooooooooooooooooooooooooooooOOOOOQOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOODOC

......................., ' .

161l1lll

VELOCITY IN FT/SEC

~@~~0G8DDDDDDDD0~~~~.......... _.f 1"'., _ .• _.f _.f _ .••_. ,nil. II " _ .•UWll _ _ _·II_.n_.•

6000 11000 16000

FIG. 2. Simulated earth model. Flat reflectors at 5 000, 10 000, and 15 000 ft. Velocities are laterally invariant except in the toplayer.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Tomographic Determination of Velocity 907

THE EFFECTS OF LIMITED-ANGLE RAYS­UNDETERMINED AND POORLY DETERMINED MODEL

FEATURES

There are limitations to the information about the subsurfacethat can be inferred from a particular set of traveltime measure­ments, even if such measurements are exact. These limitationsare analyzed here for two problems, both of which are simplerthan the actual tomographic inversion specified above. Basedon experience from several numerical tests, it is found that thelimitations described by these simpler analyses accuratelycharacterize the results of the more general tomographic algo­rithm.

The vector L\p which is the solution of equation (5) is also thevector that minimizes the quantity

(8)

The first term of expression (8) represents a linearized approxi­mation to II r(k+ 1) 11 2

• Typically, there are many other choices ofL\p for which this first term is of comparable magnitude. It isknown that all of these vectors differ from each other by linearcombinations of those eigenvectors of 4(k)'4(k) that correspondto small eigenvalues (Wiggins et al., 1976; Franklin, 1970). Thisincludes all elements in the null space of 4(k) for which theeigenvalues are zero, as well as a number of other components

COP 31.8 111.8 191.8 271.8 351.8 431.8

2888.

4888•...UJUJ 6888 •....zH

:: 8888........UJCl

18888

12888

14888

16888

RAY PATH PLOT

FIG. 3. Raypaths from one shot to the top reflector traced through the model shown in Figure 2.

COP 31.' 191.' 271.111 351.' 431.'

2••••

41•••...UJUJ 6••••....ZH

:: 881••.......UJCl

18•••

12••8

14•••

16.88

NEGATIVE SLOWNESS CHANGE, -MS/KFT

-...~~~.@J..@J.~.~.~.QQQQqqQQE;l[;J,~~~[!,J..[!,J.~.-12 0 12

FIG. 4. Difference in slownesses between the simulated earth model shown in Figure 2 and "gray" model for tomography.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

908 Bishop et al.

which in theory can be found by computing the "singular valuedecomposition" (SVD) of 4(k) (Golub and Reinsch, 1970; For­sythe, et aI., 1977). Unfortunately, the SVD of a matrix as largeas 4(k) is an extremely expensive computation. In what follows,although two variations on this general theme are pursued,SVD analysis of the large matrix 4(k) is avoided.

In the first approach, after certain simplifying assumptionsare made, an exact specification is given of those parameters ofthe model earth which cannot be determined from the data.

This approach reveals the undetermined components in thesolution of equation (3), the original, nonlinear problem. In thesecond approach, the most restrictive of the simplifying as­sumptions is relaxed, and a small model earth (151 parametersand 499 rays) is constructed for which an SVD analysis isfeasible. The SVD analysis shows only those ambiguities thatappear in the solution of the first Gauss-Newton step [equation(5)], but not necessarily in the full nonlinear problem. However,in addition to revealing those components which are impossible

COP 111 •• 1'1•• 271 •• 351••. .......... ••.•••••••••••_·(X)OOo.oo••••••••... ............

2••••

......~

11.I11.I s••••...z...:z: 8••••...L11.ICl

1••••

12•••

14•••

lS•••

• • • • • • • • • • • • • • • • • •• , •••••••••••••••••••••• 010 •••••••••• ••••• 0" ••••••••• •••••••••••••••.••••..••••••••.

.'

NEGATIVE SLO~NESS CHANGE, -"S/KFT

~@~~~~0G880DDD88GG0~~OOOO~~~_11~_W~~'~"1.'~~·$~~~_I~·~_I~'~I",.ti~'~~""'~Y~'~11~U~

-12 0 12

FIG. 5. Difference between the simulated earth model and output model after five Gauss-Newton steps.

DEPTH

i

..... - -+_._...; ·····r .... _ .•. -...."T .... -r -~,,,--- ,-- -t--_._+_..._-+-- ..--i··_..,,-······~..J················+-l i

5000 +'""<+-'-.__...._..•. _.--_.

COP 111.. 131.. 271.. 351.8 431 ••4900 r-....----.......-'I'"-----r------.,.-.......-t---r---1~_r__,.-'"t"_..;.:;...:...-__. ,

FIG. 6. Depth to the first reflector after one, three, and five Gauss-Newton steps. "Correct" answer would be flat lines at 5 000 ft.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Tomographic Determination of Velocity 909

to infer from the data, SVO also indicates which componentsare well-determined and which are poorly determined.

In both analyses, it is assumed that rays are straight, exceptfor an equal-angle reflection at one of the reflectors. It is furtherassumed that the subsurface is discretized in boxes, with aconstant velocity in each box. In the first analysis, it is alsoassumed that reflectors are flat and horizontal, so that only oneparameter is needed to characterize each of n, reflector depths.In addition, it is supposed that at least two distinct rays existwhich are completely contained in one vertical column ofboxes. Under these restrictions, the following results are provenin Appendix C.

(1) The depths ofthe flat and horizontal reflectors can bedetermined.

(2) Most of the velocity structure can be determined.Average slowness between reflectors can be deter­mined. Lateral changes in slowness, including thevertical location of these changes, can be determined.

(3) Part of the velocity structure cannot be determined.Perturbations in slowness which are laterally con­stant and have zero-mean slowness between reflec­tors have no effect on the traveltime data and cannotbe determined. Assuming that each box has at mostone reflector in it, then the null space has dimensionnz - n,. Recall that n. is the number of horizontallayers of boxes.

Several of the assumptions made in this first analysis arerather restrictive. The SVO computations described next avoidsome ofthese assumptions. Specifically, reflector depth is takento be a continuous piece-wise linear function with bends at boxboundaries, and a more realistic distribution of rays is con­sidered. The points below summarize the results of a number ofSVO calculations for a range of model choices.

There are singular engenfunctions with zero eigenvalues cor­responding to

(1) the n. - n, slowness perturbations described, and(2) edge effects in slowness and reflector depths where

there is an absence of rays or there are very few rays.

There are singular eigenfunctions with moderately small ei­genvalues corresponding to

(3) more edge effects,(4) linear variations in slowness in rows of boxes with

zero vertical mean between reflectors, and(5) sloping slowness compensating for sloping reflectors.

Small singular values corresponding to point (4) suggest thatthese variations in slowness are poorly determined. Similar butslightly larger eigenvalues associated with point (5) suggest atradeoff between event dip across the whole model and a linearvariation in slowness. This effect is seen in the example present­ed in the "Inversion algorithm" section. Figure 6 shows aninexact solution for depth which is characterized by a residualdip in the reflector of about 40 ft. Although each nonlinear passof the inversion improves the depth estimate, the algorithm isconverging to a depth function with a residual error in the slopeof the reflector.

In the model examples considered, all the other singularvalues are an order ofmagnitude larger than the values detailedabove, indicating that all other eigenfunctions of parametersare relatively well determined. This relationship has also beenobserved empirically by comparing target models with finalinversion solutions.

In summary, we found that the Jacobian matrix has a nullspace corresponding to slowness variations in rows of boxeswith zero vertical mean. The null space also contains some edgeeffects. Poorly determined quantities include linear slownessvariations in rows with zero vertical mean and reflector tilt witha compensating linear slowness variation.

The above results are derived for a much simpler inversionproblem than that encountered in the actual tomography algo­rithm. It is probable that the ray-bending and the discretizationusing a constant velocity gradient, which characterize the realalgorithm, add information to the problem which is not takeninto account in the simple analyses. This additional infor­mation may change an "undetermined" quantity into a"poorly determined" quantity, but it seems unlikely to alter thegeneral picture which these studies establish of what can andwhat cannot be inferred from tomography. Indeed, numericaltests of the actual algorithm indicate that the indeterminanciesdescribed here do obtain in practice.

CONVENTIONAL AND TOMOGRAPHIC PROCESSING OFA SEISMIC LINE OVER A PERMAFROST BODY

We now compare conventional and tomographic processingof a seismic line. The line exhibits an abrupt lateral change invelocity due to permafrost. Proper depth interpretation was animportant exploration problem on this line.

Figure 7 is a seismic cross-section (line A) from the BeaufortSea. Shown in red is an interpretation of five major reflectionevents, labeled El through E5, which extend across most of thesection. Near-surface reflection anomalies between COPs 80and 125 are due to permafrost. A time pull-up of events El andE2 is observed under the permafrost at the left end of the figureindicating a high-velocity, near-surface feature.

Further evidence for this interpretation comes from thestacking velocity analysis shown in Figure 8, This figure is acontour plot of stacking velocities after smoothing with a22-CDP filter whose weights are given by a triangle function_There is a long-wavelength lateral gradient in velocity with theleft end of the line faster than the right. The gradient is verystrong at the surface and decreases to a rather small effectbelow 2.0 s. Superimposed on this gradient is a stacking­velocity anomaly from COP 60 to COP 190. The center of theanomaly is at COP 125, corresponding to the edge of theshallow permafrost feature. The sense of the anomaly-high­velocity swing to the right and low-velocity swing to the left ofthe edge of the permafrost--<:orresponds to an abrupt lateralchange to lower velocity from left to right.

The influence of an abrupt lateral velocity change on thetraveltime and the stacking velocity for a reflection event belowthe permafrost is depicted schematically in Figure 9. In thisfigure the traveltime effect of the high-velocity layer is approxi­mated by a static speed up. Note that the influence of theanomaly extends for a distance equal to a cable length centeredon the edge ofthe layer.

To illustrate the effect of the stacking-velocity anomaly ondepth conversion, the velocities shown in Figure 8 can be used

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

co

p·

..1

Q.O

.2Q

O

LIN

ES

I 30

0..I"

..4

QO

1.0 3.0

!! C)

DI ii" ::T o 'a !. ~

FIG

.7.

Sta

cked

sect

ion

ofL

ine

A.T

he

five

even

tsus

edin

the

tom

ogra

phic

anal

ysis

are

indi

cate

dby

red

line

s.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Tomographic Determination of Velocity 911

Table 1. Root-mean-square misfit between data and theoretical times(ms), equation (7).

The exploration importance of the correct depth analysis isnoted by examining event E4. An apparent anticline existsbetween CDPs 170 and 300 on E4 which could be interpretedas a favorable structure. The closure of that structure on the leftend of the line is due to the synclinal feature around CDP 165.This raises the question of whether this closure is real or just aresidual permafrost effect.

To answer the question, the five events discussed previouslywere used in a tomographic analysis. Traveltimes as a functionof offset were digitized on shot gathers displayed on a televisionscreen at an expanded time scale. The digitizing was checked bydisplaying these time picks on both CDP and offset gathers.Data quality was generally good and the five events were easyto follow on the shot gathers. In the few areas where the datawere noisy, or the events ambiguous due to faulting, no timepicks were made.

In this example, the first-guess model chosen consisted ofvelocities that did not vary laterally and depths which wereconstant except at faults. Jump discontinuities were input atCDP positions of faults which were interpreted on the timesection (Figure 7). The theoretical times, obtained by ray trac­ing the first-guess model, differ from the data times by an rmsaverage of 46 ms across the line.

The tomography algorithm attempts to minimize the differ­ence between the theoretical times and the data times by modi­fying the velocities and depths of the model. Each iterationproduces a new model with a corresponding set of theoreticaltimes. Table 1 shows the misfit between the data times and the

to convert the time section of Figure 7 to depth (Figure 10). Thevelocity anomaly is translated into a depth anticline centeredaround CDP 100 and a depth syncline centered around CDP165 for all the events on the section. Deeper events are distortedmore than shallow ones due to the increase in the size of thevelocity anomaly with depth.

Typically, a longer smoothing function would be used tosmooth the stacking velocities for depth conversion, and thusreduce the anomaly by averaging the negative and positiveexcursions. Further, having recognized the permafrost feature,most processors would attempt to correct for the time distor­tion it introduces by doing a statics analysis. The combinationof statics corrections and long-wavelength velocity smoothingwill reduce the influence of the permafrost on the inferred depthto the reflectors below it.

This conventional processing approach has two short­comings. First, the permafrost does not produce purely staticeffects. That is, different events at the same offset have differentdelays (or speed-ups) due to raypath bending effects. Second,the lateral extent of the permafrost is a large fraction of a cablelength. Solving for long-period delays is difficult with conven­tional residual-statics programs (Wiggins et aI., 1976).

Figure 11 is the depth section after such a conventionalprocessing sequence. A standard residual-statics program(Taner et aI., 1974) was used to estimate static delays aftermoveout with a very slowly varying velocity function. Thesedelays were then applied to the traces before moveout. A subse­quent velocity analysis with a one-quarter cable-length triangu­lar smoother was used to stack the data. The same velocitymeasurements, after the application of a one-cable-lengthsmoother, were used to convert the stacked section to depth.

The red lines in Figure 11 show the depth positions of theevents interpreted on the time section (Figure 7). Much of thedepth distortion due to the velocity anomaly has now beenremoved. However, there are still indications of residual per­mafrost effects. A depth syncline persists for events E4 and E5near CDP 165. There is also a considerable pull-up under thepermafrost. This effect is seen most easily on event El.

First-guess modelFirst iterationSecond iterationThird iterationFourth iteration

46.26.35.45.15.1

FIG. 8. Contour plot of stacking velocities for line A. A stacking-velocity anomaly can be seen on the left side of the line.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

AS

TA

CK

ING

VE

LO

CIT

YA

NO

MA

LY

CA

US

ED

BY

AH

IGH

VE

LO

CIT

YL

AY

ER

!! II'

Eo

CB

A

E

VH

IGH

o

Eo

CB

A

21

\

-'"

.....-.....

......-

II' g: .g !l !!-

TIM

E

VE

LO

CIT

Y

A

--...._

--

••--

VL

OW

BC

oE

II

....

II

I~

CO

P

FIG

.9.

Ast

acki

ng-v

eloc

ity

anom

aly

caus

edby

ahi

gh-v

eloc

ityla

yer.

(a)

Ray

path

sat

vari

ous

CD

Ps;

red

indi

cate

sra

ystr

avel

ing

thro

ugh

the

high

-vel

ocit

yla

yer.

Lay

eris

assu

med

toca

use

spee

dup

<re

lati

veto

rays

whi

chm

iss

the

laye

r.(b

)M

oveo

utcu

rves

inC

DP

gath

ers.

Red

curv

esar

ebe

st-f

ithy

perb

olas

atC

DP

sB

and

D.(

c)S

tack

ing

velo

city

vers

usC

DP

show

ing

velo

city

anom

aly.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

10

02

00

30

04

00

!!"'

!l'I

~"""

!!'U

''''

''lI

It!I

IIW'

lf''

''....

,"'U.M!'I""!!IUlllll!llljW'i1iU·H,W.WCWWUC.w.w1J~:jj~;,;:~llwu;.;:l~l.lilJJ!.,....u.:.;.,.;,illi!Jjl

..:..aW1l~:'~i~!~~:;;.~i.~ilid'i-::-·-'.~;·'-;;';~.,.,

•..

..._

-.,,

~":;

·-;·

li;i

;jil

'ji-

·••

-It

IWjW

,"","

c1 3 .g •'0 ;r n c ! 3 :i" !!. 0' :=I g, < • go ~

FIG

.10

.Lin

eA

conv

erte

dto

dept

hus

ing

stac

king

velo

citi

esfr

omF

igur

e8.

Eff

ect

of

velo

city

anom

aly

isob

viou

sne

arth

ele

fted

geo

fthe

line.

CD ... Col

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

CD~'

,10

0.

.20

.0,3

00,4

QO

40

00

"~'I

S"iI

'~'!(

'i".of

::iIilY

i8

00

0

FIG

.11

.L

ine

Aco

nver

ted

tode

pth

usin

gst

acki

ngve

loci

ties

calc

ulat

edaf

ter

stat

ics

and

smoo

thed

wit

ha

long

wav

elen

gth

filte

r.

cg ... ... m .­ ~ o " ~ !!-

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Tomographic Determination of Velocity 915

theoretical times for four iterations.It is evident from Table 1 that the tomography algorithm

does modify the model to reduce significantly the misfit andthat most of the reduction occurs in the first iteration.

Figures 12 and 13 show the tomography model after the firstand fourth iterations, respectively. The slownesses (plotted incolor) are the differences between the original gray model andthe result of the corresponding iteration. Red areas indicatewhere velocity has been increased and green where it has beendecreased. The depths to events are represented by dotted blacklines while the original gray-model depths are solid lines. Multi­ple dotted lines occur for each event where faults have beeninterpreted. The highlighted dots show the estimated eventdepth across the line.

Several interesting velocity features are seen in these models.There is a fast, shallow zone between CDPs 80 and 120 whichcorresponds to the range of CDPs where the near-surface seis­mic character indicates permafrost. The fast area is smeareddown to the left and covers an area of the line greater than afocused image of the permafrost. This is due to both the limitedvertical resolution of velocity between events and to the influ­ence of the edge of the line. Velocities in the top layer tend todecrease to the right, corresponding to the lateral gradientobserved on the plot of stacking velocities (Figure 8). Theleftmost end of the line shows a region of velocity increasewhich corresponds to the edge of the data; this is probably anedge effect. The two deepest layers show velocity increases tothe right. These increases may be due to lateral changes inlithology.

What distinguishes the results of the first iteration from thefourth? Most of the same velocity and depth features appear inboth. The main differences are small, detailed changes in theparameters. One change is the improved resolution of the per­mafrost feature in the upper left end of the model.

To display the tomography results in a form comparable tothe conventional processing, the tomographic velocity field isused to create the depth section shown in Figure 14. The depthof events in this display is obtained from the tomographicvelocity field only. No direct use is made of the depth parame­ters. However, examination of the depth section indicates thatthe depths to the events are essentially the same as the estimatesfrom tomography. This implies that the velocity-depth modelfound by tomography gives vertical-incidence traveltimes con­sistent with the stacked data. The red lines on this figure arethen essentially the depth estimates from tomography.

The pull-up which lies under the permafrost feature, andwhich persists after conventional processing (Figure 11), is nowcorrected. There is also little or no evidence for structuralclosure on the left end of the line for the deeper events. Wesuggest that this apparent structure, present after conventionalprocessing, is a residual permafrost effect which has now beencorrected.

Similar analyses, both conventional and tomographic, wereperformed on a cross line (line B). The position of the lineintersection is marked on the stacked section (Figure 7). Table 2compares the depth for the first four events at the line intersec­tion. The interpretation of the depth to the fifth event on Figure11 is too ambiguous to allow a meaningful comparison.

Note that in both analyses, line A is consistently deeper thanline B indicating that the depth mistie is not due to randomerror. It is likely that tomography has not completely removedsystematic velocity-depth effects in the data. However, it has

done considerably better than the conventional processing asindicated by the improved depth ties shown in Table 2.

Some general conclusions can be drawn from this tomo­graphic analysis. Traveltimes obtained by tracing rays throughmodels constructed by this method fit the measured traveltimedata to approximately 5 ms rms. This result is within the errorin picking the recorded data. Significant shallow velocity vari­ations were imaged in areas where they were predicted by otherevidence (shallow waveform character, stacking-velocity anom­aly). Depth estimates obtained indicated little or no structuralclosure on any of the events. The depths determined by tomog­raphy tied significantly better at the line intersection, indicatingthat although some indeterminancyremains, tomography hasdone a better job of distinguishing velocity effects from depththan the conventional processing scheme discussed earlier.

CONCLUSION

The technique presented uses a general inverse method toestimate velocities and depths from CDP seismic data. Thetechnique attempts to produce a model of both the velocityfield and depths to selected reflectors which minimizes thediscrepancies between traveltimes derived from tracing raysthrough the model and those measured from the data. Theminimization is achieved by using a Gauss-Newton method tomodify a first-guess model and produce a series of model iter­ates. The first-guess model is constructed manually from wellinformation or from the results of conventional seismic pro­cessing. Use of this technique on seismic data indicates thatfirst-guess models with rms misfits in the range of 20 to 100 mscan be modified to produce models with misfits of approxi­mately 5 ms in three to five Gauss-Newton steps.

An investigation ofthe uniqueness of the models reveals thatmost features of velocity and depth are determined. However,certain velocity features are undetermined while others aredetermined relatively poorly. The only degradation in depthresolution is a possible small overall tilt of reflectors.

The tomographic analysis was performed on a pair of inter­secting CDP lines in an area with significant lateral-velocityvariations due to the presence of permafrost. The techniqueproduced models which fit the data measurements from the twolines to within 4.8 and 5.1 ms, respectively. The models includedsignificant shallow velocity variations in areas where they werepredicted by other evidence. The tomographic depth estimatesat the intersection point agreed significantly better than thecorresponding depth estimates derived from conventional pro­cessing.

ACKNOWLEDGMENTS

Some of this work was guided at an early stage by thedoctoral work of Einar Kjartansson, who is now at the Univer­sity of Utah. R. L. Parker of the Institute of Geophysics andPlanetary Physics, La Jolla, California, provided insights relat­ed to the inversion process. R. A. Kroeger developed some ofthe inital inversion code and R. C. Jones and M. L. Rathbuntranslated it into fast array-processor code. Additional pro­gramming support over the course of the project was providedby N. J. Spera.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

STH

COP

71

.75

-G.I

11

1.7

74

.1

151.

7

15

4.1

19

1.7

23

4.a

23

1.7

314

••

27

1.7

3S4.

8

31

1.7

47

4••

35

1·7

55

4.8

!l G»

-

••••

••••

••••

••••

••••

••••

••••

••••

'll·

)(

-n

.n-1

'.1

'-jA

..,-'

.ZlI

l-1

I.n

7-l.~,

-'.1

1'$

.,.'S)

-S.U

l-.

..sn

-).1

S.~.".

-l.

l'l

-1••

-••U~

I.'

••U

S1

.41

U2

.11

15

Z.,.'

).1

$4

.5n

O!

S.U

lS,.

n:n

".l

lS1

.UU

,.4

)15

,.U.'

.1.'

l',"12U.5'l'Hi.~)l

-12

01

2

FIG

.12

.Vel

ocit

y-de

pth

mod

elfo

rli

neA

afte

ron

eG

auss

-New

ton

step

.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

5TH

CO

P

71

.75

-&.0

11

1.7

74

.8

15

1.7

15

".0

19

1.1

23

".8

23

1.7

31

4.0

27

1.1

39

".8

31

1.7

"1

4.0

35

1.1

55

".8

',',

',

,.r,

'.

-'.'

,',

· ·,.

-

·',

',',

;'.

','. ••

••••

••••

••••

••••

••••

••••

••••

••--

12

H.S

&-U

.Jll

-1

'·'

-'.2

'18

-S.U

l-}

.&"'

f>·'

.lll

S-"

.WH

-S.3

12

'-4

.0;,

)1-).1

,-;

1."

;8'l.1

81

-1..

..'

-'.'2''

''"0··

"'2';

1.4

.2l.

1II

1S

2'.

"8

1).

lS4

.Ul<

l:S

.lli

S'

.•'l

l6

.81

5,,

''S'i

!'8

.U1

S'.

ille

l1

1.'

li.lelihl.5u~.1<'.H)'

12

FIG

.13

.V

eloc

ity-

dept

hm

odel

for

line

Aaf

ter

four

Gau

ss-N

ewto

nst

eps.

'"... .....

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

CO

P'

10

.0

...

.20

.0.~QO

.40

.0.

FIG

.14

.L

ine

Aco

nver

ted

tod

epth

usin

gve

loci

ties

from

the

tom

ogra

phic

anal

ysis

.

40

00

80

00

12

00

0

16

00

0

CD~ 00 III i o 'a !. l!

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Tomographic Determination of Velocity 919

Table 2. Depth comparison at the line intersection.

Tomographic processing Conventional processing

5360 + 1207080 +2009600 +320

11 620 +260Average = + 225

Event LineA Line B Difference

E1 5626 5587 +39E2 7299 7230 +69E3 9757 9713 +44E4 11912 11 814 +98

Average = + 62.5

Line A

548072809920

11 880

Line B Difference

REFERENCES

Aki, K., and Lee, W. H. K., 1976, Determination of three-dimensionalvelocity anomalies under a seismic array using first P arrival timesfrom local earthquakes; 1. A homogeneous initial model: J. Geo­phys. Res., 81. 4381-4399.

Anderson, D. L., and Dziewonski, A. M., 1984, Seismic tomography:Scientific American, 251, 60-68.

Born. M., and Wolf, E., 1959, Principles of optics: Pergamon Press Inc.Cornuelle, B. D., 1982, Acoustic tomography: Inst. of Elect. and Elec­

tron. Eng. on Geosci. and Remote Sensing, 6E-20, 326-332.Dines, K. A., and Lytle, R. 1., 1979, Computerized geophysical tomog­

raphy: Proc. of the Inst. of Elect. and Electron. Eng., 67,1065-1073.Doherty, S. M., and C1aerbout, J. F., 1976, Structure independent

velocity estimation: Geophysics 41, 850-881.Forsythe, G. E., Malcolm, M. A., and Moler, C. B., 1977, Computer

methods for mathematical computations: Prentice-Hall, Inc.Franklin, 1. N., 1970, Well-posed stochastic extensions of ill-posed

linear problems: 1. Math Anal. Appl., 31, 682-716.Gill, P. E., Murray, W. and Wright, M. H., 1981, Practical opti­

mization: Academic Press, Inc.Golub, G. H., and Reinsch, c., 1970, Singular value decomposition and

least-squares solutions: Numer. Math, 14,403-420.Hawley, B. W., Zandt, G., and Smith, R. B., 1981, Simultaneous inver­

sion for hypocenters and lateral velocity variations: an iterativesolution with a layered model: J. Geophys. Res., 86, 7073-7086.

Kjartansson, E., 1980, Attenuation of seismic waves in rocks, Ph.D.thesis, Stanford Univ.

Langan, R. T., Lerche, I., and Cutler, R. T., 1985, Tracing of raysthrough an heterogeneous media: An accurate and efficient pro­cedure: Geophysics, September.

Larner, K. L., Hatton, L., Gibson, B. S., and Hsu, I-Chi, 1981, Depthmigration of imaged time sections: Geophysics, 46, 734-750.

Lynn, W. S., and Claerbout, J. F., 1982, Velocity estimation in laterallyvarying media: Geophysics, 47, 884-897.

Marquardt, D. W., 1963, An algorithm for least-squares estimation ofnon-linear parameters: Soc. oflndustr. App!. Math., 11, 431-441.

Pollet, A., 1974, Simple velocity modeling and the continuous velocitysection: Presented at the 44th Annual International Soc. of Expl.Geophys., Meeting, Dallas.

Taner, M., Koehler, F., and Alhilali, K., 1974, Estimation in correctionof near-surface time anomaly: Geophysics, 39, 441-463.

Telford, W. M., Geldart, L. P., Sheriff, R. E., and Keys, D. A., 1976,Applied geophysics: Cambridge Univ. Press.

Varga, R. S., 1962, Matrix iterative analysis: Prentice-Hall, Inc.Wiggins, R. A., 1972, The general linear inverse problem: Implication

of surface waves and free oscillations for earth structure: Rev. ofGeophys. and Space Phys., 10,251-285.

Wiggins, R. A., Larner, K. L., and Wisecup, R. D., 1976, Residualstatics analysis as a linear inverse problem: Geophysics, 41, 922-938.

APPENDIX A

METHOD OF TRACING RAYS

where Wi is the slowness in the ith box, L\x is the x-dimension ofthe box, and i increases in the x-direction.

Within a single box, it is known that the path of a ray is a

During the development of seismic tomography we utilizedtwo different ray-tracing methods. Our experience indicatesthat the difference in the results is principally one of efficiencyand the ability to resolve features on the scale of one or twoboxes. Our initial method was an iterative one based upon theanalytic approach given in Telford et al. (1976). However, theray-tracing method which we currently use for maximum ef­ficiency and accuracy is presented in a companion paper(Langan et aI., 1985). In this appendix we give the analyticapproach and its iterative approximation.

The model of the earth used in tomography consists ofrectangular boxes, with each box having a known slownessspecified at the center and a constant gradient of velocity acrossthe box. The components of the velocity gradient in each boxare calculated using the central-difference approximation. Forexample, the x-component of the velocity gradient in the ithbox Ax is

where Vo is the velocity at the origin, A is the velocity gradient,and r is the position vector. For slowness, equation (A-2) iswritten

where Wo is the slowness at the origin.Suppose a ray enters a box at the point (x = 0, z = 0) at an

angle of entry 4>0 measured with respect to the vertical. Let theray emerge from the box at (x, z) with angle 4>. Then

(A-4)

(A-3)

(A-2)V(z) = Vo + A• r,

w(r) = 1/(I/wo + A• r),

4> = arccos [cos 4>0 - x/R],

circular arc. Thus, we have constructed a model where raybending occurs within boxes and not at box boundaries. There­fore we can use simple geometry to express the orientation andtraveltime of the ray in the box as a function of position, initialorientation, velocity gradient, and slowness at the center of thebox. Telford et al. (1976) gave the expressions for the case wherevelocity increases linearly with depth only. We present similarequations for the more general case of an arbitrary direction forthe gradient. In this case the velocity field is defined by

(A-I)Ax = (l/w i + 1 - 1/wi _ 1)/(2L\x),

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

920 Bishop et al.

and Ij1 is the angle the gradient makes with the vertical.These equations can be used to find the position of the ray

and its angle of exit from the box. For instance, if the ray exitsfrom below, the depth of exit z is known, and 4> and x thenfollow from equations (A-5) and (A-4), respectively. (The valueof x obtained in this way must lie within the box; if it does not,the assumption that the ray exits from below is in error. In thiscase, it is really 4> and z that must be computed with known x.)With the raypath determined, the path length s is given by

The traveltime is then approximated as the product of pathlength and slowness at the center of the box.

Because this analytic approach is computationally slow, weperformed our initial studies with a less precise, iterativemethod which approximates it. A tangent to the circular path isconstructed at the entry point. The length of this tangent in thebox is used as an initial estimate for arc length. An estimate ofthe change in ray orientation across the box is calculated fromequation (A-7). An estimate of the position of the chord to thearc is constructed from this angle. The length of this chord isthen used to calculate a new estimate of the arc length. If thechord and tangent exit different sides, this process is repeated.

and<P = arcsin [sin <Po + z/R],

where R is the radius of curvature given by

(A-5)

(A-6)

s = 1R(<p - 4>0) I· (A-7)

APPENDIX B

CALCULATION OF SLOWNESS AND DEPTH DERIVATIVES

Entries in the Jacobian matrix ~ikJ are of two kinds: deriva­tives of traveltime with respect to slowness, and derivatives oftraveltime with respect to reflector depth. Both are computedimmediately following the computation of the traveltimes t(pikJ).

Thus, the raypaths through the model are known when thederivatives are computed.

In the ray tracing used here, rays are continuous at boxboundaries and refract inside boxes by moving along curvi­linear paths determined by the slowness gradient. However, forsimplicity the discussion will begin with an examination of thecalculation of the slowness derivatives in a medium where thegradient is zero inside each box, and rays refract by bending atbox boundaries. Also, reflectors will consist of piece-wise linearfunctions rather than cubic splines. Some discussion of theproblem of computing the derivatives in the more general casewill then follow.

From Fermat's principle, the raypath through the medium isthat path which exhibits the minimum traveltime between thefixed source and receiver. The minimum is taken over all pathswhich lie in a certain "regular neighborhood" (Born and Wolf,1959), and it is assumed that in that neighborhood there is onlyone such path. This guarantees that the traveltime is a trueminimum rather than just an extremum.

In the finite-dimensional medium considered here, the paththrough a single box is simply a straight line. Moreover, anypossible path through the entire medium can be specified byprescribing the depths and an additional finite set of indepen­dent path parameters. The latter will be denoted by the vector qwith components qn' n = 1, 2, ... , y. (For example, one specificqn might be the distance from a box corner to the point at whichthe ray enters or leaves the box, or the distance from a fixedpoint on a reflector to the point of reflection.)

Now define a real functionJ; (q, p) to be the traveltime for theith ray along a path through the medium specified by q. FromFermat's principle, the true raypath traveltime ti(p) is given by

The minimum in equation (B-1) is taken for q in a certaindomain in gJY. Assume that the parameterization is such thatthe minimum does not occur on the boundary of the domain,and suppose that inside that domain the function J; is smooth.It follows that at the minimum

(B-4)

(B-2)

(B-3)

n = 1,2,3 ... y.

ti(p) =J;(q*(p), p).

q* = q*(p),

and equation (B-1) can be written

Differentiating equation (B-4) with respect to Pj' evaluating allderivatives at the minimum, and making use of equations (B-2)yields

ati = t (aJ; I ) oq: + aJ; = oJ;. (B-5)OPj n= 1 oqn q =q. OPj apj apj

This shows that it is not necessary to know how the pathparameters q* depend upon the medium parameters in order tofind ati/apj' To ascertain the effects of a small perturbation inslowness, only the original raypath (corresponding to the un­disturbed parameters and depths) need be known. Similarly, fora perturbation in depth, the path required is the one specifiedby the unperturbed q* and the new depth.

Obtaining closed-form expressions for the derivatives is nowstraightforward. The traveltime in a box is the product of theslowness Wkt and the path length Skt. The latter is a function ofsome of the q: (and, if the box contains a reflector, oftwo depth

aJ; _ o.aqn - ,

Let the vector of values of the components of q at theminimum be denoted by q*. The equations (B-2) provide animplicit relationship between q* and p. Assume it is possible toexpress this dependence as an explicit function of p. This isequivalent to the assumption that the matrix with entriesa2J;/aqn aqm is positive definite, and not just semidefinite, at theminimum. Thus at the minimum

(B-1)ti(p) = min J; (q, p).q

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

921

(B-IO)

lIE EML.ut'!II!NT---..:::::--....:,..--------,,­_flOW

oti (XR- Xm- 1)"'z = 2Wkl cos ~ cos e _ .U pm Xm Xm-1

FIG. B-l. Calculation of depth derivative.

( x m-I,Zpm-1 )

(B-7)

(B-6)

where the sum is taken over all boxes through which the raypasses. Using W kt = Pi in equation (B-5) then gives

oti-",- = Skt·uWkt

Tomographic Determination of Velocity

parameters), but it does not depend upon any of the slownessparameters. Therefore,

For the depth derivative calculation, let the reflector passthrough (Xm -1' Zp. m-1) and (xm , Zpm), and let the point ofreflection be (xR, ZR), with Xm- 1 ::;; xR ::;; Xm, and Zp,m-1 ::;;ZR ::;; Zpm (see Figure B-·I). To compute otdoZpm, it is necessaryto evaluate the increase in path length !J.s due to an increase indepth !J.Zpm along a path characterized by the same q* parame­ters as the unperturbed route. Consider the case where the qparameters represent the distance from a box corner to thepoint of entry into or exit from that box, or the distance from(Xm -1' Zp,m-1) to (xR, ZR)' Then the new path is identical tothe original raypath except in the box containing the point ofreflection, In that box, although the reflector is rotated clock­wise, the point of entry of the new path into the box, the pointof exit from the box, and the distance from (X"'-l' Zp, ",-1) tothe reflection point must remain intact. Note that such a pathdiffers from the actual path taken by a ray which reflects fromthe downward-rotated reflector, The proof demonstrates thatthe true raypath need not be known, because the differencebetween the traveltime along it and the traveltime along thepath described is at most of second order,

Let 9 be the angle between the incident ray and a line normalto the reflector passing through (xR , ZR)' The reflector is in­clined with respect to the horizontal by an angle P= tan - 1

(Zp", - Zp, m-1)/(X", - Xm-1)' Suppose a vertical line through(XR' ZR) intersects the perturbed reflector at (XR, ZR + !J.d).Referring to Figure B-1, it is clear that

!J.d = !J.Zpm(XR - xm- 1)/(x", - xm- 1). (B-9)

Thus, if the box has slowness W kl , the depth derivative is givenby

!J.s = 2!J.d cos Pcos e,and furthermore that

(B-8) As noted, these calculations pertain to a simplified version ofray tracing. In practice the ray tracing employed is that de­scribed in Appendix A. Nonetheless, these equations are used asshown here. The justification for this procedure ultimately liesin the success of the inversion algorithm; even with such sim­plifying assumptions, the residual is reduced to acceptablelevels.

APPENDIX C

DETERMINED AND UNDETERMINED QUANTITIES

and

column of boxes striking reflector p at depth Zp (Figure Col).Let the distance between shot and receiver be 2!J.x, let theround trip traveltime be ll, and let 91 be the angle of incidenceat which the ray encounters the reflector. Similarly, let 2!J.X2't2 , and 92 denote the analagous quantities for a second ray inthe same vertical column of boxes. Then,

Appendix C gives a proof of statements made earlier regard­ing the determination of depth and slowness in a model earth inwhich ray paths are straight, reflectors are flat, and the gradientof slowness in each box is zero. (Slowness in adjacent boxesmay differ; that is, this is not a restriction to laterally homoge­neous models.) It is assumed that traveltime measurements forthe model earth are available from sources and receivers placedat locations specified as necessary below. For simplicity, it isalso assumed that reflectors lie on horizontal boundaries be­tween boxes.

That the depth of each reflector is uniquely determined bythe data can be proven as follows: consider a ray in a single

(C-I)

(C-2)

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

922 Bishop et 81.

(C-3)

that is, L\p is a member of N(~), the null space of ~. It is easy toshow that N(~) is not empty. In particular, one element of thisspace consists of a slowness distribution L\p which is laterallyhomogeneous and has the property that the sum of the slow­ness components in a single vertical column of boxes above areflector is zero. With nR reflectors and nz rows of boxes, thedimension of N(~) is clearly at least nz - nR'

With certain distributions of rays, the dimension of the nullspace may be much larger than nz - nR • However, there is atleast one finite set of rays for which no other elements in N(~)

exist. This is proven by exhibiting a set of rays for which therow space of ~ has dimension M - (n z - nR). The null space of~, which is the orthogonal complement of the row space, thenhas dimension at most nz - nR .

A collection of rays having this property is most easily de­scribed in connection with a specific example, such as that

where the sum is over the slowness in the column of boxeswhich contains the rays. Hence,

:i _sec2 91 _ Z; + L\x~

t~ - sec2 92 - Z; + L\xr

and, consequently,

(C-4)

Thus the depth ofeach reflector is uniquely determined.What can be said about the determination of the slowness

parameters in this model? With the reflector depths inferred,they can be regarded as fixed, and in that case the traveltimefunction t(p) can be considered to depend only on the nx nz

slowness parameters. The straight-ray approximation makest(p) a linear function of the slowness parameters p. Thus t(p) canbe written as

~L\p = 0, (C-7)

t(p) = ~p, (C-5)

where the ijth entry in the N x M matrix ~ is the path length ofray i in boxj. If the data vector for this system is td , then p mustsatisfy the equation

(C-6)ELEMENTS OF SPAN OF ROWS OF A

FIG. C-l. Determination of depth in the case of straight raysand a flat horizontal reflector.

Suppose one solution of equation (C-6) is p = p*. If there isanother solution p = p* + L\p, then L\p is such that

o

o

o

o

o

o

o

FIG. C-2. Elements of the row space of ~.

0 0 0 0

0 0 0 0

0 1 -I 0

0 1~ I 1 0

0 1 1 0

0 1 1 0

C2.1

C2.3

C2.2

Tdz

1

--..,.--

\ ~ EJ2

...J,01

I

II

IZp

\1II1

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Tomographic DetermInation of VelocIty 923

shown in Figure C-2. Here, nz 0= 3, nR = I, nx = 4, and M = 12.The numbers in each box of Figures C-2a and C-2b are pro­portional to the entries in that row of Awhich corresponds tothe particular ray shown. The difference between these tworows, which is also in the row space, is shown in Figure C-2c.All the vectors seen in Figure C-3 except the last can begenerated in this fashion. This yields a total of nz (nx - I) lin­early independent vectors. In addition, using one ray in eachcolumn of boxes, elements in the row space of the type seen inthe last diagram of Figure C-3 can be created. There is one suchvector corresponding to each of the nR reflectors. Thus the total

number oflinearly independent vectors for this ray distributionis

Hence the null space has dimension nz - nR .

Note that elements in the null space affect neither the averageslowness between reflectors nor the lateral change of slownessin a single row of boxes. This is the basis for the statementsmade in the discussion of effects of limited-angle rays in thetext.

LINEARLY INDEPENDENT ELEMENTS OF SPAN OF ROWS OF A

I -I 0 00 0 0 00 0 0 0

0 0 0 0I -I 0 00 0 0 0

0 0 0 00 0 0 0

1 -I 0 0

0 I -I 00 0 0 00 0 0 0

0 0 0 00 I -I 00 0 0 0

0 0 0 00 0 0 0

0 1 -I 0

I 1 I I

I 1 1 II I I I

0 0 1 -I0 0 0 00 0 0 0

0 0 0 00 0 I -I0 0 0 0

0 0 0 00 0 0 0

0 0 1 -I

FIG. C-3. Linearly independent elements which span the row space of A.

Dow

nloa

ded

10/0

6/13

to 1

07.2

02.7

4.25

0. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/