GPH 52 1 Zhu Dimensional Velocity

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GEOPH YSICS, VOL. 52 , NO. i jj.&NUPiiW i987); P; 37-50, ii PIGS., 3 TABLES .

Two-dimensional velocity inversion

synthetic seismogram computation

Tianfei Zhu* and Larry D. Brown*

and

ABSTRACT

A traveltime inversion schem e has been developed to

estimate velocity and interface geometries of two-

dimensional media from deep reflection data. The veloc-ity structure is represented by finite elements, and the

inversion is formulated as an iterative, constrained,

linear least-squa res problem which can be solved by

either the singular value truncation method or the

Levenberg-Marquardt method. The damping factor of

the Levenbe rg-Marqu ardt method is chosen by the

model-trust region approach. The traveltimes and de-

rivative matrix required to solve the least-squares prob-

lem are computed by ray tracing. To aid seismic inter-

pretation, we also include in the inversion sch eme a fast

algorithm based on asymptotic ray theory for calculat-

ing synthetic seismograms from the derived velocity

model. Nu merical tests indicate that the inversionscheme is effective, and that the accuracy of inversion

results depends upon both noise in the data and the

aperture of recording used in data acquisition. Two real

examples demonstrate that the new inversion scheme

produces velocity models fitting the data better than

those estimated by other approaches.

INTRODUCTION

Results from d eep reflection profiling have shown that the

continental crust is typically laterally heterogeneous due to

nonhorizon tal interfaces and/or la teral velocity variations

(e.g., Schilt et al., 1979 ). The velocity s tructure of these hetero-

geneous media from the deep reflection data must be deter-

mined in order to exploit them. A conventional approach for

estimating subsurface velocities is to treat s tacking velocities

from normal-moveout (NMO) analysis as root-mean-square

(rms) velocities and then compute the corresponding interval

velocities by using the Dix equation (Dix, 1955; Taner and

Koehler, 1969; Al-Chalabi, 1974). However, this approach

may produce unacceptable errors in the interval velocities be-

cause it assumes a horizontally homogeneous medium and

even then is valid only for small source-receiver offset.

A m ore flexible approach was proposed by Sattlegger (1965)

who modeled the subsurface as a stack of dipping, c onstant-

velocity layers and estimated both layer velocities and inter-face dips and depths by classical least-squares inversion. While

this approach may be useful for data collected with sm all

offset, for deep reflection data recorded with large offset the

effect of lateral velocity variations could be so significant that

the constant-velocity layer model wou ld no longer be appro-

priate.

A method for modeling lateral velocity variations w as pro-

posed by Aki et al. (1977 ). They divided a horizontally layered

medium into rectangular blocks, and then estimated a con-

stant perturbation for each block by inverting trave ltime d ata.

With this block model, however, it is difficult to accommodate

nonhorizon tal interfaces comm only detected by deep reflec-

tion su rveys. This m odel also introduces artificial dis-

continuities cau sing problems for ray tracing. Hawley et al.(I 98 I) sugges ted using an interpolation function to vary veloc-

ities smoothly between blocks, but suc h smoothing may de-

crease the resolution of vdoeity estimates.

Since asymptotic ray theory implies smooth velocity vari-

ations within each layer, it is more app ropriate to represent

such a med ium by finite elements. In the inversion schem e

described here, we divide each layer of a medium into finite

elements, assigning each nodal point of the elements a velocity

(node velocity). Velocity within an element is linearly interpo-

lated from its node velocities. With this velocity mo del, no

additional smoothing is necessary for ray tracing. Since the

finite-element model includes triangular, rectangu lar, and

trapezoidal elements with adjustable boun daries, it is not difi-

cult to incorporate nonhorizontal interfaces whose dips and

depths can change during iteration. This is desirable because

in our inversion sch eme interfaces are not fixed as they are in

the scheme of Aki et al. (1977); instead, interfaces are esti-

mated, a long with the node velocities, from reflection travel-

time data. Furthermore, the finite-element model is flexible

and it is possible to approxim ate a velocity structure from a

few large elements.

Seismic velocities and reflector depths were also determined

simultaneously from shallow reflection data by Bishop et al.

Manu script receivedby the Editor December31, 1985; evisedmanuscript eceived une6, 1986.*Institute for the Study of the Continentsand D epartment of GeologicalSciences, orn ell University, thaca, NY 14853.

#P 1987 Society of Exploration Geophysicists. All rights reserved.

37



38 Zhu and Brown

(1985) using a rectangular block model with smoothing. In

addition to employing a different velocity model, the emphasis

of the present study also differs from that of Bishop et al.

(13%); Whtle they cm~ eLl’d 7-d !arg seismic tomogra phy prob-

lems. the main purpose of this study is to develop a flexible

inversion technique for determining accurate velocities from

several selected field reco rds. Such velocities are often useful

for interpretational purposes such as calibrating the velocity

profiles routinely produced in data processing by the conven-

tional method.

The inversion is explicitly formulated as an iterative, con-

strained. linear least-squares problem that can be solved by

either the singular value truncation (SVT) method (Wiggins,

1972; Lawson and Hanson, 1974), or the Levenberg-

Marquardt (L-M) method (also known as the damped least-

squares method) (Levenberg, 1944; Marquardt, 1963-). Suc-

cessful application of the L-M method req uires an appropriate

damping factor (L-M param eter). In this study, we introduce

the model-trust region approach (More, 1977; Dennis and

Schnabe l, 198 3) to optimize the choice of the damping factor.

To aid seismic interpretation and reduce the nonuniquene ss

in traveltime inversion, we also incorporated into the routine

an algorithm for computing reflection seismograms for a two-

dimensional (2-D) medium. This algorithm is similar to an

efficient approach used by M arks (1980) and Spence et al.

(1984 ), and the width of a ray tube in the out-of-plane direc-

tion is computed by an analytic express ion, Unlike theirs,

however, our algorithm is not limited to a particular velocity

model, but is applicable to any media where the ray method is

valid. B y using this algorithm, the synthetic seismograms for

an estimated velocity model can be generated, along with the

velocity inversion, with little additional effort.

We discuss the formulation and solution of the inverse

problem, and the algorithm for computing synthetic seismo-

grams. Then we present results from num erical tests which

illustrate the capabilities and limitations of the inversion

scheme, and the effects of noise in the data and of the apertureof recording (i.e., maxim um source-receiver offset) on the final

results. In two real data exa mples, we compare the results

from our inversion scheme with those from other methods.

0 E*X

FIG. I. Finite-element velocity model and a reflected ray OE.The thick lines represent the interfaces and thin lines are div-ider boundaries of the finite elements.

MODEL AND FORMULATION

Velocity model

In the finite-element velocity model u sed in this study. each

layer of a 2-D medium (e.g., Figure 1) is divided such that an

interface always forms the boundaries of certain elements.

Thus there are two types of boundaries between adjacent ele-

ments: divider bound aries across which velocities are continu-

ous (thin lines in Figure 11,and interfaces (thick lines in F igure

1). To represent the velocity discontinuity across an interface,

two nodes with different node velocities are assigned to each

intersection point involving the interface, while only one node

is assigned to an intersection point of two divider bound aries.

The locations of interfaces may c hange during iterations of an

inversion. All nodes in Figu re 1 are labeled. For examp le, the

nodes of a typical element (F in Figu re I) a re iabeied p, 4, r,

and s. The corresponding node velocity and node coord inates

such as those for node p are deno ted a s cp and (xp, z,), respec-

tively. The velocity at a point (x, Z) within F is linearly inter-

polated from the surrounding node velocities by

2$x. Z) = A + B(x - x&J+ C(z - ZP)

+ D(s - xp) (z - “J.

where

(l)

A = PC’

c = (c, - UJ/(Z, - z&J,

:D = (c, - L',,) :

i 1(x, - x,&z, - zq) - C/(x, - xP),

an d

B =

[

f’ - up - C(Z,? zp)I’(x . XJ - D(z, - “J.//For a triangular element, D = 0. and the velocity gradient

within this element is constant. In the case where each layerforms an e lement with constant velocity. the model redu ces to

Sattle gge r’s mod el.

The interfaces within the medium are approximated by

straight lines:

zt = a/.x + b,, f = I, 2, . , L,

where L is the total number of interfaces in the medium and a,

an d h, are referred to as interface parameters. Thus the

medium is now represented by a set of model parameters

consisting of P node velocities and 2L interface parameters

which ca n be written a s a vector of length M = P + 2L ,

m = (ml, m2, . . . . 111~)~.

where T denotes matrix transpose. With this parame -terization, we can formulate the inverse problem.

Linearization of the nonlinear least-squares

problem

The associated forward problem is to calcula te reflection

traveltimes for a parameter vector m by the system of nonlin-

ear equations

T = T(m).

where T=(T,, T, ,..,, TN)’ corresponding to N (N > M) ob-

served traveltimes t = (t,, t2, , t,)‘‘ The inverse problem is



Velocity Inversion and Seismograms 39

to find an m tha t m inimizes the sum of squares of the difTer-

ences between T and t, which can be written as

min II t - T(m) II, (2)

where 11 - T(m) 11 [t - T(m)]‘[t - T(m)]. This nonlinear

least-squares problem is solved by the Gauss -Newton method

which linearizes expression (2) about a current m, to obtain

the linear least-squares problem

min IIb - J&m 11. (3 )

where b = t - T(m,) is the vector of traveltime residuals, and

d is the Jacobian matrix containing the derivatives

_?Ti7Ji, - I Illcinj. i-:, 2, . . . . N;j= :, 2, ___ ) w. (4

Then the current m,. is iteratively corrected by an amoun t Sm

solved from equa tion (3). In general, the approxim ation of

equation (3) to equation (2) is only valid for a small Sm. In

practice, however, the Sm solved from equation (3) may be

large, or even unbounded, due to the fact that the matrix J

may be singular or nearly singular (e.g.. Jupp and VozofT,

197 5; Aki et al., 19 77). To stabilize the iteration, it is necessary

to solve equation (3) subject to 11m // 5 A. Here A is a posi-

tive number and delines the radius of a model-trust region inwhich one can trust the local linear model b - JSm in equa-

tion (3) to approximate the nonlinear fun ction t - T(m) ad-

equately in eq uation (2). Thu s the inverse problem is finally

reduced to solving iteratively the constrained, linear least-

squares problem:

min (11 - JSm 11 11m 11 A) . (5 )

It is clear from equation (5 ) that the implementation of the

inverse problem involves two fun dame ntal steps: finding for-

mulas for computing T(m) and J based on the previously

described velocity model and then from the resulting J, and b

solving for the correction vector Sm which satisfies

11m 11 A. The first step is accomplished (shown next) by ray

tracing.

Chmputathr af iraveltimes and- he Jacobian- matrix

A shooting method similar to those described by Cerven$ et

al. (1977) and Julian and Gubbins (1977) is used here for

tracing a ray between two specified points. This method in-

volves repetitively tracing a ray from a source point (0 in

Figure I) with a trial takeoff angle PO , and employing a pro-

cedure for adjusting that angle until the distance between the

receiver (E in Figure I) and the end point of a trial ray is

smaller than a given tolerance. To trace a ray through themedium , a fourth-order Runge -Kutta integration routine is

used to solve the 2-D ray equations (e.g., cervenjl et al., 1977 ).

If only triangular elements are used in the finite-element

model, then there is a simple analytic solution for the ray

equations within each element and the Runge -Kutta routine

can be replaced by a more efficient ray-tracing routine similar

to that used by Miiller (1984 ). The proced ure for adjusting

takeoff angles involves solving a nonlinear e quation,

X = X(Po), (6 )

where X is the distance between 0 and the end point of a ray

starting from 0. Equation (6) is iteratively solved by the

secant method (Dennis and Schnabel, 1983):

where X, is the distance between 0 and E.

Assume that the ray OE in Figure 1 corresponds to the ith

observation and is divided into R(i) segments. Then the travel-

time along this ray is computed by

(7 )

where As, is the length of kth ray segm ent and v(k - 1) and

a(k) are the velocities at the end points of this~ ay segm ent

(Figure 1).

Com bining equations (1), (4), and (7) yields the formula for

compu ting the derivatives of the traveltime with respect to the

node velocities :

_,,= _; yAsk ’ iiu(k- )

k=, u2(k 1) av j

I 1 W )?(k) aoj 1 (8

i= 1,2,..., N; j=l,2 ,..., P,

where P is the total number of the node velocities. There is noanalytic formula such as equation (8) for the derivatives of the

traveltime with respect to the interface parameters. These de-

rivatives are compu ted by a finite-difference approxim ation

(Lines et al., 1984),

J__ _ a’& __T(m’ aj + Aaj) - T(m’ uj)1, iiccj Aa j

(9 )

i = 1, 2, __., N; j= P+ 1, P+2 ,..., M,

where Aaj is the perturbation of the jth interface param eter aj,

and m’ represents the remaining unchanged model parameters.

So far, we have obtained the basic comp utational formulas

for the traveltimes, equation (7), and the Jacobian matrix,

equations (8) and (9). Now we can proceed to solve the con-strained least-squares problem in equation (5 ).

SOCUTIUN OF THE CONSTRAINED

LINEAR LEAST-SQUARES PROBLEM

The solution of equation (5) can be obtained by using either

the SVT m ethod or the L-M method. Both methods are im-

plemented he re by using singular value decompo sition (SVD ).

The SV D of J is (Lanczos, 1961 )

J = UQy,

where u and v are N x M and M x M matrices withUTU = VTV = VVT = 1__ __ __ -M 1 and b is an M x M diagonal

matrix containing the singular values

an d

hpi, -h,,,=,..=h,=O.

Here p is the rank of a. Without considering the constraint of

11m 11 A, the generalized inverse solution of equation (5) is



40 Zhu and Brown

where 4’ is a diagonal matrix with A,; = l/hi if hi > 0,

A:, = 0; otherwise, Vj is thejth column of v and

g = yTb. (13)

In practice, some nonzero singular values can be very small

(e.g., Jupp and Vozoff, 1975) and can cause solution (12 ) to

increase and e xceed the limit in which the linear model is

valid. To stabilize the solution, the SVT method reduces the

sire of 6m by eliminating the terms associated with smallsingular values from the summation in equation (12) while the

I--M method ac hieves this by modifying I/h, to h,/(hj -t p)

with a damping factor u > 0.

1,-M method with the mode l-trust region approach

Impleme ntation of the L-M method described here is simi-

lar to the implemen tation d escribed in More (197 7) and

Dennis and Schnabel (1983). However, instead of QR and

Cholesky decompositions, we have used the more robust SVD

in this study. The extra computing time involved in the SVD

is marginal compared to the overhead of the rest of the pro-

gram and is more than compensated by the extra information

prov>ided.With equations (10) and (13) the solution of equa-tion (5) can be written as

general. p approaches unity when the agreement is good and

becomes smaller as the agreement becomes poorer. In the case

wrhere I(b( +) > I)b // (i.e., the inversion diverges), p < 0

(More, 1977). To ensure a reasonable convergence rate, we

follow Dennis and Schnabel(l983) and accept a solution when

p 2 0.000 1; otherwise we reduce A to its half-value and

repeat the current iteration. When a current solution is accept-

ed, A is updated for the next iteration using a scheme adopted

from More (1977) and Dennis and Schnabel (1983). The basicidea is that we take a large step in correcting the parameter

vector when p is large, otherwise we decrease A (hence increas-

ing u) to stabilize the solution. Thus , for large p (> 0.9) we

save the current solution 6m and recompute equations (14)

and (15) with an increased A = 2 11m 11o obtain a new solu-

tion If the ratio p associated with the new solution is less than

0.00 0 1. we drop back to the earlier solution a nd proceed to

the next iteration with the updated A = 2 (( m 11; ut if the

ratio p is again larger than 0.9, we may repeat the procedure.

This a pproach reduces the numbe r of iterations by fully u tiliz-

ing the current linear model and thus saving computer time

For 0.000 1 c: p i 0.9, we update A as follows: A = 11m 11f

p > 0.25, A = t311m 11f p < 0.25. The quantity 9 is computed

by the formula

8 = u/ ‘(I Ibi+) II2 IIb I I2+ 2-r),6m(p) = v(.A’ + ~1)~ ‘Qg. (14)

When u = 0 and 11m(O)” I A, the solution of equation (5) is

Sm(0). Otherwise, there is a unique u determined from the

equation

+(cl) = II 6W) II - A = 0, (15)

and its corresponding 6m(p) is the solution of equation (5).

In practice, it is not necess ary to solve equa tion (15)

exactly; instead we only require that 6m satisfy

0.75A < //Fm(p) /I < 1.5A. In this case, a satisfactory p c an

easily be obtained by the iteration (Dennis and Schnabel,

1983, p. 134-l 3 6)

where y = 11&m II2 + p 118m/’ (More, 1977). If 0 is outside of

the interval CO.2, .61, we replace 0 by the closer end point of

this interval. To assure convergence and efficiency of the algo -

rithm. we also adopted or modified some other techniques

described by Marc (1977) and Dennis and Schnabel (1983),

including computing upper and lower bounds on p and using

a parameter weighting matrix in equation (5). The latter is also

used in the SVT method discussednext.

The ‘+‘VT me tho d

P- I = pk (11m@ ) ~~,/A)& I~)/$‘($),

where the derivative is computed by

The controlling effect of u on the inversion results has been

discussedby various investigators (e.g., Jupp and VozoK, 197 5;

Crosson, 1976; Haw ley et al.. 1 981). The choice of p depends

upon the problem considered. An optima l u. which shou ld be

as small as possible to achieve the maximum resolution and

convergence rate yet large enough to stabilize the solution,

also varies from iteration to iteration. An effective algorithmfor selecting I_Ihas recently been developed by More (1977)

and Dennis and Schnabel (1983) and is adopted here. This

algorithm. known as the model-trust region approach, at-

tempts to find an optimal p through equation (15) by varying

A in each iteration. The A for the next iteration is estimated

based on the size of the current solution and the ratio between

the actual reduction in 11 11 nd the reduction predicted by

the !ocal !inear mode! in equation (5) is

Since it is not necessary o satisfy the constraint in equation

(5) exactly, we can also use the SVT method. To accomplish

this. we only need to select a large enough cutoff value and

eliminate from the summ ation (12) the terms associated with

the singular values smaller than the cutoff value , such that the

norm of the resulting solution is closest to b ut sm aller than A.

Lawson and Hanson (1974) describe the procedure. which

selects a cutoff value by inspection and req uires no extra func-

tion ev aluation for c ompu ting a criterion such as p in the last

section. We comp uted a series of candidate so lutions Sm k by

using each singular value A, as the cutoff value in equation

(12):

6mk = 6m km’ + V,g ,:‘h,, k= 1,2 )..., p,

where 6m” = 0. At the same time the norm of each solution11mk // and the asso ciated residual norm rli are also compu ted

by using the formulas

1Ii2

11mk-’ I I2 + (qk/hk)2 ,

an d

P = ( II h I I’ - II b(+) II%ll b ,,2 - II b - Jh 112),,

where b( + ) = t - T(m + 6m). The ratio p measures the agree- where (r’)’ = g”g. By comparing these computed quantities,

ment between the linear model and the nonlinear function. In we select an accep table solution by req uiring that its norm be




smaller than an a priori limit (an estimated magnitude of

6m 11)and the associated residual norm must be acceptably

small. In practice, we found that an acceptable solution can

often be located based on the structure of the singular values.

For a nearly singular Jacobian matrix, some of the later singu-

!ar values~are significant!y smaller than the ear!ier ones in

equation (I I), and small values are often separated from large

values by a distinct “cliff,” as observed by Jupp and Vozoff

(1975). It is natural to choose a cutoff value to eliminate these

small sing ular valu es because they cau se the solution to be

large and unstable, while contributing little to reducing the

residual norm (Ju pp and Vozoff, 197 5). A further che ck for the

selected solution is provided by the diagonal elements of the

resolution matrix and standard deviations computed for each

candidate solution, as discussed by W iggins (1972).

COMPUTATION OF SYXTHETIC SEISMOGRAMS

The algorithm used here for computing s ynthetic seismo-

grams for a velocity model is based on asymptotic ray theory.

Although this method cannot model certain types-of signals,

e.g., diffractions and c austics, it is simple a nd ha s proven ap-

plicable in a wide range of velocity structures. The principlesand limitations of asymptotic ray theory have been described

in detail by various authors (e.g., cerven$ and Ravindra , 197 1;

C‘ervenp et al., 1 977 ); here we concentrate on the formu las

relevant to comp utation of ray amplitudes. According to

cervenjr and R avindra (1971). the amplitude A(E) at the end

point E of the reflected ray OE (Figure 1) is related to the

initial amp litude A (0) at the source point 0 by

where H, is the incident point of the ray at the Ith interface

an d R, is its transmission (reflection) coefficient computed by

using the routine of Young and B raile (1976). The velocity

r(H,) and density p(H,) are evaluated on the side of the inci-

dent ray, while d(H,) an d p’H,) are on the side of the emerg-

ent ray. The geometric spreading L in equation (16) is given by

J(E)

[ 1:2 iv

L= 7

sm PO,pos o,/cosq’” (17)

where 0, and 0; are, respectively, the angles of incidence and

emergence at H,. Following Marks (1980), the Jacobian J(E)

in equation (17) is given by

J(E ) = cos P(E ) (?X/ (‘p,) (+:‘?a,) . (18)

P(E) is the angle between the ray an d the z-axis at E, X the

distance between 0 and E, y the out-of-plane coordinate, an d

a0 the takeoff angle of the ray measured from the x-z plane.Computing J(E) is a key step in generating an asymptotic

seismogram . In o ur algorithm, ‘:X,/C;&, is approxim ated by the

difference AX/A&, compu ted by using two rays with their end

points close to E. The derivative r3Ja/Oa,,s related to the width

of the ray tube in the out-of-plane direction and calcu lated by

a formula derived from the ray equations (see the Appendix)

where p is the angle between the ray and the z-axis and S the

length of ray O E. It is clear from eq uation (1 9) that ay/da, =

X when i%/?x = 0. Formula (19) is applicable to 2-D media

with arbitrary, smooth velocity distributions. A similar ap-

proach for computing L7y/Sa, was also employed by Marks

(1980) and Spence et al. (1984). Their approach. how ever, is

limited to the velocity model consisting of blocks with con-

stant velocity gradient. With the algorithm described here, the

raysamplitude in eeqqltiion~16) can_ a.sily he computed, along

with traveltimes, at minor additional cost. Neither incon-

venient out-of-plane ray tracing nor time-consum ing “dynam-

ic ray tracing” is needed. In the latter app roach, ra y ampli-

tudes are calculated by solving a set of differential equations

in addition to the ray equations (e.g., Cerven? et al., 1977 ).

After the amplitudes for all desired reflections arriving at a

particular receiver are calculated, a seismic trace is generated

in which each arrival is represented by an impulse proportion-

al to its amplitude. Convo lution of the seismic traces with an

appropriate apparen t wavelet produces the final seismogram.

ln the case where an a rrival is phase-shifted due to internal

caustics and/or complex reflection coefficients, the wavelet is

replaced by a linear combination of the wavelet with its Hil-

bert transformation (cervcny and Ravindra, 197 1).

NUMERICAL TESTS

Num erical tests on synthetic data hav e been undertak en to

study the abilities and limitations of our inversion scheme.

Results from two tests are presented. The first test was pri-

marily designed to examine the capability of the inversion

scheme in recovering laterally varying velocity structure, while

the second studied the effects of noise and the maximum

source-receiver offset (recording aperture) on the accuracy and

stability of inversion.

Test 1

The velocity model u sed to generate the synthetic data in

this test is shown in Figure 2a. The node velocities linearly

increase vertically as well as laterally. A s ynthetic seismogramfor this model is shown in Figure 2b. The CPU time for

calculating ray am plitudes is about 5 s on an SEL 32,/77. In

comp uting this seismogram, we have assum ed that the half-

space P-wave velocity is 6.5 km/s. Th e S-wave velocities and

densities used in equation (16) were computed from P-wave

velocity v by v,~ v/d and p = 0.252 + 0.378 80, respectively

(Birch, 1964 ). Note that in spite of a smaller impedance con-

trast due to the lateral velocity increase of layer 2, the ampli-

tude of the wide-angle reflection from the lower interface at

trace 10 is still considerably larger th an the o thers. The large-

amplitude, wide-angle reflections from the upper interface

start at trace 5. The phase shift of wide-angle reflections is due

to complex reflection coefficients.

The traveltimes us ed for inversion were directly calcu lated

from the velocity model. 200 reflection arrivals were computed

with about 15 km recording ape rture, and then noise, in the

form of random traveltime delays with 5 ms standard d evi-

ation, was added to these data. Tw o separate inversions were

made from the synthetic traveltime data, and the resulting

model parameters, along with the initial models, are shown in

Figures 3 and 4, respectively. While only no de velocities were

calculated in Figure 3, both node velocities and the depth and

dip of the lower interface were calculated in Figure 4. In both

cases, the rms residuals were reduc ed to the level of the noise

(defined here as the standard deviation of the random travel-

time delays) after three to five iterations. E xcellent results were



Zhu and Brown

A DISTANCE (km)

0 0 16

2.0 2.64 3.282

6

6 DISTANCE (km)

0 3 6 9 12 15

III I ISOURCE

Ll

TRACE 2 4 6 0 lb

FIG. ? . (a) Two-layer velocity model used in test 1 . The numb ers shown on the corners of the elements are the nodevelocities. (b) A synthetic seismogram from the velocity mo del.

A DISTANCE (km1 B DISTANCE (km)

0 8 16 0 8 16

3.0 3.0 3.0 20 0 2.65 3.29

(50.0) (13.6) (8.5) (0.0) (0.41 (0.3)

(35.7) (IO.51 (6.9)

2 4 3.8 3.8 38

- 4.8 4.6 4.8

= (26.31 (8.1) (5.5)

aw (4.7)

D 5.68

+

C DISTANCE (km)0 8 16

c1.97 2.64 3.32

(I.51 (0.01 (1.2)

i 4;;;

(0.3) (0.7)3.45 4.04

4.43 5.09

z (0.3) (0.2) (0.2)

w”n

(0.2)

IO

1

FIG. 3. Inversion results when only the node velocities were estimated from the synthetic data of the velocity model inFigure 2 . (a) The initial model. (b) and (c) The final m odels estimated by the L-M method and the SVT methodrespectively. The numbe rs inside parentheses ndicate the percent differences between estimated parameters and actualvalues.



Vi3locity hversibn and Seismograms

obtained by both L-M and SVT methods, except for the re-

sults shown in Figure 4c which have larger differences from

their actual values than those in Figure 4b. The SVT method,

however, does not require extra function evaluation for

choosing cutoff values, and was found very useful when only

node velocities are to be determined from the data (see, for

example, Figure 3~).

For comparison, we also computed the model parameters

by using the L-M method with a fixed u selected through a

trial-and-error process (Crosson, 1 976). The results show thatthis conventional L-M method is less effective than either the

SVT method or the improved L-M method used in this study.

For example, the maximum differences between actual node

velocities and those produced by the conventional L-M

method after five iterations for the cases n Figures 3 and 4 are

respectively about 5 percent and 8 percent, which are larger

than those shown in Figures 3b, 3c, and 4c (about 2 percent).

In addition to its strong convergence properties, the improved

L-M method is simple to use because it is no longer necessary

to choose an initial u by empirical trials as is required by the

conventional L-M method.

To estimate the reliability of an obtained so lution, the diag-

onal elements of the resolution matrix and standard devi-

ations for the solution were computed in this study using the

formula given by Wiggins (1972) and Aki et al. (1977). The

diagonais of the resoiurion matrices for the estimates shown in

Figures 3 and 4 are generally large (most of them are larger

A DISTANCE (km)

0 8

3.0 3.0

(50.0) (13.6)

(35.71 (10.5)

4 3.8 3.84.8 4.8

(26.3) (8.11

(20.0) (9.5)

8 6.0 6.0

16

3.0

(8.51

(6.9)

3.84.8

(5.51

(0.7)

6.0

C DISTANCE (km)

0 8

1.97 2.64

( 1.5) (0.0)

(1.41 (0.0)4 284 3.44

3.51 4.17

(7.61 (6 . I)

16*

3.30

(0.61

(0.5)4.064.96

12.41

than 0.7), indicating that these estimates are reasonably re-

solved. In general, this is also true for the rest of the numerical

examples presented. The standard deviations for the estimates

are small (e.g., the maxim um for the estimated node velocities

in Fig ure 4 is 3.2 percent) because of the small amoun t of

noise and large recording aperture used.

Test 2

Data for our inversion schem e were taken from deep reflec-tion results recorded with surface sources and receivers. With

such data, the stability of an inversion and the uncertainty of

estimated m odel pa rameters are sensitive to the recording ap-

erture used. The sensitivity must be ana lyzed (see, e.g., Limond

and Patriat, 1975 ); here we study this sensitivity numerically,

as well as the effect of noise in the data, by using a constant-

velocity layer model shown in Figure 5 a. Although simple, this

model is particularly useful for sma ll-aperture data. A series of

traveltime data sets was generated from this velocity model by

simulating different recording apertures, including 2.5, 5, and

10 km (IO km is now commonly used in COCO RP field sur-

veys, for exa mple). E ach set consists of 60 reflection arrivals to

which noise was added in the form of random traveltime

delays with varying standard deviations (5, 10, and 20 ms).

From these data, the velocities of all three layers and the dips

Andydepths of interfaces 2 and 3 were then calculated. All

calculations started with the initial model shown in Figure 5b,

B DISTANCE (km)

0 8 16

1.97 2.65 3.32

11.5) (0.4 1 (I.21

-Z

(2.1) (0.3) (1.0)

r 4 ‘;8”7” 345 4.044.52 5.21

: (1.81 (I.81 (2.5)

Eln (1.71

FIG. 4. Inversion results when the dip and depth of the lower interface as well as the node velocities were estimatedfrom the synthetic data of the velocity mo del in Figu re 2. (a) The flat-layered initial model. (b) and (c) The final modelsfrom the L-M method and the SVT method, respectively.



44 Zhu and Brown

an d the rms residuals were reduced to the level of noise after

three iterations.

The resulting velocity profiles are shown in Figure 6, and

the standard deviations for the estimated parameters are listed

in Ta ble 1. It is clear from these results that the effect of noise

on the accuracy of estimated parameters is largely controlled

by the ratio of the recording aperture to interface depth (A,/D).

With the 10 km aperture, the increase in noise causes only a

small increase in the errors in the estimated param eters, and

good results were obtained even when the noise level is high.

For example, the difference between the actual vj and that

estimated with 20 ms noise is 2.6 percent, and the correspond-

Ao’STANzE (km)

B0 10 0 D’STAN:E (lcrn) 10

ing standqrd d eviation is 0.17 km /s (3.4 percent) (Table 1). In

contrast, for the 5 km aperture, the errors were largely ampli-

fied, particularly for the parameters associated with the d eep

layers. The difference between the actual V, and that estimated

from data with 20 ms noise is now 14.4 percent, and the

standard deviation is 0.90 km/s (18 percent). Comparing the

results in Figures 6a and 6 b as well as their s tandard devi-

ations (Table 1) shows that the am plification of the param eter

errors due to the reduction of A/D also increases with the

depth o f a layer (defined as the depth to the lower interface).

While reducing the aperture from 10 to 5 km only slightly

increased the error in the estim ated VI, the increase in the

errors of the parameters of layers 2 and 3 due to such re-

duction is mu& iarm This indicates ihat A,0 is a critical

parameter in deep reflection surveys, especially for those de-

signed for m apping subsurface velocities. This is, of course, an

expression of the well-known fact that large offsets tend to

improve velocity resolution.

The num erical results from this test also confirm that small

recording aperture and noisy data can make an inversion un-

stable. All results in Figu res 6a and 6b were obtained by the

classical least-squares procedure (i.e., the L-M method with

u = 0). This p rocedure, however, diverged for the data w iththe 2.5 km aperture. The results in Figure 6c were obtained by

the SVT method which also failed to converge for the data

with 20 ms noise. The derivatives of traveltimes with respect

to the parameters of a deep layer approach zero when A/D is

small (e.g., Lines et al., 198 4). This corresponds to the cas e

“1 -4,o(km/a)INTERFACEi z7-F

FIG. 5 . (a) The ve locity mod el, and (b) the initial model used in

C

test 2.

A

4 6

1

Iiiiiii Dl =0.02

i D2~0.04

D3=0.06.-.-.-. -51

:iii.:I

Dl=0.15 ,i

D2~0.13 _i

D3~0.11 _JL._.-

B

VELOCITY (km/s)

d

2 4 60 0

4 4

8

E

8

12

8 8

-.._.

12

r

fi

i!

:I.

i!

t !:! D1=0.06

:I D2=0.11

ii D3~0.40. . . .

L.-.-.-..

II

16 16 i 16

Dl= 0.89

D2= 0.25

- TRUE VALUE. . . . , . WITH 10 ms NOISE

-- - WITH 5 ms NOISE -‘-‘- WITH 20 ms NOISE

FIG. 6. The estimated velocity profiles at the zero distance in Figure 5: (a) from 10 km, (b) from 5 km, and (c) from 2.5km aperture data. Som e velocity profiles, such as that es timated from data with 5 ms noise in (a), are too close to thetrue value to be shown. Dl, D2, and D 3 indicate the differences (in degree) between the true dips and those estimatedfrom the data w ith 5, 10, and 2 0 ms noise, respectively.




where the matrix J, in equation (5) is nearly singular. In this

case, as indicated by eq uation (12) the small singular values of

J will largely am plify the noise and cau se the solution to

become large and unstable. In general, the stability of an in-

version also depends upon the velocity structure involved. For

example, in spite of the fact that a small amount of noise and

a large A/D ratio were used in test 1, nonzero damping factors

were required in order to stabilize the iterations. The SVT or

L-M method was also used in the real data examples present-

ed below. Thus, an acceptable A/D for a particular case de-

pends upon the noise level and required accuracy as well as on

the velocity structure involved. In this example, if the veloci-

ties are required to be accurate to within 5 percent, then the

acceptable A/D for the data w ith 10 ms noise is about 0.5.

REAL DATA EXAMPLES

The present inversion scheme was applied to different types

of COCO RP deep reflection data to provide subsurface veloc-

ities for seismic interpretation. Presented here are examples

from COC ORP experiments in Michigan (Zhu and Brown,

1986) and Utah (Liu et al., 1986). Data used in both examples

have been extensively processed in order to increase thesignal-to-noise (S/N) ratio and data resolution. As a result,

traveltimes can be more accurately picked on the field records.

Unfortunately, such processing also destroyed the true-

amplitude relations of the data; hence no attempt was made

to compu te synthetic seismograms. Howe ver, deep reflections

with unusually strong amplitudes have been observed on

various sites (e.g., De Voogd et al., 1986 ). In these cases, syn-

thetic seismogram s could be u seful, for exam ple, in ded ucing

reflectivity of an interface.

Example 1

The data in this example were collected by COCOR P in

central Michigan near the McClure-Sparks no. 1-8 deep wellwith a recording aperture of about 7 km (Zhu and Brown,

1986 ). Two field records close to the dee p well were used in

the calculation; one is shown in Figure 7. Five reflections on

these records were selected (Figure 7) because they corresp ond

to reflectors separating seismic sequences identified on the

6

0 2 4 6

OFFSET (km)

FIG. 7. A field record from CO COR P Michigan surveys andfive reflections (dotted curves) used in the inversion. The re-flections were selected based on the seismic units identified onstacked sections (Zhu and Brown. 1986).

Table 1. Standard deviations for the parameters estimated in test 2.

Recording DataStandard deviation

aperture noise Layer velocity (km/s) Dip (degree) Depth (km)

(km) (ms)Y V2 V3 a2 a3 b2 b3

I-

1;

e;co~ fi;c+ OJ3~ 0104 O:t'r OIB 0.04

10 0.00 0.02 0.08 0.08 0.22 0.05 0.09

20 0.01 0.04 0.17 0.18 0.45 0.13 0.19

5 0.00 0.04 0.20 0.08 0.40 0.12 0.19

5 10 0.01 0.09 0.39 0.16 0.86 0.25 0.30

20 0.02 0.20 0.90 0.34 1.80 0.55 0.90

2.5 5 0.00 0.10 0.13 0.20 0.50 0.28 0.10

10 0.20 0.28 0.03 0.40 0.94 0.57 0.20



stacked sections Zhu and Brown, 1986). Traveltimes along

these reflections were then visually picked and digitized. The

accuracy of such picks depends upon the ability to identify

and c orrelate individual reflections on a record by sight. In

this example, reflections B to E can be clearly identified. The

deep reflection F , however, is less certain du e to the lower S/N

ratio. Trav eltime picks for this reflection were guided by a

smooth curve connecting the correlatable reflection segments.

The resulting velocities and depths, as well as their standard

deviations, are shown in Table 2. No significant dips were

detected. As expected, the uncertainties of the estimated pa-

rameters increase as the A/D decreases with the depth o f the

layers (Table 2).

For comparison, Table 2 also lists the results obtained with

the conventional method (Zhu and Brown, 1986). This method

estimates the interval velocities from stacking velocities by

using the Dix equation. Th e s tacking velocities were deter-

mined b y the velocity semblance technique (Taner a nd Koeh-

ler, 1969).

The resulting interval velocities and depths from these two

. .

*;;‘... ‘‘..:E. . . . . . . .

. .

%ic4 . . . . 1.. . . . . . . . .,

46 Zhu and Brown

different approaches agree reasonably well for some intervals,

but not so well for others. Higher interval velocities below

layer B derived from the stacking velocities (Table 2) may be

due at least in part to the known bias between stacking and

rms velocities, which leads to e rroneously high interval veloci-

ties (Taner and Koehler, 1969; Al-Chalabi, 1974). Traveltime

curves for both velocity models were computed and compared

to the actual traveltime picks in Figure 8. It is evident that the

velocity model estimated by the new method fits the observed

traveltimes better than the model from the conventional

method.

The resultant velocity model is consistent with the data

from the deep well and the structure imaged by the stacked

seismic sections. For examp le, the flat-layer structure indicated

by the velocity model was also imaged by the stacked sections

near the deep well (Zhu and Brown, 1986), and the calculated

depth of the lower interface of layer B matches the thickness

of Paleozoic rocks from the deep well, which penetrated about

3.7 km of Paleozoic section and 1.6 km of underlying Precam-

brian rocks (Sleep and Sloss, 1978). Based on the velocity

model, the lithologies encountered in the deep well, the seismic

character of the stacked sections, and the regional g eology, an

interpretive stratigraph ic section (Table 2 from Zhu andBrown. 1986) was constructed which provides a useful con-

straint on the seismic interpretation of the deep structures

beneath the Michigan basin (Zhu and Brown, 1986).

Example 2

FIG. 8. Comparison between the traveltimes picked from theCOC ORP Michigan d ata (dotted curves) and calculatedtraveltimes (solid curves) (a) from the con ventional method,and (b) from the new inversion scheme.

In order to improve subsurface velocity estimates by in-

creasing A/D, COCO RP sometimes also records expanding

spread profiles (ESP) on selected sites (Liu et al., 198 6). One

such ESP was recorded in the Sevier Desert of west-central

Utah, where COCO RP Utah line 1 clearly imaged a shallow,

low-angle normal fault, the Sevier Desert detachment, and a

relatively prom inent dee p reflector interpreted as the Moho

(Allmendinger et al., 1983). The maximum recording aperture

in this survey is about 32 km, or three times normal (Liu et al.,

198 6). The traveltime-offset display of the ESP data an d the

picked reflections are shown in Figure 9. Similar to example 1,

the uncertainty involved in visually picking traveltimes for

reflection E is also considerably larger than the uncertainty for

Table 2. Velocity m odels from Michigan data and the interpreted stratigraphy. Velocity and depth for the layer A wereestimated from first refractions Zhu and Brown, 1986).

Conventional Standard deviation Interpreted

method New method of the new method stratigraphy

Velocity Depth Velocity Depth Velocity Depth

Layer (km/s) (km) (km/s) (km) (km/s) (km)-------

A 2.30 0.13 2.30 0.13 Surface

low-velocity layer

6 5.31 3.49 5.62 3.72 0.01 0.01 Paleozoic

C 5.56 5.35 5.46 5.55 0.05 0.02 Upper Keweenawan

D 5.83 7.47 5.29 7.47 0.08 0.03 elastics

E 6.15 11.30 5.71 11.02 0.09 0.05 Middle Keweenawan

F 6.59 18.00 5.94 17.06 0.12 0.11 volcanic sequence




the shallow reflections, especially beyond 23 km.

Originally, the data were analyzed by assuming a horizon-

tally homogeneous medium, and the analysis included the

methods used in test 1 and Al-Ch alabi’s shifting-stack tech-

nique, which attempts to remove bias from stacking velocities

(Al-Chalabi. 197 4). The stack ing velocities used in the conven-

tional method for this example were determined by the

7‘* - X 2 technique (Dix. 195 5; Al-Chalabi, 1974 ). The re-

sulting velocities and depths, as well as their standard devi-

ations, are listed in Table 3 . The traveltime curves compu tedfrom these velocity models are compared to the observed

traveltimes in Figure 10. Again, the interval velocities deter-

mined by the conventional method are higher than those de-

termined by the other two methods, and they lead to large

discrepancies between the calculated and observed traveltimes

(Figure 1Oa).

Although the results from both the shifting-stack metho d

and ou r inversion schem e agree well with the observations for

the shallow layers. there is a large mismatch at the deepest

reflection for all the velocity models in Table 3. This mismatch

may be due to inadequate representation of the crustal struc-

ture by a 1-D model (Liu et al.. 198 6). Thus. the computation

was repeated by us ing the new inversion sch eme with a 2-Dmodel sho wn in Figure lla. Becaus e the velocities and depths

derived from I-D inversion are satisfactory for four shallow

layers. the model parameters for the upper eight elements were

fixed to be identical with those results. Only the node veloci-

ties of the two lowest elements and the depth of the lowest

interface were allowed to vary, and the resulting estimates and

their standard deviations are shown in Figure 1 a. The travel-

times calculated from the final model are now in excellent

agreeme nt with the observations (Figure lib), and the stan-

dard deviations of the estimates are also smaller than those for

the velocity and depth of the lowest layer shown in Table 3. In

this final model, velocities in the lower crust increase slowly

from south to no rth, as indicated by the node velocities in

Figure 1 a. Howev er, note that the node velocities at eitherend of the profile are significant only for interpolating velocity

values for the shaded portion of the crust where the seismic

rays have actually sampled the subsurface.

CONCL<USIONS AND DISCUSSION

We have presented a new technique for determining subsur-

face velocities from deep reflection data. Becaus e it uses a

finite-element model, this technique is simple but flexible and

can accommo date nonhorizontal interfaces as well as lateral

velocity variations. We also include an algorithm for calculat-

ing synthetic seismograms of a 2-D medium . This algorithm is

efficient ar..i subject only to the limitations of asymptotic ray

theory. Th us, the routine provides a useful tool for seismic

interpreters to model the velocity structure as well as the am-

plitude and phase response of any 2-D media with smooth

layer velocities.

The inverse problem was explicitly form ulated as an itera-

tive, constrained, linear least-squares problem. Two effective

methods. the SVT method and the improved L-M method,

-- *0 1’0 2'0

SHOT-RECEIVER OFFSET (km)

FIG. 9. COCO RP Utah ESP data and five reflections (dottedcurves) used in the inversion. Reflection A is from a basaltlayer, reflection D is from the Sevier Desert detachment, andreflection E is interpreted as the Moho reflection. Thequestion m ark indicates that reflection E is less certain be yond23 km.

Table 3. Homogeneous layered velocity models from Utah data.

Conventional Shifting stack Standard deviation

method method New method of the new method

Velocity Depth Velocity Depth Velocity Depth

Layer (km/s) (km)

Velocity Depth

(km/s) (km) (km/s) (km) (km/s) (km)--- --- --

A 2.15 1.05 2.15 1.05 2.1 1 1.05 0.01 0.00

B 4.30 2.38 3.65 2.19 3.54 2.15 0.02 0.01

C 5.15 3.67 4.84 3.39 4.82 3.40 0.03 0.01

D 6.26 5.36 5.06 4.76 5.18 4.75 0.03 0.01

E 7.83 38.87 6.21 31.32 6.21 31.18 0.43 0.59



48 Zhu and Brown

were used to solve this problem. The L-M method w as imple-

mented with the model-trust region approach. This improved

L-M method is robust and has strong convergence properties.

Also, although the SVT method is less effective than the L-M

method for simultaneously determining node velocities and

interface parameters. it is useful when on ly node velocities are

to be determined.

Experiments with synthetic data indicated that accurate ve-

locity and interface geome tries can be determined with our

inversion scheme, providing the traveltime picks are accura te

and/or recording aperture is sufficiently large. Based on ou r

experiences, accura te reflection arrivals can be picked from

data when the S/N ratio is high. For the noisy data, large

uncertainties may be involved in such picks. Theoretically, we

may improve the inversion results for noisy data by increasing

the recording aperture . In practice, however, such improve-

ment is limited due to the fact that the S/N ratio also de-

creases as the recording aperture increases, and 2-D and 3-D

variations become more significant.

Computations with real data suggest that while a simple

A

0 z a 8

---Observed

-Calculated

w9

2E

12

t

-‘-.------W

15VOFFSET (km)

B

I50 IO 20 30

OFFSET (km)

C

15WOFFSET (km)

FIG. 10. Com parison of the traveltimes picked from ESP d ata (dotted cu rves) and calcu lated traveltimes (solid curves)(a) from the conventional methodd,~(b)rom shifting-stack method, and (c) from the new inversion scheme using a 1-Dmodel.

A FINITE-ELEMENT MODEL

0.mt 05

2.153.40

4.75

B 0~. -Observed

-Calculated

3

OFFSET (KM)

FIG. 11. ESP velocity results from the new inversion scheme using a 2-D model. (a) Velocity structu re determined bythe inversion. Shaded area indicates the portion of the crust where seismic rays have actually sam pled the subsurface.The numbe rs shown in the center of the eight upper elem ents indicate the velocities (km/s) of the elements. Thenumb ers in the corne r of the two lowest elements are nod al velocities, and numb ers inside parentheses are theirstandard d eviations (km/s). (b) Com parison between the observed traveltimes (dotted curv es) and the calcu latedtraveltimes (solid curves) from the 2-D model.




constant-velocity layer model is useful for data with a small

recording aperture, the heterogeneity effect may be so signifi-

cant on the wide-aperture data that a 2-D, or even a 3-D,

analysis technique is necessary in order to find a realistic sub-

surface velocity structure . Although this study was limited to

2-D media, the technique can be generalized to 3-D media.

Howe ver, the computing cost involved will be correspondingly

higher. The technique can also be modified to solve a large

seismic tomograp hy problem. This modification may include

using a more efficient ray-tracing routine by limiting the ve-locity model to triangular elements and incorporating auto-

matic traveltime-picking techniques. It is also necessary to

approxim ate the interfaces by more flexible functions than for

the linear function used here. The latter is approp riate only

when the data have limited lateral extent.

ACKNOWLEDGMENTS

We tha nk Prof. J. Oliver and J. McB ride for reading the

manuscript and making valuable suggestions. We thank two

rcviewcrs for their suggested revisions to the manu script. Th e

comp utations and data processing were performed on

CO CO RP’s MEG ASE TS’” system. The field data were col-

lected by crew 683 4 of Petty-Ray Exploration Service of Geo-

source, Inc. This work was supported by National Science

Foundation grants EAR 82-12445 and EAR 8 3-13569. Insti-

tute for the Study of the Continents (IN STO C) Contribution

no. 40.

“ Seiscom Delta Inc.

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50 Zhu and Brown

APPENDIX

DERIVATION OF FORMULA (19)

The ray equations for a Z-D medium w ith velocity indepen-

dent of the y-axis are

NI.+/s = sin p CDS ,

di,/ds = cos fi>

da/ds = u, sin a/([: sin p),(A-1)

an d

dp/ds = (6, sin fi - oI cos p cos a)io,

where s is the arc length along a ray, a and j3 are the azimuth

and declination of the ray, and u, and U, are the derivatives of

velocity c with respect to x and 2, respectively. The corre-

sponding ray c oordinates are (s. a,, PO), a, and PO are the

azimuth and dec lination at the source point, often referred to

as the takeoff angles. For our purp ose, we only need to con-

sider the rays with small a. Thus, the fourth equation in equa-

tions (A-l) can be written da/ds = u,a/(u sin p), which has the

solution

where

a = a0 E(s, a,, PO), (A-2)

Differentiating the second equation in equ ations (A-l) with

respect to a0 and us ing equation (A-2) yield

d i v-2

i is ?a,= sinpcosa(E +a,,g] + sinucosflg. (A-3)

Integrating equation (A-3) an d letting a, (hence a) approach

zero, we obtain formula (19):

sdJ:/cYa,= sin J3E(s,0, p,) ds.

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GPH 52 1 Zhu Dimensional Velocity