The inverse problem of seismology

Elsevier Publishing

Company,

THE INVERSE

Amsterdam

PROBLEM

- Printed

in The Netherlands

OF SEISMOLOGY

M. GI:RVER Itlstitlrte of‘ Ph~jsics of‘ (Received

August

the Earth. MOSCON~(U.S.S.R.)

25, 1971)

ABSTRACT Gervcr. hf.. 1972. The inverse problem of seismology. Trctorwph.mics. 13(1-4): 483-496.

In: A.R. Ritsema

(Editor).

Tl~r Crpper Marztle.

The common scheme for the solution of the inverse problem in seismology is described. Some shortcomings of this scheme are enumerated. A computer experiment is described which would help to find and correct these shortcomings. The question is investigated what information on the structure of the earth can be obtained from each of the major characteristics: travel-times, dispersion curves and spectral data. If only these characteristics are used, the solution of the inverse problem is not unique. Two additional spectral characteristics for the model example of the wave equation are determined. The use of the analogous characteristics in the real inverse problem probably will make its solution unique.

INTRODUCTION

Scheme

for the solution of the inverse problem in seismology

The following scheme for the solution established during the last ten years.

of the inverse problem in seismology has been

Seismic records are used for the determination of the major characteristics of the earth: travel-time curves of body waves, dispersion curves of surface waves, amplitude-distance curves and eigenfrequencies From these characteristics earth in the following way. Every major characteristic observed characteristics computed

of the earth’s free oscillations. conclusions is represented

can be drawn about the internal

structure

in two ways: observed and theoretical.

are derived from seismic observations.

The theoretical

of the The

ones are

from the chosen model of the earth.

As a rule, the model of the earth is described by three functions:

p(r), a(r) and b(r);

p(r) is the density, a(r) and b(r) are the velocities of P and S-waves. It is usually assumed that the parameters depend on radius r only, i.e. that there are no lateral inhomogeneities. To compute the theoretical characteristics we have to specify not only p(r), a(r) and b(r), but also the equations of seismic waves. Usually the system of elasticity equations is assumed. The model is successful if the observed and the theoretical characteristics coincide.

484

Shortcomings of the described schenw Shortcomings in carrying out the scheme (1) The solution

depends heavily on the particular

ed for study. This dependence

has a tendency

of inversion. (3) The danger of circular arguments: influenced

by the particular

theoretical

class of the theoretical

to be forgotten

the determination

in the discussion of the results

(2) It is impossible

is

models assumed a priori. of the major observed cha-

base.

Shortcomings of {he scheme itself (1) The major characteristics used in inversion are to some extent arbitrarily their list is apparently

select-

of observed characteristics

(3) None of the algorithms which are used for construction racteristics has a theoretical

models

chosen and

incomplete. to determine

whether our tinal solution is really correct.

THECOMPUTEREXPERIMENT

The idea of the experiment Imagine a computer experiment with the following program: its input is a model of the earth, a model of the seismic source and, of course, the coordinates of the stations and the time interval of the observations. The output is a set of seism@@,~ms. The program imitates what happens in the earth at the time of an earthquake. We nyw begin the experiment without any prior assumptions about which model was fed into computer. We first determine the major observed characteristics from the computed seismdgrams. Then we calculate the model by the scheme described above. At the end, we h&F%‘$6 great advantage that we may compare the solution with the model that was fed irito the computer. In the first attempts, the result of the calculation may differ greatly from the model fed into the computer.

This, in particular;2an

be due to the above-men~loned~ijhortcomings.

Are the correct major characteristics determined from the seismograms? Let us modify the experiment. Compute the major observed characteristics from the seismograms generated by the computer for some model, atid then compute them directly (as theoretical ones) for the very same model. These characteristics should be the same, but it is not clear whether they will be the same. The doubt is cast by the above-mentioned absence of a theoretical basis for seismogram processing.

THF INVERSE PROBLEM OF SEISMOLOGY

A circular argument in the comparison There is a reason to determine

485

of obsenaed and theoretical

the observed characteristics

retical ones. Often the reverse order is used in seismological theoretical

eigenfrequencies

of a spherical harmonic frequencies

where the spectral peaks are located correspond

where n is the number

and so on, is Fourier transformed.

earth. But their numbers II, k are still unknown. frequency

It is assumed that the

to the eigenfrequencies

of the

Then for every observed eigenfrequency

ollx- is found and its numbers tz. k are attributed

to the

6.

Then it is asserted that the observed and theoretical sequently

the theo-

and k is the number of an overtone. Then, the seismogram, after

filtering, smoothing

observed frequency

before calculating

practice. For example, the

unk of the earth are first calculated,

preliminary

&, a nearby theoretical

eigenfreyuencies

the model corresponding

to the theoretical

frequencies

are close, and that con-

w,,~ is close to that of the real earth.

This reasoning is an example of a circular argument. Two rlecessary coFlditioFls for algorithFFls

Two kinds of algorithm are used in the inverse problem: for the transformation mograms into the major observed characteristics, and for the direct transformation earth’s model into the theoretical characteristics.

of seisof the

Summarizing the two preceding sections. we can impose on these algorithms the two following conditions: ( 1) they must be mutually independent; and (7) for the same model they must lead to the same characteristics. These conditions are obviously necessary for a successful solution of the inverse problem by the scheme described above. But apparently they are not sufficient, because the list of the major characteristics seems to be incomplete. THE MAJOR CHARACTERISTICS

Let us see what information

on the model can be obtained

from a single major charac-

teristic if it is given exactly and completely. Travel-time

curves for body waves aFld ItelocitjlPdepth

In the absence of waveguides, the velocity-depth by the Herglotz-Wiechert

method.

distribution.

distribution

can be determined

In the presence of waveguides, the solution

uniquely

is non-unique

(Gerver and Markushevitch. 1965, 1966, 1967a.b): an infinite set of models corresponds to the same travel-times. We may, however, derive some general properties of these models. One example is shown in Fig.1. It was shown in Gerver and Markushevitch (1967a) that all velocity-depth structures corresponding to the same travel-times lie inside the shaded exactly

area in Fig. 1.

486

II. GERVI

K

Fig. 1, Representation in a plane of velocity models (for two low-velocity channels). All models lie along the curves or within the “giraffe-like” (shaded) area. There exists at least one model that passes through any point of the shaded area.

The travel-times for reflected waves or for those of a deep-focus earthquake give the following additional information: (1) the focal depth h independently of the earth model used; and (2) some integral property of the model F(G) = mesb,.v
(Gerver and Kazdan, 1967, 1968) for Love waves only.

Here also the solution is non-unique: may correspond

in the presence of waveguides, a whole set of models

exactly to the same fundamental

branch of the dispersion curve. The

same integral property of the model can be determined of waveguides, this property leads to a single model.

uniquely,

however. In the absence

Two approaches to the inverse problem We can respond to the described non-uniqueness a compact description

in two ways: (1) finish the solution by

of all possible models; or (2) look for some additional

major charac-

teristics that will allow us to make the solution unique. Amplitude-distance curves cannot be used for this purpose since, in the absence of errors, they are essentially equivalent to travel-time curves. The question -what information the eigenfrequencies can give- will be considered below.

THE INVERSE

PROBLEM

OF SEISMOLOGY

487

The inverse problem for the wave equation in a spherica&) symmetrical layer (I) Statement

of problem.

The connection

with the problem

of torsional oscillations.

Let R 1 < r < R2 be a spherical layer with a variable density p(r), which depends on radius r only. and does not depend on time r, latitude 0 and longitude the wave equation ,.. p(r)V-

@. Let us consider

in this layer:

AV= F

J-et the source F be g(t, 19,@)6 (r-R2)/

r2 where g(t. 0, $) is a finite function,

i.e., it is

different from zero only in the limited domain: O
V(t, r, 0, @) satisfy the conditions

av

Vlt
ar

r=R,,R

2

=0

i.e., there is no motion until the origin time and the boundaries

of the layer are free.

Knowing p and F, one can determine V uniquely. The inverse problem is to find p(r), if the function V is known at a finite number of points of observation on the surface r = R2 during a finite time interval 0 < t < T. We assume that F is unknown also. Now we assume that the number of points of observation and the duration of observation Tare large enough. It seems that the formulated problem is closely connected with the problem of torsional oscillations of the earth. The main difference is that in the last problem the medium is characterized by two functions p(r) and p(r). We may suppose that the results obtained for the wave equation will be true for some combination of functions the acoustical density u = (p/~)“~ as a function of the argument: 4 7 = 7(r) = s

[P @j//J

(II) S-models. Theorem

(XII“zdx on finding p(r) from frequencies

The initial plan of solution of uniqueness

and stability

p and ,Uas for example

of the problem formulated

for the class of functions

w,,, , .... w,,~, where K = c,,S~

in (I) was to prove the theorem

p(r) that is wide enough and simple

at the same time, and then to extend the theorem to arbitrary functions We have started with the class of S-models, defined as follows: The whole layer R 1 < r < R2 is divided into S concentric ri_l


p(r).

layers:

1,2 ,..., S;rO =R,,rs=R2

In each layer the function

p(r) is constant

and equal to pi, where pi-‘/’ = rj

rj__r, so that:

488

s’

P

I:*

(r)

&=

,

In other words, the S-model is a piecewise constant

function

p(r) containing

S segments:

the time of propagation of an impulse through each segment (along the radius) is the same and equal to 1. The following uniqueness theorem for S-models is true: Denote by W,,k the eigenfrequencies

for our model equation,

earth. The a,& thus are the eigenfrequencies p(r)&AU=O:

FrzR

I

=zrzR

Fix any number n of spherical harmonics Wnl

in the boundary

as was done above for the

value problem:

=O and consider the sequence of frequencies:

2w nZ, ..., W,,k>...

Is it possible to find another S-model 6(r) with the same eigenfrequencies the answer is negative. Theorem. As a rule, the S-model p(r) is uniquely

determined

onk? As a rule

by the finite number of fre-

quencies:

where K = c,,S*, c,~is a constant

depending

on n.

(Each S-model can be considered as a point p = {pr , . . ., p,} of the S-dimensional space. There is a (S-1)-dimensional algebraic surface A with the following property. If points p and 6 do not belong to A, then they cannot have the same frequencies

anr,

.. .. oa.

Points

belonging to A may perhaps correspond to the set of frequencies wnl, .... a,~, but not m&e than 2s points. This is the exact sense of words f‘as a rule” in statement of the theorem.) p(r).

It is possible to find an S-model which is arbitrarily close to practically any function That is why the following generalization of the last theorem seemed to be true: Any

model p(r) is determined

uniquely,

or in any case uniquely

where n is fixed. But we can show that this generalization substitutions: R*n+l fin) 1.2

=

._ R2n+ 2

_.?_~___ 1 _R2n+l

1

as a rule, by the sequence:

of the theorem is incorrect.

Use the following

THF INVERSF

PROHI.EM

489

OF SEISMOLOGY

Take the string with linear density pll(x), x E lo.11 ; and consider for this string the following boundary

value problem:

The numbers mnh- were proved to be the eigenfrequencies It is known that properties of a string are not determined eigenfrequencies.

To determine

in problem A. uniquely by its spectrum of

the density of a string uniquely

one must know not one

but two sets of eigenfrequencies, corresponding to the two homogeneous boundary problems: with the same conditions at one end, and with different conditions at the other. For examples. the second problem for problem A may be: y ” + w2p,$

)y = 0

We denote the eigenfrequencies

in problem (B) by v,~~. There is an infinite

set of strings

p,,(x) with the same frequencies allk in the problem A. Their eigenfrequencies fir7,,.in problem B do not coincide with the v,,~. The frequencies ?j,7k interlace with w,~x_and except for satisfying some known asymptotic

relations,

they may be arbitrary.

Thus we have a

method to construct the strings e,,(xj setting up the sequencies fink rather arbitrarily. By the reverse substitution for each string p,,(x) constructed as above. we obtain the whole set of models i(y) whose eigenfrequencies w,~~ are the same as those of the beginning model p(r). (111)Hypotheses about the determination of p(r) j?onz W17k (a) We saw that generally speaking p(r) can not be determined

from the frequencies

(-+,I, ‘+7,,2, . . . . ‘h,k, ... Let us fix not one but two order numbers H: n = nI and tz = rz2. We may suppose that the frequencies w17rk and U,12k, k = I,?, . determine p(r) almost uniquely. Hypothesis 1. Most probably only one other function p(r) has all the same eigenfrequenties w,* ,k and w,,k exist .)

as p(r) has. (In section (b) it is explained why two solutions probably

52.GERVF:K

490

Hyporhesis

2. The theorem about the almost unique determination

ofp(r)

quencies o,*~ (n = 0,1,7.3, ... . k = 1 J.3 ...) must also be true. ank can be determined in the following way: let us examine the boundary

from all freproblem:

Z=rZ’ r= R,.R,

then w,~, q12, . . .. are the eigenfrequencies of this problem. Let us assume that n changes continuously, assuming not only non-negative integers but all real values. Then tink for any fixed integer k has some locus on the n-w plane. By definition it is the k-th branch of the dispersion curve. Let us make one further step: assume that all the branches of the dispersion curve are given. Hypothesis

J’. In such a case it is indeed possible to determine

Probably it does not depend on the boundary

conditions:

p(r) almost uniquely.

the condition

that the bound-

aries are free probably may be replaced by the simpler condition that they are fixed. Then we have the following: &pothesis 4. Let A&) be the eigenvalues of the boundary value problem: y" t [Q(r)-$1

y=O;Y(R,)=Y(R2)=0

with a continuous real parameter t. Then only one other function actly the same eigenvalues hk({) correspond. (b) Let us examine the latter hypothesis

,6(r) exists to which ex-

in detail. The substitution

x = In (r/R ,),

X=ln(R2/R1): Z(x) = r-l/* y(r), leads to a boundary

q(x) = p(r)?,

s t ?4= f

value problem:

z” t [A&x) .-- {] z = 0,

Z(0) = Z(x) = 0

The eigenvalues will be denoted by hk (0, though it would be more correct to write X/J{-‘/). Here then is an equivalent

formulation

of the hypothesis

4 from section (a): only one

function Q(x) exists which is different from q(x) and to which exactly the same eigenvalues A, ({) correspond. One part of the formulation has become quite evident after our substitution: in any case one function g(x) does exist: 6(x) = 4(X-x) The question whether all hypotheses

are true remains open.

THE INVERSE

PROBLEM

(c)We have made all the substitutions Z” + [b(x)

491

OF SEISMOLOGY

and reduced the problem to the equation

- I] Z = 0 in order to discuss one important

Let us assume that q(x) is, in fact, determined all the branches h,({)of

question.

almost uniquely

(within symmetry)

the dispersion curve X(c). It seemed to be sufficient

finite though perhaps a great number q(x) not exactly but approximately

of branches A,({): k = I, 2,3, ...K in order to determine (and, as before, within symmetry).

It is this last statement that is of practical interest and not so much the “pure” of uniqueness, formulated above. (IV) Rejection dispersion curve

from

to know a

of the theorem about definition

theorem

of q(x) from the first K brarlches of the

This theorem is wrong. For any K > 0, several different functions q(x) can have exactly the same first K branches of the dispersion curve. The corresponding examples are of a rather artificial nature. They are built in approximately the same way as the symmetrical strings in Gerver ( 1970a,b and in press) with the same spectrum of frequencies in the problem FR (one end is free, the other is rigid). We shall not discuss them here in detail; the shape of the q(x) for which X,(c) coincides for several first k is shown in Fig.2.

Fig.?. Examples curves.

of density

The constructed

distributionsq(r)

functions

having

identical

oscillate strongly,

sets of lower-order

branches

of dispersion

their variation increases infinitely

with K.

This may be avoided if we lessen the amplitude of oscillations. but then the functions q(x) will practically turn into constants and therefore will become alike, which is of no interest. The conclusions is that to make the theorem correct one must impose on q(x) some a priori limitations. Judging by analogous theorems for the one-dimensional wave equation Gerver (1970a.b and in press) these limitations in terms of p(r) must be the following: for some positive constants C, and C2 : C;’
< Cl,

var p(r) < C2

492

M (;f:RL’ER

(V) Necessity of the exact description of the class of models for which the inverse problem is being solved The contents

of par. II and IV show the importance

of the exact description

of the class

of models for which the inverse problem is solved. Restricting equation

ourselves to S-models, we could have the theorem of uniqueness

for the wave

from par.11. For arbitrary models, not necessarily belonging to S-models. this theo-

rem is wrong. There is nothing strange in the fact that the solution of the inverse problem, being unique in a narrow class, becomes non-unique in a wider class. However, in practice, in solving the inverse problem of seismology, such a possibility is often ignored. For example: a set of models is investigated by the scheme described at the beginning. Each model is represented by a great number of homogeneous

layers. The models for which the difference

between the computed and observed major characteristics is small are included in the solution. No objections arise if we remember that all the models were layered and if we do not represent our conclusions as something proven for the real earth. If the model was parametrized in another way, we could come to different conclusions. A general and doubtless trivial statement is that when the class of the examined models widens, the set of solutions may widen too. This expanded set may include solutions which do not fit to the regularities noticed for a narrower set. Practically, whereas par.11 warns us against too narrow a class of models, par.IV warns us against too wide a class: even in the simplified problem there is no hope for a unique and stable solution without a priori limitations

on the class of possible models.

After this philosophical discussion, we return to our concrete problem for the wave equation. We shall outline the approach to its possible solution and introduce some additional major characteristics. (VI) New major characteristics (a) Summary of par.11 -V. If some a priori limitations are imposed on p(r) one can probably determine p(r) uniquely from the known frequencies w,k. They should be known for at least two different n, while k must vary from 1 to some number K which is large enough -the larger the more exactly we want to determine p(r). The proof of such a theorem with not very strong limitations on p(r) is doubtless of great theoretical interest. If this theorem is really true, it is especially interesting out how the accuracy of the determination of p(r) depends on K.

to find

There are two difficulties, however. First, such a theorem is rather complicated; second, and more important, it is very difficult to determine o,k for large k from observations. It may even be impossible and anyhow the methods described in the paragraph: “Two necessary conditions for algorithms” are not suitable for this. (b) We could try the following. Add to the list of major characteristics two spectral ones: (1) amplitude factors D,, (see below); and (2) the eigenfrequencies nnk in the problem with the boundary conditions:

THE INVERSE

au

ar

r=R,

PROBLEM

=uI

r=R,

493

OF SEISMOLOGY

=o

The following theorems are true: (A) Let tink be the eigenfrequencies p(r)&

NJ=

in the boundary

au ar

l-R,

for the wave equation:

0

problem with free surfaces:

=au ar

=() r=R 2

and let qnk be the eigenfrequencies

for the same equation

in the boundary

problem with

one surface free and the other fixed:

au ar

,.=R,

=q

=o .r=R,

n is the number of a spherical harmonic and k is the number of an overtone. Fix arbitrarily H=H,,. Then the density p(r) is determined uniquely by two sequences: ‘+,k and nnOk k- = 1, 7-,3, . . (B) To determine approximately p(r) with an accuracy E, it is sufficient to know a finite set of frequencies Wn,h_ and nnOh_: the number k varies from 1 to some K,(e).* Let us compare these two theorems with the hypothesis formulated in the beginning of this section on the determination

of p(r) with an accuracy E from wn,h_ and wli,/i.

k = 1, .... K(E).

The theorems are already proved, but the hypothesis

is not and it may be wrong. Besides,

it seems likely, that Ko(e) is less than K(e). Naturally,

that there is still a danger that K,(c)

it is a weak point in our considerations

also will be too large. (c) The natural question is: if all that is true, how can we determine servations? The answer is: to determine

T?,~ rather than Wnk seems to be not much more

difficult because of the following. The frequencies seismogram’s Fourier transform the Fourier transform Fourier transform

Wnk are associated with peaks of the

since it is a mathematical

ir,r(z) of some functions

of the same functions.

n,rk from the ob-

u,(t).

fact that W17kare the poles of

And v,!~ happen to be zeros (of the

Let us define u,(t)

and iT,(z) more specifically.

< p(r) < C,. “3~ p(r) < C2 for some positive constants Cl and Cz. * The limitations on p(r):C;’ “To find p(r) with an accuracy E” means to find p0 (r) such that lb0 --pII < E. The norm lip0 -PII is an integral one; for example in Lz

: R2 w- &91 2dr lb” -PII2 =JRI [PO

\I. (;ERVl,

494

Let cf, (t, r, f3) be the fundamental p(r)+_

a@ = !(C+,

+II<0

=o,

solution

R

of our equation:

6(f) !$,

;:I r=R,,2,

t>o

Then : @(C r. 0) = nTo (a,([, r)Pn(cos fl),

where P, are Legendre polynomials

and:

Take : u,,(v)

=

C

&k(r) --in Wnk

k=l

a,,kt;

u&)

= U,(t&,);

D,,

=D,k(&)

Thus un(f), except for the constant factors P,(l)/IIP,jja are the coefficients of the developing of @(?,R2, 0) into a set of Legendre polynomials. Using Gerver (1970a, b and in press),

one can show that:

11 1-Z*h;k

u”,(z)=47(O) k= t

1

_z2,w’

nk

(d) Note that we determined in section c the amplitude factors Dnk. These factors as well as the frequencies 7),& together with the frequencies w,,k determine p(r) uniquely. To find p(r) with accuracy f it is sufficient to know Wnok and D,ek for k = 1, .... KO(e) Note: One can determine W,& and D,, from the next one-dimensional boundary Z”+

[&p(r)-y]Z=O,

Z=rZ’I

r=R, ,

&&k are the eigenfrequencies in this problem, and D,, of this problem by the formulae: D

Q.)

rrk

-

znk(‘)zd$z ) rR2 ii&k 11 R,

where IIZnkl12 =

J Z$&Mr)dr

R,

problem:

are connected

with the eigenfunctions

THEINVERSEPROBLEMOFSEISMOLOGY (e) We hope to determine

495

mnk, nnk and D,,

from the solution

V(f, R2. 8. 4) of the

equation:

fw=“yQg(t,

P(r)p

0, @)

because of the following. It is known from the theory of spherical functions sphere is uniquely

represented

where F, is a homogeneous

that any function F(B, 4) on the

in the form:

harmonic

polynomial

of the n-th power on the unit sphere.

Let us represent the source g(t, 0, @) and the solution

V(t, R2, 8, @) in this form for every

t:

go,0, t4 = C g,(t,

6 $1,

v(t. Rz, 0, @>=

n

c v,# 6 44 II

We can show that:

Let the points of observation v,(t,

ep +i> = vnj(t),

be (ej, #j), j = 1, .... N. Put:

g,(t, ei3 3) = gni(t)

Suppose that the functions &i(Z), j = 1, .... N have no common real zeros. Then (see the analogous theorem in Gerver (1970a, b and in press) we can determine uniquely “n,k

1 %,,k

and Dnok3 k =

1, 2, .... from V,rei(t), j = 1, .... N, &(O, T), where T is determined

as follows: R, T, = j-

Pl”(r)dr

R, I = the duration

of the source (see par.l), T > 4Ti + I.

(f) Thus the determination of w,,~~_, n,.@ and Dnok, k = I,?, . ., is reduced to the finding of the coefficients Vnd. It is rather easy to find them after some additional conditions imposed on the source; however, the number N of points, where the solution is known, must be big enough. Future investigation is needed to show whether these limitations are necessary, and what are the errors in consecutive determination of coefficients Vnj of spectral characteristics and, finally, what are the consequent errors of the function p(r).

496

M (;I:RVI~l<

(‘ONCLUSIONS

Our first conclusion earth structure

is that the characteristics

used now for the determination

do not ahow us to obtain a unique solution,

of the

even if these characteristics

are known exactly. Our second conclusion determination

is that there is the possibility

from observations

of some additional

to make the solution unique by a

characteristics

such as nnk and/or

D,2k.

This possibility more complicated: anelesticity,

has been explored only for the wave equation. it involves three unknown

the earth’s rotation,

functions,

The real problem is much

not to mention

the effects of

and gravity.

It seems worth while to carry out the computer experiment, described in the beginning. This could be started for the same wave equation, since it already invokes all major characteristics used in seismology:

travel-time curves, dispersion curves, amplitude-distance

curves

and spectral data. Such a formal and cautious treatment of this well-developed field may seem strange. If the earth were not so big, the results could be directly checked. In the absence of this possibility,

however, there is no other way to reach a true solution of the inverse problem

of seismology. REF’ERENCES Gerver, M.L., 1970a. Inverse problem for the equation of a string. Izv. Akad. Nauk. S.S.S.R., Fiz. Zemli. 8: 3-20 (in Russian). Gerver, M.L.. 1970b. Inverse problem for the one-dimensional wave equation. Geophys. J. R. Astron. SOL. 21: 331-357. Gerver, M.L., in press. Inverse problem for the one-dimensional wave equation with unknown source of vibration (in Russian). Gerver, M.L. and Kazdan, D.A., 1967. On finding the function p(x) according to the eigenvalue s = s(p) of the equation .I”’ + &p(x)-s] y = 0. Mat. Sb., 73 (115): 227-235, (in Russian). Gerver, M.L. and Kazdan, D.A., 1968. Finding the velocity profile from dispersion curve. Questions of uniqueness. Vychisl. Seismol., 4: 78-94 (in Russian). Gerver, M.L. and Markushevitch, V.M., 1965. Investigations of non-uniqueness in determining seismic wave velocity from travel-time curves. Dokl. Akad. Nauk S.S.S.R., 183(6): 1377-l 380 (in Russian). Gerver. M.L. and Markushevitch, V.M., 1966. Determination of seismic wave velocity from the traveltime curve. G’eophys. J.R. Asrrorz. Sot.. 11: 165-173. Gerver, M.L. and Markushevitch, V.M., I967a. Determination of seismic-wave velocity from travel-time curves. Vychisl. seismol., 3: 3-5 1 (in Russian).* Gerver, M.L. and Markushevitch, V.M., 1967b. On the characteristic properties of travel-time curves. Geophys. J.R. AsrronSoc..

13: 241-246.

Gerver. M.L. and Markusheviteh, V.M., 1968. Characteristics of travel-time curves from surface sources. Vvchisl. Seismol., 4: 15-63 (in Russian).*

l

Gerver and’Markushevitch,

Selections.from

Computation

1967a and 1968 appear in English translation in: V.I. Keilis-Borok (Editor), Seismology. Plenum Press, New York, N.Y., 1971.