Some inverse problems of magneto-telluric sounding for obliquely incident plane waves, I

30 REFERENCES 1. NEMIROVSKII A.S. .and YUDIN D.B., Complexity of problems and the efficiency of optimization 1979. methods (Slozhnost' zadach i effektivnost' metodov optimizatsii), Nauka, Moscow, 2. NEMIROVSKII A.S. and YUDIN D.B., Informational complexity of mathematical programming, Izv. Akad. Nauk SSSR, Tekhn. kibernetika, No.1, 88-117, 1983. 3. NESTEROV YU.E., A method of solving convex programming problems with convergence rate O(i/h*).Dokl. Akad. Nauk SSSRm 269, No.3, 543-547, 1983. 4. NEMIROVSKII A.S., A unit vector method of smooth convex minimization, Izv. Akad. Nauk SSSR, Tekhn. kibernetika, No.2, 18-29, i982. Translated

U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain

.,Vo1.25,No.2,pp.30-37,198s

by D.E.B.

0041-5553/85 $lO.OD+O.CC Pergamon Journals Ltd.

SOME INVERSE PROBLEMS OF MAGNETO-TELLURICSOUNDING FOR OBLIQUELY INCIDENT PLANE WAVES, I,*

E.Y. BIDAIBEKOV

and V.G. ROMANOV

Inverse problems of magneto-telluric sounding are formulated, and are reduced to converse problems for a non-stationary system of Maxwell equations with Cauchy data and a concentrated source on the boundary. The structure of solutions of the latter problem is analyzed, and the expressions needed to study the inverse problems are given. In the present paper we study the unique solvability of some particular inverse problems of magneto-telluric sounding (m.t.s.), when the required parameters of the medium depend only on the space variable z. The general statement of these inverse problems is as follows: the parameter-s of the medium are assumed known in the half-space 60 and unknown in the halfspace ZTO, where they have to be determined from observations of the reflection coefficients of obliquely incident sinusoidal plane waves. Notice that inverse problems of m.t.s. were studied in various formulations in /l-4/. The results below are a direct development and continuation of the study method that we used in /4/.

1. Statement

of inverse

problems.

We take the model of an unbounded medium, filling the entire space of variables 5, y, Z. The properties of the medium filling the half-space 60 are assumed known, and are characterized by the constants e-eo->O, p=pc->O, a=~,->O. In the domain z>O the medium is simply anisotropic; the parameters E, p are unknown, depend only on the one coordinate z, and have the form

The parameters e, l&o can have a finite discontinuity in the plane z=O. The plane electromagnetic wave is written as the superposition of a plane-polarized and a normally polarized wave, when the electric and magnetic field-strengths E and H areindependentofthe y coordinate; in the case of parallel polarization they have the form E=(E,.O,E~), In /4/ we H=(O.Hz.0). and in the case of normal polarization: E=(O,&,O).I-I-(Ht,O,H,). studied the inverse problem of m.t.s. for an iostorpic medium, or more precisely, the problem in the case of E(z)>O, a(z)>0 (p=1) for 530 of finding the parameters of the medium parallel polarization. In the present paper, to find the parameters, we use either one polarization separately, or both polarizations together. The system of Maxwell equations is written in either case as (1.1)

.&kE + ffl, n++ 31

lZh.vychisl.Mat.mat.Fiz.,25,3,370-380,198s

33

v=I corresponds to parallel, and r=2 to normal, polarization. Here, the index 1n the domain z
Here,

n>(s. i.t)=II,"exp (/,[+r/:+i.,I.).y=l. 2. equation

P. 9, Al satisfy the dispersion

det(pl,+~,‘~,+ilz'q+.4,‘)=0, 4,', i= 1.2.3. have to be identity matrix, and the matrices t.=l.,~~-. (I=~,~I,-. o=llca-. Given 11.E., the dispersion equation has

in which I, is the third-order evaluated under the condition two solutions: q=-kq-

and q==-I]-.

~)-=(/)(~~,,-+a~-)~c,~--i,,~)

‘:.

Henceforth, inthelast equation we shall mean the branch of the square root that takes values on the real semi-axis. The vector n," is an eigenvector of the matrix Since there jlpl,+/l ,‘A, +d?‘q+.rll’/l. values of g. there are correspondingly two eigenvectors in each type of polarization. in the case of normal polarization, the wave incident from --m on the z=O plane and complete field in the domain 60, due to the incident wave, are found in the same way with parallel polarization /4/, i.e., IIV=lY,inc +R‘(h,. where R', R’ tion, while

are

the reflection

coefficients

p)II‘

ref,

with parallel

the field

n"

can be written

transposition),

and

II' satisfies

> + (pl,+h,A*‘+A,‘)

AZ,%

as

2,

in the case of parallel

polarization

v:!

v=l, 2,

the system liv-0,

and the radiation condition at +a. From the continuity condition for the tangential i--l,

polariza-

).=I , _, 3

Wb, z, t)-fi*(L, 2, p) exp (pt+Lz), 3’=IlE,E,R,(IT, iiqE,R,R,jl’ (where T denotes

and normal

v=l, 2,

IF ref -~,‘-exp(pt+stl-+h,l),

290,

are two Hence, the as

v=l, 2,

respectively

II",inc =II,,'-exp (pl-zn-+A,.r),

In the domain

positive

we obtain

of equations

00,

(1.2)

Es-1 x-o=E~+Ix-o,

components

H,-I.-,=Hi+l,_,.

the equations

(-L,++

(1.3)

11-

I,($ and in the case of normal

E,+ -

polarization,

(1.4) the equations

(1.5)

--+.+++L+) 'fw,,P). I-0 0

(1.6)

Here and henceforth, the superscripts + and - denote respectively the limiting values of the functions from the side of domain 00 and zO, p,(z)>O, o,(z)>O, i=l,2, and that we have e,(z)=eo--

&(z) (l+z)“’

P‘(z)=F’o+

Ii(z) (i+z)“.

lJ,(z)==oo+

it (2) (l+a)~

7

+=-i,

2,

a>&

32 where

'k,(z),&z,,

8,(z) are sufficiently

smooth bounded

functions

in the domain

ZSO.

More-

over, let ee)O, p,,>O, aa~uo->O. The constants s:(r),i(z), O,(z) are so,PO,00 and functions not assumed known. If in system (1.2) we put e-e,l~, p=pLd~, o-ffoZs it will have a solution W-rr,,'exp(r&,

Rvl=rr~rVeap(-nz),

v=i, 2,

in which 9, n.,",II,," are obtained from the expressions giving n-.n,,"-, IL:‘-. if we replace eo-,pO-,oO- in them by &. poloa. The radiation condition for system (1.2) , which is needed to isolate the unique solutions of (1.11, can be stated as I&o, Z'rn. Consider

the vector

functions l%llE, &R*II',

connected

with the function

n"

WlEIa,

R#,

by the equations

\ @(A,, 2, P)-~~wfi(L,

2,P),

where T’(z)=

and let us introduce g(s) is given by

h

h

0 I-h-’

0 h

instead of the variable

r the variable

s by the relation

z-g(s), where

I,.) (1.5)

*- I [el(E)~l(E)l"dE. . Then, putting p,,O‘,i=l. 2,

'(h,.s,p)=fi(L. g(s),P). and retaining the same notation for the functions after making the replacement r-n(s) 7 we obtain from (1.2) the system pLY+K~+D*(A,,s)*'-o,

in which the matrices,

W=llA,"A?*A,'11', v-i(2,

sc,

(1.9)

K,D"--D‘(X,,s) have the form

dp = Z-

q?‘.

b=%, For later convenience, we put e;=%- --1,ao-==O, and we replace the parameter h, by A. Below, we assume throughout that Imh==O. Then, from (1.3), (1.51, we obtain the boundary conditions for system (1.0) at the left-hand end, i.e., with s=O: [(h,lp)(A,'+Az')++(h,-'lp) (n*'-n,')+l.~,=-~, p=(pz+Lz)",

(1.9)

k-_(l.hWlet(0))“‘,

[(ho/p) (A,‘+A~‘)++(h,-‘lp)

(,\z’-A,‘)‘]

mo=1.

(1.10)

Henceforth we omit the superscipt +, since we shall only be concerned with the solution for S>O. If, in system (1.81, we now put er=so,h-p0.a,=o,,i=l.2, then, with v=l , it has the solution

and with

~=2,

33 x=I)(F,+tU)The radiation can be stated

condition as

is obviously

=(/‘?+‘)pd,+2A~qo)‘;.

satisfied .\'-0,

by the solutions

A"

, and for system (1.8) it

.s+m.

(1.11)

In just the same case as for the Put G,(A)-(p:p-=6+fl, lpl>O, Ip+2a,l>O, Rex>O). hh-0. case of parallel polarization /4/, we can show that system (1.8) has solutions A", \'==I, 2 which satisfy the radiation condition (1.X), are analytic in the domain G,(A) (see LemmH 1 of /4/) and for the vector L", formed by its first two components, we have as IpI+m uniformly 1, J ( the estimate (see Lemmas 2 and 3 of /4/) with respect to variables

IL’-MyI-O(I), where

0

II II

y”=y”=

1

From these results

e-“‘,

and the matrices

II II

L’=

W-M’(1,

y;

s. p)

have

the form

we have:

Lemma 1. then system (1.8) has in the domain If e,(z).Irr(z)=C'[O,m)* o,(z)=C[O, -). i-1.2, z>o the solution A", satisfying the radiation condition (1.11) and, for any fixed I (Im1=0), is analytic with respect to p in the domain G,(li). For this solution, uniformly with respect to the variable pc&,‘(h), G,“(k)==(p: Rex)& lpl>& lp+Z~,(>b>O). we have the estimate (1.12)

IL'(A,O,p)-U'o., P)l'O(l). Put

%’a, PI -- [ %’@,P) - [ If

x"(h,p)PO,

then the solution

$L(;\,‘.+*,‘.)+~w-A,~*,

+.+n,:y+y(n*‘*-a,‘.)

of system

M(h, s,p)-[x’(h,

(1.8) under p)l-‘a”‘@,

1,

] ‘_O *.

conditions

s. p).

(1.9)-(1.11)

v=l. 2.

is (1.13)

If %'(A,P)‘O, then the inhomogeneous problem (1.8)-(1.11) is in general unsolvable, and the corresponding homogeneous problem has non-zero solutions. Since %'(A,p).v-1, 2, are analytic %'(A,p) can vanish in functions in the domain G,(A), then, for any fixed A, the functions any finite domain belonging to G,"(h). only at a finite number of points. Lemma

2. The function %‘(h,p), ~=l. 2, is analytic in the domain G(I.,)==-c,(5)\I', r-=(p: admits analytic continuation with a side cut along I', and for any 00 has in the domain G,"(~)-G,'(~)\(p:-80
I~~~PIW~),

%'(A,p)=2(h,+h,-‘)+o(u1Pl). From

(1.4),

h”=[~,(O)/el(O)l”‘.

(1.6) we have

[ (ho/p) (.\,‘+.\A’) -(k-‘/p)

(A,‘-_\,‘)]

[ (ho/p) (‘\,*+_\rz) - (h,_‘lp) (‘\,z-,I,‘) From

(1.13)-(1.15)

and Lemmas

.=,,=R’(h, p). l._,=H’(A,

p).

(1.14)

(1.15)

1 and 2, we have:

The solution Theorem 1. A"(h,s, p) of problem (l-8)-(1.11) and the reflection coefficient R’(h, p) are, in the conditions of Lemma 1, analytic functions of the variable p at all the and they admit of analytic continuation with points of the domain G(h), at which %'(A,p)#O, x*(&p) are poles of the functions a side cut along r. The zeros of the function N0.V z,P), H’(k P). using these properties of the reflection coefficient !?‘(A,p), we can state the following Given a finite set of values h=ii,,j=l.&...,r. the values of R‘(&, p,,) general inverse problem. which converges to a point p, at which the function are known in a sequnece pLI,f-i,2,..., e,(z),u,(z),o,(z) in the is analytic; we wish to find from these values the quantities R'(5, P) domain z>O ($30). USSR 25:2-C

34 of Rep,,fi, <=l{cy, there is a number DO such that lA"(A,s,p)<(lpl+ l)C,s=[O, -). In view of the above properties of the functions N(h, s, p) andR'(A, p)we can pass to their Laplace prototypes U&s, t),II'@, 1) in the space of generalized functions, and replace the inverse problems considered by the equivalent inverse problems for the hyperbolic systems of equations

[

a 1u-=-o, 13ii+Ka+DyL,s) as

GO,

v=l,2,

with the initial conditions (1.17) with boundary

conditions (1.18)

and with the auxiliary

conditions

(1.19)

Here, J,is the Bessel

function of 1st kind of 1st order, n-h,+h,-‘.

m-ho-h,,-‘.

and

ho-_[p,(O)/e,(O)I".

Expressions (1.18), (1.191 are obtained from (l-9), (l.lO), (1.14), (1.15) by means of an inverse Laplace transformation in the space of generalized functions and subsequent differentiation with respect to t. Our general statement of the problem is given concrete form in each of the cases below in order to reduce the information needed to find the required parameters of the medium. 2. Structure of the solution of thedirect problem. We write (1.16) as a system of two equations, excluding the third equation for each Y. We then obtain

where

the operator

and matrices

of the system

L,, is given by

K,.D'=D"(s)

and

p"(s,t) are

functions, their The solutions of direct problems (2.11, (1.171, (1.18) are generalized singular part being a wave travelling in the domain G,P{(s, t): s>O, 00) along the characteristic In view of the later importance of these solutions and their issuing from the origin. derivatives, we shall write them down, using the method of isolation of singularities. Let e(z),r(s)HF+*[O, -), a(a)EC'+'[O,-) for some r>O. If we introduce unknown functions V*hp&/8P, k-0, l,...,r, Vvo-~, system (2.1), i.e.

L,P"+J. ai P*(r,t-7)V*(h,s,?)dr=O, . with the initial conditions

v-&2,

they will satisfy

(2.2)

35 (2.3)

VV"I*,PO and the boundary

conditions

v=l, 2.

L"V*L(-l)YP+"(1), of problems

We write the solution

v--t

(2.4)

(2.2) -_(2.4) as

1+1 Kl &(A, I) 6(*+1--I’(t-s)

+wyh,

s, f) ;

(2.5)

4-8

6”’ is the j-th derivative of the e-function and 6'"'=6. j=O, l,...,rfi. Substituting here, for v“ from (2.5) into (2.2)-(2.4) and equating coefficients of the singularities, we obtain the recurrence system fez a”‘: (I,-&)

(I,-K,)a”-0.

a”’ +D*a’“+Ko

d - a”=O, ds

(2.6al

,_-t (~,-K,)a”+Daav~~-~+K~~~~~~~‘+~z~~P(I*,

(ma,"+nu,'")._0-(-1)‘,

s, O)a’,‘-‘-‘-0,

(ma,"+n~,").==,,=O.

(2.6b) (2.ia) (2.7b)

-(-i)"a~“1,4-0, and the system of relations

for

%Ci
W":

I P(s, t-r)W-(R,s.

T)dr+cpv*(L, s)d(t-s)

f

. $“(A, s, t-r) WY

e (t-r)

(2.8)

-0,

I,0-O, (2.9) jL(A(t-r))[W,‘L-(-I).W,“l._“d~]-O, .

in which (2.10a)

$*(A, s, I)

=-h’g$P(s,:)a”-*+‘-‘,

(2.1Ob)

*+1 Y”(A,t)== r(*h+l-‘J:‘+l+‘l 1*

(At) (at”-(-i)“a,*‘)._o.

We also require that cp,"'(h, s)=O. Then, relations unique function a”‘, i-0, I,...,k+l, pvL,$“*. with respect system of algebraic equations

(2.1O.c)

(2.6), (2.71, (2.10) fully define the to a;’ . Eqs.(2.6) are a recurrence

(2.11a) (2.iib)

and for

at"', they are a recurrence

system of differential $a,vo+[Doavo]r-O.

equations: (I.i%a)

(2.12b) From

(2.7) we can find the initial data for (2.12):

(-1)’

(2.13a)

36

(2.13b) i4.2,. (ai'-(-l)"al"'].-,,

. . , k+i.

Thus

the algorithm for finding a,",a;‘ is as follows: we put a,"'== 0; from (2.12) we obtain a linear differential equation for al*; on solving this equation under condition (2.13), we find the vector ay": we then find 2,"' from (2.111,and the vector a" from (2.12), (2.13). etc. Notice the smoothness properties of the vectors a". Obviously, a"" is a function of class c'+* in the domain ~20. Further, a"EC'+'-L[0,-),k~l.2~...,r+i.The reduction in the smoothness of a"' as k increases is due to the presence ofthedifferentiation operator in (2.11). We now find functions WL. Integrating the equations of system (2.8) along the characteristics, we have the system of integral equations

(2.14a)

‘b’“(i.*E, 1-f) j,w+,+,d+J,

(2.14b)

h’lPO.,‘& x-4 j’,.‘,, t-s

53 r-E)

W‘“(h, t, r,)dr,

dr+ ] 1 2’ t-r-r+,

,+.-,+a dr==O

From (2.9) we obtain (2.15)

It follows from the system of linear integral equations (2.141, (2.15) that functions w'h(h, t-=3. for a,t) are uniquely defined and continuous in the domain t>a>O. Obviously, WV’=0 In short, the structure of the function P(h, S,t) is given by (2.5), in which a”&?+-‘. Wb(h, s,t)-0, while the function W'* is continuous in the domain f>s>O and i-0, 1.. . . , Ml, tts,k=O. 1, . . . . r. A; this follows from the Notice that a",W"'" are analytic functions of the parameter properties

of the solution

of problem

(1.8)-(1.11).

We now give the expression needed to study inverse problems. It follows from (1.19) and (2.5) that we have the representation for inverse problems .+, (2.16) where

(2.18)

( yyx,

t)+ j .

J,(r(t-r))[~~'-(-1)'W,~l.-,dr).

37 Notice that both polarization are connected via the parameters of the medium and hence the data of inverse problems in the two polarizations depend on one another. Obviously,

mt'"(s) --~tl"(g), ~,~~(~)--_oL,"'(J), 0"'--a'", ((:'--(I". We easily obtain from Eqs.(2.9), (2.18): (2.19a)

(2.19b)

Equation (2.14a) gives 8

( )

cp**a$

--zw,~*I,_,25 $"%E,T--f)Ii--r+tdT2 112

a

~(a-, E, T-~,)%‘“(a,

f, n)dr,

.

(2.20)

dr.

II

e--r+!

I

Further, using (2.17) and (2.7), we find that -Ye

h.

=-

1 -g*

Cl(O) lh [

a,” (0)-

Ccl03

i

-a%, 4

i+a*

(-I)’

G+(o) - -

R

G"(O)=

--,

a\*

(2.21)

m

i

--ma”,

4

i-1 i

J.c(-i-iJ~'-'-') (0)[a;'-(-i)'a,“].-o, a,“@, 0)- - nu "(h)++x 4 I-0 L-I

i 4

a,"(A,O)----ma "(A)-(-l)'$E

I,(‘-‘-” (0) [a;‘-

(A'-'-', I-"

(-i)*a,“l._.),

2Ctek+l

1. TIKBONOV A.N., On a mathematical proof of the theory of electromagnetic soundings, Zh. vych. Mat. i. mat..Fiz., 5, No.3, 545-548, 1965. 2. GUSAROV A.A., On the uniqueness of the solution of the inverse problem of magneto-telluric sounding for two-dimensionalmedia, in: Mathematical models of problems of geophysics (Matem. modeli zadach geofiz), Izd-vo MGU, Moscow, 1981. 3. KBRUSLOV E.YA. and LEVCBENKO E.P., On the restoration of the parameters of a medium from the results of electromagnetic sounding, Preprint, Fiz.-tekhn. in-ta nizkikh t-r Akad. Nauk UkSSR, Iharkov, No.20-82, 1982. 4. ROMANQV V.G. and BIDAIBEKOV E.Y., On the problem of magneto-telluric sounding for obliquely incident plane waves, Preprint VTs SO Akad. Nauk SSSR, Novosibirsk., No.427. 1983.

Translated by D.E.B.