existence, the uniqueness and the stability of the solution, which is nearest the initial guess in the L z norm, is described. 1. INTRODUCTION. Determining of the ...
THE
INVERSE
GRAVIMETRIC AND
PROBLEM:
STABILITY
OF THE
EXISTENCE,
UNIQUENESS
SOLUTION
CTIRAD MATYSKA Department o f Geophysics and Meteorology, Faculty o f Mathematics and Physics. Charles University, Prague*)
Pe3toMe: Onucatt Memob peey~npu3aquu o@amno~ saOattu O:la tmetuueeo epaeuma~¢uounoeo nod.q 3em~u c,tumaa, ,~mo u~eecmna naqadbttaa otteu~a pacnpeOe.~enun ndomuocmu e 3erase. ~mom memot) npeOocmaaum 603MOgtOuoemb na(Imu peutenue, Komopoe oOno~na~tuo, yemo(~uueo u a6Jtu~u nauaJtbuo~ oqeuKu. S u m m a r y : The paper is concerned with the mathematical U'operties o f the density diso'ibu tion within the Earth obtained by inverting o f the external gravity field, provided an Earth's reference density model, used as the initial guess, is available. The method o f regularization, which proves the existence, the uniqueness and the stability o f the solution, which is nearest the initial guess in the L z norm, is described.
1. I N T R O D U C T I O N Determining of the density distribution within the Earth is one of the main factors needed for understanding the Earth's dynamic processes. An information about these processes is thus contained in the external gravity field [4]; therefore we have to ask the question whether we are able to solve the inverse gravimetric problem for the Earth as a whole [8]. Unfortunately, the mathematical solution of this problem is unsatisfactory from many points of view: it is not unique, stable and, in general, it is hard to match the data exactly. That is why, in "the popular mythology", this problem was said to be meaningless. On the other hand, the inversion of the external gravity field is not only the way to obtain some information about the density distribution since we can take a model based on seismic velocity determinations in the Earth as the initial guess [ 1, 2]. Utilization of an apriori information and constraints to density models hold the key to more correct formulation of the inverse problem. The aim of this study is to give a mathematical description of using this initial guess and of choosing a set of admissible density models in order to obtain a solution of the inverse problem which is unique, stable and, in a certain sense, nearest to the guess. 2. M A T H E M A T I C A L F O R M U L A T I O N OF THE I N V E R S E PROBLEM L e t Up be t h e p o t e n t i a l o f t h e e x t e r n a l g r a v i t y field w h i c h is d e r i v e d f r o m o b s e r v a t i o n s . L e t t h e initial guess ~0 o f t h e d e n s i t y d i t r i b u t i o n w i t h i n the E a r t h b e given that (2.1)
~
(Up-
Uo) 2 d r
< oo,
' . a\rh
*) Address: V Hole~ovi~k~ch 2, 180 00, Praha 8. 252
Studia geoph, et geod. 31 [1987~
The Inverse Gravimetrie Problem... where U 0 is the potential of the density distribution 60, E3 is the whole threedimensional Euclidean space, r e E3 and ~2 is the domain filled by the Earth. Condition (2.1) means that there is no term of the order 1/r (r =-Irl)in the spherical harmonic analysis of the potential U - U v - Uo. We thus arrive at the condition for the density Q which generates the potential U (2.2)
f ~ ( r ) dr = 0
(the meaning of this condition will be explained in the next section). The inverse problem (IP): Let us determine the density distribution 0 within the domain f2 (within the Earth) which satisfies (2.2) and generates the potential U outside the Earth. Difficulties will arise if we try to solve the IP. Some of them can easily be demonstrated: Let U be extended into f2 by the (unknown) internal potential which is generated by the (unknown) density distribution 0. Similarly, let ~ be extended into E 3 x f~ by zero. Then the Laplace-Poisson equation (in E3) holds for the potential U: (2.3)
av=4~C~,
U(r)-,0
if
r-~Go,
where G is the gravitation constant. In the next text, we shall be using the Fourier transformation in the form (2.4)
f(k) : (2r0 -3/2 ~ E f ( r ) exp ( - i k . r ) d r ,
(2.5)
f(r) = (2~)-3/z f e / ( k ) e x p (ik. r) d k ,
the symbol • denotes the scalar product of vectors in E 3. Applying formally Fourier's. transformation to (2.3) we obtain (k -- lk]) (2.6)
-kZ~(k)=4~GO(k)
in
E3.
In the spaces of Fourier's images of the density and its potential (see the more exact specification in the next section), it is easy to obtain the spectral representation A(k) of the operator .~, which maps the density distribution onto its potential, by putting ~(k) = 1, hence (2.7)
A(k) = - 4 n G / k z ,
This means that the general dependence U = ~(~o) is at the spectral domain specified by /.~(k)= A(k)O(k). In order to avoid speculations about the equivalency of Eqs. (2.3) and (2.6) and to put thinking about the problem on firmer ground, we shall assume (2.6) as the primary equation and we shall work only with transfer function A (represented Studla geoph, et geod. 31 [1987]
253
C. Matyska
by A(k) at the spectral domain). To ensure the existence of the Fourier image O(k), we shall confine ourselves to such choice of ~, which guarantees the existence o f O(k). Then the question to be explored is if A(k) O(k) is the Fourier image of a real function, too. (If k ~ 0 then A ( k ) ~ oe and thus, in general, certain difficulties arises.) If k ~ o% A(k) ~ 0, consequently, the inverse transfer function diverges at high wave numbers k. This means that the inversion is meaningless at high wave numbers. In the next section we shall show, however, that these difficulties can be coped with. We shall reformulate the IP now provided the set M of admissible density distributions is chosen. In general, there may be no Q e M generating the potential U outside the Earth. Therefore, we shall try to determine Q~e M for ~ ( ~ ) to be nearest to U in the sense of the space LZ(Ea \ f2) metric.*) The new formulation of the IP is thus the following: IP: Let U - Up - U o be the function prescribed on E 3 \ f2 satisfying (2.1). Let M be the set of admissible density distributions ~ on f2, such that the Fourier transformations 0 exist for all ~ ~ M (extended by zero into E3 \ ~2) and X(O) - A(k) ~(k) is the Fourier transformation of a real function ~{(0), which restriction on E a \ f2 is from L2(E3 ", ~). Let us determine 0s ~ M such that the functional
[
(2.8)
F(O) = -
•/
[U -- -,~(0)]2 dr 3\~
attains its minimum on M in ~. The new formulation of the IP having been accepted, the appropriate choice of the set M has to be dealt with for the case to be stated more crisply. This will be analysed in detail in the next section.
3. EXISTENCE, UNIQUENESS AND STABILITY OF THE SOLUTION L e m m a 1: Let M be a non-empty, closed, bounded and convex set in space L2(O) such that condition (2.2) holds f o r all ~ ~ M. There is then at least one solution of the IP and there is only one solution ~,~ such that (3.1)
II~,,IIL2(m
< II~stk=~
f o r all solutions ~ • Ore.
P r o o f : It is clear that F(0) defined by (2.8) is the convex functional since .~ is the linear operator. It remains to be proved that F: L2(f2) ~-. E 1 is continuous on M which ensures that the solution of the IP exists [3]. F(ff) is continuous if X: L2(f2) *) This is the space of all functions which can be square integrated in E 3 \ f2 with the norm
ll011=L=(Ez\r~) 254
= [
g2 dr
J~ 3\.Q Studio geoph, el; geod. 31 [1987]
7he Inverse Gravimetric P r o b l e m . . . ~-~ LZ(Ea) is continuous, and • is continuous if it is bounded. From (2.7) it follows that it is sufficient to prove that the set {I,~ [A(k) 0(k)] z dk; ~ ~ M and extended by zero into E a \ ~} is bounded for some 6. (Ba is the ball in E a with center zero and radius 6.) If k is sufficiently small, we can write
(3.2)
e x p ( - i k , r) - l - i k . r .
Eqs (2.2) and (3.2) yield
O(k) - f E30(r)(1 -
(3.3)
ik. r)dr = -ik.
f Sr) r
d r =- k . ~ .
Since £2 is bounded in E a and M is bounded in L2(£2), it is true that the set {i~(Q); 0 E M} is bounded in Ea, for
l¢l-J-ifoc,(r)rdr[ 0 b e g i v e n , let us d e n o t e M = {0 ~ WI'2(Q); (2.2) is true and II~/Iw,,2(m < c} .**) • ) zA(o)(t)=Oifr~andzA(o)(r)=A(o)(r)ifr~E a\~. • *) The scalar product in Sobolev's space W1,2(~Q) is d¢fi~ed by
(f,g)w (~ is complexly conjugate to 9). Studio geoph, et geod. 31 [1987]
'. f/Ofo ~) =
dr+
V f . V,j dr
255
C. Matyska
Then F(O) attains its minimum on M. I f we confine ourselves only to the set M n n [Ker (Z.,i)] ± (_L denotes the orthogonal complement in LZ(12)), the solution on this set is then evidently unique and it is stable in the following sense:
where Q1 and Pz are the solutions corresponding to U 1 and U2, respectively. ] ' r o o f : Accordingto Rellich's theorem [ 6, 7] M is even compact in LZ(f2) which yields the existence of the solution. It remains to prove the stability. We can immediately obtain the proof of the stability by means of the Ivanov et al. theorem [9, page 36], for .~: L2(f2)w-~ L2(E3)is injective and continuous operator on the set M n [Ker (Z.~)] ±, which is also compact in LZ(O).
4. C O N C L U S I O N S
Assuming an initial guess of the Earth density distribution to be available we have shown how to choose the set M of admissible density distributions so that the solution of the inverse gravimetric problem is able to exist on M and even the solution which is nearest the initial guess in the U-norm can be unique. The only condition, which turned out to be fulfilled exactly, was the one described by (2.2). This means that the initial guess has to be good enough to yield the accurate total mass of the Earth. In order to obtain the stability of the solution, we had to use regularization, which is based on the limitation of the possible derivatives of the solution so that the functions of M cannot too much oscillate. We have shown that it is sufficient to take .functions which are bounded in the norm of Sobolev's space W ~'2. We may conclude that the problem is correctly posed, therefore, a numerical approach is worth doing - see [8], where the method to make it is suggested. Acknowledgements: I wi~h to thank Prof. K. P ~ and Dr. Z. M a r t i n e c whose discussions of the inverse gravimetric problem stimulated this study. I am also indebted to my father Dr. J. M a t y s k a for his advice concerning the properties of function spaces. Received 28. 1. 1986 (revised 10. 9. 1986)
Reviewers: O. Man, M. Burda
References [ 1] K. E. B u l l e n : The Earth's Density. Cha!cman and Hall, London 1975. [2] A. M. D z i e w o n s k i , D o n L. A n d e r s o n : Preliminary Reference Earlh Model. Phys. Earth Plan. Inter., 25 (1981), 297. [3] S. FuEik, J. N e ~ a s , V. S o u E e k : ~ v o d do variaEniho po~tu. SPN, Praha I972. German translation: Einftihrung in die Variationsrechnung, Teubner, Leipzig I977. I[4] F. J. L e r c h , S. M. KI o s k o , G. B. P a t e l : A Refined Gravity Model from Lageos (GEM-L2). Geoph. Res. Lett., 9 (1982), 1263. 256
Studia 9eoph. et geod. 31 [1987]
The Inverse Gravimetric P r o b l e m . . .
[5] J. M a t y s k a : Personal Communication. 1985. [6] J. Ne~as, I. H l a v ~ e k : Mathematical Theory of Elastic and Elastico-plastic Bodies: An Introduction. Elsevier, Amsterdam etc. 1981. Czech translation: l]vod do matematick6 teorie pru~n~Pch a pru~2ng ptastick?~ch tNes. SNTL, Praha 1983. [7] O. P i r o n n e a u : Optimal Shape Design for Elliptic Systems. Springer, Berlin etc. 1984. [8] K, P ~ , Z. M a r t i n e c : Constraints to the Three-dimensional Non-hydrostatic Density Distribution in the Earth. Studia geoph, et geod., 28 (1984), 364. [9] B. K. I/Iaattoa, B. B. Bacl,IH, B. II. T a u a ~ a : Teopn~ JitIue~ubix ~ieKoppeKTtti,ix 3ajlaq 14 ee np~noz
