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Geophys.J . R . astr. SOC. 1967) 13, 247-276.

Numerical Applications of a Formalism for Geophysical

Inverse ProblemsG . E. Backusand J. F. Gilbert

Sumniary

A gross datum of the Earthis a single measurable n umber describing someproperty of the whole Earth, such as mass, moment of inertia, or thefrequency of oscillation of some identified elastic-gravitational normalmode. We prove tha t the collection of Eart h models which yield thephysically observed values of any independent set of gross Earth data iseither empty o r infinite dimensional. W e exploit this very high degree ofnon-uniqueness in real geophysical inverse problems t o generate com puterprogra ms w hich iteratively produce Earth models to fit given gross Ea rthda ta and satisfy other criteria. We describe techniques for exploring thecollection of all Earth models whichfit given gross Earth data . Finally,we apply the theoryto the normal modesof elastic-gravitational oscillationof the Earth.

1. IntroductionAny single number which describes some property of the whole Earth will be

called a gross da tu m of the Earth. Examples of gross da ta are the Earths mass, itsmoments of inertia, its Love numbers, the frequencies of oscillation of its elastic-gravitational normal modes, the quality factors(Qs) of those normal modes, therotational splitting parametersof the normal modes, the travel time ofS o r P wavesfrom a pa rticular source to a particular receiver, and the coefficients in the sphericalharmonic expansion of the Earths external gravitational potential.

Hu m an limitations a re such that a t any given epoch only a finite num ber of grossEarth da ta will have been measured. This paper is a discussion of the extent to whichthese finitely ma ny gross d ata can be used to determine the Earths internal structure.The general procedure to be described is applicable to any finite set of gross Earthda ta which vary continuously in response t o variations in the Earths internal structure.Fo r concreteness an d because of the authors present interests, detailed calculationswill be given only for the m ass, the m om ent of inertia, an d the frequencies of the elastic-gravitational normal modes.

It will be assumed that the Earth is invariant under all rigid rotations about itscentre, that the angular velocity of steady rotation of the Earth is zero, and that thestatic stress field an d dynam ic stress-strain relation a t every point in the Ea rth areisotropic. Each of these assumptions is false, bu t probably to a n extent which onlyslightly perturbs the theoretical calculationsof normal modes (Toksoz & Ben-Menahem 1963, Backus& Gilbert 1961, Ha st 1958). The problem o f determiningthe deep asphericities an d a nisotropies from measured gross E arth d at a is an inverseproblem of the general sort to be discussed in the rest of this paper, but we shall notconsider the details here.

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D o wnl o a d e d f r om
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248 G . E. Backus and J. F. Gilbert

The problem we shall conside r is the following: supposea non-rotating, spherical,isotropic Earth of radiusa has densityp(r), bulk modulus ic(r), and shear modulusp(r), all functions onlyof r , the radial distance fro m the centre. Suppose a finitenumber J of gross Earth dataE l, E,, . , EJ have been measured, and that these datadepend only o n the functionsp , K, p. Given the observed values ofEl , .. , E,, whatcan be said about the unknown functionsp, K , p ?In carrying out the quantitative discussion, we shall use the radiusor the Earthas the unit of length (6-371megametres), the Earths mean densityp as a unit ofdensity (5-517g/cm3), and27r(7rpC)-%as a unit of time (5844seconds). HereC isNewtons universal constan t of gravitation. Then th e unit of velocity is 1.090 km /sand th e unit of pressure o r elastic modulus is 65.57 kilobars.

2. The nature of the inverse problem

An ordered triple of functions,m = ( p , K , p), defined on O b r G 1, will be calledan Ea rth model. Fo r the m om ent we put no restrictions whatever on the functionsp, K, p except th at they b e real-valued, piecewise continu ous functions ofr, the radialdistance from the centre of the Earth.If in=(p , I C , p ) an d m=(p , ti, p ) are twoEarth models andb and 6 are two real numbers, we shall use the symbolbm+bmto denote the Earth model(bp+bp, bti +b ti, b p +b p ) . With linear combina-tions of Earth models being thus defined, Earth models can be thought of as pointsin an infinite-dimensional linear space9 1 , the space of all conceivable Earth models.On zx3l we introduce an inner product as follows:

(ni, m)= d r r z [ p ( r ) p ( r ) + K( r ) K ( r ) + , u ( r ) , u ( r ) ] , (1).i.and a norm llrnll= (m, m)*. With these definitions,91 becomes an inner productspace, which can be com pleted t o a H ilbert space (Riesz& Nagy 1955). In(1) threepositive weighting functions which vary withr may be introduced if some particularshell of the Earth is to be emphasized. We use unit weightsso as to emphasize allvolumes of the Earth equally.

A gross datum of the Earth, suchas the squared eigenfrequency of the normalmode o S , , is a real nu mb erE which can be calculated fo r each Ea rth modelm n W.A gross datum ca n be thou ght of as a rule which assigns to every pointm n CiJX(thatis, to every E art h model)a real numberE(m). I n j i n e , a gross Earth d atum is a real-valued function on the space911. Since 91 is itself a function space, real-valued func-tions on $331are com mo nly called functionals. In general,a gross Earth datum willbe a non-linear functional onzx31 However, the mass and moment of inertia areclearly linear functionals on91: the mass is

E , =47t d r r 2 p ( r )iand t he mom ent o f inertiais

1E,= j / d r r 4 p ( r ) .n

0

If measurement shows that the valueof a particular gross Earth datumE for thereal Ea rth isEo, then we know th at the pointme n 9ll which describes the real Earthlies on the hypersurface iniV1defined by the equation

E (m)= Eo . ( 2 )

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Numerical applications of a formalism 249

If E is a linear functional, the hypersurface(2) will be a hyperplane; this hyperplanewill contain 93s origin [the Earth model(0, 0, O)] if and only if E o =O . If E is non-linear, in general (2) will bea curved surface. If we have measuredJ gross Earth da tafor the real Ea rth an d found their values to beEjo , = 1, .. , J , then we know that thepoint me hich describes the real Earth lieson the intersection in9X of the J hyper-

surfaces

From the measurementsElo, ..., E we can deduce nothing whatever about theEarth, except what follows from theJ equations (3).

The inverse problem for the gross Earth dataEl, .. , E , consists in trying tounderstand wh at is the totalityof Ea rth m odels which lieon all the J hypersurfaces(3).If we are fortun ateor shrewd in our choice of whichJ gross data t o m easure, thenallthe different Earth models which satisfy(3) may share some common property. Fo rexample, they may all havea low-velocity zone in the upper mantle;or they may allbecome essentially the same when we take running averages of theirp , K and p over

some fixed depth intervalH . In the first example, we can definitely assert that theEarth has a low-velocityzone in the upper mantle. In the second example, we canclaim to knowp , K and p as functions of radiusr , except for unresolved de tails whosevertical length scale isH or less.

Of course, ifE l 0 , . , E, are real measurements they will conta in errors of observa-tion (such as incorrect identification o f a mode) an d will be perturbed by the rotation,asphericity, an d anisotropy of the Earth. Therefore in principle it is possible th atthe set of Earth models which satisfy equations(3) will be empty, that is, thatnospherical Earth modelm is capable of producing the observed values of the grossEarth dataE l, . , E,. As far as we know, all of the theoretical workso far publishedon the inverse normal mode problem has been addressed t o the question of whetherequations (3) have even one solutionm when Elo, ..., E, are the squared raw fre-quencies obtained from the real Earth and uncorrectedfor rotation, asphericity,and anisotropy. As yet no exact solutions have been published.The published Earthmodels have maximum errors between0.5 and 1.5% for the normal modes withperiods longer tha n ab out300s (Pekeris & Jarosch 1958, Altermanet al. 1959, Sat0et al. 1960, Pekeris et al. 1961a, Bolt & Dorman 1961, MacDonald& Ness 1961,Alsop 1963a, Landism anet al. 1965). Slichter (1967) believes th at many of the longerperiods of spheroidal modes are now measured with an error no larger than0.1 /,.

E j ( m ) = E j o , j = 1, ..., J . (3)

3. Non-uniquenessin the inverseproblem

Th e linear spaceW of all conceivable Earth modelsm is infinite dimensional, a ndequations (3 ) place only finitely many restrictionson m. Intuitively it is reasonablethat if equations (3) have any solution at all, they will have an infinite-dimensional(usually curved) manifoldof solutions. Th at this intuitive argument needs someexamination is indicated by the following example: fo r anym = ( p , K , p ) define

1

E(m) 4 + [ rr [ ( p - 3)2+(K- 2)2+ p - )]. (4)J0

Then the equationE(m)=4 imposes only one conditionon the infinite-dimensionalspace of ms, and yet there is exactly onem which satisfies that condition, namelym = ( p , K, p ) with p ( r ) = 3 , 1 c ( r ) = 2 ,p ( r ) = 1 for O

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Numerical applicationsof a formalism 251

theorem in implicit function theory (Graves1946) that for any sufficiently small..., aK there is exactly one set of values ofa l , ..., uJ such tha t them defined

by (8) satisfies (3) exactly. Thereforea J + l , ..., aK are the K - J independent para-meters of aK - J dimensional familyof solutions(8 ) of (3). SinceK can be arbitrarilylarge, we ar e justified in asserting th at the totality of solutions of(3) near any solution

m, is infinite-dimensional if the differential kernelsdllo, .., A,' a t m, are linearlyindependent.W ithout this hypothesis of linear independence serious comp lications ensue which

we shall no t examine, but which can lead to failure of the argument. In the num ericalexamples to be discussed in this paper we have computedA t o , ., .XJo explicitlyand verified their linear independence by Gram-Schmidt orthogonalizing them.Physically, whenAle, . , 4,' are linearly dependent the gross Eart h dataEl, . , E ,are not independent, at least to first order in variations from the Earth modelm,.Such a special situation is possible, but rare.The more usual situation is that~ h ' ~ ' , A,' are almost but not quite linearly dependent. Th at is, the determinantof the J x J matrix (.kjo, ka) is not zero bu t is very m uch sm aller tha n the pro ductof its diagonal elements. This approximate linear dependence calls for great carein num erical calculations and e rror estim ates, but do es not affect the validity of th eforegoing proof of non-uniqueness.

Geometrically, the linear dependence ofA l l o , .., A,' means that one of theJhypersurfaces(3) is tangent a tm, to some linear combination of the others. Exam ina-tion of the case J = 2 makes clear that this tangency permits the set of solutions of(3) to be infinite dimensionalor finite dim ensionalor t o consist ofm, alone.

One physical reason for the infinite-dimensional non-uniqueness of the inverseproblem (3) becomes clear on examining(6). Only finitely many gross Earth dataEl0, .. , E,' have been measured. There is a lengthH such that none of the kernelsPi, K j , M i , with j = 1,

.. , J , changes appreciably in a lengthH. Then if top , K , p

we add perturbations6p , 6 ~ ,p whose Fou rier series contain no wavelengths greaterthan H , the perturbatio ns produced inEl, . , E j by 6p , 6 ~ ,p will be extremely small.With only finitely many gross da ta we can no t expect to resolve details of arbitrarilysmall vertical scale; our vertical resolution is finite. This remark is sufficientlytautological t o be w ithout much geophysical interest.A geophysicaliy mo re interestingquestion is whether there is any o ther source of th e non-uniqueness in(3) besides thefinite resolving power inherent in a finite set of gross Ea rth data. We shall see tha tin general thereis.

Suppose that for theJ gross Earth d ataEl,. , E j we have found an exact solutionm, of (3). Suppose tha tA l o , .., A J o , he differential kernels ofEl, .., EJ a t m,,are linearly independent. Simply to know th at there are other exact solutions of(3)is unsatisfying. W e would like t o know w hat they loo k like an d how they differ fromm,. One app roa ch to this problem is as follows: Letn be any Ea rth model orthogonalto all of A',', ..., A J o ; hat is, for anyj among 1, ..., J , suppose

( A j , , ) = 0. (10)

Then according to( 5 ) if p is any small numb er the Ea rth modelm, +/In is a solutionof (3), a t least to first order inp. It is no t generally a n exact solution unlessp = O .

We c an find a n exact solution w hich is nearlym, +Bn when /? s small. To startthe procedure, letm,', ..., mJObe chosen to satisfy(7). We have seen tha t for anysufficiently sm allp there is exactly one solution of(3) which has the form

Since this so lution is determined byp, the coefficientsaj are functionsof p, as is thesolutionm itself. Fo r anyp, let m(p) be the solution of(3 ) which has the form(11).

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252

Let ~ h ' ~ ( j ? )e the differential kernel of E j at m(p), so ,le,(0)=,41jo. If we change pto B+Sp we will change m to m +6m where, from (1 I ) ,

G . E. Backus and J. F. Gilbert

J

k = I6 m = 2: m,''6rL+n6B (12)

and 6ak= ( ~ ~ c t , / c f g ) s g + O [ ( s t ) ' ) ~ ] .according to (5 ) ,

Then E,(m) will change to Ej(m +am ) where,

Ej(m +am) = Ej(m) +( c +Y,( t)'), dm) cl1SmiI. (13)

If both m and m + 6 m are exact solutions of (3), then from (12) and (13) we haveJ

k = lX (ytLj(P),mk0)6%k+ ( . / i j ( p ) ,n)6B+&!16mll=O.

Dividing by S/3 and letting Sp approach zero, we have

with j = 1 , ..., J .Assume that near m, the differential kernels d t j depend continuously on m. Then

the J x J matrix (.M,(fi), mko)depends continuously on p. Since it is the identitymatrix when p=O, it is non-singular for all sufficiently small p. Therefore for allsufficiently small p, equations (14) can be solved for da,/dp. The numerical procedurefor constructing the one-parameter family m(P) of exact solutions of (3) is now evident.We start with m ( 0 )= m,, and a1 =. = IJ O i n (1 1). Using ddj(0) we find dcr,/d/3a t p=O from (14), and then take a small step At)' in p. With A a j = A p ( d a j / d p )weconstruct m(AP) from ( 11 ) . For m ( A P )we find the differential kernels A j ( A P )asdescribed in Appendix B. Then we use (14) to obtain du , /dp at A P , and proceed toanother step in p. Notice that the curve m(P) thus constructed in the space 9Jl doeshave the form m, +Bn+O(P2) when p is small, because, according to ( I 0), (I l), and(14), at p=O we have cr,=...=aJ= dcr,/dp= ...= cr,/dj?=O.

The result of the foregoing discussion can be summarized as follows: letJLlo, .., A',' be the differential kernels of E l , ..., E , at m,, assumed linearly inde-pendent. Let n be any Earth model orthogonal to all of ..., A J o n the senseof (10). Then there is a unique one-parameter family m ( p )of Earth models (a uniquecurve in the space 931) which starts at m, when p=O and leaves m, i n the directiondescribed by n.

The key relations in these remarks are (10). Any model near m, which satisfies(3) exactly has the form m , +n + erms of second order in n, where n is an Earth modelsatisfying (10). Conversely, any Earth model n satisfying (10) generates a one-para-meter family of exact solutions or (3) which, for small p, have the form m, + In+ O (f i2 ) .

4. Resolving powerin the inverse problem

We define the mean vertical wave number k, of a function f(r) by the equationIt is now possible to give a quantitative discussion of the question of resolving power.

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(A definition in terms of the Bessel, Legendre, Chebyshevor Fourier expansioncoefficients o f f involves less com puting in wha t follows. We use(1 5 ) because of itsform al simplicity and intuitive appeal.) Then the resolving power with which a grossEarth datum E looks a t the densityp is simply the mean vertical wave number ofP ( r ) n the differential kernel ofE a t mo. This idea could be elaborated by considering

mean wave numbers in different regions, but we content ourselves here with thecoarser discussion.If we assume tha t theP and S wave velocities,yP(r) nd vs(r),are known exactly,

so th at the inverse problem is to find the densityp from (3), then we must recall that~ = p ( v , ~ - - - $ o ~ ~ )nd p = p v s 2 , so that, from (6),

1

0

where Pj= j + K j ( v p 2 - $ v S 2 ) + M j v s 2 . (16)

Then the resolving power of a gross Earth d atu mE is the mean vertical wave num berof P.In the restof this section, we will in fact assume thatup and v, are known, in order

to simplify the discussion. The required modifications for the more general problemwill be apparent. We point ou t tha t all the results abo ut non-uniqueness provedhitherto in this paper applyin toto to the case whereup and cS are known and onlypis sought.

We have defined the resolving power of a single gross Earth da tum . Now we mustdefine the resolving power of a finite set of gross Earth dataE l , . , E j at a pointm oin 91 where their differential kernels areP I , .., P J [see (16)]. One definition whichat first looks reasonable is as follows:if we have available measurem ents ofE l, .. , E j ,then we have available a measurement of the gross Earth parameter

J

j = 1E = x u j E j ,

where a l , ..., aJ are any constants whatever. The differential kernel ofE a t m, isJ -

j = 1

If k p s the mean wave number ofP then this resolving poweris available t ous fromou r data. The best resolving power available from allour data is the largest valuethat k pcan take asa l , ..., aJ in (17) take all their possible values. This maxim um wewill call K ( E , , ..., EJ) . I t seems at first to bea reasonable definition of the resolvingpower of ou r data. For normal modes of oscillation it corresponds to the assertiontha t we can expect to see unambiguous detail inp whose vertical wavelength is approxi-mately the sho rtest wavelength of any of the squared wave functionsor linear combina-tions of squared wave functions whose eigenfrequencies we have measured. It iseasy to computeK 2 ( E l , .., E j ) , since it is A, he largest root of the equation

wheredet ( A , - B,) = 0

d P, d P jA , = d r r 2 - -/ dr dr0and

1

B i j= / d r r 2F iP j .0

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254

Furthermore, lower bounds forK ( E , , . , E,) are easily obtained. The mean wavenumber of any particular linear combination

G . E. Backus and J. F. Gilbert

is such a lower bound. Estimates of such lower bounds can be obtained by G ram-Schmidt orthogonalizing the sequencePI,..., p , relative to the inner product(1 )an d inspecting the w avelengths of the oscillations in the resulting functions. In otherwords, we expect detail inp t o be meaningful if its vertical scale is not sm aller than theshortest wavelength available in the Gram-Schm idt orthogona lization ofP,, ., P,.

In general this expectation will be disappointed. A little consideration shows th atth e real resolving power of ou r da tais obtained froma second definition. Supposewe have a densityp o ( r )which satisfies(3 ) exactly. Am ong all perturbationsSp whichcan be added top o without affecting the validity of(3), which is that one,f ( r )=S p ,whose mean vertical wave numberk, is least (whose mean wavelength is longest)?Since,to first order inp ,p o+pf satisfies(3), we cannot tell from ou r data how muchof .f is present in the true p . And among all such undetectable perturbationsS p inp o ,f has the longest mean wavelength. Clearly we want to kno w w hatf is, and w hatk, is. Equally clearly, the real resolving power of our da ta is no tK ( E , ,. , E,) but k,

is simply the demand thatfor j = 1 , ..., J we have

The demand thatp o+pf satisfy (3) to first order in1

Thus the problem isto find thatfwhich minimizes thek, of (15) subject to the constraints

(18). This is a classical variationa l problem, The method of Lagrange multipliersshows that there are Lagrange multipliersR, M ~ ,.., aJ such that

whilef s finite a t r = 0, and

df- odr

at r = 1. F or any fixed1 etfj be chosenso as to be finite atr = 0, to satisfy(20) a t r = 1,and to satisfy

d dd r d r

- - 2 -fi +Af, = Fj

in O < r < 1 . Then the solution of(19) and (20) isJ

f = c Y j f Jj = 1

where the constantsyj are determined by the requirements1

J d r r 2 f 2= 10

1and, for k = l , .., J ,

j = 1 y j. d r r2 - f iF k =0 .0

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The existence ofa nonzero solutionyl, ..., J of (22) is a condition onI , namely

det(i r rfj p) = 0.0

The smallest1, which satisfies this condition is the desired minimumk,, as can beseen on multiplying (19) byr2f, integrating fro m0 to 1, and applying(15), (18) an d (21).

We will see from a numerical example later in this paper thatkf can indeed beconsiderably smaller thanK ( E 1 , , E j ) . T h at is, the real resolving power of a finiteset of norm al m odes can be considerably poorer tha n the resolving power estimatedfrom the vertical wavelengths present in the eigenfunctions of the normal modes.In the light of th e foregoing discussion, this result is no t surprising. Speakingvery loosely, what we know fromEl0 ,..., EJo are the coefficients in the expansion

where Pl, ..., pJ ,P J + &... is some sort of a complete set onO < r < I whose firstmembers are P1, ..., P J . Among the remaining functionspJ+l, .. may be somewhose vertical wavelength is longer than the shortest of the waveleng ths ofpl, ., p J .Fo r example, the whole set might be sinnnr, n = 1 ,2 , . , and p jmight be sinn( j +1)rfo r j = , . . . The procedure described abo ve fo r calculatingk , andf is essentiallya constructive procedure for finding the lowest such gap among the vertical wavenumbers m ade available byPi, ..., P J . The next gap can be com puted by adjoining.f to the set H,, .., PJ and calculating a new functiong which minimizesk g 2 ubject

1

to (18) and the new constraint d r r 2 f g = 0 .

J0Some of the modifications made possible in the foregoing discussion by thepresence of discontinuities inp , K and ,u will be discussed elsewhere.

5. Finding solutionsto the inverse problem

Since the set of solutionsm of (3) is usually infinite dimensional, any practicalnumerical technique f or finding one such solution m ust somehow remove this seriousambiguity. When there arev functions of radius to be determined an dJ Earth dataavailable, previous systematic procedures fo r dealing with the am biguity have involved

subdividing a part or all of the Earth radially intoJ / v or fewer shells in which th efunctions are assumed constant or linear (Dorman& Ewing 1962, Anderson&Archambeau 1964, Anderson 1967, Landismanet ul. 1965, Verreault 1965b).

The question arises whether itis no t possible to mak e mo re explicituse of the verylarge extent of the non-uniqueness in the inverse problem. Fo r example, the ideas inSection 4 might be pursued. GivenP and S velocity distribution sup@)and us(r),we might seek the smoothestp(r ) which solves equations(3). Th at is, we might seekto minimize

1

[ d r r 2(dpldr)

kp2= 1J drrp0

subject to the co nstraints(3) onp. All the structural d etail in such a smoothest solutionwould be real, in the sense that, roughly speaking, the shortest vertical wavelength

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256 G. E. Backus and J. F. Gilbert

Present in such a solu tion would be longer than the longest vertical wavelength presen t'n any acceptable perturbation to the solution. An iterative numerical technique canbe set up, using Lagrange m ultipliers an d the differential kernels ofE l , . ., E,, whichProduces this smoothest solution of(3). The technique is essentially the same asanother which we shall descr ibe in detail later in this sec tion,so we postpone the details.

Another useto which the non-uniqueness of the inverse problem can be p ut is totest the validity of guesses ab ou t the internal structure of the Earth. Suppose againthat up(r)and u s @) are known, and th at a guessp G ( r ) s put fo rth concerning the internaldensity of the Ea rth . Th e modelm, = p,, K ~ ,, ) with IC, pc(up2 -&') andF G = ~ ~ u ~ ~ay not solve(3). Yet we can always ask, what is the modelm closestto mG n th e least squares sense which does solve(3). Tha t is, we can seek to m inimize

1

IIm -m,l12 = j d r r 2 (p - - p c ~ ~K - G12 + p- G ) 2 ~ (24)0

Subjectto the constraints(3). In general this problem will have a unique solutionm,and if m, is at all close tom then m can be found numerically by a simple iterativeprocedure. This particular application of the non-uniqueness of the solutions of(3)Produces a simple, systematic scheme for generating exact solutions of(3). Everyreasonable guess will generate sucha solution. The scheme is very close t o the

procedure we have used to generate solutions of(3), so we describe it in

Suppose, for generality, that we are trying to obtainp , K , ,u from El0, ..., EJo.w h e n uP(r)and us(r) are already kn own the required modifications in o ur procedurewill be obvious, and we have in fact used both procedures. Ifm(")isa good approxima-tion to a n exact solutionm of (3) we may w rite, to first order inm-m'"),

detail.

Ej(m)=Ej(m("))+ ( J f ) , -m(")) (25)

for j = 1, . . Here A , ( " ) s the differential kernel ofE j a t m("). hen to first orderin m-rnc") we have, from (3 ) and (25),

( A ? ) , - m("))=Ejo -Ej(m(")) (26)

fo r = 1, . , . Th us we seek to minimize(24) subjectto the constrain ts (26). Intro-ducing Lagrange multipliersu j , we conclude that

J

k = 1m-m,= X ffk. ,dk(n), (27)and that the@k are determined from(26) as the solution of th e linear system

J

k = 12 uk(.,d?), ,,dfl)) = E; - j(m("))- -aC'"),m, - m(")) (28)

fo r j = 1, ..., . When we have solved this system we define

If (25) were exact, we would haveEj(m("+'))=Ej0, and ,("+I) would be the exactt o our problem. Because of the non-linearity of(3), usually we will have only

IEj(m("+)) - j o -g Ej(m("))- jOSO ha t m("+') ill be more nearly a solu tion of(3 )than ism("). I t seems likely that fairlyweak hypotheseson the continuity of the differential kernels ofE l, .. , EJ as functions

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of m will suffice to ensure th atif mG s sufficiently close to the solutionm of our problemand if we take m(')=m,, then as n approaches infinitym(")converges to m.

The numerical procedure we have actually used to generate solutions of (3)is likethe foregoing except that at the nth stage we try to minimizem-rnc") rather thanm-m,, subject to the constraints (26). In all of the numerical w ork to be reported

here we have assumed that the Ear th hada rad ius of 6371 km, consisted of a fluid coreof radius 3473 km an d outside the corea solid mantle. We have described an y Ea rthmodel numerically by giving the values of p ,K and p at 66 core points,98 mantle points,an d 4 crust points spaced fo r Gauss-Legendre integration, and we have obtainedintermediate values ofp , K , and p by linear interpo lation. Thu s we have in factconfined our attention to a 438 dimensional subspace of%I]. ince we always haveJ 4 38, this limitation affects none of the foregoing arguments. The problemattempted has been to findp and K in the core andp , K , and p in the mantle from (3).

By introducing in to the inner products (1) and (24)and the differential kernelsAterms involving the depths of the discontinuities inp and K, it is possible to let thedepths of the discontinuitiesas well as their magnitudes be determined by the datain (3). Th e necessary formalism is presented in A ppendixC. In this paper the depthsof the discontinuities were assumed to be known exactly, and only their magnitudeswere determined by (3). Ina subsequent paper we will report o n calculations in whicha solid inner core was used and the depths as well as the magnitudes of thediscontinuities were assumed at the outset to be unknown.

Th eie is, of course,no reason in principle why one must postulate discontinuitiesin trying to solve (3). However, whenall th e observational dat a in(3) are at very lowfrequencies an d w henm, is continuous it seems most unlikely that am ong th e infinitelymany solutionsof (3) the iteration procedure will select a discontinuous one.High-frequency dat a assureus tha t the real Ea rth ha s at least one discontinuity on the scaleon which we work here. Ideally we would like to know th e whole manifold (3), bu tlimitations of time and ingenuity makeit possible forus to explore only a small partof it. We have elected to tryto arrange th at t he small part we d o explore is near thereal Earth,so we have explicitly introduced a discontinuity a t the core-mantle boun daryin our starting modelsm,.

6. Numerical experiments with raw data

In a first applicationof the foregoing procedure we took as gross Earth dataE jothe Earth's mass (Gutenberg 19-59), the Earth's moment of inertia (Jeffreys 1963,King-Hele et al. 1964), and the squared frequencies observed by Slichter (1966) foroSo, $,, o S , , ..., oS,. Here ,Sl denotes the (n+ 1)th term in a list of all spheroidalmodes of angular orderI , arranged in orde r of increasing fiequency. Similarly,,7;denotes a toroidal mode. F orm, we used Gutenberg's u p and ti S as reported byBullard (1957), smoothedso as to have no low velocity zone, and Bullen's densitymodel A , slightly modified to agree with recent satellite measurem ents of the m omentof inertia. Thism, will be called the smoothed Gutenberg model in what follows.We performed seven successively mo re dem anding calculations: in the first, onlymass, mom ent, andoSo were used; in the Ith we used mass, mom ent,oSo, oS2, . oS,.All of p , K, p were regarded as unknown. In each of the seven calculations only oneiteration was performed. Th e quality of the Earth modelm so produced was measured

by its r.m.s. erro rem,defined byJ

j = 3( J - ) 8, = 2 (I - j(m)/EjO)'.

Since Elo and E z o , the mass and moment, were always fitted exactly, we did notinclude them in our measurementof error. Tab le 1 gives the r.m.s. errorsE~ and el

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258 G . E. Backus and J. F. Gilbert

Table 1

T1~er.m.s.rrorsEoand E l fo r the starting model (smoothed Gutenberg)and the first iterate, using as gross data the mass, moment, and

Slichters observed raw frequencies fo roSo, o S 2 ,o S 3 ,..., ,S1

I 0 2 3 4 5 6 7EO x 102 1.1 1.0 1.0 1.0 0.9 1.0 1.0El x 104 2.6 1.7 3.1 17.0 25.0 3200 -

for m, and the first iterate in each of the seven calculations. A t1=7 the first iteratefo r the density was negative and th e computer program was unab le to produce eigen-frequencies. T ha t is, the procedure described in Section5 was unable to produce amodel to fit Slichters data foroSo, oS2 ,oS,, ..., oS,.

In a second experiment we used as E arth dat a mass, mo ment, an d Slichters(1966)observations ofoSo, oS2q, = 2,3 , .. , 9. We used only even angular ordersso asto reach moderately high frequencies without excessive com puter time, an d we om ittedo S2as the mode most heavily contam inated by the ellipticity an d rotation of the Ea rth.Again all ofp , K, u were regarded a s unknown. In the core we took form, the smoothedGu tenberg mode l already men tioned , while in the man tle we took velocity and densitydistributions of th e formA+&, with A and B chosenso tha t the mass and moment ofinertia of the whole Earth w ere correct,, while the corresp ond ing mom ents ofu pand usagreed with those for th e Gutenberg velocities. Thism, we will call quadraticGutenbe rg. Taking thism, as a zeroth iterate we had, for the r.m.s. erro rE, at thenth stageof iteration the resu lt given in Table2. The r.m.s. error had dropped by thethird iteration toa value which was below the estimated erro rsof observation (Slichter1966) bu t it remained essentially unchanged throu gh six furth er iterations. We regardthis as an unsuccessful outcome.

Table 2

The r.m.s. error E , of the n-th iterate , starting with the quadraticGutenberg model and usingas gross data the mass, moment, andSlichters observed rawfrequencies fo roSo, oS4,o S s ,o S 8 ,. , oSls

n 0 1 2 3 4 5 6 7 8 9& x 104 200 8 11 5 4 4 4 6 3 4

The negative results of these two experimentson the real raw data imply one or

(i) there is an undiscovered conceptual or numerical error in our iteration

(ii) the models were too far from the real Earth to permit the linearization (26);(iii) the raw observational data cannot be obtained exactly from any spherical,

isotropic, no n-rotating Ea rth m odel of radius 6371 km with a fluid core ofradius 3473 km inside a solid mantle.

The last possibility might be the result of contamination by asphericity, anisotropyan d rotation, o r the misidentification of one or m ore m odes.It seems unlikely th atany inner core mod e was included in ou r list of observations, because their surfaceamplitudes are small (Alsop 1963b); but if an inner core mode was mistaken for amantle mode, we might expect difficulty from our failure to include an inner core inou r models.

more of th e following conclusions:

procedure;

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7. Numerical experiments with artificial data

In the hopeof understanding our lack of success with real data, we replaced thereal gross Ear th d ataby artificial gross da ta generated from a particular mo del Eart hwhich we shall call Model I.It has the correct mass and m oment, fits56 of Slichters(1966) spheroidal frequencies, includingoSo, oSz,oS3, . , with a maximumerror of 1-3%, and fits the lowest ten,T frequencies (Benioffet at. 1961) witha maximum error of0.9%. For all these 66 modes the r.m.s. error of ModelI is

FIG. 1. The dimensionless density p ( r ) for Model I (solid line), the initial guess(dotted line), and the thirteenth iterate (dashed line), using 20 gross Earth data cal-culated from Model I, and permitting p, K and p to vary during iteration. Thecentre of the Earth is at bottom, the surface at top. Values of p range from

0 at left to 3 dimensionless units at right.

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260 C . E. BdCkUS and J. F. Gilbert

FIG. . The dimensionless compressional velocitiesc P ( r ) for the same numericalexperiment as in Fig. 1 . Full scale for up is 67r dimensionlessunits.

0.3%. We took as our observed gross Earth data,Ejo , the mass, moment, andvarious calculated eigenfrequencies of Model1. In a first experiment we usedoSo, Sz, o S , , ..., o S , , and took for mG, ur starting model, the smoothed Gutenbergmodel. All of p , K, p were allowed to vary. One iteration reduced the r.m.s. errorfrom 1.2 x

In a second experiment we used the same gross Earth d at a and tookm, to be thequad ratic Gutenberg model, a poore r starting point. All ofp , K, p were allowed tovary. The r.m.s.error E,, after n iterations is given in Table3.

In a third experiment we used asgross Earth data twenty numbers, the mass, themoment, and the squared eigenfrequencies, computed from ModelI, of the eighteenmodes oSo, So, 2So, S, ,2S1, .., ,S1, ,S2, ..., ,S2. Th e starting modelm, was

to 1.9 x

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Numerical applicationsof a formalism

Table 3The r.m.s. error E, , of the ntli iterate, starting with the quadraticGutenberg model and using as gross data the mass, moment, and

frequencies calculated fro m Mo delI fo r oSo, oS2, oS3, ..., $18n 0 1 2 3

&,x 104 200 50 16 0.6

261

the quadratic Gutenberg model. All of p , K, p were allowed to vary. The r.m .s. errorE, after n iterations decreased steadily from 1.75 x for E ~ ~ .Fig. 1 showsp ( r ) for Model I, the starting model, and the thirteenth iterate. Fig. 2shows up(r) for these same three models, and Fig.3 shows us(r) for them. Model I

for E~ to 1.1 x

FIG. 3. The dimensionless shear velocities vs(r) for the same numerical experi-ment as in Fig. 1 . Full scale for u s is 6n dimensionless units.

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262 G. E. Backus an d J. F. Gilbert

and the thirteenth iterate have the same gross Earth data El0 ,. ., EZo0 o eight signifi-cant figures.

In a fourth experiment we used the same gross Earth data, but we permitted onlyp to vary during our attempts to fit the data. We required that the starting model,m,, and all subsequent iterates have the same up(r)and us@) as Model I, the source of

the ' observed ' gross Earth data being used. The initial guessed density was obtainedessentially by demanding that the Brunt-Vaisala frequency vanish in a fluid core andsolid mantle and that the mass and moment of inertia be correct. The r.m.s. error E ,after n iterations decreased steadily from &,=0.9 x and thensteadily but more slowly to = 3.6 x l o V 6 . In Fig. 4 are shown the densities formodel I and the 1lth iterate.

to & 6 = 1.6 x

FIG. . The dimensionless densityp ( r ) for ModelI (dashed line) and the eleventhiterate (solid line) using thesame gross data and the same units on horizontal

and vertical axes as in Fig. 1 . Only p was varied during iteration.

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8. ConclusionsWe regard the experiments with artificial Earth data as successful. Since they were

reasonable imitations of the unsuccessful experiments performed with the real data,there is some suggestion that our difficulties with the real data lay not in the inversiontechnique nor the initial guess, but in contamination of the real data by rotation,asphericity, anisotropy, mode misidentification and other effects. The evidence isinadequate to justify a firm conclusion, but we intend to keep the suggestion in mindas a guide to future work. In a subsequent paper we propose to correct the real dataas far as possible for rotation and the known asphericities, to introduce an inner coreand variable core boundaries, and to learn what we can about the possibility of modemisidentification. In this latter connection, a world-wide array of long-period seismo-meters (tiltmeters, gravimeters and strain gauges) would be extremely helpful, althoughsome mode-identifying criteria can also be obtained by observing correlations amongcomponents of strain, gravity and tilt at a single station (Gilbert & Backus 1965,Smith 1966). Admittedly a world-wide array of strain-gauges, tiltmeters and gravi-meters would be expensive to establish and maintain, but it might be a good investmentin earthquake prediction. Press (1965) has shown that the Alaskan earthquake of1964 produced a permanent strain change of lo-' at Hawaii which could be explainedapproximately by treating the stressed source region before the earthquake as a disloca-tion loop. Such a dislocation would strain the whole Earth and redistribute its mass,so it ought to produce over the whole Earth changes in gravity of about one part in10' as well as the observed changes in tilt and strain. A world-wide network of long-period seismometers appears to be capable of detecting and locating such localiredaccumulations of large stress as occurred prior to the Alaskan earthquake. It is to behoped that if such a network is established, the data it produces will be made availablefor work on normal modes.

Our third experiment with artificial gross Earth data from Model I leads to aninteresting conclusion about resolving power. Model I and the thirteenth iterate hadthe same values for mass, moment, and the eigenfrequencies of oS o , So, 2S0, S1, zS1,

..., 7S2, correct to eight significant figures. The twenty grossEarth data used to find the thirteenth iterate included frequencies of modes with highradial orders. It is widely believed that if such data can be observed they will giveunambiguous results about the deep interior of the Earth. The Gram-Schmidtorthogonalization of the sequence of differential kernels Ply P,, . P Z Dproducesfunctions the nineteenth of which is shown in Fig. 5. In the mantle this kernel haswavelengths shorter than 900 km. Our naive expectations about resolving power [asmeasured by K( E , , ..., E J ] might lead us to expect that the data permitted us todetermine the density structure except for details with wavelengths shorter than 900 km.In fact the density difference between Model I and the thirteenth iterate, as shown inFig. 1, has wavelengths of 1800 km in the mantle.

By contrast with the third experiment, the fourth numerical experiment withartificial gross Earth data produced a very good fit between the eleventh iterate and theModel I which was the source of the data. This fit is seen in Fig. 4. In the third experi-ment all of p , K, p were to be determined from (3), while in the fourth experimentu p and v s were assumed known and onlyp was to be determined from (3). The samegross Earth data were used in both experiments. Is it possible that those gross dataare sufficient to determine p uniquely (except for small-scale ambiguities due to lack ofresolving power) if up and us are given, but are not adequate to determine all three ofp , v pand us? We have no idea whether this suggestion is correct, but a very naiveargument may be advanced in its favour. Borg (1945) has shown that when a Sturm-Liouville problem on a finite interval is reduced to its canonical form as a Schrodingerequation for a wave function $(x) on the interval 0

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r I I I 1

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