When instead of the function y =U DUX only its values yi = y(P, ) , at a finite number n<- of discrete points are given, we obviously are facing an underdeter- mined problem, which we may solve by a minimum norm principle Ilxll=min , or by regularization. As a matter of fact we can remove in this case the spherical approxi- mation for the earth surface and even the points P, may be of arbitrary position. In this way we consider instead point observations yi = Ei ( y ) s y (4 ) where y = DUX, or even different differential operators Dk at the various points, giving rise to differ- ent types of observed functions ytk) = DkUx (gravity anomalies, geoid heights, gravity gradients, etc.). If y is the vector of the observables y i we can set
where Ei is the "evaluation functional" associated with the point Pi , which maps the corresponding function into a real number. In the special case when Dk = I (identity operator) we have a typical interpolation problem: Determine a function y=Ux from its point values y, =y(P,)=(Ux)(P,) at discrete points P, , i=l,. . .,n , of its domain of definition. In this sense eq. (5.43) represents a "generalized" interpolation problem.
A formulation of the model (5.431, which is more convenient for its conversion into a "spectral" form, is based on the factorization
where Lk is an operator "confined to functions on the unit sphere, corresponding to a particular case of the operator A=U -l DU of the continuous case (eq. 5.15). For the sake of notational convenience we will denote the operator associated with an obser- vation at a point 6 by Li rather than Lk , where the point-index i is replacing the operator-type-index k . Of course identical operators may appear at different points and we may even have more than one operators (and observations) at the same point.
Introducing the evaluation operator E and the "combined" differential operator L by means of
the model can be written in the form y=EULx=Ax .
Using the spherical harmonic representation (5.8) in the "matrix" form
we can transform the model into a "spectral" form by taking into account that
Thus the matrix-spectral form of the model becomes
This is an underdetermined problem and we may consider a minimum norm solu- tion, satisfying
which can be determined by applying the solution (3.113)
provided that all the sums resulting from the presence of infinite dimensional matrices converge.
We shall examine the possibility of applying the inner product (5.1 1) for the domain space X = L 2 ( o ) , with respect to which the spherical harmonics are orthonormal yielding <ei ,ek >=aik and Q =I . We leave to the reader the general case where the matrix A is defined as in eq. (5.50). Instead we will consider only the simple case where Di =I Li =I , so that y=Ux and the observations are carried on an (ap- proximately) spherical earth, identified with the Bjerhammar sphere. For such a spherical approximation we may omit the upward continuation ( y = x ) and write the model in the form y = Ex , or explicitly recalling (5.47)
y = E x = E e x =
Erie . - . En (e,, )
A minimum norm solution will have the form i = x e n m i n m = e i , where ( Q = I )
n,n
The matrix to be inverted has elements
and we have a serious problem! Even without the spherical approximation, with
6 -(Ai,Oi,ri), Pk -(ak,Ok,lj,), we arrive at
To give a mathematical description of this problem (within the spherical approxi- mation) we consider a single observation, which we write as
where, in view of (5.47), the ~ 0 x 1 vector eq consists of the components of a new function
which (if it exists) has the property
The function e,: is called the representer of the functional Ei (see e.g. Taylor and Lay, 1980) and its norm in L2 (o) is given by
This means that there exists no element e,: E L2 (o) such that Ei (x)=<e6 ,x> . In fact this possibility is restricted to the class H * of functionals F on a Hilbert space H , such as L2 (o) , which are bounded, i.e., I F(x)l<Cllxll for some constant C . The bounded functionals are also continuous and vice-versa. H * is called the dual of the Hilbert spaces H .
Since the functionals Ei at hand cannot be changed, the only way out of the prob- lem is to abandon L2 (0) for the sake of another Hilbert space H k , large enough to accommodate the representers of all E, , i.e. such that Ei E H i .
Since the inner product < , > of H k is different from that of L2 (o) , the spheri- cal harmonics are not any more orthonormal, in general, but instead
so that
Comparing < f , g > k =f Qg with < f , g >=f g , we see that we have switched to a
"weighted inner product, where the weight matrix has to be selected in such a way that the series implicit in f TQg converge.
On the other hand, in order to make lle4 Ilk =ll<ep, ,e4 >, finite, it is sufficient to
00
include in (5.60) a factor k: - 0 ( n - l ) so that lle,: 11; =zk,~ (2n+l)<- . Both re-
n=O
quirements < f , g >k =f Qg < - and llec I1 =<m can be fulfilled by using a diagonal weight matrix, with diagonal elements which depend only on the degree n
With this choice < f , > =x k i fnm ,, , <en, ,ePq > =6,,,6,, k: and if we in-
nm
troduce
we have for f , g~ L2 (0) the representations
where the "weighted spherical harmonics - en, =-en, 1 form an orthonormal system k n
We take H , to be the Hilbert space where the functions Cm form a complete or- thonormal system, which means that f E H , whenever
With a choice such that k i +O (n+m) it holds that
nrn nm
which means that ll f 112 <- 3 Il f 11: <- , i.e., f E L~ (0) f E H k , so that L2 (0) c H , . We have thus enlarged the space X so that it can accommodate the representers of the evaluation functionals. Indeed
where the representer of Ei is eq =Egnm (P, ) Znm with norm
nm
This means that e,: E H , and consequently Ei E H i , i.e. the evaluation functionals, when considered to act on elements of H k are bounded (continuous) functionals.
We may now return to our norm minimization problem, which with X = H k can be restated as
The minimum norm solution will have the form
and thus
where we have introduced the notation
and we should recall that Z =[. -Znm . . .] . The two matrices required for the solution have elements
If we introduce the two-point function
the elements of the matrices K and k , are simply given by
The function k ( P , Q ) has the "reproducing" property
nrn P4
nrn
and it is therefore called the reproducing kernel of H k , which becomes a "reproduc- ing kernel Hilbert space" (RKHS).
The generalization to a non-spherical approximation, with observations carried on the surface of the earth, or even at any point outside the earth, leads to exactly the same results, the only difference being that the reproducing kernel takes the form
The generalization to observables yi = (DkUx)(<) = (U&x)(e) , with even different operators Dk and L, = U D,U for different functionals Ei , i.e. a combination of different types of observables, is also easy. The only difference is that the elements of the relevant matrices will be instead
where F, = E, Dk are functionals resulting from the application of an operator Dk on k(P, Q) of eq. (5.82), viewed as a function of one of the points only while the other is held fixed, and next evaluating the resulting function at a particular point Pi .
The functions k and k Q , defined by k (Q)= k(P, Q) and k Q (P)= k(P, Q ) , are elements of H ( k E Hk , k Q E Hk ). Conveniently identifying operators by point rather than type indices, we can rewrite (5.83) in the more rigorous form
Usually the values of the observables y are not available and we have instead ob- servations b=y+v affected by errors. Then a two-stage solution is equivalent to re- placing y by b in equation (5.74). This follows from the fact that the operator A in y=Ax is surjective, i.e. R(E)=R(A)=Rm ( y , b ~ Rm ) and the application of the
least-squares principle vTPv=min in the first stage simply gives ?=O and f = b . Of the resulting estimate i(P)=kFK-lb is affected by the true values of the errors pres- ent in b .
To avoid absorbing the errors, a hybrid norm minimization, or regularization ap- proach may be used, where the estimate 2 of the model b = ~ x + v = i ? % + v should satisfy
The solution is given by
or in terms of the notation introduced above
; ( ~ ) = k ; ( K + & P - ~ y l b .