APPROXIMATION

SUMMER SCHOOL
PROCEEDINGS
LOCAL GRAVITY FIELD
APPROXIMATION
Beijing, China
Aug. 2l - Sept. 4,1984
REPRINT
Edited by K.P. Schwan

C. C. Tscherning
Copenhagen, Denmark
LOCAL APPROXIMATION OF THE GRAVITY
POTENTIAL BY LEAST SQUARES
COLLOCATION.

ABSTRACT
The theory behind the use of least squares collocation (LSC) for
the determination of an approximation (7) to the anomalous gravity poten-
tial, T, and the steps leading to the implementation of the method are
reviewed. LSC is described as an approximation method in a reproducing
kernel Hi l bert space (RKHS) of harmonic functions.
The first step in the implementation is the selection of an appro-
priate inner product or reproducing kernel for the RKHS. It is explained
how an isotropic kernel may be selected so that it represents the main
features of an empirically determined covariance function, which here is
defined completely without reference to probabilistic concepts. The
choice of this type of kernel assures, that T will have the smallest possible
difference from T in a least-squares sense.
The use of LSC, for the determination of 7 and contingently of a
set of related parameters, necessitates that a set of linear equations is
soived with as many unknowns as the number of observations plus the num-
ber of parameters. It is explained how (and under which conditions) the
number of observations can be limited by constructing local solutions,
valid for small, but overlapping areas.
Furthermore a smoothing of the gravity field will have the effect
that fewer observations are needed in order to achieve a given resolution.
This may be achieved by subtracting out the contribution from local topo-
graphic masses and more or less well-known geological structures. Alter-
natively or in combination with this the method of mixed collocation may
be used. Here a RKHS is conctructed, which gives the (theoretical ) possi-
bility of estimating anomalous densities.
Finally, the implementation of LSC on a computer is described by
dividing it into separate steps of which several may be accomplished
using published software. Each step is illustrated by describing the ac-
tions taken when determining a quasi-geoid for the Nordic countries.

1 . INTRODUCTION
The gravity potential of the Earth (W) is equal to the sum of the
gravity potential (V), produced by the attraction of the density distri-
bution (p) of the Earth, and the centrifugal potential (G),
The potential will vary with time due to for example changes in the densi-
ty distribution or variation in the rotation axis and rotational velocity
of the Earth.
It is possible to subtract from W a normal or reference potential,
U, and to take time variations into account, so that the anomalous poten-
tial,
becomes a harmonic function outside the surface of the Earth, and so that
it fulfils certain regularity conditions at infinity. We have also then
presupposed, that the effect of masses outside the Earth's surface (at-
mosphere, Moon, planets etc.), has been taken into account. The determina-
tion of an approximation to T (denoted 7) is then equivalent to the de-
termination of an approximation to W.
Since T is harmonic, we have to our disposal the whole arsenal of
methods for solving elliptic partial differential equations. However,
most of the available methods can not directly be used considering the
data available for gravity field determination. Some methods require
that mean values are formed, that data is given in a regular grid or re-
duced to a common reference surface. Furthermore, the data available are
of many different kinds, contain errors and may depend on parameters like
these relating a local geodetic reference system to a global, geocentric,
reference system.
Here we will describe the properties of a method, which may be used
in nearly a1 l situations, namely the method of least squares col location,

LSC. We will restrict our scope of interest to the determination ci a
local approximation to T. LSC may be used for the global determination
of T, but requires, when used for this purpose, that a number of restric-
tions are put on the kind of data used.
The method has been extensively discussed in scientific papers and
textbooks in recent years, most extensively in Moritz (1980). Here the
main emphasis was put on theoretical and methodological aspects. I will
here try to broaden the scope by including material concerning the prac-
ticaI implementation of the method, and have therefore had to limit the
treatment of the theoretical aspects.
In section 2 the mathematical background will be outlined; mainly
results will be given. But all the important definitions and equations
will be collected, and some of the implications of these equations will
be pointed out.
The approximation 7 determined by LSC will have the minimal norm
in between the set of functions, which agrees with the observations. But
which norm should be used? This problem is treated in section 3, based On
considerations concerning primarily the practical implementation and the
numerical stability.
In the process of determining 7 a set of linear equations must be
solved. The number of unknowns is equal to the number of observations
pllis the (generally small) number ofparameters used. We must therefore
limit the number of observations, and in section 4 we discuss how this is
possible. Here we also discuss which parameters it may be useful (or ne-
cessary) to determine as a part of the LSC procedure.
If we are able to smooth the gravity field. then fewer observations
are needed in order to achieve a given resolution. This can be done by
subtracting out the contribution from IocaI topographic masses and more
or less well-known geological structures. In section 5 we discuss how the

topography and geology is cr may be taken into account.
The implementation of the method on a computer is straightforward.
Many subroutines are now available, which may be put together to form a
nice LSC computer program. In section 6 is described how this can be done.
However, not everything can be left to the computer. Still in some places
human judgements are necessary. This is illustrated by going through the
sequence of decisions and computations, which was necessary during the
process of determining the quasi-geoid for the Nordic countries.
No other implementations are described. because all other so far
discussed in the literature seem to follow almost the same principles.
I have tried to describe LSC in as general a framework as possible.
The formulae are given in spherical or no approximation. The use of pla-
nar approximation is avoided, because it - despite the simplicity of many
formulae - in my opinion has a number of important limitations.
Also the probabilistic model for LSC is not treated. LSC may be ba-
sed on purely functional analytic developments.where onedoes not have to
invoke the stochastic saints (or devils).
I will suppose, that the reader is familiar with the basic concepts
of physical geodesy, corresponding to the textbook "Physical Geodesy" by
Heiskanen and Moritz (1967), chapters 1-7. References to this book and to
"Advanced Physical Geodesy", (Moritz, 1980) wi l l be abbreviated "PG" and
"APG", respectively.

2. MATHEMATICAL BACKGROUND
2.1 Reproducing Kernel Hi l bert Spaces
The mathematical model behind LSC takes advantage of concepts well-
known from finite dimensional Euclidean spaces: existence of base vectors
coordinates, inner product, angles and distances. Also here linear map-
pings may be represented by matrices.
All these concepts become meaningful in linear vector spaces of
functions called reproducing kernel Hilbert spaces, RKHS. The anomalous
potential T will itself be an element of such a space. In fact any harmo-
nic function will be element of some RKHS. Furthermore approximations T
may be determined based on observations of e.g. values of T or of its de-
rivatives in discrete points. If T is an element of the space it may
easily be proved that arbitrary good approximations will be obtained if
the amount of (appropriate) data increase in a regular manner. However,
even if T is not an element, it may be possible to show that T may be
approximated arbitrarily well.
Since the objects we will deal with are functions defined in a sub-
set of a three-dimensional Euclidean space R', the letter n will be re-
served for such a set. Points in n will be denoted P, Q and the boundary
of n is denoted the functions will be denoted f, g or h and for real
numbers we reserve the letters a, b and c. Points in the n-dimensional
Euclidean space will be denoted X and y with coordinates {xi} and Iyil
with respect to the canonical basis.
A Hilbert space is a complete, linear vector space with an inner
product. Let us denote the space H and use the usual symbols ( , ) and
11 (1 forthe inner product and the norm, respectively. Then for f, g, hcH
we have
(h, af + bg) = a(h,f) + b(h.g) (linearity)
(f,g) = (g,f 1 (symmetry
llf (lZ = (f ,f
(positivity)
)zo

Since T always may be considered an element of a Hilbert space with a
countable basis we need only consider such so-called separable Hilbert
spaces. We will denote the elements of such a basis f, , f,, f, 9 ..a .
Since we from this may construct an orthonormal basis, we will suppose
that it has this property, i.e.
(2.1 )
For any f~Hwe may construct its expansion as a series with respect to
this basis. The coefficients - or coordinates - ai will be equal to the
projection of f on the base vector fi,
ai = (f, fi) . (2.2)
In a separable Hilbert space the expansion with respect to the basis con-
verge in norm towards f,
n
lim (1 f - z a. f. ( ( = O ,
1 1
n- m i =o
but we do not necessarily have
The inner product of two elements of H may be calculated as the scalar
product of their coordinates
Also any element is uniquely determined by its coordinates
and
(g,fi) = (f,fi), i = 0,1, ..., m =) g = f

Example 2.1:
Consider the linear vector space of functions harmonic outside a
sphere with radius R and regular at infinity. We use here usual spherical
coordinates = (geocentric) latitude, x longitude and r distance from
the origin. A (possible) inner product is
i.e. the integral of the product of f and g evaluated on the surface of
the set of harmonicity. (If f or g is not defined here, the integral
should be interpreted as the limit of the integrals over spheres with
radius R+d, d - 0.)
(2.5)
The orthonormal basis is the usual fully normalized solid spherical
harmonics, V. .:
1J
where P. . are the fully normalized associated Legendre functions, see
1J
PG(eq. (1.77)). (We have put F.. =
1J
The remark, that H consist of functions regular at infinity have as
a consequence that only base functions with subscript i > 1 can be used.
A1 1 elements are harmonic functions produced by density (anomaly) distri -
butions with zero total mass and center of gravity at the origin of the
coordinate system.
Mappings from a vector space to R are denoted functionals. A linear

functional, L, fulfil
L(af+ bg)= aL(f)+ b~(g) (2.7)
Functionals are used to express the relationship between a quantity we
want to determine and the observations we have. The gravity itself in a
point P
is a non-linear functional. Other quantities related to the anomalous po-
tential may be expressed using l inearized functional S, normally obtained
by considering only the zero and first order term of a Taylor-expansion.
The equation relating the gravity anomaly ~g in a point to T is a good
example (see PG, section 2-13),
A linear functional is bounded, if there exist a positive constant, M, so
that
IL(f) Is M.11 f ll
(2.10)
holds for all ~EH. Such functionals form a linear vector space H*, and a
norm may be introduced by selecting the maximal value of M in eq. (2.10),
i.e.
Hence
l]Lll* =sup ILOJ (f f Q).
fEH I l f II
lL(f)I c I l f 11.11 L ll* 3
(2.12)
an equation which is very useful when determining upper bounds for errors
of approximation.

In a Hilbert space the linear functionals possess a very simple represen-
tation similar to what is well-known from R! To each L~H*there exist a
unique element (EH so that
~(f) = f~,f}. (2.13)
l is denoted the Riesz representer of L. We can use this to define the
inner product in H*, for two functionals L, and L, with representers
and L,, by
(L, L,),= (L,,~J - (2.14)
In this manner H* has been equipped with an inner product, and it becomes
hereby a Hilbert space. (It is easily shown that
(L.L)* = II L 11:
as defined in eq. (2.11), see e.g. Tscherning (1978, p. 176)).
If the evaluation functionals
Lp(f) = f(P) (2.15)
are bounded, then there exist a so-called reproducing kernel. Correspond-
ing to LP we have its representer, which we will denote Kp(Q). Then
(Kp(Q), f(Q)) = f(P) . (2.16)
If P varies, we have a mapping
K(P,Q): flx fl - R ,
the reproducing kernel,
(K(P,Q), f(Q)) = f(P). (2.17)
(It is called a kernel, since it when used in spaces with inner product
like in example 2.1 will look like the kernel of an Integral equation).
In a RKHS eq. (2.3) holds. In order to see this we use eq. (2.12)
and the condition that the evaluation functionals are bounded. Then

which tend to zero when n go to infinity.
as
The reproducing kernel may be expressed using an orthonormal base
Using eq. (2.3) we have
which proves eq. (2.18). This equation is very useful when constructing
closed expressions for K(P,Q) based on a known orthogonal basis.
Example 2.2:
We consider nearly the same Hilbert space as in example 2.1 but we
permit functions which are not regular at infinity, i.e. harmonics of zero
and first degree are included. Then
From PG(eq. (1 -82 ' ) ) we have
[cosjk cosjkl+ sinjxsinjx' 1 ,

with
cos+ = sin; sini' + COS~COS~' COS(X' -X)
& is the spherical distance between P and Q.
Since
Then using eq. (2.6) and (2.19) we have
1
(a 2 + b 2 - 2ab cosdi i=o
a>b, it is easily found (see PG(p.35)) that
K(P.Q) =
R2((rr1)2 - RL)
((r r ')'+Rb- 2Ra r r ' COS$)^^
The reproducing property is then
=f(i,~,r),
which is the well-known Poissons integral.
Using the reproducing kernel, the representer of a linear func-
tional is easily obtained,
L(f = L(f(P), K(P,Q)) = (f (P), L(K(P, -1) ,
i.e. the representer is L(K(P;)) which we in the following will denote
K(P,L). If two functionals are applied on K(P,Q), we will use
L,L (K(P,Q)) = K(L,, L, ) .

Since the inner product of two functionals per definition is equal
to the inner product of their representers we have
(Lr, L2 1, = ( K(P, L1 1, K(P, L2 l)= Ll(K(P,Q), K(P,L, 1)
and \/L 1: = K(L,L). Hence, the inner product of two functionals is obtained
by applying the functionals on the reproducing kernel. (Note here
the similarity with finite dimensional Euclidean spaces, where the reproducing
kernal corresponds to the unit matrix, I. Here the inner product
of two functionals (represented by transposed vectors) are also obtained
by multiplying these on I from left and right, respectively).
2.2 Approximation in a RKHS
Given a set of linear independent elements gi E H, i=l , . . . ,n it is
possible to find a unique "best" linear approximation f to a function EH
in the sense that f -f has the smallest possible norm. This means, that
for any set bi, i=l ,..., n
n n
I l f - f /I = I l f - aigillzll f- z bigi ll.
i =l i=l
Corresponding to the functions gi there exist an orthonormal set of
functions gi* , and it is easily seen that a best linear approximation is
given by
i.e. equal to the sum of the projections of f on the elements of an ortho-
normal basis spanning the same subspace as spanned by the elements gi.
The difference between f and f is orthogonal on all gi* because

= (f, gi*) - (f, gi*) ' 0
and must therefore always be orthogonal on a1 l elements gk. f is the pro-
jection of f on this subspace,
see Fig. 1.
span {gi, i=l, ..., n } ,
-
Figure 1. Construction of f as a projection.
We can use this to write down a system of normal equations which
directly will determine the constants Isi) in eq. (2.26). We must have
and in matrix form

Note than in order to find 7 we must require that f is an element
of the same Hilbert space as the functions gi,so that (f ,gi) can be calcu-
lated and also that it really is possible to calculate these inner pro-
ducts.
Ordinary inner products like the one used in example 2.1 can there-
fore not be used to construct best approximations to the anomlalous poten-
tial T, since its values are known only in discrete points or as mean va-
lues of "blocks". Furthermore its values are not given on a sphere, but on
the much more complicated Earth's surface.
On the other hand, had we used for gi the representers of the func-
tional~ corresponding to the observations gi = K(P.Li) , where K(P,Q) is
the reproducing kernel of a Hilbert space having T as an element, then a
best linear approximation can be found.
Eq, (2.28) becomes
where ILif 1 are the observed quantities. In this case we even achieve
that
i.e. there is an exact agreement between the observations and values com-
puted using the best approximation f.

The fulfilment of this condition is the basis fcr least square c31-
location. Here a function f is obtained so that eq. (2.29) is fulfilled
and so that f has the minimum norm in between the elements of a RKHS
which fulfil this equation. Since we do not require that l( f-f 1) is mini-
ral (or that it can be computed) we do not even need to require that f is
an element of the RKHS, H. Only the linear functionals associated with
the observations must be elements of H*. (On the other hand li fcH, then
11 f-7 11 will be minimal).
The situation is shown in Fig. 2, where must be an element of an
affine subspace, ? E A={gI Lig = Lif, i=l, ..., n l , where again Lif are the
observed vaiues.
Figure 2. The construction of 7 as the intersection
of two subspaces. Note f supposed to be ln H in the fiqure.
f must be the element in the affine subspace, which has the shortest di-
stance from zero, i.e. it must be located on the n-dimensional subspace
orthogonal to the aff ine subspace. This subspace must also be orthogonal
on the subspace A. = I h 1 Lih = 0, i=l. ..., n 1 , parallel to the affine
subspace A.
If we regard the representers K(Li,?), then if h€Ao
i .e. the functions K(Li ,P) span the subspace orthogonal to Ao,

l
n
A = {g = C ai K(Li,P), {a. } E R"
0
1
i=l
l
The function f must therefore be equal to the intersection between A. and
A, so
n
f (P) = z ai K(Li,P) (2.30)
i=l
and
{L,? 1 = {K(Li ,L.) J 1 [a. J 1 = {Lif 1,
{a.} J = {K(L.,L.)~-'
1 J { ~ ~ f }
A further condition for finding i is then naturally that I K(Li,L. )l is
J
positively definite, or that the functionals as elements of H* are line-
arity independent.
It can easily be proved rigorously (cf. Tscherning, 1975, p. 89)
that 7 has the minimum norm. In fact the norm is
Using 7, the approximate value corresponding to any functional LEH* can
be calculated,

If f EH we may also calculate an upper limit for the error of prediction
using eq. (2.12).
I
T
L -L?/= ~ IL(~-?) 1 = (L(f)- C K(L,Li) 1 {K(L.,L.) }-l 1L.f }
1 1 l
= L - . L i 1' L i L j 1 ' L 1 } ) (f) l
An alternative expression for the upper limit of the error is found in
(Krarup, 1978, eq. (12)). However, such equations have a limited use since
in all cases llT 11 must be known.
Example 2.3:
Consider the case where we have only one observation Lpf =f (P). Then

and then
1
l = L i L b . . l (b.) ={b.] ,
J J 1J J 1
as we should expect.
Example 2.5:
Consider a Hilbert space H* spanned by a stationary stoch 'rocess
X(t), where t is an element of an index set. Let us suppose that the
process has a covariance function
(2.35)
where P is the probability measure related to the process. Then C(t) will
be the reproducing kernel of the space, H, dual to H *. Linear prediction
of a quantity is then obtained using eq. (2.30) and (2.31), see e.g.
Parzen (1959).
Example 2.6:
Consider the Hilbert space spanned by the solid spherical harmonics
given by eq. (2.6 ), i > o, but with the inner product
The functions V.. will still be orthogonal, but not orthonormal:
13

Hence
A closed expression for this kernel is found in (Tscherning, 1972, p.28).
Example 2.7:
Consider the Hilbert space spanned by the solid spherical harmonics
U. ., i 2 0, but with the Dirichlet inner product
1J
= vf-vgdn
4nR3
In this case K(P,Q) = G(P,Q) + N(P,Q), where G is Green's and N is Neumannls
function, see (Garabedian, 1964, section 7.3). Using Green's identity,
and the fact that F
aN
and G are zero at the boundary, we get
This shows, that in this case the reproducing property is equivalent to
the fact that the functions N and G are these which solve the boundary
a f
value problems where either c of f are known.
Again, V.. will be an orthogonal but not orthonormal set of base
13
functions. We must therefore calculate the norm of V..:
1J

so that
Closed expressions can be found in (Tscherning, 1972).
The norms used in examples 2.2, 2.6 and 2.7 are all rotational inva-
riant, which means that if two functions f and g become identical after a
rotation of the coordinate system, then (If (1 = )(g 11. It may be proved,
that all reproducing kernels of such spaces have the form
where R > 0 and a. are positive constants (the so-called degree-variances)
1
Such kernelsmay in many cases be expressed in a closed form, which make
these very useful in actual calculations. However, as a consequence the
set of harmonicity, fi , will have to be a set in R3 outside a sphere to-
tally included in the Earth. T itself will then = be an element of this
kind of space, but arbitrarily good approximations to T exist in the
space , as a consequence of Runges Theorem, see Krarup (1969, p. 54).
Thls is the justification for using this so-called Bjerhammar sphere as
the sphere bounding the set of harrnonicity for the approximation 7.
However, reproducing kernels in RKHS which has the correct set of
harmonicity may be given explicitly, see (Krarup, 1978, section 3) and
the following example:
Example 2.8:
Consider the set of bounded, harmonic density functions with the same

support n o = Cn as the Earth Is density distribution. We nay equip this
linear vector space with the inner product
The value in a point Q of the potential T, generated by p, is the value
of a linear functional, NQ, applied on p, which can be expressed very
simply using the inner product
where G is the gravitational constant.
The harmonic function k(P,Q) = G// P-Q I is the Riez-representer of
this functional.
If Q varies in n, we have an operator N from the set of density
distributions to the set of functions harmonic in n.This will be a linear
vector space and we may introduce an inner product by
(Tl, T2)H ' (PI
which makes N an isometric mapping. In this manner, a Hilbert space is
constructed. Since
with T = N(p) we get
KD(P,Q) = k(P,N) = N(- G 1
I P-Q l

If no is a sphere, then KD(P,Q) may be expressed in the form (2.39)
with
(The proof of this is left as an exercise to the reader).
2.3 Use of the empirical covariance function
In all Hilbert spaces described in section 2.2 each individual ele-
ment will be treated in the same manner. Contingently available informa-
tion about T's actual smoothness properties can not be used. The question
then is, whether it is possible to construct a RKHS, so that approxima-
tions to T, only, becomes as "good" as possible.
In such a space a computed value, e.g. T(P), must be equal to a line-
ar combination of the observed values, which we here will suppose also are
values of T in certain points, Qi,
n
(We use the subscript P on b in order to stress, that the coefficients de-
pend on P. )
and
The error of prediction is
The idea is now to determine the constants bpi so that the mean value of
2 computed for repeated configurations of the point set tP,Qi, i=l , . . .,n 1
ep becomes minimal. The repetitions are obtained e.g. by taking all configurations
which are formed by a rotation around the origin or by the rota-

ticn around a line through the origin
Put
and
In order to express this we introduce the averaging operator, M, and
M iT(Qi) T(Qk) 1 = Cik
M (T(P) T (Qi) 1 = Cpi
M tT(P)' 1 = C,
where C is the covariance function of the anomalous potential.
The definition of the covariance function is not limited to value5
of T in points. General covariances between values of linear functionals
can be defined also. Due to the linearity of the averaging operator, M,
the covariance between two values of linear functionals L, and L, are
C(L,(T), L, (T)) = C(L, ,L, = L, (Lz (C(P.Q))).
This is the so-called "law" of covariance propagation.
Example 2.9:
We will work in spherical approximation, where the Earth is approxi-
mated by a sphere with radius R, G:= s the geodetic latitude, and r=R+h
h the ellipsoidal height. Then
GM
T(7.A. r) = I (;i \ p. .(sinv)cc. .cosj~+3. . sinj Al. (2.45
i =2 j=o '3 13 1J
Suppose repetitions are formed byall rotations around the origin. Then
C(P,Q) = M (T(P), T(Q) I =
where a is the azimuth from P to Q.

with
As shown in PG (section 7-3) for r=rl=R,
i.e. C only depends on the spherical distance between P and Q and on r,rl.
The general expression is
compare eq. (2.39).
The average m; of e; becomes in general
If we want to minimized this expression, then
so that
with

It is easily seen that T(Q.) = T(Q.), i-l, ..., n, so this procedure
1 1
has given us a LSC-solution. We only need to prove that 7 is minimized in
some norm. But this will be the case, if the covariance function is the
reproducing kernel of some Hilbert space of harmonic functions.
This is also easily seen, if M is the average of all configurations
obtained by rotations around the origin as in example 2.9.
Since C(P,Q) in eq. (2.48) is the reproducing kernel, then the base
vectors are
(If oi=O, then we exclude the (2i+l)-dimensional subspacespannedby V .. from
1J
H). We can then find the norm and inner product if we know the expansion
of a function with respect to the basis V. ..
1J
Suppose
then
Hence
In this case we see that, unfortunately,
i.e. T is not an element of the space for which C(P,Q) is the reproducing
kernel.

We are then not able to use the expression eq. (2.34) for the error
bound. However expressions for the mean square error can be calculated,
using eq. (2.45) and (2.46). For an arbitrary functional L and observa-
tions Li(T) we have
After some reductions (see PG(p. 269)) we get
m( = c(L,L) - {C ( L , L ~ I tC(Li,pl ~
tC(Lk,L) (2.53)
2.4 Treatment of noise and parameters
The data we have will generally not be directly related to T through
a llnear functional. It contain measurement errors (ei) and may be affec-
ted by parameters such as an incompletely known relationship between a lo-
cal geodetic datum and a geocentric, correctly oriented datum. Let us de-
note the observations xi, the m parameters {X. )=X and suppose the obser-
J
vations are related to the parameters through a vector AL. Then
Let us suppose, that the noise vector e has the variance-covariance ma-
trix D. A LSC solution may then be obtained so that the sum of three
T
quantities are minimized, namely 117 /I2, eTD-'eand X PX, where P is a
positive definite matrix expressing some a priori weights of the parame-
ters,
T
117 11' + e T D"e + X PX = min . (2.55)
The solution may be found using Lagrange multipliers as described in APG
(Chapter 29 and 30). We regard
(where A is the m ~n matrix formed by the vectors AL), and the differen-

tlal
T
d 0- (T-k IK(Li,P) i .dT) t (eTg-l -k T )de t (xTP-kTA)dx. (2.57)
The condition do= 0 gives 3 equations
i - k T i ~ ( ~ i . t ~ = ) o
and
eT g-l - kT = 0
XTP - kA = D .
This gives
T i = k { K(L~,P) }
so that with
Also e = Dk, so
and
X-AX = (C t D) k
where ? = C +D.
With
1 T
X=P- A k
we get flnally
X = P-~A~c-'(~-Ax)
= ( ~ T t - 1 ~ ~
A T--' C 'X
i = {K(L~,P) )T (X-AX)

The fulfilment of the last two equations are only a necessary cond:.
tion for a minimum. However, it is possible to show that we have obtained
a minimum.
It is still possible to write down expressions for maximal error
bounds similar to eq. (2.34), now using both the norm in the reproducing
kernel Hilbert space, and the norms implicitly introduced in the n-dimen-
sional space of measurements by D and the m-dimensional space of parame-
ters by P, see APG (p.128). For the mean square error of prediction we
find with
that
and
2.5 Consequences of the mathematical model
In order to use LSC we must select an appropriate RKHS. This is done
by chosing a reproducing kernel as discussed in the next chapter. But what
are the consequences of these choices in case we for example need to or
want to use spherical or planar approximation? Something which look nume-
rically or computationally nice may become a failure in practice.
We must also be able to compute the quantities I K(Li,L.) l for all
J
kinds of linear functionals, which correspond to the data types we en-
counter in practice as well as the vectors AL. Aiso the noise variance-
covariance matrix must be available. However in practice we generally on-
ly know the variance8 for groups of measurements and no information is
available for the covariances.
Also systems of linear equations must be solved, with dimension as
large as the number of observations plus the number of parameters. How we

may be permitted to reduce this number by oniy considering data in a li-
mited neighbourhocd of the area of interest is discussed in chapter 4.

3. CHOICE OF REPRODUCING KERNEL HILBERT SPACE AN0 THE EVALUATION OF
K(LI,L, ).
3.1 Choice of norm and reproducing kernel
In section 2.2 and 2.3 we discussed two possibilities for chosing
the norm. Either a norm could be selected so that a certain kind of ma-
thematically defined smoothness was obtained, or a norm adapted to T
could be used, which assured that the mean square error of prediction was
minimized.
Alternatively, the choice of norm could be based on considerations
related to the ease of evaluating the quantities K(Li,L.) and K(P,Li)
J
used when constructing f. Also it is important to assure, that numerical
difficulties do not occur during the solution of the normal equations
(2.31). This could happen if data are clustered, since the inner product
of two functionals of the same type, but associated with two different
points, becomes equal to the square of the norm when the points converge
towards each other,
The angle between the functionals and the correlation becomes zero and
one, respectively
* K(Lp.LQ)
c0s(Lp,LQ) = 7 - 1 , P- Q.
I I LP ll* IILQ /I* (K(L~,L~)K(L~.L~))'
This phenomenon can be compensated in practice by using reference
fields of higher and higher degree.

Example 3.1 :
Figure 3. Typical variation of C ($) = K(AgpPg )
A g Q
as a function of spherical distance 4 between P and
Q for r =rt=R using a function of the type given by
eq. (2.48). 4, is the correlation distance C(+,) = i C(0).
Suppose we have observed the first m coefficients Li (T) = ai of T's
expansion with respect to a set oforthonormal base function ii and that
we have n further observations Li,,(T). We also suppose, that no errors
are present, and put T= To +T, where
Then K(Li,L.)= 6 .. i,j < m and K(L. L. ) =Li,fj, i ~n and j c m
J ,U' J' 1+m
The normal equations matrix may then be partitioned in 4 parts so that

Then
where
and
Hence
This means, that in case we have a set of potential coefficients as ob-
servations a collocation solution is obtained by uslng (1) U+To as a re-
ference fleld,(2) a reproducing kernel, where the corresponding products
of the base functions fi(P)fi(Q), i m have been eliminated.
It is easily shown, that if the coefficients have a non-zero error,
then the products will not be eliminated, but enter with a much smaller
weight, see (Tscherning, 1974, section 2.3).

The modified reproducing kernel K,(P,Q) of example 3.1 wlll in
many cases imply a smaller correlation between point-related linear functionals
of the same kind. This can be studied for kernels of the form given
by eq, (2.39) and is illustrated in Fig. 4 and 5. Here are shown the
(smallest) distance +land $o , in which the correlation becomes 0.5 and 0,
respectively,
1
for the gravity anomaiy functional eq. (2.9) (r=R) (see Fig.
3). Values are given for varying maximal degree and order n of an errorfree
reference field and for three different types of norms corresponding
to oi = A/ik, k=2, 3 and 4, A > 0, and RE-,R=1500 m.
200 r
m/
!
l00 4
;lkLL
I
1 k=3 0% 1 O . l t OS k=4 G,
Figure 4. The correlation distance +, for c'(bgp, "g0) as a
function of the maximal degree and order m of a reference
fleld for three different types of norms corresponding to
al .: A/ik, k.2, 3 and 4. r=r1 = R + 1500m.
Even in the case that the coefficients ai of example 3.1 are not
available, we will be able to construct a reference field T, (cr a se-
quence of reference fields Ti) using LSC, which represents these coeffi-
cients. For example mean values of gravity data of blocks of varying size
can be used as described in (Tscherning, 1978b). This means, that we in
general always will be able to "decorrelate" the data we have, if we nave

Figure 5. The location of the first zero point$,for
C(agp,ag )as a function of m for three different types
Q
of norms. r = r' = RE = R + 1500 m.
enough data to form good mean values. - And it is exactly in such a case
where it in practice may be necessary to decorrelate the data, in order
to counteract a numerical difficulty. On the other hand, we also see from
Fig. 4 and 5, that the speed with which oi tends to zero for i -- influ-
ences +l strongly. And this speed is directly related to the type of norm
used, see examples 2.6, 2.7 and 2.8 and Table 1.
In a space with a rotational invariant norm, it is also possible to
change the correlation of the point related functionals by changing the
radlus of the Bjerhammar-sphere. In order to see this, we modify the general
expression of a rotational invariant kernel eq. (2.39) by introducing
the radius of the mean earth sphere, RE. Then
(3.5)
* REP i+~
K(P,Q) = , E "i($)2i*2 (X Pi (COS*) = ,E oi(7,) Pi (COS*),
1 =o r r 1 =o

where ojf = ( R )2it2 oi. If R - RE, then more "weight" is put on the high
l %
order degree variances. The correlation distance will decrease as shown
in Fig. 6. (This is caused by the fact, that the iegendre polynomials
Pi (t) have i zero-points approximately evenly distributed from - 1 to 1 . )
Figure 6. The variation of the correlation distance 4, forchanging
value of R E -R and for varlous values of m, the maximal degree and
order of the reference field used. Degree-variances for which
a. = A/il are used.
1

Lelgemann (1981) has used the kind of relationship shown in Fig. 6
to select a value of R optimal for a given data spacing. This was used to
assure the numerical stability in a situation where rather weak norms we-
re used, which imposed strong correlations between the linear functionals
associated with the observations. However, the development of algorithms
for selecting an optimal value of R generally presupposes, that data-
points are uniformly distributed.
Butthe data-spacing needed to achieve a certain mean error may va-
ry in an area, due to the changing smoothnessof the gravity field. And
data is not spaced uniformly in practice - normally data will be lacking
at lakes.
Therefore, the norm or other parameters like R should not be cho-
sen based on considerations related to the data density. Such considera-
tions should influence the choice of reference field instead, see also
Chapter 4.
Should the norm then be selected, so that it is as easy as possible
to evaluate K(Li ,L.) ? If the answer is yes, then kernels as given in
J
example 2.2. could be selected as done e.g. by Lelgemann (1978). Or flat
earth approximations could be used, cf. Jordan (1972). The ultimate consequence
of this principle is not to use LSC at all, but to select the simplest
possible base functions gi, T - n
= zi,, aigi. where {ai) are determined
so that L,(:) = Li(T). Then normal-equations will in this case not be
symmetric.
If the functions gi are selected so that g. = K(L. .P) for some ker-
1 1
nel K, then 7 will be a LSC-solution anyway. This is what sometimes hap-
pens, since it in practice has been found, that the use of such base func-
tions give good results. An analysis of point mass or surface layer model-
ling methods will show, that in many cases LSC-solutions are obtained. We
will see in the following example which reproducing kernels implicitly
are used in point mass modelling.

Example 3.2:
Suppose that point mass functions l/j P-Q; I are used to model not T,
but the gravity anomaly function ag multiplied by r, which is a harmonic
function. The points Q; are all located in the depth D = RE- R and all ob-
servations are in points Qi on the Earth's surface with the same latitude
and longitude as the points Q;. Then
This is also a collocation solution and we will find the reproducing kernel.
We must have
which also must be equal to
This shows, that point masses should not be used to model gravity anoma-
lies, since they contain harmonics of degree zero and one. However, we may
subtract these harmonics from the point mass functions, and if we do this
we see that R. = JRE'R and oi = l/((i-1)'R).
Hence, the use of point mass rnodeiling of gravity anomalies cor-
respond in this case to the use of a kernel with degree-variances which
tend to zero like i-'.
If instead f = xi!, ai/ I P-Q; 1 , then we must have

so that a. = R /(i-l), i $ 1. This shows, that we in this case implicitly
1 E
use a stronger norm. Note, that a similar analysis can be made of the so-
called Dirac-approach, (Bjerhammar, 1976), see Tscherning (1983b).
Even the mathematically simple norms have corresponding kernels, the eva-
luation of which involves as an intermediate step the computation of loga-
rithms or square-roots. It is then generally worth-while to tabulize the
functions for varying values of t, =l-cos+ and sl = l - (R2/(rr1)j, see
e.g. Sunkel(1979). Weachieve hereby that theevaluation of all reprodu-
cing kernels takes the same time. The only argument then left for using e.g.
point-masses is that they may be given a physical interpretation, and that
the Earth's potential maybe behaves like a linear combination of point mass
potentials.
In fact, there is in this sense a physical interpretation of all the (ro-
tational invariant) norms for which oi = A i k , see Table 1.
Table 1. Relationship between rotational invariant norms, the asymptotic
behavior oi = A i k , for i - -, and the physical interpretation of the norm.
Norm expressible as integral LSC corresponds to the modelling of
k of m'th-order derivatives of T based on gravity data using:
T over
n W
m m
2 0
1 0 mass-quadrupoles
mass-dipoles
point-masses
mass-lines
In general an even value of k corresponds to an integration over O using
the (2-k)/2 order derivatives and the value 1 smaller corresponds to the

i~tegration of the same derivatives over the boundary, compare e.g. Fred~n
(1982). As shown in example 2.8 are the kernels and norms also related to
kernels and norms of Hilbert spaces of harmonic density distributions. If
we use integration over the m'th derivatives for the potentials, then we
must use the (m-2) 'th derivatives for the densities.
However, it is still difficult to say what is the most suitable non to be
used for the modelling of T. Will it be best to use point-masses, mass-li-
nes or mass-multipoles, when the data primarily are gravity observations ?
It becomes even more difficuIt, if we have a mixture of data involving
both zero, first and second order derivatives of T.
Since spline functions have been successfully used in many fields of science,
it has been argued that the corresponding harmonic splines should be used
for gravity field modelling, see e.g. Freden (1981). LeIgemann (1981). Es-
pecially Freden argues, that various types of (surface) integrals of higher
order derivatives should be used. But what should be the maxim61 degree ?
One could here argue (cf. Tscherning (1973, p. 159)) that one should go so
high up, that the l inear functionals corresponding to derivatives of the
highest order which in practice are measured (i .e. second order) should
have finite norm in H*. We should, following this line of thought, use a
norm, where the corresponding degree-variances go asymptoticallq to
zero like i-5-'. On the other hand, if a Bjerhamar-sphere is used, R < RE,
then the linear functionals associated with derivatives of any order and
evaluated in points with r > R will be elements of H*. This is because for
an arbitrary polynomial p(i) the product p(i) wil! go so fast to
zero, that the sum of the products always will be finite.
Between all the mathematically reasonable possibilities for a norm, we
could finally make a choice: namely selecr the one m03t closely resembling
an empirical covariance function for T. This possibility will be discussed
in the following section.

Furthermore the values available for the estimation will not be associated
with the grid points (qi,+) or (qj,5). These values must be predicted
using a suitable reproducing kernel, see Goad et al. (1984).
However, if a sufficimtly dense and regular data coverage is available, then
can be used, where the spherical distance between Pi and Qi falls within the
i'th sampling interval used.
A
Generally the functions C(+,%) are not equal - the covariance function will
be anisotropic. A measure for the anisotropy, introduced by R.Forsberg (19843
is the ratio between the maximal and the minimal correlation distance +lk. An
isotropic covariance function will have index 1.
Forrunatly, the main an-isotropies may be eliminated by subtraction of the
attraction of the topography. Also a more isotropic field is obtained if a high
order reference field is subtracted out as described in example 3.1. Such a
procedure is not required in order to evaluate eq.(3.9), but the error esti-
mates calculated using eq.(2.51) will be much better if they are evaluated
using a rotational invariant representation of the covariance function.
If we have a global estimate of the covariance function using gravity data,
where oi(a) = oi (i-1)2/R2, cf. eq.(2.47). Hence a harmonic analysis of a
set of estimated values c (+ ) will enable us to determine the first n
Ls2 q
degree-variances, where n depends on the sampling interval n/w. This is a
quite uncertain procedure. The first set of values estimated by Kaula (1959)
did for example include negative values of oi.
In Tscherning & Rapp (1974) another procedure was followed. Since at that
time the data included an estimate of C (01, the l0 X l0 mean gravity an-
A g
omaly covariance function and values ofoi, i < 21, (based on satel!ite de-
termined potential coefficients), a simple polynoinial expression for oi and

a value of R was found so that C($) given by eq.(3.5) fitted these data the
best possible. The following constants 'were determined,
Since 1974 other estimates have been made on a slightly more complex form,
for example presented as sums of expressions like (3.12) using for each
expression a different value of R and A, see e.g. Moritz (1977), Jordan
(1978), Jekeli (1978) or APG (Sect~on 23).
The main reason behind the development of new estimates for ai has been
that the model eq.(3.12) implies a rather large value of the global mean
square variation of the horizontal gravity gradients of approximately 3500
EZ (1 E = IO-' s-~) at the Earths surface, see (Tscherning, 1976). This
value is clearly too high in areas with topographic height variations below
500 m. However, in mountanous areas and at the deep ocean trenches the
value is many times too small. So it may still be a valid estimate of the
global mean square variation.
Estimates of the type given by eq.(3.12) or linear combinations of such
estimates then correspond to the use of a norm integrating the squaresum
of up to second order derivatives at the boundary of the Bjerhammar-sphere,
thus also fulfilling - at least partially - the mathematical requirements
discussed in section 3.1.
Local covariance functions are mainly computed using data from which the
contribution from a m'th degree reference field To, cf. example 3.1, have
been subtracted. In this case the covariance function will consist of two
parts, where the first part represents the nolse in the coefficients of the
reference field, and the second part uses a model like (3.12) for i > m,
m
R; it1 m R; it1
K1 (P.9) = i=2 ui (F) Pi(cOs$) f i=i+l (II (F) Pi (cos+), (3.13)

where d (C. ) and d (S. . ) are the error variances of the estimates of
1J 1J
the coefficients. (Values of of are given e.g. in (Tscherning, 1982, Se-
ction 4.2))-
Since the coefficients C.. and S.. of highest degree generally will have the
1J 1J
highest error, they may for some areas contain nearly no information. This
can be checked using Fig. 5, which shows the value of +o for an error free
reference field To. Conversely the value of +, can be used to fix the value
of m to be used in eq. (3.13).
The fit to the estimated values, e.g. C (+ ), q = O ,..., W, can then be
A9 9
made using the model (3.12), with A and R as free parameters. This can be
done in an iterative process, see e.g. Tscherning (1972, App. 3), Schwarz
& Lachapelle (1981 ), Lachapelle & Schwarz (1981 ). A somewhat simpler proce-
dure would be to use an estimate of the correlation distance +, and the gra-
phs of Fig. 6 for the determination of R.
The procedure proposed for model ling a local covariance function presupposes
implicitly that eq.(3.13) is a valid model for the estimate (3.10). This is
clearly not so, except if the gravity field outside the local area behaves
in the mean like inside the area. It is therefore important to subtract out
not only a reference field To, but also local topographic effects as des-
cribed in Chapter 5. Still this may not be satisfactory, having as a conse-
quence that the "law" of covariance propagation can not be expressed in a
simple manner as used in for example eq. (3.1 l), through a simple modificati~n
of the degree-variances. This should of course be checked if possible by
computing empirical covariance values for several different quantities. At
sea this might be possible using gravity anomalies together with sea-surface
heights obtained from satellite radar altimetry, but treated as if ~t was
geoid undulations.

A check of the consistency of covariances derived from gravity data and in-
dependently from deflections of the vertical using a model based on eq.(3.12)
and (3.13) in the New Mexico test area (Schwarz, 1983) showed a good agree-
ment between the estimated and the model values. However, even if the agree-
ment is not satisfactory, LSC-solutions may still be computed because K1 (?,Q)
given by eq.(3.13) is a valid reproducing kernel. Contingent inconsistencies
will primarily show up in the calculation of the error estimates eq.(2.53).
Let me finally explain why degree-variance models like (3.12) have been se-
lected. The reason for the factor (i-l) is naturally that it disappears if
the gravity anomaly functional is applied one time. But the main reason is
that the infinite series can be computed using closed expressions. First it
is easily seen, cf. (Tscherning, 1976) that
where a1 l k. are different integers. Then (cf. Tscherning & Rapp (1974, Se-
J
ction 8)) all infinite sums,
F. = 2 l/(i - h.) sit' pi(t),
3
J i=n
(3.16)
n > k., 0 c S < 1 are equal to closed expressions. Hence, covariance func-
J
tions modelled by expressions similar to (3.12) may be computed using clo-
sed expressions.
The above described procedure for estimating and representing a covariance
function then leads both for local and global covariance functions to the
implicit selection of a mathematically well-defined norm. Fine results have
been obtained in practice, see e.g. Schwarz (1983), but some problems have
been found in areas with large height variations. These problems will be
discusses in ChaDter 5.

3.3 Derivation of expressions for K(Li,L.) based on an isotropic reprodu-
J
cing kernel (covariance function).
In order to determine T using LSC we must be able to evaluate K(Li,L.)
J
for the functionals associated with the observations. We will here regard
the functionals associated with the gravity anomaly, Ag, the gravity dis-
turbance, 69, the height anomaly, 5, the deflections of the vertical, 5, Q,
the vertical gravity gradient and the torsion balance observations. We will
only consider functionals evaiuated at a point, since the evaluation of mean
values normally require the use of a numerical integration procedure based
on the weighted sum of point values. In some cases it may even be satisfac-
tory to substitute the mean value functional by a point functional associa-
ted with a point at a higher altitude as proposed in Tscherning & Rapp (1974,
Section 10).
The functionals will be evaluated in two points P, Q, respectively, and we
will with each point associate a local cartesian coordinate system. The
coordinates associated with P, Q, are denoted (X, , X, , X, ) and (y, , y, , y3 ),
respectively. Initially we will suppose that the first axis points east, the
second north and the third in the direction of the radius vector. (For points
with latitude equal to + or -90' the third axis may be selected so that it
is lying in the Greenwich meridian plane). We may keep the origin at the
gravity center or move it to P, Q, respectively. Then we have in spherical
approximation and with v = loUl the normal gravity (or reference gravity
lo(U + To)( 1,
aT aT
, 6g=--:--
ar ax, '
and the torsion balance observations

a 2 T a i aT
T,, =- =- (-
ax, ax, ar rcoscp K)
T, , a2 T
= -- =-(--) a l aT
ax, ax, ar r a?
a 2 T az T 1 1 a a
TA 2 7 - = 7 ( - - -(costp -T) +
1 a2T
ax, ax, COST arp acp WC)
a2T 2 a 1 aT
2T,, = 2-= --(
ax, ax, r acp rcos~ z)'
(3.21 )
a2 T aZ T
Then using T,, = - v - all second order derivatives can be expressed
as linear combinations of the quantities given in eq.(3.17) - (3.21).
Now suppose K(P ,Q) is a rotational invariant reproducing kernel or covariance
function. It is then given on the form (2.21), i.e. it depends only on
t = cos$, r and rl.
The computation of derivatives with respect to x3 or y, of, or the application
of the gravity anomaly functional eq.(2.9) on, K(P,Q), K(P,Q)/r,
K(P,Q)/rl or K(P,Q)/(rrl) will give as a result that the degree-variances
oi are multiplied by factors which include (i+l), (i+2), (i-l), I/r and 1/r1.
Hence these operations will result in a new rotational invariant expression,
which here will be denoted C, for which a closed expression can be found if
degree-variance models like these discussed in section 3.2 are used. Following
Krarup & Tscherning (1984) we will now show that K(Li,L.) can be expres-
J
sed in all cases using the derivatives of C with respect to t = cos+, C', C",
C"' etc.
The "trick" to be used is as follows. First the two local coordinate systems
are rotated around the third axis, so that the first axis points in the di-
rection to the other point. Secondly, all the horizontal derivatives are ev-
aluated, and finally transformed back to the original system.
Let us denote an arbitrary set of new horizontal coordinates z, and z, and
the rotation angle B. Then with

cos0 -sin0
R(B) = {sins
and a simple calculation using the chain rule will show that
9op
The vector (T, , ZT,,) is hence transformed by a rotation the angle 28, while
(T,, , T,, ) and (-rl, -5) are transformed by a rotation the angle B.
Let us then suppose that we have rotated the coordinate system associated
wlih F and Q the angles 90'-a and 90'- a', respectively, where a and a'
are the azimuths from P to Q and from Q to P, respectively, see Fig. 7.
Pole Fig. 7. The angles a and a'.
P'
P' and Q' are the projections
on the unitsphere of P, Q,
respectively .
Then again P = (0, 0, r) in the X-system and Q = (0, 0, r') in the y-system.
If we denote the coordinates of a point X in the y-system by X and vice-versa
then with s = sin+ ,
-
and ,? = X. The same equations may be used for y.

We may express C using some more convenient variables,
then f(u,v):= C(t, r, r'). (Note that in (P,Q) u = trr' and v = +(rr1)' ).
Since we now have C expressed in usual cartesian coordinates through u and
v, we may easily compute the horizontal derivatives in the two systems, and
subsequently insert the actual values of P and Q's coordinates, 2 X 2 of
which are zero. However, first we compute some auxilliary expressions.
2E= -t, ay,. -1,
ay, ay 3 ' ay,
and put
cijnm:= - a - a C.. , i, j,n,m=O, 1, 2or3,
ayn ay, 1J
where one or more of the derivatives are deleted if a subscript is zero
and two subscripts may be replaced by A (cf. eqh(3.20)). Then
C.= 1 f ,,,i y +fO1xiIyI2 and
C ij . = f 20ij y y + f 11 (;x+;.x)~YI~
ij ji +f02xi~j~~~4+~ijfOl~~12
The evaluation of the derivatives in P and subsequently in Q will take place
following "(=)". Then using a/au = l/(rrl)a/at,
Cl = flOyl (=) flOrls = (s/r)C1 . (3.26)
.

For the derivatives of first order with respect to yi we have
so that
+ Xi( lY l' (fllxj + f02Yj 1x1' ) + 2Yj ly IfOl 1,
(=) -2 tsr' fZ0 + (sr')Lr~f~~ = (S C"' - 2 ts C1')/(?r:),
but
Cao ('1 0 .
A few simple derivations give
C1212 (=) (t C" - S' cut )/(v' 1'
The interchange of P and Q will then give all other needed equations. (Due
the the symmetry it only means that r will be interchanged with r').
With C:= K(P.Q) we then have with 7' the normal gravity in Q,

The equations where ~ g 69 , or T,, occur are easily obtained using
C = K(w,Q), K(A~,Q) or K(T,, ,Q).
Closed expressions for degree-variance models based on eq. (3.74) are then
easily obtained since the derivatives of F. (eq. (3.16)) with respect to t
J
are also closed expressions, see (Tscherning, 1976, section 3). Furthermore
note that the components of the vectors (-",-c), (T,, ,T,, ) and (TA, 2T,, )
rotated the angles 90'-a or 180°-2a a1 l have rotational invariant covariance
expressions. Hence, these rotated quantities should be used if empirical
covariance values are estimated using a rotational invariant covariance
function model K(P,Q).

4. CHOICE OF DATA AND PARAMETERS - STEPWISE COLLOCAT!ON
4.1 Stepwise collocation
The main drawback of the LSC method is the fact that a system of li-
near equations with as many unknowns as the number of observations plus
the number of parameters must be solved. And the normal equation matrix
will generally contain very few zero elements.
Could we then not simply just use the observations close to the
point in which we wanted to compute a certain quantity? Unfortunately,
this is only possible if we use LSC as an interpolation tool. As soon as
we want for example to compute from surface gravity data the height ano-
maly or gravity above the surface of the Earth, then we know that in prin-
ciple the boundary value problem for the harmonic function T must be solv-
ed first. And this requires that data covering the whole surface are avail-
able. Contingently mean values can be used in areas far away from the
point or area of interest.
On the other hand we know that the error due to lack of data far
away will be approximately constant. This means that if observed height
anomalies or deflections are available, then we will be able to determine
the error. For LSC we simply use these data as observations. However, at
present we have for example no observed gravity values in space, so the
errors committed when computing such quantities can not be eliminated in
a simple manner.
In M3ritz (1973) it is prcposed to use a stepwise procedure, where
the matrix C = {K(Li, L. is partitioned in blocks corresponding to a
3
partition of the observations in two groups, X,, x2. Let us put

Then
and with
then
1
bl = C;1 (xl - CI2 (Cz2 - CZ1 c;,' c ~ ~ ~ - ~ c;; ( X xl)) ~ - C ~ ~
b2 = (Cz2 - CZ1 cl-,' ~ ~ (x2 ~ - CZ1 l c;; x1 - ) ~
i1(p) = clpcTj x1 (4.1)
where tLilis the vector of functionals associated with xi,i=12. We see,that
this corresponds to the use of a stepwise procedure, where in the second
step a new covariance function or reproducing kernel,
The justification for the use of such a procedure was earlier, that
the soiutions bl and b2in this way could be computed also on computers,
where all operations had to be executedwith all the variables simultaneous-
ly inthe fast memory (core store) of the computer. In fact, no savings oc-
cur, except in cases like in ex. 3.1, where K2(P, Q) may be computed directly.
The data set xl should therefore always be selected, so that the sub-
traction of ?, and the Kernel KZ as much as possible resembles the situa-
tion where a part of T1s spherical harmonic expansion is known.
The procedure may be repeated, so that the approximation valid for a
local area finally becomes equal to a sum of approximations ii.
Example 4.1
For a local f0 X fo area with gravity observations spaced 015 apart,
and where the best possible solution is wanted, the following procedure
could be used. As a first step an approximation To is used based on e.g.

the Rapp (1978) set of coefficients complete to degree and order 180.
This means that wavelengths down to about l0 are reprensented by To. As a
second step 5' x 5 ' mean gravity values are predicted in a 2' X 2' area with
the smaller area in the middle. A system with 576 unknowns must then be
solved in order to obtain 7, using these data. Finally solutions valid
for each of the 36 5' X 5' subblocks could be determined, using e.g. a
2! 5 overlap area. Each system of equations will in this case have 400 un-
knowns.
Contingent disagreement between e.g. the height anomalies at the boun-
dary between two solutions may be elimated to a certain extend, by using
computed values in one solution as artificial observations in the neigh-
bouring solution.
This procedure very much resembles the procedure used when evaluating
Stokes or Vening-Meinesz integrals, where mean values of larger and
larger blocks are used the longer the distance from the block to the
point of evaluation.
4.2 Data selection
Frequently we are in a situation, where we have more observations than
actually needed in one region, but too few in another region. The lack of
data can no method remedy. But an effective data selection procedure can
easily be designed:
First we need a rule of thumb relating the mean of the estimated er-
ror to the mean data spacing. We will consides a situation where we hae
Only one data type with isotropic covariance function C(+). For this pur-
pose let us suppose that to a good approximation for4 c +o
Then with only one observationwe have the following mean square er-
ror of prediction

This linear relationship is also found in model studies cf.(Tscher-
nlng, 1975a, Fig. 5a). The mean error in a square area with side length
d with one data point in the middle becomes
ior a data spacing with a mean distance d between the points one
will find for d < 2+,
Example 4.2
Suppose CO = 625 mga1 2 , = 10'
and 6 = '3mgal.Thend =4'.
A mean data spacing may then be determined [f G, CO and are given,
and a preliminary solution may be determined using the corresponding obser-
vations. This solution may then be used to predict values in the data
points not used. If some discrepances are too large (e.g. > 3 F), then
these values can be used as additional data points, if wanted. Alternati-
vely an upper limit for the absolute error, emax, is selected and the first
data spacing is chosen so that Gd = emax/3. Such a procedure will limit
the number of data considerably, see e.g. Goad et al. (1984). However, the
data density may still be so large that a stepwise procedure is needed.
Using a given value of emax, know maximal values of CO and minimal

values $, for various areas, the maximal blocksize can be selected, e.g.
j0 X iO, so that the normal equations always can be solved with a reason-
able effort.
ExampIe 4.3
For the computation of the Nordic geoid the mean error in the geold
should not exceed 0.5m.This correspond to that deflections of the verti-
cal should be computed with a mean error of * 1". Gravity anomalies must
then be computed with an error of f 6.6mgal. In the Nordic area cf.
(Tscherning, 1983, Table 3) the maximal value of CO is 2576mga12 and the
minimal value of $, is 3' (in the same area).
Then
d . 6.6 .-3'/(0.3 -2576') = l! 3
Unfortunately, in this area the actual value of d is 5'. However,
if we by eliminating topographic effects can smooth the gravity field so
that CO becomes 500mga1 2 and = 5' (which is a realistic possibility),
then the existing data spacing is satisfactory.
4.3 Choice of parameters
The data we have will generally be associated with a point with
known coordinates. These coordinates may be taken as parameters in a pro-
cedure called integrated geodesy, see e.g. Krarup (1980). We will here
suppose that this is not necessary.
The most important parameters are associated with the sometimes
not well known relationship between a local geodetic datum and a geocen-
tric, correctly oriented reference system. Such a connection is mainly
given by one or more sets of translation, scale and rotation parameters,
valid for an area or parts of an area.
Also the relationship between a local hight datum and a continentai
(global) datum is frequently uncertain and may be modelled using one pa-
rameter. The parameters defining the normal potential U, such as the semi-

major axis and GM, also may need improvements, which may be expressed
through pararreters. However, they should not be updated us~ng only local data.
Both ag and q are relative observations, depending on the value of
a gravity base station (or network) and a longitude reference station.
Biases in these stations must be modelled. Especially longitude biases
may cause large distortions in computed geoids, see Tscherning (1983 a).
Seasurface heights obtained by satellite altimetry are for each e.g.
(100km)part of a track equal to the geoid height plus a bias. This bias
should also be taken as a parameter.
Finally, also an unknown density used in terrain reductions may be
modelled, see section 5.2.
Table 2. Relationship between the most important parameters and
the various data types
Parameter from
Doppler Altimetry 6g r1
Location of
horizontal datum
origin X
Location of ver-
tical datum origin X
Reference ellipsoid
and normal field
parameters X X x x X X
Longitude origin X
Gravity base1 ine X X
Density used in
terrain reductions (X) (X) X X (X) (X)
The elements AL(k) of the vector AL are in most cases equal to zero
or one. Only the elements associated with the location of the origin of
the local geodetic datum are a little more complicated, see Tscherning
(1978b. Appendix 2).

5. SMOOTHING M E GRAVITY FIELD, AND USE OF TOPOGRAPHiC AND GEOLOGICAL
DATA
5.1 Subtraction of the contribution from a spherical harmonic expansion
The use of a spherical harmonic expansion is discussed in example 3.1
for the situation, where we had coefficients without errors. The corre-
sponding changes in the loaction of the first zero point $o and of the cor-
relation distance +, are illustrated in Fig. 4 and 6 for the gravity anc-
maly covariance function.
Table 3. Smoothing of T using a 20-degree exact reference field,To.
(Based on results from Tscherning and Rapp (1974, Table 9 and 11)) .
Variance Correlation Variance Correlation
Function of 5 distance of 5 of Ag distance of Ag
m2 rnga12
As seen from Table 3, the height anomaly variance decreases propor-
tionally as the square of the correlation distance. However, the square of
the correlation distance for gravity anomalies decreases much more than
the variance of the anomalies. In both cases, the maximal prediction error
(e.g. predicting using no observations at a1 l ) decreases.
If we use the "rule of thumb", eq. (4.6), then we will after a removal
of a 20-degree reference field need the same data spacing as before in or-
der to achieve the same prediction error for height anomalies. But for the
gravity anomalies, we need a much denser data spacing in order to achieve
the same gravity prediction error. The main effect of using a reference
field seems therefore to be that the maximal error is reduced. This con-
clusion can also be drawn from eq. (2.34), since ]IT - Tojl c iiTl1 independently
of the norm used.

The rather small decrease in the variance of ag does not mean that
it is not worthwhile to svbtract out To. The decrease in the value of
has as earlier mentioned the effect, that the normal equations eq. (2.31)
get a better numerical conditioning.
However, it would in some situations be preferable, that T could be
smoothed in such a manner, that the variance of the various quantities
decreases, and the correlation distance increases. This situation occurs,
when the numerical conditioning is satisfactory, but data is not spaced
so densely that a needed quality of 7 can be achieved.
5.2 Removal of topographic effects.
The removal of topographic effects achieves the goal just mentioned.
This is because this damps the coefficients of very high degree and order.
The subtraction of topographic effects only causes a small decrease in the
magnitude of the low order coefficients, as illustrated in Table 4.
Table 4. Srnoothlng of the function T given by the Rapp (1978)
coefficient set after subtraction of O the potential T of the
rock-equivalent topography wlth lsostatic compensatih a 20 km
depth.
Function Variance of 5 (m 2 ) Variance of Ag (mgal' )
916 552
To
T~
10 1 14
To - T~ 925 432
In Table 4 is used the spherical harmonic expansion of the potential
of the condensed rock-equivalent topography (Rapp, 1982). Note, that no
smoothing occurs for C ! But Ag issmoothed considerably. A slightly larger
smoothing of Ag would have been obtained, if the potential of the isosta-
tic-reduction potential had been used, cf. Lachapel le (7976).
When calculating topographic effects, the purpose was earlier to

enable the reduction of observed quantit'ies from the point of observarion
to a point on the ellipsoid or geoid. This justified the use of Stokes
integral formula for the solution of the boundary value problem.
The use of Molodenskys theory, PG (Chapter 8), or of collocation does
not require the reduction of observations to the geoid. Instead it was of
main importance. as pointed out in PeIlinen (1962) and Tscherning (1979),
that TM and tnereby T -TNis a harmonic function.
Since this is the only condition, we may very well use a model of the
topography, wh~ch is different from the real topography. This has big ad-
vantages in a situation where 7 is being determined for an areas with very
scarce topographic mapping, such as in Greenland, see Forsberg and Madsen
(1981).
If a reference field of maximal degree m, To, is used, then the topo-
graphic effects up to the same degree are included. This does not mean,
that To includes totally the effect of the topography expanded to the
degree m in spherical harmonics. The potential of this expansion will also
include coefficients of degrees higher than m.
However, the topographic heights to be used Iocally should refer to
th~s expansion, see Fig. 8. The consequence of this is, that it is possible
in many cases to disregard the isostatic compensation, because this
generally is a regional effect, and not a pointwise effect. We call this
residual terrain modelling (RTM) in contrast to topographic-isostatic
modelling (TIM), the use of which requires that also To is modified, see
Lachapelle (1976).
Fig. 8. Residual terrain modelling (PTM) with rectangular prisms. Masses
above the mean elevation surface are removed while valleys are filied
(with negative mass).

The local topography may be represented by a digital topographic map,
givlng the topographic point or mean neights in a regular grid. The effect
of these blocks may be easily calculated as described in e.g.iorsbergand
Tscherning (1981). It is possible to present the effects of blocks far
away from the point of evaluation by the effect of a cylinder having the
same total mass, and to make other approximations in order to facilitate
the computation of the effects. However, the possible use in the future of
fast fourier techniques may make such considerations less important.
In Table 5 are given examples of the smoothing achieved in various
areas of the United States, based on results obtained in Forsberg (1984).
Table 5. Smoothing of free-air gravity anomalies referring to a
reference field T of maximal degree 180, caused by the removal
of topographic ef?ects, TM. (T1 : = T - To).
Height (m) ~i (mgal) Anisotropy index
Area st.deviation based on based on based on
T1 T1- TM T1 T1- TM T1 T 1 - T ~
Colorado
37O< < 41'
-:09O< X

The table shows, that not cnly does the value of CO decrease in
mountaneous areas, but $, increases and the anisotropy index decreases.
The field becomes considerable much smoother in the sense we want and also
more isotropic. Error estimates computed using eq. (2.53) will then be-
come more reliable.
The result shown in Table 5 have been obtained using a constant den-
sity p, for the topographic masses. We may as mentioned in section 4.3
introduce a parameter pi for the different regions, expressing the vari-
able (mean) density. The gravity observation equation then is
where L (Ti) is the gravity computed from the i 'th region (or for various
A9
layers) using the density po, and TD=T - TM. The procedure for the deter-
mination of pi using eqJ2.58) corresponds very closely to the usual regres-
sion procedure used to determine the Bouguer density, see (Sunkel and
Kraiger, 1983).
5.3 Mixed collocation
We have not seen how it is possible to use topographic information
in a smoothing procedure. However, we may also use the topographic infor-
mation in order to modify the Hilbert spaces with a rotational invariant
norm, as discussed in Sanso' and Tscherning (1952)
The use of a rotational invariant norm in an area with varying topographic
heights causes some problems in practice. Since the norm is associated
with a Bjerhammar-sphere, then the radius of this sphere can maximally
be as large a RE + h
min
. ,where h,nin is the smallest height in the
area. Let us then consider an example, where we have only one gravity observation
and want to predict the height anomaly in the same point. Then
The function K( 3, ~ g will ) not change as much with the height as

does K(A~. Ag ), which decreases considerably for increasing height.
K(A~,w) = i - l z R 2i+2
(F ) (5.3)
i=2 ' 'I-
This means, that the same value of ~g will predict a larger value of
c whenwe are at high altitudes than at low altitudes, see Fig. 9 . In
fact, we should expect a nearly constant mean square variation of the pre-
dicted height anomalies, when we follow the Earths surface, maybe except
at the highest peaks.
1..
r-R (km)
0 1 2 3
Fig. 9. Height anomaly 5 predicted from a gravity anomaly
equal to 100 mgal in the same point for.varying height r-R
of the point above the Bjerhammar-sphere. The covariance
function model from (Tscherning, 1982) is used.

We can repair this by constructing a n2w Hilbert space of harmonic
functions, having a set of harmonicity nearly like T itself. We use the
direct sum of two Hilbert spaces, one having a rotational invariant norm
11 JJR and one given as the linear space spanned by the potential of mass
elements M. with constant mass p i, filling out (a part of) the space be-
1
tween the Bjerhammar-sphere and the Earth's surface. (The elements may
very well extend into the Sjerhammar-sphere, see Fig. 10).
Fig. 10. Filling out the space between the
Bjerhammar-sphere and the Earth's surface
with prisms, which very well may extend into
the sphere.
Sphere
We denote the potential of the mass elements by ii, and the volume
of the elements by vi, and define for TM = T, + ...... + Tn,
where ii is the indicator function for the i'th block. (Ii(?) = 1 when P
in the block and otherwise = 0). The reproducing kernel simply becomes
(p, = 1 used)

This kind of col location using an "external " and "internal" norm 1s
called mixed ccllocatlon. The "mixed" kernel will simply be
where K is the reproducing kernel for the space, HR, with the rctational
li
invariant norm. Hence,
If KR(P, Q) has degree-variances given by eq. (2.43), and if we fill
out the space between the Bjerhammar-sphere and the Earth's surface with
smaller and smaller blocks, then K will in the limit be equal to the kernel
Kg given by eq. (2.42). The Hilbert space, which has KD as a Kernel, has
the advantage that T is an element of the space.
Note, that the use of the density functional, for r < R is permitted,
if a covariance function model for density anomalies has been selected.
It must also be possible to compute density values associated with (pro-
ducing) a contingent reference field. Procedures for these purposes have
been developed in Tscherning (l977), Jordan (1978) and Tscherning and Sun-
kel (1981).
Unfortunately, the mixed collocation method has not yet been imple-
mented. Here it would be important to assure, that the density anomalies
implicitly determined for each block by the solution (5.7) is within rea-
sonable limits. This may e.g. be done by changing the norm in HR by a
scale factor until realistic values (hopefully) are found.

6. IMPLEMENTING THE LSC-METHOD ON A COMPUTER
lhe process of constructing an approximation 7 , the determination of
various parameters and the subsequent prediction and error estimation
phase may be divided in a number of steps. In most of these steps a com-
puter will be able to do the hard work. In fact, without the computer LSC
would probably never have been tried in practice.
The mosx time-consuming steps are these associated with the evaluation
of the contribution from a set of harmonic coefficients, L(To), the
computation of contingent terrain contributions L (TM), the evaluation of
the covariances K(Li, L. ), K(L,Li) and the solution of the normal equa-
J
tions.
Since the normal-equation coefficient matrix E = IK(Li, L.)+O. .) will
J 11
be positive definite (if no observations occur twice), the Cholesky's
method for reducing and solving the equations can be used. This method factorizes
t into the product of an upper triangular matrix, U, and ~ts transposed,
T
U ,
C
T
= U -U. (6.1)
The solution to
is found from
which gives
(uT)-' C a = (uT)-' X,
U is denoted the reduced matrix, and (6.4) the back-substitution.
The algorithm giving the elements of U is
which for i = j specializes into

The columns of U may be computed one after another. This means, that
if new columns are added to C (e.g. because of new observations are obtain-
ed), then the already reduced part need not to be recomputed. The algo-
rithm simply starts with the last unreduced column. This gives very good
restart capabilities, if for example the computer breaks down during the
execution of the algorithm.
The algorithm may be modified, so that it directly produces the so-
lutions to the equations (2.58) and (2.59). Regard the extended ?-matrix
equation
then
Hencewe will by using the algorithm get a wrong result. because the
quantity P has a wrong sign.This can easily be repaired, by changing the
minus sign in eq. (6.6) when the subscript k is smaller than n (the dimen-
sion of c ) , and equal to the numbe,- cf observations. Furthermore, if we add
(1) a new row witi? elements K(L,ii), i =l ,..., n, AL(j) , j =l ,..., m, (2) a
new column identical to the transposed of the new row and (3) a new diagonal
element equal to -K(L,i), the mojiiiea Cholesky algorithm will also deliver
the (negative of) the square of the error mL eq. (2.61) as the bottom dia-
gonal element of the reduced matrix. For further details see Tscherning
(1978b, App. l ).

Having clarified this important point, I will descrlbe how LSC have
been implemented and used for the computation of the Nordic quasi-geoid
(Tscherning, 1982, 1983, 1983a):
(1) Set goal for f, e.g. as good as posslble using all available data
or by fixing a value for G ( 5). This last possibility was chosen for the
Nordic geoid, where S (-2 - CO): = 0.5m. and CO is the height anomaly in a
central point in the area. (In this manner, errors in the reference system
parameters would not play a role). This goal may be reached safely if &g)
= 7mga1, corresponding to G (?, , ; ) = 1 ".
(2) Specify reference system parameters and the relationships of the systems
to an optimal geocentric system, by giving a-priori values of datumshift
parameters etc. Specify also which parameters must be updated/corrected.
For the Nordic geoid initially no parameters were updated. But
later it was found that the longitude origin had to be corrected.
(3) Specify set of potential coefficients (maximal degree m) to be used,
(To), and an error model for the coefficients to be used in eq. (3.13). For
the Nordic area, the R;pp (1978) model, m = 180, was used, since it was
fomd, that it agreed best with the data in the area as compared to other
sets of coefficients, see (Tscherning and Forsberg, 1982 ).
(4) Compute, from a small gravity data sample, CO and $i, for the whole
area. (The contribution from To must be subtracted from these data). For
the Nordic area CO = 982mga12 and $, = 7.3' was found.
(5) Determine preliminary estimate of needed mean data spacing, using eq.
(4.6). If the actual data spacing is smaller than required,then smoothing
using topographic data must be used. If the number of observations needed
is too large to permit an easy handling ~n the computer (depending on speed
and size of the fast memory), then the area must be subdivided in overlap-
ping blocks. If these blocks have an extend smaller than laO/m,the smaI1-
est wavelength represented in To, then an improved reference field To + Tl
should be constructed using mean values as described in section 4.1.

In the Nordic area, eq. (4.6) gives as resuit, that 166 gravity obse -
vations are needed per 1°x 1°/cosrp block. This amount of gravity data was
available nearly everywhere, see (Tscherning, 1983, Fig. l), or have subsequently
become available. It was decided to use nearly quadratic blocks
with side-length 2' and an overlap area of 4'.
(6) Subtract if needed the effect of TM. This has not yet been used, but r
will have to be used in the Nordic area in order to achieve the goal of
0.5malso in the high mountains of Norway and Sweden.
(7) Select preliminary covariance function, e.g. the model given by eq.
(3.13). This was not done for the Nordic geoid.
(8) Compute empirical covariance functions for ail areas, e.g. using predicted
gravity values in eq. (3.10). Select covariance function model for
each area based on the empirical values. For the Nordic area, eq. (3.10a)
2 2
was used giving values of CO ranging from 2578mgal to 142mgal and values
of between 3' and 16' were found, see (Tscherning, 1983, Table 3).
(9) Tabulize the covariance function for each block. This was done using
the method described in Sunkel (1979).
(10) Now one block must be regarded after the other (if more than one, ob-
viously). Compute from the actual value of CO and the needed gravity
data spacing, d, for the block. Extract from the data files values as close
as possible to a regular grid with mesh width d. Also other data types may
be used.
For the Nordic geoid the data selection was not done this way. In-
stead a data spacing of d = 3' was used in areas with moderatly varying
topography and d = 2' in mountainous regions. Deflections were used also.
(1 1 ) Compute coefficients of upper-triangular part of normal equation matrix
(K(Li, L.) l and parameter vector A . For the Nordic geoid the algol-
J
program system described in Tscherning (1978b) was used in this and the
following steps. An updated version of the FORTRAN program described in
Tscherning (1974) could also have been used.

(12) Reduce normaI equations, cf. eq. (6.5).
T
(13) Compute right-hand side xi - Li(To) - Ai Xo, where X contains the
0
apriori values of the parameter vector X.
(14) Solve the equations, cf. eq. (6.4) using the modified Cholesky's al-
gorithm. (ail and X are obtained. Add X to Xo.
(15) Predict ~ (7) in data points, which contingently have not been used
(by computing the sum of K(Li. L) multiplied with ai, add L(T,,,), and L(To)
and execute a datum transformation back to the system in which the observation
is given).
(16) Inspect the residuals, and select as additional data all for which
IL(T) - ~(7) - L(To) - L(T~)~ > 3 F(L). Verify that the differences are
not caused by large data errors. Then go back to (11) if any data are left.
During the computation of the Nordic geoid, this procedure lead to
the discovery of several large errors in the deflections of the vertical
in Denmark, Norway and North Cfrmany.
(17) If parameters are determined, which basically are related non-linea-
rily to the data and the absolute value of X is larger than a suitable
bound, then go back to (13). (The bound could e.g. be 0.01" for a datum
rotation component).
(18) Predict if possible other test values to check, whether the goal has
been reached. In the Nordic area, Doppler derived geoid heights and ap-
proximate geoid heights obtained from satellite altimetry were predicted.
The difference between predicted and observed Doppler-derived geoid heights
showed large differences, which sometimes had a systematic character de-
pending on the time of the observation. This made it impossible to us2
these observations, and it was later found that the differences were caused
by the varying solar activity, see (Tscherning and Goad, 1984). It was also
found, that the solution had an cast-west bias, which could be attributed
to a basic uncertainly in the longitude origin, see Tscherning (1983a). The

computations were repeated from step (ll), where the vector AL! sorrespon-
ding to a shift in the longitude origin was added to the reduced normal-
equations, which had been saved on magnetic tape. Step (12) then only in-
cluded the reduction of this new column.
(19) If the comparison shows, that the goal is not reached, then the only
solution is to smooth, i.e. return to (6) - or to make more observations!
For the Nordic area, the result in 3 blocks were not satisfactory, and to-
pographic effects need to be computed for these areas. (In one block, the
effect illustrated in Fig. 9 was probably causing a part of the error).
(20) Men more than one solution has been computed, then compare the so-
lutions at their common boundary. If the discrepancies are too large, the
overlap area must be extended, or predicted values from one solution used
as observations in the neighbouring solution, taking into account the
error-correlations between the values.
In this manner the discrepances between overlapping regions, which
initially amounted to up to 0.4m, were reduced to 0.2mfor the Nordic geoid.
The error correlations (but not the variances) were disregarded.
(21) Predict needed quantities and contingently their error estimates.
In some of the steps, program modules (FORTRAN Subroutines or Algol proce-
dures) have or may been used. Several of these modules have been published.
In Table 6 are listed the most versatile ones, i.e. COVA (Tscherning and
Rapp, 1974) which has been widely used is for this reason not listed in
the table, because it can not be used with torsion balance data, while
COVAX (Tscherning, 1976) is listed because it can handle such data types.
From the tab!e it Is seen that for nearly all of the 21 steps !isted
a published program or subroutine is available. A user may from these mo-
dules easi!y design a new LSC-program. However, in some cf the steps not
mentioned in Table 6, we would benefit from the assistance from the com-
puter. The selection of data associated with points as close as possible

to a regular grid is a difficult task even for a geodesist, but an easy
task for a computer. On the other hand such functions mav be an integra!
part of a gene~al data base management system, see e.g. Fury (1981),
Tscherning (197ac), Carozzo et al. (1982).
For those, who do not need the most advanced tools (such as the abi-
lity to use or predict gravity gradients) the FORTRAN program GEODETIC
COLLOCATION (Tscherning, 1973) or later versions may still be useful.
The program, or program modules, have recently been used successfully at
a number of places, see Benciolini et al. (19831, Arabelos (!980), and
liein and Landau (1983). who have developed an extremely versatile system.

Software hzs also been developed at several other places, such as
at ETH (Gurtner, 1983) and NGS (Goad et al., 1984). Also several program
systems exist for topographic reductions.
Table 6 Survey of the most versatile published software for LSC in Algol
(A) or FORTRAN (F).
Function Program (P) or Language Reference Used in step
Subroutine (S) F 4
Specify reference GEODETIC COLLO- X (1) (2)
system and datum CATION (MAIN)(?)
Datum transforma- ITRAN (5) X (1) (13)(15)(21)
t ion
Evaluate normal GRAVC, NORMAL (S) X (1)(3) (13)(15)(21)
field contribution dnpot X (9)
Specify and load STORECILOADCS (S) X (3) (3
potentialcoeffic.GEO0.COLL. (P) X (1
Evaluate potential GPOTDR (S) X (3 (13)(15)(21)
coeff. contribution
gpotdr (S) X (8)
Determine covari ance "Bjerhammar- X (7) (7) (8)
function param. A,R. sphere" (P)
Compute K(Li, Lj ) COVAX (S) X (2) (11)(15)(2i)
cova/b/c (S) X (6)
Tabulize K(Li, Lj ) COVNET (p) X (4) (9)
Reduce normal eq. NES (S) X (1) (12)
Solve reduced eq. NES (S) X ( 1) (14)
Predict (L(T)) ?RED (S! X (1) (15)(18)(21)
Estimate error mL PRED+NES (S) X (1) (21 )
Compute terrain TC (P) X (10) (6)(15)(21)
effects
References: (l
1 Tscherning (l974), (2) Tscherning (1976). (3) Tscherning
et al. (1983), (41 Sunkel (1979), (6) ischerning (1976a), (7) Tscherning
(1972). (8) Tscherning and Poder (1982), (9) Tscherning (1976b), (10) Fors-
berg (1984).

7. CONCLUSION
I have in the previous chapters tried to present LSC as a versatile,
powerful and easily implementable method. I have not - and will not -
claim that it is the E method. A comparison with other methods is pre-
sented in (Tscherning, 1981) and my conclusion there was that all methods
having a sufficiently solid theoretical basis would give comparable re-
sults. This has subsequently been confirmed to a large extend through the
comparison of various methods for gravity and deflection computation using
the same data in New Mexico, cf. Schwarz (1983). But more comparisons
should be made using other data types, such as heignt anomalies and gravi-
ty gradients.
Oneadvantage,which LSC has, as compared to many other methods,is the
ability to compute error estimates. At least the estimates are able to
show where no data were used.
The estimates may be used in feasibility studies, see e.g. Schwarz
(1981 ), and in the plann~ng of data collection campaigns, see Tscherning
(1975 a, 1983 a). But how good are the error estimates? Which percentage of
theerrors shouid we expect to be for example within the interval
-3 .mL( IL(T) - ~(i)l( 3 .mL,
where mL is the estimate eq. (2.51)? The estimates may be very bad, indeed,
as illustrated in Tscherning (1980).
Here we would have been able to benefit from probabilistic models,
such as these introduced in (Lauritzen, 1973) and further developed by
Moritz, see APG (sections 33 - 38). However, I have purposely avoided the
introduction of probabilistic concepts here, because I wanted to show that
collocation may be described cons~stently without such concepts. In fact,
the inability to justify the exploitation of properties of importance in
the theory of stochastic processes, such 2s .;??'iionarity, have been used
by some colleagues as an excuse for not h:ving to deal seriously with LSC.

The ease of using LSC will hopefully not lead to the situation as we
see with least squares adjustment, which sometimes is used uncritically.
Both methods will give answers to even inappropriate questions. In LSC the
danger of numerical instability does exist. Also despite much progress in
the treatment of the convergence problem for collocation, (Tscherning,
1978a) ,(Sans0 and Tscherning, 1980), (Krarup, 1981 ) , a completely satisfac-
tory solution has not yet been found.
However, the problems associated with the implementation of colloca-
tion seems to me to be most challenging. Will mixed collocation work? Nil1
we be able to estimate realistic mass densities? Is it possible to esti-
mate reliable gravity gradients? And does the covariance modelling proce-
dure (section 3.2) always give reasonable models for the quantities not
used for the modelling? Is the "law" of covariance propagation violated?
There are still many problems to be solved, but until now the efforts
have been worthwhile considering the practical results which have been ob-
tained (e.g. in Greenland) using collocation in situations where probably
no ordinary method would have worked.

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EXERCISES:
Orthonormalize the functions 1, X, x2 , with respect to the inner
product
J -1
fgdx =(f,g).
The three functions form the orthonormal base in a three-dimensional
Hilbert-space, H, equipped with this inner product. What is the re-
producing kernel of this space ?
Give the expression for the Riesz-representer of the following func-
tional~ in H*, the space dual to the space H in El :
L(f) = f(t), L(f) = f(O),
df
Find the elements in H (defined in El ), which have the minimum norm,
and which fulfil one of the following conditions:
(a) f(1) = 0, (b) f(O) = 1. (c) fl(0) = 1,
(d) f"(0) = 0, (e) f(0) = 1 and f(1) = -l .
The function g(x) = -2x + 1, is an element of H (defined in El).
Compute the norm of g and write down its Fourier expansion as an ele-
ment of H.
Suppose the function g in E4 has been approximated using the solution
to E3 (e). What is the upper limit for the error as computed using
eq.(2.34) ?

Let H(n) be the Hilbert space of functions harmonic outside a sphere
with radius R and regular at infinity equipped with the norm
1 1
[fir = K[ F(vf)z dn.
Find the reproducing kernel corresponding to this norm. (Hint: use
Green's first identity ).
Show that the degree-variances for the kernel
where no is a sphere with radius R are given by eq. (2.43).
Let now H be a Hilbert space of functions harmonic outside a sphere
with radius R and regular at infinity equipped with the norm
Compute the norm of the solid spherical harmonics V.. given by eq.(2.6).
1J
Uhat is the reproducing kernel of this space ?
In a Hilbert space of regular harmonic functions the reproducing kernel
i S
1 RZ it1
K(P3Q) = m Pi (COS&)
i =Z
Compute the norm of the a functional, eq.(2.9), for r = -/? R , R =
6370 km. What is the inner product of two ag functionals, where the
spherical distance between the points of evaluation is 45' or go0,
respectively and r = r'.
E10: f is an element of a RKHS with kernel K(P,Q). is the collocation
determined approximation to f using the errorf ree observations Li (f j .
i = I ,... ,n. Show that f is the "best" linear approximation to f com-
puted using the base functions K(P,Li), i = l, ..., n, in the sense
discussed in section 2.2.
Ell: The base functions K(P,Li) of E10 have a matrix of inner products gi-
ven by

I
with Cholesky decomposition C = U U, where U is an upper-triangular
matrix. Show that the vector of functions
form an orthonorrnal basis in the n-dimensional subspace.
Show that if we use the set of observations
then the collocation solutions for increasing sets of observations,
L, (f), Ln+2(f 1. etc., have the permanence property, namely that the
coefficients obtained in the n'th approximation do not change in the
following approximations.
E12: Two sea-surface heights (5, and 5, ) obtained by satellite radar
altimetry are treated as geoid helghts with the same bias X.
where nl is the error. Let us suppose that
and the variance of the noise is 0.01 mz . Suppose P = 0 .
Then

&rera!n? X?:? ci3~ X and (L., . b2 1.
For :he geoid height L9(:j ?n a fioint Q we heve
iomcute the predicted value LQ(T) and the mean square error of
prediction.
E 13: Show that the estimate of the error of predicilon may be computed
by performins the Cholesky reduction of a (n+l ) X (n+l ) matrix
where C is the n X n covariance matrix of the observations, Cp the
veczor c i covariances between the observations and the predicted
quaniity and CO is the variance of the quantity.
Show that the modified Cholesky reduction (5 6) in a s~milar manner
may be used in case a set of parameters have been estimated.
E 14: Use the fact that the covariance function of the form (2.39) fulfil
Lapiaces equation in both P and Q to derlre an equation relating the
rovarlance function of the gravity disrurbance and the covariance
functions of :he deflections of the vertical. Hint: the relationship
merely expresses the well-known differential equation for the Legendre-
polynomials Pn(t).