212 - School of Earth Sciences - The Ohio State University

Reports of the Department of Geodetic Science
Report No. 2 1 2
A FORTRAN IV PROGRAM FOR THE DETERMINATION
OF THE ANOMALOUS POTENTIAL
USING STEPWISE LEAST SQUARES COLLOCATION
by
C. C. Tscherning
The Ohio State Univer&y
Department of Geodefic Science
1958 Neil Avenue
Columbus, Ohio 4321 0
July, 1974

Reports of the Department of Geodetic Science
Report No. 212
A FORTRAN IV PROGRAM FOR THE DETERMINATION OF THE
ANOMALOUS POTENTIAL USING STEPWISE LEAST SQUARES COLLOCATION
by
C. C. Tscherning
The Ohio State University
Department of Geodetic Science
1958 Neil Avenue
Columbus, Ohio 43210
July, 1974

Abstract
The theory of sequential least squares collocation, as applied to the
determination of an approximation ?? to the anomalous potential of the Earth T,
and to the prediction and filtering of quantities related in a linear manner to T,
is developed.
The practical implementation of the theory in the form of a FORTRAN
IV program is presented, and detailed instructions for the use of this program
are given.
The program requires the specification of (1) a covariance function
of the gravity anomalies and (2) a set of observed quantities (with known stan-
dard deviations).
The covariance function is required to be isotropic. It is specified
by a set of empirical anomaly degree-variances all of degree less than or
equal to an integer I and by selecting the anomaly degree-variances of degree
greater than I according to one of three possible degree-variance models. The
observations may be potential coefficients, mean or point gravity anomalies,
height anomalies or deflections of the vertical. A filtering of the observations
will take place simultaneously with the determination of ?.
The program may be used for the prediction of height anomalies,
gravity anomalies and deflections of the vertical. Estimates of the standard
error of the predicted quantities may be obtained as well.
The observations may be given in a local geodetic reference system.
In this case parameters for a datum shift to a geocentric reference system
must be specified. The predictions will be given in both the local and the geo-
centric reference system.
N
T may be computed stepwise, i. e. the observations may be divided
in up to three groups. (The limit of three is only attained when potential
coefficients are observed, in which case these quantities will form the first set
of observations. ) Each set of observations will determine a harmonic function
and Twill be equal to the sum of these functions.
N
The function T determined by the program will be a (global or local)
solution to the problem of Bjerhammar, i. e. , it will be harmonic outside a
sphere enclosed in the Earth, and it will agree with the filtered observations.

Foreword
This report was prepared by C. C. Tscherning, Research Associate,
Department of Geodetic Science, and the Danish Geodetic Institute. This work
was sponsored, in part, by the Danish Geodetic Institute, and in part by the
Air Force Cambridge Research Laboratories, Bedford, Massachusetts, under
Air Force Contract No. F19628-72-C-0120, The Ohio State University Research
Foundation Project No. 3368B1, Project Supervisor, Richard H. Rapp. The
Air Force Contract Monitor is Mr. Bela Szabo and Dr. Donald H. Eckhardt is
Project Scientist.
Funds for the support of certain computations and program developments
described in this study were made available through the Northern European
University Computer Center, Lyngby, Denmark and through the Instruction and
Research Computer Center of The Ohio State University.
A travel grant was supplied by the NATO Senior Fellowship Scheme.
The reproduction and distribution of this report was carried out through
funds supplied by the Department of Geodetic Science. This report was also
distributed by the Air Force Cambridge Research Laboratories as document
AFCRL-TR-74-0391 (Scientific Report No. 17 under contract no. P19628-72-
c-0120).
The writer wishes to thank Dr. R. H. Rapp for valuable suggestions and
critic ism and Jannie Nielsen for relieving me of most of my household respon-
S ib ilities in the period when this report was being prepared.
iii

Table of Contents
1. Introduc tion
2. The Basic Equations
2.1 Least Squares Collocation
2.2 Equations for the Covariance of and Between Gravity
Anomalies, Deflections of the Vertical, Height Anom-
alis and Potential Coefficients.
2.3 Stepw ise Collocation
Data Requirements
3.1 Observed Quantities
3.2 The Reference Sys tems
3.3 The Covariance Function
4. Main lines of function of the IWRTRAN IV Program.
5. The Storage of the Normal Eq~~ations and the Function of the
Sub-Programs.
5.1 The S torage of the Normal Equations
5.2 Solution of the Normal Equations and Computation of the
Estimate of the Error of Prediction, Subroutine NES.
5.3 Transformation BetweenReference Systems, Subroutine
ITRAN.
5.4 Computation of the Normal Gravity, the Normal Potential
and the Contributions from the Potential Coefficients. Sub-
routines GRAVC and IGPOT.
5. 5 Coinpu ta tion of Euclidian Coordinates, Convers ion of
Angles to Radians, Subroutines EUCLID and RAD.
5.6 Subroutines for Output Management and Prediction Statis-
tics, HEAD, OUT, COI\IPA.
5.7 Subroutines for the Computation of Covariances, PRED
illld SUMK.
6. Input Spec ifieations
7. Output and Output Options
8. Recornmendat ions and Conclus ions.
References
Appendix
A. The FORTRAN program.
B. An input example.
C. An output example.
iv
Page
1
3

1. Introduction.
The theory of least squares collocation has been discussed extensively by
Kramp (1969), hloritz (1972, 1973), Lauritzen (1973), Grafarend (1973) and
Tscherning (1973). Collocation was originally introduced by Krarup (1969) as a
method for the deter~nination of the anomalous potential using different kinds of
observations. Primarily Moritz (1972) has extended the theory to a wider field.
This report will only consider the use of least squares collocation for the determination
of the anomalous potential, T and the estimation of quantities dependent
on T.
We will regard the problem of the determination of T as being equivalent
to the solution of the Bjerhammar-problem, i. e. the determination of a function,
harmonic outside a sphere totally enclosed in the Earth and regular at infinity,
which agrees with observed values of e. g. gravity anomalies and deflections of
the vertical.
It is not required, that the observations and the solution agree exactly.
The observations will contain a certain amount of "noise", the magnitude of which
is specified by an estimatecl stmckird error.
Least squares collocation will filter out some of this noise, and the
solution will agree cxactly with t,he filtered observations.
N
Having determined a solution, T, to the Bjerhammar problem, this
function can naturally he used to compute geoid heights (or more correctly,
height anomalies), gravity anomalies or deflections of the vertical in points in
the set of harmonicity. Hence, by solving the Bjerhammar problem, we have
implicitly also solved e. g. the problems of interpolation or extrapolation (pre-
diction) of gravity anomalies or deflections and the problem of astrogeodetic or
astrogravimetric geoid computation.
An Algol-program, which used this approach was published in Tscherning
(1972). The program could only handle a very limited amount of observations.
In the FORTRAN IV program presented in this report, we have taken advantage
of the availability of a computer (IBhI System 370), which has large core storage
and fast peripherial units (disks), so that very large amounts of data can be
treated. Thus, the use of FORTRAN IV, which does not have variable dimension-
ing of amays, has required that certain (arbitrary) limits have been put on e.g.
the number of observations, which the program can handle.

h section 2 we will present the basic equations of least squares collocation as
applied to the Bjerhammar problem. We will also discuss the method of stepwise
collocation, which differs solnewhat from the p ocedure described by
Moritz (1973). All the observed quantities must be in the same reference system.
This requirement is discussed in section 3. The main lines of function of the
FORTRAN IV program is described in section 4. The most important details
are given in the following section, which especially discusses the subprograms
used. Input and output options are described in sections G and 7 respectively,
and the final section 8 contains some recommendations and conclusions. The
FORTRAN program, an input and an output example are contained in an appendix.
r
References are given by author name and year, with one exception:
Heiskanen and Moritz, Physical Geodesy, will be referenced only by PG,
because references to this book occur frequently.

2. The Basic Equations
There are two ways of approach to least squares collocation. A mathe-
matical (functional analytic) and a statistical. The mathematical approach is the
most well founded and without dark spots. But its appreciation requires a math-
matical background, which not yet is common among geodesists. The statistical
approach is with a first glance less difficult and gives a sufficient insight. This
means, that a geodesist, well educated in the theory and application of least
squares adjustmcnt, will be able to use the method.
We will, without hereby having questioned the intellectual ability of the
reader, use the statistical approach in the following presentation of the basic
equations of least squares collocation.
2.1 Least sauares collocation.
Let us suppose, that T is an element of a sample space H of functions
harmonic outside a sphere totally enclosed in the Earth. We will denote the
probability measure of H by @. Let the random variables Yp be the mappings,
which relate a function in H to the value of the function in the point P, i.e.
Yp (T) = T(P). The variables Yp will then form a stochastic process with the
set of harnzonicity as index set (provided P fulfills some basic requirements see
e. g. Grafarend (1973)). The covariance between two random variables Yp , YQ
will be denoted cov(Tp , TQ ) , because Yp(T) = T(P) and YQ (T) = T(Q). It is equal
to:
We will require, that the variance cov(Tp , Tq) is finite.
Example. Following Meissl (1971), the probability measure 9 may be
defined by specifying the distribution of the random variables, Y, , which maps
m
T into its coefficient v& in a development of T as a series in solid spherical har-
monics. Let us suppose that this random variable has a Gaussian distribution
with mean value zero and variance oe(T, ~)/m only depending of the degree
4, we will then have

where $ is the spherical distance between P and Q, r and r' the distance of P and
Q from the origin, R the radius of the (l3jerhammar)-sphere bounding the set of
harmonicity and Pi (cos Jr) the Legendre polynomial of degree 1.
The constants ol(T, T) are called the (potential) degree variances. The CO-
variance function will be isotropic, i. e. invariant with respect to rotations of
the pair of points P and Q around the origin*
From the random variables Yp we may form a second order stochastic
field. This field consists of all random variables, which are linear combinations
or limits of linear combinations of a finite number of these random variables and
which have finite variance, (cf. e.g. Parzen (1967), page 260).
Their covariances can all be derived from the covariance (1). Let us
regard
Then
And generally for quantities si and s , where
we have
(We have here implicitly presupposed, that cov (Tp, SJ) regarded as a function of
P and Yj (cov(Tp, TQ )) regarded as a function of Q, are elements of the sample
space H, i. e. that they are harmonic. This will be proved below).
The equation (3) is the so called law of propagation of covariances,
Moritz (1972, page 97).

We will denote
(4) clj = cov(s,, sj), C = {clj ] a qxq matrix,
(5) cp1 = cov(Tp , si), Cp= {cP1 ] a q-vector and
(We will below use subscripted quantities in brackets, { ] to denote vectors or
matrices. In case the limit(s) of the subscript(s) are not obvious, the upper
limit(s) will be indicated by subscripts, i, e. C = {cil l,,, ).
We will have to regard one more kind of random quantities (independent
of the above discussed), namely the random noise, n. A random variable will
be associated with each of the random varial~les Ys. They are all supposed to be
Gaussian distributed with mean value zero, known variance (denoted 0:) and uncorrelated.
The covariance matrix, which hence is a diagonal matrix, will be
2
denoted D= {dlj 3, dil = OS
Following Moritz (1972), the basic equation of llobservationfl is
where X is the measurement or observation, S' the corresponding l1signalU and n
is the noise. X, S' and n are q-vectors, where q is the number of observations.
The n-vector X comprises n parameters, and A is a known qx m matrix.
Let us now assume, that we want to estimate the outcome S of a stochastic
variable Ys , given a set of observed quantities X, Denoting C = C+D we obtain from
Moritz (1972, eq. (2-38) and (2-35))
where the superscript T means transposition.
The corresponding estimate of the error of estimation m?(of S) and Exx
(of
X) are, cf. Moritz (1972, eq. (3-38) and (3-33))

T-1
with h, = C. C and C,. is the variance of S.
The program presented in this report can only handle the non-parametric
(i.e. X= 0) case. But the general equations are presented here, so that we later
on can point out the main changes, which will have to be made in order to
incorporate the parameters X.
The special case we will consider here can then be described by the follow-
ing equations :
c:.
T --l
(12) s = C. C x and
T
The filtered observations E'are obtained from (12) by substituting C for
The equations (12) and (13) differ from the equations given by PG(eq. (7-63)
and (7-64)) only in that C has been substituted for C and that we are not restricted to
consider only gravity anomalies.
The quantities we want to consider here are potential coefficients, gravity
anomalies, deflections of the vertical and height anomalies. They are all (at
least in spherical approximation) expressible as either linear combinations or
limits of linear combinations of values of the anomalous potential. We will pre-
suppose, that the variances of the corresponding stochastic variables all are
finite. Equation (3) is hence valid for these kinds of quantities.
The value of the Laplace operator A, applied on T and evaluated in a point
P in the set of harmonicity,
is related to a stoclmstic variable, YATp. This variable will also belong to the
stochastic field (variance zero) and we will have for an arbitrary stochastic
variable Y, :

Hence, the covariance between a quantity S and the value of the anomalous
potential in P is a harmonic function (regarded as a function of P).
Let us now assume, that we want to estimat the value of T in a point P
from a set of observations X= {X, 1, i=l,. . . ,q. We then have from (12),
Introduc ting the solution vector
we have
N
(17) T (P) = C: b = [cov(~, , S, )IT [b, } = 1 cov (Tp, S, )bl
Using (11) we see that
Q
i. e. ?? (P) is a harmonic function.
By also requiring, that the functions in the sample space H are regular at
infinity, it can be shown, that ?(P) is regular at infinity as well.
We have then obtained a solution to the problem of Bjerhammar, if we
2
can prove, that the g, = S, = X, for us, (or d,,) = 0. But this is easily seen, '
because
a
This fact makes available an easy test of a collocation program. The used obser-
vations are predicted and it is checked, that the predictions agree with the obser-
ved values (and that the estimates of the error of prediction are zero).
9
i= l

2.2 Equations for the covariances of and between gravity anomalies,
deflections of the vertical, height anomalies and potential coefficients.
The relation (3) between the signal and the anolnalous potential has been
given in Tscherning and Rapp (1974), eq. (30)-(33)) in spherical approxima~ion
for gravity anomalies, height anomalies and deflections of the vertical. We
have for the height anomaly in P
the latitude component of the deflection of the vertical
the longitude component of the deflection of the vertical
the point (free-air) gravity anomaly
and the mean (free-air) gravity anomaly
where r is the distance froin the origin, rp the latitude, X the longitude, y the
reference gravity and A the area over which the mean gravity value is computed.
We may, as explained in Tscherning and Rapp (1974, section 10) repre-
sent mean gravity anomalies by point anomalies in a certain height above the
center of the area A. For this reason we will not in the following distinguish be-
tween mean and point gravity anomalies. The program is able to use all the
quantities (20)-(24) as observed quantities for the computation of ?. The same
kind of quantities may also be predicted by the program.
One more kind of quantities, potential coefficients, can be used, though
only as observed quantities. The given coefficients will generally be the
coefficients of the potential of the Earth, W, expanded in spherical harmonics
and not the coefficients of the ano~nalous potential. Denoting the normal potential
by U we have

and for W and U expanded in fully normalized spherical harmonics
Wa
+-
2
(r*sin~)~ and
W
kM al J24 - w2
(26) U(P) = ( 1 - )C () J = = ~
a= l
P,Q(COS a))+, (r sin Q)',
where w is the speed of rotation of the Earth, 6 = 90' - 9, kM the product of the
gravitational constant and the total mass of the Earth, the coefficients and
-
Cam the potential coefficients, and X(cos8) an associated Legendre polynomial,
normalized so that
The coefficients J8a in (26) are given in PG, (eq. (2-92)).
For the potential coefficients we then have the following equation
where wis the surface of the sphere with radius equal to the semi major axis a
and with center in the origin. (We are now denoting two quantities by W , but since
they are used in a different context, we hope, that no confusion is caused).
In the program it is possible to use one of three different kinds of
(isotropic) covariance functions, which we below will distinguish by a subscript k,
k = 1, 2 or 3. They are all specified by a so called anomaly degree-variance
model, i.e. by the coefficients a ,,~ (Ag, Ag) of degree A greater than a constant I
of the covariance function of the gravity anomalies developed in a Legendre
series:

where r' is the distance of Q from the origin, R the radius of the Bjerhammar
A
sphere, Jr = the spherical distance between P and Q and a~ (Ag, Ag) are empirically
determined coefficients.
The three different kinds of covariance functions correspond to three of
the five anomaly degree-variance models discussed in Tscherning and Rapp
(1974, section 8). The models are for l

Denoting
we will have
We rearrange (33):
From this covariance function all the other covariance functions can be derived
using the "law of propagation of covariances", eq. (3) and the equations (20)-
(24) and (28) relating the observed quantities to the anomalous potential.
Due to the linear relationship (39) we generally have for two arbitrary
randoin variables Y and Yj
where S, = Yi (T) and sJ = YJ (T).
For either si equal to Agp or

For the other part, cov; (si , sJ) we have (from Tscherning and Rapp 11974
equations (145)-(150) and (50)))
with
Covariance functions, cov(si, sj ), where either si or sj is a deflection
component can not explicitly be found in Tscllerning and Rapp (1974). But using
the equations (20)-(24) we get (K = K(P, Q) = cov, (Tp Tq )):

Applying these equations on cov, (Tp, TQ), (39) shows, that the quantities we need
to determine are (apart from the derivatives of t with respect to the latitude and
longitude) :
The last term in each of the equations (51)-(53) are identical to Tscherning and
Rapp (1974, eq. (108)-(110), (118)-(120) and (133)-(135)) for k= 1,2,3.
For the first term in each of the three equations we have, using (41), (42)
and (43):
I

with pi(t) = DtP (t) and ~,"(t.) = D:P,(~).
A a
The sums (54) - (56) are evaluated in the program (subroutine PRED) using
the recu sion algodhm given in Tscherning and Rapp (1974, section 9).
To avoid numerical prohleins for P near to Q, the following expressions
are used for the evaluation of t and the derivatives of t with respect to and cc':
(denoting: dX=X-X/a~~dd~=co-~'):
I
(57) t=cos$=sina0sir,~ +cos~*cosco'~cos(d~)
=cos (dq) - cos cp ' cos * (1 - COS (dX))
= 1 - 2 (S in 2 (&o/2)+ cos cos c; sin 2 (dA/2))
(58) Dpt =cos'p * sin cpl- sin rp COB COS (dX) = -sin(dv)+Zcos q'* sin9 sin 2 (d~/2),
(59) Dmtt = sin(dcp)+2cosrp singis in 2 (dh/2) and
(60)
D' ~t =cos cp * cos cpt+sh~*shv'* cos(d~)xos(d~~)-(l-cos(dX)) *sins* sinv'
cpcp
r
We will now introduce a compact notation for the normalized surface
harnlonics, which will facilitate the presentation of the covariances between the
coefficients of T developed in spherical harmonics and other quantities. Denoting

and the coefficients of T developed in spherical harmonics by vAE we have
where s =a in this integration and P is on the surface of the sphere of radius a.
From equations (28) we get
(2 1+1)' for m = 0 and Aeven,
(63) vAm = a * kM* for m> 0,
1
for m < band m = 0 and J! uneven
We will now compute the covariances covk(vR,, S!) where S, is either v,!,
!Q, ?')Q , A&, or CQ . These covariances are not exylicitly used in the program,
so we will not distinguish between the different covariance models, but denote
the degree-variances by uQ(T, T), uA(T,Ag) and uA(Ag, Ag).
From eq. (62) and (33) we get
Using the well lcnown summation formulae, (PG(~-81'))
and the orthogonality propcrty of the surface harmonics, we get

, 1
v, (0: A') dw
Again using (62) and the orthogonality property we see, that
-
Ra a+1
%(T,T)(~) forl=A andj=m
otherwise.
Thus, the covariance of two different anomalous potential coefficients is zero
and their variance is equal to the degree-variance multiplied by a constant
depending on R, a and the degree A.
The other covariance functions can be derived using (66) and (20)-(23):
and

2.3 Stcpwise collocation.
The solution of the normal equations (8) and (9) may be a difficult numeri-
cal task even when using a large computer, when the number of unknowils is
greater than a few thousands.
Moritz (1973) uses the term sequential collocation when the observations
are divided in two or more groups and when the corresponding normal equations
then are solved by inverting only the submatrices containing tlae covariances
between the observations within one group.
Let us first consider an example, where the observations have been
divided in two groups containing m, and m, observations respectively:
where X, is a m, vector and X, a m, vector of observations. The covariance
matrix is then divided in four subma1;rices accordingly:
and the vector of the covariances between the quantity S to be predicted and the
observations becomes
Hence, according to Moritz (1973, eq. (1-22)) we have
Let us regard the case where we want to estimate T(P). Denoting

we see that
(80) bg = cll~~~dlx, and
N
Hence, we are by using stepwise collocation, getting an estimate T which is
equal to the sum of two other estimates.
The first estimate ?,(P) is computed by (76) and (77) from the observations
X, using the original covariance functions. The residual observations d, % (78) is
then computed. Then the second estimate ?,(P) can be obtained from (80) and (81)
using the covariances of the residual observations, dlCZ2 and dlCp, (79). (Supposing
the "noise" matrix D to be zero we easily see
which is nothing but the i, jth element of the matrix dlC2,. )
The formulae (76)-(81) .can be generalized as to describe a partition of the
observations into more than two groups. Such equations can for example, .be
found in (1973, eq. (5-1)-(5-14)). We will here use a slightly different
type of general equation.
and with
For a partition in k groups,

we have the estimates (P) and based on the residual observations dI- ,xi
(Le. on all sets of observations with subscript less than i):
and the final es tinmtes

(As usual, the linear equations corrcsponding to (80a) are denoted the normal-
equations and b, the solution to the norinal equations. j
One of the main advantages achieved by using stepwise collocation is
according to Moritz (1973, page 11, that the normal equation matrices to be in-
verted are smaller than the original c matrix. Thus, as may be realized from
equations (79) and (go), the total storage requireinents are not diminished. So,
when using a computer, which has peripheral storage units with fast access,
stepwisc solution of the nornlal equations is of no real advantage.
Thus, a cons iderable simpiification of the computations may be achieved
when the residual covariances ~ J C l and d jCpi (79a) can be computed analytically.
In this case, only the bl vectors (80a) are needed for the representation of and
for the computation of p edictions. The residual covariance inay naturally be
computed analytically, when the datasets are ~ncor~elated, i. e. when diC = CIi .
But the possibility for analytical covariance also exists, when the first dataset
x1 consists of potential coefficients.
The matrix Cl, is in this case a diagonal matrix with diagonal elements
equal to the sum of the q~~antities given in (67) and the error variances of the
observed coefficient, o&. We will suppose, that all the variances are the same
for the same degree, R, and denote this variance by o:(R).
The elements of the vector b, are equal to the observed coefficient divided
by the corresponding diagonal element. Let us then suppose, that potential
coefficients up to degree I have been observed. The estimate 5 is then (cf. eq. (66),
(67) and (77)):
r

Using equations (6G), (67)-(79) and (85b) we easily see, that the residual observations
(78), d,x, are nothing but the original
,-., observations, but now referring to a
higher order reference field, U1 = U + T1.
For the i, j'th element of dlCzz we get from (GG) and (79), supposing e.g.
that S,, = T(p), S,, = T(Q) and that G ~, xz, are the corresponding observed values
(and P different from Q):
with
This quantity is zero for o?(~) equal to zero. The covariance function (84) is
in this case a local I'th order covariance function, cf. Tscherning and Rapp
(1974, Section 9).
It is supposed in the program, that the error variances a: can be either
a,
disregarded or that they only depend on the degree. The program will, in the
latter case, require that the quantity

(86) dluA (Q, Ag) = dlo (T, l') B
&- -- q2
a X"
is specified. The quantity will be treated as if it was an empirical degree-variance.
We have here seen, how we in one case explicitly can derive expressions
for the covariance, covjd,~, , c'r,~,~ ). Anolher method would he simply to estimate
a covariance fmc tion for the rcs idual. observations d1 x, . However, the program can
only use three types of covariance fmct ions, and they are all isotropic.
We are then restricted either to divide the observations x in groups of
quantities, which are nearly uncorrela.ted or to find a kind of observations, which
we can treat like the potential coefficients.
The potential coefficients are (in a general sense) weighted mean values of
the anolnalous potential. Mean gravity anomalies are also mean values, weighted
with a function, which is equal to one in the considered area and zero outside.
A mean gravity znolnaly fieid will represent an amount of information which
is elal to the znlount of information contained in a set of potential coefficients 01
degree less than or equal to an integer I. The magnitude of I will depend on the
size of the area over which the mean anomaly is computed. 1 will be large mhen
the area is small and small when the area is large. ( I will be zero when the
area is the whole Earth and infinite when we are dealing with points). An estimate
of the degree may be hund in the following way: The total number of equal area
mean anomalies of a particula size is theoretica!ly equal to the total area of the
Earth divided by the area of the basic mean anomaly. Let us call this number N.
For the,perfect recovery of N r quantities we need a set of coefficients of degree
up to N5. This method of estimation will give us NZ 202 for l0 equal area anomalies.
The degree may also be estimated in a more empirical way. This can be
done by first estimating the empirical covariance function of a set of residual observations
(gravity anomzlies) cl,%. The first zero point of the empirical. covariance
ft~nction (regarded as a function of the spherical distance $) will then
give a reasonable estimate of the degree (cf. Tscherniag and Rapp, (1974, Section
9))s
As an example, the program described in this report was used to compute
residual point gravity ano~nalics in a 2" 30 x 3" 40' square in the state of Ohio,
U. S.A. The data set x did consist of three groups. The set X, was a set of po-
tential coefficients of degree upto and inclusive of 20, given by Rapp (1973, Table
G). X, consisted of 157 1" x 1' mean gravity znomalies surrounding the area and
r;, was a set of 420 pojllt gravity anomalies, spaced zs uniform as possible with

a distance of 7%' in latitude and 10' in longitude between the points. The data-
sets X, and X, was regarded as errorless.
The covariance function recommended by Tscherning and Rapp (1974)
was used (i.e. given by eq. (32) with B = 24 and A, = 425 mga1 2 ). The covariances
d1CeiaJ can then be computed using a corresponding local 201th order covariance
function (i. e. with degree-variances of order up to and inclusive of 20 equal to
zero), cf. (84).
The function ??, is then computed without actually solving any normal
equations. We have (cf. (83a) and with of( R) = 0):
We will now, as mentioned above represent the mean gravity anomaly as a point
anomaly in a certain height 11' above the center of the area. Let us denote this
point by Q and its distance from the origin r'. (We have in this case used
hf=rf - R, = 10.5 ltm, cf. Tscherning and Rapp (1974, Section 10)).
The residual anomaly is then, cf. eq. (78a) and (71):

Figure 1: Empi?-ically determined covariance
values of res ikml point gravity anomalies compa
ed with the local 205'th order covariance
function,page 25,for varying spl-lerical distance $.
+ empirically determined values
- graph of the local covariance function
r

i. e. , the mean anomaly computed with respect to the higher order reference
field UT= US Tl. (The program does not use this equation for the computation of
the res idual anomaly. The contribution from is evaluated us ing the actual
length of the gradient of U1 , see the description of the subroutine IGPOT, Section
5.4).
N
The residual obsclvations dlx2 was then used to determine T, , this
time by actually solving a set of normal equations, obtaining the solution vector
b2*
The residual point gravity anomalies were then computed by
d,x,=x,- d,~:, b, - * bl
T
cf. (78a). The term CL? bl is again here the change due to the higher order
reference field U1 .
The empirical covariance hnc tion was computed using d, x3 by taking the
sample mean of the products of the residuals sampled according to the spherical
distance between the points of observation. The size of the sample interval m7as
1
72'. The covariance function
with R/R, = 0.9998 was found to have the same zero point as the empirical covariance
function, see Figure 1.
We then see, that the two mentioned methods give nearly the same es-timate
of the integer L This agreeinent should merely be taken as an illustration and not as
a proof. It shows one of the inany kinds of coinputaticns the FORTRAN program can
perform.
The choice of a proper covarinnce function is a delicate task, but we point
out that the presented program may use three different degree-varianc e models
and hence be usefd in test computations using different covariance functions.
2 5

-.
The program may compute Tin up to three steps. The different possibil-
ities are illustrated in Table 1. Note, that potential coefficjents always form a
separate data set, which will be the data set X,.
Number
of steps
Table 1
rU
Different: options for the Computation of T.
Dataset May Contain:
1 xa I X3

3, Data Requirements.
In this section the data requirements will be discussed. The precise
specifications are given in Section 6.
Three types of information are needed for the determination of T: obser-
vations, information about the reference system of the observations and a co-
variance function.
The observed quantities we want to use are (a) potential coefficients,
(b) point or mean free air gravity anomalies or measured gravity values,
(c) height anomalies and (d) deflections of the vertical.
The potential coefficients available will generally all be of a degree less
than 25. There has then only been reserved space in core store for up to 625
coeflicients.
The program accepts potential coefficients, which are fully normalized
and multiplied by 106. The coefficients can naturally only be used, when a value
of kM and ihe semi major zxis a art! specified.
An observation (different from a potential coefficient) will be given by
(1) the geodetic latitude and longitude, (2) a potential difference, (a geopotential
number, for example), conve ted into a metric quantity e. g. by dividing the dif-
ference with the reference gravity and (3) the measured quantity. The height
above the reference ellipsoid is regarded as unknown except, naturally, when
it itself is the observed quantity.
A11 measured gravity values will have to be given in the same gravity
reference system or a correction must be known. Measured gravity values are
converted to free-air anomalies. The orthometric height must hence be known.
The geodetic latitude (which is used to evaluate the normal gravity) would princi-
pally have to be given in a geodetic reference system consistent with the gravity
reference system (i.e. with the same flattening and semi major axis as used for
the computation of the coeffic icnts in the e'xpression for the normal gravity). But
the variation of the normal gravity with respect to the latitude is so small, that
this requirement can be neglected here. The point or mcan gravity anomalies will
all have to be free-air anomalies. They must all refer to the same normal gravity
field. LE they are not all given with respect to the same gravity base reference
system, the correction to be applied for the conversion must be known.

A mean gravity anoinaly will be represented as a point gravity anomaly at
a point of a certain height, h, above the ccnter of the area over which the mean
value is computed. This height is specified by the ratio, RP between the sum of
this height and the mean Earth radius R, and the mean Earth radius, i.e.
A height anomaly will have to be given in the same reference system as the
geodetic latik~de and longitude. This will generally require, that a height anomaly
obtained through an absolute position determination and given in a geocentric reference
system must be transferred back to a local geodetic refe ence system,
before it can be used in the program.
We are, with observed deflections of the vertical, faced with a complicated
problem. The deflections are equal to the difference between the astronomical
coordinates of a point on the geoid and the geodetic coordinates of a point on the
reference ellipsoid (multiplied with cosine to the latitude for the longitude difference).
We have herehy implicitly int~oduced assun~ptions about the mass densities
in between the geoid and the astronomical station. To avoid this, the deflections
should have been given at the proper height (i.e. the height of the observation
stations).
Thus, heights of astronomical stations are seldoin found recorded together
with the deflections. But if the heights are actually recorded, the program will
treat the deflections as quantities, which have not been reduced to the geoid.
The astronomical coordinates may carry systematical errors due to systematic
differences between star catalogues or due to the neglect of corrections
for polar motion. The observations may be corrected for ltnown systematical
errors, if they can be specified in the same way as a datum shift, i. e. by specifyr
ing the corrections in the latitude and the longitude com~onents at a certain point.
Systematic errors in the height anornalies may be corrected in the same manner.
In Section 2.2 we mentioned, that the equations which related the obser-
vations and the anomalous potential was given in spherical approximation. This
means, e.g. that all points onthe surface of the Earth are regarded as lying on
the mean Earth sphere. This fact has been used in the program to speed up the
computations. This is done by using the fact that the quantities

will be the same for a group of input data.
Data which actually are observed above the surface of the Earth must be
grouped so that they all refer to a sphere with radius equal to a mean height of
the points plus the mean Earth radius, R,. As for mean gravity anomalies, the
height is specified in the program through the value of the quantity RP, (ey. (87)).
The standard deviations of observations, different from potential coef-
ficients,will have to be given in meters for the height anomaly, in mgal for gravity
observations and in arc sec for deflection components. The standard deviations
may be specified (1) individually for the single observations, (2) for a group of
observations or (3) as being zero for all observations.
3.2 The Reference Systems.
We have to know the parameters specifying the geodetic coordinate system.
The program requires the semi major axis, a, and the flattening, f, to be specified.
The gravity formulae may then either be given (1) through the values of kX4,
U, a, and f, (2) by specifying that the international gravity formulae and the Pots-
clgm referencc system has bccn used or (3) that the Szcdetic Refererce Systerr,
1967 has been used. One of the three excludes the others.
The covariance functions which can be used, will all have the degree-
variances of degree .zero and one equal to zero. This implies, that we, in the
computations, have to use the best possible kM value and a geodetic coordinate
system which has origin coinciding with the gravity center of the Earth, Z-axis
parallel to the mean axis of rotation and Z-X-plane equal to the mean Greenwich
meridian plane.
We will also require the global mean value of the gravity anomalies, the
height anomalies and the deflections to be zero. This requirement implies, that
we have to use the best possible semi-major axis. The geodetic latitude and
longitude may then be transformed into such a reference system by specifying the
new kM, a, f values, the translation vector, the scale change and the three rota-
tion angles for the rotations around the X, Y and Z axes respectively.
-
The approximation T will be given in the same reference system as the one
specified through the transformat ion parameters, i. e. in a geocentric reference
sys tem. Predictions will be given in both the original and the new reference
sys tem .

3.3 The Covariance Function.
We explained in Section 2.2 how an isotropic covariance function can be
specified through (1) a set of empirica.1 anonia7.y degree-variances of degree less
than or equal to an integer I, bl(ag, Ag) and (2) an anomaly degree-variance model
for the degree-variances o,,~ (Ag, Ag) for R greater than I.
The values of the empirical degree-variances will depend on the radius of
the Bjerhanlmar sphere, R. We have, therefore chosen to specify these quantities
on the surface of the mean Earth9 i. e. the quantities
must be given together with the ratio R/R,. The quantities (88) must be given in
units of mga12.
The anomaly degree-variance model is for k= 1 and 2 specified through
the constants A, and A, (eq. (30) and (31)) and for k= 3 through the constant A3
and the integer B (eq. (32)).
Thus, in the program the models are specified not through the constants
A,, k= 1,2 or 3, but through the variance of the point gravity anomalies on the
surface of the Earth.
(29):
This qualltity is then used for the determination of A,. We have from
Let us now, for example regard model 2, i. e. a2,p, (Ag, Ag) = A,@- 1) . Then
(1- 2)

I 2 $+2
h
(90) A,= (cov(Ag~ , Agp ) - 7 4 (Ag, Ag)
!A
A=O
The infinite sum may be computed by the formula given in Tscherning and Rapp
(1974, Section 8), and A, (and in the same way A, or A,) can then be found.
We will finally mention, that the ratio R/R, is used by the program for the
computation of the radius of the Bjerhammar sphere.

4. Main Lines of Function of the Program.
In Section 2 we mentioned that the program could be used to estimate ?
N
from maxiinally three sets of observations X, , x, and X,. T would then be equal
N N
to the sum of up to three harmonic functions, T, , T, and ?, . The limit of three
was only attained, when potential coefficients formed the first set of observations,
X1 *
We will here clescribe the function of the program, when we are in this
situation, i.e. when we have three datasets X,, x, and x3, and X, is a set of
potential coeffic ients .
The flow of the program is illustrated in Figure 2. Several logical vari-
ables determine the flow. The logical variable LPRED will e. g. be "false" until
the estimation of? is finished and will have the value fltrue", when predictions
are computed.
The program will start by initializingdiffermt variables. It will require
illformation about the reference systems of the observations and use this infor-
mation to select e. g. the proper formulae for the normal g~avity.
When the reference sys tem is not geocentric or when the normal gravity
does not correspond to a proper kM value, the necessary transformation elements
and the kM value must be given.
The next step is then to read in the observations X,, the potential coef-
ficients. The normal equations (12) will not have to be solved in this situation,
cf. Section 2.3. ?, is represented by (83).
The following two steps, where we explicitly use the equations for collocation,
will be denoted Collocation I and Collocation 11. We will first have to specify the
covariance function and observations used in Collocation I:
The covariance hnc tion for the residual anomalies d, X, must be specified
through the selection of an anomaly degree -var ianc e model and contingently by
specifying a set of low order empirical. deg ee-variances.
The observations X, (and later X,) may be subdivided in different files
according to format, kind of observed quantity etc. Each single observation is
first transformed to a geocentric reference system (if necessary). Then the
residual observation is computed, by subtracting the contribution from Tl from
the observed value. After the input of a file, the value of a local variable LSTOP
will be input, which will signify if more files belonging to X, will have to be input.

Flow-Chart of Program.
Legend:
T = true
F= false
Figure 2
The main flow is determined by the values of the following logical variables:
LTRAN - coordinates and observations must be transformed to a geocentric
reference system and gravity observations must refer to a gravity
formulae consistent with the refe ence sys tem.
LPOT = potential coefficients from first set of I1observed" quantities.
LCREF= second set of observations (or third when LPOT is true) will be
used, and the harmonic function computed by Collocation I will be
r
used as an improved reference field. LCREF is initialized to be
false and will get its final value after Collocation I is finished.
LPRED = predictions are being computed.
LGRID = the predictions are computed in the points of a uniform grid.
LERNO = the error of prediction inus t be computed.
LCOMP= compare observed and predicted values (an observed value is input
together with the coordinates of the point of prediction).
\input operation /
/butput operation\
reference system of observations
I
\input LTRAN, LPOT]
~~~~F
arame ters
I T
ut potential coefficients/
I
SpecjIy kind of degree-
ariance model to be used
1

Figure 2
(cont'd)
~pecl~6C"i~r.s in the
elected
\ Specify kind an1 formkt of file
of obsei-vations or file of coor
-.
1
F&~R ID)
+T
[-{reference svs tern I
I~onqmte contribution from reference ]
V-
],F k l f i e l d given by potential coefficients I
A
ut u so utions
1~ {&/I and the residual observations when I l
ution from Collocation I ancl
1
Pu tput p ediction l
Y
rid or file ?

After the last file has been input, the vector d, ,u, is stored on a disk.
The coefficients of the nornml equations are then computed and stored on the
disk as well.
A subprogram NES, which only uses a limited amount of core store, will
then compute the solution vector b, and in this way 'i;, is determined.
The solution vector may be output on punched cards, so that the function
N
T, can he retrieved without computing the coefficients of (and solving) the normal
equations.
Collocation I is now finished. It is then possible either to start the compu-
tation of predictions or to start Collocation 11. -4 logical variable LCREF is used
to distinguish between the two situations. Thus, LCREF will have to be true in
this case, because we have decided to describe the situation, where three data-
sets are used.
The covariance function of the residual observations d,% is then first
specified. It is done in the same way as for the covariance function used in
Collocation I, though the kind of anomaly degree-variance model used will have
to be the same.
The different files of the dataset X, can then be input. Each observations
is first transformed to a geocentric reference system. Then the contribution from
F, and F, is computed, so that finally &X, can be stored. The coefficients of the
new normal-equations can then be computed and the equations solved. Again, here
the solution b, may be output on punched cards. (In case b, or b, had been com-
puted in previous runs of the program, their respective values would have been
input and the coefficients of the normal equations are then not computed.)
When the equations have been solved, the reduced normal equation matrix is
retained on a disk, so that errors of estimation can be computed, using equation (13).
The estimate of the anomalous potential is then, cf. eq. (82a)
rU N
T(P) = Tl (P) + ?'a (P) + ?,(P),
with T, (P) given by eq. (83),

cf. eq. (81a).
The prediction of a height anomaly, a gravity anomaly or a deflection
component can now be computed using eq. (82a). The computation is based upon
exactly the same type of information as was used for the computation of F, and
and T, , i. e. geodetic latitude and longitude, and a height. The program itself
may generate lists of coordinates. Such a l.ist is generated, when the logical
LGRID is true. The list will consist of coordinates of points lying in a grid. The
grid is specified by its south-west corner, and the number and magnitude of the
grid increments in northern and eastern direction. The heights of the points are
specified by the ratio RP (equation (87)).
The prediction of a quantity, e.g. a gravity anomaly will then be computed
by first determining the difference between the anomaly given in a geocentric
reference system and the reference system of the observations, Ago. The contri-
bution && is then evaluated from ?, and the contributions from ?, and F, using
(81a), i. e.
The predicted value is then, (cf. eq. (82a)):
Predictions of other quantities are computed in the same way. A special facility
for the comparison of observed and predicted quantities can be used, when the
logical LCOMP is true. The differences between observed and predicted quantities
are in this case, computed together with their mean value and variance. A sampling
of the differences is done in intervals of a specified magnitude.
The processing time (or more correctly, the central unit processing time)
will valy depending on (1) the covariance function used, (2) the number of obsema-
tions and (3) the number of cluantities to be predicted and estimates of errors to be
computed. The program has been used for a variety of test computations, though
never with more than 500 observations. The used processing times for anumber

Model
of situations are presented in Table 2. The computations were all made on the
IBM system/370 inodcl 165 computer of the Instruction and Research Computer
Centcr, Ohio State University. The so called Fortran H-extended compiler
(BM (1972)) was used for the compilation of the program. The normal equations
were stored on an LBM model 3330 disk.
Table 2
Examples of Processing Times for Different Input Data and Covariance Functions.
Potential Coefficients of Degree up to 20 Used.
Collocation I
Collocation II**
Irde r of local
zovar. fct. I
110
110
1 GO
160
110
110
**Normal equations not computed and solved in Collocation I.
Predictions
Number of
Total pro-
cessing time
m sec
0 43
0 43
2 31
3 22
2 57
4 09
2 54
4 09

5.
The Storing of the Cocffic ients of the Normal Eq~mtions and the Function
of the Subprograms .
We will in this section discuss in more detail the function of an important
part of the main-program and the different subprograms which have been used.
However, the most detailed clcscription is hund in the appendix, where the FOR-
TRAN program, which includes a large number of comment statements, is repro-
duc ed .
5.1 The storing of the normal equations.
The IBM system 370 model 165 computer of the Instruction and Research
Computer Center of the Ohio State University makes available a 630K !byte) core
storage for a usual program. Let us suppose, that we have used 180 I< for the
storing of the program and variables different from the coefficients of the normalequations.
We are then left with 450K bytes, which can be used to store these quantities.
When the coefficients are represented as double p ecision variables (8 bytes),
it is then possible to store 450* 1024/8=57600 coefficients in the core.
A system of equations with N unknowns, and a full symmetric coefficient
matrix plus a constant vector of length N+l will totally occupy (N+2)* (~+3)/2
8 byte positions. This implies, that we maximally can sdve a system of eqmtioxls
with 336 unknowns, if we want to store all the coefficients in the core.
The solution to the problem is naturally to divide the upper (or lower) tri-
angular part of the matrix in blocks, which then are stored on a disk and later
read into core storage when needed. The subdivision in blocks can be made in
several ways. h case we wanted to compute the inverse matrix, the optimal
subdivision seeins to be a subdivision in squares submatrices, as used by Karki
(1973). It is unnecessary to compute the inverse matrix for our purpose. The
solution vector b (16) and the estimate of the error of prediction (13) may both be
computed without using the inverse matrix. It is enough to compute the so .called
reduced matrix L',
where L is a lower triangular matrix, cf. Poder and Tscherning (1973). The
T
computation of L is most easily programmed, when the upper triangular
-
part of
C is subdivided in blocks, which contain a number of consecutive r columns, stored
in a one-dimensioned array with the diagonal element having the highest subscript
cf. Figure 3.

Number of last row stored
in block
Block Number
Figure 3. Blocking of 400 X 400 matrix
It is necessary, that two blocks can be stored sin~ultaneously in the core storage,
i.e. the maximum block size is then 225K or (450/2) * 1024/8 = 28800 double precision
coefficients. This number is then also the upper limit for the dimension
of the normal equation matrix, N. (Another limit is set by the magnitude of the
disk unit used. For the I13M model 3330 disk used here, N will have to be less
than circa 5000).
We have in this program edition preferred to limit the total storage re-
quirements to 252K (which for the present operative system gives a reasonable
turn-around time). Thus, 90K can be used for the storing of the required two
blocks and for the buffer area necessary for the transfer between core store
and disk.

On a disk it is practical to block data in groups which occupy an integer
number of tracks. We have then chosen to work with data partitioned into blocks
of size 4800 *8 bytes, covering three tracks and to use a buffer area of 1200* 8
bytes. The total area occupied in core storage is hence 2*(38400) + 9600 bytes
or nearly 86K. (The disk discussed is, as mentioned above, an BM model 3330
disk). Figure 4 shows the number of tracks used as a function of the number of
observations, N.
Tracks
Figure 4. Number of tracks used on an IT3M model 3330 disk unit for a varying
number of uulmowm, N (12800 bytes used on each track).
40

When a coefficient of the normal equation matrix C (eq. 12) is computed
(subroutine PRED) it will first be stored in an array C of dimension 4700 (the
last 100*8 bytes are used to hold two catalogues). Where the array C is filled
up with as many columns as possible, the content will be transferred to the disk
and stored in a direct access dataset (see IBM (1973, page 67)). The constant
vector of observations, X is stored together with the coefficients, as if it was an
extra column. The ficticious diagonal element of this column will contain the
normalized square sum of the observations.
As mentioned above, the last 100*8 bytes of a block are used to hold two
catalogues. The first catalogue contains the subscripts of the diagonal elements
of the columns stored in the block. The second contains the subscript of the
last zero element encountered in a column, starting the inspection of a column
from the top. This catalogue may especially be used when C is a sparse matrix
(e. g. when potential coefficients are not necessarily stored). In this program C
will always be a full matrix, so the catalogue entries are just equal to zero.
(Their value may be changed by the subroutine NES, in case singularities are
encountered).
5.2 Solution of the Normal Equations and Computation of the Estimate of the
Error of Prediction, Subroutine NES.
The equations are, as mentioned in Section 5.1, solved by first computing
the upper triangular matrix L? (cf. (91)). This method is the well known Cholesliyls
factorization method.
b
We obtain, from (15) and(91) by a left multiplication with L-~
T
The algorithm for the computation of the elements of L , 1 1 is
and nearly exactly the same for the computation of the elements b: of (92):

i.e. the algorithm (93) will compute (92), when x is regarded and stored as an
extra column of the matrix E. When b' has been computed, b can easily be
obtained from (92) by a so callcd back-substitution procedure. We note, that
we have (cf. eq. (6) and (13)):
Then, using the algo ithm (93) for j = m+l with C , ~ subst.ituted for ciJ, we will
obtain the quantity L-'C, for i= l, . . . ,m. By defining an element c,+~,,+~= CSs
and using (9 3) for i= m+l we will have computed the quantity G", (13).
The sub
outine NES uses these algorithms for the computation of the vector
b and the quantity 1%:. The elements of c are stored in the positions on the disk,
where earlier the coefficients of the upper triangular part of Cwere stored.
The matrix @ is theoretically, always positive definite. Thus, mis takcs may
be made, which make E non positive definite. The Choleskys algorithm (93) will
not work in this case, because the diagonal element of L', Jll, is computed by taking
the square- root of equation (93), where both sides have been multiplied with Jii.
The occiireuce of a negative quantity
r
will not stop the execution of the program. NES will regxrd the column and corres-
ponding row as deleted, and bl will be put equal to zero.
Choleslyts method is very favorable numerically. But the proper use of
the method requires that the sum of the products II,, R,, in (93) are accun~ulated in
a variable, which in this case would be in quadruple precision. The final product
sum would then have to be rounded properly to double precision. Unfortunately
rounding is not
r
done by simply requiring the quadruple precision variable to be
stored in a double precision variable, but supplementary statements have to be
used. Thus, the solution vector by is heye obtained by computations performed
in double precision only, which in this case anyway, gives a satisfactory number of
significant digits.
'N \3
The solution vector b, is obtained in 0.71 Lz0; seconds, where N is the
dimension of the normal equation matrix.

5.3 Transformation Between Reference Systems, Subroutine ITRAN.
We pointed out in Section 3, that it was necessary to transform the co-
ordinates and measurements into a geocentric reference system. This transfor-
mation is performed for the coordinates, the deflections and the height anonlaly
by the subroutine ITRAN.
The subroutine uses the euclidian coordinates X, Y, and Z for a point
with geodetic latitude and longitude X and with the ellipsoidal height equal to zero.
XI Yl, Z, by
These coordinates are then transformed into geocentric coordinates,
where (AX, AY, AZ) are the coordinates of the center of the old reference ellipsoid
given in the new coordinate system, AL the scale change and (c,, c,, c,) the three
infinitesimal rotations around the X, Y, Z axes respectively .
The new geodetic latitude 9 and longitude X, of this point is computed
using the ite ative procedure given in PG (p. 183). This computation will also
furnish us with the change in the height anomaly, which is identical to the height
of the point (Xl, Y,, Z1) above the ncw reference ellipsoid.
The change in the deflection components are then determined using the
differences col - cp and X, - X.
A contingent correction for systematical errors in the deflections or the
height anomaly (cf. Section 3.2), specified by the changes 65,, 6~,, 65, in a
point with coordinates cp, Xo, is conlputed by the subroutine using the equations
given in PG (eq. (5-59)).
5.4 Computation of the Normal Gravity, the Normal Potential and the Contri-
butions from the Potential Coefficients. Subroutines GRAVC and IGPOT.
The normal gravity may be given in two ways. Either by a gravity formula
or by specifying a norinn1 gravity field, from which the gravity formulae then can
be derived, (cf. PG, Chapter 2). The only gravity formulae which can be used is
the international gravity formulae, PG (cq. (2-126) and (2-131)). The normal
gravity fields which can bc used, are those for which the reference ellipsoid is an
equipo tential surface, i. e. it is specified by the values of kM, a, f and U.

We may need to kuow the reference gravity in two situations. Firstly
when free-air anomalies are computed using measured gravity values and secondly,
when we want to compute the change in the gravity anomalies due to the use of a
new reference sys tern.
The subroutine GRAVC will compute and store the constants (PG, Section
2.10) necessary for the computation of the normal gravity in one or two reference
systems. The constants used to compute the value of the normal potential (J,,,
ns 5 in eq. (26)) and the change in the latitude component of the deflection of the
vertical 5 due to the curvature of the normal plumbline (PG(5-34)) are computed as
well. When the height exceeds 25 km, the derivatives of the series (26) with
respect to the latitude and the distance froln the origin, will be used for the compu-
tation of the normal gravity and the change in 5. Thus, this method of computation
can, unfortunately, not be applied when the international gravity formula is used.
The values of the normal gravity, the normal potential and the change in S
are computed by calling separate entries to the subroutine. The subroutine IGPOT
computes the value of the potential W(P) and the three components of thz gradient
of the potential, the value of kM, a and w , cf. eq. (25).
Let us, as usual, define V by
The coefficient modification method is used for the computation of the values
of V and the gradient of V. This method uses the fact, that the derivative of a har-
monic function with respect to euclidian coordinates again is a harmonic function.
The potential coefficients of the (three) new harmonic functions D,TJ, DyTJg and DZV
are computed by means of a recursion algorithm given in James (1969, eq. (3) and
(4)). The recursion algorithm is identical to the algorithm used for the evaluation
of the values of the solid spherical harmonics. This fact simplifies the computations
very much. It furthermore makes it unnecessary to store the three sets of modified
potential coefficients. They are conqmted for each call of the subroutine from the
original potential coefficients. The algo ithm may easily be modifjed to compute
higher order derivatives (without extra storage requirements). Thus, the algorithm
may also be used in case the program is extended to use second order derivatives
as observed quantities.
The value of the potential and the gradient is used to compute the residual
observation d,%i :
The value of the potential is used together with the value of the normal

potential (as computed by GRAVC) to compute a residual height anomaly. The
gradient is used to compute the residual gravity anomalies and deflection compo-
nents :
A
where y, is the normal gravity, qj is the geodetic latitude plus a correction for
the curvature of the plumbline and X, the longitude.
The components of the gradient used in (97a) are evaluated in a point with
height equal to the orthometric height plus the distance h. between the reference
ellipsoid and an equipotential surface of U1 = U + T1 with potential equal to the
potential of the normal potential, U on the ellipsoid. The separation h, is computed
by evaluating Tl /y oil the eliipsoicl. The other gradieiits are evaluated iil the height
equal to the orthome tric height.
5.5 Computation of, Euclidian Coordinates, Conversion of Angles to Radians,
Subroutines EUCLID, RAD.
The subroutine EUCLID computes the euclidian (rectangular) coordinates for
a point with geodetic coordinates p, X , h (ellipsoidal height) given in a reference
system with semi-major axis a and second eccentricity e by the equations PG (5-3)
and (5-5).
RAD converts angles given in either (1) degrees, minutes, arc seconds,
(2) degrees and minutes, (3) degrees or (4) (400) grades into units of radians.
Other options may easily be added.
5.6 Subroutines for Output h'lanagement and Prediction Statistics, HEAD, OUT
and COMPA.
The output requirements are discussed in Section 7. The main require-
ment is, that a determination of ?must be as well documented as possible. ?!
may be computed in several ways, cf. Table 1. This implies, that the output may
vary in just as many ways.

An array 013s is used for the storage of the observed quantities, the resid-
ual observations, the contributions from the different sets of observations and the
predicted quantities. The storage sequence of these quantities is determined by
BEAD, which a,lso will print proper headings. The coordinates of an observed or
predicted quantity and the quantities stored in OBS are printed on the line printer
by OUT, which also will punch a part of this information when requested (see Sec-
tion 7).
COMPA uses the content of OBS for the computation of prediction statistics.
The difference between observed and predicted quantities are sampled in classes
defined by a specjfied class width. The number of differences in each class is
printed by COVA after the final predictions have been computed. The sampling is
done separately for Ag, 5 and v. No sampling is done for 5.
5.7 Subroutines for the Computation of Covariances, PRED and SUMK.
PRED computes :
(a) the vector dim ,Cs or dl-, C l , (cf. eq. (78a) and (82a)).
(h) a colunm of the upper triangular part of the normal equation matrix
(eq. (80a)) diCii or
t Sxi~ di S = d,~: bi , (cf. cq. (8Ia)).
(C j Ihe pi.06~~
The subroutine may theoretically work even when the observations (different
from potential coefficients) are divided in more than two groups, as long as the
total number of observations do not exceed 1600 minus the number of groups minus
one.
When the observed quantity is a pair of deflections of the vertical it is very
easy to compute the two corresponding colunms of the upper triangular part of C
at the same time. This is due to the similarity of the equations for the covariaaces
(44)-(50) for 5 and v. This fact is used in the subroutine.
We mentioned in Section 3.1, that ic would facilitate the computations if the
observations were grouped according to cominon characteristics, i. e. , e. g. gravity
amoinalies on the surface of the Earth in the first groilp, deflections in the second
group, gravity anonxilies in 10 kinf S height in the third group, etc. The group
characteristics (the type of observation and the quantity RP, ($7)) are stored in two
arrays INDEX and P, which will also contain the subscripts of the first observation
in the group within the total set of observations and a quantity related to the square
root of the varimce of the observations. This quantity is used for the scaling of
the normal equations (in which all the diagonal elements will be equal to me).

The covariances are computed using the equations given in Section 2.2.
Thus, for degree-variance model 3, the covariances will be evaluated using the
subroutine SUMK as well. This subroutine computes the sum of the infinite series
CD Q, m
1 a 1 1-1 1 p - 2 - S Dt Pa(t) and -
S k t L (a+B) 2 @+B) D: pkt) ;
L
which are needed for the computation of the quantities cov:(si, sJ), cf. eq. (39) and
Tscheming and Rapp (1974, eq. (130) - (135)).

6. Input Spec if icatioils .
We can divide the input data in different (sometimes overlapping) groups:
(A) Data (generally true/false values of logical variables) determining
the flow of the program (LTllAN, LPOT, LCREF, LGRID, LERNO,
LSTOP),
(B) Data specifying selected input/output options,
(C) Data specifying the reference systems used for coordinates and obser-
vations,
(D) Data specifying the degree-variance model used,
(E) Data used for the determination of T (i. e potential coefficients, gravity
values, deflections, etc. ) and solutions to normal equations, and
(F) Data used to specify which quantities we want to predict.
The input flow is roughtly sketched in Figure 5. The position of the
integers i-5 in the diagram indicates the beginning of the input data beionging to
one of the 5 groups described below:
up to
one
repetition
Figure 5: The Input Flow.
@ Input data of category
(A)-(F) mentioned in text.
Up to 8 repetitions, when the input data
has significantly different characteristics
Unlimited number of
repetitions

The input consists exclusively of data on 80-column punch-cards. We will
describe the content of each card, but not the format of thc card. Instead, the for-
mat statement number will be given (in brackets) together with a two to five digit
number, e.g. 3.013. This number will be used to identify the corresponding card
shown in the input example, Appendix B.
We will divide the data in 5 categories:
(1) Data of type (A), (B) and (C), i. e. data describing the reference
systems used,
(2) Data of type (A), (B) and (E), where the data of type (E) are the
potential coefficients,
(3) Data of type (A), (B) and (D), i. e. data related to the degree-
variance models,
(4) Data of type (A), (B) and (E), where the data of type (E) are obser-
vations of gravity anomalies, measured gravity values, height anom-
alies and deflections of the vertical,
(5) Data of type (A), (B) and (F).
The first digit in the identifying number will be the number of the category
to which the card belongs. The other digits are used to indicate to which group
or subgroup within the category the card belongs. In case an input situation de-
pends on the content of e. g. the card 3.01 and there are two different input pos-
sibilities, the two cards will have the numbers 3.011 and 3.012 respectively.
(Data of type (A) and (B) will, as mentioned above, in many cases be the true or
false value of a logical variable. The function of a logical variable can be explained
by writing e. g. : LE is a logical variable, which is true when XXX and false other-
wise. Thus, we will in several cases below simply write LE = XXX.)
Categoq 1. (The numbers in brackets are, as mentioned above, the corresponding
---- - format statement numbers).
1.0 (105) The (true or false) values of five logical variables: LTRAN= the
observations must be transformed to a new reference system, LPOT=
potential coefficients are part of input data (observation data. set XI),
LONEQ = output the coefficients of the normal equations on the line
printer, LLEG= output a legend of the tables, which will be printed and
LE = the standard deviations of the observations to be input (otherwise
they are set equal to zero).

1.1 (103) A text of maximally 72 characters included in apostrophes, which
identify the referace sys tem of the observations.
1.2 (120) The semi-major axis (meters) and the inverse of the flattening of
the Geodetic Reference System. The value of two logical variables,
LPOTSD = the gravity are given in the Potsdam system and LGRS67 =
the gravity data are given in the Geodetic Reference System, 1967.
In case the gravity data are not in the Potsdain system or in GRS 1967, input of:
1.21 (121) The product of the gravity constant and the mass of the Earth
(kM) in units of me ters 3 / sec 2 .
When LTRAN is true input of:
l. 3 (131) The new semi-major axis (meters), the new kM (meters 3 /sec 2 ),
the inverse flattening, the translation vector (dX, dY, dZ) (meters), dL =
one minus the scale factor, the three rotation angles (c,, c,, c,) (arc sec)
and the value of a logical variable, LCHANG, which is true when the
deflections and the height anomaly are to be corrected for a systematic
error. (The correction must be given as a change 65,, 6q0, 65, at a
point with coordinates p, X,, cf. Section 5.2).
When LCHANG is true, input of:
1.31 (133) cp, and X. in degrees,minutes and arc seconds, 65, , 6q0 in arc
seconds and 6C0 in meters.
Category 2. Data of this category are only input, when LPOT (card 1.0) is true.
The values of kIvI and a, input on card 2.1, will have to be the best
available estimates, cf. Section 3.2. They must be identical to
the values input on card 1.3, when LTRAN is true.
2.0 (103) A text of maxiinally 72 characters, describing the source of the
potential coefficients.
2.1 (137) kM (meters 3 /sec"), a (meters), the normalized coefficient c2*,
multiplied by 106, the maximal degree of the coefficients and the value of
a logical variable, LF&I, which is false, when the coefficients E!,, are
punched on a separate card, and C,, , S,, on the same card in a sequence
increasing with i and j, and true w11en the coefficients are punched in the
same scqxnce, on a number of cards, but with a fixed number of coef-
ficients on each card. The first coefficient will, in both cases, have to

e C,,, (even when this is zero) and all cards must have the same format,
as given by 2.1. All the coefficients have to be fully nornlalized and
multiplied by 10~.
2.2 (103) The format of the cards on which the coefficients are punched
(in brackets).
2.11 (format as given by 2.2). When LFM is false, the coefficients with c,O
on one card and Clj and SIS on one card.
2.12 (format as given by 2.2). When LFM is true, the coefficients in a se-
quence increasing with i and j on a number of cards.
Catezory 3. We can select one of three anomaly degree-variance models, by
-- --- giving the variable KTY PE the value 1,2 or 3, cf. Section 2.2
eq. (30), (31) and (32).
When KTYPE is equal to 3:
3.01 (107) IK = the variable B in equation (32).
The degree-variance model is then specified by giving
(a) the ratio R/R, between the radius of the Bjerhammar sphere and
radius of the mean Earth,
(b) the variance of the gravity anomalies on the surface of the Earth
(VARDGB), (from which the constant A, in the equations (30)-(32)
are computed, cf. , e. g. equation (go)),
(c) either the "order" Ll'vfAX of the local covariance model to be used
or a zero, which will indicate, that empirical anomaly degree-
variances are used, and in this case
(d) the empirical anomaly degree-variances, given on the surface of
A
the Earth, aE(Ag, Ag), (88).
a
When IMAX is equal to zero, input of the maximal degree, IMAXO of the degree-
var iances &; (A g, A g).

3.12 (103) The forinat of the degree-variances. These must be punched on
one or more cards, sequentially from degree 2 to IMAXO.
3.13 (format as given on card 3.12). The quant,ities &;(ng, Ag) in units of mga12.
------ Category 4. hput of up to 9 clatasets with ~i~gnificantly different characteristics.
One dataset can, for example, be two separate.datasets punched
differently, but both being gravity anomalies on the surface of the
Earth. Another clataset may consist of mean gravity anomalies, all
with the same format. Before each separate dataset, there will
be input of 2 or more cards specifying the characteristics of the
dataset.
All the records in a dataset must, be punched im the same way.
There are the following restrictions (or options): A station number
may be punched. In this case it must be the first datafield on the
card and maximally occupy seven positions. The next two data-
fields must contain the latitude and the longitude (in an arbitrary se-
quence). When the height is given, it must be punched in the next
datafield.
The following (up to four) datafields will have to contain the observed
quantity (or quantities in case of pairs of deflecti n components) and
P
its standard deviation. When the observation is a pair of deflections,
they have to be punched in the same sequence as the latitude and the
longitude are punched. In t,he last datafield the value of the logical
STOP has to be punched, generally false ( = blank), but true for
the last record in the dataset.
4.0 (103) The format of the records (in brackets).
4.1 (202) INO=l when a station number is punched, 0 otherwise, ILA = the
number of the datafield occupied by the latitude, ILO = the number of the
datafield occupied by the longitude (ILA and ILO will be equal to 1, 2 or
3), am integer IANG specifying the units used for the latitude and the
longitude (1 for degrees, minutes, arc. sec., 2 for degrees and minutes,
3 for degrecs and 4 for 400-grades), IH= 0 when the height is zero and
not punched and otherwise the number of the datafield in the record in
which the height is punched (generally 3 or 4), IOBSl= the datafield number
of the first observation in the record, IOBSB = the clatafield nunlber of the
second observation (zero when there is only one observations), the

value of an integer XI?, specifying the kind of observation: 1 for height
anomalies, 2 for measured gravity or gravity anomalies (point or mean),
3 for pairs of deflections, 4 for the latitude component 5 and 5 for the
longitude component 7) .
Then the ratio RP (87) between the sphere on which the observations are
situated atid the mean Earth radius and finally 5 logical variables: LPUNCH =
punch observations together with the difference between the observed quantity
and a possible contribution from the potential coefficients and a contribution
from Collocation I, LWLONG = longitude is measured positive towards west,
LMEAN= the gravity is a mean value, LSA = the standard deviations are the
same for all observations, LKM= true when the height is in units of kilo-
meters and false when the height is in meters.
When the observation is a height anomaly it will have to be given in units
of meters, and when it is a deflection component in arc seconds. But
when the observation is a gravity anomaly or a measured gravity quantity,
it is possible to specify tivo constants DhI and DA, which when DA is first
added and the sum multiplied with DM will bring the observed quantity into
units of mgal.
4.11 (203) DM, DA and a logical variable LMEGR, which is true, when the ob-
servation is a measured gravity value.
When LSA is true, the records of observations will not contain a standard
deviation of the observed quantity, and the standard deviation will then have
to be input separately:
4.12 (212) the standard deviation of the observations. Then the observations
are input record after record, not exceeding a total number of 1598.
4.2 (format given on card 4.0). Input as specified on card 4.1, last record
with LSTOP equal to true.
WhenLSTOP is true, a logical variable with the same name (i.e. LSTOP) is input.
It is true when the last dataset is the final dataset used in Collocation I or 11.
The final card will hence have the value true (T) punched in the first datafield. On
this card, in this case, the values of two other logical variables can be punched.
4.3 (230) The value of LSTOP, and when LSTOP is true, the values of two
logical variables, LRESOL = input the solutions to the norinal equations
(they must then have been produced in a previous i-un of the program) and
LWRSOL= punch the solutions to the normal equations.

When LSTOP is false, the input process will he repeated from card 4.0. When
LSTOP is tme and LRESOL is true, input of the solutions to the normal equations:
First an identification card is read, then the solutions:
4.31 (361) Input of solutions, i.e. the cards produced in a previous mn of
the program, where the logical variable LWRSOL was true.
When the set of observations is the first one (the variable LC 1 is false),
input of a logical variable LCREF, else jump to 5.0.
4.4 (230) LCREF = a new set of observations, which will be used in collocation
11, will have to be input,
For LCREF= true, jump back to 3.1.
Category 5. Data specifying quantities to be predicted. This specification will
naturally have to be done in much the same way, as when the obser-
vations were specified. We need coordinates and some variables,
which specifies tile type of quantity to be predicted. There are then
two possibilities, which are distinguished by the true and false value
of the variable LGRID.
When LGRID is false, we will proceed in exactly the same way as
above, dealing with data of Category 4. The quantities to be pre-
dicted will be specified by a list of coordinates and 2 or more
cards specifying format and type of quantity to be predicted.
The list of coordinates may in fact be a list of observed quantities,
which we want to coinpare with the quantities to be predicted. If
this is the case, a logical variable LCOMP has to be true.
When LGRLD is true, the predictions will have to take place 1n
points which form a grid. The south-west corner of thegrid
will have to be specified together with the distance between
the mesh points in northern and eastern direction and the num-
ber of mesh points having the same longitude and the same latitude.
5.0 (200) Input of the logical values of LGRID, LERNO = estimate of error of
prediction is nmted and LCOMP = compare predicted and observed quan-
tities.
First time LCOMP is true, two constants used to specify the sampling width for
a frequency distribution of the difference between observed and predicted gravity

anomalies (VG) and deflections (VF) must be input (e. g. equal to 2.0 mgal and
0.5 arc. sec. ). (The differences are sampled in 21 groups. )
5.01 (203) VG and VF.
When LGRID is true:
5.02 (201) Coordinates (latitude and longitude in degrees and minutes) of the
south-west corner of the grid, the increments in latitude and in longitude
(minutes), the number of increments in northern and in eastern direction,
the value of the R P giving the type of quantity to be predicted (see 4. l),
RP (see 4. l), L&AP= print the predicted quantities on the line printer
with all values which are predicted in points with the same latitude on
one line and all values predicted in points with the same longitude above
each other, LPUNCH = punch coordinates, predicted quantity and when
LERNO is true the estimated error, LiLIEAN= gravity to be predicted
is a mean value.
When LGRID is true, jump to card 5.1.
Now, when LGRID is false, we may input lists of coordinates just as above:
5.030 as 4.0 (format of records)
5.031 as 4.1, with the following changes; When LCOMP is true, LPUNCH will
mean the same as in 4.1 and the error of prediction will be punched when
LERNO is true. When LCOhlP is false, the predicted quantity as given
in the new and in the old reference system will be punched together with
the error of prediction, when LERNO is true. The logical variable LSA
has no function in this phase of the computations.
When no observed quantity is contained in the record, both IOBSl and
IOBS2 will have to be put equal to zero. Thus, in this case the record
will have to contain the height. (The program requires the presence of
at least one datafield be!xeen the datafields occupied by the latitude and
the longitude and the datafield occupied by the logical variable LSTOP).
When IKP= 2 (we are predicting gravity quantities):
5.033 input as specified by 5.030.

5.1 The value of LSTOP, true when no more quantities are tb be predicted.
When ISTOP is false, jump to 5.0.
An input example is printed in Appendix B.

7. Output and Output Options.
The output from the FORTRAN program has been designed with the purpose,
that the determination of ? and subsequent predictions should be as well documented
as possible. This means, that nearly everything, which is used as input also will
be output.
There are a few exceptions:
(a) data of type (A) and (B) (cf. Section 6) are not printed,
(b) the potential coefficients are not printed,
(c) a measured gravity value is not printed, but the corresponding free-air
anonlaly is,
(d) more than two decimal digits of coordinates given in minutes or seconds
and of observations are generally not reproduced.
With these exceptions all input of type (C) to (F) are printed with proper headings
02 the !ize printer.
We will now distinguish between non-optional and optional output. The out-
put can be made on h 7 0 units, unit 6 the line printer and unit 7 the card punch. Non-
optional output is output on the line printer exclusively.
-A program identification is printed giving date of program version.
-The used mean Earth radius and the reference gravity used on the sphere in
equations (20)-(22) is printed.
-The equatorial gravity and the potential of the reference ellipsoid as computed
from the constants specifying the reference systems.
-The residual observations di-,X,, and if meaningful: the contribution from the
datum transformation, from the potential coefficients, the first dataset (Collocation
I) and the second dataset (Collocation IT) and the sum of these contributions,
-The pedicted quantity and if meaningf~tl: the contributions from the datum transformation,
the potential coefficients, Collocation I and 11.
-The solutions to the normal equations,
-The estimaLcd variance of the residual observed and predic ted quantities, and
-Error messages in case e. g. certain array limits are exceeded.

Optional output on the line-printer:
-A legend of the labels of observations and preclic tions, (LLEG - tme),
-The difference between observed and predicted quantities (LCOMP = true),
-The estimated error of prediction (LERNOZ true),
-A primitive "map" of the predictions (LGRID = true and LMAP = true). (The
predicted quantities multiplied by 100 will be printed with the values predicted
in points with the same latitude on one line and the values predicted in points
with the same longitude above each other, see the 'tmaptf, Appendix C, page 125. )
-Mean value and variance of difference between observed and predicted quantities
and table of distribution of the differences samples according to specified sample
width (LCOhZP is true).
Optional output on the card punch:
-The solutions to the normal equations b, and b, (LWRSOL = true)
-the observed quantities and the residual observations (LPUNCH= true),
-the predicted quantities, the estimated error (LERNO = true), the difference
between observed and predicted quantities (LCOMP = true), and when LCO&ZP
is false, the predicted quantity in the original and the new reference system.
The solutions to the normal equation can be used as input to the program,
cf. Section 6, input specification No. 4.31.
An example of the output on the line printer is given inn Appendix C.

8. Recomn~endations and Conclusions.
A development of a computer program as the one presented here is a task,
which can be continued for years. But at some point it is necessary to stop and
present a fully documented program version, even if it is obvious that improve-
ments can be made.
Most of the recent ideas and investigations in the field of least squares
collocation are used in the program. Hence, the program may principally be
used for
N
-the determination of an approximation to the anomalous potential, T and
-prediction and filtering of gravity anomalies, deflections and height
anomal ie S.
The determination of may be improved in several ways. The program
should be changed so that other types of data as e. g. density anomalies, satellite
orbit perturbations, and gravity gradients can be used as observations and pre-
dictions. The program should also be able to predict potential coefficients. The
covariance models, which can be used in the program are all isotropic. The use
of a non-isotropic covariance model may improve the determination of y.
In Section 3.2 it was pointed out, that the data had to be given in a geo-
centric refe ence system. Thus, the necessary translation parameters may be
estimated by including these quantities as parameters X, cf. eq. (7) and (g),
Tscherning (1973) and Moritz (1972, Section 6).
It is also possible to add new data to an original set of observations, with-
out having to conlpute and invert the full covariance matrix. This type of compu-
tation is denoted sequential collocation cf. Moritz (1973). This feature may very
easily be incorporated in the program, especially because of the flexible design
of the subroutine NES (cf. Section 5.2 and the comments given to the subroutine
in Appendix A).
The determination of potential coefficients and datum shift parameters may
also be incorporated without difficulties. But the other proposed improvements
can not be made before the theoretical background and the necessary algorithms have
been developed.

References
Grafarend, Erik: Gcodetic Frediction Concepts, Lecture Notes, International
Summer School in the Mountains, Ramsau, 1973 (in Press).
Heiskanen, W. A. and 1-1. Moritz, Physical Geodesy, W. H. Freeman, San Fran-
cisco, 1967.
(BM): IBM OS FORTRAN IV (H Extended) Compiler,Programmer's Guide,
SC28-6852-1, 1972.
(BM): IBM System/3GO and Sys tem/370 FORTRAN IV Language, GC28-6515-9,
1973.
Jamcs, R. W. : Transformation of Spherical Harmonics Under Change of Reference
Frame, Geophys. J.R. astr. Soc. (1969) 17, 305-316. J
Karki, P. : A FORTRAN IV program for the Inversion of a Large Sylnlilet.ric
Matrix by Block Partitioning, Department of Geodetic Science, OSU,
Report No. 205, 1973.
Krarup, T. : A contribution to the Mathematical Foundation of Physical Geodesy,
Danish Geode tic Institute, Meddel else No. 44, 1969.
Lauritzen, S. L. : The Probabilistic Background of some Statistical Methods in
Physical Geodesy, Danish Geodetic Institute, Meddelelse No. 48, 1973.
Meissl, P. : A Study of Covariance Functions Related to the Earth's Disturbing
Potential, Dep. of Geodetic Science, OSU, Report No. 151, 1971.
Moritz, H. : Advanced Least-Squares Methods, Dep. of Geodetic Science, 'OSU,
Report No. 175, 1972.
Moritz, H. : Stepwise and Sequential Collocation, Dep. of Geodetic Science, OSU,
Report No. 203, 1973.
Parzen, E. : Statistical Inference on Time Series by Hilbert Space Methods, I,
Tech. Rep. No. 23, Stanford. Published in Time Series Analysis Papers,
Holden Day, San Francisco, 1967.

Poder, K. and C. C. Tscherning: Choleskyls Method on a Computer, The Danish
Geodetic Institute, Internal Report No. 8, 1973.
Rapp, R. H. : Numerical Resu1.t~ from the Combination of Gravimetric and Satellite
Data Using the Principles of Least Squares Collocation, Dep. of
Geodetic Science, OSU, Report No. 200, 1973.
Tscherning, C. C. and R. H. Rapp: Closed Covariance Expressions for Gravity
Anomalies, Geoid Undulations, and Deflections of the Vertical implied
by Anomaly Degree Variance Models, Department of Geodetic Science,
OSU, Report No. 208, 1974.
Tscherning, C. C. : An Algol Program for Prediction of Height Anomalies, Gravity
Anomalies and Deflections of the Vertical, The Danish Geodetic Institute,
Internal Report No. 2, 1972.
Tscherning, C. C. : Determination of a Local Approximation to the Anomalous
Potential of the Earth using llExactll Astro-Gravimetric Collocation,
Lecture Notes, International Summer School in the Mountains, Ramsau,
1973 (h Press).

Appendix
A. The FORTRAN 1V program.
B. An input example.
C. An output exanzple.

Appendix A.
The program is written in the language FORTRAN IV, cf. BM (1973). It may
be run on an 113M model. 370 computer equipped with an BM model 3330 disk
unit. The program may be compiled and executed using the catalogued pro-
cedure FORTXCLG, cf. IBM (1972, p. 89) using the following job control
language statements:
// EXEC FORTXC LG, PARM. FORT='OPT(2)',
// TIME. FORT=(, 3O), REGION=2521<
//FORT. SYSIN DD *
program statements
//GO. FT GGFOOl DD DSN=DASET, CNITSYSDA,
// SPACE=(12800,920), DISP=(, DELETE), DCB=(DSORG=DA, BUFNO=l)
//GO. SYSIN DD *
/ *
//
input data

a:
C PPOGRAM GCODETIC CO!-LJCATlC!hS VERSION 20 APR, 19749 FORT'ZAN TV9 (?FP
C 36Q/?O), PPnGKLCiYkD 6V C,CoYCUERh%NG~ DANISH GFOOETlC INSTITUTE/ CFP,
Q GEODETIC SCItNit? OSL~,
C SHE PROGRAM COVPUTES AN APPK3XIMAT13Y TO THE AQ9MALOUS POTFYTIAL '-IF
C THE EARTH USIhG S7Er'WISE LEAST SQUAXE-F COLLOCATlONe THE MFTHrIT) 9FL)U.I-
6 KES THE SPFClF1CATlWi OF (PI @!vE Oh TWC $CYD I N A SPFCIAL CASE TPQFE)
C SETS OF QYSEKVED OUAiJTITIES WITH KVfJH1\l STANDARD OFVIATIflNS AQD (2 1 PNF
C OR TWO CDVARIkk'CE FUYCTIONS.
C THE CfiVkRIANCC FUYC7ICINS USFD ARE ISOTROPICe THEY ARE qPFCIF1 ED SY A
C SET OF EMPIRICAL A~P4ALY DEGREE-VARIAYCES OF PEGRFF L E S THAY AY
C IKTEGEW VPQIPFLE IMbXt RYD EY A ANOMALY DFS!iFE-VA?IA?4CE MOOFL FR9 THE
C DEGPEF-VAFIbhCES @F DFGPEF CRFATHFR THAN IMAX.
& THE OF. SERVATIONS MAY 3F POTFIJTIAL COEFFICIFYTS 9 bIEAN 9R PUInlT GSPVITY
C AMOMALIEC, HFlGHT ANOWLLIES P44D IJEFLECTIPYS CF TUE VERTICAL. B FIL-
C TtRING OF THF '-IMSF9VATiW!S 1AYFS PLACE I/rULTAYE3USLY WITH THF DETER-
C MIIJATIOV OF Tt*E AM3VALOIJS FfJTFUTIAL.
C THE DETFRMIkATIL't: I S VPCIF 1N A AI',LVREQ GF STEPS Ff'UAL TO THE NUYSCQ OF
e stvs CF oe:Enva-rrrr%s, WHEN POTENTI AL COFFFIC~ENTS AFE USFP~ WI LL THF-
C ESE FflRM A SFPEFbTE SFT AFID THE TOTAL YLVMYE9 OF SETS EIAY Ir\l THIS CASE
C AMnUNT TQ THRIF,
C EACH DATASET IFLCH STFP) WILL DETERMIUE A )-;IAQMOYIC FUYCTIDYy AVD THE
C PPiOMALI3US POSENTIAL NILL 5E EQUAL TO THE SUY OF TFEFSF (?lAXIMbLLY
C TtiRFF 1 FUhCT EDhS.
C POTENTIAL CnEFFLCIENTS WILL DETERVIWF A FUNCTIOFl FQUAL TO THE CVFFFH-
C ClENTS MULTIPLIED BY THE CUC.RFSDONDIi'4G SOLID SPHERICAL r-'AQMOUICC. THE
C UP TO TWO cFT? UF DATA DIFFFFENT FRPM POTFNTIAL COEFFICIFVTS WILL
C FACH RE U5FD TO DETERPIXE CONSTAYTS E( 11, THE COF'?ESPr]Yrll?lG HAQMnYIC
C FUNCTIOVS 4RE THFX FQUAL TO THEESE COlrSTAYT5 MULTIPLIFr3r SY TYE COVA-
C RlANCF EEThFEN THE PESERVAT IDNS A\D THE VALUE OF THE AVOMALOUS POT-
C FNYI4L IN A PCtLhTp Ps
C %"F MAIN FUUCTlrN PF TUF PqOOPAM IS, RESIPES THE C'IMPYTATION OF THF
C COYSTANTS E( I S, TYE PREDICTI~Y C:F THE GU~TITIE? FTA, GELTA e, usr
C AND ETA 191 POINTS Q. TH-rE PREDICTED VALUE IS ECUAL TO THE PRgDUCT SUY
C OF B( I ) ANP TEE CDVPRIAYCE EETWEFN OBSERVATION Y0.I AND TtJE OUAVTITY
C TO @F PRFDICTED.
C
C REF(A1: TSC"tRNINGpC*C- AND R-HeRAPP: CLOSEP COVARIANCE FXPUESSIOt'IS
C F09 t-SAVITY ANOMALIES, GEOIO L!YDULATIPY!Ss AhD DEFLECT1 nUq PF
C THE VFRT IC AL IMP LI ED EY A'V3MALY DEGREE-VAR IAVCE M'J9EL5, DEP-
C ARTMFNT OF GFOGETIC SC lENCFt TPF OHIO STATE UY IVE9SITY 9
C REPORT NO* 708 t 1974
C
e
REF(R): TSCHERNIr\;G,CoCe: A FORTRAN 1V "RCCRAP FTIR
nF T ~ E ANOMALOUS POTENTIAL USIQG STEPWISE
THt DETFRMINATIPN
LEAST S Q U A ~ F CPL- ~
C LDCATlON t DEFAFTME'!T OF GEODFT IC SC IFUCFI THE OHID STATE UY I-
C VESSITY, 9FPDRT NO* 2129 19749
& REFf C): HFISKANFN b.1.A- AND H.MC1R IT?: PHYSICAL GEOCIFSYp 1067.
C REF(D1: JAMESr ReW*: G E D ~ H Y S ~ J ~ R ~ A S T R ~ S O17, C ~ I 305- ~ ~ Q ~ ~ )
c
IMPLICIT S Y T E G E P ( ~ ~ J ~ K ~ M LflGICAL(L)tREAL I Y ) ~
*R(A-HqO-Z)

COMMQY/PR/SIGMA(250)TC,IGMAn(250) vi3( 16C!r))~P(42)r
*SINLAT( 1600) rCOSLAT(160r)) IPLAI (l6nn) ,RLflKG( 16,001 vC~~SLAPTSI'JLAPT
*RLATP TELONGPTQPTPPETkP9 PREDP~PWOLflNFCflT L?

C MFNSION OF TFE bRKAY NPL (IN 1HF 60i4elOU RLQCK /NFSnL/),
DEFINE FILF 8 (5?O932OOtU,IQ?
e
C lNITlALIZAYlON OF VbRIAPLESt WHICH ARE IY CnMMOhJ BLCCKSo
DO = O*ODO
- oi = n,nw
D2 Z00D0 D3 = 3aOGO
P(19 = DO
P(21B = DO
COSLATt 1600 J = D1
SINLATflt,OO) = U0
RLkTd Pt~QOI = DO
RLONG(16001 = DO
CLA = PO
WP(11 = RF+*S/GP
W = S E**2 /GlA*2O6?6t,806LO
WP(2) = D1
WP(3) = W
WP(41 = I4
WP(51 = W
BT = 0
1P = 0
LNFRtL'3 = LT
LCPEF = LF
LC1 = LF
LC2 = LF
INDEX (I 1 = 0
D@ 12OG I = 1, 250
1700 SIGMAO(11 = D0
C
C HEADINGS AND DEFINING CPNSTANTS.
klR1TE(6?104)
104 FORMAT( a 1GFODETIC COLLOCATTONIVERS~OY 20 APR
MRITE(hrll3)
1974aF//%
113 FORPAT( "Oh'C!TF THAT THE FUhCT IONALS ARE I N SPqFRPCAL APPPf7X%YATIflNB
*/B MEAN RADIUS = RE = 6371 KM AND MEAV GRAVITY 981 KGAk USEDe*)
e
C INPUT OF 5 LOGICAL VARIABLfSv LTRAN = CFORGINATFS ARE T f SE T4AVS-
C FORMED TO NEN RFFERENCE SYSTEM? LPOT = POTEYTIAL CDrFFICIE\!TS AQc TO
BE USED AS FIRST SET OF OBSERVATlOQTt LONEQ = OUTPUT COEFFICIFUTS 3F
C NCRKAL FQUATIGNS ON UNIT 69 LLEGK = OUTPUT LEGEND VF TAPLES ?F 09SFP-
C VATIOYS OR PREDPCTPONS AND LE = TAKE ERRORS CIF OFSERVATIORS
C CCUNTe
READ(~~LO~)LTRANILPOT,LUNEQ~LLEL~LE
B05 FORMAT (5LF')
LNTRAN = .NOT.LTi?AN
LNPOT = .NflT.LPOT
INTP bC-
IF (eNQT-LE) WRLTE(6tll8)
118 FORMAT l ' FF.RORS IN I1bSERVATlCINS ARE NOT TAKFN iNTO ACLflUVT- '1

C
I F (LLFG) WRHTF(6sll4)
114 FORMAT ( 'C'LEGEND OF TABLES OF OSSFRVbTIOVS ARD PRFDTCTlnNq :' ,/ 9
*' 06s = OPSEKVED VALUE (klHFsl BOTH COMPDWFYTC '1F DFFLECTIqYS AI;FB/e
* @ OBSERVEI ETA RFLOW KC l ) p DIF =THE DIFFEQFVCF RETWEFRI 03SF9VFD8/9
*I AND PPFr ICTFD VALUF p WHFV PREDICTIONS ARE: CPYDUTFD AKD FLCC ' / v
Ss THF KFrIlCjrJPL (>RSFPVATIO:4, ER? = FSTlrlA7EI) FPRO'I OF PRF~)ICTICYV/P
4'3 TRA = CPVTPIZUTIPY FROM DATUM TSAYSFGRMATIC'Yp POT = CnYT?I-'/v
*' BUTInY FPOV PflTEYTlAL CDEFFICIEYTSv COLL = COYTQTEUTI~U "3!4'/9
*' COLLCICATJO'I DETERWINEP PAiiT UF FSTIflATE? WHF'! THFFC ONLY '+AS",
* @
* v
PEFY USkD ONE SET OF GBYESVDTIONC (DIFFERENT FsfJ'4 P'JTgCOCF!=,) '6,
COLLI = CPk'TXIkUTLON FPPF ESTIVATF OF ANflXbLOUS POT. DFTFR-@/v
* @ MIYFL'I FSOM FIRST SFT OF Oi3SE9VATIONS~ COLLZ = C3YTRIBUTIT)".'/r
* @ FPOV ESTIYATE CJRTAIYED FROM SECDVD 5ET PF ClFSFRVATITlYS, PQED='/,
S' PREDICTFD VALUE IY YEW REFFREYCE TYSTFMI WHFY PRF9ICTI7YS APF'/*
*l COMPUTFCI ANO FLSE Tt-IF SUlZ OF THE CONT%IBUTI7NS FRfkl THF PDT.s/t
*' COEFFIC IEhTS AKD F1 QST TSTlb4LiTE OF ANDMDLCJIJS POTENTIAL. A W J '/p
*' PREC-TRA = PREDICTIPN PR ?Ui+ OF CONTRIEUTIONS IN THF nLr PE-@/?
*' FERENCt SYSTEM.')
C.
C. INPUT OF PATA OF REFERFYCF SYSTEM.
WKITF (6,106)
106 FORgAT('OoEFEQEh!CE SYSTEM:')
C INPUT OF TFXT RESCiZIRIFiG REFERENCE SYSTEH I FqkX.72 CHARCICTLFS
READ( 'T9103)FMT
WRITE(hqFf1T)
C IRPUT OF SFYI-MPJOR AXIS (METFRS)? l/FLATTENlYG, VALUE OF TWO LOGICAL
C. VA41AbLESI LPCjTSD = GQAVITY I Y DOTSEAM SYSTFM AVC LGRS67 = GPAVITY RF-
C FER TT; GRS 1967,
READ( ~ ~ ~ ~ ~ ) A X ~ / P F O ~ L P O T S D / P L G X S ~ ~
120 F@RMAT(FlO.LyFlPe5,2L2)
F1 = DI/Fn
E21 = Fl*(P2-F1)
IF (LPOTSn.OR.LGRSh7) 50 TO 1021
C. IKPUT OF GM CIF- PEFERENCF-SYSTFM OF DPSERVATIDNS.
REAL( 51 121 )CM1
121 FOKMAT (Dl5.8)
1021 IF(.Y3Tg LCTRSh7) CALL CSAVC(AX~,FI,GM~V~TLFOTSDIUREFOT~R~F)
I F (LCFS67) CALL GRAVC(6378160.0C0,D1/2?8.24?17DO13.4~603D14~~~
*LFyU?FFOTGQEF)
WRITF (h,177)AXlpFO,GP,EFpUCiEFO
122 FOPMAT(O0A -'7F10.17' H V T
4' 1/F =@TF1O.F/ p
*@
REF.GQAV1TY AT FQUATOQ ='?F12.2~' MGAL'/r
8' POTF?

C
C INPUT OF TFXT DESCRIBING SOUrlCE OF THE POTENTIAL COEFFICIENTS (MAX, 73
C CHARACTF? S ).
RFAD( S9lO3) FMT
WPITE(h,l30)
130 FORYAT(@OSOURCE OF THF POTFNTIAL CDFFFICIEYTS USED:')
WRITE(h,FMT)
C
C lNPUT OF GM (P'ETERS**3/SEC*+2) A (METEQS) 9 THE WOPVALIZFD CnFFFICI FNT
C C'F DEGREE TWO ARD OPCEP ZERI: (THF FCOYD DFCRFF ZDklAL YA9!lORIC) MEL-
C TlPLlED 3Y ZeODhT THE MAXIMAL DEGhEF DF THE CnFFFlCIFYT', A LC

1050 N - O
WR17E (69lOQ8
109 FPRMAT(@OCTART rF CTLLOCATICY 1:
C INPUT CIF TEF INBECiEP KTYPE DETEGPlINIING TYPE OF DEGREE-VARIANCE Yf'PFL
C USFD FOR PFGREE-VPRZANCES OF PEGKEE GREATHER THAY IVAX (SEE BELOW)
C KTYPE MAY L!? FCUAL TO le 2 AND 3 CORRESPONDING TO THE DEGREE-VAQIANCF
C FIU.lELS l* ? AKD 3, CF. REF(B)? SECTION 7a2s
WEAD(59102)KTYPE
LKFQl = KTYPEeFQeI
LKEQ3 = KTYPE.EQ.3
LKNEl = .VOT.LKFWl
I F (KTYDf c.LT.31
READ( 5 p l Q 7 ) IK
B07 FnRr44T ( 14.1
GO TO 1036
IF tKTYPEeLEOO aOX. KTYPE-GE.4) GO TO 9999
C INlTIkLIZPIT ICJY CIF:
TKO = 1K-f
IKI = IK+l
IKZ = IKt.2
VRQIARLFS I N COMMON BLOCK /SCK/.
C
TKA = IK4YIK1
1026 WRITEBh,141)
141 FORMAT( 'OTHF
A*(T-l)/sl
MODF L A'JOMALY DEGPEE-VARIAVCES ARE EQUAL TOR/*
GO T9 (lO379l03EplO39) ,K%YPE
P037 WRTTE(6rl42)
142 FORMAT(B+6~9Xo'I.'1
cc7 Ti) Inno
1038 WhITF(bpl43)
143 FORMhTI'+OqYX,VI-Z). '1
GO TO 1COO
1039 kJRITE(69144!%K
144 FnR.XATIs+"9X9' (( I-21*(
C
1000 CXR = Dn
DO 1035 I = 1 9 250
1035 S I GMA I 1 = L0
sues = DO
SUMSlG = DO
SUKRS? = DO
IV1 = -1
MAXC1 = li
C
I + ' 9 1 4 P ' ) 1 *'l
C IWUT OF C:I%STANTS USF-0 FOR THE FINAL SpEClFICATION OF T1iF DFG9EE-VAK-
C IANCE FEODFL* R = RATIT! bET,+EFN THE BJERclAVMAR SPHERE RADIUS AKn THE
C MEAN FARTu RADlUS9 VFRDC2 = VARlANCE OF TKE GRAVITY ANDMALJES AN9 IMAX
C WHICH IS FOUPL TO THE (3FDFR OF TUE LPCAL COVORIANCE FUNCTICIN USF9.
C
C
TtJUS IMAX=n IkPlCATES9 THAI WE ARE USLNG A GLOBAL COVA9IANCF F2YCTION.
I N THIS LAST CA$€ 4 SET DF EClPI9ICAL ANOMALY DFG9EE-VA'IIAVCFS '3F ORDER
C UP TO IMAXO CA% DE INPUT, THEESE DEGPEE-VARIANCES WILL HAVE 10 BE

C GIVFN ON THE SUPFACE OF TPE MEAN FAXTHv CFoRFF(E)rEQo(PfJ)o
RCAD( SplGl)R~VARDS2~IPAX
101 F!lRMAT(k9e6,F70?r
I F (R.GT.Dl mnR,
14)
VkFDG2.LT~GOmDRmlMAX.LT,O) GO TO 0999
LZFP7 = IMAX .YEm
IMAXl = I&!AX+l
0
IF(LZEQO)GO TO 1040
C INPUT OF TPF FCRMAT OFT THF PIGHEST DFGREF OF9 AND THF EMDIPICAL ANR-
C KALY DEGREF-VARIAKCES
102 FORMAT ( I 2 9
READ( 5tlC2I IMAXO
I V UYITS OF MGbL**2.
IPLlAX = IRAXO
IMAX1 = IVAX+1
READ( S, 102
P03 FUFt/lkT (QAP
Fb4T
READ(5,FMTI (SIrlNA(I)t I = 39 IMAXl)
C ROTE TVAT THE DFCRFC-VARIAVCE OF OF'OEP I I S STORFD IN SIGMA (I+I
C
DO 1001 I = 3r IMAXl
1001 SUMSIS = SUMS IG + S IGYA ( I)
1040 I F (Itl#rX1+ISoLT.250) GO TD 1002
WRITE (6,1FE)
l08 FnRMAT(' SUSSCRIPTS OF ARPAY 5IGMA EXCEEDS APYAY LIMITt STflP*O
GO TO 99Q0
C
1007 S = R W
S2 = S*S
RL = (Dl-k)*(Dl+R)
S1 = RL
RLNL = DLDG(QL)
I F (IMEX.LT.3) GO TO 1004
R I = GFLOPT( It4AX)
D0 1003 J = 39 IPAX
GO TO (1101~110?~1103)~KTYPE
1101 R& = (FI-Fl)/RI
GO TO 1105
1102 RA = (RI-?l)/(RI-P29
GO TO 1105
1103 RA = (RI-Cl)/((EI-@2)*(kI+IK))
1105 SUMRST = (SUMRST+RA)*S
1003 R I = RI- D1
C
SUMRST = SUKRST*52
1004 DIF = VARTGZ-'UMS I G
I F (DIF,LT,PO) GP Tfl 9999
C CCIMPUTATION OF THE NORFALlZlNG CCYSTANT A9 CFo REF,(B) FARAo 3030
G@ TO ( I l ~ 6 r 1 1 0 7 ~ 1 1 ~ 8 ~ ~ K T Y P E
1106 RA = KLYL+5/qL--SZ/DZ
GO TO 111P
1107 RA = D1/RL-S2*RLNL-SZ-S-n1
1,

C WHEN PRF-DICTIPYS SHALL RE YADF I N PDINTS C'F A UPIFOFV GRID, LFKND IS
C TFUFTW"tN TpE tS7 IMAFFD ERRPR OF PKEPICTICIY StfALL PE C'3KPUTED. LCnMP
C IS TRUET WHCN PEiSE'?VTD k\D FPFDICTED VALUES SUALL BE CnMPAIREDm
2000 PEbO( "200)LGP
200 FURHAT(3L3)
InrLFF~NO~ LCOMP
1 F ( .NDTo (LFRkO.AMD.LkESOL)
LERlJO = LF
1 GO TO 3002
WRITE (h?Z?t)
226 FORMAT(' *** ERROR WILL NOT BE COMPUTEDl F.EQUIRED YFQ YOT STPREDe
****l)
2002 LCOMP = LCOMP .AND. (,?;QT.LGRID)
I F (LCOM.CIRe(
LCOM = LT
oNOTmLCORP) GO TO 2005
C lNPUT OF SCALE FACTPQS, USED FOR TPBLE OF DlSTRIRUTIOY OF DIFFEnFNCE?
C SEE SUBROUTIYF CO14PA.
RFAD( ~T?O?J)VGTVF
CALL CfiWPA(L1G1VF
C
C
VC, IN FAGAL APD VF IN ARCSFC.
2005 LNFRNO = .NOT.LERNO
LMAP = LF
LMEGR = LF
DM = D1
@A = D@
I F (.NQT.LGRID )GC TO 2000
C
C INPUT c c n ~ n r n \ n u ~ ~ h i+ uIES kr r ( LLTITUEEpLDYGITLIDE IN PEG9EE.S A';? DFi. OF r.7 IF:;-
C TFS) OF S3UTH-WEST CORYER OF GYID? MAGYITUDE CIF GYID IVCPEPFYTS I N
C N09THFKN ANP 'ASTERY DIRECTIOYS (V1YUTES)r YUWBFP nF IYCSFYFYTC IN
C
C
TPE SAME DI'ECTIONS? THE VALUE OF IKP: ( l FOR ZETAp 2 FOR DFLTb
G, 3 FOR KSI, 4 FCR FTA AKD 5 FOR (KSI?ETA)) RP = THE '?ATIT BCTWFEN
C THE RADIUS OF THE SPHERE OV WHICH TYE POINTS PF PREDICTICX A9E 'ITUA-
C TED ANG RE? TUE VALUE OF LMAP, WUIC" I S TRUE 9 Wk'FN TVE PREDICTIO'IS
C SHALL 6E PR INTFD AS A PSI XI TIVE MAP? THE VALUE OF LPUYCI-I, WHICH I S
C TRUE WHEY THE PREPlCTIfl!'IS SHALL 9E PUVCHFD AND THE VALUE OF LucAY,
C TRUE WHEN THE PRREDlCTED QUANTlTIES ARE MFAN VALUE' (BECAUSE THIS IF+
C PLIES? THAT TPFY ARE REPRESENTED AS POINT VALUFS I N A CFRTAIV YFIGHT).
READ( 5,201) IDLACTSLACTIDLOC?SLGC?GLA~GL@?NLA~NLO? IKP~RPTLMAP,
*LPUNCH?LPEAN
201 FORMAT(?( I5,Fb-2)
I F INLA.GT.19
LWLONG = LF
*OR- NLO.GT-19 .OR. IKPeEQ.51 LMAP = .FALSE,
LGRP = IKP,EQe2
NAI = 0
NOI = O
IOBS2 = 0
I H = 0
H = DC1
I F (.NPT,LMFAK)
LKM = H-GE.1.OD4
H = (RP-Dl)*RE

C
H0 = H
OBS(P) = K
IF (LKM) nescl, = M*I,OD-~
LSTUP = LT
JANG = 2
NO = Q)
2001 SLAT = NAI*LLAaSLAC
SLOY = NC!I*GL@+SL@C
ISLA = SLAT/60+OR13-2
]I SLO = SLOId/hO+n. 10-2
lDLAT = IDLAC+ISLA
IDLOV = IDLCC+ISLO
SLAT = SLAT-604ISLA
SLON = SkON-bO*
NE = NC+l
U = H0
ISLO
IF (NO1 .EGO
NO1 = "\iOl+B
GO TD 2004
2003 NO1 = 0
NLO) GO TO 2003
NAI = (:,A I+P
2004 IF[NAIeNE*O ,CR.
GD TO zon7
C
KQI oNEo 1)GO TO 3031
C INPUT OF OVE PATA-SET OF OtjSERVATIONS 09 COOR.DINkTES OF P?EDICTID%
C POINTS. ALL RECORCIS YUST EE PUNCPCD K'\! THE SAME WAY, THERE ARE THE
C FOLLOWING CESTR~CT~TJNS ANb UPTIDNS: k STATION NUM9F9 MAY FE USFDp BUT
C IT MUST OCCUPY THE FIRST GATAFIELD ON THF RECCtRD. THE TiJ@ NEXT DATA-
C FIELDS MUST CCWTA1N TUE GEODETIC LATITUDE AND LOVGITCIDE ( IN P% \:R.FI-
C TRARY OP9ER). IN CASE THE HEIGHT IS GIVEN* MUST IT BE PUNCHFD I? TPE
C NEXT DATAFIELP. THE FDLLOkIING UP TO FOUR DATPFIELDS WILL WAVE T'3
C CONTAIN THE OESERVED GUANTlTY (G!? QUANTITTEZ WHFN A PATT nF DEFLECTI-
C ONS ARE EbSFRVFD) AND CONTINGENTLY THE STANOPRLi DEVIATIPMS ('IIIHFN LE IS
C TRUE AND LSA 15 FALSE), A LIST OF C30RUIYATES OF PC:INTS WILL VAVF TO
C CONTAIN A UEIGL'T CR A FlCTICIDUS OBSERVATION. TL'F LAST DATAFKFLD HAVF
C TO HOLD THE VALUE 13F A LOGICAL VARIA3LE LSTClP? TSUE FOS THE LAST
C RECORD IN THE FILE AND FALSE (I.E. BLANK) OTHERWISF,
C
C INPUT OF THE FORMAT OF THE RECORDS HnLDING THE OBSESVATIDN @R THE CO-
C OR.DINATES OF THE PREEICTIPN PGlNT,
2006 READ( 5,103) FKT
C IYPUT OF VARIABLES SPECIFYIYG THE CCXTENT OF THF RFCOQDS. INO = Z t
C WHEN THE STATION NUMGER IS PUVCHEDT O DTHFRWISE, ILA* IL'3 THE VUKBEQ
C OF THE DATAFIELDS OCCUPIEO FY THE LATITUDE ANC! THF LONGI'IUDE 2FCDFC-
C TIVELY? IANG SPECIFYING UNITS OF AhlGLES (1 FCI? DEGREES, MINUTES* ARC,-
C SECONDSp ? FOR DEGREES, MI%I.!TES, 3 FUR GEGREFS AND 4 FnR 400-GRADFSIr
C Iu = THE NUMBEk f3F T U E DATAFIELD WLDlNG TUE VEIGPT (PE90 WHEN WO
C HEIGHT IS C3NTAINED)r IOPS1 IOES2 = THE D4TAFIELD NUMSER OF THE FIRST
C AND THF SFCOYC' OBSERVATION, RESPFCTHVELY (ZERfl WHEY N3 FIRST OR SFCOVD

C OBSERVATIDNI* IKP* SPECIFYIYG THE KIVD OF OOSE?VATION~ (1 F99 ZFTA? 2
C FOR MEASU2EG GRAVITY* POINT OR MEAN GRAVITY ANDMALIFSI 3 F9R KSIT 4
C FUR ETA AF!D 5 FPR PAlR nF CIFFLECTIONS (K.SIVETA1 39 (ETA,YSII?(IN THE
C SAME nRDER AS THE LATITUDE AND THE LONGlTUDT)
C TUEN TuF. PATIP RP (CF.REF(B)r EQ-(07)) EETWEFN THE SPHERE ON WMICH Tt-!E:
C OBSERVATIONS ARE SITUATED AYD RE, THE VALUES OF 5 LDGICAL V69IABLFS:
C LPUNCH = PUNCH 09S. Oi? PRFOIClED VALUE AY@ CCNTIMGFNTLY TI!EI9 DIFFE-
C RENCE* LWLONG = LONGITUDE I S PCJSITIVE TOWARDS WEST* LMFAN = THE DYED-
C lCTED OR OBSERVED QUANTITY IS A PEAN VALUE* LSA = ALL OESFRVED CUAN-
C TITIES 'JkVE TUE SC.HE STANDARD DEVIATIONS AND LKM = THE HEIGHT IS IN
C LNITS OF KILUMETERS.
READ( 592OZ) INOr ILA* ILO* IANGTIH*IOBS~~
IOBS~~IKP*RP*LPUYCHTLWLONG
*,LMEAVrLSA,LKl-l
202 FORMAT(t13~F10.795L2)
GM = Cj1
DA = DO
LGRP = IKP-EQ.2
I F (LGRP I REAP( 59203) CMvDA*LF!F!GQ
C LrZiEG9 IS T9UF WHEN THE MEASURED GqAVITY VALUE I S INPUT. OM ArJD DA ARF
C AN ADDITIVE ANC! A MULTIPLICATIVE CONSTANT* RESPECTIVFLY? WHICH CAN BF
C USED TO CI?NVERT IKPUT' VALUES TO MGAL PK CORRECT FDR A SYSTEMATICAL
C ERROR-
203 FORMAT( 2F10.21LZI
C INPUT OF STANDARD DEVIATLON.
I F (LSAI REAG(5*212)WM
212 FORMAT( F6.2)
C
2007 LRFPEC = IYP,FQ-5
C INITIALIZATInN OF VARIABLES IN COMMON BLOCK /PR/.
LONEC9 = sNOToLREPEC
LNKSIP = IKPoNEo3 .AND. LONFCO
LNETAP = IKPoNE-4 .AND- LONECO
LDEFVP = IKP oGT- 2
LNDFP = ,biOT,LDEFVP
LNGR = .NOToLGRP
C LREPEC IS TPUF, WHEN TNO COLUF!YS CAV BE COHPUTED AT THE SAME TIME.
C LNSKIP? LNETAP IS TRUE? WWP\I THE OBSERVATION OR REQUESTED PREDICTlON
C IN P IS NnT KS1 PESF. ETA.
LZETA = IKP-EQ.1
LKSIP = ,MOT,LNKSlP
IF (RPoLTeR) R.P = D1
C CHECK? THAT POINT I S NOT INSIDE THE BJFRHAMMAR-SPHERE,
C
C CONPUTATION OF CCINSTANT'S USED TO NORMAL I ZE THE NORMAL-EGUAT IONS 09
C SCALE TEE FSTIMATES OF THE ERRCihS OF PREDICTION.
NI = MP,XCl
P(JQ) = D1
P(JR+l) = RP
INDEX( JR+1) = IKP
PW = D1

COSLAP = Dl
SINLAP = DO
RLATP = P0
RLOQGP= Gn
CALL PREP(S~SRFFPUO~A~ISBI~QQ~JRY~~~~IYAXIPLF~LF~LT)
PW = C(MAXC1)
IF (PW.GT,CbO)
P(JR1 = PW
PW = DSCQT(PW1
W = WPtIKP)
IF (LNGRI W = W*RP*!?P
I F BLZETAI W
PhO = PN*k
= W*RP
PW2 = PWO~~PWO
IF (LCREFI JVL = -1
C
C OCTFUB OF HEACIYG AYD INIIIALIZATION OF VARIABLES.
LINVPF = LONEC9c+OPe (IOBS2~EC80).r!t?~ (IO8SleLT.InBS2)
LOUTC = LNFCB02 ,LCOb4P
I F (LEFAR) WRITF(bp205)
205 FORMAT( 'OTHE FCLLDWING QUAKTITIES APE MFAN-VALUES9 AND ARE RFPRESE
*NTFD AS PCINl VPLUFS IF! A HEIGHT Yeo)
I F (RP eGTol*03G0eAND. (LKCIPeGPebGRP) ePNDe(LTP.ANeCReLPnTIeANDe
*LPOTSD) WF 1TF (6 pZO4)
204 FOQMAT('O** WARNING: THE HEIGHT MAY BE TCO BIG FO? THE COMPUTATION
* DF'pBpt THE REFERENCE GRAVITY OR THE CYANGE IN LATITUDE **'I
C
CALL EEAD i iKP ,LONECOt FiiO, XP 1
C INlTlALlZATIf!Y OF LOGICAL VAP IAELES USED TO DFTFRMIYE WulCu QUAYTITIFS
C WE WILL HAVE TO ADD TOGETHEQ TO FOPE THE FlYAL DUTPUT OR TO QETFPMINE
C WHICH QUANTITIES W1 LL €F INDUT,
LADRA = TH.hEeIA
LADDBC = IE.NE, IC1
LbDDBP = 1SsNFe 1P
LADGP R = LADDBP ,AqD .LREPEC
LTYB = LTRAl!.ANDe(IT.YEeIG)
LTEB = LTRANeANDe(IT.FQeIBI
LOEl = LESAND-( ,NOT-LSA .ANGeLO?iECO
LOE2 = LEeAh1De( ekCITeLSA) .ANDeLPEFEC
C K1 "AS PEEN IIVITIAL.IZEC PY THE CALL OF 'HEAD'.
C BER OF QUANTITIES READ IN TO TUE ARRAY oes.
IF (LOF11 K9 = Kl+P
I F (LOEZI K 1 = K1+2
C
I F (LGRID) GO TO 2031
C
IT IS EQUAL TO THE NUM-
IF (LCOMP) CALL COMPB~IKPpLNG~~LPEPFC~LOh'ECO)
C
C lWUT OF COClQDINATES OF OBSEQVATInN 09 PRFDICTION POINTS 4YD CflNTIN-
C GFNTLY THE OESFRVED QUANTITIES AND THFIR STANDARD DEVIATIflNS.
%J = IANG*2+INO-1

2033 GO Tfl(?O249?025 920269 2027~20?Pv?029~?07E 9?079) 9 IJ
2024 RFLD(5,FMT) IDLATTMLAT~SLAT~~~LONTMLON~SLO~~(DE~~
119 I=lvKl TLSTOP
GOTO 2030
2025 READ( 5PFMT)NOrIDLATtMLATtSLATTIDLnNTMLr)Y tSL3Vt (OBS( 1) ~1=1 ~ q lT l
*LSTOP
GO TO 7070
2026 READ(~*FPTTIIJLAT~SLAT~~DLON~SLDN~~OES(I
)rI=lyKl)rLSTOP
GOTO 2030
2027 HEAD( 5rFM7 )NOIIDLAT?SLATT ]:DLC!N*SLf)fVy (OBS(1) tI=lyKl 1 9LSTP0
GDTO 7030
2028 READ(5rFMT)SLAT ,SLOYv (OBS(I 1 v]:=lyKl TLSTOP
GO TO 2030
2029 READ( ~,FMT)~!R~SLATTSLCN;* (C:BS( 1) 9 I=ltKl) TLSTOP
C
2030 IF(1LA .LT. ILOIGC TO 2031
C CORRECTING IYVERTED ORDER 3F LAT* AXD LONG*
I = IDLAT
IDLAT = IfLGN
IDLON = I
1 = MLAT
MLAT = MLPN
MLOV = I
A0 = SLAT
SLAT = SLDN
SLON = A0
C
2031 CALL RAD: ICLAT~~!LATqSLAT~RLAT?t IAYG1
I F (LWLZNG-AVD* IAYG.LE.2) IDLON = -1DLON
I F (LWLI)NG.AYD. IANG.GT.2) SLOY = -SLOY
CALL RAD(IDLON~MLONrSLOXrRLOYGP~1ANG)
COSLAP = rcm (Q LATP
SINLAP = GSINIRLATP)
I F (LGRIG) GO TO 2049
C
I F (Lot31 WflBS(Y+l) = ORS(K1)
IF (LEE21 NOBS(N+l) = QaS(K1-l)
I F (LOF2) RG5S(N+2) = OBS(K1)
I F (IH.KE.0) GO TO 2050
I F (LRFPEC) CBS(12) = ObS(2)
OBS(2) = neS(1)
ORS(1) = E0
GO TO 2051
1050 OES(l?) = OeS(3)
I F (LWLf'NG-AhDe LKPedT*3) OBS( 12) = -ClPS( 17)
C CCRRFCTING TUF OR SERVATION f;Y AN ADO ITlVE AND MULTIPLICATIVE COVSTANT
C (IS USED ONLY FOR GRAVl TY OBSERVATIONS).
2051 ORS(2 1 = OSS(2)*DM+!?A
H = ORS(1)
I F (LKM) H = H*1.0D3
C CONVERSION OF PEIGHT INTO METERS.

2049 I F dLNGR? GO
C
70 2056
IF (".LT,25,0E3 .AND*( -NFToLPOTSDI ICALL EUCLID(COSLAP9 SINLAP*
*RLflNGPsH~EZlqPXIl
CALL ?G9AV(SINLAPpHpO 9CREFl
C COMPUTING THE GQbVl TY APiOMALY e
BF 6LPFGR) ObS(7) = QBS12)-GRFF
C
2056 %F (LINVDEI GO TO 2032
OBB = DES(l2)
OBS[%2) = DSS(2)
065(2) = CiEl
IF (eNOTmLE.9E.LSA) Gn 7C' 2032
MM = WOBSINsl)
WCBS(%+l b = WCPSIN+ZI
WORS(Y+21 = MM
C
70'2
C
OESiIP.1 = DO
IF (LREPEC9 PHS(IE1.I = DO
IF ILNTP) GC) TO 2055
IF ILNTRAY) GO -r3 2053
CALL EUCL ID(COZLAP~SI!JLAP~RL~NGP~DOI€~~~AX~ I
CALL 7RAVS(SI?~iLPPqCflSLAPqRLAT'PtRLONGPqIKPqITIi
IF {LNGRI GO TO 2053
GREFP = GREF
IF (HeGE.25 -003 I CPLL EUCLID(COSLAP~SINLAP,PPLONI;P~w~E22~AX2 1
CALL Ri:~RkViS1VLb.PpHr15rGktF)
OBS(IT9 = GREF1-GREF
IF (LPOTSD) OBS(IT1 = OES(1T)-13.700
GP.EF1 = GREF
C
2053 1F (LNPOTI GO TO 2055
IF (LVGRS GO TO 2054
C
C FfiR GRkVI TY ANCIMALlES WE USE? THAT THE NEW REFs POTENTIAL GIVES A PFT-
C TER ESTIMkTE CF THE GEOMETRIC HEIGHT? SO THAT THE NEW REF. GQAVITY CAN
C BE COMPUTED AT THIS HEIGHT,
CALL EUCLIG(CFSLAPtSINLAP~RLOI\iGPtDO~E22qAX21
CALL SPOT (RLATP ICOSLAP~RLCVC~P~~~
1 1 9 GRCF9LiREFO?DO)
H = H+OBSI11I
2054 CPLL EUCL ID(COSLAP9 SI NLAP 9RLOrYGP ?H, E 22qAX2I
IF QLZFTA) CALL UREFER(UREFtl5,Hl
IF (LkSIP) CALL CLAT(CLAv159p9RLATPI
CALL GFCJT (RLATPpCOSLAPvRLOIVGPv IKPIIPvGREFvUQEFTCLA)
IF (LADPBP) OLS(IB9 = OSS(IBI+OBS(IP)
IF (LADRPR) OPS(IBl1 = C)bS(IE1)+OSS(IP1)
2055 IF (,rYOT,LCREFl GO TO 2052
C
C THE VARIAFLE IVP IS USED TO INDICATE q THAT THE DEGREE- VARIANCE5 STCRED

C I N SIGMk WILL WAVE T@ BF CUAYCED FPPM COLLOCLTIDN I Tn COLLOCATION 11,
C THE VALUE IS TRANSFFR.RER TO PRED BY THF CCWMOY PLflCK /PQ/.
CALL P9FD(S?rFREFRpUROpAR,090~2p
1v1 = -1
flES(IC1) = PRFDP*W
I F (LAG~JRC) OEsZ(IR) = OFS(16)+08S(lCl)
I F (LOYECO) GO TP 2052
OBStIC11) = FRFTAPWW
IF (LADOBC) oDscrei I=
2052 I F (LPPEL) GO TO 3021
ossc rei)+ossc rc111
C
C STORING COOKRINATFS AND R IGHT-HAYD
C COLUVNS AYD IC TLJE STATIORS.
N = N+1
SIDE OF NOilYAL-EQmp N CnUUTS THE
-
IC = 1C+1
IF (LTKB) OES(1U) = OFS(l!3)-GES(IT)
1F (LTER) OI.S(I1)) =+JFS( IT)
IF (LKSO) 06S(3) = CSS(2)-OBS(lU)
OEl = VBS(K2)/PWO
B(Y) = OS1
IF (LSA) KOBS(Y) = W!-!
SSORS = SS05S+OF1**?
I F (LRNECC) GP TC ?C33
C
I F (LTNS) IfGC(IU1) = T~ES(IBl)-OBS(ITl)
IF (LTEB) OES(IU1) = -ObS(ITl)
I F (LK30) OSS(13) = OBS(12)-ORS( IU1)
OB2 = OBE(K21)/PWG
SSOGS = SSf€S+Clb2*=+2
N = N+1
B(N) = P82
I F (LSA) WOBS(Y = \$M
2033 I F (N,LE115?E) GO TO 2060
C CHECK OF hUMFER GF 05SEQVATIONS NOT EXCEEDING 4R9AY LIMIT
WRITE(6p?29)
229 FORMAT( ' NCMEER OF OESERVATLDNS TOO BIG* *** STOP ***'l
C
2060 COSLAT ( IC = COSLAP
SINLAT(IC1 = SIRLAP
RLAT( 1C) = RLATP
RLORG ( 1C 1 = RLONGP
CNR =RLATF*IC+ChR
C OUTPUT OF flB5ERVATIVNS.
CALL fIUT(Y9r IDLATTMLATTSLATT IDL~bI~ML~h'psL~NpLON~~n)
IF(.NOT~LSTOP)GO TO m23
C
230 FORMAT(3L2)
READ( 59230) LST@P,LRFSrL,LWRSOL
C
IF (LRESOLmAND. LWRSOL I G(3 TO 9999
~ O B S R ~ Y ~ F T I M A X ~ R ~ L T ~ L F T L ~ )

C
C FSTABLIWHVG A CATALnGUE OF THE ObSE9VAYIPNSrMRXlrZlALLY 9 SETC ALLf'bJEDI
INDFX(JP1 = IC+I
$R = JRt-2
1F IHNDFX(J2-3) .NEe IKP *OR* (DAGS(P(~R-~)-RPP~GTC~~OD-?)I
9Gll TO 3034
JR = JP-2
IYOEX( JP,-l 1 = IYDEX[JP+l)
2094 I F [(JR-II) .LTC 191 GO Tfl 7035
HRITF(h,298)
298 FORMAT ( @ DPSFRVATSONP ASRANGED TOO COMPLICATED B
GrJ 10 q9G5
2035 HF (,YDT,LST3P) GO TO 3006
C
C FKD OF INPUT CF OFSF9VATIfVJS. M = NgMBER OF CibZFSVATTDYS, lO3S =
C RUPLFP CF I'tLSFQVATI PI\' PC1 KT?.
100s = IC-1'0
N = N-ISP
"41 = Y+l
S(Nl+iSO) = SSOPS
C
I F I.YnT.LF) CO TO 2036
WRlTE(bp2Q7)IROFS(I+ISOlr I = 1 9 NI
297 FOEPATIPO~JR"d9AKD DTVlATIFYS OF TUE PESFRVATIONS 1Y T W SP:IF SEOUF
*NCE'/tV AYO l",' THE SAME UYITS AS THE OSSESVATLflNS:'/yf' '9lCFE*2))
C
2036 IF (LQFSGL) GO TO 3?2[
C
I F (LDEFF) GP Tfl 3027
LDEFF = LT
NT = 3
IPLMC = L700
C I N THIS P9RGPAbl-VFRSIONt THE ARRAY C HAS DIYENSIOV IDEYC AYD ITS
C VALUES AQE STbRfG G? RETRIVED FPOM UNIT 8 3Y NT READ 07 WqlTE ?PERA-
C TIORS.
C
C SETTING UP A CPTPLOGUE OF THE NORMAL FQCIATIONS
C Nb I S TPE PFCflFD ?:UEIGESr IC COUNTS THE YUMBER nF COLUMVS WIT'JRN A
C RECORD* &"!D TPIS h:UME\FR I S STORFD IY NCAT(101)$. YCAT [ I ) WILL CnYTAIY
C THE SUbSCQIpT CF THE GIAGONAL FLEMENT OF COLUFY I-Hp AWD ALL THE
C FLEPFRTS @F ISZF WILL 6F ZEP.0 BECAUSE WE WORK WITH A FULL MATRIX,
C NBL( I) COYTAIVS JHE NUMBER OF THE LAST COLUYN I N RECORD 1-1 NOTE
C TPAT MAXIIYALLY GP CPLUMNS MAY BE STORED I N ONE RECORD,
2037 YET = I
YEL(1 )=C,
N5 = 1
IC=O
If=O
C
DO 2304 I = I p N1

C
IC=IC+l
ISZE( IC )=Q
NCAT( ICI-11
H1 = Il+I
1?=11+1+1
IF(BIZ.LE.IDIMC),AND,(I,NE~Nl~.ANDo(IC.LEoQU GO TO 2304
NCATB IC+P) = I1
IFBIeYFoYll GO TO 2303
12=I2+Nl+l
IF(l2.LE.IbIMC) GO TO 2301
C SECURIXG THAT THF LAST COLLUYN + ON€ MORE CAN BE STORED IN THF SAME
C RECORD.
NCAT( lnO)=IC-l
WRITF( bfYeT+?)C3tNCATIISZE
NET = I',E;T+NT
NE=NB+l
NBLW )=]-l
NCAT(2)=Nl
I IL =N1
I[ c=1
C MAXC I S THE SUSSCRIPT OF THE DIAGCNAL ELEMFVT OF THE CAST COLUYnS, AN13
C MAXC2 IS THc SUbSCRIPT PF THE FICTICIDUS CIAGfGUAL FLEMENT OF THE RIGHT
C HPND SlDE (IN WHICH THE SUARE-SUM OF THE NORMALIZED DBSERVATIOVT I S
C STOREDJ.
2301 MAXC = 11-YI
MAXC2 = I1
C STORING THF RIGHT- HAND SIDE c
00 2307 J=ltNP
2302 C(MAXC+J) = b(J+ISO)
C
2303 I1=0
NCAT( 100) =IC
IQ = NET
WRITE(B'IQ)C~NCAT~ISZE
NBT = NBT+NT
I[ C=O
ME=!\1!3 +l
NBL(NHl=I
I F (NbeLEo309) GO TO 2304
WRITF(6r29q)
299 FORMAT(' RESERVED AREA PN FILE 8 TOO SMALL')
GO TO 9999
2304 CONTI VUF
C
MAXBL=NR-l.
MAXPLT = (MAXBL-1) *NT+1
C
C CPMPUTATIDN OF ELEMENTS OF NORMAL EQUATIONS (FQUAL TO TUE COVADIANCE
C BETWEEN THE DBSERVATIONS) c TNE COEFFlCIEYTS ARE STPRED IN TUE ONE-OI-

C MFNSIOhAL ARRAY C, CnLLUMN AFTFR CQLLUHNpTME DIAGONAL ELEMENT HAVING
C THE HElGHEST SUPSCRIPBe
C
G lNlTIALlZ ING VARIABLF?:
C N I IS Ih MA%& THE SUEISCRIPT OF THE FIRST FLEWFNT OF COLUMN NC IQ AQRAY
C Ce I N THE SUbFGUTINF PKEEp NI I S THE SUZSCEIPY f9F THF FLFMFNTS 3F THF
C COLUMN, Nb
C STOKED AYD
15 THF NUMbFK OF THE BLQCk IN HHICtl BEF COVARIANCFS ARE
I1 IS TEE NUMBER RF TUE LAST CPLUMQ STORED IN BWE ELDCK,
C HCYEXT IS THE NUMPER OF THE FIRST COhUY'lsd HITHI3 A GROUP OF DATA WITH
C THE SAME CHARACTEXISTICS, (THE CWARACTERISTlCS ARE GIVEN BY TPE ARRAYS
C IhDEX AYD P (SUBSCRIPTS JC AND JC+BI).
NI = 1
NC = 1
4C = II
IVP =-l
N@=E
19=PF3LI2)
RfAD(E'NTIC39NCAT,HSZE
FINlj(8@19
NBT = l
ICNEXT = ISO+l
C
DO 3100 3C = l
ICC = IC+lSr7
9 IOBS
c
COSLAP = CLSLAT ( I CC 1
SINLAP = SIYLAT (ICC 1
RLATP = RLAT ( ICC 9
RLONGP = RLONC( ICCI
IF( ICC .NF. TCNEXTIGO 70 3003
IKP = INDEXIJC+I)
C INITIALIZATION OF VARIABLES IN COMMON BLOCK /PR/,
PW = P(JC)
RP = P(JC+l)
PCNEXT = INDEX( JC)
4C = J6+2
LREPEC = TKP ,EQe 5
e
LPNECO = ,NOTeLREPEC
LGRP = IKP.EQ.2
LNGR = .NOT,LGKP
LNKSIP = IKP.NE-3 -AND,
LNETAP = IKP-RIE .4 ,AND,
LOEFVP = IKP .GTe 2
LNDFP = ,NOT,LDEFVP
PWO = PW*WP(BKPI
LONFCO
LONECQ
IF (LNGRI PM0 = PWO*RP*RP
IF (IKPeFGoPl PKO = PkO*RP
3003 LIRST=LF.,EPEC eAh'D. (KC eEQe 11 1
C AS WE FOR LREPEC=TRUE ARE COMPUTIIKG TWT) CCLUMNS AT THF SAME TIMEo WE

l
C PLIST, 1N CASE THt SFCONC) CCLUYN I S THE FIRST ONE I N THE NFXT RECVPP
C STORE Tt'l S ONE TEMPORARY 1Y ARRAY B. THE PRLlCLEM WILL 7NLY OCCIJR WHEX
C WE AYE SETTIYG UP THE NflRMALEQUATIONS* LBST = B-STORE*
C
C
CALL P?ED(S~SREF~UOtAtIStIS3~IIt ICtQCt IXAXltLFtLBSTtLT)
NE = N1-l
DIA = C ( W )
I F (LE) C(ND) = DIA+(WOGS(NR-l)/PWO)**2
IF (LOYECfl) GO TO 3020
I F (LE) DIA = DIA+(wOtiS(YKI/PWO)**2
I F (LEST) b(NQ) = DIA
I F (.NCT,LbST) C(NI+NCI = DIA
C THE PPECEDING STATEMENT ASStIRES* THAT TPE DIAGONAL ELFYENT CflRREZPON-
C DING TO ETAP E',ECPP.',ES EQUAL TO TcJkT OF KSIP-
NC = YC+l
N I = hl+NC
C 3020 I F (NCrnLToIl*PNO-KC-LT.h) GO TO 3100
C
C STORING TPE COtFFICIErVTS OF THE NOSMPL-EB* ON FILE 8, RECORD N?.
10 = YBT
KRITE (8' 1Q)CtNCATrTSZE
I F (.NOT-LOkECI GP TO 3200
C
C OUTPUT OF CnEFF IC IFYTS OF NORMAL-EQUATIONS,
WS 1TE (h,3hO)NB
380 FORMAT('OCDEFFIC1ENTS 3F NORMAL-EQUATIONS* BLOCK 'tI4r/l
I1 = NI- l
I F (NQ.ECot4AXf?LI I1 = MAXC2
C
WPITE(6p361)(C(KI* K = LT 11)
381 FORMAT( ' ' 9 10F8 04)
3200 N9T = YbT+YT
NE,=9!3 + 1
NJ=1
I F (NC-NE-NI 11 = NBL(NE+l)
I F (NR.GT,t+AXBLI GO TO 3201
READ( IQICl
READ( IQIC2
RFAD(F'IQ)C3*KCAT*ISZE
C WE HAVE TI) READ ThE WHOLE CONTENT OF RLXK NB INTO ARRAY C, RFCAUSE
C WE MUST BE SUPE TUAT TUE RIGUT-PAS0 SIDE (WuICu ALREADY I S STORFDI
C I S PLACFD COPSFCTLY.
FIND(PtNB1
C
3701 I F (oNCTeLSST) GO TO 3100
C
00 1202 K=l*NC

CBSERVATlCNS AND KORP CF APPROXIMATION
C
C STORING SOLUTIONS DIVEDED BY A CDMMON FACTOR.
N1 = P
JRNEXT = 1SO+I
JP = I1
DO 3032 I = 1 9 HOBS
IF ((I+ISCl).VE*JRKEXTI GO TO 3030
IKP = INDEX(JR+l)
LONFCf9 = 1KP ,LT, 5
JRNEXT = INDEX( JR)
PW = PIJR)
4R = JR+2
4030 IF (LCIWECC) GO TO 303%
B(NI+ISfl) = C(MAXC+YIi/PW

NI = NI+1
3031 RfYI+T5O) = C(MAXC+YI)/PW
3032 N I = NI+1
IF (LCI) GO TO 3116
C
LC1 = LT
C INPUT OF LCQEF. WHICH I S TRUE WHEN ONE MORE SET OF OBSERVATION7
C SPALL BE IYPUT ANT) USED FOR THE ESTlMATIOU OF QNF MORE HARYOYIC FUYC-
C TION.
READ( 59230) LCPFF
IF (.NOT,LCRFF) GO TO 3110
C
C STORIYG AdAY THE WECESSARY CONSTANTS FOR COLLOCATION I.
SR = S
SREFR = S
URO = U0
ICtRSR = IPBS
AR = P
IMAXlR = IMAX1
NIR = N1
C INITIALIZING VAKIAFiLE? FOS STAR1 OF COLLOCATION 11.
I S = INAX+3
I1 = 22
JR = 22
IC = nlIR+2
N = 1C
IS0 = IC
IhDEX(2li = I C
WRITFtbv345)
345 FOFMAT('0STAST OF COLLOCATION II:'/)
GO TO 1OOO
C
C INITEALIZIYG VARIABLES FOP. PRFnICTIO%. HAXC1 IS THE SUSSCRIPT 3F THE
C FISST ELE%FNT I& THE- COLUMN FORKING THE QIGHT-HAND SIDE,
3110 LPEFD = LT
LNEQ = LF
LE = LF
LC2 = LCREF
MAXCl = MAXC+l
IkDEX(41) = 0
JR = 41
WRITE(69344)
344 FORMAT('1 PREDICTIDI\IS:t/)
GCI TO 2000
C
3021 NI = MAXCl
COLL PPED(S?SPEFvUO?A?ISv IS09 1 1 9 J13BSvhlv IMAXI vLT*LFTLFYNO)
I F (LYFRNO) GO TO 3b22
C
C(MAXC2) = D1

C
e
C
C
iF (LCOXP) CALL COPiPCfiUi
CALL OUT(KO~IDLAT~PLAT~SLATPIPLON~MLOY~SLONTLPNEC~I
I F (LMAPI IMAP(NO1 = OBS(IUl*100
IF(LGR1D ,AND- NAI oLF* NLAI GO TO 2001
KF (,NOT,LMAPI GO TO 3045
K = NLA+1
WRITE(6r360)IDLACySLACo I@LOCpSLCC,GLA~GLC!
360 FORMAT( "MP PF PREDICTIONS: @ / ' COORDINATES OF SOUTH-WEST CORNER "/
*c LATITUDE LOVGITUDE AND SIDE LENGTH LAT. LONG, sf
g 0, U Y D M M M V / ~ ( I ~ T F ~ . ~ ) T ~ F ~ ~ ? T / ! )
DO 3044 J = le K
WRlTE(bp350)
350 FDRMAT ( W @ P)
NAP = NO-NLO
WRITE(6r352)(IMAP(IIe I = NAP9 NO)
352 FORMAT(' '92015)
3044 NO = QAP-1
e
3045 I F t,NOT.LSTDP) GO TO 2023
C

C
C 9999 STOP
f h'D
READ( 59230) LSTO?
I F (,NOT.LSTGP) GO TO 2000
IF (LCOM) CALL OUTCOM
SUERDUTINF YES(f'1YtI IFC? I I FRrLPSIFW)
C THE SUB99UTIVF WILL? USING THE CH3LFSKYS PETFOD:
C (1) CnKPUTF THE REDUCED MATDIX L COYQESPOhDl hG Tfl A SYMMFTSIC POS ITIVF
C DEFIYITE (NN-l)*(NN-l) MATRIX At WHFN THE JIFC COLUPNS AND IIFR
C P,C,WS OF L ARE YFU'OWNT (L*LT = AT LT THE TRAN5PnSFD f3F L)
C (2) COMPUTE THE PEDL~CED NY-1 VECTOR (L**-~)*Y.
C (3) CCJMPUTF THE DIFFFRENCE PW = YfiJ-YT*( A**-l)*Ym
C (4) SGLVE THE ECVATIOYS LT*X = (L**-l)';YT (THF SO CALLED BACK-S'3LUTION)
C
C THE REDUCEG SfWS AV0 COLUPNS C!F L (THERE VAY BE hCNE)t THF CORFFS-
C PCWDING UYqEDUCFD UDPEQ TPIkNCtlLAR DART OF At THE NN-VECT39 FORMED
C CY Y AVD YY Fn9MS AN UPPFR TFIAkGULAk ?JY*NV-MATkIX,
C THE MATRIX IS 5TOKED COLUKYVIZE I N NT*PAXSL FECDQPS OF A DIRECT ACCESZ
C FlLE (UNIT KUF:EER P ) . THE YT RECQRDS (YT*12F00 BYTE?) CONTAI'JS AS MA'VY
C CCLUPMS AS POssIFLf l N THE FIRST E*( NT-l)*lh00+8*1500 RYTFS CnQSF;SPqN-
C DlNG TO TEF DIMENSION OF THF ARRAYS C AYP C?. TPE LAST PO0 RYTFS
C OF THE NT RECC:PDS HOLDS TWO CPTALOGES (INTEGER ARRAYS) NCAT AND ISZE
C (NRCAT? IRSZF RESPECTIVELY). kiE WILL CAii THE UT REC0qD5 A BLOCY,
C WHEN THE CGNTEqT OF A ?,LOCK HAS 5FFN TRAhSFERSFD €*G* TO C, Y C ~ T T
C ISZEq WE HAVE THF FCILLOI4Ih:G SITUATIQh', THE COLUMVS ARE STODFP Ih C
C FROM THE FIRST ELEMEQT DlFFFRFYT FRnEl ZERC TC THE DIAGfIYAL ELf'+f'NT-
C NCAT(1) IS TPr SUESCRIPT CF T U E DIAGONAL FLEMEWT CIF COLUMN 1-1 CV@
C ISZE( I ) IS THF YUMBER OF IGU3PEC (SAVFD ZEQOES IV COLUMN I, YCAT (100)
C IS THE hUK6Ek OF CDLUMNS STCISED I N THE PECO?D.
C N6L( I ) 1s FQUAL TO THE NUMSER OF THF LAST COLUMN STORE0 IN DEC09D 1-1.
C (1) TO (3) ABOWE NILL ALWAYS PE EXFCUTED? BUT (4) WILL QNLY 9F EX€-
C CUTE@ h'+% TUE LOGICAL LBS 1s TRUE* T U E EXECUTIDY OF (11, (2) FUD (4)
C I S EQUIVALEYT TO THF SCLUTIOU OF THE EaUATIUNS A*X=Y. THE SnLUTI7YS
C WILL 3E ST3XED I N THE ARRAY C I N TPE POSITIONS ORIGIYALLY OCCUPIED BY
C Y AND WILL 6E TRANSFERRED TO FAIN TH90UGH THE COMMON BLDCK.
C I N CASE A Y;UF'ERICAL SIh'GULAFilTY CCCURS IN COLUKF' &UMBER JP* THE COLUMN
C IS DELETED 8Y CHANGlNG THF CATALOGUE ISZE AVCl TFJE ELEPEVTS IY RE\-+' J D
C AND THE JD'TH t-LFMEYT OF T U E SOLUTION VECTOR IS PUT EQUAL TO ZE4O.
IMPLICIT INTECER(I?JrK?M?W) REAL *S(A-H?~-Z)TLCGICAL(L)
COMM9N /NESnL/C(4700) ?NCAT( 100) ?ISZE ( ~C)O)?NEL(~~O)?MAXBLTID
COKMOQ /C"W/CRl (1hOO)
DIMENSION CR(4700)?h1RCAT(100) qIRSZF(100)
FQUIVALtNCE (CR ( 1 1 ?CRl( 1)
NT = 3
C NOTEv THAT IY CASE YT I S CHANGEDT WILL IT ALSO BE YECESSAPY Tn CHANGE

C THE DIFSEVSIOPJ OF THE ARQAYS C At'lD CK,
C
N = M".!
IFR=lI LFR+B
IFC=IIFCcl
IF ( I F P e G T o I F C ) WRITE (6910)
10 FORKAT(' ERPOP IY CALL* IFReGT.IFC 1
e
C RfGUCTIT3d OF COLUMNS IFC TT) M, ELEMENTSy WHICH ARF ALREAnY REnUCEP9
C ARE STOkED IN CR [EXCEPT FOR JBL=KBL)e ELEMENTS9 WHICH AqE GOING TO BF
C REDUCED* ARE STERED IN C*
C
C FIND FIRST ACTUAL RECORD AND RDH/COLUMN.
JP r==o
200 JBF=$tiF+1
IF(NBL(JEF+l) .LT.TFC) GO TO 2CO
11 = t\lDhlJEF9
JF = IFC-IL
JTF = JBF*NT-NT+1
JTL = JTF
1D = JTF
FI&D(E*ID)
C
KBF = 0
201 K6F = KEF +l
IF (NBL[KEF+l l.LT,IFR) GO TO 201
KLO = NBL(KBF)
KF = IFR-KLG
KFO = KF
KTF = (KBF-l)*NT+l
C RFAD RECORD JPL FROM FILE. THE ARRAY C WILL CONTAIN AT LEAST ONE UNRE-
C DUCED COLUMN.
DO 280 JBL=JBF,PAXbL
READ( 8'ID)CVNCAT,ISZE
FIND( Fi8KTF1
NC=NCAT( 10@)
JO=NBL($BLI
K1 = KLO
ID = KTF
DO 270 KBL = KBF,
LREC=KbL-EQoJBL
JBL
READ( 6' I D )CF.TURCATT IRSZE
NR=NYCAT( 100)
KO=KL
K L =KL + NR
MR=MAXO(KO~ IIFR)
C
C IN ORDER TO MINIMIZE T"E NUMBER OF TRAYSPORTS TO AND FROM FILF PT WF

C DO NOT [GFS'ERALLY) COMPUTF PLL TFJF REDUCFrJ ELEMENTS OF llNE COL'JMYT BUT
C nwy THF FLFVTYTS IV PON K O + ~ TO KL. (KC IS THE NUMBER OF T ~ E LAST
C COLUMY IY THE PREVIOUS RECORD* KL THE NUMBER OF THF LAST I Y THF ACTUAL
C RECORD. 1
C
U0 760 J=JFtNC
ISZ=ISZE(J)
C WE CHECH THAT THFRE ARE tLEMENTS (UNRFDUCED) DIFFERENT FROM ZEQn IY
C COLUMN J HITP YUSSCPIPT GREATMEY THAN OR FQUAL TO KLm
1F (#L.LE.ISZ) GO TO 260
C THE SUBSCRIPT CIF TPE FLEMENT IN COLUPN JtPnW K I S NCAT( J)+K-ISZF( J ) c
C TPE DIFFERENCE NCAT(J)-ISZE(J) I S STOqEG IQ THE VARIABLE ICO.
C 3HF SAME DIFFERENCF FOR COLUMY (ROW) K I S STORED IN IRO. T U E ELFMFYT
C JUST BFFORF THE FIRST ELEMEYT TD E€ RFDUCFD WILL FEYCE YAVF 7U3SCSIPT
C I=ICO+MAX(KO, IIFRTISZ) . K G I S TPE ABSOLUTt ROW NUMBER-
ICO = M A T ( J)-1x2
JD=JO+J
NRO=MINO( JDTKLI-KO
I F (%R.GE.ISZ) KFS =

C
G
IF( .QT!T.LRFC.ORa(JD.l\lF.KD,.oREJn,EQ.Nl Lt7 To 360
C TEST OF tdUMERLCAL STAGILlTY
QC12 = QCI/C(I)**?
IF (QC12 ,GY, 0,lD-16) GO TO 251
WPITE(6p20lJDpGC12
20 FORMAT( "UMMERlCAL SINGULARITY IN ROW N0.'9H5? @
9 TEST QUANTITY ='P
C
C
C
*Dk7alO)
lSZE( J)=JD
TVE COLUMN I S DFLFTEDa
GO TO 260
251 C(I) = l7SQP,T((P
260 KF = b
270 COhTINUE
C REDUCFD ARRAY C BACK TO FILE, FIRST CPLUMY I N YEXT RECORD I S NPW TPE
C FIRST STOKFD,
I D = JTL
EUT FIRST REDUCED COLUPSN I S AGAIN CCDLU!43 KFO IN QECe J6F
WRITE(P81D)C9NCATpISZE
JTL = JTL+NT
JF=1
KF=KFn
C
IF(JSL.NEcMAXRL)
PW = QC1
GO TO 280
lFBINOTeLGS) GC TO 280
C
C BACK-SDLUTICN. NCTE TPAT TUF YAPIASLF l[ AT TuIS MPPENT I S THE SUB-
G SCRIPT OF THE DiAGOYAL ELEMEVT CIF THF CGLUMN CONTAINIKG THE RIGHT- HAY0
C SIDE, 1 WILL SUCCESSIVELY TAKE THE VALUE OF THE SUBSCRIPT qE THE ELF-
C MENT If% RCJW M OF Wl[S COLUMN, AND THE SnLUTIOY WILL BE STOYED I N C(I)a
C
M = N
KTL = (MAXEL-L) *NT+l
I D = KTL
DO 277 #B= lpMbXBL
READ( R1ID)CRtNi?CAT9TRSZE
KTL = KTL-KT
ID = KTL
I F (KTL.GT,O) FIND( E0 ID)
NR=NRCAT( 100)
C THE COLUMN COPJTAIYIY!G THE RIGHT-HAYD SIDE I S SKIPPED,
IF(KSeEQa1lNR=FJR-1
K1 =NR+1
IF(N2,FQ.O) GO TO 277
DO 276 K=l,NR
C WE STEP BACKWARD (M) FRCM CC'LUMbi N-19 TAKIhlG P E NR COLUMNS CRnM EACH
C RFCORO SUCCF SSIVELY.

I=I-P
IC=I
M=M-P
IR=NRCAT(Kl)
Kl=Kl-1
IRSZ=IkSZF(KlI
MR=M-IRSZ
I F (MR.GT.0) GO TO 373
C I N CASE A COLUKN HAS BEEN DELETED7 THE UNKNCIWY IS PUT EQUAL TO ZERO*
C( I ) = OaCIDO
Gn TO 276
C TWE (UN)KN@WN C( I ) IS COMPUTED (ONE EQUATICIN WITH ONE UNKNOWY).
273 C1 = C(I)/CR(IS)
C(I1 = C1
C
IF(MP*FO.l) GO TO 276
D@ 274 VP=?TttR
C 1R I S TliE SUG5CRIPT OF THE ELEMENTS OF COLUMN M, RUVYIVG FnPM THE
C DIAGnYAL-1 UP TO TVE SUEZCRIPl OF TUE FIRST ELEMENT DIFFERENT FPOM
C ZERO. 1C IS THE SUBSCRIPT OF THE ELEMEYTS OY THF P IGHT-HAUD S IDF I N
C THE SAME POW AS CK( IR).
IR=IR-l
IC=IC-1
C TPE CONTR IBUTIGN FP.OM THE ( UR)KNOWN C( I 1 IS SUBTRACTED FROM TCF QUAN-
C TITIES ON THE QIGHT-HAND SIDE.
274 C(IC) = C(IC1-CI*CK(IR)
C
276 CONTINUE
277 CONTINUF
C ThE SOLUTIOYS ARE YOW ALL STORED I N C FROM C(YCAT(Y)+l) TO C(LtCAT(M+l)
C -1) - THEY ARE TRANSFERRED TO 'MAIN' THROUGH TWE COMMOY BLOCK NESOL.
280 CONTIYUE
RETUQN
END
C
SUESOUTINF ITSAN(DX,DY,DZ,EPSl,EPS2~EPS3,0L,AX2,F22,RLATO,
*RLO~GO~DKSIO~DETAO~@ZFTPO~LCHANG)
TRANSFORMATIGN OF GEODETIC CQUROINATES AND CORRESPONDING DEFLECTIONS
C OF THE VERTICAL AND HEIGHT ANOMALY7 WHEN THF COORDINATE SYSTEM IS
C TRANSLATE9 PY (DX,DYIDZ 17 ROTATED (EPSl ,EFS2,EPS3) ASOUYD THE XwY 72
C AXES RESPECTIVELY AND MULTIPLIED LY l+DL.
C WHFN LCHAYG 1S TRUE, THEkE I S FUTHERMOEE A CHAYGF IY THE ASTRnV3MICAL
C COORDINATES AND IN THF HEIGHT SYSTEM, GIVEN AS A CWAYGF OF THF D€-
C FLFCTIOYS OF THF VERTICAL AND THE HEIGHT ANPMALY I N A POINT HAVING
C COORDINATFS RLATO AND RLONbO*
IMPLICIT INTEGER( I) 1 REAL *P(A-'-',P-Z)
COMM3Y /OESER/flRS (20)
COMMON /EUCL/XyY 7 Z ,XY 9XY2 ,@I ST3tDIST2
LfIGICAL(L1

c
C CF. PFF(C) PAGE ?On9 FORPULA (5-59)
COSDLO = LCOS (!?LONGp-RLOYOO
S INOLC = GSIN(RL0YGP-RLDNGO)
GO TO (61 *75rh2,63r62)p IKP
61 DZETA = (-( CC!SLAO*S IULk P-S TNLAr)*COZ LAP*CClSnLn) WKSI 0-CflSLAP*S IKDLn
**@ETAO)*AY?/YADSFC+ (S IF\ILAO*SIFYLAP+COSLAO*COSLAP*COSDLO)*DZET~O
GO Tfl 69
62 DKSI = (CRSLAP*CDSLhO+S INLAP*SINLCO*C~ISGLO) *nKSIO-5 INLAP*SIh!DLCl*
*DETAn-(S IULAn*COSLAP-CnSLAO*S IYLAP*C@SSLL)*RbDSEC*DZFTAO/AXZ
I F (IKP.NF.5) GO TO 69
63 DETA = S I&LAO*S IKDLO*~ETAO+COSDLO*DETAO+COSLAO*SIQDLCl*RADSFC
**DZETAO/AX?
C
69 GO 10 (71~75,72,73~72)tIKP
71 ORS(IT1 = DZETA+DH
GO TO 75
72 OBS(IT1 = DKSI-DLA
I F (IKP.NE.5) GO TT, 7 5
73 OES( I T 1) = DFTA-DLOKPSLPP
75 RETV9Y
END
SUBROUTlNF GRAVC(AXTFTGM~ IT LFOTSDTUKEF~GAklMA)
C THE SUkROUTINE COMPUTES FIRST BY TuE CALL OF G2AVC TuE CPNSTANTS TO EE
C USED IY Tt1E F3EPULA F09 TPE YDRMAL GRkVITYy THE FOYMULA F04 THE VCQMAL
C POTENTIAL An4D THF CHA!.IGE IN LATITUDE WITH YFIGHT. CDYSTAQTS RFLATEn T9
C TWO DIFFERENT REFERENCE FIELDS MAY BE USED. THEY ARE ST07ED TV THE
C ARRAY FG I N THE VARlABLES SUBSCRIPTFD FRCY 1 TO 15 FnQ TYF FIQST
C FIELD AND FRCM 16 TO 30 FOP THE SFCClNP ONF. THE APP.AY FJ COWTAIXS TFE
C ZOYAL HASMONICSI W17H SIGN OPPClSITE T r TVE USUAL CONVEYTICNST CF
C RFF(C)r EQ. (2-921.
IMPLICIT INTEGF9( I) 7 REAL *e(A-HtM-Z)
COMPON /DCON/DO ID1 y D2 ,D3
9 LOGICAL(L1
COMMON /EUCL/XTYTZ~XY TXY~,DTSTO,DIST~
DlMENSION FG(3O) vFJ(3O) tLP(2)
D4 = 4.ODfi
D5 = 5.000
OMEGA? = (0.7292115150-4)**2
LP( l+I/l5 1 = .NOT.LPOTSD
E2 = F*(D?-F)
AX2 = AX*AX
FG(I+14) = AX
C
FG(I+15) = GM
I F (LPflTSD) GO TO 1501
DO 1550 J = 1 7 15
1550 FJ(J+I) = DO

C FG( I+8) CfJNTAINS THE THlSCl OERIVATIVE OF THE NORMAL GRAVITY.
FG(I+E)
RETURN
= D4*FG(I+6)/(AX*Cl?)
C
EYTRV RGRAV (5 INLAPqHq IIGREF )
IF (H.GT.75.003 .AY[). LP(1+I/lS) GO 70 1504
C COMPUTATICN f3F THE REFEPENCE GFbVITY I N UNITS OF MGAL* CF. REF.(CI
C PAGE 77 AND 79. H PUST PE
SIN2 = SINLPP*SINLAP
1N UklTS OF MFTFKS.
GSEF = FG( I+1 l* (Dl+FG( 1+2)*SIU2+FG( I+~)J~SI~\:~*ZINZI
* +(FG( I+S)+FG(1+5)*SlN2+( FG( I+61-FG(T+Y )*H)*H)*H
RETURN
C
ENTRY CLbT(CLA7 I~HIRLATP)
LKSI = eTFUE.
IF (H.GT.~~ID~ .AYD. LFI~+I/~~)) GO TO 1504
C CCIMPUTATIOM OF THE CHbr\lGE I N THE n1RECTlON OF THE GRAPIFNT WITH HElGHT
C CF* 4EF(C) EGO (5-34).
CLA =-FG( l+l?I*H*(Z+XY)
LKSI = .FPLSE.
RFTURN
/DIST2
C
ENTRY USEFE:9(URFF7 I qH)
C COMPUTATIOk nF THE VALUE OF
C (6-14) AYD lh-15)0
LZETA = .TRUE.
THE NOSPAL POTENTIAL* CF. '?EF(C)t EOe
IF (Dh5S(H).GTo1.00-3)
UREF = FC(I+7)
RFTUQN
GCI TO 1503
C
1504 LZETA = .FALCE.
C
1503 T = Z/DISTQ
U = XY/DISTD
AX = FG(I+14)
GM = FG(1+15)
A1 = DO
A0 = DO
B1 = C10
B0 = DO
DAl = 00
DAO = D0
S = AX/DISTO
TS = T*S
S2 = S*S
K = 11
C1 = 12eOGO
CO = 1l.On0
C
C SUMMATION OF LEGENDRE-SERIES REFRESEIVTIW THE NORMAL POTENTIAL. B0

C WILL HOLD TEE DEi71VATIVF WITH RESPECT TO THE BISTANCF FROM THE ORIGIN
C AND DAO BL9E CT?IVATIVF WITH RFSPFCT TO THF LATITUDE AFTER THE FINAL
c EFCU~SIn"J qTFr..
DO 1553 J = P p 11
FJK = FJ(K91)
c3 = I-~?-DI/CO
c4 = CO*S?/C1
A2 = A1
A% = kO
A 0 = C3*TS*Al-C4*A2+FJK
I F CLZETA) GO TO 1555
C
DA2 = DL'1
DAl =: DAO
DAn = C3* I S*AS+TS*@AI 1-C4WA2
R2 = B1
BP = 50
B0 = C?*TS*B1-C4*BZ+FJK*CO
1555 C1 = CO
CO = CO-D1
1553 K = K- l
C
IF ILZETA) GO TO P554
C GP IS THE riFR IVATlVE OF THF N(jQMAL GSAVITY WITH 9F5PECT 13 PIST9
C (THE DXSTAhCF FROP THE ORIGIN) AYD GL I S THE DERIVATLVE WTTH OEfPFCT
C TO THE LATITGCEq CF. REF(C) EQ- (h-20') A\' (6-221.
GL = [GM*PPO/DISTZ-fMEGAZ*DlSTF*T )*U
GP = -GM*SP/DIST2+0MEGAZ*DI STO*U*U
GZ = T*GR+lJ*GL
GREF = DS€'QT( GR *GR+GL*GL)
IF [LKSI) CLA =-~DARSIN~-GZ/GREF~-RLATP)*206?64080600
LKSI = .FALSE*
GREF = GREF*I,OD5
RETUQV
I554 UREF = GM*AO/@I STO+OM€SAZ*XY2/D2
RETUSN
END
SUBROUTINF IGPBT( GMpAXqCOFFTbqrYMAX)
@ THE SUBROUTINF CQWUTFS THE VALUE OF THF GQAYITATIflYAL POTFNTIAL AND
C THE THREE FEKST GRDER DFRIVATIVFS 1N A POINT P HAVING GEnOETIC CIRG-
C DiNATES RLATP,RLONGP Ahlj EUCLIDIAY COOSDIYATES XqYtZ. THE PllTEUTlAL I5
C REPRESENTED BY A SERIES IN QUASI-'L'ORMALIZEO CPUERICAL 'JARPnNICS qUAVING
C COEFFICIEUTS OF HIGHEST DEGkEF NPAX, ALL STORFD IN THE ARR4Y CqFF.
C THE ALGOQITHM USFD IS GESCRIBED IN SFF(D).
IMPLICIT Ib

DIMEN'lPN F(55)rG(55)rA(25)~6(25)rC@FF(M)
C
C TPE ARRAYS P
C HARMONICS OF
C THE ORDFR-1.
AND '8 AKF USED TO KnLD THE VALVES OF THE SOLIP SPHF9ICAL
CjIFFEQENT DEGREES, THE SUBCCRIPT flF A nR B I S EOU4L TO
C THE ARRAY CCFF COhTAlNS THF FULLY RORMALIZED COEFFICIENTS OF TYC PPT-
C ENTIAL LEVFLRPED IN A SERIES OF NORMALIZED COLID 5PHFRIChL HAR~lOidlC~
C UP TO AND IVCLIJSIVC DEGREr YMAX AND FOUF UUMMY CPFFFICIFVTS FQUAL TO
C ZERO. TIE CDEFFICIFNTS WILL BE QUAZI-Y7QMALlZED BY THE TUBQrjUTIVE,
C F AND G CO:iTA 1% SSk'LikEE-ROOT TAHLtS USED IN THE aECURSI0Y PQ3-
C CEGURE. GP I S TPF PRCOUCT OF TPF GRAVITATIONAL CONSTANT AND THF MPSS
C PF THE EARTH (METFRS**~/S~C,'J*~) T AX THE SFM I-MAJCIP AXlS (METERS r
C OMEGA2 THE SOUARF flF TUE SPEED @F ROTATION.
C
C
INITIAL12 LYC COYSTAYTS,
CjMEGA2=(O -72921 1515r)-4) **2
RADSEC = ?C16764 ,bO6DO
DO=O,l)fiO
Dl=l. 000
D2 = 2001jO
NMAXl=YMAX+I
N3=hMAX+3
N4 = 2*NtJAX+3
C THE ZERO URGE9 COtFFICIENT I S PUT EQUAL TO ZERO AND TWE CPNTRIQUTIPN
C FROM THIS TERM IS FIRST USFD? WHEY THE CDVTRIBUTTCIY FRDM ALL OTPEq
C DEG9EFS HAS bFEN ACCUMULATEb.
CCFF! 1) = no
C
C SETTING UP A SCUAKE-ROOT
F(1) = Do
G(1) = PO
D0 1031 N = LT N4
DN = DFLOAT(hf)
TA6LE.
F(N+l)
1031 G(N+l)
C
= DSORT(DY*(DN-01))
= DSQRT( DN)
C THE COEFFICIENTS A9F GOOIh'G TO BE QUASI-NORMALIZED.
IJ = 1
DO 1033 I = IT NPAX
DN = G(2*(1+1))*1.0G-h
NZ = 2*I+1
D0 1032 J = 1 9 V?
1032 COFF(IJ+J) = COFF(IJ+J)*DN
1033 IJ = 1J+N2
SQ2=G (3
RETURR
C
EYTRY GPflT(RLATD~CflSLAPrRLONGPTIKPTIPrGRFFrURFFrCLA)
C THF SUBROUTIVE IS HERE USFD TO COMPUTF HE IGHT-AYOMALIES GPAVITY-AUD-
C MALIES AYD DEFLECTIOYS (KSIy ETA OR (KSI9FTA))r CORRESPOYGING T9 THF

C VALUF OF IKP = L e a a p 5 m ?HE RESULT IS STORFD IN OBSlIPI AYD 08S(IP+101
C FOR IKP= 5,
XP% = IP
%F (IKP,f0.54 IPB = IP+PO
U=X/D I STO
V=Y/D I STD
W=Z/D I[ STO
ADIQS=AX/DISlCI
POT-DO
DX=DO
DY=G0
BZ=PO
C A l l ) ES NOW TPE VALUE OF THE SOLID SPHeHAqMONIC OF DFGqFF 0.s
A(l l )=@l
B(l1 = CO
FACT- PI
ADIV?I=DI
c
DO 7002 I = 2 9 N3
A(l)=C0
7061 B(I)=PO
e
C
C
DO XI 10 I=l VNMAXI
61 = D1
Al=DO
- B1 = Cic?
B2 no
Db, l =D 0
DP1=DO
062=DO
DX O=D 0
DY O=D n
DZO=DO
POTO=DO
C = no
IS=~I-11*(1-11+1
01
- 62
CO = S02
A2=Aa 1)
DAZ=CCFF ( IS1
BPLUSJ=I
PPLUSI=I+l
IMINJ1=I+L'
SS=IS+l
b.RFbC=ADIVR I/FACf
DO 7005 J = I p l PLUS1
IPLUSJ-I PLUSJ+1

IMIYJl~IMINJl-1
JPLUSl=J+l
ALFA = F( IPLUSJ )*Cl
ALFA? = ALFA*C2
BETA = F( lkIhJ1 )*CO
GAMMA=(;( IPINJl) *G( IPLUSJ)
DAO=DAl
DAl=DA3
DC2 = COFF(1S)
DBO=Dbl
DEl=DB2
D82 = COFF(IS+l)
ALBO=ALFAWEO
BEB?=E ETA*DP2
ALAO=ALFA2*0AO
BEA2=3ETA*DA2
nXO=DXO+( ALAO-EEA2) *AJ+(ALt30-E!EBZ)*BJ
DYO~YO+(~LAO+PFA~)*QJJ(ALBO+~EB~)~AJ
DZO=DZO+GAKMA*( DAl*AJ+DEl*E? J)
IS=lS+2
C1 =Dl/CO
C2 = D2-C
CO = Ctl
7005 C = D1
C
POT=POT+AFFAC*PtlTO
FACT=FACT*I
ARFAC=ADIVR I/FACT
C CXTDYTDZ IS THF VALUE OF THE FIRST DE9IVATIVES OF THE PQTENTIAL WIT U
C RESPECT TO XyYyZ.
DX=DX-AP FAC*DXO
DY=DY-AEFAC*DYO

D P=DZ-AD FAC*D ZO
C
7010 ADIVR I=ADIVRI*ADIVR
C
C CONTRIEUT 19'4 FROM RCITATIONAL POTENTIAL.
R!JTX=OY EGAZsX
ROTY = QMFLA2*Y
ROTFOl = OMEGA2 /G3*XY 2
POT = (D 1+POT )*GM/G ISTOsRSTPOT
AO=GM/DI ST2
DX = (U-DX/D2 )*An-RnTX
DY = (V-DY/D? 1 * AO-SOTY
DZ = (W-DZl*AO
IF (IKP.LF.21 GP = DSGRT(UX*DX+0Y*DY+DZ*DZ)
C GP I S THE GPAVITY I N P.
CO T!3 (7VO6?7nO7,?On8 ,?O09e?n08), IKP
7006 ORS(IP1 = (POT-USEF)/GP
QETUqN
7007 BBS(IP1 = GP*1.005-GKEF
PETCQN
7008 O BS(IP1 = (DATANZ(DZ*DSQRT(DX*DX+DY*DY))-RLATPMADS€C+CLb
IF (IKPeLTe4I RETURN
7009 ObS( ID1 = (DATAN2( GY pGX) -RLONGP)*RADSEC*COSLAP
KFTURN
END
SUBR3UTINE EUCL ID(C3SLAPTSIYLAPTRLO'\fG*H*F2rkX)
C COMPUTATIflN OF EUCLIDIAV CnnSOlNATES X,Y,Z 9 PISTANCF AND SQUAqE OF
C DISTANCE FFQM Z- AXIS XYT XY? AND DISTANCF AND SCUPRE OF DISTAYCF FROM
C THE ORIGIX DlSTD AND DIST2 FP.CM GRCPETIC COCKDINATFS REFERRlYG TO AN
C ELLIPSOID PAVING SEMI-MAJPR AXIS EQUAL 10 AX AND SECOND EXCFNT2ICITY
C EZ. IMPLICIT INTEGER( I I JIK~KIN) T PEAL *8(A-H,O-Z 1
COMMON /FUCL/X~Y,Z?XYvXY2,DlSTO,DIST2
DN = AX/DSQRT( 1 .f)DO-E?*SINLAP**2 I
Z = ((leODO-E2I*DN+H)*SINLAP
XY = (DN+HI*COSLAP
XY2 = XY*XY
DIST2 = XY2+ZeZ
DlSTO = DSQRT(DIST2 1
X = XY*DC@S(RLONG)
Y = XY*DSIY(PLONG)
RETURY
END

SUPROVTINF RADI IDfGTM INTSECTRApIANC)
C TC'E SUEPDUTINE CDMVtR.TS FOR IANG = lTZT3r4 ANGLFS IN (1) DEGSFFST MI-
C NUTEST SECOYDST (2) GEGREES? MINUTEST (3) DECREES AND (4) 400-DFGRFFS
C TO RADIANS.
IMPLI C1 T INTFGER( I J,K,M,N) ,RE'AL *R IA-HTn-Z)
1 = 1
I F (IDFG ,LT* 0 *AVE. lANG .LT* 5) 1 = -1
GO TO ( l r 2 ~ 3 ~ 1IANG 4 )
1 SF =I>* IDEG*3A0O+M IN*6O+SEC
GO TO 5
2 SF=l* IDEG*3600+SFC*60
GO TO 5
3 SE = SEC*3600
GO TO C;
4 S€ = SEC*324n
5 RA= 1*SE/206264.t?06DO
RF TUF? N
END
SUEROUT I%E HEAD ( IKP 1LO?4FCf?~P~O(RP)
C OUTPUT OF HEADIMLS Aid@ INITIALIZATIOW OF THE FOLLOWIP?IG VAKIAr3LE.5: IAT
C I R ~ I P T I Il~IL1~IE~l~I~I.~lTl~I21~131~IC1~ICll
T ~
(ALL SUSSCYIPTC @F DIF-
C FERENT CUTPUT CUANTITIES), KZ - K4 (SUESCRIPT D3UXDARlCS FnR OUTPUT
C QUANTITIES), K1 = UPPE9 LlMlT FOP WANT1TTC.S READ IVTO 08s.
IMPLICIT LOGICAL( L)
COMMDU/OUTC/KZIK~~K~T lU~K21, IU19 IAYGTLPU%CHTL~UTCTLYTQA~!TLVE~%R
**LK30
COMMON /CHEACI/IA~IBTIH,I~~ITT IAlql61, IOlTITlT IClT ICllTKl, ICiESlq
*ICbS2 1 L PUTi LCl, LC? LCFiEF 1 LKM
LTRAN = .NOT.LNTOAN
LERNO = .%OT.LNERNO
K 1 = 0
IF (I@PSl.h!E.O) K1 = 1
GO TO (2008~2009~2010~2011~2012) ,1Kp
2008 WRlTE(6~204)
204 FORMAT('0
GO TO 2013
NO L AT l TUDE LONG lTUDE H ZETA (M)')
2009 WRITE (69705 1
205 FORMAT ( '0
* 1
GO TO 2013
NO LATITUDE LONG1 TUDE H DELTA G (VGALl8
2010 WR ITE (6,206 I
206 FORMAT('@ NO LATI TUDE LnNGl TUDE F, KSI (ARCSEC) @ B
GO TO 2013
2011 WRITE(6r207)
207 FORMA7 ( '0 kD LATITUDE LONGITUDE H ETA (ARCSFC)')
GO TO 2013
2012 WRITE(6r20R)
208 FORMAT('@ NO LAT I TUDE LO%G ITUDE U KSI /ETA (ARCSE

*Cl0)
I F (I3ES2 ,YF. 01 K1 = 2
2013 I F (IH.UE.0) K1 = K1+1
WRITF(hq267)"WGvRP
267 FDKt'AT('+\59X9' STeVAR* ='qF6*49\ RR4TIO R/RE =v 9F10.79'9' 1
C
GO Tc? (2O%P92Ol59ZOZO,ZOZlI pIkNG
2nlA WRTTF(6r2fi9)
209 FORKAT(Q D M S E M S M' 1
GO TO 2022
7019 WRITF( 692101
210 FORMAT( n M D M M~ 1
GO TO 2022
2020 kQlTF(6r211)
711 FORMAT(' DFGFEFS DFGPFES M' 1
GO TO 3022
7071 WRITF(h*712)
212 FOkMAT ( g GRAGES G kA@F S M' 1
C
2022 I F (LKM) WRITE(h,213)
213 FORMAT( Q + 93779 'I:' 1
C
C WF NOW COYPUTE THE SUPSCRIPT CF THE DIFFFQFYT 6UhkTITIESq WHICH WILL
C FE STOhED FOP LATER OUTPUT 1R TFF AEkAY OF!. THF DIFFFRENCE BETNFFN
C T U E 0BCEF;VLTTl'Pt GIVFV IN TV€ PRIGlNAL AtID TVE NEI.1 REF,SYSTFPl IV
C OBS(1Tlc THE COYTRILUTIPY TO THF RFF.POT. FFDM THE HASM!l'IIC FXPAVSION
C I N OBS(IP1 9 THE CUYTRIBVTIEIY FKOM COLL.1 IU DBS( IC1) AY@ FK3Y COLLtlI
C 1N GRS(IC2).
IC1 = 11
IF (LTRAV) GO TO 2105
IF( LPOT 1 GO TO 2102
IF (LC11 GO TO 2201
r B = 4
GO TO 2104
1201 IC1 = 5
I F (LC21 GO TC 2101
1e=5
WRITE( h925O)
250 FnRMAT('+\b4X9 ' PRED'I
GO TT) 2104
C
2101 IB=7
1C 2=6
WRITE (6,291
251 FORMAT( '+ "63x9 ' COLLI COLL2 PRED 8 1
GO TO 2104
C
2102 IP=5
1F (LC%l CC' TO 2202
liB = 5

WRITF(hq245)
245 FORMAT('+'T64XT'
GO TD 2104
C
2703 IC1 = 6
POT '1
IF (LC2) G@ TO 2103
I E=?
WRlTE(A,252)
252 FORMAT( '+' ?64X1 ' POT COLL PREDt)
GO TO 3104
C
2103 IC2=7
IB=R
3RITF (4~2")
253 F3RMAT( '+'?04X? ' POT
2104 K3 = I?-4
1u = It:
COLLl CULL2 PSFD')
GO TO 2110
C
2105 IT=5
IF(LP3T) C3 70 2107
IF (LCl) CO To 2205
IR = 3
WP 1TE (6,246
246 FC?RM&T('+'T~LXT ' TRP '1
GO TO 2103
C
2205 IL1 = 6
IF (LC2) GO TO 2106
1B=7
WRlTE(6~254)
254 F ~ R ~ ~ T ( ' + '' T ~ TRA ~ X T PRED PRED-TRP'
GQ TO 2109
C
2106 IC2=7
I R=€?
WRITE(6t255)
255 FORMAT( '+'~63X, ' TRA CRLLl COLL2 PRED PRED-TQb '1
GQ TO 2109
C
2107 IP=6
IF (LC1) GO TO 2208
IR = 6
WRITE (69247)
247 FORMAT( '+'~63X? ' TPA POT PET-TRA')
GO TO 2109
C 230P I C 1 = 7
IF (LC2) GO TC 2108
l B =8

2%
WP IT€ (692561
FFP/(AT ( 'a ' r63Xe ' TRA
GO T7 310"
Pc',? COkk PRFD PRED-?RA 1
C
2108 IC2=R
16=9
WRITE (6923)
257 FCRMAT['+'F~~X,'
2109 K3=19-3
IiU = 1591
C
2110 IF (LC21 1A = 1C2
TSA POT COLLl COLL2 PRED PRED-TSAE9
IF (,SOT,LCPFF) 1A = 1C1
C
IF [LDUTC) GO TO 2125
IF(LEF'4r-l)
K?=1
Gn 10 2112
GO TO 3135
C
2112 Y?=l
W!? IT' (6,2601
260 FORPAT( b+*r43X~ U F R R m )
G9 TO 2135
2125 IF(K3.GT.O)
K 2 =2
GP T3 3127
WRITE (6,361)
26! FqD,h'AT! + Q r4?Xr 8
GO Trl 2135
ces 1
C
2127 IF (LFSPIO) GO TD 7128
K2=3
WKITE(bp262)
262 FORMAT ( '+' ~43x9 ' OBS DIFe 1
GO TO
e
2128 K?=4
2135
MRITE (693631
263 FOKMAT('+',43X,' 06s DIF ERR 1
C
2135 K 4 = K?
LK30 = K3,GT.O
IF (Lowcm C.O
IP1 = IG+lO
IU1 = IU+lO
IAI = 1A+1n
171 = IT+lo
ID1 = IP+lO
ICll = IC1+10
K4 = ?*KZ-1
K31 = K2+ln
TO 2111

C
2111 RETUQY
E NI3
SUHROUTINE neT( m, IDLAT ,fJILATT SLAT, InLor\l,MLar?!,s~c~,Lnuc)
C THE 5l;BQnUTINF NF ITFS ON UNIT 6 (3.) STATION NUMPER, (2) CRn?DINATFSy
c (2) DESERVED VALUE (IN C~QI(;.REF.SYSTEM), (4) CIFFERENCE SETWFFY nYsEe.-
C VED AND P9ECICTFD VALUE, (5) EKSOR OF PREPICTIO!Jr (6) TGAUSF3RWTI!7V
C VALUE, (7) SPEERICAL HARMONIC SEPIES COST~I5UTI@U, (F) QFYULT OF CULL
C I AND (Q) COLL.11, (10) Sub4 RF QUANTITIES (7)-(9) AXD (11) S?rK DC (6)-
C (9) - ALL IF MEP.NINGFULL. I?\! LAZE WE AkE DE4LIYG WITP A P4IR PF DE-
C FLECTIONS, (LCNC = FALSE) 9 TPE CEF.RES?ONDING QUA?;TITEY FOR ETA ARE
C WRITTEN A LIVE fiLLOW,
C WHEN LPUNCH IS TRUE, THE FOLLOWIXG OF TWE ARGVE "4fWTITJUF;r) QUAVTITIES
C ARE WRITTEN ON UNIT 7: (1) AND (21, AQrf WHFN LOUT€ I S TRUE: (3) - (E)
C AND ELSE (11 1, (10) AWD (5)
IMPLICIT Ib!TEGER( l J9K9MTN) + LOGICAL( L) 9REAL *8(A-H,O-Z
COMfXlN/DUTC/I 7117 I27I47 1219 131,IANG~LPUNC~,LEETC~LNTKAV,LhEQVP
* 9 LK.30
COMMClN/DB SE ?/Of3 S ( 20
nlKENSInN ObN(10)
I F (DAP(S(SLAT) .LT, 0.1G-h) SLAT = O.ODO
I F (OPBS(SLDN) .LT, 0-1s-6) SLOY = 0.090
C THIS IS DONE IV ORDER TO AVOID PKINTIXL OF SlGY. WuEV TVE ARC-SFCVND
C PART I S NEAR TO ZER3r(OR ZEZO IS REPRESENTED AY A SMALL YEGATI\IE NUM-
C BER).
I F (OKS(l).G€,1.004) OLS(1) = 9949,900
I F (ORS(l)eLFe-leOD3)nP~S(l) = -999-99D0
I F (*NRTeLPUNCU) GO TCl E010
I F (LDUTC) GO TD 8007
OBN(1) = OBS(1)
10 = 2
OEN(2) = GES(14)
I F (LNTRAN) Gn TD 8031
OBN(3 1 = DES( 14-1
IQ = I0+1
8031 I F (LYERND) GO TO 8032
10 = 104-1
OBN(10) = OES(I)
8032 I F (LCNC) GO TO 8034.
I0 = I0+1
OEN(Io1 = 06SII31)
I F (LNTRAN) GC 70 P033
I0 = I041
nBN(I0) = ChS(14+Q)
8033 I F (LNERNO) GP TO 8034
I0 = IO+l
OBN(I0) = OBS(I21)

RL2 = RL2-DP/IK-S*T/IKl-(O3*T*T-D1I*S2/(n2*IY2I
I F (LND) GO TO 905
DRL2 = DRL2/S-Dl/LKl-D3*S*T/IK2
IF (LNDDI GC TO 905
DDRL2 = DORLZ/S2-D3/IK2
905 RETURN
END

Appendix B.
An iuput example.
The digital numbers written to the left of the input data correspond to the
numbers identifying the input specification given in Section G.
The input example will produce the output shown in Appendix C. It corresponds
to the situation described in Section 4, i. e. where three datasets of observations
X,, X,, and X, are used for the determination of ?.

Z CCOOOCCC
rzl I I I I
c3 GGOOCC00
2 I I I I I

G I-- a?
C B I
c N
m N Ccl
Orla-N
rid d

F T T
f -00 0.10
b15~20X,I3~13,Fh~2~I5pI3pF6o?,?F?e2pLl)
1 2 3 1 0 4 5
40216 EJER SAVNEHQEJ
51.0 F F F F F
55 58 34.68 9 44 54e9CJ -5e13 3-61
40606 VlGERLOESE
F
F T T
54 42 54-22 11
(Ih,2( I3,F6-2) 12F7.2rL2)
1 2 3 2 4 5 0 2100 F F F
1-00 981000-00 T
4413 54 51.49 9 59-55 6-94 517.5P
3163 54 50-23 12 00.01 17-63 510.18T
F
T T F
54 30eOO
F
T T F
9 00.00
54 30.00
F
T T F
9 00.00
54 30.00
F
T T F
9 00.00
54 30.00
T
9 00.00

Appendix C.
An output example.
The output has been produced by the FORTRAN program using the input given in
Appendix B.

GFODETPC COLLOCATIONIVERSION 20 APR 1974-
MOTE THAT THE FUNCTIONALS ARE I N SPHERICAL APPROXIMATION
MEAN RADIUS = RE = 6371 KM AND MEAN GRAVITY 981 KGAL USED.
LFGEND OF TASLFS OF DBSFRVATl nYS AND DQEnlCTIONS:
OLS = Ot.SFRVFD VALUE (WHEN POTH COMPnNfNVS nF DFFLECTlONS ARE
OPSFKVEC ETh PFLOH KSI), GIF =THE LIFFERENCE EETWEEN OUSERVED
bYD PP.EO 1CiED VALUE V W E N PREDICT IONS ART CTtMPUlED AN0 €LSE
TPC PE5ICIJAL nE7FDVATIOY. ER9 = FSTIMATFD FR40Q OF P3EOICTION
1Q.A = CO';TP.IYUT19Y F90'4 OCTUN TRAYSFOPMATIO", PflT = C3NTRI-
PUTION FoPM POTENTIAL CnFFFIClFNTS. CDLL = CnYT91BUTION FROM
COLLOCATlPk DFTFRMIIIED PART OF ESTIHATEr WHEN THERE ONLY HAS
BEEN USED OYE SET OF OBSFRVATlflNS (DIFFERENT FRnM PDT.COEFF.1
COLLl = CCNTRleUTlON FROM ESTIMATE OF ANOMALOUS POT. OETER-
MINED FPDM FIRST SET OF OBSFP.VATlnNS, CCtLL2 = COUTRI3UTION
F909 FSTIYATF ORTPINEI? FROM SFCONP SET OF OBSERVATIW4S I PRED=
PREDICTFD VALUF- IN NFU PEFEaENCE SYSTEM, MHEN PkfDlC.TIONS ARE
COYPUTEO AkIl ELSF THE SUM nF THE CONTPltUTlONS FROM TtIF POT.
COEFFICIENTS AN0 FIRST ESTlMATE OF AYOMALDUS POTENTIALI AND
PRED-TRA = PREDlCTInN nR SUM @F CONTRIBUTIONS IN TWE OLD RE-
FERENCE SYSTEM.
REFFREYCE SYSTEM:
EUROPEAN DATUM. 1050.
A = 6378388.0 M
1/F 297.00040
sEF.GRdVITY AT EQUATOR = 978049.00
POTFUTl AL AT REF .ELL. = 62639787.00
MGAL
M**2/SEC+*Z
GWAVI TY FORMULA:
INTERNA TI OKAL GRAVl TV FORMULA* POTSDAM SYSTEM.
NEW A NEW G# NEW 1/F
6378143.0 0.390601 ZO+l5 298.25000
EPSl €PS2 EPS3
1.57 0.19 1.05
NEU REF. GFAVITY AT EQUATOR= 97R032.67 MGAL
NEW POTENT 1AL AT ELLIPSOID = 62636916.84 M**Z/SEC**2
DEFLECTIONS AND HFIGHT ANOMALIES CHANGED I N
LAT I TUOE LONGITUDE BY DKSI DETA DZETA
55 30 0.0 10 0 0.0 -1.00 1-00 0.0
SOURCE OF TPE PPTEVTIAL COEFFICIENTS USED:
R.H.QAPP, THE GEOPOTENTIAL TO (14r14) ETC-r IUGG. LUCERNEI 1967-
G M A COFF (5) MAX .DEGREE
0.39860122D+15 6378143.0 -484.1803 8

START OF COLLOCATION I:
WE MODEL ANOMALY DEGREE-VAWIANCFS ARE EQUAL TO
A - - l 2 + 2411.
PATIO R/RE L 0.999800
VARIAUCE OF POINT GRAVITY ANOMALIES = 16~1.00 MGAL**Z
1HE FACTOR A P 412.08 MGAL**2
12 EHPIPICAL ANOMALY DEGREF-VARIANCES FOR DEGREE > l*
(IN UNITS OF WGAL**2):
0.0 0.0 0.0 0.0 0.0 0.0 0.0 15.1 17.7 13.7 8.4
OBSERVATIONS:
THE FOLLOWING QUANTITIES ARE MEAN-VALUES, AND ARE REPRESENTED AS POIYT VALUES I N A HEl6Wf R.
LIT1 TUOE
D M
56 30.00
56 30.00
56 30.00
55 30.00
55 30.00
55 30.00
54 30.00
54 30.00
54 30.00
H DELTA G (HGPL1
M OBS DIF
0.0 1P.00 -5.26
0.0 10.00 -13.19
0.0 27.00 3.93
0.0 26.00 3.50
0.0 12.00 -10.43
0.0 6.00 -16.32
0.0 5.00 -16.67
0.0 10.00 -11.61
0.0 2.00 -19.51
S1At;DAtlD DEVIATIO?4S OF THE OBSERVATIONS IN THE SAME SEQUENCE
AND IN THE SAME UNITS AS THE OBSERVATIONS:
4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00
SOLUTIONS TO NORMAL EQUATIONS:
1PE SOLUTIONS HAVE BEEN COMPUTED IN A PREVIOUS RUN.
NUMEER OF EQUATION = 9
NORHALI ZED SOUARE-SUM OF OBSERVATIONS E 0.178534D+Ol
NORRALI ZED DIFFERENCE BETWEEN SQUARE-SUM OF
OBSERVATIONS AND NORM OF APPROXIMATION P 0.2196870+00
ST-VAR. 27-98. RATIO RIRE 1.0015554s
TRA POT POT-TRA
-6.69 16-57 23.26
-6.69 16.50 73.19
-6.69 16.37 23.m
-6.47 16.02 22.50
-6.48 15.96 22-43
---5.48 15.84 22.32
-6.25 15.42 21.67
-6.25 15.36 21.61
-6.25 15.25 21.51

START OF COLLOCITPOM 1 X r
RATIO RfRE
VAR~ARCE 0~ POlNT GRAVITY ANOMALIES
THE FACTOR A
90 OECREE-VARIANCE% EQUAL 80 ZERO
DBSERVAT IONS:
LAT l TUDE
D 1 4 5
55 58 39-65
LAT ITUOE
O M S
55 52 38.51
LATITUDE
D *
56 4.90
56 7.10
54 51.49
54 50.23
LONG1 TUOE
O H S
9 49 54.90
LONG1 TUDE
o n s
12 50 38-93
LONG1 TUOE
0 H
10 0.34
11 57.00
Q 59.55
12 0.01
H #SI/ETA (ARCSEC) STaVAR, 1.95, RATIO RfRE = 1.0000000~
TRA POT COLL PRED PREO-YRA
H EETAIM)
H OUS DIF
0.0 20.40 -0.07
H DELTA G (HGALI
H 005 DIF
70.60 37.49 21.61
0.0 -5.70 -19.90
6.94 17.20 5.69
17.63 14.89 8-68
ST.VAR. = 0.37, RATIO R/RE = 1.onoonno.
TRA DOT COLL PQED PRED-TRA
22.55 46.20 -3.18 43.02 20.47
ST.VAQ. = 13-04. RATIO R/RE = l-0000000*
TRA POT COLL PRED PRED-TRA
-6.60 16.32 -7.04 9.28 15-88
4-61 16.25 -8.66 7.59 14.20
-6.33 15.61 -10.44 5.18 11.51
-6.33 15.52 -15.64 4.12 6.21
STANDARD DEV1ATION5 OF THE DRSERVATIONS IN THE SAME SECUFNCE
AND IN THE SAME UNITS AS THF OBSERVATIONS:
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.0 0 -20
0.20 0.20 0.20
COEFFIC IEVTS OF NDRMAL-EPUATIOYSI BLOCK 1
SOLUTIONS TO NORMAL EQUATIONS:
KUMEER t3F FCUATIONS = 13
NORMAL1 ZED SOUARE-SUM OF QBSERVATlONS P 0.1991610+02
NORMAL1 ZED DIFFERENCE BETWEEN SQUARE-SUM OF
OBSERVATIONS AND NORM OF APPROXIMATHO% ~-0.4244679+01

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U . 0 EI u o .
LU
U1 ZOQ o m W zoo cm > moo am
z W 0 > WII 0
W
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