Geoid determination by

least-squares collocation using




Geophysical Department

University of Copenhagen

Juliane Maries Vej 30

DK-2100 Copenhagen OE



Abstract: GRAVSOFT is a package of FORTRAN programs for gravity field modelling. The specific use of the programs for the determination of the empirical covariance function and its analytic modelling and geoid determination using least-squares collocation is described. An example using data from New Mexico, USA, illustrates the use of the method.


The purpose of these notes are to provide a guide to gravity field modelling, and especially to geoid determination, using least-squares collocation (LSC). The use of the GRAVSOFT package of FORTRAN programs will be explained. The readers are supposed to be familiar with LSC as described in Moritz (1980). However the theory will be reviewed briefly in order to fix the terminology.

So, for example when using the term geoid, we will mean the quasi-geoid, i.e. the so-called height anomaly evaluated at the surface of the Earth.

A general description of the GRAVSOFT programs are given in Tscherning et al (1994), which is updated regularly when changes to the programs have been made. It is available on computer-readable form as a part of the GRAVSOFT-files.

The general methodology for (regional or local) gravity field modelling is as follows:

A: Transform all data to a global geodetic datum (GRS80/WGS84)

B: Use the remove-restore method. Here the effect of a spherical harmonic

expansion and of the topography is removed from the data and subsequently

added to the result. This is used for all gravity modelling methods

including LSC. This will produce what we will call residual data.

C: Determine at statistical model (a covariance function) for the residual data in

the region in question.

D: Make a homogeneous selection of the data to be used for geoid determination

using rule-of-thumbs for the required data density, check for gross-errors

(make contour map of data), verify error estimates of data,

E: Determine using LSC a (regional) gravity field approximation. Compute estimates

of the geoid heights and their errors.

F: If the error is too large, and more data is available add new data and

repeat E.

G: Check model, by comparison with data not used to obtain the model.

The whole process can be carried through using the GRAVSOFT programs GEOCOL, EMPCOV, TC (TC1), TCGRID, COVFIT, SELECT and GEOIP. GRAVSOFT includes data from New Mexico, USA., which can be used to test the programs and procedures.

They have here been used to illustrate the use of the programs. The programs have been used for geoid determination by means of least-squares collocation in several countries and regions (Andreu & Simo, 1992, Arabelos, 1989, Ayhan, 1993, Benciolini et al. 1984, Catalao & Sevilla, 1994, Gerrard, 1990, Gil et al. 1993, Sevilla & Rodriques, 1993, Tscherning, 1985, Tscherning & Forsberg, 1986).

2. Theory.

The anomalous gravity potential, T, is equal to the difference between the gravity potential W and the so-called normal potential U, T = W-U. T is a harmonic function, and may as such be expanded in solid spherical harmonics

and GM is the product of the gravitational constant and the mass of the Earth, and

The functions Ynm(P) are orthogonal base-functions in a Hilbert space with an isotropic inner-product, harmonic down to a so-called Bjerhammar-sphere totally enclosed in the Earth. T will not necessarily be an element of such a space, but may be approximated arbitrarily well with such functions. This justifies the equality sign in equation (1).

The space will have a so-called reproducing kernel, which is a function of two points in space P, Q. (The coordinates of Q will be distinguished from these of P with an '). The kernel will be

where is the spherical distance between P and Q, Pn the Legendre polynomials and n are positive constants, the so-called (potential) degree-variances.

It is well -known, that if the degree-variances are selected equal to simple polynomial functions in the degree n multiplied by exponential expressions like qn, where q < 1 then K(P,Q) may be represented by a closed expression. The most simple example are where n=qn, where we get well-known expressions related to the reciprocal distance.

In a reproducing kernel Hilbert space (RKHS) one may determine approximations to the elements from data for which the associated linear functionals are bounded. The relationship between the data and T are expressed through functionals Li, so that

where yi is the i'th data element, Li the functional, ei the error, Ai a vector of dimension k and

X a vector of parameters also of dimension k. AiTX may for example express the effect of a datum-shift or of a bias and tilt in the data.

For the data we will meet (cf. Table 1) we have

or mean values of theses quantities. (Mean values will in practice be represented a means of point-values). The point Q is a point with latitude and longitude equal to P, but having ellipsoidal height equal to the orthometric height of P.


Table 1. GRAVSOFT data from the New Mexico Area to be used in LSC geoid test.

Type Format Error File name

Free-air gravity number, , , height, g (mgal) 0.2 nmfa

Defl. vertical number, , , height, , (arcsec) 0.5 nmdfv

DTM Grid label, grid data (m) 25 m nmdtm


The linear functionals are then

shown here in spherical approximation. (This approximation is only used on the residual quantities, see section 3).

An optimal approximation to T using error-free data in a geocentric system may then be obtained using that the observations are given by, Li(T) = yi. The "optimal" solution is the projection on the n-dimensional sub-space spanned by the so-called representers of the linear functionals, Li(K(P,Q)) = K(Li,Q). The projection is the intersection between the subspace and the subspace which consist of functions which agree exactly with the observations, Li(g)=yi. (See Figure 1).

Figure 1. Collocation estimates the "point" with shortest distance from zero agreeing with the observations, and is simultaneously the orthogonal projection of T on the subspace spanned by the basefunctions K(Li,P), if T is an element of the space.



Based on the gravity field model given by equation (7), any quantity associated with a linear functional, L, may then be computed

If the data contain noise, then the elements ij of the variance-covariance matrix of the noise-vector is added to K(Li,Lj). The solution will then both minimalize the square of the norm of T and the noise variance. If the noise is zero, the solution will agree exactly with the observations. This is the reason for the name collocation. Upper limits for the approximation error may be calculated if the norm of T is known,

This is not very useful, since the upper limit generally is very pessimistic (see Tscherning, 1984).

If we want to minimalize the mean-square error, the reproducing kernel must be selected equal to the so-called empirical covariance function, COV(P,Q). This function is equal to the reproducing kernel given in eq. (2), and having the degree-variances equal to

Here the normal equation matrix

result in predictions

and error estimates

In the diagonal of the normal equation matrix we find the sum of the data variance and the noise variance. We can say that we here have obtained a balance between the signal and the noise.

The covariances are computed using the "law" of covariance propagation, i.e. COV(Li,Lj) = Li(Lj(COV(P,Q))), where COV(P,Q) is the basic "potential" covariance function. COV(P,Q) is an isotropic reproducing kernel with the degree-variances given by eq. (10).

Example. If we want to derive the gravity anomaly covariance function for gravity anomalies in two points P and Q then we must appy the functionals given in eq. (6) on K(P,Q). This is

The quantities COV(L,L), COV(L,Li) and COV(Li,Lj) may all be evaluated by the sequence of subroutines COVAX, COVBX and COVCX which form a part of the programs GEOCOL and COVFIT for the functionals given in eq. (6). For further details see Tscherning(1976, 1993).

3. The remove-restore method.

The least-squares collocation solution is giving the minimum mean square error in a very specific sense, namely as the mean over all data-configurations which by a rotation of the Earth's center may be mapped into each other. So if this should work locally, we must make all areas of the Earth look alike, see from the gravity field standpoint.

This is done by removing as much as we know, and later adding it. We obtain a field which is statistically more homogeneous than before.

First we may remove the contribution Ts from a known spherical harmonic expansion like the OSU91A field, complete to degree N=360 (Rapp et al.(1991)). Secondly we may remove the effect of the local topography, TM, using Residual Terrain Modelling (RTM, see Forsberg & Tscherning, 1981). We will then be left with a residual field, with a smoothness in terms of standard deviation of gravity anomalies between 50 % and 25 % less than the original standard deviation.

The residual quantities are then

Exercise 1.

Compute residual gravity anomalies and deflections of the vertical using the OSU91A spherical harmonic expansion and the New Mexico DTM, cf. Table 1. The free-air gravity anomalies are shown in Fig. 2.

The program GEOCOL may be used to subtract the contribution from OSU91A. A job-file with the name jobosu91.nmfa is shown in the Appendix 1. An example of a run is shown in Appendix 2. The coefficients of the OSU91A model are found in the file osu91a1f. The difference files should be denoted nmfa.osu91 and nmdfv.osu91. The gravity anomaly differences are shown in Fig. 3.

The RTM contribution may be computed and subtracted using the program tc1. First a reference terrain model must be constructed using the program TCGRID with the file nmdtm as basis. A jobfile to run tc1 named jobtc.nmfa is found in Appendix 1. The result should be stored in files with names nmfa.rd and nmdfv.rd, respectively. The residual gravity anomalies are shown in Fig. 4.

The degree-variances will be changed up to the maximal degree, N, of the spherical harmonic series, contingently up to a smaller value, if the series is not agreeing well with the local data (i.e. if no data in the area were used when the series were determined). The first of N new degree-variances will depend on the error of the coefficients of the series. We will here suppose that the degree-variances at least are proportional to the so-called error-degree-variances,

The scaling factor must therefore be determined from the data (in the program COVFIT, see later).

4. Covariance function estimation and representation.

The covariance function to be used in LSC is equal to

where is the azimuth between P and Q and , are the coordinates of P. Note that this is a global expression, and that it will only dependent on the radial distances r, r' of P and Q and of the spherical distance between the points.

In practice it must be evaluated in a local area by taking a sum of products of the data grouped according to an interval i of spherical distance,

where is the interval length (also denoted the sampling interval size). Note that two intervals may be merged, so that the sampling interval becomes the double. In the program EMPCOV this may be done a number of times, as an aid in selecting the right size of the sampling interval.


where in actual calculations the covariance will refer to the mean height of the area.

In practice data directly related to T will not be available, but we may have geoid heights, gravity anomalies and deflections of the vertical. Covariance functions of geoid heights and gravity anomalies are estimated like for T, but the representation as a Legendre-series, cf. eq. (4) will be different. In spherical approximation we have already derived

where R is the mean radius of the Earth.

Exercise 2.

Compute the empirical gravity anomaly covariance function using the program EMPCOV for the New Mexico area both for the anomalies minus OSU91A and for the anomalies from which also RTM-effects have been subtracted (input files nmfa.osu91 and nmfa.rd). A sample input file to EMPCOV is called empcov.nmfa, see Appendix 1. A sample run is shown in Appendix 3. The estimated covariances are shown in Fig. 5.

Note the correlation distance 1, i.e. the distance where the covariance becomes equal to 50% of the variance C0.

What is the mean height to which the calculated covariances refer ?

We see here, that if we can find the gravity anomaly degree-variances, we also can find the potential degree variances. However, we also see that we need to determine infinitely many quantities in order to find the covariance function.

The solution to this problem is to use a so-called degree-variance model, i.e. a functional dependence between the degree and the degree-variances. In the program COVFIT, three different models (1, 2 and 3) may be used. The main difference is related to whether the (potential) degree-variances go to zero like n-2, n-3 or n-4. The best model (see Tscherning & Rapp, 1974) is of the type 2,

where RB is the radius of the Bjerhammar-sphere, A is a constant in units of (m/s)4, B an integer. If a spherical harmonic series expansion is used, B is typically put equal to a small number like 4, while in the original work it was put equal to 24, so that the low-degree degree-variances could be modelled appropriately.

The actual modelling of the empirically determined values is done using the program COVFIT. The factors , A and RB need to be determined (the first index N+1 must be fixed).

The program makes an iterative non-linear adjustment for the Bjerhammar-sphere radius, and linear for the two other quantities (see Knudsen, 1987). Unfortunately the iteration may diverge (e.g. result in a Bjerhammar-sphere radius larger than R). This will normally occur, if the data has a very inhomogeneous statistical character. Therefore simple histograms are always produced together with the covariances (in EMPCOV) in order to check that the data distribution is reasonably symmetric, if not normal.

Exercise 3.

Compute using COVFIT an analytic representation for the covariance function. An example of an input file is found in covfit.nmfa, see Appendix 1. An example of a run is shown in Appendix 3. Gravity error-degree-variances for the OSU91A coefficients are found in the file edgv.osu91. The estimated and the fitted covariance values are shown in Fig. 5.

Subsequently use COVFIT to tabulate the auto-and cross-covariance functions for residual gravity, deflections and geoid heights. (Job file covfit.nmta, see Appendix 1). (Values at the altitude of 1700 m are shown in Table 2.)

Table 2. Table of model-covariances of height anomalies, gravity anomalies and deflections of the vertical. Height 1700 m. Model degree-variances equal to A/((i-1)(i-2)(i+4)). Error degree-variances used from degree 2 to 360 with scale factor 0.207. Error degree variances from OSU91A (file osu91.edg). Depth to Bjerhammar-sphere = -6061.00 m, variance of point gravity anomalies at 0 height = 196.20 mgal2, the factor A, divided by RE2 is = 452.52 mgal2

Covariances between quantities KP and KQ, Evaluated in P,Q, having spherical distance PSI and an azimuth equal to zero. Values of KP or KQ used signify 1=height anomaly, 3=gravity anomaly, 6= component of deflection of the vertical in the direction between P and Q. Units: m, mgal and arcsec.

KP= 1 3 3 6 6 6

KQ= 1 3 1 6 3 1

PSI(ddmm.m) m2 mgal2 m*mgal arcsec2 arcsec*mgal arcsec*m

0 0.00 0.06524 142.22765 2.22634 3.17297 0.00000 0.00000

0 3.00 0.06411 131.37725 2.12867 2.81328 -7.00904 -0.08223

0 6.00 0.06096 104.86753 1.86979 1.95443 -11.71086 -0.14717

0 9.00 0.05640 73.50534 1.52289 0.97988 -13.58973 -0.18664

0 12.00 0.05112 44.55151 1.15654 0.13173 -13.29683 -0.20119

0 15.00 0.04573 21.11034 0.81785 -0.49983 -11.66534 -0.19571

0 18.00 0.04069 3.91214 0.53348 -0.90540 -9.36200 -0.17628

0 21.00 0.03630 -7.42000 0.31480 -1.11005 -6.85612 -0.14870

0 24.00 0.03270 -13.73315 0.16276 -1.15150 -4.45955 -0.11789

0 27.00 0.02994 -16.04179 0.07157 -1.07052 -2.36722 -0.08771

0 30.00 0.02794 -15.37807 0.03139 -0.90642 -0.68736 -0.06091

0 33.00 0.02661 -12.71152 0.03039 -0.69476 0.53671 -0.03925

0 36.00 0.02577 -8.90302 0.05627 -0.46611 1.31012 -0.02360

0 39.00 0.02528 -4.67778 0.09734 -0.24525 1.67322 -0.01405

0 42.00 0.02496 -0.61127 0.14338 -0.05094 1.69044 -0.01013

0 45.00 0.02469 2.87494 0.18609 0.10414 1.44048 -0.01095

0 48.00 0.02434 5.50874 0.21941 0.21302 1.00791 -0.01533

0 51.00 0.02384 7.15599 0.23955 0.27394 0.47589 -0.02200

0 54.00 0.02315 7.80293 0.24487 0.28952 -0.07969 -0.02969

0 57.00 0.02224 7.53425 0.23562 0.26588 -0.59476 -0.03726

1 0.00 0.02115 6.50831 0.21361 0.21158 -1.02003 -0.04376

1 3.00 0.01990 4.93120 0.18173 0.13657 -1.32262 -0.04848

1 6.00 0.01855 3.03134 0.14354 0.05124 -1.48616 -0.05103

1 9.00 0.01717 1.03626 0.10285 -0.03458 -1.50975 -0.05124

1 12.00 0.01581 -0.84740 0.06328 -0.11229 -1.40588 -0.04924


The numerical evaluation of the expression for the covariance function is rather time-consuming because it involves the summation of a Legendre-series up to degree N, here equal to 360. However, the covariances of geoid heights, gravity anomalies and deflections of the vertical may be tabulated. This option is available in COVFIT and in GEOCOL. However, the selection of the optimal table entries is complicated. The solution recommended for the moment is trial and error. The tool here is COVFIT, which may compute the differences between tabulated and correctly evaluated quantities.

Exercise 4.

Use COVFIT to compute differences between tabulated and analytically calculated values. For tabulation as a function of height use three intervals between 1500 m and 2600 m. For the tabulation in spherical distance use three equidistant intervals bounded by 0" to 60", 60" to 360" and 360" to 3600" and with 20 sub-intervals in each main interval. An input example is provided in the jobfile covfit.nmtt, see Appendix 1.

The covariances may also be tabulated in 1-dimension, i.e. only as a function of spherical distance. This is very useful if we deal with ocean data (at height zero).

5. LSC geoid determination from residual data.

We now have all the tools available for using LSC: residual data and a covariance model. The rest is to establish the normal equations eq. (11), solve the equations, eq. (12) and compute predictions and error estimates, eq. (12) and (13). This may be made using GEOCOL.

However, as realized from eq. (8) we have to solve a system of equations as large as the number of observations. This is one of the key problems with using the LSC method. The problem may be reduced by using means values of data in the border area. Also, if the observations are clustered, we may not need all observations. Rules for the necessary data density (d) as a function of the correlation length 1 of the covariance function are given in Tscherning(1984, p. 330). Suppose we want to determine geoid height differences with an error of 10 cm over 100 km. This corresponds to an error in deflections of the vertical of 0.2". This is equivalent to that we must be able to interpolate gravity anomalies with a mean error of 1.2 mgal. The rule-of-thumb is

Exercise 5.

Use the residual gravity variance C0, and the correlation distance determined in exercise 3 for the determination of the needed data spacing.

Then use the program SELECT for the selection of points as close a possible to the nodes of a grid having the required data spacing, and which covers the area of interest. The area covered should be larger than the area in which the geoid is to be computed. Data in a distance at least equal to the distance for which gravity and geoid becomes less than 10 % correlated, cf. the result of exercise 3. Denote this file nmfa.rd1.

When data have been selected (as described in exercise 5) it is recommended to prepare a contour plot of the data. This will show whether the data should contain any gross-errors. LSC may also be used for the detection of gross-errors, see Tscherning (1991).

An input file for the program GEOCOL must then be prepared, or the program may be run interactively. However, in order to compute height-anomalies at terrain altitude, a file with points consisting of number, latitude, longitude and altitude must be prepared. This may be prepared using the program GEOIP, and input from a digital terrain model.

Exercise 6.

Prepare a file named nm.h covering the area bounded by 33.0o and 34.0o in latitude and -107.0o and -106.0o in longitude consisting of sequence number, latitude, longitude and height given in a grid with 0.1 degree spacing. Use the program GEOIP with input from nmdtm. This will produce a grid-file. This must be converted to a standard point data file (named nmh2) using the program GLIST.

When using GEOCOL there must then be specified

the coordinate system used (GRS80),

the spherical harmonic expansion subtracted (and later to be added),

the constants defining the covariance model and contingently its tabulation

the input data files (nmfa.rd or nmfa.rd1 if a selected subset is used)

the files containing the points in which the predictions should be made (nm.h2).

Several options must be selected such as wether error-estimates should be computed or whether we want statistics to be output. The program may also produce a so-called restart file. This file is an ASCII-file which contains input to GEOCOL which enamles the calculation of predictions only. But it has the advantage that it may be used on different computers.

Exercise 7.

Run the program GEOCOL (geocol10) with the selected gravity data for the prediction of geoid heights and their errors in the points given by nm.h2. The result should be output to a file named nm.geoid. Predict also residual deflections of the vertical (nmdfv.rd) and compare with the observed quantities. Make sure a restart file is produced.

A model input file is found in jobnmlsc, see Appendix 1. An example of a run where all data in a sub-area are used is found in Appendix 5.

When the LSC-solution has been made, the RTM contribution to the geoid must be determined. Here the program tc1 may be used with the file nm.h defining the points of computation. The LSC determined residual geoid heights and the associated error-estimates are shown in Fig. 6 and 7.

Exercise 8.

Compute the RTM contribution to the geoid using tc1 and add the contribution to the output file from exercise 7, nm.geoid.

If mean gravity anomalies, deflections or GPS/levelling determined geoid-heights were to be used, they could easily have been added to the data. It would not be necessary to recalculate the full set of normal-equations. Only the columns related to the new data need to be added. Likewise, an obtained solution may be used to calculate such quantities and their error-estimates.

Exercise 9.

Compute a new solution with the same observations as in exercise 7, but add as observation one of the predicted residual geoid heights. Define the error to be 0.01 m. Recalculate the geoid heights and the error-estimates. Use the possibility for re-using the Cholesky-reduced (see Tscherning, 1984) normal-equations generated in exercise 7. Verify that the error-estimates, which now are equivalent to error-estimates of geoid height differences, have a magnitude smaller than the one specified in exercise 5. (Error-estimates corresponding to one observed geoid height are shown in Fig. 8).

The use of deflections and geoid heights (e.g. from satellite altimetry) may require that parameters such as datum shifts and bias/tilts are determined. These possibilities are also included in GEOCOL (see Tscherning, 1984).

6. Conclusion.

We have now went through all the steps from data to predicted geoid heights. The exercises describes the use of gravity data only, but observed mean gravity anomalies, GPS/levelling derived height anomalies as well as deflections could have been used as well.

The difficult steps in the application of LSC is the estimation of the covariance function and subsequent selection of an analytic representation.

The flexibility of the method is very useful in many circumstances, and is one of the reasons why the method has been applied in many countries. If the reference spherical harmonic expansion is of good quality, only a limited amount of data outside the area of interest are needed in order to obtain a good solution. But if this is not the case, data from a large border-area must be used so that a vast computational effort is needed to obtain a solution. This may make it impossible to apply the method.

A way out is then to use the method only for the determination of gridded values, which then may be used with Fourier transform techniques (Schwarz et al., 1991) or Fast Collocation (Bottoni & Barzaghi, 1993).

Acknowledgement: This paper was prepared for the International School for the determination and Use of the Geoid, Milano, Oct., 1994. Thanks to the US National geodetic Survey for the permission to use the New Mexico data. Also many colleagues have contributed to or inspired the delvelpment of the programs used: D.Arabelos (Thessaloniki), T.Krarup (Horsens), K.Poder (Oelstykke), H.Suenkel (Graz), R.H.Rapp (Columbus, Ohio).


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Appendix 1.

Here are shown examples of job-files used when solving the excercises. They should be regarded as a help when running the programs interactivally.

Job to subtract OSU91A contribution from gravity. Execution of the job is shown in appendix 2. jobosu91.nmfa.
f T f F f t f
3.98600500E+14 6378137.0 -484.1655 360 f f T F
F F t
-1 2 3 3 4 5 0 13 -1 0.00 t F F F F f t f F t

Job to compute and subtract RTM effects from gravity. jobtc.nmfa.

-1 4 1 1 2.67
31.52 34.98 -107.98 -105.02
18.0 999.0 2.67
2 2

Job to compute empirical covariance function of gravity. empcov.nmfa. Execution of job is shown in appendix 3.

5.0 96 3 F T F
4800 9 T 3 2 0 5.0 f
3 3 0

Job to determine parameters for covariance kernel. covfit.nmfa. Execution of job is shown in appendix 4.

4 f f t f
-6.150 200.0 360 f t t
0 2 0.27 F
10 0.5 0.2 0.1
1 1 000 000 T
18 3 3 1700 1700 1 1.0 t
177.8 177.8 40.0 42.0 0.2 100.0 103.0 0.2

Job to tabulate covariance functions. covfit.nmta. Result is found in Table 2.

1 f f t f
-6.061 196.2 360 f t t
0 2 0.207 F
25 3 0 0 0.0 t
1 1 1700 1700 F 3 3 1700 1700 f 3 1 1700 1700 f
6 6 1700 1700 f 6 3 1700 1700 f 6 1 1700 1700 t

Job to control tabulation error. covfit.nmtt.

3 f f t f
-6.061 196.2 360 f t f
0 2 0.207 F
3 1600.0 2600.0 4
60.0 300.0 1750.0 4000.0
10 20 30 40
25 3 0 0 0.0 t

Job to predict geoid from residual gravity and compare predicted and observed deflections of the vertical. jobnmlsc. Execution of job is found in appendix 5.

F T T F f F t
t f f f f
F T f
3.98600500E+14 6378137.0 -484.1655 36 f f T F
osu91a1f (use local file name !!)
-6.061 196.20 360 F f T f
0 4 0.207227
1 2 3 3 4 7 0 -13 -1 1700.00 f F F t F f t t F t
32.5 34.5 -107.5 -105.5
T f
F t f
1 2 3 3 5 0 0 11 -1 1700.0 t f F F F F f T F T
33.0 34.0 -107.0 -106.0
F t t
1 2 3 3 4 5 6 -5 -1 1700.0 f F F F F F T T F T
33.0 34.0 -107.0 -106.0

Updated 1997-07-04 by cct