1. Geoid determination by 3D least-squares collocation
C.C.Tscherning
Niels Bohr Institute, University of Copenhagen
C.C.Tscherning, University of Copenhagen,

2. Purpose:
Guide to gravity field modeling, and especially to geoid determination, using least-squares collocation (LSC).
DATA 
GRAVITY FIELD MODEL
EVERYTHING =
Height anomalies, gravity anomalies, gravity disturbances, deflections of the vertical, gravity gradients, spherical harmonic coeffients
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3. the term geoid = the quasi-geoid,
Important:
the term geoid = the quasi-geoid,
i.e. the height anomaly evaluated at the surface of the Earth.
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4. The use of the GRAVSOFT package of FORTRAN programs will be explained.
A general description of the GRAVSOFT programs are given in:
Forsberg, R. and C.C.Tscherning: An overview manual for the GRAVSOFT Geodetic Gravity Field Modelling Programs. 2.edition. Contract report for JUPEM, http
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5. We will only consider 3D applications.
Gravsoft
We will only consider 3D applications.
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6. All programs in FORTRAN77.
Have been run on many different computers under many different operating systems.
Available commercially, but free of charge if used for scientific or educational purposes.
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7. General methodology for (regional or local) gravity field modelling :
Transform all data to a global geocentric geodetic datum (ITRF05/GRS80/WGS84), (but NO tides, NO atmosphere) GEOCOL
“geoid-heights” must be converted to height anomalies N2ZETA
Use the remove-restore method.
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8. Remove-restore method
The effect of a spherical harmonic expansion and of the topography is removed from the data and
subsequently added to the result. GEOCOL, TC,
TCGRID
This is used for all gravity modelling methods including LSC.
This will produce what we will call residual data.
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9. Thereby we have an analytic representation of the covariance function.
Determine a Reproducing Kernel so that it agrees with a covariance function for the residual data in the region in question.
EMPCOV, COVFIT
Thereby we have an analytic representation of the covariance function.
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10. Select
Make a homogeneous selection of the data to be used for geoid determination using rule-of-thumbs for the required data density, SELECT
If many data select those with the smallest error X Selection of points O closest to the middle. 6 points selected X o o o x X o x o o x x
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11. verify error estimates of data, GEOCOL.
Errors
check for gross-errors (make histograms and contour map of data), GEOCOL
verify error estimates of data, GEOCOL.
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12. Gravity field approximation and datum
Determine using LSC a gravity field approximation, including contingent systematic parameters such as height system bias N0. GEOCOL
Compute estimates of the height-anomalies and their errors. GEOCOL
If the error is too large, and more data is available add new data and repeat.
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13. Restoring and checking.
Check model, by comparison with data not used to obtain the model. GEOCOL.
Restore contribution from Spherical Harmonic model and topography. GEOCOL, TC.
Convert height anomalies to geoid heights if needed N2ZETA.
The whole process can be carried through using the GRAVSOFT programs
Compare with results using other methods !
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14. They have here been used to illustrate the use of the programs.
Test Data and programs
GRAVSOFT includes data from New Mexico, USA, which can be used to test the programs and procedures. (Files: nmdtm, nmfa, nmzeta etc.)
They have here been used to illustrate the use of the programs.
If pyGravsoft has been loaded on your computer, programs are found in directory src, executables in bin and data in data. Documentation in doc.
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15. Anomalous potential.
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 approximated arbitrarily well by a series in solid spherical harmonics, Snm
GM is the product of the gravitational constant and the mass of the Earth and the fully normalized spherical harmonic coefficients.
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16. Coordinates used.
GEOCOL accepts geocentric, geodetic and Cartesian (X,Y,Z) coordinates but output only in geodetic.
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17. Solid spherical harmonics.
where a is a scale factor (generally the semi-major axis) and Pnm the Legendre functions.
We have orthogonality:
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18. The functions Ynm(P) are ortho- gonal basefunctions in a Hilbert
Bjerhammar-sphere
The functions Ynm(P) are ortho-
gonal basefunctions 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. In spherical approximation the ellipsoid is replaced by a sphere with radius 6371 km.
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19. Spherical approximation
NOT used when evaluating an EGM.
When used: r=Mean radius+h.
Geodetic latitude = geocentric latitude.
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20. Reproducing Kernel
where ψ is the spherical distance between P and Q, Pn the Legendre polynomials and σn are positive constants, the (potential) degree-variances. P r Q ψ r’
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21. Inner product, Reproducing property
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22. Closed expression – no summation to
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) given by a closed expression. Example:
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23. Hilbert Space with Reproducing Kernel
Everything like in an n-dimensional vector space.
COORDINATES:
ANGLES  between two
functions, f, g
PROJECTION f ON g:
IDENTITY MAPPING:
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24. Data and Model
In a (RKHS) approximations T from data for which the associated linear functionals are bounded.
The relationship between the data and T are expressed through functionals Li,
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.
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25. Data types GEOCOL codes: 11 12 13 16 17 Also gravity gradients,
along-track or area mean values.
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26. Test data
GRAVSOFT data from the New Mexico Area to be used in LSC geoid examples.
Type
Format (see Forsberg & Tscherning, 2008)
Error
File name
Free-air gravity (Fig. 2)
#, latitude, longitude, altitude, Δg
0.2 mgal
nmfa
Deflections
#, latitude, longitude, ξ, η
0.5 arcsec
nmdfv
Height anomalies
#, latitude, longitude, altitude, ζ
0.02 m
nmzeta
Digital height model
Grid-label, data in N-S, E-W
5 m
nmdtm
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27. Linear Functionals, spherical approximation
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28. Best approximation: projection.
Ti pre-selected base functions:
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29. Collocation
LSC tell which functions to select if we also require that approximation and observations agree and
how to find projection !
Suppose data error-free:
We want solution to agree with data
We want smooth variation between data
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30. Projection
Approximation to T using error-free data is obtained using that the observations are given by, Li(T) = yi
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31. LSC - mathematical
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.
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32. Collocation solution in Hilbert Space
Normal Equations
Predictions:
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33. Statistical Collocation Solution
We want solution with smallest “error” for all configurations of points which by a rotation around the center of the Earth can be obtained from the original data. And agrees with noise-free data.
We want solution to be linear in the observations
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34. Mean-square error - globally
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35. Global Covariances:
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36. Covariance – series development
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37. Collocation Solution
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38. Noise
If the data contain noise, then the elements σij of the variance-covariance matrix of the noise-vector is added to K(Li,Lj) = COV(Li(T),Lj(T)).
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.
BUT THE METHOD IS ONLY GIVING THE MINIMUM LEAST-SQUARES ERROR IN A GLOBAL SENSE.
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39. Minimalisation of mean-square error
The reproducing kernel must be selected equal to the empirical covariance function, COV(P,Q).
This function is equal to a reproducing kernel with the degree-variances equal to
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40. A simultaneous estimate of T and of the parameters X are obtained as
Parameter estimate
A simultaneous estimate of T and of the parameters X are obtained as
where W is the a-priori weight matrix for the parameters (Generally the zero matrix).
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41. The associated error estimates are with
the mean square error of the parameter vector
and the mean square error of an estimated quantity
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42. N0 estimation Height system bias may be determined.
The determination of a datum-shift requires data which covers a large area,
If the area is not large, this will be reflected in large error estimates
This will be illustrated using 2920 gravity data and 20 height anomalies from the New Mexico test area.
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43. Covariance Propagation
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 harmonic for either P or Q fixed.
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44. Covariance of gravity anomalies
Appy the functionals on
K(P,Q)=COV(P,Q)
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45. Evaluation of covariances
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.
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46. Remove-restore.
If we want to compute height-anomalies from gravity anomalies, we need a global data distribution.
If we work in a local area, the information outside the area may be represented by a spherical harmonic model. If we subtract the contribution from such a model, we have to a certain extend taken data outside the area into account.
(The contribution to the height anomalies must later be restored=added).
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47. Change of Covariance Function
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48. We obtain a minimum mean square error in a very specific sense:
Homogenizing the data
We obtain a minimum mean square error in a very specific sense:
as the mean over all data-configurations which by a rotation of the Earth's center may be mapped into each other.
Locally, we must make all areas of the Earth look alike.
This is done by removing as much as we know, and later adding it. We obtain a field which is statistically more homogeneous.
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49. but center of mass may have changed !!!
Homogenizing (II)
1.We remove the contribution TEGM from a known spherical harmonic expansion like the EGM96 to N=360 or EGM2008 to N=2190.
2. We remove the effect of the local topography, TM, using Residual Terrain Modelling (RTM): Earths total mass not changed,
but center of mass may have changed !!!
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.
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50. Residual quantities
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51. Exercise 1.
Compute residual gravity anomalies and residual height anomalies using the EGM96 spherical harmonic expansion and the New Mexico free-air gravity anomalies in data/nmfa and the height anomalies in data/nmzeta.
The Python module GEOEGM may be used to subtract the contribution from EGM96. Result files: data/nmfa-egm96.dat and data/nmzeta-egm96.dat.
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52. Input-file: data\nmfa
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53. GEOEGM with nmfa
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54. Output-files Output from run on screen (last part) and in geoegm.log:
COMPARISON OF PREDICTIONS AND OBSERVATIONS
DATA TYPE = 13
NUMBER:
OBSERVATIONS PREDICTIONS DIFFERENCE
MEAND
ST.DEVI
MAX
MIN
DISTRIBUTION OF DIFFERENCES, UNITS:
OUTSIDE
GEOCOL TERMINATED AT:
Wed Jul 23 11:18:
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55. Output-file contoured with GMT.
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56. Exercise 2. Residual topography removal
The RTM contribution may be computed and subtracted using the TC module.
First a reference terrain model must be constructed using the program TCGRID with the file data/nmdtm as basis,
The result should be stored in files with names nmfa-egm96-tc.dat.
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57. Residual topography removal
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58. Residual topography removal
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59. Smoothing or Homogenisation
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60. Consequences for the statistical model.
The degree-variances will be changed up to the maximal degree, N, sometimes 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,
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61. Error-degree-variances
The scaling factor α must therefore be determined from the data (in the program COVFIT, see later).
Note that the error-degree variances refer to the mean –earth sphere.
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62. Covariance function estimation and representation.
The covariance function to be used in LSC is equal to
where α is the azimuth between P and Q and φ, λ are the coordinates of P. The spherical distance ψ is fixed.
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 ψ between the points.
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63. Global-local evaluation
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,
Δψ is the interval length (also denoted the sampling interval size).
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64. In spherical approximation we have already derived
Covariance
In spherical approximation we have already derived
where R is the mean radius of the Earth.
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65. Exercise 3. Hand calculation of covariances.
The following data must be used, with format: number, latitude, longitude, altitude and gravity anomaly in mgal.
‑2.0
‑4.0
‑6.0
‑8.0
‑9.0
‑7.0
‑5.0
‑3.0
‑1.0
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66. Exercise 3.
Compute the empirical gravity anomaly covariance function using the program EMPCOV for the New Mexico area both for the anomalies minus EGM96 and for the anomalies from which also RTM-effects have been subtracted (input files nmfa-egm96.dat and nmfa-egm96-tc.dat).
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67. Exercise 3.
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68. Exercise 3: Part of empcov.log.
2920 VALUES INPUT FROM FILE data/nmfa-egm96-tc.dat
NUMBER OF OBS 1= MEAN = VAR. =
HISTOGRAM, USING BIN SIZE=
OUT ' 10OUT
PSI COVA( 1, 1) PROD. STDV OF COV..
O M (UNIT)**2 NUMB (UNIT)**2
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69. Empirical and analytic Covariances
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70. Degree-variances
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
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71. Model-degree-variances
Use a degree-variance model, i.e. a functional dependence between the degree and the degree-variances.
In 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 is of the type 2,
where RB is the radius of the Bjerhammar-sphere, A is a constant in units of (m/s)2, B an integer.
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72. Instead of A the gravity variance C0 at zero is used.
COVFIT
The actual modelling of the empirically determined values is done using the program COVFIT. The factors a, A and RB need to be determined (the first index N+1 must be fixed).
Instead of A the gravity variance C0 at zero is used.
The program makes an iterative non-linear adjustment for the Bjerhammar-sphere radius, and linear for the two other quantities
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73. Divergence ?
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.
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74. Covariance table from EMPCOV: data/covtable_nmfa.txt
Exercise 4.
Compute using COVFIT an analytic representation for the covariance function.
Input are: empirical error-degree variances from EGM96 in data/egm96.edg
Covariance table from EMPCOV: data/covtable_nmfa.txt
Mean altitude and variance of gravity anomalies
Area boundaries and data spacing.
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75. Exercise 4.
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76. Exercise 4 - bottom of covfit.log.
TAU(J) USED IN THE CX MATRIX E E E+01
RESULTS IN VARIANCE OF GRAVITY ANOMALIES:
1'TH ROW OF INVERSE MATRIX E E E-01
2'TH ROW OF INVERSE MATRIX E E E-01
3'TH ROW OF INVERSE MATRIX E E E+00
STD.DEV E E E+03
STD.DEV.*RMS E E E+03
RESULTS IN VARIANCE OF GRAVITY ANOMALIES: MGAL**2.
N RATIO AA A RE-RB VARG IT
D D
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77. Table of model covariances at h=1700 m.
Ψ (deg) ζ,ζ Δg,Δg Δg,ζ ξ,ξ ξ,Δg ξ,ζ
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78. Conversion of geoid heights to height anomalies
In 3D LSC all data must refer to points outside the surface of the Earth.
Geoid heights must be converted to height anomalies.
GRAVSOFT module N2ZETA can be used. It implements the equation:
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79. LSC geoid determination from residual data.
We now have all the tools available for using LSC: residual data and a covariance model.
1.establish the normal equations,
2.solve the equations, and
3. compute predictions and error estimates.
This may be done using GEOCOL.
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80. Equations
However we have to solve a system of equations as large as the number of observations. GEOCOL has been used to handle observations simultaneously.
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.
Globally gridded data can be used (sphgric)
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81. Necessary data density (d)
Function of correlation length of the covariance function.
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
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82. Exercise 5. Data density.
Use the residual gravity variance C0, and the correlation distance determined in exercise 4 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.
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83. Data selection.
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.
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.
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84. An input file for the program GEOCOL must then be prepared,
Exercise 6.GEOCOL INPUT.
An input file for the program GEOCOL must then be prepared,
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 GLIST, and input from a digital terrain model (nmdtm). A file data/nm.h2 has been prepared.
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85. Exercise 6.
Prepare a file named nm.h covering the area bounded by 33.0o and 34.0o in latitude and o and o 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.
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86. GEOCOL INPUT/SPECIFICATIONS.
the coordinate system used (GRS80),
the spherical harmonic expansion subtracted (and later to be added),
the constants defining the covariance model
the input data files (nmfa-egm96-tc.data and nmzeta-egm96-tc.dat)
the files containing the points in which the predictions should be made (nmzeta-egm96-tc0.dat and nm.h2).
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87. GEOCOL OPTIONS
Several options must be selected such as whether error-estimates should be computed or whether we want statistics to be output.
The Cholesky-reduced normal equations can be re-used if the program is run with the same observations as in an earlier run.
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88. Exercise 7.
Run the program GEOCOL (geocol17) with the selected gravity data for the prediction of height anomalies and their errors in the points given by nmzeta.
Compare input and predicted values. Remember to use negative data codes (-13 and -11) since EGM96 has already been subtracted.
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89. Re-use the 2920 reduced normal equations.
Exercise 8.
Make a second run where the reduced height anomalies are used as a second data-set.
Re-use the 2920 reduced normal equations.
This will determine the height-system bias N0. (If the Python module GEOCOL is used, see following).
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90. C.C.Tscherning, University of Copenhagen, 2008-09-10 90

91. In geocol.log we see the height system bias N0:
Exercise 7.
In geocol.log we see the height system bias N0:
ELEMENTS OF (AT*C**-1*A)**-1.
0.0085
CORRELATION MATRIX:
1.0000
PARAMETER TYPE ESTIMATE ERROR ESTIMATE (FOR TILT: ZERO POINT
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92. Exercise 7.
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93. Exercise 7.
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94. Exercise 7+8 continued.
In geocol.log is a comparison of the input height anomalies with the predicted.
COMPARISON OF PREDICTIONS AND OBSERVATIONS
DATA TYPE = 11
NUMBER:
OBSERVATIONS PREDICTIONS DIFFERENCE
MEAND
ST.DEVI
MAX
MIN
DISTRIBUTION OF DIFFERENCES, UNITS:
OUTSIDE .
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95. Use tc with the file nm.h defining the points of computation.
Exercise 9. RESTORE.
When the LSC-solution has been made, the RTM contribution to the geoid must be determined.
Use tc with the file nm.h defining the points of computation.
The LSC determined residual geoid heights and the associated error-estimates are shown above.
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96. Exercise 8.
Compute the RTM contribution to the geoid using tc 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.
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97. Large difference – possible gross error.
Error detection:
LSC filters the data .
We may as done in geodetic network adjustment inspect the residuals by using LSC for the prediction and comparison with the data used to determine the model.
Large difference – possible gross error.
Outlier
Predicted
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98. Exercise 10.
Detection of suspected gross-errors may be done by comparing the differences between observed quantities and predicted quantities to the estimated error. Use GEOCOL (cf. Exercise 6) for this purpose by predicting reduced gravity anomalies (data/nmfa-egm96-tc0.dat) and comparing these with values predicted from an identical file, but named data/nmfa-egm96-tc.dat. A file name for a file to hold suspected gross errors must be input.
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99. The exercises describes the use of gravity data and height anomalies,
Conclusion (I)
We have now went through all the steps from data to predicted geoid heights.
The exercises describes the use of gravity data and height anomalies,
and the determination of a height-system bias.
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100. Conclusion (II)
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.
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101. This may make it impossible to apply the method.
Conclusion (III)
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 or Fast Collocation.
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