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Geoid determination by 3D least-squares collocation by C.C.Tscherning Niels Bohr Institute, University of Copenhagen. C.C.Tscherning, University of Copenhagen, 2008-09-10 1. Purpose:.

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slide1
Geoid determination by

3D least-squares collocation

by

C.C.Tscherning

Niels Bohr Institute, University of Copenhagen

C.C.Tscherning, University of Copenhagen, 2008-09-10 1

purpose
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

C.C.Tscherning, University of Copenhagen, 2008-09-10 2

quasi geoid
Quasi-geoid:

Important:

the term geoid = the quasi-geoid,

i.e. the height anomaly evaluated at the surface of the Earth.

C.C.Tscherning, University of Copenhagen, 2008-09-10 3

gravsoft
Gravsoft

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, 2008. http

C.C.Tscherning, University of Copenhagen, 2008-09-10 4

gravsoft1
Gravsoft
  • We will only consider 3D applications.

C.C.Tscherning, University of Copenhagen, 2008-09-10 5

fortran 77
FORTRAN 77
  • 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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 6

general methodology
General methodology
  • 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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 7

remove restore method
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 8

covariance
Covariance

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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 9

select
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 XSelection of points O closest to the middle. 6 points selected

X

o

o

o

x

X

o

x

o

o

x

x

C.C.Tscherning, University of Copenhagen, 2008-09-10 10

errors
Errors
  • check for gross-errors (make histograms and contour map of data), GEOCOL
  • verify error estimates of data, GEOCOL.

C.C.Tscherning, University of Copenhagen, 2008-09-10 11

gravity field approximation and datum
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 12

restoring and checking
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 !

C.C.Tscherning, University of Copenhagen, 2008-09-10 13

test data and 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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 14

anomalous potential
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 15

coordinates used
Coordinates used.

GEOCOL accepts geocentric, geodetic and Cartesian (X,Y,Z) coordinates but output only in geodetic.

C.C.Tscherning, University of Copenhagen, 2008-09-10 16

solid spherical harmonics
Solid spherical harmonics.
  • where a is a scale factor (generally the semi-major axis) and Pnm the Legendre functions.
  • We have orthogonality:

C.C.Tscherning, University of Copenhagen, 2008-09-10 17

bjerhammar sphere
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 18

spherical approximation
Spherical approximation
  • NOT used when evaluating an EGM.
  • When used: r=Mean radius+h.
  • Geodetic latitude = geocentric latitude.

C.C.Tscherning, University of Copenhagen, 2008-09-10 19

reproducing kernel
Reproducing Kernel

where ψis 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’

C.C.Tscherning, University of Copenhagen, 2008-09-10 20

inner product reproducing property
Inner product, Reproducing property

C.C.Tscherning, University of Copenhagen, 2008-09-10 21

closed expression no summation to
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:

C.C.Tscherning, University of Copenhagen, 2008-09-10 22

hilbert space with reproducing kernel
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:

C.C.Tscherning, University of Copenhagen, 2008-09-10 23

data and model
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,

Lithe functional, ei the error,

Ai a vector of dimension k and

X a vector of parameters also of dimension k.

C.C.Tscherning, University of Copenhagen, 2008-09-10 24

data types
Data types

GEOCOL codes:

11

12

13

16

17

  • Also gravity gradients,
  • along-track or area mean values.

C.C.Tscherning, University of Copenhagen, 2008-09-10 25

test data
Test data

GRAVSOFT data from the New Mexico Area to be used in LSC geoid examples.

C.C.Tscherning, University of Copenhagen, 2008-09-10 26

linear functionals spherical approximation
Linear Functionals, spherical approximation

C.C.Tscherning, University of Copenhagen, 2008-09-10 27

best approximation projection
Best approximation: projection.

Ti pre-selected base functions:

C.C.Tscherning, University of Copenhagen, 2008-09-10 28

collocation
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

C.C.Tscherning, University of Copenhagen, 2008-09-10 29

projection
Projection

Approximation to T using error-free data is obtained using that the observations are given by, Li(T) = yi

C.C.Tscherning, University of Copenhagen, 2008-09-10 30

lsc mathematical
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 31

collocation solution in hilbert space
Collocation solution in Hilbert Space

Normal Equations

Predictions:

C.C.Tscherning, University of Copenhagen, 2008-09-10 32

statistical collocation solution
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

C.C.Tscherning, University of Copenhagen, 2008-09-10 33

mean square error globally
Mean-square error - globally

C.C.Tscherning, University of Copenhagen, 2008-09-10 34

global covariances
Global Covariances:

C.C.Tscherning, University of Copenhagen, 2008-09-10 35

covariance series development
Covariance – series development

C.C.Tscherning, University of Copenhagen, 2008-09-10 36

collocation solution
Collocation Solution

C.C.Tscherning, University of Copenhagen, 2008-09-10 37

noise
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 38

minimalisation of mean square error
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

C.C.Tscherning, University of Copenhagen, 2008-09-10 39

parameter estimate
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).

C.C.Tscherning, University of Copenhagen, 2008-09-11 40

error estimates
Error-estimates
  • The associated error estimates are with
  • the mean square error of the parameter vector
  • and the mean square error of an estimated quantity .

C.C.Tscherning, University of Copenhagen, 2008-09-11 41

n 0 estimation
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.

C.C.Tscherning, University of Copenhagen, 2008-09-11 42

covariance propagation
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 43

covariance of gravity anomalies
Covariance of gravity anomalies

Appy the functionals on

K(P,Q)=COV(P,Q)

C.C.Tscherning, University of Copenhagen, 2008-09-10 44

evaluation of covariances
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 45

remove restore
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).

C.C.Tscherning, University of Copenhagen, 2008-09-10 46

change of covariance function
Change of Covariance Function

C.C.Tscherning, University of Copenhagen, 2008-09-10 47

homogenizing the data
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 48

homogenizing ii
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 49

residual quantities
Residual quantities

C.C.Tscherning, University of Copenhagen, 2008-09-10 50

exercise 1
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 51

input file data nmfa
Input-file: data\nmfa

C.C.Tscherning, University of Copenhagen, 2008-09-10 52

geoegm with nmfa
GEOEGM with nmfa

C.C.Tscherning, University of Copenhagen, 2008-09-10 53

output files
Output-files

Output from run on screen (last part) and in geoegm.log:

http://cct.gfy.ku.dk/geoidschool/appendix2.txt

COMPARISON OF PREDICTIONS AND OBSERVATIONS

DATA TYPE = 13

NUMBER: 2920

OBSERVATIONS PREDICTIONS DIFFERENCE

MEAND 9.181986 12.113828 -2.931842

ST.DEVI. 30.405342 23.100298 21.283325

MAX 162.500000 77.947778 126.430437

MIN -58.700000 -28.049925 -74.792095

DISTRIBUTION OF DIFFERENCES, UNITS: 5.000000

17 21 40 66107169225261317309324279221149110 73 59 29 25 24 19 76

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 OUTSIDE

GEOCOL TERMINATED AT:

Wed Jul 23 11:18:40 2008

C.C.Tscherning, University of Copenhagen, 2008-09-10 54

output file contoured with gmt
Output-file contoured with GMT.

C.C.Tscherning, University of Copenhagen, 2008-09-10 55

exercise 2 residual topography removal
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 56

residual topography removal
Residual topography removal

C.C.Tscherning, University of Copenhagen, 2008-09-10 57

residual topography removal1
Residual topography removal

C.C.Tscherning, University of Copenhagen, 2008-09-10 58

smoothing or homogenisation
Smoothing or Homogenisation

C.C.Tscherning, University of Copenhagen, 2008-09-10 59

consequences for the statistical model
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,

C.C.Tscherning, University of Copenhagen, 2008-09-10 60

error degree variances
Error-degree-variances

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

Note that the error-degree variances refer to the mean –earth sphere.

C.C.Tscherning, University of Copenhagen, 2008-09-10 61

covariance function estimation and representation
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 62

global local evaluation
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,
  • Δψis the interval length (also denoted the sampling interval size).

C.C.Tscherning, University of Copenhagen, 2008-09-10 63

covariance1
Covariance
  • In spherical approximation we have already derived
  • where R is the mean radius of the Earth.

C.C.Tscherning, University of Copenhagen, 2008-09-10 64

exercise 3 hand calculation of covariances
Exercise 3. Hand calculation of covariances.

The following data must be used, with format: number, latitude, longitude, altitude and gravity anomaly in mgal.

11 56.0 10.0 0.0 4.0

12 56.1 10.0 0.0 2.0

13 56.2 10.0 0.0 0.0

14 56.3 10.0 0.0 ‑2.0

15 56.4 10.0 0.0 ‑4.0

16 56.5 10.0 0.0 ‑6.0

17 56.6 10.0 0.0 ‑8.0

18 56.7 10.0 0.0 ‑9.0

19 56.8 10.0 0.0 ‑7.0

20 56.9 10.0 0.0 ‑5.0

21 57.0 10.0 0.0 ‑3.0

22 57.1 10.0 0.0 ‑1.0

23 57.2 10.0 0.0 1.0

24 57.3 10.0 0.0 5.0

25 57.4 10.0 0.0 4.0

C.C.Tscherning, University of Copenhagen, 2008-09-10 65

exercise 3
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).

C.C.Tscherning, University of Copenhagen, 2008-09-10 66

exercise 31
Exercise 3.

C.C.Tscherning, University of Copenhagen, 2008-09-10 67

exercise 3 part of empcov log
Exercise 3: Part of empcov.log.

2920 VALUES INPUT FROM FILE data/nmfa-egm96-tc.dat

NUMBER OF OBS 1= 2920 MEAN = 0.3066 VAR. = 173.53

HISTOGRAM, USING BIN SIZE= 5.000

0 0 0 0 3 9 34 49118248372887357343233127 59 35 18 18 10 0 0

OUT-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9' 10OUT

PSI COVA( 1, 1) PROD. STDV OF COV..

O M (UNIT)**2 NUMB (UNIT)**2

0 0.00 175.509014 3132 4.9 0.330449

0 2.00 145.042029 4762 3.4 20.163033

0 4.00 119.317243 9487 2.3 45.463833

0 6.00 90.458667 14003 1.8 75.421001

0 8.00 67.998329 17995 1.5 101.052568

0 10.00 48.805606 21731 1.3 119.723639

0 12.00 31.054065 25167 1.2 136.020523

0 14.00 18.040267 28540 1.1 149.844158

C.C.Tscherning, University of Copenhagen, 2008-09-10 68

empirical and analytic covariances
Empirical and analytic Covariances

C.C.Tscherning, University of Copenhagen, 2008-09-10 69

degree variances
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

C.C.Tscherning, University of Copenhagen, 2008-09-10 70

model degree variances
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 71

covfit
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

C.C.Tscherning, University of Copenhagen, 2008-09-10 72

divergence
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 73

exercise 4
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 74

exercise 41
Exercise 4.

C.C.Tscherning, University of Copenhagen, 2008-09-10 75

exercise 4 bottom of covfit log
Exercise 4 - bottom of covfit.log.

TAU(J) USED IN THE CX MATRIX 0.10E+01 0.10E+01 0.10E+01

RESULTS IN VARIANCE OF GRAVITY ANOMALIES:

1'TH ROW OF INVERSE MATRIX 0.4132E-01 -0.1243E-01 -0.3080E-01

2'TH ROW OF INVERSE MATRIX -0.1243E-01 0.1211E-01 0.4449E-01

3'TH ROW OF INVERSE MATRIX -0.3080E-01 0.4449E-01 0.1782E+00

STD.DEV. 0.576779E-01 0.732173E+05 0.334610E+03

STD.DEV.*RMS 0.329951E-01 0.418845E+05 0.191416E+03

RESULTS IN VARIANCE OF GRAVITY ANOMALIES: 334.36 MGAL**2.

  • N RATIO AA A RE-RB VARG IT
  • 360 0.5721D+00 0.2837 0.6654D+06 -792.72 334.36 10

C.C.Tscherning, University of Copenhagen, 2008-09-10 76

table of model covariances at h 1700 m
Table of model covariances at h=1700 m.

Ψ (deg) ζ,ζΔg,Δg Δg,ζξ,ξξ,Δg ξ,ζ

0.00 0.0476 174.15 2.058 3.878 0.000 0.000

0.05 0.0463 139.00 1.885 2.749 -11.167 -0.092

0.10 0.0430 90.20 1.535 1.307 -13.884 -0.146

0.15 0.0387 53.43 1.167 0.318 -13.132 -0.167

0.20 0.0342 27.60 0.837 -0.309 -11.188 -0.167

0.25 0.0298 10.08 0.566 -0.678 -8.886 -0.153

0.30 0.0260 -1.15 0.358 -0.860 -6.597 -0.132

0.40 0.0201 -10.61 0.112 -0.857 -2.730 -0.084

0.50 0.0167 -9.71 0.038 -0.593 -0.233 -0.044

0.60 0.0150 -4.53 0.062 -0.264 0.885 -0.021

0.70 0.0141 0.91 0.115 0.004 0.939 -0.014

0.80 0.0133 4.40 0.152 0.153 0.377 -0.019

0.90 0.0120 5.20 0.152 0.174 -0.353 -0.028

1.00 0.0102 3.75 0.117 0.100 -0.910 -0.036

1.10 0.0081 1.11 0.061 -0.014 -1.117 -0.038

1.20 0.0061 -1.51 0.003 -0.121 -0.961 -0.035

C.C.Tscherning, University of Copenhagen, 2008-09-10 77

conversion of geoid heights to height anomalies
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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lsc geoid determination from residual data
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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equations
Equations
  • However we have to solve a system of equations as large as the number of observations. GEOCOL has been used to handle 100000 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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necessary data density d
Necessary data density (d)
  • Function of correlation lengthof 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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exercise 5 data density
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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data selection
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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exercise 6 geocol input
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 84

exercise 6
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 85

geocol input specifications
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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geocol options
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 87

exercise 7
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.

C.C.Tscherning, University of Copenhagen, 2008-09-10 88

exercise 8
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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exercise 71
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

1 11111 -0.783996623 0.092367316

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exercise 72
Exercise 7.

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exercise 73
Exercise 7.

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exercise 7 8 continued
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: 20

OBSERVATIONS PREDICTIONS DIFFERENCE

MEAND -0.897275 -0.897275 0.000000 0.184852

ST.DEVI. 0.159149 0.156162 0.015303 0.000101

MAX -0.632800 -0.635039 0.038347 0.185066

MIN -1.267500 -1.246579 -0.028575 0.184632

DISTRIBUTION OF DIFFERENCES, UNITS: 0.010000

0 0 0 0 0 0 0 1 3 1 10 2 2 0 1 0 0 0 0 0 0 0

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 OUTSIDE

.

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exercise 9 restore
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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exercise 81
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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error detection
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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exercise 10
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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conclusion i
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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conclusion ii
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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conclusion iii
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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