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

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

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  1. 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

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

  3. 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

  4. 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

  5. Gravsoft • We will only consider 3D applications. C.C.Tscherning, University of Copenhagen, 2008-09-10 5

  6. 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

  7. 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

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

  9. 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

  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 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

  11. 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

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

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

  14. 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

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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. Inner product, Reproducing property C.C.Tscherning, University of Copenhagen, 2008-09-10 21

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

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

  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, 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

  25. 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

  26. 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

  27. Linear Functionals, spherical approximation C.C.Tscherning, University of Copenhagen, 2008-09-10 27

  28. Best approximation: projection. Ti pre-selected base functions: C.C.Tscherning, University of Copenhagen, 2008-09-10 28

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

  30. 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

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

  32. Collocation solution in Hilbert Space Normal Equations Predictions: C.C.Tscherning, University of Copenhagen, 2008-09-10 32

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

  34. Mean-square error - globally C.C.Tscherning, University of Copenhagen, 2008-09-10 34

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

  36. Covariance – series development C.C.Tscherning, University of Copenhagen, 2008-09-10 36

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

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

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

  40. 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

  41. 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

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

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

  44. Covariance of gravity anomalies Appy the functionals on K(P,Q)=COV(P,Q) C.C.Tscherning, University of Copenhagen, 2008-09-10 44

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

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

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

  48. 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

  49. 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

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

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