1 / 79

Geoid determination by least-squares collocation using GRAVSOFT

Geoid determination by least-squares collocation using GRAVSOFT. C.C.Tscherning, University of Copenhagen, 2005-01-28 1. Purpose:.

moral
Download Presentation

Geoid determination by least-squares collocation using GRAVSOFT

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Geoid determination by least-squares collocation using GRAVSOFT C.C.Tscherning, University of Copenhagen, 2005-01-28 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, 2005-01-28 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, 2005-01-28 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 http://cct.gfy.ku.dk/gravsoft.txt which is updated regularly when changes to the programs have been made. C.C.Tscherning, University of Copenhagen, 2005-01-28 4

  5. FORTRAN 77 • All programs in FORTRAN77. • Have been run on many different computers under many different operating systems. • Available commercially, but free charge if used for scientific purposes. C.C.Tscherning, University of Copenhagen, 2005-01-28 5

  6. General methodology • General methodology for (regional or local) gravity field modelling : • Transform all data to a global geocentric geodetic datum (ITRF99/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, 2005-01-28 6

  7. 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. (Files with suffix *.rd). C.C.Tscherning, University of Copenhagen, 2005-01-28 7

  8. Covariance Determine at statistical model (a covariance function) for the residual data in the region in question. EMPCOV, COVFIT C.C.Tscherning, University of Copenhagen, 2005-01-28 8

  9. 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, 2005-01-28 9

  10. 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, 2005-01-28 10

  11. 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, 2005-01-28 11

  12. 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, 2005-01-28 12

  13. Test Data • GRAVSOFT includes data from New Mexico, USA, which can be used to test the programs and procedures. (Files: nmdtm, nmfa, nmdfv etc.) • They have here been used to illustrate the use of the programs. C.C.Tscherning, University of Copenhagen, 2005-01-28 13

  14. 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 expanded in solid spherical harmonics, Ynm • GM is the product of the gravitational constant and the mass of the Earth and the fuly normalized spherical harmonic coefficients. C.C.Tscherning, University of Copenhagen, 2005-01-28 14

  15. Coordinates used. GEOCOL accepts geocentric, geodetic and Cartesian (X,Y,Z) coordinates but output only in geodetic. C.C.Tscherning, University of Copenhagen, 2005-01-28 15

  16. Solid spherical harmonics. • where a is the semi-major axis and Pnm the Legendre functions. • We have orthogonality: C.C.Tscherning, University of Copenhagen, 2005-01-28 16

  17. 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, 2005-01-28 17

  18. 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, 2005-01-28 18

  19. Inner product, Reproducing property C.C.Tscherning, University of Copenhagen, 2005-01-28 19

  20. 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, 2005-01-28 20

  21. 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, 2005-01-28 21

  22. 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, 2005-01-28 22

  23. Data types GEOCOL codes: 11 12 13 16 17 • Also gravity gradients, • along-track or area mean values. C.C.Tscherning, University of Copenhagen, 2005-01-28 23

  24. Test data C.C.Tscherning, University of Copenhagen, 2005-01-28 24

  25. Linear Functionals, spherical approximation C.C.Tscherning, University of Copenhagen, 2005-01-28 25

  26. Best approximation: projection. Ti pre-selected base functions: C.C.Tscherning, University of Copenhagen, 2005-01-28 26

  27. 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, 2005-01-28 27

  28. 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, 2005-01-28 28

  29. 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, 2005-01-28 29

  30. Collocation solution in Hilbert Space Normal Equations Predictions: C.C.Tscherning, University of Copenhagen, 2005-01-28 30

  31. 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, 2005-01-28 31

  32. Mean-square error - globally C.C.Tscherning, University of Copenhagen, 2005-01-28 32

  33. Global Covariances: C.C.Tscherning, University of Copenhagen, 2005-01-28 33

  34. Covariance – series development C.C.Tscherning, University of Copenhagen, 2005-01-28 34

  35. Collocation Solution C.C.Tscherning, University of Copenhagen, 2005-01-28 35

  36. 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, 2005-01-28 36

  37. 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, 2005-01-28 37

  38. 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, 2005-01-28 38

  39. Covariance of gravity anomalies Appy the functionals on K(P,Q)=COV(P,Q) C.C.Tscherning, University of Copenhagen, 2005-01-28 39

  40. 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, 2005-01-28 40

  41. Remove-restore (I). 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, 2005-01-28 41

  42. Change of Covariance Function C.C.Tscherning, University of Copenhagen, 2005-01-28 42

  43. Homogenizing the data • 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, 2005-01-28 43

  44. Homogenizing (II) • 1.We remove the contribution Ts from a known spherical harmonic expansion like the OSU91A field, EGM96 or a GRACE model complete to degree N=360 • 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, 2005-01-28 44

  45. Residual quantities C.C.Tscherning, University of Copenhagen, 2005-01-28 45

  46. 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 http://cct.gfy.ku.dk/geoidschool/nmfa.pdf • The program GEOCOL may be used to subtract the contribution from OSU91A. • Job-files: http://cct.gfy.ku.dk/geoidschool/jobosu91.nmfa • http://cct.gfy.ku.dk/geoidschool/jobosu91.nmdfv C.C.Tscherning, University of Copenhagen, 2005-01-28 46

  47. Output-files Output from run: http://cct.gfy.ku.dk/geoidschool/appendix2.txt OSU91: http://cct.gfy.ku.dk/geoidschool/osu91a1f Differences: http://cct.gfy.ku.dk/geoidschool/nmfa.osu91 http://cct.gfy.ku.dk/geoidschool/nmdfv.osu91 Difference map: http://cct.gfy.ku.dk/geoidschool/nmfaosu91.pdf C.C.Tscherning, University of Copenhagen, 2005-01-28 47

  48. Residual topography removal 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, http://cct.gfy.ku.dk/geoidschool/nmdtm • A jobfile to run tc1 • http://cct.gfy.ku.dk/geoidschool/jobtc.nmfa • The result should be stored in files with names nmfa.rd and nmdfv.rd, respectively. • The residual gravity anomalies • http://cct.gfy.ku.dk/geoidschool/nmfard.pdf C.C.Tscherning, University of Copenhagen, 2005-01-28 48

  49. Smoothing or Homogenisation C.C.Tscherning, University of Copenhagen, 2005-01-28 49

  50. 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, 2005-01-28 50

More Related