1. C.C.Tscherning, University of Copenhagen, Denmark. Developments in the implementation and use of Least-Squares Collocation. IAG Scientific Assembly, Potsdam, Sept. 2013

2. LEAST-SQUARES COLLOCATION Developed 1960+ by H.Moritz and T.Krarup for gravity field modelling. Optimal (best) linear approximation method in reproducing kernel Hilbert-Space of harmonic functions or in an equivalent stochastic process. Minimum norm solution All types of Data (linearized) can be used and ”predicted” Errors can be estimated, Parameters may be solved for. Convergence of method proven for increasing number of data in ideal situation. IAG Scientific Assembly, Potsdam, Sept. 2013

3. LEAST-SQUARES COLLOCATION OPTIMAL IAG Scientific Assembly, Potsdam, Sept. 2013 EGM08GOCE Dir R3 RPM all Mean0.380.28-0.120.36-0.68 St.devia.20.0411.8911.9715.9510.87 Mean0.020.010.000.02-0.03 St.devia0.580.52 0.590.50

4. LEAST-SQUARES COLLOCATION – Problems. Hilbert space of harmonic functions outside a (Bjerhammar) sphere needed in order to make the reproducing kernels or covariance functions computable. But data at poles inside sphere ! IAG Scientific Assembly, Potsdam, Sept. 2013

5. LEAST-SQUARES COLLOCATION – Solution. Use coefficients of an EGM as observations and change to spherical approximation. Predict equivalent data at poles at 20 km altitude. Equivalent to Remove/restore of EGM. Side effect: Data variance decreases and data are decorrelated ! IAG Scientific Assembly, Potsdam, Sept. 2013

6. LEAST-SQUARES COLLOCATION BASIC EQUATION IAG Scientific Assembly, Potsdam, Sept. 2013

7. LEAST-SQUARES COLLOCATION SOLUTION OF EQUATIONS As many equations as observations (+parameters) Multi-processing enables large systems to be Cholesky reduced. Results from 2.40 GHz Intel® Computer IAG Scientific Assembly, Potsdam, Sept. 2013 N3797122464 Processors22 4 Time (s)440136391

8. GLOBAL LEAST-SQUARES COLLOCATION IAG Scientific Assembly, Potsdam, Sept. 2013 ModelN, Number of observations EstimateError-estimate EGM960.1110.036 EGM20080.1000.012 GOCE TIM2  10000000 0.1050.015 422190.1200.054 1642120.1060.028 ~650000?0.014

9. GLOBAL LEAST-SQUARES COLLOCATION IAG Scientific Assembly, Potsdam, Sept. 2013

10. LSC COVARIANCE FUNCTIONS IAG Scientific Assembly, Potsdam, Sept. 2013

11. LSC COVARIANCE FUNCTIONS Locally empirically estimated functions may be represented by reproducing kernels using basic model from 1974. Model is not optimal since it is isotropic. Recent development Reguzzoni & Gatti: Anisotropic covariance modelling based on locally adapted coefficient variances in gravity field estimation. Shows improvements compared to isotropic models !! IAG Scientific Assembly, Potsdam, Sept. 2013

12. Local LSC COVARIANCE FUNCTIONS IAG Scientific Assembly, Potsdam, Sept. 2013

13. LSC Software First general program written in Algol, 1972. General FORTRAN program written in 1974 for 3D LSC, geocol, now geocol19, by C.C.Tscherning et al.. in 1986 2D LSC, gpcol, by R.Forsberg. Both programs available for scientific or teaching purpose free of charge. Programs also developed at TUGraz, POLIMI, UHannover. IAG Scientific Assembly, Potsdam, Sept. 2013

14. LSC ERROR ESTIMATES + IMPROVEMENTS IAG Scientific Assembly, Potsdam, Sept. 2013

15. Example of local improvement of error estimates IAG Scientific Assembly, Potsdam, Sept. 2013 Gravity anomalies from GOCE Tzz & EGM2008 to 512. (ITG- Grave2010c to 36 subtracted everywhere), units: mgal.

16. Differences and LSC error estimates IAG Scientific Assembly, Potsdam, Sept. 2013 Differences gravity fromGOCE Tzz- EGM2008 to 512 and LSC error estimates.

17. Tzz RMS in 1 deg. blocks and scaled error estimates IAG Scientific Assembly, Potsdam, Sept. 2013 Tzz RMS (E.U.) and and scaled error estimates (mgal).

18. Global scaling using Tzz RMS in 1 deg. Blocks. IAG Scientific Assembly, Potsdam, Sept. 2013 GOCE Tzz (-ITG-GRACE2010c to 36) RMS (E.U)

19. Global scaling of LSC error estimates in 1x1 deg blocks. IAG Scientific Assembly, Potsdam, Sept. 2013 Scaled error-estimates, mgal.

20. Conclusion (1)LSC not anymore restricted due to large number of observations if multiprocessing can be used. (2)Analytic ellipsoidal or an-isotropic kernels under development. (3)Software available in GRAVSOFT package. (4)The scaling of LSC derived error-estimates improves the error estimates, so that the variation of the error due to changing local signal standard deviation is seen. Thanks to those who have contributed to the development: D.Arabelos, M.Reguzzoni, F.Sansò, R.Barzaghi, R.H.Rapp, P.Holota, H.Sünkel, P.Knudsen, R.Forsberg, M.Veicherts, B.Sørensen, G.Moreaux, IAG Scientific Assembly, Potsdam, Sept. 2013