1. 1 Least Square Modification of Stokes’ Formula vs. Remove- Compute-Restore Technique Lars E. Sjöberg Royal Institute of Technology Division of Geodesy and Geoinformatics SE-100 44 Stockholm SWEDEN E-mail: lsjo@kth.se

2. 2 Overview General modification of Stokes The RCR-technique The LSq modification A numerical comparison in Sweden Additive corrections Further comparisons Concluding remarks

3. 3 Stokes’ formula (ideal case without errors) The same result for any choice of modification parameters No truncation error

4. 4 General modification of Stokes’ formula Geoid estimator (with errors) Deterministic choices of parameters (e.g., Molodensky 1962, Wong and Gore 1969) Stochastic choices of parameters based on optimization: spectral combination (Sjöberg 1980, Wenzel 1981); LSq modification (Sjöberg (1984, 1991, 2003d)

5. 5 The RCR - technique The IAG Geoid School practices (only) this method; in the past with L=0, and some corrections are ignored or simplified The modification by Vanicek-Kleusberg uses L = M and Molodensky type of modification parameters (i.e. minimum truncation error bound)

6. 6 THE RCR Technique (cont.) * Must be used with careful consideration of all corrections. * Big workload to compute all corrections. * If L = 0, the truncation error is usually large. Example: If M=360 and integration radius less than 20 degrees, the error exceeds 1 cm. * The rigorous dwc correction is likely to be ill-conditioned for dense data. In practice it is approximated with large error. * The atmospheric correction may introduce a large bias (Sjöberg 1999). * The residual gravity anomaly is advantageous for interpolation and integration.

7. 7 Least Squares Modification with Additive Corrections (LSMSA) Uses the rigorous rcr formula above, but direct and indirect effects are combined into additive corrections (Sjöberg 2003a).

8. 8 Least Squares Modification of Stokes’ Formula with Additive Corrections (CONT.) Uses the general modification formula above. Parameters are based the global mean square error vs. errors of truncation, gravity anomaly and EGM. No corrections to surface gravity anomalies. Additive corrections added to Stokes’ integral.

9. 9 KTHNKG 2004 GPS/levelling residuals for the KTH and NKG models after 1-parameter transformations. (The scales are given by the 5 cm arrows to the south-east.)

10. 10 Discussion of figures The figures show the residuals of gravimetric geoid models vs. GPS/leveling geoid model after 1-parameter fits. As the majority of the 195 GPS points are located at low elevations, the topographically related corrections are not very significant. The rms residuals after 1-par. (4- parameter) fits are 40 (32) mm and 23 (20) mm for RCR - and LSq - methods, respectively.

11. 11 Discussion of figures (cont.) As the residuals in LSq are small, the larger residuals for RCR are likely to be due to errors in RCR rather than GPS/levelling. There are obvious long-wavelength errors in the RCR residuals. Assuming that the standard errors of GPS and levelling are 15 and 7.5 mm, simple error propagation with the 4-parameter rms residual 20 mm yields the standard error of 11 mm for geoid height of the LSq method.

12. 12 The additive corrections The combined topographic effect (Sjöberg 1977, 2000, 2007) The downward continuation effect (Sjöberg 2003b)

13. 13 The additive corrections (cont.) * The total atmospheric effect (Sjöberg 1999) * The total ellipsoidal effect (Sjöberg 2003c, 2004)

14. 14 Comments to LSq Modification with Additive Corrections Optimized matching of various error sources Numerically efficient The simple formula for combined topographic effect is exact The dwc effect is stable No bias in atmospheric effect

15. 15 Height anomalies by LSMSA Almost the same as for geoid, but there is no topographic effect and the dwc effect differs slightly (e.g., Ågren et al. 2009 and Sjöberg 2013).

16. 16 Height anomalies: the dwc effect

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18. 18

19. 19 From Yildiz et al. (2013): Comparison of RCR and LSMSA techniques to quasi-geoid determination over the Auvergne test area; JGS 2(1): 53-64 (open access)

20. 20 KTH vs. RCR in Konya mountain area, Turkey; Abbak et al. (2012)

21. 21 KTH vs. RCR in Konya mountain area, Turkey; Abbak et al. (2012), cont. (7-parameter fitting with GPS-levelling)

22. 22 CONCLUDING REMARKS The RCR method should not be used without modifying Stokes’ function. The RCR method requires accurate corrections to both gravity anomaly (direct effects) and geoid height (indirect effects). Only geoid corrections apply to LSq with additive corrections. The dwc effect to gravity anomaly is ill- conditioned for dense data. The dwc effect on the geoid height is more stable.

23. 23 CONCLUDING REMARKS (CONT.) The LSq modification is based on a minimum error solution. The LSq modification is more computer efficient. The transition from geoid determination to quesigeoid determination is small: the combined topographic effect disappears and the dwc correction changes slightly; otherwise the same formulas.

24. 24 Concluding remarks (cont.) For detailed theoretical comparison LSMSA-RCR: see Sjöberg (2005) LSMSA is now used for official geoid models in Sweden and Estonia. LSMSA is the best known geoid model over Tanzania, Moldova, Auvergne in France, Konya Closed Basin in Turkey, etc. LSMSA is likely to become the NKG 2014 geoid model.

25. 25 References Ågren J, Sjöberg L E, Kiamehr R (2009) The new gravimetric geoid over Sweden. JAG 3: 143-153 Abbak et al. (2012) Comparison of the KTH and remove-compute-restore techniques to geoid modelling in mountainous area. Compute&Geosciences 48: 31-40 Molodensky et al. (1962) Methods for Study of the External Gravitational Field and Figure of the Earth.), Israel program for Scientific Translations, Jerusalem, Israel Sjöberg L E (1977) On the Errors of Spherical Harmonic Developments of Gravity at the Surface of the Earth. Department of Geodetic Science, Report No 257, OSU, Columbus, August 1977. 74 pp. Sjöberg (1980) Least Squares Combination of Satellite and Terrestrial Data in Physical Geodesy. Presented to Presented at the Int. Symp. Space Geod. Appl. November 1980; Ann. Geophys. 37(1981): 25-30 Sjöberg (1984) Least squares modification of Stokes' and Vening Meinesz' formulas by accounting for truncation and potential coefficient errors. Manusc. Geod 9: 209-229 Sjöberg LE (1991) Refined least squares modification of Stokes’ formula. Manuscr Geod 16: 367-375 Sjöberg LE (1999) The IAG approach to the atmospheric geoid correction in Stokes formula and a new strategy, J Geod, 73(7):362-366 Sjöberg (2000) On the topographic effects by the Stokes-Helmert method of geoid and quasi-geoid determinations. J Geod 74(2):255-268 Sjöberg (2003a) A computational scheme to model the geoid by the modified Stokes's formula without gravity reductions. J Geod 77: 423-432 Sjöberg (2003b) A solution to the downward continuation effect on the geoid determined by Stokes' formula. J. Geod. 77: 94-100

26. 26 References (cont.) Sjöberg (2003c) The ellipsoidal corrections to order e2 of geopotential coefficients and Stokes' formula. J Geod 77: 139-147 Sjöberg (2003d)A general model of modifying Stokes' formula and its least-squares solution. J Geod 77: 459-464 Sjöberg ( 2004) A spherical harmonic representation of the ellipsoidal correction to the modified Stokes formula. J Geod 78: 180-186 Sjöberg (2005) Discussion on the approximations made in the practical implementation of the remove-compute-restore technique in regional geoid modelling. J Geod 78: 645- 653 Sjöberg (2007) The topographic bias by analytical continuation in physical geodesy. J Geod 81: 345-350 Sjöberg (2013) The geoid or quasigeoid- which reference surface should be preferred for a national height system? J Geod Sci 3: 103-109 Sjöberg (2014) The KTH approach to modelling the geoid- extended lecture notes. (In preparation.) Yildiz H et al. (2013) Comparison of remove-compute-restore and least squares modification of Stokes formula techniques to quasi-geoid determination over the Auvergne test area. JGS 2(1): 53-64 Wenzel H-G (1981) Zur Geoid bestimmung durch Kombination von Schwereanomalien und einem Kugelfunktionsmodell mit hilfe von Integralformeln. ZfV 106(3): 102- 111 Wong L, Gore R (1969) Accuracy of geoid heights from modified Stokes kernels. Geophys J R astr Soc 18: 81-91

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