1. AGU Joint Assembly / May 23-26, 2006 The cost of assuming a lateral density distribution in corrections to Helmert orthometric heights 1 Department of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton, New Brunswick E3B 5A3 Robert Kingdon 1, Artu Ellmann 1, Petr Vanicek 1, Marcelo Santos 1

2. AGU Joint Assembly / May 23-26, 2006 Orthometric Height Bouguer Shell Terrain Geoid Plumbline

3. AGU Joint Assembly / May 23-26, 2006 Evaluating Mean Gravity e.g. Helmert’s method: Surface GravityCorrective Terms Converting Helmert mean gravity to rigorous mean gravity: (for Bouguer plate and normal gravity) Additional Corrective Terms Each correction takes the form: Corrections may be made for topographical or non-topographical masses. We will only discuss corrections for topographical effects.

4. AGU Joint Assembly / May 23-26, 2006 Models of Topography Bouguer Plate/Shell (used by Helmert) Terrain + Bouguer Plate/Shell 2D Density Distribution (used for rigorous corrections) 3D Density Distribution (more rigorous corrections?) Real Topodensity Distribution

5. AGU Joint Assembly / May 23-26, 2006 Rigorous with 2D Density Distribution Rigorous with 3D Density Distribution Shortcomings of the 2D Density Model Residual Anomalous Density

6. AGU Joint Assembly / May 23-26, 2006 Additional Correction Accounting for 3D Density Distribution

7. AGU Joint Assembly / May 23-26, 2006 Test Area Test area: 50° to 51° N latitude, and 236° to 237° E longitude. Heights from 0 (green) to 2862 (white) m.

8. AGU Joint Assembly / May 23-26, 2006 Simulation B1 Values from 0 (red) to 5.7 (blue) cm, 1 cm contours. Cone

9. AGU Joint Assembly / May 23-26, 2006 Simulation B2 Values from 0 (red) to 6.9 (blue) cm, 1 cm contours. Slab

10. AGU Joint Assembly / May 23-26, 2006 Simulation B3 Values from 0 (red) to 7.9 (blue) cm, 1 cm contours. Wedge

11. AGU Joint Assembly / May 23-26, 2006 Simulation A1 Values from -1.2 (green) to 1.2 (red) cm, 1 cm contours. θ=45°

12. AGU Joint Assembly / May 23-26, 2006 Simulation A2 Values from -1.7 (green) to 1.6 (red) cm, 1 cm contours. θ=30°

13. AGU Joint Assembly / May 23-26, 2006 Simulation A3 Values from -2.3 (green) to 2.6 (red) cm, 1 cm contours. θ=15°

14. AGU Joint Assembly / May 23-26, 2006 Summary of Results Simulation Max (cm) Min (cm) Height where cm level is reached (m) % of points with cm level corrections (%) B1: Cone5.70240311.2 B2: Slab6.90105113.0 B3: Wedge7.90107312.1 A1: θ=45°1.2-1.2225012.6 A2: θ=30°1.6-1.7180312.6 A3: θ=15°2.6-2.3154313.5

15. AGU Joint Assembly / May 23-26, 2006 Conclusions – Shortcomings are largest for regional phenomena. – Shortcomings are only significant close to their source. – More realistic estimates of shortcomings reach up to ~3 cm, but are only > 1 cm (in the test area) at elevations above ~1500 m. – There do exist limited areas where these shortcomings may be significant, in terms of rigorous orthometric heights. Effects on e.g. geoid heights have yet to be evaluated.

16. AGU Joint Assembly / May 23-26, 2006 Acknowledgements We would like to acknowledge NSERC and for their funding of this research.