1. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 1 Summary of Last Presentation Reference system Conventional reference system Reference frame –Space fixed –Earth fixed: CTS ITRF WGS84 NAD

2. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 2 Problem with Ellipsoid Heights They are new: easily obtained since the 80s with GPS, not accurate enough before. By comparison, triangulation exists since around 1500. They do not represent what most users want which is to see that “water flows downhill!” => need potential based height.

3. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 3 Basic Definitions Gravitational potential of the Earth=Newtonian potential generated by the mass distribution within the Earth: Gravity potential of the Earth: Gravity vector: x y z P l Q O

4. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 4 Gravity Vector-Astronomical Quantities Λ Φ g x y z plumb line level surfaces W = const. P

5. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 5 Geoid and orthometric Heights The geoid is a particular level surface chosen to be close to the average surface of the oceans: oceanic floor geoid ocean Earth surface H P 1 (<0) H P 2 (>0) P1P1 P2P2

6. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 6 Approximations of V Zero level: First order: Leveled ellipsoid

7. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 7 Ellipsoidal Coordinates for Potential U P β u F2F2 F1F1 U depends on 4 constants: - ω is known - U 0 is computed by imposing: mass of the ellipsoid = mass of Earth - a and E are derived from satellite observations choosing Earth J 2

8. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 8 Different Types of Geoid –Current NGS definition of geoid: The equipotential surface of the Earth's gravity field which best fits, in a least squares sense, global mean sea level –The definition of the geoid is complicated by the permanent deformation of the Earth caused by the presence of the Sun and the Moon. Consideration of these permanent tidal effects has led to the definition of three types of geoids and three types of reference ellipsoids [Ekman, 1989, 1995; Rapp et al., 1991; Bursa, 1995a]. The three geoids are described as follows: 1.Tide-free (or nontidal)—This geoid would exist for a tide-free Earth with all (direct and indirect) effects of the Sun and Moon removed. 2.Mean—This geoid would exist in the presence of the Sun and the Moon (or, equivalently, if no permanent tidal effects are removed). 3.Zero—This geoid would exist if the permanent direct effects of the Sun and Moon are removed, but the indirect effect component related to the elastic deformation of the Earth is retained.

9. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 9 Tidal Effects Gravitational Potential Tidal effects: –permanent / periodic –direct / indirect Nomenclature –no effect => free –only direct-permanent => zero-tide –all permanent =>mean x y z B l P O d r

10. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 10 Geoid in the US NGS Definition: The equipotential surface of the Earth's gravity field which best fits, in a least squares sense, global mean sea level. NGS Warnings: Even though we adopt a definition, that does not mean we are perfect in the realization of that definition. For example, altimetry is often used to define "mean sea level" in the oceans, but altimetry is not global (missing the near polar regions). As such, the fit between "global" mean sea level and the geoid is not entirely confirmable. Also, there may be non-periodic changes in sea level (like a persistent rise in sea level, for example). If so, then "mean sea level" changes in time, and therefore the geoid should also change in time. These are just a few examples of the difficulty in defining "the geoid". Latest models: GEOID03, USGG2003, GEOID99, G99SSS Presentations: http://www.ngs.noaa.gov/GEOID/PRESENTATIONS/ http://www.ngs.noaa.gov/GEOID/PRESENTATIONS/

11. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 11 Leveling ellipsoid geoid terrain The process of precise leveling is to measure height differences and to sum these to get the heights of other points.

12. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 12 Different Height Systems

13. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 13 Orthometric Height How to compute gravity inside the Earth? –Assume a simple terrain model: Bouguer plate (standard density 2.67g/cm 3 ) –Perform the following steps: Remove the Bouguer plate Do the free air anomaly Restore the Bouguer plate –Final formula: Helmert heights:

14. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 14 Pizzetti Projection earth’s H h N P  PoPo QoQo Q surface geiod ellipsoid

15. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 15 Deflections of the Vertical terrestrial equator equipotential surface through P 1 CTP φ P1P1 φ Ф CTP P2P2 A CT P T CTP ZeZe ZaZa α ellipsoidal surface normal astronomic normal CTP ZeZe α ε η A CTP ZaZa ξ T θ

16. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 16 Methods for Geoid Computations

17. Catherine LeCocq SLAC USPAS, Cornell University Large Scale Metrology of Accelerators June 27 - July 1, 2005 Height Systems 17 Three Components of the Geoid N GM = long wavelength calculated from a geopotential model N  g = medium wavelength computed with Stokes N T = terrain correction