1. Lecture 21 – The Geoid
2 April 2009
GISC-3325

2. Class Update Read Chapter 10 of text.
Deadline for Reading Assignments (2) is 16 April 2009.
Good set of definitions are available at:
Good article on latest U.S. geoid model

3. Geoid
The equipotential surface of the Earth's gravity field which best fits, in a least squares sense, global mean sea level.
Dependent upon the irregular distribution of masses of the Earth.
It is the surface to which heights refer.
There are two “implementations” of geoid modeling: gravimetric and hybrid.

5. Geoid Model and Horizontal Datum
NAD 83
Earth-centered (geocentric) ellipsoid. GRS-80.
Heights determined by GPS are computed with respect to it.
NAD 27
Fitted to the reference ellipsoid.
NOT geocentric.
No geoid model associated with NAD 27.

6. NAVD 88
Based on a minimum-constraint adjustment of Canadian-Mexican-U.S. leveling observations.
The height of the primary tidal benchmark at Father Point/Rimouski, Quebec, Canada, was held fixed as the constraint.
This constraint satisfies the requirements of shifting the datum vertically to minimize the impact of NAVD 88 on U.S. Geological Survey (USGS) mapping products

8. NAVD 88 datum + global geopotential
A “recent” study by Rapp (1996) compares ITRF93 GPS positions and a global geopotential model against the NAVD 88 vertical datum.
Rapp found a mean offset for the NAVD88 datum of -27 cm when computed with a set of 397 GPS points.
In sense that NAVD 88 is beneath global geoid model.

9. Marine Geoid
Height of the sea surface caused by both gravity and the active ocean circulation.
Topex/Poseidon launched 1992.

10. SST Equations a = c*(delta_t / 2) h = N + Hbar + a
“a” distance from satellite to sea surface
“C” is speed of light (electromagnetic energy)
“delta_t” is roundtrip time of signal
h = N + Hbar + a
“h” ellipsoid height
“N” is geoid height
“Hbar” is difference between mean instantaneous sea level and geoid
“a” height observation from satellite

11. Sea Surface Topography SST
SST: Deviation of the mean sea surface from the geoid.
Differs from geoid by 1-2 meters due to
Salinity difference
Large-scale differences in atmospheric pressure
Strong ocean currents
Use of SST yields precision of ~ 2 meters.

12. Gravity Datums Potsdam Gravity Reference System 1909
Standard until IGSN adopted (60 years)
+/- 3 milliGals
International Gravity Standardization Net 1971
Worldwide network: 24,000 gravimeter, 1,200 pendulum and 10 absolute measurements.
Collected over twenty years.
Adjusted by a small Working Group of the International Association of Geodesy.
Datum is determined not by an adopted value at a single station, but by the gravity values for 1854 stations obtained from a single least squares adjustment of absolute, pendulum and gravimeter data.
Standard error +/- 50 microGals

13. Surface Gravity measurements
Absolute
Formerly pendulum now falling mass
Relative
Each meter has own calibration value.
Observations made at a number of points with the differences the measurement.
Require observations be made in loops that start and end on same point to account for drift in instrument.

15. Gravity Measurement Reductions
Gravity anomaly is the difference between the actual acceleration of gravity at a point on the surface of the earth and the computed normal acceleration of gravity of the same point on the level ellipsoid.
Free-air - Only accounts for the elevation of the station not mass between the station and the geoid.
Bouguer – Accounts for the variations of gravity due to differences in the density and mass of underlying materials.

16. Geoid Height
Distance from the ellipsoid to the geoid measured along the normal to the ellipsoid.
Geoid height, geoid separation or geoid undulation

17. Stokes function
Theorem is not valid if masses exist outside the geoid. Hence the need to reduce gravity measurements to anomalies.
While the function looks different from version in text, it is just reorganized. Also uses 1/sin rather than csc (these are equivalent).

18. U.S. Geoid models Gravimetric geocentric geoid Hybrid
based on Earth Gravity Model, DEM data, and gravity measurements.
Current model USGG 03 (beta version USGG 09 is posted to NGS site)
Hybrid
based on Gravimetric Model with datum transformations plus GPS on benchmarks
Current model is Geoid 03 (beta version Geoid 09 is posted to NGS site)

19. Gravimetric geoid data used
2.6 million terrestrial, ship, and altimetry- derived gravity measurements
30 arc-second Digital Elevation Data
A1-arcsecond DEM for the Northwest USA (NGSDEM99)
The EGM96 global geopotential model

20. Geoid 03
In addition to the gravimetric geoid model USGG2003, the GEOID03 model consisted of the following input: NAD 83 GPS heights on NAVD 88 leveled bench marks

21. USGG 09 Characteristics
One arc-minute model (2 km by 2 km nodal spacing)
Based on the EGM08 reference model.
Model is complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree and order 2159.
Improved surface gravity and terrain data.

22. USGG 09 evaluation
Has significantly reduced long wavelength errors EGM 09.
More accurate than previous models:
new computation method
new satellite gravity model
improved altimetric gravity anomalies.
In comparison with GPSBM implied geoid undulations, the improvement goes from 9.1 cm to 7.3 cm for USGG2003 and USGG2009, respectively.

23. Geoid 09 Builds on gravimetric model.
Incorporates National Readjustment of 2007
modified most GPS-derived coordinates (including heights) at the cm- to dm-level.
Changes in vertical component mostly around 2 cm, with few changes exceeding 10 cm.
Produced by tailoring USGG2009 to fit the 12,715 points where both GPS-derived ellipsoid height and NAVD 88 differentially- determined heights were available.

26. 14,308 points

27. GPS-derived heights Differential leveling is too expensive.
Accurate GPS height determinations can only be achieved using differential methods.
GPS baselines result from the combination of data observed simultaneously at at least two sites. Common errors cancel.
GPS observations only yield ellipsoid heights.
We must apply a geoid model to approximate NAVD 88 heights.

29. http://www. ngs. noaa. gov/PUBS_LIB/NGS59%20-%202008%2006%209-FINAL-2

30. Note on hybrid geoid modeling
Regional gravimetric geoids and quasi-geoid models are now commonly fitted to GPS leveling data which simultaneously absorbs leveling, GPS and quasi-geoid errors due to their inseparability.

31. NAVD 88 via OPUS
OPUS provides an Ortho Hgt (NAVD 88) by applying the Geoid 03 value to the el hgt at the computed point. Peak-to-peak error in Ortho Hgt takes the el hgt peak-to-peak error and adds an error estimate for geoid model.