Lecture 21 – The Geoid

2 April 2009

GISC-3325

Class Update

Read Chapter 10 of text. Deadline for Reading Assignments (2) is 16

April 2009. Good set of definitions are available at:

http://www.geod.nrcan.gc.ca/hm/gloss_e.php#qg Good article on latest U.S. geoid model

http://www.ngs.noaa.gov/GEOID/USGG2009/tech.html

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.

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.

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

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.

Marine Geoid Height of the sea surface caused by both

gravity and the active ocean circulation. Topex/Poseidon launched 1992.

SST Equations a = c*(delta_t / 2)

“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

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.

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

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.

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.

Geoid Height Distance from the ellipsoid to the geoid

measured along the normal to the ellipsoid. Geoid height, geoid separation or geoid undulation

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).

U.S. Geoid models

• Gravimetric geocentric geoid

– 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)

Gravimetric geoiddata 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

Geoid 03

In addition to the gravimetric geoid model USGG2003, the GEOID03 model consisted of the following input: 14185 NAD 83 GPS heights on NAVD 88 leveled bench marks

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 2190 and order 2159.

• Improved surface gravity and terrain data.

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.

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.

14,308 points

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.

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

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.

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.