Dru Smith
Updated: May 12, 1997
Although the conversion between ellipsoidal height and orthometric height requires "the" geoid undulation, it must be pointed out that there is no one specific definition of "the" geoid in global use today. As such, we at NGS recognize the need to define what we mean when we speak of geoid undulations in our GEOID96, G96SSS and MEXICO97 geoid undulation models.
To put it simply, three things must be defined when one wishes to speak of a geoid undulation:
1) What ellipsoidal reference system will be used?
2) Which equipotential surface is the geoid?
3) What permanent tide system will your ellipsoid/geoid be referred to?
What ellipsoidal reference system will be used?
There are a number of different ellipsoidal reference systems in use today. For gravimetric geoids, NGS uses the geodetic reference system GRS-80, with the center coinciding with the origin of the ITRF94(1996.0) reference frame. For GEOID96, whose purpose is direct conversion between NAD 83 ellipsoid heights and NAVD 88 orthometric heights, we use the NAD 83 ellipsoid.
To reference a geoid undulation, any ellipsoidal reference system may be used, as long as one has a clear understanding of the degree-zero terms induced by your choice. For example, degree zero (i.e. 'constant') terms are induced if:
1) The GM value of the reference ellipsoid differs from the GM value assumed for the Earth
2) The normal potential (U0) of the ellipsoid differs from the true potential (W0) of the geoid.
In addition, degree 1 terms (geographically dependent, long wavelength "tilts") are induced if the origin of the ellipsoid does not coincide with the center of mass of the Earth.
The table below describes the reference fields used in our recent models:
| Geoid Model | GEOID96 | G96SSS | MEXICO97 |
| Reference Ellipsoid | NAD 83 | GRS-80 | GRS-80 |
| Origin | NAD 83 (86) | ITRF94 (1996.0) | ITRF94 (1996.0) |
| a (meters) | 6378137.000 | 6378137.000 | 6378137.000 |
| GM (1014 m3/s2) | 3.986005 | 3.986005 | 3.986005 |
| J2 | 1.08263 x 10-3 | 1.08263 x 10-3 | 1.08263 x 10-3 |
| omega (10-11 rad/sec) | 7292115 | 7292115 | 7292115 |
| U0 (m2/s2) | 62636860.850 | 62636860.850 | 62636860.850 |
What equipotential surface is the geoid?
For all its history, "the" geoid has never had one unified, uncontested definition. There are, however, certain agreements:
1) The geoid is one of the infinitely many equipotential surfaces surrounding the Earth, with some potential value of W0.
2) It has some connection to mean sea level (both the connection, as well as mean sea level itself have debatable definitions)
To clearly identify our choice for the geoid, NGS uses a tool known as the "best fitting global ellipsoid". This should not be confused with the reference ellipsoid! The best fitting global ellipsoid is a geometric ellipsoid which has the property of "best fitting" (in the least-squares sense) the global geoid. This is done through satellite altimeter measurements of the Earth's sea surface. Certain assumptions must be made:
a) Average dynamic topography (geoid/sea surface separation) is zero, globally.
b) Non-global altimeter missions yield sufficient information on the global geoid.
With a long enough time period, altimeter measurements are averaged, and a geometric ellipsoid is fit to the data, yielding an "a" and "f" value. Again, these are not related to the reference field (above). If we then take our best estimate of the GM value for the Earth, as well as our best estimate of the spin of the Earth (omega), we now can compute U0 (normal potential) for this "best fitting" ellipsoid. Finally, with the assumption that the "best fitting" ellipsoid fits to the geoid, we assume no degree zero separation between this best fitting ellipsoid and the geoid. This means that the GM of the Earth and the GM of the ellipsoid are identical, as well as the U0 and W0 being equal. Therefore, given U0, we also have W0, and have thus defined our geoid!
The table below describes the best fit ellipsoids, and potential of the G96SSS and MEXICO97
geoid models. GEOID96 uses GPS on level benchmarks, in addition to gravimetric information,
and therefore cannot be simply assigned a value of W0 (and may not truly be an equipotential
surface).
| Geoid Model | G96SSS | MEXICO97 |
| "Best fitting" a (meters) | 6378136.590 | 6378136.460 |
| "Best fitting" f | 1/298.25722917701 | 1/298.25765 |
| "Best" GM (1014 m3/s2) | 3.986004415 | 3.986004415 |
| "Best" omega (10-11 rad/sec) | 7292115 | 7292115 |
| W0 (m2/s2) | 62636855.678686 | 62636856.852138 |
At the time of computing G96SSS, our understanding of the best fitting ellipsoid yielded the
values in the table above. With the MEXICO97 model (and further models in the near future),
we are consistent with the values for the best fitting ellipsoid as defined by the
NIMA/GSFC EGM96 team.
Therefore, a correction must be applied to G96SSS to yield gravimetric geoid
undulations that use are consistent with our current knowledge of a best fitting ellipsoid.
Specifically, the separation between a surface whose potential is W0=62636855.678686 and one
whose potential is W0=62636856.852138 is the correction. Over the United States, Alaska,
Hawaii and Puerto Rico this separation is 12 cm. It is necessary to subtract 12.0 cm from the
G96SSS values to obtain the geoid undulation between the best-fit global geopotential surface
and the GRS-80 ellipsoid (when both are expressed in a tide-free system).
What permanent tide system will your ellipsoid/geoid be referred to?
The masses of the Sun and Moon, as well as their influence on the crust of the Earth must be defined to understand what potential field is used when choosing a W0 value for the geoid. Basically, one has three choices when defining a "permanent tide system". These three choices, and the masses which generate time-independent ("permanent") potential are:
1) Mean tide system: Mass of the deformed Earth and Mass of the Sun/Moon/etc
2) Zero tide system: Mass of the deformed Earth
3) Non-tidal system: Mass of the undeformed Earth.
The Earth's deformation is due to the pull of the Sun and Moon. More complete details may be found at our permanent tide web page.
The tide system of GEOID96, G96SSS and MEXICO97 is non-tidal.
Got a question about this page?
Contact us in the USA at 301-713-3202 or at
dru@ngs.noaa.gov or
dennis@ngs.noaa.gov
.