# Geoid Modelling - Public Models

## 1. The Public Geoid Models for Canada

The actual shape of the Earth is neither a sphere nor an ellipsoid; it is a geoid. The geoid is the equipotential (level) surface that represents best, in a least square sense, the global mean sea level.  However, the geoid does not coincide exactly with mean sea level because, just like land, oceans have a permanent topography albeit ranging from -2 m to 2 m globally.  The geoid is expressed in potential unit of m2/s2.  Thus, two equipotential surfaces which differ by 1 m2/s2 have a geometric separation of about 0.1 m because gravity is approximately 9.8 m/s2 (0.1 m = 1.0 m2 s-2 / 9.8 m s-2).

The geoid does not only represent the actual physical shape of the Earth, but it is also a reference surface for elevations.  An elevation above the geoid is referred to as an orthometric height.  The geoid can not be access physically because it is potential afterward.  However, it can be represented geometrically by a model that gives the separation between the geoid and a reference ellipsoid. This separation is referred as the geoid height (N).  It is positive when the geoid is above the ellipsoid and is negative when the geoid is below the ellipsoid.

Naturally, the importance of determining the geoid accurately increased with the advance in space-based positioning (e.g., GPS) where heights are given with respect to a reference ellipsoid.  The geoid model allows the conversion of ellipsoidal heights (h) to orthometric heights (H): H = h - N.  Figure 1 depicts the relation in heights between the topography, geoid and ellipsoid.

Figure 1: Relation between the topography, ellipsoid and geoid
larger image
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Over the last 20 years, Natural Resources Canada has published five (5) geoid models, which are summarized in Table 1.  These geoid models have improved over the years by implementing better theory; by conducting gravity surveys to fill large areas void of data; by collecting more precise Digital Elevation Models (DEM); and by embedding more accurate global gravity models derived from satellite observations and dedicated satellite gravity missions.  These enhancements brought significant changes to each published model.  Table 2 shows improvements between each published model.

Table 1: Recent geoid models published at Natural Resources Canada.
Model: CGG2010
Region: N10°/N84°/W170°/W10°
Horizontal Resolution: 2’ x 2’
Equipotential  Surface: 62636855.69 m2/s2
Global Gravity Model: GOCO01S/EGM08 (2-2190)
Error weighted combination
Stokes kernel: Degree-Banded (Degree: 120, Cap: 6°)
Transition from degree 90 to 150
Reference: Huang and Véronneau (in preparation)

Model: CGG2005
Region: N20°/N84°/W170°/W10°
Horizontal Resolution: 2’ x 2’
Equipotential  Surface: 62636856.88 m2/s2
Global Gravity Model: GGM02C + EGM96
Stokes kernel: Degree-Banded (Degree: 90, Cap: 6°)
Reference: Véronneau and Huang, 2007

Model: CGG2000
Region: N20°/N84°/W170°/W10°
Horizontal Resolution: 2’ x 2’
Equipotential  Surface: 62636855.8 m2/s2
Global Gravity Model: EGM96
Stokes kernel: Modified spheroidal (Degree: 30, Cap: 6°)
Reference: Véronneau, 2001

Model: GSD95
Region: N41°/N72°/W142°/W46°
Horizontal Resolution: 5’ x 5’
Equipotential  Surface: 62636860.85 m2/s2
Global Gravity Model: OSU91A
Stokes kernel: Standard
Reference: Véronneau, 1996

Model: GSD91
Region: N41°/N72°/W142°/W46°
Horizontal Resolution: 5’ x 5’
Equipotential Surface: 62636860.85 m2/s2
Global Gravity Model: OSU91A
Stokes kernel: Standard (planar)
Reference: Véronneau and Mainville, 1992

The Stokes kernel is the equivalent of a weight function.  It assigns the weight to gravity measurements according to the distance from the computation point and between the global gravity model and the surface gravity data.  For example, the long wavelength components of CGG2010 up to degree 90 (~450 km) are defined entirely by GRACE and GOCE satellite data, which are part of the GOCO01S and EGM08 global models.  The terrestrial gravity data are smoothly introduced into the model from degree 90 and contribute entirely from degree 120.  GRACE and GOCE are dedicated satellite gravity missions.

Table 2: Comparison of recent published geoid models in Canada
Models Nodes Min.
(m)
Max.
(m)
Mean
(m)
St. Dev.
(m)
GSD95 – GSD91 428544 -9.230 10.740 0.048 0.706
CGG2000 – GSD95 419616 -6.450 3.160 -0.408 0.456
CGG2005 – CGG2000 Canada 2892270 -1.609 0.797 -0.256 0.264
N.A.* 9216000 -2.918 2.528 -0.226 0.378
CGG2010 – CGG2005 Canada 2892270 -0.807 1.462 0.085 0.064
N.A.* 9216000 -1.818 4.284 0.092 0.124

*: North America

## 2. The validation of the public geoid models in Canada

The most common technique to validate geoid models is by comparing them to geoid heights derived from GPS measurements and levelling observations.  The former gives the ellipsoid height (h) while the latter gives the orthometric height (H).  The difference between the two heights is the geoid height (N). The discrepancy ε (= h – H – N) would be equal to zero if GPS and levelling observations would be errorless and if the reference system of the levelling network would be the same as the geoid model.  Furthermore, the reference ellipsoid for GPS data and the geoid model must be in the same reference frame:

In the above equation, we have the ellipsoidal height and geoid model in the NAD83(CSRS) reference frame and the orthometric height and geoid model are realized with respect to an equipotential surface W0 for the vertical reference system. The reference frame for the ellipsoidal heights and the geoid model could also be an ITRF realization. Figure 2 [JPEG, 90.9 kb, 1263 X 942, notice] illustrates in a simple way the geocenter difference between NAD83(CSRS) and ITRF while Figure 3 [JPEG, 2.3 Mb, 2700 X 1766, notice] show the actual differences for the ellipsoidal heights in NAD83(CSRS) and ITRF97 over North America.

For the validation of the geoid models, the orthometric heights are derived from the Sep12 adjustment of the federal first-order levelling network. It is a minimum constrain adjustment. The only fixed station is located in Halifax, Nova Scotia. The reference system is the equipotential surface W0 = 62636856.0 m2/s2. It is the same reference as the Canadian Geodetic Vertical Datum of 2013 (CGVD2013). Overall, Sep12 should be very precise locally, but it still includes some remaining systematic errors of unknown origin that accumulate over long distances.

Since 1986, NRCan has conducted GPS surveys at benchmarks across the country for the purpose of validating geoid models in Canada.  All these "GPS on BMs" surveys are put together in an adjustment to create the SuperNet.  The accuracy of the ellipsoidal heights varies with the epoch of observations.  Surveys prior to 1994 might have accuracy at the decimetre level while the most recent surveys should have accuracy better than 2 cm.

The complexity in using this validation approach is that the discrepancy ε does not separate the errors from the geoid model, GPS measurements and levelling observations.  Furthermore, the discrepancy might be caused by the instability of the benchmarks, i.e., markers may move between the epochs of levelling and GPS observations.  It is not rare that levelling observations were conducted some 30 years prior to the GPS measurements.

Table 3 shows the comparison of the five (5) latest published geoid models at NRCan to a common GPS on BMs data set in Canada. The comparison includes only the benchmarks located on the main land because the levelling networks for Newfoundland, Prince Edward Island and Vancouver Island are independent. These independent networks are not tied to the marker in Halifax.

For the public geoid models, the comparison could be at best a constant because the reference system of these models is not the same as Sep12. However, we can observe that the standard deviation of the discrepancies has decreased significantly since the realization of GSD91 in 1991. It went down from 79.0 cm to 13.5 cm (66.7% confidence level).

Table 3: Comparison of geoid models to GPS on BMs, where the levelled orthometric heights are from the Sep12 adjustment and the ellipsoidal heights are from SuperNet v3.3n
Model No Min (m) Max (m). Mean (m) St. Dev. (m)
GSD91 2445 -3.980 3.985 -1.289 0.790 (0.695*)
GSD95 2445 -2.018 0.236 -1.188 0.413 (0.144*)
CGG2000 2449 -1.049 0.494 -0.340 0.225 (0.087*)
CGG2005 2449 -0.743 0.330 -0.197 0.140 (0.084*)
CGG2010 2449 -0.706 0.355 -0.162 0.135 (0.074*)

*: Standard deviation after filtering out systematic errors

## 3. The next public geoid model

The next geoid model publicly available will be in 2013. This model will be the realization of the future Canadian Geodetic Vertical Datum of 2013 (CGVD2013). This new vertical reference system will be defined by the geopotential surface W0 = 62636856.0 m2/s2, which represents by convention the mean coastal sea level for North America. The next model will include these improvements:

1. Longer time series for the GRACE and GOCE data;
2. Focus on the Canadian Arctic;
3. Inclusion of new Digital Elevation Models in some regions;
4. Updated set of gravity data over the USA; in particular for the Great Lakes; and
5. Resume research in providing accuracy of the geoid model.