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.

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.

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.

*: North America

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:

h_ITRF97 – H_Rimouski – N_{ITRF97, Rimouski} = ε

In the above equation, we have the ellipsoidal height and geoid model in the ITRF97 reference frame and the orthometric height and geoid model have the mean water level at Rimouski for the vertical reference system.  The reference frame for the ellipsoidal heights and the geoid model could also be NAD83(CSRS). 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 Nov07 adjustment of the federal first-order levelling network.  It is a minimum constrain adjustment.  The only fixed station is located in Rimouski, Québec.  The reference system is the mean water level at the tide gauge in Rimouski.  It is the same reference as the North American Vertical Datum of 1988 (NAVD88).  Overall, Nov07 should be very precise locally, but it may include unknown systematic errors that accumulate over long distances.

Heights from CGVD28 cannot be used to validate geoid models.  The reference system of CGVD28 is not a single equipotential surface.  CGVD28 is constrained to a series of tide gauges across Canada to which the mean sea level is at different elevations.  Furthermore, CGVD28 omits a series of systematic corrections.  The most important one is that it does not consider the actual gravity field.

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 Rimouski reference system.

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 Nov07.  The models do not represent the equipotential surface going through the mean water level in Rimouski.  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 78.6 cm to 13.1 cm (66.7% confidence level).

*: Standard deviation after filtering out systematic errors

The next geoid model publicly available will be in 2013.  This model will be the realization of the future vertical reference system in Canada.  Even though the reference system (reference surface) remains to be defined, the 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.