Geoid Modelling
Geoid modeling is the representation of the geoid with respect to a reference ellipsoid. The separation between the geoid and the reference ellipsoid is the geoid height (N). The reliability of the geoid model will depend on the accuracy and horizontal resolution (spacing) of the geoid heights.
The development and analysis of an accurate geoid model encompasses four (4) essential elements: Theory, Data, Coding and Processing, and Validation.
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The Theory is the foundation on which is based the development of a geoid model. There are several theoretical approaches in developing geoid models. Currently, the method used at Natural Resources Canada is Helmert-Stokes (Vaníček and Martinec 1994). This method consists in solving the Stokes integral through Helmert's second condensation method where all masses (topography and atmosphere) above the geoid are condensed as a thin layer onto the geoid. The theoretical development of the Helmert-Stokes can achieve a geoid model with accuracy better than the centimetre level.
Figure 1: illustrates the Stokes-Helmert approach
- The Data allows the realization of the geoid model. The fundamental data are gravity. The Stokes integral requires a global set of homogeneous and accurate gravity measurements with their position (φ, λ, H). The collection of such a set is difficult; however, this problem can be averted by using global gravity models. These models are derived from satellite-only data (e.g., GGM02S) or a combination of satellite and terrestrial gravity data (e.g., EGM96). Recently, these datasets have improved significantly with the success of dedicated satellite gravity missions (CHAMP and GRACE). The up-coming GOCE gravity mission (to be launched in 2007) has for objective to achieve cm accuracy of the geoid at the spatial resolution of 100 km. Currently, the global gravity models define the long wavelength components (> 450 km) and contribute gravity field outside North America in the development of the geoid for Canada. The terrestrial gravity data for North America and surrounding oceans contribute the short wavelength components of the geoid models. Because terrestrial gravity data remain too sparse in some regions, high-resolution Digital Elevations Models (DEM) allows the enhancement of the highest frequencies of the geoid model. Also they contribute the determination of different terrain reductions, which are applied to the gravity measurements and the geoid model. Finally, a model of the topographical density is necessary in order to do proper terrain reductions. Current geoid models published at NRCan omit this dataset and assume a mean topographical density of 2.67 g cm-3.
- The Coding and Processing allows the transformation of the theory into programmable source code and the computation of each component forming the geoid model, respectively. A geoid model cannot be solved as theoretical proposed because the integral equations cannot be evaluated analytically. Therefore, a series of assumption and approximation are applied to the equations when converting them to source code. On the other hand, the efficiency of the processing depends on two elements: speed of the computer and space on hard disks. In the development of a geoid model, there are several components that can be quite computational intensive (e.g., terrain corrections, downward continuation). Furthermore, DEM’s can take large disk space when its spatial resolution is 0.75" (~25 m) for a country as vast as Canada.
- The Validation is the analysis of the quality of the geoid model. The validation of the geoid models can be internal or external. The internal validation consists in evaluating the accuracy of the geoid model from error propagation based on the theoretical method. The difficulty is to assign a realistic error model to the input data. Furthermore, the error model does not take into consideration unknown systematic errors and omissions. In general, results from internal analysis tend to be somewhat optimistic. On the other hand, external validation used independent datasets. The most commonly used dataset is GPS-Levelling. The geoid heights (N) can be determined from the GPS ellipsoidal heights (h) and the levelled orthometric heights (H): NGPS/Lev = hGPS - HLev. The difficulty in this technique is to separate errors coming from GPS, levelling and the geoid model. In addition, vertical motion of the benchmark could also be a major factor when GPS measurements are collected several years after the levelling surveys. Other independent techniques are astro-geodetic deflections of the vertical and satellite altimetry data. The latter allows the validation of the marine geoid model. Finally, the validation can be done by comparing geoid models with other models determined from other institutions, especially with the US National Geodetic Survey (NGS). This comparison is not truly independent because the models could be based using similar theory and data sets. However, discrepancies between the models can raise questions about our theoretical approach, data, and source coding.
