3.3 Geodesy
3.3.1. Height Systems
Introduction.
The determination of height requires the definition of an equipotential reference surface. The definition is made by assigning values to a number of reference points. These values are based on observations of mean sea level for a given epoch at a number of tide gauges. Time depended land movements should be taken into account to prevent inconsistencies between different tidegauge mean values and the levelling lines connecting the tidegauge stations. In this way, it is expected that different areas will use the same reference surface.
Several investigations have showed considerable height differences between height datums from island to island, from region to region and from continent to continent. For instance, the hydrostatic levelling between Denmark and Germany showed a height datum difference of 7.6 cm (Andersen et al., 1990). The reasons for these inconsistencies could be attributed to changes of sealevel, to the stationary part of the sea surface topography, and to changes in coastal currents. Due to datum differences a number of serious problems are caused in geodetic and other applications.
The height datums are physically realized on land. However, we also need a height reference surface at sea. If a gravitational model is available, this could be used at any point to calculate the geopotential W. Then the geoid is found to a good approximation along the vertical at a distance (W  W_{0})/g where W_{0} is the value of the potential of the geoid. The calculation may be improved by augmenting the gravity model with known regional gravity information. This procedure may be executed in points with known orthometric height and thereby used to determine , the offset of a local height datum with respect to a global datum (e.g., Rapp and Wang, 1994). Using a gravity model alone, we expect to have a global height datum realized with an error below 10 cm (e.g., Rummel et al., 1995).
The goal of this simulation study is to investigate the improvement in the vertical datum control using the GOCE simulated gravity field model. The quality of the gravity field model is expressed through a variance covariance matrix of the spherical harmonic coefficients. From this matrix the precision of derived quantities like geoid undulation differences, gravity anomalies etc. can be expressed.
In order to be able to assess the improvement of the accuracy of the vertical datum transfer expected from the GOCE mission, experiments were conducted using the simulated coefficient error estimates for the GOCE computed by Visser (personal communication, 1998) as well as the error estimates of the coefficients of the EGM96 geopotential model (Lemoine et al., 1996).
Estimation of the accuracy of vertical datum transfer
For the experiments concerning the accuracy of the datum transfer, a "MonteCarlo" type simulation was planned and implemented using the FORTRAN program "HARMEXG", see Chapter 2. The computation of the geoid heights needed in the following experiments was based on the expansion of EGM96 spherical harmonic model to degree 200. Random errors (perturbations) were added to the coefficients of EGM96 model.
The random errors were computed taking into account the errors of the coefficients of EGM96 and the simulated GOCE error model. For the estimation of the accuracy of the vertical datum transfer from one point to another, 50 pairs of geoid heights were computed. The standard deviation of the differences between the heights of each of the 50 pairs was considered as the best way to express the result of the accuracy. 50 perturbations were needed for the computations of the 50 geoid heights in each point were computed so that they had a mean value equal to zero and a standard deviation equal to the given error estimate of the corresponding coefficient in the case of EGM96, or to the simulated error estimate in the case of GOCE.
According to the discussion in Chapter 2 the simulated GOCE error estimates were multiplied by a scale factor. The scale factor was computed as the local gravity variance divided by the global gravity variance as computed from the coefficients of the spherical harmonic model used. Then it was used either as a constant scale factor for all degrees, or as a linear factor being equal to 1 at degree 0 and equal to the scale factor at the maximum degree used.
For the computation of the local variance of the gravity anomaly a 5' by 5' grid covering an 1 deg by 1 deg. area (169 point values) around the point under consideration was computed using the GPM98A model.
In the following simulation computations, the vertical datum was transferred from the tidegauge station of Amsterdam (Longitude = 4.9333800^{o}, Latitude = 52.378378^{o}, standard deviation of gravity anomalies of 3.43 mGal) to other places of the world (see Figure 3.3.1.1). The results of these computations for the various error models mentioned previously are shown in Table 3.3.1.1.?N are the geoid undulation differences between Amsterdam and the other places, computed from GPM98A.
From Table 3.3.1.1 it is clear that the accuracy of the vertical datum transfer does not depend on the geoid undulation difference between the starting and the end point. These results confirm the discussion of Chapter 2. The accuracy estimation based on the EGM96 error model does not depend on the location. The accuracy estimation based on the simulated GOCE error model seems to be correlated with the latitude with the exception of Peiraias. The constant scaling of the GOCE error degreevariances leads to a very optimistic accuracy estimation, while the linear scaling gives more realistic results, taking into account the local variation of the gravity anomaly field. Using GOCE the result of the accuracy estimation is about 10 times better than the corresponding one, using EGM96 error model.
Table 3.3.1.1 Accuracy of the vertical datum transfer from Amsterdam positioned at
(Longitude = 4.9333800^{o}, Latitude = 52.378378^{o}, N=43.997 m) to different places of the world. The first GOCE column are the results using the global geoid variation. The last two GOCE columns takes into account the actual local gravity variation.



local std 
?N 
EGM96 
GOCE 
GOCE 
GOCE 
Amsterdam 
Long. 
Lati. 
(mgal) 
(m) 


const. scal. 
Linear scal. 
Tasiilaq Greenland 
37.7 
65.6 
18.9 
5.69 
0.534 
0.069 
0.036 
0.091 
New York 
74.0 
40.7 
19.14 
78.46 
0.572 
0.060 
0.029 
0.074 
Peiraias 
23.6 
37.9 
18.96 
5.34 
0.550 
0.062 
0.032 
0.073 
Osaka 
135.4 
34.6 
24.74 
6.84 
0.573 
0.054 
0.028 
0.069 
Los Angeles 
118.5 
34.0 
41.29 
78.68 
0.572 
0.059 
0.048 
0.088 
Hong Kong 
114.4 
22.4 
9.08 
45.29 
0.563 
0.056 
0.014 
0.064 
Madras 
80.3 
13.2 
19.25 
134.34 
0.472 
0.056 
0.022 
0.069 
Rio de Jane. 
43.2 
22.9 
20.06 
48.51 
0.470 
0.059 
0.027 
0.073 
Cape Town 
18.4 
33.9 
14.42 
12.76 
0.610 
0.057 
0.020 
0.069 
Melbourne 
144.9 
37.8 
11.83 
39.21 
0.551 
0.060 
0.019 
0.073 
Table 3.3.1.2 Accuracy of the vertical datum transfer from Peiraias ( N = 38.6 m) to islands of the Aegean Sea. The first GOCE column are the results using the global geoid variation. The last two GOCE columns takes into account the actual local gravity variation



local std 
?N 
EGM96 
GOCE 
GOCE 
GOCE 
Peiraias 
Long. 
Lati. 
mgal 
(m) 


const. scal. 
Linear scal. 
Lemnos 
25.26 
39.87 
19.52 
2.57 
0.486 
0.065 
0.049 
0.095 
Hios 
26.13 
38.42 
19.76 
0.99 
0.576 
0.049 
0.037 
0.078 
Rhodos 
28.23 
36.45 
86.64 
17.73 
0.474 
0.078 
0.172 
0.199 
Souda 
24.08 
35.48 
47.71 
14.11 
0.591 
0.0 45 
0.066 
0.069 
It is also interesting to have results from the transfer of datum from a tide gauge station to islands lying in relatively small distance like e.g. is the case of Peiraias and the islands in the Aegean Sea, where the variance of the gravity field is relatively high. In Table 3.3.1.2 the results of the vertical datum transfer from Peiraias to four islands are shown.
As it was expected, in the case of EGM96 the accuracy estimation is similar to that of Table 3.3.1.1. It is remarkable that in the case of Rhodos, the GOCE error model reflect the very high variation of the local gravity anomaly field.
Conclusions
The goal of this study was to assess the improvement of the accuracy of vertical datum transfer due to the high accuracy and spatial resolution gravity field model that we expect from GOCE. The assessment was based on computations made using the existing EGM96 error model and simulated errors for GOCE.
The results of computations using the GOCE error model showed that vertical datum transfer is possible with an accuracy of the order of 6 cm. This value increased to 7 cm, when the simulated GOCE errors were scaled using a linear factor. On the other hand the accuracy was increased to about 3 cm when the same scale factor was used as a constant but this improvement is obviously due to the fact that all error degree variances are multiplied with the same constant. Considering these results, the case of GOCE error model scaled linearly seems to be the more realistic.
Comparing the results obtained with the EGM96 error model with that obtained using the GOCE, the accuracy in the last case was almost ten times higher. This shows the importance of having available in the near future a gravity field model comprising the high accuracy and resolution expected from the GOCE mission.