3.3.2 Influence on levelling by GPS
Introduction
The errors in the Spherical Harmonic Coefficients (SHC) of the global gravity field models are inherent. For any local area of the Earth these errors affect the accuracy of the global geoid model for that area, i.e. the accuracy of the model reference surface for the height systems.
The purpose of this investigation is to assess the effect of such error on the accuracy of the height difference estimation for two locations in the area as used in GPS levelling. Thus, indirectly, the relative geoid height error is the relative error in the height reference surface, and, thus, the relative error in levelling by GPS. The investigation is conducted both for varying distances and for areas with different types of variability of the gravity field (low and high).
We have chosen The following test areas:
Low variability: Denmark and adjacent areas (53^{o}N  58^{o}N and 8^{o}E  16^{o}E);
High variability: The High Alps (46^{o}N  48^{o}N and 6^{o}E  16^{o}E).
Furthermore, the overall purpose of the study is to assess how the future improvement in the accuracy of SHC will improve the accuracy of the relative heights. The present study is based on the expected SHC error statistics of GOCE mission and on the error degree variances of EGM96.
The relative accuracy of height differences
The computer program HARMEXG.F (see chapter 2.1) was used to generate 20 realizations of noise in geoidal heights caused by the errors in SHC, and for the test areas. In details, SHC of EGM96 have been perturbed 20 times according to the expected SHC error statistics of the GOCE mission. Subsequently, the reference geoid (EGM96 without noise) was computed and subtracted from the above 20 realizations of noisy geoid models. This yields 20 realizations of noise according to the expected SHC error statistics of the GOCE mission. The same procedure was repeated for EGM96 error statistics. For both areas each realization of noise is determined on a regular grid in geographical coordinates (spacing: dlat x dlon= 0.25^{o} x 0.50^{o}) was used.
In each of the test areas we have chosen (somehow arbitrarily) the central parallel and the central meridian to represent the geoidal height difference. The difference are taken with respect to the easternmost grid point along the parallels and the southernmost grid point along the meridians. In all cases we choose 3 distances:
(1) short: 25 km;
(2) medium: 100 km
(3) long: 500 km/300km/or other (depending on the size of the area).
Subsequently, for these 20 differences we compute the mean, the variance and the standard deviation, presented in Table 3.3.2.1 and 3.3.3.2.
In order to illustrate the range of the scatter in the geoidal height differences which are caused by different realizations of noise, and for different distances, a figure for each central parallel are shown.
Table 3.3.2.1. Denmark and the adjacent areas (53^{o}N  58^{o}N and 8^{o}E  16^{o}E). Area type: low variability of the gravity field. Statistics of the error in the geoidal height differences for different distances. Based on 20 realizations of noise. <TBODY>
central parallel 
distance: 25 km 
distance: 100 km 
distance: 500 km</TBODY> 
<TBODY>

GOCE 
EGM96 
GOCE 
EGM96 
GOCE 
EGM96 
mean value (m) 
0.000 
0.007 
0.003 
0.032 
0.003 
0.051 
variance (m^{2}) 
0.0002 
0.0374 
0.0009 
0.1404 
0.0008 
0.1958 
standard deviation (m) 
0.015 
0.193 
0.031 
0.375 
0.028 
0.443</TBODY> 
<TBODY>
central meridian 
distance: 25 km 
distance: 100 km 
distance: 500 km</TBODY> 
<TBODY>

GOCE 
EGM96 
GOCE 
EGM96 
GOCE 
EGM96 
mean value (m) 
0.003 
0.028 
0.010 
0.109 
0.023 
0.121 
variance (m^{2}) 
0.0005 
0.0063 
0.0076 
0.1618 
0.0047 
0.0330 
standard dev. (m) 
0.021 
0.079 
0.087 
0.402 
0.068 
0.182</TBODY> 
Table 3.3.2.2. The High Alps (46^{o}N  48^{o}N and 6^{o}E  16^{o}E). Area type: high variability of the gravity field. Statistics of the error in the geoidal height differences for different distances. Based on 20 realizations of noise. <TBODY><TBODY></TBODY>
<TBODY>
central parallel 
distance: 25 km 
distance: 100 km 
distance: 225 km 


GOCE 
EGM96 
GOCE 
EGM96 
GOCE 
EGM96 
mean (m) 
0.020 
0.018 
0.035 
0.037 
0.071 
0.075 
variance (m^{2}) 
0.0033 
0.2158 
0.0087 
0.7769 
0.0051 
0.4419 
standard dev. (m) 
0.057 
0.465 
0.093 
0.881 
0.072 
0.665</TBODY> 
central meridian 
distance: 25 km 
distance: 100 km 
distance: 225 km 

mean (m) 
3 
0.025 
26 
17 
23 
11 
variance (m^{2}) 
4 
116 
51 
1336 
32 
2078 
standard dev. (m) 
19 
108 
71 
366 
56 
456 
Conclusion
The main idea behind this investigation was to assess how GOCE mission will improve the absolute accuracy in the determination of the relative heights. Thereby, the influence of the relative error in the reference surface on the levelling by GPS is also assessed indirectly. It was decided to conduct an error study for the height reference surface (the geoid) and to compare these results to those for the existing global model(s) EGM96. The test areas which were chosen for the investigation had different type of the variability of the gravity field (high, low). The conducted error study was ensamble statistics (20 realizations of noise) for different distances along the central parallel in each test areas were chosen).
The tables are in principle a measure of accuracy of the model to determine the height differences. A lower mean value (in the absolute sense) is better. (For a perfect geoid means that the estimated mean value should be zero.) The two types of the area are to some degree reflected in these mean values. For the low variability gravity field GOCE mission yield a mean value which is 610 times lower then EGM96. For the medium variability (not shown) gravity field the corresponding improvement factors are 220.
In the case where the high frequencies dominate the gravity signal the results of GOCE and EGM96 will be similar. However, when combined with local information about topography whihc are responsible for the hgh frequencies in the gravity signal, GOES results will again be significantly better than those obtained using EGM96
The standard deviation is a measure of the spread of different realizations of noise in a local area around the mean value. By inspecting the figures one must conclude that GOCE mission will considerably improve the precision, i.e. there is systematically less spread then for EGM96. This is also reflected in Tables 3.3.2.1 and 3.3.2.2 where the improvement in standard deviation is in general by factor 10 (also for the high variability gravity field).