**WP 3.3.2 Influence on Levelling by GPS **

Gabriel Strykowski, Geodynamics Division, KMS

Draft Report KMS

**1. 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, medium and high). We have chosen 3 test areas: **(1)** the low variability: Denmark and
adjacent areas (53^{o}N - 58^{o}N and 8^{o}E - 16^{o}E); **(2) **the medium variability: The Central Scandinavia (60^{o}N - 66^{o}N and 12^{o}E - 18^{o}E); **(3)** the 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 (which
is the expected outcome of the GOCE mission) 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.

Acknowledgement: R. Forsberg and prof. C.C.Tscherning contributed with ideas and discussion.

**2. The relative accuracy of height differences - an error study**

The study consist of 2 parts:

(1)* Computing realizations of noise in geoidal heights.* The computer program HARMEXG.F (see WP 2.1) was used to generate 20 realizations of noise in
geoidal heights caused by the errors in SHC, and for the three test areas (see Introduction). In details, SHC of EGM96 have been perturbed 20 times according
to the expected SHC error statistics of the GOCE mission. For each of the three areas this yields 20 realizations of the geoid model contaminated by the noise.
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 all
three 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.

(2) *Assessment of the relative accuracy of height systems. *In each of the three areas we have chosen (somehow arbitrarily) the central parallel and the central
meridian to represent the geoidal height diferences. The diferences are taken with respect to the eastermost 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 and (3) long: 500 km/300km/or other (depending on
the size of the area). [In determining the geoidal height differences we use grid points along parallels/meridians which distance from the initial grid point is
closest to the above distances.]

Referring to (1), for each of the two types of error statistics (GOCE, EGM96), and for each distance, the above procedure yields 20 geoidal height differences caused by the SHC error. (If the geoid was perfect, i.e. no noise, the difference caused by SHC error would be zero). Subsequently, for these 20 differences we compute the mean, the variance and the standard deviation. The results are presented in Table 1- Table 3. 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, we bring in sec. 4 one figure for each central parallel/meridian and for each type of error statistics.

**3. The results**

**Table 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.

central parallel |
distance: 25 km |
distance: 100 km |
distance: 500 km |

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 |

central meridian |
distance: 25 km |
distance: 100 km |
distance: 500 km |

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 deviation (m) | 0.021 | 0.079 | 0.087 | 0.402 | 0.068 | 0.182 |

**Table 2. **The Central Scandinavia (60^{o}N - 66^{o}N and 12^{o}E - 16^{o }E). Area type:medium variability of the gravity field. Statistics of the error in the geoidal height
differences for different distances. Based on 20 realizations of noise.

central parallel |
distance: 25 km |
distance: 100 km |
distance: 300 km |

GOCE | EGM96 | GOCE | EGM96 | GOCE | EGM96 | |

mean value (m) | -0.000 | 0.039 | 0.004 | 0.145 | 0.007 | 0.157 |

variance (m^{2}) |
0.0002 | 0.0316 | 0.0016 | 0.3267 | 0.0019 | 0.3321 |

standard deviation (m) | 0.014 | 0.178 | 0.040 | 0.572 | 0.043 | 0.576 |

central meridian |
distance: 25 km |
distance: 100 km |
distance: 500 km |

GOCE | EGM96 | GOCE | EGM96 | GOCE | EGM96 | |

mean (m) | -0.010 | 0.026 | -0.012 | 0.071 | -0.008 | 0.083 |

variance (m^{2}) |
0.0032 | 0.0571 | 0.0094 | 0.3215 | 0.0066 | 0.2705 |

standard deviation (m) | 0.056 | 0.239 | 0.097 | 0.567 | 0.081 | 0.520 |

**Table 3.** 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.

central parallel |
distance: 25 km |
distance: 100 km |
distance: 500 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 deviation (m) | 0.057 | 0.465 | 0.093 | 0.881 | 0.072 | 0.665 |

central meridian |
distance: 25 km |
distance: 100 km |
distance: 225 km |

GOCE | EGM96 | GOCE | EGM96 | GOCE | EGM96 | |

mean (m) | -0.003 | -0.025 | -0.026 | -0.017 | 0.023 | 0.011 |

variance (m^{2}) |
0.0004 | 0.0116 | 0.0051 | 0.1336 | 0.0032 | 0.2078 |

standard deviation (m) | 0.019 | 0.108 | 0.071 | 0.366 | 0.056 | 0.456 |

**4. Figures**

Figure 1a. . Relative error in geoidal height differences for 20 realizations of "GOCE error statistics" SHC noise along the central parallel in Denmark and the
adjacent areas (53^{o}N - 58^{o}N and 8^{o}E - 16^{o}E). Area type: low variability gravity field.

Figure 1b. . Relative error in geoidal height differences for 20 realizations of "EGM96 error statistics" SHC noise along the central parallel in Denmark and the
adjacent areas (53^{o}N - 58^{o}N and 8^{o}E - 16^{o}E). Area type: low variability gravity field.

Figure 1c . Relative error in geoidal height differences for 20 realizations of "GOCE error statistics" SHC noise along the central meridian in Denmark and the
adjacent areas (53^{o}N - 58^{o}N and 8^{o}E - 16^{o}E). Area type: low variability gravity field.

Figure 1d . Relative error in geoidal height differences for 20 realizations of "EGM96 error statistics" SHC noise along the central meridian in Denmark and the
adjacent areas (53^{o}N - 58^{o}N and 8^{o}E - 16^{o}E). Area type: low variability gravity field.

Figure 2a . Relative error in geoidal height differences for 20 realizations of "GOCE error statistics" SHC noise along the central parallel in The Central
Scandinavia (60^{o}N - 66^{o}N and 12^{o}E - 16^{o}E). Area type: medium variability gravity field.

Figure 2b . Relative error in geoidal height differences for 20 realizations of "EGM96 error statistics" SHC noise along the central parallel in The Central
Scandinavia (60^{o}N - 66^{o}N and 12^{o}E - 16^{o}E). Area type: medium variability gravity field.

Figure 2c . Relative error in geoidal height differences for 20 realizations of "GOCE error statistics" SHC noise along the central meridian in The Central
Scandinavia (60^{o}N - 66^{o}N and 12^{o}E - 16^{o}E). Area type: medium variability gravity field.

Figure 2d. Relative error in geoidal height differences for 20 realizations of "EGM96 error statistics" SHC noise along the central meridian in The Central
Scandinavia (60^{o}N - 66^{o}N and 12^{o}E - 16^{o}E). Area type: medium variability gravity field.

Figure 3a . Relative error in geoidal height differences for 20 realizations of "GOCE error statistics" SHC noise along the central parallel in The High Alps
(46^{o}N - 48^{o}N and 6^{o}E - 16^{o}E). Area type: high variability gravity field.

Figure 3b. Relative error in geoidal height differences for 20 realizations of "EGM96 error statistics" SHC noise along the central parallel in The High Alps
(46^{o}N - 48^{o}N and 6^{o}E - 16^{o}E). Area type: high variability gravity field.

Figure 3c . Relative error in the geoidal height differences for 20 realizations of "GOCE error statistics" SHC noise along the central meridian in The High Alps
(46^{o}N - 48^{o}N and 6^{o}E - 16^{o}E). Area type: high variability gravity field.

Figure 3d . Relative error in the geoidal height differences for 20 realizations of "EGM96 error statistics" SHC noise along the central meridian in The High
Alps (46^{o}N - 48^{o}N and 6^{o}E - 16^{o}E). Area type: high variability gravity field.

**5. Conclusions**

The main idea behind WP 3.3.2 is 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. Moreover, the three test areas which were chosen for the investigation had different type of the variability of the gravity field (high, medium, low). The conducted error study was ensamble statistics (20 realizations of noise) for different distances and for two perpendicular azimuthal directions (the central meridian and the central parallel in each of the three areas were chosen).

The conclusions are:

(1) **Mean values** in Tables 1-3 is in principle a measure of accuracy of the model to determine the height differences. The lowe the mean value (in the absolute
sense) the better. (For a perfect geoid means that the estimated mean value should be zero.) The three 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 6-10 times lower then EGM96. For the medium variability gravity
field the corresponding improvement factors are 2-20. However, for the high variability gravity field the results of GOCE and EGM96 are similar. (In some
cases the mean value gets slightly worse for GOCE.)

(2) **Standard deviation** is a measure of the spread of different realizations of noise in a local area around the mean value. By inspecting figures 1a-1d, 2a-2d
and 3a-3d 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 1-3 where the improvement in standard deviation is in general by factor 10 (also for the high variability gravity field).

Last update 1999-06-07 by cct.