GNSS in Practical Determination of Regional Heights - Global Navigation Satellite Systems: Signal,Theory and Applications - page 151

where the observations matrix is = − A , A is the coefficients matrix, and X consists the

unknown parameters. The a-priori root mean square error of Δ H of a baseline of S km is

Δ Δ hN

mm = that m 0 is the a-priori RMSE of unit observation. The unknown parameters
from the solution of matrix system in Equation 20 is calculated as

0( km )

( T T

XA AAP = A (21)

where P includes the weights of Δ H observations. Hence the adjusted orthometric height
differences are:

* HHv

ΔΔ = + (22)

The success of the method can be assessed at the test points where GNSS and levelling
observations exist, and in the evaluations the orthometric heights of the test points are
compared with their observed orthometric heights.

Furthermore, combining the height sets using the method of least squares, weights of each
set are essential to correctly estimate the unknown parameters. Improper stochastic
modelling can lead to systematic deviations in the results. Therefore, for the purpose of
estimating realistic and reliable variances of the data sets, and therefore constructing the
appropriate a−priori covariance matrix of the observations, variance component estimation
techniques can be included in combining algorithms of the heights. Numerous solution
algorithms suggested for variance component estimation problems can be found in various
literature published on the subject however, Rao's Minimum Norm Quadratic Unbiased
Estimation is commonly used one of these methods (Rao, 1971.). Sjöberg (1984), Fotopoulos
(2003) and Erol et al. (2008) can be referred to for further readings and practicing variance
component estimation techniques in the adjustment.

3.2.3 Case study: Local Çankırı geoid

Suggested data combination methods related to local improvement of regional geoids are
exemplified and tested in a numerical case study in this title. These results are also included
by Erol et al. (2008) to provide a detailed investigation on local performances of the various
regional models and their improvement capabilities. The local area covers 154 km x 198 km,
and the number of reference benchmarks used in the tests is 31. The GNSS positions of the
benchmarks were determined with static measurements using dual frequency GNSS
receivers. The accuracies of the latitudes and longitudes in ITRF96 datum is ±1.5 cm, and for
the accuracy of ellipsoidal heights is reported as ±3.0 cm (Erol et al., 2008). The adjustment
of levelling observations revealed the orthometric heights of the benchmarks with ±2.5 cm in
TUDKA99 datum. As can be seen in Figure 14, the benchmarks have quite poor density and
non-homogeneous distribution over the area. The approximate density of the benchmarks is
1 point per 900 km 2 . When the poor density of the benchmarks and rough topographic
pattern of the area (the heights of the region change between 41 m and 2496 m) are
considered, alongside the levelling technique the regional geoid model or its locally
improved version can be applied to obtain regional orthometric heights from GNSS. As a
result of this the density and distribution of the reference benchmarks do not allow
determination of local GNSS/levelling geoid. According to Large Scale Map and Spatial
Data Production Regulation of Turkey, legalized by July 2005, the density of the geoid