GNSS in Practical Determination of Regional Heights - Global Navigation Satellite Systems: Signal,Theory and Applications - page 144

regions, revealed the statistics in Tables 3 and 4. As is seen from the Table 3 for Istanbul
area, the accuracy of the fifth order polynomial in terms of RMSE of geoid heights at the test
points is ±4.4 cm with a coefficient of determination of 0.992. The geoid height differences of
the polynomial model and observations at the benchmarks are mapped in Figure 10a. The
test statistics of the polynomial model for Sakarya local geoid are summarized in Table 4
that the evaluation of the model at the independent test points revealed an absolute
accuracy of ±20.4 cm in terms of RMSE of the geoid heights. Although the qualities of
employed reference data in computations of both local geoid models are comparable (see
section 3.1.1), the polynomial surface model revealed much improved results in Istanbul
territory than Sakarya. The reasons of low accuracy in local geoid model of Sakarya territory
can be told as sparse and non-homogeneous distribution of geoid reference benchmarks and
rough topographic character of the territory that makes difficult to access for height
measuring. Hence the GNSS/levelling benchmarks whose density and distribution are very
critical indeed for precise modelling of the local geoid, are not characterize sufficiently the
topographic changes and mass distribution in Sakarya (compare point distribution versus
topography in Figure 8). Figure 10b shows the geoid height differences of the polynomial
model and observations at the benchmarks for Sakarya.

5th order polynomial ANFIS

TG03

Reference BMs Test BMs Reference BMs Test BMs

Minimum -11.2 -11.5 -10.5 -9.7 -32.5
Maximum 11.4 11.5 12.4 9.5 30.0
Mean 0.0 0.0 0.0 0.0 -0.3
RMSE 4.2 4.4 3.6 3.5 10.8
R 2 0.993 0.992 0.996 0.995 0.960
Table 3. Statistical comparison of applied approximation techniques in Istanbul local geoid
(units in centimetre, R 2 unitless)

4th order poly nomial ANFI S

TG03

Reference BMs Test BMs Reference BMs Test BMs

Minimum -52.0 -36.3 -39.7 -35.4 -53.8
Maximum 82.7 24.1 42.1 19.0 64.3
Mean -0.3 -7.5 0.0 -11.0 -4.4
RMSE 22.7 20.4 12.0 18.9 18.6
R 2 0.923 0.905 0.978 0.913 0.945
Table 4. Statistical comparison of applied approximation techniques in Sakarya local geoid
(units in centimetre, R 2 unitless)

Nonlinear regression structure of ANFIS and its resulting system, based on tuning the
model parameters according to local properties of the data may reveal improved results of
surface fitting. However one must be careful whilst working with soft computing
approaches and pay attention for choosing appropriate design of architecture with optimal
parameters such as: (e.g. in ANFIS) the input and rule numbers, type and number of
membership functions, efficient training algorithm. Since the prediction capabilities of these
algorithms vary depending on adopted architecture, use of unrealistic parameters may