Global Positioning System Reference
In-Depth Information
In modelling Istanbul and Sakarya local geoids using the ANFIS approach, training data
(the geoid reference benchmarks) were used to estimate the ANFIS model parameters,
whereas test data were employed to validate the estimated model. The input parameters are
the geographic coordinates of the reference benchmarks, and the output membership
functions are the first order polynomials of the input variables. As the number of the output
membership functions depends on the number of fuzzy rules, in computations, the latitudes
and longitudes were divided into 5 subsets to obtain 5 x 5 = 25 rules in Istanbul, and 4
subsets to obtain 4 x 4 = 16 rules in Sakarya. In both case studies, we adopted the Gaussian
type membership function as suggested by Yılmaz (2010). After determining the ANFIS
structure, the parameters of both the input and output membership functions were
calculated according to a hybrid learning algorithm as a combination of least-squares
estimation and gradient descent method (Takagi & Sugeno, 1985). Using the determined
ANFIS model parameters for Istanbul and Sakarya data, separately, the geoid heights both
at the reference and test benchmarks were calculated. In addition, the statistics of the geoid
height differences between the model and observations were investigated in each local area.
In the test results for Istanbul local geoid with ANFIS (Table 3), the geoid height residuals at
the test benchmarks vary between -9.7 cm and 9.5 cm with a standard deviation of ±3.5 cm.
As the basic statistics in Table 3 provides a comparison between the performances of two
methods in Istanbul, ANFIS has a 20% improvement in terms of RMSE of geoid heights
comparing the 5 th order polynomial model. As the RMSE of the computed geoid heights for
the reference benchmarks and the test benchmarks are close values, we can say that the
composed ANFIS structure is appropriate for modelling the Istanbul data. The coefficient of
determination (R 2 ), as the performance measure of ANFIS model is 0.996.
However, in Sakarya, the ANFIS method did not reveal significantly superior results from
the 4 th order polynomial at the test points with the geoid height residuals between -35.4 cm
and 19.0 cm with root mean square error of ±18.9 cm. The improvement of the model
accuracy with ANFIS method versus the polynomial is around 7%, considering the RMSE of
geoid heights. On the other hand ANFIS revealed much improved test statistics at the
reference benchmarks than the polynomial. The inconsistency, observed between the
evaluation results at the reference and test benchmarks for ANFIS model may indicate an
inappropriateness of this model for Sakarya data. Figure 11 maps the geoid height
differences of ANFIS model and observations at the benchmarks in Istanbul and Sakarya.
In addition to the evaluation of surface approximation methods in modelling local
GNSS/levelling geoids in case study areas, TG03 model was also evaluated at the reference
geoid benchmarks. The statistics of geoid height differences with 0.3 cm mean and ±10.8 cm
standard deviation for Istanbul, confirms the reported accuracies of the model by TNUGG
(2003) and Kılıçoğlu et al. (2005). Conversely, the validation results of TG03 model in
Sakarya GNSS/levelling benchmarks revealed the differences of geoid heights with -4.4 cm
mean and ±18.6 cm standard deviations. Considering these validation results, although the
performance of TG03 model seems low by means of RMSE of geoid heights, they revealed
approximately 44% of improvement when comparing to the performance of previous
Turkish regional geoid TG99A in the same region (see the results of TG99A validations in
Sakarya region by Kılıçoğlu&Fırat (2003)).
In the conclusion of this section, the Istanbul and Sakarya local GNSS/levelling geoid
models by ANFIS approach can be observed in the maps depicted in Figures 12 and 13.
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