Extended Abstract
Introduction
Development of reliable models for estimation and prediction of changes inTotal Electron Content (TEC) of the ionosphere is still considered to be a real challenge for geodesists and geophysicists. This ispartly due to the nonlinear behavior of the physical and geophysical parameters affecting the TEC variations, as well as the difficulty in accurate measurement of some of these parameters. Due to its specific nature, as well as its physical and geophysical properties, quantity of TEC hasspatio-temporal variations, which can be attributable to daily, and seasonal variations, various anomalies, or periods of solar activity. Total Electron Content is the quantity which can be used to study ionospheric activities, as well as the spatio-temporal variations in electron density of this layer. In fact, TEC is the total number of free electrons in the path between the satellite and the receiver in a one square meter column. The measurement unit of TEC is TECU, which is equivalent to 1016electrons/m². Due to inappropriate spatial distribution of GPS receivers and their limited number, as well as observationaldiscontinuity in the time domain, TEC values and electron density obtained from theGPS measurements will be spatiallyand temporallyconstrained. In order to calculate TEC value in areas lacking observation or appropriatestation distribution, TEC value obtained from GPS measurements must be interpolated or extrapolated in a suitable manner.
 
Materials and Methods
By combining wavelet localization features with standard neural networks, Wavelet Neural Networks (WNN) have emerged as a new mathematical method for modeling and predicting the behavior of different phenomena.In WNNs, the output parameter is usually calculated by the following equation:
(1)               
 
 
wherex is the inputobservations vector,  is a the multi-variablewavelet whichcan be calculated by the tensor productof m (basic function of single variable wavelets), ë is the number of neurons in the hiddenlayer, and ù shows the network weight. Unlike the Backpropagation (BP) algorithm, PSO is a global search algorithm that can optimize the initial weights and introduce the appropriate structure for the network. Equations used in this algorithm are as follows:
                                                                                                                                                                                                    (2)
 
                                                                                                                                                                                                         (3)
In which, shows the initial weight, represents the particle’s velocity i in repetition t, c1 and c2, indicate the particle acceleration coefficients,  is the current position of particle i in repetition t and gbest represents the best particle position. The present study took advantage of a smoothing algorithm to determine STEC observations. Observed STEC values are as follows:
                                                                                                                                                                                                           (4)
 
To obtain TEC value along the zenith, the following mapping function can be used:
                                                                                                                                                                                                        (5)
Which we will have:
                                                                                                                                                                                                          (6)
Elev. in relation (6) is the satellite’s elevation angle.
 
Results and Discussion
Observations of 37 Iranian GeodynamicNetworkson 2012.08.11 (DAY 224) were used to evaluate the efficiency of WNN and PSO training algorithm in modeling and predictingspatio-temporal variations of TEC in Iran. Of the 37 stations, 5 were used as test stations, 2 were used to evaluate the wavelet neural network, and the rest were used to train the network. Four different combinations of input observations are examined in this paper. Number of input observations selected from the Iranian Permanent Geodynamic Network(IPGN) to train the WNN using PSO algorithm was25, 20, 15 and 10, respectively.Table 1 shows the characteristics of different combinations evaluated in this paper.
Table 1. Characteristics of the observations used in the different combinationsevaluated
 
 
To evaluate the accuracy of the results obtained from IRI and WNN model, all results were compared with TEC observations obtained from GPS. Table 2 shows the correlation coefficient for different scenarios.
 
Table 2. correlation coefficient for different scenarios
 
 
According to Table (2), the first scenario in WNN method with GPS hasthe highest correlation coefficient. Even when the number of observations in the databasedecreases in the third scenario, theWNN method still has a higher correlation coefficient compared to the IRI2012 model. In the fourth scenario, the correlation coefficient for WNN method is reduced to some degree. The average relative and absolute error values at the 5 test stations were calculated for the four different scenarios and presented in Table3.
 
Table 3. Comparison of mean relative error and absolute error values
at 5 test stations for four different scenarios.
 
 
Statistical analysis of relative and absolute error showssuperiority of WNN method in TEC modeling as compared to the IRI2012.
 
Conclusion
To model total electron content of the ionosphere, 4 combinations of observations were evaluated. 25, 20, 15 and 10 stations were used to train the wavelet neural network. 300, 240, 180, and 120 observations(latitude and longitude, observation time)were considered in the database, respectively.Results of the analysis indicated that with a decrease in the number of observations in the database, the absolute and relative error increase, while correlation coefficient decreases. This decrease was not evident before 180 observations, but relative and absolute errorreached up to twice their values with 120 observations. It should be noted that even with 120 observations (10 stations for training), results of the wavelet neural network model are more accurate than the results of the IRI2012 model.