## 2 Materials and Methods
### Artificial Neural Network (ANN) Model  
Interpolation of wet troposphere delay according to the ANN model was evaluated in terms of different seasons, altitudes, station distributions, and numbers of independent variables. In the ANN model, any number of parameters can be included in the network in order to estimate the dependent variable. The weights are calculated iteratively for each of these parameters until the optimal model is calculated between the input and output data. For training of the network, the contribution of each of the input parameters to the model is determined by regression analysis. Therefore, optimization of the network can be improved by subtracting the parameters which have no significant contribution from the input matrix. In this context, six different artificial neural network models were designed for the interpolation of wet delay (Figure 1). In the first model, troposphere delays were interpolated by using latitude, longitude, meteorological parameters (water vapor pressure, temperature, pressure and weighted average temperature) and altitude information. Afterwards, other artificial neural network models were designed by subtracting the parameters one by one from the model. In summary, the input parameters of the model were determined as latitude, longitude, altitude, pressure, temperature, weighted mean temperature, and water vapor pressure, while the output was ZWD for the training process of the network.  
The principle of artificial neural networks is to generate the output variable corresponding to the input data [19]. The weight values vary, according to the examples, and go through an iterative process until the optimum output value is achieved. The fact that artificial neural networks produce  
Figure 1: Artificial neural network (ANN) models.  
the most accurate weights means that they are capable of making generalizations about the model that the samples represent. The learning process has two stages. In the first step, the output generated by the network is determined by the sample data set supplied to the network. In the second step, the weights of the network connections are changed according to the accuracy of these output values. In general, the artificial neural network model is determined according to the topology, aggregation, and activation functions used as a result of linking process elements, learning strategies, and learning rules. The selection of these functions is based on the trial and error method required for the solution of the problem. The wet troposphere delay varies significantly with the height of the station, the water vapor pressure, the temperature, and the weighted mean temperature. Based on these parameters, a three-layer feed-forward neural network model was designed. In addition, the sigmoid function is usually used as the activation function in the multilayer sensor model (Equation (6)). In this study, the sigmoid function was used for ANN model design. Throughout the training process, biases of the network were adjusted by means of Levenberg-Marquardt backpropagation to minimize an appropriate performance function based on RMSE.  
\[F(x)=\frac{1}{1+e^{-x}} \tag{6}\]