# Gridded Visibility Products over Marine Environments Based on Artificial Neural Network Analysis
## 2 Data and ANN Model
### ANN Model  
Artificial neural networks have been studied since the 1940s, and have been widely applied in the military, medical science, and broader economy, etc. This study employs a BP (back propagation) network to infer Vis data, which was first proposed by [PERSON] and [PERSON] in 1986 [20]. The BP network is one of the classical artificial neural networks, and can simulate any non-linear input-output relationship.  
There are three layers in a BP network: the input layer, the hidden layer, and the output layer. The neurons are connected with others in the adjacent layers, but the neurons in the same layer are not connected with each other. The input and output layers are comprised of a single layer, but the hidden layer could have a different number of layers, and the number of neurons in each layer is also different depending upon the layer properties. We plot the topological structure of the BP network, which includes only one layer of the hidden layer (Figure 3).  
From the structure chart, it is evident that the signal is transmitted from the input layer to the output layer. The output value of each node in the hidden layer depends on the input value, the weight of each node in the input layer influencing the node value in the hidden layer, and the threshold value and activation function of each node in the hidden layer.  
\[X_{j}^{\prime}=f\left(\sum_{i=1}^{n}{\left({\omega_{ij}*X_{i}}\right)-\Theta_ {j}}\right) \tag{3}\]where \(X^{\prime}_{j}\) is the output value of the \(j\)th neuron in the hidden layer when the input value was input, \(\omega_{ij}\) is the weight where the \(i\)th value in the input layer influences the \(j\)th value in the hidden layer, \(X_{i}\) is the \(i\)th value in the input vector, \(\Theta_{j}\) is the threshold of the \(j\)th neuron in the hidden layer, and \(f\) is the activation function of the \(j\)th neuron in the hidden layer. The signal transmission rule between the input layer and hidden layer is the same as the transmission rule between the hidden layer and the output layer.  
The ANN model constructed in this study is trained based on a BP algorithm [21]. When there is an error between the actual output value and ideal output value, which form the output value in the training data set, the error would be transmitted from the output layer to the hidden layer and input layer to amend the weight and threshold of each neuron. The training process is complete only when the error is less than the convergence threshold. Artificial neural network technology is highly suitable for simulating the non-linear relationship between atmospheric visibility and the influencing factors.  
Figure 3: Topological structure of the back propagation (BP) network, which includes only one layer of the hidden layer.  
Figure 2: The technical flow chart to find the best R.  
This study constructs the BP neural network system with three layers based on the MATLAB toolbox. The number of nodes of hidden layers is considered to be 5, the maximum number of iterations is 100, the study rate is 0.1, and the convergence threshold is taken as 0.00001. Considering that Vis changes with monthly features [22], this paper inferred Vis for one month using ANN with the training data for that month. To test the fitting effect of the ANN model, the data from January 2010-2016 in the Indian Ocean (22.5\({}^{\circ}\) E-127.5\({}^{\circ}\) E, 70\({}^{\circ}\) S-25\({}^{\circ}\) N), representing a 7-year period, was employed to train the parameters of ANN, and the data from January 2017 was employed to test the parameters of the trained ANN model. To avoid the contingency of training results of neural networks due to the scarcity of training data, 30 experiments were repeated. Since the referred Vis grade based on ANN is not an integer, we defined its grade to the nearest grade. An average relative error of each observation point is defined as:  
\[\delta=\frac{|V_{c}-V_{r}|}{V_{r}}. \tag{4}\]  
where \(\delta\) is the relative error, \(V_{c}\) is the referred Vis, and \(V_{r}\) is the measured Vis.