## 2 Data and Methods
### Implementation of the Regression NN  
The goal of this study is to estimate Arctic and Antarctic SIT from the selected set of input features. Targeting continuous values of SIT based on the relationship between the input features represents a regression task. Linear regression models typically adjust a number of model parameters to a set of training data in an iterative process by minimizing a cost function, which eventually converges to an optimal fit. The Gradient Descent method is a common technique to find the optimum solution. It computes the local gradient with respect to the model parameters and a cost function, following the direction of descending gradient until reaching convergence [49]. In a first attempt, a simple regression model without hidden layers was selected, but the model did neither converge nor generalize well on the test set. This implied that the input features are not linearly separable, but rather non-linearly related. Neural networks (NN) can manage complex regression tasks and they are more adequate than traditional approaches when dealing with non-linear relationships between the variables. In its basic structure, a NN consists of an input layer, an output layer, and interposed hidden layers, with each layer consisting of a number of neurons [50]. The observations for training are assigned to the input layer with the number of neurons being the number of input features. The output layer of a regression task has a single neuron that represents the retrieved continuous target parameter. At least one hidden layer between the input and output layer makes the model different from a simple regression framework by enabling the network to learn the relationships that are contained in the dataset. In this work, the networks were built as sequentially dense layers, i.e., each neuron of the previous layer is fully connected to all neurons of the following layer. The specific model set-up was adjusted during the training and Figure 3 illustrates the final network architecture for the thin and full-range model. Similar to the coefficients in a linear regression, each connection (lines) between neurons represents the weight of the output of the neuron in the previous layer. The output value \(Y\) of each neuron is determined by forward propagating the weighted sum of the inputs coming from the neurons \(i\) of the previous layer with the weights \(\omega\) and an additive bias \(b\)  
\[Y=f\Big{(}\sum_{i}(\omega_{i}*input_{i})+b\Big{)}\quad\text{with}\quad f(x)= max(0,x). \tag{1}\]  
The activation function \(f\) of each hidden layer introduces the non-linearity between the input features and the target variable, without which the regression network would be entirely linear. ReLU (Rectified Linear Unit), representing the activation function defined above, is a commonly used function in regression tasks, and it has the advantage of being computationally efficient and it does not saturate for positive values [51]. The weights and biases of the neurons are updated according to the final error of the cost function via back-propagation. The number of hidden layers and neurons per layer can be increased to capture more feature interactions, depending on the complexity of the problem. The specific model set-up was adjusted during the training. Adaptive Moment (Adam) estimation was selected as an optimizer [52]. Model parameters (\(\omega\), \(b\)) were randomly initialized and a small learning rate of 0.001 was chosen to obtain smooth convergence. The entire feature set was split into a training set (80%) and a test set (20%). The training set was further split into a reduced training set (80%) and validation set (20%). The performance of the different models was quantified using the Mean Absolute Error (MAE) as a cost function, being defined as the average sum of absolute differences between ground truth and predicted SIT. The MAE value indicates the training and validation loss at each epoch of the training.