## 3 Methodology
### Parenter Estimation
#### 3.3.2 SVR Model  
The main goal of support vector regression (SVR) is predicated on the concept of dimensionality transformation, where it projects the original dataset into a higher-dimensional feature space. This process is achieved by utilizing a kernel function that implicitly performs this mapping. In this elevated feature space, SVR aims to construct a linear regression  
Figure 8: Shown here are the results of the training stage of the selected regression model (SVR model) used to predict missing data in well logs. The model was trained using data from 10 well logs in the area, split into 70% for training data (test) and 30% for test data (blind). model that embodies the simplest possible form to mitigate overfitting and ensure robust generalization capabilities. The formulation of the SVR model is centered on identifying a linear equation that encapsulates the relationship between the input vectors \(\mathbf{x}\in\mathbb{R}^{n}\) and the output variable \(y\in\mathbb{R}\) (petrophysical parameters like porosity). The linear model is given by:  
\[y=f(\mathbf{x})=\mathbf{w}^{T}\mathbf{x}+b, \tag{19}\]  
where \(b\) signifies the bias term and \(\mathbf{w}\) represents the weight vector. Together, they define the slope and position of the regression hyperplane within the feature space. The optimization of the SVR model involves adjusting the values of \(b\) and \(\mathbf{w}\) to minimize the cost function \(R\), which is described as:  
\[R=\frac{1}{2}\|\mathbf{w}\|^{2}+C\cdot\frac{1}{l}\sum_{i=1}^{l}\mathcal{L}_{ \epsilon}(y_{i},f(\mathbf{x}_{i})), \tag{20}\]  
where \(\mathcal{L}_{\epsilon}\) represents the \(\epsilon\)-insensitive loss function. This loss function only penalizes predictions that deviate from the actual values by more than a margin of \(\epsilon\), thus allowing for some errors within this margin, which contributes to the model's ability to ignore outliers and prevent overfitting [48; 49; 50]. The kernel choice, particularly the Radial Basis Function (RBF) denoted by \(K(\mathbf{x}_{i},\mathbf{x}_{j})=\exp\left(-\frac{\|\mathbf{x}_{i}-\mathbf{x} _{j}\|^{2}}{2\sigma^{2}}\right)\), is instrumental in defining the SVR's performance [51]. The RBF kernel is favored due to its proficiency in dealing with non-linear relationships and its requirement for only one tuning parameter, \(\sigma\)[52]. This parameter influences the flexibility of the decision boundary: a small \(\sigma\) results in a highly flexible boundary, whereas a large \(\sigma\) leads to a smoother decision function that can approximate a linear boundary. The RBF kernel has the unique advantage of potentially transforming the training data into an infinite-dimensional space, facilitating the SVR to handle complex non-linear relationships, which is often the case in predicting petrophysical properties such as porosity. The efficacy of SVR, therefore, is dependent on the optimal configuration of its learning parameters--primarily the RBF kernel parameter \(\sigma\), the error margin \(\epsilon\), and the regularization parameter \(C\). These parameters are typically tuned through a systematic search, often employing a logarithmic grid that spans a broad range of potential values: \(\sigma\) usually varies within \(2^{-5}\) to \(2^{5}\), \(\epsilon\) may be set between 0.01 and 0.6 to capture the permissible deviation in the predictions, and \(C\) could range from \(2^{-5}\) to \(2^{15}\), balancing the trade-off between the model's complexity and the degree to which deviations beyond \(\epsilon\) are penalized. This meticulous tuning process is important for developing an SVR model that can accurately predict porosity from well-log inputs, encompassing the necessary generalization to yield reliable predictions for unseen data. In this study, initially, the testing and training of the model were conducted using well-log inputs. Subsequently, the developed model was applied to a 2D scenario. The petrophysical inputs logs utilized included compressional velocity (\(V_{P}\)), shear-wave velocity (\(V_{S}\)), and bulk density (\(\rho\)).