## 3 Methodology
### Evaluating Electromagnetic Profile Using the Gaussian Process  
This work uses Gaussian process machine learning (GPML) MATLAB codes for the GP regression, previously written by [PERSON] and [PERSON], which can be found in [36]. Note that the MATLAB version used by this work is R2019b. As mentioned before, the Gaussian process (GP) acts as the alternative approach to the numerical forward modeling in the SBL application for evaluating EM profiles at various \"untried\" hydrocarbon depths. Suppose all nine CST datasets, \(\mathcal{D}_{1},\mathcal{D}_{2},\dots,\ \mathcal{D}_{9}=\{(x_{i},y_{i})\}_{i=1}^{N}\), where \(N=73\), are treated as the training datasets for the GP modeling. It is also possible to simply write the \(d\)-dimensional input variable vector and its corresponding output, which is magnitude of E-field vector, as \(x_{i}\in X\) and \(y_{i}\in Y\), respectively. Generally, in this work, \(X\) is the \(N\)-by-\(d\) matrix and \(Y\) is the \(N\)-by-\(1\) column vector, where \(x\in\mathbb{R}^{d=2}\) and \(y\in\mathbb{R}^{d=1}\). In other words, \(x=(s,h)\), where \(s\) and \(h\) represent the offset (i.e., source-receiver separation distance) and the depth of hydrocarbon, respectively. In order to evaluate or estimate the EM profiles at various \"untried\" depth inputs conforming to the testing input, \(x_{*}=(s_{*},h_{*})\), the posterior distribution of the testing output \(y_{*}\) is defined in Equation (1). This work's interest was to predict the depth parameter of hydrocarbon along the offsets from 11,840.80 m to 19,004.97 m. Thus, here, only the testing input variable of hydrocarbon depth, \(h_{*}\), is varied accordingly, while the \(s_{*}\) remains unchanged. Note that the interested hydrocarbon depths in the GP regression are from 200 m to 1000 m with an increment of 20 m each (including the \"tried\" depths). \[(y_{*}|X,Y,x)\,\sim G(m_{*},\kappa_{*}), \tag{1}\]  
where the predictive mean, \(m_{*}\), and the predictive variance, \(\kappa_{*}\), are expressed in Equations (2) and (3), respectively. \[m_{*}=K(X,x_{*})^{T}\Big{[}K(X,X)+\sigma_{N}^{2}I_{N}\Big{]}^{-1}Y, \tag{2}\]  
and  
\[\kappa_{*}=k(x_{*},x_{*})-K(X,x_{*})^{T}\Big{[}K(X,X)+\sigma_{N}^{2}I_{N} \Big{]}^{-1}K(X,x_{*}). \tag{3}\]  
Based on Equations (2) and (3), \(K(X,X)\) is the covariance matrix with a dimension of \(N\)-by-\(N\) amongst the \(X,K(X,x_{*})\) is the covariance matrix with a dimension of \(N\)-by-\(1\) between the \(X\) and \(x_{*}\), \(k(x_{*},x_{*})\) is the covariance of the \(x_{*}\), \(I_{N}\) is an \(N\)-dimensional unit matrix, and \(\sigma_{N}^{2}I_{N}\) is the noise covariance matrix. All these covariance matrices are composed of squared exponential (SE) terms, where each element of the matrices is computed using Equation (4) as follows:  
\[k\big{(}x,x^{\prime}\big{)}=\sigma_{f}^{2}\Bigg{[}exp\left(-\frac{\left|x-x^{ \prime}\right|^{2}}{2\ell^{2}}\right)\Bigg{]}, \tag{4}\]  
where \(\sigma_{f}\) represents the signal variance and \(\ell\) represents the isotropic length scale.These two parameters are known as the hyperparameters of the SE covariance function, \(\theta=\left\{\sigma_{f},\ell\right\}\). SE is a very famous covariance function used in GP regression. According to [37], GP with SE is infinitely differentiable. This feature is very important for the derivative-based approach. Here, it must be noted that all the computations of Equations (1)-(4) require the estimate of \(\theta\). These hyperparameters are numerically estimated by minimizing the negative log marginal likelihood (NLML) using a gradient-based optimizer. The NLML equation is defined in Equation (5) as follows:  
\[NLML=-log[p(Y|X,\theta)]=\frac{N\log(2\pi)}{2}+\frac{\log[K(X,X)+\sigma_{N}^{ 2}I_{N}]}{2}+\frac{Y^{T}\big{[}K(X,X)+\sigma_{N}^{2}I_{N}\big{]}^{-1}Y}{2}. \tag{5}\]In Equation (5), a logarithmic identifier is used to simplify the integral expression involved in the marginal likelihood method [25].