## 3 Methodology
### Predicting Depth of Hydrocarbon  
The fitted GP model is denoted as \(Y_{fit}(s,h)\), where \(h\) here includes all \"tried\" and \"untried\" depths. This means that, for every \(h\) (i.e., 200 m-1000 m with a gap of 20 m), the GP gives estimates of the EM profile, \(Y_{fit}(s,h)\), at distances from 11,840.80 m to 19,004.97 m. Assume that the observational dataset \(Y_{obs}(s,\blacksquare)\) is observed at \(N=73\) distance points and the hydrocarbon resides underneath the seabed but with \"unknown\" depth. To find out the depth of hydrocarbon of the observational dataset \(Y_{obs}\), the least square criterion is formed between the \(Y_{fit}\) and \(Y_{obs}\). Mean squared error (MSE) is used as the loss function in the optimization algorithm. In general, MSE measures the average amount of squared differences between \(Y_{fit}\) and \(Y_{obs}\). The loss function \(L\big{(}h_{j}\big{)}\) is expressed in Equation (6) as follows:  
\[\big{\{}L\big{(}h_{j}\big{)}\big{\}}_{j=1}^{M}=\frac{1}{N}\sum_{i=1}^{N}\Big{[} Y_{fit}\big{(}s_{i},h_{j}\big{)}-Y_{obs}(s_{i})\Big{]}^{2}, \tag{6}\]  
where \(L\big{(}h_{j}\big{)}\big{|}j=1,2,\ldots,M\) and \(M=41\). Equation (6) is a function of \(h\). The loss function is minimized over the \(h\) with the minimizer serving as an estimate of the hydrocarbon depth. By using Equation (6), the \(L\big{(}h_{j}\big{)}\) between the \(Y_{fit}\) and \(Y_{obs}\) are calculated. Here, the least \(L\big{(}h_{j}\big{)}\) is then identified. As for now, the minimizer of the least \(L\big{(}h_{j}\big{)}\) serves as the best estimate of hydrocarbon depth for \(Y_{obs}\). Thus, in this work, the respective minimizer is treated as the initial guess of depth. Then, the iteration index \(j\) is set as zero and the initial guess is denoted as \(h_{0}\). Next, for the optimization using the GD method, the first-partial derivative of the \(L\big{(}h_{j}\big{)}\) with respect to \(h_{j}\) is required. The partial derivative of the loss function is defined in the equation below. The derivation of Equation (7) is provided in Appendix B (in Equation (A1)):  
\[L^{\prime}\big{(}h_{j}\big{)}=\frac{2}{N}\left(\sum_{i=1}^{N}\Big{[}Y_{fit} \big{(}s_{i},h_{j}\big{)}-Y_{obs}(s_{i})\Big{]}\right)\cdot Y^{\prime}_{fit} \big{(}s_{i},h_{j}\big{)}, \tag{7}\]  
where \(Y^{\prime}_{fit}\) is the derivative of the GP using the SE covariance function with respect to \(h_{j}\). It must be noted here that the forward GP model is defined by estimations composed by the predictive mean as shown in Equation (2). To find out the \(Y^{\prime}_{fit}\), the predictive mean \(m_{*}\) is differentiated with respect to \(h_{\text{j}}\) as follows:  
\[m^{\prime}_{*}=K^{\prime}(X,x_{*})^{\top}\Big{[}K(X,X)+\sigma_{N}^{2}I_{N} \Big{]}^{-1}Y, \tag{8}\]  
where \(K^{\prime}(X,x_{*})\) is the derivative of covariance function between \(X\) and \(x_{*}\) with respect to \(h_{\text{j}}\). As mentioned earlier, each element of the covariance matrix is computed using the SE covariance function (Equation (4)). Thus, the derivative of SE between \(X\) and \(x_{*}\) with respect to \(h_{\text{j}}\) is defined in order to find the \(K^{\prime}(X,x_{*})\), as expressed in Equation (9). Its derivation is provided in Appendix B (in Equation (A2)):  
\[k^{\prime}(x,x_{*})=\sigma_{f}^{2}\cdot\bigg{(}\frac{|x-x_{*}|}{\ell^{2}} \bigg{)}\cdot exp\bigg{(}-\frac{|x-x_{*}|^{2}}{2\ell^{2}}\bigg{)}. \tag{9}\]Next, the following step is finding the correction or update of \(h_{\rm j}\). It is denoted as \(h_{\rm j+1}\). By referring to the GD algorithm, the equation of \(h_{\rm j+1}\) is expressed as follows:  
\[h_{\rm j+1}=h_{\rm j}-\alpha\cdot L^{\prime}\big{(}h_{\rm j}\big{)}, \tag{10}\]  
where \(\alpha\) is the learning rate, which governs the step size of the optimization. Normally, as a start, a geometric sequence with a common ratio of 10 is used to indicate which minimizer can give a significant optimization.