## 5 Multi-Source Fusion Bayesian Compressed Sampling (MSF-BCS)
### Determination of the Optimal Location for Fusion Sampling  
The concept of information entropy was first introduced in information theory by [PERSON] [41] to quantify the uncertainty of random processes. A random process with large information entropy is considered to have high uncertainty. The procedures in Sections 4 and 5.1. are combined with information entropy to determine the optimal location for MS-BCS to perform sampling. The uncertainty of the sparse data simulation \(\left(q_{\text{c1N}}\right)_{\text{cs}}\) cross section can be quantified using information entropy. \(H\left(\mathbf{\hat{F}}^{III}\right)\) represents the information entropy of the \(\left(q_{\text{c1N}}\right)_{\text{cs}}\) cross-section obtained by 2D BCS using geotechnical data from previous CPT soundings, and \(H\left(\mathbf{\hat{F}}^{III}_{\text{new}}\right)\) represents the new information entropy of the \(\left(q_{\text{c1N}}\right)_{\text{cs}}\) cross-section obtained by MS-BCS co-sampling with Section 4 process. Determining the optimal sampling location for fusion is the location that maximizes \(\Delta\) between \(H\left(\mathbf{\hat{F}}^{III}\right)\) and \(H\left(\mathbf{\hat{F}}^{III}_{new}\right)\) and can be mathematically viewed as an optimization problem [20]. When MS-BCS is performed along the horizontal direction at a given location \(j\) (\(j=0,\ 1,\ \cdots,\ N_{x_{2}}\)), it can be calculated using the following equation:  
\[\Delta=H\left(\mathbf{\hat{F}}^{III}\right)-H\left(\mathbf{\hat{F}}^{III}_{\text{new}} \right)=\frac{1}{2}\sum\limits_{i=1}^{N_{x_{1}}}\text{ln}\left(1+\tau^{*} \sigma_{\hat{F}^{III}_{\text{cj}}}^{2}\right) \tag{20}\]  
\(\Delta\) considers the optimal locations of MS-BCS. From the equation, \(\Delta\) achieves its maximum when the term \(\sum\limits_{i=1}^{N_{x_{1}}}\text{ln}\left(1+\tau^{*}\sigma_{\hat{F}^{III}_{ \text{cj}}}^{2}\right)\) is maximized. Where \(\tau^{*}\) and \(N_{x_{1}}\) are considered constants for different potential CPT-sounding locations. It is worth noting that the above procedure has already been computed in MS-BCS with the process of Section 4 for co-sampling, and \(\sum\limits_{i=1}^{N_{x_{1}}}\sigma_{\hat{F}^{III}_{\text{cj}}}^{2}\) represents the sum of the variances of \(\mathbf{\hat{F}}^{III}(\left(q_{\text{c1N}}\right)_{\text{cs}})\) interpreted along the depth at a given location \(j\). The optimal location for fusion sampling is the \(\mathbf{\hat{F}}^{III}\) location with the largest sum of variances (with the largest uncertainty), which is consistent with engineering judgments [42]. As MS-BCS proceeds, the uncertainty in the process typically decreases, defining reliability level \(\left(\text{COV}_{\text{M+N}}\right)\) to interpret \(\hat{F}^{III}\). It is worth noting that the coefficient of variation \(\text{COV}_{\text{M+N}}\) reflects the reliability of the interpretation results and is used to compare with the target reliability level \(\text{COV}_{\text{T}}\). \[\text{COV}_{\text{M+N}}=\frac{1}{N_{x_{1}}N_{x_{2}}}\sum\limits_{i=1}^{N_{x_{ 2}}}\text{COV}_{\hat{F}^{III}_{\text{cj}}}=\frac{1}{N_{x_{1}}N_{x_{2}}}\sum \limits_{i=1}^{N_{x_{1}}}\sum\limits_{j=1}^{N_{x_{2}}}\frac{\sigma_{\hat{F}^{III} _{\text{cj}}}}{\mu_{\hat{F}^{III}_{\text{cj}}}}\times 100\% \tag{21}\]  
where N (N \(=0,\ 1,\ \cdots,\ n\)) represents the number of fusion sampling performed. One sampling is performed by MS-BCS to trigger the adaptive process, and \(\Delta\) is combined with the COV\({}_{\rm M+N}\) result to determine the optimal location for sampling until MS-BCS is completed along the horizontal direction. For the real data case in Section 6 to be able to determine the stop sampling in a self-adaptive manner, this study considers setting the target reliability level (COV\({}_{\rm T}\)) as the threshold for stop sampling.