# Projection Pursuit Multivariate Sampling of Parameter Uncertainty
## 3 Case Studies
### Synthetic Five-Variate Case  
To further investigate the efficiency of each sampling technique, we present another case study where the sampling space is considered to be five-dimensional. We use the petroleum reservoir's OOIP as a variable to be sampled. The formula for calculating the OOIP is given as follows:  
\[\text{OOIP}=\text{CAT}\cdot\text{NTG}\cdot\phi_{net}(1-\text{S}_{w}), \tag{6}\]  
where C is the constant that accounts for units and is considered to be one; A represents the area of the deposit; T is the thickness of the deposit; NTG is the net oil to gross volume; \(\phi_{net}\) is the net porosity and S\({}_{w}\) is the water saturation. As given in Equation (6), the value of OOIP is calculated as a function of five variables. To assess the quality of the  
Figure 6: Four sets of samples with \(n=20\) realizations generated by each sampling technique in a two-dimensional sampling space where the random variables are independent. sampling designs generated by each technique, the underlying (or truth) distribution of each OOIP variable is simulated using 10 million realizations generated by MCS. The selected distributions and their parameters for each variable are contained in Table 1.  
To make the results comparable to the ones shown in the study presented by [PERSON] and [PERSON] [1], we use the same distribution types and their parameters given in that paper. In the first part of this case study, we consider that all of the OOIP variables are \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Distribution** & **Parameters** \\ \hline A & Triangular & \(a=2\), \(b=4\), \(c=6\) \\ T & Gaussian & \(m=10\), \(\sigma=1\) \\ NTG & Uniform & \(a=0.6\), \(b=0.8\) \\ \(\phi_{ntd}\) & Triangular & \(a=0.15\), \(b=0.25\), \(c=0.35\) \\ S\({}_{w}\) & Triangular & \(a=0.15\), \(b=0.2\), \(c=0.3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parametric distributions and their parameters used for each OOIP variable. Figure 7: Four sets of samples with \(n=20\) realizations generated by each sampling technique in a two-dimensional sampling space where the random variables are positively correlated. uncorrelated. Given the nature of sampling, it is clear that if one takes a larger sample of realizations, the accuracy of the parameter estimation will increase. Therefore, we consider several scenarios where the sample sizes \((n)\) are relatively small \((10\leq n\leq 100)\), medium \((100<n\leq 1000)\), and large (1000 < n \(\leq\) 10,000). For each case, 100 sets of the predetermined sample size are generated using MCS, CLHS, maximin LHS, LHSMDU, and PPMT, that is, each of the OOIP variables is sampled from their underlying CDFs and the empirical OOIP CDF is then constructed for every sample size and every sampling technique. The efficiency of the sampling designs generated by each technique is assessed based on a criterion that is similar to the Kolmogorov-Smirnov \(D\) statistics [35], which is given as follows:  
\[e=\max\lvert F_{ref}^{-1}(p)-F_{emp}^{-1}(p)\rvert, \tag{7}\]  
where \(e\) is the error value indicating the maximum discrepancy between the empirical CDF generated by the particular sampling technique and the underlying CDF; \(F_{ref}^{-1}(p)\); \(p=0.1,0.2,\ldots,0.9\) are the quantile values read from the underlying CDF; and \(F_{emp}^{-1}(p)\); \(p=0.1,0.2,\ldots,0.9\) are the quantile values read from the empirical CDF. Figure 8 shows the underlying OOIP CDF and the randomly selected empirical CDFs of the OOIP generated by each sampling technique. It can be seen in Figure 8 that only the empirical CDFs of the OOIP generated based on the small sample sizes are shown; this is because the greatest discrepancy between the sampling techniques should be observed in cases where the sample size is small.