## 3 Results  
### Gaussian Process Modeling and Sampling Algorithm Verification  
We present a test problem to illustrate and verify our Gaussian process modeling with the variability-based sampling and bisection algorithm. Equation (7) is the mathematical form of the given mean function \(q\) and is expressed as follows:  
\[q(x)=\sin\left(2(x-0.9)^{4}\right)\cos\left(x-0.2\right)+\frac{1}{2}(x-0.5)^{3}, \tag{7}\]  
The linear slope between the starting and ending points of the function is removed since locally varying spatial variation is the main focus. We decide to use Equation (7) that has a higher spatial variation frequency in the region \(x\in[-1,\ 0]\) when compared to that in region \([0,\ 1]\) to test whether our algorithm captures the correct behavior of spatial roughness variation when its characteristics are not uniform over the entire domain. A Matern covariance function with the parameters: \(\
u=2\), \(\phi\) = 2, \(\theta\) = 0.4 is used to generate the spatial variation. We generate realizations of the random process and sample these realizations at a total of 61 measurement locations. The sampled data is used as the input for estimation, wherein we try to reconstruct the parameters of the actual stochastic process from which the data originated. We assign the following prior PDFs for the unknown parameters:  
\[ \begin{array}{l}\alpha_{i}\sim Uniform\ [-10,\ 10],\\ \
u\sim Uniform\ [1,\ 3],\\ \phi\sim Exponential\ (1),\\ \theta\sim Exponential\ (1),\\ \end{array}  \tag{8}\]  
The prior PDFs can be specifically selected to incorporate any knowledge regarding the values of the unknown parameters. Otherwise, they can be set as uniform distribution in the absence of the aforementioned information. We use 500 realizations of the given random process that are sampled at the given locations to generate the joint posterior distribution of the parameters and select 1000 samples from it using the Markov chain Monte Carlo method with a burn-in of 30,000 and a thinning factor of 10 [17]. Figure 2a shows 10 realizations from the given stochastic process. The sampled points in the 1 st (circles), 2 nd (triangles), and 3 rd (rectangles) runs of non-uniform sampling are also shown in Figure 2b. Fourteen points from the 1 st sampling based on the local variability effectively capture the high frequency spatial variation near the left sidewall. However, they miss a few details near the left sidewall as well as near the right sidewall. The lack of sampling points is compensated in the 2 nd and 3 rd bisection steps by placing additional sampling points and reducing the deviation between the mean function calculated from all realizations and the estimated mean function. Total sampled points in the 2 nd and 3 rd sampling from the bisection method are 20 and 31, respectively. To visualize the effectiveness of variability-based sampling and the bisection algorithm, the root-mean-square error (RMSE) between the estimated mean and the mean from realizations is calculated and compared in two different sampling cases: proposed variability-based sampling and uniform sampling. The result is shown in Figure 2c. The x-axis denotes the number of sampling points and the y-axis denotes the RMSE value. Given the efficient placement of measurement locations, the RMSE in non-uniform sampling is considerably lower than that in the uniform sampling result with an equal number of sampling points. Additionally, while the RMSE is reduced with increases in the sampling locations, the RMSE decreases significantly faster in the variability-based method. The RMSEs in both methods are identical when the sampling locations correspond to almost half of the total measurement locations. Therefore, we conclude that variability-based sampling and the bisection algorithm is a more attractive option for the effective allocation of computing power and resources. Figure 2d shows 10 sampled realizations from the estimated Gaussian process model.