# Optimization Method for Improving Efficiency of Thermal Field Reconstruction in Concrete Dam
## 3 Optimization Method for Thermal Field Reconstruction
### Interpolation Algorithm Evaluation and Selection  
#### 3.1.1 Interpolation Algorithm Comparison  
Numerous interpolation algorithms have achieved successful applications in the reconstruction of concrete thermal fields. Among these, Kriging [33], Natural Neighbor [34], and Inverse Distance Weighting [35] are three widely used algorithms. Natural Neighbor and Inverse Distance Weighting are convex methods. The estimated value of the convex method is always between the minimum and maximum of the measured values. The estimated value of the non-convex method (e.g., Kriging) can be outside the range of the observed values. The convexity of the estimators may depend on factors such as weights assigned for samples and the difference between the ranges of predictors for the samples and the unobserved locations. The estimated surface of the Natural Neighbor is continuous and smooth except at the data points where the derivative is discontinuous. Inverse Distance Weighting works well with well-spaced data but not spatial clustering samples. When employing the DTS system for monitoring the temperature in concrete dams, it is essential to evaluate the performance of various algorithms in conjunction with the specific optical fiber layout schemes. The interpolation principle and calculation procedure of each one is shown in Figure 4.  
Figure 4: _Cont_. 1. Kriging  
Based on the regionalized variable theory, the Kriging interpolation algorithm takes the variation function as the main tool and obtains the estimated value under the premise of ensuring that the estimated value meets the unbiased condition and the minimum variance condition [43]. If the measured value at the monitoring point position \(p_{i}\) is \(f(p_{i})\), then the estimated value at the point \(y\) to be interpolated is obtained by the following equation:  
\[f(y)=\sum_{i=1}^{m}\phi_{i}f(p_{i}). \tag{4}\]  
where \(\phi_{i}\) is the weight of position \(p_{i}\), which considers the distances between the points to be interpolated and the monitoring points, as well as their spatial distribution. 2. Natural Neighbor  
Based on the Delaunay triangulation and Voronoi diagram, the Natural Neighbor interpolation algorithm calculates the interpolation result of the point \(y\) according to the contribution rate of each Natural Neighbor point [44]. Assuming the Natural Neighbor points of point \(y\) count \(m\), namely, \(p_{1},p_{2},\cdots,p_{m}\), the interpolation algorithm is as follows:  
\[f(y)=\sum_{i=1}^{m}w_{i}f(p_{i}). \tag{5}\]  
where \(f(p_{i})\) is the value at the Natural Neighbor points \(p_{i}\), and \(w_{i}\) is the weight coefficient occupied by the Natural Neighbor point \(p_{i}\) and is determined according to the following principle. The point \(y\) to be interpolated is taken as a new node, added to the generated Voronoi diagram, and a new Voronoi diagram corresponding to the new node is obtained. If the total area enclosed by the Voronoi cell with the point \(y\) in is \(s\), the Voronoi cell with the new node in it can be divided into \(m\) parts by the original Voronoi cell of its Natural Neighbor points. The areas of each part are denoted as \(s_{1},s_{2},\cdots,s_{m}\). The weight coefficient for each part is obtained by dividing the area of each part by the total area. \[w_{i}=\frac{s_{i}}{s}. \tag{6}\]  
Figure 4: Interpolation principle and calculation procedures of Kriging, Natural Neighbor, and Inverse Distance Weighting. The interpolation of each region is not affected by the other regions, and the temperature is interpolated in different regions to complete the reconstruction of the concrete thermal field. 3. Inverse Distance Weighting  
For the Inverse Distance Weighting interpolation algorithm, the distance between the point to be interpolated and the monitoring points is weighted, setting as \(\beta_{i}\). The closer a monitoring point is to the point to be interpolated, the greater the weight given to this monitoring point will be [35]. The weight contribution is inversely proportional to the distance between the two points.