## 2 Basic Principles and Methodologies
### The Coefficient of Heating Effect
#### 2.3.1 RBF Approximation and the Adaptive Hybrid Kernel Method  
The solvability of linear equations requires that the number of grids must not exceed the number of acoustic propagation paths [40] when the general domain discretization assumption is used to solve reconstruction problems, thus restricting the reconstruction performance of the algorithm. The RBF-based reconstruction algorithms do not have this strict constraint [34; 44], allowing the number of ROI grids to be selected according to actual needs through RBF approximation. The ROI is divided into N grids. Directly determining the distribution of the reciprocal of sound velocity f is difficult due to the complexity of its functional expression. However, f can be expanded into a linear combination of kernel functions through RBF interpolation [44; 45]:  
\[f(x,y)=\sum_{i=1}^{N}\lambda_{i}\phi_{i} \tag{6}\]  
where \(\lambda_{i}\) represents the respective coefficient when the nonlinear function forms a linear combination \(\phi_{i}\) represents a nonlinear function with an independent variable of Euclidean distance r. The expression of \(\phi(r)\) in two dimensions [45] is as follows:  
\[\phi(r)=\phi(x,y)=\phi\bigg{(}\sqrt{\left(x_{i}-x\right)^{2}+\left(y_{i}-y \right)^{2}}\bigg{)} \tag{7}\]  
where \(\left(x_{i},y_{i}\right)\) represents the coordinates of the known points, which are the centers of the N grids, \(\left(x,y\right)\) represents the coordinates of the target point to be expanded, and \(r\) is the Euclidean distance between the known point and the target point. In the context of acoustic pyrometry, the RBF approximation is equivalent to the process of interpolating the unknown acoustic velocity field prior to reconstruction. However, this preliminary interpolation may not entirely align with the inherent field characteristics, resulting in deviations of the reconstructed values from the true values. Consequently, the type and shape parameter of the RBF are critical for the accuracy of the reconstruction [16]. The commonly used Gaussian (G) and Multiquadric (MQ) radial basis functions were selected for acoustic reconstruction [45; 46]:  
\[ \begin{array}{ll}Gaussian:&\phi_{G}(r)=\exp(-\varepsilon^{2}r^{2})\\ Multiquadric:&\phi_{MQ}(r)=\sqrt{1+\varepsilon^{2}r^{2}}\end{array}  \tag{8}\]  
where \(\varepsilon\) is the shape parameter. For uniformity and convenience, \(\varepsilon\) is defined as the coefficient of the Euclidean distance \(r\), as shown in Equation (8). The coefficient of linear combination can be calculated according to the known points:  
\[ \begin{bmatrix}\phi_{1}(x_{1},y_{1})&\phi_{2}(x_{1},y_{1})&\cdots&\phi_{N}(x_{1},y_{1})\\ \phi_{1}(x_{2},y_{2})&\phi_{2}(x_{2},y_{2})&\cdots&\phi_{N}(x_{2},y_{2})\\ \vdots&\vdots&\ddots&\vdots\\ \phi_{1}(x_{N},y_{N})&\phi_{2}(x_{N},y_{N})&\cdots&\phi_{N}(x_{N},y_{N})\\ \end{bmatrix}\begin{bmatrix}\lambda_{1}\\ \lambda_{2}\\ \vdots\\ \lambda_{N}\\ \end{bmatrix} = \begin{bmatrix}f_{1}\\ f_{2}\\ \vdots\\ f_{N}\\ \end{bmatrix} \Rightarrow\mathbf{\phi}\mathbf{\lambda}=\mathbf{x} \tag{9}\]  
Substitution of Equation (9) back into Equation (3) yields the following result:  
\[\mathbf{Ax}=\mathbf{b}\Rightarrow\mathbf{A\phi}\mathbf{\lambda}=\mathbf{b}\Rightarrow\mathbf{B}\mathbf{ \lambda}=\mathbf{b} \tag{10}\]  
The problem of finding \(\mathbf{x}\) in Equation (3) is transformed to finding \(\mathbf{\lambda}\) in Equation (10). The matrix \(\mathbf{B}\) containing RBF information replaces the original coefficient matrix \(\mathbf{A}\), recovering the null-space component and reducing the ill-conditioned nature of the problem. Specifically, \(\mathbf{\lambda}\) in Equation (10) is solved by the reconstruction algorithm; then, the reciprocal of sound velocity matrix \(\mathbf{x}\) is obtained from Equation (9), and the average temperatures of the ROI grids are reconstructed through Equation (1). Unfortunately, under the strong temperature gradients, the existing RBF-based reconstruction algorithms encounter significant challenges in achieving precise measurement outcomes.