## 2 Basic Principles and Methodologies  
### Basic Principle of Acoustic Pyrometry  
Acoustic pyrometry (AP) is a noninvasive temperature measurement technique. The region of interest (ROI) is first discretized by meshing, and the temperature and sound velocity within each grid are treated as unity values. Acoustic transceivers are then placed at the edge of the ROI, and the time of flight (TOF) of multiple acoustic propagation paths in the ROI is measured. Based on the dependencies of sound velocity and temperature, the grid-averaged temperature is solved by a reconstruction algorithm. Theoretically, the sound velocity depends on the gas temperature [3]:  
\[c=\sqrt{\frac{\gamma R}{M}T}=Z\sqrt{T} \tag{1}\]  
where \(c\) is the sonic velocity (m/s), \(\gamma\) is the adiabatic gas constant, \(R\) is the molar gas constant with a value of 8.314 J/(mol\(\cdot\)K), \(M\) is the average molecular weight of gas (g/mol), \(T\) is the absolute temperature (K), and \(Z\) is a coefficient determined by \(\gamma\), \(M\) and \(R\). The acoustic time of flight (TOF) can be obtained via integration along the acoustic path \(l\)[3]:  
\[t=\int\frac{1}{c}dl=\int fdl \tag{2}\]  
where \(f\) is the reciprocal of sound velocity at specific ROI point (s/m), which is termed \"slowness\", and \(t\) is the TOF. The average temperature of the acoustic path can be obtained by calculating the reciprocal of sound velocity via the TOF. Similarly, the average temperature of the grid can be obtained by solving the linear equations subsequent to the grid division. \[ \begin{bmatrix}l_{11}&l_{12}&\cdots&l_{1N}\\ l_{21}&l_{22}&\cdots&l_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ l_{M1}&l_{M2}&\cdots&l_{MN}\end{bmatrix}\begin{bmatrix}f_{1}\\ f_{2}\\ \vdots\\ f_{N}\end{bmatrix} = \begin{bmatrix}t_{1}\\ t_{2}\\ \vdots\\ t_{M}\end{bmatrix} \Rightarrow\mathbf{A}\mathbf{x}=\mathbf{b} \tag{3}\]  
where \(l_{\bar{y}}\) is the length of the \(i\)-th acoustic path through the \(j\)-th grid; \(f_{j}\) is the reciprocal of sound velocity in the \(j\)-th grid of the ROI; \(t_{i}\) is the TOF of the \(i\)-th acoustic path; \(M\) and \(N\) are the number of acoustic paths and the number of ROI grids, respectively; \(\mathbf{A}\) is the coefficient matrix; \(\mathbf{b}\) is the TOF matrix; and the unknown \(\mathbf{x}\) is the reciprocal of sound velocity matrix. Based on the dependence of the acoustic velocity on temperature, the temperature distribution can be reconstructed by measuring the TOF between pairs of acoustic transducers in the ROI. In fact, the number of grids that each acoustic path passes through is less than N. Therefore, the coefficient matrix \(\mathbf{A}\) in Equation (3) contains many null-space components, resulting in the ill-conditioned nature of the problem, which is difficult to solve accurately via general methods [40]. When acoustic waves propagate through a nonuniform temperature field, they bend because of the change in propagation velocity [21]. In early studies, researchers simplified the acoustic paths into a linear model of the connection between acoustic transducers, resulting in a deviation from the true value and a maximum reconstruction error of 20% based on the linear propagation model [41; 42]. According to [PERSON]'s principle, the differential equation satisfied by acoustic wave propagation in a two-dimensional temperature field can be derived from the Euler formula [21; 34]:  
\[y^{\prime\prime}=\frac{1+y^{\prime 2}}{2T(x,y)}\left(y^{\prime\prime}\frac{ \partial T(x,y)}{\partial x}-\frac{\partial T(x,y)}{\partial y}\right),\ \left\{ \begin{array}{l}y_{\text{SA}}=y(x)|_{x=y_{ \text{SA}}}\\ y_{\text{EB}}=y(x)|_{x=x_{\text{EB}}}\end{array} \right. \tag{4}\]  
where \((x,y)\) represents the 2D coordinates of discrete points; \(y^{\prime}\) and \(y^{\prime\prime}\) represent the first-order and second-order derivatives of \(y\), respectively; SA and EB are the two acoustic transducers. The target acoustic path curve we need is the acoustic path starting at the transceiver SA and ending at EB.