## 2 Research on 3D Temperature Field Reconstruction Algorithms  
### Basic Principles of Acoustic Measurement in 2D Temperature Fields  
The functional relationship between the sound propagation speed (_c_) in a gas medium and the temperature of the gas medium (_T_) is given by the following equation [3]:  
\[c=\sqrt{\frac{\gamma R}{M}T}=Z\sqrt{T} \tag{1}\]  
where \(\gamma\) is the ratio of the specific heat at constant pressure to the specific heat at a constant volume of the gas, \(R\) represents the gas constant, and \(M\) denotes the molar mass of the gas [33]. Moreover, \(Z\) represents a constant determined by the gas composition; \(T\) symbolizes the gas medium temperature; and c signifies the sound propagation speed. Since \(Z\) can be approximated as a constant during calculation, the speed of sound is a single-valued function of the temperature of the medium [34]. This means that acoustic temperature measurement centers on calculating the gas temperature by measuring the sound wave propagation time or frequency, independent of the radiation absorption coefficient. In some cases, radiation processes in high-temperature gases may cause energy dissipation that affects the amplitude of the sound wave (but usually does not significantly affect the speed of sound). Multicomponent gas effects: Changes in absorption coefficients may be related to gas composition and temperature, and changes in gas composition may indirectly affect the speed of sound. However, this effect is generally small, and we have previously demonstrated that the error due to this effect does not exceed 2%. Once the gas composition is determined, the TOF, or the time required for the sound wave to travel through the medium, can be measured by determining the distance between two acoustic transceivers. To reconstruct the 2D cross-sectional temperature distribution, the temperature field is discretized and subsequently divided into \(N\) = n \(\times\) n non-overlapping pixel regions. For the \(i\)-th acoustic temperature measurement path, the following equation can be obtained [4]:  
\[TOF_{i}=\int\frac{1}{c}dl_{i}=\sum_{j=1}^{N}H_{ij}x_{j},i=1,2,\ldots M \tag{2}\]  
where \(TOF_{i}\) represents the total TOF of the acoustic wave in each pixel along the \(i\)-th path, and \(l_{i}\) indicates the total length of the \(i\)-th path [35]. By completing a full acoustic wave transmission and reception measurement process, a system of linear equations is obtained as follows [8]:  
\[\left\{ \begin{array}{l}H_{11}x_{1}+H_{12}x_{1}+\cdots+H_{1N}x_{N}=TOF_{1}\\ H_{21}x_{1}+H_{22}x_{1}+\cdots+H_{2N}x_{N}=TOF_{2}\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ H_{M1}x_{1}+H_{M2}x_{1}+\cdots+H_{MN}x_{N}=TOF_{M}\end{array} \right. \tag{3}\]  
where \(N\) represents the total number of pixels in the reconstruction region, and M indicates the total number of acoustic wave measurement paths passing through the temperature field section. Moreover, \(H_{ij}\) denotes the length of the i-th path passing through the j-th pixel, and \(x_{i}\) symbolizes the reciprocal of the acoustic wave velocity within the i-th pixel [36]. The linear equation system can be represented in matrix form as follows [11]:  
\[H\cdot x=TOF \tag{4}\]  
where \(H\) is an M \(\times\) N matrix with elements \(H_{ij}\), representing the lengths of the acoustic wave paths passing through each pixel in the temperature field; \(x\) denotes an N-dimensional column vector with elements \(x_{1}x_{2}\),..., \(x_{N}\), indicating the reciprocal of the acoustic wave velocity at each pixel; and TOF signifies an M-dimensional column vector with elements \(TOF_{1}\), \(TOF_{2}\),..., \(TOF_{M}\), defining the measured time-of-flight values for each acoustic wave measurement path [37]. The reciprocal of the velocity within each of the N pixels, denoted by vector x, can be obtained by solving the above matrix equation.