## 3 Results and Discussion  
### Initial Settings  
According to previous studies, the least square QR decomposition (LSQR) algorithm has universal and comprehensive reconstruction performance [18; 19; 20; 21], and requires far less time for parameter selection compared to Tikhonov regularization iterative methods and the truncated singular value decomposition regularization (TSVD) algorithm [1; 16]. Consequently, the LSQR algorithm serves as the base algorithm for reconstruction in this study. Since LSQR is a generalized algorithm, a detailed description is not provided here. A classical layout scheme [14] was used in the research on two-dimensional acoustic reconstruction, as shown in the descriptive diagram in Figure 6. The ROI is a 10-m square that is equally divided into \(N=n\times n\) grids (small blue squares). The ROI predefines the location of twelve acoustic transducers (red pentagrams), which are evenly spaced along the four boundaries of the ROI, to ensure that the acoustic paths between the acoustic transceivers cover the entire ROI in a more complete way. Considering the engineering limitations, the number of installed acoustic transducers is restricted. Twelve transceivers and a greater number of grids were chosen for better data density, uniformity, accuracy, resolution, and field-wide distribution. The preset two-dimensional temperature phantom in the ROI is denoted as \(T(x,y)\). After the reconstruction, the results were compared with the preset true values and analyzed. The acoustic refraction effect was taken into account and the actual acoustic path curves were calculated for the construction of the coefficient matrix \(A\) in Equation (3). The simulation data used in this study were generated based on predefined temperature phantom models rather than real experimental measurements. And the simulation data were generated as follows:  
Initially, the preset temperature field and the ROI configurations are known. The acoustic path curve within the ROI is calculated from Equation (4), resulting in the acquisition of the coefficient matrix \(A\). The TOF matrix \(b\) can then be derived through Equation (2). Subsequently, the ill-conditioned linear equation \(Ax=b\) in Equation (3) is solved, thereby reconstructing the average grid temperature within the ROI. To investigate the effect of the intrinsic characteristics of the reconstructed temperature field (\(C_{\text{HE}}\)) on reconstruction fidelity, three temperature phantoms of varying complexity were selected for reconstruction, namely single-peak symmetric _Tfield_1, double-peak symmetric _Tfield_2 and triple-peak asymmetric _Tfield_3. The selected temperature phantoms are the results of placing several ideal inverse-quadratic heat sources at different locations within a two-dimensional space, as described in:  
Figure 6: 2D ROI configuration. \[ \begin{split} Tfield1&=\frac{1000}{\text{coeff}((x-5)^{+}+(y-5)^{ +})+1}\\ Tfield2&=\frac{1000}{\text{coeff}((x-2)^{+}+(y-2.5)^{ +})+1}+\frac{1000}{\text{coeff}((x-7)^{+}+(y-7.5)^{2})+1}\\ Tfield3&=\frac{1000}{\text{coeff}((x-2)^{+}+(y-8)^{ +})+1}+\frac{1000}{\text{coeff}((x-9)^{+}+(y-6)^{+})+1}+\frac{800}{\text{coeff} ((x-9)^{+}+(y-3)^{+})+1}+\frac{800}{\text{coeff}((x-4)^{+}(y-3)^{+})+1}\end{split}  \tag{12}\]  
where \(\text{coeff}_{\text{HE}}\) is a subfactor of the coefficient of the heating effect in the temperature fields, taken as 0.1, 0.2, 0.5 and 1. Altering the \(\text{coeff}_{\text{HE}}\) can easily modify both the heat source influence range and the gradient of the temperature phantom. A \(\text{coef}_{\text{HE}}\) of 0.05 is commonly used by many researchers to conduct AT simulations [14; 34; 46; 48]. Within the ROI, the temperature contour maps of the three temperature phantoms under four \(\text{coeff}_{\text{HE}}\) values are shown in Figure 7, and their corresponding \(\text{C}_{\text{HE}}\) values are presented in Table 1.  
Figure 7: The temperature contours of the three temperature phantoms under four \(\text{coeff}_{\text{HE}}\) values.