## 2 Basic Principles and Methodologies
### The Coefficient of Heating Effect  
The domain discretization assumption, combined with the limited number of acoustic transceivers, results in sparse valid data and an ill-posed AP problem. Moreover, as the complexity or temperature gradient of the temperature phantom rises, the ill-conditioned nature of the reconstruction problem becomes more severe. Since most of the existing acoustic temperature distribution reconstruction algorithms are developed and modified with relatively uniform temperature distribution environments [11; 25; 26; 27], a quantitative evaluation criterion is needed to assess the reconstruction difficulty of the temperature phantoms. Consequently, considering the gradient and complexity of the temperature phantom, a dimensionless parameter \(C_{\text{HE}}\) was proposed to assess the reconstruction difficulty as follows:  
\[T_{\text{mid}}=(T_{\text{max}}-T_{\text{min}})/2,\ P_{\text{TD}}=\frac{T_{ \text{max}}-T_{\text{min}}}{T_{\text{max}}+T_{\text{min}}}\cdot\frac{S_{\text{HT }}}{S_{\text{ROI}}},\ C_{\text{HE}}=n_{\text{HS}}\cdot P_{\text{TD}} \tag{5}\]where \(T_{\rm max}\) and \(T_{\rm min}\) are the maximum and minimum temperatures within the ROI, respectively. \(T_{\rm mid}\) is the middle temperature of the temperature phantom, and regions above this temperature are considered to be high-temperature regions; \(S_{\rm HT}\) and \(S_{\rm ROI}\) represent the area of the high-temperature regions and the overall area of the ROI, respectively. \(P_{\rm TD}\) (Proportion of Temperature Difference) is a dimensionless proportion of the temperature change within the heating zone, representing the relative temperature gradient within the ROI and the extent of the high temperature regions. \(n_{\rm HS}\) is the number of heat sources within the ROI, representing the complexity of the temperature phantom. Thus \(C_{\rm HE}\) (Coefficient of heating effect) is a dimensionless parameter that simultaneously takes into account the number of heat sources, the temperature gradient and the heating range, -representing the comprehensive complexity as well as the reconstructing difficulty of the temperature phantom within the ROI range. It is well-established that the reconstruction accuracy of acoustic tomography is influenced by multiple factors [16], including reconstruction algorithms, parameter selection, domain discretization, and the intrinsic characteristics of the reconstructed temperature field (\(C_{\rm HE}\)). To exclusively investigate the influence of \(C_{\rm HE}\) on reconstruction performance, a systematic review of 15 articles [4; 6; 11; 14; 17; 20; 28; 32; 33; 34; 35; 36; 37; 38; 39] published since 2015, denoted in citation order as Ref 1 to Ref 15, was conducted. In these studies, after finalizing grid configurations, researchers applied proprietary enhancement methodologies to reconstruct multiple temperature phantoms. Based on the phantom designs and region-of-interest (ROI) selections reported in these studies, the \(C_{\rm HE}\) values for all temperature fields were calculated. Subsequently, the root mean square error (RMSE) of the reconstruction results in each study was correlated with its corresponding \(C_{\rm HE}\) value, as illustrated in Figure 1.  
As shown in Figure 1, the RMSE exhibits a significant positive correlation with the \(C_{\rm HE}\) of the temperature phantom in all 15 papers [4; 6; 11; 14; 17; 20; 28; 32; 33; 34; 35; 36; 37; 38; 39], indicating that the intrinsic characteristics of the reconstructed temperature field (\(C_{\rm HE}\)) have a direct effect on reconstruction fidelity. Furthermore, the temperature phantoms used in all 15 papers exhibit \(C_{\rm HE}\) values below 5 and demonstrate adequate reconstruction accuracy, with RMSE values below 8.5%.