## 1 Introduction  
The temperature distribution in a furnace has a direct effect on fuel combustion efficiency and equipment safety during the operation of industrial combustion devices, reflecting the emission evolution and combustion process [1, 2]. The analysis of the temperature field in furnaces helps to improve combustion efficiency, reduce pollutant emissions, and prevent security incidents [3, 4, 5]. 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. AP has a wide measurement range, quick dynamic response and good environmental adaptability. In addition, acoustic transducers are smaller and easier to install than other noncontact sensors. Therefore, AP is widely used to obtain the temperature distribution in boiler furnaces [1; 3; 6; 7] and gases [8; 9; 10]. Currently, researchers are focusing their investigations on improving the reconstruction algorithms for acoustic pyrometry, including the iterative method [1; 11; 12; 13], the truncated singular value decomposition (TSVD) [14; 15; 16; 17], and the least square method (LSM) [18; 19; 20; 21]. Many researchers have devoted their efforts to improving the reconstruction performance of AP based on different reconstruction algorithms, yielding remarkable outcomes. The main objective of a reconstruction algorithm is to mitigate the difference between the theoretical data and actual data and to use this information to correct the value of the average temperature value inside each grid. These outcomes have been applied for industrial temperature monitoring. [PERSON] [3] proposed and applied two classes of acoustic reconstruction algorithms to both simulated and experimental data measured in power plants of the Italian National Electricity Board. [PERSON] [22] proposed a temperature monitoring method based on acoustic pyrometry for the early prediction of temperature anomalies in stored grain. By improving the iterative method and the time delay estimation method, [PERSON] [1] used acoustic computed tomography technology to obtain the temperature profile of a furnace cross-section in a domestic 600-MW coal-fired boiler. In addition to temperature measurements, the achievements in the field of acoustic pyrometry are also used in other related industrial applications. [PERSON] [7; 23] adopted acoustic pyrometry to monitor the changes of the flue gas temperature near the heat surface and thus proposed a new clean factor to monitor the ash fouling in a domestic 330-MW unit manufactured by the Babcock & Wilcox Company. [PERSON] [24] applied acoustic tomography combined with radial basis function artificial neural network for coordinate estimation to locate the leakage from a water-cooling wall tube. However, most of the existing acoustic temperature distribution reconstruction algorithms are developed and modified with relatively uniform temperature distribution environments [11; 25; 26; 27]. Unfortunately, under strong temperature gradients, these reconstruction algorithms are almost incapable of performing accurate measurements [28; 29]. This phenomenon can be attributed to two primary factors. First, the domain discretization assumption eliminates the details of continuously varying temperatures within the ROI, only resulting in limited grid-averaged values. Furthermore, the domain discretization causes remarkable discretization errors in high gradient regions. Second, the sparse data from limited transducer placements results in an ill-posed AP problem, exacerbating the difficulty of reconstructing temperature fields with high gradient variations and increasing the grid reconstruction errors.