## 4 Results and Discussion
Figure 6: Average reconstruction error rates regarding TOF data for experimental condition 2. Figure 8: Multi-layer reconstruction of temperature field based on Tucker decomposition acoustic thermometry for experimental condition 2. Figure 7: Modeled temperature field for experimental condition 2. Comparing the reconstructed temperature field at z = 1 m, obtained via Tucker decomposition and shown in Figure 8, with the reconstructed temperature field obtained via acoustic thermometry and shown in Figure 5b, it can be seen that Tucker decomposition-based reconstruction yields more accurate and clearly differentiated results. In addition, it can sufficiently reflect the main temperature region and capture the fine details around the edges, thereby reducing the number of errors introduced compared to acoustic thermometry interpolation and minimizing edge errors. Compared to traditional acoustic thermometry interpolation, Tucker decomposition improves the accuracy and quality of the temperature reconstruction. As evidenced by Figures 7 and 8, the measurement points are evenly distributed at a height of z = 1 m due to the arrangement of the measurement points, reflecting that used in actual power station boilers for acoustic thermometry. Consequently, the temperature field reconstruction at z = 1 m yields the best results. However, some information loss occurs during temperature reconstruction on the other planes, leading to errors in certain details, although the main temperature characteristics are retained. Although placing the measurement points on a single plane introduces some errors, the proposed reconstruction algorithm maintains a high level of accuracy and reliability. Compared to traditional acoustic thermometry, the reconstruction algorithm significantly improves the reconstruction accuracy for a single plane and overcomes the limitations of traditional acoustic thermometry, which is restricted to the measurement of a single plane. It is also worth noting that more temperature information is acquired when generalizing the measurement data from a single plane to other planes. Figure 9 illustrates the reconstruction errors and time consumption for the five distinct experimental conditions. Notably, the reconstruction time for all five conditions is approximately 4 s, and the choice of conditions has a minimal impact on the reconstruction time. Regarding the reconstruction accuracy, for these experimental conditions, within the range of boundary conditions present in the prior dataset, the reconstruction error rate ranges from 4% to 9%. However, it is approximately 8.4% for conditions beyond the range of the prior dataset. Figure 9: Time consumption and error rate during the reconstruction of the experimental working conditions. In the case of the boiler model examined in this study, the design condition corresponds to a first air velocity of 27 m/s. Deviations in the first air velocity from the design condition lead to a more complex temperature distribution. Experimental conditions 3, 4, and 5 exhibit notable deviations from the design condition in terms of the first air velocity; specifically, conditions 4 and 5 exhibit boundary conditions that exceed the range of the prior dataset. Consequently, the reconstruction error rates for conditions 4 and 5 increase slightly, but they remain within a relatively small range, at approximately 8.4%.