## 4 Results and Discussion
When reconstructing complex temperature fields using traditional acoustic thermometry interpolation, some essential features can be preserved, but many details are lost, resulting in a considerable number of errors. The data in Table 3 reveal that the error rates in traditional acoustic reconstruction for the temperature distributions at z = 1 m in the five experimental conditions gradually rise with the increase in the primary air velocity. The temperature field at z = 1 m for experimental condition 2 exhibits the characteristics of a skewed single-peaked symmetric temperature distribution, as observed in Figure 4(a). The acoustic thermometry interpolation method performs well in reconstructing symmetric temperature distributions due to its characteristics. Therefore, for experimental condition 2, the reconstructed temperature field at z = 1 m preserves the main features, being consistent with the modeled temperature field. However, the formation of a four-corner cut circle and the single-peaked symmetric temperature distribution are no longer observed as the primary air velocity continues to rise, leading to an increased number of reconstruction errors. Figure 5: Temperature distribution at z = 1 m in experimental condition 2. (**a**) Numerical temperature field; (**b**) reconstruction of complex temperature field. Reconstruction of a Multi-Layer Temperature Field in Acoustic Thermometry Based on Tucker Decomposition  
As shown in the flow chart of the Tucker decomposition reconstruction algorithm, displayed in Figure 1., the first step involves performing Tucker decomposition on the prior dataset _TR_ to obtain A, followed by unfolding A along the 1-mode direction to obtain \(A_{1 unfold}\). Considering experimental condition 2 and the arrangement of the acoustic thermometry measurement points shown in Figure 4a, the acoustic thermometry path position information \(M\) and the TOF vector _t\({}_{M}\)_ for 24 paths are obtained. Using Equation (10), _f\({}_{1x}\)_ is calculated; then, Equation (8) is applied to obtain the 3D TOF data to be reconstructed for the four cross-sections in experimental condition 2. Figure 6. presents the errors in the reconstructed TOF data compared to the simulated TOF data derived from the model. The blue clusters represent the reconstruction errors comprising 4 \(\times\) 1944 TOF values, while the orange scattered points indicate the average reconstruction errors in the TOF data at the four height planes. As can be seen in the graph, the maximum reconstruction error rate in the TOF is approximately 5%, while the average reconstruction error rates in the TOF data at the four height planes range between 0.82% and 1.27%. In particular, the average reconstruction error rate at the z = 1 m height plane is significantly lower than those at the other three height planes. The main reason for this is that the acoustic thermometry path position information M is located at the z = 1 m height plane, and this represents the richest feature information. The Tucker decomposition reconstruction algorithm performs well in reconstructing the TOF data for experimental condition 2. Using Equations (4) and (5) and the algebraic reconstruction technique (ART), it is possible to reconstruct the TOF data for the four planes. This, in turn, provides the 2D reconstructed temperature distribution for experimental condition 2 at different heights (1 m, 1.15 m, 1.3 m, and 1.45 m). To provide a visual representation, Figures 7 and 8 display the model temperature distribution and the corresponding reconstructed temperature distribution at these heights, respectively. The root means square error (RMSE) for the temperature reconstruction in experimental condition 2 is 6.4035%, and the reconstruction time is 4.38 s. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Experimental Condition** & **1** & **2** & **3** & **4** & **5** \\ \hline Error \% & 18.61 & 16.11 & 28.47 & 31.57 & 31.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reconstruction errors for the five operating conditions.