## 1 Introduction  
In the context of carbon reduction, the increasing share of renewable energy sources [1] has led to the transformation of coal-fired power plants, shifting their role from providing base-load power to offering system regulation capabilities and ensuring security. Consequently, these changes have led to higher demands regarding the flexibility and deep load adjustment capabilities of coal-fired power stations [2]. To achieve optimal combustion in power plant boilers and to ensure stable combustion during deep load adjustments, accurate and reliable flame temperature distribution measurement is crucial [3]. Currently, temperature measurement in thermal power plants relies on both contact and non-contact methods, with acoustic temperature measurement representing one of the major non-contact techniques. This method exhibits strong resistance to vibration and ash interference in complex furnace environments, coupled with distinct advantages such as long-term stability and cost-effectiveness [4]. Acoustic temperature measurement is a temperature sensing method that uses the sound wave propagation time to determine the temperature distribution [5, 6, 7]. Extensive research has been conducted on various related aspects, such as inverse matrix algorithms. For example, [PERSON], [PERSON] et al. proposed an inverse matrix algorithm based on singular value decomposition (SVD), which can effectively handle high-dimensional data but performs poorly in noisy environments [8]. [PERSON], [PERSON] et al. further improved the regularization algorithm, which improves the stability of the algorithm [9]. In addition, [PERSON] conducted measurements of sound wave propagation times to reconstruct both temperature and velocity fields in the furnace using the algebraic reconstruction technique (ART) [10]. Finally, the SIRT algorithm developed by [PERSON], [PERSON] et al. achieves a better balance between accuracy and speed but with higher computational complexity [11]. In terms of optimizing the mesh, [PERSON] et al. proposed an adaptive mesh method, which significantly improves the reconstruction efficiency [12]. For the acoustic line bending problem, [PERSON], [PERSON] et al. proposed an improved algorithm based on ray tracing, which can better handle nonlinear acoustic wave propagation [13]. Finally, for coupled temperature field measurements, [PERSON] et al. proposed a multi-physics field coupling method, which is capable of reconstructing temperature and flow velocity fields simultaneously [14]. However, despite these advancements, several critical limitations remain in the current state of knowledge: The acoustic measurement of three-dimensional (3D) temperature fields in power plant boilers is restricted by the limited number of measurement locations [15], resulting in a predominant focus on two-dimensional (2D) temperature fields [16]. While acoustic thermometry provides accurate results for single-peaked symmetric or skewed temperature fields [17], it struggles to accurately reconstruct complex temperature fields that arise during deep load adjustments in tangentially fired furnaces [18; 19]. In such cases, the temperature distribution deviates from simple single-peaked patterns, leading to significant reconstruction errors due to the need for interpolation and extrapolation [20]. Existing methods, such as tensor train (TT) [21] and hierarchical Tucker (HT) decomposition [22], can efficiently handle higher-order tensors but are often impractical for industrial applications due to their high computational requirements and structural complexity [23]. To address these limitations, this study proposes a novel multi-layer temperature field reconstruction algorithm that integrates Tucker decomposition with acoustic temperature measurement techniques. Tucker decomposition is an ideal choice due to its balance between handling complex data structures [24] and maintaining data interpretability [25].