## 3 The Left Rank of the Arc Sparse Array under the Condition of Weld Scattering  
### The Rank of the Receiving Matrix of the Arc Sparse Array under the Condition of Weld Scattering  
In the multi-frame imaging of the ultrasonic guided wave receiving aperture in a single plate without a weld structure, the signals received by the array are relatively straightforward and do not yield complex signals. However, in the presence of weld scattering in welded structures, the ultrasonic signal is interfered with, thereby affecting the acoustic signal and subsequently impacting the quality of imaging. Arc sparse array multi-frame imaging refers to a scenario where the transmitting aperture of the transducer remains fixed, while the signals received by the array elements within the receiving aperture can be characterized by the amplitude matrix and the time-of-flight matrix. In the imaging of ultrasonic guided wave arrays within a plate, the arc sparse array evolves from the arc array by undergoing radial movement. Essentially, the arc array serves as the reference array for the arc sparse array, as depicted in Figure 1. When the array element m emits sound waves and the number of receiving aperture array elements is \(S\) (\(1\leq s\leq N\)), which is equivalent to \(S\) in Figure 1, the wave amplitude matrix \(A_{B-ms}\) can be defined as follows:  
\[ \begin{array}{llll}A_{B-ms}&=[a_{1}\cdots a_{k}\cdots a_{s}]^{\text{T}}\\ &=\left[\begin{array}{cccc}a_{b1}&\cdots&a_{bk}&\cdots&a_{bs}\\ \vdots&&\vdots&&\vdots\\ a_{kb}&\cdots&a_{kk}&\cdots&a_{bs}\\ \vdots&&\vdots&&\vdots\\ a_{sb}&\cdots&a_{sk}&\cdots&a_{ss}\end{array}\right](1\leq s\leq N)\end{array}  \tag{1}\]  
The transit matrix \(T_{B-ms}\) is as follows:  
\[ \begin{array}{llll}T_{B-ms}&=[t_{1}\cdots t_{k}\cdots t_{s}]^{\text{T}}\\ &=\left[\begin{array}{cccc}t_{11}&\cdots&t_{1k}&\cdots&t_{1s}\\ \vdots&&\vdots&&\vdots\\ t_{m1}&\cdots&t_{mn}&\cdots&t_{ms}\\ \vdots&&\vdots&&\vdots\\ t_{s1}&\cdots&t_{sn}&\cdots&t_{ss}\end{array}\right](1\leq s\leq N)\end{array}  \tag{2}\]Once the sound wave emitted by array element m traverses various scatterers within the sound field, it is received by different array elements within the array receiving aperture via distinct propagation paths. The column vectors within the wave amplitude matrix \(A_{B-ms}\) symbolize the equivalent wave amplitude received by different array elements for the scattered echo originating from the same scatterer. Conversely, the row vector represents the equivalent wave amplitude received by the same array element for the scattered echo generated by different scatterers. It is evident that the row vectors of the wave amplitude matrix \(A_{B-ms}\) exhibit linear independence, indicating no linear relationship between them. Similarly, the column vectors also demonstrate linear independence. This same linear relationship can be observed in the time-of-flight matrix \(T_{B-ms}\). Given that the transmitting aperture remains constant, it is worth noting that both the wave amplitude matrix \(A_{B-ms}\) and the time-of-flight matrix \(T_{B-ms}\) coexist and correspond to each other. Furthermore, the rank of the wave amplitude matrix \(A_{B-ms}\) is equivalent to that of the time-of-flight matrix \(T_{B-ms}\). Thus, we can establish the following relationship:  
\[R(A_{B-ms}^{T})=R(T_{B-ms}^{T})=s \tag{3}\]  
The rank of the wave amplitude matrix is identical to the rank of the time-of-flight matrix, which is equivalent to the number of receiving array elements within the receiving aperture. Hence, in this context, the wave amplitude receiving matrix and the time-of-flight receiving matrix are collectively referred to as the receiving matrix. In the presence of a welded structure during detection, the wave amplitude receiving matrix \(A_{B-WS}\) of the ultrasonic guided waves arc sparse array in the plate can be represented as follows:  
\[