## 3 The Left Rank of the Arc Sparse Array under the Condition of Weld Scattering
### Left-Rank Arc Sparse Array and Left-Rank Receiving Aperture  
Based on the previous discussion, it becomes evident that when the number of array elements within the receiving aperture, as depicted in Figure 1, is equal, the left array receiving aperture, right array receiving aperture, and inner array receiving aperture all have the same structured receiving matrix. This implies that the rank of the identical receiving matrix corresponds to the left-rank receiving aperture, right-rank receiving aperture, and inner-rank receiving aperture simultaneously. More specifically, the left array receiving aperture corresponds to the left-rank receiving aperture, with its rank referred to as the left rank. The array associated with it is known as the left-rank arc sparse array. Similarly, the right-rank receiving aperture corresponds to the right-rank receiving aperture, with its rank termed as the right rank, and its associated array referred to as the right-rank arc sparse array. Likewise, the inner-rank receiving aperture corresponds to the inner-rank receiving aperture, with its rank denoted as the inner rank, and its associated array called the inner-rank arc sparse array. In this discussion, we will focus solely on the left-rank arc sparse array. The left-rank receiving aperture is positioned on the left side of the arc sparse array. Following the emission of the sound wave signal by the emitting array elements, the array elements on the left side sequentially receive the sound wave signal. When the signal is emitted from a specific emitting aperture within the arc sparse array, and the rank of the receiving matrix is 1, it corresponds to \(N\) receiving apertures. This implies that among the \(C_{N}^{1}(N=14)\) receiving apertures, there is only one left-rank receiving aperture. Similarly, when the rank of the receiving matrix is two, indicating a correspondence to \(C_{N}^{2}\) receiving apertures, there is still only one left-rank receiving aperture. Likewise, when the rank of the receiving matrix of the arc sparse array is three, namely, in the \(C_{N}^{3}\) receiving apertures, there is also only one left-rank receiving aperture. Similarly, when the rank of the receiving matrix of the arc sparse array is m, among the \(C_{N}^{m}\) rank receiving apertures, there exists only one left-rank receiving aperture. This signifies the presence of a sole left-array receiving aperture, as depicted in Figure 1.  
The distinctive arrangement of array elements within the left-rank receiving aperture, inner-rank receiving aperture, and right-rank receiving aperture determines the disparity in the signals they receive. These three components complement each other and are integral to ultrasonic guided wave imaging in the left-rank sparse array welding structural plate. Ultrasonic Guided Wave Multi-Frame Imaging in the Welding Structural Plate Based on the Left-Rank Arc Sparse Array  
When conducting imaging using the left-rank arc sparse array of ultrasonic guided waves in a plate, it is crucial to consider the characteristics of the sound field. By performing a scan across the imaging area, the process of \"specific array element emission, full aperture or sub-aperture reception\" can be achieved. When the array receives signals with the full aperture, it obtains \(N\) sets of \(N^{2}\) imaging signals. These imaging signals encompass the scattering information from all scattering points within the array's sound field imaging area. Through subsequent post-processing, involving \"relative phase-shifting and superimposing\" for each scattering point, the corresponding scattering information is extracted. This process enables the complete imaging of all scatterers present throughout the entire scanning area. Let us consider the presence of a scatterer, denoted as \(P(x_{p},y_{p})\), within the imaging area of the arc sparse array, as illustrated in Figure 2. The scatterer is positioned in the far field of the array elements.