## 2 Materials and Methods
### Dort-Music  
Let the ultrasound signal excited by the \(m\)-th driving element be received by the \(n\)-th sensing element. The form of the multi-dimensional state matrix is:  
\[T_{mn}(\omega)=U_{m}^{a}(\omega)/U_{n}^{s}(\omega) \tag{9}\]  
Assume the number of scatterers (damage) \(K\) in the structure is less than the number of transducer array elements \(M\), perform singular value decomposition on \(T\):  
\[T=\hat{G}^{s}\hat{V}\hat{G}^{aH} \tag{10}\]  
In this context, \(\hat{V}\) represents the singular values corresponding to the number of scattering sources and scattering intensity, while \(\hat{G}^{s}\) and \(\hat{G}^{a}\) are the left and right singular vectors of the singular value decomposition.  
\[\hat{G}^{s}=\left[\hat{G}_{1}^{s}\quad,\quad\dots,\quad\hat{G}_{K}^{s},\hat{G} _{K+1}^{s}\quad,\quad\dots,\quad\hat{G}_{M}^{s}\right] \tag{11}\]  
\[\hat{G}^{a}=\left[\hat{G}_{1}^{a}\quad,\quad\dots,\quad\hat{G}_{K}^{a},\hat{G} _{K+1}^{a}\quad,\quad\dots,\quad\hat{G}_{M}^{a}\right] \tag{12}\]  
The singular vectors \(\left[\hat{G}_{1}^{s}\quad,\quad\dots,\quad\hat{G}_{K}^{s}\right]\) and \(\left[\hat{G}_{1}^{a}\quad,\quad\dots,\quad\hat{G}_{K}^{s}\right]\) span the signal subspace \(\mathbf{S}\), which is related to the propagation of guided wave fields from the scattering source to the sensors and from the actuators to the scattering source, respectively. Meanwhile, the vectors \(\left[\hat{G}_{K+1}^{s}\quad,\quad\dots,\quad\hat{G}_{M}^{a}\right]\) and \(\left[\hat{G}_{K+1}^{a}\quad,\quad\dots,\quad\hat{G}_{M}^{a}\right]\) are associated with noise in the measurements and span the noise subspace \(\mathbf{N}\). According to the symmetric properties of the \(T\) matrix, the singular vectors of the signal subspace and the noise subspace corresponding to the same singular value are orthogonal. To this end, we establish the guided wave propagation Green's function vector related to the spatial position \(\overset{\rightarrow}{r}_{s}\) in the structure:  
Figure 3: Flowchart of the frequency-domain DAS method.  
\[\begin{split} g^{a}\left(\overrightarrow{r}_{s}\right)& =\left[g\left(\overrightarrow{r}_{s},\overrightarrow{s_{1}^{a}}\right), \ldots g\left(\overrightarrow{r}_{s},\overrightarrow{s_{M}^{a}}\right) \right]^{T}\\ g^{s}\left(\overrightarrow{r}_{s}\right)&=\left[g \left(\overrightarrow{r}_{s},\overrightarrow{s_{1}^{a}}\right),\ldots g \left(\overrightarrow{r}_{s},\overrightarrow{s_{M}^{a}}\right)\right]^{T} \end{split} \tag{13}\]  
In the inspection space, if a certain point (\(\overrightarrow{r}_{s}\)) is exactly at the location of a damage scattering source, then the corresponding guided wave propagation Green's function vector \(g\left(\overrightarrow{r}_{s}\right)\) must be orthogonal to the Green's function vector \(\hat{g}\) obtained from the singular value decomposition of the noise subspace. The imaging condition for the corresponding MUSIC method can be expressed as:  
\[I_{3}\left(\overrightarrow{r}_{s}\right)=\frac{1}{\sum_{m=K+1}^{M}\hat{g}_{m }^{a}\cdot g^{a}\left(\overrightarrow{r}_{s}\right)\sum_{t=K+1}^{M}\hat{g}_{ s}^{s}\cdot g^{s}\left(\overrightarrow{r}_{s}\right)} \tag{14}\]  
The flowchart of DORT-MUSIC imaging method is shown in Figure 4.