# Virtual Source Array-Based Multiple Time-Reversal Focusing
## 5 Robust VSA-Based Multiple TR Focusing  
The approach described in this section partially retains [PERSON]'s application [10] of SVD to matched-field processing (MFP) and the stability of TR focusing in a fluctuating ocean environment [7]. However, it extends to VSA-based multiple TR focusing, rather than MFP or conventional TR focusing. As investigated in Section 4.2, the performance of the adaptive VSA is significantly degraded in the presence of sound speed mismatch owing to the fact that the constraint matrix, \(\mathbf{M}\), partially contains the characteristic of the homogeneous field with constant sound speed, \(c\). In practice, the constraint matrix, \(\mathbf{M}\), can be decomposed into several component matrices using SVD with a best rank \(K\) approximation,  
\[\mathbf{M}\cong\mathbf{U}\Lambda\mathbf{V}^{\dagger}, \tag{8}\]  
where \(\mathbf{U}\) is an \((M\times K)\) matrix whose columns are left singular vectors, \(\Lambda\) is a \((K\times K)\) matrix whose off-diagonal entries are zeros and diagonal elements are the singular values of \(\mathbf{M}\), and \(\mathbf{V}\) is a \((P\times K)\) matrix whose columns are right singular vectors. Indeed, the first singular vector corresponding to the largest singular value is sufficient to approximate the columns of constraint matrix, \(\mathbf{M}\); thus, the backpropagation vector, \(\mathbf{w}\), is reformulated with the response vector, \(\mathbf{c}=[1,0,\cdot\cdot\cdot,0]^{\mathrm{T}}\),  
\[\mathbf{w}=\mathbf{R}^{-1}\mathbf{U}\left(\mathbf{U}^{\dagger}\mathbf{R}^{-1} \mathbf{U}\right)^{-1}\Lambda^{-1}\mathbf{V}^{\dagger}\mathbf{c}. \tag{9}\]  
Figures 8 and 9 provide a clear understanding of VSA-based multiple TR focusing using SVD method in sound speed mismatch environment. The selected locations were changed to \(\vec{r}_{s1}=(2.2\ \mathrm{km},20\ \mathrm{m})\) and \(\vec{r}_{s2}=(2.3\ \mathrm{km},40\ \mathrm{m})\) to show the dramatic results of VSA-based multiple  
Figure 7: (Color online) Simulated adaptive VSA-based multiple TR focusing using MCM method. Each panel represents a backpropagated field corresponding to a different single snapshot time. (**a**) For isovelocity SSP; (**b**) For downward-refracting SSP. The numbers here correspond one-to-one with the time series in Figure 5.  
TR focusing using SVD method, and the results were compared to VSA-based single TR focusing results at a selected location, \(\vec{r}_{s2}\) = (2.3 km, 40 m). Figure 8: (Color online) Comparison of the simulated time series for the weight vector, \(\mathbf{w}\), with VSA-based single TR focusing (dot-dashed) and VSA-based multiple TR focusing using singular value decomposition (SVD) (solid) in sound speed mismatch environment. (**a**) Time series at \(\vec{r}_{s1}\) = (2.2 km, 20 m); (**b**) Time series at \(\vec{r}_{s2}\) = (2.3 km, 40 m). Figure 9: (Color online) Comparison of VSA-based single TR focusing and VSA-based multiple TR focusing using SVD method in sound speed mismatch environment. Each panel represents a backpropagated field corresponding to a different single snapshot time. Selected locations are \(\vec{r}_{s1}\) = (2.2 km, 20 m) and \(\vec{r}_{s2}\) = (2.3 km, 40 m). (**a**) VSA-based single TR focusing; (**b**) VSA-based multiple TR focusing using SVD method. In the case of VSA-based single TR focusing (dot-dashed), pulse compression is achieved at the selected location, \(\overline{r}_{s2}\) = (2.3 km, 40 m), as expected (see Figure 8b), but because the field is not propagated to the location, \(\overline{r}_{s1}\) = (2.2 km, 20 m) (see Figure 9a), incomplete pulse compression and signal spread over time are observed (see Figure 8a). Note that the normalization of the axis amplified low-amplitude noise (dot-dashed in Figure 8a). On the other hand, using the SVD method, pulse compression is fully achieved at both selected locations, \(\overline{r}_{s1}\) and \(\overline{r}_{s2}\), as shown in Figure 8 (solid), which provides robustness to sound speed mismatch environment.