# Virtual Source Array-Based Multiple Time-Reversal Focusing
## 4 Simulations in Waveguide
### Adaptive VSA  
Figure 5 shows the simulation result of the adaptive VSA. To emphasize the relationship between sound speed mismatch and the performance changes in multiple focusing, the time series at the selected locations are normalized with respect to its own maximum as well as the existence of noise is not considered in numerical simulations. The weight vector, \(\mathbf{w}\), is obtained from Equation (4); then, it is time reversed and backpropagated. Panels (a) and (b) in Figure 5 show the simulated time series at the selected locations. A comparison between the two different profiles (dot-dashed and solid) provides strong evidence that the adaptive VSA with multiple constraints is sensitive to sound speed mismatch (solid) because the residual signal, denoted by a rectangular box, is still observable compared to the isovelocity case (dot-dashed).  
In fact, the weight vector, \(\mathbf{w}\), with multiple constraints, \(\mathbf{f}=[1\;\;1]^{\mathrm{T}}\), can be expressed as the sum of \(\mathbf{w}_{1}\) and \(\mathbf{w}_{2}\) with constraints \([1\;\;0]^{\mathrm{T}}\) and \([0\;\;1]^{\mathrm{T}}\), respectively. Thus, the weight vector, \(\mathbf{w}=\mathbf{w}_{1}+\mathbf{w}_{2}\), can focus distortionless response at the selected locations while simultaneously placing the null at different locations, as shown in Figure 6, providing time series as a function of depth at range of 2.3 km, which corresponds to Figure 5b. For isovelocity (see Figure 6a), it is clear that the residual signal is mitigated as seen in the inset at the bottom right, which is magnified to highlight the location where the null is to be placed. However, imperfect pulse compression is achieved when the actual SSP in the  
Figure 4: (Color online) Simulated VSA-based multiple TR focusing using superposition 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 3.  
water column is different from the assumed sound speed, \(c\), beyond the VSA owing to the failure of null placement (i.e., sound speed mismatch) (see Figure 6b).  
Figure 7 provides more insight into how nulls are placed in multiple locations through field propagation. Interestingly, in the case of the isovelocity SSP (see Figure 7a), nulls are placed at the  
Figure 5: (Color online) Simulated adaptive VSA-based multiple TR focusing for different types of profiles. (**a**) Time series at \(\overline{r}_{s1}\) = (2.2 km, 50 m); (**b**) Time series at \(\overline{r}_{s2}\) = (2.3 km, 50 m). While pulse compression is fully achieved for the isovelocity with a sound speed of \(c\) = 1500 m/s (SSP 1, dot-dashed), the residual signal (denoted by a rectangular box) is still observable for the downward-refracting SSP (SSP 2, solid).  
Figure 6: (Color online) Time series as a function of depth at a range of 2.3 km, which corresponds to Figure 5b. (**a**) For isovelocity SSP; (**b**) For downward-refracting SSP. The null location, denoted by a rectangular box, is magnified and the residual signal is highlighted at the bottom right.  
corresponding locations, \(\vec{r}_{s1}\) and \(\vec{r}_{s2}\) (i.e., 1 and 2\({}^{{}^{\prime}}\)) as the field is propagated. However, in the case of the downward-refracting SSP (see Figure 7b), it is clearly shown the nulls are placed at different locations due to the sound speed mismatch.