# Virtual Source Array-Based Multiple Time-Reversal Focusing
## 4 Simulations in Waveguide
### Superposition Method  
First, we take a look at the case of the superposition method which is the most intuitive method of simultaneous multiple focusing. The sampled transfer functions between the TRM and VSA are time delayed corresponding to a line-of-sight distance between the VSA and selected locations \(\vec{r}_{s1}\) = (2.2 km, 50 m) and \(\vec{r}_{s2}\) = (2.3 km, 50 m), respectively, and then superimposed prior to backpropagation from the TRM. Figure 3 shows the simulated time series at two selected locations for two different profiles, i.e., the isovelocity with a sound speed of \(c\) = 1500 m/s (SSP 1, dot-dashed) and the downward-refracting SSP (SSP 2, solid). Since the sampling frequency is 7500 Hz and fast Fourier transform size is 8192, the time duration is 1.09 s long. This time duration is not sufficient to prevent aliasing on the time axis as the pulse propagates (e.g., \(\frac{2200\text{m}}{1500\text{m/s}}\approx 1.5\) s). Thus, a random time shift was applied to the time series results. Not surprisingly, imperfect pulse compression is achieved at both selected locations, \(\vec{r}_{s1}\) (see Figure 3a) and \(\vec{r}_{s2}\) (see Figure 3b), owing to the residual term of the phase-conjugated field,  
\[p(\vec{r})=\sum_{m}[g(\vec{r}_{m}|\vec{r}_{s1})+g(\vec{r}_{m}|\vec{r}_{s2})]^{ *}g(\vec{r}|\vec{r}_{m})=[\mathbf{g}(\mathbf{r}_{\text{TRM}}|\vec{r}_{s1})+ \mathbf{g}(\mathbf{r}_{\text{TRM}}|\vec{r}_{s2})]^{\dagger}\mathbf{g}(\vec{r }|\mathbf{r}_{\text{TRM}}), \tag{7}\]  
where the residual term can be presented as \(\mathbf{g}(\mathbf{r}_{\text{TRM}}|\vec{r}_{s1})^{\dagger}\mathbf{g}(\vec{r}_ {s2}|\mathbf{r}_{\text{TRM}})\) in the case of the phase-conjugated field at location \(\vec{r}_{s2}\).  
Figure 4 shows snapshot results of field propagation when the superimposed vector is backpropagated from the TRM, and each number corresponds to a time series at multiple locations in Figure 3. Panels (a) and (b) in Figure 4 represent the case of isovelocity SSP and downward-refracting  
Figure 3: (Color online) Simulated VSA-based multiple TR focusing using superposition method. The simulated time series are normalized with respect to its own maximum. (**a**) Time series at \(\vec{r}_{s1}\) = (2.2 km, 50 m); (**b**) Time series at \(\vec{r}_{s2}\) = (2.3 km, 50 m). The residual signal (denoted by a rectangular box) is clearly observed for the cases of isovelocity with a sound speed of \(c\) = 1500 m/s (SSP 1, dot-dashed) and downward-refracting SSP (SSP 2, solid).  
SSP, respectively. As the synchronized transfer functions corresponding to the two locations are superimposed and backpropagated, it can be seen that two fields with the appropriate distance are propagated for simultaneous multiple focusing. Thus, the residual term occurs as fields 1 (left) and 2\({}^{{}^{\prime}}\) (right) pass through multiple locations. Also, as shown in the lower panel, the focused fields (i.e., 2 and 1\({}^{{}^{\prime}}\)) are slightly shifted down by the sound speed structure, but are still included in the focal resolution. The solid line indicates the depth of the focal location (i.e., 50 m).