# Search Space Reduction for Localization and Tracking of an Acoustic Source
## 3 Three-Dimensional Localization and Tracking  
Calculations of the 2D initial guess with the KRAKEN model require a reduced amount of computational time because bottom properties and sound speed are known; the next step for 3D localization and tracking consists in generating estimates of source bearing. To this end, one can rely on ESP for the generation of replicas; the source at depth \(z_{s}\) is idealized as rotating around the VLA along a circle with a radius \(r_{s}\) and \(N\) different bearings, and each bearing defines a different source-array configuration relative to the bathymetry and corresponds to a different set of replicas, to be calculated either through 3D or \(N\times 2\)D modeling. Thus, bearing estimation can take place in the search space \((r_{s},z_{s})\times[\phi_{\min},\phi_{\max}]\) with \(\phi_{\max}-\phi_{\min}=360^{\circ}\). To reduce calculations as much as possible, the VLA is considered to be always at the origin; the bathymetry is also corrected, so the bottom depth at the VLA corresponds to the tuned value indicated in Table 3. As shown in Figure 6a, this approach produces a butterfly-like shape when the Bartlett estimator is displayed in polar coordinates. The butterfly's \"wings\" point in the directions where bearing estimation is more reliable, although the alignment of isobaths makes the estimation ambiguous. In the case of Figure 6a, the upper wing is aligned along \(\phi_{up}\approx 89^{\circ}\), while the lower wing aligns at a sort of antipode at \(\phi_{down}\approx 250^{\circ}\), which in fact has a higher Bartlett power. Thus, blind acceptance at this stage of the highest Bartlett would provide a source bearing, which deviates almost \(180^{\circ}\) from the true bearing. Similar results can be seen in Figure 6b,c; for Figure 6b, the wings tend to be wider, and the Bartlett curves practically overlap each other, while in Figure 6c, the wings are somehow between the two cases of the stationary sources but with a Bartlett power that increases as the source-array distance increases. Results for the moving source are remarkable because of the constant wing alignment, a feature of Figure 6c that is consistent with the description of source motion presented in Section 2. While the Bartlett estimators shown in Figure 6 do not resolve source bearing, they drastically reduce the search space from \((r_{s},z_{s})\times[\phi_{\min},\phi_{\max}]\) to \((r_{s},z_{s})\times[\phi_{up}\pm\delta\phi]\cup[\phi_{down}\pm\delta\phi]\). Since the error in the estimation of source depth can be expected to be much smaller than the error in source range, the strategy used to remove bearing ambiguity consisted in calculating additional replicas in the search space \([r_{s}\pm\delta r]\times z_{s}\times[\phi_{up}\pm\delta\phi]\cup[\phi_{down} \pm\delta\phi]\), where \(\delta r\) is to be chosen to be sufficiently \"small\"; a faulty choice of \(\delta r\) can be detected when the localization result is found at one extreme of the search space. This strategy was applied to all files, and in each case it produced a \"wing motion\", in which the estimator wings extend or contract as range changes (see Figure 7). Bearing ambiguity was removed in all cases by noticing that the region \(\phi_{up}\pm\delta\phi\) exhibits the highest Bartlett power at a range \(r_{*}\in[r_{s}\pm\delta r]\) and a bearing \(\phi_{*}\in[\phi_{up}\pm\delta\phi]\). \begin{table}
\begin{tabular}{l c c c} \hline
**Source** & **Frequency (Hz)** & **File** & \((r_{s},z_{s})\) **(m)** \\ \hline Stationary & 331 & 1–5 & (5380, 64) \\ Stationary & 171 & 1–4 & (5530, 75) \\ Stationary & 171 & 5–10 & (5530, 74) \\ Moving & 170 & 1 & (5590, 73) \\ Moving & 170 & 2 & (5650, 67) \\ Moving & 170 & 3 & (5710, 64) \\ Moving & 170 & 4 & (5800, 64) \\ Moving & 170 & 5 & (5890, 63) \\ Moving & 170 & 6 & (5980, 65) \\ Moving & 170 & 7 & (6130, 65) \\ Moving & 170 & 8 & (6190, 64) \\ Moving & 170 & 9 & (6310, 62) \\ Moving & 170 & 10 & (6400, 62) \\ \hline \end{tabular}
\end{table} 
Table 4: Two-dimensional initial guess.