## 3 Coherent Image Superposition
#### Ideal Sas Beampatterns  
An imaging example with four antenna positions is used to illustrate our method. The number of sensors, respectively sensor elements \(N_{s}\) is 128, the center-frequency is \(f_{0}=400\) kHz with a bandpass sampling frequency \(f_{s}=200\) kHz, which is two times the pulse bandwidth. The sensor spacing is \(\frac{3}{2}\).  
Figure 3 shows the different beampatterns of a single antenna and the coherent processing from all antenna positions. The beampatterns are numerically calculated using a point-scatterer at a range of 10 m. Beamforming and image reconstruction is then carried out in a lateral range (x) of \(\pm 1\) m and a depth range (z) of \(\pm 0.05\) m where the depth range is integrated and then plotted along the lateral range. Figure 3a shows the beampattern for an array of 128 elements directly in front of the point-scatterer. In comparison, Figure 3b shows the distinctly improved beampattern of the coherent sum from all antenna positions. In total, four positions with an overlap of 50% of the array length (\(L\)), hence a complete aperture of three times the array length, is used. The beampattern shows a significantly improved resolution with even smaller sidelobes compared to the single beampattern. This illustrates of course the ideal case where scans are pixelwise correctly summed.  
The following illustrates this SAS image reconstruction using two different range step-widths. Figure 4c shows the case with \(r_{w}=\frac{c}{2f_{0}}\). Random pixel-errors are introduc  
Figure 3: The synthetic aperture sonar (SAS) beam pattern from four adjacent positions and signal parameters \(f_{s}=200\) kHz, \(f_{0}=400\) kHz and pulse length \(T_{p}=\frac{1}{100\,\mathrm{kHz}}\). The comparison demonstrates a significant gain in resolution.  
The described baseband processing leads to an extremely low oscillation. As a consequence, the erroneous coherent summation still achieves a gain in resolution within certain limits of voxel/pixel errors. The image reconstruction (Figure 4c) still shows a bound region of the point scatterer. The counterpart uses an odd step-width \(r_{w}=1.4\cdot\frac{c}{2f_{0}}\) where the image reconstruction (Figure 4d) and its phase structure is an unstructured region.  
The general principle demonstrated by the beampattern of the point-scatterer motivates that a significant SAS gain from the coherent imaging can still be achieved with pixel/voxel errors, which are possible due to localization inaccuracies when just using registration to derive the spatial relations between multiple scans.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Antenna Position** & **Pixel-Error x/z** \\ \hline (1) \(-\)L & \(-2\)\(-1\) \\ \hline (2) \(-\)L/2 & \(1\)\(2\) \\ \hline (3) \(+\)L/2 & \(2\)\(0\) \\ \hline (4) \(+\)L & \(1\)\(2\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Pixel-error artificially added in our SAS reconstruction comparing different range steps. The results are illustrated in Figure 4.  
Figure 4: Comparison of erroneous SAS superposition using step-widths \(r_{w}=\frac{c}{2f_{0}}\) and \(r_{w}=1.4\cdot\frac{c}{2f_{0}}\). Even with erroneous superposition (Table 1) a reasonable reconstruction of a point scatterer is possible compared to the oscillating phase, which leads to a degraded SAS superposition.