## 3 Results and Discussion
The size of the acoustic footprints of individual MBES beams in the along-track direction, for a flat seafloor, is approximately given by the following:  
\[a_{x}=\frac{2\cdot D}{\cos\beta}\cdot\tan\frac{\psi_{x}}{2}, \tag{21}\]  
where \(a_{x}\)--the width of the footprint in the fore-aft direction; \(D\)--the mean depth [m]; \(\beta\)--the beam pointing angle [\({}^{\circ}\)]; and \(\psi_{x}\)--the along-track beamwidth of the MBES [\({}^{\circ}\)]. The length of the MBES footprints in the across-track direction is approximately given by the following:  
\[a_{y}=\frac{2\cdot D}{\cos^{2}\beta}\cdot\tan\frac{\psi_{y}}{2}, \tag{22}\]  
where \(a_{y}\)--the width of the footprint in the across-track direction; \(\psi_{y}\)--the across-track beamwidth of the MBES [\({}^{\circ}\)]. The variable MBES resolution values as a function of the beam pointing angles (deviated from Nadir) are presented in Table 2.  
Many studies have been carried out on the performance of multi-beam echosounders available on the market in recent years. Their results and conclusions have been published, inter alia, in [42; 43; 44; 45; 46; 47; 48]. The publications contain test results for multi-beam echosounders in terms of accuracy requirements and MBES detection capabilities. Many publications are devoted to the depth measurement uncertainty and the error sources that accompany all multi-beam systems. These phenomena have been widely described in [49; 50; 51; 52; 53]. Some  
Figure 3: Estimated time of survey conducted with a multi-beam echosounder as a function of vessel speed and MBES swath angle (water basin 5 \(\times\) 2 Nm). sources [54; 55; 56] focus on the analysis of the accuracy and detection capabilities of the external beams of the multi-beam echosounder. On the basis of research conducted by [PERSON] [57] in a basin with a depth of 8-10 m, the useful angular sector of MBES is 140\({}^{\circ}\) (maximum \(\theta=160^{\circ}\)). It remains a matter of choosing the appropriate survey speed so that the bathymetric data obtained in this way ensure full bottom coverage, are error-free, and enable the detection of underwater objects. The speed can neither be arbitrary nor the maximum one. Full bathymetric coverage is a criterion required by the IHO standard S-44 [58]. In the case of a multi-beam echosounder, it is vital to achieve full coverage in the direction of the vessel's movement (the so-called along-track coverage). Hence, it is very important to determine the maximum vessel's survey speed that will meet the condition of full coverage in the along-track direction. The maximum speed, Vmax, of the vessel surveying with a multi-beam echosounder under the condition of full coverage is calculated according to [50; 57] the following:  
\[V_{max}\leq c\cdot\frac{\psi_{x}}{2}\cdot\cos\left(\frac{\theta_{max}}{2} \right), \tag{23}\]  
where \(\psi_{x}\)--along-track beamwidth of the MBES [\({}^{\circ}\)]; \(c\)--speed of sound in the water column [m/s]; and \(\theta_{max}\)--maximum swath angle of the MBES [\({}^{\circ}\)]. The latest models of multi-beam echosounders have very narrow along-track beamwidths. Norbit Winghead i80S has along-track beamwidth of \(\psi_{x}=0.5^{\circ}\). Teledyne Marine SeaBat T-51-R is characterized by a beamwidth of \(\psi_{x}=1^{\circ}\), and for the frequency of 700-800 kHz, the beam is even narrower and amounts to \(\psi_{x}=0.5^{\circ}\). Sonic 2026 produced by R2 Sonic, on the other hand, declares the narrowest acoustic beamwidth of \(\psi_{x}=0.45^{\circ}\). Such narrow beams mean that in order to maintain full bottom coverage and detect underwater objects on the bottom, the maximum surveyed speed should not be exceeded. For a basin with a depth of 10 m, the MBES swath angle \(\theta_{max}=140^{\circ}\) and the along-track beamwidth \(\psi_{x}=0.5^{\circ}\), the hydrographer should conduct the survey at a speed of \(V_{max}=4.3\) kts. This means that it will take on average 4 days and 4 h.