# Assessing the Performance of the Multi-Beam Echo-Sounder Bathymetric Uncertainty Prediction Model
## 4 Results and Discussion
### Comparison between Modeled and Measured Uncertainties
Uncertainties in the angle of impact of the incoming sound wave with the MBES,
3. non-zero along-track beam opening angle. The first and third contributions are equal for both amplitude and phase detection. However, the expression for the second term differs. In the case of using amplitude detection, the contribution of the across-track angular measurements' uncertainty to the depth uncertainty is a constant term equaling \(\frac{1}{12}\Omega_{R}\)[6; 7]. Comparing the measured and predicted uncertainties suggested that this term requires modification. An additional point to clarify is the term referred to as range sampling resolution used to calculate the contribution of the range measurements to the bathymetric uncertainty. This term is defined as \(\delta r_{s}=\frac{c}{f_{s}}\) with \(f_{s}\) the sampling frequency. The reported value for \(f_{s}\) is the output sampling frequency, different from the one used for applying the bandpass filters and beamforming with the latter being higher. Knowledge of this value is required for correct calculation of the contribution of the range measurements to the bathymetric uncertainty. If the sampling frequency used for the bottom detection would be in the order of MHz, the contribution of the range error would become negligible. For the present contribution, lacking information on the sampling frequency of relevance, the maximum output sample rate of EM2040c equaling 60 kHz was used for the predictions, corresponding to a contribution varying from 0.009 m at nadir to 0.0008 m for the most outer beam to the bathymetric uncertainty. Figure 4: Predictions of total bathymetric uncertainty for a water depth of 30 m and a pulse length of 134 μs using the number of phase samples obtained from Equation (5) (gray) and extracted from the data (black). Shown with green markers are the measured vertical uncertainties. The transition from the amplitude to phase detection is shown by changing the square marker to an asterisk. Shown in Figure 6 are the modeled and measured depth uncertainties for the area indicated by the dashed line in Figure 1f for a water depth of 30 m and pulse lengths of 27 us (a), 54 us (b), and 134 us (c). Compared to the shallower water depth, for the two pulse lengths of 27 us and 54 us, the discrepancy between the modeled and measured uncertainties increases. This can partly indicate that the prediction model cannot fully capture the depth dependency of the bathymetric uncertainties. For the pulse with the longest duration, relatively good agreement is obtained between the predicted and measured uncertainties for the middle sector of the beams. With regards to the variations of the bathymetric uncertainty with the beam angle, in contrast to the data acquired at a water depth of 10 m, the measured uncertainties show a pronounced dependency on the beam angle. This behavior is captured to a limited extent by the prediction model. Figure 5: Bathymetric uncertainty derived from the measurements (blue markers) and those predicted for a water depth of 10 m and a pulse length of (**a**) 27 us, (**b**) 54 us, and (**c**) 134 us. The change from amplitude to phase detection is shown by switching from square markers to asterisks with the same color. The location of this switch was similarly applied in the modeling. Improving Model-Data Agreement by Accounting for the Most Recent Insights into the Contributors to MBES Depth Uncertainties  
In order to improve the agreement between the predicted and measured uncertainties, in particular for deeper waters, the contribution of the following uncertainty sources is added to the existing bathymetric uncertainty prediction model:  
1. Doppler effect: Since the MBES is constantly moving, the received signal is affected by a Doppler frequency shift. This frequency shift has an impact on the beam steering, resulting in uncertainties in the steering angle. An uncertainty in the steering angle gives rise to uncertainty in the estimated depths.