# Assessing the Performance of the Multi-Beam Echo-Sounder Bathymetric Uncertainty Prediction Model
## 4 Results and Discussion  
For the calculation of the vertical uncertainties, the size of the surface patch on the bottom is of importance. If a too large surface patch is considered, the variations of the measured uncertainties within a patch cannot not be solely associated with the uncertainties inherent to the MBES as the small-scale roughness of an order higher than that used for the fit will affect the vertical uncertainties. One the other hand, if a too small surface patch is considered, the number of the measurements falling within a patch is small, and a robust estimate of the variance component Equation (4) cannot be obtained. Therefore, an optimal size for the surface patch is required. To this end, five different patch sizes of \(2.5^{\circ}\), \(2^{\circ}\), \(1.5^{\circ}\), \(1^{\circ}\), and \(0.5^{\circ}\) in the across-track by seven pings in the along-track directions were considered. Shown in Figure 2 is the mean value of the vertical uncertainty over the swath (from \(-65^{\circ}\) to \(65^{\circ}\)) for different patch sizes at a water depth of \(10\,\mathrm{m}\) (gray) and \(30\,\mathrm{m}\) (black). It was seen that the largest decrease in the vertical uncertainty occurred by decreasing the size of the patches in the across-track direction from \(2.5^{\circ}\) to \(2^{\circ}\) for both water depths. For the deeper water, a further decrease in the patch size to \(1^{\circ}\) resulted in a further decrease in the vertical uncertainty. However, decreasing the patch size further to \(0.5^{\circ}\) increased the vertical uncertainty. For a water depth of \(10\,\mathrm{m}\), the vertical uncertainties were almost equal for the patch sizes of \(2^{\circ}\) and \(1.5^{\circ}\) followed by an increase in the vertical uncertainties for a further decrease in the patch size. The observed increase in the uncertainties for too small patch size indicated that the number of measurements within the patches was not enough for the extraction of the required statistics. Based on this result, patch sizes of \(1.5^{\circ}\) by seven pings and \(1^{\circ}\) by seven pings were chosen for the measurements at a water depth of \(10\,\mathrm{m}\) and \(30\,\mathrm{m}\), respectively.  
Figure 1: Study areas: (**a**) the North Sea and (**b**) southwest of The Netherlands showing the Oosterschelde estuary (Eastern Scheldt) and the location of the two datasets. Bathymetry maps of the surveyed area for (**c**) a \(10\,\mathrm{m}\) water depth with the grid cell size of \(0.25\,\mathrm{m}\)\(\times\)\(0.25\,\mathrm{m}\), (**d**) a \(30\,\mathrm{m}\) water depth with the grid cell size of \(1\,\mathrm{m}\)\(\times\)\(1\,\mathrm{m}\), (**e**) zoomed in on the black rectangle shown in (**c**), and (**f**) zoomed in on the black rectangle shown in (**d**). The bathymetry maps correspond to the measurements with a pulse length of \(54\,\mathrm{\SIUnitSymbolMicro s}\). The black rectangles in (**e**,**f**) indicate the areas considered for the depth uncertainty calculation.  
The measured bathymetric uncertainty was derived by fitting a bi-quadratic or linear function to the measurements within each surface patch. The degree of the fit function (bi-quadratic or linear) was chosen based on the curvature, which is a measure of the surface patch deviation from a flat plane. For the data acquired, it was seen that the absolute curvature larger than 0.2\(\,\mathrm{\SIUnitSymbolOhm}\) corresponded to gradual slopes. Therefore, for a surface patch with curvature smaller than this threshold, a linear function was used, otherwise a bi-quadratic fit was employed.