## Author Contributions

The first two authors have equally contributed to the work. Conceptualization, G.S. and D.C.; methodology, G.S., D.C., and P.L.; software, G.S.; validation, G.S., D.C., and P.L.; formal analysis, G.S., D.C., P.L., A.O., and F.R.; investigation, G.S., D.C., P.L., A.O., and F.R.; resources, G.S., D.C., and P.L.; writing—original draft preparation, G.S. and D.C.; writing—review and editing, G.S., D.C., A.O., and F.R.; visualization, G.S., D.C., and P.L.; supervision, G.S. and D.C.; project administration, G.S. and D.C.; funding acquisition, G.S., D.C., A.O., and F.R.

**Figure 1.**
Angle of the Pricipal Component (PC) (**A**–**C**) and frequency (**D**–**F**) for the monochromatic (**A**,**D**), bichromatic (**B**,**E**), and reflective (**C**,**F**) 1D cases versus time: First (second) mode is in solid (dashed) lines. Red lines are for the mean angular frequencies.

**Figure 1.**
Angle of the Pricipal Component (PC) (**A**–**C**) and frequency (**D**–**F**) for the monochromatic (**A**,**D**), bichromatic (**B**,**E**), and reflective (**C**,**F**) 1D cases versus time: First (second) mode is in solid (dashed) lines. Red lines are for the mean angular frequencies.

**Figure 2.**
Angle of the Empirical Orthogonal Function ((**A**–**C**), wavenumber from phase fitting **D**–**F**), depth from phase fitting (**G**–**I**), function fitting (**J**–**L**), and windowing (**M**–**O**, with ${w}_{t}=40\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$) for each of the monochromatic, bichromatic, and reflective 1D cases: First (second) mode is in solid (dashed) lines. Blue lines are for the exact depth.

**Figure 2.**
Angle of the Empirical Orthogonal Function ((**A**–**C**), wavenumber from phase fitting **D**–**F**), depth from phase fitting (**G**–**I**), function fitting (**J**–**L**), and windowing (**M**–**O**, with ${w}_{t}=40\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$) for each of the monochromatic, bichromatic, and reflective 1D cases: First (second) mode is in solid (dashed) lines. Blue lines are for the exact depth.

**Figure 3.**
Angle of the PC before centering (circles) and after centering around $t={t}_{0}=10\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ (triangles) for $\Delta t=1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ and ${R}_{t}=3\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$: Red denotes the point of interest, and blue indicates the neighbour points used.

**Figure 3.**
Angle of the PC before centering (circles) and after centering around $t={t}_{0}=10\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ (triangles) for $\Delta t=1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ and ${R}_{t}=3\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$: Red denotes the point of interest, and blue indicates the neighbour points used.

**Figure 4.**
Evolution of the Root Mean Square (RMS) error in h, ${\mathrm{E}}_{h}^{\mathrm{RMS}}$, as a function of ${t}_{\mathrm{max}}$ for the monochromatic case using phase fitting (without windowing).

**Figure 4.**
Evolution of the Root Mean Square (RMS) error in h, ${\mathrm{E}}_{h}^{\mathrm{RMS}}$, as a function of ${t}_{\mathrm{max}}$ for the monochromatic case using phase fitting (without windowing).

**Figure 5.**
Bathymetries (in meters) for the analysis of linear waves (**A**) and nonlinear waves (**B**): The white strip next to the shore highlights $h=0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

**Figure 5.**
Bathymetries (in meters) for the analysis of linear waves (**A**) and nonlinear waves (**B**): The white strip next to the shore highlights $h=0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

**Figure 6.**
Initial snapshots for linear synthetic wave trains W1 (**A**), W2 (**B**), and W3 (**C**) and their superposition WS (**D**): Spatial domain is $200\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (in the alongshore and cross-shore directions), and pixel intensity is a linear function of the modelled free surface elevation.

**Figure 6.**
Initial snapshots for linear synthetic wave trains W1 (**A**), W2 (**B**), and W3 (**C**) and their superposition WS (**D**): Spatial domain is $200\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (in the alongshore and cross-shore directions), and pixel intensity is a linear function of the modelled free surface elevation.

**Figure 7.**
Angles ${\alpha}_{t}$ (**A**) and ${\alpha}_{x}$ (**B**) of the first PC and EOF corresponding to linear propagation of W1 for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ and $\Delta x=4\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$: The explained variance is above $99\%$.

**Figure 7.**
Angles ${\alpha}_{t}$ (**A**) and ${\alpha}_{x}$ (**B**) of the first PC and EOF corresponding to linear propagation of W1 for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ and $\Delta x=4\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$: The explained variance is above $99\%$.

**Figure 8.**
Recovered k (**A**) and h (**C**) and the corresponding local relative errors, ${\u03f5}_{k}$ and ${\u03f5}_{h}$ (**B**,**D**, in %), obtained using the phase fitting method for the first EOF corresponding to linear propagation of W1 and for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, and ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

**Figure 8.**
Recovered k (**A**) and h (**C**) and the corresponding local relative errors, ${\u03f5}_{k}$ and ${\u03f5}_{h}$ (**B**,**D**, in %), obtained using the phase fitting method for the first EOF corresponding to linear propagation of W1 and for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, and ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

**Figure 9.**
Phase fitting without windowing of the three modes of the linear polychromatic wave field WS for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, and ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$: ${\alpha}_{x}$ (**A**–**C**) and ${\u03f5}_{h}$ (**D**–**F**).

**Figure 9.**
Phase fitting without windowing of the three modes of the linear polychromatic wave field WS for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, and ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$: ${\alpha}_{x}$ (**A**–**C**) and ${\u03f5}_{h}$ (**D**–**F**).

**Figure 10.**
Initial snapshots for synthetic nonlinear wave trains W1 (**A**) and WS (**B**) for $\mathrm{F}=2.5$: Spatial domain is $200\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (in the alongshore and cross-shore directions), and pixel intensity is a linear function of the modelled free surface elevation.

**Figure 10.**
Initial snapshots for synthetic nonlinear wave trains W1 (**A**) and WS (**B**) for $\mathrm{F}=2.5$: Spatial domain is $200\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (in the alongshore and cross-shore directions), and pixel intensity is a linear function of the modelled free surface elevation.

**Figure 11.**
Results for ${\alpha}_{x}$ (**A**–**C**) and ${\u03f5}_{h}$ (**D**–**F**) obtained with the phase fitting method without windowing from the nonlinear polychromatic wave field WS with $\mathrm{F}=1.0$ for $\Delta t=0.532\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1.1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, and ${t}_{\mathrm{max}}=90\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$.

**Figure 11.**
Results for ${\alpha}_{x}$ (**A**–**C**) and ${\u03f5}_{h}$ (**D**–**F**) obtained with the phase fitting method without windowing from the nonlinear polychromatic wave field WS with $\mathrm{F}=1.0$ for $\Delta t=0.532\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1.1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, and ${t}_{\mathrm{max}}=90\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$.

**Figure 12.**
Results for ${\u03f5}_{h}$ obtained with phase fitting (**A**), function fitting (**B**), and windowing (**C**, with ${w}_{t}=60\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$) from the first mode for the nonlinear polychromatic wave field WS with $\mathrm{F}=1.0$, $\Delta t=0.532\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1.1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, and ${t}_{\mathrm{max}}=150\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$.

**Figure 12.**
Results for ${\u03f5}_{h}$ obtained with phase fitting (**A**), function fitting (**B**), and windowing (**C**, with ${w}_{t}=60\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$) from the first mode for the nonlinear polychromatic wave field WS with $\mathrm{F}=1.0$, $\Delta t=0.532\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1.1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, and ${t}_{\mathrm{max}}=150\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$.

**Figure 13.**
Propagation of the errors in k and $\omega $ to water depth h when using the dispersion relation.

**Figure 13.**
Propagation of the errors in k and $\omega $ to water depth h when using the dispersion relation.

**Figure 14.**
Propagation of the errors ${\u03f5}_{h}$ in the bathymetry inversion for three time windows.

**Figure 14.**
Propagation of the errors ${\u03f5}_{h}$ in the bathymetry inversion for three time windows.

**Figure 15.**
On top (bottom) are results from cBathy (uBathy). From left to right: measured bathymetry with the CRAB (**A**,**E**, in m), inferred bathymetry (**B**,**F**, in m), error of the inferred bathymetry (**C**,**G**, in m), and histogram of the errors for the pixels (**D**,**H**).

**Figure 15.**
On top (bottom) are results from cBathy (uBathy). From left to right: measured bathymetry with the CRAB (**A**,**E**, in m), inferred bathymetry (**B**,**F**, in m), error of the inferred bathymetry (**C**,**G**, in m), and histogram of the errors for the pixels (**D**,**H**).

**Table 1.**
Wave conditions in the seaward boundary for the 1D examples: For each wave train (two at most), ${\mathrm{T}}_{j}$ is the period, ${a}_{j}^{0}$ is the wave amplitude at $x=0$, and ${\mathrm{dir}}_{j}$ is the direction of wave propagation (+, rightwards).

**Table 1.**
Wave conditions in the seaward boundary for the 1D examples: For each wave train (two at most), ${\mathrm{T}}_{j}$ is the period, ${a}_{j}^{0}$ is the wave amplitude at $x=0$, and ${\mathrm{dir}}_{j}$ is the direction of wave propagation (+, rightwards).

Cases | ${\mathbf{T}}_{1}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathit{a}}_{1}^{0}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{cm}\right]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathbf{dir}}_{1}$ | ${\mathbf{T}}_{2}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathit{a}}_{2}^{0}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{cm}\right]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathbf{dir}}_{2}$ |
---|

monochromatic | $5.1$ | $3.0$ | + | − | − | − |

bichromatic | $5.1$ | $3.0$ | + | $8.3$ | $1.0$ | + |

reflective | $5.1$ | $3.0$ | + | $5.1$ | $1.0$ | − |

**Table 2.**
Summary of the results of the Principal Component Analysis (PCA) obtained for the 1D examples.

**Table 2.**
Summary of the results of the Principal Component Analysis (PCA) obtained for the 1D examples.

| Mode | ${\mathit{\sigma}}^{2}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{T}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ |
---|

monochromatic | 1 | $99.5$ | $5.1$ |

bichromatic | 1 | $87.6$ | $5.1$ |

2 | $12.1$ | $8.3$ |

reflective | 1 | $99.7$ | $5.1$ |

**Table 3.**
Wave conditions in the seaward boundary for the analysis of synthetic 2D cases: For each wave train, $\mathrm{T}$ is the period, $\mathrm{A}$ is the wave amplitude in deep waters, $\theta $ is the angle with respect to the shore normal in deep waters, and $\phi $ is a phase lag.

**Table 3.**
Wave conditions in the seaward boundary for the analysis of synthetic 2D cases: For each wave train, $\mathrm{T}$ is the period, $\mathrm{A}$ is the wave amplitude in deep waters, $\theta $ is the angle with respect to the shore normal in deep waters, and $\phi $ is a phase lag.

Wave Train | $\mathbf{T}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{A}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{cm}\right]\phantom{\rule{0.166667em}{0ex}}$ | $\mathit{\theta}\phantom{\rule{0.166667em}{0ex}}\left[{}^{\xb0}\right]\phantom{\rule{0.166667em}{0ex}}$ | $\mathit{\phi}\phantom{\rule{0.166667em}{0ex}}\left[{}^{\xb0}\right]\phantom{\rule{0.166667em}{0ex}}$ |
---|

W1 | $7.945$ | $10.0$ | $-16.588$ | $39.0$ |

W2 | $12.00$ | $6.0$ | $+0.0$ | $0.0$ |

W3 | $5.022$ | $2.0$ | $+26.079$ | $108.7$ |

**Table 4.**
Relative errors for $\omega $, ${\u03f5}_{\omega}$ (in %), as a function of $\Delta t$ and ${R}_{t}$ for $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, corresponding to linear propagation of W1.

**Table 4.**
Relative errors for $\omega $, ${\u03f5}_{\omega}$ (in %), as a function of $\Delta t$ and ${R}_{t}$ for $\Delta x=2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, corresponding to linear propagation of W1.

${\mathit{R}}_{\mathit{t}}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{\Delta}\mathit{t}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ |
---|

$\mathbf{0.25}$ | $\mathbf{0.50}$ | $\mathbf{1.0}$ | $\mathbf{2.0}$ |
---|

$0.5$ | $-0.066$ | $-0.049$ | — | — |

$1.0$ | $-0.101$ | $-0.092$ | $-0.051$ | — |

$2.0$ | $-0.105$ | $-0.105$ | $-0.095$ | $0.024$ |

$4.0$ | $-24.7$ | $-48.4$ | $-72.7$ | $-55.4$ |

**Table 5.**
Relative RMSE for k, ${\u03f5}_{k}^{\mathrm{RMS}}$ (in %) for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, as a function of $\Delta x$ and ${R}_{x}$ for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ and ${R}_{t}=1.0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ corresponding to linear propagation of W1.

**Table 5.**
Relative RMSE for k, ${\u03f5}_{k}^{\mathrm{RMS}}$ (in %) for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, as a function of $\Delta x$ and ${R}_{x}$ for $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ and ${R}_{t}=1.0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ corresponding to linear propagation of W1.

${\mathit{R}}_{\mathit{x}}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{\Delta}\mathit{x}\left[\mathbf{m}\right]$ |
---|

1 | 2 | 4 | 10 |
---|

2 | $0.583$ | $0.578$ | — | — |

4 | $0.542$ | $0.530$ | $0.508$ | — |

8 | $0.466$ | $0.468$ | $0.487$ | — |

12 | $1.279$ | $1.002$ | $4.579$ | $0.741$ |

16 | $5.534$ | $5.453$ | $6.091$ | $0.732$ |

**Table 6.**
Summary of the results of the PCA obtained using phase fitting (without windowing) for linear wave propagation and $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2.0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1.0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, and ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$: Relative RMS errors ${\u03f5}_{k}^{\mathrm{RMS}}$ and ${\u03f5}_{h}^{\mathrm{RMS}}$ are given for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Next to the retrieved period, the corresponding wave field is indicated.

**Table 6.**
Summary of the results of the PCA obtained using phase fitting (without windowing) for linear wave propagation and $\Delta t=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, $\Delta x=2.0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, ${R}_{t}=1.0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$, and ${R}_{x}=8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$: Relative RMS errors ${\u03f5}_{k}^{\mathrm{RMS}}$ and ${\u03f5}_{h}^{\mathrm{RMS}}$ are given for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Next to the retrieved period, the corresponding wave field is indicated.

| Mode | ${\mathit{\sigma}}^{2}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{T}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathit{\u03f5}}_{\mathit{\omega}}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathit{\u03f5}}_{\mathit{k}}^{\mathbf{RMS}}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathit{\u03f5}}_{\mathit{h}}^{\mathbf{RMS}}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ |
---|

| 1 | $99.2$ | $7.952$ (W1) | $-0.092$ | $0.468$ | $1.083$ |

monochromatic | 1 | $98.7$ | $11.993$ (W2) | $0.062$ | $0.526$ | $1.085$ |

| 1 | $99.9$ | $5.022$ (W3) | $-0.003$ | $4.232$ | $18.572$ |

| 1 | $71.0$ | $7.955$ (W1) | $-0.122$ | $1.432$ | $3.182$ |

polychromatic | 2 | $25.3$ | $11.949$ (W2) | $0.423$ | $2.703$ | $5.826$ |

| 3 | $2.9$ | $5.028$ (W3) | $-0.124$ | $6.665$ | $27.554$ |

**Table 7.**
Results of the PCA obtained for nonlinear wave propagation of the monochromatic W1 case with different $\mathrm{F}$ factors: Next to the retrieved period, the corresponding wave field is indicated (when applicable).

**Table 7.**
Results of the PCA obtained for nonlinear wave propagation of the monochromatic W1 case with different $\mathrm{F}$ factors: Next to the retrieved period, the corresponding wave field is indicated (when applicable).

Factor $\mathbf{F}$ | Mode | ${\mathit{\sigma}}^{2}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{T}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ |
---|

$0.25$ | 1 | $98.7$ | $7.953$ (W1) |

$1.0$ | 1 | $94.6$ | $7.948$ (W1) |

2 | $3.9$ | $3.961$ ( — ) |

$2.5$ | 1 | $87.6$ | $7.955$ (W1) |

2 | $8.4$ | $3.958$ ( — ) |

3 | $2.0$ | $2.637$ ( — ) |

**Table 8.**
Summary of the results for nonlinear wave propagation of the polychromatic WS case with different $\mathrm{F}$ factors: Relative RMSE, ${\u03f5}_{h}^{\mathrm{RMS}}$, are given for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Here, “p” and “f” stand for phase and function fitting of the wavenumber. Next to the retrieved period, the corresponding wave field is indicated (when applicable).

**Table 8.**
Summary of the results for nonlinear wave propagation of the polychromatic WS case with different $\mathrm{F}$ factors: Relative RMSE, ${\u03f5}_{h}^{\mathrm{RMS}}$, are given for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Here, “p” and “f” stand for phase and function fitting of the wavenumber. Next to the retrieved period, the corresponding wave field is indicated (when applicable).

Factor $\mathbf{F}$ | Mode | ${\mathit{\sigma}}^{2}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{T}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ | ${\mathit{\u03f5}}_{\mathit{h}}^{\mathbf{RMS}}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ |
---|

$\mathbf{p}$ | $\mathbf{f}$ |
---|

| 1 | $58.6$ | $7.954$ (W1) | $19.0$ | $10.1$ |

$0.25$ | 2 | $36.2$ | $11.882$ (W2) | $54.5$ | $23.4$ |

| 3 | $3.1$ | $5.023$ (W3) | $24.8$ | $10.7$ |

| 1 | $52.5$ | $7.951$ (W1) | $13.5$ | $11.7$ |

$1.0$ | 2 | $34.2$ | $11.870$ (W2) | $37.1$ | $25.2$ |

| 3 | $4.2$ | $4.779$ ( — ) | — | — |

| 1 | $45.4$ | $7.964$ (W1) | $30.3$ | $31.7$ |

$2.5$ | 2 | $31.6$ | $11.849$ (W2) | $36.4$ | $39.0$ |

| 3 | $6.9$ | $4.788$ ( — ) | — | — |

**Table 9.**
Results for the first mode for nonlinear wave propagation of the polychromatic WS case with different $\mathrm{F}$ factors, total video length ${t}_{\mathrm{max}}$, and window width ${w}_{t}$: Relative RMSE, ${\u03f5}_{h}^{\mathrm{RMS}}$, is given for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Here, “p” and “f” stand for results using phase and function fitting for ${t}_{\mathrm{max}}$, respectively. The number of sub-videos are included in parentheses.

**Table 9.**
Results for the first mode for nonlinear wave propagation of the polychromatic WS case with different $\mathrm{F}$ factors, total video length ${t}_{\mathrm{max}}$, and window width ${w}_{t}$: Relative RMSE, ${\u03f5}_{h}^{\mathrm{RMS}}$, is given for $h\u2a7e0.75\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Here, “p” and “f” stand for results using phase and function fitting for ${t}_{\mathrm{max}}$, respectively. The number of sub-videos are included in parentheses.

${\mathit{t}}_{\mathbf{max}}$ | $\mathbf{F}$ | ${\mathit{\u03f5}}_{\mathit{h}}^{\mathbf{RMS}}\phantom{\rule{0.166667em}{0ex}}[\%]\phantom{\rule{0.166667em}{0ex}}$ |
---|

p | f | Windowing, ${\mathit{w}}_{\mathit{t}}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]\phantom{\rule{0.166667em}{0ex}}$ |
---|

$\mathbf{30}\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{40}\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{60}\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{80}\phantom{\rule{0.166667em}{0ex}}$ | $\mathbf{90}\phantom{\rule{0.166667em}{0ex}}$ |
---|

| | | | (112) | (94) | (56) | (18) | (1) |

| $0.25$ | $19.0$ | $10.1$ | $28.0$ | $13.1$ | $13.6$ | $16.2$ | $19.0$ |

$90$ s | $1.00$ | $13.5$ | $11.7$ | $28.1$ | $10.3$ | $10.8$ | $14.1$ | $13.5$ |

| $2.50$ | $30.3$ | $31.7$ | $86.1$ | $91.0$ | $83.8$ | $30.8$ | $30.3$ |

| | | | (225) | (207) | (169) | (131) | (113) |

| $0.25$ | $18.4$ | $8.2$ | $24.8$ | $11.6$ | $12.0$ | $11.9$ | $11.0$ |

$150$ s | $1.00$ | $13.0$ | $10.1$ | $24.2$ | $8.2$ | $8.8$ | $8.6$ | $7.0$ |

| $2.50$ | $27.5$ | $26.5$ | $99.1$ | $52.5$ | $43.2$ | $30.9$ | $15.1$ |

**Table 10.**
Summary of the results for the field site video analysis.

**Table 10.**
Summary of the results for the field site video analysis.

| cBathy | uBathy |
---|

percentage of points | $60\%$ | $84\%$ |

average error (bias) | $-0.50\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $-0.27\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ |

RMS error | $1.38\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $1.29\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ |