# Unmanned Aerial Vehicle Depth Inversion to Monitor River-Mouth Bar Dynamics
## 3 Depth Estimation Method
### Estimation of Wave Celerity and Direction
By letting \(F_{\text{n}}\) (\(c\), \(\theta\)) be the right-hand side of Equation (3), we can rewrite the said equation as:  
\[t_{\text{n}}=F_{\text{n}}(c_{0},\theta_{0})+\left(\frac{\partial F_{\text{n} }}{\partial c}\right)_{0}dc+\left(\frac{\partial F_{\text{n}}}{\partial \theta}\right)_{0}d\theta+v_{\text{n}}, \tag{4}\]  
Figure 6: The circular array for estimating wave parameters (\(N=8\)). Point \(A\) indicates the analysis point, and \(B_{\text{n}}\) is the \(n\)-th reference point along the circumference of radius \(r\). where \(v_{\text{n}}\) is the residual, and the subscript 0 indicates the initial estimate. The \(N\) equations can be cast into a single matrix equation as  
\[\textbf{Y}d\textbf{X}=\textbf{L}-\textbf{V} \tag{5}\]  
with  
\[\textbf{Y}=\left[ \begin{array}{cc}\left(\frac{\partial\textbf{F}_{1}}{ \partial\theta}\right)_{0}&\left(\frac{\partial\textbf{F}_{1}}{\partial c} \right)_{0}\\ \vdots&\vdots\\ \left(\frac{\partial\textbf{F}_{N}}{\partial\theta}\right)_{0}&\left(\frac{ \partial\textbf{F}_{N}}{\partial c}\right)_{0}\end{array} \right],\ \ \ d\textbf{X}=\left[ \begin{array}{c}d\theta\\ dc\end{array} \right],\ \ \ \textbf{L}=\left[ \begin{array}{c}t_{1}-F_{1}(c_{0},\theta_{0})\\ \vdots\\ t_{\text{n}}-F_{N}(c_{0},\theta_{0})\end{array} \right],\ \ \ \textbf{V}=\left[ \begin{array}{c}v_{1}\\ \vdots\\ v_{N}\end{array} \right]\]  
Since the propagation time variation along the circumference could be produced by noise signals, we introduce a weighting matrix **P** into Equation (5). The weighting matrix is based on the cross-correlation coefficient \(R_{\text{n}}\) of two signals at \(A\) and \(B_{\text{n}}\) at the time lag \(t_{\text{n}}\) and can be expressed as:  
\[\textbf{P}=\left[ \begin{array}{ccc}\frac{R_{1}}{NR}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\frac{R_{N}}{NR}\end{array} \right]\text{ with }R=\frac{1}{N}\sum_{n=1}^{N}R_{ \text{n}}, \tag{6}\]  
where \(R\) represents the cross-correlation coefficient averaged over the circumference. Hence, we give a higher weight to the propagation time estimated with higher cross-correlation. By introducing **P** into Equation (5), the normal equation that minimises \({}^{t}\textbf{V}\textbf{PV}\) can be expressed as:  
\[(tYPY)dX=tYPL \tag{7}\]  
The values of \(c\) and \(\theta\) are obtained by solving Equation (7) for \(d\textbf{X}\) and updating them successively until both \(\left\lvert dc\right\rvert\) and \(\left\lvert d\theta\right\rvert\) produces less than 0.01 m/s and 0.01\({}^{\circ}\), respectively. Figure 7 illustrates an example of the estimation of a wave-celerity vector field for Case 3. The estimation was performed with \(r=20\) m and \(N=8\), and the initial estimates were given as \(c_{0}=6\) m/s and \(\theta_{0}=90^{\circ}\), which is perpendicular to the shoreline (Figure 7a). After the iterative solutions of Equation (7), the wave-celerity vector field converges to exhibit wave refraction over the river-mouth bar (Figure 7b). The estimation does not work due to wave breaking and superposition of refracted waves in the vicinity of the river mouth. Moreover, the dispersion relation is significantly affected by the river and tidal flows in the area. Therefore, the depth inversion from wave information is not possible in this area. Nevertheless, we can obtain a reasonable distribution of wave-celerity vectors of swells in a broad region apart from the river mouth. Figure 7: Computation of wave celerity vectors. (**a**) Initial estimate and (**b**) converged result.