# Unmanned Aerial Vehicle Depth Inversion to Monitor River-Mouth Bar Dynamics
## 3 Depth Estimation Method
### Estimation of Wave Celerity and Direction  
Despite complex optical processes on the wavy sea surface, pixel intensity and water level signals show similar temporal characteristics, but the intensity signals are affected by different factors [11]. Therefore, it is common to use a bandpass filter to remove low- and high-frequency components; the former involves light changes due to clouds, whereas the latter involves changes due to wind waves, the camera auto-iris control, and camera movement. [PERSON] et al. (2009) [11] used a bandpass filter (0.05-0.5 Hz) to eliminate the noise signals. [PERSON] et al. (2020) [19] further narrowed the band to 0.05-0.2 Hz to target swells. We follow the latter approach and filter signals out of 0.05-0.2 Hz. There are two popular methods to extract wave parameters from coastal video imagery: temporal and spectral methods. Each method has its advantages and disadvantages. [PERSON] and [PERSON] (2018) [23] compared these methods and confirmed that both, in general, estimate bathymetry with similar accuracy. They also suggested that the temporal method works better for cases with a limited number of wave components. [PERSON] et al. (2009) [10] pointed out that the temporal method is robust for various waveforms; intensity signals are often far from being sinusoid in the nearshore area. Given its advantage for narrow-band waves, we employed the temporal method in the present study. The temporal method estimates the signal propagation time based on cross-correlation between intensity signals at two points. The wave celerity can be simply calculated by taking the distance between the two points divided by the propagation time. This can be  
Figure 5: The parameter \(G\) as a function of water depth and wave frequency. extended for a two-dimensional case by cross-correlation analysis, both in the cross-shore and alongshore directions. We take a different approach to realise the robust estimation of a wave-celerity vector field over non-uniform coastal bathymetry. By letting \(A\) be an analysis point at which the wave celerity is estimated, we can compare wave signals between \(A\) and \(B_{\text{n}}\) aligned along a circumference of radius \(r\) with \(A\) at its centre (\(n=1\) to \(N\)). Figure 6 illustrates the circular alignment for the case of \(N=8\). \(B_{1}\) is placed such that \(AB_{1}\) is parallel to the horizontal axis of the orthogonal image, and \(B_{2}\),... \(B_{N}\) are aligned in the counter-clockwise direction from \(B_{1}\) with the angle \(\theta_{\text{n}}\) being \(\angle B_{\text{n}}AB_{1}\). We employed a circular array to determine the measurement distance \(r\) and the number of reference points \(N\). We will later discuss how the two independent parameters affect the estimation accuracy. First, we evaluate the propagation time \(t_{\text{n}}\) from \(A\) to \(B_{\text{n}}\) as time lag to maximise the cross correlation of bandpass-filtered signals at the two points. When a wavefront propagates with the wave angle \(\theta\), the distance between the wavefront passing \(A\) and \(B_{\text{n}}\) is equal to \(r\text{cos}(\theta-\theta_{\text{n}})\). Therefore, the propagation time can be expressed as  
\[t_{\text{n}}=\frac{r\cos(\theta\theta_{\text{n}})}{c} \tag{3}\]  
Equation (3) suggests that \(t_{\text{n}}\) is represented by a sinusoidal function of \(\theta_{\text{n}}\) when a monochromatic or narrow-band wave propagates over the array. The cross-correlation analysis is affected by noise signals, so it may produce an abnormal time lag. To eliminate this, we can obtain a regression sine curve and exclude reference points where \(t_{\text{n}}\) deviates significantly. The deviation mainly occurs when the array overlaps with land area or surf zones. Next, we estimate \(c\) and \(\theta\) from the \(N\) equations using the non-linear least-squares method. Given initial estimates \(c_{0}\) and \(\theta_{0}\), the true values can be expressed as \(c=c_{0}+dc\) and \(\theta=\theta_{0}+d\theta\), respectively.