## 3 Methods
### Noise Removal and Signal Range Selection  
The waveform includes background noise and random noise caused by environmental and systematic factors. Background noise is long-term, low-frequency noise that is mainly related to solar radiation and detector dark current. In contrast, random noise is short-term, high-frequency noise that is mainly caused by random fluctuations inherent in the acquisition [36]. Systematic background noise tends to be uniformly distributed throughout the duration of the signal and is traditionally removed by applying a threshold [23; 36]. However, because the amount of background noise can slightly vary for each waveform, applying a constant threshold to all waveforms acquired in a single flight may not be appropriate. Accordingly, a threshold value for each waveform, such as the mean value at the non-signal range [29] or a wavelet adaptive threshold [37], is commonly used. Because the main contribution to DN in the non-signal range occupying most of the waveform is background noise, we assumed that the most frequently recorded DN value in each waveform would be reflecting the background noise. The background noise is removed by subtracting the mode of an individual waveform, as follows:  
\[y^{*}=y-mode\left(y\right), \tag{2}\]  
Figure 3: Workflow of progressive Gaussian decomposition (PGD) process. where \(y\) and \(y^{*}\) are the original waveform and the waveform with background noise removed, respectively. Low-pass filtering techniques are mainly used to remove random noise [38; 39]. In this study, the waveform is smoothed through Gaussian filtering to remove random noise:  
\[y^{\prime}_{t}=\frac{1}{\sqrt{2\pi\sigma_{\mathrm{G}}^{2}}}\sum_{k=1}^{w}y^{*}_ {k}\;exp\left(-\frac{\left(t-k\right)^{2}}{2\sigma_{\mathrm{G}}^{2}}\right), \tag{3}\]  
where \(y^{\prime}_{t}\) is the Gaussian filtered value at time \(t\), \(w\) indicates the number of waveform samples, and \(\sigma_{\mathrm{G}}\) is the standard deviation of the Gaussian distribution. The larger \(\sigma_{\mathrm{G}}\) is, the more smoothing is performed, which may be effective in removing noise, but a return signal with low intensity may be ignored. In this study, \(\sigma_{\mathrm{G}}\) was \(1\). Figure 4 presents an example of a waveform signal before and after noise removal. In a waveform with thousands of samples, the range over which the signal reflected from the actual targets is recorded occupies a short part of all the recording samples [25; 27]. The range depends on the flight altitude, transmission angle, topographic elevation, and water depth. Therefore, if the actual signal range can be determined, the waveform processing can be more efficient. Moreover, the generation of noise-induced outlier points can be avoided. In the criterion suggested in [40], signal range was detected if the waveform was at least triple the standard deviation of noise for a \(5\) ns duration. Another study [27] assumed the last \(10\%\) of the waveform as the non-signal range and estimated the mean and standard deviation of the amplitude of this interval as the noise level. The interval with an amplitude greater than the mean plus three standard deviations was selected as the signal range. The Seahawk system records \(2400\) waveform samples over a period of \(1.5\)\(\upmu\)s, during which no reflection signal is returned for the first \(0.11\)\(\upmu\)s. In this study, the first \(0.1\)\(\upmu\)s (\(160\) bins) of the Seahawk waveform was considered as the non-signal range, and the signal range was selected based on the standard deviation \(\sigma_{N}\) in this period. The start and end points of the signal range, \(t_{S}\) and \(t_{E}\), respectively, are determined as follows (Figure 5). \[t_{S}=min\big{\{}\ t\ \big{|}\ y^{\prime}_{t+1}-y^{\prime}_{t}>3\sigma_{N}\ \big{\}}, \tag{4}\]  
\[t_{E}=max\big{\{}\ t\ \big{|}\ y^{\prime}_{t+1}-y^{\prime}_{t}<-1.5\sigma_{N}\ \big{\}}, \tag{5}\]  
Figure 4: Examples of noise removal: (**a**) original bathymetric waveform; (**b**) waveform after background and random noise removal.