# Retrieval of Nearshore Bathymetry around Ganquan Island from LiDAR Waveform and QuickBird Image
## 3 Materials and Methods
### Data Processing and Method
#### 3.2.3 LiDAR Waveform Processing  
1. Depth extraction from waveform data  
Extracting the signals of the water surface and the bottom part from an echo waveform is the key to retrieving bathymetry based on the LiDAR waveform. The retrieving methods and parameters of the sounding system were not disclosed, but they seem to apply the commonly used peak detection method [9; 18] and mathematical fitting method [12; 38; 39]. A burr phenomenon always exists in the echo waveform signals arising from noise. A big error occurs when the peak detection method is directly used to extract the time in which the echo signals return to the water surface and the bottom. The method of filtering, denoising, and processing affects the response of the water component to the seabed waveform. Although the water component is not the focus of our extraction, it has great influence on the seabed waveform. By analyzing the radiation transmission of LiDAR in water bodies [11], we find that the received echo signals approximate a Gaussian distribution [21; 39]; thus, we use the multi-Gaussian function fitting to extract the target echo signal. \[f(t)=\sum_{i=1}^{n}a_{i}exp\Big{(}-((t-b_{i})/c_{i})^{2}\Big{)}+d, \tag{9}\]where \(f(t)\) is the fitting curve expression based on the Gaussian function, \(a_{i}\), \(b_{i}\), and \(c_{i}\) are the parameters of the function, \(i\) is the number of Gaussian function, \(d\) is the correction parameter, and \(t\) is time. We construct the objective function as  
\[F(a_{i},\ b_{i},c_{i}d)=\frac{1}{N}\sum_{t=t_{bg}}^{t_{end}}\left(f(t)-P_{T}(t) \right), \tag{10}\]  
where \(t_{bg}\) is the start time of the echo signal reception, and \(t_{end}\) is the end time of the echo signal reception. The Levenberg-Marquardt algorithm is used to solve the optimization problem as follows:  
\[\overline{F}=min(F(a_{i},\ b_{i},c_{i}d)). \tag{11}\]  
Given a reasonable initial value, we can derive the coefficient \(a_{i}\), \(b_{i},c_{i}d\) after optimal operation and then use the peak detection method to extract the peak position of the fitting curve for the echo signals of the water surface and the bottom part as \((t_{1},p_{1})\) and \((t_{2},p_{2})\), respectively. The expression of the peak detection method is shown in Equation (12). The difference between the \(x\)-coordinates of the two peak values is the time difference of the return of echo signal from the water surface and the bottom. The bathymetry can be calculated as shown in Equation (13). \[k=find(diff(sign(diff(f^{\prime}(t))))<0)+1, \tag{12}\]  
\[Z=c_{w}(t_{2}-t_{1})\cos\theta_{w}/2, \tag{13}\]  
where \(f^{\prime}(t)\) is the sum of the Gaussian function used to iterate and fit the LiDAR echo signal, \(diff(\cdot)\) is the difference between two consecutive points, i.e., \(diff(f^{\prime}(t))=f^{\prime}(t+1)-f^{\prime}\); \(sign(\cdot)\) is a sign function to depict whether the input value is negative, positive, or zero, in which the return value is \(-1\), \(1\), or \(0\), respectively, \(find(\cdot)\) returns the indices of the elements to satisfy the expressions inside it, \(c_{w}\) is the speed of light in water, and \(\theta_{w}\) is the local incidence angle in water. The blue points in Figure 5 represent the echo signals of LiDAR, while the red curve represents the fitting curve based on the Gaussian function. 2. Datum conversion  
The bathymetry obtained by resolving the LiDAR waveform corresponds to instantaneous water depth. The bathymetric data need to be corrected by using tide data to derive the water depth value for the acquisition time of the optical remote sensing image. The relationship of instantaneous sea  
Figure 5: LiDAR echo signal detection based on the fitting of Gaussian function. surface, mean sea level, depth, and water bottom is shown in Figure 6. However, the LiDAR data were acquired from 11:00 a.m. to 2:00 p.m. on 19 January 2013, and the collection time of each point differed. Therefore, performing a tide correction for each control point and validation point is necessary to eliminate the errors caused by different tides.