# Retrieval of Nearshore Bathymetry around Ganquan Island from LiDAR Waveform and QuickBird Image
## 3 Materials and Methods
### Data Processing and Method
#### 3.2.2 Water Depth Inversion Model  
1. Principle of bathymetry based on passive optical remote sensing  
Sunlight has a perspective in a water column. The signal received by the optical remote sensor contains reflection information of the sea bottom to the sunlight, and this information is the physical basis of the passive optical remote sensing inversion of shallow water depth. In other words, the information reflected from the seabed to the optical remote sensor is the direct reflection of underwater terrain, that is, the information source of optical remote sensing bathymetry. The attenuation coefficient of sunlight in water determines the penetrable depth of the light in water. The smaller the attenuation coefficient of light in water is, the better its transparency in water is. This phenomenon is the basic principle of water depth inversion.  
In the process of radiation transmission, a series of complicated changes take place in the light after sunlight passes through the atmosphere and the water column, then to the bottom of the water, before finally entering the satellite sensor. As shown in Figure 4, the satellite sensor receives the radiation signal \(L_{t}\) mainly in four parts.  
\[L_{t}=L_{p}+L_{S}+L_{w}+L_{b}, \tag{6}\]  
Figure 4: Interaction between electromagnetic wave and water.  
where \(L_{p}\) is the atmospheric radiation, \(L_{S}\) is the reflection radiance of the water surface, \(L_{w}\) is the water column radiance, and \(L_{b}\) is the bottom radiance. The radiative transfer equation of electromagnetic waves in water is the physical basis of water depth inversion. [PERSON] [28; 29] and [PERSON] [30] developed the water depth radiation transfer equation according to Beer's law.  
\[L=L_{\infty}(1-exp(-gz))+A_{d}exp(-gz), \tag{7}\]  
where \(L=L_{t}-L_{p}-L_{s}\) is the radiative brightness value after atmospheric correction and radiative brightness correction of the water column in deep water, \(L_{\infty}\) is the radiation brightness value at infinite depth, \(g\) is the two-way diffuse attenuation coefficient in the water body, \(z\) is the water depth value, and \(A_{d}\) is the reflected energy value of the bottom.  
2. Remote sensing bathymetric model  
The radiation transfer equation and the basic principle of water depth inversion led to the rapid development of the bathymetric method, which mainly comprises the theoretical analytical model [31], single-band water depth inversion model [32; 33], multi-band log-linear model [34; 35; 36], and logarithmic conversion ratio model [37]. [PERSON] [4] improved the method on the basis of the classic Stumpf model to significantly improve water depth inversion ability. Considering the risk that the denominator of the Tian model may be equal to zero, we added the constant factor \(a\) to the bathymetric model. The \(a\) value may not increase model accuracy, but it can increase the stability of the model to a certain extent. The passive optical remote sensing bathymetric model used in this study was named ILCRM, expressed as follows:  
\[Z=a_{0}\frac{\ln(mR_{w}(\lambda_{i})+a)}{\ln(nR_{w}(\lambda_{j})+a)}+a_{1}, \tag{8}\]  
where \(Z\) is the water depth, \(R_{w}(\lambda_{i})\) and \(R_{w}(\lambda_{j})\) are the reflectance data of the blue and green bands, respectively, \(a_{0}\) and \(a_{1}\) are regression coefficients, \(m\) and \(n\) are the adjustment factors, and \(a\) is the added constant factor (i.e., users can customize its value, such as setting it to 1.01). Because blue and green bands have strong transmission ability to water body, the signals received by sensors contain seabed information; thus, the two bands were used in this study. This method belongs to the classification of semi-empirical and semi-analytical models. A part of the water depth obtained from LiDAR waveform data was used as the control point to calculate the model parameters of \(a_{0}\), \(a_{1}\), \(m\), and \(n\). The other part of the water depth data was used as the validation point to evaluate the accuracy of the bathymetric model.