# Spaceborne GNSS-R Observation of Global Lake Level: First Results from the TechDemoSat-1 Mission
## Data and Methods
### Lake Level Retrieval  
The time delay of the specular reflected signal with respect to the direct signal determines the retrieval of a relative surface height [22]. There are multiple methods to track the specular point from the waveform extracted from DDM at central Doppler, e.g., using a given point of leading edge maximum derivative (LED), using the maximum of the waveform (MAX), and using the half or a fractional power point (HALF) [23]. In this study, the HALF method is implemented since previous studies have shown that this method is simple, computationally efficient, and more precise than other point tracking techniques [24].In order to determine the delay at the specular point more accurately (to overcome the coarse sampling of the waveform), the delay waveform is interpolated using a spline interpolation method and the interpolation factor is set as 1000, the same as the process for the observation of sea surface height in [3].  
The WGS84 ellipsoid reference is used to measure lake water level from TDS-1. The water level is calculated using the following Equations (1)-(3):  
\[\Delta\tau=\tau_{obs}-\tau_{ref} \tag{1}\]  
\[\Delta h=\Delta\tau\text{c}/2\cos\theta \tag{2}\]  
\[\tau_{obs}=\tau_{window}+\tau_{gp} \tag{3}\]  
Figure 1: Global lakes (area greater than 500 km\({}^{2}\) and elevation lower than 3000 m each) used for water level retrieval in this study. There are 351 lakes (blue solid circles), with 284 in the Northern hemisphere and 67 in the Southern hemisphere. Red solid triangles represent the overlapped 89 lakes in the Hydroweb data. All the lakes are overlapped with the Cryosat-2 data.  
where, \(c\) is the speed of light. As shown in Figure 2a, \(\theta\) is the incidence angle of each specular point. \(\tau_{obs}\) and \(\tau_{ref}\) are the observable time delay with respect to the direct signal and the time delay with respect to the direct signal of the hypothetical specular point on the WGS84 ellipsoid reference, respectively. \(\Delta h\) is the altimetry of the specular point over lake surface relative to the WGS84 ellipsoid reference. As shown in Figure 2b, \(\tau_{window}\) and \(\tau_{sp}\), are the delay of DDM window and the delay of specular point within the DDM window, respectively.  
The TDS-derived lake water level (\(\Delta h_{TDS}\)) is susceptible to large errors due to measurement noise, orbit uncertainty, ionosphere/troposphere delay, and limited receiver bandwidth, which may lead to a few outliers. Therefore, we filter the raw results using the following criterion to remove the nonphysical outliers [3]:  
\[\Delta h_{TDSman}-3\times\left(\Delta h_{CSman}-\Delta h_{CSmin}\right)\leq \Delta h_{TDS}\leq\Delta h_{TDSman}+3\times\left(\Delta h_{CSmax}-\Delta h_{CSman}\right) \tag{4}\]  
where \(\Delta h_{TDSman}\) is the mean TDS-1 lake level. \(\Delta h_{CSman}\), \(\Delta h_{CSmin}\), and \(\Delta h_{CSmax}\) are the mean, minimum, and maximum lake levels, respectively, of each lake from CryoSat-2.  
The altimetry of each specular point with respect to the WGS84 ellipsoid reference is converted to that with respect to the GGM02C. The lake level is finally calculated as the average of the altimetry values of those specular points which fell into the lake during a given time period.