# Airborne GNSS Reflectometry for Water Body Detection
## 4 Segmentation of the GNSS Signal
### Change Detection  
We show in Figure 5 the architecture of the change point detector. The amplitudes of the direct and reflected GNSS signals are estimated with an Extended Kalman Filter (EKF) that uses the in-phase components of the signals \(y_{k}\) as observations based on the model proposed in [44]. A Kalman filter and a CUSUM algorithm are used to detect a change at instant \(t_{I,i}^{a}\). \(l\) is the satellite and \(i\) is an instant of time. \(y_{k}\) and \(x_{k}\) are respectively the state and measurement vectors. \(\hat{x}_{k}\) represents the estimated GNSS signal amplitudes.  
Figure 4: Signal model in a working window. \(m_{1}\) and \(m_{2}\) represent the mean values of the GNSS signal amplitudes when the ellipses are on land and on water, respectively.  
Figure 3: Satellite footprints displacement from one area to another. The area is located at (50.888653\({}^{\circ}\)N, 1.870741\({}^{\circ}\)E). \(m_{1}\) and \(m_{2}\) represent the mean values of the GNSS signal amplitudes when the ellipses are on land and on water, respectively.  
is the innovation of the EKF and \(S_{k}\) is the covariance of the innovation. When a change is detected an alarm is generated to initialize the Kalman filter.  
We show in Figure 6 the GNSS signal amplitude and the filter innovation as a function of time. The CUSUM detection algorithm uses an integration of the innovation process in order to detect the changes in the GNSS measurements. At a transition of the signal, between two stationary pieces, the innovation increases and then decreases (or vice versa) to produce a peak. When a change occurs in the mean of the innovation, the mean value after the change is either \(\mu_{1}^{+}=\mu_{0}+\
u\) or \(\mu_{1}^{-}=\mu_{0}-\
u\) with \(\
u\), the dynamic of the change, assumed to be known. In this work we use a two-side CUSUM algorithm [47]. The CUSUM detection process is defined by  
\[g_{k}^{+} = \left(g_{k-}^{+}+\epsilon_{k}-\mu_{0}-\
u/2\right)^{+} \tag{2}\] \[g_{k}^{-} = \left(g_{k-}^{-}-\epsilon_{k}+\mu_{0}-\
u/2\right)^{+}\] (3) \[\hat{t}_{l,i}^{a} = min\{k:\left(g_{k}^{+}\geq\lambda\right)\cup\left(g_{k}^{-}\geq \lambda\right)\} \tag{4}\]  
where \((X)^{+}=max(X,0)\) and \(t_{l,i}^{a}\) is the \(ith\) instant of change detected for satellite \(l\). Before the change, the mean of the innovation process is \(\mu_{0}=0\), and \(g_{k}^{+}\) and \(g_{k}^{-}\), the integration of the innovation process, evolves as a Gaussian random walk. After the change instant \(w+t\) defined in Figure 4, \(g_{k}^{+}\) and \(g_{k}^{-}\) are monotonic increasing functions. In our implementation, we assume that the innovation before and after the change is distributed according to a Gaussian distribution notice \(N(\mu,\sigma)\) where \(\mu\) and \(\sigma\) are, respectively, the mean and the standard deviation of the distribution. The normalized innovation process \(\epsilon_{k}S_{k}^{-1/2}\) is distributed according to \(N(0,1)\) before the change and \(N(\
u,1)\) after the change.  
Figure 5: Architecture of the change point detector.  
Figure 6: GNSS signal amplitude and filter innovation as a function of time.  
According to the transition model defined in Figure 4, the theoretical value of the detection threshold \(\lambda\) can be defined as \(\lambda=\frac{\Delta t\,\
u}{2}-\frac{\
u}{2}\) with \(\
u=m_{2}-m_{1}\). In practice, \(\
u\) is a parameter defined by the user. It represents the minimum change dynamic that we want to detect. In our application, \(\Delta t\), the length of the transition area in the change model, is defined with the satellite elevation angle, as well as the speed and height of the airborne carrier.