# Circular Regression Applied to GNSS-R Phase Altimetry
## 3 Linear-Circular Regression Applied to GNSS-R Altimetry
### Initialization and Assessment  
Figure 3 shows a theoretical representation of \(W\left(Y\right)\) as a function of the slope \(\beta\). We observe that the contrast function \(W(Y)\) has several local maxima and a global maximum corresponding to \(\beta\)th, the value to estimate. The Newton-Raphson algorithm is a local maximization technique. The estimate \(\hat{\beta}\) thus depends on the initialization \(\hat{\beta}^{i=0}\). In order to reach the global maximum, we propose to initialize \(\hat{\beta}^{i=0}\) with an additional search step.  
Due to the periodicity of the angular data, the local maxima of \(W(Y)\) are distributed periodically, with a period denoted \(\beta_{T}\). This period only depends on the \(x_{s}^{el}\) parameter. The period \(\beta_{T}\) can thus be obtained by numerically solving the following unnoisy formula, where the measurement component is canceled (which corresponds to \(\beta\)th = 0):  
\[\sum\limits_{k=1}^{N}\cos\left(-\arctan^{*}\left(\frac{\sum\limits_{k=1}^{N} \sin\left(-\beta_{T}x_{s,k}^{el}\right)}{\sum\limits_{k=1}^{N}\cos\left(-\beta_ {T}x_{s,k}^{el}\right)}\right)-\beta_{T}x_{s,k}^{el}\right)=0 \tag{26}\]  
which can be rewritten as:  
\[\sum\limits_{k=1}^{N}\cos\left(\beta_{T}\left(\overline{x_{s}^{el}}-x_{s,k}^{el }\right)\right)=0 \tag{27}\]  
where \(\overline{x_{s}^{el}}\) is the mean of \(\left\{x_{s,k}^{el}\right\}_{k=1:N}\).  
The period \(\beta_{T}\) can be obtained as the minimum value that solves this equation. \(\dot{\beta}^{i=0}\), the initialization of the Newton-Raphson algorithm, can then be obtained by realizing a first fast global maximization of \(W\left(Y\right)\) with a search step of \(\beta_{T}\).  
The expected accuracy of the proposed method is the same as the variance of the classic linear regression. For our model, the theoretical variance of the estimate is:  
\[var\left(\hat{h}\right)=\frac{\lambda}{4\pi}\ \frac{{\sigma_{s}}^{2}}{\sum \limits_{k=1}^{N}\left(\overline{x_{s,k}^{el}}-\overline{x_{s}^{el}}\right)} \tag{28}\]  
The standard deviation \(\sigma_{s}\) of the noise is defined as in the linear domain by approximating the von Mises distribution by a wrapped normal distribution. It is shown in [30] that \(\sigma_{s}\) can be linked to \(\kappa_{s}\) using the following expression:  
\[{\sigma_{s}}^{2}=-2\ln\left(\frac{I_{1}(\kappa_{s})}{I_{0}(\kappa_{s})}\right) \tag{29}\]  
with \(\kappa_{s}\) the concentration parameter associated with the von Mises noise measurement of the observed phase delay.  
Figure 3: \(W\left(Y\right)\) as a function of \(\beta\).  
We can observe in Expression (28) that the variance of the estimate decreases (so the accuracy of the estimation increases) when the denominator increases. The denominator increases as the data are observed over a larger span of satellite elevation. In practice, in order to increase this span, large observation durations are needed. This is a major drawback of the proposed method. However, we show that it is possible to fuse several reflections observations using the contrast function defined in Equation (25). In this case, the denominator of Expression (28), and the estimation accuracy, is increased by using GNSS satellites that have different elevations.