# Stochastic Modeling of Smartphones GNSS Observations Using LS-VCE and Application to Samsung S20
## 2 Stochastic Modeling of Smartphone Observations Using Least-Square Variance Component Estimation
### Application of LS-VCE Method to the Noise Assessment of the Smartphone Observations
 \tag{7}\]  
in which \(\otimes\) denotes the Kronecker product, the satellite number \([1]^{G}\) and \([1]^{R}\) are assumed to be the reference satellites for GPS (G) and GLONASS (R) constellations, respectively; \(\rho^{GPS}\) and \(\rho^{GLO}\) are the correlation between the GPS code and carrier phase observations and the GLONASS code and carrier phase observations, respectively. The components of \(Q^{DD}_{GPS}\) and \(Q^{DD}_{GLO}\) matrices (\(\sigma^{2}_{[i]^{R}}\)) can be defined as a function of the satellite elevation angle. However, smartphones are equipped with linearly polarized antennas due to the low cost and limited space, resulting in degraded GNSS measurement quality compared with the high-end geodetic GNSS receivers. Therefore, the traditional elevation-dependent weighting method is not appropriate when employing the GNSS observations obtained from Android smartphones. Instead, the C/N0-dependent weighting method is a better option. In this contribution, we will use a combination of C/N0 and elevation-dependent weighting methods. The C/N0/elevation-based stochastic model of GNSS observations (\(\sigma^{2}_{[i]}\)) is of the form  
\[\sigma^{2}_{[i]}=\frac{10^{-\frac{C/N0}{10}}}{\sin E_{i}} \tag{8}\]  
in which \(C/N0\); is the carrier-to-noise ratio value and \(E_{i}\) is the satellite elevation angle. In Equations (6) and (7), \(a^{sys}_{type}\) is a constant needed to be determined for each device, each constellation (_sys_: GPS and GLONASS), and each observation type (_type_: code and phase). We aim to estimate these unknown values using the LS-VCE method for the Samsung S20 device. We use the following values as the primary values denoted as \"nominal stochastic model\":  
* For GPS: \(a^{GPS}_{code}=10^{5}\), \(a^{GPS}_{phase}=10^{3}\) indicating that code observables are 10 times noisier than the carrier phase observables, and \(\rho^{GPS}=0\) indicating there is no correlation between the GPS code and phase observables;
* For GLONASS: \(a^{GLO}_{type}=2\times a^{GPS}_{type}=2\times 10^{5}\) indicating that GLONASS observables are \(\sqrt{2}\) times nosier than the GPS observables and \(\rho^{GLO}=0\) indicating there is no correlation between the GLONASS code and carrier phase observables. We will also estimate these values for the two systems and two observation types as well as possible correlations (i.e., \(a^{GPS}_{code}\), \(a^{GPS}_{phase}\), \(\rho^{GPS}\) for GPS and \(a^{GLO}_{code}\), \(a^{GLO}_{phase}\), \(\rho^{GLO}\) for GLONASS). The unknown vector of the stochastic model is then \(\sigma=\left[a^{GPS}_{code}\;a^{GPS}_{phase}\;\rho^{GPS}\;a^{GLO}_{code}\;a^{GLO }_{phase}\;\rho^{GLO}\right]^{T}\) and their corresponding cofactor matrices are provided in Table 1. The LS-VCE estimates can then be obtained as \(\hat{\sigma}=\left[\hat{a}^{GPS}_{code}\;\;\;\hat{a}^{GPS}_{phase}\;\;\;\hat{ \rho}^{GPS}\;\;\hat{a}^{GLO}_{code}\;\;\;\hat{a}^{GLO}_{phase}\;\;\;\hat{\rho}^ {GLO}\right]^{T}=N^{-1}l\).