# 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  
To assess the noise characteristics of the GNSS smartphone observables, the LS-VCE method can be applied to the double-differenced (DD) observations of two smartphones. For zero and short baselines (up to 10 m as used in this contribution), the DD atmospheric (ionospheric and tropospheric) delays can be ignored. In this case, the geometry-based model is of the form [29]  
\[ \begin{split} E\left(\Delta\
abla\Phi^{\varsigma k}_{a,b,j}\right)& =\Delta\
abla\rho^{\varsigma k}_{a,b}+\lambda_{j}\Delta\
abla \text{N}^{s,k}_{a,b,j}\\ E\left(\Delta\
abla P^{s,k}_{a,b,j}\right)&=\Delta \
abla\rho^{s,k}_{a,b}\end{split}  \tag{4}\]  
where \(\Delta\
abla\) represent the DD operator, \(a\) and \(b\) represent the base and rover stations, respectively; \(s\) and \(k\) denote the satellites, \(P\) and \(\Phi\) stands for the pseudorange (code) and carrier phase observations in meter, respectively; \(\Delta\
abla\rho^{s,k}_{a,b}\) is the DD receiver-satellite range, \(\lambda_{j}\) is the corresponding carrier phase wavelength (m) on the \(j\)th frequency and \(\Delta\
abla\text{N}^{s,k}_{a,b,j}\) is the DD ambiguity. The unknowns are the baseline components between the reference and rover receivers as well as the ambiguous term \(\Delta\
abla\text{N}^{s,k}_{a,b,j}\). In this contribution, we focus on the noise assessment of GNSS observations from the Samsung S20 device. Samsung S20 is a dual-frequency smartphone supporting L5/E5a frequencies for GPS and Galileo, respectively. To collect data for this study, we placed two Samsung S20 devices on top of a geodetic pillar with known coordinates, resulting in a very short baseline of approximately 5 cm. However, for some reason, they could not log Galileo observations as well as E1 and E5a frequencies. In addition, the number of GPS satellites with L5 observations is only one or two during the whole time of data collection. Therefore, in this contribution, we will employ the single-frequency GPS and GLONASS observations (\(j=1\)). In this case, the unknowns are \(\Delta\
abla\text{N}^{s,k,GPS}_{a,b,1}\) and \(\Delta\
abla\text{N}^{s,k,GLO}_{a,b,1}\) (considering the known coordinates of the geodetic pillar). Following [25; 26], when dealing with DD observation types of one epoch, the structure of the stochastic model is as follows:  
\[Q^{DD}_{y}=\text{blkdiag}\Big{(}Q^{DD}_{GPS},Q^{DD}_{GLO}\Big{)} \tag{5}\]  
in which blkdiag constructs a block diagonal matrix from input arguments and \(Q^{DD}_{GPS}\) and \(Q^{DD}_{GLO}\) are the two covariance matrices of the DD GPS and GLONASS observations, respectively. Assuming \(n_{1}\) and \(n_{2}\) visible satellites for GPS and GLONASS, respectively, \(Q^{DD}_{GPS}\) and \(Q^{DD}_{GLO}\) can be expressed as follows:  
\[Q^{DD}_{GPS}= \begin{bmatrix}a^{GPS}_{cubg}&\rho^{GPS}_{cps}\\ \rho^{GPS}&a^{GPS}_{phase}\end{bmatrix} \otimes 2 \begin{bmatrix}\sigma^{2}_{[1]^{G}}+ \sigma^{2}_{[2]^{G}}&\sigma^{2}_{[1]^{G}}&\cdots&\sigma^{2}_{[1]^{G}}\\ \sigma^{2}_{[1]^{G}}&\sigma^{2}_{[1]^{G}}+\sigma^{2}_{[3]^{G}}&\cdots&\sigma^{ 2}_{[1]^{G}}\\ \vdots&\vdots&\ddots&\vdots\\ \sigma^{2}_{[1]^{G}}&\sigma^{2}_{[1]^{G}}&\cdots&\sigma^{2}_{[1]^{G}}+\sigma^ {2}_{[n_{1}]^{G}}\end{bmatrix}  \tag{6}\]  
and  
\[Q^{DD}_{GLO}= \begin{bmatrix}a^{GLO}_{cubg}&\rho^{GLO}_{cD}\\ \rho^{GLO}&a^{GLO}_{phase}\end{bmatrix} \otimes 2 \begin{bmatrix}\sigma^{2}_{[1]^{R}}+ \sigma^{2}_{[2]^{R}}&\sigma^{2}_{[1]^{R}}&\cdots&\sigma^{2}_{[1]^{R}}\\ \sigma^{2}_{[1]^{R}}&\sigma^{2}_{[1]^{R}}+\sigma^{2}_{[3]^{R}}&\cdots&\sigma^{ 2}_{[1]^{R}}\\ \vdots&\vdots&\ddots&\vdots\\ \sigma^{2}_{[1]^{R}}&\sigma^{2}_{[1]^{R}}&\cdots&\sigma^{2}_{[1]^{R}}+\sigma^ {2}_{[n_{2}]^{G}}\end{bmatrix}