# Stochastic Modeling of Smartphones GNSS Observations Using LS-VCE and Application to Samsung S20
## 1 Introduction
[18] developed the Smart-PPP software application for smartphones, which achieved sub-meter float positioning accuracy in horizontal directions by utilizing a C/N0-based weighting model. The weighting model was based on an exponential function that took into account various factors such as the instantaneous C/N0 of the observations, a maximum C/N0 value for cutoff, and the corresponding pseudorange noise at maximum C/N0. Further details on this weighting model are available in their study [18]. [PERSON] et al. [19] also proposed a combined elevation angle and C/N0 weighting method for smartphone-based GNSS PPP by normalizing the C/N0-derived variances to the scale of the elevation-angle-derived variances. Their findings showed that this method improved the three-dimensional positioning accuracy by 22.7% and 24.2% in an open-sky area, and by 52.0% and 26.0% in a constrained visibility area, compared with the elevation-angle-only and C/N0-only weighting methods, respectively. [PERSON] et al. [20] also proposed a new weighting model that considers the variation range of C/N0, providing a better weighting model than the traditional weighting model, thus improving the positioning accuracy. The results indicated that the L5/E5a/B2a signals could generally obtain higher IAR fix-rate and positioning accuracies than the L1/E1/B1 signals. To date, the existing stochastic models are mainly either elevation-dependent or C/N0-based weighting models. Our contribution proposes a more reliable stochastic model that combines both elevation-dependent and C/N0 weighting methods. This approach takes into account the impact of both elevation angles and C/N0 records on different GNSS constellations and observation types (i.e., pseudorange and carrier phase observations) to adjust the stochastic model accordingly. The parameters in such a stochastic model can be estimated by using the least-squares variance component estimation (LS-VCE) method. The LS-VCE method is a well-established method for the estimation of unknown (co)variance components [21; 22; 23; 24]. The method has been successfully employed to assess the noise characteristics of the high-end geodetic receivers [25; 26; 27]. In [PERSON] and [PERSON] [28], the LS-VCE method was also applied to stochastic modeling of noisy smartphone GNSS observations but the effect of the elevation angle and C/N0 on different GNSS constellations and different observation types was not investigated. In this study, the weighting coefficients will be estimated separately for each constellation and each observation type. We will also estimate any possible correlation between the GNSS pseudorange and carrier phase observables. This paper is organized as follows: The LS-VCE method is briefly reviewed in Section 2. How to apply the LS-VCE method to noise assessment of GNSS smartphone observations is also explained in this section. The reliable stochastic model will then be applied to the single-frequency PPP (SF-PPP) model in Section 3. In Section 4, the performance of the SF-PPP model applying the obtained reliable stochastic model is investigated using the GNSS observations from the Samsung S20 device in a kinematic mode. Finally, some conclusions are made in Section 5.