# Stochastic Modeling of Smartphones GNSS Observations Using LS-VCE and Application to Samsung S20
## 1 Introduction  
The widespread demand for precise position information in mass-market applications has led to a significant increase in the use of low-cost and ultra-low-cost receivers. In recent years, many smartphones have also been equipped with advanced carrier phase tracking measurement technology to support precise position determination. During the \"Google I/O\" conference in May 2016, Google announced that the raw GNSS measurements, such as pseudorange, carrier phase, Doppler shift, and carrier-to-noise density ratio (C/N0) observations, would be available on Android Nougat (version 7) or higher operating systems. With the availability of several hundred smartphone models in the market that can provide raw GNSS observations, there is an increasing demand for research on improving smartphone positioning performance [1, 2]. While the GNSS signals can be effectively used for outdoor localization purposes, signal attenuation or blockage prevents their employment in indoor settings. As a result, various alternative technologies, including Frequency modulation (FM), Ultra-wideband (UWB), Radio Frequency Identification (RFID), Wi-Fi, Bluetooth, and cellular networks as well as inertial sensors, are being utilized to meet the growing demand for indoorpositioning [3; 4; 5; 6; 7; 8]. In this contribution, however, we focus on the application of the GNSS technique for outdoor positioning. The precise point positioning (PPP) method is a single-receiver global navigation satellite system (GNSS)-based precise positioning technique. It is different from the real-time kinematic (RTK) technique, which typically requires a base station [9; 10]. Currently, there is a high demand for improving PPP accuracy with smartphones, especially in kinematic mode. Since the quality of smartphone observations is not as good as that of high-end geodetic receivers, there are several issues that must be addressed. Choosing appropriate stochastic models is crucial for achieving high-precision positioning where the stochastic model describes the statistical properties of observables. The elevation-dependent weighting functions are generally employed to build a weighting model for the high-end geodetic receivers' observations. However, the elevation angle is not the only parameter that can be used to create a stochastic model for smartphone observations. The C/N0 value can also be effectively used to assign proper weights to the smartphone GNSS observations based on signal strength. The C/N0 refers to the ratio of the carrier power and the noise power per unit bandwidth [11]. It can therefore be considered a powerful indicator of the GNSS signal strength in the sense that a larger C/N0 indicates a stronger signal while a lower C/N0 shows a weaker signal. Recent studies have shown that the C/N0-dependent weighting method is superior to the elevation-dependent weighting method when using the GNSS observations obtained from Android smartphones. We will mention a few of these studies in the following. [PERSON] et al. [12] investigated the quality of raw GNSS observations of smartphones in terms of the C/N0, noise, tracking capability of the carrier phase, and velocity estimation. They found that there is a stronger correlation between the pseudorange accuracy and C/N0 rather than the satellite elevation angle. Therefore, using an elevation-dependent weighting model is not suitable for low-cost receivers, and a C/N0 weighting model is a better choice. [PERSON] et al. [13] also proposed a C/N0 weighting model as an alternative to an elevation-dependent weighting model based on factors such as multipath noise, chipping length, or wavelength and C/N0. This weighting model was later utilized by [PERSON] and [PERSON] [14]. Employing the C/N0 weighting model instead of an elevation-dependent weighting model has been also suggested in other studies [15; 16; 17]. [PERSON] et al.