# Noises in Double-Differenced GNSS Observations
###### Abstract
The different characteristics of the measurement noise for the GPS, GLONASS, Galileo, and BeiDou systems [12] as well as for individual blocks of satellites within a given system [13] make the development of a stochastic model for a multi-system solution difficult and complex. The existing stochastic models for GNSS observations mainly use the dependence of observation variance in the satellite elevation [14; 15; 16], Signal-to-Noise (SNR) [17; 18], and Carrier-to-Noise density Ratio (C/N) [19; 20; 21] parameters as well as others parameters: the ionosphere scintillation [22], receiver tracking error [23], or multipath effect [24]. However, these models do not take into account the characteristics of the type of observation noise and only describe its dependence on selected parameters. In this work, an attempt was made to perform a multidimensional analysis of the GNSS measurement noise, the aim of which was to fully describe the stochastic parameters of individual types of observations. In particular, different types of carrier phase and code observations for the GPS, GLONASS, and Galileo systems were analyzed. The analysis of the measurement noise characteristics was performed with the use of three statistical tools: the autocorrelation function (ACF), the Lomb-Scargle (L-S) periodogram method, and the Allan variance (AVAR) method. These data analysis tools allow noise-type observations to be studied and do not include an analysis of the dependence of the noise level on SNR, C/N0, satellite elevation, and receiver type, which may be part of the further studies that take all of these factors into account. The ACF is a well-known analytical measure for time-domain signals, which plays a significant role in the time series models for the identification and evaluation of the correlation degree. In statistics, the autocorrelation of random processes is the Pearson correlation coefficient between the time series and its copy shifted by the time lag, which is expressed as a function of the delay. The ACF is defined for stationary processes, but in practice, it is often used, even if the data do not meet the stationarity conditions [25]. In this study of the time series for the GNSS measurements time series, the ACF is used to determine the parameters of the temporal correlation of observations, which, in the form of empirical models (e.g., exponential function [26]), is applied to the stochastic model of the data. Analyses of the residuals of double-differenced carrier phase GPS observations show the existence of significant temporal correlations for both medium-long and zero-short baselines [27; 28]. The obtained time correlation values differ significantly for a given measurement set; however, the time correlation is clearly dependent on the measurement interval data recorded at a higher frequency and that are more time-correlated. In many types of time series of natural phenomena, noise can be described by a power-law noise process [29]. This kind of noise has been identified in, e.g., music [30], economics [31], geophysical data [32], electronic devices [33], astronomy [34], and GNSS positions [35]. There are several ways to determine this kind of noise, most of which are based on spectral analyses or maximum likelihood estimation [32; 36]. A commonly used approach for power-law noise identification is to calculate the power spectrum of the data and then estimate the spectral index using linear regression [29] from the log-log plot of the spectrum. The slope of the fitted straight line corresponds to the kind of noise in the analyzed data. To estimate the power spectrum of unevenly sampled data or data with gaps, such as in our case, the Lomb-Scargle periodogram method is particularly useful [37; 38]. AVAR was originally invented to analyze the stability of atomic clocks in the 1960s [39]. Standard variance measures the total signal power, and it does not allow for the signal stability to be characterized, which is in contrast to Allan variance.