# Comparison of Enhanced Noise Model Performance Based on Analysis of Civilian GPS Data
###### Abstract  
We recorded the time series of location data from stationary, single-frequency (L1) GPS positioning systems at a variety of geographic locations. The empirical autocorrelation function of these data shows significant temporal correlations. The Gaussian white noise model, widely used in sensor-fusion algorithms, does not account for the observed autocorrelations and has an artificially large variance. Noise-model analysis--using Akaike's Information Criterion--favours alternative models, such as an Ornstein-Uhlenbeck or an autoregressive process. We suggest that incorporating a suitable enhanced noise model into applications (e.g., Kalman Filters) that rely on GPS position estimates will improve performance. This provides an alternative to explicitly modelling possible sources of correlation (e.g., multipath, shadowing, or other second-order physical phenomena). noise models; stochastic process; GPS error; Ornstein-Uhlenbeck process; sensor fusion; embedded computing 2  
It would be ambitious to model and distinguish between all these processes; alternatively, one can consider their cumulative effect on the time series [13; 14]. Irrespective of the source, a robust signal-processing algorithm needs some model of noise processes. Such a model unavoidably contains assumptions on the nature of the noise, e.g., amplitude and correlation in the time domain, which must be experimentally validated. Often, the measurement errors are modelled by independent, identically distributed random Gaussian variables (Gaussian white noise). While this model is sufficient for many applications, it is generally accepted that a linear combination of coloured noise (whose power spectrum follows a \(P\sim f^{-\kappa}\) law with a characteristic exponent \(\kappa\)) is a more suitable approximation [15]. With the assumption of white noise, the time correlation in GPS datastreams leads to biased and inconsistent parameter estimates [16], especially for sampling rates above \(5\,\mathrm{Hz}\). Moreover, it is unrealistic that measurements from different satellites would have the same accuracy; therefore, stochastic models assuming independent and constant variance result in unreliable statistics, and stochastic modelling is needed. Directly estimating statistical parameters of the GPS time series by explicitly taking into account the heteroscedastic space- and time-correlated error structure of the observations could result in improved accuracy at minimal computational cost. Noise in long GPS time series has been analysed for geodetic purposes by [17; 18], who found \(-1\leq\kappa\leq 3\) in the relevant frequency band. Their results are, however, not conclusive, as many spectra would fit this range, e.g., white noise (\(\kappa=0\)) or flicker noise (\(\kappa=1\)). Many processes that these GPS observations are expected to capture, e.g., earthquakes or the wobble of the Earth around its axis, have similar spectra [19]. Consequently, the physical process being observed by GPS and the inherent noise of a GPS signal are present simultaneously. The noise amplitude changes with time and, for older data, can be a considerable fraction of the signal [20]. In this study, we analyse time series of GPS measurements with duration between a day and a week at \(1\,\mathrm{Hz}\) sampling frequency. Observations were made in the Southern Hemisphere with stationary units. The recordings show significant autocorrelation incompatible with a Gaussian white noise model. The observed presence of correlations implies that there are some stochastic equations governing the dynamics of the signal. There are two major sources that plausibly introduce correlations: (a) the movement of GPS satellites and, thus, their constellation, and (b) the algorithms coded in the GPS emitters and receivers. Both sources are, to a high degree, deterministic; thus, in a perfect environment, they would produce a strongly correlated signal.