GNSS Pseudorange Measurement Noise Identification by Measurement Difference Method
Oliver Kost, University of West Bohemia; Jindrich Dunik, Honeywell International, Advanced Technology Europe and University of West Bohemia; Ondrej Straka, University of West Bohemia; Ondrej Daniel, Huld
Knowledge of an appropriate system model is a crucial prerequisite for the optimal design of all signal processing and decision-making algorithms. The model typically consists of deterministic and stochastic parts. While the deterministic part usually arises from the first principles based on physical, kinematic, and mathematical laws governing the system behaviour, the stochastic part (i.e., noise model) is identified from the measured data.
Global navigation satellite system (GNSS) measurement processing is at the core of any modern navigation system. To obtain an accurate and consistent batch (or snap-shot) estimate of the position accompanied by the estimate error covariance matrix, the noise variance of the pseudorange measurements must be known.
The GNSS measurement noise is typically assumed to be a normally distributed random variable with time-correlated and white components . The particular parameters of the noise description depend not only on the GNSS satellite software and hardware (e.g., clock, ephemeris), actual signal-in-space conditions (e.g., ionosphere, troposphere, scintillation, multipath, satellite elevation), and receiver hardware and software (e.g., oscillator, antenna, tracking loops, signal processors) but also on the correction models (e.g., ionosphere and troposphere models) and the respective residuals. Thus, the noise properties are significantly time and spatially varying and task-specific. Therefore, their correct specification is not a trivial problem and has been solved for the last decades.
From the perspective of a navigation system designer, we can distinguish three main classes of noise models;
(i) Models found in standards and literature (standard-based models),
(ii) Models provided by the receiver manufacturer (receiver-specific models),
(iii) Models identified from data (data-specific models).
The standard-based models are found by combining the first principles of modelling and identification using the massive volume of offline recorded data. The models available in the literature and GNSS-related standards [1-3] are designed for general working conditions (e.g., single or dual frequency, utilisation of environment correction models, satellite elevation). They must be carefully selected for the particular task.
The receiver-specific models come from an analysis of carrier-to-noise ratio and are easy to use, but details about the models' construction may not be available . These models may not consider receiver external factors (such as complete signal-in-space induced noise). As such, such incomplete noise models can lead to an inaccurate or inconsistent estimate of navigation parameters.
The data-specific models are identified from an available set of data offline or online (e.g., correlation or maximum-likelihood methods, Bayesian, covariance matching, or adaptive estimation methods [5-10]), possibly without deep physical insight. Although these methods can provide accurate models considering all error sources, the state-of-the-art methods are based on various limiting assumptions or approximations or require specification of the state (and parameters) dynamics making them specific for a particular application.
An analytical method for the model identification of the GNSS receiver measurement noise respecting actual working conditions with proven properties is missing.
Therefore, this paper proposes an extended and efficient version of the measurement difference method (MDM)  to be purposely designed for GNSS pseudorange measurement noise identification. The proposed version is based on such residual maker matrix design that projects a sequence of available pseudorange measurements into a linear combination of (directly unavailable) measurement noise realisations. Consequently, the pseudorange noise variance, estimated by the least-squares method, can be shown to converge to actual values with increasing amounts of data. The method is illustrated using simulated data to validate the method's theoretical properties and real data to demonstrate the method's applicability.
Compared to the original MDM, the proposed extended version identifies the measurement noise properties only without any assumption on object state dynamics. Moreover, the version is analytically derived and tailored to estimate the varying pseudorange measurement noise variance as a function of a satellite elevation. The paper is accompanied by an exemplary implementation of the proposed method.
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