Fast error analysis of continuous GNSS observations with missing data |
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Authors: | M. S. Bos R. M. S. Fernandes S. D. P. Williams L. Bastos |
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Affiliation: | 1. CIIMAR/CIMAR, University of Porto, Rua dos Bragas 289, 4050-123, Porto, Portugal 2. University of Beira Interior, IDL, R. Marquês d’ávila e Boloma, 6201-001, Covilh?, Portugal 3. Delft Earth-Oriented Space Research (DEOS), DUT, Kluyverweg 1, 2629 HS, Delft, The Netherlands 4. National Oceanography Centre, Liverpool, 6 Brownlow Street, Liverpool, L3 5DA, UK 5. Faculty of Science, DGAOT, University of Porto, Rua do Campo Alegre, 4169-007, Porto, Portugal
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Abstract: | One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires $mathcal{O }(n^3)$ operations for $n$ observations. Previous research by the authors has shown that this amount of operations can be reduced to $mathcal{O }(n^2)$ for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present a Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series.Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data. |
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