Fast approximation algorithm to noise components estimation in long-term GPS coordinate time series

Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise...

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Published in:	Journal of geodesy Vol. 95; no. 2
Main Authors:	Tehranchi, R., Moghtased-Azar, K., Safari, A.
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2021 Springer Nature B.V
Subjects:	Algorithms Approximation Components Computation Earth and Environmental Science Earth Sciences Geodetics Geophysics/Geodesy Global positioning systems GPS Methods Noise Original Article Parameters Positioning systems Software Storage requirements Time series Noise estimation GPS time series Variance component estimation (VCE) Wavelet domain
ISSN:	0949-7714, 1432-1394
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Abstract	Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise content of GPS coordinate time series. For large-scale data, namely over a wide range of spatial and temporal scales, the previous methods’ efficiency could significantly improve. Meanwhile, the estimation method, including repeated inversion of large matrices, has led to intensive computations and large storage requirements. By quantifying the REML estimator by decorrelation property of discrete wavelet transformation, the current research has offered FREML (fast REML) for accurate and fast approximation of noise content. For evaluating the method’s efficiency, 360 synthetic daily time series with different lengths N = 2048 , 4096, and 8192 observation epochs were used. The time series composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The FREML algorithm’s outcomes were compared with existing software that uses a maximum likelihood approach to quantify the uncertainties (e.g., Hector). The results indicated that both methods provided equivalent results for noise components, unknown parameters (rate, offset, and transient displacement), and their uncertainties. Moreover, the FREML method reduced the computation time by a factor of 2–14 compared to Hector software, depending on the amount of data and missing epochs. For more assessment of the method, the FREML method was applied to the 36 real time series with noise models as (i) white plus flicker noise and (ii) combination of white, flicker, and random walk noises. The results demonstrated that the two methods’ outcomes were close, and the FREML method speeded up the estimation of noise and unknown parameters.
AbstractList	Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise content of GPS coordinate time series. For large-scale data, namely over a wide range of spatial and temporal scales, the previous methods’ efficiency could significantly improve. Meanwhile, the estimation method, including repeated inversion of large matrices, has led to intensive computations and large storage requirements. By quantifying the REML estimator by decorrelation property of discrete wavelet transformation, the current research has offered FREML (fast REML) for accurate and fast approximation of noise content. For evaluating the method’s efficiency, 360 synthetic daily time series with different lengths N = 2048 , 4096, and 8192 observation epochs were used. The time series composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The FREML algorithm’s outcomes were compared with existing software that uses a maximum likelihood approach to quantify the uncertainties (e.g., Hector). The results indicated that both methods provided equivalent results for noise components, unknown parameters (rate, offset, and transient displacement), and their uncertainties. Moreover, the FREML method reduced the computation time by a factor of 2–14 compared to Hector software, depending on the amount of data and missing epochs. For more assessment of the method, the FREML method was applied to the 36 real time series with noise models as (i) white plus flicker noise and (ii) combination of white, flicker, and random walk noises. The results demonstrated that the two methods’ outcomes were close, and the FREML method speeded up the estimation of noise and unknown parameters. Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise content of GPS coordinate time series. For large-scale data, namely over a wide range of spatial and temporal scales, the previous methods’ efficiency could significantly improve. Meanwhile, the estimation method, including repeated inversion of large matrices, has led to intensive computations and large storage requirements. By quantifying the REML estimator by decorrelation property of discrete wavelet transformation, the current research has offered FREML (fast REML) for accurate and fast approximation of noise content. For evaluating the method’s efficiency, 360 synthetic daily time series with different lengths N=2048, 4096, and 8192 observation epochs were used. The time series composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The FREML algorithm’s outcomes were compared with existing software that uses a maximum likelihood approach to quantify the uncertainties (e.g., Hector). The results indicated that both methods provided equivalent results for noise components, unknown parameters (rate, offset, and transient displacement), and their uncertainties. Moreover, the FREML method reduced the computation time by a factor of 2–14 compared to Hector software, depending on the amount of data and missing epochs. For more assessment of the method, the FREML method was applied to the 36 real time series with noise models as (i) white plus flicker noise and (ii) combination of white, flicker, and random walk noises. The results demonstrated that the two methods’ outcomes were close, and the FREML method speeded up the estimation of noise and unknown parameters.
ArticleNumber	18
Author	Tehranchi, R. Moghtased-Azar, K. Safari, A.
Author_xml	– sequence: 1 givenname: R. surname: Tehranchi fullname: Tehranchi, R. organization: School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran – sequence: 2 givenname: K. orcidid: 0000-0002-8107-6888 surname: Moghtased-Azar fullname: Moghtased-Azar, K. email: moghtased@tabrizu.ac.ir organization: Department of Geodesy, Faculty of Civil Engineering, University of Tabriz – sequence: 3 givenname: A. surname: Safari fullname: Safari, A. organization: School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran
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Snippet	Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of...
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SubjectTerms	Algorithms Approximation Components Computation Earth and Environmental Science Earth Sciences Geodetics Geophysics/Geodesy Global positioning systems GPS Methods Noise Original Article Parameters Positioning systems Software Storage requirements Time series
Title	Fast approximation algorithm to noise components estimation in long-term GPS coordinate time series
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