Evaluating stochastic models for estimating site velocity from daily and weekly GNSS time series in the stable region of the South American plate
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Haroldo Antonio Marques
,
Lucas Gonzales Lima Pereira Calado
,
João Francisco Galera Monico
,
Heloisa Alves Silva Marques
,
Maurício Carvalho Mathias de Paulo
and
Gabriela de Oliveira Nascimento Brassarote
Abstract
Accurate estimation of GNSS station velocities requires a precise characterization of stochastic noise in coordinate time series. This study evaluates stochastic models for estimating site velocities using weekly network and daily PPP GNSS solutions from 74 stations in the stable South American mid-plate. The functional model incorporated seasonal components, while the stochastic model was based on noise variance estimates. The Non-Negative Least Squares Variance Component Estimation method was applied to estimate noise amplitudes, classifying noise as a combination of white and colored components. Additionally, the spectral index was refined using Maximum Likelihood Estimation. Results show that most time series are best described by a combination of white and flicker noise, with differences in spectral properties between weekly and daily solutions. The impact of these models on velocity uncertainties was assessed, showing that neglecting an appropriate noise model can lead to overestimated uncertainties. These findings contribute to improving GNSS-based velocity estimations for geodynamic studies.
Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award Identifier / Grant number: 304773/2021-2
Award Identifier / Grant number: 431559/2018-0
Funding source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Award Identifier / Grant number: 495252/2020-00
Award Identifier / Grant number: 817759/2023-00
Acknowledgments
The authors would like to express their gratitude to the Nevada Geodetic Laboratory (http://geodesy.unr.edu/) and SIRGAS-CON (https://www.sirgas.org/en/sirgas-realizations/sirgas-connetwork/) for providing free access to the GNSS time series database. We also extend our thanks to the Graduate Program in Cartographic Sciences at FCT-UNESP.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: Marques (grant 431559/2018-0) is funded by the Brazilian National Council for Scientific and Technological Development (CNPq) (https://doi.org/10.13039/501100003593). Monico also acknowledges CNPq for support through grant 304773/2021-2. Calado (grant 817759/2023-00 and 495252/2020-00) is funded by Coordination of Superior Level Staff Improvement (CAPES) (https://doi.org/10.13039/501100002322).
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Data availability: Not applicable.
References
1. Marotta, GS, França, GS, Monico, JFG, Bezerra, FHR, Fuck, RA. Strain rates estimated by geodetic observations in the Borborema Province, Brazil. J South Am Earth Sci 2015;58:1–8. https://doi.org/10.1016/j.jsames.2014.12.006.Search in Google Scholar
2. Assumpção, M, Dias, FL, Zevallos, I, Naliboff, JB. Intraplate stress field in South America from earthquake focal mechanisms. J South Am Earth Sci 2016;71:278–95. https://doi.org/10.1016/j.jsames.2016.07.005.Search in Google Scholar
3. Yang, K, Weiping, J, Fan, L, Li, C. A method of strain rate calculation based on GNSS time series and its accuracy analysis. Geomagn Aeron 2023;63:41–50. https://doi.org/10.1134/S001679322260045X.Search in Google Scholar
4. Altamimi, Z, Rebischung, P, Métivier, L, Collilieux, X. ITRF2014: a new release of the International Terrestrial Reference Frame modeling nonlinear station motions. J Geophys Res Solid Earth 2016;121:6109–31. https://doi.org/10.1002/2016JB013098.Search in Google Scholar
5. Altamimi, Z, Rebischung, P, Collilieux, X, Métivier, L, Chanard, K. ITRF2020: an augmented reference frame refining the modeling of nonlinear station motions. J Geod 2023;97:47. https://doi.org/10.1007/s00190-023-01738-w.Search in Google Scholar
6. Williams, SDP. The effect of coloured noise on the uncertainties of rates estimated from geodetic time series. J Geod 2003;76:483–94. https://doi.org/10.1007/s00190-002-0283-4.Search in Google Scholar
7. Zhang, J, Bock, Y, Johnson, H, Fang, P, Williams, S, Genrich, J, et al.. Southern California permanent GPS geodetic array: error analysis of daily position estimates and site velocities. J Geophys Res Solid Earth 1997;102:18035–55. https://doi.org/10.1029/97JB01380.Search in Google Scholar
8. Wei, WWS. Time Series Analysis: Univariate and Multivariate Methods, 2nd ed. Boston: Pearson; 2006.Search in Google Scholar
9. Amiri-Simkooei, AR. Least-squares variance component estimation: theory and GPS applications [Ph.D. thesis]. Delft University of Technology; 2007.10.54419/fz6c1cSearch in Google Scholar
10. Williams, SDP. CATS: GPS coordinate time series analysis software. GPS Solut 2008;12:147–53. https://doi.org/10.1007/s10291-007-0086-4.Search in Google Scholar
11. Bogusz, J, Klos, A. On the significance of periodic signals in noise analysis of GPS station coordinates time series. GPS Solut 2015;20:655–64. https://doi.org/10.1007/s10291-015-0478-9.Search in Google Scholar
12. Dach, R, Lutz, S, Walser, P, Fridez, P. Bernese GNSS software version 5.2. University of Bern: Bern Open Publishing; 2015. https://boris.unibe.ch/72297/.Search in Google Scholar
13. Sánchez, L, Drewes, H, Kehm, A, Seitz, M. SIRGAS reference frame analysis at DGFI–TUM. J Geod Sci 2022;12:92–119. https://doi.org/10.1515/jogs-2022-0138.Search in Google Scholar
14. Zumberge, JF, Heflin, MB, Jefferson, DC, Watkins, MM, Webb, FH. Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res Solid Earth 1997;102:5005–17. https://doi.org/10.1029/96JB03860.Search in Google Scholar
15. Blewitt, G, Hammond, W, Kreemer, C. Harnessing the GPS data explosion for interdisciplinary science. EOS 2018;99. https://doi.org/10.1029/2018EO104623.Search in Google Scholar
16. Li, X, Huang, J, Li, X, Shen, Z, Han, J, Li, L, et al.. Review of PPP–RTK: achievements, challenges, and opportunities. Satell Navig 2022;3:28. https://doi.org/10.1186/s43020-022-00089-9.Search in Google Scholar
17. Davies, P, Blewitt, G. Methodology for global geodetic time series estimation: a new tool for geodynamics. J Geophys Res Solid Earth 2000;105:11083–100. https://doi.org/10.1029/2000JB900004.Search in Google Scholar
18. Teunissen, PJG. Towards a least-squares framework for adjusting and testing of both functional and stochastic models. The Netherlands: Delft University of Technology, Faculty of Aerospace Engineering; 2004.Search in Google Scholar
19. Leick, A, Rapoport, L, Tatarnikov, D. GPS satellite surveying, 4th ed. New Jersey: John Wiley & Sons; 2015.10.1002/9781119018612Search in Google Scholar
20. Bock, Y, Melgar, D. Physical applications of GPS geodesy: a review. Rep Prog Phys 2016;79:106801. https://doi.org/10.1088/0034-4885/79/10/106801.Search in Google Scholar PubMed
21. Teunissen, PJG, Montenbruck, O, editors. Springer Handbook of Global Navigation Satellite Systems. Switzerland: Springer Handbooks; 2017.10.1007/978-3-319-42928-1Search in Google Scholar
22. Kreemer, C, Hammond, WC, Blewitt, G. A robust estimation of the 3-D intraplate deformation of the North American plate from GPS. J Geophys Res Solid Earth 2018;123:4388–412. https://doi.org/10.1029/2017JB015257.Search in Google Scholar
23. Gravelle, M, Wöppelmann, G, Gobron, K, Altamimi, Z, Guichard, M, Herring, T, et al.. The ULR-repro3 GPS data reanalysis and its estimates of vertical land motion at tide gauges for sea level science. Earth Syst Sci Data 2023;15:497–509. https://doi.org/10.5194/essd-15-497-2023.Search in Google Scholar
24. Hackl, M, Malservisi, R, Hugentobler, U, Wonnacott, R. Estimation of velocity uncertainties from GPS time series: examples from the analysis of the South African TrigNet network. J Geophys Res Solid Earth 2011;116. https://doi.org/10.1029/2010JB008142.Search in Google Scholar
25. Langbein, J. Methods for rapidly estimating velocity precision from GNSS time series in the presence of temporal correlation: a new method and comparison of existing methods. J Geophys Res Solid Earth 2020;125. https://doi.org/10.1029/2019JB019132.Search in Google Scholar
26. Bianchi, MB, Assumpção, M, Rocha, MP, Carvalho, JM, Azevedo, PA, Fontes, SL, et al.. The Brazilian seismographic network (RSBR): improving seismic monitoring in Brazil. Seismol Res Lett 2018;89:452–7. https://doi.org/10.1785/0220170227.Search in Google Scholar
27. Teunissen, PJG, Amiri-Simkooei, AR. Least-squares variance component estimation. J Geod 2007;82:65–82. https://doi.org/10.1007/s00190-007-0157-x.Search in Google Scholar
28. Amiri-Simkooei, AR. Non-negative least-squares variance component estimation with application to GPS time series. J Geod 2016;90:451–66. https://doi.org/10.1007/s00190-016-0886-9.Search in Google Scholar
29. Bos, MS, Montillet, JP, Williams, SDP, Fernandes, RMS. Introduction to geodetic time series analysis. In: Montillet, JP, Bos, M, editors. Geodetic time series analysis in earth sciences. Cham: Springer Geophysics. Springer; 2020.Search in Google Scholar
30. Kaczmarek, A, Kontny, B. Identification of the noise model in the time series of GNSS stations coordinates using wavelet analysis. Remote Sens 2018;10:1611. https://doi.org/10.3390/rs10101611.Search in Google Scholar
31. Amiri-Simkooei, AR, Tiberius, CCJM, Teunissen, PJG. Assessment of noise in GPS coordinate time series: methodology and results. J Geophys Res Solid Earth 2007;112. https://doi.org/10.1029/2006JB004913.Search in Google Scholar
32. Williams, SDP. Offsets in global positioning system time series. J Geophys Res Solid Earth 2003;108. https://doi.org/10.1029/2002JB002156.Search in Google Scholar
33. Métivier, L, Collilieux, X, Lercier, D, Altamimi, Z, Beauducel, F. Global coseismic deformations, GNSS time series analysis, and earthquake scaling laws. J Geophys Res Solid Earth 2014;119:9095–109. https://doi.org/10.1002/2014JB011280.Search in Google Scholar
34. Nikolaidis, R. Observation of geodetic and seismic deformation with the global positioning System. San Diego: University of California; 2002.Search in Google Scholar
35. Gelb, A, editor. Applied Optimal Estimation. London, England: MIT Press; 1974.Search in Google Scholar
36. Brown, RG, Hwang, PYC. Introduction to Random Signals and Applied Kalman Filtering with MATLAB Exercises, 3rd ed. New York: John Wiley & Sons; 1996.Search in Google Scholar
37. Teunissen, PJG. Dynamic data processing: recursive least-squares, 2nd ed. The Netherlands: TU Delft OPEN; 2024. https://books.open.tudelft.nl/home/catalog/view/180/334/567.Search in Google Scholar
38. Mao, A, Harrison, CGA, Dixon, TH. Noise in GPS coordinate time series. J Geophys Res Solid Earth 1999;104:2797–816. https://doi.org/10.1029/1998JB900033.Search in Google Scholar
39. Montillet, JP, Bos, MS, editors. Geodetic Time Series Analysis in Earth Sciences. Switzerland: Springer Geophysics; 2020.10.1007/978-3-030-21718-1Search in Google Scholar
40. Agnew, DC. The time-domain behavior of power-law noises. Geophys Res Lett 1992;19:333–6. https://doi.org/10.1029/91GL02832.Search in Google Scholar
41. Hosking, JRM. Fractional differencing. Biometrika 1981;68:165–76. https://doi.org/10.1093/biomet/68.1.165.Search in Google Scholar
42. Bos, MS, Fernandes, RMS, Williams, SDP, Bastos, L. Fast error analysis of continuous GNSS observations with missing data. J Geod 2012;87:351–60. https://doi.org/10.1007/s00190-012-0605-0.Search in Google Scholar
43. Langbein, J. Improved efficiency of maximum likelihood analysis of time series with temporally correlated errors. J Geod 2017;91:985–94. https://doi.org/10.1007/s00190-017-1002-5.Search in Google Scholar
44. Gobron, K. Statistical analysis of vertical land motions and sea level measurements at the coast [Ph.D. thesis]. Earth Sciences, Université de La Rochelle, Université de Liège, Faculté des Sciences; 2021. Available from: https://theses.hal.science/tel-03566564v1.Search in Google Scholar
45. Kehm, A, Sánchez, L, Bloßfeld, M, Seitz, M, Drewes, H, Angermann, D, et al.. Combination strategy for the geocentric realization of regional epoch reference frames. J Geophys Res Solid Earth 2022;127. https://doi.org/10.1029/2021JB023880.Search in Google Scholar
46. Blewitt, G. An improved equation of latitude and a global system of graticule distance coordinates. J Geod 2024;98:6. https://doi.org/10.1007/s00190-023-01815-0.Search in Google Scholar PubMed PubMed Central
47. Santamaría-Gómez, A. SARI: interactive GNSS position time series analysis software. GPS Solut 2019;23:52. https://doi.org/10.1007/s10291-019-0846-y.Search in Google Scholar
48. He, X, Bos, MS, Montillet, JP, Fernandes, RMS. Investigation of the noise properties at low frequencies in long GNSS time series. J Geod 2019;93:1271–82. https://doi.org/10.1007/s00190-019-01244-y.Search in Google Scholar
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