alex boava meza prof. dr. edvaldo simões da fonseca junior · 2019. 12. 5. · behavior modelling...
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Behavior Modelling of Multi-Annual GPS Stations Using Time Series
Analysis
Alex Boava Meza
Prof. Dr. Edvaldo Simões da Fonseca Junior
Escola Politécnica
Universidade de São Paulo
Mestrado em Engenharia de Transportes – Informações Espaciais
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1. INTRODUCTION
2. TIME SERIES ANALYSIS
– Parametric method
– Non-parametric methods
3. RESULTS
4. DYNAMIC MODELLING
5. CONCLUSION
SUMMARY
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• What is deformation?
• What is modelling?
• Modelling methods in Geodesy
– Congruence
– Kinematic
– Dynamic
INTRODUCTION
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TIME SERIES ANALYSIS
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• Time series analysis?
– According to Morettin and Toloi (2018)....
TIME SERIES ANALYSIS
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Spectral-DomainTime-Domain
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• Aims to define certain components that, in general, compose a Time
Series (TS)
Parametric Method
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TrendTS = Sazonality+ + Stochastic
Polynomial Regression
𝑇 𝑡𝑖 =
𝑘=0
𝑚
𝛽𝑘𝑡𝑖𝑘
Trigonometric Functions
𝑆 𝑡𝑖 =
𝑘=1
𝑠
[𝑎𝑘 cos 2𝜋𝑓𝑘𝑡𝑖 + 𝑏𝑘𝑠𝑖𝑛(2𝜋𝑓𝑘𝑡𝑖)]
White noiseARMA
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• Concerns to the spectral-domain, also called frequency-domain.
• The analysis can be:
1. Univariate
– Autocorrelation (ACF) and Partial-Autocorrelation (PACF);
– Fast Fourier Transform;
– Periodograms; etc....
2. Bivariate
– Cross-correlation;
– Cross-spectral correlation; etc...
Non-parametric Methods
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Mestrado em Engenharia de Transportes – Informações Espaciais
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SOME RESULTS….
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•
• Binomial Model:
Trend
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𝑇𝑆𝑓 𝑡𝑖 = 𝛽0 + 𝛽1𝑡𝑖
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The main frequencies in a time series can be identified with PERIODOGRAMS.
Sazonality
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𝑆 𝑡𝑖 =
𝑘=1
𝑠
[𝑎𝑘 cos 2𝜋𝑓𝑘𝑡𝑖 + 𝑏𝑘𝑠𝑒𝑛(2𝜋𝑓𝑘𝑡𝑖)]
Using the selected frequencies, parameters 𝑎𝑘and 𝑏𝑘 were estimated and subject to t-test. Iftheir value were lower than 1.96, it wasdiscarded.
Period (days)
1461,00 365,25 182,63
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• ARMA (q,p) – Best fit -> ARMA (1, 3)
Stochastic
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Result - Residuals
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Mean = 0.5 mmStd. = 7.8 mm
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DYNAMIC MODELLING
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• According to Welch and Heunecke (2001), the definition of a
model for a transfer function is feaseble if the input (𝑦𝑘) and the
output (𝑥𝑘) quantities are measured. And they pose one way to
model the system using ARMA (q, p):
• 𝑥𝑘 - Altimetric coordinate
• 𝑦𝑘 - River level
• 𝑥𝑘 = 𝑎1𝑥𝑘−1 + 𝑎2𝑥𝑘−2 +⋯+ 𝑎𝑞𝑥𝑘−𝑞 + 𝑏0𝑦𝑘 + 𝑏1𝑦𝑘−1 +⋯+ 𝑏𝑝𝑦𝑏−𝑝
Dynamic Modelling
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Autoregressive Moving Average
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Dynamic Modelling
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• In order to validate the hypothesis that the river level has influence in the station
coordinates, several time series cross-correlation were calculated between:
ALTIMETRY and/or RIVER LEVELS
Dynamic Modelling
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cm
Lag = 1 day
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Dynamic Modelling
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RIVER LEVELLag = 65 days
cm
STATIONSLag = 40 days
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• ARMA (p,q) – Best fit -> ARMA (2, 1) with lag = 6
Dynamic Modelling
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met
ers
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• By this work, it cannot be said that the river levels in Amazonian region
strictely affect station coordinates;
• Nevertheless, the RIVER x STATION system has some similarities and river
data will be used in the modelling process.
• The portryed methods of analysis showed to be robust for the porpuses of
this work;
• The main problem to be attacked is the existence of gaps in the time
series. Which may lead to a bad estimation of parameters.
CONCLUSION
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• CHINARRO, D. System Engineering applied to Fuenmayor Karst Aquifer (San Julián de Banzo, Huesca) and Collins Glacier (King George Island, Antartica). 2014. Tese (Doutorado em Ciência da Computação e Engenharia de Sistemas) - Departamento de Computação e Engenharia de Sistemas, Universidade de Zaragoza, Zaragoza.
• GÜLAL, E.; ERDOĞAN, H.; TIRYAKIOĞLU, I. Research on the stability analysis of GNSS reference stations network by time series analysis. Digital Signal Processing, v. 23, n. 6, p. 1945-1957, 2013
• WELSCH, W. M., HEUNECKE, O. Committee of FIG (2001) Report of Ad-Hoc Committee of Working Group 6.1. No. 25 Models and Terminology for the Analysis of Geodetic Monitoring Observations. FIG Guide, 2001.
• MORETIN, P. A.; TOLOI, C. M. C. Análise de séries temporais: modelos lineares univariados. 3ª ed. São Paulo:Blucher, 2018, p. 474.
REFERENCES
Escola Politécnica
Universidade de São Paulo
Mestrado em Engenharia de Transportes – Informações Espaciais