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Most Global Positioning System (GPS) position time series contain annual and semi-annual periods that are routinely modelled as two periodic signals with constant amplitude and phase-lag. However, the amplitudes and phase-lags of seasonal signals vary slightly over time. Also, time series contain specific colored noise. Which methods should we employ to detect time-varying seasonal signal in GPS position time series with different noise levels? Do these methods artificially absorb some part of the power?
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Modelling of Time-Varying Seasonal
Signals in GNSS Time Series
Anna Klos1), Machiel S. Bos2), Janusz Bogusz1)
1) Military University of Technology, Warsaw, Poland
2) University of Beira Interior, Portugal
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
1
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
2
Motivation:
1. Most Global Positioning System (GPS) position time series contain
annual and semi-annual periods that are routinely modelled as
two periodic signals with constant amplitude and phase-lag.
2. The amplitude and phase-lag of seasonal signals vary slightly
over time.
3. Time series contain specific colored noise.
4. Which methods should we employ to detect time-varying
seasonal signal in GPS position time series with different noise
levels?
5. Do they artificially absorb some part of the power?
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
3
Methods:
1. Moving Ordinary Least-Squares: MOLS
a) split the time series into segments of 3-year which overlap
with a shift of 1 year,
b) model the seasonal signals with WLS,
c) and then, use linear interpolation to generate a single time
series.
2. Wavelet Decomposition: WD
a) use the 7th and 8th levels of Meyer’s wavelet
b) to model time-varying signals with periods between 128
and 512 days, which include the annual and semi-annual
periods.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
4
Methods:
3. Singular Spectrum Analysis: SSA
a) employ a 3-year window (2-, 3- and 4-year windows were
tested under different noise levels)
b) to model the time-varying seasonal signal in GPS position
time series with annual and semi-annual periods.
4. Chebyshev Polynomials: CP
a) choose degree 4 (degrees from 1 to 10 were tested)
b) to model time-varying signals with annual and semi-annual
periods.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
5
Methods:
5. Kalman Filter: KF
a) employ the KF, with a correct tuning of the variances in the
filter, as described by Didova et al. (2016),
b) i.e., use a third order autoregressive model to mimic
a power-law noise.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
6
Data:
1. daily GPS time series processed at the JPL/NASA from 174
stations,
2. time span longer than 13 years,
3. outliers & offsets removed,
4. Maximum Likelihood Estimation (MLE) applied to estimate the
character of series (used to create a synthetic set),
5. MOLS employed to estimate of how one may be misled when a
constant amplitude is assumed.
Please, note: In this presentation we focus on the noise character
which may be found in both horizontal and vertical position
changes.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
7
Data:
Standard deviations (mm) of the annual amplitudes estimated with
MOLS for 3-year segments in vertical direction.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
8
Data:
STDs are smaller than 0.5 mm for around 30% of stations, while for
around 15% of stations they are larger than 1.0 mm.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
9
Synthetic series:
1. 500 synthetic time series generated, a length of 6000 days (16.4
years),
2. no gaps introduced,
3. a pure flicker noise assumed with the amplitudes between 1 and
25 mm/yr0.25,
4. the annual and semi-annual signals were simulated with the mean
amplitudes of, respectively, 3.0 and 1.0 mm, and various phase-
lags between 1 and 6 months,
5. the modelled variations in the amplitude were chosen to have
standard deviations of 1.0 and 0.5 mm for annual and semi-
annual signals.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
10
Synthetic series:
a. a very low noise level, that is 1 mm/yr0.25,
b. a high noise level, that is 10 mm/yr0.25.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
11
Synthetic series:
a. a very low noise level, that is 1 mm/yr0.25,
b. a high noise level, that is 10 mm/yr0.25.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
12
Synthetic series:
1. a case of a very low noise level, that is 1 mm/yr0.25,
2. WLS performs the worst,
3. SSA and KF obtain the lowest misfit values,
4. SSA and KF explain 99.1-99.5% of seasonal signal variance.
Method
Trend
uncertainty
(mm/yr)
κ𝝈(mm/yr-k/4)Misfit
(mm)
No seasonal assumed 0.475 -1.76 3.39 2.39
WLS 0.061 -1.23 1.47 0.56
MOLS 0.027 -1.05 1.08 0.24
CP 0.031 -1.07 1.12 0.27
KF 0.020 -0.98 0.96 0.16
SSA 0.021 -0.99 0.98 0.16
WD 0.030 -1.07 1.07 0.24
Ideal 0.022 -1.00 1.00
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
13
Synthetic series:
1. a case of a very high noise level, that is 10 mm/yr0.25,
2. WD absorbs some part of the noise,
3. SSA and KF explain 49-77% of seasonal signal variance
(decreasing with increasing noise level).
Method
Trend
uncertainty
(mm/yr)
κ𝝈(mm/yr-k/4)Misfit
(mm)
No seasonal assumed 0.294 -1.07 11.18 2.44
WLS 0.221 -1.00 9.95 1.11
MOLS 0.205 -0.98 9.63 1.31
CP 0.209 -0.98 9.67 1.29
KF 0.209 -0.98 9.71 0.73
SSA 0.191 -0.96 9.35 1.08
WD 0.175 -0.94 9.00 1.53
Ideal 0.222 -1.00 10.00
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
14
Synthetic series:
For our study, we define:
Signal = amplitude of estimated annual period
Noise = amplitude of the estimated power-law noise
Ratio = Signal / Noise
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
15
Synthetic series:
1. not estimating a seasonal signal gives the largest misfit
2. WLS produces misfits that are equal to the standard deviations of
the estimated variations in the annual signal
3. SSA and KF have excellent
performance
for high SNR
4. CP absorbs noise
for high noise levels
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
16
Real GPS time series:
Ratio 0.02-0.05:
17 (N), 12 (E) and
34 (U) stations > KF
Ratio 0.05-0.10:
110 (N), 108 (E) and
120 (U) stations > KF & SSA
Ratio > 0.10 (ideal case):
the rest of stations
> KF, SSA & CP
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
17
Conclusions:
1. Estimating a varying seasonal signal always results in lower noise
amplitudes and lower spectral indices compared to estimating a
constant seasonal signal.
2. The most accurate estimates of the variations are given by the
SSA and KF methods.
3. There are advantages of using SSA rather than KF.
4. WD and CP have trouble in separating the seasonal signal from
the noise for high noise levels.
5. For real GPS data, SSA and KF can model 49-84% and 77-90% of
the variance of the true varying seasonal signal, respectively.
Motivation Methods Data Synthetic series Real GPS time series Conclusions
IAG-IASPEI Joint Scientific Meeting, July 30 – August 4, 2017, Kobe, Japan
18
Acknowledgments:
Anna Klos is supported by the National Science Centre, Poland, grant
no. UMO-2016/23/D/ST10/00495.
Machiel Simon Bos is supported by national funds through FCT in the
scope of the Project IDL- FCT-UID/GEO/50019/2013 and grant
number SFRH/BPD/89923/2012.
JPL time series were accessed from
ftp://sideshow.jpl.nasa.gov/pub/JPL_GPS_Timeseries/repro2011b/.
The map was drawn in the Generic Mapping Tool (Wessel et al.,
2013).
Thank you!
ResearchGate has not been able to resolve any citations for this publication.
12 (E) and 34 (U) stations > KF Ratio 0.05-0
  • Gps Real
  • Series
Real GPS time series: Ratio 0.02-0.05: 17 (N), 12 (E) and 34 (U) stations > KF Ratio 0.05-0.10: 110 (N), 108 (E) and 120 (U) stations > KF & SSA Ratio > 0.10 (ideal case): the rest of stations > KF, SSA & CP 18
The map was drawn in the Generic Mapping Tool
  • Wessel
The map was drawn in the Generic Mapping Tool (Wessel et al., 2013).