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Benchmarking the algorithms to detect seasonal signals under different noise conditions

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Abstract

Global Positioning System (GPS) position time series contain seasonal signals. Among the others, annual and semi-annual are the most powerful. Widely, these oscillations are modelled as curves with constant amplitudes, using the Weighted Least-Squares (WLS) algorithm. However, in reality, the seasonal signatures vary over time, as their geophysical causes are not constant. Different algorithms have been already used to cover this time-variability, as Wavelet Decomposition (WD), Singular Spectrum Analysis (SSA), Chebyshev Polynomial (CP) or Kalman Filter (KF). In this research, we employed 376 globally distributed GPS stations which time series contributed to the newest International Terrestrial Reference Frame (ITRF2014). We show that for c.a. 20% of stations the amplitudes of seasonal signal varies over time of more than 1.0 mm. Then, we compare the WD, SSA, CP and KF algorithms for a set of synthetic time series to quantify them under different noise conditions. We show that when variations of seasonal signals are ignored, the power-law character is biased towards flicker noise. The most reliable estimates of the variations were found to be given by SSA and KF. These methods also perform the best for other noise levels while WD, and to a lesser extend also CP, have trouble in separating the seasonal signal from the noise which leads to an underestimation in the spectral index of power-law noise of around 0.1. For real ITRF2014 GPS data we discovered, that SSA and KF are capable to model 49-84% and 77-90% of the variance of the true varying seasonal signals, respectively.
Benchmarking the Algorithms to Detect Seasonal Signals Under Different Noise Conditions
1 2 1
Anna Klos , Machiel Simon Bos , Janusz Bogusz
Contact e-mail: anna.klos@wat.edu.pl
1) Military University of Technology, Faculty of Civil Engineering and Geodesy, Warsaw, Poland
2) University of Beira Interior, Instituto D. Luis, Covilha, Portugal
Acknowledgements
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).
AUCK
0.0
0.5
1.0
1.5
2.0
Standard deviations (mm)
ULAB
1996 2000 2004 2008 2012 2016
-20
-15
-10
-5
0
5
10
15
20
Detrended Up (mm)
AUCK
Time (years)
-15
-10
-5
0
5
10
15
2000 2004 2008 2012 2016
ULAB
Time (years)
Detrended Up (mm)
-6
-4
-2
0
2
4
6
8
2000 2004 2008 2012 2016
Synthetic series (mm)
Time (years)
-20
-15
-10
-5
0
5
10
15
20
2000 2004 2008 2012 2016
Time (years)
Synthetic series (mm)
0.1 1 10 100 1000
Frequency (cpy)
-3
-4
-2
-1
0
1
2
3
1 mm/yr0.25
log(Power(mm /cpy))
2
synthetic
WLS
MOL S
CP
KF
SSA
WD
-3
-2
-1
0
1
2
0.1 1 10 100 1000
10 mm/yr0.25
Frequency (cpy)
synthetic
WLS
MOL S
CP
KF
SSA
WD
log(Power(mm /cpy))
2
0.5 0.2 0.1 0.05 0.02
Signal to noise ratio
Mist std (mm)
0
1
2
3
1 2.5 5 10 25
Power-law noise amplitude (mm/yr )
κ/4
KF
SSA
CP
WLS
No seasonal
0
10
20
30
0.02 0.06 0.10 0.14 0.18 0.22
Count
Signal to noise ratio
North
0
10
20
30
Count
East
0.02 0.06 0.10 0.14 0.18 0.22
Signal to noise ratio
0
10
20
30
Count
0.02 0.06 0.10 0.14 0.18 0.22
Signal to noise ratio
Up
-2
-1
0
1
2
0.1 1 10 100 1000
Frequency (cpy)
AUCK
log(Power(mm /cpy))
2
GPS
WLS
MOLS
CP
KF
SSA
WD
INTRODUCTION
The coordinate time series determined with the Global Positioning System (GPS) contain annual and semi-annual periods that are
routinely modeled by two periodic signals with constant amplitude and phase-lag. However, the amplitude and phase-lag of the
seasonal signals vary slightly over time. Various methods have been proposed to model these variations such as Wavelet
Decomposition (WD), writing the amplitude of the seasonal signal as a Chebyshev polynomial that is a function of time (CP),
Singular Spectrum Analysis (SSA), and using a Kalman Filter (KF). Using synthetic time series, we investigate the ability of each
method to capture the time-varying seasonal signal in time series with different noise levels. We demonstrate that the precision by
which the varying seasonal signal can be estimated depends on the ratio of the variations in the seasonal signal to the noise level.
For most GPS time series, this ratio is between 0.05 and 0.1. Within this range, the WD and CP have the most trouble in
separating the seasonal signal from the noise. The most precise estimates of the variations are given by the SSA and KF
methods. For real GPS data, SSA and KF can model 49-84% and 77-90% of the variance of the true varying seasonal signal,
respectively.
→ Fig. 1 A total of 174 GPS stations are used in this
research. The color of the circles indicates standard
deviation (mm) of the annual amplitudes estimated with
MOLS for vertical component. Stations AUCK (Auckland,
Australia) and ULAB (Ulaanbaatar, Mongolia), which we
focus on in this research, are also marked.
REAL GPS DATA
We employed daily GPS time series processed at the JPL/NASA from 174 stations with a time span longer than 13 years. Outliers
were removed using the median criterion. Epochs of offsets were taken from the information provided by JPL. Additional offsets were
estimated using the Sequential t-test algorithm with a segment length of 100 days and a confidence level of 95%. Gaps in the data
ranged between 0.1 to 11% of the entire time series. The SSA method described below requires that these missing data are filled.
Therefore, gaps were interpolated with a linear interpolation which is the simplest and most often employed to interpolate any
missing value.
METHODS
We applied the following methods to model the time-varying seasonal signal:
1. Moving Ordinary Least Squares (MOLS): 3-year segments ( ),Figure 5
th th
2. Wavelet Decomposition (WD): 7 and 8 levels of Meyer’s wavelet,
3. Singular Spectrum Analysis (SSA): 3-year window,
4. Kalman Filter (KF): third-order autoregressive noise was added, as suggested by Didova et al. (2016),
5. Modeling the seasonal amplitudes with polynomials (CP): degree of 4.
1
→ Table 1 Mean trend uncertainty, spectral index
κ, noise amplitude σ and a misfit estimated from
500 simulations of the synthetic time series of the
length of 6000 days (16.4 years). Various methods
were employed. The values within the brackets
were obtained from the time series with linearly
interpolated data gaps. The noise amplitude of the
synthetic flicker noise (κ=-1) was equal to
0.25
10 mm/yr .
→ Fig. 2 Example of a few
synthetic time series
created for different noise
levels employed in this
research. Left panel: a
very low noise level that is
0.25
1 mm/yr . Right panel: a
normal noise level, that is
0.25
10 mm/yr , which is the
most common for GPS
position time series.
Various colors mean the
consecutive simulations.
↑ Fig. 5 Variations in annual and semi-annual signals for
stations AUCK (top panel) and ULAB (bottom panel)
estimated with MOLS for vertical component. The annual
amplitude varies in each segment from 0.3±0.1 mm to
2.6±0.1 mm for AUCK and from 0.1±0.1 mm to 1.1±0.1 mm
for ULAB.
SYNTHETIC GPS TIME SERIES
We generated 500 synthetic time series without gaps with a length of 6000 days: 16.4 years (Figure 3). We assumed a pure flicker
0.25 -κ/4
noise (spectral index of -1) with the amplitudes between 1 and 25 mm/yr which covers the range of 7 to 21 mm/yr that we found in
0.25
the real GPS JPL time series. The noise amplitude of 1 mm/yr is an ideal situation. The annual and semi-annual signals were
simulated in all-time series with mean amplitudes of, respectively, 3.0 and 1.0 mm, and various phase-lags between 1 and 6 months
and added to pure flicker noise. The modeled variations in the amplitude of the seasonal signal were chosen to have standard
deviations of 1.0 and 0.5 mm for annual and semi-annual signals, respectively, to mimic the mean values of real time-varying signals .
To investigate the effect that data gaps may have on the precision of each approach, we also simulated time series with missing data
that varied from 4 to 16% of the total length of data, with a mean of 8%. These missing data were filled using linear interpolation.
RESULTS FOR SYNTHETIC SERIES
1. WLS performs worse than any of the methods that try to model the varying seasonal
signal.
2. The ability of MOLS to separate noise from the annual signal decreases when noise
levels increase.
3. WD absorbs a part of the noise which results in an underestimation of the spectral index.
4. CP absorbs noise for high noise levels which makes it worse than WLS.
5. SSA and KF have excellent performance for high signal to noise ratios in capturing the
varying seasonal signal, but the precision of SSA deteriorates for higher noise levels. KF
suffers from the same problem but to a lesser extent.
→ Fig. 3 Standard deviation of the estimated varying seasonal signal minus the
synthetic one (the misfit) as a function of the power-law noise amplitude in the time
series. The top axis (signal to noise ratio) notes the corresponding ratio of standard
deviation of the estimated annual amplitudes to the noise amplitude. WD and MOLS
were not included for a better clarity of a plot.
← Fig. 4 PSDs of synthesized time series and
residuals after applying the WLS, MOLS, WD, KF,
SSA and CP methods for two levels of noise: 1
0.25 0.25
mm/yr and 10 mm/yr . Left panel: When the flicker
noise amplitude is very low relating to the size of the
variations in the seasonal signal, estimating a constant
seasonal signal performs worse than any of the
methods. Right panel: When normal noise levels are
used, the varying seasonal signal can no longer be
estimated so precisely, as it absorbs some part of the
noise. PSD was estimated with Welch periodogram.
Method
Trend uncertainty (mm/yr)
κ
Misfit (mm)
No seasonal
assumed
0.294 (0.301)
-1.07
11.18
2.44 (2.42)
WLS
0.221 (0.228)
-1.00
9.95
1.11 (1.09)
MOLS
0.205 (0.211) -0.98
9.63
1.31 (1.31)
CP 0.209 (0.215) -0.98
9.67
1.29 (1.27)
KF
0.209 (0.215)
-0.98
9.71
0.73 (0.74)
SSA
0.191 (0.195)
-0.96
9.35
1.08 (0.96)
WD
0.175 (0.180)
-0.94
9.00
1.53 (1.52)
Actual
0.222
-1.00
10.00
(mm/yr-κ/4)
σ
RESULTS FOR 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
↑ Fig. 6 Histograms of the signal to noise ratio for the 174 GPS stations, for the North, East
and Up components.
Fig. 7 PSD of AUCK vertical time series and that of the residuals of MOLS, CP, KF, SSA,
and WD estimated with Welch periodogram.
Please, see: Klos A., Bos M.S., Bogusz J. (2018): Detecting time-varying
seasonal signal in GPS position time series with different noise levels.
GPS Solutions, doi: 10.1007/s10291-017-0686-6.
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