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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

Misﬁt 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.