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Math Geosci

https://doi.org/10.1007/s11004-018-9760-z

Noise-Dependent Adaption of the Wiener Filter for the

GPS Position Time Series

Anna Klos1·Machiel S. Bos2·

Rui M. S. Fernandes2·Janusz Bogusz1

Received: 4 January 2018 / Accepted: 16 July 2018

© The Author(s) 2018

Abstract Various methods have been used to model the time-varying curves within

the global positioning system (GPS) position time series. However, very few consider

the level of noise a priori before the seasonal curves are estimated. This study is the

ﬁrst to consider the Wiener ﬁlter (WF), already used in geodesy to denoise gravity

records, to model the seasonal signals in the GPS position time series. To model the

time-varying part of the signal, a ﬁrst-order autoregressive process is employed. The

WF is then adapted to the noise level of the data to model only those time variabilities

which are signiﬁcant. Synthetic and real GPS data is used to demonstrate that this

variation of the WF leaves the underlying noise properties intact and provides optimal

modeling of seasonal signals. This methodology is referred to as the adaptive WF

(AWF) and is both easy to implement and fast, due to the use of the fast Fourier

transform method.

Keywords Wiener ﬁlter ·GPS ·Noise analysis ·Adaptive ﬁlters

1 Introduction

The global positioning system (GPS) position time series features seasonal signals

that are routinely modeled as sinusoids with a constant amplitude over time (Blewitt

and Lavallée 2002; Ding et al. 2005; Santamaría-Gómez et al. 2011; Bogusz and

Figurski 2014). The environmental loading effects contribute largely to the annual

and semi-annual amplitude variations (van Dam and Wahr 1987;vanDametal.2001,

BAnna Klos

anna.klos@wat.edu.pl

1Faculty of Civil Engineering and Geodesy, Military University of Technology, Warsaw, Poland

2Instituto D. Luis, University of Beira Interior, Covilhã, Portugal

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2012); however, they are not the only causes of seasonal changes recorded by the GPS

receivers. Other contributors are systematic errors, either due to variations in satellite

orbits (Ray et al. 2008) or of a numeric origin (Penna and Stewart 2003), which may

sometimes be as large as the loading effects. Klos et al. (2018c) demonstrated that

the seasonal signals in loading models do not remain constant over time. This means

that the seasonal changes in the GPS position time series may be also time-variable

(Freymueller 2009; Chen et al. 2013; Bogusz et al. 2015a; Gruszczynska et al. 2016).

Therefore, amplitudes given as constants over time, typically derived with the weighted

least-squares method, do not provide the most accurate description of them.

Beyond the time variability of seasonal changes, the noise present in the GPS

position time series is another issue that needs inclusion in order to achieve reliable

modeling of the observations. This has already been widely described in numerous

studies by Zhang et al. (1997), Mao et al. (1999), Williams et al. (2004), Bos et al.

(2010) and Klos et al. (2016), that a power-law process being close to ﬂicker noise

with a spectral index of −1 is the optimum noise model to describe the stochastic

part of the GPS data. The impact of this phenomena on the velocity uncertainties may

be reduced by spatial ﬁltering using different methods, such as stacking or principal

component analysis (PCA), which help to estimate a common mode error (CME; Dong

et al. 2006; Bogusz et al. 2015b; Gruszczynski et al. 2018). Some authors observed

that the instability of monuments may be another contributor to site-speciﬁc noise.

If signiﬁcant enough, it will change the character from ﬂicker to random-walk noise,

with a spectral index of −2(Beavan2005).

The power-law character of the GPS position time series may be incorrectly inter-

preted if some of the processes are mismodeled or incorrectly modeled during the time

series analysis. As an example, the character of ﬂicker noise may be under- or over-

estimated if too many or too few frequencies are modeled and removed. Removing

the autocorrelation from the stochastic part of the time series results in an artiﬁcial

improvement in a velocity uncertainty of up to 56% (Bogusz and Klos 2016;Klos

et al. 2018e). Besides this, the undetected offsets in the time series can lead to an

artiﬁcial shift in the character of the stochastic part from ﬂicker to random-walk noise

(Williams 2003a), which leads to unrealistic overestimates of velocity uncertainty

(Williams 2003b).

All of the above indicate that the time-varying curves need to be properly modeled

during the GPS time series analysis, including the site-speciﬁc noise level, in order to

produce reliable velocity estimates. Different methods have been employed to estimate

the time-varying curves. Santamaría-Gómez et al. (2011) used a non-linear iterative

least squares method to remove all signiﬁcant peaks from the GPS position time

series. They noted that the effect of the remaining peaks was insigniﬁcant to the

noise analysis. Xu and Yeu (2015) and Gruszczynska et al. (2016) proposed using the

singular spectrum analysis (SSA) approach to model the time-varying curves. Chen

et al. (2013) compared the estimates obtained with the SSA and Kalman ﬁlter (KF)

techniques, and emphasized that the SSA is much faster than the KF and features a

lower computational cost. They also found that both methods gave a similar response.

Klos et al. (2018b) emphasized that the reliability of seasonal signals estimated with the

SSA, KF, wavelet decomposition (WD) and Chebyshev polynomials (CP) depends on

the level of the noise present in the data. The SSA is based on selecting the frequencies

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of interest from the power spectrum of the data, with no separation between signal and

noise. The KF, when not properly tuned, begins to model the seasonals together with

the noise; the WD removes most of the power from the frequency band of interest;

while the CP absorbs noise when the order of the polynomial is incorrectly chosen.

Thus, the time-varying curves estimated with any of these methods change, depending

on the noise level, as these methods absorb some part of the power of noise into the

seasonal estimates.

In this study, the merits of the Wiener ﬁlter (WF) (Wiener 1930) to model the vary-

ing part of the seasonal signal are investigated. The WF is used to analyze the geodetic

time series, but is only applied to gravity records, with Li and Sideris (1994) applying

the WF to ﬁlter the noise out from the gravity series. Kotsakis and Sideris (2001)

emphasized that the WF was an efﬁcient tool for denoising the gravity data in the fre-

quency domain. Migliaccio et al. (2004) used a time-wise WF to estimate the gravity

potential, taking the colored noise into consideration. Reguzzoni and Tselfes (2009)

applied an iterative procedure based on the WF to remove the colored noise from the

gravity ﬁeld and steady-state ocean circulation explorer (GOCE) data. Liu et al. (2010)

performed Wiener-based ﬁltering for unconstrained solutions of the gravity recovery

and climate experiment (GRACE) satellite mission. Sampietro (2015) employed the

WF to determine the depth of the Moho from the GOCE gravity ﬁeld model. Klein-

herenbrink et al. (2016) applied the WF to analyze the gravity ﬁeld from the GRACE.

However, no attempts were made to employ the WF to ﬁlter the noise or seasonal

signals from the GPS position time series.

TheworkbyKlosetal.(2018b) is continued in this study, and the problem of

reliable modeling of time-varying seasonal curves with the noise level present in the

data being taken into consideration is addressed. The time-variable curves are provided

by a ﬁrst-order autoregressive process which, when properly tuned, enables modeling

the seasonal amplitudes varying over time. This process is employed for residuals with

time-constant curves already removed, ensuring a better expression of time variability.

Then, the WF is adapted to the level of noise that characterized the data, assuming the

proper power spectral density function of the noise, and applied to the original time

series. By using an inverse fast Fourier transform (IFFT) and the level of noise present

in the observations, the seasonal signal provided by the autoregressive process reﬁned

by the level of noise is estimated. The entire methodology is named the adaptive WF

(AWF), as the ﬁltering of seasonalities is adapted to the level of noise. Within this

study, annual and semi-annual variations are analyzed, although this method could

be successfully employed for any other type of oscillations or higher harmonics in

the annual changes, which must also introduce additional correlations if not removed

(Bogusz and Klos 2016).

The analyses start with a synthetic dataset, proving the effectiveness of the AWF

in modeling the time-varying curves. The maximum likelihood estimation (MLE) is

then used to characterize the noise content of a set of 385 international GPS service

(IGS) permanent stations, which is applied to construct the AWF for real data. The

next step is to estimate the time-varying seasonal curves for a real dataset. The ﬁnal

step is to cross-compare the estimates provided by the AWF with those obtained by

the KF or SSA, and, in this way, proving that the AWF was able to separate the signal

from the noise effectively. Only the GPS position time series are focused on, although

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there was no apparent reason why this methodology could not be successfully applied

to any other type of geodetic data, such as Zenith total delay or integrated water vapor

series, as well as tide gauges or gravity records.

2Data

This section provides a description of all the real and synthetic datasets. For the

synthetic series, two different datasets are considered. Synthetic series 1 has a constant

noise character, where different time-varying amplitudes of the seasonal signals are

added. Synthetic series 2 is based on different noise characters: different amplitudes

and spectral indices of noise, with the same time-varying seasonal signal.

2.1 Synthetic Series 1

The GPS position time series are best characterized by a power-law noise close to the

ﬂicker noise (Williams et al. 2004; Bos et al. 2010; Klos and Bogusz 2017). Several

studies have been carried out that prove that an incorrect assumption in the noise type

may cause over- or under-estimation of the uncertainty of velocity, leading to incorrect

interpretations (Bos et al. 2010; Langbein 2012;Klosetal.2016). Therefore, to test

the impact of the noise character and its level on the estimates of seasonal signals,

a series of 100 synthetic 22-year sets, equivalent to the longest time series available

from the IGS, are produced. A pure ﬂicker noise with a spectral index of −1 and

an amplitude of 8 mm/year0.25 is assumed, which agrees with the average amplitude

of the power-law noise found in the vertical changes in the GPS position time series

(Santamaría-Gómez et al. 2011). The amplitudes of the seasonal signals are set to

vary over 3.0–10.0 mm for the annual signal and over 1.5–4.0 mm for the semi-annual

signal. Their variation over time is expressed by standard deviations of 1.0 and 0.5 mm

for the annual and semi-annual periods, respectively. The standard deviation of the

seasonal signals indicates the value of which the amplitude of the seasonal signal may

differ from year to year. Phase lags vary between 1 and 6 months within the synthetic

dataset.

2.2 Synthetic Series 2

Five hundred synthetic time series for a 22-year period are generated, in which a

pure ﬂicker noise, spectral index of −1, and various noise amplitudes from 1 to

25 mm/year0.25 are assumed, covering the low and high noise levels present in the

GPS position time series. The annual and semi-annual signals of the amplitudes, 3.0

and 1.0 mm, respectively, and the standard deviations of 1.0 and 0.5 mm are added,

which were also used by Klos et al. (2018b).

2.3 Real GPS Position Time Series

The IGS GPS position time series from 385 permanently working stations (Rebischung

et al. 2016) are employed, with 8 years being the minimum time span of observations.

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Fig. 1 Geographical distribution of the 385 ITRF2014 IGS GPS stations employed in the analysis. The

length of each series is marked in a speciﬁc color

The series appear in the latest release of the International Terrestrial Reference System

(ITRF2014; Altamimi et al. 2016) (Fig. 1). The offsets are removed using the epochs

deﬁned by the IGS and supported by additional epochs detected with the sequential T

test analysis of regime shifts (STARS; Rodionov and Overland 2005). To remove the

outliers, the interquartile range (IQR) approach (Langbein and Bock 2004) is used.

The time series are characterized by a small percentage of gaps, at 3.5% for the entire

data. Since the modiﬁed MLE for time series with missing data (Bos et al. 2013)is

used, no interpolation is performed, excluding small gaps lasting a few days. The north

and east components are omitted and only on the vertical changes are analyzed, as the

time variability of the seasonal signals is the greatest for the "Up" component (Klos

et al. 2018b).

3 Methodology

To estimate the time-varying seasonal signals, the AWF is employed. It is adapted to the

noise level of the observations, in this way ensuring the best separation between noise

and signal. In this section, a detailed description of the AWF algorithm is provided,

as well as the MLE algorithm and the methods used to compare the seasonal signals

estimated with the AWF.

3.1 Adaptive Wiener Filter

Following Davis et al. (2012), the seasonal signal sican be modeled as a periodic

time-constant signal sc

iand a random process sr

i

si(a+δai)cos (ω0ti)+(b+δbi)sin (ω0ti)

[acos (ω0ti)+bsin (ω0ti)]+[δaicos (ω0ti)+δbisin (ω0ti)]

sc

i+sr

i,

,(1)

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where aand bare constants. The variable ω0is the normalized angular frequency of

the annual signal (ω0πf0

2fswhere fsis the sampling frequency, equal to 1 day in this

study). The random variables δaiand δbiare the ﬁrst-order autoregressive, AR(1),

processes

δaiφδai−1+vi

δbiφδbi−1+wi.(2)

Here, viand wiare the two Gaussian variables with standard deviations σvand

σw(mm), respectively. If φ1, then δaiand δbibecome the random-walk variables

(Davis et al. 2012). However, this implies that the amplitude of the varying seasonal

signal is increasing over time, which is not supported by observations. Therefore, in

the presented case, the φdeﬁned for the AR(1) process is slightly smaller than 1, that

is 0.9999.

The least-squares method is an estimator used for sc

i. However, since sr

iis a ran-

dom process, the least-squares method cannot provide estimates for it. Therefore, the

problem of time-varying seasonal curves is reduced to estimating sr

ifrom the noisy

observations from which sc

ihas been removed with the least-squares. Note that during

the least-squares estimation it is possible to include offsets and other deterministic

signals in order that the residuals only contain noise and the seasonal signal. Next,

assume that δbi0, so that the autocovariance γfor sr

iis

γsr

i,sr

i+kcov (δaiδai+kcos (ω0ti)cos (ω0ti+k))

cov δaiδai+k

1

2[cos (ω0(ti+ti+k)) + cos (ω0k)].(3)

While the AR(1) process is time-invariant, sris not due to the modulation with the

cosine. Therefore, by introducing the average autocovariance function γ

γ(k)lim

T→∞

1

T

T

0

γdt1

2

σ2

v

1−φ2φkcos (ω0k),(4)

where σ2

v/1−φ2φkis the autocovariance of the AR(1) process, dependent on lag

k. An example of the autocovariance function is given in Fig. 2.

Using the Wiener–Khinchin theorem, the average autocovariance function [Eq. (4)]

can be employed to compute the one-sided spectral density function S(ω) with the unit

mm2/rad

S(ω)2σ2

v

π1

1−2φcos (ω+ω0)+φ2+1

1−2φcos (ω−ω0)+φ2.(5)

An example of S(ω) for the annual signal, with a period of 365.25 days, is shown

in Fig. 3. In addition to S(ω), also provided is the power spectral density of the power-

law noise process, W(f), plotted in grey, to compare the behavior of the variations in

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Fig. 2 An example of the average autocovariance function γ(k)

Fig. 3 An example of power

spectral density function S(f)of

the time-varying annual signal.

The power-law noise is plotted

in grey

both. The closer the value of φis to 1, the sharper the peak of the annual signal. The

assumption so far is σw0. Now, σ2

vcan be replaced by (σ2

v+σ2

w) to also include

σw.

By employing all the information provided above, the WF can be constructed.

Assuming the time series xiwith its Fourier transform X(ωj)

XωjF(xi).(6)

Next the optimal ﬁlter ωjis deﬁned in the frequency domain as

ωj

Sωj

Sωj+Wωj.(7)

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The power spectral density of noise W(ωj) is employed to adapt the WF for the noise

present in the time series being considered. For the GPS observations, for which the

power-law noise model is preferred, the one-sided power spectral density is estimated

with (Agnew 1992)

Wωj

σ2

pl

π2sin ωj

2κ

≈

σ2

pl

πωκ,(8)

where κis the spectral index of power-law noise and ωjis the normalized angular

frequency. The spectral index of 0 means pure white noise, while values of −1 and

−2 mean pure ﬂicker noise and random-walk noise, respectively. σpl, the standard

deviation of the power-law noise, with units of mm, is the constant estimated with

the MLE. For example, assuming ﬂicker and white noise generates the following

one-sided power spectral density

Wωj2σ2

pl 2sin ωj

2−1

+2σ2

w,(9)

with σwbeing the standard deviation of the white noise component estimated with

the MLE. The noise model employed to create the WF is estimated from the real

observations to provide the best separation between signal and noise.

The estimated varying seasonal signal siis computed with the inverse Fourier

transform

siF−1ωjXωj.(10)

Both the forward and backward Fourier transforms have been implemented using

the FFT. So far, it is assumed that the annual signal only varies over time. To derive the

time-varying semi-annual signal, a period of 182.63 days, the two spectra are added

to obtain the total power spectral density function, as given in Fig. 4.

To obtain the total time-varying seasonal signal estimates with the AWF, the constant

seasonal signal derived with the weighted least-squares is added to the variations

estimated with Eq. (10), derived with the proper noise model assumed.

3.2 Maximum Likelihood Estimation

The MLE has been already widely used to estimate the time-constant seasonal oscilla-

tions and trends simultaneously with the character of the noise (Williams et al. 2004;

Bos et al. 2008; Langbein 2012). In this analysis, the GPS vertical position time series

are described as a sum of the initial value, trend and seasonal components of the fre-

quencies of interest, as in Bogusz and Klos (2016). The stochastic part that remains

after the deterministic model has been removed, named residuals, is characterized by

the white plus power-law noise model with spectral index κand the standard deviation

of noise, which can then be re-estimated to the amplitudes of the individual contribu-

tors. Such a model is assumed in the MLE analysis for the real GPS data in order to

estimate the parameters of noise to construct the AWF.

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The MLE is also applied in this study to assess the character of the residuals: to

estimate the parameters of power-law noise, when the AWF-derived seasonal curves

are removed from the synthetic and real series. On this basis, the parameters of noise

are compared between original series, using both synthetic and real data, and their

residuals to deﬁne the effectiveness of the AWF.

3.3 Comparison to Other Algorithms

To compare the time-varying seasonal signal obtained using the AWF, two other meth-

ods, namely the KF and SSA are employed. These approaches were used by Klos et al.

(2018b) to evaluate the reliability of the seasonal curve estimates. It was emphasized

that both the KF and SSA were able to provide reliable seasonal signals under different

noise levels.

In this study, the parameters of the KF and SSA estimated for both approaches by

Klos et al. (2018b) are applied. For the KF, the traditional KF is tuned to include the

third-order autoregressive process, which mimics the power-law noise present in the

residuals. For the SSA, the 3-year windows are employed. See Klos et al. (2018b)for

more details.

4 Results

The methodology described above is applied to analyze the synthetic and real datasets,

and to estimate the time-varying curves that are optimally separated from the noise

affecting the observations. The comparison between the AWF estimates of the seasonal

signals and those derived with the KF and SSA algorithms is presented.

Fig. 4 An example of the power

spectral density function S(f)of

the time-varying annual and

semi-annual signals. Power-law

noise is plotted in grey

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4.1 Synthetic Series 1

The series of 100 synthetic data sets with pure ﬂicker noise, spectral index of −1 and

amplitude of 8 mm/year0.25, is subjected to estimates of seasonal oscillations using the

AWF. The different cases are summarized in Table 1and shown in Fig. 5. For all cases,

synthetic series 1 is used with spectral index κ−1 and a 2.0-mm standard deviation

of noise, to which different seasonal time-varying amplitudes are added. The AWF is

created in ﬁve different versions, assuming different noise levels. The too-low noise

levels are described by cases 1 and 2, the ideal case, which agrees with the simulated

parameters, by case 3, and too-high noise levels by cases 4 and 5.

Different levels of noise are assumed when the AWF is created, in order to inten-

tionally impose improper noise levels while estimating the seasonal oscillations. The

too-low noise levels (cases 1 and 2), with a too-low spectral index or too-low ampli-

tude of noise, result in the mean variance of differences between the synthetic and

estimated curves of 0.97 and 0.78 mm2, respectively. The too-high noise levels (cases

4 and 5), with too-high amplitude of noise or too-high spectral index, result in the

Tab le 1 A set of parameters assumed to assess the performance of Wiener-based ﬁltering to estimate the

time-varying seasonal signals

Method Case Standard

deviation (mm)

Amplitude

(mm/year−κ/4)

Spectral index

κ

Variance

(mm2)

AWF 1 0. 5 2 −1.0 0.97

22.0 4 −0.5 0.78

3 (ref) 2.0 8 −1.0 0.70

42.0 18 −1.5 0.67

53.0 12 −1.0 0.69

For all cases, the mean variance (mm2) of the differences between synthetic seasonal curve and the one

estimated with the AWF is provided

Fig. 5 Left panel: Synthetic series 1 and synthetic seasonal signal plotted in, respectively, grey and yellow.

Right panel: Power spectral density generated for synthetic series 1 and the residuals after AWF curves are

removed

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mean variances of 0.67 and 0.69 mm2, respectively. When the proper noise level is

assumed during the creation of the AWF (case 3), the mean variance is 0.70 mm2.

The differences between individual variances in each case, from 1 to 5, depend

on the amplitude of the seasonal signal being synthesized. If it is assumed that the

time series being analyzed has a large amplitude of the annual signal and a small

amplitude of the semi-annual signal, then estimating the seasonal signal with cases 4

and 5 allows for a minor time variability in the annual signal, which is only higher than

the assumed noise level. In these cases, the semi-annual signal will probably not be

detected due to its too-small amplitude. In this way, the resulting seasonal signal will

have a small time variability in the annual signal, which will not be fully modeled and

include no semi-annual curve. Setting too-low noise levels (cases 1 and 2) results in

the artiﬁcial estimation of the time variability, which does not necessarily arise from

the seasonal signals, but can stem from the noise character of the analyzed time series.

In this case, both the annual and semi-annual signals are detected, as they are larger

than the assumed noise level, but are in fact modeled with a too large time variability.

As a consequence, a part of the autocorrelation originally included in the noise is

transferred to the estimates of the seasonal signals, resulting in biased estimates of the

noise type. This can be also easily noticed in Fig. 5, where different assumptions of

the AWF result in dissimilar estimates of the residuals: the time series for which the

seasonal model is removed.

To summarize Table 1, the large mean variances in cases 1 and 2 arise from the fact

that part of the time variability which characterizes the noise is modeled and included

in the seasonal signals due to the too-low noise levels employed during the creation

of the AWF. On the other hand, the underestimated mean variances from cases 4 and

5 prove that too-high noise levels, set incorrectly during the AWF creation, do not

permit the modeling of the full-time variability of the seasonal signal being simulated.

4.2 Synthetic Series 2

This synthetic series is compared with the KF and SSA approaches, both applied with

the parameters described in the previous section and by Klos et al. (2018b). In this

case, seasonal signals with an equal time variability are assumed, but with different

underlying noise levels added. This provides an insight into the effectiveness of the

AWF for various noise conditions. The results of this analysis are presented in Table 2

and compared with the KF- and SSA-derived estimates. Two characteristic noise levels

are listed: a low noise level, meaning a noise amplitude of 1 mm/year0.25, which is

very rarely met in the GPS position time series, and a high noise level, with a noise

amplitude of 10 mm/year0.25, a level of noise usually found in the vertical changes of

GPS stations.

For the low noise level, the misﬁts between the synthesized and estimated seasonal

curves are the greatest when no seasonal signal is assumed in the estimates. All three

methods tested give similar misﬁt results, of 0.17 or 0.16 mm. The misﬁt created when

the seasonal signals are derived is then transferred to the residuals. For these, a noise

analysis with the MLE is performed and the spectral indices and the amplitudes of

power-law noise are provided in Table 2. The estimates of trend uncertainty (mm/year),

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Tab le 2 The results of the noise analysis, trend uncertainties and misﬁts between the synthesized and estimated seasonal curves for the different approaches: KF, SSA and

AWF, from 500 simulations of synthetic series 2

Method Trend uncertainty (mm/year) Spectral index κAmplitude (mm/year−κ/4) Misﬁt (mm)

Low noise-level

No seasonal assumed 0.475 −1.76 3.39 2.39

KF 0.020 −0.98 0.96 0.16

SSA 0.021 −0.99 0.98 0.16

AWF 0.022 −0.99 0.96 0.17

Actual 0.022 −1.00 1.00 –

High noise level

No seasonal assumed 0.294 −1.07 11.18 2.44

KF 0.209 −0.98 9.71 0.73

SSA 0.191 −0.96 9.35 1.08

AWF 0.224 −1.00 9.92 0.67

Actual 0.222 −1.00 10.00 –

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spectral index κand the amplitude of the power-law noise (mm/year−κ/4) are delivered.

Two different noise levels are presented: (1) the low noise level of spectral index −1

and noise amplitude 1 mm/year0.25 and (2) the high noise level of spectral index −1

and noise amplitude 10 mm/year0.25. The actual parameters that should be obtained

from each method are presented in the ‘Actual’ row. From Table 2, it is clear that the

largest amplitude and spectral index of noise are found for the case when no seasonal

is assumed, as the seasonal signals which are not modeled would be transferred to

the residuals as the long-term correlation. The KF, SSA and AWF all resulted in

similar spectral indices and amplitudes of power-law noise, being close to the actual

synthesized value. Using Eq. (29) of Bos et al. (2008), the trend uncertainty is estimated

for noise parameters present in the residuals. Due to the fact that the KF-, SSA-

and AWF-delivered curves provide similar misﬁts and parameters of noise, the trend

uncertainties estimated here are almost equal to the actual value.

For the high noise level, the estimates of the seasonal curves provided by the dif-

ferent methods produce a larger misﬁt than the one observed for the low noise level.

Here, when no seasonal was assumed, a misﬁt of 2.44 mm is obtained. The seasonal

curves estimated with the SSA give a mean misﬁt of 1.08 mm, while the KF and

AWF, respectively, allow computing a seasonal curve which differs by only 0.73 and

0.67 mm from the synthesized one. Although the misﬁts are similar, the real impact of

the different methods may be observed for the noise parameters delivered for the resid-

uals. The lowest amplitude of the power-law noise is found for the residuals when the

SSA-delivered curve is removed. The AWF gives the amplitude closest to the actual

synthesized one. Similar results are observed for spectral indices, for which both the

SSA and KF model a part of the noise along with seasonal oscillations. For the AWF,

the estimated spectral index is equal to that synthesized, due to the optimal separation

between seasonal signal and noise.

The above analyses performed for the synthetic datasets demonstrate that the AWF

can distinguish the seasonal signal from noise, and, in this way, prevent the noise

power being transferred to the seasonal estimates.

4.3 Noise Analysis of Real Data

The values of the trends and amplitudes of annual and semi-annual seasonal signals

and noise parameters are also estimated simultaneously for a set of 385 IGS stations

using the MLE approach. Only the vertical time series are chosen for analysis, as

they feature the greatest variability in seasonal oscillations over time. The seasonal

amplitudes range between 0 and 11 mm for the annual signal, and between 0 and

3 mm for the semi-annual curve (Fig. 6). The greatest amplitude in the annual signal

occurs in the data for the IMPZ (Imperatriz, Brazil) station. The amplitude of the

annual curve is much greater for Asian stations than for any other region. However,

the amplitudes are very consistent regionally. For Europe, the smallest amplitudes are

found for the coastal stations, while the inland ones are characterized by amplitudes

of between 3 and 6 mm. The amplitudes of the semi-annual signal are the smallest for

Europe, Australia, Oceania and Southern parts of both North and South America. The

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greatest amplitudes of the semi-annual signal occur for central Asia, the northern part

of North America and a few stations situated in Antarctica.

The draconitic signal for the 351.60-day period, which has been already found to

be present in the GPS position time series (Amiri-Simkooei 2013), is estimated in this

study to vary between 0.2 and 1.7 mm for the set of stations examined. This is a clear

improvement over the previous estimates delivered from earlier reprocessing of the

GPS observations, and which has also been stated by Amiri-Simkooei et al. (2017).

The parameters of the stochastic part, or so-called noise, are examined, assuming

the combination of the power-law and white noises during the MLE analysis. These

parameters include the fraction of the power-law noise, meaning a percentage of the

power-law noise contribution in the considered combination, the spectral index and the

amplitude of power-law noise. These parameters are then used a priori in the creation

of the AWF to ﬁlter the seasonal curves from the real GPS observations.

The fraction of the power-law noise is large and equal to almost 1 (or 100%) for the

majority of stations with latitudes higher than 40° (Figs. 7and 8). For the remaining

stations, the percentage of the contribution of the power-law noise is much smaller.

It drops dramatically for latitudes lower than 10°. These results are consistent with

those published in Williams et al. (2004), who analyzed the latitude dependencies of

white and ﬂicker noises. They showed that white noise is much greater for lower than

for higher latitudes.

Fig. 6 The estimates of the amplitudes of the annual and semi-annual seasonal oscillations in the vertical

direction with the MLE method for the 385 IGS stations used in this study

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Fig. 7 The estimates of noise parameters delivered with the MLE for the GPS height residuals. These noise

parameters are used further to create and run the Adaptive Wiener Filter

Fig. 8 Parameters of the stochastic part plotted with respect to the latitude of the station. The amplitudes of

white (left) and power-law noise (middle) are given, along with a fraction of the power-law noise estimated

for the combination of the white plus power-law noises (right)

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4.4 The AWF-Based Estimates of Seasonals from the Real Dataset

As demonstrated for the synthetic datasets, assuming incorrect parameters of noise may

lead to incorrect estimates of the seasonal signals. Therefore, to create the AWF ﬁlter

for the real dataset, noise parameters estimated individually for each IGS GPS station

are employed, as presented in the previous section. To provide the time-changeability

of the seasonal curves, the AWF is run with the AR(1) process, as described in the

Methodology section. The residuals are analyzed after the AWF seasonal curve is

removed, using the MLE algorithm to compare the noise parameters of the original

and ﬁltered residuals.

Figure 9presents the detrended time series in the vertical direction for six GPS

stations for the original GPS time series and the seasonal curves estimated with the

AWF. These stations are characterized by large seasonal variations, of up to several

millimeters when different time spans are compared.

After the time-varying curves are removed from the IGS dataset, the parameters of

the noise created for the original residuals are compared with those for which the AWF

seasonal signals are removed. The differences between the fraction of power-law noise,

spectral indices and the standard deviations of noise (Fig. 10) are cross-compared. The

differences in the fraction of power-law noise are in the range −0.3 to 0.1, meaning

the maximum change in the fraction of the power-law noise is 30%, towards ﬂicker

noise. This is only observed for the ZHN1 station (Honolulu, Hawaiian Islands). For

the majority of stations, the differences in the fraction of the PL noise lay in the range

−0.1 to 0.1.

The differences in the spectral index of the power-law noise lay in the range −0.3

to 0.1. The maximum difference is found for the NAUR (Nauru, Australia), MCIL

(Marcus Island, Japan) and MERI (Merida, Mexico) stations, which are characterized

by an abnormally low fraction of power-law noise of less than 20% for the original

residuals. The spectral indices of the power-law noise lay in the range −0.1 to 0.1 for

the majority of stations, meaning no signiﬁcant change in the character of the noise

when the two types of residuals are compared.

The differences in the standard deviations of noise when the AWF curves are

removed vary between −0.25 and 0.25 mm for the majority of stations, which are

relatively low values, meaning no change in the power of the stochastic part when the

seasonal signals are estimated and removed. This also means that the AWF method

provides the optimal separation between signal and noise, with no absorption of noise

into the seasonal estimates.

5 Conclusions

The GPS position time series are affected by the time-varying seasonal curves arising

from the environmental loadings, systematic errors or numerical artifacts. Whether

they are affected more by the former or the latter, these seasonal changes have to

be modeled on a station-by-station basis using site-speciﬁc information on the noise

level, as undetected correlations present in the series will affect the noise parameters

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Fig. 9 Detrended "Up" component for six permanent stations, i.e. RAMO (Mitzpe Ramon, Israel), NICO

(Nicosia, Cyprus), NEAH (Neah Bay,USA), MAS1 (Maspalomas, Spain), LPAL (La Palma, Canary Islands)

and YAR2 (Yaragadee, Australia)

estimated from the data and, therefore, also the uncertainties of the estimated velocities

(Klos et al. 2018b).

Various methods have been used to model the time-varying curves, such as the

SSA, KF or WD. In addition to these methods, the AWF method is introduced in this

study, which has been used in geodesy only to denoise gravity records.

Klos et al. (2018c) showed how the power spectral densities change when the envi-

ronmental loading models are removed directly from the GPS position time series.

The CANT (Santander, Spain), IRKT (Irkutsk, Russia), VARS (Vardo, Norway) and

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Fig. 10 The differences in the fraction of power-law noise, spectral index of power-law noise and standard

deviation of noise (mm) when the AWF seasonal curve is removed from the GPS data. The differences

are estimated between the original (see Fig. 7) and the ﬁltered residuals (parameters for the original minus

parameters for the ﬁltered residuals)

GUAT (Guatemala City, Guatemala) stations were indicated. In this study, the differ-

ences between the noise parameters delivered for the residuals determined upon the

constant amplitude assumption and the AWF-derived curves being removed are esti-

mated. These differences in the spectral index vary from −0.06 for CANT, −0.02 for

IRKT to −0.03 for GUAT, meaning a change towards white noise. Almost all of the

385 series analyzed show no evident change between the spectral index estimated for

the AWF-based residuals and the original residuals. This means that the AWF approach

maintains the noise properties of the signals and removes only the time-varying curves

which are greater than the noise level.

Among others, Gu et al. (2016) presented the power spectra for the GPS position

time series and noticed the 4th and 5th harmonics for a tropical year of stacked PSDs,

while Bogusz and Klos (2016) showed the signiﬁcance of even the 9th overtone.

Khelifa (2016) presented the seasonal curves of 118 and 59 days estimated for the

Doppler orbitography and radiopositioning integrated by satellite (DORIS) time series.

The presented method can be extended to model these higher frequencies.

The proposed AWF is easy to implement and is a computationally low method

that allows us to determine the seasonal oscillations, including information on the

level of noise. If assumed properly during the creation of the AWF, the noise present

in the data will remain intact. Not only the level of noise but also its type has to

be included. Different geodetic observations are characterized by different kinds of

noise. As an example, zenith total delays are well-approximated by the autoregressive

process (Klos et al. 2018d), while DORIS observations are represented by pure power-

law noise (Klos et al. 2018a). For all these cases, the AWF constitutes an alternative

approach to how to split the original geodetic signal into seasonal signals and noise.

Also, the AWF provides a good separation between both, which means no inﬂuence

is observed by the noise on the seasonal estimates.

Acknowledgements This research is ﬁnanced by the National Science Centre, Poland, grant no.

UMO-2016/23/D/ST10/00495 under the leadership of Anna Klos. Machiel Simon Bos is supported by

national funds through FCT in the scope of the project IDL-FCT-UID/GEO/50019/2013 and grant no.

SFRH/BPD/89923/2012. IGS time series were accessed from ftp://igs-rf.ensg.eu/pub/repro2. Maps were

drawn in the Generic Mapping Tool (Wessel et al. 2013).

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-

tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,

and reproduction in any medium, provided you give appropriate credit to the original author(s) and the

source, provide a link to the Creative Commons license, and indicate if changes were made.

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