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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 theWF 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.
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
δaiφδai1+vi
δbiφδbi1+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
1φ2φkcos (ω0k),(4)
where σ2
v/1φ2φkis 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
12φcos (ω+ω0)+φ2+1
12φcos (ωω0)+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
21
+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
siF1ωjXωj.(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/yearκ/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/yearκ/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/yearκ/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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... This software uses maximum likelihood estimation (MLE) to estimate these parameters and related uncertainties. White noise and power-law noise close to flicker noise with a spectral index of 1 were considered as the optimum noise models to describe the stochastic part of the time series (He et al., 2017;Klos et al., 2018Klos et al., , 2019Williams et al., 2004). ...
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Obvious seasonal crustal vertical deformation largely related to mass redistribution on the Earth’s surface can be captured by Gravity Recovery and Climate Experiment (GRACE), simulated by surface loading models (SLMs), and recorded by continuous Global Positioning System (GPS). Vertical deformation time series at 224 GPS stations with more than four-year continuous observations are compared with time series obtained by GRACE and SLMs with the aim of investigating the consistency of the seasonal crustal vertical deformation obtained by different techniques in mainland China. Results of these techniques show obvious seasonal vertical deformation with high consistency at almost all stations. The GPS-derived seasonal vertical deformation can be explained, to some content, by the environmental mass redistribution effect represented by GRACE and SLMs. Though the mean weighted root mean square reduction is 34% when remove the environmental mass loading from the monthly GPS height time series (up to 47% for the mean annual signals), systematic signals are still evident in the residual time series. The systematic residuals are probably attributed to GPS related errors, such as draconitic errors, while the leakage errors in the GRACE data processing and unmodeled components in land water storage should be considered in some regions. Additionally, the obvious seasonal residual perturbations in Southwest China may be related to the leakage errors in the GRACE data processing and large uncertainty in the land water storage in SLMs, indicating that GPS observations may provide more realistic mass transport estimates in Southwest China.