A MATLAB-based Kriged Kalman Filter software for interpolating missing data in GNSS coordinate time series

GPS Solutions (2018) 22:25
https://doi.org/10.1007/s10291-017-0689-3
GPS TOOLBOX
A MATLAB‑based Kriged Kalman Filter software for interpolating
missing data in GNSS coordinate time series
Ning Liu1 · Wujiao Dai1 · Rock Santerre2 · Cuilin Kuang1
Received: 6 May 2017 / Accepted: 26 November 2017 / Published online: 9 December 2017
© Springer-Verlag GmbH Germany, part of Springer Nature 2017
Abstract
GNSS coordinate time series data for permanent reference stations often suffer from random, or even continuous, missing
data. Missing data interpolation is necessary due to the fact that some data processing methods require evenly spaced data.
Traditional missing data interpolation methods usually use single point time series, without considering spatial correlations
between points. We present a MATLAB software for dynamic spatiotemporal interpolation of GNSS missing data based on
the Kriged Kalman Filter model. With the graphical user interface, users can load source GNSS data, set parameters, view
the interpolated series and save the final results. The SCIGN GPS data indicate that the software is an effective tool for GNSS
coordinate time series missing data interpolation.
Keywords GNSS · Missing data · Kriged Kalman Filter · MATLAB software
Introduction
GNSS coordinate time series might include tectonic displacements and non-tectonic signals such as seasonal variations and noise. Many methods can be used to separate
useful signals from GNSS time series, such as the wavelet
transform (Collin and Warnant 1995), principle component
analysis (PCA, Dong et al. 2006), independent component
analysis (ICA, Liu et al. 2015; Peng et al. 2017; Liu et al.
2017) and spectral analysis (SA, Mao et al. 1999). Some
methods are integrated into various GNSS time series
GPS Tool Box is a column dedicated to highlighting algorithms
and source code utilized by GPS engineers and scientists. If you
have an interesting program or software package you would like
to share with our readers, please pass it along; e-mail it to us at
gpstoolbox@ngs.noaa.gov. To comment on any of the source
code discussed here, or to download source code, visit our
website at http://www.ngs.noaa.gov/gps-toolbox. This column
is edited by Stephen Hilla, National Geodetic Survey, NOAA,
Silver Spring, Maryland, and Mike Craymer, Geodetic Survey
Division, Natural Resources Canada, Ottawa, Ontario, Canada.
* Wujiao Dai
wjdai@csu.edu.cn
1
School of Geosciences and Info‑Physics, Central South
University, Changsha 410083, Hunan, China
2
Center for Research in Geomatics, Laval University,
Quebec G1V 0A6, Canada
analysis software, such as the IDL package for GPS position
time series analysis iGPS (Tian 2011) and the GPS interactive time series analysis GITSA (Goudarzi et al. 2013)
software. However, all of these methods can only process
the evenly spaced time series without missing data. Unfortunately, with permanent GNSS reference stations running
automatically, missing data are very common in GNSS
coordinate time series because of the interruption of communication links or power failures. The missing data could
be random, e.g., 3–5 days, but more often, it continues for a
long period of time, e.g., more than 3 months. In those cases,
the coordinate time series are discarded, though they may
contain useful information. Thus, missing data interpolation
is an essential data preprocessing step before the GNSS time
series analysis.
There are three types of GNSS missing data interpolation
methods: conventional interpolation methods, the singular
spectrum analysis (SSA)-based method and the iterative
empirical orthogonal function (EOF)-based method. Conventional interpolation methods include the nearest interpolation, linear interpolation, cubic spline interpolation,
polynomial interpolation and sectioned Hermite interpolation. Goudarzi et al. (2013) integrated these methods into a
GNSS analysis software, called GITSA. But they work well
only with random missing data. Therefore, time series with
continuous missing data during GNSS time series analysis are often discarded (Wang et al. 2012). The SSA-based
13
Vol.:(0123456789)
25 Page 2 of 8
method is widely used, because it does not require any prior
knowledge of the behavior of the time series. It has been
introduced to describe the change of abundance of zooplankton (Colebrook 1978). Kondrashov and Ghil (2006)
first proposed to use it for filling dataset gaps in sea-surface
temperature time series. Qiu et al. (2015) and Wang et al.
(2016) used it to interpolate GNSS missing data. Results
show that this method performs well even for GNSS continuous missing data. But the interpolated results are too
smooth and the technique cannot keep high frequency signals. The EOF-based method (Xu 2016) is another method
which does not require any prior knowledge of the behavior
of the time series and it also considers offsets and colored
noise presented in real GPS coordinate time series. But all
of these methods focus on single point time series, without
considering the spatial correlation among points in a region.
We propose a MATLAB software, called the GNSS Missing Data Interpolation Software (GMIS), for interpolating
GNSS coordinate time series with missing data using a
dynamic spatiotemporal model based on the Kriged Kalman
Filter (KKF). KKF was first proposed to deal with sulfur
dioxide distribution in the UK (Mardia et al. 1998). After
that, a Distributed Kriged Kalman Filter DKKF (Cortes
2010) and a Bayesian Kriged Kalman Filter BKKF (Sahu
and Mardia 2005; Lasinio et al. 2007; Al-Awadhi and
Alhajraf 2012) were proposed. Nowadays, KKF has been
successfully applied in environment pollution monitoring (Mardia et al. 1998; Sahu and Mardia 2005), climate
(Lasinio et al. 2007), wind farm (Qing et al. 2012) and dam
deformation monitoring (Dai et al. 2016). However, KKF
has been rarely applied for the analysis of GNSS coordinate
time series.
We first review briefly the KKF model and introduce the
interpolation method for GNSS missing data based on KKF.
The installation and introduction of this software are followed. Finally, a SCIGN GPS missing data interpolation
experiment using this software is presented.
KKF model
The spatiotemporal observation Z(s, t) in a KKF model can
be decomposed into
(1)
where s = {s1 , s2 , .., sn } is the position, t = {1, 2, … , m} is
T
the time,
operator. The spatial field
[ and (∗) is the transpose
]T
𝐡(s) = h1 (s)h2 (s) … hp (s) can be obtained by the universal
kriging (Mardia et al. 1998). 𝜶(t) represents the state vector
of the system and 𝜀t (s) is the observation noise.
Z(s, t) = 𝐡(s)T 𝜶(t) + 𝜀t (s)
13
GPS Solutions (2018) 22:25
For dimension reduction, we consider the generalized bending energy matrix B (Mardia et al. 1998; Bookstein 1989)
describing the degree of spatial variation, which is as follows:
)−1
(
T −1
−1
F
F T Σ−1
F
F
Σ
−
Σ
B = Σ−1
𝜍
𝜍
𝜍
𝜍
(2)
w h e[r e (∗)−1 i s t h e ]i n v e r s e o p e r a t o r ,
( )T ( )T
( )T
F = f s1 ; f s2 ; … ; f sn
is the trend matrix. The
[
]T
chosen space trend field f (s) = f1 (s)f2 (s) … fq (s) could be
constant, linear or cubic polynomials. Σ𝜍 is the covariance
matrix of (the )residuals after removing a space trend, and its
elements Σ𝜍 ij can be calculated through the fitted semi(
)
variogram model 𝛾𝜍 si , sj :
)
(
)
(
( )
Σ𝜍 ij = C − 𝛾𝜍 si , sj = C − 𝛾𝜍 si − sj
(3)
where C is the sill of the semi-variogram model, and ‖ ∗ ‖
is the distance operator. The semi-variogram model can be
obtained by fitting the empirical semi-variogram value 𝛾̂𝜍 (d):
m
𝛾̂𝜍 (d) =
]2
1 ∑[
Dt (s) − Dt (s + d)
2m t=1
(4)
∑∗
where ∗ (∗) is the sum operator, (∗)2 is the square operator and d is the lag distance. Dt (s) = Z(s, t) − f (s)𝛽t is
the residual after removing a space
( trend,
)−1 and coefficient 𝛽t can be calculated by 𝛽t = F T F F T Zt , where
[ (
) (
)
(
)]T
Zt = Z s1 , t Z s2 , t … Z sn , t . More often, the lag
distance is divided into some groups from 0 to maximum
lag distance for semi-variogram calculation, and one can
calculate an empirical semi-variogram value in each group
for the same lag distance, so d is the middle lag distance of
each group.
In order to measure the spatial variation degree, the spectral
decomposition of B is performed:
B = UEU T , B𝐮i = ei 𝐮i (i = 1, 2, … , n)
(5)
(
)
where the column vectors of U = 𝐮1 𝐮2 … 𝐮n( are eigen-)
vectors and the diagonal elements of E = diag e1 , e2 , .., en
are the corresponding eigenvalues, and diag(∗) is diagonal
matrix operator. Bookstein (1989) and Sahu and Mardia
(2005) reported that smaller eigenvalues are associated with
larger spatial variation and larger eigenvalues represent more
local spatial variations. To reduce the dimension, one only
needs to take the first p spatial variation, which can be cal∑p
∑
culated while the ratio r = i=1 ei / ni=1 ei is greater than
a certain proportion, where usually 90 or 95% are chosen.
Spatial field 𝐡(s) can be obtained following the work of
Mardia et al. (1998) and Sahu and Mardia (2005):
hj (s) = fj (s)
hk (s) = ek 𝝈 𝜍 (s)T 𝐮k
j = 1, 2, .., q
k = q + 1, … , p
(6)
Page 3 of 8 25
GPS Solutions (2018) 22:25
vector with its elements
(where)𝝈 𝜍 (s) stands
( for covariance
)
𝝈 𝜍 (s) i = C − 𝛾𝜍 s − si , and i = 1, 2, … , n.
Now, one can write (1)–(6) into matrix forms:
[ ( ) ( )
( )]T
𝜺t = 𝜀t s1 𝜀t s2 … 𝜀t sn
[ ( ) ( )
( )]T
H = 𝐡 s1 𝐡 s2 … 𝐡 sn
(7)
and the observation equation of KKF can be expressed by:
(8)
where 𝜺t ∼ N(0, R), N(∗, ∗) represents the normal distribution. Let 𝜶(t) evolve in time with a first-order autoregressive [AR(1)] model and form the state transition equation
of KKF:
Zt = H𝜶(t) + 𝜺t
(9)
where 𝛷 is the state transition matrix and the system noise
is 𝜼t ∼ N(0, Q). Equations (8) and (9) form the KKF model.
More details on KKF model are discussed in Mardia et al.
(1998) and Sahu and Mardia (2005).
𝜶(t) = 𝛷𝜶(t − 1) + 𝜼t
Missing data interpolation method
The key issue of dynamic spatiotemporal interpolation of
missing data using KKF is how to continuously perform the
process while missing data arise randomly and continue for
a certain period of time. Shumway and Stoffer (1982, 2010)
established a Kalman Filter observation equation for missing
data. Suppose at time t the KKF observation equation can
be written as follows:
(
Z1t
Z2t
)
=
(
)
( 1)
H1
𝜺t
𝜶(t)
+
H2
𝜺2t
(10)
where Z1t , H 1 and 𝜺1t are the observed data, the spatial field
of the observed data and the noise of the observed data,
respectively. Z2t , H 2 and 𝜺2t are the missing data, the spatial
field of the missing data and observed noise of the missing
data, respectively. The distribution of observation noise can
be expressed by:
( 1)
(
)
R11 R12
𝜺t
∼
N
0,
(11)
𝜺2t
R21 R22
Replace Z2t and H 2 with 0 vector and 0 matrix, so the
missing data Z2t are not involved in the estimation of 𝜶(t).
Then, replace R12 and R21 with 0 matrix. It shows that the
replaced 0 vector is uncorrelated with observed data Z1t .
Using the Kalman recursion of the updated observation
equation to estimate 𝜶(t), then the missing data Z2t can be
interpolated by Z2t = H 2 𝜶(t).
In simple terms, the procedure of the above-mentioned
interpolation method involves three steps: (1) interpolate
unobserved data Z2t by the universal kriging only using the
KKF observation equation; (2) forecast Z2t by the AR(1)
model only using the KKF state transition equation; and (3)
combine the interpolated result in the spatial domain with
the forecasted result in the temporal domain to generate the
spatiotemporal prediction result.
The effect of missing data interpolation by KKF is closely
related to the precision of spatial
{ field H and the}estimation of Kalman
parameters
𝜃 = â 0|0 , P0|0 , 𝛷, R, Q , where
(
)
𝜶(0) ∼ N â 0|0 , P0|0 . The fitted semi-variogram model
reflects the precision of spatial field H . There are three main
reasons for incorrectly fitted semi-variogram model: (1) an
improper space trend field f (s) is chosen, which results in
non-stationary residuals; (2) the measurement sites are so
sparse that observed values cannot grasp the spatial correlation; and (3) observations in the spatial domain are uncorrelated in itself. If the semi-variogram model is poorly fitted,
using inaccurate spatial field H will lead to filter divergence.
In this case, a unit matrix, instead of the spatial field, which
shows no correlation in the spatial domain, is a better choice.
Shumway and Stoffer (1982) proposed a method, which
combines Expectation Maximum (EM) estimation and
Kalman smoothing, to estimate Kalman parameters 𝜃 . If the
data contain missing values, this method can also reasonably estimate 𝜃 . But Kalman smoothing needs to save the
filtered state covariance matrix Pt|t , smoothed state covariance matrix Pn|t , and cross-covariance Pt|t−1 for all time
t = {1, 2, … , m}, which results in three p × p × m matrices.
If the monitoring time m is long, this computation load is
very large. The experiment of Shumway and Stoffer (1982)
only involved 28 issues of data, which is suitable for using
Kalman smoothing. But for GNSS data, the monitoring often
continues for a few years, even tens of years. Compared with
Kalman smoothing, Kalman filtering, which only saves the
filtered state covariance matrix Pt|t and cross-covariance
Pt|t−1 of the previous time, needs less computation time
and computer memory. Although the accuracy of Kalman
smoothing is better than Kalman filtering, GMIS software
still keeps Kalman filter option for computers with lower
computational abilities. Table 1 shows the steps of the GNSS
missing data interpolation method based on a KKF model.
Software platform and installation
The GMIS software was developed using a MATLAB
graphic user interface. MATLAB provides excellent matrix
manipulation ability and powerful graphic capability. Also,
MATLAB has integrated a lot of useful data analysis algorithms and developers can use it directly and scan its source
code. Moreover, MATLAB graphic user interfaces can
compile source code into an executable program, which
can be used in most operating systems, such as Windows
13
25 Page 4 of 8
GPS Solutions (2018) 22:25
Table 1 Steps of GNSS missing data interpolation-based KKF model
Establishment of spatial field H
(1) Analyze GNSS data spatial distribution characteristics and choose suitable space trend field f (s)
(2) Obtain residuals after removing space trend and calculate empirical semi-variogram value
(3) Choose a suitable semi-variogram model to fit experimental semi-variogram value, then construct the spatial field
Estimation of KKF model parameters 𝜃
(4) Use expectation maximum (EM) estimation with Kalman filter (or Kalman smoothing) to estimate KKF model parameters 𝜃
Dynamic spatiotemporal GNSS missing data interpolation
(5) t = 1
(6) Check whether data contain missing data. If yes, then replace corresponding missing value Z2t , spatial field H 2 and covariance matrix R12 and
R21 between observed data and missed data with 0 vector or 0 matrix
(7) Use the Kalman filter equation to estimate state vector 𝜶(t) at time t
(8) Interpolate missing data at time t : Z2t = H 2 𝜶(t)
(9) t = t + 1
(10) Repeat steps (5)–(9) until all missing data are interpolated
and Linux. So, MATLAB is a suitable platform to develop
our software for GNSS missing data interpolation. In addition, GMIS software needs extra ‘variogramfit.m’ and
‘fminsearchbnd.m’ toolboxes, which can be downloaded
from http://cn.mathworks.com/matlabcentral/fileexchange
for free. These two toolboxes have been included in the
GMIS software.
The GMIS software and its source codes are publicly available through the Internet. The software can be
downloaded from http://faculty.csu.edu.cn/daiwujiao/en/
lwcg/35988/content/11646.htm or http://www.ngs.noaa.
gov/gps-toolbox/GMIS.htm. There are two ways to use
this software: (1) check that your computer has an installed
MATLAB Compiler Runtime (MCR) for version 8.1, then
double-click the executable program to run the software; (2)
add the source code folder into the MATLAB search path,
type GMIS in the MATLAB command window, then the
main interface of the GMIS software will pop up.
Software introduction
The software is composed of three graphic user interfaces
(GUI): GMIS, SemiVarigFit and InterpolationMode. GMIS
is the main GUI, which is used to load input GNSS data,
save interpolated GNSS results, invoke SemiVarigFit GUI
and InterpolationMode GUI, and view source and interpolated GNSS time series.
GMIS allows only two native formats as input and output
data files, which are the GNSS missing time series format
(GMS) and the MATLAB mat-file format (MAT). One GMS
data file only records one GNSS site data. The GMS file
is an ASCII text containing header block and data block.
Header block records the name, longitude and latitude of
GNSS sites, start date and end date of interpolation, etc. The
13
Load Data
(GMIS GUI)
Calculate empirical semi -variogram
values
(SemiVarigFit GUI)
Fit empirical semi -variogram
values
(SemiVarigFit GUI)
Interpolate missing data
(InterpolationMode GUI)
Save interpolated result
(GMIS GUI)
Fig. 1 GMIS software operation flowchart
data block records source GNSS time series or interpolated
GNSS time series in the north, east and up directions. The
input MAT file is a MATLAB mat file containing a structure. All GNSS site data are saved in this structure. This
structure with an arbitrary name contains 7 fields, and the
field names are fixed. Field ‘day’ saves monitoring time.
Fields ‘dn,’ ‘de’ and ‘du’ show GNSS data in the North,
East and Up directions, and missing values are substituted
by NaN (Not a Number). Position coordinates information
is shown in the ‘x’ and ‘y’ fields, in which ‘x’ represents the
Page 5 of 8 25
GPS Solutions (2018) 22:25
Fig. 2 Interpolation effects for
four sites (BLYT, SIBE, LNMT
and GNPS) with seriously
missed data in the GNSS coordinates time series
Original data
0
5 SIBE-N
0
-5
-10
-15
BLYT-N
-10
GPS series(mm)
-20
0
-10
0
-10
-20
-30 SIBE-E
30
SIBE-U
20
10
0
-10
5 GNPS-N
0
-5
-10
-15
-20
BLYT-E
-30
30 BLYT-U
20
10
0
5
LNMT-N
0
-5
0
0
-20
-10
-20
LNMT-E
-30 GNPS-E
30
GNPS-U
20
10
0
-40
20 LNMT-U
10
0
-10
Filter and Interpolated data
2005-09-01 2006-05-02
2005-09-01 2006-05-02
Time(Year-Month-Day)
longitude and ‘y’ the latitude. The last field ‘site’ is used to
save the site name. The output MAT data file contains all the
fields of the source data file, but with 4 added fields to save
the final results. The added fields ‘interp_dn,’ ‘interp_de’
and ‘interp_du’ save interpolated GNSS data, and the added
‘VarigPar’ field saves the fitted semi-variogram model result.
It is worth noting that the first day value of all input GNSS
time series in the three directions should be translated to
zero, because the initial state of Kalman filter is set by a zero
vector in this software. More GMS and MAT data file details
can be found in the user manual.
Empirical semi-variogram value calculation and its
semi-variogram model fitness for three directions are carried out in the SemiVarigFit GUI. In SemiVarigFit, users
can choose spatial trends for three directions, set the maximum lag distance for semi-variogram calculation and set
number of groups that the distance should be divided into.
Once these parameters are set, clicking the ‘Plot-Exp-Variog
Value’ button will bring empirical semi-variogram value
and draw it into the left box panel. Note that the maximum
distance is an important parameter for semi-variogram calculation, because sometimes the spatial correlation is local
in scale, and the use of the default value, which may be
a large scale, will result in a disorganized empirical semivariogram value. Users can decrease the maximum distance
gradually to obtain a better fitted effect of semi-variogram
model into the three directions. According to the empirical semi-variogram value in the left box panel, set a proper
initial range, sill and nugget parameters (Cressie 1993), and
choose a suitable semi-variogram model. If users want to
use the number of observations in each lag distance group
as the weight function for fitting semi-variogram model,
just choose ‘cressie85’ (Cressie 1985) or ‘mcbratney86’
(Mcbratney and Webster 1986) for the weight function.
After the fitting parameters are all properly assigned, click
‘Fit Exp-Varigo Value’ button to fit the empirical semi-variogram value, which can be shown in the left box panel. Once
the semi-variogram model is fitted, the fitted result will be
saved automatically.
Some configurations of KKF can be assigned in the
InterpolationMode GUI, e.g., the number of Expectation
Maximum (EM) iterations, the option to use unit matrix
instead of an inaccurate spatial field, and an option to use
‘EM + Kalman filter’ mode or ‘EM + Kalman smooth’
mode. If the user is not satisfied with the fitted effect of
the semi-variogram model in one of the three directions,
13
25 Page 6 of 8
GPS Solutions (2018) 22:25
Fig. 3 Interpolation effects with
the introduction of artificial
gaps for four slightly missed
data at GPS sites 7ODM,
BSRY, DVPB and HOGS
Original data
Artificial gaps
20 7ODM-N
10
5
0
0
-10
-5
0
0
-20
BSRY-N
-20
-40
-60 7ODM-E
40 7ODM-U
GPS series(mm)
Filter and Interpolated data
-40 BSRY-E
20 BSRY-U
20
10
0
0
-20
-10
20 DVPB-N
40
10
HOGS-N
20
0
0
0
-20
-40
-60
-80 HOGS-E
20
0
-20
-40
-60 DVPB-E
20 DVPB-U
10
0
-10
10
HOGS-U
Time(Day)
0
-10
2005-09-01 2006-05-02
2005-09-01 2006-05-02
Time(Year-Month-Day)
he can use a unit matrix instead of the spatial field in the
InterpolationMode GUI. When all parameters or options are
assigned, the spatial field will be calculated automatically
and the GMIS software is ready to interpolate missing data.
A software flowchart is shown in Fig. 1. More details about
the software can be found in the user manual showing how
to use this software with two step-by-step examples.
Table 2 RMS for the interpolation results of artificial gaps in four
GPS sites 7ODM, BSRY, DVPB and HOGS with slightly missed data
(unit: mm)
North
East
Up
7ODM
BSRY
DVPB
HOGS
2.6
3.2
9.9
0.8
0.7
2.8
1.6
1.1
6.3
0.6
0.7
2.1
GPS missing data interpolation experiment
This section presents the GNSS missing data interpolation
results using the GMIS software as an example. The data
that are used, containing GPS coordinates from daily solutions for 209 sites, come from the Southern California Integrated GPS Network (SCIGN: http://www.scign.org/) from
January 1, 2005, to January 1, 2007, for a total of 731 days.
Before the experiment, GPS offsets and some gross errors
are removed and the first day value of all site series in the
three directions are translated to zero. Furthermore, in order
to show the efficiency of the GMIS software, we also remove
100 days of data for four GPS sites, and use the GMIS software to fill these artificial gaps.
13
Some parameters of the fitting semi-variogram model
and configurations of KKF can be seen in the SCIGN
example in the user manual. The interpolation effects for
the four sites with seriously missed data are displayed in
Fig. 2. Figure 3 shows the interpolated result for artificial
gaps of four slightly missed GPS sites, and the interpolated results fit well with the original data. Table 2 shows
the root mean square (RMS) of artificial gaps interpolated
results. The largest RMS is 9.9 mm in the Up direction
of 7ODM site, the smallest RMS is 0.6 mm in the North
direction of HOGS site, and the average RMS is 2.7 mm.
We also compared the computation time of the
‘EM + Kalman filter’ mode and that of the ‘EM + Kalman
smooth’ mode. All computations were carried out in
GPS Solutions (2018) 22:25
MATLAB on a Windows 10 system with a 2 GHz Intel processor and 4 GB memory. The computations related to the
‘EM + Kalman filter’ mode cost 3264 s (54.4 min) and 5574
(92.9 min) seconds for ‘EM + Kalman smooth’ mode.
Conclusions
We presented a MATLAB software, called GMIS, for
GNSS missing data interpolation based on a dynamical
spatiotemporal model KKF. This software can serve as
a GNSS preprocessing tool for software which requires
evenly spaced GNSS coordinate time series. The GMIS
software and all source codes are available from the compressed files. If there are any questions, suggestions or
corrections about GMIS software, please send them to the
following e-mail address: nliucsu@csu.edu.cn.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 41674011) and the State
Key Development Program of Basic Research of China (Grant No.
2013CB733303).
References
Al-Awadhi FA, Alhajraf A (2012) Prediction of non-methane hydrocarbons in Kuwait using regression and Bayesian kriged Kalman
model. Environ Ecol Stat 19(3):393–412
Bookstein FL (1989) Principal warps: thin-plate splines and the
decomposition of deformations. IEEE Pattern A 11(6):567–585
Colebrook JM (1978) Continuous plankton records: zooplankton and
environment, North-East Atlantic and North Sea, 1948–1975.
Oceanol Acta 1(1):9–23
Collin F, Warnant R (1995) Application of the wavelet transform for
GPS cycle slip correction and comparison with Kalman filter.
Manuscr Geod 20(3):161–172
Cortes J (2010) Distributed Kriged Kalman filter for spatial estimation. IEEE Autom Control 54(12):2816–2827
Cressie N (1985) Fitting variogram models by weighted least
squares. Math Geol 17(5):563–586
Cressie N (1993) Statistics for spatial data. Wiley, New York
Dai WJ, Liu N, Santerre R, Pan JB (2016) Dam deformation monitoring data analysis using space–time Kalman filter. ISPRS Int
J Geo-Inf 5(12):236–251
Dong D, Fang P, Bock Y, Webb F, Prawirodirdjo L, Kedar S,
Jamason P (2006) Spatiotemporal filtering using principal component analysis and Karhunen–Loeve expansion
approaches for regional GPS network analysis. J Geophys Res
111(B3):1581–1600
Goudarzi MA, Cocard M, Santerre R (2013) GPS interactive time
series analysis software. GPS Solut 17(4):595–603
Kondrashov D, Ghil M (2006) Spatio-temporal filling of missing
points in geophysical datasets. Nonlinear Process Geophys
13(2):151–159
Lasinio GJ, Sahu SK, Mardia KV (2007) Modeling rainfall data
using a Bayesian Kriged-Kalman model. In: Upadhy SK, Singh
U, Dey DK (eds) Bayesian statistics and its applications, 1st
edn. Anshan, Tunbridge Wells, pp 1–24
Page 7 of 8 25
Liu B, Dai WJ, Peng W, Meng XL (2015) Spatiotemporal analysis
of GPS time series in vertical direction using independent component analysis. Earth Planets Space 67(1):1–10. https://doi.
org/10.1186/s40623-015-0357
Liu B, Dai W, Liu N (2017) Extracting seasonal deformations of the
Nepal Himalaya region from vertical GPS position time series
using independent component analysis. Adv Space Res. https://
doi.org/10.1016/j.asr.2017.02.028
Mao A, Harrison C, Dixon TH (1999) Noise in GPS coordinate time
series. J Geophys Res 104(B2):2797–2816
Mardia KV, Goodall C, Redfern EJ, Alonso FJ (1998) The Kriged
Kalman filter. Test 7(2):217–282
Mcbratney AB, Webster R (1986) Choosing functions for semi-variograms of soil properties and fitting them to sampling estimates.
Eur J Soil Sci 37(4):617–639
Peng W, Dai WJ, Santerre R, Kuang CL (2017) GNSS vertical coordinate time series analysis using single-channel independent
component analysis method. Pure Appl Geophys 174(2):723–
736. https://doi.org/10.1007/s00024-016-1309-9
Qing XY, Yang FW, Wang XY (2012) Short-term wind speed forecasting for multiple wind farms using Bayesian Kriged-Kalman
model. Proc Chin Soc Electr Eng 32(35):107–114
Qiu RH, Cheng YY, Wang H, Wang XM, Cao BQ (2015) The interpolation application of interval quartering algorithm of singular
spectrum analysis iterative in GPS coordinate time series. J Geod
Geodyn 35(6):1017–1020
Sahu SK, Mardia KV (2005) A Bayesian kriged Kalman model
for short-term forecasting of air pollution levels. Appl Stat
54(1):223–244
Shumway RH, Stoffer DS (1982) An approach to time series smoothing and forecasting using the EM algorithm. J Time Ser Anal
3(4):253–264
Shumway RH, Stoffer DS (2010) Time series analysis and its applications: with R examples. Springer, New York
Tian Y (2011) iGPS: IDL tool package for GPS position time series
analysis. GPS Solut 15(3):299–303
Wang W, Zhao B, Wang Q, Yang XM (2012) Noise analysis of continuous GPS coordinate time series for CMONOC. Adv Space Res
49(5):943–956
Wang XM, Cheng YY, Wu SQ, Zhang KF (2016) An effective toolkit for
the interpolation and gross error detection of GPS time series. Surv
Rev 48(348):202–211
Xu C (2016) Reconstruction of gappy GPS coordinate time series using
empirical orthogonal functions. J Geophys Res 121(12):9020–9033
Ning Liu is a master student in
the School of Geosciences and
Info-Physics at the Central South
University, China. His research
focuses on spatiotemporal
Kalman Filter model and its
application in deformation
monitoring.
13
25 Page 8 of 8
GPS Solutions (2018) 22:25
Wujiao Dai is a professor in the
School of Geosciences and InfoPhysics at the Central South University, China. He received his
Ph.D. from CSU in 2007 and
worked as a research assistant in
Hong Kong Polytechnic University for 3 years. His research
mainly focuses on GNSS data
processing and deformation
monitoring.
Rock Santerre is a full professor
of Geodesy and GPS in the
Department of Geomatics Sciences and a member of the
Center for Research in Geomatics at Laval University. Since
1983, his research activities have
been mainly related to high-precision GPS for static and kinematic positioning. Dr. Santerre is
13
the author and coauthor of more than 150 publications, and he holds
three patents related to GPS equipment.
Cuilin Kuang is an associate professor in the School of Geosciences and Info-Physics of
Central South University, China.
He received his Ph.D. from
Wuhan University in 2008. His
research mainly focuses on
GNSS data processing and
deformation monitoring.