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Journal of Geodesy (2023) 97:109
https://doi.org/10.1007/s00190-023-01789-z
ORIGINAL ARTICLE
Kalman filter-based integration of GNSS and InSAR observations for
local nonlinear strong deformations
Damian Tonda´s1·Maya Ilieva1·Freek van Leijen2·Hans van der Marel2·Witold Rohm1
Received: 21 November 2022 / Accepted: 4 October 2023
© The Author(s) 2023
Abstract
The continuous monitoring of ground deformations can be provided by various methods, such as leveling, photogrammetry,
laser scanning, satellite navigation systems, Synthetic Aperture Radar (SAR), and many others. However, ensuring sufficient
spatiotemporal resolution of high-accuracy measurements can be challenging using only one of the mentioned methods.
The main goal of this research is to develop an integration methodology, sensitive to the capabilities and limitations of
Differential Interferometry SAR (DInSAR) and Global Navigation Satellite Systems (GNSS) monitoring techniques. The
fusion procedure is optimized for local nonlinear strong deformations using the forward Kalman filter algorithm. Due to
the impact of unexpected observations discontinuity, a backward Kalman filter was also introduced to refine estimates of
the previous system’s states. The current work conducted experiments in the Upper Silesian coal mining region (southern
Poland), with strong vertical deformations of up to 1m over 2 years and relatively small and horizontally moving subsidence
bowls (200m). The overall root-mean-square (RMS) errors reached 13, 17, and 35mm for Kalman forward and 13, 17, and
34mm for Kalman backward in North, East, and Up directions, respectively, in combination with an external data source -
GNSS campaign measurements. The Kalman filter integration outperformed standard approaches of 3-D GNSS estimation
and 2-D InSAR decomposition.
Keywords Integration ·Kalman filter ·InSAR ·GNSS ·Mining deformations
1 Introduction
Over the years, the geodetic capabilities for determining land
surface changes have developed significantly and also have
found application in areas particularly affected by mining
activities. Nowadays, leveling, gravimetry, photogramme-
try, Light Detection and Ranging (LiDAR), Interferometric
Synthetic Aperture Radar (InSAR), and Global Navigation
Satellite Systems (GNSS) techniques are the most widely
adopted methods for ground deformation monitoring. How-
ever, providing a high-accuracy spatiotemporal system for
3-D terrain motion measurements can be challenging using
only one of the mentioned methods. To provide a con-
BDamian Tonda´s
damian.tondas@upwr.edu.pl
1Institute of Geodesy and Geoinformatics, Wrocław University
of Environmental and Life Sciences, Grunwaldzka 53, 50-357
Wrocław, Poland
2Department of Geoscience and Remote Sensing, Delft
University of Technology, Stevinweg 1, 2628 CN Delft, The
Netherlands
tinuous ground monitoring service for ongoing long-term
movements, the GNSS and InSAR methods are commonly
implemented (Shen et al. 2019; Del Soldato et al. 2021).
We performed a thorough review of the strengths and
weaknesses of both techniques, which is available in
Appendix A (Table 5). In summary, the InSAR is a few-day
periodic, one-dimensional, high spatial resolution, remote
sensing technique (no ground-based equipment required),
whereas the GNSS is a continuous, three-dimensional,
station-dependent resolution measurement technique (requires
ground-based equipment). The InSAR one-dimensional Line
of Sight (LOS) observations taken using different geometries
could not be directly decomposed into all three coordi-
nate components (North, East, Up) without introducing bias
(Brouwer 2021). Based on the presented analysis, we demon-
strate that the combination of GNSS and InSAR as two
complementary methods allows to establish an overall sys-
tem for the determination of three-dimensional displacement
values and movement rates. In this study, we focus on exploit-
ing strengths and reducing weaknesses of GNSS and InSAR
techniques by providing a new methodology of integra-
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109 Page 2 of 23 D. Tondaś et al.
tion involving Kalman filter algorithms for nonlinear strong
ground displacements that prevent the use of multi-temporal
InSAR (MT InSAR) techniques (e.g., Przyłucka et al. 2015).
Due to the near-polar orbit of the satellites, the limited
sensitivity of the northern component is one of the main prob-
lems of conducting three-dimensional InSAR monitoring.
Hu et al. (2014) provide a systematic review of the methods
for mapping 3-D displacements using InSAR measurements
pointing out the strengths and weaknesses of each approach.
The presented by Hu et al. (2014) classification has three
groups: (I) a combination of multi-pass LOS and azimuth
measurements from at least two different geometries (e.g.,
Ng et al. 2011; Yang et al. 2017), (II) integration of GNSS
and InSAR data (e.g., Bozso et al. 2021; Liu et al. 2019),
and (III) prior information assisted approaches (e.g., Mohr
et al. 1998; Kumar et al. 2011). The authors highlight that
the methods from the first group are suitable for investigating
the large displacements caused mostly by natural phenom-
ena (e.g., earthquakes, volcanic eruptions). The methods of
GNSS and InSAR integration (group II) are sufficient for
both large-scale and local displacements; however, the final
outcome is highly dependent on the number and distribu-
tion of the GNSS receivers. The third group, related to prior
data, can be applied only in combination with external data
sources, e.g., glacier or landslide movement models.
In the presented study, we refer directly to the (II) group
of the mentioned methods—the integration of GNSS and
InSAR techniques. Currently, the fusion process involves dif-
ferent approaches and assumptions, such as linear prediction
for large-scale displacements or spatiotemporal interpola-
tion for rapid nonlinear movements. For instance, Fuhrmann
et al. (2015) presented an integrated method using level-
ing, GNSS (76 stations) and InSAR (ERS-1/2 and Envisat
data from ascending and descending tracks) to determine
the linear velocities over a larger area (around 32 000 km2).
The nonlinear displacements (located mostly in regions with
man-induced surface deformations) were excluded from the
investigation. The first step of the proposed method relied
on determining the offset and trend between the InSAR data
and the other two techniques. In the second step, local tan-
gent rates were estimated using the least-squares adjustment
(LSA) methodology. The precision of the obtained results
was estimated as 0.30 mm/a in the horizontal and 0.13 mm/a
in the vertical direction.
Another concept applies linear interpolation and predic-
tion based on the specific spatial and temporal domains of
the GNSS and InSAR data (Liu et al. 2019). The authors
used a dynamic filtering method that relied on a spatiotem-
poral Kalman filter to predict the large-scale deformations
in the area of Los Angeles, USA. The research presented in
this study was based on simulated and real data. The real
observations comprised 15 InSAR images from the Envisat
descending orbit and data from 35 GNSS receivers. Due to the
single geometry orbit, horizontal and vertical deformations
were not resolved, and GNSS coordinates were projected
into the LOS direction. The quality of the deformation field
model was established using the leave-one InSAR image-out
validation method (one InSAR image was taken as the refer-
ence and compared with the corresponding image generated
from the spatiotemporal deformation model). The root mean
square (RMS) between the interpolated and the reference
InSAR image did not exceed 7.2 mm in the LOS direction,
whereas the maximum deformation did not exceed 100mm.
More recently, a new type of procedure to integrate Dif-
ferential InSAR (DInSAR) and GNSS results was presented
in the study of Bozso et al. (2021). The paper shows a method
for the estimation of 3-D deformation parameters and rates
using Sentinel-1 SAR data and GNSS epoch observations
converted to the Sentinel-1 LOS domain. The Kalman filter
algorithm was tested on a landslide area in Hungary with
the assumption of linear velocities in the LOS direction. A
network of geodynamic benchmarks, consisting of a pair of
corner reflectors (aligned with the ascending and descend-
ing Sentinel-1 geometry) and a GNSS antenna adapter for
campaign measurements, was created in the study area. To
determine the 3-D deformation parameters, the GNSS obser-
vations were projected onto the radar LOS vectors. The
presented approach was based only on linear velocities cal-
culated from two GNSS campaigns. As a model of the system
noise, the apriorivalues were assumed as 2mm and the a
posteriori errors of the NEU components were 16, 2, and
3 mm, respectively. However, this way of uncertainty assign-
ment is not rigorous.
The Kalman filter is already widely used in the analysis
of InSAR (Shirzaei and Walter 2010; Dalaison and Jolivet
2020) and GNSS (Pacione et al. 2011; Bian et al. 2014)
time series; however, only a few studies (Bekaert et al. 2016;
Liu et al. 2019; Bozso et al. 2021) use it for the integra-
tion of these two types of measurement. The strength of
properly parameterized Kalman filtering is the ability to esti-
mate the state of a system with high accuracy, even when
the measurements are noisy, incomplete, and determined in
different spatiotemporal domains. Moreover, we can con-
clude, based on the methods presented in the literature, that
nowadays the GNSS-InSAR integration is mostly applied in
areas with high-spatial resolution GNSS receivers and lin-
ear motions (Liu et al. 2019). However, for local nonlinear
strong deformations, current integration techniques are not
well developed (Ng et al. 2011; Bozso et al. 2021) and are
not robust against different levels of observation noise. A
sufficient nonlinear ground movements monitoring system
can be important for early warning issues, understanding
the deformation process, assessment of risk and damage,
and evaluation of mitigation measures. Moreover, due to the
ambiguous nature of observations, MT InSAR techniques
such as Persistent Scatterer Interferometry (PSI) or Small
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Kalman filter-based integration of GNSS and InSAR Page 3 of 23 109
Baseline Subset Interferometry (SBAS) have a limited capa-
bility to measure rapid nonlinear deformation phenomena,
greater than several tens of centimeters per month (Przyłucka
et al. 2015; Pawluszek-Filipiak and Borkowski 2020).
Given the literature review, presented above and the need
to find a way to monitor local nonlinear strong deformations,
we come up with the following research questions:
– What functional and stochastic model will be required
to integrate on the observation level GNSS and InSAR
data to retrieve consistent 3-D deformations in the under-
ground mining region?
– How to overcome limitations of DInSAR measurements,
unwrapping problems, varying, and unstable coherence,
given non-existing MT InSAR in the area of interest?
– How to utilize the GNSS point measurements as anchor
points to constrain the DInSAR field observations with
decomposition into 3-D deformation?
The study presented in this manuscript extends the
Kalman filter algorithm described by Bozso et al. (2021),
using GNSS observations from permanent ground
stations and DInSAR deformation time series derived from
Sentinel-1 satellites. Our algorithm is able to ingest the time
series of GNSS topocentric coordinates with gaps of sev-
eral months, and noisy time series of DInSAR ascending
and descending LOS velocities subject to troposphere arti-
facts or SAR phase unwrapping errors. The latter errors
appear in different situations but mostly either in the case of
increased phase noise due to decorrelation effects (e.g., dense
vegetation, snow), severe weather conditions, strong and
abrupt displacement behavior, or a combination thereof. The
observation uncertainties are rigorously determined from the
parameter estimation process for the GNSS data and from
coherence values for DInSAR velocities (see maps of mean
cross-correlation coefficient in Appendix B (Fig. 7)). To ver-
ify the fusion results, a quality analysis based on independent
measurements was performed. The verification procedure
was implemented using the external survey data source -
GNSS campaign points named also as epoch-based mea-
surements. The proposed methodology is tested in an area
of intensive underground works within the Upper Silesian
Coal Basin (USCB) in Southern Poland—one of the biggest
deposits in Europe. The surface deformations in the area
have been monitored in recent years by different geodetic
approaches, revealing a complex regime of multiple dynamic
subsidence patches with a substantial range of displacement
values reaching in some cases 1.5 m per year and non-
linear character of the deformations, corresponding to the
distributed underground mine works along the longwalls—
among the others, Del Ventisette et al. (2013), Przyłucka
et al. (2015), Ilieva et al. (2019), Pawluszek-Filipiak and
Borkowski (2020), Kope´c(2021). In Sect. 2, we introduce
our methodology for the integration of DInSAR and GNSS
measurements. The results of our algorithm are reported in
Sect.3. In Sects. 4and 5, we present the discussion and con-
clusions of our work.
2 Methodology
The paper presents an original methodology for the inte-
gration of two different techniques, optimal for nonlinear
motions, conducted for an area affected by underground min-
ing works. In the approach, we processed geodetic data types
namely SAR and GNSS observations, aiming to achieve
comprehensive 3-D monitoring of the ongoing ground defor-
mations. The estimation workflows are presented in Sects. 2.1
and 2.2. Further, to compare the DInSAR and GNSS time
series, a procedure unification of both data sets is introduced
(Sect.2.3). Section2.4 shows a new method for integration
of DInSAR and GNSS data using a Kalman filter. Finally,
to verify the obtained results, the quality analysis based on
independent data set is performed (Sect.2.5).
2.1 DInSAR processing
In the current study, a consecutive cumulative DInSAR
approach for surface deformation monitoring is applied
(Fig.1a), where the second (secondary) image of each inter-
ferogram is used as a first (primary) image in the next
interferometric pair, forming continues time series (Fig. 1b).
The consecutive DInSAR method provides a simple and fast
tool for monitoring by adding new data to the subsidence
stack every time a new SAR image is released. It is also rela-
tively easy to automate the process. A serious disadvantage is
that this approach does not eliminate the contribution of the
interferograms with lower quality, e.g., due to temporal or
geometric decorrelation (Hanssen 2014). On the other hand,
in the case of atmospheric artifacts registered in one SAR
image, caused by a higher percentage of water vapor which
slows down the radar signal, they will be canceled out by
adding the affected image with a negative sign in the fol-
lowing pair. However, the atmosphere delay artifacts from
the first and last image will persist in the results. The DIn-
SAR processing in this study is accomplished with the open
science Sentinel Application Platform (SNAP), provided by
the European Space Agency (ESA), following the standard
DInSAR procedure, outlined below.
Sentinel-1 data covering the area of interest acquired in
Interferometric Wide swath (IW) mode with VV polarization
for the time span of the GNSS measurements (Sect.2.2)are
used. Sentinel-1 is a constellation of two satellites (A and
B) maintained by ESA that have mounted C-band (wave-
length of ∼5.55 cm) SAR sensors and provide a 6-day revisit
time within the study period. The generated SAR images are
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109 Page 4 of 23 D. Tondaś et al.
Fig. 1 Principle of the consecutive cumulative DInSAR technique for
surface deformation monitoring applied in the current study. aAn
example of real data from the Sentinel-1 A and B satellites’ positions
(perpendicular Baselines) within a month time span showing the suc-
cessive formation of DInSAR interferograms using common images
between the pairs. bRelative displacements (gray solid lines) for the
time of acquisition of the primary and secondary images forming each
interferogram, and the cumulative displacement along the LOS (black
solid line)—an example from Sentinel-1 A and B ascending data, that
correspond to the time scheme above in (a)
freely available on the ESA’s Sentinel hub (https://scihub.
copernicus.eu). The precise Sentinel-1 orbit auxiliary files
prepared by Copernicus Precise Orbit Determination (POD)
Service are downloaded automatically from Copernicus Sen-
tinels POD Data Hub (https://scihub.copernicus.eu/gnss) and
are applied to the SAR data to minimize the satellite orbital
error. An external Digital Elevation Model (DEM), namely
the 1-sec distribution of the Shuttle Radar Topography Mis-
sion (SRTM) Height file (Jarvis et al. 2008)(http://srtm.csi.
cgiar.org), is used for coregistration of the secondary image
to the geometry of the primary one. During this procedure,
a 21-point truncated sinc interpolation is applied. Then, the
secondary image is corrected for range and azimuth shifts
by implementing the Network Enhanced Spectral Diversity
(NESD) approach (Fattahi et al. 2017). The step is needed
due to the specific format of the Sentinel-1 data which are
acquired with the TerrainObservation with Progressive Scans
SAR (TOPSAR) technique by switching the SAR antenna
between neighboring sub-swaths on a number of bursts. The
NESD algorithm aims to correct the shifts in the overlap
areas of adjacent bursts. The interferogram is computed by
subtracting the flat-earth phase, while the SRTM is used
again to subtract the stable topography component resulting
in the wrapped interferogram. A Goldstein non-adaptive filter
(Goldstein and Werner 1998) was applied to reduce speckle
noise, to ease the two-dimensional spatial phase unwrap-
ping, which solves for the 2πambiguity. The Statistical-Cost,
Network-Flow Algorithm for Phase Unwrapping (SNAPHU)
(Chen and Zebker 2001) integrated in SNAP was applied, and
the values of the resulted unwrapped interferometric phase
(φd) are transformed to metric units based on the relation:
dLOS =− λ
4πφd,(1)
where dLOS is the relative displacement projected on the
radar LOS, and λis the wavelength of the transmitted radar
signal mentioned above.
The quality of the processed interferometric phase is rep-
resented by the cross-correlation coefficient (γ) between the
two SAR images of each interferogram expressed as a level
of coherence at each pixel (Hanssen 2014):
γ=N
n=1y(n)
1y∗(n)
2
N
n=1y(n)
12N
n=1y(n)
22,(2)
where y1and y2are the complex values at the observed pixel
of the primary (1) and secondary (2) SAR images, and Nis
the number of pixels in a surrounding estimation window.
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Kalman filter-based integration of GNSS and InSAR Page 5 of 23 109
The total coherence (γtot) depends on the influence of
several decorrelation factors (Hanssen 2014):
γtot =γgeom .γDC .γvol .γthermal.γtemporal.γpr ocess ing ,(3)
where γgeom presents the decorrelation due to geometry dif-
ferences between the two SAR images, γDC —differences
in the Doppler centroids between the two images, γvol —
the effect of the scattering medium on the radar signal,
γthermal—noise due to the characteristics of the system,
γtemporal—decorrelation due to the physical changes of
the scattering characteristics of the ground over time, and
γprocess ing —influence of the used processing algorithms.
Some of these effects are limited either by providing bet-
ter orbital tracks or limiting the time span between the SAR
acquisitions, while others are highly dependent on the system
characteristics, e.g., used wavelength. The lower coherence
in bigger areas of the interferogram impedes the proper solv-
ing of the phase ambiguities resulting in jumps in the time
series (Sect.3.1).
To determine the error values of a particular pixel, the
coherence coefficient (Eq. 3) was used to estimate the phase
noise (σφ ). The phase variance can be established using the
probability density functions (PDF) depending on multilook
levels (Hanssen 2014). The phase variance resulting from
γtot <1 can be defined as:
σ2
φ =+π
−π
[φ−E{φ}]PDF(φ)dφ, (4)
In the presented study, the multilooking level is equal to 1
(L=1), and thus, the phase variance can be expressed as:
σ2
φ, L=1=π2
3−πarcsin(|γtot|)+arcsin2(|γtot|)
−1
2
∞
k=1
|γtot|2k
k2,(5)
where kis the index of summation. The final interferomet-
ric phase errors in the ascending and descending directions
(σdA
LOS,σ
dD
LOS) are transformed to metric units based on the
relation:
σdLOS =λ
4πσφ, L=1.(6)
Since the unwrapped displacements are relative, all maps
of displacement must be related to a common reference.
Instead of one reference point, a set of points is used by
applying the method proposed in Ilieva et al. (2019). The set
of points is chosen based on stable coherence characteristic
over time for both orbits—ascending and descending. These
points are also away from local nonlinear strong deformation
zones, and the time span for points selection is the same as
the study time (here 2018–2020). A correcting surface for
each map is interpolated between the pixels that preserved
the highest coherence (more than 90%) over a year.
As it was mentioned before, the DInSAR displacements
are calculated in LOS. In order to be able to compare the
3-D GNSS data with the DInSAR results, the heading and
incidence angles are also extracted from the SAR data. The
values of the incidence viewing angle (θinc) are exported
for each interferogram. The satellite’s heading angle can be
treated as a constant value for each geometry, since the vari-
ation is usually within 1 deg over the extent of a SAR image
and hence has a negligible influence (Fuhrmann and Garth-
waite 2019). Thereby, in the current study, a mean value of
the heading angles (α) for the two SAR images participating
in an interferogram formation is taken for further calculation.
Thus, the deformation vector measured in the LOS direction
can be described by three-dimensional components:
dLOS =sin(θinc)sin(α) −sin(θinc)cos(α) cos(θinc)
⎡
⎣
nS
eS
uS
⎤
⎦,(7)
where dLOS,nS,eS, and uSare the LOS, North, East, and
Up deformations, respectively, and index Sis the indication
of the SAR technique.
2.2 GNSS processing
The study is focused on the ground changes above the
Rydułtowy coal mine, located in the south-western part of
USCB. Rydułtowy is the oldest continuously operational
mine in Upper Silesia (since 1792) with a planned lifetime
of exploitation up to 2040 (https://pgg.pl). The mining area
covers 46 km2with operational resources of around 80.4 mil-
lion tons and a production capacity of 9000–9500 tons per
day. The longwall method of coal extraction is used in this
mine with the depth of the seams between 800 and 1200m.
The work in the region of interest E1 of the mine is ongo-
ing on several walls at 4 seams, each with a thickness of
about 18 m ( ´
Cwi¸ekała 2019). The investigated area covers
two walls, with a spatial extent of 2 km2, and a moving,
nonlinear deformation bowl with a diameter of 200m.
The coordinates of the permanent GNSS stations were
estimated in post-processing mode using a double difference
technique. The solution was based on the Global Positioning
System (GPS), the Russian Global Navigation Satellite Sys-
tem (GLONASS), and European Galileo observations. The
estimation of the coordinates was performed with a daily
computational interval using Bernese GNSS software v.5.2
(Dach et al. 2015). The processing scenario was based on
the final precise GNSS products. The mentioned products
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109 Page 6 of 23 D. Tondaś et al.
included the orbits and clocks of the multi-GNSS satel-
lites obtained through the Center for Orbit Determination
in Europe (CODE, Dach et al. (2016)). To ensure a stable
reference, 14 stations belonging to the International GNSS
Service (IGS, https://igs.org) network and the European Ref-
erence (EUREF) Permanent Network (EPN, https://www.
epncb.oma.be) were included in post-processing calcula-
tions.
In the research area affected by mining deformations
(Fig.2), two types of permanent GNSS stations can be dis-
tinguished: geodetic and low-cost receivers deployed under
the EPOS-PL project (Mutke et al. 2019,https://epos-pl.eu).
One geodetic receiver was located on the building “Ignacy”
Historical Mine (RES1). The site was not placed directly
above the longwall panels; however, ground deformations
were expected at this location as well. The second type of
permanent station was equipped with five low-cost receivers
(PI02, PI03, PI04, PI05, PI16). These stations were installed
above the longwall of “Rydułtowy” mine currently exploited.
The beginning of collecting observations was 11.02.2019;
however, due to technical problems, the low-cost time series
are not complete (with a maximum break of around 9 months
for the PI02 station). In general, the low-cost station can be
defined as an inexpensive, light, small, and low-power con-
sumption device which is able to obtain position accuracy at
sub-centimeter level in static positioning. The overall price
of a low-cost GNSS instrument was variously defined as less
than 500e(Cina and Piras 2015) or below 1 000$ (Jackson
et al. 2018). The most important aspect of using low-cost
stations in ground deformations monitoring is to maintain
the quality of position determination. Various studies con-
ducted for different types of low-cost receivers indicated
sub-millimeter or millimeter differences compared to tradi-
tional geodetic GNSS stations (Cina and Piras 2015; Hamza
et al. 2020,2021).
In addition to the mentioned permanent stations, inde-
pendent epoch-based measurements were also considered in
this study. The GNSS campaign network contains over 120
well-stabilized and repetitively measured points developed
by the Military University of Technology in Warsaw (MUT)
across the Upper Silesia region. In the area of interest, 42
campaign points were located over underground “Ryduł-
towy” coal seams (Fig. 2). The MUT team processed the
epochs of observations together with the GNSS data from
the local network of the Continuously Operating Reference
Station (CORS) collected in the GNSS Data Research Infras-
tructure Centre (Mutke et al. 2019,http://www.gnss.wat.edu.
pl). The measurements at selected epoch-based points were
completed within one-to-two surveying days. Data were col-
lected at 1 or 15s intervals with a time span of 3.2 to 5.5
h. The processing strategy followed the EPN guidelines. It
is also consistent with the MUT analysis for operational
EPN products and was used successfully in the EPN Repro2
project (Araszkiewicz and Völksen 2017). To provide consis-
tency with the ITRS/ETRS realization, the estimated results
were combined with the MUT-PL solution created within
the EPOS-PL project (Mutke et al. 2019). The details about
GNSS equipment and the execution of the measurements are
presented in Table 1.
The results obtained for the permanent and campaign
GNSS points were determined in the ITRF2014 reference
frame Altamimi et al. (2016). In order to ensure consistency
with the InSAR data, which assume unchanging positions
with respect to the reference points, the ITRF2014 coordi-
nates and uncertainties were transformed to the ETRF2000
reference frame using the 7-parameter approach (Altamimi
2018). Afterward, for a better understanding of the local
displacements, the geocentric ETRF2000 coordinates were
converted to the topocentric reference frame (Tao et al. 2019).
An illustrative way to present the position variation of a
GNSS station over time is to take the first computational
epoch as the reference value. To avoid adopting an outlier
as an initial value in the time series of the permanent sta-
tions, the averaged coordinates (NG,EG,UG) from the first
five computational observations were assumed as reference
values. The reference values for the positions of the cam-
paigned GNSS points (NC,EC,UC) were determined as the
first epoch that is common in time with the data from the
permanent stations. To establish the uncertainties for the per-
manent and epoch-based topocentric coordinates, the errors
conversion was defined as:
⎡
⎣
σ2
NσNE σNU
σNE σ2
EσEU
σNU σEU σ2
U
⎤
⎦=F⎡
⎣
σ2
X00
0σ2
Y0
00σ2
Z
⎤
⎦Ft,(8)
where σN,σ
E,σ
Uare the transformed topocentric GNSS
variances, σNE,σ
NU,σ
EU are the covariances, and σX,σ
Y,σZ
are the errors of the geocentric ETRF2000 coordinates. Fis
the rotation matrix determined as:
F=⎡
⎣
−sin(B)cos(L)−sin(B)sin(L)cos(B)
−sin(L)cos(L)0
cos(B)cos(L)cos(B)sin(L)sin(B)
⎤
⎦,(9)
where B,Lare the longitude and latitude in ETRF2000 of
the reference point.
2.3 Unification of the DInSAR and GNSS
deformations dimensions
The DInSAR measurements enable spatial continuous mon-
itoring of large-scale subsidence areas. However, the radar
technique acquires the observations in ascending and descend-
ing 1-D LOS directions. Whereas, the GNSS method pro-
vides determination of the displacements in 3-D, but only on
sites where the antennas were located. Therefore, to perform
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Kalman filter-based integration of GNSS and InSAR Page 7 of 23 109
Fig. 2 Map of the study area including undergoing mining exploita-
tion panels at seam 703/1 (gray rectangles), locations of permanent
GNSS stations (green circles and orange triangles), and campaign sur-
vey points (purple pentagons). The map in the bottom right corner shows
the locations of IGS/EPN reference stations (red squares)
Table 1 Classification of GNSS infrastructure deployed in Upper Silesia under the EPOS-PL project (names of the stations are consistent with
Fig. 2)
Name Receivers Antennas (Type) Execution of measurements
Permanent geodetic RES1 Leica GR30 LEIAR20 (LEIM) Continuously, beginning 10.08.2018
Permanent low-cost PI02-PI05,PI16 Piksi Multi HXCGPS500 (NONE) Continuously, beginning 11.02.2019
Campaign 42 points Trimble 5700 TRM39105.00 (NONE), TRM41249.00 (NONE) Twice per year, beginning 04.2018
a unification of both techniques, it was necessary to overlay
the DInSAR rasters with the GNSS locations and to focus
our study on the intersecting pixels.
As it was mentioned in Sect.2.1 (Eq. 7), theoretically the
DInSAR LOS could be decomposed to its 3-D components
nS,eS, and uSif a sufficient number of observations are avail-
able. In the common case two DInSAR data sets, namely
from ascending (Asc or A) and from descending (Desc or D)
satellite geometry, can be used for the decomposition of LOS
vector in horizontal and vertical projections (Fig.3). Due to
the nearly north–south trajectory of the SAR satellites, the
system has limited sensitivity to the ground movements in
this direction. That is why often the parameter nSis neglected
so the large noise driven by the north component of the LOS
displacement is minimized in the decomposed in 3-D LOS
vector. In this case, the measurements from the two DIn-
SAR geometries (dA
LOS and dD
LOS) could be enough to define
the other two unknowns—eSand uS, though the pattern
of the horizontal movement will be incomplete and espe-
cially the up component will be biased (Brouwer 2021). To
minimize this problem, in this study the GNSS North dis-
placement (NG) was implemented instead (Samsonov and
Tiampo 2006; Jun et al. 2012). Within this procedure, it was
kept in mind that the two sets of observations have differ-
ent temporal sampling. The GNSS technique works in an
absolute mode with respect to the initial time that defines
123
109 Page 8 of 23 D. Tondaś et al.
Fig. 3 Scheme demonstrating
the proposed method for
unification of the DInSAR and
GNSS observations (top part of
the graph, described in Sects.2.1
and 2.2) in absolute temporal
domain, realized through
DInSAR decomposition (bottom
left part—Sect.2.3), including
quality analyses of the results
(bottom right part) presented in
Sect.2.5
the start of the GNSS time series (Sect.2.2). On the other
hand, the DInSAR method provides only the relative defor-
mation referring to the initial time the first (primary) image
of the pair forming each interferogram. To unify the data in
a common temporal domain, the GNSS NGcomponent was
divided into intervals corresponding to the time span of the
DInSAR interferometric pairs (Fig. 1). The origin for each
interval was established at the beginning of the period con-
verting the absolute temporal GNSS NGmeasurement to a
relative one. Then, it was introduced as the known parameter
nGin the DInSAR LOS decomposition of the DInSAR eS
and uSestimation:
eS
uS=−sin(θ A)cos(α A)cos(θ A)
−sin(θ D)cos(α D)cos(θ D)−1
dA
LOS
dD
LOS−nGsin(θ A)sin(α A)
nGsin(θ D)sin(α D),(10)
where θAand θDare the incidence angles, and αAand αD
are the heading angles of the ascending and descending DIn-
SAR observations, respectively. As a final step, the relative
DInSAR parameters (eS,uS) are converted to the cumulative,
absolute temporal domain using equations:
ES(t)=
t
i=1
es(i), (11)
US(t)=
t
i=1
us(i), (12)
where tis the epoch of the DInSAR measurement. As a
consequence of the relative character of the DInSAR observa-
tions, every potential outlier propagates into the subsequent
epochs in the absolute time series. The maps of DInSAR
cumulative East and Up deformations over the subsidence
bowl located in the study area are available in Appendix
C (Fig. 8). To test the impact of outliers for the DIn-
SAR displacements (ES,US), a statistical analysis of the
accuracy was performed. For the validation, we used an exter-
nal data source from the campaigned GNSS measurements
(NC,EC,UC) (Sect.2.5).
2.4 DInSAR and GNSS data integration
The process of integrating DInSAR and GNSS observa-
tions is based on the Kalman filter (Kalman 1960) algorithm
(Fig. 4). To estimate the unknown observation parameters,
namely NEU displacements and rates, and the uncertainties
of the system, the process is built as a recursive two-step pro-
cedure. During the first stage, termed as the time-update step,
the information about the system from the previous epoch
(t−1) is used to predict the parameters in the current epoch
(t). In the second step, named as the measurement-update
step, the modeled values, as well as the uncertainties, are
corrected by the real observations and measurement noise
registered in the epoch t.
2.4.1 Time-update segment of the data integration
The time-update part of DInSAR and GNSS integration con-
tains the dynamic model for time-varying parameters:
xt|t−1=t|t−1xt−1|t−1,(13)
Pt|t−1=t|t−1Pt−1|t−1T
t|t−1+St,(14)
where xt|t−1and Pt|t−1are the predicted state vector and
error variance matrix, respectively, xt−1|t−1and Pt−1|t−1are
the state vector and error variance matrix in the previous
epoch, while Stis the system noise matrix, and t|t−1is the
transition matrix for the current moment. The 3-D descrip-
tion of the ground position, common to both techniques, is
considered in the following calculations by the state vectorx,
123
Kalman filter-based integration of GNSS and InSAR Page 9 of 23 109
Fig. 4 Scheme of the approach
for the two-step Kalman filter
algorithm (bottom left part of
the graph) applied for
integration of DInSAR and
GNSS observations (upper row
of the graph), including
verification of the quality based
on the external GNSS campaign
measurements (lower right part)
while the velocities of the ground deformations are sampled
to daily intervals (t=1 day). The processing computations
are conducted in the topocentric reference frame, and the ini-
tial values of the NEU positions and rates in the state vector
(x0|0) are equal to zero. As initial values for the errors of the
parameters (P0|0), the system noise values (St) are taken.
In our approach, the zero-mean acceleration model is
introduced as the system noise matrix Teunissen (2009):
St=σ2
0
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
4t41
2t30000
1
2t3t20000
00
1
4t41
2t300
00
1
2t3t200
0000
1
4t41
2t3
0000
1
2t3t2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(15)
The σ0parameter, representing the initial standard devi-
ation, was defined empirically, and the detailed results are
presented in Sect.3.2. The transition matrix (t|t−1)ofthe
dynamic model describes the evolution of the state parame-
ters in time and can be obtained as a first-order differential
equation:
t|t−1=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1t0000
010000
001t00
000100
00001t
000001
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(16)
2.4.2 Measurement-update segment of the data
integration
In the measurement-update step, to obtain a filtered system
with error variances, the prediction is combined with the
actual observations in terms of minimum mean square error:
Kt=Pt|t−1AT
t(AtPt|t−1AT
t+Rt)−1,(17)
xt|t=xt|t−1+Kt(zt−Atxt|t−1), (18)
Pt|t=(I−KtAt)Pt|t−1,(19)
where Ktis the Kalman gain matrix, xt|tis the system of
the filtered state vector (NF,vNF,EF,vEF,UF,vUF), and
Pt|tis the variance measurement-updated matrix and Ithe
identity matrix. The observations are introduced with the zt
vector (Eq. 20), while Atis the projection matrix (Eq. 21)
and Rtis the measurement noise matrix (Eq. 22):
zt=NGEGUG
dA
LOS
T
dD
LOS
TT
.(20)
In the vector of the observations (Eq. 20), three ele-
ments (NG,EG, and UG) refer to the topocentric position
of the GNSS station received by the processing with Bernese
software (Sect.2.2). The two last elements (dA
LOSanddD
LOS)
are the differential LOS displacements obtained from the
DInSAR ascending and descending geometries, respectively
(Sect.2.1).
The DInSAR observations were divided by the time factor
T, which corresponds to the period of the interferogram
time span acquisition. This interval Tis equal to 6 days,
or 12 days in the case of a missing Sentinel-1 image. In
the projection matrix (At), the number of rows is equal to
the number of measurements, while the number of columns
123
109 Page 10 of 23 D. Tondaś et al.
determines the number of parameters:
At=
⎡
⎢
⎢
⎢
⎢
⎣
100000
001 0 00
000 0 10
0sin(θ A)sin(α A)0−sin(θ A)cos(α A)0cos(θ A)
0sin(θ D)sin(α D)0−sin(θ D)cos(αD)0cos(θ D)
⎤
⎥
⎥
⎥
⎥
⎦
.
(21)
The two last rows are related to the conversion of NEU
coordinates into LOS geometry (Eq. 7). It should be high-
lighted that in the proposed model, GNSS observations
describe positions, while the radar observations are consid-
ered as velocity measurements. This approach allows us to
preserve observations in their original domains: GNSS as an
absolute estimate of coordinates and DInSAR as a relative
LOS distance change over revisit time. The measurement
noise matrix (Rt) contains the correlated variances of the
GNSS topocentric coordinates (σNG,σEG,σUG,σNE
G,σNUG,
σEUG,Eq.8), whereas estimates from DInSAR in ascending
and descending geometries are assumed to be uncorrelated
(σdA
LOS,σ
dD
LOS,Eq.6):
Rt=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
σ2
NGσNE
GσNUG00
σNE
Gσ2
EGσEUG00
σNUGσEUGσ2
UG00
000σ2
dA
LOS
0
0000σ2
dD
LOS
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(22)
The presented Kalman filter algorithm can be applied in
areal-time mode and run anytime when a new observation
is available. However, this method works only in a forward
direction, so the detection and elimination of outliers that
potentially occurred in the past is not possible. In order to get
better estimates of the forward results, the backward Kalman
filter could be applied (Fig. 4).
2.4.3 Kalman backward algorithm
The Kalman backward algorithm, named also as smoothing
processing, relies on the results from the forward application
of the Kalman filter, but can also be used parallel to a real-
time filter Verhagen and Teunissen (2017). The backward
filter runs recursively with t={N−1,N−2,...,0}, where
Nis the current epoch. The smoothed state values (
xbt|N) and
error variance–covariance matrix (Pb
t|N) are equal to:
xbt|N=xt|t+Lt(
xbt+1|N−xt+1|t), (23)
Pb
t|N=Pt|t+Lt(Pb
t+1|N−Pt+1|t)LT
t,(24)
where
Lt=Pt|tT
t+1|tP−1
t+1|t.(25)
The Kalman backward filter is mainly used in the post-
processing mode. The smoothing algorithm cannot be run
separately—the results from the forward filter are required
in every recursion. Finally, the statistical accuracy analysis
was performed for the filtered state vectors of forward (NF,
vNF,EF,vEF,UF,vUF) and backward (NB,vNB,EB,
vEB,UB,vUB) Kalman filters with reference to the external
data source (Sect.2.5).
2.5 Quality analysis
To verify the obtained results, a quality analysis based on
an independent data set was performed. Five GNSS cam-
paign measurements (04.2019, 08.2019, 11.2019, 06.2020,
and 01.2021) covering the time span and the study area of
the DInSAR imagery and the GNSS permanent observations
were used in the validation process. In order to co-locate
the low-cost receivers with the nearest campaign points (see
Fig. 2), a cross-reference was performed. Due to the non-
identical locations between the epoch-based points and the
permanent GNSS sites, the closest possible site was selected
for verification. The distances between the chosen campaign
points and the low-cost stations ranged from 25 to 80m. The
coordinates estimated by the Kalman backward filter for the
first epoch of campaign measurements were considered as
a reference and subtracted from each subsequent campaign
measurement. The outcome of this procedure is a set of resid-
ual values for each of the campaign measurement epochs
(except the first one).
Due to the insufficient number of samples (Chai and
Draxler 2014), the calculation of any statistic based on
five values was not robust. That is why another approach
for validation was proposed. The differences in coordinates
(ei,i=1,2,...,n) between the campaigns at the measured
sites are compared with the displacements for the correspond-
ing intervals (n) from the analyzed data sets, namely DInSAR
and GNSS (Sect.2.3), Kalman forward and Kalman back-
ward filter (Sect.2.4). The results from the comparison are
presented in Tables 2,3,4. Based on the calculated sets of
differences (n) in NEU directions, the overall RMS errors for
each of the methods were estimated:
RMS =
1
n
n
i=1
e2
i.(26)
123
Kalman filter-based integration of GNSS and InSAR Page 11 of 23 109
3 Results
In our study, we proposed two methods for combining DIn-
SAR and GNSS data mentioned in Sects.2.3 and 2.4.The
effectiveness of each of them is observed by testing the qual-
ity of the results for the study case of Rydułtowy coal mine.
The two first approaches based on a separate comparison
are presented in Sect.3.1—once only with the introduced N
component of the displacement from the permanent GNSS
measurements instead of the omitted one in the DInSAR 2-D
decomposition (Sect.2.3). The assessment of the reliability
of the second method—the forward–backward Kalman filter
assimilation, is reported in Sect. 3.2, and the methodology of
estimation is presented in Sect.2.4.
3.1 Unified DInSAR and GNSS time series
By applying the methodologies proposed in Sect.2.3,we
observed all permanent stations, chosen in this study (see
Table 1). We compared their GNSS and DInSAR time series
with the overlaid data from the GNSS epoch-based measure-
ments (if available) in order to assess the efficacy of these
two techniques. The results from the comparison for three
low-cost stations: PI16 (Fig. 5.1—first row), PI03 (Fig. 5.2—
second row), PI02 (Fig. 5.3—third row), located just above
active extraction panels (see Fig. 2), and one geodetic per-
manent station RES1 (Fig. 5.4—fourth row), placed in the
vicinity of the extraction panels, are presented below.
The PI16 deformation, visible in Fig.5.1, demonstrates
that the epoch and permanent GNSS estimates follow a
similar pattern—the North component (Fig. 5.1.a) shows dis-
placement in the first 9 months of the observations and then
stabilizes around 0.04 m. The East component (Fig.5.1.b)
is demonstrating similar 0.04 m deformation for the first 6
months of 2020 and stabilizes afterward. The Up component
(Fig. 5.1.c) has a similar deformation cycle, with the first sig-
nificant period lasting for 9 months in 2019 introducing 0.30
m vertical subsidence, while the second stage comprises the
entire 2020 and shows rather smaller deformation reaching
the level of −0.40 m.
The GNSS observations are in disagreement with the
DInSAR results for the East component (Fig. 5.1.b) with
exceeding deformation, which is exhibited with a sharp
inflexion in the very first months of the observations in 2019.
Another discrepancy is introduced in 06.2020 shifting the
East DInSAR deformation time series by 0.08 m, which can
be associated with an unwrapping error. Such an error can be
associated with strong decorrelation related to severe weather
conditions or with greater than 1 cycle displacement. It has
to be pointed out that the Up component (Fig.5.1.c) shows
much better alignment with GNSS but slightly underesti-
mates overall deformation (0.35 m instead of 0.40 m).
A similar effect is observed in the results for station PI03
(Fig.5.2), which is located above another extraction wall
to the west of the main area of interest (Fig.2). The esti-
mated, from the GNSS permanent observation, deformation
for the North component (Fig.5.2.a) has two clearly defined
sections—almost no deformation until 03.2020 and then a
rapid change to −0.20 m in the next year, until 03.2021.
The East component (Fig. 5.2.b) shows marginal positive dis-
placements in the first nine months of the measurements
(03.2019–12.2019) up to 0.02 m, followed by a shift in
the western direction (12.2019–03.2021). During the second
period, some slow small variations in a positive direction are
also detected, but the total site displacement at the end of the
period in 03.2021 is recorded at −0.04 m. The Up compo-
nent (Fig. 5.2.c) shows, after a short 3-month non-deforming
period (03.2018−06.2019), a fast pacing negative deforma-
tion accumulated to 1.00 m in 21 months (06.2019–03.2021).
The DInSAR results demonstrate a discrepancy in the East
component in reference to the GNSS solution, starting from
03.2020, which coincides with the acceleration in the North
component (Fig.5.2.a), but also with the results for station
PI16 (Fig. 5.1.b). The Up (Fig. 5.2.c) shows a similar DIn-
SAR pattern in comparison with the GNSS trend—after a
non-deforming period there is rapidly developing deforma-
tion accumulating up to 0.70 m sink at the end of 12.2020.
It is 0.20 m less than what is visible in the GNSS data, most
likely associated with the unwrapping problem in the DIn-
SAR processing.
Station PI02 is located in the northern part of the extrac-
tion wall No.VII-E-E1 (see Fig. 2), and hence, the vertical
deformation presented in Fig. 5.3.c appears in the very begin-
ning of the period-05.2019 and is rather linearly incrementing
until 09.2020 when it reaches −0.20 m. There is an agree-
ment between the permanent GNSS (light blue), the epoch
GNSS (pink), and the DInSAR-based (dark blue) estimates.
Interestingly, the East component from DInSAR (Fig. 5.3.b)
demonstrates variations in the order of 0.04 m. However, the
final East value is similar to the one from the GNSS obser-
vation (−0.04 m).
Substantially different results, in comparison with the low-
cost (epoch-based) GNSS observations, are derived from the
geodetic permanent station RES1. As shown in Fig.5.4, the
completeness of the GNSS data is close to 100%. The coor-
dinates variability is in the range of 0.001 and 0.002 m for
the horizontal components (Fig. 5.4.a and 4.b), whereas for
the vertical component it is between 0.003 and 0.004 m
(Fig. 5.4.c). The accumulated displacement of North direc-
tion is 0.12 m (Fig.5.4.a), and the Eastward movement
comprises two periods of −0.015 m and 0.020 m and finally
reaches 0.0 m (Fig.5.4.b), while the Up component drops
to −0.12 m (Fig. 5.4.c). The DInSAR results underestimate
the vertical deformation by −0.06 m and interestingly show
similar variability in the Up component as in the East com-
123
109 Page 12 of 23 D. Tondaś et al.
Fig. 5 Displacement values of PI16 (1), PI03 (2), PI02 (3), and RES1
(4) stations, determined in North (a), East (b),andUp(c) directions.
The graphs present the results of the GNSS permanent data (light blue
dots), DInSAR data (dark blue dots), and campaign GNSS measure-
ments (pink dots with error bars)
ponent (compare Fig. 5.4.c and Fig. 5.4.b), most likely again
due to the unwrapping problem.
3.2 Kalman filtering application
The methodology for the application of the Kalman filter
algorithm for DInSAR and GNSS integration is outlined in
Sect.2.4. However, before executing the estimation process,
it was necessary to provide aprioriknowledge related to the
dynamics of the deformation. In the presented study, we used
the zero-mean acceleration model to represent the noise of
the system (Eq. 15). To provide the most optimal Kalman
system noise value (σ0), the RMS errors for 4 permanent
stations were analyzed with respect to the GNSS epoch mea-
surements. Among the nine tested levels (10, 5, 1, 0.5, 0.1,
0.05, 0.01, 0.005, and 0.001 mm/day2), the lowest errors for
both the Kalman forward and Kalman backward solutions
were obtained for acceleration level at 0.05 mm/day2with
89 and 91mm RMS errors, respectively. For the Kalman for-
ward filter, the largest RMS was calculated for σ0=0.001
mm/day2(205 mm error), while for the Kalman backward
filter the worst result equals the 138mm error obtained for
σ0=10 mm/day2.
The application of the methodology to the permanent
GNSS station and DInSAR data shows significantly differ-
ent results (Fig.6) than DInSAR data and GNSS data alone
123
Kalman filter-based integration of GNSS and InSAR Page 13 of 23 109
(Fig.5). It has to be also noted that the integrated results
are available for all six components: N,E,U,vN,vE,vU;
therefore, for each station, the results are presented in six
panels—first row for the deformations and the second row
for the velocities.
It is clearly visible in Fig.6.1 that the Kalman forward
model (green solid line) well traces the estimated PI16 dis-
placements in the epoch solution of the North component
(Fig. 6.1.a, top row). Similarly to Fig.5.1.a, the positive
deformation is estimated to be 0.04 m at the last period
of 03.2021. It is worth to note that the GNSS permanent
station demonstrates periods with noisy results in 06.2020.
However, it does not have an impact on the Kalman filter,
which produces stable outputs. Both East and Up compo-
nents (Fig.6.1.b and 1.c, top row panels) resulting from the
forward filter show a strong linear extrapolated trend from
06.2019 until 12.2019. The velocities (Fig.6.1, bottom row)
that are taken from the DInSAR observations (dark blue dots)
show very noisy output (Fig.6.1.b and 1.c). It is interesting
to observe the fast re-convergence of the Kalman estimates
to the GNSS results once the GNSS data are available again
(after the data gap 06.2019–12.2019). The Kalman derived
displacements align well with the epoch solution except for
the last height values for 12.2020 (Fig. 6.1.c, top row), which
shows −0.35 m instead of −0.40 m. The backward Kalman
filter applied as a smoothing solution aligns well with the
GNSS observations in all three components (Fig. 6.1, top
row panels), tracing the permanent GNSS solution.
Figure 6.2 demonstrates the time series of the fast deform-
ing station PI03. The overall Kalman forward solution (green
solid line) is tracing the GNSS observations that transpire
as linear extrapolation sections visible in the long GNSS
data gaps (06.2019–12.2019; 04.2020–06.2020). However,
a quicker return to the tracing data trend is notable in the
case of the East component (Fig. 6.2.b, top row) on 10.2019,
two months before the GNSS data reappear in the solution. At
the same time, the velocity calculated from the DInSAR data
is much less noisy than the rest of the time series (Fig. 6.2.b,
bottom row). It is also remarkable that the velocities of the
Up component (Fig. 6.2.c, bottom row) show significant neg-
ative DInSAR values with noise low enough to make the trend
estimation viable (up to 2mm/day). The smoothed solution
(backward Kalman filter-red line) shows no significant jumps
or breaks, tracing the GNSS estimates closely.
The relatively slow deformation process at station PI02
with the final accumulated vertical displacement of −0.20 m
is manifested in Fig.6.3.c. The Kalman forward filter (green
line) shows dependence on the GNSS data (light blue dots).
However, the estimated trends in all components—North,
East, and Up (Fig.6.3, top row panels) indicate a high level
of agreement with the epoch measurements (pink bars). It is
worth to mention that the East and Up components are visible
using information derived from the DInSAR observations
(Fig.6.3.b and 3.c, bottom row) that happens to be less noisy
in winter (09.2019–03.2020). Similarly to the results of the
previous sites, Kalman filter outputs are not sensitive to the
outliers in the GNSS results (light blue dots in the top row—
close to 06.2020).
Figure 6.4 presents the Kalman filter estimates for the
RES1 station that is away from the major deformation zone
and for which almost the entire GNSS series is completed. It
is clear that the Kalman filters, both forward and backward,
are tracing the GNSS observation curve, while the DInSAR
observations for this particular point are not as critical (no
GNSS data gaps). Moreover, evaluating the bottom panels of
Fig.6.4, it is clear to see that the breaks in 2018 and 2019
brought a lot of noisy data that were difficult to process (see
Fig. 5.4). Surprisingly, the Up component (Fig.6.4.c) demon-
strates sections of different deformation velocities aligned
well with the East component inflection points (03.2019 and
12.2019). The overall quality of Kalman retrieval at the RES1
station could be taken as a reference for other sites.
3.3 Intersecting results
The results presented in the previous sections demonstrate
capabilities for deformation monitoring by DInSAR and
GNSS data separately, or in an integrated concept based
on the Kalman filter algorithm. The subsequent validation
is based on the residual values for the GNSS perma-
nent sites located in the vicinity of the GNSS campaign
points. The choice of the residual analysis against statisti-
cal evaluation is dictated by the small number of low-cost
stations (only four) in the vicinity of campaign sites, and
only five overlapping dates within the time of the perma-
nent and campaign points installation. The low-cost stations
used in the intersect comparison are PI02, PI04, PI05, and
PI16, with residuals in North (Table 2), East (Table 3),
Up (Table 4) directions, calculated as a difference with the
values for these three components measured at the clos-
est points of the GNSS campaign network. What can be
definitively observed in all Tables is that the RMS (Eq. 26)
for the GNSS observations is similar to the Kalman for-
ward and backward model, regardless of the station and
site.
In Table 2, the total deformation in the North direction is
relatively small and does not exceed 0.10 m (point PI02). The
overall RMS for all calculated residuals is 0.013 m regardless
of the technique applied to obtain these deformations. It is
worth to notice that Kalman forward and backward demon-
strates availability for all reference epochs.
In Table 3, the total deformation in the East direction does
not exceed 0.11 m (point PI04). The discrepancies between
campaign points and low-cost observations do not exceed
0.09 m; however, the uncertainty of the campaign points is
0.02 m. The largest residuals are recorded for the DInSAR
123
109 Page 14 of 23 D. Tondaś et al.
Fig. 6 Displacement values (top rows) and rates (bottom rows) of the
PI16 (1), PI03 (2), PI02 (3), and RES1 (4) stations, determined in
North (a), East (b), and Up (c) directions. The charts present results
of the GNSS permanent data (light blue dots), campaign GNSS mea-
surements (pink dots with error bars), DInSAR data (dark blue dots),
forward Kalman filter (green lines), and backward Kalman filter results
(red lines)
123
Kalman filter-based integration of GNSS and InSAR Page 15 of 23 109
Table 2 North residuals for low-cost permanent stations in the vicinity of campaign points—see Fig. 2, for location
ID Dist. to camp.
pt. [m]
Max. defo.
[mm]
Date InSAR
[mm]
GNSS
[mm]
Kalman forward
[mm]
Kalman backward
[mm]
PI04 80 29 04.19 – −1−10
08.19 – – −39 −38
11.19 – 37 37 36
06.20 – −12 2
01.21 – – −34 −34
PI02 25 −96 04.19 – −10 0
08.19 – – −8−13
11.19 – – 22 4
06.20 – −9−9−9
01.21 – – 5 5
PI05 47 84 04.19 – −1−20
08.19 – – 12 19
11.19 – −14 0
06.20 – – 11 12
01.21 – – 23 23
PI16 62 90 04.19 – −10 0
08.19 – – 3 4
11.19 – −6−5−7
06.20 – −14 −15 −15
01.21 – −6−5−5
Overall RMS – 13 13 13
The missing data for the InSAR column are the result of the applied 2-D methodology of decomposition
result up to the maximum of −0.052 m (PI04, 01.2021). It
is also worth to mention that the Kalman filter provides the
best quality solution across all reference epochs regardless
of the presence or absence of GNSS data (see PI02, PI05).
In Table 4, the Up direction deformation is substantial
with amounts up to −0.388 m (point PI16), and discrep-
ancies between campaign points and low-cost observations
up to 0.16 m. Note that the uncertainty of campaign points
is 0.03 m. The largest residuals are recorded again for the
DInSAR results up to the maximum of −0.142 m (PI04,
06.2020). For the PI04 station, the distance to the clos-
est campaign point is equal to 80m and the residuals are
142, 77, 91, and 91mm for InSAR, GNSS, Kalman for-
ward, and Kalman backward, respectively. It is also worth
to note that the Kalman filter provides a solution across
all reference epochs regardless of the completeness of the
GNSS data. The residuals for PI04 are higher than half
of the deformation, ranging from 0.003 m to 0.142 m,
whereas the maximum recorded deformation was −0.201
m. In the case of PI16, we may observe growing resid-
uals with increasing deformation (maximum −0.388 m).
4 Discussion
In our study, we developed a novel Kalman filtering approach
combining high-temporal resolution displacement time series
(GNSS) with a high-spatial resolution measurements (InSAR)
to provide consistent 3-D coordinates and velocities for the
location of the GNSS stations. It is classified as one of the
most promising solutions postulated by Hu et al. (2014)for
local and regional scale deformations. In the presented study,
we used data from six permanent GNSS stations for a rel-
atively small extraction field (2 km2). For comparisons, in
Bian et al. (2014) 19 receivers were distributed over 8000
km2large-scale coal mining area in China with an average
distance between the monitoring points of about 35 km, while
in Liu et al. (2019), the number of GNSS sites is 35 on over
2000 km2. As the local nonlinear strong deformations were
investigated in a relatively small area with quite large number
of GNSS receivers, we can identify and correct the number
of DInSAR processing issues that limit the quality of defor-
mation estimates.
The DInSAR time series of displacements can be affected
by outlier values related to the limitations of the technique,
123
109 Page 16 of 23 D. Tondaś et al.
Table 3 East residuals for low-cost permanent stations in the vicinity of campaign points—see Fig. 2, for location
ID Dist. to camp.
pt. [m]
Max. defo.
[mm]
Date InSAR
[mm]
GNSS
[mm]
Kalman forward
[mm]
Kalman backward
[mm]
PI04 80 −109 04.19 −50 3 0
08.19 3 – 12 15
11.19 −27 0 −2−3
06.20 −26 35 34 34
01.21 −52 – −20 −20
PI02 25 −33 04.19 −20 −10
08.19 20 – 6 7
11.19 25 – −21
06.20 18 −8−8−8
01.21 −8– −4−4
PI05 47 −68 04.19 −40 −10
08.19 −2– −12 −12
11.19 −30 −39 −37 −38
06.20 −3– −50
01.21 33 – 13 13
PI16 62 −36 04.19 −21 −20 0
08.19 −27 – 9 9
11.19 −43 9 8 7
06.20 −21 11 11 11
01.21 24 9 9 9
Overall RMS 24 18 17 17
e.g., decorrelation in vegetated areas, local atmospheric
effects, or other phase unwrapping problems (Osmano˘glu
et al. 2016; Fattahi and Amelung 2014). The unwrapping
procedure is a non-deterministic problem that has many
equally correct solutions and can only be solved under certain
assumptions. Nevertheless, the high subsidence rate pro-
vides a strong interferometric signal. Thereby, the fringe rate
observed in interferograms increases, making it more dif-
ficult to unwrap properly (Osmano˘glu et al. 2016). In the
presented study, the largest discrepancy between the GNSS
and DInSAR results occurred for the PI03 station (max. dis-
agreement equal to 0.30 m for the Up component), which
corresponds directly to the most prominent observed vertical
rate of deformation (−1.00 m over 2 years of GNSS moni-
toring).
Another major issue related to the transformation of
ground deformations measured in the DInSAR slant direc-
tion to LOS-decomposed results can introduce a significant
distortion into the three-dimensional representation of the
displacement depending on the applied method. As the
authors presented in Wright (2004), the a posteriori Up errors
decreased from 286 to 4mm after neglecting the North com-
ponent, which is the result of constraining the null space of
the solution (Brouwer 2021). In our study, in the LOS decom-
position into East and Up components, we applied the North
displacement from the GNSS data sets (Eq. 10). To test the
impact of the North component magnitude on the presented
results, we introduced two separate decomposition processes
(with and without North) for the PI03 station, which is char-
acterized by the largest GNSS North deformation (−0.20
m). Taking into account the cumulative DInSAR ascending
(−0.63 m) and descending (−0.45 m) displacements, the
Up component becomes −0.72 m after decomposition with-
out North (North component equals 0), while with North
is −0.77 m (North component from GNSS). Therefore, the
impact of the North GNSS information is equal to 6% (0.05
m). For the East deformation, the 0.106 m is obtained after
decomposition without North (North component equals 0),
and 0.101 with North (North component from GNSS); there-
fore, the impact is 5% (0.005 m). However, the final Kalman
filter solution is fully constrained in all components, which
results in the same accuracy for GNSS and Kalman filter
solution for the North component (0.013 m) (see Table 2).
We successfully overcome the geometrical and technolog-
ical limitations of InSAR, but also we had faced challenges
related to GNSS processing. One of the issues with low-cost
123
Kalman filter-based integration of GNSS and InSAR Page 17 of 23 109
Table 4 Up residuals for low-cost permanent stations in the vicinity of campaign points—see Fig. 2, for location
ID Dist. to Camp.
pt. [m]
Max. defo.
[mm]
Date InSAR
[mm]
GNSS
[mm]
Kalman forward
[mm]
Kalman backward
[mm]
PI04 80 −201 04.19 3 −20 0
08.19 110 – 84 85
11.19 45 24 24 23
06.20 142 77 75 75
01.21 138 – 91 91
PI02 25 −203 04.19 14 2 1 0
08.19 2 – −27 −20
11.19 42 – 2 38
06.20 21 31 29 28
01.21 7 – 36 36
PI05 47 −270 04.19 6 2 1 0
08.19 −4– 6 −15
11.19 35 35 26 20
06.20 63 – 14 7
01.21 8 – −18 −18
PI16 62 −388 04.19 16 6 1 0
08.19 4 – −22 −31
11.19 36 −3−7−8
06.20 20 −32 −33 −34
01.21 −30 −52 −56 −56
Overall RMS 52 35 35 34
sensors is the incomplete coordinate time series recorder by
the GNSS receivers. In the presented literature review for
studies using permanent GNSS observations (Fuhrmann et al.
2015; Liu et al. 2019; Tao et al. 2019), there were no signif-
icant interruptions (longer than a month). However, in the
current study, as we used low-cost receivers, we observed
up to nine months of breaks in GNSS data flow. Such data
gaps have a great impact on the continuous monitoring of
nonlinear deformations. The missing GNSS time spans can
be filled by DInSAR data using a Kalman filter approach
(Eqs. 20–22). However, calculations provided only in the for-
ward mode can introduce significant leaps and overestimates
in the time series. Such an example is visible for the PI16
station around 12.2019 Fig.6.1.a) in North direction or PI02
station around 04.2020 in the Up direction Fig. 6.3.c). Hence,
to smooth and eliminate unexpected discontinuity of the data,
we introduced a backward Kalman filter algorithm (Eqs. 23–
25). The applied backward filter significantly reduced the
impact of gaps in the GNSS data. In our case, we have run
the backward filter only at the last epoch; however, subse-
quent backward filter runs are possible and would reduce the
systematic error introduced by GNSS data gaps.
Finally, one of the significant problems that many defor-
mation studies face is validation. In Bozso et al. (2021),
the a posteriori errors of the Kalman filter algorithm were
assessed to be 16, 2, and 3mm in North, East, and Up direc-
tions, respectively, given the standard deviation of GNSS
and InSAR dLOS to be 2mm apriori. No input noise esti-
mation was performed, and no external data were applied for
validation. The fusion of GNSS and InSAR in the paper of
Liu et al. (2019) generated spatiotemporal deformation time
series, however, the distribution of GNSS receivers and the
sampling interval of InSAR interferograms had an impact on
the integration accuracy. The quality of the deformation field
model, established using the leave-one-out method, did not
exceed 7.2 mm in the LOS direction. In contrast to the above,
in the current paper, the verification procedure was performed
using the external data source - GNSS campaign measure-
ments. Due to the non-identical location of the campaign
points and permanent GNSS stations, the closest possible
locations were selected for verification. The most significant
residual values are presented in Tables 2,3,4and correspond
to the longest distance between these points (station PI04 and
the closest campaign point at 80 m distance). For the Kalman
backward approach, the maximum values are -38, 34, and
123
109 Page 18 of 23 D. Tondaś et al.
91mm for N, E, and U, respectively. The overall RMS val-
ues of all stations by the Kalman backward filter are equal
to 13, 17, and 34mm for N, E, and U, respectively. These
values are well below the campaign measurement uncertain-
ties where the maximum values reach 102mm, 75mm, and
98mm for N, E, and U, respectively.
5 Conclusions
The paper presents an original methodology for the inte-
gration of two different techniques, optimal for nonlinear
motions, conducted for an area affected by underground
mining works. The algorithm is able to ingest the noisy
GNSS topocentric coordinates with significant gaps, as well
as non-corrected for the troposphere and unwrapping errors
DInSAR ascending and descending velocities. The observa-
tion uncertainties are rigorously determined for GNSS during
the parameter estimation process, and for DInSAR error cal-
culated from coherence coefficient values.
The presented methods are tested and verified for the
“Rydułtowy” mine (2 extraction walls with a total area of
2km
2), located in the south-western part of Upper Sile-
sia in Poland, where multilevel coal exploitation induces
rapid ground deformations up to 1.00 m over 2 years. The
cross-reference of the data sets is presented for low-cost
GNSS stations PI16, PI03, PI02, and geodetic GNSS sta-
tion RES1 (Figs. 5,6). The nearest GNSS campaign points
to the permanent stations PI02, PI04, PI05, and PI16 were
used for validation (Tables 2-4). The lowest overall RMS
errors were reached for the Kalman backward approach
(13, 17, and 34mm for the N, E, and U, respectively),
whereas the highest values were obtained for the sole use
of the DInSAR technique (24 and 52mm for the E and
U, respectively). The missing North displacements data for
the DInSAR are a result of the limited sensitivity of radar
observations to the deformations in the along-track direc-
tion.
It can be concluded that the introduced Kalman forward
and backward methods of integration provide around 1.5
times better results for the East and Up components of the
deformation vector than the standard DInSAR decomposi-
tion alone. Simultaneously, the RMS uncertainties of Kalman
filters in the case of the North displacements are on the same
level as for GNSS data sets (13mm). What is worth to note,
the overall RMS errors were calculated based on residuals
from only common epochs of all methods (e.g., the mea-
surements from 08.2019, 11.2019, and 01.2021 have to be
omitted due to technical problems of PI02 and PI05 GNSS
stations, see Tables 2-4). One of the most significant advan-
tages of the Kalman fusion, compared to the standard GNSS
or DInSAR solution, is the completion of potential breaks in
the time series of observations (see the GNSS data in Fig.6).
Another benefit of integration is the preservation of the
data in their original domains: GNSS as an absolute 3-D
estimate of coordinates and DInSAR as a relative 1-D LOS
distance change over revisit time. Moreover, the proposed
measurement-update model allows us to extract separate
NEU parts from ascending and descending DInSAR obser-
vations avoiding the decomposition process (see Eq. 21).
Otherwise, using the standard decomposition approach, to
obtain the absolute DInSAR time series, the conversion from
relative to the cumulative domain is required. The incorrect
identification and elimination of any outlier may bias sub-
sequent epochs in the data set (see Fig. 5). The difficulty
of removing effects of the local atmospheric conditions and
other phase unwrapping problems can be reduced by thresh-
olding of system noise level in the Kalman filter time-update
segment (see Eq. 15). Subsequently, the fusion processing
can be easily extended (adding additional rows in Eqs. 20,
21,22) by data from other sources, e.g., leveling, or LiDAR,
to create a framework for multi-sensor ground deformation
monitoring system.
In the further research steps, the knowledge gained about
radar phase discontinuities in co-located GNSS-DInSAR
points can be extended to surrounding pixels. The developed
methodology of integration creates the foundation which can
allow nonlinear areal modeling. Currently, numerous empir-
ical models and influence function models can be used to
predict mining-induced subsidence, e.g., the Probability Inte-
gral Method, the Logistic Model, or Knothe theory. The
Knothe model is a commonly applied method in the case of
coal mining deformation monitoring and can be adopted in
the integration approach together with improving the fusion
process by other ground monitoring techniques.
Acknowledgements This study was started in 2016 in the frames of the
EPOS-PL project POIR.04.02.00-14-A003/16 and continued in 2021-
2022 within the EPOS-PL+ project POIR.04.02.00-00-C005/19-00 that
were funded by the Operational Program Smart Growth 2014–2020,
Priority IV: Increasing research potential, Action 4.2: Development
of modern research infrastructure in the science sector. The presented
investigation was accomplished as part of a scientific internship at Delft
University of Technology (TU Delft), Netherlands, conducted within
the GATHERS project, funded by the European Union’s Horizon 2020
research and innovation program under grant agreement No 857612.
The presented paper was performed thanks to the institutions which
provided the access to their databases: Military University of Technol-
ogy (raw GNSS observations from low-cost stations: PI02, PI03, PI04,
PI05, PI16, and calculated results of estimation from GNSS campaign
measurements), Central Mining Institute (detailed information about
coal extraction panels), International GNSS Service and the European
Reference (EUREF) Permanent Network (raw GNSS observations from
reference stations). The authors gratefully acknowledge the Wrocław
Center of Networking and Supercomputing (http://www.wcss.wroc.pl/)
computational grant using MATLAB Software License No. 101979 and
computational Grant No. 170.
Author Contributions All authors contributed to the design of the
research plan. D.T., M.I., F.v.L., and W.R. formulated the manuscript.
D.T. and M.I computed the GNSS and DInSAR data. H.v.d.M., F.v.L.,
123
Kalman filter-based integration of GNSS and InSAR Page 19 of 23 109
W.R, and D.T. developed the methodology for the integration process.
D.T. implemented the integration algorithm. D.T. and M.I. created the
figures. W.R. supervised the project.
Data Availability The GNSS, Campaign, and DInSAR datasets asso-
ciated with this study are publicly accessible in the Zenodo repository
(https://doi.org/10.5281/zenodo.7319132).
Declarations
Conflict of interest The authors have no conflicts of interest that are
relevant to the content of this article.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing, adap-
tation, distribution and reproduction in any medium or format, as
long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indi-
cate if changes were made. The images or other third party material
in this article are included in the article’s Creative Commons licence,
unless indicated otherwise in a credit line to the material. If material
is not included in the article’s Creative Commons licence and your
intended use is not permitted by statutory regulation or exceeds the
permitted use, you will need to obtain permission directly from the copy-
right holder. To view a copy of this licence, visit http://creativecomm
ons.org/licenses/by/4.0/.
A Appendix
See Table 5.
Table 5 Table presenting advantages and disadvantages of GNSS and DInSAR use for deformation monitoring
Advantages Disadvantages
GNSS - Continuous monitoring enables observation of ground
movements in 3-D space (Bian et al. 2014)
- Deformation can only be observed at the point where the GNSS
antenna is located Tao et al. (2019)
- To minimize latency, the calculations can be implemented in near
real-time (NRT) (Branzanti et al. 2013;Šeginaetal.2020),
ultra-fast NRT (Tonda´setal.2020)orreal-time(RT)(Hada´s
2015; Han et al. 2018; Tao et al. 2019) modes
- Acquisition, installation, and maintenance of the GNSS
equipment may involve high costs
- Position of the GNSS antenna is precisely determined with
millimeter-level accuracy (Tonda´setal.2023;Shenetal.2019)
- Technical issues related to, e.g., power loss may introduce
significant interruptions in the time series of observations Tonda´s
et al. (2023)
- Nowadays, to improve the spatial density, low-cost receivers are
increasingly used (Cina and Piras 2015; Hamza et al. 2020,2021)
- At least several dozen GNSS receivers are needed to acquire a
ground system for monitoring the horizontal and vertical
movements across an entire mine area (Bian et al. 2014; Tao et al.
2019)
InSAR - InSAR provides a better overview of local terrain changes
compared to the point-based character of the GNSS
measurements (Del Soldato et al. 2021)
- There is a few days latency in acquiring a new image - between 6
and 42 days for the different satellite missions currently in
operation (Imperatore et al. 2021)
- The density of spatial coverage ranges from 5 to 250km with
ground resolution from 0.5 to 20m (Imperatore et al. 2021)
- Limited sensitivity to changes in the northern component of the
ground displacement is attributed to the near-polar orbit of the
satellites
- The satellite radar missions provide data on a regular basis of
several days time span, collecting images of the Earth since 1992
(ERS-1 satellite) (Imperatore et al. 2021; Del Soldato et al. 2021)
- Significant loss of coherence (decorrelation) occurs in vegetated
areas depending on the used radar wavelength (more severe for
small wavelengths, e.g., X-band and C-band data, less for large
wavelengths, e.g., L-band)
- Observations, products, and software related to some SAR data
are freely available and do not require ground infrastructure,
which significantly reduces the cost of ground deformation
monitoring (e.g., Sentinel-1 satellite mission) Karamvasis and
Karathanassi (2020)
- Improper removing of local atmospheric conditions can be
manifested as additional radar signal (Jolivet et al. 2014)
- The radar imagery of surface acquired from ascending and
descending orbits enables decomposition of LOS observations
into east–west and vertical components (Eriksen et al. 2017;
Fuhrmann and Garthwaite 2019)
- Phase unwrapping problems arise due to temporal and spatial
decorrelation (Zebker and Villasenor 1992; Ahmed et al. 2011)
- Deformations are acquired only in the 1-D LOS direction toward
and away from the satellite
123
Kalman filter-based integration of GNSS and InSAR Page 21 of 23 109
C Appendix
See Fig. 8.
Fig. 8 DInSAR maps of ground
deformations in East (a)andUp
(b) directions including
undergoing mining exploitation
panels at seam 703/1
(rectangles) and locations of
permanent GNSS stations (pink
circles). The maps were
developed using the
decomposition procedure
(Eq. 10) neglecting the North
component. The time series of
relative East and Up movements
were converted to the
cumulative domain using
Eqs. 11 and 12, respectively
123
109 Page 22 of 23 D. Tondaś et al.
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