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Evaluation of Tidal Effect in Long-Strip DInSAR Measurements Based on GPS Network and Tidal Models

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A long-strip differential interferometric synthetic aperture radar (DInSAR) measurement based on multi-frame image mosaicking is currently the realizable approach to measure large-scale ground deformation. As the spatial range of the mosaicked images increases, the nonlinear variation of ground ocean tidal loading (OTL) displacements is more significant, and using plane fitting to remove the large-scale errors will produce large tidal displacement residuals in a region with a complex coastline. To conveniently evaluate the ground tidal effect on mosaic DInSAR interferograms along the west coast of the U.S., a three-dimensional ground OTL displacements grid is generated by integrating tidal constituents’ estimation of the GPS reference station network and global/regional ocean tidal models. Meanwhile, a solid earth tide (SET) model based on IERS conventions is used to estimate the high-precision SET displacements. Experimental results show that the OTL and SET in a long-strip interferogram can reach 77.5 mm, which corresponds to a 19.3% displacement component. Furthermore, the traditional bilinear ramp fitting methods will cause 7.2~20.3 mm residual tidal displacement in the mosaicked interferograms, and the integrated tidal constituents displacements calculation method can accurately eliminate the tendency of tidal displacement in the long-strip interferograms.
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Citation: Peng, W.; Wang, Q.; Cao, Y.;
Xing, X.; Hu, W. Evaluation of Tidal
Effect in Long-Strip DInSAR
Measurements Based on GPS
Network and Tidal Models. Remote
Sens. 2022,14, 2954. https://
doi.org/10.3390/rs14122954
Academic Editors: Massimo Fabris
and Mario Floris
Received: 24 April 2022
Accepted: 16 June 2022
Published: 20 June 2022
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remote sensing
Article
Evaluation of Tidal Effect in Long-Strip DInSAR Measurements
Based on GPS Network and Tidal Models
Wei Peng 1,2 , Qijie Wang 3,4, *, Yunmeng Cao 5, Xuemin Xing 1and Wenjie Hu 1
1
Hunan International Scientific and Technological Innovation Cooperation Base of Advanced Construction and
Maintenance Technology of Highway, Changsha University of Science & Technology,
Changsha 410114, China; pengwei@csust.edu.cn (W.P.); xuemin.xing@csust.edu.cn (X.X.);
huwenjie@stu.csust.edu.cn (W.H.)
2School of Traffic & Transportation Engineering, Changsha University of Science & Technology,
Changsha 410114, China
3School of Geosciences and Info-Physics, Central South University, Changsha 410083, China
4Hunan Key Laboratory of Remote Sensing of Ecological Environment in Dongting Lake Area,
Changsha 410007, China
5GNS Science, Lower Hutt 5040, New Zealand; y.cao@gns.cri.nz
*Correspondence: qjwang@csu.edu.cn; Tel.: +86-13808425350
Abstract:
A long-strip differential interferometric synthetic aperture radar (DInSAR) measurement
based on multi-frame image mosaicking is currently the realizable approach to measure large-scale
ground deformation. As the spatial range of the mosaicked images increases, the nonlinear variation
of ground ocean tidal loading (OTL) displacements is more significant, and using plane fitting to
remove the large-scale errors will produce large tidal displacement residuals in a region with a
complex coastline. To conveniently evaluate the ground tidal effect on mosaic DInSAR interferograms
along the west coast of the U.S., a three-dimensional ground OTL displacements grid is generated by
integrating tidal constituents’ estimation of the GPS reference station network and global/regional
ocean tidal models. Meanwhile, a solid earth tide (SET) model based on IERS conventions is used
to estimate the high-precision SET displacements. Experimental results show that the OTL and
SET in a long-strip interferogram can reach 77.5 mm, which corresponds to a 19.3% displacement
component. Furthermore, the traditional bilinear ramp fitting methods will cause 7.2~20.3 mm
residual tidal displacement in the mosaicked interferograms, and the integrated tidal constituents
displacements calculation method can accurately eliminate the tendency of tidal displacement in the
long-strip interferograms.
Keywords: ground tidal deformation; differential InSAR; GPS; tidal models
1. Introduction
In coastal interferometric synthetic aperture radar (InSAR) deformation measurements,
the ground tidal displacements along the LOS direction can reach 2~4 decimeters, and it
usually reaches centimetres in a single-frame differential InSAR (DInSAR) interferogram
with a spatial range of 100~250 km [
1
]. Currently, the tidal effects are generally ignored in
InSAR deformation measurements, while a best-fitting ramp is applied to remove the spatial
residual large-scale errors in differential interferograms [
2
4
]. The traditional bilinear ramp
fitting method can eliminate tidal displacements in most instances, such as inland areas
or small-range DInSAR measurements because the magnitude of the ocean tidal loading
(OTL) displacement within the SAR image is minor and the spatial variations of the solid
earth tide (SET) displacements tend to be linear ramps. However, with the increasing
spatial range of a long-strip differential interferogram based on mosaicked multi-frame
images, the SET displacement can reach 3 decimeters, and the OTL displacement from
the coastline to inland reaches several centimetres with significant nonlinear variations
in complex coastline areas. If the bilinear ramp fitting method is applied to the coastal
Remote Sens. 2022,14, 2954. https://doi.org/10.3390/rs14122954 https://www.mdpi.com/journal/remotesensing
Remote Sens. 2022,14, 2954 2 of 14
long-strip interferograms, the residual of the tidal displacement can be up to centimetre-
level [
5
,
6
]. Therefore, the long-strip DInSAR measurements covering the coastal areas with
significant tidal variation need to initially be assessed [7].
The ocean tidal models and GPS precise point positioning (PPP) methods are the most
commonly used analysis and correction methods for studies of ground OTL displacements
in InSAR measurements [
8
]. Without considering the atmospheric delay and other large-
scale errors, DiCaprio et al. simulated the OTL displacements in the ERS and ENVISAT
data based on the FES2004 ocean tidal model, demonstrating the importance of correcting
the nonlinear OTL displacement in a DInSAR interferogram [
9
,
10
]. If atmospheric delay
and other large-scale errors are considered, they need to be settled by external correction
data or two-dimensional image separation methods, which are limited to the accuracy
of atmospheric delay and other large-scale error correction [
11
,
12
]. Yu et al. simulated
the OTL displacement in global Sentinel-1 data based on the FES2014b model and anal-
ysed the comprehensive impact of the OTL effect and atmospheric delay error [
13
]. Wu
et al. compared the differences of the ocean tide models in OTL correction of the InSAR
measurements [
14
]. However, the global ocean tide models are ocean grids with a low
spatial resolution (0.125
or 0.5
) [
10
,
15
], which need Green’s function to point by point
inefficiently calculate the OTL displacements of the loading point [
16
,
17
]. For the tidal
effect evaluation of the DInSAR measurements with a large number of pixels, obtaining
spatial high-precision and high spatial-resolution coastal ground tidal displacement grid is
extremely important.
According to tidal theory, the SET effect is relevant to the tidal force and the Earth’s
internal structure, and it can be accurately modelled using the SET model of the 2010 IERS
Convention [
18
,
19
]. In contrast, the expressions between OTL displacement and ocean tidal
height, geographical location and coastline shape are more complicated, and the spatial
OTL displacement based on this method needs to be further validated [
20
]. The OTL
displacement can generally be decomposed into eight main tide constituents (semi-diurnal
constituents M2, N2, S2, K2 and diurnal constituents Q1, O1, P1, K1) according to the
harmonic analysis method, and the GPS precise point positioning (PPP) technique has
been shown to accurately determine the tide constituents displacement of semi-diurnal
principal lunar M2, semi-diurnal elliptical lunar N2, diurnal principal lunar O1 and diurnal
elliptical lunar Q1 in the north (N), east (E) and up (U) directions [
21
23
]. As a dense
globally distributed OTL displacement measurement method, it has been used to evaluate
the spatial accuracy of the OTL models [
24
]; on the other hand, it can be used to improve
the spatial OTL displacements in complex coastline areas. Therefore, the OTL displacement
in the three-dimensional directions can be estimated from the PPP time series and tidal
models, taking these high-precision OTL displacements and SET displacements based on
the SET model, we proposed a three-dimensional ground tidal displacement grid, which
can convert to any line of sight (LOS) direction of pixels in the differential interferogram to
analyse the ground tidal deformation.
In this paper, Section 2introduces the acquisition method of the high-precision three-
dimensional tidal displacements, Section 3introduces the evaluation and correction of
the tidal displacement in mosaicked long-strip differential interferograms using the tidal
estimation of the kinematic PPP and tidal models.
2. Materials and Methods
2.1. Traditional Tidal Models
The OTL displacement is usually calculated by the convolution integral of Green’s
loading function and the global ocean tidal height provided by ocean tidal models. The
main principle is to use Green’s function to describe the superimposition of the deformation
field on the loading point caused by the global ocean tidal height variation, which can be
expressed as [25]:
dOTL_model (L,B,t) = xSρR2H(L0,B0,t)G(θ,β)sin θdθdβ(1)
Remote Sens. 2022,14, 2954 3 of 14
where
dOTL_model (L
,
B
,
t)
is the ocean tide displacement of the model at the calculation
point,
ρ
is the seawater density,
R
is the radius of the earth, and
L0
and
B0
are the spherical
coordinates of the loading point, respectively.
H(L0
,
B0
,
t)
is the instantaneous tide height
provided by the tide models.
G(θ
,
β)
is Green’s function, and
θ
and
β
are the spherical
distance and azimuth between the calculation point and the loading point, respectively.
The spatial estimation accuracy of this traditional method strongly depends on the spatial
resolution and accuracy of the tide model, which still has deficiencies. Therefore, it is
necessary to determine its spatial estimation accuracy in complex coastline areas.
2.2. Spatiotemporal Modelling of OTL Displacement Estimated from PPP Time Series
According to the 2010 IERS Convention, the OTL displacement consisting of
M
tidal
constituents can be written as a tidal harmonic function:
dOTL
m,n(t) =
M
m=1
fmAmcos(ωm,nt+χm+µmΦm,n)(2)
where
Am,n
refers to the amplitude, and
Φm,n
represent phase delay,
ωm,n
is the angular
frequency,
n
is GPS sites,
fm
is the node factor,
χm
and
µm
are the initial astronomical angle
and astronomical angle, respectively.
Based on Formula (2), the amplitude and phase delay can be estimated from the
kinematic PPP coordinate time series of the GPS network, which are spatial location-related
parameters that can be constructed as a phasor:
Phasorm(x,y)=Phasor1m
Phasor2m=Am(x,y)cos(Φm(x,y))
Am(x,y)sin(Φm(x,y))(3)
where,
(x,y)
is the longitude and latitude. In terms of spatial dimension, the observations
of GPS reference sites are independent. PhasorPPP
m,nmodelling follows the spatial tendency
of constituents
m
can eliminate spatial random errors and predict high-precision and
high-resolution OTL displacements.
PhasorPPP
m,n=b0+fLSSV M PhasorModel
m,n(4)
where
fLSSV M
is the least squares support vector machine (LSSVM) based on the polynomial
kernel function [
26
],
b0
is a constant deviation, and
Phasor Mo del
m,n
is the phasor of the
m
tidal
constituents of the OTL model at the location of
n
GPS sites with the order from large to
small, and the tidal constituents displacements of any tide loading point
p
in the range of
the GPS reference sites network can be predicted. In coastal areas with a high site density
of GPS networks, the expression of the site location and the constituents can be determined
based on the Formula (4).
After that, the spatiotemporal modelled M2, N2, O1 and Q1 tidal constituent displace-
ments of a GPS reference station network and P1, K1 and K2 tidal constituent displacements
of an ocean tidal model are composed to obtain spatial higher precision OTL displacements.
According to Equations (2) and (3), the tidal constituents’ displacement is determined based
on a GPS reference sites network and a tidal model can be blended to estimate the OTL
displacement dOTL measured in long-strip differential interferograms.
dOTL
m1+m2,p(t) =
4
m1=1B1
m1B2
m1"Phasor1PPP
m1,p
Phasor2PPP
m1,p#+
3
m2=1B1
m2B2
m2"Phasor1Model
m2,p
Phasor2Model
m2,p#(5)
B1=fcos(ωt+χ+µ)
B2=fsin(ωt+χ+µ)(6)
Remote Sens. 2022,14, 2954 4 of 14
where,
m1
is Q1, O1, N2, and M2 constituents estimation of a GPS reference station network,
m2
is P1, K1, and K2 constituents estimation of a global/regional ocean tidal model,
B1
and
B2are constants in the spatial domain.
2.3. Tidal Displacements in the Long-Strip Differential Interferogram
Traditional InSAR measurements believe that the spatial large-scale ground tide effect
can be ignored in the course of multiple imaging ground deformations measured by SAR
satellite systems and that the spatial relative distance between the sensor and the target
object in an InSAR image does not affect by the tidal deformation of the OTL and SET.
Based on this, the usual differential interference model does not consider the ground tidal
phases, and its displacement expression of the pixel in the LOS direction of the differential
interferogram can be expressed as [27]:
dx,y=dde f o
x,y+datmospheric
x,y+dtopo
x,y+dorbit
x,y+ε(7)
where
x
and
y
are the longitude and latitude of pixels in the interferogram after geocod-
ing,
dde f o
x,y
,
datmospheric
x,y
,
dtopo
x,y
,
dorbit
x,y
, and
ε
are the ground deformation, atmospheric delay
error, topography residual error, orbital error, and other residual signals, respectively. To
analyse the ground tidal displacement, the interferogram without spatial large-scale land
subsidence and earthquakes were selected based on the PPP coordinate time series of
the GPS sites network, so the
dde f o
x,y
mainly contains the ground tidal deformation in the
coastal areas. The atmospheric delay error
datmospheric
can be reduced by using the ICAMS
advanced atmospheric correction method based on the ECMWF ERA-5 global atmospheric
model [
28
]. If the ground tidal deformation is not considered, the residual large-scale error
is usually believed to approach the linear plane, and fitting these errors with a bilinear
function model yields:
dx,ydatmospheric
x,y=dlargesc ale_error s
x,y+ε=a0+a1x+a2y(8)
where
a0
,
a1
,
a2
are the coefficients to be sought to fit the plane. For the multi-frame
mosaic interferograms in the coastal ground areas, it is erroneous to eliminate the spatial
nonlinear ground tidal deformation by the bilinear function. Considering the ground tidal
displacement, the large-scale error fitting model can be expressed as:
dx,ydatmospheric
x,ydOTL
x,ydSET
x,y=dlargesc ale_error s
x,y+ε=a0+a1x+a2y(9)
where the SET displacement
dSET
x,y
can be calculated by using the SET method in the
2010 IERS Convention, its accuracy is submillimetre, and the OTL displacement
dOTL
x,y
can
be calculated and mutually validated by the GPS network and ocean tidal models.
2.4. Tidal Data Analysis and Processing
The global OTL displacements are usually calculated using an ocean tidal model-based
loading Green’s function method. Taking the FES2014b model as an example, the up (U),
north (N) and east (E) OTL displacements
dOTL
n,p
of point
p
with an interval of 300 s in the
year 2019 can be obtained by using the ocean tidal model-based loading Green function
method, and the variation of the OTL displacements in three-dimension is evaluated by
the standard deviation (StdDev) SOTL
p, which can be expressed as [12]:
SOTL
p=v
u
u
t1
N1
N
n=1dOTL
n,pdOTL
p2(10)
where
N
is sample numbers of OTL displacement time series,
dOTL
p
is the average of OTL
displacement time series. According to the OTL StdDev values of
p
pixels in spatial large-
Remote Sens. 2022,14, 2954 5 of 14
scale differential interferograms, the magnitude and spatial relative variation characteristics
of the ground OTL effect can be preliminarily evaluated. The OTL displacement varies
greatly in some coastal areas around the world, including the west coast of the United
States, the west coast of Europe, the northeast coast of South America, and the south coast
of Africa. (in the red box in Figure 1).
Remote Sens. 2022, 14, x FOR PEER REVIEW 5 of 15
where N is sample numbers of OTL displacement time series, OTL
p
d is the average of
OTL displacement time series. According to the OTL StdDev values of p pixels in spatial
large-scale differential interferograms, the magnitude and spatial relative variation char-
acteristics of the ground OTL effect can be preliminarily evaluated. The OTL displacement
varies greatly in some coastal areas around the world, including the west coast of the
United States, the west coast of Europe, the northeast coast of South America, and the
south coast of Africa. (in the red box in Figure 1).
Figure 1. The standard deviation map of global ground OTL displacement with an interval of 300 s
in the U, N, and E directions based on the FES2014b model.
Due to a large amount of Sentinel-1 SLC ascending data, the dense number of GPS
continuous reference stations and a larger OTL effect on the west coast of the U.S., the
tidal effect in long-strip DInSAR measurements can be evaluated using the tidal estima-
tions of the GPS network and tidal models. To measure the larger-range ground defor-
mation, nine-frame Sentinel-1 SLC images with a range of 250 km on track 137 were mo-
saicked as a long-strip differential interferogram (see Figure 2), and the imaging time of
the long-strip SAR images was approximately 01:59:30 UTC. The length of the mosaicked
image is more than 1600 km, covering a variety of terrains such as the ocean and flat and
mountainous areas. There is lower coverage of forest in the southern area, and the coher-
ence of SAR images is high, which is often used to analyse earthquakes, landslides, and
other geological tectonic activities. The specific InSAR data processing is as follows:
(i). Image registration, interferogram generation, removal of the flat phase using SRTM
with a 30 m resolution, phase unwrapping follows the minimum cost flow method,
and geocoding is performed on the Sentinel-1 SLC image using GAMMA software
[29]. The precise orbital file is added in the data processing, and the plane fitting is
not implemented until the ground tidal displacement correction.
(ii). The 29 differential interferograms are selected based on the principle of a small base-
line [30].
(iii). The differential interferograms of the nine adjacent frames are mosaicked to obtain
the long-strip differential interferograms.
Figure 1.
The standard deviation map of global ground OTL displacement with an interval of 300 s
in the U, N, and E directions based on the FES2014b model.
Due to a large amount of Sentinel-1 SLC ascending data, the dense number of GPS
continuous reference stations and a larger OTL effect on the west coast of the U.S., the tidal
effect in long-strip DInSAR measurements can be evaluated using the tidal estimations
of the GPS network and tidal models. To measure the larger-range ground deformation,
nine-frame Sentinel-1 SLC images with a range of 250 km on track 137 were mosaicked as a
long-strip differential interferogram (see Figure 2), and the imaging time of the long-strip
SAR images was approximately 01:59:30 UTC. The length of the mosaicked image is more
than 1600 km, covering a variety of terrains such as the ocean and flat and mountainous
areas. There is lower coverage of forest in the southern area, and the coherence of SAR
images is high, which is often used to analyse earthquakes, landslides, and other geological
tectonic activities. The specific InSAR data processing is as follows:
(i).
Image registration, interferogram generation, removal of the flat phase using SRTM
with a 30 m resolution, phase unwrapping follows the minimum cost flow method,
and geocoding is performed on the Sentinel-1 SLC image using GAMMA software [
29
].
The precise orbital file is added in the data processing, and the plane fitting is not
implemented until the ground tidal displacement correction.
(ii).
The 29 differential interferograms are selected based on the principle of a small
baseline [30].
(iii).
The differential interferograms of the nine adjacent frames are mosaicked to obtain
the long-strip differential interferograms.
Remote Sens. 2022,14, 2954 6 of 14
Remote Sens. 2022, 14, x FOR PEER REVIEW 6 of 15
In this study, 1038 GPS reference stations of the Plate Boundary Observation (PBO)
network were selected (see Figure 2), and the time range of the GPS observations is from
the year 2014 to 2019. All observations of 1826 days are processed using the kinematic PPP
processing module of Bernese V5.2 to obtain the PPP coordinate time series in the U, N,
and E directions with 600 s as the sample interval. The three-dimensional coordinate time
series are pre-processed by outlier (>3σ) elimination and wavelet filter denoising, then the
ground OTL displacement can be estimated from the kinematic PPP coordinate time series
of a GPS reference station network.
3. Results
3.1. The OTL Estimation Based on Kinematic PPP and Ocean Tide Models
After kinematic PPP data processing, the three-dimensional ground displacement
time series is fitted by the tidal harmonic function to obtain the amplitude and phase delay
parameters of the tidal constituents, which can be constructed as a phasor. Meanwhile,
the offshore grid of the FES2014b global model is replaced by the higher spatial resolution
of the regional tidal model osu.usawest, which is compared with the phasor of the tidal
constituents measured by kinematic PPP (see Figure 3). The spatial precision of the con-
stituents phasor estimated based on the kinematic PPP technique is different, where Q1,
O1, N2, and M2 tide constituents have higher estimation accuracy, and their spatial trend
is consistent with the FES2014b+osu.usawest model. Although the P1, K1 and K2 constit-
uent tendencies are similar to those of the tidal model, the random error in the spatial
dimension is large. The SAR satellite’s revisit period is an integral multiple of the constit-
uent S2 (12 h) period, the displacement of constituent S2 at multiple imaging times is the
same, which is offset in the differential interference.
Figure 2.
Distribution of 1038 continuous GPS sites and multiple Sentinel-1 SLC image ranges on the
west coast of the U.S.
In this study, 1038 GPS reference stations of the Plate Boundary Observation (PBO)
network were selected (see Figure 2), and the time range of the GPS observations is from
the year 2014 to 2019. All observations of 1826 days are processed using the kinematic PPP
processing module of Bernese V5.2 to obtain the PPP coordinate time series in the U, N,
and E directions with 600 s as the sample interval. The three-dimensional coordinate time
series are pre-processed by outlier (>3
σ
) elimination and wavelet filter denoising, then the
ground OTL displacement can be estimated from the kinematic PPP coordinate time series
of a GPS reference station network.
3. Results
3.1. The OTL Estimation Based on Kinematic PPP and Ocean Tide Models
After kinematic PPP data processing, the three-dimensional ground displacement
time series is fitted by the tidal harmonic function to obtain the amplitude and phase delay
parameters of the tidal constituents, which can be constructed as a phasor. Meanwhile,
the offshore grid of the FES2014b global model is replaced by the higher spatial resolution
of the regional tidal model osu.usawest, which is compared with the phasor of the tidal
constituents measured by kinematic PPP (see Figure 3). The spatial precision of the con-
stituent’s phasor estimated based on the kinematic PPP technique is different, where Q1,
O1, N2, and M2 tide constituents have higher estimation accuracy, and their spatial trend is
consistent with the FES2014b+osu.usawest model. Although the P1, K1 and K2 constituent
tendencies are similar to those of the tidal model, the random error in the spatial dimension
is large. The SAR satellite’s revisit period is an integral multiple of the constituent S2 (12 h)
period, the displacement of constituent S2 at multiple imaging times is the same, which is
offset in the differential interference.
Remote Sens. 2022,14, 2954 7 of 14
Remote Sens. 2022, 14, x FOR PEER REVIEW 7 of 15
Figure 3. Comparative analysis of the phasor of tidal constituents estimated by the kinematic PPP
technique and FES2014b+osu.usawest models.
Figure 3.
Comparative analysis of the phasor of tidal constituents estimated by the kinematic PPP
technique and FES2014b+osu.usawest models.
Remote Sens. 2022,14, 2954 8 of 14
According to the spatial characteristics of each tidal constituent’s phasor of the tidal
model, the phasor determined by kinematic PPP is fitted by Formula (4), and the pre-
dicted Q1, O1, N2, M2 tidal parameters are used to replace the corresponding tidal har-
monic parameters of the tide model to form the three-dimensional ground OTL displace-
ments. The StdDev map difference between the proposed tide calculation method and the
FES2014b+osu.usawest model is calculated in Figure 4. The coastland areas A and B within
the two rectangles in Figure 4have larger differences between the ground OTL estimations
of the proposed OTL calculation method and the FES2014b+osu.usawest model. Its maxi-
mum StdDev value of the displacement difference is 1.93 mm, and its corresponding that
of the vertical OTL displacement is 17.9 mm, which suggests that the OTL displacement
difference between the two methods reach 10.7%. Likewise, the StdDev values of the north
direction are 0.37 mm, 3.61 mm, and 10.3%, respectively, and those of the east direction are
0.91 mm, 5.85 mm, and 15.6%. The differential interferograms in Section 3.2 will be used to
verify the improvement of the correction accuracy of the proposed method in coastland
area A.
Remote Sens. 2022, 14, x FOR PEER REVIEW 8 of 15
According to the spatial characteristics of each tidal constituent’s phasor of the tidal
model, the phasor determined by kinematic PPP is fitted by Formula (4), and the predicted
Q1, O1, N2, M2 tidal parameters are used to replace the corresponding tidal harmonic
parameters of the tide model to form the three-dimensional ground OTL displacements.
The StdDev map difference between the proposed tide calculation method and the
FES2014b+osu.usawest model is calculated in Figure 4. The coastland areas A and B
within the two rectangles in Figure 4 have larger differences between the ground OTL
estimations of the proposed OTL calculation method and the FES2014b+osu.usawest
model. Its maximum StdDev value of the displacement difference is 1.93 mm, and its cor-
responding that of the vertical OTL displacement is 17.9 mm, which suggests that the OTL
displacement difference between the two methods reach 10.7%. Likewise, the StdDev val-
ues of the north direction are 0.37 mm, 3.61 mm, and 10.3%, respectively, and those of the
east direction are 0.91 mm, 5.85 mm, and 15.6%. The differential interferograms in Section
3.2 will be used to verify the improvement of the correction accuracy of the proposed
method in coastland area A.
Figure 4. StdDev maps of the difference of the OTL displacements estimated by the proposed tide
calculation method and FES2014b+osu.usawest model in U, N, and E directions, and the areas with
lager difference (red box).
Furthermore, a three-dimensional ground OTL displacements grid with a spatial res-
olution of 1 × 1 is formed by integrating the spatiotemporal modelled Q1, O1, N2, and
M2 constituents of a GPS reference station network and the P1, K1, and K2 constituents of
the FES2014b+osu.usawest model. For the long-strip differential interferogram of Senti-
nel-1 data covering the west coast of the United States, the StdDev values are 1.89 mm,
18.16 mm, and 10.4% in the LOS direction (ascending; the incident angle is 39°; the time
interval is 12 days), which means millimetre to centimetre OTL displacement residuals in
a differential interferogram and time-series InSAR measurement (see Figure 5).
Figure 4.
StdDev maps of the difference of the OTL displacements estimated by the proposed tide
calculation method and FES2014b+osu.usawest model in U, N, and E directions, and the areas with
lager difference (red box).
Furthermore, a three-dimensional ground OTL displacements grid with a spatial
resolution of 1
0×
1
0
is formed by integrating the spatiotemporal modelled Q1, O1, N2, and
M2 constituents of a GPS reference station network and the P1, K1, and K2 constituents of
the FES2014b+osu.usawest model. For the long-strip differential interferogram of Sentinel-1
data covering the west coast of the United States, the StdDev values are 1.89 mm, 18.16 mm,
and 10.4% in the LOS direction (ascending; the incident angle is 39
; the time interval is
12 days), which means millimetre to centimetre OTL displacement residuals in a differential
interferogram and time-series InSAR measurement (see Figure 5).
Remote Sens. 2022, 14, x FOR PEER REVIEW 8 of 15
According to the spatial characteristics of each tidal constituent’s phasor of the tidal
model, the phasor determined by kinematic PPP is fitted by Formula (4), and the predicted
Q1, O1, N2, M2 tidal parameters are used to replace the corresponding tidal harmonic
parameters of the tide model to form the three-dimensional ground OTL displacements.
The StdDev map difference between the proposed tide calculation method and the
FES2014b+osu.usawest model is calculated in Figure 4. The coastland areas A and B
within the two rectangles in Figure 4 have larger differences between the ground OTL
estimations of the proposed OTL calculation method and the FES2014b+osu.usawest
model. Its maximum StdDev value of the displacement difference is 1.93 mm, and its cor-
responding that of the vertical OTL displacement is 17.9 mm, which suggests that the OTL
displacement difference between the two methods reach 10.7%. Likewise, the StdDev val-
ues of the north direction are 0.37 mm, 3.61 mm, and 10.3%, respectively, and those of the
east direction are 0.91 mm, 5.85 mm, and 15.6%. The differential interferograms in Section
3.2 will be used to verify the improvement of the correction accuracy of the proposed
method in coastland area A.
Figure 4. StdDev maps of the difference of the OTL displacements estimated by the proposed tide
calculation method and FES2014b+osu.usawest model in U, N, and E directions, and the areas with
lager difference (red box).
Furthermore, a three-dimensional ground OTL displacements grid with a spatial res-
olution of 1 × 1 is formed by integrating the spatiotemporal modelled Q1, O1, N2, and
M2 constituents of a GPS reference station network and the P1, K1, and K2 constituents of
the FES2014b+osu.usawest model. For the long-strip differential interferogram of Senti-
nel-1 data covering the west coast of the United States, the StdDev values are 1.89 mm,
18.16 mm, and 10.4% in the LOS direction (ascending; the incident angle is 39°; the time
interval is 12 days), which means millimetre to centimetre OTL displacement residuals in
a differential interferogram and time-series InSAR measurement (see Figure 5).
Figure 5.
The OTL displacement StdDev along the LOS direction (Incident angle is 39
) of the ascend-
ing (Heading direction is
13
) and descending (Heading direction is 193
) Sentinel-1 measurements
with an interval of 12 days based on the proposed OTL calculation method.
Remote Sens. 2022,14, 2954 9 of 14
3.2. Assessment and Removal of Tide Displacements in a Long-Strip Differential Interferogram
For the 29 long-strip differential interferograms, the spatial variations in OTL and SET
and their superposition ground displacement in the long-strip differential interferograms
are calculated. The results in Figure 6reveal that the ground tidal displacements in the
long-strip differential interferograms can reach 77.5 mm; the largest SET displacement is
78.9 mm and the largest OTL displacement is 41.9 mm, which illustrates that the magnitude
of the tidal displacements can match the atmospheric delay error in some differential
interferograms. If the interferograms are fitted by the bilinear ramp, the largest tidal
displacement residual generated in interferogram 20180906-20181012 is 20.3 mm and that
of interferogram 201800801-20180813 is the smallest, which is 4.9 mm.
Remote Sens. 2022, 14, x FOR PEER REVIEW 9 of 15
Figure 5. The OTL displacement StdDev along the LOS direction (Incident angle is 39°) of the as-
cending (Heading direction is 13°) and descending (Heading direction is 193°) Sentinel-1 measure-
ments with an interval of 12 days based on the proposed OTL calculation method.
3.2. Assessment and Removal of Tide Displacements in a Long-Strip Differential Interferogram
For the 29 long-strip differential interferograms, the spatial variations in OTL and
SET and their superposition ground displacement in the long-strip differential interfero-
grams are calculated. The results in Figure 6 reveal that the ground tidal displacements in
the long-strip differential interferograms can reach 77.5 mm; the largest SET displacement
is 78.9 mm and the largest OTL displacement is 41.9 mm, which illustrates that the mag-
nitude of the tidal displacements can match the atmospheric delay error in some differen-
tial interferograms. If the interferograms are fitted by the bilinear ramp, the largest tidal
displacement residual generated in interferogram 20180906-20181012 is 20.3 mm and that
of interferogram 201800801-20180813 is the smallest, which is 4.9 mm.
Figure 6. The SET and OTL displacements in 29 long-strip differential interferograms and the error
analysis of residual tidal displacement produced by bilinear fitting ramp, and the interferogram
with largest residual tidal displacement (red box).
According to the comparison of the tidal displacement residuals in Figure 7, the long-
strip differential interferogram is fitted by the bilinear fitting method, and the tidal dis-
placement residual in the near coastline area can reach 20.3 mm (area B in Figure 7(a4))
and 7.2 mm using splice bilinear ramp fitting (area A in Figure 7(a5)); therefore, the tradi-
tional plane-fitting methods cannot eliminate the SET and OTL displacements. For the
area far from the coastline, both plane fitting methods make minor errors (area C in Figure
7(a5)). However, since the tidal displacement residual remains a long-wavelength signal,
it has a limited impact on the spatial small-scale differential interferogram (area B in Fig-
ure 7(a5)).
Figure 6.
The SET and OTL displacements in 29 long-strip differential interferograms and the error
analysis of residual tidal displacement produced by bilinear fitting ramp, and the interferogram with
largest residual tidal displacement (red box).
According to the comparison of the tidal displacement residuals in Figure 7, the
long-strip differential interferogram is fitted by the bilinear fitting method, and the tidal
displacement residual in the near coastline area can reach 20.3 mm (area B in Figure 7(a4))
and 7.2 mm using splice bilinear ramp fitting (area A in Figure 7(a5)); therefore, the
traditional plane-fitting methods cannot eliminate the SET and OTL displacements. For
the area far from the coastline, both plane fitting methods make minor errors (area C in
Figure 7(a5)). However, since the tidal displacement residual remains a long-wavelength
signal, it has a limited impact on the spatial small-scale differential interferogram (area B in
Figure 7(a5)).
In a large-scale spatial signal, the external correction of atmospheric delay error mainly
corrects the terrain-related atmospheric delay error [
31
]. The spatial variation of SET
displacement is not affected by the ground geographical environment factors, and its spatial
variations are close to the uniform curve plane, while the OTL displacement is affected by
the shape of the coastline, and the spatial variation shows the trend of gradual weakening
from the coastline to inland, and the other large-scale errors can be neglectable based on
ground deformation analysis of the GPS sites network in this study. The atmospheric delay
error (especially for long-wavelength and topography-related atmospheric delay errors)
Remote Sens. 2022,14, 2954 10 of 14
and tidal displacements in the long-strip differential interferograms can be corrected using
the atmospheric delay, SET and OTL correction models, and then bilinear ramp fitting was
performed to eliminate the other large-scale error residuals (Figure 8(a1–d3)).
Remote Sens. 2022, 14, x FOR PEER REVIEW 10 of 15
Figure 7. OTL displacement (a1), SET (a2) and their superposition displacement (a3) based on the
differential interferogram acquisition on 6 September 2018 and 12 October 2018; (a4) residual tidal
displacement generated by the bilinear ramp fitting method; and (a5) residual tidal displacement
from the multi-frame mosaic bilinear ramp fitting method.
In a large-scale spatial signal, the external correction of atmospheric delay error
mainly corrects the terrain-related atmospheric delay error [31]. The spatial variation of
SET displacement is not affected by the ground geographical environment factors, and its
spatial variations are close to the uniform curve plane, while the OTL displacement is
affected by the shape of the coastline, and the spatial variation shows the trend of gradual
weakening from the coastline to inland, and the other large-scale errors can be neglectable
based on ground deformation analysis of the GPS sites network in this study. The atmos-
pheric delay error (especially for long-wavelength and topography-related atmospheric
delay errors) and tidal displacements in the long-strip differential interferograms can be
corrected using the atmospheric delay, SET and OTL correction models, and then bilinear
ramp fitting was performed to eliminate the other large-scale error residuals (Figure 8(a1–
d3)).
Figure 7.
OTL displacement (
a1
), SET (
a2
) and their superposition displacement (
a3
) based on the
differential interferogram acquisition on 6 September 2018 and 12 October 2018; (
a4
) residual tidal
displacement generated by the bilinear ramp fitting method; and (
a5
) residual tidal displacement
from the multi-frame mosaic bilinear ramp fitting method.
Figure 8.
(
a1a3
) The long-strip differential interferograms after the correction of (
b1b3
) atmo-
spheric delay error, (
c1c3
) SET and OTL, and (
d1d3
) bilinear ramp elimination. (
e1e3
) The
SET and OTL displacements, whose spatial trend variations are similar to the long-strip differential
interferograms after the atmospheric delay is corrected.
Remote Sens. 2022,14, 2954 11 of 14
By analysing the long-wavelength signals of the long-strip differential interferogram
in the results in Figure 8, the atmospheric delay is the most vital component in the long-
strip differential interferogram, but its magnitude is independent of the spatial scale of
the differential interferogram, and the OTL displacement increases with the spatial cover
range of the differential interferogram. Therefore, the SET and OTL effects have become the
major signals in large-range differential interferograms in coastal areas. The topography-
related signal in the long-strip differential interferogram corrected by the atmospheric
correction algorithm is weakened, and the StdDev values of the differential interference
diagram are reduced by 38.1~50.3% (Figure 8(b1
b3)). The spatial trend signal in the
long-strip differential interferogram with the atmospheric delay correction is similar to
the tidal displacement of the SET and OTL effects (Figure 8(e1
e3)). Further correction
of the SET and OTL displacements further reduces the StdDev value of the differential
interferogram by 3.9~19.3% (Figure 8(c1
c3)), and its magnitude depends on the relative
spatial variation in the ground tidal displacement. The residual large-scale signal in the long
strip differential interferograms is inconsistent with the spatial characteristics of the OTL
and approaches the linear plane, which can be eliminated using a bilinear fitting function
(Figure 8(d1
d3)). After that, the trend signal in the long-strip differential interferogram is
basically eliminated.
The difference between the proposed tidal method and the traditional plane fitting
method for OTL displacement correction of the long-strip interferograms is compared. As
shown in Figure 9, the pixels of the long-strip differential interferograms were reordered
following the tendency of the OTL displacements from small to large, the proposed OTL
calculation method can effectively correct the OTL displacement, and the fitting line of the
displacement residuals in the interferograms is close to zero and it is basically consistent
with that of the PPP displacement residuals, which indicated that the displacement residuals
are mainly related to the random noise. Moreover, the original differential interferogram
with atmospheric delay error and bilinear ramp correction the OTL corrections in the inland
area, but it produces larger errors in the near coastline areas (see Figure 9).
Remote Sens. 2022, 14, x FOR PEER REVIEW 12 of 15
The difference between the proposed tidal method and the traditional plane fitting
method for OTL displacement correction of the long-strip interferograms is compared. As
shown in Figure 9, the pixels of the long-strip differential interferograms were reordered
following the tendency of the OTL displacements from small to large, the proposed OTL
calculation method can effectively correct the OTL displacement, and the fitting line of
the displacement residuals in the interferograms is close to zero and it is basically con-
sistent with that of the PPP displacement residuals, which indicated that the displacement
residuals are mainly related to the random noise. Moreover, the original differential inter-
ferogram with atmospheric delay error and bilinear ramp correction the OTL corrections
in the inland area, but it produces larger errors in the near coastline areas (see Figure 9).
Figure 9. The comparison of the displacement residuals in the long-strip differential interferograms
after the OTL and SET displacements corrected using the proposed tidal method (interferograms-
atmospheric-SET-OTL-bilinear fitting ramp) and traditional bilinear ramp fitting method (interfer-
ograms-atmospheric-bilinear fitting ramp).
To further compare and analyse the OTL correction difference between the proposed
method and the FES2014b+osu.usawest model, the coastal area A with a larger OTL dis-
placement difference in Figure 4 was selected to validate the improvement of the pro-
posed tidal method. As the results are shown in Figure 10, the absolute value of the fitting
line slope of the proposed OTL calculation method is smaller than that of the tidal model
in the interferograms 20180906–20181012 and 20180918–20181012, so the tendency of the
OTL displacement can be effectively eliminated that suggests the OTL correction im-
provement of the proposed tidal method in complex coastline areas.
Figure 9.
The comparison of the displacement residuals in the long-strip differential inter-
ferograms after the OTL and SET displacements corrected using the proposed tidal method
(interferograms-atmospheric-SET-OTL-bilinear fitting ramp) and traditional bilinear ramp fitting
method (interferograms-atmospheric-bilinear fitting ramp).
To further compare and analyse the OTL correction difference between the proposed
method and the FES2014b+osu.usawest model, the coastal area A with a larger OTL dis-
placement difference in Figure 4was selected to validate the improvement of the proposed
tidal method. As the results are shown in Figure 10, the absolute value of the fitting line
slope of the proposed OTL calculation method is smaller than that of the tidal model in
Remote Sens. 2022,14, 2954 12 of 14
the interferograms 20180906–20181012 and 20180918–20181012, so the tendency of the OTL
displacement can be effectively eliminated that suggests the OTL correction improvement
of the proposed tidal method in complex coastline areas.
Remote Sens. 2022, 14, x FOR PEER REVIEW 13 of 15
Figure 10. The comparison of the displacement residuals in a complex coastline area of the long-
strip differential interferograms after the tidal displacements was corrected using the proposed tidal
method and FES2014b+osu.usawest model.
4. Discussion
We evaluated the ground tidal displacement in the long-strip differential interfero-
gram based on the tidal estimation of the GPS continuous reference station network and
the tidal model. For inland areas more than 200 km away from the coastline of the west
coast of the U.S., the error produced by bilinear ramp fitting is much less than in coastal
areas, which has limited influence on the studies of InSAR ground deformation monitor-
ing; For most coastal areas, there is little difference between the traditional ocean tide
model and PPP tidal displacements, and previous studies have proved the OTL displace-
ments in the InSAR measurements can be corrected based on a global ocean tide model
[12,13]. However, Figure 4 and Abbaszadeh et al. both indicated that in some localized
coastal areas, such as Area A, the maximum StdDev differences between the tidal constit-
uents’ displacement time series of recent global ocean tide models are 1~2 mm [22], which
may be introduced tidal displacement residuals into the OTL correction of the InSAR
ground deformation measurements. To solve this problem, this paper has shown that the
PPP tidal displacements can further improve the spatial accuracy of OTL displacements
in Area A, and the residual tidal displacement of a few millimetres can be eliminated with
the comparison of the commonly used FES2014b+osu.usawest model as shown in Figure
10.
For the long-strip DInSAR measurements, the magnitude and nonlinear variation of
tidal displacement are related to the size of the interferogram, shape of the coastline, sat-
ellite flight direction, incident angle and imaging time. The OTL effect on the differential
interferogram with wider coverage and complex coastline shape was analysed above (see
Figures 810). In addition to improving the spatial accuracy, the advantages of the method
in this paper also can quickly predict the tidal displacement at any time and in any direc-
tion according to the system parameters of SAR platforms. If there are dense GPS contin-
uous reference stations in the coastline area that has a larger inter-model discrepancy,
Figure 10.
The comparison of the displacement residuals in a complex coastline area of the long-strip
differential interferograms after the tidal displacements was corrected using the proposed tidal
method and FES2014b+osu.usawest model.
4. Discussion
We evaluated the ground tidal displacement in the long-strip differential interferogram
based on the tidal estimation of the GPS continuous reference station network and the
tidal model. For inland areas more than 200 km away from the coastline of the west coast
of the U.S., the error produced by bilinear ramp fitting is much less than in coastal areas,
which has limited influence on the studies of InSAR ground deformation monitoring; For
most coastal areas, there is little difference between the traditional ocean tide model and
PPP tidal displacements, and previous studies have proved the OTL displacements in
the InSAR measurements can be corrected based on a global ocean tide model [
12
,
13
].
However, Figure 4and Abbaszadeh et al. both indicated that in some localized coastal
areas, such as Area A, the maximum StdDev differences between the tidal constituents’
displacement time series of recent global ocean tide models are 1~2 mm [
22
], which may
be introduced tidal displacement residuals into the OTL correction of the InSAR ground
deformation measurements. To solve this problem, this paper has shown that the PPP
tidal displacements can further improve the spatial accuracy of OTL displacements in Area
A, and the residual tidal displacement of a few millimetres can be eliminated with the
comparison of the commonly used FES2014b+osu.usawest model as shown in Figure 10.
For the long-strip DInSAR measurements, the magnitude and nonlinear variation
of tidal displacement are related to the size of the interferogram, shape of the coastline,
satellite flight direction, incident angle and imaging time. The OTL effect on the differential
interferogram with wider coverage and complex coastline shape was analysed above (see
Figures 810). In addition to improving the spatial accuracy, the advantages of the method
in this paper also can quickly predict the tidal displacement at any time and in any direction
according to the system parameters of SAR platforms. If there are dense GPS continuous
reference stations in the coastline area that has a larger inter-model discrepancy, such as
Remote Sens. 2022,14, 2954 13 of 14
coastal areas of Western Europe and East Asia, the spatial accuracy of OTL corrections
may be improved by integrating the tidal constituent of the GPS network and tidal models,
which will be validated in future studies.
5. Conclusions
In this study, we presented a tidal displacement calculation method for long-strip
differential interferograms, which integrates tide displacement of the GPS network com-
posed of 1038 continuous sites, the FES2014+osu.usawest model and the SET model in the
2010 IERS Convention. Based on the long-strip differential interferogram of Sentinel-1 SLC
images on the west coast of the U.S., the experimental results show that (1) according to
the spatial variations of the phasor, the tidal constituent parameters estimated from the
PPP coordinate time series of the GPS reference station network can effectively validate
the spatial accuracy of the OTL displacement along the LOS direction in the inland area
and improve the OTL estimations in the complex coastline area by 10.4%. (2) The SET
and OTL effects become the main spatial large-scale signals in the long-strip differential
interferograms, the superposition displacement of SET and OTL effects can reach 77.5 mm,
the tidal displacement residuals generated by bilinear ramp fitting can reach 20.3 mm,
and that generated by the splice-frame bilinear ramp fitting method can reach 7.2 mm
with displacement shift where the boundary is mosaicked. (3) The proposed tidal dis-
placement method can effectively eliminate the tendency of tidal displacement in complex
coastline areas.
Author Contributions:
Conceptualization, W.P. and Q.W.; methodology, W.P.; software, W.P.; val-
idation, W.P., Q.W., Y.C., X.X. and W.H.; formal analysis, W.P.; investigation, W.P.; resources, W.P.;
data curation, W.P. and Y.C.; writing—original draft preparation, W.P.; writing—review and editing,
Q.W., Y.C., X.X. and W.H.; visualization, W.P., X.X. and W.H.; supervision, Q.W.; project administra-
tion, Q.W.; funding acquisition, Q.W. All authors have read and agreed to the published version of
the manuscript.
Funding:
This research was funded by Hunan Key Laboratory of remote sensing of ecological
environment in Dongting Lake Area (No. 2021-010) of China, Scientific research projects funded by
the Department of education of Hunan Province of China (No. 21A0006), Natural Science Foundation
of Hunan Province, China (No. 2022JJ40472), and Open Fund of Hunan International Scientific and
Technological Innovation Cooperation Base of Advanced Construction and Maintenance Technology
of Highway (Changsha University of Science & Technology) (No. kfj210802).
Data Availability Statement:
The GPS data were provided by Scripps Orbit and Permanent Array
Center (SOPAC) at http://sopac-csrc.ucsd.edu (accessed on 15 October 2020). The Sentinel-1 dataset
was provided by European Space Agency (ESA) at https://scihub.copernicus.eu (accessed on 6
November 2020). The FES2014 model was available online https://www.aviso.altimetry.fr/en/data/
products/auxiliary-products/global-tide- fes.html (accessed on 6 February 2020).
Acknowledgments:
The NLOADF program was used to produce the OTL correction, and the solid
earth tide program was provided by Dehant. The atmospheric delay maps were provided by
the ICAMS advanced atmospheric correction method based on the ECMWF ERA-5 global atmo-
spheric model.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Xu, X.; Sandwell, D.T. Toward Absolute Phase Change Recovery With InSAR: Correcting for Earth Tides and Phase Unwrapping
Ambiguities. IEEE Trans. Geosci. Remote Sens. 2020,58, 726–733. [CrossRef]
2.
Bähr, H.; Hanssen, R.F. Reliable estimation of orbit errors in spaceborne SAR interferometry. J. Geod.
2012
,86, 1147–1164.
[CrossRef]
3.
Du, Y.; Fu, H.; Liu, L.; Feng, G.; Peng, X.; Wen, D. Orbit error removal in InSAR/MTInSAR with a patch-based polynomial model.
Int. J. Appl. Earth Obs. 2021,102, 102438. [CrossRef]
4.
Kowalczyk, K.; Pajak, K.; Wieczorek, B.; Naumowicz, B. An Analysis of Vertical Crustal Movements along the European Coast
from Satellite Altimetry, Tide Gauge, GNSS and Radar Interferometry. Remote Sens. 2021,13, 2173. [CrossRef]
5. Francis, O.; Mazzega, P. Global charts of ocean tide loading effects. J. Geophys. Res. 1990,95, 11411. [CrossRef]
Remote Sens. 2022,14, 2954 14 of 14
6.
Melachroinos, S.A.; Biancale, R.; Llubes, M.; Perosanz, F.; Lyard, F.; Vergnolle, M.; Bouin, M.N.; Masson, F.; Nicolas, J.; Morel,
L.; et al. Ocean tide loading (OTL) displacements from global and local grids: Comparisons to GPS estimates over the shelf of
Brittany, France. J. Geod. 2008,82, 357–371. [CrossRef]
7.
Wu, K.; Ji, C.; Luo, L.; Wang, X. Simulation Study of Moon-Based InSAR Observation for Solid Earth Tides. Remote Sens.
2020
,
12, 123. [CrossRef]
8.
Jolivet, R.; Agram, P.S.; Lin, N.Y.; Simons, M.; Doin, M.P.; Peltzer, G.; Li, Z. Improving InSAR geodesy using Global Atmospheric
Models. J. Geophys. Res. Solid Earth 2014,119, 2324–2341. [CrossRef]
9.
DiCaprio, C.J.; Simons, M. Importance of ocean tidal load corrections for differential InSAR. Geophys. Res. Lett.
2008
,35, L22309.
[CrossRef]
10.
Lyard, F.; Lefevre, F.; Letellier, T.; Francis, O. Modelling the global ocean tides: Modern insights from FES2004. Ocean Dyn.
2006
,
56, 394–415. [CrossRef]
11.
Peng, W.; Wang, Q.; Cao, Y. Analysis of Ocean Tide Loading in Differential InSAR Measurements. Remote Sens.
2017
,9, 101.
[CrossRef]
12.
Peng, W.; Wang, Q.; Zhan, F.B.; Cao, Y. Spatiotemporal Ocean Tidal Loading in InSAR Measurements Determined by Kinematic
PPP Solutions of a Regional GPS Network. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 2020,13, 3772–3779. [CrossRef]
13.
Yu, C.; Penna, N.T.; Li, Z. Ocean Tide Loading Effects on InSAR Observations Over Wide Regions. Geophys. Res. Lett.
2020
,
47, e2020GL088184. [CrossRef]
14.
Wu, Z.; Jiang, M.; Xiao, R.; Xu, J. Ocean tide loading correction for InSAR measurements: Comparison of different ocean tide
models. Geod. Geodyn. 2022,13, 170–178. [CrossRef]
15.
Egbert, G.D.; Erofeeva, S.Y. Efficient Inverse Modeling of Barotropic Ocean Tides. J. Atmos. Ocean. Technol.
2002
,19, 183–204.
[CrossRef]
16.
Shum, C.K.; Woodworth, P.L.; Andersen, O.B.; Egbert, G.D.; Francis, O.; King, C.; Klosko, S.M.; Le Provost, C.; Li, X.; Molines,
J.-M.; et al. Accuracy assessment of recent ocean tidal models. J. Geophys. Res. 1997,102, 125–173. [CrossRef]
17.
Thomas, I.D.; King, M.A.; Clarke, P.J. A comparison of GPS, VLBI and model estimates of ocean tide loading displacements. J.
Geod. 2007,81, 359–368. [CrossRef]
18. Dehant, V.; Defraigne, P.; Wahr, J.M. Tides for a convective Earth. J. Geophys. Res. 1999,104, 1035–1058. [CrossRef]
19.
Petit, G.; Luzum, B. IERS Conventions (2010), Technical Report DTIC Document; International Earth Rotation and Reference Systems
Service: Frankfurt, Germany, 2010; No. 36; p. 180.
20.
Lu, F.; Konecny, M.; Chen, M.; Reznik, T. A Barotropic Tide Model for Global Ocean Based on Rotated Spherical Longitude-
Latitude Grids. Water 2021,13, 2670. [CrossRef]
21.
Bos, M.S.; Penna, N.T.; Baker, T.F.; Clarke, P.J. Ocean tide loading displacements in western Europe: 2. GPS-observed anelastic
dispersion in the asthenosphere. J. Geophys. Res. Solid Earth 2015,120, 6540–6557. [CrossRef]
22. Abbaszadeh, M.; Clarke, P.J.; Penna, N.T. Benefits of combining GPS and GLONASS for measuring ocean tide loading displace-
ment. J. Geod. 2020,94, 63. [CrossRef]
23.
Yuan, L.; Chao, B.F. Analysis of tidal signals in surface displacement measured by a dense continuous GPS array. Earth Planet. Sci.
Lett. 2012,355–356, 255–261. [CrossRef]
24.
Wei, G.; Wang, Q.; Peng, W. Accurate Evaluation of Vertical Tidal Displacement Determined by GPS Kinematic Precise Point
Positioning: A Case Study of Hong Kong. Sensors 2019,19, 2559. [CrossRef] [PubMed]
25.
Agnew, D.C. NLOADF; a program for computing ocean-tide loading. J. Geophys. Res. Solid Earth
1997
,102, 5109–5110. [CrossRef]
26.
Suykens, J.A.K.; Vandewalle, J. Recurrent least squares support vector machines. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
2000,47, 1109–1114. [CrossRef]
27.
Hooper, A.; Bekaert, D.; Spaans, K.; Arıkan, M. Recent advances in SAR interferometry time series analysis for measuring crustal
deformation. Tectonophysics 2012,514–517, 1–13. [CrossRef]
28.
Cao, Y.; Joónsson, S.; Li, Z. Advanced InSAR Tropospheric Corrections From Global Atmospheric Models that Incorporate Spatial
Stochastic Properties of the Troposphere. J. Geophys. Res. Solid Earth 2021,126, e2020JB020952. [CrossRef]
29.
Wegnüller, U.; Werner, C.; Strozzi, T.; Wiesmann, A.; Frey, O.; Santoro, M. Sentinel-1 Support in the GAMMA Software. Procedia
Comput. Sci. 2016,100, 1305–1312. [CrossRef]
30.
Berardino, P.; Fornaro, G.; Lanari, R.; Sansosti, E. A new algorithm for surface deformation monitoring based on small baseline
differential SAR interferograms. IEEE Trans. Geosci. Remote Sens. 2002,40, 2375–2383. [CrossRef]
31.
Bekaert, D.P.S.; Walters, R.J.; Wright, T.J.; Hooper, A.J.; Parker, D.J. Statistical comparison of InSAR tropospheric correction
techniques. Remote Sens. Environ. 2015,170, 40–47. [CrossRef]
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The orbit error caused by the inaccuracy of the orbit state vector can lead to fringes in differential interferograms, which can impede the estimation of deformation in differential SAR interferometry (DInSAR) applications. Usually, a set of polynomial coefficients for an entire SAR image is obtained for orbit error removal. However, the orbit error plane is influenced by overfitting in the case that the SAR satellites do not have a precise orbit. In this paper, a patch-based polynomial method is proposed to fit the orbit error plane. The new method divides an SAR image into several overlapping patches in the azimuth and range directions. Every patch obtains its own polynomial coefficients, and an iterative least-square method is used to mosaic the orbit plane. This method is tested and validated via a simulated dataset and then applied to ALOS1/2 PALSAR and Sentinel-1A datasets. The accuracy of deformation is evaluated by in situ GPS datasets. The results show that the patch-based method can fit the orbit phase plane more accurately than the traditional polynomial model with millimeter-level displacement improvement, especially in the margin areas of ALOS1/2 and for the wide-coverage Sentinel-1A datasets. Moreover, in the MTInSAR parameter calculations, the new method improves the accuracy of mean velocity calculations for ALOS1 time series, with a reduction of RMSE from 4.47 mm/yr to 3.17 mm/yr. Additionally, the new method reduces the spatial correlation of the residual topographic phase, with a mean value reduction from 0.32 m to 0.13 m.