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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+µm−Φm,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,y−datmospheric

x,y=dlarge−sc 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,y−datmospheric

x,y−dOTL

x,y−dSET

x,y=dlarge−sc 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

N−1

N

∑

n=1dOTL

n,p−dOTL

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-

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 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.

Remote Sens. 2022, 14, x FOR PEER REVIEW 11 of 15

Figure 8. (a1−a3) The long-strip differential interferograms after the correction of (b1−b3) atmos-

pheric delay error, (c1−c3) SET and OTL, and (d1−d3) bilinear ramp elimination. (e1−e3) The SET

and OTL displacements, whose spatial trend variations are similar to the long-strip differential in-

terferograms after the atmospheric delay is corrected.

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 topogra-

phy-related signal in the long-strip differential interferogram corrected by the atmos-

pheric correction algorithm is weakened, and the StdDev values of the differential inter-

ference 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 inter-

ferogram is basically eliminated.

Figure 8.

(

a1−a3

) The long-strip differential interferograms after the correction of (

b1−b3

) atmo-

spheric delay error, (

c1−c3

) SET and OTL, and (

d1−d3

) bilinear ramp elimination. (

e1−e3

) 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 8–10). 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 8–10). 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.

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