A Review on Modeling Environmental Loading Effects and Their Contributions to Nonlinear Variations of Global Navigation Satellite System Coordinate Time Series

Zhao Li; Weiping Jiang; Tonie van Dam; Xiaowei Zou; Qusen Chen; Hua Chen

doi:10.1016/j.eng.2024.09.001

Engineering ›› 2025, Vol. 47 ›› Issue (4) :29 -41. DOI: 10.1016/j.eng.2024.09.001

Research Precise Positioning and Geoinformation Science—Review

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A Review on Modeling Environmental Loading Effects and Their Contributions to Nonlinear Variations of Global Navigation Satellite System Coordinate Time Series

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Abstract

Nonlinear variations in the coordinate time series of global navigation satellite system (GNSS) reference stations are strongly correlated with surface displacements caused by environmental loading effects, including atmospheric, hydrological, and nontidal ocean loading. Continuous improvements in the accuracy of surface mass loading products, performance of Earth models, and precise data-processing technologies have significantly advanced research on the effects of environmental loading on nonlinear variations in GNSS coordinate time series. However, owing to theoretical limitations, the lack of high spatiotemporal resolution surface mass observations, and the coupling of GNSS technology-related systematic errors, environmental loading and nonlinear GNSS reference station displacements remain inconsistent. The applicability and capability of these loading products across different regions also require further evaluation. This paper outlines methods for modeling environmental loading, surface mass loading products, and service organizations. In addition, it summarizes recent advances in applying environmental loading to address nonlinear variations in global and regional GNSS coordinate time series. Moreover, the scientific questions of existing studies are summarized, and insights into future research directions are provided. The complex nonlinear motion of reference stations is a major factor limiting the accuracy of the current terrestrial reference frame. Further refining the environmental load modeling method, establishing a surface mass distribution model with high spatiotemporal resolution and reliability, exploring other environmental load factors such as ice sheet and artificial mass-change effects, and developing an optimal data-processing model and strategy for reprocessing global reference station data consistently could contribute to the development of a millimeter-level nonlinear motion model for GNSS reference stations with actual physical significance and provide theoretical support for establishing a terrestrial reference frame with 1 mm accuracy by 2050.

Keywords

Environmental loading / Global navigation satellite system / Nonlinear variations / Time series analysis / Surface mass distribution / Green’s function / Spherical harmonic function

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Zhao Li, Weiping Jiang, Tonie van Dam, Xiaowei Zou, Qusen Chen, Hua Chen. A Review on Modeling Environmental Loading Effects and Their Contributions to Nonlinear Variations of Global Navigation Satellite System Coordinate Time Series. Engineering, 2025, 47(4): 29-41 DOI:10.1016/j.eng.2024.09.001

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1. Introduction

The global navigation satellite system (GNSS) reference station coordinate time series accumulated over the past 30 years contains rich geophysical information, providing valuable basic data for geodetic and geodynamic research. Significant nonlinear variations exist within the GNSS coordinate time series, and establishing precise nonlinear motion models is crucial for determining accurate positions and velocities of reference stations, as well as building and maintaining a dynamic terrestrial reference frame. Moreover, this enhances the applicability of the GNSS coordinate time series to a wide range of cutting-edge geophysical issues, including plate tectonic movements, ice and snow mass changes, as well as sea-level variations, highlighting its scientific and application value [1].

Surface mass redistribution in the atmosphere, oceans, and terrestrial water is one of the primary causes of nonlinear motion at GNSS reference stations [2], [3]. Since the late 1980s, many scientists have studied the contributions of environmental loads resulting from variations in atmospheric pressure, continental water storage, and ocean bottom pressure (OBP) to GNSS coordinate time series [4], [5], [6], [7], [8], [9]. Surface displacements caused by atmospheric, hydrological, and nontidal ocean loads have been confirmed to be strongly correlated with the vertical displacements of GNSS stations [1], [4], [5], [6], [7], [8], [10], [11], [12], [13], [14], [15], [16], [17], [18].

Continuous developments in the accuracy of surface mass distribution products, performance of Earth models, and precise data-processing techniques have significantly advanced research on the effects of environmental loading on nonlinear variations in GNSS coordinate time series. However, discrepancies remain between environmental loading and GNSS reference station displacements. For example, environmental loading displacement can only partially account for the annual and semiannual amplitudes of the vertical displacement [7], [14]. Systematic differences in vertical annual phases have been observed, and the correlation with horizontal reference station displacements is relatively low [7], [15]. Moreover, the contributions of environmental loads to the nonlinear variations in the GNSS coordinate time series estimated in different studies differ significantly. This raises the following questions: apart from the three well-known environmental loads, are there other load types or unknown geophysical effects that contributed to seasonal displacements at GNSS reference stations? What causes the systematic annual phase differences and low horizontal correlation? Current environmental load models are based on an elastic Earth model with a homogeneous crustal structure; but are they adequate? Among the various environmental loading products, which is the most effective in correcting the nonlinear motion of GNSS reference stations? How can the impacts of GNSS technology-related systematic errors caused by unmodeled or imperfectly modeled factors be minimized, as these errors can couple with the environmental loading effects, resulting in erroneous geophysical interpretation of the nonlinear motion observed at GNSS reference stations? Currently, the complex nonlinear motion of reference stations remains a major factor limiting the accuracy of the terrestrial reference frame, thus addressing these questions is crucial.

This study reviews recent developments in modeling environmental loading effects and their contributions to the nonlinear variations in GNSS coordinate time series, examines the causes of the inconsistency, and proposes potential solutions. The remainder of this paper is organized as follows: Section 2 illustrates the approaches used to model environmental loading effects, surface mass loading products, and associated service organizations; Section 3 summarizes recent breakthroughs in applying environmental loading to rectify nonlinear variations in global and regional GNSS coordinate time series; and Section 4 consolidates the scientific challenges identified in existing studies and offers insights into future research directions. This work contributes to the development of a nonlinear motion model with genuine physical significance for reference stations at the millimeter level, providing a theoretical foundation for establishing an international terrestrial reference frame (ITRF) with an accuracy of 1 mm.

2. Environmental load modeling methods and current data products

2.1. Environmental load modeling method

The displacement of GNSS reference stations caused by environmental loading effects is typically modeled using two methods: the load Green’s function approach and the spherical harmonic function method. The load Green’s function was developed by Farrell based on Longman’s Earth load theory [19], [20], [21]. The static Green’s function for a spherically symmetric earth under a unit-point mass load is theoretically derived for a given earth model. The surface deformation caused by such a load can be determined by convolving the variations in surface mass, such as atmospheric, hydrologic, and oceanic masses, with the load Green’s function, as expressed in the following equation [21]:

(1)

u = \int_{Ω} m a s s G_{r} (Θ) d Ω

where r is the Earth’s radius; mass represents surface mass variations, typically presented as gridded data;

Θ

is the angular distance; Ω denotes the region of the surface mass variations;

G_{r} (\cdot)

denotes the Green’s function, with distinct numerical values for the horizontal and vertical components; and

u

denotes the surface displacement calculated based on the corresponding component of the Green’s function.

The spherical harmonic function method expands the global mass load into spherical harmonics. The environmental loading displacement in the topocentric coordinate system is obtained by integrating and summing over the spherical harmonic domains based on the location of the station and applying Farrell’s definition of load Love numbers. The mathematical formulas for calculating the vertical and horizontal elastic deformations of the Earth’s surface using the spherical harmonic coefficients provided by the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field model are as follows [22]:

(2)

Δ u (θ, λ) = r \sum_{l = 1}^{N_{m a x}} \sum_{m = 0}^{l} \frac{h_{l}^{'}}{1 + k_{l}^{'}} {\bar{P}}_{l, m} (\cos θ) (C_{l, m} \cos (m λ) + S_{l, m} \sin (m λ))

(3)

Δ n (θ, λ) = - r \sum_{l = 1}^{N_{m a x}} \sum_{m = 0}^{l} \frac{l_{l}^{'}}{1 + k_{l}^{'}} \times {\frac{\partial}{\partial θ} \bar{P}}_{l, m} (\cos θ) (C_{l, m} \cos (m λ) + S_{l, m} \sin (m λ))

(4)

Δ e (θ, λ) = \frac{r}{\sin θ} \sum_{l = 1}^{N_{m a x}} \sum_{m = 0}^{l} \frac{l_{l}^{'}}{1 + k_{l}^{'}} {\bar{P}}_{l, m} (\cos θ) m ({- C}_{l, m} \sin (m λ) + S_{l, m} \cos (m λ))

where

θ

and

λ

are the colatitude and longitude of the point, respectively;

Δ u (θ, λ)

Δ n (θ, λ)

, and

Δ e (θ, λ)

represent the vertical, north, and east displacement of the Earth’s surface, respectively;

N_{m a x}

is the maximum degree for GRACE monthly gravity field;

l

and

m

are the degree and order, respectively;

{\bar{P}}_{l, m} (\cos θ)

is the fully normalized associated Legendre function;

l_{l}^{'}, h_{l}^{'}

, and

k_{l}^{'}

are the load Love numbers of the degree

l

;

C_{l, m}

and

S_{l, m}

are the time-variable gravity field spherical harmonic coefficients provided by GRACE, including the high-frequency atmospheric model, geoid model, and high-frequency ocean model for calculating atmospheric pressure loading (ATML), hydrological loading (HYDL), and nontidal ocean loading (NTOL) displacements.

The surface mass distribution can also be used to calculate environmental loading displacement using the spherical harmonic function method. First, the spherical harmonic coefficients corresponding to surface mass changes are calculated using the spherical harmonic coefficient fitting method, as shown in Eq. (5) [22]:

(5)

\{\begin{matrix} Δ C_{l, m} \\ Δ S_{l, m} \end{matrix}\} = \frac{3 (1 + k_{l})}{4 π ρ_{a v e} r (2 l + 1)} \int_{E a r t h} m a s s {\bar{P}}_{l, m} (\cos θ) \{\begin{matrix} \cos (m λ) \\ \sin (m λ) \end{matrix}\} \sin θ d θ d λ

where

Δ C_{l, m}

and

Δ S_{l, m}

are the expanded spherical harmonic coefficients;

r

and

ρ_{a v e}

are the Earth’s radius and water density, respectively; and

k_{l}

is the load Love number of the degree l .

Subsequently, according to Eqs. (2), (3), (4), the vertical (

Δ u

), northward (

Δ n

), and eastward (

Δ e

) displacements caused by surface mass changes are calculated using the spherical harmonic function method. It is worth noting that the Green’s function is continuous on a global scale, but the numerical calculation is performed discretely. The Green’s function method is preferred to the spherical harmonic method for determining the impact of local environmental loading. However, as the degree approaches infinity, the Green’s function and spherical harmonic methods become mathematically equivalent [23].

2.2. Surface mass distribution products

There are three well-known types of environmental loads: ATML, HYDL, and NTOL [24]. Calculating the GNSS reference station displacement caused by these environmental loads necessitates surface mass distribution products, primarily atmospheric surface pressure (SP), terrestrial water storage (TWS), and OBP grids. SP and TWS products are typically obtained from reanalysis datasets, which employ data assimilation to integrate various surface and upper-air observations with numerical model results. This process yields gridded, continuously long-time span, and post-processed historical information datasets with high spatiotemporal resolution [25]; consequently, this process has been widely applied in fields such as climate change, marine science, and hydrology [26], [27], [28], [29], [30].

The global surface mass products used for ATML primarily include datasets from the National Center for Environmental Protection (NCEP), Modern-Era Retrospective Analysis for Research and Applications (MERRA), Global Earth Observing System Forward Processing Instrumental Team (GEOSFPIT) and European Centre for Medium-Range Weather Forecasts (ECMWF), providing SP grids. The NCEP and MERRA have two versions, namely NCEP-R-1, NCEP-R-2, MERRA-Land, and MERRA-2. The most commonly used ECMWF products include the ECMWF re-analysis Interim (ERA-Interim) and fifth generation of ECMWF atmospheric reanalysis of global climate data (ERA5). Among these, ERA5 is the most accurate, benefiting from a combination of hybrid four-dimensional variational assimilation (4D-Var) and ensemble data assimilation (EDA) techniques [31]. This may be because more historical observational data have been assimilated into ERA5, improved surface parameterization and cloud precipitation schemes, as well as the revised schemes of ice/snow albedo and roughness parameterization have also significantly enhanced simulation effects [32].

The global surface mass distribution products used for HYDL primarily include TWS such as soil moisture (SM) and snow data provided by the NCEP, MERRA, ECMWF, Global Land Data Assimilation System (GLDAS), Famine Early Warning System Network Land Data Assimilation System (FLDAS), WaterGAP Global Hydrology Model (WGHM), and Land Surface Discharge Model (LSDM) [33]. Regional products include data from the North American Land Data Assimilation System (NLDAS), National Climate Assessment-Land Data Assimilation System (NCA-LDAS), FLDAS-Central Asia, FLDAS-Forecast, and China Land Data Assimilation System (CLDAS). Among these, NLDAS currently operates in near real-time (∼4 days lag) on a 0.125° grid with an hourly time step over Central North America [34]. The NCA-LDAS product covers the contiguous United States and parts of Canada and Mexico, with the same spatial resolution as NLDAS, and the temporal resolution is one day [35]. FLDAS-Central Asia provides near real-time products covering Central Asia (with a time delay of approximately one day) at a spatial resolution of 0.01° [36], while FLDAS-Forecast covers Africa and the Middle East with a spatial resolution of 0.25° and forecasting duration of five months [37], [38]. CLDAS is a near real-time product developed by the China Meteorological Administration (CMA), featuring a time lag of two days [39]. It covers the Asian region (0–65°N, 60°E–160°E), with temporal and spatial resolutions of one hour and 0.0625°, respectively. Except for WGHM, which provides groundwater data, all other products include only TWS in the surface components. Moreover, apart from WGHM and LSDM, which include SM, snow, rivers, and lakes [40], other products offer only specific components of TWS, such as SM, snow depth (SD), or snow water equivalent (SWE). Hence, the current HYDL displacement modeling for GNSS reference stations typically utilizes these two hydrological variables.

The OBP data used for NTOL are obtained from the Estimating the Circulation and Climate of the Ocean (ECCO) developed by the National Oceanographic Partnership Program (NOPP) in the United States. ECCO is based on the global circulation model of Massachusetts Institute of Technology (MIT). The latest version is ECCO-V4r4; however, the product coverage period is limited to 1992–2017. Therefore, the near real-time OBP products provided by ECCO-Kalman Filter/Smoother (ECCO-KFS) and ECCO2 are typically used to correct NTOL in the GNSS coordinate time series. Compared with the traditional Kalman filter solution applied by ECCO-KFS, ECCO2 has a higher spatial resolution (0.25° × 0.25°), and the data assimilation process considers the influence of sea ice, making it theoretically more accurate and reliable.

In addition to ECCO, the Ocean Model for Circulation and Tides (OMCT), Max Planck Institute Ocean Model (MPIOM06), and Global Ocean Reanalyses and Simulations model (GLORYS2v3) also provide OBP data. The OMCT model released by the German Research Centre for Geosciences (GFZ) covers the entire globe with a spatial resolution of 1.875° × 1.875° and a temporal resolution of six hours. The model used by OMCT to estimate OBP differs from that of ECCO as it is driven by GRACE data [41], whereas ECCO uses an iterative optimization algorithm. The weak correlation between OMCT and ECCO over a broad oceanic region may be because of the different submarine Digital Elevation Model (DEM) used [42]. Except for global OBP, the Proudman Oceanographic Laboratory in the United Kingdom has also developed a Proudman Oceanographic Laboratory Storm Surge Model (POLSSM) for the European region, predicting hourly sea-level heights by obtaining sea surface wind pressure and atmospheric pressure from the UK Met Office. The POLSSM model covers marine areas from 48°N to 63°N and 12°W to 13°W, featuring spatial and temporal resolutions of 0.11° × 0.16° and one hour, respectively [43].

In addition, since 2020, the Gravity Information Service (GravIS) provided by GFZ has released two sets of Level-3 products based on the latest GRACE and GRACE-Follow On (GRACE-FO) data, including TWS and OBP [44], [45]: GFZ GravIS RL06 and combination service for time-variable gravity field solutions (COST-G) GravIS RL01. The time span covers the period from 2002 to 2023. The latest global TWS dataset, the global land water storage data set release 2.0 (GLWS: 2.0; excluding Greenland and the Arctic region), is based on the integrated Kalman filter method, integrating surface mass changes obtained from GRACE/GRACE-FO and WGHM. This dataset considers the uncertainties in data and models. However, it covers only the period from 2003 to 2019 [46].

Table 1 lists details of the surface mass distribution products. Owing to discrepancies in data sources, data assimilation systems, and methods used in various reanalysis products, these products exhibit different spatial and temporal distribution characteristics [47]. Numerous studies have assessed the applicability of surface mass distribution products in different regions [7], [28], [47], [48], [49]. These studies revealed significant discrepancies among hydrological models from different institutions and even different versions from the same institution. Consequently, the applicability of the environmental loading displacement corrections for the nonlinear motion of GNSS reference stations varies across regions [6], [7], [15], [29]. In addition, the simulation accuracy of some products needs improvement. For example, the hydrological models derived from reanalysis data are typically based on the upscaling of basin-scale hydrological models. However, consideration of runoff generation and confluence mechanisms in different climates, ecosystems, and topographical regions is limited, leading to uncertainties in model simulation results [50]. Besides, owing to geographical constraints, the paucity of surface observation sites at high latitudes, particularly in polar regions, and the scarcity of valid surface observation data lead to spatial instability when using observation data for quality assessment of numerical analysis products [51], [52]. Therefore, comparing existing reanalysis data with different temporal and spatial resolutions and selecting appropriate surface mass distribution products, particularly hydrological models, for environmental load modeling, are crucial for providing numerical references to analyze inconsistencies between environmental load prediction results and GNSS coordinate time series. This can contribute to the development of a more accurate environmental loading displacement model for global and regional GNSS reference stations.

2.3. Environmental loading products and services

In addition to using surface mass distribution products and GRACE time-varying gravity field products to calculate the environmental loading displacement of GNSS reference stations based on the Green’s function approach or the spherical harmonic function method, several well-known organizations provide environmental loading products directly, as summarized in Table 2. The Global Geophysical Fluids Center (GGFC) provides a global grid of nontidal atmospheric, oceanic, and HYDL displacements under the Center of Earth (CE), Center of Figure (CF), and Center of Mass (CM) reference frames, with the Preliminary Reference Earth Model (PREM) as the elastic Earth model. Users can also obtain environmental loading displacement time series for selected GNSS stations by entering their latitudes and longitudes. Input data for HYDL are SM and SWE from the National Aeronautics and Space Administration (NASA) GLDAS dataset. The time span for HYDL product ranges from 1948 to 2012, with a time resolution of one month. The ATML product covers the period from 1948 to 2015, with a time resolution of six hours. The input data for ATML is the NCEP SP. The input data for NTOL are derived from ECCO’s OBP dataset, with a time span and resolution of 1993–2014 and 12 hours, respectively.

GFZ released ATML, NTOL, and HYDL daily products considering lakes and rivers from 1976 to the present with a spatial resolution of 0.5° and loading forecast products with a forecast duration of six days under the CF and CM reference frames [53]. Input data for HYDL product are obtained from the LSDM, while ATML and NTOL use SP data from ECMWF and OBP from MPIOM06, respectively.

EOST at the University of Strasbourg published a global grid of environmental loading displacements and loading displacement time series for ITRF sites under the CF and CM reference frames [54], offering a wider range of products than GGFC and GFZ. TWS data applied for HYDL are obtained from GLDAS, ECMWF, MERRA, and GRACE global mascon solutions. SP data used for ATML are obtained from ECMWF and MERRA-2, while the input data for NTOL product is OBP from ECCO, including ECCO (KF080i) and ECCO2.

The International Mass Loading Service (IMLS) also provides global ATML, NTOL, and HYDL displacement grid products based on various reanalysis data sources and loading displacement time series for specific ITRF sites. ATML and HYDL products are obtained from the MERRA-2 and GEOSFPIT redistribution products, while NTOL is modeled based on OBP provided by the MPIOM06. The environmental loading products released by the above four services (GGFC, GFZ, EOST, and IMLS) are obtained based on the Green’s function approach.

2.4. Environmental loading correction method for GNSS coordinate time series

Data preprocessing is necessary before performing environmental load correction on GNSS coordinate time series. First, the environmental loading displacement time series for a given GNSS station is obtained by interpolating the global load grids provided by selected institutions or directly extracting the loading time series of the station if it belongs to an ITRF site. For a more rigorous approach, users can also calculate the environmental loading time series of the station themselves, following the method described in Section 2.1. Second, the three types of loading displacements (ATML, NTOL, and HYDL) are interpolated or averaged separately for the epochs of all selected GNSS coordinate time series. Third, linear trends from the GNSS coordinate time series should be removed. Users should pay careful attention to the presence of any offset or earthquakes during the time span, as these can significantly affect the accuracy of the obtained linear trend. Subsequently, the predicted environmental loading displacement can be subtracted from the remaining nonlinear GNSS coordinate time series, resulting in a load-corrected reference station displacement. Note that for the HYDL time series obtained from GLDAS, an apparent linear trend should be removed before performing environmental load corrections because it is unclear whether this linear trend represents a true hydrological signal [15]. For other HYDL products, the linear trend is relatively small, and users should evaluate whether to remove this trend before implementing HYDL correction on the GNSS coordinate time series.

After obtaining the load-corrected station displacement, weighted root mean square (WRMS) reduction, correlation analysis, and noise analysis can be used as metrics to assess the quality of different environmental loading models and their impact on GNSS coordinate time series through inter-comparison between various loading models and cross-comparison between loading models and GNSS coordinate time series [7], [15], [55], [56].

3. Recent progress on addressing environmental loading effects in GNSS coordinate time series

3.1. Impact of HYDL on nonlinear variations of GNSS coordinate time series

HYDL primarily refers to the solid Earth deformation caused by changes in TWS, including groundwater, soil moisture, and surface water (comprising liquid water, snow, and glacier ice) [24]. There are three primary methods for obtaining HYDL displacements for GNSS reference stations: GRACE-derived surface deformation in the spectral domain; convolving global or regional hydrological models, such as ERA5 and GLDAS, with Green’s function; and directly obtaining the HYDL displacements of the station through global grids or station lists released by different institutions [15], [57], [58].

Extensive research has confirmed a strong correlation between surface deformation caused by HYDL and the vertical displacements of GNSS reference stations [55], [59]. After correcting the global GNSS height time series using the GRACE-derived HYDL displacement, the average annual amplitude reduction rate of the stations reached 47% [59]. For regions with pronounced seasonal water variations, such as the Amazon Basin and the Himalayas, the HYDL-induced displacements derived from GRACE exhibited good consistency with the horizontal and vertical seasonal deformations of GNSS reference stations [60], [61]. However, owing to the spatial and temporal limitations of GRACE, the GRACE-derived HYDL displacement could reflect only the effects of large-scale monthly variations in TWS and fail to precisely capture local details. This makes it unsuitable for correcting nonlinear variations in daily or even higher-frequency coordinate time series of GNSS reference stations in small-scale regions. For example, the presence of nearby glaciers, snow, and other factors leads to significant discrepancies between the displacements of GNSS reference stations on the western coast of the Southern Amazon Basin and those derived from GRACE [62]. When applying the GRACE-derived TWS to small-scale regions such as China's Taiwan, the resulting leakage errors could significantly underestimate the water variation amplitudes [63].

Compared with GRACE, the spatiotemporal resolution of global and regional hydrological models has increased owing to the advancement of data assimilation techniques and the launch of numerous remote sensing satellites. Numerous hydrological models are widely used for modeling the HYDL effect on GNSS time series but have primarily focused on surface displacements caused by SM and SD [6], [13], [17], [53], [64], [65], [66], [67]. In China, HYDL predominantly affects the vertical components of GNSS reference stations, among which the displacement caused by SM is significantly larger than that caused by SD, exhibiting distinct latitudinal distribution characteristics [15]. In particular, considering water level changes in rivers and lakes significantly improves the consistency between vertical displacements of reference stations and HYDL displacement (0.86 vs 0.40) [17], [65]. Furthermore, the regional hydrological model NLDAS-2 demonstrated an improvement of over 10% in correcting the vertical nonlinear variations of GNSS reference stations in the mountains of the northwestern and southeastern parts of the NLDAS-2 Noah spatial coverage within the Plate Boundary Observatory network compared with the GLDAS model [68]. On a global scale, the HYDL displacements obtained from LSDM and GLDAS exhibited significant differences in regions such as the Rocky Mountains, Amazon Basin, La Plata River Basin, and Himalayan Plateau, with magnitudes of up to 16 mm along the Amazon River channel [53]. The MERRA-Land product performed best in reducing the weekly scatter of global GNSS reference station coordinate time series [6], whereas NCEP performed the worst. However, MERRA-2 demonstrated better performance in correcting the nonlinear motion of the global daily GNSS coordinate time series [66]. The latest ERA5, released in 2020, exhibited a significantly higher spatiotemporal resolution than other reanalysis products. Because the ERA5 hydrological model achieved the highest correlation of 0.9 with global positioning system (GPS)-derived integrated water vapor, it is expected to further enhance the performance of correcting nonlinear variations in global GNSS daily coordinate time series [67].

Overall, significant differences exist among the global and regional hydrological models provided by different institutions, resulting in varying contributions of HYDL to GNSS coordinate time series. Correction effects also vary across regions [6], [15], [17], [59], [66], [67]. Integrating GRACE-derived HYDL displacements from various institutions can effectively suppress or even eliminate noise, ensuring strong global applicability and providing users with more accurate hydrological load results across different scales [69]. When calculating HYDL displacements based on surface mass distribution products, only SM and SD are typically considered, neglecting influences such as groundwater and rivers. By contrast, the GRACE-derived HYDL displacements represent the contribution of the total TWS, including groundwater. Users should select appropriate methods and products based on the specific geographical location of the reference stations for HYDL modeling to study the nonlinear motion of GNSS reference stations.

3.2. Nonlinear displacement of GNSS reference stations caused by ATML

ATML primarily refers to the elastic response of the Earth caused by the redistribution of SP [70]. Currently, the ATML displacements of GNSS reference stations are typically obtained based on the load Green’s function approach, either by convolving SP grids with Green’s function (Section 2.1) or obtaining data directly from global grids or station lists released by different institutions (Section 2.3). ATML is a major factor contributing to the nonlinear motion of GNSS reference stations. Depending on the force source, ATML can be divided into two parts: Tidal ATML resulting from air temperature changes, primarily manifesting as oscillations at diurnal (S1), semi-diurnal (S2), and higher harmonic frequencies. The amplitude of the vertical surface deformation caused by tidal ATML is of the same magnitude as that of some of the tidal components in ocean tidal loading. Nontidal ATML (NTAL) resulting from extreme weather events, ocean–land redistribution, and changes in atmospheric moisture is the most significant contributor to ATML [7], [71], [72], with the potential to cause seasonal surface displacements of up to 24 mm (as observed at station ARTU, Russia). NTAL displacement is strongly correlated with global and regional GNSS coordinate time series [5], [7], [73]; however, the latest third global GNSS data reprocessing (repro3) campaign did not consider its impact. When applying ATML corrections to the GNSS coordinate time series, users should first confirm whether atmospheric tides are corrected during data processing. If atmospheric tides are considered through modeling, the calculated ATML should only be the nontidal component. Otherwise, both tidal and nontidal components of the ATML must be corrected.

Over the past century, scholars have increasingly studied the contribution of ATML to the nonlinear variations in GNSS coordinate time series [4], [7], [11], [29]. The seasonal vertical displacement caused by ATML exhibits remarkable regional characteristics, with significantly larger magnitudes in high-latitude regions than in low-latitude regions. The loading displacements of continental stations in the Southern Hemisphere are only 1/3 to 1/2 of those in the corresponding latitudinal regions of the Northern Hemisphere [5], [7], [11], [73], [74], [75]. After ATML corrections, 71% of global international GNSS service (IGS) stations exhibited scatter reduction in the GNSS height time series [76], and the annual amplitude was reduced by up to 1.6 mm [77]. Li et al. [7] evaluated the performance of global SP data with different temporal–spatial resolutions to correct nonlinear variations in the global GNSS height time series under the ITRF2014 frame. They found that the ERA-Interim product significantly outperformed NCEP-R-2 and MERRA in reducing global GNSS vertical scatter (90.3% > 89.1% > 86.4%). In addition, the ATML displacements obtained by different models significantly vary in coastal regions owing to the influence of the applied land–sea masks, inverted barometer (IB) effect, and atmospheric SP grids [7]. Moreover, significant discrepancies in SP were observed at higher elevation regions among different institutions, resulting in different ATML displacements [7], [29], [56]. ATML modeling methods that consider topography can better explain nonlinear variations in GNSS coordinate time series (74% vs 48% [12]).

On a regional scale, existing studies have primarily focused on the United States and China. Martens et al. [29] demonstrated that applying ATML corrections significantly reduced the scatter of GNSS time series in the neighboring regions of the United States and Alaska (5%–30%); however, the correction effects varied slightly for five selected GNSS data products, with an average difference of approximately 4%, confirming the correlation of GNSS data-processing strategies with ATML models. In China, the ATML-induced vertical crustal deformation in parts of East, North, and Northeast China was twice that in the Qinghai–Xizang Plateau region (∼20 mm vs ∼10 mm [72]), whereas the vertical scatter of the station coordinate time series from the Crustal Movement Observation Network of China (CMONOC) was reduced by up to 24% after ATML correction [17]. Moreover, differences in correcting the nonlinear variation in CMONOC height using ATML products from different organizations, such as IMLS, GFZ, and EOST, reached up to 15% [56]. The NASA-released product is more appropriate for environmental loading studies in China; however, most of the applied products currently come from the GFZ. Moreover, Hu et al. [78] found that the NTAL products published by the GFZ, EOST, and IMLS could not effectively correct seasonal variations in GPS height time series in the Yunnan region. Therefore, further detailed studies are necessary to evaluate the applicability of different ATML products in correcting the nonlinear motion of global and regional GNSS reference stations.

3.3. Spatiotemporal variation characteristics of NTOL displacement at GNSS reference stations

Crustal deformation caused by changes in OBP owing to mass redistribution within the ocean driven by atmospheric circulation is called NTOL [79], [80]. The NTOL modeling method is similar to that used for ATML. Specifically, the load Green’s function approach is applied to obtain NTOL displacement for GNSS reference stations, either by convolving OBP grids with Green’s function or directly obtaining data from publicly released NTOL products. NTOL-induced vertical displacements for global GNSS reference stations are correlated with their proximity to coastlines. In coastal regions (< 50 km), NTOL displacement is significantly larger than that at inland stations (> 500 km) [81], [82]. For example, the vertical crustal deformation near the island of Gotland in the Baltic Sea caused by a 1 m water layer was approximately 21 mm, whereas the vertical deformation near the west coast of Latvia reached 16 mm [83]. By contrast, the maximum vertical displacement at the inland stations resulting from NTOL was approximately 2 mm [80]. On a regional scale, the average vertical NTOL displacement of the CMONOC reference stations in China's mainland was approximately 2–3 mm, with a maximum of 3.5 mm in the Jiaodong and Liaodong Peninsula regions [10]. The scatter of the GNSS height time series in the southern part of the North Sea was reduced by 20%–30% after NTOL and ATML corrections [16]. In particular, the correlation coefficient between the NTOL displacement and subdaily (3 h) GNSS height for coastal stations in the North Sea in Europe was up to 0.7 [84], [85], while the vertical NTOL displacement caused by storm surge events reached up to 20 mm [86].

Wind, atmospheric circulation, and other environmental factors cause significant regional differences in NTOL displacements of near-coast GNSS stations [87]. For example, the maximum amplitude of vertical NTOL displacements at GNSS stations along the west coast of the Atlantic Ocean reached up to 13.1 mm, whereas that along the east coast was typically around 2.9 mm [82]. The vertical NTOL displacements at the GNSS stations near the Upernavik Isstrom, West Greenland, were less than 2 mm, while those along the coast of the southeastern Greenland Ice Sheet exceeded 10 mm [8], [88]. Discrepancies were observed in the NTOL displacements obtained from different OBP models. Those based on high spatiotemporal resolution regional products exhibited superior performance in correcting nonlinear variations of GNSS coordinate time series compared with global products. For example, the ECCO model is more suitable than OMCT for correcting global GNSS height time series. This can be attributed to the higher spatial resolution of ECCO and the assimilation of in-situ measurements into the ECCO model [12], [42]. Compared with the global ECCO model, the high spatial and temporal resolution POLSSM model improved the vertical scatter of GNSS stations in the North Sea region by approximately 11% [16]. A high-resolution regional model for Australia also captured high-frequency dynamic changes in the ocean that were not captured by the global model [86], [89], [90].

In summary, the contribution of NTOL to the nonlinear variations in GNSS coordinate time series was smaller than those of HYDL and ATML. However, NTOL-induced displacements at coastal GNSS stations can reach the millimeter or even centimeter level, with distinct spatiotemporal distribution characteristics. Currently, few types of OBP products are used to obtain NTOL solutions with lower spatial and temporal resolutions, and also with inconsistencies observed among products. In the future, it will be necessary to improve the spatiotemporal resolution of global and regional OBP models, conduct thorough quality assessments of various NTOL products, and establish a more accurate NTOL displacement model to study nonlinear variations in global and regional GNSS coordinate time series data. In addition, the IB effect commonly used in ATML modeling may couple with NTOL in coastal regions, resulting in poor corrections for nonlinear variations in coastal GNSS coordinate time series [54], [70]. The contributions of ATML and NTOL to the nonlinear motion of GNSS reference stations in coastal regions should be further evaluated.

3.4. Integrated effects of HYDL, ATML, and NTOL

Currently, the most recognized environmental loads include HYDL, NTAL, and NTOL. Ignoring environmental loading effects can bias the estimation of reference station velocities [91], [92]. Analyzing the temporal and spatial characteristics of reference station displacements caused by environmental loading based on surface mass distribution and environmental loading products released by various institutions can mitigate the nonlinear motion of GNSS reference stations, enhancing the accuracy of their position and velocity estimates. Dong et al. [13] confirmed that the three environmental loads could account for approximately 40% of the seasonal signal in global GPS height. Jiang et al. [15] compared the contributions of different environmental loading models to global IGS station coordinate time series. They identified that differences in hydrological models, grid interpolation, and topography corrections primarily caused discrepancies in environmental load-induced displacements at reference stations. Klos et al. [93] found that correcting environmental loads using EOST products could reduce the scatter of coordinate time series for most IGS reference stations worldwide by at least 25% and significantly decrease station velocity uncertainties. Comparing the contributions of 18 environmental load products provided by the EOST, GFZ, and IMLS to the nonlinear variations of global GNSS coordinate time series, He et al. [66] further confirmed the existence of discrepancies among different models provided by the same institution and even the same model provided by different institutions. After loading correction using an optimal combination model, namely ECMWF_IB + MERRA-2 + ECCO, 84.6% of the reference stations exhibited a seasonal amplitude decrease in the vertical components. The latest results also indicate that the three environmental loads could explain approximately 43% of the annual variations in the height time series of 900 IGS reference stations obtained from the third global GPS reprocessing campaign [57]. Considering the environmental loading effects could further contribute to the geodetic reference frame. In particular, researchers have found that correcting the HYDL effect could significantly reduce the annual signal in the scale obtained from the very long baseline interferometry (VLBI) technique during the generation of the global reference frame, International Terrestrial Reference System (ITRS) realization of Deutsches Geodätisches Forschungsinstitut der Technischen Universität München (DGFI-TUM) released in August 2016 (DTRF2014). The annual signal at the origin of DTRF2014 disappeared after considering the NTAL and HYDL effects, while the annual signal at the scale obtained from VLBI and satellite laser ranging (SLR) also decreased significantly [94].

Studies on the contribution of environmental loads to regional GNSS coordinate time series primarily focused on China's mainland. Wang et al. [10] found that the vertical annual amplitude could be reduced by approximately 37% after applying three environmental loading corrections. Discrepancies between the vertical component of the loading displacement obtained from the GFZ and EOST for the Finnish reference station in Antarctica reached 3 mm [95], while the IMLS product from NASA performed the best in correcting the nonlinear variation of the CMONOC height time series within China's mainland. In particular, the performance difference among the environmental loading products provided by the GFZ, EOST, and IMLS was up to 20% [56]. Hu et al. [78] thoroughly analyzed the impact of environmental loading on the height time series of 27 GPS reference stations in Yunnan. They confirmed that HYDL primarily caused the seasonal motion of most stations in Yunnan, while implementing NTOL and NTAL corrections increased the nonlinear variations of the station coordinate time series. Moreover, the latest results indicate that the annual amplitudes of 260 CMONOC stations and 210 national GNSS reference stations from the China Modern Geodetic Datum Infrastructure Construction (CMGDIC) Project in China's mainland decreased by approximately 50.3% after correcting the three environmental loading effects, with the largest scatter reduction of 2.7 mm [96]. This highlights the strong correlation between environmental loads and the regional GNSS reference station coordinate time series.

In summary, the three well-known environmental loads could account for approximately 50% of the vertical annual amplitudes of GNSS reference stations under ideal conditions, and their contribution to the horizontal component was less than 20% [97]. Differences in input data and assimilation methods used in the surface mass products provided by various institutions cause inconsistencies among different environmental loading products, with uneven contributions to the nonlinear variations in global and regional GNSS coordinate time series. Integrating various surface mass distribution products and environmental factors (including rainfall, topography, climate, and geographical location) to establish an optimal global and regional environmental loading model remains crucial for understanding the geophysical processes causing the complex nonlinear motion of reference stations and providing more accurate data for Earth science studies, such as earthquake prediction, sea-level changes, ice–snow mass variations, and extreme weather monitoring.

4. Summary and Outlook

Previous research has confirmed a strong correlation between vertical nonlinear motions of GNSS reference stations and surface displacements induced by environmental loads, including ATML, HYDL, and NTOL. However, significant differences remain, among which only part of the annual and semiannual amplitudes of the vertical components of the stations can be explained. Moreover, the correlation between environmental load-induced displacements and horizontal components remains relatively weak. This paper summarizes recent advancements in applying environmental loads for correcting nonlinear variations in global and regional GNSS coordinate time series, identifies current challenges in this field, and proposes future research directions, focusing on the following four aspects:

(1) Nonrigorous environmental load modeling method currently used. Publicly available environmental load-induced displacements are established based on an elastic Earth model and a set of global mean Green’s functions [57], failing to accurately reflect actual scenarios. Considering the influence of mantle viscoelasticity, Earth exhibits viscoelastic characteristics [61], [98]. Recent studies have shown that incorporating the viscoelastic effects of the asthenosphere (upper mantle) could significantly improve the consistency between the GPS-observed ocean tidal loading displacement at the semi-diurnal (M2) frequency and the model prediction value by approximately 31% on the Ryukyu Islands and in the western coastal area of Kyushu, compared with the results obtained from the regional NAO.99Jb ocean tide model and purely elastic PREM Green’s function [99]. These studies have confirmed the importance of considering the asthenospheric anelasticity effects of the Earth. In addition, a systematic phase difference was observed between the displacements at the CMONOC reference stations in China and the predicted environmental loading displacements [17]. Therefore, using the viscoelastic Earth model to establish environmental loading models can help mitigate differences between observed and model-predicted values. On the other hand, the global mean Green’s function is derived from the Earth’s average structural response (e.g., the PREM Earth model). However, for surface loads with a horizontal scale smaller than 2500 km², such as strong local hydrological signals related to intense precipitation and river flooding, the Earth’s response is highly sensitive to uneven crustal structures [100]. This sensitivity hinders the application of the global mean Green’s function to obtain realistic local environmental loading displacements [101]. Further investigation is necessary to determine the best Earth model for environmental loading studies. In particular, considering regional geographical characteristics to establish local Green’s function could contribute to the development of high-precision environmental loading models. This can further enhance the consistency between environmental loading predictions and the nonlinear displacements of GNSS reference stations.

(2) Lack of surface mass distribution data with high spatiotemporal resolution and reliability. Existing global surface mass distribution products provide maximum temporal and spatial resolutions of 1 h and 0.1°, respectively, making it challenging to identify local environmental change signals. The spatial resolution of regional models reaches as fine as 0.01°. However, the available product types are limited, primarily covering specific regions such as North America, Asia, the North Sea in Europe, and Australia. In addition, numerous sudden natural and manmade events at different scales can cause mass changes. However, owing to limited in-situ data, mass distribution models may fail to capture this type of mass change. Moreover, inconsistencies exist among surface mass distribution products provided by different institutions. In particular, significant differences were observed among the hydrological models. These products also lack precision information; thus, users must assess their reliability and applicability to global and regional studies based on external data sources. For example, the SP difference in Mount Roraima (South America) could exceed 360 mbar (1 mbar = 100 Pa) between MERRA-2 and ERA-Interim [7]. The GLDAS and FLDAS products do not provide hydrological data for the Antarctic region, thus researchers typically use ERA5 products for geophysical applications in the polar regions. In addition, the GLDAS products exhibit anomalies in SWE data for Greenland and the Arctic region, requiring data exclusion from these areas when calculating environmental loading displacements. Performing quality evaluations for various mass distribution products and integrating regional and global products to establish new surface mass distribution models with high spatiotemporal resolution and reliability could more accurately determine environmental loading effects on nonlinear variations of GNSS coordinate time series. For example, Section 3.1 illustrated that the regional hydrological model NLDAS-2 was superior to the global GLDAS model within the NLDAS-2 spatial coverage. In this case, the HYDL displacements calculated from NLDAS-2 were incomplete because the HYDL outside the area was neglected. By substituting the GLDAS product within the NLDAS-2 Noah spatial coverage with the NLDAS-2 product, a new integrated product was generated to represent the complete global HYDL impact. Owing to the advantages of NLDAS-2, the new integrated product outperformed the NLDAS-2 and GLDAS models. A similar approach can be used to model other types of environmental loading effects. The same strategies as integrating GRACE-derived HYDL displacements from various institutions (Section 3.1) can also be extended to integrate surface mass distribution products to improve spatiotemporal resolution and reliability.

(3) Neglect of other environmental loading factors and geophysical effects. In recent years, significant changes have been observed in cryospheric elements such as glaciers, ice sheets, and permafrost in regions such as the Arctic, Antarctic, and Qinghai–Xizang Plateau owing to global warming [102], [103]. Mass changes in cryospheric elements, including glacier retreat [104], [105], intensified melting, and ice sheet disintegration [106], result in ice loads, also classified as environmental loads [107]. However, existing studies primarily focused on short-term (monthly or 1–2 years) vertical nonlinear variations in GNSS reference stations caused by glacier mass changes in Greenland [74], [88], [108], [109], [110], [111]. Research on the contribution of ice-loading effects to GNSS coordinate time series for larger ice-covered areas, such as Antarctica and the Qinghai–Xizang Plateau, remains limited. Moreover, GNSS reference stations used for monitoring snow and ice mass changes are typically installed near relatively stable coastlines. Owing to glacier movement, reference station displacements caused by ice load are closely related to the proximity to the ice load center. Therefore, the current GNSS analysis is limited to qualitative assessments of linear velocity changes in reference stations caused by ice loads, with faster surface uplift rates observed at shorter distances from the ice load center [64], [108], [112]. Lastly, cryospheric regions (high mountains and polar areas) typically undergo prolonged winters with snow accumulation and warm summers with intense melting, making them ideal regions for investigating seasonal HYDL effects [64], [108]. Nevertheless, most existing environmental loading research has been conducted in plains and low- and mid-latitude regions [17], [58], [63]. Future studies should focus on integrating modern geodetic technologies, such as GNSS and GRACE, to conduct a refined inversion of glacier/ice sheet mass variations. This can provide a better understanding of ice-loading effects on the long-term linear and nonlinear motion of GNSS reference stations in ice-covered regions such as the Arctic, Antarctic, Greenland, and Qinghai–Xizang Plateau. In addition, hydrological load considering barystatic sea-level changes is essential for meeting global mass conservation requirements [113]. Therefore, further investigation is necessary to evaluate the effects of barystatic sea-level loading on the nonlinear displacement of reference stations near coastlines (< 100 km) or on islands. Moreover, artificial mass-change effects represent a significant component of environmental loads. For example, many artificial dams have been constructed to generate electricity and manage water supply, creating large HYDL signals in the surrounding areas.

Variations in surface temperature can also cause notable periodic movements at GNSS reference stations [13], [97], [114], [115], [116]. For instance, the thermal expansion effect of GNSS monuments (TEM) made of concrete or metal results in vertical annual amplitudes of up to 6.6 mm at reference stations [116]. Nonseasonal temperature variations may cause vertical displacements of approximately 3 mm in specific areas [9]. The presence of soil layers on the Earth’s surface can also introduce a phase lag in the thermal expansion effects [114]. These factors may interact with the three known types of environmental loads, reducing the correlation between GNSS coordinate time series and environmental loading-induced displacements [117]. However, current research does not focus on addressing the impact of nonseasonal temperature-induced horizontal displacements and other geophysical factors, such as changes in water levels and poroelastic strains in the lithosphere and soil layers, on the displacements of global GNSS reference stations. A comprehensive analysis of unmodeled environmental loading factors (particularly ice load and the artificial mass-change effect) and other geophysical factors contributing to nonlinear GNSS reference station displacements is a critical research area in geodesy. For instance, extending the three-dimensional bedrock thermal expansion model to a full-spectrum model could be valuable for evaluating the impact of nonseasonal temperatures on the horizontal displacement of reference stations [115], [117]. Superimposing surface displacements caused by different factors is essential for establishing a millimeter-level nonlinear motion model for reference stations with practical physical meaning. In addition, this is crucial for a more profound understanding of geophysical processes and mass dynamics on the Earth’s surface and within the Earth’s interior system.

(4) Coupling of GNSS technology-related systematic errors. Limitations in the natural environment, technological manufacturing, and data-processing levels result in unavoidable systematic technical errors in GNSS observations. For example, the draconitic signal caused by GPS satellite orbit errors manifests as a seasonal signal with a period of approximately 351 days in the reference station coordinate time series. Unmodeled or imperfectly modeled diurnal and semi-diurnal atmospheric or oceanic tides can accumulate as long-period signals, and different constraints on the fixed or relaxed reference station coordinates and satellite orbits can lead to systematic errors in GNSS coordinate time series [1], [118], [119]. These spurious periodic signals can couple with real environmental loading signals, causing incorrect interpretations of nonlinear motions at reference stations. For instance, an insufficiently precise GNSS data-processing strategy has led to significant discrepancies between GNSS observations and HYDL displacements inverted from GRACE in Europe [120]. Neglecting higher-order ionospheric delays caused spurious annual and semiannual signals in the north–south component of the GNSS coordinate time series [121], [122]. The amplitudes of multidraconitic signals were reduced after implementing minor ocean tide corrections [123]. In October 2019, the IGS Analysis Centers initiated repro3 based on the latest version of the International Earth Rotation Service (IERS) Convention to establish the new ITRF 2020 [124]. Compared with the second global GNSS data reprocessing (repro2), which contributed to ITRF2014, significant improvements have been made in GNSS data-processing methods and models, such as the introduction of a new mean pole tide model, solar radiation pressure model, and the Finite Element Solution 2014b (FES2014b) ocean tide model [125]. Analyzing the spatiotemporal characteristics of spurious surface displacement caused by various GNSS-related systematic errors based on repro3 and determining an optimal data-processing model and strategy for consistently reprocessing the global IGS reference station data is crucial for reducing the impact of technically related systematic errors on real environmental load signals. For example, although the ocean tide model was updated to FES2014b, it still includes only 11 primary components. The FES2014b model can be modified by considering the impacts of other minor ocean tides, which have been confirmed to partially reduce draconitic harmonics [123]. This can help avoid the overinterpretation of seasonal signals in GNSS coordinate time series [112], [125], enhancing its applicability in global change studies, including sea-level variations, snow and ice mass changes, and extreme weather monitoring.

CRediT authorship contribution statement

Zhao Li: Writing – original draft, Resources, Methodology, Investigation. Weiping Jiang: Writing – review & editing, Validation, Supervision, Methodology, Funding acquisition, Conceptualization. Tonie van Dam: Writing – review & editing, Validation, Conceptualization. Xiaowei Zou: Writing – original draft, Investigation. Qusen Chen: Writing – review & editing, Investigation. Hua Chen: Writing – review & editing, Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We thank the National Meteorological Information Center of CMA for providing the Chinese Meteorological Administration Reanalysis (CMA-RA) product, which serves as a valuable tool for analyzing historical atmospheric conditions. We express our gratitude to the National Earth System Science Data Center (NESSDC) for data resources and model analysis services. We are also indebted to NCEP, MERRA, ECMWF, GLDAS, ECCO, GGFC, GFZ, EOST, and IMLS for their collective efforts in maintaining and advancing global Earth science and geodetic research. This work was supported by the Basic Science Center Project of the National Natural Science Foundation of China (42388102), the National Natural Science Foundation of China (42174030), the Special Fund of Hubei Luojia Laboratory (220100020), the Major Science and Technology Program for Hubei Province (2022AAA002), and the Fundamental Research Funds for the Central Universities of China (2042022dx0001 and 2042023kfyq01).

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