Volume 128, Issue 9 e2023JB026988
Research Article
Open Access

Introducing the Idea of Classifying Sets of Permanent GNSS Stations as Benchmarks for Hydrogeodesy

A. Klos

Corresponding Author

A. Klos

Faculty of Civil Engineering and Geodesy, Military University of Technology, Warsaw, Poland

Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany

Correspondence to:

A. Klos,

[email protected]

Contribution: Conceptualization, Methodology, Software, Validation, Formal analysis, ​Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration, Funding acquisition

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J. Kusche

J. Kusche

Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany

Contribution: Conceptualization, Methodology, Writing - original draft, Writing - review & editing, Supervision

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G. Leszczuk

G. Leszczuk

Faculty of Civil Engineering and Geodesy, Military University of Technology, Warsaw, Poland

Contribution: Software, Formal analysis, Data curation, Writing - original draft

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H. Gerdener

H. Gerdener

Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany

Contribution: Software, Formal analysis, Data curation

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K. Schulze

K. Schulze

Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany

Contribution: Software, Formal analysis, Data curation

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A. Lenczuk

A. Lenczuk

Faculty of Civil Engineering and Geodesy, Military University of Technology, Warsaw, Poland

Contribution: Software, Validation, Formal analysis, Data curation, Writing - review & editing

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J. Bogusz

J. Bogusz

Faculty of Civil Engineering and Geodesy, Military University of Technology, Warsaw, Poland

Contribution: Conceptualization, Writing - review & editing, Supervision

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First published: 15 September 2023


We propose a novel approach to classify sets of Global Navigation Satellite System (GNSS) permanent stations as benchmarks for hydrogeodesy. Benchmarks are trusted sets of GNSS stations whose displacements are classified as significantly and positively correlated with hydrospheric changes and identified in a three temporal-scales: short-term, seasonal and long-term. We use 63 vertical displacement time series processed at the Nevada Geodetic Laboratory for the period 1998–2021 from stations located within Amazon basin and show that estimates of trends and annual signals, including the annual phase maximum, are very coherent with water surface levels provided by altimetry missions. We compute vertical displacements from Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On gravity missions and predict those also from Global Land Water Storage (GLWS) v2.0 data set which values are produced by assimilation of GRACE into WaterGAP Global Hydrological Model (WGHM). We divide vertical displacements from the three data sets into the pre-defined temporal-scales of short-term, seasonal and long-term, using non-parametric wavelet analysis. For each temporal-scale, correlation coefficients are computed between GNSS-measured and GRACE-derived/GLWS-predicted displacements. We present the benefits of applying high-resolution GRACE-assimilating hydrology model to benchmark GNSS stations, which are particularly evident when using spherical harmonic coefficients higher than 120. Their increase causes the number of stations included in the benchmarks to rise by up to 15% for short-term. Benchmarking allows hydrogeodesy to take advantage of a broader set of GNSS stations that were previously omitted, such as earthquake-affected sites and those where a possible poroelastic response is observed.

Key Points

  • Displacements measured by Global Positioning System (GPS) correlate well with surface water levels derived from radar altimetry missions

  • Trusted sets of GPS stations are classified at the three pre-defined temporal-scales: short-term, seasonal and long-term

  • Benchmarking allows to include more GPS stations in hydrogeodetic analyses

Plain Language Summary

Displacements of the Earth's crust measured by permanent Global Navigation Satellite System (GNSS) ground stations are used for many geophysical interpretations. However, it is common to omit the evaluation of the sensitivity of the system to the measurement of displacements from different sources, assuming in advance 100% sensitivity of the system to a given effect. Consequently, the fact that at a given station several effects can be recorded simultaneously is overlooked. This is particularly evident in earthquake-affected areas, where GNSS stations are excluded from most analyses of non-tectonic effects. We solve this problem and propose to divide GNSS stations into trusted sets, which we call benchmarks. Benchmarking is performed by indicating the stations that are certain to register a given effect in three pre-defined temporal-scales: short-term, seasonal and long-term. We present the analysis for the Amazon area, known for its large hydrosphere-related signal, and demonstrate that the benchmarking allows for the inclusion of GNSS stations that were previously omitted in analyses of this type.

1 Introduction

Global water cycle induces changes in water storage on temporal-scales from days to decades with spatial scales of a few to 10,000 km. Its monitoring in recent years has been successfully carried out using geodetic techniques, creating a brand new field of research called hydrogeodesy. Hydrogeodesy is a rapidly growing field of geodesy that aims to quantify water resources on, above and below the Earth's surface and monitor their movement using geodetic techniques. Of all the geodetic techniques, the Gravity Recovery and Climate Experiment (GRACE, Tapley et al., 2004) mission and its Follow-On mission have proven to be the most effective in monitoring global and basin-wide changes in total water storage (TWS) (Eicker et al., 2016; Humphrey et al., 2023). However, GRACE's inability to determine water storage variables with a spatial scale of less than 300 km has led to the need to search for alternatives.

The Global Navigation Satellite System (GNSS) has found its place in hydrogeodesy (White et al., 2022). Point-wise GNSS-observed displacements from a dense networks of permanent stations are used to accurately estimate changes in water storage with spatial scale competitive with GRACE (e.g., Argus et al., 2017; Knappe et al., 2019). However, because the system was originally dedicated to navigation purposes, changes in water storage may be masked by other effects to which GNSS is also sensitive. This is particularly apparent for regions where water storage is not changing significantly or in those where other processes, such as tectonics, dominate. A perfect example of this is the Pacific region or any other earthquake-prone area, for which we observe a characteristic nonlinearity in the form of a logarithmic and/or exponential function present in displacement time series for many GNSS stations (e.g., Altamimi et al., 2016; Klos et al., 2019; Tobita, 2016). Nonlinearity indicates that an earthquake has occurred at that location and will mask other effects, including water storage variations. The second example is the area of Europe, where a few thousand GNSS stations are available (Blewitt et al., 2018), and for which the displacement time series are dominated by the Glacial Isostatic Adjustment (GIA) effect, the non-tidal atmospheric effect and GNSS-related systematic errors and signals (Kierulf et al., 2021; Kreemer & Blewitt, 2021). In Eastern Europe, water storage variations are clear and present in GNSS-observed displacements (e.g., Liu et al., 2020; Springer et al., 2019), however, few GNSS stations with sufficient quality are available. Elsewhere in Europe, water storage variations are small in comparison to other signals and errors specific to GNSS (Klos et al., 2020; Springer et al., 2019).

As a result of the aforementioned limitations, certain trends have developed in hydrogeodesy regarding the use of GNSS displacements (White et al., 2022). First, it is conventional to identify and remove stations whose displacements differ from nearby stations (Johnson et al., 2021; Lau et al., 2020). While in theory this leads to a more reliable investigation and prediction of water storage variations, it can result in a worsening of the spatial scale of the estimated changes (Argus et al., 2017; Ferreira et al., 2019b; S.-Y. Wang et al., 2022). This procedure is implemented through several approaches (e.g., Argus et al., 2017; Gruszczynska et al., 2018; Hammond et al., 2021; Jiang et al., 2017), which unfortunately ignore the fact that a GNSS station can be sensitive to changes in water storage, even though specific temporal-scales are dominated by other effects (Klos et al., 2021; Memin et al., 2020). Second, the GNSS-observed displacements provided by the International GNSS Service (IGS, Beutler et al., 1999) within a so-called global reprocessing are considered to be the best. This is because IGS uses the latest models and recommendations for processing GNSS observations, and the processing itself is performed in a consistent manner using a network solution (e.g., Rebischung et al., 2016). The term network solution refers to a processing technique where data from multiple GNSS stations around the world are combined and processed together. This approach allows for the estimation of precise positions and other parameters by taking advantage of the spatial distribution of the stations. However, it is not the best option in the context of Earth system monitoring based on GNSS data. Recent analyses have shown that GNSS station displacement time series provided by IGS have the smallest standard deviation, are spatially consistent, and have good correlation with environmental loading models, although contributions from individual analysis centers differ (Niu et al., 2022). The above makes the IGS solution considered the “gold standard.” Unfortunately, the number of GNSS stations processed by the IGS does not provide a sufficiently dense distribution of stations needed for local or regional analysis. Take, for example, the Amazon basin, which encompasses nearly 7,000,000 km2, for which IGS processes observations from only 10 stations (Figure 1). This is far too few for regional analysis. The second major drawback of using the IGS solution is that the displacement time series are released only every few years, along with a need for another International Terrestrial Reference Frame (ITRF, e.g., Altamimi et al., 2023). The previous IGS solution (repro-2, Rebischung et al., 2016) went back to 2014, while the current one (repro-3) goes back to 2020, and has been only recently extended to November 2022. This significantly limits analyses of present-day variations in water storage. The above arguments support the use of solutions that make the time series of displacements available on an ongoing basis and institutions which process observations from a larger number of permanent stations. Such a solution is the one being provided by the Nevada Geodetic Laboratory (NGL, Blewitt et al., 2018), whose suitability for hydrospheric analysis has already been proven (e.g., Klos et al., 2021; Knowles et al., 2020; Lenczuk et al., 2023). NGL solution would be also recommended since there is no dependence on the network and thereby not impacted by the trajectory of the reference stations. We point out here that we are not thereby depreciating the IGS solution. We admit that it is of extraordinary importance and is an indispensable solution for the realization of the International Terrestrial Reference System.

Details are in the caption following the image

Amazon basin with Global Navigation Satellite System stations available from different institutions and earthquakes epicenters with magnitudes greater than 6 which occurred from 1995 within or close to the Amazon basin (marked as green background color).

We take all these aspects into careful consideration and propose that GNSS stations are used in hydrogeodesy only after their sensitivity to changes in water storage is properly recognized. We argue that the GNSS can be simultaneously sensitive to changes in water storage and to other effects, even though the latter dominate the displacements on specific temporal-scales. We believe that classifying GNSS stations into trusted data sets, which we call “benchmarks,” will contribute to the wider use of GNSS in hydrogeodesy and may constitute important input for hydrological sciences. Benchmarks are identified for long-term, seasonal and short-term temporal-scales separately. We thus eliminate stations affected by non-hydrological effects just at a given temporal-scale, but still include them in analyses for other temporal-scales. We believe that such an approach may potentially improve the interpretation of work published over the past few years that has used GNSS to validate GRACE or hydrological models (e.g., Dill & Dobslaw, 2013; Doll et al., 2014; S.-Y. Wang et al., 2017; S. Wu et al., 2021; Yan et al., 2015 and many others). Many of these studies used the standard method of estimating correlation coefficients or estimated the reduction of root-mean-square values for original displacements. Dozens of papers have concluded that many GNSS stations do not capture the hydrosphere loading. This has recently changed, when the authors began to consider studying GNSS sensitivity at different temporal-scales (e.g., Klos et al., 2021; Memin et al., 2020).

To classify GNSS stations as benchmarks, we use a second release of a high-resolution GRACE-assimilating hydrological model produced at the University of Bonn (Gerdener et al., 2023) and named as Global Land Water Storage (GLWS2.0). The advantages of using assimilated models instead of GRACE or hydrological models standalone have been demonstrated, for example, in the work of Springer et al. (2019). We show that predictions of displacements computed with GLWS for GNSS locations are roughly similar to GRACE estimates, but we also note differences due to the model's ability to capture local effects; this is particularly pronounced when the degree and order (d/o) of spherical harmonics is increased from 96 to 180. Then, we demonstrate that the sensitivity of GNSS to water storage variations depends on the location but it is evident at all temporal-scales from 2 months upwards. Our analysis is presented for the Amazon basin, where displacements are driven by hydrosphere, but also tectonic and station-specific signals. Water storage variations within the Amazon basin are very frequently studied using GNSS-observed displacements and these are usually taken for analysis without initial verification of their sensitivity (e.g., Ferreira et al., 2019a; Knowles et al., 2020; Youm et al., 2023). We expect our idea to be effective not only in identifying benchmarks for hydrogeodesy, but also for studies of other environmental effects, climate change, meteorology, or other related disciplines where the use of GNSS observations finds its application or those, where GNSS is used for validation purposes. We also believe that it may increase the use of GNSS for assimilation into land surface models in the future (Yin et al., 2021). We leave these topics for further contributions. We use a uniquely large collection of GNSS displacements provided by the NGL with 63 stations situated within the confines of Amazon. We pre-process them and decompose the GNSS-observed displacements, GRACE-derived displacements and GLWS-predicted displacements into different temporal-scales, that is, long-term, seasonal and short-term, using a wavelet analysis, and we study the correspondence between the three data sets using correlation analysis. GNSS displacements correlated significantly and positively with GLWS-predicted displacements define benchmarks at three pre-defined temporal-scales.

Our paper is organized as follows. First, we introduce the reader to the data sets we use. Then we test their quality and consistency over the Amazon basin. We introduce the methodology and classify GNSS stations as benchmarks. Finally, we close the paper with a discussion of the results and summarize the study in the last section.

2 GNSS Displacements

We use the daily vertical displacement time series estimated from the XYZ coordinates as provided by the NGL (Blewitt et al., 2018) which processes only Global Positioning System (GPS) observations. GPS observations were processed using Precise Point Positioning method (PPP, Zumberge et al., 1997) in GipsyX software version 1.0. The Vienna Mapping Functions (VMF1) were used as an a priori model to evaluate the wet and dry parts of the troposphere. Zenith delays and gradients were estimated as a random-walk once every 5 min. The first order ionospheric effect was removed using a combination of carrier-phase (LC) and pseudo-range (PC), while the second order effect was modeled using IONEX data from IGRF12 (International Geomagnetic Reference Field). The Phase Center Variations (PCV) model relative to the phase center from igs14_www.atx was used. The satellite clocks were fixed to Jet Propulsion Laboratory clock products, which are given every 5 min relative to the reference clock. As for the geophysical effects models, solid Earth and pole tides were removed using the IERS Conventions (2010), keeping the permanent tide. Ocean tide loading was subtracted using FES2004. Corrections due to non-tidal environmental loading were not applied. Further details on GPS processing can be found on the NGL website provided below. We use the final daily XYZ coordinates in IGS14 frame available in txyz2 text format.

We select stations located in the Amazon basin and operated between 1998 and 2021. The Amazon basin is defined here according to the United Nations (UN) Global Compact initiative data sets. From a set of 80 GPS stations, we select those with the highest displacement quality, that is, the displacement time series is longer than 3 years, and the series itself has the least amount of missing displacements. Holding to the above guidelines, we use 63 GPS permanent stations whose length varies from 3 to 22 years, with 70% of them between 5 and 15 years (Figure 1, Table S1 in Supporting Information S1). For this set of displacement time series, outliers and jumps are removed using, respectively, three times the interquartile range rule and a database of jumps provided by the NGL. The latter is supported by manual inspection of time series. From the displacement time series, the GIA effect is removed using the ICE-6G_C model (Stuhne & Peltier, 2015). Non-tidal atmospheric and non-tidal oceanic loading effects are eliminated by using displacement predictions provided by the Earth-System-Modeling Group of Deutsches GeoForschungsZentrum Section 1.3 (ESM GFZ; Dill & Dobslaw, 2013); both environmental effects, before they are eliminated, are averaged from 3-hr to daily data by the usual procedure of averaging all observations from 1 day. No interpolation of missing displacements is performed.

3 Quality of GPS Displacements

The Earth's crust can deform at different spatial- and temporal-scales. This is a consequence not only of effects acting directly, such as tectonics but also of those acting indirectly, such as hydrosphere which masses cause loading or unloading. Both types of deformation are recorded point-wise by placing the GNSS antennas at the physical surface of the Earth. It is believed that the most stable foundation is to place GNSS antennas on concrete pillars. However, it is not a very popular solution, that is, antennas mounted on concrete pillars represent only a small fraction of all GNSS system antennas. This is because it requires the establishment of a separate station with a full infrastructure. Another, cheaper option for stabilizing an antenna is to place it on the roof of a building. So far, it has been proven that the type of antenna foundation can affect the nature of the time series of displacements over short periods (Langbein & Svarc, 2019). However, it does not transfer to the ability to measure geophysical effects.

One can now imagine, given the multitude of effects causing crustal deformation how many complex behaviors can be observed by GNSS sensors and how many complex signals are therefore present in the displacement time series. Some displacement series may surprise especially the inexperienced, so their analysis requires a trained eye of the observer, but also some experience and knowledge. It is expected that on a long-term temporal-scale GNSS displacements should behave in a linear way as a result of tectonic plate movement and post-glacial rebound (e.g., Kierulf et al., 2021; Wallace et al., 2004). However, these changes are coupled with tectonic effects, volcanic activity, human-induced crustal subsidence, or climate-induced changes in the mass loading of the environmental effects, causing large jumps, or nonlinear signals including post-relaxation curves with magnitudes that vary over the years (e.g., Brown & Nicholls, 2015; K.-H. Chen et al., 2020; Jiao et al., 2023). On a seasonal temporal-scale, we observe a year-round curve, characterized by the time of maximum occurrence and amplitude. It is generated mainly by the varying environmental loading throughout the year, but can also be contributed by draconitic period, thermal expansion of ground and monuments, systematic errors or by spurious effects (e.g., Dong et al., 2002; Penna et al., 2007; Ray et al., 2008). On the short-term temporal-scale, periodic variations occur as the overtones of seasonal changes, but they are coupled with a plethora of effects due to tidal constituents unmodelled or mismodeled during the processing, GNSS-specific signals, such as imperfect modeling of orbits, unexpected movement of the GNSS monument, errors in clocks, mismodeling of the large-scale effects, changes in GNSS processing and errors in the assumptions or background models whose predictions are used during the processing; all these effects will show up in the time series of station displacement (Amiri-Simkooei et al., 2017; Bos et al., 2015; Dong et al., 2006; Gruszczynski et al., 2018; Langbein & Svarc, 2019; Matviichuk et al., 2020; Niu et al., 2022; Ray et al., 2008; Saji et al., 2020; White et al., 2022; X. Xu et al., 2017; P. Xu et al., 2019). Superposition of the above effects means that the standard deviation of the series may change over the years, and the individual displacements may be correlated with each other. These effects also induce correlations between displacement values recorded for GNSS stations located in close proximity to each other (Dong et al., 2006).

The GNSS community is trying to do its best to map the above changes as reliably as possible with a mathematical models. It is thus common to characterize the displacement time series as the sum of trajectory x(t) and stochastic r(t) models:
The trajectory model x(t) has evolved over the years, as nicely summarized by Bevis and Brown (2014). In the early years, displacement changes were described only by a linear trend, which was called the constant velocity model. Then, this model was adjusted to include jumps and called the constant velocity model with jumps. Later, for many years, the trajectory model x(t), which contains the sum of the linear trend and the annual component along with its harmonics, was considered conventional:

Term x0 represents here the first value of the displacements estimated at t1, v is the trend (or velocity), the number of n seasonal signatures are characterized by amplitudes A and phase shifts φ. Now we know that the actual displacement values differ significantly from the conventional trajectory model and more sophisticated mathematics must be applied. Bevis and Brown (2014) noted that the polynomial trend model is efficient enough to cover nonlinear displacement behavior and called it the standard linear trajectory model. Note, however, that polynomial trend model is different from the piecewise constant velocity model, which is used for providing reference frames and assumes changes in velocity over time described by a piecewise linear function (Altamimi et al., 2016). Finally, postseismic deformation should be accommodated to account for the earthquake-related displacements at several stations around the world and this model is named as the extended linear trajectory model. It adds one or more logarithmic and/or exponential transients to the standard linear trajectory model.

The stochastic model r(t) of the displacement time series is quantified from the residuals of the displacement time series, which in common terms in the GNSS community are simply the difference between the original displacement time series and the adopted trajectory model. It is referred to as “noise” and indicates the nature of the anomalous part of the time series, conditioned by the adopted deterministic model (Bogusz & Klos, 2016). There is a general consensus that a combination of white and power-law noises lends itself best to describe the stochastic nature of GNSS displacements (e.g., Bos et al., 2008; Santamaria-Gomez et al., 2011; Santamaria-Gomez & Ray, 2021). For this particular combination, a covariance matrix has a form (Williams, 2008):
where I is the unit matrix, J represents the covariance matrix of chosen power-law noise, which differs depending on the spectral index κ. Terms urn:x-wiley:21699313:media:jgrb56455:jgrb56455-math-0004 and urn:x-wiley:21699313:media:jgrb56455:jgrb56455-math-0005 represent the amplitudes of white and power-law noises, respectively. The parameters of the noise, namely the spectral index and amplitudes, are most often determined by the Maximum Likelihood Estimation algorithm (MLE; Langbein & Svarc, 2019) and used to reconstruct the covariance matrix and estimate the errors of the parameters of trajectory model (Williams et al., 2004). Since Equation 3 is a combination of two different power-law noises, the degree of dominance of the individual components needs to be estimated. This is implemented by reformulating Equation 3 and adding term ϕ, a so-called fraction of noise (Bos, 2021):
with σ representing standard deviation of residuals.

Specific spectral index values provide hints about the underlying processes to which a station is sensitive in the short-term and help define the quality of individual GNSS stations (Langbein & Svarc, 2019). A spectral index close to 0 represents “white noise” and means that the residuals are not temporally correlated; there is no information hidden within them. A spectral index close to −1 represents “flicker noise,” which suggests that the residuals are dominated by one or more large-scale physical phenomena which are not included in the deterministic model; or the culprit may be signals and errors specific for the GNSS technique (Dong et al., 2006). A spectral index close to −2 indicates that the residuals are of a “random-walk” nature, meaning that there is no correlation between increments. Random-walk is often thought as resulting from local phenomena such as instability of ground or monument or GPS multipath (Langbein & Svarc, 2019). It can generate a trend that is physically inseparable from the trend resulting from physical processes (Bos et al., 2008; Williams et al., 2004).

Now, it is important to realize that estimating parameters of trajectory and stochastic models reliably is only possible with long time series, with the phrase “long” in the case of GNSS displacement series not yet precisely defined. At the moment, the length of the time series is a serious source of uncertainty. The longest time series of displacements is about 25 years long, but it is possible to notice for them the changing precision of observations over time; the first observations are characterized by much larger errors than the last observations. This is, of course, due to the continuous improvement of GNSS equipment or background models. With an ever-increasing observation span, we notice a progressive change in the assumptions of the stochastic model. In the early years, a combination of white noise and flicker noise was considered the most suitable model, later on, random-walk noise began to be added to this combination as well. The noise was then found to be reliably described using the fractional spectral index, close to flicker noise, estimated by the maximum likelihood estimation. Recent studies indicate that for sufficiently long time series, the nature of the stochastic part is reliably described using the Gauss-Markov process (Santamaria-Gomez & Ray, 2021). The determination of the reliable stochastic character is of great importance, since it carries over to the determination of the errors in the parameters of the trajectory model (see Equations 26–32 in Bos et al., 2008), but it is impossible without sufficiently long time series (see Equations 2 and 3 in the flagship paper of Williams et al., 2004). Interpretation of the parameters of the stochastic model should therefore be carried out with some reserve and knowledge of the length of the time series.

We now use the simplest time series model, Equation 2, to characterize the daily GPS vertical displacements over the Amazon basin. Vertical displacements are characterized by a pronounced annual signal with an amplitude of up to 30 mm for the stations closest to the Amazon (Figure 2). The largest amplitude is 30.4 ± 1.2 mm and is found for station LMA1 situated 8.5 km from the river. Annual amplitudes decrease with distance from the main river. For stations located at the border of this catchment, annual amplitudes reach several millimeters, that is, from 3 to 10 mm. The smallest annual amplitudes are observed in the western part of this region, and range from 1 to 6 mm. The spatial pattern of annual amplitudes is very consistent. The amplitudes can be easily smoothed (interpolated) across the Amazon basin to give a consistent picture of the seasonal changes. We use here the Delaunay triangulation. The interpolated values of the amplitudes allow to identify measured seasonal amplitudes that differ from the others. An excellent example of this is the LCOB station, for which the difference between the observed and interpolated amplitudes is 30%. If we remove this amplitude and interpolate again, the error is 50%. The phases of the annual sine curve measured by the GPS give a very clear, spatially consistent picture. The maximum of this curve occurs around October for stations located in the southern part of the Amazon basin, and then shifts until December as it approaches the main river. The exception are stations located in the western part of the catchment, for which the maxima of the annual curve completely disagree for stations located next to each other. However, it is worth noting that the annual amplitudes for these stations have small values, varying within 1 mm, which means they can be difficult to be identified among other signals. Besides, the phase inconsistency may also be influenced by imperfect modeling of atmospheric and oceanic effects at the land-ocean boundary. The values of measured semi-annual amplitudes are several times smaller than annual amplitudes with an average of 2 mm.

Details are in the caption following the image

Trends (a), standard deviation of long-term signals estimated with wavelet analysis from linear trend (b), annual amplitudes with annual phase (c) and noise parameters (d–f) estimated for a set of 63 daily vertical displacements recorded by Global Positioning System stations situated within the Amazon basin. The maps of trend and annual amplitude also include all stations processed by the Nevada Geodetic Laboratory (indicated as small gray dots). The trend and annual amplitude values interpolated to a regular grid are shown as background maps.

Unlike annual amplitudes, linear trends do not form a consistent spatial pattern at first glance. We now discuss them briefly, and a detailed analysis is presented below. Most of the GPS trends are in the range of −3 to 3 mm/year. Negative trends are marked at stations along the main river, with a minimum trend of −9.1 ± 0.8 mm/year measured at the LPIN station; the error of trend is estimated assuming a combination of white and power-law noises. The trends are completely divergent and can be completely different for stations in close proximity, indicating the very local nature of these changes. A pair of stations in the central part of the catchment, AMTE and LTFE, both located in the center of Tefe city at the mouth of the Tefe River into the Amazon, are excellent examples, as the trends are 3.3 ± 3.2 and −3.2 ± 2.2 mm/year, respectively (Figure 2). Further, we estimate standard deviations between linear trend and long-term nonlinear signals estimated with wavelet analysis, described further. Many displacement time series are characterized by a nonlinearity in the sense that a change between uplift and subsidence appears several times over the observation span. The greatest nonlinearity is found in the displacements for stations along the main river up to its mouth, which may be due to the influence of the hydrosphere, whose loads fluctuate significantly due to the occurrence of climate-induced dry and wet periods (Espinoza et al., 20122014). We note nonlinearity also for several stations in the west and in the south of the catchment area. This can be explained by earthquakes of magnitude M > 6 (National Earthquake Information Center [NEIC]) occurring in the Amazon basin, which can affect a set of 21 GPS stations located less than 1,000 km from NEIC-reported epicenters (Figure 1). At these earthquake-affected stations, in addition to nonlinearity, we also find other characteristic phenomena, such as large standard deviation of displacements and inconsistency in phase between GPS-observed and GRACE-derived displacements, described further. We do not report for the Amazon area displacements with a distinctive postseismic deformation, but this is most likely due to the fact that the Amazon area is located quite far from the major earthquakes taking place in the Pacific area, along the coast of South America.

The standard deviations of the residuals of the displacements of GPS stations in the Amazon basin range from 0 to 12 mm. Extreme values occur for stations at the mouth of the Amazon. Stations located in the south and east of the basin, as well as a group of stations located in the west of the basin are characterized by small standard deviations of residuals, ranging from 4 to 8 mm. Larger values, between 8 and 10 mm, are recorded for stations located in the southwest of the basin.

Spectral indices estimated for residuals of displacements suggest that majority of stations is affected by non-white noise, especially for the central and the eastern part of the Amazon basin, where significant changes from surface water and groundwater are reported (Frappart et al., 2018; Getirana et al., 2017). The lowest values were recorded for three stations located at the main river, where it is widest, between the cities of Santarѐm and São Sebastião. Spectral index values between −1 and −2 indicate the possible influence of hydrological loading on the stochastic part of GPS displacements (Klos, Gruszczynska, et al., 2018). In particularly, that stations with indices close to −2 are also characterized by extreme trend values. Since the spatial distribution of spectral indices is very uniform, it can be assumed that the GPS technique is sensitive to hydrological loading for non-seasonal bands. The noise amplitudes are station-dependent, which means that no spatial coherence was found.

We also investigate the degree to which power-law noise dominates residuals. The dominance of power-law noise was found to be at most 10% for stations located directly adjacent to the main river. This proportion increases with increasing distance and becomes the largest, about 60%–80%, for several stations on the Amazon basin boundary. For earthquake-affected stations, the power-law noise totally dominates residuals, which is not unexpected. This is because not all earthquake-related signals are possible to be captured in a functional model, even when sophisticated models are applied (e.g., Tobita, 2016).

4 Comparison to Surface Water Levels

The surface water component is believed to account for most of the TWS values in the Amazon basin (Getirana et al., 2017). An increasing level of the surface water will cause a subsidence of the Earth's crust. It is generally hypothesized that GPS stations respond in an elastic way to the loading, that is, GPS measures crustal subsidence caused by increasing mass, and vice versa, the lower the mass, the more the GPS station will uplift. There are however already increasing reports that GPS stations will also measure a poroelastic response due to changes in groundwater component, that is, stations will react in the opposite way (Razeghi et al., 2022). To verify whether GPS stations react elastically to variations in the surface water, we use time series of surface water levels derived from the processing of observations from ERS 2 and ENVISAT radar altimetry missions, provided through the multi-institutional efforts and made freely available on-line under the name Hydroweb (Cretaux et al., 2011). Both missions have a 35-day temporal resolution which is a duration of the orbital cycle and 70 km inter-track spacing at the equator. The trends estimated for surface water levels are spatially very consistent and indicate a slight decrease in water masses in the western and southern parts of the Amazon basin and a marked increase in water resources in the northeastern part of the basin (Figure 3), which may be due to water accumulation from upstream catchments (Azarderakhsh et al., 2011). Trends measured by GPS show subsidence of the western part of the basin and uplift of the eastern part, which is not consistent with surface water levels. However, positive trends measured by GPS in the southern part of the basin are very consistent with trends in drought duration per year, which indicate that the duration of droughts is increasing from year to year in the Tapajos and Xingu basins (Chaudhari et al., 2019). Although the trends observed by GPS are not consistent with surface water, the amplitudes and phases of the annual component show great consistency. Note that Figure 3 shows the GPS observed phases of the annual minimum, not the maximum, as in Figure 2. This is intentional in order to remain consistent with surface water indications, that is, maximum loading causes maximum crustal subsidence. The largest amplitudes are recorded for stations along the main river and its southern tributaries with the maximum occurring in February. These results coincide with those of Marengo (2005), who indicated maximum precipitation in this region during the months of December–January–February. Annual phases of surface water levels and those measured by GPS change from January to September, with a distinctive latitudinal pattern, which means that GPS responds elastically to surface water loading in the seasonal temporal-scale. Surprisingly, the spatial agreement of the determined standard deviations of the residuals of the surface water series and GPS is also quite good, indicating the sensitivity of GPS to surface water changes over short-term temporal-scales.

Details are in the caption following the image

Comparison of the trends (a), standard deviations of residuals (b) and annual amplitude and annual phase (c and d) of Global Positioning System (GPS) displacements with water levels measured in the Amazon basin by radar altimetry missions. It should be noted that the Hydroweb and GPS trends should have opposite signs assuming an elastic response of the Earth's crust on which the GPS stations are founded. Different scales are applied for trends, standard deviations and annual amplitudes. The GPS annual phases are presented as phases of the annual minimum to be consistent with the surface water level, that is, maximum levels of surface water cause maximum subsidence of the Earth's crust.

5 Neighboring GPS Stations Along the Main River

We now look at the impressive collection of GPS stations placed along the main river, at a maximum distance of 20 km from the shore (Figure 4). For 6 locations more than one GPS station is available, these are so-called “co-located stations.” We will start by discussing the trends, especially for the neighboring stations, for which the values did not agree with the overall pattern. The first example is the AMTE and LTFE stations, mentioned earlier, for which the trends do not match at all. However, if we look at the time series of displacements, we find that the LTFE station was the first to operate at the site, in the period 2007–2014, and later in 2014 the AMTE station was established; AMTE is still an active station. Displacements measured at the LTFE station coincide with periods of several floods that hit the Amazon with two of them called record floods, occurred in 2009 and 2012 (J. L. Chen et al., 2010; Satyamurty et al., 2013). The impact of these two floods on the GPS displacements is evident for all stations operating in the region; a slow subsidence of all stations can be noticeable until 2015–2016 (Figure 5), when the Amazon was hit by record heat and drought during El Niño phenomena (Jimenez-Munoz et al., 2016). As a result of these droughts, we observe a sudden crustal uplift of several millimeters for all stations in 2015, followed by a noticeable positive trend, indicating further slow uplift in this part of the basin. We now look at the displacements measured at the LMA1 and LPIN stations. Both stations are characterized by extreme negative trend values, respectively, −8.7 ± 1.8 and −9.1 ± 0.8 mm/year. The trends at both stations can be explained by exceptionally short time series, about 3 years. Displacements at both stations were measured during a period 2007–2011 of extreme Amazon flood in 2009, which caused massive subsidence of the Earth's crust. We note here that the geodetic community believes that short time series should not be taken into account when determining trends. On the one hand, because the value of the trend itself is extremely sensitive to the changing length of the time series, and on the other hand, because the length of the series will affect the determined value of the trend uncertainty (compare Bos et al., 2008, Equations 26–32). We are aware of this problem. However, we decide to use all time series, since removing the short ones will lead to a deterioration in the spatial resolution of the changes obtained. We note, however, that short time series should be treated with great care and analyzed extremely carefully.

Details are in the caption following the image

Comparison of parameters estimated for Global Positioning System (GPS) daily vertical displacements for neighboring stations situated along the main river. Figures (a–c) present trends, (d) presents annual amplitudes (as color-coded dots) and phases (as lines), (e) presents spectral indices. For 6 locations where 2 or 3 GPS stations are measuring, we artificially slightly moved the dots for better visibility. Small dots in Figures (a–d) in the background indicate estimates for surface water level, as in Figure 3, with the scale agreeing with Figure 3.

Details are in the caption following the image

Vertical displacements measured by Global Positioning System (GPS) stations along the main river, as included in Figure 4. Vertical dashed line indicates an abrupt uplift of all GPS stations due to exceptional drought in 2015. Note, that for better visibility, we present monthly time series instead of daily time series. Procedure of averaging is described further in the text.

For now, we divide GPS stations into two groups: those that measured before 2015 and those that measured after 2015, and plot the measured trends separately (Figure 4). As a reminder, 2015 is a year in which we note a significant uplift in the displacements due to the extreme drought (Figure 5). For the period 2006–2015, we observe a significant subsidence of the region along the main Amazon River. In the middle part of it, we find two extreme trend values, measured for GPS stations with 3-year long time series, mentioned earlier. We suppose that longer time series would give a picture consistent with the other stations. This subsidence is due to several floods that hit the region and brought with them enormous loading on the Earth's crust. For the period 2015–2021, we get a coherent uplift of the area around the main river, consistent with observed increasing surface water level. This may be due to the significant loss of water during the 2015 drought and the inability to renew resources with new rainfall after that period. It is noteworthy that there is also a clear change in trends from negative to positive in the measured surface water levels in the middle of the river, which is mapped by the measured GPS trends. This leads us to conclude that the measured GPS trends may be due to changing surface water level loading in the region. The amplitudes of the annual component are spatially consistent, regardless of the length of the time series or the period over which the stations operated.

Spectral indices of noise show a clear presence of non-white noise, indicating the possible existence of short-period signals coming from the hydrosphere. All estimates indicate the existence of power-law noise between flicker and random-walk noises. The two stations LMA1 and LPIN stand out for their rather high values of spectral indices, but this is probably related to too short time series. Very interestingly, the spectral index estimates are also very consistent for neighboring stations, forming a clear spatial pattern. The general consensus is that in this region the type of noise is station-specific rather than region-specific, but in this case the consistent spatial pattern shows that the hydrosphere can have a strong influence on the nature of the residual GPS displacements in this region. We leave this topic for further contribution.

6 Averaging of GPS Displacements

We resample the daily GPS displacements into monthly samples to be consistent with the GRACE-derived and GLWS-predicted displacements we use further. Trends and seasonal components remain intact during this procedure, but short-term signals below 1 month are removed (Figure 6). This is very evident in the power spectral density plots, where the frequency response of monthly and daily displacements match each other, that is, we see identical signal power values for long periods and for seasonal periods, as well as for all periods up to 6 cycles per year (cpy), which is a Nyquist frequency of monthly samples. For daily samples, signal power can be estimated up to 182 cpy. The slope of the power spectral density plots is directly interpreted as spectral indices of the power-law model. It is evident, that the monthly series contain power-law noise, similar to that present in the daily series. The daily series, however, have white noise at low periods, which is missing in the monthly series due to their averaging.

Details are in the caption following the image

(a) Vertical displacements of AMUA station, located 13.5 km from the main river. (b) Power spectral density plots of these. Daily samples are shown in blue, while samples averaged over a month are shown in orange.

Figure 7 and Table S2 in Supporting Information S1 shows that averaging procedure does not change the parameters of the trajectory model, that is, the trends and the amplitudes and phases of the annual component, which is encouraging because it means they can be interpreted independently of the sampling interval. What changes is the degree of dominance of power-law noise in the combination of white noise and power-law noise. This modification induces a change in the amplitude of power-law noise, according to Equation 4. For the monthly time series, the residuals are completely dominated by power-law noise, while for the daily time series, power-law noise must be supplemented by a white noise component to reliably represent changes in the spectrum in periods larger than 6 cpy.

Details are in the caption following the image

Parameters of Global Positioning System displacement estimated for daily time series and those but averaged to monthly samples. A 1:1 line indicating a perfect agreement is included in each plot.

7 Reference Hydrosphere-Induced Displacements

We use gravity field changes available from GRACE and GRACE Follow-On (GRACE-FO) missions from April 2002 to December 2019 available as spherical harmonic coefficients up to degree and order (d/o) 96 provided as release-06 (RL06) product by the Center for Space Research (CSR) at the University of Texas, Austin (Bettadpur, 2018; Save, 2019). We remove the north-south stripes observed in GRACE data reflecting the noise increasing at higher degrees of spherical harmonic coefficients by the anisotropic decorrelation filter DDK3 (Kusche, 2007; Kusche et al., 2009). We subtract static gravity field with GGM05C model (Ries et al., 2016). The degree-1 and degree-2 terms are replaced using Technical Note 14 (TN14) with Sun et al. (2016) and Satellite Laser Ranging (SLR) estimates (Cheng & Ries, 2017), respectively. The C30 coefficients are also replaced for GRACE-FO periods with the ones derived from SLR using Technical Note 14 (TN14). We also apply GIA corrections using A et al. (2013) model. We do not fill in missing values between the two GRACE missions or single missing months for each mission.

The spherical harmonic coefficients Cnm and Snm are then used to estimate the monthly vertical (radial) displacement dr for GPS locations using (Farrell, 1972; L. Wang et al., 2017):
where urn:x-wiley:21699313:media:jgrb56455:jgrb56455-math-0008 represent fully normalized Legendre functions of degree n and order m, θ is colatitude, λ is longitude, R is the radius of the Earth, hn and kn are load Love numbers of degree n that describe the response of the elastic Earth to surface loading as given in Farrell (1972).

Since spatial scale of spherical harmonic coefficients truncated to d/o 96 is around 200 km, we also use another representation of gravity field changes available from GRACE and GRACE-FO missions in a form of mass concentration blocks (mascons). It is declared that mascon solution corresponds to a spherical harmonic coefficients of d/o 120 which means a spatial scale of 166 km (Save et al., 2016; Scanlon et al., 2018). We use a set of TWS values defined as global mascons on a 0.25° per 0.25° grid for the number of 180 months from April 2002 to December 2019; the 163 months from the GRACE and the 17 months from the GRACE-FO mission. We use RL06 solution provided by the CSR (Save, 2020; Save et al., 2016).

We also use monthly grids of TWS changes provided by the University of Bonn as the high-resolution GLWS data set (Gerdener et al., 2023). These TWS values are obtained by first assimilating TWS anomalies from GRACE/-FO into the WaterGAP Global Hydrological Model (WGHM; Muller Schmied et al., 2021), and subsequently accumulating all updated individual storages in the model, that is, canopy, snow, soil, surface water body, and groundwater storage. The resulting TWS grids thus benefit from the high spatial scale of the WGHM model (0.5° per 0.5°), driven by precipitation and radiation data, but still incorporate the mass change information contained in the GRACE data. GLWS TWS data are provided as a global 0.5° per 0.5° grid for the period of 2003–2020.

Gridded TWS values for GRACE and GLWS are transformed into spherical harmonic coefficients using (Wahr et al., 1998):
where urn:x-wiley:21699313:media:jgrb56455:jgrb56455-math-0011 and urn:x-wiley:21699313:media:jgrb56455:jgrb56455-math-0012 represent dimensionless coefficients of surface mass change, denotes the basic area element and is equal to sin θd θd φ, ρw = 1,000 kg · m−3 and ρe = 5,496 kg · m−3 are constant values for the density of water and the average density of the Earth, respectively. Equations 6 and 7 are used to derive spherical harmonic coefficients from mascon-based TWS values from GRACE and from gridded TWS values from GLWS model. In both cases, however, spherical harmonic coefficients are truncated to different d/o. TWS variations from mascon-based solution are transferred to spherical harmonic coefficients up to d/o 120. TWS grids from the GLWS model are transferred to spherical harmonic coefficients of d/o 96 and d/o 120 to stay consistent with GRACE solutions, but also to d/o 180. The latter corresponds to spatial scale of 111 km and is used to demonstrate benefits of using high-resolution GLWS model and force a more detailed comparison between GLWS and GPS. The spherical harmonic coefficients Cnm and Snm are then applied in Equation 5 to estimate monthly vertical displacements for GPS locations.

Surface loading in the Amazon basin is predominantly driven by heavy seasonal rainfall, which is unevenly distributed over the large basin and modulated by El Niño-Southern Oscillation (ENSO) events. In the presence of a huge tributary network and over 250,000 km2 floodplains, this leads to a huge but complicated pattern of inundation and, consequently loading and vertical displacements. The ability of hydrological models to represent these patterns correctly depends on whether and how detailed important water storages like wetlands, rivers and groundwater are represented in the model, and how well such representations were calibrated with data for example, from gauging stations. WGHM does represent floodplains and rivers as storages but these are generally static in extension. In the assimilation of GRACE data into GLWS we have noticed that the river storage often seems to absorb unmodelled signals which could be related to unmodelled floodplain inundation (Gerdener et al., 2023). However, river storage in a routed 0.5° model corresponds to grid cell shares and cannot be evaluated directly against gauge water levels, so it is difficult to assess the realism of such updates. We hypothesize that GLWS could, due to its improved level of detail when compared to GRACE data, be beneficial in simulating loading in stations particular at distances close to the river network.

8 Consistency Between GRACE and GLWS for the Amazon Basin

Gerdener et al. (2023) compared trends, annual, and anomalous water storage signals across GRACE/-FO, GLWS and the WaterGAP model, with generally good agreement over the Amazon basin. As expected, the hydrological model exhibits much weaker linear trends as compared to satellite observation (which we ascribe to problems with forcing data and process representations e.g., for wetland inundation, tropical soils hydraulic conductivity, etc.), and the assimilation has not been able to fully mitigate this in GLWS. Annual amplitudes of GRACE-derived water storage are slightly higher than GLWS but exhibit much less spatial detail (cf. Figure 2 in Gerdener et al., 2023), with the phase of all three data sets remarkably close. On a Hövmöller diagram (Figure 3 in Gerdener et al., 2023), it becomes obvious that GLWS inherits many anomalous events in the tropical belt from assimilating GRACE data, which are likely related to ENSO (Barbosa et al., 2019; Vilanova et al., 2021).

In a nutshell, from the construction of these data sets our hypothesis is that GRACE-simulated loading may well fit better to GNSS at longer temporal-scales, but on annual and shorter temporal-scales GLWS is supposed to improve over GRACE/-FO due to its much improved spatial scale (which is to say, that the information contributed by hydrometeorological forcing data and hydrological model parameterization can be trusted at these spatial and temporal-scales). Results presented in Figure 8 seem to confirm this hypothesis. We notice that GRACE-derived displacements appear generally larger as compared to GLWS, which is likely partly due to a generic lower signal in the assimilation, and partly due to the higher spatial scale it has been evaluated here (which can be expected to be relevant in particular for stations located close to rivers, as mentioned above). Figure 9 compares GNSS-observed, GRACE-derived and GLWS-predicted displacements at the same degree of truncation (96) and this proves that the latter both fail to simulate the huge vertical displacement along the main river branch correctly, which we ascribe indeed to the insufficient spatial scale. We notice that for example, GNSS stations AMCR and AMPR are only few km distant from a river branch (see Supporting Information S1) and we emphasize that even at 0.5° (spatial scale of WaterGAP and GLWS), rivers and thus riverine surface loads cannot be represented geometrically correct. Of course, any mismatch of the local Earth's structure from the assumed PREM-type response (i.e., the load Love numbers in Equation 5), for example, caused by thick, loosely consolidated sediments dominating the upper layers of the crust, is bound to create differences between GRACE and GLWS displacement and GNSS observations (Bevis et al., 2005). Krüger et al. (2002) observed significant North-South gradients of sedimentary thickness and Moho depth even within the Amazon region.

Details are in the caption following the image

Plots present displacement time series derived from Gravity Recovery and Climate Experiment (GRACE), predicted by Global Land Water Storage (GLWS) and observed by Global Positioning System (GPS) for selected locations. Note that the axis boundaries are set to the time range and GRACE/GLWS values for better visibility. Map presents standard deviations of GRACE-GLWS differences.

Details are in the caption following the image

Trends and annual amplitudes (left and middle columns, respectively) estimated for monthly vertical displacements measured by Global Positioning System (GPS), derived from Gravity Recovery and Climate Experiment (GRACE) and predicted by Global Land Water Storage (GLWS), respectively. The right column shows their histograms. Note the change in scale between the GPS and GRACE/GLWS for the trends. The upper right histogram shows the percentage coverage of GPS observations by GRACE and GLWS.

9 Pre-Defined Temporal-Scales

Wavelet analysis is a very common technique for time-frequency analysis of non-stationary signals. Researchers have already extensively demonstrated its usefulness in time series analysis in hydrology, such as to show the variability of series over time or to improve hydrological models (see Labat et al., 2004, Rathinasamy et al., 2014, or Sang, 2013 and the references within). It has also been widely used for analyzing nonlinearities on long-term temporal-scales, modeling seasonal signals, or de-noising GNSS displacements (Carbonari et al., 2023; Ducellier et al., 2022; Khelifa et al., 2013; Klos, Bos, & Bogusz, 2018; H. Wu et al., 2015).

Wavelet analysis is a type of window analysis with a changing dimension of the window, depending on which information (time or frequency) one wants to obtain with greater reliability. Using a window with high temporal resolution, we get accurate low-frequency information, while using a window with low temporal resolution, we get accurate high-frequency information. Variable time-frequency resolution allows wavelets to be used to study signals that have high- and low-frequency components. Basis functions, or wavelets, are created from the main wavelet, called the mother wavelet, by shifting and scaling it. The wavelet transform thus involves dividing the signal into smaller parts and then comparing them with the shifted and scaled wavelet. From this, wavelet coefficients are estimated, which represent how much the original signal corresponds to the mother wavelet (Box & Jenkins, 1970):
where * corresponds to the conjugated complex. A mother wavelet ψ(t) is scaled by a and shifted by τ to reliably model original signal at all frequency levels:
The wavelet transform is not based on the time-frequency relationship, but on the time-scale relationship. To relate scale and frequency to each other, one must use the concept of center frequency, which is described by the formula (Misiti et al., 2000):
where a is the scale index, T represents sampling interval, fc is the mean frequency and fa is a pseudo-frequency related with scale index a.

In time series analysis, a discrete wavelet transform is used more frequently than a continuous wavelet transform from Equation 8. It is very often applied to decompose the series y(t) into a set of frequency components. Suppose we have a time series y(t) of length N, which can be decomposed into p frequency components. N and p are related to each other by the following relation: N = 2p. The decomposition of y(t) can now be understood as band-pass filtering, which produces two sets of coefficients at each step: approximation coefficients and detail coefficients. In the first step of decomposition, we decompose the original signal y(t) and obtain the approximation and detail coefficients for it at level one. The next step is to divide the coefficients of the first approximation into two parts using the same scheme, replacing the original signal by the first approximation and forming the coefficients of the next approximation and the next detail, and so on.

We use the wavelet analysis with Meyer mother wavelet (Meyer, 1990) to decompose the monthly vertical displacement time series into four levels with pre-defined period bands (Table 1), the wavelet analysis is implemented according to the Matlab procedures. We use the Meyer wavelet because it is one of several that allow decomposition of the series and their subsequent reconstruction without loss of information. This means that the decomposed signals add up to the original signal. It is also a wavelet that we have used in several of our previous works and proved its effectiveness for modeling geodetic time series (Bogusz, 2015). The advantage of wavelet spectral decomposition is that it is sensitive to gradual or episodic changes of the spectral content of the time series; something that is indeed observed with vertical GNSS time series (e.g., Bevis & Brown, 2014).

Table 1. Period Bands Recovered With Wavelet Analysis Using Meyer Mother Wavelet
Decomposition level Period band Description of temporal-scale
D1 From 2 to 5 months Short-term signals
D2 From 4 to 9 months
D3 From 7 months to 1.4 years Seasonal signals
D4 From 1.1 to 3.0 years Long-term signals
A4 From 3.0 years upwards

Since wavelet analysis is a non-parametric method, wavelets are fitted into all observations consecutively. A problem can therefore arise when fitting in the first and last observations, and this is called the edge effect (Goglewski, 2020). We remove this effect by extrapolating the displacement time series before and after the actual observations. A 5-year extension is added to either end of the time series prolonging each time series by 10 years in total. Extrapolation is performed using a least-squares time series model including trend, annual and semi-annual signatures to which white noise is added. The latter has the variance of the residuals of the displacement time series; the original time series minus the functional model. Then, we decompose the time series using wavelet decomposition. Four levels of decomposition are recovered. This number is dependent on the length of the time series and sampling interval. Period bands of each decomposition level are estimated by relating scale of wavelet to the frequency using central frequency of the wavelet and sampling period, Equation 10, but also confirmed using frequency analysis. After recovering the period bands, the displacement time series are shortened again, leaving the original data length. Wavelet decomposition enables the capture of signals at predefined periods, and then compare the signals at various decomposition levels (Klos et al., 2021).

We now estimate the percentage of variance of displacement time series explained by short-term, seasonal and long-term temporal-scales (Figure 10):
where Var(temporal-scale) represents the variance of the signal of the temporal-scales, as defined in Table 1, and Var(y(t)) represents the variance of the original displacement time series. Displacements derived from GRACE and predicted by GLWS contain a similar amount of time series variance on short-term temporal-scales, up to a maximum of 20%. A slight difference is noticeable for GLWS when increasing the degree and order of truncation of the spherical harmonic coefficients from 120 to 180. The seasonal temporal-scale explains the largest amount of variance in the GRACE-derived and GLWS-predicted displacements, up to 80%. The variances for long-periods are also similar, we see a slight difference for GRACE-derived displacements for stations located in the western part of the catchment, whose variance explained by this band reaches 40%, but this comes at the expense of a smaller variance explained by the seasonal signal.
Details are in the caption following the image

Percentage of variance of displacement time series explained by different temporal-scales whose periods are provided in Table 1.

This is somewhat different for GPS displacements, where the explained variance forms a spatial pattern for individual temporal-scales. For the short-term temporal-scales, stations located in the western and central parts of the catchment explain about 20% of the total variance of displacements, while stations located in the eastern part of the catchment explain at most 10% of the variance. For the seasonal temporal-scale, we see a clear dominance of this band for stations located in the central and eastern parts of the catchment area, where the Earth's crust loading due to surface water and groundwater changes, mentioned earlier, is dominant. Displacements for stations in the western part are mostly generated by long-term temporal-scales, and the variance explained by them reaches up to 85% in the most extreme cases. This pattern correlates well with the occurrence of earthquakes shown in Figure 1 and indicates that earthquake-induced nonlinearities dominate for these stations.

10 Classifying GPS Stations as Benchmarks

Monthly GPS vertical displacements are compared with monthly vertical displacements derived from GRACE truncated to d/o 96 and d/o 120 and predicted for GLWS hydrological model truncated to d/o 96, d/o 120 and d/o 180 at different decomposition levels. We employ Pearson's correlation coefficient, the significance of which is tested using Student's t-test with the probability of observing the null hypothesis below 0.05 (Table 2). Finally, GPS stations whose displacements are positively and significantly correlated with displacements predicted by the GLWS hydrological model truncated to d/o 180 at each level of decomposition are classified as the benchmark sets.

Table 2. Percentage of Global Positioning System (GPS) Stations Whose Displacement Time Series Are Significantly and Positively Correlated With Displacements Derived From Gravity Recovery and Climate Experiment (GRACE) and Displacements Predicted by the Global Land Water Storage (GLWS) Model at Different Time Resolutions and for Different Truncations of Spherical Harmonics
Percentage of GPS stations correlated positively and significantly with different data sets
Short-term (%) Seasonal (%) Long-term (%)
GRACE d/o 96 20 93 60
GRACE d/o 120 25 95 63
GLWS d/o 96 15 100 55
GLWS d/o 120 30 100 58
GLWS d/o 180 30 100 60
  • Note. Significance is estimated according to the Student's t-test. The probability of observing the null hypothesis is set at 0.05.

Figure 11 presents a set of correlation coefficients estimated between GPS displacements and GRACE-derived and GLWS-predicted displacements at different temporal-scales. GPS displacements are positively correlated with GRACE-derived displacements truncated to d/o 120 for stations situated in the central part of the basin. Stations from the southern part of the catchment only correlate positively with GLWS-predicted displacements truncated to d/o 120, and this is even more apparent when we increase the number of spherical harmonic coefficients included in the truncation. This is also readily apparent in the histogram plots, where more GPS stations correlate positively with GLWS-predicted displacements than with those derived from GRACE. On the seasonal temporal-scale, the spatial distributions of correlation coefficients look very similar. Differences are observed for the histograms, especially for displacements predicted by GLWS and truncated to d/o 180, for which we note a significant decrease in the number of negatively correlated stations and a significant increase in the number of positively correlated stations with a sharp peak around 0.7. This means that GPS is sensitive to small-scale changes in the hydrosphere, which are modeled by GLWS but not observed by GRACE due to its coarse spatial scale. On a long-term temporal-scale, the correlation coefficients between GPS and GRACE-derived displacements do not form any obvious spatial pattern. It looks as if the correlation coefficients were generated completely at random for each station. However, the correlation coefficients between GPS and GLWS-predicted displacements form a systematic spatial pattern that can be interpreted. Stations located in areas where surface water-induced and groundwater-induced changes predominate have a distinctive positive correlation between GPS and GLWS-predicted displacements. In contrast, for stations located in the western part of the basin GPS and GLWS-predicted displacements correlate negatively, which is due to the occurrence of earthquakes in these regions, to which GPS is sensitive, but GLWS model does not take them into account during modeling.

Details are in the caption following the image

A set of correlation coefficients estimated between Global Positioning System (GPS) measured displacements and displacements derived from Gravity Recovery and Climate Experiment (GRACE) and predicted by Global Land Water Storage (GLWS) for d/o 120 and d/o 180. Histograms on the right present correlation coefficients between GPS displacements and GRACE/GLWS truncated to d/o 96, 120 and 180. Note that correlation coefficients between GPS and GRACE truncated to d/o 96 are not presented on the maps.

We verify the above conclusions using Euclidean distance and median absolute error (MAE) values (Figure 12). We could also use other measures of correlation besides Person's correlation, such as Spearman's rank correlation coefficient, but with data between which there is a linear or nearly linear relationship, both determinations of correlation coefficients will give identical values (Kozak et al., 2012). For short-term and long-term temporal-scales, we note a gentle linear relationship between correlation coefficients and Euclidean distances along with median absolute error. This demonstrates the correctness of determination of correlation coefficients and the validity of their use for benchmarking; for higher values of the correlation coefficient, the correspondence between the series is greater, which is equivalent to smaller differences between the two data sets, that is, smaller Euclidean distances and median absolute errors. This relationship is unfortunately not fulfilled for the seasonal temporal-scale, for which a high correspondence is expected between GPS and GLWS-predicted displacements in the Amazon area. For high correlation coefficients, we note extreme values of Euclidean distance and median absolute errors. However, if we look at the individual series, we see that the culprit for the extreme values of Euclidean distance and median absolute error is the amplitude of the annual component, which is sometimes up to twice as high for the GPS displacements as for GLWS-predicted displacements. The phase shifts between the annual components for GPS displacements and GLWS-predicted displacements agree perfectly, which generates large values of correlation coefficients.

Details are in the caption following the image

Euclidean distance (mm) and median absolute error (mm) between Global Positioning System (GPS) and Global Land Water Storage (GLWS) truncated to d/o 180 plotted against correlation coefficients for three predefined period bands.

The above considerations lead us to classify GPS stations as benchmarks, which is based on the correlation coefficients between GPS and GLWS-predicted displacements. We perform this task for each decomposition level to highlight the similarity of the time series in each temporal-scale (Figure 13).

Details are in the caption following the image

Global Positioning System (GPS) stations classified as benchmarks for various levels of decomposition based on the correlation coefficients between GPS displacement time series and displacements predicted by Global Land Water Storage (GLWS) hydrological model retrieved up to d/o 180; details are provided in Tables S3 and S4 in Supporting Information S1. Significance of correlation coefficients is estimated using t-Student test. GLWS model is gapped, since it is retrieved for the GPS epochs. Results are intentionally presented separately for each decomposition level to show the variation for each time resolution and the similarities in the displacement series.

The GPS displacements are sensitive to changes in hydrosphere masses over the entire frequency range. For the highest frequencies, included in detail D1, that is, periods between 2 and 5 months, as many as 20% of GPS stations are significantly and positively correlated with the GLWS model truncated to d/o 120, meaning that for these stations, the model explains part of the signal in the shortest time window. The benefits of using the high-resolution GLWS model are very apparent; changing the number of spherical harmonics from d/o 120 to d/o 96 reduces the number of stations included in the benchmark for the D1 detail from 20% to 11%, and switching from the high-resolution GLWS model to GRACE reduces this number even further, to 3%. Stations included in the D1 benchmark form a spatial pattern—positively correlated with GLWS are series from stations located right at the Amazon River, and those in the northwestern part of the catchment. For periods between 4 and 9 months (detail D2), we note a positive correlation between GLWS truncated to d/o 120 and GPS for 60% of the stations. These are stations located in the northern and southern parts of the Amazon basin. Changing the truncation of spherical harmonic coefficients for the GLWS-predicted displacements from d/o 120 to d/o 96 reduces the number of stations included in the benchmark to 25%, and using GRACE-derived displacements results in 40% of the stations included in the D2 benchmark. For seasonal periods (detail D3, periods from 7 months to 1.4 years), as described before, 98% of GPS displacements are positively and significantly correlated with the GLWS-predicted displacements. This indicates that the hydrological signal dominates the GPS displacement time series located in the Amazon basin. Other signals contributing to the annual component may not play a significant role. For higher periods (greater than 1.1 years), we note a significant correlation of the GPS displacement time series with the GLWS-predicted displacements for more than 50% of the stations, indicating that the multi-year signals as well as the long-term trends present in the GPS displacements are due to changes in the hydrosphere. This result may be helpful in future analyses of stations responding elastically and poroelastically to loads caused by changes in hydrosphere masses (Larochelle et al., 2022).

Further advantage of estimating the sensitivity of GPS displacements to hydrological loading on different temporal-scales is that one can indicate stations affected by earthquakes, but still include them in the analyses in hydrogeodesy. Previously, stations from earthquake-affected areas were completely excluded from analysis in hydrogeodesy. Figure 12 shows that while some stations in the western basin are missing from the maps on long time scales exceeding 1.1 years (detail D4 and approximation A4), meaning that they are negatively or insignificantly correlated with GLWS-predicted displacements, some are positively and significantly correlated with GLWS-predicted displacements on both short-term (detail D1 and D2) and seasonal (detail D3) time scales. This means that displacements from these stations, despite clear signals related to tectonic deformation, still also contain a signal related to hydrosphere loading.

In addition to analyzing correlation coefficients, we also study the root-mean-square values for the GPS displacements classified as the benchmarks and their time series (Figure 13). From the time series plots for the D1 and D2 details, it is evident that the GPS displacements have comparable root mean square values to the displacements predicted by GLWS. Despite this, hydrosphere signatures are clearly distinguishable in the GPS displacements; displacements from GPS stations located in the central and western parts of the catchment are positively and significantly correlated with the GLWS. The map of root-mean-square values for the GPS displacements yields a very consistent spatial pattern: stations located at the mouth of the Amazon River have the largest scatter, while stations in the central and western parts of the catchment have a small scatter.

For the seasonal temporal-scale (detail D3), GPS displacements are characterized by distinctly dominant and very consistent annual signals. Annual amplitudes estimated for GPS displacements are larger than those estimated for the displacements predicted by GLWS. Both types of amplitudes have the same phase shift. The map of root-mean-square values is spatially coherent, with the highest values for stations located right at the mouth of the Amazon. Root-mean-square values decrease with increasing distance from the river, and we found the smallest values, that is, less than 5 mm, for stations in the southern and western parts of the basin.

For long-term temporal-scales, that is, detail D4 and approximation A4, more than 50% of the stations, evenly distributed across the Amazon basin, are classified as the benchmarks. For periods from 1.1 to 3 years (detail D4), the spatial pattern for root-mean-square is consistent, with multi-year signals having the greatest amplitude for stations located near the basin itself. This is due to the presence of a strong signal in 2015–2016, indicative of drought-induced uplift of GPS stations. During this period, a record drought was recorded due to the El-Nino effect (Espinoza et al., 20122014). This signal is consistent with the signal present for this period in the displacements predicted by the GLWS. However, the inter-annual signal determined for the GLWS-predicted displacements has three times less magnitude. Another drought occurred in late 2017 and early 2018, that is, the time series of GLWS-predicted displacements indicate a significant uplift of all stations due to the loss of hydrospheric masses. A similar effect is evident for GPS displacements during this period, but its amplitude is three times smaller.

For the longest periods, that is, more than 3 years (A4 approximation), we can also find similarities between GPS displacements and GLWS-predicted displacements for the stations classified as the benchmark. We observe a significant uplift and then subsidence of GPS stations in 2009–2010, also present in the GLWS-predicted displacements. All displacements indicate a slow subsidence, from 2006 until 2015, followed by a slow uplift. Since this is evident in both the GPS displacements and GLWS-predicted displacements, we hypothesize that this may be due to long-term slow drainage of the Amazon basin from 2015 to the present, possibly caused by intense deforestation of the Amazon basin, which triggered a significant decrease in precipitation (Zemp et al., 2017).

11 Impact of Other Contributors on Benchmarks

We must be aware that sometimes the classification of stations into benchmarks for different temporal-scales can be affected by other signals to which GPS is also sensitive. Non-tidal atmospheric and oceanic loadings can be present in all temporal-scales considered in this analysis, that is, from detail D1, up to the A4 approximation (Klos et al., 2021). They can be modeled using the latest geophysical models. We remove both loadings from the GPS displacements using predictions available from the ESM GFZ service. In our recent study (Klos et al., 2021), we showed that there is no significant difference between GPS displacement time series reduced by different types of atmospheric and oceanic loading models, and no single model provider should be preferred. Therefore, we assume here that the type of both models does not significantly affect the results. It is obvious that there may be some NTOL and NTAL residuals left in the GPS displacements due to the model not being perfect enough, but in principle this mismodeling can happen with other models as well. We also admit that for the Amazon region we are analyzing, the impact of the NTOL model residuals will be insignificant; GPS stations are located in the interior of the continent at a distance of more than 100 km from the coastline. The influence of the residuals of the NTAL model may be the greatest for details D1 and D2. Other time scales should not be affected by the residuals of the NTAL model, because the hydrological effect prevails in detail D3, and the NTAL model practically does not contain long-term components, so the signals present in D4 and A5 will be insignificant.

Signal multipath, which is an error characteristic to the GPS system, and which results from signal reflections from various terrain obstacles around a given permanent station, will be present in the daily displacements. Since we use averaged daily samples into monthly samples, we assume that this effect has been averaged out from the displacements. If not, its residuals may be present in the averaged series and visible in the D1 detail. Since we use correlation coefficients between GPS and GLWS-predicted displacements we assume this signal not to be included in the displacements for benchmarks at the D1 decomposition level.

The common mode error is determined commonly from the GPS displacements, since there are no predictions of this error (Dong et al., 2006). Stacking of time series, or principal component-based methods, are used to determine it. Our previous work has shown that common mode error, despite its commonly believed origin from GPS systematic errors, can also come from unmodeled large-scale effects such as the atmosphere or hydrosphere (Gruszczynski et al., 2018). We removed the atmospheric effect using ESM predictions from the GFZ. Thus, the common mode present in the Amazon region could be due to the hydrospheric effect or GPS systematic errors. Since the two are indistinguishable, we leave common mode error without modeling and trust that examining the cross-correlation between the GPS and GLWS-predicted displacements at each decomposition level will allow us to eliminate stations for which systematic effects dominate in short time periods; these stations will not be included in the benchmarks at the D1 and D2 decomposition levels.

The draconic signal occurs in the GPS displacement time series due to the repeatability of the GPS satellite constellation relative to the Sun. This is a typical error associated with GPS characteristics, and its period of 351.6 days has been reliably determined in previous analyses based on sufficiently long displacement time series (Amiri-Simkooei et al., 2017). Its amplitude was estimated from a global set of stations at −6.3 to 6.7 mm for vertical displacements (Amiri-Simkooei, 2013). The draconic period is very close to the tropical year (365.25 days) and is therefore very difficult to distinguish from it without having a long time series. Referring to the Rayleigh criterion, the distinction between the above two periods in the frequency domain can only be made for continuous time series longer than 25 years. Since such time series from GPS are recently rarely available, the draconic year is added alongside the tropical year to the deterministic time series model (Bogusz & Klos, 2016), Equation 2 and its amplitude is determined by the same methods as the other series parameters. However, some believe that such a determination may bias into the deterministic model, especially for short time series. Therefore, it is accepted to say that the draconitic period can be added to the deterministic model, but only for series over 15 years. The length of the displacements we use varies from 3 to 22 years, but only the series of one station, that is, the RIOP station, has a length of more than 15 years. The amplitude of the draconic oscillation for this station is 0.8 mm. Therefore, we leave the draconitic year unmodeled. We acknowledge, however, that when using longer series for future classifications, the draconitic period should be removed before including the station in a benchmark data set, especially in the seasonal frequency band, detail D3.

Thermal bedrock expansion is another contributor to the seasonal frequency band. This component contributes to the series with the annual component; to distinguish it models should be used. According to one of them, published by X. Xu et al. (2017), the annual component caused by thermal bedrock expansion in the Amazon region vary between 0 and 1 mm. Comparing these values with those of the annual amplitudes we estimated for the displacement time series, it is nearly 20–40 times smaller. We therefore assume that it has an insignificant impact on the benchmark results. However, we note that when classifying sets of stations as benchmarks for regions where the annual amplitude caused by the thermal bedrock expansion is larger, such as the area of Asia or North America (X. Xu et al., 2017), this effect should be removed from the time series because it can affect the determination of benchmarks for seasonal band (detail D3), especially for stations where the hydrospheric effect is not dominant.

The final contributor to the displacement time series is the effect of post-glacial rebound, causing significant uplift of the Earth's crust in certain regions, observable as long-term trends in the time series. This effect dominates over long periods, especially for the northern part of North America and Scandinavian regions. The most reliable way to remove it is to use GIA models, which predict how much the crust is uplifting based on the Earth's rheology and assumptions about the Earth's visco-elastic response. In this study, we use ICE-6G_C model, to model and remove the effect of post-glacial rebound. This model is consistent with the GIA model applied during GRACE processing, so we assume that if there are any residuals left in the A4 approximation due to model inadequacies, they will be identical in GRACE as well as in GPS. At the same time, we point out that the effect of post-glacial uplift for the Amazon region is insignificant, and even failing to remove it will not affect the determination of the benchmarks at the A4 decomposition level.

12 Summary and Conclusions

We present a novel approach for the wider and more reliable use of displacements measured by GPS stations (or GNSS stations in a wider sense) in hydrogeodetic analyses. The approach is based on classifying GPS displacements as trusted sets of stations. In other words, we have identified stations for further analysis that other scientists can use directly without worrying about hydrospheric changes there. We specify three temporal-scales to facilitate even more advanced comparisons. For classification, we use displacements predicted by the high-resolution hydrological model GLWS v2.0 based on assimilation of GRACE observations into the WaterGAP hydrological model. We demonstrate that their use is much more effective than the use of GRACE-derived displacements, especially in mapping local changes in the hydrosphere, falling below the nominal spatial scale of GRACE. Our results have important implications for hydrology with a special emphasis on hydrogeodesy. We show that individual GPS stations are sensitive to hydrological loading in different temporal-scales, which also means that a station may be classified into a benchmark in one temporal-scale and not be classified in another. We show that GPS stations rejected from previous hydrogeodetic analyses due to other important contributors, such as earthquakes, can still reliably represent hydrological changes for other temporal-scales. We argue that benchmarks allow for a better understanding and interpretation of hydrological signals, as spatial smoothing of GPS displacements or blind removal of GPS stations is no longer required. Benchmarking can add stations for future inversion studies (e.g., Argus et al., 2017; White et al., 2022) or can help to remove those strongly affected by GPS-specific signals and errors. Note that we present our analysis for vertical displacements, since the influence of the hydrosphere is greatest for them. However, the description regarding the effects occurring in GNSS displacements applies to both types of displacements, that is, vertical and horizontal, and the reliability of the presented method will be comparable for them. We leave detailed discussion of this topic for future contributions. We also note that the comparison of GPS displacements with those predicted by GLWS is still not fully satisfactory on short-term temporal-scale. In order to improve it, developments in the GPS background models, which significantly influence the nature of the series on the short-term temporal-scale, should be pursued. Either way, benchmarking will work well for any GPS solution. Benchmarks allow a better comparison with other geodetic techniques whose sensitivity to hydrosphere effects is still being studied, such as Interferometric Synthetic Aperture Radar (e.g., Castellazi et al., 2018). They can also be extremely useful in validating future GRACE-like gravity missions (such as MAGIC future ESA Earth Observation mission), whose assumptions aim to provide TWS anomalies with higher spatial and temporal-scales.


This research is performed in a context of DFG NEROGRAV Project which aims at developing methods for validating GRACE/-FO under Grant KU1207/29-1. AK is supported by the National Science Center, Poland, Grant UMO-2022/45/B/ST10/00333. HG and KS are supported by the DFG GLOBALCDA project (KU 1207/26-2) and JK by DFG under SFB1502/1-2022-PN450058266. GL, AL and JB are funded from the Military University of Technology Faculty of Civil Engineering and Geodesy.

    Data Availability Statement

    We are grateful to UN Global Compact initiative for providing the Interactive Database of the World's River Basins through: http://riverbasins.wateractionhub.org. GRACE-observed TWS series and GRACE-derived spherical harmonics have been downloaded through: http://icgem.gfz-potsdam.de/series, NGL displacement time series have been downloaded through: http://geodesy.unr.edu/NGLStationPages/GlobalStationList. Detailed information on their processing can be accessed through: http://geodesy.unr.edu/gps/ngl.acn.txt. GIA model is available through the website: ftp://podaac-ftp.jpl.nasa.gov/allData/tellus/L3/pgr/GIA_n100_mass_0km.txt. Low-degree corrections for GRACE are available through: https://podaac-tools.jpl.nasa.gov/drive/files/allData/gracefo/docs/. Hydroweb data set can be accessed through: https://hydroweb.theia-land.fr/.