Quantification of uncertainty in surface mass change signals derived from Global Positioning System (GPS) measurements poses challenges, especially when dealing with large datasets with continental or global coverage. We present a new GPS station displacement dataset that reflects surface mass load signals and their uncertainties. We assess the structure and quantify the uncertainty of vertical land displacement derived from 3045 GPS stations distributed across the continental US. Monthly means of daily positions are available for 15 years. We list the required corrections to isolate surface mass signals in GPS estimates and screen the data using GRACE(-FO) as external validation. Evaluation of GPS time series is a critical step, which identifies (a) corrections that were missed, (b) sites that contain non-elastic signals (e.g., close to aquifers), and (c) sites affected by background modeling errors (e.g., errors in the glacial isostatic model). Finally, we quantify uncertainty of GPS vertical displacement estimates through stochastic modeling and quantification of spatially correlated errors. Our aim is to assign weights to GPS estimates of vertical displacements, which will be used in a joint solution with GRACE(-FO). We prescribe white, colored, and spatially correlated noise. To quantify spatially correlated noise, we build on the common mode imaging approach by adding a geophysical constraint (i.e., surface hydrology) to derive an error estimate for the surface mass signal. We study the uncertainty of the GPS displacement time series and find an average noise level between 2 and 3 mm when white noise, flicker noise, and the root mean square (rms) of residuals about a seasonality and trend fit are used to describe uncertainty. Prescribing random walk noise increases the error level such that half of the stations have noise

© 2023 California Institute of Technology. Government sponsorship acknowledged.

For more than 2 decades, the Gravity Recovery and Climate Experiment (GRACE) space gravity mission and its nearly identical successor mission, GRACE-Follow on (GRACE-FO), have provided mass change estimates through tracking the time-variable part of the Earth's gravity field (Landerer et al., 2020). Mass change products are typically given on a monthly basis and have been used to study a variety of critical climate-related factors (Tapley et al., 2019), such as sea level rise (Frederikse et al., 2020), ice mass change (Velicogna et al., 2020), prolonged drought periods (Thomas et al., 2014), and regional flood potentials (Reager et al., 2014). The measurement geometry of GRACE(-FO) limits the study of geophysical processes to spatial scales of

The spatial resolution of gravity maps derived from satellite measurements is limited by sampling at altitude. Fusion with external geodetic data sources, however, can improve spatial resolution over what can be achieved only with satellite gravimetry. GPS position time series have been used widely to study the elastic response of Earth's surface to mass loading (e.g., Argus et al., 2017; Fu and Freymueller, 2012) and can provide information at short wavelengths (

GPS displacements between two epochs have many different signals embedded in them, i.e., those related to non-tidal atmospheric and oceanic loading, solid Earth phenomena such as tectonics and glacial isostatic adjustment, and others related to surface mass changes. With the proper treatment (see Sect. 2) GPS stations can capture local surface mass changes. We are interested in isolating the signals that reflect the Earth's elastic response to mass variations; thus, we apply a set of corrections to GPS vertical displacement estimates, and then we screen the data for outliers or potential errors. The data screening process checks for consistency between GPS and GRACE(-FO) vertical displacement estimates (similar analysis has been performed by Yin et al., 2020; Blewitt et al., 2001; Van Dam et al., 2001; Becker and Bevis, 2004; Davis, 2004; Tregoning et al., 2009; Tsai, 2011 and Chew et al., 2014) and identifies outliers that statistical tests fail to pick up (He et al., 2019).

The last step is to estimate uncertainty in the screened dataset. Since our purpose is to isolate surface mass load signals, we define

Error sources include errors driven by satellite antenna phase center offsets (Haines et al., 2004; Santamaria-Gomez et al., 2012), atmospheric pressure models (Kumar et al., 2020), non-tidal ocean loading (Jiang et al., 2013), satellite orbits (Ray et al., 2008; Amiri-Simkooei, 2013), Earth orientation parameters (Rodriguez-Solano et al., 2014), and tectonic trends and post-seismic relaxation after earthquake activity (Ji and Herring, 2013; Crowell et al., 2016).

The GPS position time series have common mode displacements (Tian and Shen 2016), including both a common mode error strongly varying each day and a common mode signal associated with seasonal water fluctuations. Wdowinski et al. (1997) first defined common mode error to be a series of rigid-body translations that reflect an error in the position of all geodetic sites in an area relative to an absolute reference frame; by removing the mean position (or stack) of all sites in an area, scientists recover more accurate estimates of relative position contained in the data. Dong et al. (2006) and Serpelloni et al. (2013) defined common mode error in a more sophisticated manner using principal or independent component analysis such that they remove spatially correlated, temporally incoherent error. Independent is different than principal component analysis in that it finds the maximum independence of the components instead of minimum correlation (Milliner et al., 2018; Liu et al., 2015). Common mode displacements includes both error (such as that associated with error in satellite orbits) and signal (such as the seasonal oscillation of elastic vertical displacement in elastic response to seasonal fluctuations in mass between the hemispheres) (Sun et al., 2016).

Considering the increased number of GPS stations and the limitations posed by the existing methodologies, Kreemer and Blewitt (2021) used a robust methodology to estimate the common spatial components of GPS residuals (i.e., the remaining signals of a time series after subtraction of a trajectory model). A trajectory model is a model consisting of an offset, a rate, and a sinusoid with a period of 1 year (Bevis and Brown, 2014). The so-called common mode component (CMC) imaging technique was originally introduced by Tian and Shen (2016) and quantifies the spatial correlation of the residuals (position or vertical displacement time series anomaly with respect to a trajectory model) of unequal-length time series using information from neighbor stations. It is important to note that CMC reflects both spatially correlated noise and spatially correlated signals, including elastic displacements, that a trajectory model fails to describe.

Spectral analysis of the residuals (with respect to a trajectory model, see Eq. 2) is an alternative way to estimate the noise level of vertical displacement series for each GPS station. The spectrum of the residuals can be approximated by white or colored noise (flicker, random walk, power-law approximation, generalized Gauss–Markov, etc.) or by a combination of white and colored noise (Williams et al., 2004; Bos et al., 2008; Klos et al., 2014). A summary of the different noise models and their power distribution can be found in He et al. (2019). Several standard GPS time series analysis packages are available to perform such an analysis, e.g., Create and Analyze Time Series (CATS) (Williams, 2008) and Hector (Bos et al., 2013). Various studies in the past suggested that the residuals are better described by a combination of white and flicker noise (see e.g., Klos et al., 2014; Argus et al., 2017), with the latter contributing the most (Argus and Peltier, 2010). Recently, Argus et al. (2022) showed that the longer the time series the more the spectrum of GPS residuals converges with the noise model of random walk.

Here, we outline a comprehensive framework for processing large datasets (continental and/or global) of GPS time series to derive estimates that only reflect surface mass signals for use in a joint inversion with GRACE(-FO) measurements. We lay out the corrections required to capture local surface mass changes (Sect. 2.1). Our interest is to make the process as automated as possible, and thus we set a number of evaluation metrics to detect outliers among all candidate (for the joint inversion) sites. Stations flagged as outliers are further evaluated for extra corrections (e.g., offsets, poor site maintenance). Finally, we assign weights to each GPS vertical displacement record. We test the most popular methodologies to quantify the error, considering time correlation, spatial correlation, and/or white noise (Sect. 3). Note that for spatially correlated noise the commonly used PCA/ICA is not as applicable to our use case because our dataset extends over very large spatial areas (continental). CMC imaging (Kreemer and Blewitt, 2021) fits our needs better. We build on the existing CMC algorithm to remove hydrology signals from the error estimate by deriving surface loading signals from a hydrology model and removing them from the GPS vertical displacements (see Sect. 3 for more details). The final product is a new dataset with GPS vertical displacement estimates that reflect elastic mass variations and their uncertainties.

We analyze positions of 3054 GPS sites as a function of time from 2006 to 2021 estimated by scientists at the Nevada Geodetic Laboratory (NGL) (Blewitt et al., 2018). Technologists at Jet Propulsion Laboratory (JPL) first estimate satellite orbits, satellite clocks, and positions for a core set of roughly 50 sites on Earth's surface (Bertiger et al., 2020). NGL uses JPL's clock and orbit products and performs point positioning to a total of about 18 500 GPS sites distributed across the world. Following the International Earth Rotation Standards (IERS) (Luzum and Petit, 2012) NGL's positions are corrected for solid Earth, ocean, and pole tides. NGL's positions in International Terrestrial Reference Frame 2014 (ITRF2014) (Altamimi et al., 2016) are more accurate than NGL's previous estimates of positions in ITRF2008. NGL estimates GPS wet tropospheric delays each day using the ECMWF weather model (Simmons et al., 2007) and the VMF1 tropospheric mapping function (Boehm et al., 2006). We input the NGL position time series, derive the displacement relative to a reference epoch, and then follow Argus et al. (2010, 2017, 2021) to isolate the part of GPS displacements reflecting solid Earth's elastic response.

Construct time series of elastic displacement uninterrupted by offsets due to antenna substitutions or earthquakes that pass through a specific reference time (such as 1 January 2014) by eliminating data before and/or after an offset.

Identify and omit GPS sites recording primarily (i) poroelastic response to change in groundwater, (ii) strong volcanic fluctuations, and (iii) post-seismic transients following Argus et al. (2014a, 2017, 2022). In the western US, GPS sites responding to groundwater change have maximum height around April when water is maximum, subside in the long term faster than 1.8 mm yr

Remove non-tidal atmospheric (NTAL) and non-tidal oceanic (NTOL) mass loading by interpolating global grids of elastic displacements calculated by the German Center for Geoscience (GFZ) (Dill and Dobslaw, 2013) following the method of Martens et al. (2020).

Remove glacial isostatic adjustment as predicted by model ICE-6G_D (VM5a) (Peltier et al., 2015, 2018; Argus et al., 2014b).

Remove interseismic strain accumulation associated with locking of the Cascadia subduction zone using an upgrade of the model of Li et al. (2018). The model is a superposition of

Average the daily estimates of GPS vertical displacements into monthly means centered at the center of each month from January 2006 to June 2021.

A total of 2705 (or 88.8 %) of the GPS stations remain after the choice of reference epoch, the 3

We compare GPS observations of vertical displacement against GRACE(-FO) estimates of solid Earth's elastic vertical displacement from terrestrial water, snow, and ice.

To compare to GRACE(-FO), we analyze JPL's 3° mascon solution (Release 6, Watkins et al., 2015; Wiese et al., 2016). The effect of glacial isostatic adjustment is removed from GRACE(-FO) products using ICE-6G_D model estimates (Peltier et al., 2018). The geocenter motion (degree 1) coefficient is using the technique of Sun et al. (2016) (Technical Note 13). Values of C20 (Earth's oblateness) and C30 (for months after August 2016) are substituted with SLR data (Loomis et al., 2019). We calculate solid Earth's elastic response by using the loading Love number of the Preliminary Reference Earth Model (Wang et al., 2012).

Estimates of GPS positions in ITRF2014 (Altamimi et al., 2016) are relative to the center of mass (CM) in the long term but relative to the center of the figure (CF) in the seasons (because ITRF2014 does not allow seasonal oscillations of CM). We therefore remove the long-term rate of CM relative to CF to transform the GRACE estimates in the long term from CF to CM (but do not remove seasonal oscillations of CM relative to CF so as to preserve the ITRF seasonal frame relative to CF). The annual signal of the geocenter (as realized by ITRF 2014) projected on the up component in North America on average explains 3 % of the GPS vertical displacement signal and can explain up to 20 % for certain sites.

GRACE(-FO) vertical displacement monthly estimates are derived as follows (e.g., Davis et al., 2004):

GPS vertical displacement estimates are evaluated against the ones derived from GRACE(-FO) to assist in identifying outliers or further corrections that may be needed. We employ a number of different metrics to evaluate the agreement between the two datasets and to determine whether to include it in the joint solution or not. Similar to Yin et al. (2020) we quantify correlation and variance reduction between GPS and GRACE(-FO) vertical displacements. The structure of surface mass periodic signals (e.g., annual cycles, trends) as picked up by the two measurement techniques also entails critical information regarding mis-modeled offsets and is evaluated as well.

This process flags sites that need correction and corroborates joint inversion's hypothesis (Argus et al., 2021) that a basic level of agreement is needed for the GPS data to be used to infer surface mass change.

First, we specify the level of agreement between the datasets by estimating the Pearson correlation coefficient between GPS and GRACE(-FO) time series. On average correlation is 62 %, but stations located on the west coast exhibit agreement higher than 80 %, which in most cases is driven by the larger annual signal amplitude there. A more detailed look into the correlation metric is performed to evaluate the agreement of GPS/GRACE(-FO) in retrieving the seasonal cycle amplitude in different watersheds. We fit and remove a trajectory model

We classify stations in watersheds and plot the GPS–GRACE(-FO) correlation coefficient (

In order to study the agreement between GPS and GRACE(-FO) in more detail, we split the time series of each station into non-overlapping intervals of 36 months and fit Eq. (2) for each station during each time window. Different time lengths of the GPS series may lead to misinterpretation of the geophysical content. For example, a station that has records only for the first 13 months out of the total 36-month window may reflect different fit constituents compared to a neighbor station with full records if the actual behavior of Earth's response changes during the 36-month window. Although in our dataset this case is rare, we proceed with deriving the rate (slope) and the annual cycles only for stations that have records for at least 28 out of the 36 months. We did not interpolate the series during the GRACE(-FO) gap; thus, the last time window reflects trends estimated using only GRACE-FO and GPS time series between June 2018 and 2021. As expected, GPS rates feature higher spatial variability than GRACE(-FO). However, both techniques capture large-scale quasi-periodic variations every 3 years (Fig. 3), agreement that is noteworthy. The effect of this metric to detect outliers is pronounced when the two techniques show flipped trends.

Regions with pronounced trend disagreement include the following.

Rates of vertical displacements derived by GPS and GRACE. The rates are calculated every 36 months (3 years) between 2006 and 2021.

Similarity in both amplitude and phase between two quantities is quantified via the variance attenuation factor (Gaspar and Wunsch, 1989; Fukumori et al., 2015).

The higher the agreement in phase and amplitude between GPS and GRACE(-FO), the closer the metric gets to 100 %.

Variance reduction between GPS and GRACE(-FO) vertical displacements.

We also compared the annual amplitudes of GPS and GRACE(-FO) vertical displacements (cosine and sine components in Eq. 2). This analysis was not informative for the presence of outliers or errors in the current data sample studied.

Overall, the screening process not only assisted in outlier detection, but it also allowed for a deeper look into the structure of vertical displacement periodic signals. We identified the need for antenna offset corrections (in sites located in the Great Lakes region); removed sites affected by glacial isostatic adjustment and interseismic modeling errors; and sites located at the Parkfield segment of San Andreas Fault.

With the updated dataset we are now ready to proceed with the uncertainty quantification of the GPS vertical displacement time series. We apply different error characterization schemes consisting of a root sum square of a random error, white noise error, power-law noise error (flicker noise and random walk), and spatially coherent error.

Residuals

The power distribution of residuals (and its agreement with noise models) is another popular way to quantify uncertainty of GPS time series (e.g., Klos et al., 2019; Argus et al., 2022). Typically, GPS series are evaluated for white, flicker, and random walk noise or a combination of them. Hector software (Bos et al., 2013) is used to estimate full noise covariance information by means of a maximum likelihood estimator. The covariance matrix

In this study, we consider three cases:

white noise (WN),

a combination of WN and flicker noise (WN

a combination of WN, FN, and random walk noise (WN

The common mode component (CMC) is derived following the processing scheme suggested by Kreemer and Blewitt (2021), which can be summarized as follows.

Input GPS displacement time series (referenced to September 2012) for

Derive each station's residuals by removing the trajectory part of the series (

Quantify the correlation coefficient

The median absolute deviation (MAD) is the absolute deviation around the median. For example, for a residual series res(

Derive the median slope estimator (ccs) using the Theil–Sen median trend. The ccs is the median trend of the

Derive the zero-distance intercept

Construct CMC: calculate the cumulative (

CMC is limited in providing a realistic error approximation in that the technique cannot isolate spatially correlated noise from signal (e.g., hydrology signals not described by the trajectory model are present in the residuals fed into CMC). Under the realistic assumption that a component of the high-frequency signal contained in CMC reflects real hydrological processes, we remove the contribution of surface hydrology using Global Land Data Assimilation System (GLDAS) (Rodell et al., 2004) vertical displacement estimates. GLDAS does not model deep groundwater and open surface water, so these signals remain in the residual (Scanlon et al., 2018). Vertical displacement estimates driven by surface hydrology are derived similarly to GRACE(-FO) (Sect. 2.2). We use Noah v2.1 monthly estimates of soil moisture storage given at 0.25° grids (Beaudoing and Rodell, 2020), convert the fields from terrestrial water storage (kg m

Vertical displacement uncertainty of each station is estimated by means of all the different approaches discussed in Sect. 3. Mean (

Different uncertainty quantification cases.

Noise amplitudes of GPS time series estimated using different techniques.

Probability density function of vertical displacement estimate uncertainty.

RMSE and WN exhibit a smooth transition among the regions, which indicates the presence of a spatially coherent regime signal mostly driven by hydrology (Fig. 6). The combination of WN

We obtain the relative likelihood of each uncertainty quantification method by estimating the probability density function (PDF) (Fig. 7). White noise has a flat power spectrum, having the same amplitude across frequencies. Estimating a best fit for a flat spectrum does not allow for capturing the long tail skew of the residuals (low frequency), which are biased towards their mean. Thus, the amplitude of white noise is smaller compared to the rest of the techniques (Table 1). Flicker and random walk noise models add to the long tail of the power distribution: that is, they allow more low-frequency noise, which explains the higher amplitude of the uncertainty when these two noise types are considered.

RMSE and WN show a 50 % probability of a station having an uncertainty (

The data product described in the paper is available on Zenodo (DOI:

GPS-derived vertical displacements are very useful for supplementing GRACE(-FO) gravity products to infer mass change signals at spatial scales smaller than what can typically be achieved with current satellite gravimetry alone (i.e.,

Several uncertainty quantification schemes have been tested to prescribe weights on GPS vertical displacement estimates that are needed for a joint inversion with GRACE(-FO) data. The average noise level indicated by RMSE is 2.8 mm. White noise average is 2.5 mm. The errors increase when lower frequencies are included in the noise estimation. When we account for flicker noise, one-third of the sites exhibit noise levels of up to 3 mm. The average noise increases significantly in the presence of random walk, as more power of the lower frequencies gets into the estimations, and the distribution of noise is more dispersed. In this case, half of the stations are prescribed with

The supplement related to this article is available online at:

AP outlined the methodology, performed the analysis, and wrote the paper. DFA processed the GPS time series and outlined the methodology to isolate surface mass loading from GPS time series. FWL and DNW helped with the GPS and GRACE(-FO) data screening and provided critical feedback and ideas. ME helped with the uncertainty quantification schemes. All authors reviewed and substantially edited the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

Maps were made with the Generic Mapping Toolbox (Wessel et al., 2019). We thank Corne Kreemer (UNR) for his feedback and Mike Heflin (JPL) for his insights on draconitic errors.

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA; grant no. 80NM0018D0004) with support from the GRACE-FO Science Team grant (80NM0018F0585).

This paper was edited by Kirsten Elger and reviewed by two anonymous referees.