Analysis of The Itsg-Grace Daily Models in The Determination of Polar Motion Excitation Function
Published Online: Oct 10, 2023
Page range: 105 - 121
Received: Mar 13, 2023
Accepted: Jul 10, 2023
DOI: https://doi.org/10.2478/arsa-2023-0008
Keywords
© 2023 Aleksander Partyka et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.
Mass redistributions, along with movements of the solid part of the Earth, such as tectonic plate movements and earthquakes, cause changes in the Earth’s rotational motion. The motion of the Earth’s axis of rotation relative to its surface, namely, polar motion (PM), is caused mainly by the global mass distribution of the Earth’s atmosphere, ocean, land hydrosphere, and cryosphere (Lambeck, 1980; Gross, 2015; Bizouard, 2020).
Disturbance of the Earth’s PM resulting from the changes in mass distribution of its surficial fluid layers (atmosphere, ocean, and continental hydrosphere) can be described with effective angular momentum (EAM) functions, being a modified linearized Liouville equation expressed as two equatorial components (χ1, χ2) and one axial component (χ3), and can be derived from observational data (Munk and MacDonald, 1960; Brzezinski, 1992). The χ1 and χ2 components of the EAM function describe the PM excitation caused by perturbing forces, while χ3 is related to the day-length fluctuations caused by these forces. The EAM function, depending on the factor disturbing it, is expressed as atmospheric angular momentum (AAM), oceanic angular momentum (OAM), or hydrological angular momentum (HAM). Analyses of the sources of terrestrial hydrological signals in PM excitation indicate that HAM can explain some of the changes in PM excitation after the effects of AAM and OAM are removed (Chen and Wilson, 2005; Jin et al., 2010; Sliwinska et al., 2022). HAM can be estimated from global models of the continental hydrosphere, measurements of changes in the Earth’s gravitational field, and climate models (Seoane et al., 2011; Meyrath and van Dam, 2016; Göttl, 2018).
Previous studies have shown that hydrological excitation is important in determining PM excitation (Dobslaw et al., 2010; Meyrath and van Dam, 2016). The impact of HAM signals on PM excitation from changes in mass distribution has been little studied than the impact of AAM and OAM signals, particularly if we consider oscillations with a period shorter than annual (Chen et al., 2000; Nastula et al., 2019). It has previously been shown that, at seasonal time scales, PM excitations from geophysical fluids (i.e., atmosphere, ocean, and hydrosphere) agree well with geodetic observations of PM in the χ2 component (Göttl et al., 2018, Chen et al., 2012). For the χ1 component, considerable discrepancies were shown between the PM excitations and the geodetic observations (Seoane et al., 2011; Chen et al., 2012; Wińska et al., 2017).
Since the Gravity Recovery and Climate Experiment (GRACE) mission began in 2002, scientific teams at the Center for Space Research (CSR, Austin, USA), GeoForschungsZentrum (GFZ, Potsdam, Germany), and Jet Propulsion Laboratory (JPL, Pasadena, USA) have been providing GRACE-based monthly gravity field solutions (Tapley, 2004). These data can be used to interpret PM excitation perturbations caused by changes in the mass distribution of continental hydrosphere, focusing on different oscillations and periods. Compared with HAM from geophysical models of the hydrosphere, GRACE-based HAM has been shown to be more consistent with the hydrological signal in PM excitation obtained from geodetic observations (Chen et al., 2012; Nastula et al., 2019; Sliwinska et al., 2020; Winska et al., 2017). In addition to monthly gravity field solutions, the Institute of Geodesy at Graz University of Technology (ITSG) also produces daily solutions, providing the only available GRACE-based data at daily resolution (Kurtenbach et al., 2012). Daily solutions based on the GRACE data have not yet been considered for tackling the components of HAM.
The main aim of the current study is to assess the usefulness of the daily gravity field models from the GRACE mission delivered by the ITSG as ITSG-Grace2016 (abbreviated here as ITSG 2016) and ITSG-Grace2018 (abbreviated here as ITSG 2018) to determine the equatorial components (χ1 and χ2) of HAM. These are compared with HAM derived from the hydrological signal in geodetically observed PM excitation (called geodetic residuals [GAO]) and HAM based on hydrological land surface discharge model (LSDM). We analyzed HAM at daily temporal resolution, which allows us to determine oscillations with higher frequencies than the more commonly used monthly data. This provides an opportunity to investigate the existence of a sub-monthly signal in the HAM data.
The structure of this paper is as follows. Section 2 describes data and methodology: section 2.1 presents the ITSG 2016 and ITSG 2018 daily gravity field models and describes the method of computing HAM excitation from the ITSG solutions, section 2.2 includes characteristics of the geodetic residuals and sources of the data, section 2.3 describes features of HAM based on LSDM, and section 2.4 explains the steps of the calculations performed.Section 3 presents the results of our study: section 3.1 shows the time series comparison, section 3.2 presents amplitude spectra of the series, and section 3.3 includes validation of series with root mean square error (RMSE), relative explained variance (Varexp), and correlations. Finally, section 4 provides a discussion of the results and presents the conclusions of our study.
Daily ITSG 2016 and ITSG 2018 gravity models were used to calculate χ1 and χ2 of HAM (Table 1). We selected these two models because of the identical availability of their data during our chosen period, giving rise to the possibility of comparing them. These models are characterized by an increase in temporal resolution to 1 day compared with more common monthly data. The increase in resolution was made possible by the application of a Kalman smoother estimation procedure (Kurtenbach et al., 2012).
Models used to produce the ITSG 2016 and ITSG 2018 series
| ITSG 2016 | ITSG 2018 | |
|---|---|---|
| Earth rotation | IERS 2010 | IERS 2010 |
| Moon, sun, and planets ephemerides | JPL DE421 | JPL DE421 |
| Earth tide | IERS 2010 | IERS 2010 |
| Ocean tide | EOT11a | FES2014b, co-estimated |
| Pole tide | IERS 2010 (constant mean pole) | IERS 2010 (linear mean pole) |
| Ocean pole tide | Desai 2004 (IERS 2010) | Desai 2004 (IERS 2010, linear mean pole) |
| Atmospheric tides (S1, S2) | van Dam, Ray (2010) | AOD1B RL06 |
| Atmosphere and ocean dealiasing | AOD1B RL05 | AOD1B RL06 |
| Relativistic corrections | IERS 2010 | IERS 2010 |
| Permanent tidal deformation | Included (zero tide) | Included (zero tide) |
Based on information from TU Graz website (
ITSG 2016 and ITSG 2018 differ in terms of the background models used to produce them, for example, the ocean tide, pole tide, ocean pole tide, atmospheric tides, atmosphere and ocean de-aliasing, and sub-monthly continental hydrology (Table 1) (Mayer-Gürr et al., 2016, 2018). ITSG 2018 is the most recent realization of the GRACE solution provided by ITSG.
The ITSG data used for our calculations were taken from Graz University of Technology (TU Graz) resources (
The χ1, χ2 components of HAM excitation were calculated from the ITSG solutions based on their proportionality to changes in the coefficients AC21, AS21 of the geopotential (Gross, 2015):
In the ΔC21, ΔS21 coefficients used in our study, the impact of atmospheric and oceanic nontidal mass variations was removed using GRACE Atmosphere and Ocean De-Aliasing Level-1B (AOD1B) data provided by GFZ. Consequently, GRACE-based HAM series included not only signals from the continental hydrosphere, but also some contributions from glacial isostatic adjustment (GIA) and barystatic sea-level variations resulting from inflow of water from land into oceans (so-called sea-level angular momentum [SLAM]). Nevertheless, GIA should affect only trends in PM excitation series, which are not analyzed in our study. To keep the highest consistency between GRACE-based HAM and HAM computed from hydrological models, one should add the contribution of SLAM to the series calculated from the models.
The HAM series were evaluated based on comparison with the geodetic residuals, which represent the difference between geodetic angular momentum (GAM) obtained from precisely measured pole coordinates and the sum of AAM and OAM derived from geophysical models as follows:
GAO reflects mainly the impact of the land hydrosphere on PM excitation, but similar to HAM obtained from GRACE ΔC21, ΔS21 coefficients, it also includes SLAM and GIA signals.
The following data sets were used to calculate GAO:
χ1 and χ2 equatorial components of the GAM series—EOP 14 C04 (IAU2000A), derived from the International Earth Rotation and Reference System Services (IERS) ( χ1 and χ2 equatorial components of the AAM series provided by GFZ and based on the European Centre for Medium-Range Weather Forecasts (ECMWF) model, available at a temporal resolution of 3 h (data available at χ1 and χ2 equatorial components of the OAM series provided by GFZ and based on the Max Planck Institute Ocean Model (MPIOM), available at a temporal resolution of 3 h (data available at
ITSG-based series were compared with the sum of HAM and SLAM based on the LSDM hydrological model. The LSDM model was processed with the atmospheric and oceanic data from the ECMWF and MPIOM models, respectively (Dill et al., 2008). HAM based on LSDM was calculated by GFZ from terrestrial water storage and included the representation of soil moisture, shallow groundwater, snow coverage, and surface water stored in rivers and lakes. SLAM was calculated by GFZ on the basis of the spatially variable sea-level variations as inferred from globally integrating atmospheric and terrestrial masses in LSDM and ECMWF with consideration of self-attraction (Dobslaw and Dill, 2018). Global mass conservation was maintained in the AAM, OAM, HAM, and SLAM series produced by GFZ. The HAM and SLAM series used in our study were taken from GFZ resources (
Our calculations and analyses were performed according to the following steps (Figure 1):
application of a Gaussian filter with full width at half maximum (FWHM) equal to 2 to remove oscillations with periods shorter than 1 day (because the data we used have temporal resolution from 3 h to 1 day); interpolation of all time series to the same moments of time in order to obtain uniform data; removal of annual, semiannual, and terannual oscillations from the time series in order to analyze nonseasonal variations; this was done by fitting the second-order polynomial and sum of sinusoids with periods of a year, half a year, and a third of a year to the series; application of a fourth-order Butterworth filter with bandpass from 4 days to 120 days to the series in order to narrow our analysis to oscillations with periods below 120 days; use of fast Fourier transform to calculate the amplitude spectra of the series; and evaluation of ITSG-based HAM and HAM + SLAM by calculating RMSE, Varexp, and correlation coefficients with GAO as a reference.
Figure 1.
Methodology flowchart, which shows all the calculation steps
Varexp describes the variance compatibility between the reference series (
Figure 2 shows that the GAO series has the highest amplitude and the ITSG-based series have considerably lower amplitudes than the GAO series. However, it is worth highlighting that the amplitudes of the ITSG-based series are clearly higher than those of the HAM + SLAM series. We can also see in Figure 2 that the ITSG 2016-based series has higher amplitudes than the ITSG 2018-based series. This is confirmed by the statistics shown in Table 2, which include standard deviation (STD), minimum (min), and maximum (max) values for the ITSG-based series, HAM + SLAM, and GAO. The ITSG 2016-based series has higher STD, higher maximum, and lower minimum values than the ITSG 2018-based series. This shows that the ITSG 2016-based series is closer to GAO than the ITSG 2018-based series in terms of time series variability.
Figure 2.
Comparison of the (a) χ1 and (b) χ2 components of nonseasonal series of GAO, HAM + SLAM, and HAM obtained from ITSG 2016 and ITSG 2018
Comparison of STD, min, and max values from χ1 and χ2 nonseasonal series of GAO, HAM + SLAM, and HAM obtained from ITSG 2016 and ITSG 2018
| STD [mas] | Min [mas] | Max [mas] | ||||
|---|---|---|---|---|---|---|
| Series | χ1 | χ2 | χ1 | χ2 | χ1 | χ2 |
| ITSG 2016 | 5.48 | 7.28 | −15.91 | −23.15 | 19.46 | 24.25 |
| ITSG 2018 | 3.90 | 5.05 | −12.04 | −14.11 | 14.03 | 17.84 |
| GAO | 13.71 | 21.76 | −67.01 | −76.18 | 42.90 | 76.10 |
| HAM + SLAM | 1.99 | 2.24 | −6.65 | −7.38 | 6.68 | 7.18 |
Amplitude spectra of all considered series are plotted in Figures 3 and 4. In addition, Figures A1 and A2 of the Appendix show the same spectra, but with GAO excluded for better visibility.
Figure 3.
Comparison of (a) χ1 and (b) χ2 amplitude spectra series of GAO, HAM + SLAM, and HAM obtained from ITSG 2016 and ITSG 2018 from 20 to 120 days
Figure 4.
Comparison of (a) χ1 and (b) χ2 amplitude spectra series of GAO, HAM + SLAM, and HAM obtained from ITSG 2016 and ITSG 2018 from 2 to 20 days
Figure 3 presents the amplitude spectra of all series, which were computed for the periods between 20 and 120 days (Figure 3a and b for χ1 and χ2, respectively). Figure 4 focuses on the period between 2 and 20 days for better visibility of oscillations with shorter periods (Figure 4a and b for χ1 and χ2, respectively). Figure 3 shows that the GAO spectrum has the highest amplitudes and the HAM + SLAM series has the lowest amplitude, well below that of the other series. The ITSG-based series have several peaks in common with the GAO series. In χ1, the common peaks are found at 32.78 days, 98.34 days, and around 61 days (60.88 days for GAO, 62.37 days for ITSG). In χ2, the common peaks are found at 56.82 and 77.48 days.
Figure 4 shows the common peaks for ITSG 2016 and GAO around Day 14. For χ1, the peaks are at 13.67 days in GAO and 13.74 days in ITSG 2016. For χ2, the peaks are at 13.67 days in GAO and 13.53 days in ITSG 2016. These peaks may be related to the Earth’s tide as they are positioned very close to the value of fortnightly ocean lunar tide with a period of 13.66 days. This value equals half of the Moon’s declination changes with a period of 27.33 days (Bizouard, 2020; Sidorenkov, 2009).
In this section, we evaluate the ITSG-based HAM series and the HAM + SLAM series by analyzing RMSE, Varexp, and correlations. These statistical indicators allow intercomparison of the time series and will help determine the similarities between the ITSG 2016-based series, ITSG 2018-based series, or the HAM + SLAM series and the GAO reference series.
The RMSE values of ITSG 2016-based series, ITSG 2018-based series, and HAM + SLAM series are shown in Figure 5. In χ1, the ITSG 2016-based series and ITSG 2018-based series have the same RMSE value (13.0 mas). Moreover, HAM + SLAM has an RMSE of 13.7 mas, which is slightly higher than for the ITSG-based series. In χ2, the RMSE values are higher and have a wider range than in χ1. The lowest value is RMSE of ITSG 2016-based series (19.4 mas) and the highest is RMSE of HAM + SLAM series (21.6 mas), which is the same as in χ1.
Figure 5.
RMSEs of ITSG 2016-based series, ITSG 2018-based series, and HAM + SLAM calculated using GAO as the reference series. The exact RMSE values are given above the bars for each series.
All Varexp values are shown in Figure 6. In χ1, ITSG 2016 and ITSG 2018 have Varexp of 10% and HAM + SLAM has an explained variance of 0%. In χ2, the explained variance values of all series are highly variable. ITSG 2016 has the highest variance (21%), whereas ITSG 2018 has a lower explained variance (12%). HAM + SLAM has the lowest explained variance (1%), which is also found for χ1.
Figure 6.
Varexp of ITSG 2016-based series, ITSG 2018-based series, and HAM + SLAM calculated using GAO as the reference series. The exact explained variance values are given above the bars for each series.
The final part of the analysis considers the correlation between all considered series. Correlations of ITSG-based series with reference series are statistically significant (the critical value of the correlation coefficient for 100 independent points is 0.17, and in our case, there are 500 independent points).
Figure 7 shows the calculated correlations for each time series. The ITSG-based series are better correlated with GAO in χ2 than in χ1, a result which is similar to previous studies (Seoane et al., 2011; Chen et al., 2012; Winska et al., 2017). If we compare the correlation coefficients of the ITSG-based series and the HAM + SLAM series with GAO, we observe a stronger correlation for the ITSG-based series. For χ1 (Figure 7a), the series have the following correlation coefficients: ITSG 2016, 0.35; ITSG 2018, 0.35; and HAM + SLAM, 0.07. For χ2 (Figure 7b), the series have the following correlation coefficients: ITSG 2016, 0.51; ITSG 2018, 0.41; and HAM + SLAM, 0.12. The correlations between ITSG-based series and GAO are similar for χ1; however, ITSG 2016 is better correlated with GAO than ITSG 2018 for χ2.
Figure 7.
Correlations between ITSG 2016-based series, ITSG 2018-based series, GAO, and HAM + SLAM in (a) χ1 and (b) χ2
The advantage of ITSG 2016 over ITSG 2018 in χ1 may be due to the difference in the versions of the AOD1B models. The AOD1B RL05 model was used in ITSG 2016 and the AOD1B RL06 model in ITSG 2018. Compared to AOD1B RL05, changes such as increased spatial and temporal resolution, changing the ocean model from the Ocean Model for Circulation and Tides (OMCT; Dobslaw et al., 2013) to MPIOM (Jungclaus et al., 2013), and improved long-term consistency (Dobslaw et al., 2017a, b) have been made in AOD1B RL06. The correlations we examined after swapping AOD1B releases in the ITSG series may indicate better consistency of AOD1B RL05 with GAO compared to AOD1B RL06 (Table A1).
On comparing the ITSG-based series with HAM + SLAM, it is clear that ITSG 2018 is better correlated with HAM + SLAM than ITSG 2016 (Figure 7). The correlation values for χ1 (Figure 7a) are 0.24 for ITSG 2016 and 0.36 for ITSG 2018. The correlation values for χ2 (Figure 7b) are 0.25 for ITSG 2016 and 0.33 for ITSG 2018. There are no significant differences between χ1 and χ2 within each series, with values of 0.24 and 0.25 for χ1, and χ2, respectively, for ITSG 2016, and values of 0.36 and 0.33 for χ1 and χ2, respectively, for ITSG 2018.
The main aim of this study was to evaluate the usefulness of ITSG daily gravity field models in the determination of HAM at nonseasonal time scales, which allows oscillations to be determined with higher frequencies than the more commonly used monthly data. A comparison was made with the hydrological signal in the PM excitation (GAO) and HAM + SLAM excitation series obtained from the LSDM hydrological model.
Determination of oscillations with higher frequencies provided an opportunity to investigate the existence of a sub-monthly signal in the HAM data. The results are promising, but not entirely satisfactory. Although ITSG-based HAM series have several times smaller amplitudes than GAO, they are still higher than HAM + SLAM obtained from LSDM. Analysis of amplitude spectra for oscillations with periods between 2 and 120 days showed that the ITSG-based HAM series, despite smaller amplitudes, have several common peaks with GAO. Of special note is the joint peak visible for ITSG 2016-based HAM and GAO around 14 days, which may be caused by the Earth’s tides.
Our study showed that in the case of the χ2 component determined from ITSG-2018, deterioration in compliance was observed with GAO compared with the ITSG-2016-based series (lower correlation, lower relative explained variance, and higher RMSE). These differences might result from changes in processing between the ITSG 2016 and ITSG 2018 models.
The fact that the correspondence with GAO is higher after using the previous ITSG release (ITSG 2016) is a little unexpected. One of the factors why the level of agreement with GAO changed after the transition from the old and the new ITSG solution may have been the use of different AOD1B releases in ITSG 2016 and ITSG 2018 to eliminate atmospheric and oceanic nontidal contributions. To deepen this issue, we removed AODB RL06 from ITSG 2016 (instead of AODB RL05) and AODB RL05 from ITSG 2018 (instead of AODB RL06) and computed correlations with GAO (Table 3). It turned out that changing the AOD1B product had no noticeable effect on the correlations for χ1, but it affected the χ2 correlations to a greater extent. What is interesting, the change of AOD1B data from new to old in ITSG 2018 improved correlation coefficients with GAO, while the change of AOD1B data from old to new in ITSG 2016 worsened correlation coefficients with GAO. The ITSG 2018-based series with the AOD1B RL05 application appeared to achieve the highest correlation with GAO of all the series in χ2. It is a little puzzling that using the more recent AOD1B data does not provide improved consistency with GAO. Nevertheless, it should be kept in mind that in our study, we analyzed de-seasoned series, which were additionally filtered with the Butterworth filter to isolate oscillations with periods between 4 and 120 days. Therefore, this conclusion may not be valid for shorter oscillations as AOD1B data were originally processed in 3- and 6-h temporal resolution.
Correlation coefficients between GAO and ITSG-based HAM series computed from ITSG 2016, ITSG 2018, ITSG 2016 with changed AOD1B data (AOD1B RL06 instead of RL05), and ITSG 2018 with changed AOD1B data (AOD1B RL05 instead of RL06)
| ITSG 2016 | ITSG 2016 RL06 | ITSG 2018 | ITSG 2018 RL05 | |
|---|---|---|---|---|
| χ1 | 0.35 | 0.31 | 0.35 | 0.32 |
| χ2 | 0.51 | 0.30 | 0.41 | 0.56 |
In conclusion, it can be stated that daily GRACE solutions provided by ITSG can be useful in determination of HAM at nonseasonal time scales. Although they do not ensure full compliance with GAO, their use is a much better alternative than the usage of hydrological models.
The correlations between ITSG-based HAM series and GAO are statistically significant and higher than between HAM + SLAM and GAO. The ITSG-based series have a correlation of the same value in χ1 (0.35). In χ2, the ITSG 2016-based series has a higher correlation with GAO than the ITSG 2018-based series (0.51 and 0.41, respectively).
This study showed that the GAO series have the largest amplitudes and the highest STD (13.71 mas for χ1 and 21.76 mas for χ2) of all series, while the HAM + SLAM series have the smallest amplitudes and the lowest STD (1.99 mas for χ1 and 2.24 mas for χ2). The ITSG-based series have smaller amplitudes than GAO, but larger amplitudes than HAM + SLAM. The STD analysis showed that the ITSG 2016-based series is closer to GAO (5.48 mas for χ1 and 7.28 mas for χ2) than the ITSG 2018-based series (3.90 mas for χ1 and 5.05 mas for χ2).
In general, agreement between ITSG-based HAM and GAO, expressed by the higher correlation coefficient, lower RMSE, and higher Varexp, is better in χ2 than in χ1, which is consistent with the findings from other studies (Göttl et al., 2018; Nastula et al., 2019; Seoane et al., 2011).
It should be recalled that, although GRACE was operational between 2002 and 2017, our study focused on analysis for the period between 2004 and 2011. This was because some gaps in GRACE data are present at the beginning and at the end of the mission, which also affect the daily solutions.
Because ITSG daily gravity field models (ITSG 2016 and ITSG 2018) can be useful to determine the equatorial components (χ1 and χ2) of HAM at nonseasonal time scales, studies exploiting GRACE data for interpretation of PM excitation should also be continued with inclusion of data from the mission successor, GRACE Follow-On.