Keywords

1 Introduction

Mass redistribution within the Earth’s fluid envelope including the atmosphere, the oceans, and the terrestrial water storage elastically deforms Earth’s crust hence inducing displacements of geodetic markers in excess of 1 cm at sub-daily to seasonal and even inter-annual timescales. This contribution focuses on high-frequency deformation induced by atmospheric tides. Unlike ocean tides that are mostly excited by the gravitational pull of the Moon, atmospheric tides are mostly excited by the Sun, in particular, the periodic absorption of infrared radiation by water vapor in the troposphere and ultraviolet radiation by ozone in the stratosphere, as well as large-scale latent heat release. Inspired by Ray et al. (2021), we have calculated the response of atmospheric pressure anomalies at different frequencies along the \(-169\,^{\circ }\)W meridian that intersects the least with land (see Fig. 1). We note sharp spectral lines at integer overtones of the solar diurnal wave S\(_{1}\) as well as its side-bands, and the fact that the highest power spectral density (PSD) values are found in the equatorial belt. The high-frequency waves to which the atmospheric pressure response is the strongest are the S\(_{1}\) and the S\(_{2}\). These variations are not artefacts, rather manifestations of a well-studied phenomenon called atmospheric tides, which are responsible for high-frequency peak-to-peak pressure anomalies in excess of 500 Pa. Atmospheric tides may be studied by manifestations thereof, which are mainly in atmospheric density and its spatial gradients. For space geodesy, atmospheric tides induce temporal variations in (i) the gravity field (e.g., Boy et al. 2006); (ii) the deformation of Earth’s crust due to the loading exerted by the atmospheric mass (e.g., Petrov and Boy 2004); (iii) the coefficients with which we describe how refraction affects signals traversing Earth’s electrically neutral atmosphere (e.g., Jin et al. 2008); and (iv) the motion of Earth relative to its spin axis, that is, polar motion and \(\mathit {UT1}\mbox{--}\mathit {UTC}\) or length-of-day (e.g., Girdiuk 2017), as well as components of Earth’s nutation (e.g., Schindelegger et al. 2016). While atmospheric tides are responsible for crustal deformation at the sub-cm-range (see Fig. 2), they should be considered during space geodetic data analysis to mitigate aliasing artefacts. Herein, for the waves that induce the largest mass anomalies in the sub-diurnal frequency band we assess the extent to which the predictions of tidal mass loads and the associated crustal displacements differ depending on the numerical weather model (NWM), as well as how much they differ as a function of time. We retrieve mass anomalies from five state-of-the-art NWMs, namely, ERA5, MERRA2, JRA55, ECMWF’s IFS, and ICON.

Fig. 1
figure 1

Power spectral density for ERA5-derived total atmospheric pressure along the \(-169^{\circ }\)W meridian, which crosses the least amount of land according to ERA5. Light pixels indicate latitudes where atmospheric pressure has a strong response at the corresponding frequency

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Fig. 2
figure 2

Radial harmonic atmospheric tidal loading displacement amplitudes employing ERA5 fields spanning the period 1979.0–2022.0, in the center of mass isomorphic reference frame. Shown are the strongest waves in the diurnal (a), semi-diurnal (b), and ter-diurnal band (c and d)

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In this contribution, we assess the extent to which displacements induced by harmonic atmospheric mass variation driven primarily by solar irradiance vary (i) over time, and (ii) between mass variation models. Section 2 describes the atmospheric and oceanic tidal amplitudes from NWM and a barotropic ocean tide model, and presents a relative comparison. Section 3 describes how tidal displacement amplitudes differ between models. Section 4 outlines the estimation of partial tide modulations and discusses the associated estimates from a filter solution. Finally, we summarize our work and draw conclusions in Sect. 5.

2 Atmospheric and Oceanic Tidal Mass Anomalies

To predict loading-induced site displacements, accurate knowledge of the instantaneous mass anomaly is required. In the atmosphere, mass anomalies are inferred from surface pressure anomalies. Ocean mass anomalies are typically obtained by sea-surface heights deduced from the analysis of satellite altimetry observations or by running a model that solves hydrodynamic equations numerically (Dobslaw and Thomas 2005). Since the magnitude of pressure fluctuations depends on the altitude (e.g., for S\(_{1}\)) and the orography of the models, we calculate the pressure at a reference orography employing the three-dimensional atmospheric density which at a given site and epoch is a function of temperature, pressure, specific humidity and geopotential (Dobslaw 2016). In this work, we calculate atmospheric density variations from two operational models, ECMWF’s IFS (three-hourly fields on 9 km grids) and DWD’s ICON (three-hourly fields on 13 km grids), and three reanalysis models, ECMWF’s ERA5 (hourly fields on 31 km grids, Hersbach et al. 2020), NASA’s MERRA2 (hourly fields on 50 km grids, Gelaro et al. 2017), and JMA’s JRA55 (three-hourly fields on 55 km grids, Kobayashi et al. 2015). Although a considerable fraction of the observations assimilated in these NWM is identical, the underlying data assimilation system as well as the spatio-temporal resolution are largely different ranging from meso-\(\upbeta \) to meso-\(\upgamma \) scale.

Following Balidakis et al. (2022), we have estimated harmonic amplitudes based on several batches of the aforementioned NWM so that we may assess the extent to which atmospheric forcing variations project into harmonic sea-surface heights predicted by the barotropic Tidal Model forced by Ephemerides (TiME, Sulzbach et al. 2021), where self-attraction and loading effects of the ocean mass are rigorously considered. In particular, we focus on the following waves: \(\uppi _{1}\), P\(_{1}\), S\(_{1}\), K\(_{1}\), \( \operatorname *{\psi }_{1}\), M\(_{2}\), T\(_{2}\), S\(_{2}\), R\(_{2}\), K\(_{2}\), T\(_{3}\), S\(_{3}\), R\(_{3}\), S\(_{4}\), S\(_{5}\), and S\(_{6}\).

We have estimated atmospheric forcing harmonics and performed TiME simulations (Sulzbach et al. 2021; Balidakis et al. 2022) where we varied the atmospheric forcing by adopting the following scenarios

  1. 1.

    ERA5a: ECMWF’s ERA5 (1979.0–1982.0);

  2. 2.

    ERA5b: ECMWF’s ERA5 (1989.0–1992.0);

  3. 3.

    ERA5c: ECMWF’s ERA5 (1999.0–2002.0);

  4. 4.

    ERA5d: ECMWF’s ERA5 (2009.0–2012.0);

  5. 5.

    ERA5e: ECMWF’s ERA5 (2019.0–2022.0);

  6. 6.

    ECMWF’s IFS (2019.0–2022.0);

  7. 7.

    NASA’s MERRA2 (2019.0–2022.0);

  8. 8.

    JMA’s JRA55 (2019.0–2022.0); and

  9. 9.

    DWD’s ICON (2019.0–2022.0).

The choice of three-year batches is not random. The precision of the harmonic amplitudes increases with the data span. We estimated harmonics for the 16 waves of interest employing variable data spans: from the theoretical minimum of one year of hourly data up to two decades. We found that the increase in precision is quadratic for all diurnal waves as well as for S\(_{2}\) within the first three years and linear afterwards. For instance, the global RMS between S\(_{1}\) harmonics estimated based on two decades of ERA5 and only one year of ERA5 is 3.5 Pa, on average. The RMS between the 20-year estimates and three-year estimates is 1.7 Pa, suggesting that the ability to predict pressure anomalies of a three-year estimate is about twice as good compared to an one-year estimate. While we have assessed the temporal tidal variations in atmospheric and ocean bottom pressure from some of the other reanalysis NWM in the framework of (Shihora et al. 2023), here we opt to work with ERA5 driven by its higher reliability (e.g., Ray et al. 2023). Due to the fact that the wind stress contribution to the ocean tide excitation process is considerably smaller in comparison to the contribution of pressure, wind stress harmonics were not used to force these TiME experiments. To evaluate the differences between harmonics estimated employing different data sets, we calculate the RMS which is defined by (e.g., Shihora et al. 2022):

(1)

where \(\zeta _j^k\) is the complex representation of the mass anomaly corresponding to wave j from data set k, which in turn runs over the harmonic amplitude estimates from different data sets up to K. Harmonics with the superscript \(\mathit {ref}\) stem from the analysis of either longer time series or the combination of the different ensemble members. We have calculated the atmospheric and oceanic mass anomaly RMS upon varying (i) the temporal range, and (ii) the NWM based on which the amplitudes are estimated. For the former, hereinafter “epoch” ensemble, we have employed harmonics from scenarios 1–5, and for the latter, hereinafter ”NWM” ensemble, we have employed harmonics from scenarios 5–9. For scenarios 1–5, we choose ERA5 since it features the highest spatio-temporal resolution among the reanalysis NWM within our ensemble. While the assimilation system employed for the production of these data sets does not change, the type, quality, and number of observations ingested within IFS model cycle 41r2 does vary. Together with differences in the input data, climate-related low-frequency changes in parameters such as water-vapor (trends between \(-0.10\,\mathrm {kg}\,\mathrm {m}^{2} \mathrm {a}^{-1}\) and \(0.13\,\mathrm {kg}\,\mathrm {m}^{2} \mathrm {a}^{-1}\) based on ERA5 from 1979 onward) and ozone (trends between \(-0.68\,\mathrm {DU}\ \mathrm {a}^{-1}\) and \(0.41\,\mathrm {DU}\ \mathrm {a}^{-1}\) based on ERA5 from 1979 onward) concentration give rise to the differences illustrated in the 1st column of Fig. 3. For the “NWM” ensemble, climate-related variations should not contribute to the differences we observe, rather only discrepancies induced by the assimilation system and the physical formulation of the NWM. The TiME configuration (\(\frac {1}{12^{\circ }}\) mesh, see Balidakis et al. 2022) is identical for all nine scenarios and reflects only differences in the harmonic atmospheric forcing (see 2nd column of Fig. 3). By and large, the assumption we make is that discrepancies within the “epoch” ensemble are due to climate change and availability of observations, whereas discrepancies within the “NWM” ensemble are due to differences in the data assimilation system, spatial resolution, and model physics.

Fig. 3
figure 3

Atmospheric pressure (1st row) and TiME-derived sea-surface height (2nd row) harmonic RMS following Eq. 1 (all waves), where the data span (left) and the model (right) of the atmospheric forcing have been varied. Atmospheric tidal loading displacement harmonic RMS are also shown in the radial direction (3rd row)

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Harmonic atmospheric pressure discrepancies from the “epoch” and “NWM” ensemble are illustrated in Fig. 3. For the “epoch” ensemble, the average RMS is 10.5 Pa. We find large discrepancies in excess of 30 Pa in the Bering Sea and the North Atlantic Ocean. We also find RMS values over 20 Pa over the North American Great Plains. For the “NWM” ensemble, we observe an average RMS of 11.3 Pa with spatial clusters exceeding 25 Pa over the ocean, similar to the “NWM” ensemble, in the Bering Sea and North Atlantic Ocean as well as the land clusters exceeding 30 Pa in the Andes and the Amazon catchment. The ensemble member responsible for the increased RMS over the equatorial land regions (South America, Central Africa, and Indonesia) is JRA55. Moreover, we observe slightly larger inter-model discrepancies in some regions with steep orographic gradients, which we attribute in large part to the representativeness of the lower-resolution models since the procedure to interpolate pressure given model or pressure level data is very accurate (Dobslaw 2016).

The RMS of the sea-surface height harmonic amplitudes is shown in the 2nd row of Fig. 3. Based on our simulations, we find that varying the NWM (5.2 mm, on average) has a larger impact on the sea-surface height in comparison to varying the period based on which the atmospheric tidal amplitudes were estimated (1.8 mm, on average). We observed deviations in excess of 50 mm at Timor Sea, Bristol Channel, and Hecate Strait. Further, the largest deviations we observe upon varying the time span of the forcing data exceeds 1 cm only at Sea of Okhotsk and the ice shelf at Ross Sea.

The atmospheric forcing harmonic discrepancies do not correlate to the TiME-derived sea-surface height predictions in the spatial domain (see 1st and 2nd row of Fig. 3), since excitations at the frequencies considered here typically lead to hemispheric waves with largest amplitudes at the coasts.

Moreover, we have calculated the pair-wise RMS of the harmonic variations (see Fig. 4). For atmospheric pressure, we find the largest discrepancies between early ERA5 data and other non-ECMWF models, namely MERRA2 and JRA55, and the best agreement between IFS, ICON, and recent ERA5 data. For sea-surface heights predicted by TiME, we observe large discrepancies between early ERA5 data and all non-ERA5 experiments. The differences between IFS and late ERA5 data are among the smallest, which justifies the use of current IFS and late ERA5 data in the same analysis; this is certainly due to the fact that the ERA5 system is very close to the operational IFS, also in terms of in situ data assimilation. While the largest discrepancies are found in the “NWM” ensemble, several pairs of the “epoch” ensemble have higher RMS.

Fig. 4
figure 4

Sorted harmonic discrepancies’ RMS for the pressure anomalies (1st row) and the sea-surface height variations (2nd row), following Eq. 1. RMS for the radial displacements is shown in the 3rd row. In purple shown are the pairs of the “epoch” ensemble and in green shown are the pairs of the “NWM” ensemble

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3 Tidal Loading Displacements

Approximating the loading mass anomalies as an infinitesimally thin layer, we calculate loading displacement variations by convolving the mass anomaly harmonic amplitudes with load Green functions, following Dill and Dobslaw (2013).

We note that atmospheric loading mass anomalies over the oceans as the atmospheric contribution to ocean tides is not considered, but will be explicitly treated with a global ocean tide model (Sulzbach et al. 2021) for all relevant frequencies, thereby making any assumption about an inverse barometric response of the ocean superfluous. We have calculated displacements induced by the atmospheric harmonic mass loads described in Sect. 2. In Fig. 2 we present the radial harmonic atmospheric tidal loading displacement amplitudes of the strongest atmospheric tidal waves that belong to the diurnal, semi-diurnal and ter-diurnal species, in the center of mass isomorphic frame we calculated employing ERA5 hourly fields spanning the period 1979.0–2022.0. We note that the radial displacements in the ter-diurnal band are about one order of magnitude smaller in comparison to the horizontal displacements in response to the S\(_{1}\) and S\(_{2}\) waves, and another three times smaller in comparison to the associated radial displacements. Moreover, we observe that while the spatial pattern for the S\(_{1}\) and S\(_{2}\) in-phase and quadrature amplitudes is degree-one and degree-two sectorial harmonic for all coordinate components, for the upper and lower sideband of S\(_{3}\) it is tesseral of degree/order 4/3 for the radial and eastward component and 5/3 for the northward component.

We find the largest RMS in the radial coordinate component, since the radial Green’s function assigns a considerably larger weight to mass anomalies in comparison to the tangential Green’s function, especially in the near-field. The largest discrepancies are found between MERRA2 and ERA5a for S\(_{1}\) (0.16 mm) and S\(_{2}\) (0.18 mm for MERRA2-ERA5a). The individual RMS for all other waves is relatively high for P\(_{1}\), K\(_{1}\), M\(_{2}\), R\(_{2}\), K\(_{2}\), S\(_{3}\), however, below 0.05 mm for all pairs. We note that in the ter-diurnal band, while the upper sideband of the S\(_{3}\), R\(_{3}\), and its lower sideband T\(_{3}\) feature larger and more spatially coherent mass anomalies than S\(_{3}\), the corresponding displacements are not as large. The “typical” wavelength of a ter-diurnal wave is shorter than that of a diurnal approximately by a factor of three, and also shorter than a semi-dirunal approximately by a factor of 3.2. Moreover, the Stokes coefficients are reduced by a factor \((2n + 1)^{-1}\), where n denotes the expansion degree. So, if the wavelength is shorter, the weight typically assigned to the coefficients is smaller by a factor of 3/1 and 3/2, in comparison to the diurnal and semi-diurnal band, respectively. Also, relatively small ter-diurnal amplitudes are partly due to the fact that in the CM a positive mass load placed at a spherical distance larger than about \(70^{\circ }\) will induce an upward displacement and the harmonic amplitudes of R\(_{3}\) and T\(_{3}\) feature a tesseral spherical harmonics pattern of degree/order 4/3. For the eastward coordinate component, the inter-model wave-wise discrepancies are a function of the displacement amplitude, that is, we find RMS up to 0.10 mm for S\(_{1}\) (MERRA2-ERA5a), up to 0.04 mm for S\(_{2}\) (JRA55-ICON), and up to 0.02 mm for S\(_{3}\) (MERRA2-ERA5a). For all other waves, the highest global RMS is below 0.03 mm. Similar to the EW component for NS the largest differences are detected in S\(_{1}\) (up to 0.09 mm for MERRA2-ERA5a), S\(_{2}\) (up to 0.03 mm for MERRA2-JRA55), and S\(_{3}\) (up to 0.02 mm for JRA55-IFS).

The highest RMS is found between MERRA2 and early ERA5 batches and the lowest RMS is found between harmonics estimated based on late ERA5 batches, IFS, and ICON data. For the principal waves S\(_{1}\) and S\(_{2}\) the RMS between the models that have the best agreement is about six times smaller in comparison to the RMS between the models that have the largest discrepancies. However, the dynamic range for the S\(_{2}\) harmonic displacement amplitudes in the tangential components is three time smaller (0.5 dB) in comparison to that of S\(_{1}\). We observe no relation between the dynamic range and the signal aimplitude. Our results indicate that while temporal variations in harmonic amplitudes are non-negligible, differences in the data assimilation system as well as its input employed by the weather models utilized herein give rise to even larger discrepancies; for the “epoch” ensemble, the spatially average RMS over all waves is 0.07/0.03/0.03 mm and for the “NWM” ensemble the RMS is 0.10/0.05/0.03 mm for the radial/eastward/northward coordinate components, respectively.

4 Estimation of Partial Tide Modulations

Changes in the atmospheric tide exciting mechanisms and forcing agents including but not limited to periodic solar irradiance variability (e.g., Schwabe cycle, seasonal, Carrington rotation), variations in the concentration of water vapor and ozone, deep convective activity in the tropics, fluctuations in Earth’s ionosphere and magnetic field are responsible for modulations in the harmonic amplitudes thereof. Section 4 describes their quantification given pressure data.

The first step to our investigations is to vary the data span utilized for the estimation of harmonic amplitudes, given a long time series. For example, given ERA5 atmospheric pressure time series at the site that on average has one of the largest S\(_{2}\) amplitudes over land (equatorial South America), we have estimated a set of tidal harmonics by different segments of the time series; we have varied both the starting point and length of the time series. The amplitude differences associated with the phase deviations feature peak-to-peak variations of 20 Pa. The shorter the time series employed to calculate the tidal harmonics, the larger the temporal harmonic modulations; analyzing data that span shorter than a decade yields amplitude differences that change as much as 20 Pa within 70 years. In essence, the phase estimate can change as much as \(10^{\circ }\) given only a decade of data. Given the S\(_{2}\) beating frequency this means that the maximum of the S\(_{2}\) wave will occur 20 min later than a couple of decades ago.

Below, we present the estimation of harmonic modulations \(\mathbf {x} = \begin {bmatrix} {\mathbf {x}}_{\mathbf {1}} & {\mathbf {x}}_{\mathbf {m}} & {\mathbf {x}}_{\mathbf {M}} \end {bmatrix}\) parameterized as low-degree B-spline functions by introducing normal equations of monthly intervals, given only three data blocks \({\mathbf {y}}_{\mathbf {m}}\), \(m\in [1, M]\), however without any loss of generality. The solution is given by \(\mathbf {x} = {\mathbf {N}}^{-1}\mathbf {u}\), where the normal equation system is constructed as follows

$$\displaystyle \begin{aligned} {} \begin{aligned} \mathbf{N} &= \begin{bmatrix} {\mathbf{N}}_{\mathbf{1}} & &\\ & {\mathbf{N}}_{\mathbf{m}} &\\ & & {\mathbf{N}}_{\mathbf{M}} \end{bmatrix} \\ &+ \begin{bmatrix} {\mathbf{P}}_{\mathbf{1}{\text -}\mathbf{m}}^{\mathbf{r}} & -{\mathbf{P}}_{\mathbf{1}{\text -}\mathbf{m}}^{\mathbf{r}} & \\ -{\mathbf{P}}_{\mathbf{1}{\text -}\mathbf{m}}^{\mathbf{r}} & {\mathbf{P}}_{\mathbf{1}{\text -}\mathbf{m}}^{\mathbf{r}} + {\mathbf{P}}_{\mathbf{m}{\text -}\mathbf{M}}^{\mathbf{r}} & -{\mathbf{P}}_{\mathbf{m-M}}^{\mathbf{r}}\\ & -{\mathbf{P}}_{\mathbf{m}{\text -}\mathbf{M}} & {\mathbf{P}}_{\mathbf{m}{\text -}\mathbf{M}}^{\mathbf{r}} \end{bmatrix} \\ &+ \begin{bmatrix} {\mathbf{P}}_{\mathbf{1}}^{\mathbf{a}} & & \\ & {\mathbf{P}}_{\mathbf{m}}^{\mathbf{a}} & \\ & & {\mathbf{P}}_{\mathbf{M}}^{\mathbf{a}} \end{bmatrix} \\ \mathbf{u} &= \begin{bmatrix} {\mathbf{u}}_{\mathbf{1}} & {\mathbf{u}}_{\mathbf{m}} & {\mathbf{u}}_{\mathbf{M}} \end{bmatrix}^\top + \begin{bmatrix} {\mathbf{P}}_{\mathbf{1}}^{\mathbf{a}} & {\mathbf{P}}_{\mathbf{m}}^{\mathbf{a}} & {\mathbf{P}}_{\mathbf{M}}^{\mathbf{a}} \end{bmatrix}^{\top} {\mathbf{x}}_{\mathit{ref}}, \end{aligned} \end{aligned} $$
(2)

where \({\mathbf {N}}_{\mathbf {m}} = {\mathbf {J}}_{\mathbf {m}}^{\top }{\mathbf {P}}_{\mathbf {m}}^{\mathbf {o}}{\mathbf {J}}_{\mathbf {m}}\) and \({\mathbf {u}}_{\mathbf {m}} = {\mathbf {J}}_{\mathbf {m}}^{\boldsymbol {\top }}{\mathbf {P}}_{\mathbf {m}}^{\mathbf {o}}{\mathbf {y}}_{\mathbf {m}}\), is the normal equation system, \({\mathbf {J}}_{\mathbf {m}}\) is the design matrix following observation equation 2 from Balidakis et al. (2022), and \({\mathbf {P}}_{\mathbf {m}}^{\mathbf {o}}\) is the associated observations’ weight matrix. \({\mathbf {P}}_{\mathbf {m}}^{\mathbf {r}}\) is the matrix that controls the relative constraints, and \({\mathbf {P}}_{\mathbf {m}}^{\mathbf {a}}\) is the matrix that controls the absolute constraints for data block m.

Stochastic equivalence constraints stabilize the solution. Relative constraints control how consecutive parameters, that is unknowns referring to one quantity (e.g., the in-phase harmonic amplitude of one wave) at different times, vary under the Markov assumption. They read \(\hat {x}(t + \mathit {dt}) = \hat {x}(t) + w(t)\mathit {dt}\), where \(\hat {x}\) denotes the a posteriori value of an unknown parameter set up to be estimated at epoch t, and w controls the weight of each pseudo-observation (\(\sigma _{\mathit {rc}}^{-2}\)). We note that imposing very tight relative constraints effectively yields harmonic amplitudes with no temporal variability. In this work, the variations of harmonic amplitudes are parameterized as random walk processes. Absolute constraint observation equations, \(\hat {x}(t) - x_{\mathit {ref}} = 0 \pm \sigma _{ac}\), where \(x_{\mathit {ref}}\) is a reference value herein derived from a least-squares adjustment involving the full-length time series, and \(\sigma _{ac}^{-2}\) controls the weight of the related pseudo-observation, may be applied as well.

However, it is apparent that a large number of waves, a large number of estimation intervals, as well as a large number of data batches will render the above procedure impractical. For instance, selecting 16 waves and monthly estimation intervals for 70 years will flood the parameter space with more than 26,000 elements. To this end, we have resorted to sequential methods where the dimension of the parameter space depends only upon the wave ensemble. We have adopted the Dyer-McReynolds implementation of the square root information filter (Bierman 1977).

Figure 5 illustrates the results of the S\(_{1}\) and S\(_{2}\) phase upon varying the process noise relative to the measurements’ noise. As expected, injecting relatively little process noise in the filter yields practically time-invariant harmonics. However, allowing for enough process noise reveals at first inter-annual variations and eventually seasonal modulations at annual frequencies and overtones thereof. In particular, the annual amplitude of the S\(_{2}\) amplitude and phase modulations for a site in South America is 6.1 Pa and 4.7°, respectively. The S\(_{a}\) modulations for the S\(_{1}\) wave at that site are 5.4 Pa and 5.8°, respectively. We note that the time-independent amplitude of the S\(_{1}\) and S\(_{2}\) is 48.4 Pa and 204.9 Pa, respectively. Furthermore, given the non-negligible deviations from a linear long-term signal evolution, quantifying temporal tidal variations with a first-order polynomial is an oversimplification since varying the interval under consideration will yield considerably different results. We note that temporal variations in harmonic amplitudes may be due to significant changes in the quality, type, and volume of observations assimilated into a NWM. Comparing the evolution of NWM harmonics with the evolution of harmonics from in situ barometers could shed light on whether the former are true or artefacts (Ray 2001; Schindelegger and Ray 2014), what cannot be done solely by studying the NWM-driven harmonics.

Fig. 5
figure 5

Temporal S\(_{1}\) (top) and S\(_{2}\) (bottom) pressure phase variation estimates from varying the filter process noise (\(\sigma _{\mathit {pn}}\)) in relation to the measurement noise (\(\sigma _{\mathit {nm}}\)) in equatorial South America

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We run the procedure described above to the displacements induced by the temporally variable atmospheric pressure harmonics to calculate the associated displacement fields. Our results suggest that given the nominal accuracy of state-of-the-art space geodetic observations (3 ps for VGOS group delays, 5 mm for GNSS P3 observations, and mean-RMS of 20 mm for LAGEOS SLR normal points with 120 s integration interval), the temporal modulations in the tidal atmospheric loading corrections will probably have little impact on the data analysis procedure as well as the products.

5 Conclusions

Atmospheric tides induce atmospheric surface pressure variations that displace the Earth’s crust vertically in the sub-cm range. The most prominent periods are 12 and 24 h, but also other relevant lines with neighboring frequencies have been identified. Accounting for such systematic effects in space geodetic data analysis potentially decreases aliasing and facilitates the detection of spurious signals. Herein, we have assessed the extent to which the NWM-predicted tidal atmospheric loading displacements vary temporally as well as across different models. We have utilized mass anomaly fields from five NWM (ERA5, ICON, IFS, JRA55, and MERRA2) from four meteorological agencies (DWD, ECMWF, JMA, and NASA-GMAO) with spatial resolutions ranging between 55 km and 9 km. We have used a square-root-information filter to estimate tidal mass anomaly amplitude modulations, which we have employed to calculate high-frequency tidal loading displacements.

Do atmospheric tidal loading displacement signals vary in time? For the “epoch” ensemble, the RMS for the radial/east/north component ranges between 0.06/0.02/0.02 mm and 0.14/0.06/0.07 mm, respectively. On average, harmonics estimated based on data sets that are close in time show a better agreement. We have also found annual and semi-annual modulations in the mass anomaly fields if we allow in the filter settings for enough process noise relative to the observation noise.

Do models agree in the prediction of tidal atmospheric loading displacements? For the “NWM” ensemble, the RMS for the radial/east/north component ranges between 0.05/0.02/0.02 mm and 0.28/0.10/0.08 mm, respectively. The largest disagreements are found between MERRA2 and early ERA5 data sets. On average the discrepancies between MERRA2 and early ERA5 data sets are 3–12 times larger compared to the RMS between recent ERA5 data and IFS or ICON.

By and large, switching from one NWM to another induces larger tidal harmonic amplitude differences than temporal modulations within a single NWM. However, both inter-model and temporal differences have a global RMS below 0.3 mm for radial displacements, which while it might exceed 10 % of the effect, it is still below the uncertainty of state-of-the-art geometric space geodetic observations. In line with Ray et al. (2023) who did not find systematic errors in ERA5-derived atmospheric tides and in absence of more accurate NWM data, we recommend the application of time-invariant harmonic atmospheric loading displacement corrections derived from multi-year ERA5 data as, e.g., made available recently by Sulzbach et al. (2022).