TY - JOUR
AU - , van Dam, Tonie
AB - SUMMARY We study fluctuations in the degree-2 zonal spherical harmonic coefficient of the Earth's gravity potential, C20, over the period 2003–2015. This coefficient is related to the Earth's oblateness and studying its temporal variations, ΔC20, can be used to monitor large-scale mass movements between high and low latitude regions. We examine ΔC20 inferred from six different sources, including satellite laser ranging (SLR), GRACE and global geophysical fluids models. We further include estimates that we derive from measured variations in the length-of-day (LOD), from the inversion of global crustal displacements as measured by GPS, as well as from the combination of GRACE and the output of an ocean model as described by Sun et al. We apply a sequence of trend and seasonal moving average filters to the different time-series in order to decompose them into an interannual, a seasonal and an intraseasonal component. We then perform a comparison analysis for each component, and we further estimate the noise level contained in the different series using an extended version of the three-cornered-hat method. For the seasonal component, we generally obtain a very good agreement between the different sources, and except for the LOD-derived series, we find that over 90 per cent of the variance in the seasonal components can be explained by the sum of an annual and semiannual oscillation of constant amplitudes and phases, indicating that the seasonal pattern is stable over the considered time period. High consistency between the different estimates is also observed for the intraseasonal component, except for the solution from GRACE, which is known to be affected by a strong tide-like alias with a period of about 161 d. Estimated interannual components from the different sources are generally in agreement with each other, although estimates from GRACE and LOD present some discrepancies. Slight deviations are further observed for the estimate from the geophysical models, likely to be related to the omission of polar ice and groundwater changes in the model combination we use. On the other hand, these processes do not seem to play an important role at seasonal and shorter timescales, as the sum of modelled atmospheric, oceanic and hydrological effects effectively explains the observed C20 variations at those scales. We generally obtain very good results for the solution from SLR, and we confirm that this well-established technique accurately tracks changes in C20. Good agreement is further observed for the estimate from the GPS inversion, showing that this indirect method is successful in capturing fluctuations in C20 on scales ranging from intra- to interannual. Obtaining accurate estimates from LOD, however, remains a challenging task and more reliable models of atmospheric wind fields are needed in order to obtain high-quality ΔC20, in particular at the seasonal scale. The combination of GRACE data and the output of an ocean model appears to be a promising approach, particularly since corresponding ΔC20 is not affected by tide-like aliases, and generally gives better results than the solution from GRACE, which still seems to be of rather poor quality. Time-series analysis, Satellite geodesy, Satellite gravity, Time-variable gravity, Earth rotation variations, Global change from geodesy 1 INTRODUCTION Variations in the low-degree spherical harmonic coefficients of the Earth's gravity potential are related to large-scale mass redistribution within and on the Earth. Monitoring these variations can thus be used to study the mass transports occurring in the Earth system. Of particular importance is the coefficient of degree-2 and order-0, C20 (also known as J2, where |$J_2 = -\sqrt{5} C_{20}$|⁠), which is related to the Earth's oblateness (e.g. Lambeck 1988). Studying fluctuations in C20, denoted ΔC20, is an important task, particularly since C20 is sensitive to surface mass transports between high (>35°) and low (<35°) latitude regions, and hence to the melting of polar ice sheets. Exploring its temporal behaviour is thus essential in view of climate-related mass redistribution caused by global warming. Different techniques to measure ΔC20 exist. Traditionally, satellite laser ranging (SLR) has been used for this purpose, and from the analysis of SLR data it is known since a long time that the main features in ΔC20 are a linear trend, generally attributed to glacial isostatic adjustment (GIA), superimposed by seasonal variations that are mostly caused by mass redistribution within the atmosphere, oceans and the terrestrial water storage (e.g. Yoder et al. 1983; Cheng et al. 1989; Nerem et al. 1993; Cheng et al. 1997; Cheng & Tapley 1999). Using an SLR series covering the period 1976–2011, Cheng et al. (2013a) recently showed that the long-term rate in ΔC20 has been decreasing during the last decades, and that over the considered time span, the long-term behaviour appears more quadratic than linear in nature. Cheng et al. (2013a) suggest that accelerated ice mass loss from glaciers and ice sheets are responsible for this effect, and the importance of ice melting processes on decadal and longer-period C20 variability was recently confirmed by Seo et al. (2015). Variations in C20 also comprise interannual fluctuations with periods of 4–6 yr, which are related to El Niño Southern Oscillation (ENSO) events and can largely be explained by atmospheric, oceanic and hydrological models (e.g. Cox & Chao 2002; Dickey et al. 2002; Cheng & Tapley 2004). With the launch of the GRACE satellites in 2002, a new source for studying the Earth's time-variable gravity field became available. In particular, GRACE provides estimates of ΔC20, but due to orbital geometry and the relatively short distance between the two satellites, ΔC20 from GRACE is generally believed to be of rather poor quality (e.g. Chen et al. 2005a). Furthermore, GRACE-derived ΔC20 is affected by large tide-like aliases, whose origins are still not fully understood, so that users are advised to replace the GRACE estimate by a solution from SLR (Cheng et al. 2013a). However, using a methodology originally introduced by Swenson et al. (2008) to infer estimates of geocentre motion, Sun et al. (2016) recently derived a new ΔC20 series from a combination of GRACE and the output of an ocean model, and found good agreement between their solution and ΔC20 as measured by SLR. Furthermore, GRACE data is useful to calculate regional influences on ΔC20, such as variations driven by Greenland or Antarctic ice mass loss (e.g. Nerem & Wahr 2011). Besides these space-geodetic measurements, there exist indirect ways to infer ΔC20. For instance, Earth rotation theory predicts that variations in C20 caused by surface mass redistribution are directly linked to corresponding excitations of length-of-day (LOD). Thus, ΔC20 can be computed from measured fluctuations in LOD, provided it is possible to remove the effects that are not related to surface mass redistribution from the observations, such as those of atmospheric winds and oceanic currents. This can be done using various global geophysical circulation models, and several authors derived ΔC20 from observed changes in LOD using this approach (e.g. Chen et al. 2000, 2005b, 2016; Chen & Wilson 2003, 2008; Gross et al. 2004; Bourda 2008; Meyrath et al. 2013). A further indirect method to obtain ΔC20 estimates is based on the observation that changes in the Earth's shape can be related to corresponding changes in the Earth's surface mass load through elastic load Love number theory (Farrell 1972). The latter can then be used to derive fluctuations in the gravity potential (Wahr et al. 1998). Hence, measurements of the Earth's time-variable geometrical shape can be used to calculate variations in its gravity field, and it was first proposed by Blewitt et al. (2001) to use the global GPS network for that purpose. Their initial approach has been modified and extended in various ways, and a number of studies successfully used GPS data to investigate fluctuations in the low-degree spherical harmonic gravity coefficients, including C20 (e.g. Blewitt & Clarke 2003; Gross et al. 2004; Kusche & Schrama 2005; Wu et al. 2006; Lavallée et al. 2010; Jin & Zhang 2014). While several possibilities to infer ΔC20 exist, estimates from different sources are not always in agreement with each other, as each technique has its individual strengths and weaknesses. Assessing different ΔC20 solutions therefore remains an important task that helps to validate the individual techniques and to increase our understanding of the large-scale mass redistribution processes that drive the observed C20 variability. In this study, we investigate and compare fluctuations in C20 over the period 2003–2015 estimated from six different sources, including SLR, GRACE, measured LOD variations and global geophysical fluids models. In contrast to previous studies, we additionally include a new ΔC20 series that we derive from the inversion of global crustal displacements from the IGS repro2 combined GPS solutions using a set of modified basis functions similar to Clarke et al. (2007). Furthermore, we also analyse a ΔC20 series resulting from the combination of GRACE and the output of an ocean model, as was recently proposed by Sun et al. (2016). To each of our time-series we apply a sequence of different trend and seasonal moving average filters in order to decompose them into (1) a seasonal component that contains all the signals related to seasonality; (2) an interannual component describing the interannual and long-period signals; and (3) an intraseasonal component incorporating short-period irregular and non-seasonal fluctuations, not captured by the other components. For each component, we then perform a comparison analysis, with the objective to assess the level of agreement between the different estimates, as well as to identify and explain discrepancies. Finally, we estimate the noise variances for each source and component using an extended version of the three-cornered-hat method. 2 DATA SETS AND PROCESSING In this section, we briefly describe the different ΔC20 estimates we use in this study. 2.1 SLR We use an SLR estimate of ΔC20 obtained from the Center for Space Research (CSR) at the University of Texas,1 consisting of solutions from five geodetic satellites (Starlette, Ajisai, Stella, LAGEOS-1, LAGEOS-2), each with a spherical shape that simplifies the modelling of non-gravitational forces (Cheng et al. 2011, 2013b). Except for Ajisai, the satellites have dense metal cores and very low area-to-mass ratios, which further reduces the impact of the non-gravitational force modelling errors. The background gravity models used in the analysis include solid Earth and ocean tides, solid Earth and ocean pole tides, and atmosphere/ocean de-aliasing. These models are consistent with those for the GRACE Release-05 products. 2.2 GRACE The GRACE ΔC20 estimate we use is from the latest Release-05 from CSR.2 The atmosphere/ocean de-aliasing product, that is removed during the GRACE level-2 data processing, is added back to the solution, so that it contains the full mass signal. 2.3 GRACE/OBP Based on a method originally proposed by Swenson et al. (2008) to derive degree-1 coefficients, Sun et al. (2016) recently developed a new methodology to obtain an estimate of ΔC20 from a combination of GRACE, the output of an ocean model and a GIA model. Sun et al. (2016) estimate the oceanic contribution to the degree-1 coefficients and to ΔC20 using an ocean model, and combine those with the non-zonal degree-2 and higher degree coefficients from GRACE to finally obtain an estimate of ΔC20. The time-series we use is available at http://www.citg.tudelft.nl/c20. Note, however, that in contrast to the description in Sun et al. (2016), the implementation parameters to produce this series were modified according to an end-to-end simulation, and not in order to produce a solution that is close to SLR (Sun, personal communication, 2016). Hence, the resulting ΔC20 estimate is essentially independent from SLR. In the following, we will refer to this solution as GRACE/OBP, in order to clarify that it is not exclusively derived from GRACE data. 2.4 LOD We use the IERS online tool3 to obtain daily LOD excitations, χ3. This tool calculates the excitations based on the widely used IERS 08 C04 series (Bizouard & Gambis 2011), which is derived from a combination of different space geodetic techniques and spans the time frame from 1962 to the present. The effects of zonal gravitational tides are removed during the calculation. The LOD excitations induced by surface mass redistribution, |$\chi _3^{\text{mass}}$|⁠, are related to corresponding C20 variability via \begin{equation} \Delta C_{20} = - \frac{3}{2 \sqrt{5}} \, \frac{C_m}{1.093 R^2 \, M} \, \chi _3^{\text{mass}}, \end{equation} (1) where R (=6378136.6 m) and M (=5.9737 × 1024 kg) are the Earth's radius and mass, respectively, and Cm (=7.1236 × 1037 kg m2) is the axial principal moment of inertia of the Earth's crust and mantle (Chen et al. 2005b, 2016; Gross 2007). The factor 1.093 accounts for the effect of core decoupling as well as for the yielding of the solid Earth to surface mass loading, including the effect of mantle anelasticity (Gross 2007, 2015). We mention that due to varying approaches, the numerical value for this factor differs at the level of about 2 per cent among different studies (see, e.g. Dickman (2003, 2005) for a discussion). To isolate the excitations caused by surface mass redistribution, contributions from other sources must be removed from the observations. Atmospheric and oceanic motion effects are modelled using a data set from the German Research Center for Geosciences4 (Dobslaw et al. 2010). The wind terms are calculated by integrating 6 hourly operational data from the European Center for Medium-Range Weather Forecast (ECMWF) at 25 pressure levels with the top level being at 1 hPa. The contribution of oceanic currents to LOD excitations is derived from the Ocean Model for Circulation and Tides (OMCT; Dobslaw & Thomas 2007), forced by operational atmospheric data from the ECMWF. We note that the hydrological motion part is negligible, as its effect is about three orders of magnitude smaller than corresponding contributions from the atmosphere and oceans (Dobslaw et al. 2010). Hence, we only need to subtract the effects of atmospheric winds and oceanic currents from the observed LOD excitations. Besides motion effects from the atmosphere and oceans, it is known that observed LOD excitation series contain strong interannual and decadal variations, which are thought to primarily result from interactions between the Earth's core and mantle (e.g. Hide et al. 1993; Ponsar et al. 2003; Gross et al. 2005). In particular, an important oscillation with a period of about 5.9 yr has been identified and investigated by various authors (e.g. Abarca del Rio et al. 2000; Mound & Buffett 2006; Holme & de Viron 2013; Duan et al. 2015). We remove the long-term signals, including the 5.9 yr oscillation, by applying a high pass Butterworth filter with cut-off frequency 1/4 cpy. This filtering process also eliminates possible effects due to Earth internal mass transports from the observations, so that the residual LOD excitations can be assumed to be entirely caused by surface mass redistribution. They are transformed into corresponding C20 variations by means of eq. (1). 2.5 GPS The GPS data we use are derived from the IGS repro2 combined solutions (Rebischung et al. 2016) and contain daily station displacements, together with their full variance-covariance matrices until GPS week 1831. Each station position time-series has a reference position and velocity, occasional discontinuities and, in some cases, post-seismic deformation models removed. Discontinuities are related to earthquakes, equipment changes, environmental condition changes around the stations or other unknown causes. They have been identified by visual inspection of the station position time-series as detailed in Altamimi et al. (2016). For stations affected by large post-seismic deformations, those have been corrected using the parametric models used in the ITRF2014 computation (Altamimi et al. 2016). We form weekly averages from the daily station displacement fields, where for each week we only take into account stations for which at least four daily solutions are available, and we propagate the covariance matrices accordingly. We further concentrate on the period from 2003 onwards (i.e. GPS weeks 1199–1831). It follows from loading theory that the displacements in north, east and radial direction of a point at the Earth's surface with latitude θ and longitude ϕ, denoted ΔN, ΔE and ΔH, related to variations in the Earth's surface mass load are given by \begin{eqnarray} \Delta N(\theta ,\phi ) = \frac{3 \, \rho _w}{\rho _e} \, \sum _{l = 1}^{\infty } \sum _{m = 0}^l \sum _{\Lambda \in \lbrace C,S\rbrace } \frac{L_l^{\prime }}{2l + 1} \, T_{lm}^{\Lambda } \, \partial _{\theta } Y_{lm}^{\Lambda } (\theta ,\phi ), \nonumber\\ \Delta E(\theta ,\phi ) = \frac{3 \, \rho _w}{\rho _e \cos (\theta )} \, \sum _{l = 1}^{\infty } \sum _{m = 0}^l \sum _{\Lambda \in \lbrace C,S\rbrace } \frac{L_l^{\prime }}{2l + 1} \, T_{lm}^{\Lambda } \, \partial _{\phi } Y_{lm}^{\Lambda } (\theta ,\phi ), \nonumber\\ \Delta H(\theta ,\phi ) = \frac{3 \, \rho _w}{\rho _e} \, \sum _{l = 0}^{\infty } \sum _{m = 0}^l \sum _{\Lambda \in \lbrace C,S\rbrace } \frac{H_l^{\prime }}{2l + 1} \, T_{lm}^{\Lambda } \, Y_{lm}^{\Lambda } (\theta ,\phi ), \end{eqnarray} (2) where |$Y_{lm}^{\Lambda }$| denote the degree-l order-m (normalized) spherical harmonic functions, ∂θ and ∂ϕ stand for the partial derivatives with respect to θ and ϕ, and |$T_{lm}^{\Lambda }$| are the coefficients of the spherical harmonic expansion of the function describing the variations in the surface mass load as a departure from a reference value and expressed as the equivalent height (in metres) of a column of seawater with density ρw (=1025 kg m−3). The symbol ρe refers to the mean density of the Earth (=5517 kg m−3), and |$H_l^{\prime }$| and |$L_l^{\prime }$| stand for the degree-dependent load Love numbers of degree l, which for l = 1 are dependent on the chosen reference frame (Blewitt 2003). In this study, displacements are modelled in the centre of figure (CF) frame, so that accordingly the values −0.268 and 0.134 are adopted for |$H_1^{\prime }$| and |$L_1^{\prime }$|⁠, respectively. Note that the summation in the third equation begins at l = 1 if one assumes that the total surface mass is conserved, as this assumption implies that |$T_{00}^C$| vanishes. Truncating the series in the above equations at a certain degree l = N, it is thus possible to invert measured crustal displacements for the low-degree coefficients of the surface mass load changes, which in turn can be related to the coefficients of the Earth's gravity potential (e.g. Wahr et al. 1998). In particular, estimated |$T_{20}^C$| can be used to recover ΔC20. However, when using GPS for this purpose, a general problem is the sparse and geographically uneven distribution of GPS sites over the Earth, particularly over oceanic areas, leading to unstable inversions and unrealistic results (e.g. Wu et al. 2002). Therefore, most authors impose further constraints in their inversion scheme and/or additionally include data from other sources, such as GRACE or oceanic models (e.g. Kusche & Schrama 2005; Wu et al. 2006; Mendes Cerveira et al. 2007; Jansen et al. 2009; Rietbroek et al. 2009, 2012, 2014). For the inversion of our weekly GPS displacements, we follow the approach of Clarke et al. (2007), who proposed to overcome the limitation of sparse and uneven data distribution by using a set of modified basis functions as an alternative to standard spherical harmonics. As Clarke et al. (2007) have shown, their basis functions give a precise and accurate fit to modelled surface mass loads and are less subject to aliasing errors than standard spherical harmonics. A further advantage of this technique is that no additional constraints are required during the inversion, nor is the incorporation of any information from models or other satellites necessary. Following Clarke et al. (2007), we first restrict the spherical harmonics to the continental areas, by considering the functions |$\bar{B}_{lm}^{\Lambda }(\theta ,\phi ) := C(\theta ,\phi ) \cdot Y_{lm}^{\Lambda }(\theta ,\phi )$|⁠. Here, C denotes the ‘land function’ that takes the value 1 over land areas and 0 over the oceans (e.g. Munk & MacDonald 1960; Lambeck 1980). We then calculate the spherical harmonic coefficients of the functions |$\bar{B}_{lm}^{\Lambda }$| up to degree and order 25, which is enough to capture the coastlines of all major landmasses. Since these functions are not mass-conserving (i.e. their degree-0 coefficient does not vanish), we complement them with a spatially uniform layer over the oceans that has the right thickness to compensate for this. Note that this is a somewhat simplified approach as compared to the functions originally used by Clarke et al. (2007), who apply the passive-ocean concept of Clarke et al. (2005) over the oceans. We denote the degree-25 expansion of the resulting mass-conserving functions by |$B_{lm}^{\Lambda }$|⁠, and their degree-a order-b spherical harmonic coefficients by |$q_{lm,ab}^{\Lambda ,\lambda }$|⁠, where λ ∈ {C, S}. These functions will serve as basis functions for describing the load. Degree-25 expansions of the corresponding basis functions for the induced north, east and radial displacements, denoted |$B_{lm}^{\Lambda ,N}, B_{lm}^{\Lambda ,E}$| and |$B_{lm}^{\Lambda ,H}$| are then derived as \begin{eqnarray*} \!B_{lm}^{\Lambda ,N} (\theta ,\phi ) = \frac{3 \, \rho _w}{\rho _e} \, \sum _{a = 1}^{25} \sum _{b = 0}^a \sum _{\lambda \in \lbrace C,S\rbrace } \frac{L_a^{\prime }}{2a + 1} \, q_{lm,ab}^{\Lambda ,\lambda } \: \partial _{\theta } Y_{ab}^{\lambda } (\theta ,\phi ), \nonumber\\ \! B_{lm}^{\Lambda ,E} (\theta ,\phi ) = \frac{3 \, \rho _w}{\rho _e \cos (\theta )} \, \sum _{a = 1}^{25} \sum _{b = 0}^a \sum _{\lambda \in \lbrace C,S\rbrace } \frac{L_a^{\prime }}{2a + 1} \, q_{lm,ab}^{\Lambda ,\lambda } \: \partial _{\phi } Y_{ab}^{\lambda } (\theta ,\phi ), \nonumber\\ \! B_{lm}^{\Lambda ,H} (\theta ,\phi ) =\frac{3 \, \rho _w}{\rho _e} \, \sum _{a = 1}^{25} \sum _{b = 0}^a \sum _{\lambda \in \lbrace C,S\rbrace } \frac{H_a^{\prime }}{2a + 1} \, q_{lm,ab}^{\Lambda ,\lambda } \: Y_{ab}^{\lambda } (\theta ,\phi ). \end{eqnarray*} The inversion is finally realized by solving the following equations in a weighted least-squares sense for each of the GPS weeks 1199–1831 \begin{eqnarray*} \Delta N(\theta ,\phi ) = \sum _{l = 0}^{N} \sum _{m = 0}^l \sum _{\Lambda \in \lbrace C,S\rbrace } t_{lm}^{\Lambda } \, B_{lm}^{N,\Lambda } (\theta ,\phi ) \\ &&+ \vec{e_n} \cdot (t_x,t_y,t_z)^T - \vec{e_e} \cdot (r_x,r_y,r_z)^T, \nonumber\\ \Delta E(\theta ,\phi ) = \sum _{l = 0}^{N} \sum _{m = 0}^l \sum _{\Lambda \in \lbrace C,S\rbrace } t_{lm}^{\Lambda } \, B_{lm}^{E,\Lambda } (\theta ,\phi ) \\ &&+ \vec{e_e} \cdot (t_x,t_y,t_z)^T + \vec{e_n} \cdot (r_x,r_y,r_z)^T, \nonumber\\ \Delta H(\theta ,\phi ) = \sum _{l = 0}^{N} \sum _{m = 0}^l \sum _{\Lambda \in \lbrace C,S\rbrace } t_{lm}^{\Lambda } \, B_{lm}^{H,\Lambda } (\theta ,\phi ) + \vec{e_h} \cdot (t_x,t_y,t_z)^T, \end{eqnarray*} where we additionally include translation and rotation parameters tx, ty, tz and rx, ry, rz, respectively, and |$\vec{e_n},\vec{e_e},\vec{e_h}$| denote unit vectors in a local spherical frame (Kusche & Schrama 2005; Lavallée et al. 2006). Measured displacements at the available GPS stations in north, east and radial directions are used as observations, and the corresponding covariance matrices for the weighting. Note that since our basis functions can only model displacements induced by land loads, we do not include any oceanic stations in our inversion. Referring to investigations of Clarke et al. (2007) and Wei & Fang (2016), we set the inversion degree N to 6. Once the coefficients |$t_{lm}^{\Lambda }$| have been determined, they can be used to describe the load in terms of the functions |$B_{lm}^{\Lambda }$|⁠, whose spherical harmonic coefficients |$q_{lm,ab}^{\Lambda ,\lambda }$| can then be used to recover the coefficients |$T_{lm}^{\Lambda }$| (Clarke et al. 2007). In particular, |$T_{20}^C$| can be derived as \begin{eqnarray*} T_{20}^C = \sum _{l = 0}^{6} \sum _{m = 0}^l \sum _{\Lambda \in \lbrace C,S\rbrace } q_{lm,20}^{\Lambda ,C} \, t_{lm}^{\Lambda }. \end{eqnarray*} Using the formula given in Wahr et al. (1998), we convert |$T_{20}^C$| to weekly ΔC20 estimates, and finally form monthly averages by summing five consecutive values with the respective weights \begin{eqnarray*} \frac{1}{\frac{1}{2 \sigma _1^2} + \frac{1}{\sigma _2^2} + \frac{1}{\sigma _3^2} + \frac{1}{\sigma _4^2} + \frac{1}{2 \sigma _5^2}} \, \left(\frac{1}{2 \sigma _1^2}, \frac{1}{\sigma _2^2},\frac{1}{\sigma _3^2},\frac{1}{\sigma _4^2},\frac{1}{2\sigma _5^2}\right)\!, \end{eqnarray*} where σi are the uncertainties of the coefficients that were estimated during the inversion and have been propagated accordingly. 2.6 Geophysical fluids model We finally use a ΔC20 estimate that is derived from different global geophysical fluids models. For the sum of atmospheric and oceanic effects on ΔC20, we use the GAC files calculated from the GRACE AOD products that use the OMCT ocean model and atmospheric data from the ECMWF. These products are also used to remove atmospheric and oceanic contributions from the GRACE measurements during the dealiasing process (Dobslaw et al. 2013). The hydrological contribution to ΔC20 is modelled using monthly grids of land water content from the GLDAS water storage model (Rodell et al. 2004), obtained from the Jet Propulsion Laboratory.5 Note that these GLDAS grids do not include groundwater changes. Furthermore, no data is available for areas below a southern latitude of 60°, and the Greenland ice sheets have been masked out. Values over ice covered regions cannot be expected to be realistically modelled by GLDAS, as it does currently not feature a dynamic ice model. We form the sum of atmospheric, oceanic and hydrological effects and impose global mass conservation by considering a uniform layer of water over the oceans that has the right time varying thickness in order to keep the total mass constant, and we include the effect of this layer in our final ΔC20 estimate. We refer to this estimate as GAC+GLDAS in the following of this paper. 2.7 Processing and filtering Since the temporal sampling is not the same for all our series and some of them are given at slightly irregular intervals, we first linearly interpolate each time-series to the midpoints of each month. We then restrict our series to the period 2003 January–2015 January, which is the time span where all our data sets overlap. We finally remove a mean and a linear trend over this period. Trends in our ΔC20 series reflect decadal scale and secular signals, which are largely attributed to GIA (e.g. Cheng et al. 2013a). Investigating these signals requires longer time-series than those considered here and is thus beyond the scope of the present study (e.g. Cheng & Tapley 2004; Cheng et al. 2013a). Before analysing the time-series, we decompose them into different components, where we adopt an additive model and assume that a series yt can be written as \begin{equation*} y_t = L_t + S_t + I_t, \end{equation*} where Lt is the interannual component, containing long period and interannual signals, St is the seasonal component, containing all the signals related to seasonality and repeating in seasonal cycles, It is the intraseasonal component, containing all the signals not captured by Lt and St, and hence mostly short-period fluctuations that are of random, irregular or non-seasonal behaviour. To estimate the different components, we apply an iterative approach that uses different trend and seasonal moving average filters. A similar technique is also the basis for the popular X11 method and its extensions, which is a widely used seasonal adjustment procedure that was initially developed by the US Bureau of the Census (e.g. Shiskin et al. 1965; Ladiray & Quenneville 2013). More precisely, we apply the following steps to each of our time-series. We use a centred, symmetric 13-month moving average with respective weights (1/24, 1/12, …, 1/12, 1/24) to get an initial estimate |$\widehat{L_t}$| of the interannual component Lt. The magnitude of the filter's frequency response is shown in Fig. 1 (left). We repeat the first/last available moving average value six times at the beginning/end of the series, in order to avoid losing data at the series ends. Figure 1. Open in new tabDownload slide Magnitude of the frequency response of the simple 13-term moving average filter (left), the symmetric S3×3 filter (middle) and the symmetric 13-term Henderson filter (right) we use for the time-series decomposition. Figure 1. Open in new tabDownload slide Magnitude of the frequency response of the simple 13-term moving average filter (left), the symmetric S3×3 filter (middle) and the symmetric 13-term Henderson filter (right) we use for the time-series decomposition. We subtract this estimate from the initial series and apply a seasonal S3×3 moving average filter to the residual series to obtain an initial estimate |$\widehat{S_t}$| of the seasonal component St. To prevent data loss, we use asymmetric weights at both ends, where we use the values listed in Ladiray & Quenneville (2013). The frequency response of the symmetric S3×3 filter is displayed in Fig. 1 (middle). |$\widehat{S_t}$| is then adjusted to add to zero over a 12 month period by applying a centred, symmetric 13-month moving average. We subtract the estimated seasonal component |$\widehat{S_t}$| from the original series and apply a 13-term Henderson moving average filter to obtain the interannual component Lt. Again, we use asymmetric weights at the beginning and the end of the series, where we apply the weights originally derived by Musgrave (Ladiray & Quenneville 2013). As can be seen in Fig. 1 (right), the frequency response of the Henderson 13-term filter is closer to the ideal low-pass form than that of the simple 13-term moving average filter used in the first step, however, as it does not eliminate seasonality, it can only be used to estimate the interannual signals in deseasonalized series. The seasonal component St is obtained by applying a seasonal S3×3 filter to the original series, from which the interannual component has been removed. Again, St is adjusted so that the sum over each 12 month period is approximately zero by using a centred 13-term moving average. Finally, we subtract the interannual and the seasonal component from the original series to obtain the intraseasonal compo-nent It. 3 RESULTS After applying the above described sequence of filters to our time-series, each of them is decomposed into an interannual, a seasonal and an intraseasonal component. In Table 1, we display the RMS variability of the different components for the various sources. Also shown is the variability of the original series, that is of the sum of the three components. Table 1. RMS variability of the original ΔC20 series, as well as of the seasonal, intraseasonal and interannual components. . Original . Seasonal . Intraseasonal . Interannual . . series . component . component . component . . (×10−10) . (×10−10) . (×10−10) . (×10−10) . SLR 1.12 1.02 0.30 0.34 GRACE 1.69 1.19 0.90 0.69 GRACE/OBP 1.27 1.17 0.29 0.37 LOD 0.84 0.56 0.40 0.39 GPS 1.03 0.96 0.26 0.25 GAC+GLDAS 1.07 1.00 0.23 0.24 . Original . Seasonal . Intraseasonal . Interannual . . series . component . component . component . . (×10−10) . (×10−10) . (×10−10) . (×10−10) . SLR 1.12 1.02 0.30 0.34 GRACE 1.69 1.19 0.90 0.69 GRACE/OBP 1.27 1.17 0.29 0.37 LOD 0.84 0.56 0.40 0.39 GPS 1.03 0.96 0.26 0.25 GAC+GLDAS 1.07 1.00 0.23 0.24 Open in new tab Table 1. RMS variability of the original ΔC20 series, as well as of the seasonal, intraseasonal and interannual components. . Original . Seasonal . Intraseasonal . Interannual . . series . component . component . component . . (×10−10) . (×10−10) . (×10−10) . (×10−10) . SLR 1.12 1.02 0.30 0.34 GRACE 1.69 1.19 0.90 0.69 GRACE/OBP 1.27 1.17 0.29 0.37 LOD 0.84 0.56 0.40 0.39 GPS 1.03 0.96 0.26 0.25 GAC+GLDAS 1.07 1.00 0.23 0.24 . Original . Seasonal . Intraseasonal . Interannual . . series . component . component . component . . (×10−10) . (×10−10) . (×10−10) . (×10−10) . SLR 1.12 1.02 0.30 0.34 GRACE 1.69 1.19 0.90 0.69 GRACE/OBP 1.27 1.17 0.29 0.37 LOD 0.84 0.56 0.40 0.39 GPS 1.03 0.96 0.26 0.25 GAC+GLDAS 1.07 1.00 0.23 0.24 Open in new tab It can be seen that in general, the seasonal signals are largely dominant and except for LOD and GRACE, the RMS variability of the seasonal component is almost as high as for the sum of the three components, and corresponding variability of the intraseasonal and interannual components are relatively low. On the other hand, these two components appear to play a relatively important role for GRACE and LOD. In the following, we make a comparison analysis of the different series, where we consider each component individually. 3.1 Seasonal component In Fig. 2, we display the seasonal components of the different series. For a better visualization, the same series are shifted with respect to each other in the bottom plot. In general, we observe a very good agreement between the different estimates. The seasonal signals in most of the series seem stable over time, with similar patterns and constant amplitudes over the years. The LOD-derived estimate, however, shows some irregular seasonal behaviour and further appears to have an underestimated annual amplitude. Slight variations in the seasonal pattern can further be observed for GPS and GRACE, where the latter additionally seems to have a tendency to overestimate the amplitude, in particular after 2012. Figure 2. Open in new tabDownload slide Seasonal component of the different ΔC20 estimates. The series in the bottom figure have been shifted for clarity. Figure 2. Open in new tabDownload slide Seasonal component of the different ΔC20 estimates. The series in the bottom figure have been shifted for clarity. We calculate mean annual and semiannual amplitudes and phases by unweighted least-squares. We further calculate the percentage of variance explained by the annual and semiannual least-squares fit. The results are given in Table 2 and, as phasor plots, in Fig. 3. Figure 3. Open in new tabDownload slide Phasor plots of annual and semiannual ΔC20. Figure 3. Open in new tabDownload slide Phasor plots of annual and semiannual ΔC20. Table 2. Annual and semiannual variations, as well as percentage of variance explained by the sum of annual and semiannual signals. The phase is defined as ϕ in cos (σ(t − t0) − ϕ), where σ is the annual or semiannual frequency and t0 refers to January 1st. . Annual . Semiannual . Percentage . . Amplitude . Phase . Amplitude . Phase . of variance . . (× 10−10) . (degrees) . (× 10−10) . (degrees) . explained . SLR 1.36 ± 0.03 51 ± 1 0.31 ± 0.03 311 ± 5 94.4 GRACE 1.57 ± 0.04 40 ± 2 0.29 ± 0.04 356 ± 8 90.9 GRACE/OBP 1.59 ± 0.02 50 ± 1 0.38 ± 0.02 329 ± 3 97.3 LOD 0.63 ± 0.03 40 ± 3 0.28 ± 0.03 349 ± 6 78.7 GPS 1.26 ± 0.03 50 ± 2 0.32 ± 0.03 292 ± 6 91.1 GAC+GLDAS 1.38 ± 0.02 42 ± 1 0.18 ± 0.02 332 ± 5 97.9 . Annual . Semiannual . Percentage . . Amplitude . Phase . Amplitude . Phase . of variance . . (× 10−10) . (degrees) . (× 10−10) . (degrees) . explained . SLR 1.36 ± 0.03 51 ± 1 0.31 ± 0.03 311 ± 5 94.4 GRACE 1.57 ± 0.04 40 ± 2 0.29 ± 0.04 356 ± 8 90.9 GRACE/OBP 1.59 ± 0.02 50 ± 1 0.38 ± 0.02 329 ± 3 97.3 LOD 0.63 ± 0.03 40 ± 3 0.28 ± 0.03 349 ± 6 78.7 GPS 1.26 ± 0.03 50 ± 2 0.32 ± 0.03 292 ± 6 91.1 GAC+GLDAS 1.38 ± 0.02 42 ± 1 0.18 ± 0.02 332 ± 5 97.9 Open in new tab Table 2. Annual and semiannual variations, as well as percentage of variance explained by the sum of annual and semiannual signals. The phase is defined as ϕ in cos (σ(t − t0) − ϕ), where σ is the annual or semiannual frequency and t0 refers to January 1st. . Annual . Semiannual . Percentage . . Amplitude . Phase . Amplitude . Phase . of variance . . (× 10−10) . (degrees) . (× 10−10) . (degrees) . explained . SLR 1.36 ± 0.03 51 ± 1 0.31 ± 0.03 311 ± 5 94.4 GRACE 1.57 ± 0.04 40 ± 2 0.29 ± 0.04 356 ± 8 90.9 GRACE/OBP 1.59 ± 0.02 50 ± 1 0.38 ± 0.02 329 ± 3 97.3 LOD 0.63 ± 0.03 40 ± 3 0.28 ± 0.03 349 ± 6 78.7 GPS 1.26 ± 0.03 50 ± 2 0.32 ± 0.03 292 ± 6 91.1 GAC+GLDAS 1.38 ± 0.02 42 ± 1 0.18 ± 0.02 332 ± 5 97.9 . Annual . Semiannual . Percentage . . Amplitude . Phase . Amplitude . Phase . of variance . . (× 10−10) . (degrees) . (× 10−10) . (degrees) . explained . SLR 1.36 ± 0.03 51 ± 1 0.31 ± 0.03 311 ± 5 94.4 GRACE 1.57 ± 0.04 40 ± 2 0.29 ± 0.04 356 ± 8 90.9 GRACE/OBP 1.59 ± 0.02 50 ± 1 0.38 ± 0.02 329 ± 3 97.3 LOD 0.63 ± 0.03 40 ± 3 0.28 ± 0.03 349 ± 6 78.7 GPS 1.26 ± 0.03 50 ± 2 0.32 ± 0.03 292 ± 6 91.1 GAC+GLDAS 1.38 ± 0.02 42 ± 1 0.18 ± 0.02 332 ± 5 97.9 Open in new tab As already suggested by Fig. 2, we generally observe a strong agreement between the annual signals for the different series, with regard to amplitudes and phases. Annual amplitudes from GRACE and GRACE/OBP are in very good agreement, although their values seem to be slightly overestimated as compared to SLR. The SLR-derived annual amplitude is in remarkable agreement with the corresponding value for GAC+GLDAS. Keeping in mind that the latter estimate does not include any effects of ice melting and groundwater changes, we may conclude that these processes do not have an important impact on annual ΔC20, as the sum of predicted oceanic, atmospheric and hydrological effects largely explains the measured variations. Note, however, that the modelled annual hydrological contributions are important, as GAC alone has an annual amplitude of 0.85 × 10−10 and a phase of 30° (not shown in the table), and adding GLDAS to GAC brings both values much closer to those from the other estimates. GPS predicts an annual amplitude of 1.26 × 10−10, which is in good agreement with corresponding estimates from SLR and GLDAS+GAC. High consistency in terms of RMS variability of the seasonal component between these three sources can also be observed in Table 1. The annual LOD-derived amplitude seems largely underestimated, which was already indicated by Table 1, where it is seen that the RMS variability of the seasonal component of the LOD series is significantly lower than for the other series. Low annual amplitudes in LOD-derived ΔC20 have also been found in previous studies (e.g. Yan & Chao 2012; Meyrath et al. 2013; Chen et al. 2016) and might be related to a systematic underestimation of the effects of zonal atmospheric winds on the excitation of LOD that was reported by Chao & Yan (2010). Inadequately modelled winds are also likely to be responsible for the rather large deviations in the seasonal behaviour of the LOD series from the other sources that are visible in Fig. 2. Annual phases derived from the different sources are highly consistent with each other, with values ranging from 40 to 51°. This corresponds to a peak in C20 around mid-February (or, equivalently, around mid-August for J2) and agrees well with previously published values (e.g. Chen & Wilson 2008; Lavallée et al. 2010; Chen et al. 2016). Semiannual amplitudes from SLR, GRACE, LOD and GPS are in very good agreement, while the values predicted by GRACE/OBP and GAC+GLDAS seem over- and underestimated, respectively. The latter observation might be related to limitations of the geophysical fluids models in capturing the semiannual mass variability in the atmosphere, oceans and the terrestrial water storage, which were noted in a similar context by Yan & Chao (2012). Semiannual phase values vary between 292 and 356° and hence present larger scatter than corresponding annual phases. Note, however, that the semiannual signal itself is much weaker than the annual variation, and thus of reduced signal-to-noise ratio. We finally conclude from Table 2 that the seasonal signals in the different series can largely be explained by the superposition of a regular annual and semiannual oscillation, further confirming that the seasonal behaviour is stable over the considered time period. This holds in particular for the GRACE/OBP and the GAC+GLDAS series, where more than 97 per cent of the variance in the seasonal component can be attributed to the sum of annual and semiannual signals with constant amplitudes and phases. The corresponding value for SLR amounts to 94.4 per cent, while for GRACE and GPS we obtain only about 91 per cent, which can be explained by a slight variability in their seasonal behaviour, also observable in Fig. 2. Seasonal signals in the LOD series appear even more irregular, and only 78.7 per cent of the seasonal component can be explained by the superposition of an annual and semiannual sinusoidal curve. 3.2 Intraseasonal component We now investigate the intraseasonal component of the different series, which are shown in Fig. 4. It can be seen in the top picture that the GRACE estimate presents large discrepancies with the other series and is dominated by a strong oscillation not contained in the other estimates, which also explains the high RMS variability of the GRACE intraseasonal component given in Table 1. On the other hand, the predictions from the other sources are in very good agreement with one another, as can be seen in the middle plot. It is known that GRACE ΔC20 is contaminated by a tide-like alias of about 161 d, and a spectral analysis (not shown) readily confirms that the signal we observe has indeed this period. A least-squares fit yields an amplitude of 0.97 × 10−10 for the 161 d oscillation, which is more than three times the value of the semi-annual amplitude and almost two third of the annual amplitude, respectively, that were found in the previous section. After removing the least-squares fit from the series, we obtain a much better agreement between GRACE and the other estimates, although some discrepancies remain, as can be seen in the bottom plot in Fig. 4. We note that the GRACE/OBP series is apparently not affected by the 161 d alias. Figure 4. Open in new tabDownload slide Intraseasonal component of the different ΔC20 estimates. The middle figure lacks the GRACE series for better readability. The series in the bottom figure have been shifted for clarity. Further note that the 161 d oscillation has been removed in the GRACE estimate displayed in the bottom figure. Figure 4. Open in new tabDownload slide Intraseasonal component of the different ΔC20 estimates. The middle figure lacks the GRACE series for better readability. The series in the bottom figure have been shifted for clarity. Further note that the 161 d oscillation has been removed in the GRACE estimate displayed in the bottom figure. In Table 3 (lower left part), we display correlation coefficients between the different estimates. Further shown in the table (upper right part) are ‘normalized’ RMS (NRMS) differences, that we define as the RMS difference of two series divided by the square root of the sum of the variances of the series, that is, \begin{equation*} \text{NRMS}(A-B) = \frac{\text{RMS}(A-B)}{\sqrt{\text{Var}(A) + \text{Var}(B)}}. \end{equation*} This dimensionless value takes into account the variability of the individual series and can also be interpreted as a crude measure of the noise contained in the series. For GRACE, we additionally show the values obtained after removing the 161 d oscillation. Table 3. Correlation coefficients (lower left part) and NRMS differences (upper right part) between the intraseasonal components of the different ΔC20 estimates. Values in brackets for GRACE are those we obtain after removing the 161 d oscillation. Open in new tab Table 3. Correlation coefficients (lower left part) and NRMS differences (upper right part) between the intraseasonal components of the different ΔC20 estimates. Values in brackets for GRACE are those we obtain after removing the 161 d oscillation. Open in new tab We generally observe a remarkably good agreement between the different series. Highest consistency is observed between GRACE/OBP and GAC+GLDAS, with a correlation of 0.91 and an NRMS difference of 0.34. GAC+GLDAS also agrees well with the other series, so that we conclude again that large parts of the observed intraseasonal variations can be explained by modelled atmospheric, oceanic and hydrological effects. As already indicated by Fig. 4, the estimate from GRACE is in bad agreement with the other sources. While removing the 161 d oscillation significantly improves the consistency with the other series, it remains clearly poorer than for the other comparisons. In particular, we note that the agreement is substantially worse than for GRACE/OBP. Although Table 1 predicts a slightly overestimated variability for the intraseasonal component of the LOD series, it is interesting to see that it generally agrees well with the other estimates. Thus, even if we have seen before that the LOD-derived estimate does not accurately represent the annual signal, it seems to capture the non-seasonal signals reasonably well. In particular, we note that the second highest overall agreement is observed between LOD and SLR. We further note a good agreement between GPS and the other sources, which is an encouraging result that shows that this indirect technique is also able to capture non-seasonal C20 variability. 3.3 Interannual component The different interannual components are displayed in Fig. 5. They appear to be in good agreement with each other, although some discrepancies are visible for GRACE and LOD. Note, however, that the LOD-derived series is not really comparable to the other estimates. As mentioned in Section 2.4, we applied a high pass Butterworth filter with cutoff frequency 1/4 cpy to the LOD excitation series in order to remove the dominant long-period fluctuations that are rather related to interactions between the Earth's core and mantle than being caused by mass redistribution. By doing so, however, long-period signals with a frequency lower than 1/4 cpy (i.e. a period longer than 4 yr) induced by mass transport were removed as well, so that differences between the resulting interannual LOD-derived ΔC20 series and the other estimates are expected. Figure 5. Open in new tabDownload slide Interannual component of the different ΔC20 estimates. The series in the bottom figure have been shifted for clarity. Figure 5. Open in new tabDownload slide Interannual component of the different ΔC20 estimates. The series in the bottom figure have been shifted for clarity. High variability in the GRACE-derived interannual component was already predicted by Table 1. A possible explanation is that besides the 161 d oscillation identified in the intraseasonal component, ΔC20 from GRACE is also known to contain longer period tide-like aliases. For instance, a possible contamination by a signal with a period of about 3.7 yr has been discussed by several authors (e.g. Chen & Wilson 2012). Again, we note that the GRACE/OBP series appears to not be affected by this, and it agrees better with the other sources than GRACE. While for the seasonal and intraseasonal components, we obtained a good agreement between GAC+GLDAS and the other estimates, some discrepancies are visible for the interannual component, and it can also be seen in Table 1 that GAC+GLDAS seems to have a somewhat underestimated RMS variability. This is likely to be related to the omission of ice melting effects, and while the previous sections have shown that these processes appear to not have a large influence on seasonal and higher frequency C20 variability, it has been shown in previous studies that they are important on longer-term scales (e.g. Nerem & Wahr 2011; Seo et al. 2015). Furthermore, as variations in groundwater storage might affect the long-period signals in ΔC20, their omission in GLDAS might also have a negative impact on the interannual component from GAC+GLDAS. Also note that Earth internal mass redistribution processes, such as fluxes within the Earth's core, have been shown to influence long-period variability in C20 (Greiner-Mai & Barthelmes 2001; Dumberry & Bloxham 2004; Dumberry 2010). Obviously, those effects are not included in the GAC+GLDAS estimate, while they are sensed by the satellite-based measurement techniques. Despite all these shortcomings, it can nevertheless be seen in the figure that GAC+GLDAS captures some of the prominent features contained in the other series, as for instance the peaks in late 2009 and 2011, rather well. Again, it is worth noting that the solution from GPS seems highly consistent with the other estimates. As this technique relies on inverting surface displacements, it is insensitive to processes that do not change the geometrical shape of the Earth, so that as for GAC+GLDAS, the effects of Earth internal processes are largely missing in GPS-derived ΔC20. However, as we observe a very good agreement between GPS and SLR, we may conclude that these processes do not seem to play an important role over the considered time span. 3.4 Noise estimates using the three-cornered-hat method Having at least three different time-series of the same process, it is possible to estimate the noise level of the individual series by means of the three-cornered-hat (TCH) method, which was introduced by Gray & Allan (1974). While the method was originally used to estimate the stability of oscillators and clocks, it has become increasingly important in the geodetic literature as well (e.g. Chin et al. 2005; Koot et al. 2006; Chen & Wilson 2008; Chen et al. 2016; Ferreira et al. 2016; Yan et al. 2016). The classical formulation of the TCH method is based on the assumption that there is no correlation between the noises of the different series. However, this cannot be expected to hold in our case, as the different C20 solutions may not be completely independent from each other. For instance, some dependencies between the estimates from GRACE and GRACE/OBP can be suspected, the same holds for the series from GRACE/OBP and GAC+GLDAS, as both of them incorporate data from the OMCT ocean model. Instead of using the classical TCH method, we therefore use a generalized version that was developed by Premoli & Tavella (1993) and Tavella & Premoli (1994), which does not make the assumption of zero correlation between the noises of the different series. We will only briefly describe the basics of this method, for further details about the generalized TCH method we refer the reader, for instance, to Galindo & Palacio (1999); Torcaso et al. (2000); Galindo et al. (2001). Let be given N series {Xi}i = 1, …, N of the same process and assume that each series consists of the real signal S plus individual noise, that is, Xi = S + εi. Considering the difference between each series and one arbitrarily chosen reference series (whose particular choice does not influence the results), it is possible to set up an underdetermined linear system involving the unknown (co-)variances Cov(εi, εj) between the individual noises, and the covariances between the various difference series. This system has N free parameters, which we choose to be the covariances between the noise of each series and the noise of the reference series. To determine these parameters, one usually solves a constrained minimization problem, where different objective functions have been proposed in the literature. We adopt the objective function from Torcaso et al. (2000), which is a modification of a function initially proposed by Premoli & Tavella (1993). Once the free parameters have been obtained, estimates of the noise variances Var(εi) can directly be computed. Note that the method also allows for recovering estimates of the covariance between the different noises, which, as shown by Galindo et al. (2001), cannot always be expected to be accurate, however. For each component of our time-series decomposition, we use the TCH method to calculate estimates of the noise variances contained in the series. For the interannual component, we do not include the LOD estimate in our calculation, and we remove the 161 d oscillation from the GRACE series when calculating the noise variance for the intraseasonal component, as the TCH method assumes that each series consists of the same real signal plus individual noise. The results are given in Table 4, where we list estimates of the standard deviation of the noise (i.e. the square root of the obtained noise variances) for each source and component. Table 4. Estimates of the standard deviation of the noise contained in the different series, calculated from the three-cornered-hat method. . Seasonal . Intraseasonal . Interannual . . component . component . component . . (×10−11) . (×10−11) . (×10−11) . SLR 3.20 1.66 2.43 GRACE 5.00 5.13 4.95 GRACE/OBP 3.77 2.17 2.71 LOD 5.98 2.48 – GPS 3.75 1.84 2.83 GAC+GLDAS 3.22 1.80 3.03 . Seasonal . Intraseasonal . Interannual . . component . component . component . . (×10−11) . (×10−11) . (×10−11) . SLR 3.20 1.66 2.43 GRACE 5.00 5.13 4.95 GRACE/OBP 3.77 2.17 2.71 LOD 5.98 2.48 – GPS 3.75 1.84 2.83 GAC+GLDAS 3.22 1.80 3.03 Open in new tab Table 4. Estimates of the standard deviation of the noise contained in the different series, calculated from the three-cornered-hat method. . Seasonal . Intraseasonal . Interannual . . component . component . component . . (×10−11) . (×10−11) . (×10−11) . SLR 3.20 1.66 2.43 GRACE 5.00 5.13 4.95 GRACE/OBP 3.77 2.17 2.71 LOD 5.98 2.48 – GPS 3.75 1.84 2.83 GAC+GLDAS 3.22 1.80 3.03 . Seasonal . Intraseasonal . Interannual . . component . component . component . . (×10−11) . (×10−11) . (×10−11) . SLR 3.20 1.66 2.43 GRACE 5.00 5.13 4.95 GRACE/OBP 3.77 2.17 2.71 LOD 5.98 2.48 – GPS 3.75 1.84 2.83 GAC+GLDAS 3.22 1.80 3.03 Open in new tab The results in Table 4 largely confirm our findings from the previous sections. For the seasonal components, we obtain the lowest noise estimates for SLR and GAC+GLDAS, which can be interpreted as these estimates being closest to the common signal in all the series. In Table 2 we found indeed that annual amplitudes for those two series lie between the values for GRACE and GRACE/OBP which are higher, and those for GPS and LOD that are lower. Somewhat higher noise values are obtained for GPS and GRACE/OBP, while the highest noise estimates are observed for GRACE and LOD, meaning that their seasonal component deviates the most from the other series, consistent with the results found in Section 3.1. The noise estimate for the intraseasonal component from GRACE is significantly higher than for the other estimates, in good agreement with our finding from Section 3.2 that it deviates considerably from the other sources, even after removing the 161 d oscillation. We observe a low noise level for GPS, confirming that this estimate is in good agreement with the other sources. Low noise is also obtained for GAC+GLDAS, while the best value is seen for SLR. In contrast to the seasonal and intraseasonal component, we obtain a relatively high noise level for the interannual component from GAC+GLDAS. This is, however, perfectly consistent with our observations from Section 3.3. As before, the highest value is observed for GRACE, which again agrees well with our previous findings. The lowest noise level is predicted for SLR, while we obtain slightly higher values for GRACE/OBP and GPS. 4 DISCUSSION AND CONCLUSIONS The traditional way to measure ΔC20 is by SLR, and our analyses confirm that this well-established technique seems indeed able to give accurate estimates. For each component, the comparisons yield good results for the SLR estimate, and the noise level as calculated by the TCH method is always the lowest. Interestingly, for the seasonal and intraseasonal components, we generally obtain very high agreement between the solution from SLR and ΔC20 as predicted by global geophysical fluids models, although the latter seem to have some deficiencies in modelling the semi-annual mass variability. Some discrepancies are, however, apparent for the interannual component, likely to be related to the omission of ice melting effects, groundwater changes and Earth internal processes in the model combination we use. Omission of these factors apparently does not have a large negative effect on modelled seasonal and higher-frequency C20 variability, and their contribution to ΔC20 on these scales seems to be rather minor. Our results further demonstrate that GRACE is unable to provide accurate measurements of ΔC20, and the corresponding estimate is generally in relatively bad agreement with the other sources. This holds particularly for the intraseasonal and interannual components and is partly related to the presence of tide-like aliases that have also been identified in earlier GRACE releases. Also note that for each component, the TCH method predicts high noise variance for the GRACE estimate. On the other hand, we mostly obtain good results for the estimate derived from the GRACE/OBP combination, in particular, the agreement for this estimate is generally superior to the corresponding value for the GRACE solution. Furthermore, ΔC20 from GRACE/OBP seems to not be affected by the aforementioned tide-like aliases, so that in particular the agreement for the intraseasonal component from GRACE/OBP is significantly higher than for GRACE. We thus conclude that the approach of Sun et al. (2016) presents a promising method to incorporate GRACE data in the derivation of variations in C20. The computation of high-quality ΔC20 estimates from LOD excitations remains a challenging task. Inferring interannual variability requires the accurate separation of the strong interannual signals in LOD excitations related to core effects from those caused by surface mass redistribution, for which a deeper understanding of the former effects is needed. Furthermore, the contribution from atmospheric winds and oceanic currents needs to be removed from the observed excitations. Atmospheric winds’ effects are largely dominant at the seasonal scale, so that reliable models of the wind fields are crucial in order to recover seasonal C20 variability. Current models seem unable to do this, and LOD-derived seasonal ΔC20 presents rather large discrepancies with other estimates and contains a largely underestimated annual amplitude. For each considered component, the ΔC20 estimate we derive from global GPS measurements is in good agreement with the other sources, and we find that this is a successful method to study C20 variability on scales reaching from intra- to interannual, which is largely independent from other techniques. It should be noted, however, that this technique is only sensitive to processes that change the geometrical shape of the Earth and further relies on the assumption that the displacements measured by GPS are caused by surface mass loading. While it is known that in particular for the vertical component, large parts of the seasonal signals in the GPS position time-series are indeed caused by surface mass loading, the contribution of these effects to measured displacements at shorter and longer scales has not been investigated in detail (e.g. Tregoning et al. 2009; Li et al. 2016; Yan et al. 2016). The good agreement we obtain for our GPS-derived estimate for the interannual and intraseasonal components indicates that loading effects also play an important role at intra- and interannual timescales. Discrepancies between GPS and estimates from other techniques might nevertheless arise due to the presence of non-mass related signals in the GPS position time-series, such as those caused by thermal deformation of the GPS monuments and the bedrock to which they are attached, or those related to GPS errors (e.g. Dong et al. 2002; Ray et al. 2008; Yan et al. 2009). Studying the temporal behaviour of the zonal degree-2 spherical harmonic coefficient of the Earth's gravity potential is an important task, in particular since it reflects mass movements between high and low latitude regions, and is thus indicative of climate-induced mass redistribution related to the melting of polar ice sheets. Different measurement techniques to infer ΔC20 exist, each having its own strengths and weaknesses, so that the corresponding estimates are not always in agreement with each other. 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TI - GRACE era variability in the Earth's oblateness: a comparison of estimates from six different sources
JF - Geophysical Journal International
DO - 10.1093/gji/ggw441
DA - 2017-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/grace-era-variability-in-the-earth-s-oblateness-a-comparison-of-3V03S07x35
SP - 1126
VL - 208
IS - 2
DP - DeepDyve
ER -