Corresponding author: M. Cheng, Center for Space Research, University of Texas at Austin, Austin, TX 78759-5321, USA. (Cheng@csr.utexas.edu)
 For over three decades, satellite laser ranging (SLR) has recorded the global nature of the long-wavelength mass change within the Earth system. Analysis of the most recent time series of 30 day SLR-based estimates of Earth's dynamical oblateness, characterized by the gravitational degree-2 zonal spherical harmonic J2, indicates that the long-term variation of J2 appears to be more quadratic than linear in nature. The superposition of a quadratic and an 18.6 year variation leads to the “unknown decadal variation” reported by Cheng and Tapley (2004). Although the primary trend is expected to be linear due to global isostatic adjustment, there is an evident deceleration () in the rate of the decrease in J2 during the last few decades, likely due to changes in the rate of the global mass redistribution from melting of the glaciers and ice sheets as well as mass changes in the atmosphere and ocean.
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 The second-degree zonal harmonic of the Earth's gravity field, J2, is directly related to the Earth's dynamic oblateness. Variations in J2 are the results of both the ongoing global isostatic adjustment (GIA) following the last ice age and climate-induced mass exchange between the “tropical” area (from the equator to the 35º north and south latitudes) and the “extratropical” areas beyond these, including the melting of the polar and mountain ice sheets. The temporal variations in J2 have typically been characterized by the sum of a linear rate (), the tidal variations (), and the nontidal fluctuations () due to mass redistribution with time scales from hours to decades.
 Studying the nature of the long-term time series of J2 remains important for understanding the Earth's global mass variations, particularly those components that may be associated with global climate change. As an example, a large fluctuation in J2 around 1998 attracted significant attention and was reported as “the 1998 anomaly” by Cox and Chao . Excitation of the “1998 anomaly” has been the subject of numerous discussions: see, for example, Cazenave and Nerem  and Dickey et al. . Analysis of the variations in J2 using satellite laser ranging (SLR) data during a 28 year period from 1976 to 2003 by Cheng and Tapley  demonstrated that the “1998 anomaly” was not a unique event; it was merely an effect of the interannual variations associated with the strong El Niño–Southern Oscillation (ENSO). Similar variations were found to occur during the period of 1986–1991, and another interannual cycle was also found to start in late 2002 [Cheng and Tapley, 2004]. The superposition of the “decadal” variation with the interannual ENSO-related signal made the J2 fluctuation appear to be anomalously large during the period of 1996–2002. In addition, a significant fluctuation with a time scale greater than 20 years [Cheng and Tapley, 2004, Figure 3] was found in the detrended time series of J2 after removing the variations related to ENSO events. The cause of this decadal variation remained unknown or a mystery at that time, while the atmosphere, ocean, and surface water redistribution could explain a major part of the 4 to 6 year fluctuation.
 The long-term trend in J2 has generally been approximated by a negative linear drift due to postglacial rebound of the Earth's mantle or GIA [Peltier, 1983]. This rate has been reported from analysis of the SLR data covering periods of only 5 years up to 20 years before 1996 [Yoder et al., 1983; Cheng et al., 1989, 1997; Nerem et al., 1993]. The rates estimated in these analyses varied from −2.6×10−11/yr to −3×10−11/yr and are consistent with −3±0.5×10−11/yr from the analysis of long-term Earth rotation data by Stephenson and Morrison . However, the SLR estimates of the linear trend in J2 have recently been seen to systematically decrease as the analysis time span is increased, suggesting that the long-term change may not be linear in nature.
 This paper presents analysis of a new and longer time series of 30 day estimates of J2 using SLR tracking of up to eight geodetic satellites, spanning the interval from January 1976 to May 2011. Section 2 discusses the determination of the time series from the SLR tracking. Section 3 addresses the nature of the secular or long-term trend, where it appears that the series is well-characterized by a quadratic (rather than linear) variation superposed by an apparent 18.6 year period variation. Section 4 provides a preliminary assessment of the various contributions to the deceleration in J2, including mass redistribution in the atmosphere, ocean, mountain glaciers, and ice sheets. We show that these contributions are sufficiently large to explain the dramatic changes in the secular trend in J2 over the last decade.
2 Analysis of the Time Series of J2 Variations From SLR
 The determination and separation of the J2 signal has been well established from earlier studies, including Nerem et al.  and Cox and Chao  and improved through the use of multiple satellites in Cheng et al.  and Cheng and Tapley [1999, 2004]. The time series discussed here is based on the methodology described in the previous analysis by Cheng and Tapley , but it adopts the improved models in the IERS2010 conventions [Petit and Luzum, 2010]. The ITRF2005 reference frame for tracking stations (specifically, SLRF2005/LPOD2005 [Ries, 2008]), the ocean pole tide [Desai, 2002], the FES2004 ocean (diurnal and semidiurnal) tides [Lyard et al., 2006], and the EGM2008 gravity model [Pavlis et al., 2012] were used in this analysis. As before, up to eight satellites (LAGEOS-1 and 2, Etalon-1 and 2, Starlette, Stella, Ajisai, and BEC) are used, though only LAGEOS-1 and Starlette were available prior to 1986. Unlike the long-arc approach, the requirement for the spatial distribution of SLR data (i.e., satellites at various orbit with altitude and inclination) during a given time interval is particularly stringent for separation of the higher-degree zonal and nonzonal geopotential coefficients [Cheng and Tapley, 1999]. The estimate of J6 is highly correlated with J8 and other higher degrees. Consequently, J4 must be adjusted to account for effects of the higher-degree zonals, and the correlations indicate that J2 is reasonably well separated from the higher-degree terms with this approach. The geopotential coefficients up to degree and order 3 plus J4 and the geocenter parameters (Δx, Δy, and Δz) were estimated for each 30 day interval based on an optimal weighting procedure using the SLR data available from whichever of the eight satellites were available. The Earth's gravitational coefficient was also estimated as a single global parameter; the resulting value was identical to the value adopted in the International Earth Rotation Service (IERS) Conventions. The SLR tracking data were provided by the International Laser Ranging Service [Pearlman et al., 2002].
 As discussed by Cheng and Tapley , except for the secular (linear trend) and the tidal harmonic variations, the variations in J2 are climate related with a stochastic (nonharmonic) behavior. To deal with those signals with varying amplitude and phase, the wavelet analysis is a suitable technique for time series analysis [see, for example, Satirapod et al., 2001; Bessissi et al., 2009; Khelifa et al., 2012], which decomposes the signals into high- (named as Detail) and low-frequency (named as Approximation) components in the time domain. In order to characterize the long-term change in the time series of the J2 variations from SLR and Atmosphere and Ocean De-aliasing model (AOD), the discrete approximation of the Meyer wavelet (dmey) was applied as a low-pass filter (see supplementary material and discussion given by Cheng and Tapley ) to show the J2 variations at time scales beyond the seasonal (denoted as decadal) and the variations beyond decadal (denoted as secular or long term). The decadal band is equivalent to the 360 day sliding average used to remove the seasonal and higher-frequency signals [Cheng and Tapley, 2004]. The long-term spectral band is closer to but smoother than the results from using a Butterworth-type low-pass filter with a 20 year cutoff [Roy and Peltier, 2011, Figure 2].
 Figure 1 shows the 30 day variations of J2, along with the estimated 1-sigma uncertainty during the 35 years between 1976 and May 2011. The uncertainty is clearly reduced when more satellites become available, particularly around 1986 and 1993. Figure 1 also shows the long-wavelength signature beyond the seasonal (the decadal component). The long-term trend is clearly visible, as well as the annual and interannual fluctuations. In particular, note that interannual variations, discussed previously in Cheng and Tapley , continue beyond 2002. Superposed is a quadratic fit to the 30 day estimates, which demonstrates that the long-term trend in this time series is characterized well by a quadratic variation.
3 Secular Variation
 The first SLR-based estimate for the linear trend in J2 of −3 × 10−11/yr was reported by Yoder et al.  based on the analysis of 5.5 years of SLR tracking to LAGEOS-1. A value of −2.6×10−11/yr was obtained for using short-arc analysis of the SLR data from LAGEOS-1, collected during the 10 year interval between 1980 and 1989 [Nerem et al., 1993]. Using SLR data from eight geodetic satellites orbiting at different altitudes and inclinations spanning the interval between 1976 and 1995, a value of −2.7 ± 0.4×10−11/yr was determined for , along with other higher-degree zonal rates and dynamic model parameters (including 18.6 year ocean tide parameters) [Cheng et al., 1997]. This result was based on a long-arc dynamic analysis approach, which requires the computation of a single dynamically consistent trajectory during each 1 year time span for the lower-altitude satellites (Starlette, Stella, Ajisai, and BEC), and a single trajectory for the entire tracking history for the high-altitude satellites (LAGEOS-1 and 2, Etalon-1 and 2). Using the long-arc dynamical approach with the time span extended year by year from 1996, the estimates for show a decrease as the analysis time span is increased. However, the long-arc approach becomes impractical for understanding the nature of the J2 variations, and we turn to analysis of the time series of 30 day estimates to characterize the variations in J2 over the last 35 years.
 Estimating the linear trend from a straight-line fit to the time series of J2 variations over various time spans, we find a value of −2.8 ± 0.3×10−11/yr for the period of 1976–1995 (which is in good agreement with that from long-arc analysis), but the value decreases significantly when the time span is increased and more recent SLR data are included, indicating a significant deceleration in J2. Roy and Peltier  fit an earlier time series (an extended version of the time series described in Cheng and Tapley ) using two separate straight lines to obtain a rate of −0.9 ± 0.2×10−11/yr for 1992–2009 and −3.7 ± 0.1×10−11/yr for 1976–1992. The estimate for 1992–2009 departs significantly from what would be expected from GIA alone and is likely related to ice sheet mass loss due to global warming. Nerem and Wahr , also using the same extension of the time series described in Cheng and Tapley , show that J2 began to increase (resulting in a rate decrease) beginning in the mid-1990s, after being detrended with a rate of −3.6×10−11/yr [from Paulson et al., 2007] to represent the effect of GIA. They conclude that the contribution to from Antarctica and Greenland ice loss over the period of 2002–2010 observed by the Gravity Recovery and Climate Experiment (GRACE) is +3.7×10−11/yr.
 Figure 2 compares (1) the long-term spectral band from the variations of J2 and (2) the residual after removing a linear trend and (3) the residuals after removing a quadratic J2. After a linear fit, the residual J2 variation appears to fluctuate at a time scale longer than 20 years without any discernible periodic component, as has been shown by Cheng and Tapley . However, as shown in Figure 2, a quadratic fit yields a residual with almost two cycles of a variation with an apparent 18.6 year period. Thus, the long-wavelength variation of J2 appears to be better represented by the superposition of a quadratic and 18.6 year variation, which was not evident at the time for the 28 year time series, analyzed by Cheng and Tapley . The quadratic and 18.6 year components account for ~98% of the variance of the variations in the long-term spectral band.
 The 18.6 year variation in the long-term spectral band could be caused by a departure from equilibrium of the lunar node ocean tide or an error in the mantle anelasticity used for the solid earth tide (or both). The potential separation of the individual contributions of the ocean and solid earth tide from the observed J2 variations is important to place a correct constraint on the mantle anelasticity. This issue merits a more comprehensive analysis and will be addressed in a separate paper. After removing a quadratic and 18.6 year period variation from the decadal variation, the dominant variations are with periods of 10.4, 5.4, and 2.5 years, which also appear in the spectrum of the cold phase of the Southern Oscillation Index. This was discussed by Cheng and Tapley , though at the time the decadal variations were not understood. These variations will also not be discussed further here.
 In brief, this analysis suggests that the superposition of an 18.6 year signal and the quadratic variation explains the observed long-term variation of J2. The linear model is a good approximation during the first two decades of the LAGEOS-1 mission, but it is not valid for time spans extending beyond the late 1990s. In light of observations of accelerated ice melt [Velicogna, 2009; Jacob et al., 2012], it is reasonable to suppose that the J2 variations are better represented by a quadratic. The quadratic signature does not depend on the choice of the long-term spectral band from the wavelet filter but is intrinsic to the 36 year time series of J2 variations (Figure 1). It is a natural choice for characterizing the long-wavelength signature for the J2 variation with time spans longer than three decades. Other approaches, such as Roy and Peltier  using two straight lines, are possible, but they bring in a subjective choice as to when to break the two lines, and the residual is difficult to interpret.
 The recent ice mass loss effects appear not only in the J2 time series but also in the motion of the Earth's mean pole of rotation and figure axis (and consequently in the degree-2 order 1 gravity harmonics C21 and S21 [Cheng et al., 2011]). The change from relatively linear motion in the mean pole over the decades prior to approximately the year 2000 is reflected by the nonlinear model (cubic polynomial) recently adopted for the updated IERS Conventions [Petit and Luzum, 2010]. Similarly, it is necessary to abandon a linear model to represent the long-term behavior of J2 and adopt instead a higher-order polynomial, at least for the modern period of space observations since 1976. The values of the best-fit quadratic are given in Table 1, based on a weighted least squares fit to the 30 day estimates (a fit to the decadal component illustrated in Figure 1 is essentially identical). The estimated uncertainty of 1×10−13/yr2 for is the formal error of that fit. Only time will tell whether this representation continues to be appropriate or if the long-term variation in J2 is even more complicated. Continued monitoring of the variations in J2 is critical for understanding the nature of the surface and crustal mass redistribution at the longest wavelengths.
Table 1. Coefficients for the Polynomial Representation of the Long-term Trend for J2 in the Form of a + bt + ½ct2, Where t is Years Past 2000.0. The 1-Sigma Uncertainties are the Formal Errors From the Least Squares Fit of a Quadratic, an Annual Term and an 18.6 Year Term to the 30 Day Estimates, Using a Uniform Sigma of 1 × 10−10. Weighting the Data or Excluding the Periodic Terms Does Not Significantly Change the Estimates or the Uncertainties
−4.84169453 × 10−4
1.08263581 × 10−3
1 × 10−11
2.7 × 10−12
−5.9 × 10−12
8 × 10−13
−8.0 × 10−13
1.8 × 10−12
1 × 10−13
4 Contributions to the Deceleration of J2
 Variations in J2 are the consequence of the mass redistribution within the Earth's dynamic system (atmosphere, ocean, continental water, and glaciers) due to climate forcing. The most significant effect on the very long-term variation in J2 is expected to be linear due to GIA (denoted as ). However, recent analyses indicate that the contribution from present-day melting of the polar ice sheets and the mass component of the rise in global sea-level rise may now be comparable. Other secondary small contributions are from reservoir impoundment [Chao, 1995] and earthquakes [Gross and Chao, 2001]. In the following, we designate the linear change in J2 over a relatively short time period by , so that contributions to changes in imply an acceleration or deceleration in the secular trend in J2.
 There are several global land surface hydrology models for the land water storage variability. Those models include the NOAA Climate Prediction Center global climatological soil moisture data [Yun and van den Dool, 2004], NASA Land Data Assimilation System (GLDAS) [Rodell et al., 2004], and Water Global Assessment and Prognosis hydrology model [Döll et al., 2003]. Comparison of those hydrology models with GRACE shows agreement at the seasonal scale but regionally dependent [Li et al., 2012]. At present, those hydrology models are inadequate for studying the long-term variations in J2 because those hydrology models, such as GLDAS, often do not simulate groundwater or “permanent” snow/ice, so that some long-term trends may be missed [Rodell, personal communication, 2012].
4.1 Long-Wavelength Variations in the Atmosphere and Ocean
 For the short-period mass variations in the atmosphere and ocean, we selected the same AOD for compatibility with GRACE Release 04 processing [Bettadpur, 2007]. This data reflects the spatiotemporal mass distribution in the atmosphere and ocean derived from a three-dimensional meteorological field with 0.5º spatial resolution at 6 hourly intervals from the integrated forecast system of the European Center for Medium-range Weather Forecast data. The ocean dynamics corresponding to wind stress, atmospheric pressure, as well as heat and freshwater flux, is obtained by using these values to force the Ocean Model for Circulation and Tides (OMCT) baroclinic ocean model [Thomas et al., 2001; Flechtner et al., 2007]. The AOD data set provides the temporal variations in the gravitational geopotential coefficients to degree and order 100 associated with the mass variations in (1) the vertical integrated atmosphere (ATM), (2) the OMCT water column (OCN), (3) the OMCT ocean bottom pressure (OBP), and (4) the global combination of atmosphere and ocean (GAO) from ATM + OCN. The long-term time series of the AOD product allows the evaluation of the contribution of the atmosphere and ocean to the variations in J2. The AOD model used for GRACE has been extended to cover the same period from 1976 to the present [Flechtner et al., 2007]. The time series of the 30 day averaged J2 variations was calculated from these four model types over the same period as the SLR data span. The AOD model was not used directly in processing the SLR data, but the 30 day mean values of the AOD model are available and can be applied to the time series if desired.
 Figure 3 shows the long-term variations (represented by the long-term component from the wavelet analysis) due to the mass redistribution in the atmosphere, ocean, and the GAO in the AOD time series during the period from 1976 to 2011. The atmospheric-induced variations in J2 gradually increased from 1976 to a maximum in early 1993 and have decreased continuously since then. In response to the atmosphere loading and other forcing, the ocean mass variations in J2 has varied inversely with the atmospheric loading during the interval from 1976 through 1992. After this epoch, it has continuously increased. As shown in Figure 3, the response of the ocean to the atmosphere mass loading deviates from the conventional inverted barometer response for the multidecade time scale involved in the OMCT model, which is forced by atmospheric pressure, winds, heat, and freshwater fluxes. The deviation could be also due to the mass exchange from the atmosphere over land to the atmosphere over the ocean and ocean circulation changes. The variation of J2 increases almost linearly due to the mass variability in the combination of the atmosphere and ocean, which will be also reflected in the OBP. The mass redistribution in the global atmosphere and ocean is one of the excitation sources that could cause a decrease in . The contribution is small but not negligible, affecting at the level of +0.3×10−11/yr.
 A 33 year time series of AOD data has also been analyzed by Chambers and Willis  to study the mass fluctuations for the ocean basins on long time scales. They used the equivalent water height computed from the monthly mean of the geopotential coefficients for the OBP from the AOD model. They found that interbasin mass exchanges are a regular occurrence and can last up to two decades. In their study, the seasonal (annual and semiannual) and linear trends were estimated for each grid cell and removed. We find that the maximum drift is less than 1 mm/year from the atmosphere, but there are significant linear trends (of several mm/yr to a cm/yr) in some regions from the OCN or OBP models, as well as for the GAO coefficients. Examples are the regions in the southern ocean, as well as the region around the Mediterranean Sea. The large linear drifts in those regions could be due to model drift, as suggested by Chambers and Willis . However, the large regional drift is mainly due to the higher-degree coefficients rather than J2, as can be shown from the difference between the distribution of the drift with and without including J2 in computing the equivalent water height from the geopotential coefficients of the AOD model. The difference of the two maps indicates a negative drift of 0.2 mm/yr over the tropical ocean and a positive drift over the extratropical ocean area. Consequently, the model drift should have only a small effect on the rate estimate of J2 due to atmosphere–ocean mass variations. However, the accuracy of the AOD model at the time scale of decades should be evaluated for this kind of application. In following discussion, the oceanic contribution to J2-rate is based on the observed global sea level rise.
4.2 Mountain Glaciers
 The National Snow and Ice Data Center (NSIDC) data for the mountain glaciers were used to study the sea-level rise by Meier et al. . We used this data set to evaluate the contributions of these effects to the J2 variations, although the temporal and spatial distribution was not adequate for direct inclusion in the analysis. The data include the annual volume changes (water equivalent in km3/yr), area, and geographic location (longitude and latitude) for 49 primary glacier systems in 12 larger glacier regions spanning the period from 1961 to 2003 [Dyurgerov and Meier, 2005, Appendix 3]. The 11 glacier systems with relatively short records, as well as the Greenland and Antarctica ice sheet, were not used in this analysis.
 The melting of glaciers and ice caps (GICs), and the resulting rise in sea level, induces variations in J2 that can be computed from the surface mass density load change through a surface integral with the Legendre polynomial weighting function, P2(cos θ), expressed as follows (Chao et al., 1987, equation (9)).
where k′2 = −0.3046 is the Earth's degree-2 load Love number. Ae and Me are the mean radius and mass of the Earth. Direct contribution to the changes in (denoted as ) due to GIC melting can be evaluated from the surface integral (equation (2)) with the function determined by the ice mass change data (Gt/yr), the area (GIC size), and the geographic location. Assuming mass conservation, melting ice leads to a rise in global sea level, which will also cause a change in J2. Let denote the contribution to due to the mass component of a rise in sea level (RSL) with a uniform rate of . Over the ocean, ∆ can be expressed in terms of the uniform sea-level rise as , where ρw is the density of seawater, and F(Ω) is the ocean function [Lambeck, 1980]. Evaluation of the integral in equation (2) over the ocean leads to the contribution of the sea-level rise to expressed as:
[Chao et al., 1987; Chao and O'Connor, 1988]. Thus, GIC melting induces changes in consisting of the direct removal of the ice on land, , and the resulting rise in sea level, .
 By some reports, melting of the mountain glaciers over the past decade or so has accelerated, causing an increase in the rate of global sea-level rise (based on glacial volume change data) [Meier et al., 2007; Hirabayashi et al., 2010]. Using the NSIDC data for 36 major glacier systems (exclusive of the Antarctica and Greenland ice sheets) spanning the interval between 1976 and 2003, the direct effect on due to the melting of the mountain glaciers is calculated to be approximately +0.3×10−11/yr on average for most of that period, as shown in Figure 4. It is clear that there is a significant increase in that begins in the late 1990s. However, the uncertainty is high in models for glacier mass balance. Based on global inversion of monthly GRACE-based gravity fields, Jacob et al.  estimate that the average mass loss of the mountain glaciers contributes to an estimated rise in sea level of +0.41 ± 0.08 mm/yr for the period of 2003–2010 [Jacob et al., 2012]. This is only slightly larger than +0.34 mm/yr estimated by Hirabayashi et al., 2010 for the period 1948–1990, but slightly smaller than +0.43–0.51 mm/yr estimated by Meier et al.  for the period 1961–2002. Consequently, it could be concluded that the contribution to from mountain glaciers after 1990 may be only slightly larger than before, on the order of +0.4×10−11/yr.
4.3 Antarctica and Greenland
 Antarctica and Greenland, which hold enormous reserves of fresh water, have undergone a significant ice mass loss during the last decade [Velicogna and Wahr, 2005, 2006], and recent results indicate a likely acceleration of that mass loss [Velicogna, 2009; Chen et al., 2009; Jacob et al., 2012]. The GRACE-derived total mass loss of 143 ± 73 Gt/yr for Antarctica and 230 ± 33 Gt/yr for Greenland [Velicogna, 2009] will cause a sea-level rise of 0.4 and 0.7 mm/yr, respectively, from Antarctica and Greenland (using the factor that melting of ~362 Gt of ice leads to a sea-level rise of 1 mm [Meier et al., 2007]). Rignot et al.  use two independent techniques to determine a combined mass loss of 475 ± 158 Gt/yr, equivalent to a sea-level rise of 1.3 ± 0.4 mm/yr in year 2006. Jacob et al.  arrive at a similar conclusion for Antarctica and Greenland (1.06 ± 0.19 mm/yr). The meltwater from the Greenland and Antarctica ice sheets results in of +0.17×10−11/yr, while the direct effect of the ice mass loss, , is calculated to be +3.9×10−11/yr, assuming a uniform distribution of ice loss over the Greenland and West Antarctica areas used in the study of Wu et al. . This can be compared to James and Ivins , which presented a linear relation between sea-level rise () and zonal rates () based on a composite scenario study over Antarctica and Greenland ice sheets. The linear relation is expressed as , where F = 3.87 (×10− 11/mm) for Antarctica and F = 3.74 (×10− 11/mm) for Greenland. This relation includes the direct effect of mass loss from the ice sheets and the resulting effect on global sea level. With a value of 1.1 mm/yr sea-level rise due to mass loss from Greenland (0.7 mm/yr) and Antarctica (0.4 mm/yr), + would be approximately +4.2 × 10−11/yr.
 Analysis of the ocean mass flux measurements by GRACE [Tapley et al., 2004a, 2004b], Jason-1 altimeter measurements of global sea level, and ocean temperature measurements from autonomous ocean observing system (e.g., Argo) suggests that the mass component of sea-level rise increased at a rate of 1.3 ± 0.6 mm/yr between 2005 and 2010 [Willis et al., 2010] and at an average rate of 1.1 mm/yr between 2002 and 2010 [Chambers, 2011, personal communication]. When the contributions from all ice-covered regions are included, Jacob et al. (2012) estimate the nonsteric contribution to sea-level rise to be 1.48 ± 0.26 mm/yr. These estimates reflect all of the contributions from continental water loss, including the melting of mountain glaciers and polar ice sheets. From equation (3), the value of from sea-level rise ranging between 1.1 and 1.5 mm/yr would be +0.2×10−11/yr.
4.4 J2 Changes During the GRACE Mission
 The effects of ice mass loss on J2 should be reflected in the GRACE-derived J2 variations, but the estimates of J2 from GRACE are affected by large tide-like aliases of unknown origin. The SLR data provide a unique means of measuring the long-term variations in J2, and the monthly SLR estimates of J2 are routinely used with the remainder of the GRACE gravity field coefficients. Figure 5 compares the monthly estimates of J2 from GRACE and SLR over the period from January 2002 to March 2012. These SLR estimates are slightly different from the long-term time series previously discussed, being obtained using background gravity models identical with GRACE Release-04 (Bettadpur, 2007), including the same AOD model and the IERS conventional value for (Petit and Luzum, 2010). For this solution, a full degree and order 5 gravity field is estimated, along with the geocenter, using five satellites (LAGEOS-1 and -2, Starlette, Stella, and Ajisai). In general, this series agrees well with the long-term time series but may be somewhat more accurate during this period. After restoring the modeled AOD and the IERS conventional rate, is estimated to be +0.7 ± 0.3×10−11/yr from SLR (denoted as ) over the period of 2002–2010.
 The ice mass loss rates for Greenland and Antarctica were obtained from the surface density changes calculated from the GRACE monthly estimates for the gravity field coefficients (up to degree and order 60) based on the formulation of Wahr et al. . The C20 harmonic have been replaced by the SLR estimates, which affects (and presumably improves) the estimates of ice mass loss rate by ~20% for Antarctica and ~2% for Greenland [Schrama, 2012]. Thus, the value of +3.9×10−11/yr for the (calculated from the GRACE data-derived ice mass loss rate as discussed above and comparable with the prediction from James and Ivins linear relation) is essentially from the SLR observations. As a verification, the effect of the mass loss of the ice sheets, , can be obtained based on the total budget (James and Ivins, 1997): using the rate of +0.7 ± 0.3×10−11/yr from SLR over the period of 2002–2010, +0.2×10−11/yr for from a nonsteric sea-level rise rate of approximately 1.3 mm/yr, −3.6×10−11/yr for and +0.3×10−11/yr for the contribution of the mountain glaciers. Considering the various uncertainties, this result indicates that the deceleration of can be explained by recent melting of continental glaciers, and the SLR-derived estimates for may provide some constraints on the magnitude of recent melting of continental glaciers, but the results will depend on the accuracy of the estimate of . The IERS conventional rate of −2.6×10−11/yr determined from SLR tracking of geodetic satellites before 1996 likely represents the combination of GIA and GIC melting over the earlier period, assuming the GIA contribution to is at the level of −3.6×10−11/yr (Paulson et al., 2007).
 The observed variations in J2 from analysis of SLR data spanning the past three and a half decades provide a clear indication of the long-term large-scale mass redistribution within the Earth system. The recent acceleration in glacial melting (resulting in an increased global sea-level rise rate) appears to increasingly offset the steady GIA activity during the recent two decades, resulting in a deceleration or quadratic signature in J2 as observed from this analysis. The long-term variations in J2 are best fit by the superposition of a quadratic variation with an 18.6 year variation possibly related to ocean tide model errors. The deceleration in the change of the Earth's oblateness is attributed to the result of accelerating melting of the glaciers and ice sheets, as well as atmosphere and ocean mass redistribution. SLR-derived estimates for can provide an important constraint on studies of global ice mass loss and sea-level rise, but the results will depend on the accuracy of the estimate of the effect of GIA. Continued monitoring of the variations in J2 from SLR are critical for understanding the fundamental nature of the ongoing mass redistribution that may result from global climate change.
 This research was supported in part by NASA grants NNX12AK13G and JPL1368074. The insightful comments from the associate editor and anonymous reviewers are greatly appreciated. The authors acknowledge Frank Flechtner at GFZ for providing the long-term AOD model for this analysis, the International Laser Ranging Service for making the SLR data available for this study, and the Texas Advanced Computing Center for providing computational resources.