## 1. Introduction

A recent, combined reanalysis of satellite-based regional altimetry, interferometry, and gravimetry datasets (Shepherd et al. 2012, hereafter S2012) has estimated the mass balance of ice sheets covering Greenland (GIS), the Antarctic Peninsula (APIS), West Antarctica (WAIS), and East Antarctica (EAIS) over the period 1992–2011 as well as during various epochs within this two-decade time window (Table 1). The uncertainties ascribed to these estimates account for the limited and unique time span associated with each of the three space geodetic techniques; observation errors; and, through modeling, uncertainties in corrections for ice sheet surface mass balance and glacial isostatic adjustment (GIA). Over the period 2000–11, for example, the total estimated mass flux from these ice sheets, −298 ± 58 Gt yr^{−1} (1-*σ* uncertainty), equates to a eustatic sea level (ESL) rise of 0.83 ± 0.16 mm yr^{−1}. [We adopt the term “eustatic” as the geographically uniform shift in sea level over the oceans that would yield a volume equal to the meltwater addition. Gregory et al. (2013) have suggested the term “barystatic” for this quantity.] In this paper, we demonstrate that this estimate may be independently tested by invoking the constraint provided by the time rate of change of the zonal harmonic of Earth's gravitational potential at spherical harmonic degree 4

Ice sheet mass balance estimates (Gt yr^{−1}; S2012) and associated ^{−1} × 10^{11}).

Estimates of secular trends in the low-degree zonal harmonics of the geopotential based on satellite laser ranging (SLR) have been available for over a quarter of a century, initially at spherical harmonic degree 2 (Yoder et al. 1983; Rubincam 1984) and later at higher degrees (Cheng et al. 1989). The first analyses of the

A significant change in the secular trend of the

The spherical harmonic basis function at degree 2 and order zero has the same sign in both high northern and southern latitudes. As such, the

In this regard, the

The

## 2. Analysis and results

### a. GIA predictions

The results in Fig. 1 show the predicted GIA-induced signal in ^{20} Pa s, respectively. Our predictions of the long-wavelength zonal harmonics are insensitive to the former, and in the calculations below we explicitly investigate the sensitivity to the latter. The predictions adopt the ICE-5G model for the evolution of ice thickness over the last glacial cycle (Peltier 2004), and they incorporate a gravitationally self-consistent sea level theory (Kendall et al. 2005). This theory outputs the present-day rate of change of relative sea level and sea surface height; the spherical harmonic coefficients of the latter are proportional to the rates of change of the Stokes coefficients (Mitrovica and Peltier 1993).

In addition to the total GIA-induced perturbation in each harmonic (solid lines in Fig. 1), we decompose the perturbation into contributions from the GIS, AIS, and all other ice sheets included in the ICE-5G inventory (e.g., Laurentide, Fennoscandian, etc.). The results indicate that, although ice sheets other than the AIS and GIS dominate the GIA-induced signal in the ^{21} Pa s, which is a hard lower bound for ice age–based inferences (Peltier 2004; Nakada and Lambeck 1989; Lambeck et al. 1998; Mitrovica and Forte 2004), the GIA-induced ^{20} to 10 × 10^{20} Pa s (Fig. 2) and an independent ice history (Fleming and Lambeck 2004). These

The decomposition in Fig. 1 highlights another important advantage of using the observed

We begin by focusing on the period 2000–11 and compute a GIA correction based on a combination of our GIA modeling and the W12a Antarctic GIA model (Whitehouse et al. 2012). The results in Fig. 2 indicate that the GIA contribution to ^{−11} yr^{−1}. Moreover, using the upper- and lower-bound estimates of the W12a model (Whitehouse et al. 2012) yields an Antarctic GIA contribution to ^{−11} yr^{−1}. Thus, we estimate the total GIA contribution to be (−1.7 ± 0.2) × 10^{−11} yr^{−1}. We analyzed the SLR-derived times series for ^{−11} yr^{−1} and (3.7 ± 0.6) × 10^{−11} yr^{−1}_{,} respectively.

We note that the Antarctic ice volume change in the W12a model (Whitehouse et al. 2012) is significantly smaller than in the ICE-5G history. The difference amounts to approximately 10 m of equivalent eustatic sea level rise, or ~10% of the total non-Antarctic ice volume in ICE-5G. This reduction would have to be compensated by an increase in the excess volume of Northern Hemisphere ice relative to ICE-5G in order to maintain a fit to far-field relative sea level records (Austermann et al. 2013). This suggests that the amplitude of our computed non-Antarctic GIA contribution (Fig. 2) will be biased low by ~10%. Accounting for this effect yields GIA-corrected trends in ^{−11} yr^{−1} and (3.8 ± 0.6) × 10^{−11} yr^{−1} for the epochs 1992–2000 and 2000–11, respectively.

### b. Polar mass change contribution to

We next turn to the estimate of polar ice sheet mass balance based on the comprehensive S2012 analysis of satellite-based measurements (Table 1). What signal in ^{−11} yr^{−1}. A final, significant contribution to the ^{−11} yr^{−1}. If we assume that this rate is applicable to the 2000–11 epoch adopted in Table 1, we can augment the above estimate [(2.9 ± 0.6) × 10^{−11} yr^{−1}] to include the mountain glacier signal; in particular, we arrive at an estimate of (3.8 ± 0.6) × 10^{−11} yr^{−1}. The consistency between this estimate and our GIA-corrected SLR-derived trend, (3.8 ± 0.6) × 10^{−11} yr^{−1}, provides independent support for recent estimates of polar ice sheet (S2012) and mountain glacier (Jacob et al. 2012) mass flux over the last decade.

The ^{−1} ESL rise (yr^{−1} × 10^{11}).

Alternatively, in inferring polar ice sheet mass balance, we can avoid entirely the contaminating effect of GIA by considering the change in the trend of the ^{−11} yr^{−1}. Taking into account a change in the mountain glacier signal using the tabulations of Kaser et al. (2006) for the first epoch and (as above) Jacob et al. (2012) for the second epoch raises this value to (2.3 ± 1.1) × 10^{−11} yr^{−1}. As noted above, our direct, SLR-derived estimate of the change in trend is (5.3 ± 1.6) × 10^{−11} yr^{−1}. This analysis suggests either that estimates of melt rates during the earlier epoch are too high and/or that the uncertainty in the SLR-derived change in the ^{−11} yr^{−1} as cited in the caption of Fig. 3] is sensitive to the start time of the epoch.

## 3. Conclusions

We have demonstrated that estimates of net polar ice sheet mass balance may be independently tested by invoking the long-neglected constraint associated with the rate of change of the zonal harmonic degree-4 of Earth's potential. The

We thank Felix Landerer and two anonymous reviewers for helpful comments and suggestions and Minkang Cheng for providing the time series of SLR zonal harmonics used in the present study. The study was funded by Harvard University, Princeton University, the Canadian Institute for Advanced Research, the Natural Sciences and Engineering Research Council of Canada, and NSF Grant EAR-1014606.

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