The time-variable gravity field shows the effects of large present-day mass variations associated with hydrological phenomena and melting ice-sheets, resulting in a shaded Post Glacial Rebound (PGR) signal. A meaningful viscosity inversion based on a global scale comparison of GRACE data and PGR predictions is thus hard to obtain. We derive a weighted surface mass distribution in water equivalent, starting from an initial guess, which portrays the secular effects of present-day phenomena. The gravity field it generates is then carefully removed from GRACE data, resulting in a gravity pattern where the effects of PGR are clearer and ready to be compared with the predictions. On the basis of viscoelastic stratified Earth models and different Pleistocene deglaciation models, we show that the quality of a global preliminary viscosity inversion greatly improves.
 The Earth surface rebound caused by Pleistocene deglaciation (PGR) or by present-day ice mass melting can be observed in satellite measurements. This rebound deeply involves the physics of the Earth interior, in particular the mantle viscosity. The rebound (due to past and present-day ice melting) can be simulated within a spherical layered viscoelastic Earth model, and model prediction analysis enables a search for the Earth parameters (mantle viscosity) that best reproduce the observational data.
 A local inversion provides information only for a selected region. A global inversion, instead, provides an estimate of the global (average) effective viscosity, that can be employed in rotation-related problems, or in the development of global deglaciation models. Effective global viscosity is also crucial when studying the gravity signature of present-day global or spatially widespread phenomena, as those related to oceans, large river basins or the mass balance in glaciated regions, since the underlying PGR signal must be removed. The best PGR global signal must be used when searching for the gravity due to phenomena occurring in areas far from the Pleistocene deglaciation centres, where PGR is in any case not negligible, mostly in the low harmonic degrees. In all those cases, in fact, the use of parameters obtained from local viscosity inversions, such as over the Hudson Bay, may introduce a strong bias.
 The interpretation of the data, especially the discrimination between the Earth response to past and present phenomena, remains one of the main challenges. The GRACE time-variable gravity field shows a water mass loss in several river basins e.g., the Mississippi. Other analyses show that Greenland [Barletta et al., 2008] and the whole North Pole region are loosing mass [Morison et al., 2007] at a much higher rate than the antagonist PGR signal. These signals heavily contribute to shade the PGR in the Northern hemisphere. Even at local scale, where the PGR signal is visible and a viscosity inversion is possible, as in Hudson Bay, [Paulson et al., 2007], other clearly superimposed signals should be carefully considered or removed [Tamisiea et al., 2007]. In West Antarctica the ice melting signal in Amundsen Sea Embayment [Chen et al., 2008] is superimposed to the PGR. Therefore reliable constraints on PGR are crucial in determining the mass balance, although highly challenging to recover.
 The measured field is often cleared from disturbing signals by removing synthetic gravity patterns obtained from models. This technique is not free from drawbacks: uncertainties in modeling introduce artifacts. Moreover, unknown or unmodelled phenomena cannot be treated in this way.
 To overcome these shortcomings, we build a realistic effective mass distribution directly from time-dependent GRACE gravity data without any a priori modeling assumption. This reproduces the secular effects of present-day phenomena all over the globe, such as ice loss or snow accumulation, dynamics of hydrologic basins and even ocean phenomena. Then we apply a reasoned clearing process to make the PGR signal emerge from GRACE trend field.
2. Straight Comparison
 We perform a preliminary straight global comparison between PGR model predictions and actual GRACE gravity-trend field. We describe here the results from a meaningful but possibly simplified model.
 We use a 5 layer PREM (Preliminary Reference Earth Model) based Earth model with a lithosphere 120 km thick, an upper mantle of 400 km, a transition zone and a lower mantle. Viscosity is stratified into an upper mantle down to 670 km depth, and a lower mantle down to the core. We use the ICE-3G [Tushingham and Peltier, 1991] and ANU [Lambeck et al., 2002] deglaciation models, and various treatments of the sea-level equation: we include and neglect the effect of the sea, and we use fixed and time-variable ocean functions. We present here only the results obtained using the classical sea-level equation, since the others are quite similar. For each viscosity combination PGRij defining a PGR prediction, where ij indexes denote a selected pair of upper and lower mantle viscosity, we calculate the related χij2 by using the original level 2 GFZ Release 04 GRACE data:
where 1ℓm and 2ℓm are the Stokes coefficients trends, σ is the error on trends, and nck = ∑ℓ = 2,m6 1 is the number of Ck in the sum. The typical PGR signature is stronger in the low degree harmonic coefficients, becoming smaller than GRACE errors for degree larger than 20. For these reasons, and for a fair comparison with previous works based on a low-resolution representation of the Earth gravity field, we perform our viscosity inversion only for ℓ, m ≤ 6, but the results are substantially the same up to ℓ, m ≤ 20.
 Global scale direct comparison of model predictions and measurements leads to unsatisfactory results (Figure S1 of the auxiliary material): the minimum χ2 is larger than 30, and the differences are far too large compared with GRACE errors. The direct comparison of PGR predictions and measured time-variable gravity field provides reliable results only if the PGR signal is predominant, as it may have been before 1998, when the low-resolution gravity field was retrieved using the Satellite Laser Ranging (SLR) technique. Indeed, an inversion based on those SLR data was performed by Tosi et al.  and their results differ substantially from ours.
3. Weighted Mass Trends
 The gravity field variations can be represented either as geoid variations, or variations of (water equivalent) masses that generate the observed field. The GFZ Release 04 data quality greatly improved, but a proper spatial filtering is still needed to recover high resolution information. The correlation among the harmonic coefficients first noted by Swenson and Wahr , at the basis of their destriping technique, together with a direct analysis of the errors behavior, shows that the errors on the GRACE Stokes coefficients grow both with increasing ℓ and m. After a careful analysis, we choose an anisotropic filtering, i.e. in both θ and , only beyond a suitably chosen threshold S = . In this way, useful information can be recovered at a resolution better than 300 km from GRACE noisy data, preserving much of the signal strength (Figure 1 (left)).
 We build a suitable effective mass distribution representation (θ, ) of the GRACE field in terms of local mass distributions. Starting from the water equivalent (w.e.) representation, that in the limit of high spatial resolution represents the localized mass distribution generating the field, we process the data assuming that: 1) the field is the superposition of localized sources; 2) each positive/negative spot (mass increase or decrease) in the w.e. representation is associated to a localized mass source. The first assumption is an operational hypothesis: the signal registered by GRACE is the superposition of a number of different phenomena insisting on the same area. It is not possible to identify the different sources without additional information, so we simply represent them through a single effective mass source. The second assumption reflects the hypothesis that all surface phenomena (e.g., from hydrosphere, cryosphere) are related to localized mass redistributions, and that the w.e. representation of the field portrays the underlying mass sources with some blurring due to truncation and filtering. The quality of the reconstruction is assessed against the original GRACE field.
 The algorithm we designed and implemented automatically identifies non-overlapping signal spots, defined according to our operational definition, in the w.e. representation of the GRACE trends at the highest available resolution. Only meaningful signals are selected, testing for the signal to noise ratio. For each selected spot, from the strongest to the weakest, a first guess mass distribution source is obtained. Since the data processing (truncation, filtering) alters the spatial distribution of the signal (blurring) we try to recover localization by choosing a first-guess shape Si(θ, ) that is strictly included inside the selected spot. The signal inside each Si(θ, ) is then “sliced” in a suitable number of intervals (j = 1, 2, …, N), which define a decomposition of the spots in subspots sj(θ, ): Si(θ, ) = = 1Najsj(θ, ). The signal inside each sk belong to the k-th interval, and all aj coefficients are equal. The mass distribution is then optimized by solving an inverse problem, in order to reproduce the gravitational potential inside the selected region: this is done keeping the shape of the subspots fixed, but allowing the weights aj (the interval widths) to vary. This choice provides enough flexibility.
 Once the strongest signals have been processed, the residual field still can give useful information even if its typical strength is comparable to GRACE errors, provided that additional filtering is performed. The resulting lower-resolution residual field is processed as sketched before, and the reconstruction of GRACE field thus improves by including unprocessed widespread and weak signals that were not clearly visible at high resolution, and refining the analysis in already processed areas. In this way an effective mass distribution is built even for low resolution signals. The spots from low-resolution field may overlap with those from high-resolution one. All these steps are perfomed self-consistently, to improve the quality of the reconstruction, and this reduces the risk of overestimation.
 The process stops when all the meaningful signal has been processed, and the difference with respect to GRACE field is almost everywhere below GRACE errors. The effective localized mass distribution representation of the field M(θ, ) = ibiSi(θ, ), consisting of 131 localized sources is further refined to give (θ, ). The relative weights bi of all the selected effective masses are optimized to best fit the gravitational potential from GRACE. The weights we obtain are all very close to unity, supporting the accuracy of the previous steps. Remember that the effective w.e. masses can be associated to physical phenomena only through their geographical position. Additional investigation is required to determine the underlying phenomena. Nonetheless, the localized sources representation of the field enables the removal of the signals from selected areas, even if their physical origin is not clear, leaving the full measured signal in the untouched regions. It is worth noting that this technique does not “kill” the signal in the regions we process, since we do not remove the measured signal itself, but the signal due to an effective localized mass distribution.
4. Step by Step Non-PGR Signal Removal
 Assuming that the mass distribution (θ, ϕ) is realistic, we select different phenomena on the basis of their sources, re-generate their gravitational contribution, and subtract it from the GRACE field, preserving the PGR signal, starting from the reference field in leftmost maps of Figure 2 (step R0). Signals from areas far from PGR and signals clearly counteracting the PGR can safely be removed. However, there are others, nearby or superimposed to the typical PGR regions, which can be confused with it. The signal in the PGR areas is purposely left untouched, since it is impossible to discern solid Earth from hydrosphere phenomena on the basis of purely gravitational evidences, at the maximum spatial resolution available, without resorting to some independent modeling of surface phenomena. By observing Figure 1, we isolate in (θ, ϕ) 8 macro-regions Ri which can be removed. The definition of Ri is arbitrary, a choice made for the clarity of the discussion.
 R1: Greenland negative signal is clearly in contrast with PGR and can be safely removed, though nearby PGR regions. This induces a drastic reduction in the negative signal in the North pole area (Figure 2, -R1 step). Notice that at the Hudson Bay latitude (Figure 1) a strong low-resolution signal on far east of Siberia remains: in a non-longitudinally-resolved analysis it could be ascribed to PGR, inducing some misinterpretations (SLR-based studies may have suffered similar issues).
 R2: Signals over the sea. PGR-related signal on the sea, far from Hudson Bay, Greenland and Fennoscandia, is mostly due to the sea-level component in the deglaciation process, but (according to the PGR models) it is almost everywhere much lower than the GRACE errors and so virtually undetectable. Accordingly, the signal on sea far from PGR regions can be removed. Arctic sea shows almost everywhere a negative trend, surely opposite to the PGR signal, and it is removed. On the contrary, some positive trends around Hudson Bay and Greenland could be PGR-related, and are not removed for precaution.
 R3: Signals nearby Fennoscandia: Arctic polar circle islands (Svaalbard) and Europe are loosing mass and their signal can be safely removed, as well as the Caucasus mass increase, that reasonably does not belong to the expected typical PGR pattern.
 R4: Signals nearby Hudson Bay: Alaska and Mississippi basin are clearly in contrast with PGR, so they can be removed. The third maps from the left (Figure 2) shows the field after removing R2, R3 and R4.
 R5: Other signals from land far from the PGR-regions can be safely removed (Figure 2, -R5 step): far East of Siberia, Himalaya, Sumatra, Amazonia and all spots on land in Figure 1 (right), except those marked by HB, F and S. The latter, and the positive spot in the same location in Figure 1 (middle), that according to the ANU deglaciation model is PGR-related, are not removed.
 R6: Negative trends in Patagonia, West Antarctica and Antarctic Peninsula are clearly in contrast with PGR, so they can be removed.
 R7 and R8: There are some evident signals in Antarctica, but PGR and present-day phenomena are superimposed there. The areas undergoing mass loss (R6), signalling a possible present-day phenomenon, can be removed. It is not clear if some of the mass gain in the East Antarctica (R8) and the other negative spots (R7) clearly visible in Figure 1 (middle), possibly related to a common phenomenon, are related to PGR or to some present-day mass accumulation. This last hypothesis is not so unlikely, since East Antarctica shows a large mass increase, mainly localized at the border [Davis et al., 2005; Chen et al., 2008]. The uncertain signals (R8) are kept for precaution, while the other negative trends in Antarctica (R7) are removed.
 From now on, we will use the “cleared” field from step R7 (Figure 3 (left)).
5. Preliminary Mantle Viscosity Inference
 Even if the cleared field portrays not only the PGR, it also provides a much more reliable starting point for a global scale inversion than the uncleared GRACE field. For each viscosity combination, we apply the same analysis of section 2 and calculate the χij (equation (1)) comparing the previously modeled PGRij with the cleared GRACE data. The fit (Figure 4) greatly improves with respect to the uncleared GRACE data. We made many grid search by using PGRij models calculated varying the deglaciation models and the sea level treatment, and we combine these results in a weighted average. While a complete and accurate inversion (subject for a future work) could show a lot of interesting details, here we only aim to show the improvement in the quality of the results due to our data processing. Basically, in each case (Figure 4) we find two viscosity ranges: one with a soft lower mantle of about 3 × 1021 Pa s, in agreement, within the errors, with results from classical PGR inversions [Tushingham and Peltier, 1991; Paulson et al., 2007], and a second range around vup = 1.5 × 1021 and vlw = 1.2 × 1023 Pa s, a stiff mantle. A second stiff lower mantle range using only GRACE data is reported also by Tamisiea et al.  and Paulson et al. . The smallness of the difference among the averaged results in Figure 4 confirm the low sensitivity of the PGR to the sea-level treatment, as shown by Barletta et al. .
 We show that a careful clearing of the measured signal, with almost no a priori assumption, is feasible: it is a valid alternative to remove model-based corrections for known phenomena, and it can be used also with unknown ones. It is also a powerful technique to investigate global and local scale unknown phenomena through their gravitational signature in its own. We show that a meaningful global scale PGR inversion for mantle viscosity is not possible with the GRACE level 2 data, contrarily to what expected [Wahr and Velicogna, 2003]. It is instead possible after clearing. The global scale gravitational effect of ongoing surface mass redistribution is indeed large enough to mask the global PGR footprint.
 The PGR pattern visible in the cleared GRACE field is weaker than the PGR classical predictions (i.e., vup = 5.0 × 1020 and vlw = 2.5 × 1021 Pa s). This may be due to the global use of Earth parameter suitable to describe the PGR in Hudson Bay, and suggests that deglaciation models in Antarctica are quite inaccurate, and indeed the glacial history of Antarctica remains an open problem.
 We thank Roberto Sabadini for important discussion. This work is partially funded on PRIN 2006 (MIUR).