Present day glacier reduction in the Alps, estimated from glacier inventories and induced viscoelastic response of a stratified Earth's model, is responsible for sizable uplift rates. Patches of 0.4–0.5 mm/yr, due to ice mass loss of largest ice complexes, overprint a characteristic area of slower uplift of 0.1–0.2 mm/yr, signature of the phenomenon in the whole Alpine chain. Viscous stress relaxation in the lower crust, due to glacier mass loss after the end of the Little Ice Age, is expected to produce uplift rates of 0.32 mm/yr, leading to a total viscoelastic response up to 0.8 mm/yr. Our predictions in the western Alps show that viscoelastic response to present day glacier shrinkage forms a substantial fraction (half) of the observed uplift data. Attempts to constrain the contributions arising from active Alpine tectonics and drainage must account for this uplift signal.
 The decline of Alpine glaciers beginning in 1850 A.D. is inferred from analysis of regional and national glacier inventories, as done in the present study, or from comparison of glacier data obtained from Landsat Thematic Mapper (TM) data with previous glacier areas [Paul et al., 2002]. Alpine glacier shrinkage is consistent with worldwide large ice mass reductions [Paul et al., 2004; Haeberli et al., 1999] and can be considered an indicator of global climate change [Intergovernmental Panel on Climate Change, 2001]. Changes in volume of glacier masses in the Alps are expected to induce vertical uplift due to the Earth's elastic and viscoelastic response to surface load redistribution, as modeled in this study. A new high spatial resolution versus standard low resolution simulation for the elastic response to present day glacier shrinkage is carried out, and, separately, the viscoelastic response to the glacier mass loss after the end of LIA (Little Ice Age). Other potential sources of uplift, such as last glacial maximum (LGM) unloading and pre-LIA load conditions, are not considered, attention being focused on present day glacier shrinkage. Furthermore, as shown by Persaud and Pfiffner , the LGM unloading signal cannot explain the observed uplift pattern (A. Schlatter, U. Marti, and D. Schneider (The new national height system (LHN95) of Switzerland, EUCOR-URGENT, Annual reports 1999, subproject 1.1, 1999, available at http://comp1.geol.unibas.ch/report99/reportindex.htm, hereinafter referred to as Schlatter et al., report, 1999) in the Swiss Alps, since the isostatic LGM compensation acts on the whole Alpine region. Uplift rates from Alpine glacier wasting can be compared with those predicted for larger ice complexes, for example the Patagonian ones treated by Ivins and James , thus providing testable tools for modeling lithosphere-cryosphere interaction, within two different environments, the Alps and the Andes.
2. Alpine Glacier Mass Loss
 World Glacier Inventory (WGI) data have been used to evaluate the mass loss affecting the Alpine glaciers on a known time interval. WGI reports glacier data (surface area, length and main dimensional parameters) collected between the 50s and the 80s of the XX century: it has been necessary to apply an area reduction factor in order to estimate more recent glacier surface areas. Updated glacier areas are evaluated for the years 1996, 1997, 1998 and 1999 by applying a surface reduction factor specific for each glacier, considering the Alpine region where it is located. This factor is obtained from the available literature dealing with glacier shrinkage rate in the different Alpine regions (Table 1). Main sources for calculating the reduction factor are multitemporal national and regional glacier inventories, whose comparison permitted the evaluation of reliable glacier reduction over a time frame of one or two decades [Paul et al., 2004; Kääb et al., 2002; Biancotti and Motta, 2001]. Additionally, the numerical variation affecting the whole glacier sample is considered: glaciers indicated as “Glacierets” in the WGI are removed from the data-set. This choice is due to the authors' findings in the Italian Alps where all the glacierets reported in the WGI database at the end of the nineties disappeared. Consequently, in 1999 the glaciers considered in this study covered an area of 2215 km2. Moreover the choice of evaluating the glacier mass loss for the years 1996–1999 is also supported by the availability, for that time interval, of specific mass balance data of a representative sample of Alpine glaciers. The mass balance data published on FOG [World Glacier Monitoring Service, 2005] can be considered representative of the mass variations which affected Alpine Glaciers on the studied time frame. In fact, according to the “glacier regionalism” introduced by Reynaud et al. , glaciers located in the same region show stronger correlation among their mass balances. Since more than one mass balance for the period 1996–1999 has been published for each region, a mean mass balance value per year has been obtained by averaging the mass balances for each region. These yearly values have then been averaged over the considered four years to obtain a unique value for each Alpine region (second column, Table 1), to be used within a high spatial resolution normal mode scheme. The average of the values reported in Table 1 equals −0.71 m/yr water equivalent (1.54 Gt/yr): this value represents an estimate of the mean mass loss affecting the whole sample of Alpine glaciers and is used within the low spatial resolution calculation. In this scheme it is necessary to include the water feeding the Mediterranean sea: by applying the above reported glacier loss value, the sea-level rise (limited to the Mediterranean sea) of 0.46 mm/yr is obtained.
Table 1. Coefficient of Glacier Area Reduction Per Year and the Mass Balance, Averaged Over 1996–1999
 Uplift rates are based on a spherical and radially stratified, viscoelastic Earth model (linear Maxwell rheology, 7 layers, Table 2), similar to that used for post-seismic calculations, but tuned to the viscosity profile used by Burov et al. , and with volume averaged rigidities from PREM [Dziewonski and Anderson, 1981]. In normal mode theory for a viscoelastic Earth model, the vertical displacement is expanded in spherical harmonics. Truncation of the expansion results into information loss, determining the spatial resolution. The Alpine glaciers have characteristic lengths of about 1 km, thus a harmonic decomposition up to 40,000 degrees should be exploited. A low resolution calculation can be carried out considering only an equivalent load spread over the area of the geographical region occupied by all the glaciers; we can thus limit the harmonic expansion to about 500 degrees. The effective mass balance (sE) is smaller with respect to the real one (sR) by a factor chosen in order to preserve the total mass balance, sE = (AR/AE) × sR, for a real area AR of 2215 km2 and a mean mass balance sR of −0.71 m/yr.
Table 2. Rheologic Structure
ν, Pa s
2.97 × 1010
1.00 × 1035
5.58 × 1010
2.15 × 1019
6.81 × 1010
5.00 × 1021
7.27 × 1010
4.64 × 1020
1.09 × 1011
4.64 × 1020
2.21 × 1011
1.00 × 1021
 Results of this low resolution calculation are shown in Figure 1, where the largest elastic uplift rate is about 0.1 mm/yr and the position of the maximum is roughly located in the center of the area, as expected. Besides the uplift rate localized in the surrounding of the alpine chain, we can also take into account the effect of Mediterranean basin sinking caused by the melting water. In this calculation we used about 1.151 Gt/yr of water feeding the Mediterranean sea, modeled as a closed basin, obtaining a small subsidence down to −0.007 mm/yr in the southern Mediterranean basin. The value of 0.1 mm/yr obtained from this low resolution calculation can be compared with the present day uplift rate of 2 mm/yr obtained by Ivins and James  for the Patagonian ice fields and a sub-cratonic mantle, prediction very similar to that for an elastic model, not shown in that article. The difference can be understood by considering that in Patagonia the mass loss considered by Ivins and James , 38.4 Gt/yr for years 1990–2000, is 24 times larger than the Alpine 1.54 Gt/yr.
 The low resolution calculation of this section is appropriate for estimating long wavelength features of the vertical displacement, but to explain the observed uplift pattern (Schlatter et al., report, 1999), accurate high resolution calculations is required.
4. High Resolution Approach
 In contrast to other global problems, such as sea level computations where the load function is extended over the whole globe, Alpine loads occupy a very limited part of the sphere, thus, it is convenient to integrate straightforwardly over the load, and treat the glaciers as a discrete ensemble of point-like sources. The vertical displacement U at the observation point (θ, ϕ) can be expressed by the following sum over the N glaciers
where γn,θϕ is the angle between the n-th glacier point and the observation point, G is the Green's function, ρI is the ice density and An and sn indicate the area and mass loss rate of the n-th glacier, inferred from the regional data (Table 1). A similar use of point-like source distribution can be found for co- and post-seismic deformation problems as recently performed by Dalla Via et al. .
Le Meur and Hindmarsh  and Barletta and Sabadini  have shown that for a surface load the elastic part of the solution tends asymptotically to a non zero value for increasingly high harmonic degrees. Le Meur and Hindmarsh  have shown how to overcome the Gibbs phenomenon due to truncation by making use of the sum of Legendre polynomials series. The Green's function, thus, can be written as follows:
where hℓ are the Love numbers for the vertical deformation, is the limit of hℓ for ℓ → , Pℓ is the Legendre polynomial, a and M are the radius and the mass of the Earth. The approximate solution for one disk, based on the Green's function G given in equation (2), (r) = sDρIADG(r/a) (sD is the mass balance (−1 m/yr), AD the area of the disk) has been compared with the highly accurate solution
where the sum of the series is carried out at 40,000 harmonic degrees to resolve 1 km loads and Dℓ(R) are the disk coefficients as follows:
where x = cos(R/a) and R is the disk radius. For the function u we assume that hℓ = = −3.82 for 1536 ≤ ℓ ≤ 40,000. u and are 1D functions, since they are obtained from the 1D Green's function and from an axially symmetric load, and are thus plotted as functions of r, the distance from the center of the load. Figure 2 shows the results for four disks of area values from 6 up to 86 km2.
 For distances from the load center larger than 3R, Figure 2 shows that the approximation (solid lines), and the accurate u (dotted lines) overlap. This indicates that use of the approximated relation for uplift rates at points far from every glacier is correct, at least for distances larger than 3R. The approximated relation (r) blows up for r → 0 as it can be inferred from equation (2). When the distance of the observation point from a glacier is smaller than 3R the contribution of that specific glacier is thus treated with the highly accurate decomposition u(r). Small size loads can thus be accurately treated by means of the above computationally efficient technique, optimizing the traditional normal mode approach. To show the results over the whole Alpine chain of Figure 3, a regular 5 km spaced grid resolution has been used. Nonetheless, uplift rate can be evaluated correctly using a restricted set of points only, as benchmarks (leveling or GPS) in inverse problem schemes. In order to better comprehend the impact of the various contributions to vertical uplift rates, we have considered separately the purely elastic response to present day glacier shrinkage, and the viscoelastic one due to post-LIA.
Figure 3, in the main color scale, shows the modeled elastic uplift rates for present day mass balance of Table 1 and for the Earth model of Table 2, while the inset, red color scale, shows the measured uplift rates (referring to the same years of our input data), from high precision leveling lines, limited to Switzerland, after Schlatter et al. (report, 1999); contour (white) lines from our simulation are superimposed in the inset for comparison. Uplift rates, in the range 0.1–0.2 mm/yr, characterize the response of the whole Alpine belt to present day glacier reduction, in agreement with the findings of Figure 1, encircling patches of high uplift rates, localized over the major glacial complexes. The largest uplift rate spot, of 0.9 mm/yr, indicated by the arrow, occurs in the French Alps, in proximity of Mount Blanc Group, where the largest mass loss is located. Uplift values of about 0.4 mm/yr, a factor two lower than the maxima in the west, are located in Austria. In the inset the (white contours) modeled and measured uplift rate patterns closely resemble in shape, indicating that present day glacier shrinkage contributes a substantial fraction of observed uplift rate. The 0.1 mm/yr white contour in the north almost overprints the 0.4 mm/yr measured one, indicating a 20% contribution from present day glacier shrinkage in the periphery of the uplifting region. The largest patches of modeled uplift rate overlap the regions of largest observed ones, although the highest modeled uplift rates of about 0.4–0.5 mm/yr, dark blue patches in the main panel of Figure 3, are about a factor two lower than measured ones. Modeling fails to reproduce the northernmost part of the large measured uplift patch, right in the inset. These findings suggest on one side that glacier shrinkage sensibly contributes to the Alpine chain uplift, on the other side that there are other phenomena, such as active tectonics, drainage and erosion [Schlunegger and Mathis, 2001; Schlunegger et al., 2001], affecting both the values of the largest uplift rates and their geographical distribution. On the Alps, the uplift rate caused by isostatic rebound after LGM deglaciation, in the estimate of Stocchi et al. , depends on the ice model ranging from −0.25 to +0.25 mm/yr [Stocchi et al., 2005, Figure 8(c)] and its pattern is broad over the whole Alpine region. Thus the LGM unloading signal could not sensitively affect our uplift pattern, but modify the value of ±0.25 mm/yr almost uniformly. Yet Stocchi et al.  used a 120 km thick lithosphere and their estimates are not properly comparable with ours.
Figure 4 portrays the effects of viscous relaxation in the lower crust and in the asthenosphere on uplift rates, assuming a constant volume loss since 1850 A.D. for a total of 155 km3, equally distributed over the representative glacier distribution of 1973, and rheological parameters in Table 2. According to Haeberli and Beniston  the estimated total glacier volume in European Alps was about 130 km3 for mid-1970s and since the end of the LIA, glacierization has lost about half of its original volume, implying 130 km3 since 1850, plus 25 km3 since 1973, according to Paul et al. . The viscous uplift simulated represents an upper bound, since pre-LIA (built-up of the load) signal, not considered in the present study, would act to partially counteract our post-LIA (unloading) effects. The viscous uplift pattern is smoother than the elastic one. In the western Alps, where the elastic part is most important, the largest viscous uplift rates are concentrated as well, reaching 0.32 mm/yr. Summing up the largest elastic contribution and the geographically corresponding viscous one, an uplift of 0.7–0.8 mm/yr is obtained in the western Alps, a substantial fraction (half) of the largest observed one of 1.5 mm/yr. The estimate of the viscous contribution, on the other hand, can be affected by uncertainties in the rheological parameters, and thus must be taken with caution. Reducing the viscosities of layers 2, 3, 4 and 5 by a factor of 2 with respect to Table 2, causes an increase of uplift rates by a factor 1.5, without modifying the pattern. The dimension of the patches where uplift is concentrated, tens of kilometers, is comparable with the thickness of the crust (40 km), indicating that contributions to uplift originates from stress relaxation in the soft lower crust of 2.15 × 1019 Pa s (Table 2).
 According to our model, rapid glacier shrinkage gives a substantial contribution to Alpine chain uplift. Our specialized normal mode technique provides a high spatial resolution scheme, a necessary tool in the forward problem. This technique will be the kernel for inverse problems in future studies. Our results are essential for a correct interpretation of uplift data in the Alpine chain and for quantifying the contributions from different drivers of uplift, such as present day glacier instability, active tectonics, drainage and LGM unloading effects.
 We are indebted to Karim Aoudia for continual discussion and help. This work is supported by the project ALPS-GPSQuakenet, funded by the EU Programme Interreg IIIB “Alpine Space”. We are indebted to Erik Ivins and to an anonymous reviewer for their important suggestions. IREALP (Regione Lombadia) is thanked for financial support.