## 3 Data and Methods
### Calving Rate  
The calving rate (\(CR\)) depends on the glacier dynamic, meteorological conditions, and the features of the sea or lake where the glacier ends. First, one needs to know whether the PMG tongue is floating or resting upon ground. Critical buoyant thickness is given by [32]. \[H_{M}<\rho_{w}/\rho_{i}H_{w}\approx 1.1\ H_{w} \tag{2}\]  
where \(H_{M}\) is ice thickness at the front margin, \(H_{w}\) is water depth, \(\rho_{w}\) and \(\rho_{i}\) are the density of water and ice, respectively. Because \(H_{M}\) is 50-80 m higher than \(H_{w}\) (about 150 m, [27]), PMG tongue should rest on ground everywhere. To quantify the calving rate, we followed the approach reported in [27], based on the velocity at the glacier front. From surface velocity one can derive vertically averaged glacier velocity as:  
\[V=0.8\times V_{S} \tag{3}\]  
where \(V\) is the mean velocity in m day\({}^{-1}\), and \(V\)s is the surface velocity representative for a certain point or cross-section area of the glacier. This correction factor is suggested by [PERSON] and [PERSON] [32], as a widely used approximation in glaciology, and we used it here accordingly. The glacier flow rate is computed multiplying velocity \(V\) by the corresponding cross-section area (area-velocity method). If a cross-section is split into several sub-sections, and each subsection displays a velocity measurement or estimate, the sum of flow rates of each sub-section gives the total flow rate. Flow rates are computed at track \(B\) and at the front of the glacier. This allows two different approaches for the calving rate estimation. The first method, \(M1\) is based upon computation of glacier flow rate at the front. Given that all calving ice has to cross the front cross-section, and it has to collapse into the _Lago Argentino_, it is possible to calculate the net advance of the glacier's front as:  
\[CR_{M1}=Q_{F}-G_{F} \tag{4}\]  
where \(CR_{M1}\) is the volumetric total calving rate [m\({}^{3}\) day\({}^{-1}\)] using this method, \(Q_{F}\) [m\({}^{3}\) day\({}^{-1}\)] is the flow rate (speed) at the front of the glacier, and \(G_{F}\) [m\({}^{3}\) day\({}^{-1}\)] is the net advance of the glacier front position. Here, we embed into \(CR_{M1}\) horizontal ice ablation or melting rate at the front and net calving rate, consistently with current literature (e.g., [22]) given that separation of such components is complex. To approximate glacier's cross sections for flow rate assessment, we used the transects as reported by [PERSON] et al. [27] (see Figure 4 in [27]). SENTINEL2 images in natural color (bands 4, 3, and 2) with 10 m of spatial resolution were used to map the glacier front variations. The second method \(M2\) is based on the glacier flow rate computed at the \(B\) profile. The rationale is similar to approach \(M1\), but it is necessary to introduce some assumptions. The ice in Profile \(B\) needs on average about 4000 days to reach the front. Therefore, such analysis is only possible thanks to large stability of the PMG, as confirmed in many studies and by the analysis of the ice speed variations during seasons and years made in this study. Second, from Profile \(B\) to front, it is likely that mass loss due to ablation is not negligible. Ablation measurements are here available from Profiles \(A\) and \(B\), quite uniform,but not from Stake \(D\) (at the front) and from Profile \(L\) (above Profile \(B\)) [27]. Moreover, at Stake \(D\), ablation is likely to be very high due to the influence of the lake and the lower altitude there. In order to compute the ablation on the whole area, two subsections are used [29]: _central zone_ and _marginal and crevassed zone_. Moreover, altitude and climate data are considered when computing the ablation data as function of the altitude, to not omit climatic and glaciological factors. Nonetheless, neither with altitude nor temperature is it possible to explain the increase of ablation from Profiles \(A\), \(B\), and \(L\), to the Stake \(D\).