Text S7: Computation of subglacial water storage (S) and subglacial water storage change (ΔS) proxies
GPS-derived proxies : GPS-measured vertical ice motion is a combination of three components: the vertical component of mean bed-parallel motion, vertical strain of the ice column, and vertical motion of the ice relative to the bed (due to some combination of cavity formation and till dilation, depending on basal conditions). Ideally, these components may be separated by leveraging local knowledge of ice conditions and, critically, several proximal GPS stations (e.g.,Anderson et al., 2004; Andrews et al., 2018; Harper et al., 2007; Hoffman et al., 2011; Howat et al., 2008; Mair et al., 2002; Sugiyama & Gudmundsson, 200 4). In the absence of additional GPS stations, the vertical strain rate cannot be estimated, but during the peak of the melt season changes in vertical strain rates are small and similar to winter background strain rates (e.g., Andrews et al., 2018 ), which are accounted for with a detrending to remove the impact of bed parallel motion (e.g., Bartholomew et al., 2012; Cowton et al., 2016 ).
As such, we detrend the 6-h smoothed z data using the long-term linear trend (Figure 2c ). This limited correction introduces unquantifiable uncertainties, a particular issue with deriving uplift and basal uplift change from GPS observations. In order to calculate the rate of basal uplift, we calculate the derivative of the detrended elevation data by applying a 6-h differencing of the 15-minute dataset, as done to calculate the horizontal velocity. The detrended elevation and basal uplift rate are considered proxies for subglacial storage (S ) and subglacial water storage change (ΔS ), respectively (Figure 2c, Figure 5a ). Our GPS-derived proxy for ΔS , albeit noisy, presents peaks that sometimes align with short-term accelerations in ice speed (Figure 5a ), unlike our GPS-derived peaks in S (Figure 2c ).
Discharge-difference proxies: While calculating basal uplift and basal uplift rates from GPS requires either multiple GPS stations or assumptions about vertical strain rate and ice flow orientation, our observations also present opportunity utilize an input-output approach to assess subglacial storage (S) and subglacial water storage change (ΔS) more directly. Input-output methods seek to measure or estimate the discharge of surface meltwater entering a glacier or ice sheet catchment and the discharge of proglacial water release. They have been used on smaller alpine glaciers to capture the role of subglacial water storage and change in water storage in driving short-term ice sheet motion (e.g.Bartholomaus et al., 2008, 2011 ), to compute water balance of a small surface catchment in the Sermeq Avannarleq ablation zone of Greenland (McGrath et al., 2011), and to examine long-term storage in the Russell Glacier using surface mass balance modeling and proglacial discharge measurements (van As et al., 2017 ).
The high-quality supraglacial discharge dataset presented here (Figure S5, Table S1 ) offers a rare in situ “input” suitable for comparison with proglacial output.
However, due to the disparities in the magnitudes of supraglacial and proglacial discharge, we must modify our approach by using the normalized difference between measured supraglacial moulin input and proglacial discharge (Figure 5b, 5c ). To calculate a qualitative proxy for subglacial storage (S ), we calculate the 6h cumulative input and discharge, then normalize both measures of cumulative discharge (e.g. instantaneous input and output) between 0 and 1, then calculate the difference (supraglacial minus proglacial) between these two measurements (Figure 5b ). Inspection of this Sproxy versus ice speed suggests that local subglacial storage and horizontal ice speed are offset, with ice speed peaking several hours before peak storage; however, once the derivative (ΔS ) of the proxy is calculated, the correlation with ice speed improves (Figure 5c; Figure S7). The derivative of the storage calculation is input – output on a 6h interval, which is the same as the 6h position derivative used to calculate smoothed ice speed. The occasional temporal offset between our ΔS proxy and ice speed, as well as a secondary daily peak in both datasets, are discussed next in SI Text S8 . Note that these discharge-difference calculations represent a fleeting measure of net subglacial water storage (i.e. input minus output), not the time required for subglacial water transport. Therefore, any subglacial routing delays (which are known to range from less than 1 to multiple days in this region, Chandler et al., 2013; van As et al., 2017 ) need not be considered in meltwater S and ΔS proxies.