Quantification of seasonal and diurnal dynamics of subglacial channels using seismic observations on an Alpine glacier

Water flowing below glaciers exerts a major control on glacier basal sliding. However, our knowledge of the physics of subglacial hydrology and its link with sliding is limited because of lacking observations. Here we use a 2-year-long dataset made of on-ice-measured seismic and in situ-measured glacier basal sliding speed on Glacier d’Argentière (French Alps) to investigate the physics of subglacial channels and its potential link with glacier basal sliding. Using dedicated theory and concomitant measurements of water discharge, we quantify temporal changes in channels’ hydraulic radius and hydraulic pressure gradient. At seasonal timescales we find that hydraulic radius and hydraulic pressure gradient respectively exhibit a 2and 6-fold increase from spring to summer, followed by comparable decrease towards autumn. At low discharge during the early and late melt season channels respond to changes in discharge mainly through changes in hydraulic radius, a regime that is consistent with predictions of channels’ behaviour at equilibrium. In contrast, at high discharge and high shortterm water-supply variability (summertime), channels undergo strong changes in hydraulic pressure gradient, a behaviour that is consistent with channels behaving out of equilibrium. This out-of-equilibrium regime is further supported by observations at the diurnal scale, which prove that channels pressurize in the morning and depressurize in the afternoon. During summer we also observe high and sustained basal sliding speed, which supports that the widespread inefficient drainage system (cavities) is likely pressurized concomitantly with the channel system. We propose that pressurized channels help sustain high pressure in cavities (and therefore high glacier sliding speed) through an efficient hydraulic connection between the two systems. The present findings provide an essential basis for testing the physics represented in subglacial hydrology and glacier sliding models.


Introduction
Subglacial water flow exerts a major control on glacier and ice sheet dynamics and their response to variations in water supply (e.g. Iken and Truffe, 1997;Zwally et al., 2002;Sundal et al., 2011;Bartholomaus et al., 2011;Chandler et al., 2013;Hewitt, 20 2013; Brondex et al., 2017;Joughin et al., 2018). Water flowing at the base of glaciers modulates glacier basal sliding by lubricating the ice-bed interface. The higher the water pressure the weaker the basal friction, resulting in faster glacier sliding (Iken and Bindschadler, 1986;Schoof, 2005;Gagliardini et al., 2007). Water pressure does not simply depend on the total ter supply changes (Parizek and Alley, 2004;Palmer et al., 2011;Sole et al., 2013;Doyle et al., 2014;Vincent and Moreau, 2016). This behavior is mostly related to the pressurization of the cavity-system, causing average basal water pressure rise and 60 subsequent basal sliding speed increase (e.g. Nienow et al., 2005;Schoof, 2010;Rada and Schoof, 2018). During periods of well-developed channelized system (e.g. in summer), this behavior has also been observed because of a channelized system drainage capacity being overwhelmed by the water input changes (Bartholomaus et al., 2008;Andrews et al., 2014) causing pressurized channel flow. These studies have been capable to underline the overall differences between cavity and channel control on subglacial water pressure over different timescales. However, the lack of dedicated channels observations independent 65 of those on cavities and concomitant with glacier sliding speed measurements renders difficult a more quantitative characterization of the physics of subglacial hydrology and its link with sliding.
Here we use on-ice seismology to explore the evolution of subglacial channels over two complete melt seasons. Over the last decade an increasing number of studies have shown the high potential of analyzing high-frequency (>1 Hz) ambient seismic 70 noise to investigate turbulent water flow and sediment transport in terrestrial rivers and streams (e.g. Burtin et al., 2008Burtin et al., , 2011Tsai et al., 2012;Schmandt et al., 2013;Gimbert et al., 2014). The recent work of Gimbert et al. (2016) based on observations of Bartholomaus et al. (2015) suggests that passive seismology may help filling the observational gap on the physics of subglacial channels. Gimbert et al. (2016) adapted to subglacial channels a physical framework that describes how turbulent water flow generates seismic waves and that was initially developed for rivers by Gimbert et al. (2014). Contrary to rivers, subglacial 75 channels have the capability to be full and thus to undergo pressurized situations. By applying this modified framework to the Mendenhall glacier (Alaska) over a two-month long summer period, the authors demonstrate that one can use concomitant seismic noise and water discharge measurements to continuously and separately quantify relative changes in channel hydraulic pressure gradient and channel hydraulic radius. They inferred that channels mainly evolve through changes in hydraulic radius over long time scales (multi-weekly), whereas changes in hydraulic pressure gradient are often short-lived (sub-daily to 80 weekly). The use of such an approach to investigate channel physics on relevant glaciological timescales (e.g. diurnal and seasonal) yet remains to be conducted, and the resulting channels properties remain to be compared to other independent observations, such as basal sliding speed. This is the objective of our study.
We conduct a unique and almost uninterrupted two-years passive seismic survey on Glacier d'Argentière (French Alps), 85 together with continuous measurements of subglacial water discharge, glacier basal sliding speed and local subglacial water pressure. First, we characterize the subglacial channel-flow-induced seismic power signature and use the model of Gimbert et al. (2016) to derive timeseries of hydraulic pressure gradient and hydraulic radius. We then compare these channel properties to the other independent measurements of glacier sliding speed and basal water pressure. We also compare our seismicallyderived observations with the theory for subglacial channels physics proposed by Röthlisberger (1972) to assess the implica-90 tions of these analysis for channels physics. Finally, we investigate the equilibrium state of subglacial channels to discuss the channel-cavity interactions and their potential link with basal sliding throughout the melt season. Doing so will also allow us to discuss the applicability of such an approach to improve our general knowledge on subglacial hydrology mechanisms of mountain glaciers and ice sheets.

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Here we provide a brief background on the theoretical framework of Gimbert et al. (2016), which relates seismic noise and water discharge to subglacial channel-flow properties, and that of Röthlisberger (1972), which predicts subglacial channel hydraulic pressure gradient and hydraulic radius scaling as a function of water discharge under certain assumptions. Refer to table C1 in Appendix C for a summary of all variables, physical quantities, and mathematical functions defined in the following sections. Turbulent water flow in a river or a subglacial channel generates frictional forces F acting on the near boundaries (e.g. river bed or conduit wall), which in turn cause seismic waves with given amplitude and spectral signature (Gimbert et al., 2014). By propagating through a medium (e.g. rock, gravel or ice), seismic waves cause ground motion at any location x away from the source location x 0 (Fig. 1). The relationship between the force timeseries F (t, x 0 ) applied at x 0 in a channel and the ground 105 velocity timeseries U (t, x) measured at x can be described from Aki and Richards (2002) as where G(t) is the displacement Green's function that converts the force applied at x 0 into ground displacement at x and the notation ⊗ stands for the convolution operator. The seismic power P of such signal is defined over a time period T as (2) 110 where U (f ) = F(U (t)) is the Fourier transform of the ground velocity timeseries and f is the frequency. We note P w the seismic power induced by turbulent water flow. Based on a description of the force F (f ) as a function of flow parameters, Gimbert et al. (2014) demonstrated that P w scales as where u * is river wall shear velocity, W is river width and ζ is a function that accounts for turbulence intensity changes with 115 changes in the apparent roughness that depends on H the flow depth and k s the wall roughness size (Fig. 1).  (3)). Turbulent flow generates frictional forces F causing seismic waves and resulting in a ground velocity U that is recorded at a distant seismic station (see Eq. (1)).
To relate P w to subglacial channels properties, Gimbert et al. (2016) expressed the shear velocity as u * = √ gRS where g is gravitational acceleration, R the hydraulic radius and S the hydraulic pressure gradient. The hydraulic radius R is defined as the ratio of the cross-sectional area of the channel flow to its wet perimeter (Fig. 1). This parameter scales with flow depth for open 120 channel-flow. The hydraulic pressure gradient S is a function of both the water pressure rate of change in the flow direction and the bed slope. For free surface flow S equals channel slope. In a case of constant channel slope and channel geometry, increasing S means closed and pressurizing channel-flow. Gimbert et al. (2016) then expressed water discharge Q as a function of water flow velocity V w using the  relation V w = R 2/3 S 1/2 n with n is the Manning's coefficient (Manning et al., 1890;Strickler, 1981). To study P w for a subglacial channel flow configuration, Gimbert et al. (2016) considered that the source-to-station distance is constant, such that changes in P w are not caused by changes in source (channel) position. Gimbert et al. (2016) then assumed a constant number N of channels and thus neglected the dependency of P w on N. Here we include the dependency of P w on N by considering that all channels have equal hydraulic radius and hydraulic pressure gradient (i.e. are of similar size and position compared to the 130 seismic station) such that where β is a function of conduit shape and fullness that may be neglected (see supporting materials of Gimbert et al. (2016) for details). Combining Eqs.(4) and (5) and neglecting changes in β leads to the two following formulations for P w , 135 P w ∝ R −82/9 Q 14/3 N −11/3 (6) From Eqs. (6) and (7) two end-member cases can be evaluated. If changes in discharge occur at constant channel geometry (i.e. constant R and N) from Eq.(6) we have

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In contrast, if changes in discharge occur at constant hydraulic pressure gradient and channel number (regardless of whether the conduit is full or not) from Eq. (7) we have Beyond these end-member scenarii, one can use measurements of P w and Q to invert for relative changes in R and S using Eqs.(6) and (7) where the subset ref stands for a reference state, which has to be defined over the same time period for both Q and P w , but not necessarily for R and S. Details on the derivation from Eqs.(6) and (7) to Eqs. (10) and (11) can be found in Gimbert et al. (2016). In the following we consider N as constant to invert for R and S, and later we support that our inversions are not 150 significantly biased by potential changes in N (Sect. 6.1).

R-channels theory
To date, state-of-the art subglacial drainage models use the theories of Röthlisberger (1972) For a steady-state channel not at equilibrium with Q and that responds solely through changes in pressure gradient S (i.e. R is constant) Röthlisberger (1972)' equations show that: Further details on the derivation of these equations from Röthlisberger (1972) can be found in Supplementary Sect. S2. Later we compare our inversions of changes in R and S (using seismic observations) with changes in R and S as predicted by the 165 theory of Röthlisberger (1972) for steady-state channels at equilibrium or not at equilibrium with water discharge.
3 Field setup

Site and glaciological context
Glacier d'Argentière is a temperate glacier located in the Mont Blanc mountain range (French Alps, see Fig. 2). The glacier is c. 10 km long and covers an area of c. 12.8 km 2 . It extends from an altitude of c. 1700 m above sea level (asl) up to c. 3600 170 m asl in the accumulation zone. Its cumulative mass balance has been continuously decreasing from -6 m water equivalent (w.e) in 1975 to -34 m w.e presently compared to in the beginning of the twentieth century (Vincent et al., 2009). This site is ideal to study subglacial channels properties since it presents a typical U-shaped narrow valley (Hantz and Lliboutry, 1983) and hard bed conditions (Vivian and Bocquet, 1973), two conditions that favor a well-developed R-channel subglacial network (Röthlisberger, 1972).

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In the present study we analyze the data recorded from spring 2017 to autumn 2018 with seismometers located between 2350 and 2400 m asl (Fig. 2). This location corresponds to the cross-section No. 4 monitored by the French glacier-monitoring program GLACIOCLIM (https://glacioclim.osug.fr/). There the glacier is up to c. 280 m thick (Hantz and Lliboutry, 1983, updated from a radar campaign conducted in 2018). Subglacial water discharge is monitored 600 m downstream of the seis-180 mometers at 2173 m asl near the glacier ice fall in subglacial excavated tunnels maintained by the hydroelectric power company Emosson S.A. Subglacial water is almost entirely evacuated through one major snout, as supported by direct observations of very limited water flowing elsewhere. Thus discharge measured at this location is well representative of discharge subglacially routed under the seismometers location. Discharge measurements are conducted from mid-spring to early autumn with an ac- geometry. The minimum measurable value for water discharge is limited by the measurement accuracy and the maximum one is of 10 m 3 s −1 due to the capacity of the collector. Because sediments accumulate in the collector, flushes are recorded when the latter is emptied, causing glitches in the discharge record. We remove these glitches removing Q values that present d(Q) dt higher than 0.2 m 3 per 15 min. Within the same tunnel network, a subglacial observatory is used to measure basal sliding speed out of a bicycle wheel placed in contact with the basal ice (Vivian and Bocquet, 1973). Since August 2017 basal sliding 190 speed is measured at a time resolution of 5 s over a 0.07 mm' space segmentation. In the close vicinity a pressure sensor, of gauged type, is used to measure subglacial water pressure with 10 min time resolution and an accuracy of 400 Pa. The sensor is installed in a borehole drilled from the excavated tunnels up to the glacier bottom (see Vivian and Zumstein (1973) for details). Air temperature and precipitation measurements are obtained at a 0.5 h time step through an automatic weather station maintained by the French glacier-monitoring program GLACIOCLIM and located on the moraine next to the glacier 195 at 2400 m asl. Precipitation is measured with an OTT Pluvio weighing rain gauge with a 400 cm 2 collecting area. When air temperature is below zero, only precipitation occurrences are accurate, but not absolute values because of snow clogging.

Seismic instrumentation
We use five seismic stations installed in the lower part of the glacier (Fig. 2). The instruments belong to two seismic networks,  The raw seismic record at each station is first corrected from the sensor and digitizer responses. Then, the frequency-dependent seismic noise power P is computed using the vertical component of ground motion (see Eq. (2)). P is calculated with the Welch's method over time windows of duration dt with 50 % overlap (Welch, 1967). The longer dt, the more likely highly energetic impulsive events occur and overwhelm the background noise within that time window (Bartholomaus et al., 2015).
To maximize sensitivity to the continuous, low amplitude, subglacial channel-flow-induced seismic noise and minimize that 220 of short-lived but high energy impulsive events, we use a short time window of dt = 2 s to calculate P, and average it over time windows of 15 min in the decimal logarithmic space. We express P in decibel (dB, decimal logarithmic), which allows properly evaluating its variations over several orders of magnitude.
We reconstruct a two-year long timeseries by merging records from the five available stations into one unique record at a 225 'virtual' station. To minimize site and instrumental effects on seismic power we shift the average power at each station to a reference one taken at ARG.B01. The seismic signal at our 'virtual' station is composed of the GDA seismic signals between May 2017 end December 2017, and of the ARG seismic signals between December 2017 and December 2018 (see Fig. S1).

Evaluating bias due to anthropogenic noise
Later in section 5 we show that when water discharge Q is low (in the early and late melt season) seismic power from anthro-230 pogenic noise (P A ) is comparable to the subglacial channel-flow-induced seismic power (P w ). Here we evaluate how much P A adding to P w can bias the evaluation of scaling predictions of Gimbert et al. (2016). We calculate a synthetic seismic power P as P = P A + P w and a synthetic P w from a synthetic Q as P w = Q n with n being equal to 5 4 or 14 3 as expected from theory (see Eqs. (8) and (9)). We quantify the relative contributions of P w and P A to P through the parameter Sr, which we define as When Sr tends to 1, subglacial channel-flow-induced seismic power dominates the synthetic seismic power 235 and when Sr tends to 0 anthropogenic noise power does.
In Fig. 3(a) we show the temporal evolution of synthetic P with a constant value for P A and with a P w that responds to a synthetic evolving water supply Q. The value of P is normalized by P A , resulting in P = 0 dB in winter. For P w ∝ Q 14/3 ( Fig. 3(a), red and orange lines), P w dominates the contribution to P within c. 10 days from the onset of water supply. For 240 P w ∝ Q 5/4 ( Fig. 3(a), black and green lines) P contains both P w and P A contributions during a period that is three times longer than for P w ∝ Q 14/3 . The evolution of Sr with respect to P-P A ( Fig. 3(b)) is the same for both the constant hydraulic pressure gradient (red line) and constant hydraulic radius (grey line) scenarii. For P-P A > 2 dB, Sr is higher than 0.8, meaning that subglacial channel-flow-induced seismic power contributes by more than 80% to the synthetic seismic power. Later in Sect. 5.2 we measure P A during winter and use the condition P-P A > 2 dB to define the periods where evaluate P w directly 245 from the measurement of P and investigate the subglacial hydraulic properties.

Definition of metrics to evaluate sub-diurnal dynamics
Since the P w versus Q relationship is not unique and may vary with time (see Sect. 2), we expect that the diurnal timeseries of P w versus Q may exhibit different patterns throughout the melt season; and that these patterns reveal changes in the subglacial hydraulic properties. To systematically quantify the diurnal variability of P w , Q, R and S throughout the melt season we define 250 three metrics that we calculate on an hydrological daily basis (defined as the period between two minimum Q within a 24 h time window). To focus on the diurnal variability only, we bandpass filter our timeseries within a [6-36] h range (see Appendix Fig. A1 for details). Our first metric quantifies the diurnal variability of a given variable X during a given day and corresponds to the coefficient of variation C v defined as: with (X day ) max and (X day ) min the maximum and minimum value of X day , respectively, and X day its average. Our second metric φ quantifies daily hysteresis between P w and Q by evaluating the difference between P w when Q is rising, e.g. in the morning, and P w when Q is falling, e.g. in the afternoon. Following the approach of Roth et al. (2016) we define φ as: The larger φ, the more seismic energy is recorded during the rising discharge period with respect to the falling one. Hysteresis 260 can occur either because of an asymmetry between (P w,day ) rising and (P w,day ) falling or because of a time lag between P w and Q.
To avoid ambiguity between these two hysteresis sources our third metric corresponds to the daily time lag δt between the time t((P w,day ) max ) when P w is maximum and the time t((Q day ) max ) when Q is maximum and is defined as: We set the condition that for δt to be calculated, t((P w,day ) max ) has to correspond to both the time when P w is maximum and 265 has a null-derivative within a [-8, 8] h' time window around t((Q day ) max ). We note that a time delay of about 0.04 h is expected due to water flowing at c. 1 m.s −1 over the c. 600 m separating our seismic stations to where Q is measured (see Fig. S2 for details). This means that any values of δt greater than ± 0.04 h are not attributable only to water transfer time lags.  observations made by Vincent and Moreau (2016) over the past decade). Basal water pressure measurements (Fig. 5(c)) show that at the seasonal timescale the basal water pressure tends to be higher in winter than in summer by c. 2.5 1e +4 Pa. In summer 2017 the short-term (diurnal) variability in the basal water pressure is more pronounced than in winter, as also observed for the water discharge ( Fig. 5(b) and Fig. A1). During heavy rainfall ( Fig. 5(a)) and consequent discharge ( Fig. 5(b)), basal 295 water pressure variations are in phase with sliding speed (Fig. 5(c); e.g in August 1 st , August 7 th , August 18 th , August 30 th , September 13 th or October 2 nd of 2017). This evolution of the measured basal water pressure rather depicts a local behavior whereas changes in the basal sliding speed (Fig. 5(c)) rather represent average changes in the average basal water pressure conditions over our study area and therefore better represent the global cavity-system pressure conditions. A similar frequency-signature of the subglacial channel-flow-induced seismic noise as been observed by Bartholomaus et al. (2015), Preiswerk and Walter (2018) and Lindner et al. (2019). This frequency range is also comparable to those observed for 310 water flow in rivers (Schmandt et al., 2013;Gimbert et al., 2014). As Q increases from less than 0.1 m 3 .s −1 in early May to about 10 m 3 .s −1 end of July, P w increases by up to 30 dB (i.e. 3 orders of magnitude). Differences in relative variations of P w across stations are lower than 0.5 dB including during periods of high discharge (Fig. S2). This supports the accuracy and validity of our 'virtual' station reconstruction to study the subglacial channel-flow-induced seismic power (Sect. 4). Variations in P w follow those of Q during the melt season and over seasonal to weekly times scales ( Fig. 5(b)). Both the high sub-monthly 315 variability in Q and air temperature observed in 2017 and the rapid changes in Q occurring in fall 2017 and 2018 are also observed in the temporal evolution of P w . In winter we observe high seismic power bursts from December to mid-January occurring when Q is null but concomitantly with the beginning of heavy snowfall events. These bursts are not associated with subglacial channel-flow-induced seismic noise but likely correspond to repeating stick-slip events triggered by snow loading similar to those observed previously by Allstadt and Malone (2014). When Q is lower than 2 m 3 .s −1 during winter, early spring 320 and fall, we observe regular weekly and daily variations in P [3][4][5][6][7] Hz that superimpose to the background variations ( Fig. 5(b)).

Overview of observations
This regular pattern corresponds to anthropogenic noise, as previously observed by Preiswerk and Walter (2018)  During these periods we subtract the mean winter diurnal pattern of P A (defined between January 29 th and April 4 th 2018) from P [3][4][5][6][7] Hz to obtain P w (Fig. S3). At the diurnal scale, because P A can slightly vary from day to day depending on the anthropic activity (e.g higher anthropic activity during working days than holidays), the periods of very early and very late melt season are still strongly influenced by day-to-day changes in P A . To study diurnal changes in P w without being biased by  Fig. S3). Later in Sect. 5.4 we filter P w with a 5-day lowpass filter (i.e. removing variability lower than 5 days) when inverting for the hydraulic properties. Doing so allows to study with confidence the early and late melt-season by reducing the influence of the diurnal variability in P A on P w while keeping sub-weekly variations in P w and Q (see Fig. S4 for details).  Gimbert et al. (2016)

Analysis of seasonal changes
Seasonal scale observations and predictions of the subglacial channel-flow-induced seismic power P w versus water discharge Q are shown in Fig. 6. We find that theoretical predictions from Gimbert et al. (2016) (red and black lines) are consistent with 340 our observations (colored dots), which exhibit a general trend between that predicted at constant hydraulic pressure gradient ( Fig. 6, see black lines calculated using Eq. (7)) and that predicted at constant hydraulic radius (Fig. 6, red lines calculated using Eq.(6)). As Q increases at the very onset of the melt season (in end of April), observed P w -values follow the trend predicted under constant hydraulic pressure gradient (Fig. 6 1 ). As Q increases more rapidly from mid-May to end of June ( Fig. 5(b)), P w follows a different trend of evolving hydraulic pressure gradient (Fig. 6 2 ). The general trend from July to September 345 is then dominated by changes in hydraulic radius (Fig. 6 3 ). As Q decreases during the melt season termination, observed P w values follow the trend of evolving hydraulic pressure gradient in a similar manner as during the early melt season (Fig. 6   4 ). At the end of the melt season 2018 (Late October to November) our observations also show a trend of changing hydraulic radius although this observation is not as clear in 2017 (Fig. 6 5 ). A clear counter-clockwise seasonal hysteresis of up to 10 dB power difference is observed in Fig. 6 between P w and Q. This shows that for a similar water discharge, higher subglacial  6) and (7)) Pw versus Q daily relationships. Note that y-axis bounds differs from panel to panel. Both variables are normalized by their daily minima. (i) Daily time lag δtQ,P w between Pw,day and Qday peaks (blue lines) and daily hysteresis φ between Pw,day and Qday (red lines). Shaded lines are data of year 2017, plain ones of year 2018. Dashed lines show δtQ,P w = 0 (blue) and φ = 0 (red). Timeseries are smoothed over 5 days. Green vertical bars show times of the four selected hydrological days with the corresponding panel number. Circled numbers refer to the two phases described in the main text. measurement threshold in Q is well observable for the two years but does not bias the observed scaling of changing hydraulic radius observed during summer.

Analysis of diurnal changes
Observations and predictions of the diurnal relationship between the subglacial channel-flow-induced seismic power P w and 355 water discharge Q throughout the melt season are shown in Fig. 7. We quantify the diurnal behaviors over the two melt seasons by calculating the hysteresis amplitude φ and time lag δt (see Sect. 4.3) and through comparing our observations with the theoretical predictions calculated for four selected days (panels (a) to (h) in Fig. 7). We selected these days based on three criteria: they represent typical variations of P w and Q over their respective periods (∼ ± 5 days around their date); they show that our observations capture diurnal variations from unique days without multi-days averaging; they give a pedagogical support for the 360 reader to interpret values of the hysteresis amplitude φ and time lag δt shown in Fig. 7i. We focus on these two indicators as they allow to evaluate respective changes of P w versus Q.
The seasonal evolution of the daily hysteresis amplitude φ presents two peaks in late-May / early-June and in late-August / early-September, which are consistently observed in both 2017 and 2018 (phases 1 in Fig. 7(i)). The seasonal evolution of 365 the diurnal time lag between δt of P w to Q is similar to that of φ, with peak values at δt > 2.5 h in late-May / early-June and in late-August / early-September (Fig. 7(i)). This supports that hysteresis is mainly caused by phase difference between P w and Q rather than by asymmetrical changes P w when Q rises compared to when Q falls Q (Sect. 4.3). The variability of δt over the season is much larger than the predicted 0.04 h instrumental time lag (see Sect. 4.3), such that its evolution represents real changes in the relationship between P w and Q.

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In the early and late melt season (phases 1 in Fig. 7(i)), P w,day peaks, in average, more than 3 h before Q day (e.g. Fig. 7(e)).
These long time delay δt are concomitant to a pronounced asymmetrical shape in P w,day with a steeper rising than falling limb (e.g. Fig. 7(e)). This results to large clockwise hysteresis in P w,day versus Q day as well pictured by the high hysteresis values during these periods (φ > 1, phases 1 in Fig. 7(i)). For example, on June 10 th our observations follow the trend of 375 evolving hydraulic pressure gradient in the morning and the one of changing hydraulic radius in the afternoon and at night.
On September 8 th our observations follow the trend of changing hydraulic radius in the early morning and the one of evolving hydraulic pressure gradient in the afternoon. On the contrary to these periods, in summer (phase 2 in Fig. 7(i)), both φ and δt are low with φ 0 and 2 h > δt > -2 h. At this time, δt has a more pronounced seasonal and year-to-year variability than φ (Fig. 7(i)) with values oscillating within [2 ; -2] h and minimum values reaching δt < -4 h. In July and August (e.g. panels 380 (b) and (c) in Fig. 7), P w peaks nearly at the same time as Q with δt < 0.5 h and with an almost symmetrical diurnal evolution ( Fig. 7(i)). For both summer days (July 6 th and September 1 st ), our observations mainly follow the trend of changing hydraulic radius throughout the whole day, with a non-null hysteresis that shows that hydraulic pressure gradient may also change. This two-phases seasonal evolution shows that the early and late melt season diurnal changes in Q cause a pronounced diurnal variability in the hydraulic pressure gradient and limited diurnal changes in the hydraulic radius, whereas over the summer 385 channels show a more marked response to diurnal changes in Q through changes in hydraulic radius.

Inversions of changes in hydraulic radius and hydraulic pressure gradient
We invert for the relative changes of hydraulic radius R Rref and hydraulic pressure gradient S Sref using Eqs. (10) and (11) and our observations of timeseries of Q and P w once filtered with a 5-day lowpass filter (see Fig. S4 and Sect. 5.2 for details). In the following for the sake of readability we use the notation R, S and V to refer to R Rref , S Sref and the relative basal sliding speed V Vref .

Analysis of seasonal changes
The temporal evolution of R, S and V are presented in Fig. 8. We recall here that the changes in V can be considered as a good proxy for changes in water pressure in the subglacial cavity network (see Sect. 5.1 for details). We find that all three 395 variables show a well-marked seasonal evolution, with low values during the early and late melt season and high values in summer. However, differences between R, S and V exist over the melt season. For both years, R starts increasing from the onset of the early melt season, until reaching a maximum within two months in late-June to early-July. R is then two times larger in average than in the early melt season. In contrast, during the first weeks of the melt season 2018, S rapidly decreases (Fig. 8 1 ), concomitantly with an abrupt increase in V by a factor of 1.5 compared to winter. This shows that as the average water 400 pressure rises in cavities and enhance sliding, channels on the contrary undergo depressurization. During the melt season 2017 we do not observe such behavior possibly because of a timeserie of P w that starts about three weeks later than in 2018. The increase in S then occurs with a delay of about one month in 2018 and of about one week in 2017 compared to that in R, and S reaches a maximum in August (Fig. 8 2 ). S is at that time on average five to six times larger than in the beginning of the melt season. As S increases, V and R have already past their summer maximum. Contrary to the conclusions obtained on the 405 Mendenhall Glacier (Alaska) where S presents no significant trend over the two-month long investigated period (Gimbert et al., 2016), seasonal changes in water discharge at Glacier d'Argentière are inferred to cause changes in both R and S. From early to mid-September, R and S decrease concomitantly and reach their minimum in late October. The summer to winter transition is most pronounced for S, which decreases by about a factor of 4 within less than a month (September to October) while R decreases more gently.  Figure 9 describes how channel and cavity properties behave at the diurnal scale throughout the melt season. We quantify the diurnal behavior throughout the two melt seasons with the time lag δt between R and Q daily maxima, noted δt Q,R , and between S and Q daily maxima, noted δt Q,S . We also calculate the amplitude of the diurnal variations C v for R, S and V (see 415 Sect. 4.3 for definitions). In the same scopes as in Sect. 5.3.2 we illustrate in panels (a) to (d) in Fig. 9 the diurnal evolution of R and S for the same four selected days as in Fig. 7.

Analysis of diurnal changes
C v (R) and C v (S) both present seasonal variation, with maximum values being reached mid-summer. The amplitude of C v (S) is however up to three-times larger than that of C v (R) since C v (S) reaches up to 80 % in August while C v (R) only increases 420 up to 30 % for ( Fig. 9(f)). In contrast, the seasonal evolution of δt Q,R and δt Q,S drastically differs (Fig. 9(e)). On one hand, the temporal evolution of δt Q,R presents no marked changes throughout the season and generally remains within a range of ± 1 h (Fig. 9(e)) as highlighted by the four selected days (Figs. 9(a) to (c)). This shows that R and Q are consistently in phase on a diurnal basis throughout the melt season. On the other hand, the temporal evolution of δt Q,S presents average values of about 5 h with two peaks of δt Q,S > 8 h in June and August (Fig. 9(e) 1 ) and a period of low values ranging within [0;5] h in 425 mid-summer ( Fig. 9(e) 2 ). These changes in S are clearly observed in the diurnal snapshots (e.g. Figs. 9(a) to (d)) that show a marked increase in hydraulic pressure gradient in the morning before the rise in hydraulic radius. Such a difference in diurnal dynamics between R and S shows that channels exhibit high hydraulic pressure gradients in the early morning time while their hydraulic radius grows slowly to reach its maximum at the same time as the water discharge does. 430 We also compare in Fig. 9(f) the diurnal dynamics of channel properties to the diurnal dynamics of the average water pressure conditions in cavities by comparing C v (R) and C v (S) with C v (V). Over the melt season, C v (V) exhibits a pattern that is similar to C v (R) and C v (S), with higher values observed for the three variables in summer (> 10 %) than during the early and late melt season (< 10 %). This shows that short-term variability in channels properties (i.e. R and S) correlates well with the short-term variability in average water pressure condition in cavities. From late August to mid-September 2017, we observe that C v (S) 435 reaches up to 60 % over less than a week, followed c. a week later by a rapid rise in C v (V) (Fig. 9(f)). Röthlisberger (1972) Our seismically derived S and R values are shown in Fig. 10 as a function of relative changes in water discharge Q, along with scaling predictions calculated using the theory of Röthlisberger (1972) assuming channels at equilibrium (melt rate equals creep rate) with S∝ Q −2/11 and R ∝ Q 9/22 (Eqs. (14) and (12), green lines in Fig. 10) and channels out-of-equilibrium that respond 440 to changes in Q only through changes in S with S ∝ Q 2 and R is constant (Eq.(13), purple lines in Fig. 10). We find that R and S generally exhibit variations with Q that lie between those expected for channels at equilibrium and those expected for channels evolving at constant hydraulic radius. At low discharge ( Q Q ref < 4, Q < 1 m 3 .s −1 ) during the early and late melt season (Fig. 10 1 ) our derived changes in S and R with Q approach the theoretical prediction for channels behaving at equilibrium. At high discharge ( Q Q ref > 4, Q > 1 m 3 .s −1 ); mid-May to early October, Fig. 10 2 ) changes in S and R with changes in Q significantly 445 departs from predictions of channels at equilibrium and approaches the one of channels evolving out-of-equilibrium through changes in S solely. The transition between the two regimes herein observed is quite abrupt for S which switches from being a decreasing to being an increasing function of Q. For R, the transition is marked by a weaker dependency on Q as thi latter is high. During the period when Q/Q ref > 5, best datafit of R with Q gives R ∝ Q 0.27 ∝ Q 6/22 and for the periods when Q/Q ref < 4 it gives R ∝Q 0.36 ∝Q 8/22 . This latter scaling is similar to the predicted scaling of R ∝Q 9/22 calculated using the theory of Timeseries of R and S are calculated from 5-day lowpass filtered timeseries of Q and Pw, and are then 30-day lowpass filtered (same as in Fig. 8). Timeseries of Q is 30-day lowpass filtered. Reference values for all three variables are taken as the first day of the 2017 melt-season (May 10 th 2017). We compare our data to the predictions of Röthlisberger (1972) for subglacial channels evolving at equilibrium with Q (green lines, S ∝ Q −2/11 and R ∝ Q 9/22 ) and for subglacial channels evolving through hydraulic pressure gradient changes only (blue lines, S ∝ Q 2 and δR δQ = 0). Arrows show the direction of time. Blue shaded areas represent the period when Q is lower than 1 m 3 .s −1 . Line sections without the black edges show interpolated values of R and S using a cubic spline interpolation as in Fig. 8. 6 Discussion 6.1 Evaluating potential bias from changes in the number and position(s) of channel(s) As stated in Sect. 2, the subglacial channel-flow-induced seismic power P w depends on the number of subglacial channels N (Eqs.(10) and (11)) and on the source-to-station distance, which we both considered as constant in our analysis. Here we 455 discuss how much potential changes in N and in channel(s) positions may bias our inversions of S and R. On one hand, given the glacier configuration in our study area (250 m thick, 500 m wide Fig. 2(a)), channels-to-seismic station distance is similar regardless of whether channels are located at the glacier center or on its sides. Therefore, we do not expect changes in channel spatial positions to bias our inverted values of R and S. On the other hand, we estimate how much the observed changes in P w would require changes in N if they were to be explained only by an evolving number of channels rather than evolving S or 460 R. From Eq.(10) we have that S weakly depends on N compared to on P w and on water discharge Q. As a result, explaining the measured variations of P w while imposing S constant would require N to change by more than 4 orders of magnitude (5 41/6 ), which is unrealistic. From Eq.(11) we have that R weakly depends on N compared to on Q. As a result, explaining P w variations while imposing R as constant would require N to change by more than factor of 30 (4 −82/33 ), which is also likely unrealistic since at the onset of the melt season channels are expected to form an arterial network with few channels being kept 465 over summer (Schoof, 2010;Werder et al., 2013). Therefore, we do not expect potential changes neither in channel positions nor in N to cause significant bias in our inverted values of R and S. 6.2 Implications for inferring water discharge using seismic noise As opposed to Gimbert et al. (2016) who inferred little variations in hydraulic pressure gradient over its two-month long period 470 of survey on the Mendenhall Glacier, on Glacier d'Argentière we infer high and sustained channel pressurization over the whole summer and early fall (June-October). This has implications for the physics of subglacial channels, which we further discuss in Sect. 6.3, and also for our capacity to invert for discharge Q based on observed seismic power P. If one considers the equilibrium assumption over the melt season this yields, under Röthlisberger (1972) steady-state equilibrium assumptions, to the scaling Q ∝ P 33/31 w (see Eqs. (6) and (12)). When applied over the melt season using our observations of P w at Glacier 475 d'Argentière, this underestimates the measured discharge by more than 65%. As shown in Fig. 10, such assumption is only valid for the early and late melt season when both discharge and its variability are low. Using the approximation Q ∝ P 33/31 w may be more appropriate for periods of low melt water input and in settings with limited water input variability such as in Antarctica. If one now considers the empirical relationship Q ∝ P 11/24 w obtained from the period of channels being out of equilibrium (using Eq.(6) and R ∝ Q 6/22 , see Sect. 5.5), this leads to an uncertainty of less than 10% on the estimated water 480 discharge over the melt season at Glacier d'Argentière. We therefore suggest that the Q ∝ P 11/24 w relationship may be preferred for inverting discharge based on seismic observations during periods of high melt water input and in settings with strong seasonal variability in water input (e.g. Alpine and Greenland glaciers).

Comparison of inversions with predictions from
6.3 Implications for subglacial hydrology and ice dynamics 6.3.1 Understanding channels approaching equilibrium at low subglacial water discharge 485 During the early melt season ( Q Q ref < 4, Q < 1 m 3 .s −1 ; Figs. 5 and 10) channels are inferred to approach an equilibrium situation for which hydraulic pressure gradient scales weakly with changes in subglacial water discharge (Fig. 10). This behavior supports that the channel's hydraulic capacity is sufficient to accommodate water input at this time of the year. We propose that, at those times, changes in water supply occur at a rate that is lower than that at which channels adjust their hydraulic radius. During the early melt season, low rates in water input changes are likely caused by water supply from melt being highly 490 damped by the snow cover (Marshall et al., 1994;Fleming and Clarke, 2005). During the late melt season ( Q Q ref < 4; Fig. 10), the cause of low rates in water input is less clear. We suggest that such rates could be induced by englacial stored water being slowly released (Flowers and Clarke, 2002;Jansson et al., 2003). Because of the well-developed drainage system at those times, channels could also adjust faster their hydraulic radius than during the early melt season and therefore could behave at equilibrium for higher rates in water input than during the early melt season. 495 6.3.2 Using periods when channels approach equilibrium to estimate channel(s) size and number Using Eqs.(6) and (8) of Hooke (1984) that predict the conditions of equilibrium for steady-state channels and assuming that total discharge is equally distributed over channels of identical geometry (R-channels), we find that in our case equilibrium is predicted if the number of channels lies between 4 and 6 (using an ice thickness of 250 m, a down-glacier surface slope of per channel and thus channel-wall melt is higher (resp. lower) than the expected channel-wall creep, which violates the equilibrium condition. Our estimate of 4 to 6 channels is consistent with the numerical modelling results of Werder et al. (2013) of 4 to 5 dominant channels lying below the Gornerglestcher tongue (CH), a glacier which has a geometry similar to that of the tongue of Glacier d'Argentière (c. 500 m wide, c. 300 m maximum thickness). Further insights on the spatial evolution of the subglacial drainage system could be gained using seismic arrays to locate the source(s) of subglacial flow-induced-seismic 505 noise (Lindner et al., 2019).
We propose to estimate the absolute size of channels at the season initiation based on the channel number previously proposed. With 5±1 channels and 1 m 3 .s −1 equally distributed discharge, the average discharge per channel is of about 0.20±0.05 m 3 .s −1 (uncertainty is obtained from that on channels number). Considering that subglacial flow-induced-seismic noise is 510 likely sensitive to water flow speed on the order of 1 m.s −1 (Gimbert et al., 2016) we can estimate a minimal channel crosssection area of about 0.20 ± 0.05 m 2 , and a resulting channel radius of 0.35 ± 0.05 m (for semi-circular R-shaped channels).
We note that absolute inversions of R and S could be done by explicitly formulating the Green function G in Eq.(1), and be compared to the present estimation using channels at equilibrium. However, this is beyond the scope of this study.
6.3.3 Understanding highly pressurized channels during the summer season 515 At water discharges higher than 1 m 3 .s −1 (Fig. 5(b)) and relative changes in water discharge Q higher than 4 ( Q Q ref >4;Figs. 8 and 10) ) the hydraulic pressure gradient S in channels remains high (Fig. 10). Considering that bed slope is constant, these high S-values require channels to be full and pressurized. During these periods of high discharge, as S increases with relative changes in Q ( Fig. 10(a)) channels respond to changes in discharge in the same way as theoretically expected for cavities but not for channels by Schoof (2010). Such an behaviour is therefore opposed to the theoretical steady-state predictions of Schoof 520 (2010) and Werder et al. (2013) that instead support that channels have a water pressure decreasing as they develop over the summer.
Using Hooke (1984) and our estimate of 5 channels made in Sect. 6.3.2, we find that in our case channel-wall melt (i.e. opening rate) is expected to dominate ice creep (i.e. closing rate) for Q > 1 m 3 .s −1 (see Sect. B for details on the calculation).

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At steady-state this should either lead to channel growth and/or to an abrupt decrease in S down to free-flow situation (i.e. atmospheric pressure). These two scenarii are not observed during summer since R stays mainly constant (i.e. limited channel growth) and S presents high values supporting closed-flow over hourly timescales. We propose that the summer channel pressurization (high S) is due to channels responding to marked diurnal and short-term changes in water supply (as theoretically described in Schoof (2010)), and that channels behave out-of-equilibrium because changes in water input occur at a rate that is 530 higher than that at which channels can adjust their hydraulic radius. This interpretation is supported by our diurnal analysis on R and S evolution. In the morning, S is inferred to rise earlier than R (Fig.9), suggesting that channel-wall melt does not accommodate the increase of Q fast enough and causes pressurized flow.
As water supply increases, channels start to respond to the water input and grow by channel-wall melt leading to a delayed 535 hydraulic radius R increases compared to S (Fig. 9). At the same time the channel capacity increases with R (Röthlisberger, 1972) leading to a decrease in S before Q reaches a maximum as shown in Fig. 9. During the afternoon, as the water supply decreases, R slowly decreases by much less than a percent per hour (Fig. 9). At this rate, ice creep is capable to adjust changes in R fast enough in order to limit open channel-flow (Fig. S6). This could explain why S does not show an abrupt decreases down to the early melt season values as one would expect if open channel-flow occurs (Fig. 9). The hydraulic pressure gra-540 dient therefore builds up from day-to-day over the summer. During night-time, as Q is at its minimum, the closure rate still adjusts channel size and therefore allows R to remain nearly constant through summer. This proposed scenario is consistent with both the investigated diurnal dynamics in the hydraulic properties and may explain the unexpected pressurized channels during summer. Estimation of melt and creep rates calculated from Hooke (1984) in a similar manner as in Sect. 6.3.2 supports the plausibility of such diurnal dynamics (see Appendix Sect. B for details). Further measurements remain to be conducted on 545 glaciers with different geometries (e.g. flatter), different bed conditions (e.g soft bed glaciers) and different spatialization of water input (e.g. discrete water input through moulins) to evaluate the effect of such parameters on the subglacial hydrology dynamics. For instance, it is possible that our proposed channel's dynamic is limited to hard-bedded glaciers as soft-bedded glaciers have the capacity to store water and possibly damper the pronounced short term variability in water supply. In such setup, sediment erosion would complement ice wall melt and allow channels to be kept a much lower hydraulic pressure gra-550 dient than described in our study.

Channel dynamics, cavity water pressure and basal sliding
Our observations and subsequent analysis (Figs. 8 and 10) indicate that over the summer channels are pressurized and behave out-of-equilibrium. On the other hand, during summer the glacier sliding speed remain high, especially in 2018, (Fig. 5), which 555 shows that the average basal water pressure (which is mainly set by pressure in cavities) is also high. These concomitantly high pressures in channels and in cavities suggest that the two systems may be well connected.
During summer, because of channel-flow pressurization, the channel-system does not operate under a significantly lower hydraulic potential than that of the cavity-system. This would therefore prevent significant water flow from cavities to channels, 560 and leads to cavities that are kept pressurized. This sustained high water pressure at the glacier basis favors high glacier sliding speed over summer. Such channel-cavity-sliding link, has been previously suggested (Hubbard and Nienow, 1997;Andrews et al., 2014;Rada and Schoof, 2018) but was not based on an independent analysis of the cavities and channels hydraulic conditions as we propose here through combining seismic and basal sliding speed measurements. 565 We suggest that during these periods of pronounced short-term variability in water supply, the whole drainage system becomes well-connected although with a limited drainage capacity. Thus the channel system may participate in maintaining high pressure in cavities and thus high sliding speed during periods of high water supply variability. Short-term variability in water supply may lead to pronounced glacier acceleration even during situations of a well-developed channel network. Such subglacial hydrology/ice dynamics link deserves further investigation through combination of seismic observations and subglacial 570 hydrology/ice dynamics models (e.g. Gagliardini and Werder, 2018). Indeed a better understanding of the impact of shortlived water input on glacier dynamics is necessary as under climate warming short-term climatic variability and extreme event occurrences are expected to increase (Hynčica and Huth, 2019), potentially causing greater glacier acceleration than previously thought (e.g. Tedstone et al., 2015).

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We investigate the physics of subglacial channels and its link with basal sliding beneath an Alpine glacier (Glacier d'Argentière, French Alps) through the analysis of a unique two-year long dataset made of on-ice measured subglacial water-flow-induced seismic power and in-situ measured glacier basal sliding speed. Our study shows that the theory of Gimbert et al. (2016) is consistent with our observations and that the analysis of the seismic power measured within the [3-7] Hz frequency range allows to study the subglacial drainage properties over a complete melt season and down to diurnal timescales. 580 We quantify temporal changes in channels' hydraulic radius and hydraulic pressure gradient using the theory of Gimbert et al. (2016) and measurements of water discharge concomitant to our seismic record. Our approach allows to isolate subglacial water-flow-induced seismic power from that of other seismic sources, and makes possible observing changes at various timescales (from seasonal to hourly) and water discharge ranges (from 0.25 to 10 m 3 .sec −1 ). At seasonal timescales we sup-585 port, for the first time, that hydraulic radius and hydraulic pressure gradient both present at least a two-fold increase from spring to summer, followed by a comparable decrease towards autumn. Comparing our analysis to the theoretical predictions of Röthlisberger (1972) we identify that channel dynamics over the season is characterized by two distinct regimes yet unprecedentedly reported. At low discharge during the early and late melt season our analysis supports that channels respond to changes in discharge mainly through changes in hydraulic radius, and that the strong changes in hydraulic radius and weak 590 changes in pressure gradient are similar to those predicted by theory for channels behaving at equilibrium. We propose that, at those times, changes in water input occur at a rate that is lower than that at which channels adjust their hydraulic radius.
During the early melt season, these low rates in water input changes are likely caused by water supply from melt being highly damped by the snow cover. From this equilibrium channel-dynamics condition we are able to estimate the number of channels, which we find to be between 4 to 6, each channel having a radius of about 0.5 m in the early melt season that may go up to 2 m 595 in summer. At high discharge and high short-term water-supply variability (often during summertime) we show that channels undergo strong changes in hydraulic pressure gradient, a behavior that is not expected for channels at equilibrium. Instead, those changes in hydraulic pressure gradient are well reproduced by theory under the end-member consideration of no changes in channel geometry in response to changes in water input. We propose that, at those times, channels behave out-of-equilibrium because changes in water input occur at a rate that is much higher than that at which channels adjust their hydraulic radius.

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This interpretation is supported by R and S behaviors at the diurnal scale, which show that channels pressurize in the early morning and depressurize in the afternoon as their hydraulic radius slowly grow concomitantly with the water supply rise. At night when water discharge decreases, ice creep then allows channels to recover their initial early morning hydraulic radius.
We do not capture significant decrease of the hydraulic pressure gradient during those days, which indicates that the hydraulic pressure gradient builds up from day-to-day concomitantly to a hydraulic radius that is kept nearly constant. Channels may 605 thus remain pressurized over the whole summer because of the short-term (diurnal, rain) variability in water supply, which forces channels to respond through a transient-dynamic state. We expect our analysis of subglacial hydrology to be applicable to glaciers of similar geometry (relatively steep U-shaped valley glaciers) and similar highly variable and distributed water supply than those of Glacier d'Argentière.

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Channels behaving out-of-equilibrium during most of the melt season also has implications for the use of subglacial waterflow-induced seismic power P w to invert for water discharge Q. The empirical relationship between Q and P w that we derive during the period when channels are out-of-equilibrium allows estimating a water discharge from seismic noise with an error of less than 10 %, while an error of 65 % is obtained when assuming channels at equilibrium. Our presently proposed out-ofequilibrium relationship for inverting discharge could be applied in settings with strong seasonal variability in water supply 615 (e.g. Alpine and Greenland glaciers). During summer we also observe high and sustained basal sliding, supporting that the widespread inefficient drainage system (cavities) is likely pressurized. We propose that channels being also pressurized may help sustain high pressure in cavities and thus high glacier sliding speed.
These results demonstrate that on-ice passive seismology is an efficient tool to overcome the classical observational limita-620 tions faced when investigating subglacial hydrology processes. In this respect, our results bring new constraints on channels physics, on links between channels, cavities and sliding, and on the use of passive seismology to invert for subglacial water discharge. In the future, an essential step towards strengthening our knowledge on the physics of subglacial processes would be to assess the applicability of our findings over a wider range of glacier geometries (e.g. soft bed glaciers and ice sheets) both through extended on-site seismic survey and the use of our seismically-derived observations as constraints for subglacial 625 hydrology/ice dynamics models.
Code and data availability. Timeseries of of physical quantities shown in Figs. 5 and 8 can be found at https://doi.org/10.5281/zenodo. 3701520 (Nanni et al., 2020). The complete dataset will be made publicly available in the future. Ongoing work is taking place to meet the format and documentation required for the release for the complete seismic survey, which is expected to happen fully or partially by mid-2021. In the meantime, it is available on request from the corresponding author. The Python and SAC codes for seismic power calculation Appendix A: Frequency content of the water discharge and the subglacial channel-flow-induced seismic power We show in Fig. A1 the power spectrum of the water discharge Q (blue lines) and subglacial channel-flow-induced seismic power P w as a function of the period. We observe for both variables a well-defined peak at one day and 12 h period. This shows that these signals present a clear diurnal and sub-diurnal variability, and supports our choice to band-pass-filter these signals 635 within  h to study these short-term variabilities. Figure A1. Power spectrum of the water discharge Q (blue lines) and subglacial channel-flow-induced seismic power Pw (red lines) shown a function of the period. Both axis are in logarithmic scale (1 over the frequency.)

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with H the ice thickness, β the down-glacier surface slope, C 2 and C 3 constant. We use the values of Hooke (1984) for the two constants: C 2 = 3.731e −5 m −4/5 s −2/3 and C 2 = 5.71e −14 m −16/5 s −3/5 . For the glacier geometry we use using an ice thickness of 250 m and a down-glacier surface slope of 5°.
interpreted the results with input from F.G.. U.N. led the writing of the paper and F.G., C.V., F. W. and D. G. contributed to it. L.P and L.M were in charge of the basal sliding speed measurements. All authors participated to field installations.
Competing interests. The authors declare that they have no competing interests.