Friction and slip measured at the bed of an Antarctic ice stream

21 The slip of glaciers over the underlying bed is the dominant mechanism governing the migration 22 of ice from land into the oceans, contributing to sea-level rise. Yet glacier slip remains poorly 23 understood or constrained by observations. Here we observe both frictional shear-stress and slip 24 at the bed of an ice stream, using 100,000 repetitive stick-slip icequakes from Rutford Ice 25 Stream, Antarctica. Basal shear-stresses and slip-rates vary from 10! to 10"Pa and 0.2 to 26 1.5 m day#$, respectively. Friction and slip vary temporally over the order of hours and 27 spatially over 10s of meters, caused by corresponding variations in ice-bed interface material and 28 effective-normal-stress. Our findings also suggest that the bed is substantially more complex 29 than currently assumed in ice stream models and that basal effective-normal-stresses may be 30 significantly higher than previously thought. The observations also provide previously 31 unresolved constraint of the basal boundary conditions of ice dynamics models. This is critical 32 for constraining the primary contribution of ice mass loss in Antarctica, and hence the endeavor 33 to reduce uncertainty in sea-level rise projections. 34

Here we address this observational void by using icequakes to provide the first spatially-mapped, 59 in-situ observations of both frictional drag and slip-rate at the bed of an ice stream. These   The normal and shear-stress histograms show a bimodal distribution, with more than two thirds 119 of the icequakes having effective-normal-stresses lower than ~5 × 10 , Pa and shear-stresses 120 Manuscript submitted to Nature Geoscience: Confidential 5 lower than 2 × 10 , Pa. Conversely, the slip-rates exhibit a unimodal distribution, tailing off 121 below 0.2 #$ and above 1.5 #$ . The spatial distribution of average basal shear-stress, slip-rate and fault radius for each cluster 138 over a 7 x 6 km region are shown in Fig. 4 confined to the minority of the bed at a given point in time, yet invoke significant basal drag.

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Aseismic regions between icequake clusters likely also contribute to the basal drag, presumably  Our confidence in the frictional shear-stress and slip-rate measurements is founded partially on the uncertainty amplitudes, but also fundamentally on the agreement between the observed basal 173 slip-rates and GNSS-derived surface displacement 34 . This agreement validates assumptions of 174 slip-dominant rather than deformation-dominant flow at RIS and the use of a rate-and-state 175 model and assumptions of the icequake source properties. The small discrepancy between the 176 surface and basal slip-rates is primarily due to uncertainty, except for a minority of particularly 177 sticky-spots. These sticky-spots exhibit particularly strong frictional drag that significantly 178 inhibits local ice flow, albeit for short durations of the order of hours to days. spatially small compared to the total bed area. Our results therefore imply that certain icequake 186 clusters accommodate a considerable proportion of the total basal drag.

188
Icequake generation mechanisms 189 We propose that the icequakes are generated by at least one of two mechanisms, or sliding     Finally, these icequakes observations can aid the understanding of earthquake mechanics more 287 generally. Even the smallest magnitude icequakes ( 7 ≈ −1.5) have high signal-to-noise-ratios,               The spectrum of an earthquake contains more information than just the long-period 506 spectral amplitude. If one assumes that an earthquake's spectrum can be described by a 507 Brune model 63 then one can also measure the corner frequency, 0 , of the earthquake.

508
However, an earthquake's spectrum is also particularly sensitive to seismic attenuation. where Ω( ) is the amplitude of the spectrum for a certain frequency and is the travel-515 time.

517
To obtain an accurate measurement of corner frequency, we therefore use a linearized 518 spectral ratios method to constrain . This spectral ratios method isolates the path effects One can estimate the fault radius, , and stress-drop, Δ , of an earthquake from the 531 corner frequency.

533
The relationship between corner frequency and fault radius, , is given by 65 , where 0 Z is the spherically-averaged corner frequency for the earthquake, is the shear- (4), 553 We now have all the observable parameters required to constrain a friction model at an 554 icequake source. proposed in this study (see Fig. 5, main text), damage at the fault interface that affects the 584 frictional properties is likely less significant than at traditional earthquake fault interfaces.

585
This lack of damage is evidenced to some extent by the highly repetitive nature of the 586 icequakes 20,28 . We assume that at least part of the icequake patch is near steady-state, or 587 approximately at steady-state if it slips sufficiently fast. A caveat to this is that some of 588 the icequake patch could have remained below the steady-state sliding limit, which we do 589 not explore this here. Overall, we deem the approximation of

632
Once we know the effective-normal-stress, ,, we can find the overall shear-stress on the 633 fault, , from Equation 9.

635
We emphasize that the effective-normal-stress, ,, is the normal stress on the fault, which 636 is not necessarily equivalent to a traditionally defined glaciological effective pressure,  The second glaciologically important parameter to measure at the bed is the slip, and 648 hence the basal slip-rate. To calculate slip, we assume that while an individual icequake 649 cluster is active, all (or at least the vast majority of) slip is accommodated seismically.

650
This is likely the case for RIS, as evidenced by the close agreement between surface slip-651 rate and seismically measured basal slip-rates (see Fig. 2f). Calculating the basal slip, , 652 from an icequake is challenging because one first has to determine a method of where : is the seismic moment released by an earthquake and is the area of the fault.

657
The bed shear-modulus, %&' , is calculated by assuming a further behavior of the rate-658 and state-friction law. This behavior is that an earthquake can only nucleate if it is in the 659 unstable regime. In this study, we assume that the temporally-averaged driving shear-  where , and are constants to invert for. We use a least squares approach to minimize hold for clast-over-bedrock sliding since the shear-modulus will still be related to some 692 exponent, n, of ,, even if that exponent were ~0.

694
Equation 14 can then be used to find the slip, , associated with a single icequake, for the 695 effective-normal-stress applied to the fault at that particular time. We also calculate the 696 approximate slip-rate associated with these highly repetitive icequakes. If one assumes 697 that all the slip when an icequake cluster is active is accommodated seismically, then one The methods described above allow us to calculate the total shear-stress, , and the slip,

702
, at the bed. These two parameters can provide observational constraint on ice dynamics 703 models of ice streams.

705
A note on assumptions 706 A number of assumptions are made to make the derivation of basal shear-stress and slip from 707 icequake observations and a rate-and-state friction model mathematically tractable. There are 708 several assumptions that warrant particular emphasis. The first is the assumption that all slip at 709 an individual sticky-spot is accommodated seismically while that cluster is active. The highly 710 repetitive nature of the icequakes (see Extended Data Fig. 1 and 28 ), with approximately constant 711 inter-event times between consecutive icequakes in a cluster, is indicative of the stability of each 712 sticky-spot (see Fig. 2), justifying this assumption. Secondly, a Brune model 63 is assumed to 713 describe the earthquake source characteristics. While such a model is likely an approximation for 714 the complex physics of earthquake rupture, it is a common assumption for other earthquake 715 studies that is likely also a valid approximation for the stick-slip icequakes presented here.

716
Thirdly, we approximate that the time-derivative of the state-variable in the rate-and-state