Concern about future sea level rise motivates a study of the subglacial conditions that give rise to streaming ice . The Whillans Ice Plain (WIP) region of the West Antarctic Ice Sheet (WAIS) is notable for decelerating from previously fast streaming flow over the instrumental record . Since most ice flux in Antarctica occurs through ice streams , understanding the conditions that cause ice-stream stagnation is of basic importance in constraining the continent's contribution to future sea level rise. Although recent progress has been made in describing the relationship between basal conditions and ice stream motion , direct observation of the temporal variation in subglacial conditions during ice-stream stagnation has remained elusive.

Antarctic ice streams exhibit a wide variety of stick-slip behavior. The WIP is an extreme case wherein the entire ∼2×104 km2 ice plain undergoes tidally modulated once- or twice-daily stick-slip motions . During the time between large-scale sliding events, several ∼100 km2 regions exhibit a high degree of locking . Slip motion observed by on-ice GPS instruments (Fig. ) then begins within the locked region and propagates outward towards the ice stream margins and grounding zone .

The focus of our study is a type of seismic tremor first identified by that occurs during large-scale slip motion. We follow in attributing these tremor episodes to small repeating earthquakes at the bed of the WIP (Fig.  and Sect. ). The earthquakes are evenly spaced in time, and this even spacing gives rise to spectral peaks that are inversely proportional to the earthquake recurrence time (Fig. d) . Over the 30 min duration of a tremor episode, the earthquake recurrence time gradually changes, causing the spectral peaks to glide (Fig. a).

The goal of this study is to quantitatively describe the relationship between conditions at the bed of WIP and the tremor that occurs there. To do this, we simulate the evolution of elastic and frictional forces along a small fault patch at the boundary between the ice and bed (Sect. ). Rapid motion along the fault excites waves that are recorded at a seismometer, and we account for this wave propagation to place constraints on the source properties by matching the amplitude and other features of the tremor signals. The simulated tremor-producing fault patch is loaded with surface velocity data recorded using GPS stations on the WIP.

A basic question is whether the events occur at an ice–ice, ice–till, or ice–bedrock interface. An observation of fundamental importance in this regard is that observed seismic particle velocity amplitudes are ∼100 nm s-1, which is quite low given the source–station distance even for tremor patch sizes ∼1 m. Low seismic amplitudes could exist for a number of reasons such as low stress drop and hence slip. However, we are able to constrain slip independently without using the amplitude. We instead find that the effect of slip along a bi-material interface is of leading importance. Specifically, when seismic slip occurs at an interface between a rigid and a compliant material, more motion occurs in the compliant material. All else being equal, lower seismic amplitudes therefore constitute evidence for a highly compliant bed material.

We also report a previously undocumented phenomenon: tremor amplitudes are anomalously high during large-scale slip events that follow a skipped slip event. In Sect. , we propose two distinct mechanisms to account for the observed anomalous seismic amplitudes: stiffening of the bed and reduction of aseismic slip. Skipped events have become more frequent in the last decade . The physical conditions that result in elevated seismic amplitudes during tremor may therefore be related to the conditions that will prevail in the near-future if stagnation continues.

We examine collocated seismic and geodetic data collected at the WIP during field seasons in 2010–2011 and 2011–2012 (Fig. ) . The data show large-scale sliding events, wherein the ice surface velocity accelerates to an elevated value and then gradually decays over a period of about 15 min. Several GPS time series are plotted in Fig. c. In this figure, the data are aligned so that t=0 corresponds to the time of maximum sliding velocity for each event. GPS data are recorded at 15 samples s-1. The fastest recorded velocity observed during any of the sliding events in our data set is 7.47×10-4 m s-1 or 64.5 m d-1. We low pass filter all GPS data below 500 s.

Large-scale sliding events are observed either one or two times per day. Ocean tides in the Ross sea feature a strong diurnal component, and the timing of slip events is nearly synchronous with the high and low ocean tides . Skipped events are usually low tide events, with the subsequent slip event occurring at high tide. The event flowing a skipped event is termed a double wait time event, and double wait time events have unique properties that we describe in Sect. .

During large-scale sliding events, seismometers record tremor episodes that consist of repeating velocity pulses (Fig. c). The spacing between the pulses changes from being as fast as 30 pulse s-1 to as slow as about 1 pulse s-1. The pulse rate is approximately proportional to the ice surface velocity during a large scale slip event, suggesting that the overall ice motion is somehow driving the tremor. Additionally, at some stations the pulse rate is sufficiently low (≲1 Hz) that seismograms show individual P- and S-waves .

A remarkable feature of the observed seismograms is seen in spectrograms as the smooth variation, or gliding, of the spectral peaks during the duration of large-scale sliding (Fig. a). The lowest frequency spectral peak occurs at the inverse of the pulse rate. This fundamental frequency f0(t) is accompanied by overtones that are present at integer multiples, i.e., 2f0, 3f0, and so on. We implement a basic feature tracking algorithm to calculate f0(t) from spectrograms of several tremor episodes, and the result is shown in Fig. b.

Tremor episodes occur during all large-scale sliding events cataloged by for which seismic data are available, and tremor episodes have never been observed when large-scale sliding is not occurring. Several stations have clearly visible tremor episodes. Every station for which data are available as of December 2015, however, shows at least a brief period during which some features of tremor episodes are present. The tremor episodes are most clearly observed when more than three spectral peaks are present. Clearly observed tremor is most common at the stations BB09 and GS17, although stations BB02 and GS07 near BB09 also have multiple tremor bands. These three stations are located in regions of the WIP, where more than 90 % of sliding occurs during large-scale slip events their Fig. 3a, although not all stations located in such regions exhibit equally clear tremor. We primarily focus on data from BB09. When GPS data are not available for this station, we combine GPS data from BB02 with seismic data from BB09. These stations are located 6 km apart, and for an average rupture velocity on the order of 100–300 m s-1 , the resulting 20–60 s delay is small compared to the ∼1800 s duration of large-scale slip.

Tremor episodes have peak seismic amplitudes on the order of 400–700 nm s-1. An important terminology note is that we distinguish between high frequency particle velocity amplitudes v (“seismic amplitudes”) recorded using seismometers and ice surface velocities Vs recorded using GPS instruments. Quantifying the variation of seismic amplitudes with time is challenging because the data are noisy and broadband p. 99. We calculate the seismic particle velocity amplitude envelope using the following algorithm. We first make a vector of all peaks in the time series. We then calculate a second vector by applying the peak finder to this vector of peaks. This process is carried out repeatedly until the spacing between peaks approaches ∼10 s. The result seismic particle velocity amplitudes are shown for several tremor episodes in Fig. a. To verify that this measure of seismic amplitude has no obvious pathological behavior, we compare it to another amplitude metric in Sect. .

The repeating velocity pulses each have duration on the order of 1/40 to 1/80 s. Pulse durations were mildly undersampled during the 2010–2011 field season when the sampling rate was 200 samples s-1. During the 2011–2012 field season, data were recorded at 500 samples s-1; we refer to these latter data as the “high rate” data. To quantify the effects of undersampling, we examine 25 tremor episodes recorded at high rate at 10 different seismometers. We observe no pulse duration less than 1/100 s. Extrapolating this finding to the data recorded at low rate suggests that we may interpret the pulse duration recorded at low-rate seismometers, even if it is only recorded over 3–6 data points. Undersampling may additionally result in amplitude reduction. We find that over the tremor episodes that we examine from the second field season, the amplitude reduction that occurs by low pass filtering data below 100 Hz is 12.5 % ± 0.5 %. We therefore correct the undersampled data from the first field season by scaling their amplitudes by a factor of 1.125.

We find that tremor episodes during double wait time events have higher seismic amplitudes than tremor episodes during single wait time events. Although large scale motion of the WIP has been shown to have higher sliding velocity during double wait time events , to our knowledge, such an association has not previously been noted for tremor amplitudes. We quantify some implications of this finding in Sect. .

We interpret each individual velocity pulse during a tremor episode as the far field expression of motion on a small fault patch. We refer to this patch as the tremor producing patch. Tremor episodes are interpreted as families or swarms of repeating earthquakes where the observed pulse rate is the earthquake recurrence rate.

At least two observations support this interpretation. First, small repeating earthquakes are expected to repeat with a faster recurrence rate at higher loading velocities, and this is observed in the data from the WIP. Second, at low pulse rate, noted the presence of individually discernible events with clear P- and S-wave arrivals. This observation implies that tremor is composed of many rapidly repeating events rather than resonance of a fluid-filled crack of cavity or wave propagation effects .

Motion of the WIP is due to the longitudinal stresses that result from the push of upstream ice. This loading occurs within the ice column, which results in a net motion of the ice during large-scale slip events. We assume that the ice and the bed deform elastically, a valid approximation over the 15 min duration of a tremor episode (Fig. ) . Throughout the earthquake cycle on the tremor patch, most motion occurs on the side of the patch that has the more compliant material (Fig. b, center). We discuss elastic strain partitioning between the ice and the bed in greater detail in Sect. .

Slip δ(x,t) at a point x on the ice–bed interface is defined as the difference in displacement between the upper and lower faces of the interface. The portion of the ice–bed interface outside the tremor-producing patch experiences relatively steady slip at nearly constant shear stress. This shear stress holds the bed in an elastically deformed state but causes no accelerations there. For this reason, outside the tremor-producing patch the slip rate V(x,t)=∂δ(x,t)/∂t is equal to the surface velocity Vs. When the tremor-producing patch is locked it experiences no slip and strain accumulates in time at a rate proportional to Vs (Fig. b, left) and inversely proportional to the patch size. Accompanying this straining is an increase in stress on the patch, which ultimately fails in a small slip event than relaxes the stress.

The tremor-producing patch thus experiences variations in the basal shear stress that give rise to stick-slip oscillations (Sect. ). In this study we do not attempt to resolve the spatial details of the slip process, given that seismic wavelengths in the available data are larger than the patch size. We therefore focus on the spatially averaged slip, D(t)=1A∫tremor patchsurfaceδ(x,t)dA=1A∫t0t∫tremor patchsurfaceV(x,t′)dAdt′, where t0 is the time at the beginning of a tremor slip event. The total slip in one event is achieved after a duration of T∼1/80 s. Whereas D(t) refers to the cumulative spatially averaged slip, we use the notation D (without an explicit argument) to refer to the spatially averaged total slip in a single event.

The sliding velocity of the tremor-producing patch averaged over many slip cycles is approximately Df0 for slip pulses that are evenly separated in time by f0-1. This relationship assumes that all motion occurs during seismically expressed slip events, as we discuss in greater length in Sect. . Over multiple earthquake cycles the sliding velocity must keep pace with the surface velocity of the ice Vs, D=Vs/f0. Using observations of Vs and f0, we infer values of D≈15–75 µm (Fig. d). Seismograms sometimes show multiple tremor episodes that are simultaneously recorded at a single station. In Fig. a, for example, these tremor bands appear as low as 1–2 Hz. These tremor events have D as large as 1 mm.

We now describe the relationship between slip on the tremor-producing patch and the seismic particle velocity amplitudes recorded at on-ice seismometers. In our description, these amplitudes are influenced only by the accelerations on the tremor patch, tremor patch size, geometrical spreading, and potential bi-material effects arising from the fault being located at the interface between ice and the bed material. Attenuation is thought to be unimportant at the frequencies of interest in the present study based on the following reasoning.

The seismic quality factor Q for ice is in the range 400–1000 over all of Antarctica . For wave propagation distances on the order of the ice thickness H, attenuation becomes important only at frequencies greater than a characteristic attenuation frequency f*≡Qci/πH. The shear wave speed of ice is denoted ci. At 250 Hz, the highest Nyquist frequency in our data set, attenuation would become important if Q< 315. We therefore do not expect attenuation to be important for the frequencies that are resolved in our data set.

Far field seismic particle velocity due to fault motion in a uniform medium is given by Eq. 4.96, v(t)=M¨(t-H/ci)4πρici3H, where the seismic moment is defined as M(t)≡πGiR2D(t), for a circular fault patch with radius R and ice shear modulus Gi. Here, and in later expressions, the overdot denotes ∂/∂t. This expression for particle velocity is specific to far field S waves that propagate from the source to the station through a lobe in the far field S-wave radiation pattern. This is consistent with horizontal faults located directly below the stations. If the station is not directly over the source, or the fault has different orientation, then the station will record both P and S waves; the expression for far field P waves has the same form, but with ci replaced by the ice P-wave speed. Thus, by assuming stations directly over the source, we are taking the smallest possible value of epicentral distance times wave speed cubed in the denominator, which will maximize particle velocity for a given seismic moment.

Despite this sensitivity to wave type and radiation pattern, we do not attempt to model distinct P- and S-waves because they are not clearly present in the data during tremor events. When single basal events do show clear P- and S-wave arrivals, their relative timing suggests a hypocentral distance of about one ice thickness . For this reason, we take the hypocentral distance to be equal to the ice thickness for the remainder of this work, but with an awareness of how this assumption might bias our estimates of source parameters.

The far field seismic velocity field predicted by Eq. () does not account for the contrast in material properties in the vicinity of the bed (Table ). We consider two variations of Eq. () that account for material property heterogeneity. The first type of heterogeneity that we consider is that where the region surrounding the fault and the region surrounding the seismometer are both homogeneous but have different material properties. The far field seismic velocity is , v(t)=M¨(t-H/ci)2πzizbci2H, where the shear wave impedances are denoted zk=ρkck (no summation) and zi and zb are the shear wave impedances of the ice and bed. The amplitudes predicted by Eq. () are valid for media with gradually varying material properties and may therefore be relevant to some aspects of ice sheet seismology . The relationship Eq. (), however, cannot account for the sharp changes in material properties that are expected to occur at the ice–bed interface.

The second type of heterogeneity that we consider is that where the fault patch is located at the ice–bed interface. A rich variety of wave behavior occurs when slip occurs along a fault that separates media with different elastic properties . We neglect head wave phases whose ray path partially travels along the bed. We take the short-time limit of Eq. (26) of and find that the far field particle velocity amplitudes are sensitive to the material properties on both sides of the fault through the impedance parameter Z, Z=zbzi+zb. Seismic moment Eq. () is not well defined for slip on a bi-material interface , in which case a more useful quantity is seismic potency, P(t)≡πR2D(t). Far-field seismic particle velocity amplitudes are given by v(t)=ZP¨(t-H/ci)πciH. We have invoked a factor of 2 higher amplitudes to account for amplification at the free surface (p. 130 in ).

We find a scaling relationship for the far field particle velocity Eq. () by replacing time derivatives with the event duration T: v∼ZDciHRT2. The factor of R/T has the interpretation of a rupture velocity, and it is interesting to note that for fixed material properties, seismic amplitudes depend only on slip and rupture velocity.

We now place constraints on the bed shear modulus Gb, the effective pressure σ¯, the patch size R, and the state evolution distance L. We hold all other parameters fixed except where explicitly mentioned (see Table ). We first discuss the choice of frictional parameters.

The properties of long-wait-time tremor episodes are of particular interest as they may be indicative of prevailing near future conditions. Tremor episodes during double wait time events have higher amplitude (420 nm s-1 ± 120 nm s-1) than tremor episodes during single wait time events (380 nm s-1 ± 120 nm s-1). This difference in means is significant with 99 % confidence. As discussed later, we further verify this result by using a separate amplitude metric from the one discussed in Sect. .

The simplest explanation for this behavior is that higher amplitudes are caused by greater higher slip per event, D, as suggested by Eq. (). This explanation, however, is not supported by the data. The mean slip per event for double and single wait time events are not significantly different at the 85 % confidence level (mean 45 nm s-1 ± 17 nm s-1). We therefore consider two mechanisms for the observed seismic amplitude variation at constant slip D. First, from the analysis in Sect. , we expect that the difference in amplitudes at constant slip D may only arise from changes in the bed shear modulus. Indeed, as discussed in Sect. , higher shear modulus values are inferred for long-wait-time events. The second explanation is that the variation is due to un-modeled effects involving partitioning between seismic and aseismic slip.

Before proceeding, we first establish that the amplitude anomaly is not an artifact of the particular amplitude envelope algorithm discussed in Sect. . We examine a catalog of 40 tremor episodes, many of which do not have adequately clear spectral bands to perform a more detailed analysis but which are adequate for the analysis of amplitudes. We then calculate amplitude as the median of the absolute value of the seismic trace during the tremor episode. This metric provides a much lower amplitude estimate than the metric described in Sect.  because it includes many near-zero trace values that occur during oscillatory motion. Comparison with a t distrubtion confirms that double wait time events have higher median amplitude (41±8 nm s-1) than single wait time events (50±4 nm s-1) with 99.99 % certainty.

We have quantified the dynamics of a small fault patch at the bed of an ice stream using a spring-slider model. Motions on the fault excite seismic waves and by comparing synthetic seismograms with those that are observed we have constrained several fault zone parameters. We are able to match many of the remarkable features of the tremor episodes recorded on the Whillans Ice Plain (Fig. ), including the variation of tremor spectral peaks with sliding velocity, seismic amplitudes, and corner frequencies.

Our simulations have constant fault zone properties throughout the tremor episode. As a result of this, our simulations do not always match the observed seismic amplitudes and recurrence frequencies during an entire tremor episode (Fig. ). This self-imposed limitation may seem contradictory because we have just argued that the fault zone properties appear to change during a tremor episode (Fig. ). We accept this limitation because of the possibility that 2-D or 3-D effects are involved in the observed temporal evolution during a single event.

Tremor episodes in glaciers may also be related to the motions of fluids contained within conduits . Several lines of evidence suggest that the WIP tremor episodes are not related to such a source process. The foremost reason is that on certain seismometers, tremor episodes clearly show individual slip events with clear P- and S-wave arrivals . Additionally, there is a strong correlation between observed ice surface velocities and the tremor frequency, and this correlation has a clear interpretation in terms of repeating earthquakes. The relationship between surface velocities and hydraulic fracture resonance, in contrast, is not as clear. Finally, simple models of hydraulic fracture resonance and turbulent channel flow predict spectral signatures that are not consistent with the evenly spaced spectral peaks observed on the WIP, although complex geometrical effects may invalidate the simplifying assumptions of such models. These distinguishing criteria may be useful in analyzing several recently described data sets .

Having identified a bed shear wave speed of 75–100 m s-1 in Sect. , we rule out the possibility that the tremor producing patch is a bedrock outcrop. We consider this inferred shear wave speed to be consistent with the presence of subglacial till . Further evidence that the fault interface consists at least partly of till is given by the relatively small estimated state evolution distance L∼1 µm. We interpret this distance to be indicative of sliding against a material with µm-scale grain size, and glacial till is the most plausible such material . Laboratory experiments also show that L scales with the thickness of shear zone deformation . Micron-scale L values therefore suggest a high degree of localization of subglacial deformation.

The WIP has long been recognized as having heterogeneous basal shear resistance . Direct borehole access to the bed has measured effective pressure in the range of -30 to 200 kPa . Numerous ice flow model-based inversions of geodetic data have found localized regions of high basal shear stress in the vicinity of the seismometers that record the WIP tremor episodes . These areas of high resistance to flow are juxtaposed against an active subglacial lake system where basal resistance to flow is presumably negligible. Recent inversions with precise digital elevation data show that shear resistance in localized km-scale patches may be as high as 10–100 kPa. These patches exist within extensive regions that have zero shear resistance within the resolution of the inversion (O. Sergienko, personal communication, 2015). Our estimated ∼30 kPa effective pressure within the tremor fault zone is therefore in reasonable agreement with both borehole and geodetic stress estimates.

Our ∼3 kPa stress drop estimate is somewhat low but not unusual for glacier sliding earthquakes. On the Hans Glacier, Svalbard, finds fault radii of 28–74 m associated with stress drops 86 Pa–1.2 kPa, respectively. On upstream sites on the Kamb and Whillans ice streams, find fault radii of 5–10 m associated with stress drops of 10–100 kPa, respectively. On the David Glacier, East Antarctica, find fault radii 65–110 m associated with stress drops 100–600 kPa, respectively. The data of show spectral troughs that are likely due to free surface reflections, and so their estimates of fault size should be interpreted as lower bounds . The range of stress drops given by occurs because they do not account for bi-material effects and instead consider end-member elastic moduli scenarios.

Tremor events may provide insight into the mechanism of long term deceleration at the WIP. Double wait time events are becoming more common during WIP stagnation . The particular conditions that give rise to higher seismic amplitudes during double wait time events (Sect. ) may therefore be indicative of prevailing near-future conditions of the WIP. We now discuss the markedly different predictions of each of the two mechanisms we have identified to account for this behavior.

The two mechanisms identified to account for anomalous seismic amplitudes make different predictions about fault zone stability: a stiffening bed implies a shift towards more stable conditions (by increasing k in Eq. ), but increased seismic slip implies a shift towards less stable conditions. Independent observations from the Kamb Ice Stream seem to favor the hypothesis that during ice-stream stagnation tremor producing patches becomes more unstable to sliding. The Kamb Ice Stream has a seismicity rate that is approximately 1000 times higher than on the fast flowing, upstream part of the Whillans Ice Stream . One interpretation of this observation is that as an ice stream decelerates, subglacial conditions become more unstable in the sense defined by Eq. (). This scenario is consistent with increasing effective pressure (at constant slip, D, see Fig. ) and is therefore consistent with a water piracy-type mechanism for ice-stream stagnation .

We have conducted dynamic simulations of stick-slip motion at the bed of the Whillans Ice Plain (WIP). By comparing the simulations to data, we are able to independently infer the presence of an elastically compliant, fine-grained till layer and high pore pressures. Furthermore, seismic amplitude variations between tremor episodes may be related to long term changes occurring at the bed.