After isolating the calving events, we apply the hyperbolic method to generate a catalogue of calving locations. A hyperbola can be geometrically defined as the locus (set of points) with a constant path difference relative to two foci, as seen in Fig. . In our case, each pair of seismometers acts as foci. We need two variables to determine the path difference: the signal–arrival time lag at each pair of seismometers, and the horizontal velocity of the surface waves.

Assuming that the speed of seismic waves across Helheim does not vary horizontally, the signals from a calving event that happened exactly at the midpoint of the two seismometers (or any other point along the perpendicular bisector of the two seismometers) would arrive simultaneously at the two seismometers. Similarly, if the event happened closer to HEL1, the seismic waves would arrive slightly earlier to HEL1, and the locus of possible calving locations would instead be the set of all points with a distance from HEL1 shorter than HEL2 by a fixed length. This length is 2a (Fig. ), which is the product of the speed of the waves through the glacier (vseismic) and the time lag in signal arrival (Δt) and is defined for a hyperbola with equation x2/a2-y2/b2=1. We may use the time lag of the signal arrivals at the two seismometers (which become the foci) to determine the path difference of the signals to form the locus. One of the curves (either the left or right in Fig. ) may always be eliminated as we know to which seismometer the event occurred more closely. Each time lag therefore generates one curve that intersects uniquely with the calving front, which will give the location of the calving. If the calving front is not known, the calving event can be triangulated using additional pairings of other stations.

This method requires evaluating the time lag between the signal arrival times at each seismometer (Fig. ), and obtaining the speed of the seismic waves through the glacier. As the surface waves travel over a topography unique to each glacier, we rename the variable veff, which is the effective speed of the seismic packet over the surface of Helheim Glacier using the above assumptions.

To identify the time lag, we first try using a cross-correlation of the signals. For subpanels HEL2 and HEL4 in Fig. , cross-correlation gives 1.5 s, which is a plausible value by eye, but for subpanels HEL3 and HEL4, cross-correlation gives 2.2 s which is not plausible by eye. The signals in Fig.  do look qualitatively different for HEL3 and HEL4, and it is possible that this is what prevents cross-correlation from generating an accurate lag time. Instead of using cross-correlation, we use an automated script that searches through the signal for the first instance of a raw waveform gradient exceeding 1.44 standard deviations of all point-wise gradients at each time step of 0.025 s for the total time window in Fig. . This value of 1.44 was empirically determined as this produced the closest match to cross-correlation for signals that were qualitatively similar enough to use cross-correlation.

From particle motion plots (Fig. ), we know these signals are dominated by surface waves. We assume that the seismic wave travels at the same lateral speed from the calving epicentre to each station. The dependence of wave speed on glacier depth is not important for this method as long as the effective (surface) lateral speed to each seismometer is the same in each direction. We also assume that the glacier surface, calving epicentre and seismometers are all coplanar, so that the hyperbolas can be kept two-dimensional for simplicity. In reality, there is some elevation between the seismometers and the glacier surface, though this distance (< 300 m) is so much shorter than the seismometer separation (> 6000 m) that refraction at the ice/rock boundary is likely negligible for characterizing the hyperbola. However, this method would become more precise with three-dimensional hyperboloids instead of two-dimensional hyperbolas.

We apply a grid search (hyperparameter optimization) to find the optimal (x,y,veff) to minimize the sum of the residuals of the time lags that would occur at each (x,y) for that veff as compared to the real observed time lags at each station. We parameterize between 1.00<veff<1.40 km s-1 (step size 0.01 km s-1) and the coordinate span of the entire map in Fig.  (step size 1 pixel) for our 11 identified calving events, and get a mean veff = 1.20 km s-1 with a standard deviation σ=0.1 km s-1. The standard error for these 11 samples is therefore σ/11=0.03 km s-1. For all further plots, we therefore use veff=1.20 km s-1. We generate four hyperbolas, using HEL1-HEL2, HEL1-HEL3, HEL2-HEL4 and HEL3-HEL4 as these have the greatest distance of ice between the stations, because we require that the rock has a negligible contribution to the wave arrival times.

Once we generate four hyperbolas we may take their intersection to be an estimate of where the calving occurred. In Fig. , we show the progression of one calving event on 6 June 2015. From this, the main peak (blue) corresponding to the highest amplitude signal is taken as a representative location for the entire event for the purposes of creating a catalogue of all events from August 2014 to August 2015. Applying this method to our entire catalogue of 11 calving events yields Fig. . We also re-run our grid-search method, this time with a fixed veff = 1.20, as a check of our localization results.

The hyperbolic method and grid-search method give very similar localizations for calving events at Helheim. Qualitatively, Fig.  shows that calving propagates up-glacier, with an initial event near the calving front (red) and subsequent seismic signals originating from locations further up the glacier. The locations of events also diverge, as after the second event (yellow), the third and fourth events (green and blue) go in opposing directions. Given that the calving front depicted in grey corresponds to one day before the calving event, the fact that the first event (red) is localized so close to the calving front is a good indicator that the event is localized correctly. Similarly, the year-long catalogue in Fig.  has events being localized near the calving front. For example, the black event of 7 July 2015 is localized for both the hyperbolic method and grid-search method and is immediately adjacent to the black calving front corresponding to 9 July 2015. Moreover, local camera imagery (Fig. ) also shows substantial ice loss on 7 July 2015 on the southern half of Helheim Glacier. We are therefore confident that the hyperbolic method and grid-search method are valid methods to localize calving.

Based on Fig. , calving appears to cluster in the northern portion of Helheim Glacier. This is consistent with the topography of the bedrock at Helheim (Fig. ), where the northern half is of the order of ∼ 200 m deeper than the southern half . It is possible that the deeper the ice, the higher the freeboard of the ice front and the greater the stresses that affect the calving front. In Fig. , we see wider gaps between crevasses in the north of the glacier as compared to the south. This may also mean that the surface velocities are different in each half, which would affect the localization results. The topographic differences of both the glacier surface and ice bottom may contribute to why we see calving primarily in the northern half of Helheim.

It is possible to constrain the fault size of the rupture caused by calving. Using a shear model from , the radius r0 of a circular fault is inversely proportional to the corner frequency fc of a S wave and is given by r0=Kcβ02πfc, where β0 is the shear velocity and Kc is a constant, equal to 2.34 for Brune's source model . From Fig. , the corner frequency is approximately bounded between 5 and 10 Hz. Taking a Poisson ratio of 0.3 for ice , the ratio of the Rayleigh-wave velocity to S wave velocity is approximately 0.930 , giving a value of β0=1.29 km s-1. For this rough calculation, we assume that the corner frequency is the same for the Rayleigh and S waves. This bounds the fracture size of the calving event between 48 and 96 m. Brune's relationship does not depend on properties of the material like effective stress σ or rigidity μ. Our range of 48–96 m is considerably smaller than a typical observed calving fracture by around one order of magnitude. A fracture size of order 1 km would require a corner frequency of order 0.1–1 Hz, which we do not observe. 100 m is more of the order of a crevassing event, which also occur during/before events, so it is possible that crevassing events continue to happen during the calving event and obscure the power spectrum seen in Fig. . Both basal crevassing e.g. and surface crevassing e.g. have been suggested as calving mechanisms. Basal crevassing may be a more plausible explanation for Helheim, as found that buoyant flexure via basal crevasses was the dominant cause for calving at Helheim in 2013. Our estimated rupture sizes using Brune's model could plausibly be the size of either and, as our method assumes a planar glacier surface, we cannot distinguish whether the crevassing is at the base or the surface.

The hyperbolic method described in this paper offers some benefits to traditional seismic location techniques, which are more suited for regional seismic arrays that can distinguish between the different seismic wave types e.g.. Moreover, regional arrays do not give the kind of precision that local arrays would have, as small errors on a regional azimuth translate to a large area of uncertainty on the local glacier surface. The hyperbolic method takes advantage of the stations' proximity to calving events and does not require separating out the different wave phases, thus sidestepping the P wave identification problem that hampered localization techniques from and .

The method also offers advantages over traditional calving detection methods, which require the use of a local camera and/or satellite data to visually confirm that calving took place. As seen in , calving generates a characteristic seismic signal (Fig. ) that is easily distinguishable from signals from regional earthquakes. This is likely because higher frequency signals from regional earthquakes are attenuated by the time they reach the seismometers. This allows seismometers to be used to monitor glaciers and quickly identify calving when power in the 2–18 Hz range exceeds some ratio above the ambient noise. Importantly, this monitoring could take place year-round, during the night and also on cloudy days, making it a helpful addition to locating calving alongside satellite imagery, camera imagery and radar monitoring.

The seismic signals detected during calving events are clearly dominated by surface waves. Particle plots (Fig. ) show the characteristic elliptical shape of a Rayleigh wave. The Rayleigh waves, which are in theory parallel to the vertical axis, appear slanted in Fig. . It is possible that the mix of different wave phases (e.g. Love waves, also a surface wave) has interfered with the Rayleigh wave such that it is no longer parallel to the vertical axis. There is also a lack of linear polarization as would be expected for a P wave. Our estimated S wave velocity, using a Poisson ratio of 0.3, is 1.29 km s-1 from above. This is lower than the 1.9 km s-1 for S waves in pure ice that found. It is possible that this is due to the anisotropy of the glacier surface, such that the ice is cracked and the seismic waves do not travel through pure ice. Given our characteristic surface wave velocity of the order of 1 km s-1 with frequencies of the order 10 Hz (see Fig. ), this corresponds to a surface wavelength of order 100 m. This is small enough to be affected by crevasses along the surface of the glacier which are of similar depths . This means that we can reasonably expect these crevasses to affect the seismic wave velocity, which could slow the S waves and surface waves, making our surface wave speed of 1.20 km s-1 a plausible value.

Because we are only working with surface waves, this limits our localization technique to just the epicentre of a calving event, with no suggestion of a focal depth. This means we could not distinguish between basal or surface crevassing, even if we could estimate a rupture size in the previous section. Moreover, we have assumed a planar ice front for simplicity. It is possible that this method could be extended to determine the depth at which calving (or crevassing) occurs by using a 3-D hyperboloid instead of 2-D hyperbolas.

The calculation method we have used ignores the presence of the rock between the glacier and the seismometers, as the proximity of the seismometers to the glacier means that the time taken for the wave to propagate through rock is negligible. Our method does not take into account the refraction at the ice–rock interface. Due to the ice dominating the wave path from the source to the seismometers, we assume that the refraction has a negligible affect on the trajectory of the surface waves.

The main source of error comes from identifying the signal onset. Picking out the signal onset is not fully automated because it requires setting a gradient threshold manually, or manually checking the plausibility of cross-correlation results. Local stations that are right by the calving front are subject to much more noise than regional arrays. While some of the noise can be filtered out, a lot of the noise still occurs in the 2–18 Hz range, which also contains most of the power from the calving signal. Moreover, as the calving events occur between the stations, the signals that arrive at each station come from different directions and may not necessarily be similar in shape. As a result, cross-correlation does not always work for determining lags. We cannot cross-correlate the envelopes as this would lose resolution of the lags (the envelope is of order 5 s in Fig.  but we have lags of order 1–3 s and even a 0.5 s shift would dramatically change the hyperbola). Our empirical method of using gradients is not rigorous as it requires manual confirmation; this means the error is difficult to quantify as the true signal onset time is not known. However, the veff of the surface waves can be estimated using a grid-search method, giving plausible results. With more calving detections, the standard error of the optimized veff value will decrease. As cross-correlation does work for some events, with a sufficiently large number of calving events, we may simply discard events that do not cross-correlate correctly. This would make it possible to create an event catalogue using only automated methods.