Introduction
The calving of marine-terminating grounded glaciers is a
significant contributor to rising sea levels worldwide due to the massive
volumes of ice involved that can suddenly be discharged into the sea.
Depending on the glacier, the contribution of calving to sea-level rise can
be equal to, or even greater than, the contribution from melt processes
. However, the lack of understanding of the
physical principles that cause these events means that it is difficult to
precisely forecast their contribution to sea-level rise in the near future
e.g.. Calving glaciers can rapidly advance
and retreat in response to minimal climate signals, which can rapidly change
the sea level . A better understanding of calving
processes is vital to developing accurate predictions of sea-level rise.
The lack of understanding of why and how calving events happen makes it hard
to create a general “calving law” . There
have not been enough direct observations of smaller calving events
e.g. to identify patterns to attempt to
form a general calving law. Calving events are intermittent, though they
exhibit some seasonality due to the seasonality of the mélange ice, ocean
temperature variations and variations in basal motion due to meltwater input
. The overall unpredictability of calving
requires monitoring equipment to be deployed on a long-term basis to detect
events.
One way to monitor glaciers and detect calving is to use seismic arrays
e.g.. Calving events can
generate glacial earthquakes, with surface waves detectable at a teleseismic
range . A common automated calving
detection method is to use triggers based on the ratio of short-time-average
and long-time-average seismic signals (STA/LTA). After an event has been
detected, it can then be localized. Currently, most localization methods
require visual confirmation of the calving location, unless they are
sufficiently large to be seen by satellite imagery. Automatic methods like
STA/LTA can help narrow down the manual search in satellite and camera
imagery for calving but, ultimately, visually locating a calving event
requires clear weather and well-lit conditions . An
exception to this is terrestrial radar e.g., but radar
cannot be deployed year-round without constant refuelling and swapping out
the data drives, and also has problems seeing through atmospheric precipitation.
Recently, high-frequency pressure metres, such as Sea-Bird Electronics
tsunameters, have been deployed to monitor calving at Helheim
.
Land-based seismometers offer improvements over simple camera or satellite
imagery for detecting calving because seismic arrays are not limited to
daylight hours, are not affected by snow, can be deployed year-round without
maintenance and provide quantitative data to help estimate the magnitude of
calving events. Seismic studies of calving have been done at the regional
(< 200 km) as well as the teleseismic level. Generally, teleseismic
detections of calving are done via low-frequency surface waves
e.g., while local detections are
done at some subset of frequencies within 1–10 Hz
e.g..
Seismicity in glaciers has been observed for both basal processes (e.g. basal
sliding) and surface processes (e.g. surface crevassing) unrelated to calving
. Until recently, seismic signals
generated by glacial calving were believed to be caused either by capsizing
icebergs striking the fjord bottom ,
interacting with the sea surface or by sliding
glaciers that speed up after calving .
found that glacial earthquakes at Helheim Glacier are caused by glaciers
temporarily moving backwards and downwards during a large calving event.
found that only capsizing icebergs generate observable
low-frequency surface-wave energy, with calving events that create tabular
icebergs not generating glacial earthquakes. Basal crevassing has also been
suggested as a mechanism for calving at Helheim . It is
not yet known how to fully categorize and characterize different calving
events.
Seismic signals of calving events typically have emergent onsets (i.e. having
a gradual increase in amplitude with no clear initial onset) with dominating
frequencies around the order of 1–10 Hz e.g.. The emergent nature of the signals
makes it hard to accurately identify a P wave onset time, let alone a
S wave onset time, which hinders the traditional seismic triangulation
method that takes the difference between the P and S wave arrival times
to generate a distance to the epicentre . The other main
method involves calculating back azimuths from a ratio of easting and northing
amplitudes of P waves from a broadband seismic station
e.g.; this fails for our study due to
the proximity of our stations and the high speed of the sound waves (around
3.8 km s-1 through pure ice, e.g. ) which make the
waves arrive near-simultaneously. Another method to locate calving events,
known as beam-forming, uses the seismic signals recorded on several array
stations to determine the time delay associated with a back azimuth that
aligns the signals coherently . A more recent method for
localizing calving events is the use of frequency dispersion of surface
waves, which uses a regional array (100–200 km away) of hydroacoustic
stations to estimate a distance between the event and detector and combines this
with an azimuth (determined from the P waves) to create a unique
intersection , as the stations are sufficiently far to separate
different seismic wave components. This method has a similar precision to using
intersecting azimuths from two remote stations, which is enough to identify
at which glacier the calving occurred, but not enough to localize the event
within the glacier.
In seismology, another technique to locate the epicentre of seismic events
uses differences in signal arrival times to create a hyperbola on which the
epicentre lies. This was first used in , and
notes that this method is best for shallow events where
refraction along a bottom interface (glacier rock) is insignificant. Such a
technique has not yet been applied to localizing calving. The aspect ratio
(vertical/horizontal dimension) of Helheim Glacier is of order 0.1 and so
calving events should be sufficiently shallow to use this technique. This
method is limited by determining the relevant wave velocity. In our case,
this is empirically determined by using hyperparameter optimization, also
known as grid search . This involves exhaustively
evaluating a product space of parameters to optimize some performance metric.
In our case, we use a product space of surface velocity veff,
x coordinate and y coordinate to minimize the total residual between the
observed lags and the lag corresponding to each (veff, x, y).
The hyperbolic method is then applied to calving events using the mean
veff from the grid search to localize the epicentres of the
seismic signals generated during calving events. The grid search is then
repeated with a product space of just (x,y) with the mean veff
from the first grid search, and these localizations are compared to the
hyperbolas.
Data
Four broadband seismometers (HEL1: Nanometrics Trillium 120, HEL2-HEL4:
Nanometrics Trillium 240) with sampling rates of 40 and 200 Hz were deployed
around the mouth of Helheim Glacier (Fig. ). HEL1 and HEL2 were
deployed in August 2013, while HEL3 and HEL4 were deployed in August 2014 .
They were synchronized with Coordinated Universal Time. These stations
detected seismic activity from calving as well as distant earthquakes,
so we first inspect the frequency distributions of the signals to isolate
calving events.
The four bedrock-deployed seismometers deployed at Helheim Glacier
as shown on a Landsat-8 image from 9 July 2015. GPS coordinates are
referenced to WGS84. HEL1: 66∘19.76′ N 38∘8.79′ W.
HEL2: 66∘23.24′ N 38∘5.91′ W. HEL3: 66∘24.06′ N 38∘12.9′ W. HEL4: 66∘19.94′ N 38∘13.60′ W. The calving front is clearly visible between them. Westward is
Helheim Glacier; eastward is the mélange and Sermilik Fjord. A camera was
also set up next to HEL1.
Spectrograms for (a) a calving event at Helheim on
12 August 2014, and (b) a regional earthquake in Bárðarbunga,
Iceland on 1 September 2014. The easting amplitude of the seismometers is
used for both events. The seismogram (top) and spectrogram (bottom) of each
event share the same time axis for direct comparison. The spectrograms have a
window size of 256 points (=6.4 s).
A calving event that was observed in situ at Helheim in August 2014
(Fig. a) matches those of , and very well, both in frequency
distribution and shape, with an emergent onset and relatively high-frequency
signals (1–20 Hz). In contrast, events from regional earthquakes have much
lower-frequency signals (< 1 Hz). A M5.2 regional earthquake in
Bárðarbunga, Iceland on 1 September 2014
(Fig. b) shows that the dominant frequencies received at the
HEL seismometers are all well below 1 Hz. This means we can easily separate
calving events from regional seismic activity by using a bandpass filter
(Butterworth, two-pole and zero-phased). We use a bandpass filter between 2 and 18 Hz
based off the spectrogram in Fig. a in order to maximize the
signal-to-noise ratio. Using some threshold of STA/LTA counts, we are able to
create a catalogue of 11 calving events on which to run our hyperbolic method
algorithm. This ignores smaller calving events, which generally have
amplitudes too small to easily identify a signal onset. Calving events, with
the exception of events in January/February 2015 for which imagery is too
snow-covered to use, are confirmed with local camera imagery and MODIS
satellite imagery from the Rapid Ice Sheet Change Observatory
(RISCO).
Localization methods and results
Hyperbolic method
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.
An example of a hyperbola of equation x2/a2-y2/b2=1, with
foci at F1 and F2 with constant path difference d2-d1=2a. b can be generated by c2-a2, where 2c is the
known distance between the foci.
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.
Seismic signals for a calving event at Helheim Glacier on
26 January 2015. The signal onset times are determined using an automated
script that searches for the first instance of a gradient exceeding a
particular threshold as defined in Sect. 3.2. The differences in the wave
onset times are then used to generate a characteristic path difference for
each hyperbola.
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.
Identifying signal lags
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.
Determining seismic wave velocity with grid search
Particle plots of seismic wave arrivals for the calving event of
7 July 2015, split into radial and transverse components. The characteristic
elliptic shape of the surface Rayleigh wave is clearly visible in the radial
component of the particle plot.
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.
Localization results
The calving event from 6 June 2015, with the localizations (top
panel) and the easting amplitudes of seismometer HEL1 (bottom panel) showing
several sub-events. X indicates locations derived from using a grid
search through a lattice of all points on the map with a fixed veff=1.20 km s-1.
Catalogue of all calving events with clear signal onsets at Helheim
Glacier from August 2014 to August 2015 overlaid on Landsat-8 imagery of
Helheim Glacier. Each colour corresponds to a calving event, with only the
area of overlap of the four hyperbolas being depicted. The x's represent the
same event located using a grid-search technique.
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.
Discussion
Interpretation of 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.
Local camera imagery for the calving event from 7 July 2015. The
blue line indicates the calving front from the last image taken before the
calving event, and the black line indicates the first image taken after the
calving events. Images are taken every hour. The position of the camera is
given in Fig.
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.
The calving events from Fig. overlain with the bedrock
topography from the Multichannel Coherent Radar Depth Sounder (MCoRDS) L3
data set from NSIDC , with the calving front from
9 July 2015 in black. The topography is collated and averaged from 2008 to
2012.
A typical power spectrum for a calving event (13 August 2014), for a
3 s time window containing the highest peak amplitude of the event. The
shaded inset in the top panel shows a zoomed-in view of this window.
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.
Discussion of methods
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.