The contribution of glaciers and ice sheets to the eustatic sea-level rise (SLR) between 2003 and 2008 has been estimated to be 1.51±0.16 mm of sea-level equivalent per year (Gardner et al., 2013). Cryogenic freshwater sources were responsible for approximately 61±19 % of the total SLR observed in the same period. Iceberg calving, defined as mechanical loss of ice from the edges of glaciers and ice shelves (Benn et al., 2007), is thought to be one of the most important components of the total ice loss. For example, solid ice discharge accounts for around 32 % to 40 % of the mass loss from the Greenland ice sheet (Enderlin et al., 2014; van den Broeke et al., 2016), and iceberg calving in Patagonia dominates glacial retreat (Schaefer et al., 2015). On the other hand, several studies found that increased submarine melting is a major factor responsible for the observed rapid retreat of tidewater glaciers (e.g., Straneo and Heimbach, 2013; Luckman et al., 2015; Holmes et al., 2019). The exact partitioning between ice mass loss caused by calving fluxes, submarine melting and surface runoff changes geographically and needs to be measured separately at each location. Calving from tidewater glaciers is driven by different mechanisms, including buoyant instability, longitudinal stretching and terminus undercutting (van der Veen, 2002; Benn et al., 2007). Terminus undercutting results from submarine melting and is often considered to be a major trigger of ice breakup at the glacier front (Bartholomaus et al., 2013; O'Leary and Christoffersen, 2013). In support of this idea, the solid ice discharge from tidewater glaciers was found to be highly correlated with ocean temperatures (Pętlicki et al., 2015; Luckman et al., 2015; Holmes et al., 2019), which are expected to increase significantly as a result of climate shifts (IPCC, 2013). Thus, accurate estimates of calving fluxes from marine-terminating glaciers are crucial to both understanding glacier dynamics and predicting their future contribution to SLR and the freshwater budget of the polar seas. Obtaining these estimates requires remote-sensing techniques, which enable the observation of dynamic glacial processes from a safe distance.

Satellite imagery is an effective way to study large-scale, relatively slow changes at the ice–ocean interface, such as the disintegration of the 15 km long ice tongue from Jakobshavn Isbræ in 2003 in Greenland (Joughin et al., 2004). For fast-flowing ice masses, changes of terminus position caused by both calving and glacier flow must be clearly separated. Consequently, satellite imagery is more limited for observing calving events, which typically occur on sub-diurnal timescales and are often not greater than 1000 m3 in volume for most tidewater glaciers in Svalbard or Alaska (e.g., Chapuis and Tetzlaff, 2014). Moreover, a thick layer of clouds, fog or precipitation in the form of snow and rain often makes it difficult to track iceberg calving continuously using optical techniques, such as surface photography or terrestrial laser scanning. These difficulties provide the motivation for investigating the use of underwater noise to quantify calving fluxes.

Many different methods have been developed to measure ice discharge from marine-terminating glaciers. Passive glacier seismology, also called “cryoseismology” (Podolskiy and Walter, 2016), is probably one of the most mature, widespread and useful tools; broadband seismometers have been widely installed in remote areas near calving glaciers since the early studies performed by Hatherton and Evison (1962) and Qamar and St. Lawrence (1983). Seismic signals associated with subaerial calving originate from two main mechanisms: (1) the free fall of ice blocks onto the sea surface (Bartholomaus et al., 2012) and (2) interactions between detaching icebergs and their glacier terminus (e.g., Ekström et al., 2003; Murray et al., 2015). The latter interactions, also known as “glacial earthquakes”, are caused by large, cubic-kilometer-scale icebergs of full-glacier height, and the resulting seismic magnitude is not related to the iceberg volume in a simple manner (Sergeant et al., 2016). Higher-frequency (>1 Hz) calving seismicity from iceberg–ocean interactions, constantly detected by distant seismic networks (e.g., O'Neel et al., 2010; Köhler et al., 2015), usually peaks between 1 and 10 Hz (Bartholomaus et al., 2015; Köhler et al., 2015). Both frequency content and amplitudes of high-frequency signatures are found to be independent of iceberg volumes (O'Neel and Pfeffer, 2007; Walter et al., 2012). Bartholomaus et al. (2015) applied generalized linear models to correlate various properties of seismic signals originating at Yahtse Glacier, Alaska, with estimates of iceberg sizes divided into seven classes. In line with previous findings by Qamar (1988), they identified ice quake duration as the most significant predictor of iceberg volume. Based on these studies, Köhler et al. (2016, 2019) successfully reconstructed a record of total frontal ablation at Kronebreen, Svalbard, using seismic data calibrated with satellite images and lidar volume measurements.

Recently, Minowa et al. (2018, 2019) demonstrated the potential of using surface waves generated by falling icebergs to quantify calving flux. They found a strong correlation between calving volumes estimated from time-lapse camera images and the maximum amplitudes of the waves. Other methods for quantifying ice discharge from marine-terminating glaciers, including surface photography (e.g., How et al., 2019), terrestrial laser scanning (e.g., Pętlicki and Kinnard, 2016), ground-based radar imaging (e.g., Chapuis et al., 2010) or terrestrial radar interferometry (e.g., Walter et al., 2019), are usually used for short-term measurements.

The approach investigated here is an example of acoustical oceanography, which extracts environmental information from the underwater noise field (Clay and Medwin, 1977). Acoustical oceanography may offer some advantages over other, more well-developed methods for the study of the interactions between land-based ice and the ocean. Low-cost hydrophones are easily deployed in front of marine-terminating glaciers, and acoustic data can be gathered continuously for several months or longer with a high (>10 000 Hz) sampling rate and low maintenance. Measurements are insensitive to lighting conditions such as fog; cloud coverage; and the polar night, humidity and intensity of precipitation. Moreover, acoustic signals recorded in glacial bays and fjords also contain signatures of ice melt associated with impulsive bubble release events (Urick, 1971; Tegowski et al., 2011; Deane et al., 2014; Pettit et al., 2015; Glowacki et al., 2018). While currently no quantitative models exist to estimate melt rates from underwater noise, the potential idea to simultaneously measure submarine melting and calving, two major processes acting at the glacier–ocean interface, is worth mentioning.

Quantifying iceberg calving by “listening to glaciers” was first proposed by Schulz et al. (2008), who suggested long-term deployments of hydrophones (underwater microphones) and pressure gauges, in addition to more traditional measurements of water temperature and salinity, to study signals of ice discharge together with accompanying hydrographic and wave conditions. Following this novel idea, independent studies conducted in Svalbard (Tegowski et al., 2012) and Alaska (Pettit, 2012) showed the first waveforms and spectra of the sounds generated by impacting ice blocks. Pettit (2012) provided an explanation for individual components of the signal, including low-frequency onset, pre-calving activity, mid-frequency block impact, iceberg oscillations, and mini-tsunami and seiche action. Encouraged by these initial results, Glowacki et al. (2015) analyzed 10 subaerial and 2 submarine calving events identified in both acoustic recordings and time-lapse photography made in front of Hansbreen, Svalbard. A spectral analysis of three different calving types, called “typical subaerial”, “sliding subaerial” and “submarine” (see supplementary videos in Glowacki et al., 2015), showed that they radiated underwater noise in distinct spectral and temporal patterns, but all with a spectral peak between 10 and 200 Hz. Most importantly, acoustic emission below 200 Hz was highly correlated with block impact energy in a simple model. The dimensionless coefficient converting impact energy to acoustic energy at the calving impact point was found to be 5.16×10-10, and the power exponent was assumed to be 1. However, this earlier analysis was limited by the small number of subaerial calving events analyzed (10), lack of a full error analysis and the unrealistic assumption of simple cylindrical spreading of acoustic waves in the water column.

To address these issues, we conducted a new study covering a total number of 169 subaerial calving events observed with time-lapse photography at Hansbreen, Svalbard. Impact energies generated by falling icebergs are estimated with error bars and related to received acoustic signals. The total noise energy resulting from block–water impact is calculated using a standard sound propagation model Bellhop (Porter, 1987, 2011), which requires bathymetry data and sound speed profiles as inputs. Variability in transmission losses associated with sound wave reflections from an idealized, flat glacier terminus is also accounted for. The analysis shows that impact energy is strongly correlated with acoustic emission below 100 Hz. We present a new energy conversion efficiency calculated with this more detailed physical model and demonstrate how cumulative values of kinetic energy and ice mass loss can be found by integrating impact noise over a specified number of subaerial calving events.

Hansbreen is a retreating, grounded, polythermal tidewater glacier terminating in Hornsund fjord, Svalbard (Fig. 1). It covers an area of around 54 km2 and is more than 15 km long (Błaszczyk et al., 2013). The glacier has a 1.5 km-wide active calving front with an average height of around 30 m (Błaszczyk et 'al., 2009). The mean thickness and total volume of Hansbreen are estimated to be 171 m and 9.6±0.1 km3, respectively (Grabiec et al., 2012). The surface flow of the glacier is dominated by basal motion in the ablation area (Vieli et al., 2004) and the mean annual flow velocity near the terminus, and its calving flux is estimated to be 150 m yr-1 and 38.1×106 m3 yr-1, respectively (Błaszczyk et al., 2009). The average retreat rate of the glacier during 2005–2010, 44 m yr-1, was more than twice the rate observed between 1900 and 2010 (Grabiec et al., 2012). These characteristics are representative of Svalbard's tidewater glaciers, making the bay of Hansbreen a good study site.

Both glacial behavior and the propagation of sound are sensitive to temporal variability in thermohaline structure of water masses in the bay (Pętlicki et al., 2015; Glowacki et al., 2016). The calving activity of Hansbreen is largely controlled by melt-driven undercutting of the ice cliff (Pętlicki et al., 2015). The water temperature and salinity in the center of the bay ranged from -1.8 ∘C to more than 2.0 ∘C and from 30 PSU to almost 35 PSU during 2015 and 2016 (Moskalik et al., 2018). Significant wave height observed in the study site reached a maximum value of around 1.5 m over the period of August–November 2015 (Herman et al., 2019). A geomorphological map of the bay reveals complicated structures in the seabed created by dynamic glacial processes acting after the Little Ice Age, including terminal moraines, flat areas and iceberg-generated pits, to name a few (Ćwiąkała et al., 2018). The water depth along a transect parallel to the glacier terminus ranges from less than 20 to almost 90 m (see Fig. 5 in Moskalik et al., 2018).

The main dataset consists of more than a thousand subaerial calving events observed between 30 July and 15 September 2016, with three time-lapse cameras and two acoustic buoys deployed in the glacial bay (Fig. 1). At least 20 ice blocks calved each day. It was not always possible to unambiguously identify a calving event in both the image and acoustic datasets; the occurrence of more than one iceberg detachment between the two consecutive images resulted in ambiguity in the acoustic data. Moreover, dense fog, rain or otherwise unfavorable lighting conditions would at times obscure the terminus. From the total calving inventory, a subset of N=169 events were unambiguously matched and analyzed (Figs. 1 and 2). The observer present in the field throughout the data collection phase reported that no anthropogenic sound sources were active during the occurrence of these calving events.

Measurements of ocean temperature and salinity in the bay revealed upward-refracting sound speed profiles, with velocities changing from around 1440 m s-1 just below the surface to almost 1470 m s-1 close to the bottom (Fig. 2a–d). The sound speed gradient between the surface layer and deeper layers, which controls refraction and transmission loss, is driven by fresh meltwater and was clearly increasing during the study period. Moreover, significant differences in sound velocity profiles taken on the same day were also observed between different locations perpendicular and parallel to the glacier terminus, driven by a complex and three-dimensional distribution of the thermohaline field in the bay. The ocean depth between the locations of calving events and the two acoustic buoys varied from 10 m on underwater sills to more than 80 m in the western part of the bay near the terminus (Fig. 3). The bathymetry profiles were very different for the two buoy locations, with a more variable depth observed in the case of the buoy deployed further from the glacier cliff.

Images of the Hansbreen terminus were taken every 15 min from three locations (“Cam 1–3” in Fig. 1) continuously between 30 July and 15 September 2016 using Canon EOS 1100D cameras (4272 pixel×2848 pixel resolution and 18 mm focal length). The three cameras were not perfectly synchronized, which in fact enabled better separation of individual iceberg calving events occurring shortly after one other. Additionally, a GoPro Hero 3+ camera was placed closer to the terminus to take pictures of the narrow ice cliff segment (“GoPro” in Fig. 1). This camera took images at a much higher rate of 1 s-1 but was not always active during the deployment. Iceberg volume and drop height were estimated using images from Cam 1, which had the most perpendicular orientation to the glacier front of all the cameras. The irregular shape of the ice cliff provided registration features, which were identified in both Landsat 8 satellite images (with resolution of 15 m) and the camera images, enabling a precise localization of calving events.

Following Minowa et al. (2018), the volumes of the calved ice blocks are estimated from the area at the glacier terminus exposed by the calving event. Newly exposed areas are identified from differences between pairs of images taken by Cam 1 (see Sect. S1 in the Supplement for details). The newly exposed area in pixels squared, Aimg, is converted to its real value (in m2), Ac, using the formula 1Ac=Aimgd2F2, where d is the distance between the camera and drop location and F is the camera focal length. The camera was oriented roughly perpendicular with respect to the calving front (Fig. 1), but precise calculation of the exact angle was impossible due to the limited resolution of the satellite images and large variability in the terminus shape over the study period. Nevertheless, this uncertainty was included in the error analysis (see Sect. 4.3 for details). Guided by previous reports on iceberg dimensions observed in Svalbard (Dowdeswell and Forsberg, 1992), we assumed that the thickness of the calved iceberg is proportional to the square root of the newly exposed area. Then, the iceberg volume is given by 2V=CAc3/2, where C is a constant scaling factor, which is reported to be around 0.12 (Åström et al., 2014; Pętlicki and Kinnard, 2016). The drop height, h, is measured as a vertical distance between the sea surface and the midpoint of the falling ice block, converted from pixels to meters. Finally, the kinetic energy of the impacting ice block, Eimp, is given by 3Eimp=Mgh=Vρigh, where ρi is the ice density, set to be a constant 917 kg  m-3, g=9.81 m s-2 is the acceleration due to gravity and M is the iceberg mass. Equation (3) for Eimp is based on the assumption that there is no energy dissipated during the free fall of an iceberg. In reality, energy is dissipated though various physical mechanisms, such as friction between an ice block and glacier terminus, momentum transfer at the early stage of the water entry, drag during the immersion phase, and block disintegration, which can happen at different stages of calving. However, the details of these hydrodynamic processes lie beyond the scope of this work. Because they are not included, Eq. (3) provides an upper bound of the total amount of energy available for noise production during the block–water interaction.

The acoustic data were recorded continuously between 30 July and 15 September 2016 using two HTI-96-MIN omnidirectional hydrophones deployed at depths of 40 and 22 m, respectively, in front of Hansbreen (“A1” and “A2” in Fig. 1). The hydrophones have a sensitivity of -164 dB re 1 V µPa-1 and were sampled at a rate of 32 kHz at a resolution of 16 bit. A single mooring system consisted of an anchor, short line and acoustic buoy with a hydrophone, powered by D-cell lithium batteries. Acoustic data were stored on SD cards. The moorings were recovered in their entirety by divers. The horizontal distance between the moorings and locations of calving events ranged from 700 to 1500 m for the closer buoy and from 1800 to 2100 m for the more distant buoy.

The sound produced by calving events was identified manually, based on timing determined from the time-lapse cameras and deviations from median sound level at frequencies below 200 Hz (see Sect. S2 in the Supplement for details). Power spectral density estimates were calculated for each calving event using the Welch method with a 16 384-point fast Fourier transform, a Hamming window of the same size and a 50 % segment overlap to investigate the noise spectra (see Fig. 6). The acoustic energy of the block–water impact at the buoy, Eac,obs, was subsequently calculated by low-pass filtering the noise record at fc and then integrating the mean-square pressure, plow2 over the event duration: 4Eac,obs=4πρwc∫tstarttendplow2dt.

The sound speed, c, and water density, ρw, in Eq. (4) were set to 1450 m s-1 and 1025 kg m-3, respectively. The factor of 4π accounts for the surface area of a unit sphere, over which the noise signal must be integrated to obtain total noise energy in joules. The selection of the cutoff frequency of the filter (fc=100 Hz) is discussed in Sect. 4.2. The background noise energy, Eac,bckg, for each event was computed analogously using noise segments of the same length as the corresponding calving signal, recorded just before the ice block impact.

An overview of the temperature and salinity structure in the study site and its influence on the propagation of sound throughout the bay has been provided by Glowacki et al. (2016). In this study, temperature and salinity profiles were taken on 1, 12 and 30 August and 9 September 2016 with a SAIV SD208 CTD (conductivity, temperature and depth) probe at 11 points, located on transects perpendicular and parallel to the glacier terminus (red and blue dashed lines, respectively, in Fig. 1). Sound velocity was calculated from the CTD data according to the Chen and Millero formulae adopted by UNESCO (Chen and Millero, 1977).

Each calving event has its own, unique set of hydrographic and bathymetric data used for modeling sound propagation, determined in the following way. Firstly, a median sound speed profile was calculated from each set of profiles measured at the same day. Then, a closest median profile was assigned to each calving event according to the time of its occurrence. As a result, four consecutive median sound speed profiles were assigned to 32, 46, 61 and 30 calving events (see Fig. 2). Additional CTD casts were taken in 2017 after significant recession of Hansbreen. These profiles provided information on bottom depths in 11 additional positions located near the glacier terminus position from 2016, which are not covered by the bathymetry data (0.1 m resolution) collected during multibeam surveys (Fig. 1). We selected five bathymetry profiles, separately for two acoustic buoys, that lie along a straight line between the mooring location and CTD stations belonging to the transect that is closest to the ice cliff. Ocean depths in these sections were then interpolated into a 1 m grid using shape-preserving, piecewise cubic interpolation (Fritsch and Carlson, 1980; Fig. 3). Despite the fact that a high level of variability in the thermohaline structure is expected and there is a lack of detailed bathymetry data close to the glacier terminus, the uniquely assigned sound speed profile and interpolated bathymetry are the best available approximation of real conditions prevailing during the study period.

The impact energy model requires an estimate of the total sound energy radiated by a calving event, which can be calculated from 5Eac,imp=(Eac,obs-Eac,bckg)10-TLtot10, where TLtot=TLprop+TLrefl is total energy loss in decibels (Clay and Medwin, 1977), which includes both propagation loss computed from the Bellhop model and a contribution from energy reflected from the glacier terminus. The subtraction of Eac,bckg from the observed impact noise at the hydrophone, Eac,obs, removes background noise energy from the measurement. The factor containing TLtot transforms the corrected, observed energy into source energy at the impact location. A total loss of -10 dB, for example, corresponds to a decrease of 1 order of magnitude in received energy. Based on visual inspection of the scatterplot between Eimp and Eac,imp, we used a log-log transformation to improve linearity in this relationship. The same type of transformation was revealed by an application of the Box–Cox algorithm, which is often used to normalize regression variables (Box and Cox, 1964). The linear model of conversion between log-transformed energies is given by 6ln⁡E^ac,imp=a+bln⁡Eimp.

Having a=ln⁡η and knowing that bln⁡Eimp=ln⁡Eimpb, the power-law relationship has a final form given by 7E^ac,imp=ηEimpb.

Coefficients a and b could be easily derived from an ordinary least-squares linear regression model using log-transformed energies as variables. However, both Eimp and Eac,imp have associated uncertainties, which should be accounted for in the analysis. Therefore, to address this issue, we used the unified equations for slope, intercept and associated standard errors proposed in a model by York et al. (2004). This model belongs to the family of errors-in-variables regression models, which include all uncertainties and always give an answer that is symmetric for both choices of dependent and independent variables. Finally, to exclude outliers from the analysis, we identified all points for which uncertainty in acoustic energy calculated with Eq. (5) is not within 2 standard deviations of the modeled impact noise energy.

A total of 169 subaerial calving events were captured by time-lapse camera and unambiguously identified with acoustic events (see Sect. 2.2). Individual detachments were unevenly distributed along the active part of the Hansbreen terminus (Fig. 1). The distance to camera, drop height, exposed terminus area and estimated block volume of the calving inventory are summarized in Fig. 5.

The distance between Cam 1 and the locations of block–water impacts varies from 1700 to 2150 m, with an average of 1880 m (Fig. 5a). The drop height spans 8 to 32 m, with a mean value of h¯=18.3 m (Fig. 5b). The range of the exposed terminus is 125 to 5850 m2 of the ice cliff surface, with an average newly exposed area of 1590 m2 (Fig. 5c). Iceberg volumes were estimated from Ac using Eq. (2) and vary from 0.2×103 to 53.7×103 m3. The volume distribution is weighted toward smaller calving events, and approximately 90 % of the ice blocks have a volume of less than 20×103 m3 (Fig. 5d). This observation is consistent with previous reports on the power-law distribution of iceberg sizes in Svalbard (Chapuis and Tetzlaff, 2014), Alaska (Neuhaus et al., 2019), Greenland (Sulak et al., 2017) and Antarctica (Tournadre et al., 2016). A least-mean-squares error analysis of the power-law distribution of iceberg volumes was made using log-transformed variables. The best-fit decay exponent of 1.48 (Fig. 5e) found for the present dataset lies between the exponent of 1.69 for Kronebreen, Svalbard, reported by Chapuis and Tetzlaff (2014), and 0.85 for Perito Moreno Glacier, Patagonia, reported by Minowa et al. (2018). However, we note that some size ranges can be under- or overrepresented due to a limited number of unambiguously matched calving events (169).

Ice block volume versus drop height is shown in Fig. 5f. The highest iceberg volumes are observed for h within the range of 17 and 26 m, which corresponds well to the middle heights of the glacier terminus at the locations of calving events. Inspection of Fig. 5f shows that ice block volume is correlated with drop height; Pearson's correlation coefficient is found to be 0.47 and 0.55, respectively, for log-transformed and non-transformed variables. This is not altogether surprising because the largest blocks of ice cannot fall from the bottom of the terminus, whereas the smaller blocks of ice are not so constrained. The correlation between drop height and iceberg mass is a source of bias in the relationship between ice block volume and impact energy and must be accounted for when inverting acoustic recordings of impact noise for ice mass loss. This issue is discussed in detail in Sect. 4.5.

Figure 6 shows a comparison between power spectral density estimates for underwater noise from calving and background noise recorded by buoys A1 and A2. Spectrograms of the noise generated by a randomly selected calving event are shown in Fig. 6a and b. The computed difference in time of arrival between the two receivers was subtracted from the more distant receiver for better juxtaposition. The two primary sources of sound in the spectrograms are ice melt noise and the underwater noise of calving.

The signal of ice melt, driven by impulsive bubble release (Urick, 1971), is most pronounced between 1 and 3 kHz and corresponds well to the spectral bands reported in previous studies (Deane et al., 2014; Pettit et al., 2015). This signal remains stable during the short observation period. The underwater noise of calving is a by-product of the interaction of the falling iceberg with the ocean. The noise is evident from 2 to 8 s in the recording at frequencies below 1 kHz. The acoustic intensity varies in both time and frequency. This variability is almost certainly driven by different noise production mechanisms active at different phases of the calving event (see the high variability in power level between 2 and 4 s, for example). As pointed out by Bartholomaus et al. (2012), low-frequency seismic signals from the impact of ice blocks on the sea surface are generated by three major mechanisms: (1) the transfer of momentum from the falling block to seawater, (2) iceberg deceleration due to buoyancy, and (3) the collapse of an underwater air cavity and subsequent emergence of Worthington jets (e.g., Gekle and Gordillo, 2010). The last mechanism is only possible during total submergence of the ice block, the occurrence of which depends mainly on iceberg dimensions and drop height. Therefore, some calving events may not result in the creation of an air cavity. Moreover, falling icebergs are often fragmented or impact the water at various angles, which certainly modifies all three mechanisms of noise production. The influence of calving style on sound emission lies beyond the scope of this work but is likely a significant factor in the variability in sound generation by blocks of similar mass and drop height, as discussed in Sect. 4.4.

The unique patterns in the time and frequency distribution of calving noise potentially contain information about the details of the calving event. However, attention here is restricted to a single number, which is the time and frequency integrated energy in the sound field generated by the iceberg impact. Calculation of this number requires selection of the start and stop times of the impact noise and the frequency band over which the noise exceeds background sound levels. The significant increase in noise power accompanying calving allows easy identification of event start and stop times, and these have been selected manually for each event analyzed (see Sect. S2 in the Supplement). Figure 6c and d show a 6 s average of noise power spectral density for a calving signal (blue) and background noise recorded just before the event (red). There is a difference between the calving and background noise levels at frequencies up to 700 and 400 Hz for buoys A1 and A2, respectively. The maximum increase in received noise power from calving is approximately 40 dB for both buoys, which corresponds to a factor of 10 000 in acoustic power. The results in Fig. 6 show that the appropriate band of frequencies to consider for calving impact noise ends at around 1 kHz. However, an upper frequency limit of 100 Hz was applied in further analysis to yield the highest correlation between the impact energy and the received acoustic energy.

The variability in calving noise power across the entire dataset is shown in Fig. 6e and f. The normalized power spectral densities of calving events and background noise are plotted as blue and red dots, respectively. A normalization factor is chosen for each calving event and taken to be the highest power level in decibels during the event. The same normalization factor is used for both calving and background noise. Calving signatures are clearly distinguishable from the background noise across the entire dataset. However, the calving noise power is noticeably more variable at receiver A2 than A1. There are two possible reasons for this discrepancy. Firstly, spatial dependency of the thermohaline structure is expected to be significant along the longer propagation path to A2. Secondly, the signal-to-noise ratio for receiver A2 is lower and more variable than at A1, as a result of the shallower depth of the hydrophone (22 m at A2 versus 40 m at A1) and greater exposure to noise coming from outside the bay (see Sect. S4c in the Supplement for more details). The increased scatter in calving noise observed at location A2 resulted in a decrease in correlation between total impact energy and impact noise (see Table S1 in the Supplement), and data from this buoy are not considered further.

There are two sources of uncertainty for block–water impact energy and impact noise energy: measurement error and uncertainty in the state of the changeable environment, which is impossible to characterize completely. Estimates of these uncertainties can be made for the various stages of the analysis connecting impact noise to ice mass loss, and these are discussed below.

Estimating calving ice mass flux from calving noise is based on the idea that these two quantities are correlated. Figure 7 shows a scatterplot of impact noise, Eac,imp, against impact kinetic energy, Eimp, for the entire dataset. The dashed black line shows the result of a regression analysis of the power-law relationship shown in the figure legend. The acoustic energy generated by a calving event was calculated from the acoustic pressure time series using Eqs. (4) and (5) with manual selection of integration time (see Sect. S2 in the Supplement) and after low-pass filtering at a cutoff frequency of 100 Hz (see Sect. 4.2). The kinetic energies of the falling ice blocks were derived from Eq. (3) using their masses and drop heights estimated from the camera data (see Sect. 4.1).

The range of energy estimates is large, roughly 2.5 orders of magnitude for both, and there is clearly a strong correlation between the energies across their entire range. The regression coefficient r=0.76 was found between the log-transformed variables for p<0.0001. If uncorrected calving noise energy and signal duration are used instead of Eac,imp, the correlation drops to 0.71 and 0.61, respectively (Table S1 in the Supplement). After removing two outliers and applying an error-in-variables linear regression (see Sect. 3.5 for details), the best functional relationship between acoustic energy and impact energy was found to be a power-law relationship given by Eq. (7), where η=8×10-7±60 % and b=0.92±3 %, respectively, for the multiplication factor and exponent of the power law. For completeness, this analysis was repeated, including the two identified outliers, and the results are shown in Fig. S6 (Supplement). Glowacki et al. (2015) previously reported η=5.16×10-10 and b=1, which gives an impact energy that is 2.5 orders of magnitude higher in comparison to the results presented here (see Fig. S7 in Supplement). This discrepancy is due to the overly simplified propagation geometry assumed in the earlier study – simple cylindrical spreading loss and no sound reflection from ice terminus – which resulted in an underestimate of the impact noise energy.

The multiplication factor η can be thought of as a conversion efficiency of kinetic energy of a falling iceberg to impact noise energy. The small value of η shows that only a tiny fraction of the ice block energy is transformed into underwater sound, which then propagates from the point of impact to the acoustic receiver. A low conversion efficiency is consistent with observations reported for other physical mechanisms of underwater noise generation. For example, only ∼10-8 of the energy dissipated by a breaking surface wave on the ocean is radiated as sound (Loewen and Melville, 1991). Similarly, the conversion efficiency of the impact energy of a 1–5 mm scale raindrop falling on the sea surface to underwater impact noise is in the range 10-9 to 10-8 (see Eq. 4.6 in Guo and Ffowcs Williams, 1991, and Gunn and Kinzer, 1949).

Despite a strong correlation between impact energy and impact noise, there is also a significant scatter in impact noise energy (roughly a factor of 10) for a given value of kinetic energy. This spread in values can be only partly explained by errors in the energy estimates, which are indicated by blue whiskers in the Fig. 7. The scatter is presumably caused by differences in noise generation between individual calving events. The consequence is that estimating the impact energy of an individual calving event from the total noise energy it radiates is accompanied with significant uncertainty. However, because of the overall strong correlation between noise and impact kinetic energy, it is possible to predict the total impact energy summed over a finite number of calving events, provided the inventory is large enough. The uncertainty in individual events tends to average out if enough events are considered, as discussed in Sect. 4.5.

Figure 7 and Eq. (7) show that the relationship between iceberg impact energy and calving noise can be modeled robustly with a power-law relationship, providing a means of estimating impact energy from calving noise. Although there is significant variability in doing this on an event-by-event basis, low-error estimates of cumulative impact energy can be made using Eq. (7) if enough events are added together. Once found, the cumulative impact energy can be converted into an estimate of iceberg calving flux as follows.

The cumulative modeled ice mass loss from N observed calving events is related to the cumulative impact energy, as inferred from the acoustic signal, by 8g∑j=1NhjM^j=∑j=1NE^imp,j, where g is the acceleration due to gravity, hj is the height of the center of mass of the jth iceberg before separation from the glacier terminus, M^j is the mass of jth iceberg determined from its underwater impact noise and E^imp,j is the kinetic energy of impact of the jth iceberg. The cumulative ice mass lost through calving would be trivial to compute from Eq. (8) if the mean iceberg drop height were independent of the iceberg mass, but this is not the case (see Fig. 5f). Icebergs that extend a significant fraction of the exposed terminus height have a minimum drop height that is larger than the minimum drop height possible for smaller icebergs. For this (and possibly other) reasons there is a correlation between iceberg drop height and iceberg mass, the consequence of which is that hj cannot be moved outside the sum on the left-hand side of Eq. (8). The correlation is dealt with by introducing the mass-weighted drop height: 9h^=∑j=1NhjM^j/∑j=1NM^j.

It follows immediately from Eqs. (8) and (9) that the cumulative mass sum is given by 10∑j=1NM^j=1gh^∑j=1NE^imp,j, which provides a means of computing the calving flux, since the kinetic energy of iceberg impact can be estimated from its underwater noise using Eq. (7).

We are left with the problem of computing h^. To address this issue, a new variable α^ is defined by 11α^=h^h¯-1=∑j=1NhjM^j/1N∑j=1Nhj∑j=1NM^j, where h¯ is an observed, average drop height. Let us now assume that, for sufficiently large N, α^ can be approximated by 12α^≈∑j=1NhjMj/1N∑j=1Nhj∑j=1NMj.

The right-hand side of Eq. (12) is in terms of iceberg mass inferred from the camera observations, providing a means of computing the mass-weighted drop height, h^=α^h¯, on a glacier-by-glacier basis. The constant α^ and resulting mass-weighted average drop height are estimated to be 1.13 and 20.7 m for Hansbreen.

Equation (10) for the cumulative calving mass flux contains significant uncertainty when N is small because of the large scatter in the total underwater sound energy generated by calving events with similar impact energies (see Fig. 7), but the uncertainty reduces as N increases. How large must N be to achieve a desired degree of uncertainty? To answer this question, a Monte Carlo simulation of cumulative ice mass loss was performed using n calving events randomly selected (with replacement) from the entire inventory of calving observations (for which N=169). This selection is repeated ψmax times (ψ={1,…,ψmax}) for each n={1,…,nmax}, noting that the total number of possible sets of calving events (and associated cumulative kinetic energies and masses) is given by 13C=n+N-1n=n+N-1!n!N-1!.

From Eq. (10), the cumulative mass sum for a given number of randomly selected calving events n and iteration ψ is 14∑i=1nM^i(ψ)=1gh^∑i=1nE^imp,i(ψ), where h^ is calculated from Eqs. (11) and (12) using the N=169 observed calving events. The modeled mass M^i(ψ) in Eq. (14) corresponds to M^j in Eq. (10), where 1≤j≤169. The inferred, cumulative ice mass normalized by the cumulative ice mass measured with the camera is then given by 15βn(ψ)=∑i=1nM^i(ψ)/∑i=1nMi(ψ), where β, for a specified n and averaged over ψmax iterations, can be expressed as 16β¯n=1ψmax∑ψ=1ψmax∑i=1nM^i(ψ)/∑i=1nMi(ψ).

We set nmax to 1000 and ψmax to 10 000 to determine the statistical properties of β over a broad range of sample sizes. We note that the probability of randomly obtaining the same set of calving events is vanishingly small for the chosen ψmax (C≫10 000 for n≥3; see Eq. 13).

Figure 8 shows the mean, β¯n, and standard deviation, βn,std, of the statistical distributions of βn(ψ) computed from the Monte Carlo simulation. The correct and unbiased estimate of the mean ice mass flux ratio is β¯n=1, which is indeed the asymptotic value reached for large n. As expected, the temporal resolution of the acoustic technique increases with increasing calving activity. The estimated cumulative mass is within 20 % and 10 % of the expected value when integrating over 40 and 135 ice blocks, respectively (Fig. 8). The number of calving events required for a specified level of uncertainty translates into an observational timescale that must be met depending on calving rate. For example, at Hansbreen an uncertainty in ice mass flux of about 20 % is expected when integrating over 2 d of acoustic measurements, corresponding to a calving rate of 20 icebergs per day. The time interval required for a specified level of uncertainty will vary between glaciers and over time. For example, some glaciers calve more than 10 ice blocks hourly (e.g., How et al., 2019), leading to a relatively short time interval requirement.

We collected the underwater noise data continuously with two light moorings powered by D-cell lithium batteries. Moorings were later recovered by divers with inflatable lift bags. The relatively shallow water at locations A1 and A2 (<45 m) made diving possible. Acoustic monitoring of calving fluxes in deeper study sites will require the use of acoustic releasers in order to retrieve the recording equipment remotely from a boat or ship. In fact, an example of calving signals recorded at location A2 shows that very shallow water prevents effective transmission of the calving noise and decreases the signal-to-noise ratio at low frequencies (see Fig. 6 and Sect. S4c in the Supplement). Further issues for consideration are data storage and clock drift. Acoustic data can be stored on SD cards, which are power-efficient and capacious. The internal clock drift during the recording period is expected to be not greater than 30 s per month, given that quartz oscillators that hold stability of at least ±10 ppm are typically used in data acquisition systems. Low-cost acoustic recorders can be added to the existing monitoring programs, where heavy moorings are used to study processes taking place at the ice–ocean interface, including water circulation, heat exchange or sediment transport, for example (e.g., Straneo et al., 2019). Moreover, data collection can be made real time with a cabled or wireless link to shore, but we have not tested this possibility yet. The problem remains with automatic extraction of the calving signal from long-term, continuous acoustic recordings. Calving events are clearly distinguishable from the noise of ice melting in spectrograms of the acoustic record at frequencies below 1 kHz (see Fig. 6a), which is promising in terms of automatic event detection. However, it should be borne in mind that there are other, low-frequency sound sources active in a glacial bay. For example, calving or the disintegration of bigger icebergs could be mistaken with glacier calving events (see Richardson et al., 2010). Moreover, distinguishing between subaerial and submarine calving can be difficult, as found during passive seismic surveys (see Köhler et al., 2019). These issues await further investigation.

In addition to a minimum sample size for a specified error requirement (see Sect. 4.5), there are five other parameters that must be known to compute reliable estimates of ice mass flux from calving noise: the mass-weighted, average iceberg drop height, h^, the conversion coefficient from the newly exposed area to block volume, C, the conversion efficiency from impact to acoustic energy, η, the power-law coefficient, b, and the transmission loss from the glacier terminus to the hydrophone position, TLprop. Errors in these parameters are important because they affect the uncertainty and temporal resolution of the acoustic measurements of calving fluxes (see Figs. 7 and 8).

The problem of how site-specific these parameters are lies beyond the scope of this work, and similar studies should be performed for different tidewater glaciers to obtain quantitative answers. Nevertheless, we briefly discuss here some techniques for measuring or modeling these parameters along with environmental factors driving variability between sites. Noise energy loss is usually calculated using a standard propagation model, such as the Bellhop model used here. Propagation models require sound speed and bathymetry profiles as inputs, making hydrographic and CTD surveys an essential component of the acoustic measurements of calving fluxes. Although the thermohaline structure of a glacial bay is complex and three-dimensional (e.g., Jackson et al., 2014), patterns of temperature and salinity that are sufficiently characteristic of prevailing conditions in the bay can be identified from limited field measurements and used for propagation model inputs (Glowacki et al., 2016). Moreover, it might be possible to use signal duration instead of the corrected noise energy to estimate calving fluxes (see Sect. S5 in the Supplement) as an alternative when no information is given on bathymetry and/or sound speed profiles. We anticipate that there is a high uncertainty associated with the conversion coefficient from the newly exposed area to block volume, which likely varies between glaciers characterized by different surface velocity, thermal regime, hydrology, terminus height, etc. This parameter can be determined more accurately for a specific glacier using short-term lidar measurements or image analysis, e.g., structure from motion or stereo photography. These techniques can also provide an estimate of the mass-weighted, average drop height. The value of h^ is expected to be close to one-half of the average terminus height (in its active part) because the size of ice blocks breaking off from the top or bottom part of the ice cliff is limited (see Fig. 5f). We hypothesize that the remaining two parameters, the energy conversion efficiency and power-law coefficient, are likely stable between glaciers of similar geometry and flow dynamics.