# Measuring the thickness and Young's modulus of the ice pack with DAS, a test case on a frozen mountain lake

Destin Nziengui-Bâ, Olivier Coutant, Ludovic Moreau, Pierre Boué

► To cite this version:

Destin Nziengui-Bâ, Olivier Coutant, Ludovic Moreau, Pierre Boué. Measuring the thickness and Young's modulus of the ice pack with DAS, a test case on a frozen mountain lake. Geophysical Journal International, 2023, 233 (2), pp.1166-1177. 10.1093/gji/ggac504. hal-04929073

HAL Id: hal-04929073

https://hal.science/hal-04929073v1

Submitted on 6 Feb 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Measuring the thickness and Young's modulus of the ice pack with DAS, a test case on a frozen mountain lake

Destin Nziengui-Bâ,<sup>1,2</sup> Olivier Coutant,<sup>1</sup> Ludovic Moreau<sup>1,●</sup> and Pierre Boué<sup>1</sup>

Accepted 2022 December 21. Received 2022 December 11; in original form 2022 July 11

## SUMMARY

We explored the potential of fibre optics coupled with distributed acoustic sensing (DAS) to measure the thickness and Young's modulus of an ice layer, using the properties of guided seismic waves. During two winter seasons (2020 and 2021), an optical fibre was deployed over one of the frozen Roberts Mountain lakes (at 2400 m a.s.l) near Grenoble (France) and we measured both the continuous ambient seismic noise as well as signals generated by active sources (hammer), with a DAS interrogator. Following a Bayesian scheme, we inverted the dispersion curves of longitudinal and flexural guided waves retrieved from the analysis of active shot gathers and obtained Young's modulus $E = 3.4 \pm 0.1$ GPa and ice thickness $h = 47 \pm 1$ cm from the second-year data. The ice thickness was consistent with field measurements. Field observations of porous and/or fracture ice may explain the relatively low effective Young's modulus (relative to pure ice), which may also be affected by a snow layer not included in the model. The drastic improvements in the inversion results between the two years are related to better coupling conditions (drone deployment before early freezing), more appropriate acquisition parameters (2 m gauge length), and the upper snow layers (less thick and less heterogeneous in the second year). Moreover, we were able to use the non-dispersive low-frequency noise associated with gravity waves to estimate the lake depth $H = 5$ m which is compatible with independent observations. The use of DAS to record guided seismic waves could then appear as a relevant tool for monitoring environments like floating ice shelves and sea ice.

**Key words:** Inverse theory; Glaciology; Seismology.

**Key words:**; Cryosphere; Distributed acoustic sensing (DAS).

## 1 INTRODUCTION

In the high-latitude and high-altitude regions of the globe, ice is an effective and economical means of supporting loads for construction, resource extraction, and transportation (Masterson 2009). Crossing lakes and rivers for northern communities have been achieved using the naturally formed ice cover in winter for centuries. Ice has been used for marine traffic and freshwater transport (Prowse et al. 2009). It has also been used to support heavy, stationary and long-term loads such as oil and gas drilling rigs, and to support the installation of offshore pipelines and related facilities. Baudais et al. (1976) and Masterson et al. (1986) give good examples of offshore drilling rigs for oil and gas exploration in the Canadian Artic Islands during the 1970s and 1980s. Hydraulic and

ocean engineering must also address ice-related problems in the winter, such as how to avoid ice damage to hydraulic constructions in open rivers, lakes, and marine areas, where oil drilling platforms, lighthouses, and sea-crossing bridges must have anti-ice designs (Mulherin & Haehnel 2003; Palmer 2013). Hence, the demand for human activities that generate energy and produce raw materials requires a better understanding of the mechanical properties of ice. Furthermore, in the alarming context of global warming, understanding sea ice decline is one of the main challenges of modern climatology (Stroeve et al. 2007; Rampal et al. 2011). In this matter, a more refined description of the processes involved in sea ice evolution requires models based on sea ice properties. Regarding all the above-mentioned cases, we can observe that systematic knowledge of ice's physical and mechanical properties is a fundamental

<footnote n="1"><sup>1</sup>Institut des Sciences de la Terre, Université Grenoble Alpes, Grenoble, 38400 Saint-Martin-d'Hères, France. E-mail: destin.nziengui-ba@univ-grenoble-alpes.fr</footnote>

<footnote n="2"><sup>2</sup>FEBUS-Optics, 64000 Pau, France</footnote>
requirement for the solution of a wide range of problems in earth sciences and engineering.

Among the various methods used to investigate ice shelf properties, recording and analysing seismic waves that are trapped in the ice layer offers the possibility of inverting guided waves to retrieve first-order ice properties (Anderson 1958; Hunkins 1960; Yang & Giellis 1994; Stein et al. 1998; Marsan et al. 2012; Moreau et al. 2020a; Romeyn et al. 2021). These experiments were conducted to better understand both the propagation modes and the energy distribution among these modes, in order to infer ice properties. However, as Moreau et al. (2020b) point out, deploying a network of dozens of geophones may not always be feasible due to hostile or dangerous conditions. They propose to use a sophisticated analysis but on a limited number of sensors. Alternatively, Romeyn et al. (2021) recently proposed to use a simple, and inexpensive microphone, located above the ice sheet or along the shoreline, to infer the ice thickness from the analysis of air-coupled flexural waves. They also demonstrated the use of air-coupled flexural waves for ice thickness monitoring using an active source acquisition scheme, and suggest that icequakes produced by natural cracking can excite air-coupled flexural waves for freshwater lake ice, which raises the potential that passive recording of air-coupled flexural waves may also be used to infer ice properties.

The rapid development of optical fibres coupled to distributed acoustic sensing (DAS) interrogators allows the replacement of geophones with a cheap and light solution while still using seismic monitoring techniques. DAS repurposes optical fibres as multi-channel seismic arrays, and in contrast to conventional arrays that consist of spatially discrete electronic sensors, a DAS system utilizes a single optoelectronic interrogator unit that can sample tens of kilometres of optical fibre at submetre channel spacing (Parker et al. 2014). By providing an affordable and scalable way to deploy large aperture, continuously running and dense seismic arrays in locations difficult for traditional seismic instruments (e.g. urban areas, oceans and planets), DAS opens up unprecedented opportunities for a wide range of seismological applications (Zhan 2020). Among the various applications, we can cite Vertical Seismic Profiling (Mateeva et al. 2014; Wu et al. 2017), earthquake ground motion response (Lindsey et al. 2017; Jousset et al. 2018; Wang et al. 2018), structure imaging and monitoring (Dou et al. 2017; Zeng et al. 2017; Ajo-Franklin et al. 2019), site effect assessment (Ajo-Franklin et al. 2019; Spica et al. 2020) and ocean-bottom exploration (Lindsey et al. 2019; Sladen et al. 2019). Although there are still challenges in storing and sharing the huge amount of DAS data and understanding DAS instrument response (Lindsey et al. 2020; Paitz et al. 2021), DAS networks will inevitably complement classic seismic sensor deployment.

Interest in cryoseismology has surged since the early 2000s (Podolskiy & Walter 2016; Lipovsky 2018; Olinger et al. 2022). The remoteness and harsh environment in glacial regions make seismic networks particularly expensive. Therefore, continuously operating dense seismic networks are rare. Nonetheless, the density, channel spacing and broadband response of DAS make it a tantalizing way to study glaciers at different scales to explore such processes as iceberg calving, basal sliding, crevassing and subglacial hydrology. Optical fibres were tested as seismic sensors on a frozen lake (Lake Mendota in Madison, WI, USA) by Castongia et al. (2017) to compare the characteristics of a dense triangular seismic array with geophones. They found DAS to be qualitatively equally or better suited (in certain situations) than geophones. Experiments more specifically dedicated to cryosphere studies were proposed by Walter et al. (2020) on a Glacier in the Swiss Alps, Hudson et al. (2021)

in Antarctica or Klaasen et al. (2021) in alpine, volcano-glacial environments. By using a fibre cable layout formed of an equilateral triangle with 220-m-long sides, Walter et al. (2020) show that DAS offers a vast potential for monitoring glacier dynamics and Alpine mass movements. They were able to locate a series of glacier stick-slip events and reveal seismic phases from which the thickness and material properties of the glacier and its bed can be derived. Alternatively, Hudson et al. (2021) investigated the potential of DAS for natural microseismicity studies in Antarctica. They show that DAS alone is relatively poor at detecting and locating seismicity compared to geophone observations when deployed with an aperture less than the depth of the earthquake hypocentres being studied. However, DAS can outperform geophones in other analyses of natural microseismic sources including source mechanism inversion and measuring anisotropic shear-wave splitting parameters. In terms of the experimental setup, Klaasen et al. (2021) give some important lessons learned from the deployment of the fibre in challenging cryospheric environments. They have shown that optical fibres have the potential to reveal previously undiscovered classes of volcanic tremors in glacial environments.

For challenging and hazardous environments like frozen lakes, thin and unsafe ice shelves, or crevassed glaciers, we can take advantage of a unique property of the optical fibre cable: it is light and in many cases, it could be deployed from the air using a drone. In this study, we evaluate the feasibility of deploying optical fibres with a DAS interrogator on a frozen mountain lake, from both logistical and methodological points of view. The methodological objective was to test the ability of the fibre optic sensor to record seismic waves that are trapped inside the lake ice cover and subsequently use these guided waves to infer lake ice properties. Two short successive experiments were conducted during the 2020 and 2021 winters at a frozen lake in Chamrousse (French Alps), with quite different coupling conditions. In the first part, we detail the two experimental setups. We then apply methods similar to Moreau et al. (2020a) and Serripierrì et al. (2022) for recovering the average ice thickness and effective Young's modulus of lake ice by inverting the dispersion curves of guided seismic waves. Besides we used the low-frequency noise associated with gravity waves to measure the lake depth.

## 2 MATERIAL

The study was conducted at Robert lakes (45.13°N, 5.91°E). These are four small, very close, mountain lakes located in Chamrousse (French Alps). They have a total surface area of $2.8 \times 10^5$ m² and are located at 2000 m above sea level. We selected for our study lake III (Fig. 1a) because of its proximity to an electrified shelter. The maximum depth of this lake is about 6 m and its surface area is $3.65 \times 10^4$ m². It is easily reachable using the ski resort facilities of Chamrousse and is usually frozen from December to April. Average ice thickness ranges from 10 cm in early December to 1.5 m in early March.

### 2.1 Lake ice cover

Lakes in the mid to higher latitudes cold regions of the Northern Hemisphere are characterized by seasonal covers of freshwater ice. The freezing of lake water generally leads to the formation of an ice sheet with a distinctive stratigraphic and crystalline structure. Michel & Ramseier (1971) give an exhaustive classification of river and lake ice. In most lakes, a typical ice sheet can be divided into two major layers: snow-ice or white ice (upper layers) that directly
<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) Aerial view of the Robert lakes (lake III) in the French Alps and fibre optic cables location. The orange line corresponds to the 2020 season and the yellow line to the 2021 season. The two short red lines represent the portion of the cable considered for DAS data analysis (active part of the fibre). Along each fibre cable line, the green dot highlights the location of the source for the active seismic experiment. The related shot gathers are highlighted in Fig. 5. The black square represents the location of the DAS interrogator (inside the electrified shelter). (b) Picture of the fibre optic cable above the lake surface which was starting to freeze (2021 season). The three black arrows highlight the fibre optic cable (Photo by Eric Larose).</figcaption></figure>

overlies clear ice or *transparent ice* (lower layers) which is formed solely from the freezing of the lake water. Snow-ice is formed by the freezing of water-saturated snow. Ohata *et al.* (2016) (Figs 4 and 5) show examples of ice cores extracted from lake Abashari in Japan. They highlight two main ice layers: a whitish one (*snow-ice*) overlying a transparent one (*clear-ice* or *congelation ice*). Snow-ice is almost composed of thinner sublayers with different characteristics, and its occupancy percentage with respect to the total ice thickness is lake-dependent. Field observations at Robert Lakes have shown that the lake ice cover has almost the same characteristics except for that snow ice layers exhibit at some locations more complex transitions, with additional *hard ice* trapped between two different snow layers (Fig. 2b).

The mechanical properties of lake ice depend largely on its struc-
tural state, which is not necessarily constant with depth and may
change drastically as the ice sheet ages (Michel & Ramseier 1971),
Gow & Langston (1977). As such, lake ice is a heterogeneous or
composite material and its overall mechanical properties depend not
only on the properties of its constituents but also on the architecture
of its microstructure (i.e. the shape, size, orientation and arrange-
ment geometries of its constituents). To maximise guided wave
propagation, we targeted the *clear-ice* layer (quasi-homogeneous
and transparent layer, with macrograin) with sledgehammer blows.
We will use these guided seismic waves to infer Young's mod-
ulus and the thickness of this layer. Given the composite nature
of the ice layer, we will measure an *effective* Young's modulus
which is expected to be lower than that of pure ice. Indeed, seis-
mic waves probe the macroscopic scale effective stiffness of the ice
layer which is expected to be weaker than the small-scale speci-
mens observed with ultrasonic waves (Sinha 1989; Moreau *et al.*
2017).

## 2.2 Fibre deployment and recording

2.2.1 DAS systems: the importance of coupling

In contrast with localized buried geophones, DAS sensors are dis-
tributed along the optical fibre and rely on its coupling to the ground,
and more precisely on the friction, to record the strain along the fibre
direction. Ideally, with good coupling, both accurate amplitude and

phase information of the seismic wavefield can be recovered from
surface DAS deployments (Harmon *et al.* 2022). In the majority of
DAS surface deployments, fibres are buried in horizontal trenches or
pipes for protection (Jousset *et al.* 2018). Buried fibres in pipes have
weaker mechanical coupling to the ground. DAS strain recordings
of seismic signals from an experiment with sandbags placed every
9 m over the length of a 142.5 m fibre, with a gauge length of 10 m
compare favourably with multichannel geophone data (Spikes *et al.*
2019). Harmon *et al.* (2022) tested four different coupling strategies
for a DAS system deployed in a grassy field, including uncoupled,
pinned under tension to the ground, weighted down by carpeting,
and weighted down by a sandbag. They found that a completely
uncoupled fibre is capable of recording seismic energy up to 10 m
away from the source, while the pinned and weighted fibre record
signals over several tens of metres. Further testing of alternative
coupling methods in a variety of substrates will also allow for the
development of optimal deployment strategies.

Two advantages of fibre deployment over snow or ice are that we
are not subject to rodents attack, and the snow cover can provide
additional protection from wind and temperature fluctuations. How-
ever, large-magnitude deformation in ice can be a problem since a
fibre cannot statically elongate by more than a few percent of its
length.

2.2.2 Deployment and acquisition

We ran two short successive experiments during 2020 and 2021 win-
ters with quite different coupling conditions (detailed in campaign-
specific sections). For both experiments, we used a simple fibre
optic cable tightly coupled to a single-mode optical fibre of 9/125
µm. This cable is reinforced with Kevlar. The seismic datasets pre-
sented in this study were recorded on 04 March 2020 (Campaign
2020) and 03 February 2021 (Campaign 2021). The DAS inter-
rogator used for these experiments is a commercial A1-R provided
by FEBUS-Optics company. It was installed inside an electrified
shelter belonging to the ski resort and close to the lake of interest.

For both experiments, we recorded active and passive sources.
For active sources, we used a steel crowbar and hit the solid colum-
nar ice layer (clear-ice) below the fresh snow and snow-ice lay-
ers (explained in Section 2.1). Passive sources consisted of 1-hr
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Structure of the lake surface. (a) Lake surface cover during Campaign 2020. (b) Stratigraphic layers of the lake surface for the two campaigns. The *fresh snow/granular ice* interface is shown as the reference level. The yellow cable is the optical fibre. The targeted layer (columnar ice) is highlighted by red contours.</figcaption></figure>

recordings while walking along the fibre, or just recording ambi-
ent wavefield. During the second experiment, a strong wind was
observed which probably excited the lake surface and generated a
low-frequency wave train. Additional parameters are detailed in the
below campaign-specific sections.

## 2.3 Campaign 2020

A first exploratory attempt took place in February 2020, when the
lake was frozen and already covered by approximately 1 m of fresh
snow. The fibre cable was buried in the fresh snow, in a narrow trench
dug by hand along an ‘L-shaped’ array of approximately 200 m long
(Fig. 1a). The longest side was oriented in the east/west direction
and was about 142 m whereas the other quasi-perpendicular portion
was 48 m long (Fig. 1a). This part of the fibre was considered as the
‘active’ section of the fibre. A remaining portion of 1000 m of cable
was rolled on a cable reel and was used for background coherent
noise removal. DAS strain rate measurements were recorded for
one hour with a channel spacing of 0.8 m, a 10 m gauge length and
500 Hz of sampling frequency.

An ice core was extracted from the middle of the lake to examine
the vertical structure of the ice layers. Fig. 2b shows a profile of
the lake surface based on observations in the field. Just below fresh
snow, lake ice consists of a granular ice layer developing into a
columnar ice layer through transition layers made of *hard ice*, and
*water saturated snow-ice*. Here, we refer to *snow-ice*, the ice layers
just above the solid columnar ice layer ( granular ice, hard ice and
water-saturated snow-ice). The thickness of the columnar ice layer
(about 1 m ) has been provided by the local ice diving company
*Divextreme*. As we can see in Figs 2(a) and (b), the fibre was

placed just at the bottom of the fresh snow layer. Note that the
fibre was not perfectly coupled with the target ice layer, which
can significantly reduce the signal-to-noise ratio of the recordings.
Moreover, for active seismic, we hit the solid columnar ice layer.
This was to maximise the propagation of guided seismic waves in
the targeted ice layer, considering the complex layering of the lake
surface.

## 2.4 Campaign 2021

The second installation was achieved in mid-December 2020 in
the early freezing stage of the lake. The objective was to trap the
fibre inside the primary ice sheet to improve the coupling and then
to repeat measurements at different stages of freezing throughout
winter. Unfortunately, the ski resort was quickly locked down for
sanitary reasons and only one measurement has been performed.
At this early stage of freezing and due to the danger (not thick
enough ice), we decided to deploy the fibre across the lake using
a drone. We tested the deployment before on a wide lawn using
a nylon fishing cable of 0.5 mm diameter. The drone flew across
the lake with a light nylon cable that was then used to pull the
fibre between two fixed positions. We observed that within 3 d, the
fibre had been completely trapped in the primary ice layer. The
*active* part of the fibre was a quasi-linear portion of 320 m in the
east/west direction. A slight deviation is observed westward on the
lake shoreline (Fig. 1a). As for Campaign 2020, a profile of the lake
surface was achieved in the middle of the lake (Fig. 2b). Lake ice
consists of a columnar ice layer overlain by a granular ice layer (S2
type, according to the classification of Michel & Ramseier 1971).
We set the recording parameters to 0.1 m for the channel spacing, 2 m for the gauge length and 2000 Hz for the sampling frequency.

# 3 METHODOLOGY

## 3.1 Seismic waves in a floating ice layer

A layer of ice floating on water acts like a waveguide for seismic waves (Moreau et al. 2020a), and as such, it supports multimodal wave propagation. In the following, we use the terminology introduced in Moreau et al. (2020a) to describe the seismic wavefield in a floating ice layer. Three fundamental modes are expected to exist: the fundamental quasi-symmetric mode (QS₀), which produces mainly longitudinal motion (corresponds to compressional waves travelling in the ice plate); the quasi-Scholte mode (QS), which produces mainly flexural motion (flexural waves or bending waves); the fundamental Shear-Horizontal mode (SH₀), which produces shear horizontal motion (horizontally polarized shear waves). Longitudinal and shear waves are non-dispersive, unlike flexural waves which are dispersive, with higher frequencies travelling faster.

In our experimental setup, we study guided waves that propagate along the fibre direction and we recall that DAS only records the axial deformation of the fibre. We then expect to record the elongation due to longitudinal (QS₀) waves, and the elongation associated with flexural (QS) waves that produces elliptical particle motion (like Rayleigh waves). The sensitivity of the fibre to QS waves is equivalent to the sensitivity of a sea floor fibre that records the deformation produced by gravity waves in shallow water as observed by Sladen et al. (2019). On the other hand, the fundamental (SH₀) horizontal shear mode produces shear strain perpendicular to the fibre that the DAS should not record.

An important property of guided wave propagation is the relationship between the dispersion curves of the guided modes and the mechanical properties and thickness of the ice layer. For a uniform isotropic ice plate floating on a finite water column of depth H, the wavenumbers of longitudinal and flexural waves are obtained by solving the equations provided by Stein et al. (1998) and Romeyn et al. (2021):

$$k_{QS0} = \omega \sqrt{\frac{\rho(1 - \nu^2)}{E}} \quad (1)$$

$$D(h) \frac{k_{QS}^5}{\rho_w} + gk_{QS} = \omega^2 \left[ k_{QS}h \frac{\rho_w}{\rho} + \coth(k_{QS}H) \right], \quad (2)$$

where ω is the angular frequency, E is the Young's modulus, ρ is the ice density, ν the Poisson's ratio, h the ice thickness and D(h) = Eh³/[12(1 - ν²)] the ice bending rigidity, ρw water density, k_w = ω/c_w with c_w the speed of sound in water and g = 9.81 m s⁻² the gravitational acceleration. Eqs (1) shows that the Young modulus E of the ice layer can be constrained by estimating the non-dispersive velocity of longitudinal waves, under the assumption that ice density and Poisson's ratio are well constrained (Moreau et al. 2020a). In eq. (2) we can see that the dispersion relation of the flexural waves is controlled by the mechanical properties of ice and by its thickness. The roots of this equation can be found numerically to obtain the frequency-dependent wavenumber of QS mode. In the following, we assume that the elastic properties used in the above forward modelling scheme represent the effective elastic properties of the ice layer given its composite nature. Comparing the experimental dispersion curves (Section 3.2) with the theoretical dispersion curves will allow the determination of the physical and mechanical properties of the ice layer.

## 3.2 Extraction of dispersion curves from active sources

We here describe the processing for extracting the dispersion curves of longitudinal and flexural waves from active seismic data. One of the methods to measure dispersion curves from a set of common shot gathers is to apply a spatio-temporal Fourier transform, which yields the frequency-wavenumber spectrum (Alleyne & Cawley 1991). When several sources are available, the spectra can be averaged over them. However, for these experiments, due to the lack of source power (hammer blows) and the presence of snow layers, this method is not sufficient to provide an adequate signal-to-noise ratio in the dispersion curves to identify the various guided waves. To obtain optimal dispersion curves we used a processing based on singular value decomposition described in Moreau et al. (2017, 2020a, b), and summarized in Serripierri et al. (2022). It was first introduced in the field of medical ultrasound (Minonzio et al. 2010) and then applied in geophysics for the study of floating ice layer. We here briefly present the method. More details are available in the above-mentioned articles.

The method relies on the decomposition of the matrix of transmit-receive signals (from multiple sources) into singular values. The underlying idea is that the different levels of modal energy are distributed into the orthogonal singular vectors (Moreau et al. 2017). This processing is performed at each frequency of the spectrum and significantly enhances the identification of the dispersion branches for two reasons: (i) it is possible to separate signal from noise by applying a threshold to the singular values in the frequency domain and (ii) all modes should stand out with the same spectral intensity. The dispersion curves can then be extracted from the 2-D spectrum by identification, for each type of modes m = QS and QS₀, the frequency-wavenumber couples (f_m, k_m) with maximum amplitude.

The datasets used to compute the dispersion curves for the two campaigns are spatio-temporal strain-rate fields measured by the optical fibre during the 4 s following the active impulsive sources (hammer blow). We used 16 shots at 2 different positions in 2020 and 10 shots at one position for 2021 (green dots in Fig. 1). The stacked seismic data for the two experiments are respectively highlighted in Figs 5(b) and (d). These shots exhibit usable energy above background noise up to 50 and 200 Hz, respectively. Fig. 3 shows the frequency-wavenumber spectrum and the extracted dispersion curves. Note that for campaign 2020 we could only retrieve the flexural mode in a narrow frequency band ([0.3; 6] Hz).

## 3.3 Estimation of ice parameters with Bayesian inference

In this section, the dispersion curves extracted from Section 3.2 are inverted to infer the thickness and effective Young's modulus of the ice layer. Given a seismic dataset, d, and a set of parameters for the ice layer X, a classical cost function, χ can be defined as the weighted L2 norm between the wavenumbers calculated from the data (experimental wavenumbers in Fig. 3) and those calculated from eqs (1) and (2) (synthetic wavenumbers), such that

$$\chi(\mathbf{d}, X) = \frac{1}{2} \sum_{M=\text{QS}, \text{QS}_0} \frac{1}{\sqrt{N_M}} ||\hat{k}_M(X) - k_M(X)||, \quad (3)$$

where k_M refers to the frequency-dependent wavenumber of mode M, M = (QS, QS₀) (the hat denotes the experimental data), ||·|| refers to L₂ norm over frequencies, X = [E, h]ᵀ the variable that contains ice properties to be inverted (effective Young's modulus and ice thickness) and N_M the number of samples per mode. We assumed the following values for Poison ratio and ice density: ν = 0.33, ρ = 915 kg m⁻³ and we set H = 5 m, based on observations of
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Dispersion curves of ice obtained from the analysis of seismic shots gathers during Campaign 2020 (a) and Campaign 2021 (b). Green dots show the frequency–wavenumber couples with maximum amplitude. The retrieved seismic modes are highlighted in red.</figcaption></figure>

the lake depth made by the local diving company. Moreover, we set the following values of water properties: $c_w = 1410 \text{ m s}^{-1}$ and $\rho_w = 1010 \text{ kg m}^{-3}$. These are standard values for cold water (Timco & Weeks 2010).

Solving the inverse problem consists of finding the parameters that best explain the data, based on the cost function defined in eq. (3). The minimization of this cost function is an optimization problem which can be solved using a Bayesian approach (Tarantola 2005). The Markov Chain Monte Carlo (MCMC) algorithm is one of several methods that can be used to evaluate the Bayesian posterior distribution. The posterior distribution is an ensemble of solutions that fit the data with an acceptable level of likelihood, given the data uncertainty. In the present problem, it is assumed that measurement errors are not correlated and are random. They can thus be modelled by a zero-mean Gaussian likelihood function, such that:

$$P(\mathbf{d}|X) = \exp \left( -\frac{\chi^2(\mathbf{d}, X)}{2\sigma^2} \right), \quad (4)$$

where $\sigma^2$ represents the variance associated with the measurement error.

### 3.3.1 Prior distribution

Sinha (1989) provided simple two-terms equations for the temperature dependence of Young's modulus of various types of freshwater ice and shows that Young's modulus can be up to around 10 GPa. This defines an upper bound value, which is also found by Moreau et al. (2017). However, these values are related to small-scale specimens analysed with ultrasonic waves. Young's modulus of bigger specimens extracted in situ or measured with seismic waves exhibits lower values between 2 and 8 GPa, depending on ice composition and temperature Moreau et al. (Table 1 in 2020b). In particular, in the Van Mijen fjord, Morozov et al. (2011); Romeyn et al. (2021) reported, respectively, Young's modulus around 2.5 and 3 GPa. Alternatively, Han et al. (2016) studied the elastic properties of large columnar-grained freshwater ice in the temperature range of -0.5 to -10.0 °C and strain rates of $4.6 \times 10^{-7} \text{ s}^{-1}$ to $1.7 \times 10^{-3} \text{ s}^{-1}$. They found that the effective modulus ranged from 1.20 to 7.60 GPa.

Regarding the ice thickness, field observations at Robert lake provided by the local diving company indicated that the clear-ice layer thickness was systematically less than 1.5 m. Overall, we assume for the prior distribution, that the model parameters have equal probability over a finite range of values: $0.5 \text{ GPa} \le E \le 10 \text{ GPa}$ and $0.15 \text{ m} \le h \le 1.5 \text{ m}$.

### 3.3.2 Simulated Annealing

Determining the optimal value of $\sigma^2$ is essential for the appropriate sampling of the PDF. However, $\sigma^2$ is data-dependent and as Serripierri et al. (2022) point it out, we are here inverting the dispersion curves, so it should account for uncertainties in the dispersion curves, and not those in the measurements. Unfortunately, because of the dispersion of the flexural QS mode, there is no linear relationship between both. Hence $\sigma^2$ cannot be set a priori or it would be a coarse approximation. Moreover, MCMC methods generally require a burn-in phase before reaching the posterior distribution. To consider both issues, we precede the MCMC algorithm with a simulated annealing (SA) algorithm to determine the value of $\sigma$ and to improve convergence towards the global minimum.

SA algorithm is a stochastic global search optimization algorithm first introduced by Kirkpatrick et al. (1983) who was inspired by the annealing procedure in metallurgy. Rather than generating samples from the posterior distribution, it aims at converging towards the global optimum. Due to the way the likelihood is calculated, when the temperature is high (large value of $\sigma$), it is more likely that the algorithm accepts a worse solution. This promotes exploration of the search space and allows the algorithm to more likely travel down a suboptimal path to potentially find a global maximum. By lowering the temperature, the algorithm is less likely or will not accept a worse solution. This means that once the algorithm is in the right search space, there is no need to search other sections of the search space and should instead try to converge and find the global minimum. This slow reduction of the temperature (consequently the value of $\sigma$) is termed the ‘cooling schedule’. If a sufficiently gradual schedule is chosen, the algorithm should converge to the global optimum.
In practice, the cooling schedule can be implemented in a number of ways. As in Serripierri et al. (2022), we use a standard schedule, T, where the variance is reduced from $T_0 = 1 \times 10^{-1}$ to $T_1 = 1 \times 10^{-3}$ in an exponential function such that $T(n) = T_0(T_1/T_n)^{(n/N_{SA})}$, where $N_{SA} = 20 000$ is the number of iterations. The SA phase is stopped when the algorithm reaches a number of iterations equal to $N_{SA}$, or when it has remained stuck at the same position in the search space for more than 200 successive iterations. In both cases, the initial temperature $T_0$ was chosen high enough so that any new solution generated in a state transition should be accepted with a certain probability close to 1. We then took its value based on cost function values

Convergence analysis of the Simulated Annealing algorithm is shown in Fig. 4 (black curves). For both experiments, we can see a gradual decrease in the cost function values subsequently related to the temperature decrease (blue curve). When the initial temperature is large ($T_0 \approx 1 \times 10^{-1}$), we explore almost the whole parameter space. By decreasing the temperature, the algorithm progressively limits the search space and converges towards the global minimum. We note that 2020 results are less constrained.

### 3.3.3 MCMC: Generation of the Posterior distribution

After the SA algorithm has been completed, The MCMC algorithm starts with $N_{MCMC} = 20 000$ iterations, from the position in the search space that minimizes the cost function during the SA phase, and with a variance equal to the minimal variance reached during SA, increased by 1 per cent. This increase is to avoid the MCMC algorithm spending hundreds of iterations (like the SA) at the same position in the search space, which would be a sign of poor calibration. MCMC curves are highlighted in green in Fig. 4. After the MCMC algorithm is completed, a probability density function (PDF) is generated for Young's modulus *E* and ice thickness *h*. The resulting PDFs are shown in Fig. 6. The ice properties are then determined by computing the histograms of the two parameters.

# 4 RESULTS

## 4.1 Recorded signals

For both experiments, we recorded ambient seismic noise and additional signals generated by hammer blows. Figs 5(a) and (c) show, for both campaigns, 200 s of DAS strain-rate data recorded along the active part of the fibre (section of fibre between the red markers in Fig. 1a). A 200 Hz low-pass filter has been applied to the presented data sets. The two seismic sections include both active and passive seismic sources. We note an overall good signal-to-noise ratio for the 2021 data set compared to 2020 one, which is related to better coupling of the fibre with the targeted ice layer (columnar ice). The most striking feature in both figures is the low-frequency wave train propagating along the fibre. It is most visible in Fig. 5(c), between 330 and 620 m with waves bouncing off the lake shoreline. In Fig. 5(a) the abrupt change in apparent velocity of these waves at 1050 m is related to the change in orientation of the fibre (almost 80°). This apparent velocity change is also present in Fig. 5(c), but less pronounced because of the slight deviation of the fibre (almost 7°). We hypothesized that this low-frequency noise is probably generated by the wind blowing the lake's surface. It will be more described in Section 5.2.

Figs 5(b) and (d) display the stack of one series of active shots extracted from the recordings. These shots are used for computing the dispersion curves of the ice layer highlighted in Fig. 3. We observe that propagating waves excited by the hammer blows are resolvable from ambient noise up to about 50 m distance from the source. The longitudinal and flexural waves described in Section 3.1 are highlighted in the seismic sections. We note that we could not retrieve the longitudinal mode QS for the 2020 experiment. The seismic section in Fig. 5(d) also highlights a strong reflection at 350 m which corresponds to the lake shore. Moreover, Comparatively to the 2020 experiment, data recorded in 2021 have a higher frequency content that we interpret as being due to a shorter gauge length (2 m). Indeed, the use of a shorter gauge length the second year enables to observe shorter wavelengths associated with higher frequency content.

## 4.2 Inversion results: ice thickness and effective Young's modulus

The ice properties are determined by analysing the histograms of the samples generated by MCMC algorithm. Fig. 6(a) shows the experimental dispersion curves derived in Section 3.2 (Fig. 3) and the theoretical dispersion curves obtained from the best fit in terms of cost function:

* Campaign 2020: $E = 2.18$ GPa, $h = 0.91$ m, RMS= 0.005

* Campaign 2021: $E = 3.41$ GPa, $h = 0.47$ m, RMS= 0.03

We observe a statistically consistent fit between the theoretical and experimental dispersion curves. The lower value of the RMS in 2020 is related to the narrow spread of the frequency-wavenumber couples. We should note that compared to 2020, results from 2021 consider the two modes for the RMS calculation.

Fig. 6 shows the probability density functions PDFs and associated covariance matrices of the two parameters. 2020 results are highlighted in red and 2021 in blue. We observe that results are better constrained for the 2021 season. For the 2021 experiment, we found Gaussian distributions with the following parameters: $E = 3.4 \pm 0.1$ GPa and $h = 47 \pm 1$ cm. However for the 2020 experiment, effective Young's modulus explores the whole space parameter with a preference for lower values (2–6 GPa), and ice thickness remains bounded between 0.5 and 1.5 m.

# 5 DISCUSSION

## 5.1 Active seismic: ice properties

Ice thickness and Young's modulus are of major importance to the rheological behaviour of an ice layer deforming at the timescale of seismic wave propagation. They depend on several environmental factors. The most important ones include the ice composition, temperature patterns, structure and salinity, and the amount and quality of snow covering the ice. At first order, ice thickness controls the bearing capacity of an ice cover (Timco & Weeks 2010). The Young's modulus *E* of a material represents the ratio of the stress (*σ*) to the strain (*ε*) during elastic deformation (i.e. $σ = E \times ε$) and is typically described as the material stiffness. It is important for the transmission of elastic stresses through ice and is a crucial parameter when considering its fracture or damage.
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Convergence analysis of MCMC algorithm for Campaign 2020 (a, b, c) and Campaign 2021 (d, e, f). The figures highlight the evolution of the Cost function $\chi$ (a,d) Young's modulus (b, e), ice thickness (c, f) according to iteration number. The Simulated Annealing part is highlighted in black and the MCMC part in green. The red dashed line separates the two processing. The blue curve in the Cost function plots shows the cooling scheme.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Seismic signals recorded along the active part of the fibre for 200 seconds and seismic shots gathers. (a) 2020 season: recorded seismic ambient noise. The green arrow refers to the north-south section of the fibre and the yellow arrow to the east-west section (b) 2020 season: Stacked seismic gather (9 shots, source location: 1130 m) (c) 2021 season: recorded seismic ambient noise. Left white box highlights hammer blows and the right white box highlights the low-frequency section used for computing the frequency-wavenumber plot in Fig. 7. (d) 2021 season: Stacked seismic gather (10 shots, source location: 365 m). In all the figures, the dashed red line corresponds to the location of the lake shoreline.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Dispersion curves and inversion results. (a) Dispersion curves extracted from the analysis of the shot gathers in 2020 (red) and 2021 (blue). Dashed lines represent the best fit in terms of cost function. Note that for 2020 we only observe the flexural mode. (b) Covariance matrices and probability density functions resulting from the Bayesian inversion of the effective Young's modulus *E* and thickness *h* of ice. Results for Campaign 2020 and 2021 are, respectively, highlighted in red and blue as in (a).</figcaption></figure>

### 5.1.1 Guided waves recovery

Fig. 6(b) shows the results on the inverted ice properties. We observe that the retrieval of ice properties is better constrained for the 2021 season. The main factor which explain these observations is the ability to recover the full wavefield (i.e. with all types of guided waves). Indeed for the 2020 season, we were not able to resolve the longitudinal wave QS₀ and therefore not able to resolve all wave components necessary to constrain both Young's modulus and thickness simultaneously. The dispersion relations in eqs (1) and (2) rely on the velocities/wavenumbers of the guided waves to estimate ice properties. For example, with fixed parameters of ice density ρ and Poison's ratio ν, *E* is closely related to the velocity of the longitudinal wave QS₀ and *h* to the velocity of the flexural wave QS. We can see from eq. (2) that the quasi-Scholte QS mode is sensitive to the ice thickness *h* at order 3 (cubic polynomial form in Romeyn *et al.* 2021) while at order 1 only for Young's modulus *E*, and as Serripiërri *et al.* (2022) point it out, Young's modulus and ice thickness are strongly correlated. Thus, a large uncertainty on one of the two parameters will have the same effect on the other. Therefore, the absence of one mode (here QS₀ mode for the 2020 season) could lead to a less constrained inversion, even if the 2020 results seem to be consistent with field observations (consistent ice thickness and relatively weak effective modulus which seems to be related to the above snow layers and inhomogeneity of the ice layers). With the two types of guided waves recovered in a broader frequency band, we expect as announced better constrained results for the 2021 season compared to the 2020 one.

### 5.1.2 Effective Young's modulus

At temperatures of interest to the engineer, ice is an elasticviscoplastic material (Schwarz *et al.* 1981; Tatinclaux & Hirayama 1982). It is known that the elastic modulus of ice depends upon several factors, including temperature, density, ice type, purity, time or frequency of stress and the direction of the applied load (Gold 1977;

Sinha 1989). There is evidence that they may depend also upon grain size and stress history. Sinha (1989) studied the elasticity of different types of polycrystalline ice (ice single crystal, granular and columnar-grained ice) and he found that the elastic properties of various types of ice, despite large differences in fabric, are very similar at a constant temperature.

Both experiments found relatively low values of Young's modulus (1–5 GPa), suggesting quite porous and brittle ice that we observed both in the field (when we dug into the ice layers) and in the data where we notice a fast attenuation of the guided waves (also likely a product of the snow loading effects). As stated previously, the effective Young's modulus *E* could not be constrained for the 2020 season because we were not able to observe the longitudinal waves. As a result, the PDF of *E* tend towards lower values [1–5 GPa]. In contrast, for the 2021 season, we found a very well-constrained effective Young's modulus *E* = 3.4 ± 0.1 GPa which is in good agreement with effective modulus values of Han *et al.* (2016) who studied large columnar grained freshwater ice at similar strain rates. We suspect that the snow layer covering the ice surface could also underestimate the effective value of *E*.

Moreover, ice porosity and inhomogeneity have also an influence on the decrease of the effective value of *E*. Wang *et al.* (2008) attempts to explain the mechanism of the effective decrease in stiffness for a cracked plate compared to an uncracked plate in order to understand earlier observations made on freshwater ice. They found that the effective modulus is controlled by both the stiffness and the thickness of the grain boundaries and reported a relatively low effective Young's modulus of the lake ice specimen (4 GPa) compare to the elastic modulus of the grains (12 GPa).

### 5.1.3 Ice thickness

We remind that in our study the targeted ice layer is the solid columnar ice just below the layers of fresh snow and snow-ice layers. For the 2020 season, the PDF of the thickness is bounded between
0.5 and 1.5 m. The local company Divextreme revealed an average thickness of about 1 m. The observed thickness difference can be explained in various ways. First, drilling was performed 100 m Eastward of the fibre optic cable and we observe in the field about 20 cm variation between three different locations with thicknesses ranging from 70 to 110 cm. Secondly, this difference can be explained by the presence of a snow layer above the ice. Indeed, we suspect that the calculated thickness could be underestimated when considering this layer which is in most cases in-homogeneous. For the 2021 season, we found a thickness of $h = 47 \pm 1$ cm which is consistent with the field observations ($h = 50$ cm), given the possibility for spatially varying ice thickness between field observations and along the fibre optic cable. This improvement can be attributed to better coupling of the fibre optic cable with the columnar ice layer and the choice of more suitable acquisition parameters (2 m gauge length instead of 10 m for the 2020 season) which made enabled to highlight higher frequency guided waves (associated with shorter wavelengths).

### 5.1.4 Influence of the snow layers and ice inhomogeneity

In the field, a snow-ice layer of 10–100 cm covered the solid columnar ice layer (clear ice). In addition a layer of fresh snow between 20 and 100 cm. It is likely that the snowpack system = {fresh snow + snow-ice} above the targeted ice layer influences our estimations because guided modes propagate at a lower speed in snow than in ice. Hence, it is possible that some parameters might have been slightly underestimated or less constrained. We think that the thinner snowpack in 2021 also contributed to the improved results since the 2021 experiment was in the early season with less accumulated snow than in the 2020 experiment. However, it is difficult to explicitly constrain the influence of the snow layers on our estimations, and this is left for a dedicated study.

### 5.1.5 Limits of the forward model

Besides, the inversion method used here relies on a simple model composed of a homogeneous, isotropic layer of ice floating on a finite water column. It does not account for the inhomogeneity of the ice layer, the layer of snow covering the ice and the changes in the ice properties along the path ('active part' of the optical fibre). These three effects can bias the estimations of ice thickness and Young's modulus. From a perspective, we could consider using more complex models including these two constraints, and implementing methods such as the Finite Element Method (FEM) or the Spectral Element Method (SEM) to compute the full wavefield.

## 5.2 Ambient noise: analysis of noise recordings

Optical fibre solutions can be of great interest in recording passive noise. Serripierri et al. (2022) shows examples of natural noise recorded by geophones that could be used in the Van Mijen Fjord in Svalbard to obtain guided wave dispersion curves. In his study, Marsan et al. (2012) uses ice swell signals, providing a natural source for analysing the dispersion of flexural waves and subsequently inverting the ice thickness. Unfortunately, the noise recording periods were quite short both years (1 hr) due to the sanitary restrictions, reducing the chance to observe impulsive natural seismic sources exciting higher frequency seismic waves. In 2020, seismic noise was generated by walking on the ice layer, along the fibre cable. However, this generated only seismic waves with a frequency content below 6 Hz. During the 2021 experiment, we

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Frequency–wavenumber spectrum of the low-frequency section in the right white box in Fig. 5(c). The dashed red line denotes a non-dispersive wave propagating at the velocity $c = 7.5 \text{ m s}^{-1}$ in both directions.</figcaption></figure>

did not record high-frequency coherent noise that could be used to compute the dispersion curves of ice from the cross-correlation of distant DAS sensors. This lack of high-frequency noise can again be related to the short recording period (1 hr) and the relatively stable temperature during acquisition.

On the other hand, one striking feature in Fig. 5(c) is the low-frequency noise wave train that reflects on the shores of the lake. We interpret these waves as being flexural waves excited by the wind blowing at 20–30 km h$^{-1}$ at the lake’s surface. We could measure the phase velocity of these waves between 0.05 and 0.2 Hz using a simple F-K analysis and found a value approximately constant of $7.5 \text{ m s}^{-1}$ with no clear dispersion (Fig. 7). Marsan et al. (2012) show that for periods much longer than $2\pi\sqrt{H}$, that is 6–14 s and for a typical ice thickness of 1–5 m, the phase velocity of the flexural waves becomes independent of the ice thickness. That is to say, when flexural wave wavelengths are larger than the ice thickness, the ice layer is not resolved and these waves behave like gravity waves. We use the complete dispersion relation for flexural waves reminded in Romeyn et al. (2021) (eq. 11) and in eq. (2) which considers the water finite depth. In the shallow layer gravity wave approximation, the wave phase velocity is constant and equal to $c_\phi = \sqrt{gH}$. With the complete theory, for example Romeyn et al. (2021), the phase velocity is stationary with a value slightly lower than the gravity wave asymptotic value. Using the equation of Romeyn et al. (2021) or the gravity wave asymptotic value, and for the given frequency range, we found a water depth of 5.7 m corresponding to a phase velocity of $7.5 \text{ m s}^{-1}$. The average lake depth estimated to 4–6 m is in accordance with phase velocity and is compatible with independent observations.

# 6 CONCLUDING REMARKS

We presented a proof of concept of the use of fibre optics coupled with DAS to measure the thickness and Young’s modulus of an ice layer. During two winters (2020 and 2021) we deployed an optical fibre cable over a frozen mountain lake located in the French Alps (2400 m altitude). This survey had been recorded using a DAS interrogator enabling to turn a simple fibre optic cable into hundreds of seismic sensors. We analyse signals generated with active sources and showed that, under good coupling conditions and appropriate acquisition parameters, we can observe broad-band
propagation of longitudinal and flexural waves guided in the ice as with conventional geophones. We then extract the associated dispersion curves and invert them with Bayesian methods (MCMC) for inferring the probability density function of the targeted ice properties (thickness and effective Young's modulus).

We obtain satisfactory results for ice thickness which is consistent with *in situ* measurements. The retrieved low values of the effective Young's modulus (1–5 GPa) may be explained by the inhomogeneity/porosity of the solid columnar ice layer and the influence of the upper snow-ice layers. Further work is still needed to understand these effects on the effective Young's modulus of the ice layer that is estimated from seismic waves. Besides we were able to use the non-dispersive low-frequency noise associated with gravity waves to measure the lake depth which is compatible with on-site observations achieved by a local diving company. We are confident that the use of DAS can be complementary to the already existing methods for monitoring ice properties.

## ACKNOWLEDGMENTS

We thank Arnaud Cimolino for his help concerning the exploratory stage of the 2020 data analysis and which made it possible to better design the following experiment in 2021. The authors thanks also the staff of the Chamrousse Ski Resort and particularly P. Halot for their help and support; Eric Larose for the deployment of fibre by drone in 2021; FEBUS-Optics engineers for their help and assistance and especially Camille Jestin for her proofreading of the paper. The authors would like to thank Brad Lipovsky and the anonymous reviewer who provided comments that significantly improved the quality of the manuscript.

ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). This research was funded by the Agence Nationale de la Recherche (ANR, France) under the Grant Agreement Number 19-CE04-0011 (MONIDAS) and 17-CE01-0019 (ICEWAVEGUIDE https://icewaveguide.osug.fr/). Most of the computations presented here were performed using the GRICAD infrastructure (https://gricad.univ-grenoble-alpes.fr), which is supported by the Auvergne-Rhône-Alpes region (GRANT CPER07_13 CIRA) and the Equip@Meso project (reference: ANR-10-EQPX-29-01) of the programme In-vestissements d'Avenir.

## 7 DATA AVAILABILITY

DAS strain-rate data sets presented in Fig. 5 are available at the following link: ftp://ist-ftp.u-ga.fr/coutanto/GJI_S-22-0566L.zip (login: isterre/password: isterre). It includes for both campaigns, 200 s of ambient noise along the active part of the fibre and the seismic shot gathers.

## REFERENCES

* Ajo-Franklin, J.B. et al., 2019. Distributed acoustic sensing using dark fiber for near-surface characterization and broadband seismic event detection, *Sci. Rep.*, **9**(1), 1–14.

* Alleyne, D. & Cawley, P., 1991. A two-dimensional fourier transform method for the measurement of propagating multimode signals, *J. acoust. Soc. Am.*, **89**(3), 1159–1168.

* Anderson, D., 1958. Preliminary results and review of sea ice elasticity and related studies, *Trans. Eng. Inst. Canada*, **2**(3), 2–8.

* Baudais, D., Watts, J. & Masterson, D., 1976. A system for offshore drilling in the arctic islands, in *Offshore Technology Conference, OnePetro*.

* Castongia, E., Wang, H.F., Lord, N., Fratta, D., Mondanos, M. & Chalari, A., 2017. An experimental investigation of distributed acoustic sensing (DAS) on lake ice, *J. Environ. Eng. Geophys.*, **22**(2), 167–176.

* Dou, S. et al., 2017. Distributed acoustic sensing for seismic monitoring of the near surface: a traffic-noise interferometry case study, *Sci. Rep.*, **7**(1), 1–12.

* Gold, L.W., 1977. Engineering properties of fresh-water ice, *J. Glaciol.*, **19**(81), 197–212.

* Gow, A.J. & Langston, D., 1977. Growth history of lake ice in relation to its stratigraphic, crystalline and mechanical structure, Report, Department of Defense, Army, Corps of Engineers, Cold Regions Research and Engineering Lab Hanover N H.

* Han, H., Jia, Q., Huang, W. & Li, Z., 2016. Flexural strength and effective modulus of large columnar-grained freshwater ice, *J. Cold Reg. Eng.*, **30**(2), doi:10.1061/(ASCE)CR.1943-5495.0000098.

* Harmon, N. et al., 2022. Surface deployment of DAS systems: coupling strategies and comparisons to geophone data, *Near Surf. Geophys.*, **20**(5), 465–477.

* Hudson, T.S. et al., 2021. Distributed acoustic sensing (DAS) for natural microseismicity studies: a case study from Antarctica, *J. geophys. Res.*, **126**(7), e2020JB021493, doi10.1029/2020JB021493.

* Hunkins, K., 1960. Seismic studies of sea ice, *J. geophys. Res.*, **65**(10), 3459–3472.

* Jousset, P. et al., 2018. Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features, *Nat. Commun.*, **9**(1), 1–11.

* Kirkpatrick, S., Gelatt, C.D. Jr & Vecchi, M.P., 1983. Optimization by simulated annealing, *Science*, **220**(4598), 671–680.

* Klaasen, S.A., Paitz, P., Lindner, N., Dettmer, J. & Fichtner, A., 2021. Distributed acoustic sensing in volcano-glacial environments—Mount Meager, British Columbia, *J. geophys. Res.*, **126**(11), doi:10.1029/2021JB022358.

* Lindsey, N.J., Dawe, C. & Ajo-Franklin, J., 2019. Photonic seismology in Monterey Bay: dark fiber DAS illuminates offshore faults and coastal ocean dynamics, *Science*, doi:10.31223/osf.io/7bf92.

* Lindsey, N.J., Martin, E.R., Dreger, D.S., Freifeld, B., Cole, S., James, S.R., Biondi, B.L. & Ajo-Franklin, J.B., 2017. Fiber-optic network observations of earthquake wavefields, *Geophys. Res. Lett.*, **44**(23), 11–792.

* Lindsey, N.J., Rademacher, H. & Ajo-Franklin, J.B., 2020. On the broadband instrument response of fiber-optic DAS arrays, *J. geophys. Res.*, **125**(2), e2019JB018145, doi:10.1029/2019JB018145.

* Lipovsky, B.P., 2018. Ice shelf rift propagation and the mechanics of wave-induced fracture, *J. geophys. Res.*, **123**(6), 4014–4033.

* Marsan, D., Weiss, J., Larose, E. & Métaxian, J.-P., 2012. Sea-ice thickness measurement based on the dispersion of ice swell, *J. acoust. Soc. Am.*, **131**(1), 80–91.

* Masterson, D., 2009. State of the art of ice bearing capacity and ice construction, *Cold Reg. Sci. Technol.*, **58**(3), 99–112.

* Masterson, D., Pare, A., Gamble, R. & Bourns, M., 1986. Construction and performance of a floating ice platform built from sprayed ice, in *Proceedings, the 6th International Conference on Offshore Mechanics and Arctic Engineering*, Vol. 4, pp. 9–16, Tokyo, Japan.

* Mateeva, A. et al., 2014. Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling, *Geophys. Prospect.*, **62**, 679–692.

* Michel, B. & Ramseier, R.O., 1971. Classification of river and lake ice, *Can. Geotech. J.*, **8**(1), 36–45.

* Minonzio, J.-G., Talmant, M. & Laugier, P., 2010. Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration, *J. acoust. Soc. Am.*, **127**(5), 2913–2919.

* Moreau, L., Lachaud, C., Théry, R., Predoi, M.V., Marsan, D., Larose, E., Weiss, J. & Montagnat, M., 2017. Monitoring ice thickness and elastic properties from the measurement of leaky guided waves: a laboratory experiment, *J. acoust. Soc. Am.*, **142**(5), 2873–2880.

* Moreau, L. et al., 2020a. Sea ice thickness and elastic properties from the analysis of multimodal guided wave propagation measured with a passive seismic array, *J. geophys. Res.*, **125**(4), e2019JC015709, doi:10.1029/2019JC015709.
Moreau, L., Weiss, J. & Marsan, D., 2020b. Accurate estimations of sea-ice thickness and elastic properties from seismic noise recorded with a minimal number of geophones: from thin landfast ice to thick pack ice, *J. geophys. Res.*, **125**(11), e2020JC016492, doi:10.1029/2020JC016492.

Morozov, E., Muzylev, S., Shestov, A. & Marchenko, A., 2011. Short-period internal waves under an ice cover in Van Mijen Fjord, Svalbard, *Adv. Meteorol.*, **2011**, doi:10.1155/2011/573269.

Mulherin, N.D. & Haehnel, R.B., 2003. Ice engineering: progress in evaluating surface coatings for icing control at corps hydraulic structures, Tech. rep., Engineer Research and Development Center Hanover NH Cold Regions Research.

Ohata, Y., Toyota, T. & Shiraiwa, T., 2016. Lake ice formation processes and thickness evolution at lake Abashiri, Hokkaido, Japan, *J. Glaciol.*, **62**(233), 563-578.

Olinger, S., Lipovsky, B.P., Denolle, M. & Crowell, B.W., 2022. Tracking the cracking: a holistic analysis of rapid ice shelf fracture using seismology, geodesy, and satellite imagery on the pine island glacier ice shelf, west Antarctica, *Geophys. Res. Lett.*, **49**(10), e2021GL097604, doi:10.1029/2021GL097604.

Paitz, P., Edme, P., Gräff, D., Walter, F., Doetsch, J., Chalari, A., Schmelzbach, C. & Fichtner, A., 2021. Empirical investigations of the instrument response for distributed acoustic sensing (DAS) across 17 octaves, *Bull. seism. Soc. Am.*, **111**(1), 1-10.

Palmer, A., 2013. *Arctic Offshore Engineering*, World Scientific.

Parker, T., Shatalin, S. & Farhadiroushan, M., 2014. Distributed acoustic sensing—a new tool for seismic applications, *First Break*, **32**(2), 61–69.

Podolskiy, E.A. & Walter, F., 2016. Cryoseismology, *Rev. Geophys.*, **54**(4), 708–758.

Prowse, T.D., Furgal, C., Chouinard, R., Melling, H., Milburn, D. & Smith, S.L., 2009. Implications of climate change for economic development in northern Canada: energy, resource, and transportation sectors, *AMBIO: A Journal of the Human Environment*, **38**(5), 272–281.

Rampal, P., Weiss, J., Dubois, C. & Campin, J.-M., 2011. IPCC climate models do not capture Arctic sea ice drift acceleration: consequences in terms of projected sea ice thinning and decline, *J. geophys. Res.*, **116**(C8), doi:10.1029/2011JC007110.

Romeyn, R., Hanssen, A., Ruud, B.O. & Johansen, T.A., 2021. Sea ice thickness from air-coupled flexural waves, *Cryosphere*, **15**(6), 2939–2955.

Schwarz, J., Frederking, R., Gavrillo, V., Petrov, I., Hirayama, K.-I., Mellor, M., Tryde, P. & Vaudrey, K., 1981. Standardized testing methods for measuring mechanical properties of ice, *Cold Reg. Sci. Technol.*, **4**(3), 245–253.

Serripiërri, A., Moreau, L., Boue, P., Weiss, J. & Roux, P., 2022. Recovering and monitoring the thickness, density, and elastic properties of sea ice from seismic noise recorded in Svalbard, *Cryosphere*, **16**(6), 2527–2543.

Sinha, N.K., 1989. Elasticity of natural types of polycrystalline ice, *Cold Reg. Sci. Technol.*, **17**(2), 127-135.

Sladen, A., Rivet, D., Ampuero, J.P., De Barros, L., Hello, Y., Calbris, G. & Lamare, P., 2019. Distributed sensing of earthquakes and ocean-solid earth interactions on seafloor telecom cables, *Nat. Commun.*, **10**(1), 1–8.

Spica, Z.J., Perton, M., Martin, E.R., Beroza, G.C. & Biondi, B., 2020. Urban seismic site characterization by fiber-optic seismology, *J. geophys. Res.*, **125**(3), e2019JB018656, doi:10.1029/2019JB018656.

Spikes, K.T., Tisato, N., Hess, T.E. & Holt, J.W., 2019. Comparison of geophone and surface-deployed distributed acoustic sensing seismic data geophone and surface DAS data, *Geophysics*, **84**(2), A25–A29.

Stein, P.J., Euerle, S.E. & Parinella, J.C., 1998. Inversion of pack ice elastic wave data to obtain ice physical properties, *J. geophys. Res.*, **103**(C10), 21 783–21 793.

Stroeve, J., Holland, M.M., Meier, W., Scambos, T. & Serreze, M., 2007. Arctic sea ice decline: faster than forecast, *Geophys. Res. Lett.*, **34**(9), doi:10.1029/2007GL029703.

Tarantola, A., 2005. *Inverse Problem Theory and Methods for Model Parameter Estimation*, SIAM.

Tatinclaux, J.-C. & Hirayama, K.-I., 1982. Determination of the flexural strength and elastic modulus of ice from in situ cantilever-beam tests, *Cold Reg. Sci. Technol.*, **6**(1), 37–47.

Timco, G. & Weeks, W., 2010. A review of the engineering properties of sea ice, *Cold Reg. Sci. Technol.*, **60**(2), 107–129.

Walter, F., Gräff, D., Lindner, F., Paitz, P., Köpfli, M., Chmiel, M. & Fichtner, A., 2020. Distributed acoustic sensing of microseismic sources and wave propagation in glaciated terrain, *Nat. Commun.*, **11**(1), 1–10.

Wang, H.F., Zeng, X., Miller, D.E., Fratta, D., Feigl, K.L., Thurber, C.H. & Mellors, R.J., 2018. Ground motion response to an $M_L$ 4.3 earthquake using co-located distributed acoustic sensing and seismometer arrays, *Geophys. J. Int.*, **213**(3), 2020–2036.

Wang, Y., Ballarini, R. & Rodin, G.J., 2008. Crack-tip parameters in polycrystalline plates with soft grain boundaries, *J. Eng. Mech.*, **134**(1), 100–109.

Wu, X., Willis, M.E., Palacios, W., Ellmauthaler, A., Barrios, O., Shaw, S. & Quinn, D., 2017. Compressional-and shear-wave studies of distributed acoustic sensing acquired vertical seismic profile data, *Leading Edge*, **36**(12), 987–993.

Yang, T. & Giellis, G., 1994. Experimental characterization of elastic waves in a floating ice sheet, *J. acoust. Soc. Am.*, **96**(5), 2993–3009.

Zeng, X., Lancelle, C., Thurber, C., Fratta, D., Wang, H., Lord, N., Chalari, A. & Clarke, A., 2017. Properties of noise cross-correlation functions obtained from a distributed acoustic sensing array at Garner Valley, California, *Bull. seism. Soc. Am.*, **107**(2), 603–610.

Zhan, Z., 2020. Distributed acoustic sensing turns fiber-optic cables into sensitive seismic antennas, *Seismol. Res. Lett.*, **91**(1), 1–15.