Ambient noise seismology has revolutionized seismic characterization of the Earth's crust from local to global scales. The estimate of Green's
function (GF) between two receivers, representing the impulse response of elastic media, can be reconstructed via cross-correlation of the ambient
noise seismograms. A homogenized wave field illuminating the propagation medium in all directions is a prerequisite for obtaining an accurate GF. For
seismic data recorded on glaciers, this condition imposes strong limitations on GF convergence because of minimal seismic scattering in homogeneous
ice and limitations in network coverage. We address this difficulty by investigating three patterns of seismic wave fields: a favorable distribution
of icequakes and noise sources recorded on a dense array of 98 sensors on Glacier d'Argentière (France), a dominant noise source constituted by
a moulin within a smaller seismic array on the Greenland Ice Sheet, and crevasse-generated scattering at Gornergletscher (Switzerland). In Glacier
d'Argentière, surface melt routing through englacial channels produces turbulent water flow, creating sustained ambient seismic sources and thus
favorable conditions for GF estimates. Analysis of the cross-correlation functions reveals non-equally distributed noise sources outside and within
the recording network. The dense sampling of sensors allows for spatial averaging and accurate GF estimates when stacked on lines of receivers. The
averaged GFs contain high-frequency (

Passive seismic techniques have proven efficient to better understand and monitor glacier processes on a wide range of time and spatial
scales. Improvements in portable instrumentation have allowed rapid deployments of seismic networks in remote terrain and harsh polar conditions

The subsurface structure of ice sheets and glaciers has been characterized by analysis of seismic wave propagation in ice bodies. For example,

At the same time, a new approach appeared in seismology which explores not only earthquakes but also ambient noise sources generated by climate and
ocean activity

For the cryosphere, few studies have successfully used oceanic ambient noise at permanent broadband stations deployed on the rocky margins of glaciers
or up to 500

The underlying seismic interferometry techniques used in ambient noise studies are rooted in the fact that the elastic impulse response between two
receivers, Green's function (GF), can be approximated via cross-correlation of a diffuse wave field recorded at the two sites

In theory, the GF estimate is obtained in media capable of hosting an equipartitioned wave field, that is random modes of seismic propagation with
the same amount of energy. In practice, the equipartition argument has limited applicability to the Earth because nonhomogeneously distributed
sources, in the forms of ambient noise sources, earthquakes, and/or scatterers, prevent the ambient wave field from being equipartitioned across the
entire seismic scale

In glaciers, the commonly used oceanic ambient noise field lacks the high frequencies needed to generate GFs that contain useful information at the
scale of the glacier. To target shallower glaciers and their bed, we must work with other sources such as nearby icequakes and flowing water which
excite higher-frequency (

Few attempts have been conducted on glaciers to obtain GF estimates from on-ice seismic recordings.

As an alternative to continuous ambient noise,

In this study, we provide a catalogue of methods to tackle the challenge of applying passive seismic interferometry to glaciers in the absence of
significant scattering and/or an isotropic source distribution. After a review on glacier seismic sources (Sect.

Glaciogenic seismic waves couple with the bulk Earth and can be recorded by seismometers deployed at local

Typically on Alpine glaciers and more generally in ablation zones, the most abundant class of recorded seismicity is related to brittle ice failure
which leads to the formation of near-surface crevasses

From spring to the end of summer, another seismic source superimposes on icequake records and takes its origin in fluvial processes. Ice melting and
glacier runoff create turbulent water flow at the ice surface that interacts with englacial and subglacial linked conduits. Gravity-driven transport
of meltwater creates transient forces on the bulk of the Earth

Englacial and subglacial conduits can also generate acoustic

Finally, in Alpine environments, seismic signatures of anthropogenic activity generally overlap with glacier ambient noise at frequencies

Icequake locations (blue dots) and seismic stations (red triangles) superimposed on aerial photographs of

We use seismic recordings from three seasonally deployed networks in the ablation zones of two temperate Alpine glaciers and of the GIS. Each of the
acquired datasets presents different patterns of seismic wave fields corresponding to the three configurations investigated for GF estimate retrieval,
as defined in the introduction. All networks recorded varying numbers of near-surface icequakes (blue dots in Fig.

The Argentière seismic array (Fig.

The GIS network (Fig.

The Gornergletscher network (Fig.

We use a standardized processing scheme for computing GF estimates here. We either cross-correlate seismogram time windows, which encompass ballistic
seismic waves of the icequake catalogue, or cross-correlate continuous seismograms as traditionally done in ambient noise studies. Prior to any
calculation, seismic records are corrected for instrumental response and converted to ground velocity. Seismograms are then spectrally whitened
between 1 and 50

For icequake cross-correlation (ICC), we follow the method of

For noise cross-correlation (NCC), we use a similar protocol as the one of

Figure

Slight disparities in amplitudes of the causal and acausal parts of the GF estimates (positive versus negative times) are related to the noise source
density and distribution. Higher acausal amplitudes observed at larger distances are evidence for a higher density of sources located downstream of
the array, according to our cross-correlation definition. More sources downstream are likely generated by faster water flow running into subglacial
conduits toward the glacier icefall

The stacked section of ICCs (Fig.

Seismic phases and their velocities can be identified on the frequency–velocity diagram (Fig.

Surface waves are dispersive, meaning that their velocity is frequency-dependent, with higher frequencies being sensitive to surface layers and
conversely lower frequencies being sensitive to basal layers. Theoretical dispersion curves for Rayleigh wave fundamental mode are indicated as
black solid lines in Fig.

Sensitivity of Rayleigh waves obtained on NCC to frequencies below 5

Following

Inversion of glacier thickness using velocity dispersion curves of Rayleigh waves and the Geopsy neighborhood algorithm. Dispersion curve
measurements are obtained from f-k analysis of noise cross-correlations on eight receiver lines whose geometry is described in the bottom-right
panel.

Figure

From the eight receiver line inversions, we find average S-wave velocities of 1710

Inversion results for the ice thickness are plotted in red in Fig.

Parameter ranges and fixed parameters for grid search to invert the dispersion curves in Fig.

Errors and uncertainties on mapping the basal interface are primarily linked to bedrock velocities, as discussed above. Potentially, the bed
properties can be refined using additional measurements from refracted P waves that should be reconstructed on NCC obtained on such a dense and large
array and stacked over longer times. Ice thickness estimation is also affected by 2-D and 3-D effects as phase velocities are averaged here over
multiple receiver pairs. The confidence interval we obtain for basal depth is of a similar order to the actual variations in glacier thickness along the
receiver lines (black dashed lines). More accurate 3-D seismic models of the glacier subsurface could be obtained using additional station pairs as
discussed in Sect.

The dense array experiment of Glacier d'Argentière covers a wide range of azimuths

For each sensor position, we obtain

Anisotropy is observed to be more pronounced near the glacier margins (lines 1–2 and 8 as labeled in Fig.

Alignment of the fast-axis directions with that of ice flow appears along the central lines of the glacier (receiver lines 4–5) with anisotropy
degrees of 0.5 % to 1.5 %. This feature is only observed along the deepest part of the glacier where it flows over a basal depression. Results
are here computed for seismic measurements at 25

Generally, we observe an increase in the degree of anisotropy with frequency, which is evident for a shallow anisotropic layer. Conversely, an increase in
anisotropy strength at lower frequency would indicate a deeper anisotropic layer. At the Alpine plateau Glacier de la Plaine Morte,

As pointed out earlier, localized englacial noise sources related to water drainage can prevent the reconstruction of stable GF estimates by
introducing spurious arrivals

One of the approaches we apply here is matched-field processing (MFP)

Moreover, joint use of MFP and the singular value decomposition (SVD) of the cross-spectral density matrix allows the separation of different noise source
contributions, as in multi-rate adaptive beam forming

We briefly summarize the basics of MFP, and the details of the method can be found in

In order to calculate the MFP output, we use 24

The lower frequency bound (2.5

Figure

SVD is a decomposition of the CSDM that projects the maximum signal energy into independent coefficients

The SVD separates the wave field into dominant (coherent) and subdominant (incoherent) subspaces. It has been shown that the incoherent sources
correspond to the smallest eigenvalues

Here, we follow the approach of

Figure

The CSDM can then be reconstructed by using only individual eigenvectors as in

Note that we do not include the eigenvalues

Reconstruction of the CSDM by using individual eigenvectors (EVs) that are related to different noise sources. Each plot shows MFP output
computed using the reconstructed CSDM with individual eigenvectors as in Eq. (

Figure

This MFP-based analysis of spatial noise source distribution allows us to select the eigenvectors of CSDM that contribute to noise sources located in
the stationary phase zone (i.e. in the endfire lobes of each station path). We now reconstruct the NCC in the frequency band of 2.5–6

Figure

Unfortunately, we notice that the equalization process reduces the overall SNR of the GF estimates and does not eliminate all spurious arrivals. This might be related to the imperfect separation of different noise sources which is likely influenced by the frequency variation in the moulin contribution. For example, we still keep some contribution of the central moulin in the first eigenvector. Moreover, by removing the second eigenvector we remove not only the seismic signature of the moulin, but also the contribution of coherent far-field sources.

To further assess the isotropy of the reconstructed noise field, we use the conventional plane wave beam former

After the eigenspectrum equalization, we are able to extract a Rayleigh wave dispersion curve from the averaged seismic section obtained in
Fig.

Several additional tests could be used to further improve the SNR of the NCC and their convergence to GF. For example, a similar procedure could be performed on other days, and the eigennormalized NCC could be stacked over a few days to increase the SNR. However, we verified that the index of eigenvectors corresponding to the moulin changes over days (the moulin can be located in the first, second, third, etc., eigenvector). This is the reason why it would be useful to find an automatic criterion for the eigenvalue selection based on the MFP output. However, this is beyond the scope of this paper. Another improvement could consist of azimuthal stacking the NCC according to the direction of the noise sources, although the GIS array does not have sufficient azimuthal and spatial coverage to implement this. Moreover, we could envisage calculating a projector based on the SVD (as in MRABF) only for the time period when the moulin is active and then project out the moulin signature from the continuous seismic data.

In summary, we conclude that the CSDM eigenspectrum equalization together with beam-forming-based selection of eigenvectors is a useful method to separate seismic sources in a glaciated environment. It can further improve the GF emergence from ambient seismic noise in the presence of strong, localized englacial noise sources for imaging applications.

Contrary to ballistic waves, the likely diffuse coda arises from multiple seismic scattering

In the following, we explore the application of coda wave interferometry (CWI) on selected near-surface icequakes in Gornergletscher to estimate the
GF which could not be obtained from traditional processing of icequake cross-correlations because of lacking sources in the stationary phase zones of
the seismic array (Fig.

The strongest 720 events chosen out of more than 24 000 icequakes detected at Gornergletscher exhibit a sustained coda with approximate duration of
1.5

Figure

We first apply a standardized CWI processing scheme following

Figure

For the pair of closer stations (Fig.

Focusing on a complex scattering medium at the glaciated Erebus volcano (Antarctica),

The overall processing and technical details of coda window optimization are described in

MCMC processing involves several parameters that need to be tuned. As for traditional ICC processing, we need to define the frequency band of analysis
(here 10–30

Figure

Figure

MCMC processing coherently increases the presence of energy at zero lag time (Fig.

On the one hand, spurious arrivals at times of 0 or later could result from seismic reflections on the glacier bed beneath the stations, early aftershocks, or other noise sources if not in the stationary phase zones. A certain portion of the icequake coda may still be influenced by background noise especially at distant stations from the event source as the coda time window of one station may fall in the noise window of the further one.

On the other hand, spurious arrival contributions will not vanish in case of localized scatterers around the seismic array if the incident waves do
not illuminate the scatterers with equal power from all directions because of limited source aperture

In general, CWI must be processed on carefully selected events which show sustained coda above the background noise. We try coda wave correlations on the Argentière node grid and notice an influence of the source position on the retrieved CWCC likely because we did not select strong enough signals with sustained coda that is coherent enough across the sensors. Similarly, GF convergence does not work for weak Gornergletscher icequakes. Moreover, the abundance of seismic sources in glaciers often pollutes coda wave seismograms. We often find the situation where ballistic body and surface waves generated by early aftershocks from repetitive and subsequent events (or bed reflections) arrive at the seismic sensor only a few milliseconds after the onset of the first event of interest and therefore fall in its coda window. This typically introduces anisotropic wave fields. The brief icequake coda duration, the interevent time distribution, and the weak scattering in glacial ice impose limitations of CWI on large arrays.

To conclude, even if limited, the extraction of GF from icequake coda waves allows imaging of a glacier subsurface between station pairs. In principle, this can be done even in cases when (skewed) distribution of icequake sources or sustained noise sources does not allow for GF estimation.

The three methods proposed in this study could theoretically be applied to each one of the explored datasets but were not further tested here. The
standard processing schemes proposed in Sect.

Eigenspectral equalization of the cross-spectral matrix of ambient noise seismograms (Sect.

In the absence of distributed noise sources, the use of icequakes is a good alternative, especially in winter when the glacier freezes, preventing
generation of coherent water flow noise. As icequakes propagate to a few hundred meters, ICC studies are well suited for medium-size arrays with an
aperture typically of the order of 500

In the following we discuss the type of array deployment, geometry, and measurements suitable for structural and monitoring studies using the GF obtained by either one of the processing methods described above.

The performance of an array for imaging the structure at depth first relies on its geometry and secondly on the wave field characteristics as discussed in
Sect.

Seismic velocity measurements can additionally be complemented by other types of observations computed on the horizontal and vertical components of
the GF, such as the horizontal-to-vertical (H

Repeated analysis of cross-correlations allows us to detect changes in the subsurface properties. Seismic velocity monitoring is usually performed in
the coda part of the cross-correlation function through the application of CWI, as multiple scattered coda waves are less sensitive to the source
distribution and travel larger distances, accumulating time delays

CWI computed on cross-correlation functions could lead to the monitoring of englacial crevasses, failure of calving icebergs, glacial lake outburst
floods, break-off of hanging glaciers, surface mass balance, and bed conditions such as the evolution of the glacier hydraulic system and subglacial
till properties. Such topics are currently investigated with active seismic experiments or through the spatiotemporal evolution of passive seismicity and
associated source mechanisms

Unfortunately, the weak ice scattering limits the emergence of a coherent coda in the correlation functions which appear to still be affected by the
changing nature of primary ambient noise sources

This study explores the application of seismic interferometry on on-ice recordings to extract the elastic response of the glacier subsurface beneath one array deployment. In contrast to ambient noise studies focusing on the Earth's crust, the GF retrieval from cross-correlations of glacier ambient seismicity is notoriously difficult due the limited spatial coverage of glacier point sources and the lack of seismic scattering in homogeneous ice. We investigate the GF emergence on three particular cases. We design processing schemes suitable for each configuration of seismic deployment, wave field characteristics, and glacier setting.

In Glacier d'Argentière (Sect.

On the GIS (Sect.

In Gornergletscher (Sect.

The capability of extracting accurate seismic velocities on a line of receivers can be substantially improved when stacking the correlation functions on a large number of receiver pairs. However, MFP and CWI allow for new kinds of measurements on sparse seismic networks and enable the speedup of the GF convergence for non-idealized seismic illumination patterns which commonly arise in glacier settings.

Finally, the use of nodal sensor technology enables fast deployment of large N arrays suitable for seismic interferometry studies. This opens up new ways of characterizing and monitoring glacial systems using continuous passive seismic recordings.

There exist a wide range of seismic sources in glaciers as well as detection schemes

Seismic waveforms of surface icequakes generally exhibit a first low-amplitude P wave followed by impulsive Rayleigh waves (Fig.

The most broadly used algorithm in weak-motion seismology and on glaciers for detecting impulsive events

As icequakes usually propagate to distances of a few hundred meters before attenuating to the background noise level, the number of identified events
varies with the network configuration and the minimum number of stations to require a trigger concurrently. For the Gornergletscher study, we work with
events that have been detected by running a STA

The vast majority of icequakes recorded on glaciers are localized near crevasses that extend no deeper than

To be able to record coherent icequake waveforms with high enough signal-to-noise ratios (SNR) at couples of sensors, the network aperture should be
less than 1

In the same spectral band that is used for event detection, icequake signals are first windowed around the Rayleigh wave and cross-correlated for each pair of stations to obtain time delays. Time delay measurements are then refined to subsample precision by fitting a quadratic function to the cross-correlation function centered on its discrete maximum. The best easting and northing coordinates of the source are concurrently inverted with the apparent propagation velocity to match the time delay catalogue for preselected pairs of stations. We only consider time shift measurements at pairs of stations whose cross-correlation maximum is above 0.8. This allows us to minimize complex source and/or propagation effects on seismic waveforms and the observed arrival times that can not be fit with the oversimplified velocity model.

The inversion process is an iterative procedure using a quasi-Newton scheme

A seismic network is called an array if the network aperture is shorter than the correlation radius of the signals, which is the maximum distance
between stations at which time series are coherent, i.e. typically less than 1

Dense sensor arrays have many advantages as the SNR can be improved by summing the individual recordings of the array stations. Compared to a single sensor or couples of sensors, array processing techniques, such as beam forming, allow for time domain stacking which constructively sums coherent signals over the sensors and cancels out incoherent random noise, enhancing the signal detection capability.

Continuous data are scanned through matched-field processing (MFP) which involves time domain beam forming

MFP was successfully used by

The MFP method of

Because dispersive surface waves of different frequencies propagate at different speeds, computation of seismic velocities generally involves Fourier analysis to decompose the wave into frequencies that compose it. One can distinguish two types of wave speeds.

The phase velocity

If the harmonic waves of different frequencies propagate with different phase velocities, the velocity at which a wave group propagates differs from
the phase velocity at which individual harmonic waves travel

In ambient noise tomography, it is common to measure the group velocity of dispersive Rayleigh waves traveling in the Earth's crust and upper mantle

In glaciers, due to homogeneous ice, only weakly dispersive surface waves are recorded at on-ice seismometers. It is then difficult to use FTAN to
measure Rayleigh wave group velocity dispersion. We choose here to compute the phase velocity dispersion curve for Rayleigh waves using different
approaches. Obtaining the group velocity from the phase velocity is then straightforward while the reverse is not possible due to unknown additive
constants which arise from the integration of the phase velocity over frequency (Eq.

Frequency–wavenumber analysis (f-k) was used to compute the velocity dispersion curves at Glacier d'Argentière (Sect.

The most basic f-k processing employs a 2-D Fourier transform on both time and spatial components to construct the f-k diagram (Fig.

There exist multiple array techniques for computing frequency–velocity diagrams using spectral analysis in time and space domains. Some of them are
described in

The performance of an array for deriving phase velocity values in a wavenumber or frequency range depends on its geometry and on the wave field
characteristics (i.e. frequency range and magnitude of seismic energy with respect to attenuation). The capability to resolve phase velocity at
a given frequency depends on the array aperture (described by the array diameter

Computation of Rayleigh wave phase velocity from f-k analysis of the noise cross-correlation section obtained at the Argentière array

Same as Fig.

This method was used to compute the Rayleigh wave velocity at the GIS and Gornergletscher arrays (Sects.

Application of this method is presented in Fig.

Computation of Rayleigh wave phase velocity using Aki's spectral method.

The dispersion curve estimation can be refined by fitting the entire cross spectrum with a Bessel function, instead of fitting the zero crossings
only. We develop an approach similar to that of

The prior dispersion curve has to be as close as possible to the measured dispersion curve to avoid cycle skipping during the fitting. In our
procedure, we take as a starting model the average dispersion curve obtained from f-k analysis of the cross-correlation section computed at the
Argentière array. For sparse network configuration (as in Greenland and Gornergletscher, i.e. Figs.

Figure

In the example shown in Fig.

This method employed by

The plane wave approximation implies a sinusoidal dependence of the arrival times which depend on the source azimuth and propagation velocity

To measure seismic velocities at one station path and at different frequencies, we filter the individual correlation functions computed for each event
to octave-wide frequency ranges. The lower frequency we can resolve is determined by the icequake spectral content and is most importantly related to
the station separation as we require that at least two wavelengths are sampled. We then restrict the analysis to 15–30

We assign all cross-correlations to event azimuth bins of 5

The velocity solution estimated by this method is naturally averaged over the azimuth range and can only be considered as the average velocity in the
presence of strong azimuthal anisotropy which implies azimuthal variations in propagation velocities (Sect.

To minimize the effects of location errors or low SNR of the correlation components, we perform a jackknife test on randomly selected events to fit the sinusoid. We require that the maximum of the cross-correlation stacked over bootstrap samples (i.e. selected correlation functions that have been shifted by the inverted arrival times prior to stacking) exceeds 0.7. We require that at least 10 azimuth bins including the endfire lobes are considered in the sinusoidal fit. The final velocity at each frequency is then averaged over 200 jackknife tests.

Seismometer data from Gornergletscher and GIS are part of the 4-D glacier seismology network
(

PR and FG designed the Argentière experiment in the scope of the RESOLVE project (

The authors declare that they have no conflict of interest.

We gratefully acknowledge all the people involved in the RESOLVE project and who participated in the array deployment in Argentière and collected and processed raw formats of seismic data. We thank Olivier Laarman, Bruno Jourdain, Christian Vincent, and Stéphane Garambois for having constructed bed and surface DEMs using ground-penetrating radar for the bed and drone data for the surface. We also thank Léonard Seydoux for insightful discussions on MFP and SVD. We acknowledge the constructive comments from the editor Evgeny Podoloskiy, Naofumi Aso, and the anonymous reviewer.

This work was funded by the Swiss National Science Foundation (SNSF) project Glacial Hazard Monitoring with Seismology (GlaHMSeis, grant PP00P2_157551). Additional financial support was provided to AS by the Swiss Federal Institute of Technology (ETH Zürich). The Observatory of Sciences of the Universe of Grenoble funded the project RESOLVE for the data acquisition at Argentière. SNSF and ETH Zürich participated in the data collection in Greenland (grants 200021_127197 SNE-ETH and 201 ETH-27 10-3) and on Gornergletscher (grants 200021-103882/1, 200020-111892/1).

This paper was edited by Evgeny A. Podolskiy and reviewed by Naofumi Aso and one anonymous referee.