One of the great challenges in glaciology is the ability to estimate the bulk
ice anisotropy in ice sheets and glaciers, which is needed to improve our
understanding of ice-sheet dynamics. We investigate the effect of crystal
anisotropy on seismic velocities in glacier ice and revisit the framework
which is based on fabric eigenvalues to derive approximate seismic velocities
by exploiting the assumed symmetry. In contrast to previous studies, we
calculate the seismic velocities using the exact

One of the most important goals for glaciological research is the
establishment of a thorough understanding of ice dynamics, in which internal
deformation plays a crucial role. This deformation is predominantly evident
and described on a macro-scale (

Currently, the development and extent of fabric anisotropy are
mainly investigated by laboratory measurements on ice-core samples which
provide one-dimensional data (along the core axis,

Our main objective is to present an improved method for the estimation
of the bulk elasticity tensor, and to use this to (i) evaluate the use of the

We first present experimental measurements, a theoretical basis and a mathematical
algorithm for our new framework (

The global coordinate system {

For our analysis of seismic velocities we use fabric data from the polar EDML
ice core and the alpine KCC ice core. The EDML ice core was
drilled as part of EPICA (European Project for Ice Coring in Antarctica)
between 2001 and 2006 at Kohnen Station, Antarctica, and reaches a depth of
2774 m

Vertical and horizontal thin sections of the ice cores were prepared and measured using polarised light microscopy

Figure

The EDML fabric data

Eigenvalues

In a glacier, the fabric anisotropy also introduces an anisotropy of the elastic properties of the material. This elastic anisotropy results in a seismic anisotropy, which means the propagation of seismic waves is influenced by the fabric anisotropy. To study this connection, theoretical velocities can be calculated if the fabric anisotropy is known.

The mathematical background for the calculation of seismic phase velocities from
the elastic properties in anisotropic ice can be found in many publications

For an anisotropic elastic medium, ice behaves elastically for the propagation of seismic waves. Stress and strain are linearly connected following the generalised Hooke's law:

To apply this description to the study of large ice sheets and glaciers, we
have to consider the elastic properties of the polycrystal. The understanding
of the elastic behaviour of a monocrystal can be used together with the
fabric description to estimate the elastic properties of the polycrystal.
Different theoretical models have been developed for the estimation of the
elasticity tensor of an anisotropic polycrystal, usually making use of fabric
symmetries

For the calculation of the polycrystal elastic properties from anisotropic monocrystal properties, the concept of Voigt–Reuss bounds is
often used

Once the elastic properties for the polycrystal are known, the Christoffel
equation provides the relationship to calculate seismic velocities. For a
linearly elastic, arbitrarily anisotropic homogeneous medium, the wave
equation is solved by a harmonic steady-state plane wave and we obtain the
Christoffel equation:

Equation (

Instead of interval velocities, often the root mean square (rms) velocity

In situ temperature and density are essential when comparing seismic velocities. However, as this study is focused on the comparison of calculation frameworks that use the same elastic moduli, a temperature correction is not applied.

The

The fabric data in the standard parameterisation of second-order orientation
tensor eigenvalues are sorted into three fabric classes (cone, thick girdle,
partial girdle), where each is defined by one or two opening angles

The opening angles characterising the fabric of each sample are used to
integrate the elasticity tensor of the monocrystals,
Eq. (

From the polycrystal elasticity tensor the approximative solutions to the
Christoffel equation (Eq.

The advantages of this approach are as follows

Eigenvalues are a standard parameter for expressing the strength of fabric
and can be directly used for the

By assuming an orthorhombic symmetry the solution to the Christoffel equation can be readily found. No information on the azimuthal orientation of the ice core (relative to any seismic measurements on a glacier) is needed, although this could be considered to improve the results for girdle fabric.

However, some uncertainty is inherent in the framework:

The eigenvalues of the second-order orientation tensor do not constitute a complete and unambiguous description of the fabric. Specifically, they do not provide information on preferential orientations with regard to the coordinate system. To get a rough idea about the orientation of the fabric the eigenvectors would have to be used in addition, an approach seldom followed. Instead, the orientation of the eigenvector to the largest eigenvalue is typically assumed to correspond to the vertical, which may in fact not be the case and could introduce an unknown uncertainty.

By assuming an orthorhombic symmetry while using opening angles to describe
the

Fabric data from ice cores indicate that transitions between fabric classes
usually occur gradually, and sudden changes are only expected to occur due to
changes in impurity content or deformation regime

In this study we aim to provide a quantitative estimate of the error
introduced by the approximation of the

Flow chart to illustrate the steps for both frameworks. The workflow
for the

If not indicated otherwise, elasticity/compliance tensors and velocities are
calculated for the effective medium, which, in this study, is typically
represented by a thin section comprising a number of grains (

A data set of COF measurements from an ice core
is considered, which gives pairs of angles determining the

The phase velocities

The frameworks (

We apply the

We now assess the velocity difference between the eigenvalue and the

The evolution of the fabric of the EDML ice core becomes apparent when
assessing the eigenvalues (Fig.

The general trends in the velocities of the two frameworks are in good agreement
(Fig.

In the upper 1785 m the velocity from the

Figure

Standard deviation of mean interval P-wave velocities at vertical incidence for several depth intervals of the EDML ice core.

Summary of the results from the seismic velocity comparison between
frameworks. Values are calculated depth-profile average (with standard
deviation) and/or extreme (

(

Comparison of zero-offset velocities calculated from EDML fabric
data (without high-resolution samples) via

Comparison of P-wave velocities at vertical incidence, calculated
from EDML fabric data measured at high resolution (vertical sections) between
2358 and 2380 m depth with the fabric analyser G50. The same
variables as in Fig.

Comparison of zero-offset velocities calculated from KCC fabric data
via

Seismic P-wave velocities for KCC calculated with

We show the results of the velocity calculations for vertical incidence from
the KCC fabric data in Fig.

During typical seismic profile surveys the seismic wave will have an inclined
angle of incidence with respect to the vertical, normal to the glacier
surface. The velocities will vary depending on the incidence angle if the
medium is anisotropic and this will affect the recorded travel times

In the following we assess how the seismic velocities will
change when the ice-core fabric data and the seismic plane of incidence are
rotated with respect to each other. The zero orientation
(

The change in the P-wave velocity with an increasing angle of incidence and
rotated seismic plane as calculated with the azimuth-sensitive

Although the slower S waves are not routinely acquired during seismic imaging in polar
environments, they provide a better resolution and are of special interest
for the study of the elastic properties of ice from traditional seismic
reflection profiles

For the EDML core, no information on the core pieces' azimuth angles relative
to the ice sheet or to each other is provided. However, it is assumed that no
sudden short-scale change in the flow regime can occur. Thus, abrupt offsets
in the girdle orientation must be caused by unnoticed rotation of the core
pieces. Prior to the application of the

As the

The

By using the fabric data from thin sections we acknowledge the uncertainty which arises from sampling with a relatively small sample size. We use less than 1 % of the EDML ice core and 11 % of the KCC ice core to infer the fabric development in the ice cores. There are currently no comprehensive data available to investigate the sampling effect on real ice. As we are concerned with the comparison of theoretical seismic velocities calculated from the same fabric data, we assume that the sampling uncertainty can be neglected. For the comparison with measured seismic data, the uncertainty needs to be considered, and appropriate density and temperature corrections are required.

The observed variation in eigenvalues
in the EDML ice core (Fig.

The currently employed algorithms for the calculation of seismic
velocities in ice polycrystals on the crystal scale (including this study) do
not consider any possible effects on the grain boundaries. For laboratory
measurements the difference in stress on a polycrystalline ice sample
compared to in situ conditions can affect the degree to which grains are
bonding and thus the elasticity

The lack of knowledge about the dispersion of seismic waves in ice introduces an unknown uncertainty to the calculation based on a monocrystal elasticity tensor that was measured in the laboratory by means of ultrasonic sounding. Again, for the application of ultrasonic methods, which operate in the same frequency range, this uncertainty can be neglected. The connection of fabric and seismic velocities on the crystal scale we present here complements this advancing field of study.

We have shown in Sect.

In the case of asymmetric

An advantage of the

Potentially, our framework can be used in principle for the development of inverse methods to derive the fabric distribution from seismic velocities. Following experience from other fields of active seismology, this would first most likely require comprehensive data sets suitable for full-waveform inversion not yet available for glaciological applications, and second some simplifying assumptions on the distribution of crystal fabric, e.g. in terms of considered symmetries. The framework we presented allows us to quantify the potential effect of simplifying assumptions and could help to more accurately specify covariance matrices, thus enabling the quantification of uncertainties along with the results produced by the application of an inverse method.

The

A requirement of the

We use COF measurements on a submetre scale for our analysis of seismic
velocities. The results suggest the existence of closely spaced reflective
surfaces for elastic seismic waves (and also radar waves). The relevance of
the presented analysis for real seismic data is based on the major assumption
of a laterally extended and coherent fabric layering on the scale of the
first Fresnel zone

Far more fabric data than are
currently sampled in ice-core studies are required to pursue this hypothesis
in the future. To this end, ultrasonic methods can be applied in ice-core
boreholes

Following the findings of our study, for seismic data acquisition in the field we
recommend (1) considering polarimetric survey set-ups (with two or even more cross lines) with both reflection and wide-angle measurements and
(2) focusing on accurate travel time recordings at high frequencies of the seismic source.
This should be
supported by three-component vertical seismic profiling where boreholes are
available. Also, S waves should be acquired as they provide useful
information on crystal anisotropy due to shear-wave splitting. On the crystal
scale, we suggest to include an investigation on the possible influence of
variations in grain size for the elastic wave propagation in polycrystalline
ice, which is currently not considered for theoretical calculations, to
complement recent work on the temperature dependency of elastic properties

The presented

We found that the azimuthal change in P-wave velocity and shear-wave splitting can be
as large as

The results of our study demonstrate for the first time that
short-scale variability in anisotropic fabric as observed in these polar and
alpine ice cores causes corresponding high short-scale variability in
seismic interval velocities. Current laboratory fabric measurements from an
ice core drilled on an ice stream also show early indications of high
fabric variability and unexpected fabric types (Jan Eichler, personal
communication, 2017), offering an ideal target for extending this study to an environment with another
deformation regime. Based on the presented evidence in this study the next
steps should include the investigation of how a succession of short-scale
fabric layers could induce englacial reflections, as has been reported and
hypothesised in earlier studies

As conventional surface-based seismic surveys are not likely to resolve this
short-scale variability, ultrasonic techniques for borehole and laboratory
studies could be the solution to both issues of lost core orientation and low
resolution. For this emerging field of applications, we offer further insight
into what to expect from crystal-orientation fabric anisotropy in ice.
Equally, our results can provide context for data collected with frozen-in
seismometers in boreholes, where evidence for shear-wave splitting on
non-vertical ray paths was found (David Prior, personal communication, 2018). Lastly, we
want to highlight that, while the depth scale of the KCC ice core differs from
that of the EDML ice core by a factor of

The fabric and eigenvalue data sets for the ice cores KCC
^{®} and are available upon request.

A fourth-order tensor rotation is expressed as

The general rotation matrix in three dimensions is given by the cosines
between the axes of local

For a coordinate transformation of the monocrystal elasticity tensor

It is possible to express both rotations in a single rotation matrix

By using Voigt notation, which mathematically implies a change in base, the rotation matrix

The rotation matrix

The expressions for

From the characteristic polynomial of Eq. (

For

The algorithm is implemented in MATLAB^{®} for this study.

The study was initiated and supervised by OE. The fabric data were collected and analysed by IW (EDML) and JK (KCC), and the data collection by JK was supervised by IW. Calculations were conducted by JK and supported by discussions with AD. The paper was written by JK, with comments and suggestions for improvement from all co-authors.

The authors declare that there are no competing interests.

We would like to thank the KCC drilling team with colleagues from the Institute of Environmental Physics (Heidelberg University, Germany), the Climate Change Institute (University of Maine, USA) and the Institute for Climate and Environmental Physics (University of Bern, Switzerland). Many thanks to Daniela Jansen for suggestions for the improvement of the manuscript. Johanna Kerch was funded by the Studienstiftung des deutschen Volkes and supported by HGF grant no. VH-NG-803 to Ilka Weikusat. We gratefully acknowledge Maurine Montagnat and Huw Horgan for their helpful comments which improved the manuscript and Martin Schneebeli for editing this manuscript. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Martin Schneebeli Reviewed by: Huw Horgan and Maurine Montagnat