A preferred orientation of the anisotropic ice crystals influences the viscosity of the ice bulk and the dynamic behaviour of glaciers and ice sheets. Knowledge about the distribution of crystal anisotropy is mainly provided by crystal orientation fabric (COF) data from ice cores. However, the developed anisotropic fabric influences not only the flow behaviour of ice but also the propagation of seismic waves. Two effects are important: (i) sudden changes in COF lead to englacial reflections, and (ii) the anisotropic fabric induces an angle dependency on the seismic velocities and, thus, recorded travel times. A framework is presented here to connect COF data from ice cores with the elasticity tensor to determine seismic velocities and reflection coefficients for cone and girdle fabrics. We connect the microscopic anisotropy of the crystals with the macroscopic anisotropy of the ice mass, observable with seismic methods. Elasticity tensors for different fabrics are calculated and used to investigate the influence of the anisotropic ice fabric on seismic velocities and reflection coefficients, englacially as well as for the ice–bed contact. Hence, it is possible to remotely determine the bulk ice anisotropy.

Understanding the dynamic properties of glaciers and ice sheets is one
important step to determine past and future behaviour of ice masses. One
essential part is to increase our knowledge of the flow of the ice itself.
When the ice mass is frozen to the base, its flow is primarily determined by
internal deformation. The degree thereof is governed by the viscosity (or the
inverse of softness) of ice. The viscosity depends on different factors, such
as temperature, impurity content and the orientation of the anisotropic ice
crystals

Ice is a hexagonal crystal (ice Ih) under natural conditions on earth. These
ice crystals can align in specific directions in response to the stresses
within an ice mass. A preferred orientation of the ice crystals causes the
complete fabric to be anisotropic, in contrast to a random distribution of
the ice crystals where the ice is isotropic on the macroscopic scale. This
fabric anisotropy influences the viscosity of the ice. The shear strength is
several orders of magnitude smaller perpendicular to the ice crystal's

The influence of anisotropic ice fabric on the flow behaviour of ice can
directly be observed in radar profiles from ice domes. At ice domes and
divides a prominent feature of flow conditions is a Raymond bump

A second prominent feature in radar data is the basal layer. Before the
advent of multi-static, phase-sensitive radar systems, the basal layer had
usually been observed only as an echo-free zone (EFZ). The onset of it was
connected to the appearance of folds in ice cores on a centimetre scale

With increasing computational power the incorporation of anisotropy into ice flow models becomes feasible in three dimensions as well as on regional scales. However, to include anisotropy in ice-flow modelling, we need to understand the development and the distribution of the anisotropic fabric; i.e. we have to observe the variation in the COF distribution over depth, as well as the lateral extent. To extend our ability to determine the influence of these properties on ice flow and map them laterally beyond the 10 cm scale of ice cores, we have to advance our knowledge of the connection between microscale properties and macroscale features on the scale of tenths of a metre to hundreds of metres observed with geophysical methods like radar and seismics.

The standard method of measuring the COF distribution is to analyse thin
sections from ice cores under polarised light. The anisotropy is then
normally given in the form of the sample-averaging eigenvalues of the
orientation tensor

Further, the anisotropic fabric has an influence on the wave propagation of
seismic waves. Hence, by analysing COF-induced reflections and travel times
the anisotropic fabric on the macroscale can be determined. Not only the
longitudinal (P) pressure waves but also the transverse waves, i.e. the
horizontal (SH) and vertical (SV) shear waves, can be analysed here for the anisotropic
fabric. One of the first studies of seismic anisotropy in the
context of ice crystal anisotropy was the PhD thesis of

These methods have one shortcoming. They limit the analysis of anisotropy of
seismic waves to the analysis of the travel times, i.e. seismic velocities.
The influence of anisotropy has not only been observed in seismic velocities.
Englacial reflections were also observed in seismic data from Antarctica

One way to improve the analysis of seismic data is to apply full waveform inversion algorithms, i.e. the analysis of the complete observed wave field and not only quantifiable characteristics such as reflection strength or travel times, which is gaining more and more importance in applied geophysics in general. If we want to be able to investigate and understand the influence of the anisotropic ice fabric on the seismic wave field and develop ways to derive information from travel times and reflection signatures about different anisotropic ice fabrics from seismic data, we need to be able to derive the elasticity tensor for different COF distributions.

In this paper we extend the analysis of seismic velocities beyond cone fabrics and derive the elasticity tensor, which is necessary to describe the seismic wave field in anisotropic media. The description of seismic wave propagation in anisotropic materials is based on the elasticity tensor, a 4th-order tensor with 21 unknowns in the general case of anisotropy. If the elasticity tensor is known, seismic velocities, reflection coefficients or reflection angles can be calculated. From ice core analysis one normally gains the COF eigenvalues describing the distribution of the crystal orientations. Hence, we first need a connection between the COF eigenvalues and the elasticity tensor.

We present a framework here to derive the elasticity tensor from the COF
eigenvalues for cone as well as different girdle fabrics. We derive opening
angles for the enveloping of the

The different ice crystal distributions as used for the calculation
of seismic velocities and reflection coefficients. Given are the sketches for
the enveloping of the

The ice crystal is an anisotropic, hexagonal crystal with the basal plane
perpendicular to the ice crystal's

Different fabric distributions were discussed by

The standard method of measuring COF distributions is by analysing thin
sections from ice cores under polarised light using an automatic fabric
analyser

Another possibility to describe the anisotropic fabric is to calculate the
spherical aperture from the orientation tensor. Hence, the

Wavefront of a P wave travelling in isotropic ice fabric (dashed
line) and in a vertical single maximum (VSM) fabric (red line), i.e. a
vertical transversely isotropic (VTI) medium. The solid arrow shows the group
velocity with group angle

The propagation of seismic waves is influenced by the anisotropic material,
affecting seismic velocities, reflection coefficients and reflection
angles, among other properties. The propagation of wavefronts in the
anisotropic case is no longer spherical. Figure

For an anisotropic medium the linear relationship between tensors of stress

To determine seismic velocities in anisotropic media, a solution for the wave
equation needs to be found. Given here is the wave equation for homogeneous,
linear elastic media, without external forces and with triclinic anisotropy:

Finally, three non-trivial solutions exist for this eigenvalue problem, giving the three phase velocities and vectors for the quasi compressional (qP), the quasi vertical (qSV) and the quasi horizontal shear (qSH) wave. The phase vectors are orthogonal to each other. However, qP and qSV waves are coupled, so the waves are not necessarily purely longitudinal or shear waves outside of the symmetry planes. Therefore, they are additionally denoted as “quasi” waves, i.e. qP, qSV and qSH waves. As the following analyses are mostly within the symmetry planes, the waves will from now on be denoted as P, SV and SH waves. Nevertheless, outside of the symmetry planes this term is not strictly correct.

To be able to find analytical solutions to the Christoffel matrix, the
anisotropic materials are distinguished by their different symmetries.
Additionally, to simplify calculations with the elasticity tensor, we will use
the compressed Voigt notation

From the analysis of ice cores we determine the COF eigenvalues which describe the crystal anisotropy over depth. The propagation of seismic waves in anisotropic media can be calculated from the elasticity tensor. Hence, a relationship between the COF eigenvalues and the elasticity tensor is needed.

For the following derivation of the elasticity tensor we will use two opening
angles for the description of the fabric that envelopes the

The two opening angles determine the kind of fabric
(Table

We will use a measured monocrystal elasticity tensor here to calculate the
elasticity tensor for the different observed anisotropic fabrics in ice from
the COF eigenvalues. For monocrystalline ice the components of the elasticity
tensor have been previously measured by a number of authors with different
methods. For the following calculations we use the elasticity tensor of

When the COF eigenvalues are derived, the information on the fabric
distribution is significantly reduced, especially as the corresponding
eigenvectors are normally unknown. Hence, it is not possible to determine the
elasticity tensor with at least five unknowns directly from the three COF
eigenvalues. Therefore, we first subdivide the observed anisotropies into
different fabric groups (cone, thick girdle and partial girdle fabric) by
means of the eigenvalues. Afterwards, we determine their opening angles
(Sect.

To differentiate between cone and girdle fabric

Girdle fabrics classified as HTI media are within the
[

In the next step the remaining, unknown opening angle for the different
fabrics needs to be calculated from the eigenvalues, i.e.

The elasticity tensor of the polycrystal can now be derived using the
measured elasticity tensor for a single ice crystal and the derived angles

To obtain the elasticity tensor of the anisotropic polycrystal

Steps for calculation of elasticity tensor
(Eq.

The different rotation directions to calculate the polycrystal elasticity
tensor

By comparing the individual components of the elasticity tensor derived
following

To be able to calculate the opening angels from the COF eigenvalues, the
fabrics are classified into the different fabric groups based on their
eigenvalues: cone, thick girdle and partial girdle fabric
(Table

For the calculation of the anisotropic polycrystal from the monocrystal
neither grain size nor grain boundaries are considered.

The resultant polycrystal elasticity tensors depend of course on the choice
of the monocrystal elasticity tensor. Different authors have measured

From the derived elasticity tensor we can now calculate velocities and reflection coefficients. Many approximations as well as exact solutions exist for the calculation of velocities and reflection coefficients for different anisotropic fabrics. They are mostly limited to certain symmetries.

In the case of velocities, most studies have been performed on VTI media

For the calculation of the reflection coefficient we use exact

For the special case of wave propagation in ice with a developed cone fabric
anisotropy,

Using the derived elasticity tensor, we are now able to calculate velocities
for different COF distributions. We use the equations derived by

From these phase velocities we have to calculate the group velocities for the
calculation of travel times. The calculation of the group velocity vector

Phase (dashed lines) and group velocities (solid lines) over the
corresponding phase

Figure

P-wave phase velocities over phase angle

By deriving the elasticity tensor for different fabrics, the group and phase
velocities of the P, SH and SV wave for these fabrics can now be calculated.
Figure

Figure

The partial girdle (

It should also be noted that for a thick girdle with

The higher velocities calculated with the equations of

The calculation of reflection coefficients for different incoming angles is
already rather complicated for layered isotropic media given by the Zoeppritz
equations

In the following, we use equations derived by

To calculate the P-wave reflection coefficient for the bed reflector with an
overlaying cone fabric, i.e. VTI media, we use the equations given by

Reflection coefficients for the boundary between an isotropic
(upper) layer and a partial girdle fabric (lower) layer with different
opening angles

With the equations given in Appendix

The reflection coefficients are given for angles of incidence between
0 and 60

P-wave reflection coefficients for ice–bed interface with different
bed properties as a function of phase angle of incidence

The largest magnitude of reflection coefficients can be observed for the
SVSV reflection (Fig.

For englacial reflections caused by changing COF, the variations in the
reflection coefficient with offset are very small: the PP-reflection
coefficient for the transition from isotropic to VSM fabric
(

P-wave velocity, S-wave velocity and density for different bed
scenarios and isotropic ice as given in

Of special interest is the determination of the properties of the ice–bed
interface from seismic data. It is possible to determine the bed properties
below an ice sheet or glacier by analysing the normal incident reflection
coefficient

Exact solutions are calculated using the equations given by

The differences between the isotropic (solid lines) and anisotropic
reflection coefficients (dashed lines) are small (

The observable differences of reflection coefficients for an isotropic and a
VSM-fabric overburden are

We presented an approach to derive the ice elasticity tensor, required for
the calculation of seismic wave propagation in anisotropic material, from the
COF eigenvalues derived from ice-core measurements. From the elasticity
tensors we derived seismic phase and group velocities of P, SH and SV waves
for cone, partial girdle and thick girdle structures, i.e. orthorhombic
media. Velocities we derived for different cone fabrics agree well with
velocities derived for cone fabric using the already-established method of

We used the elasticity tensor to derive the reflection signature for englacial fabric changes and investigated the influence of anisotropic fabric on the reflection coefficients for basal reflectors. We found that the reflection coefficients and the variations of the reflection coefficients with increasing offset are weak for the transition between different COF distributions: they are at least an order of magnitude smaller than reflections from the ice–bed interface. Thus, either significant changes in the COF distribution or extremely sensitive measurement techniques are needed to observe englacial seismic reflections. The influence of anisotropic ice fabric compared to the isotropic case for the reflection at the ice–bed interface is so small that it is within the measurement inaccuracy of currently employed seismic AVO analysis. An important result is that the difference between exact and approximate calculations of reflection coefficients for the ice–bed interface is larger than the influence of an anisotropic ice fabric above the bed. This implies that exact calculations are necessary if the fabric above the bed is in the focus of AVO analysis.

Better results in the calculation of the elasticity tensor could probably be
gained by calculation of the opening angles directly from the

The following equations give the connection between the eigenvalues

For a cone fabric the angle

For a thick girdle fabric the angle

For a partial girdle fabric the angle

Here the rotation matrix for the elasticity tensor and compliance tensor
following

The rotation matrix for the elasticity tensor is

The components of the polycrystal elasticity tensor as derived from
Eq. (

The components of the polycrystal compliance tensor as derived from
Eq. (

To be able to calculate velocities for partial girdle
fabric, the calculation of phase velocity for orthorhombic media derived by

The components of the group velocity vector are given by

Outside the symmetry planes of the HTI media the component

The reflection coefficients as derived by

We thank T. Bohlen and his group of the Geophysical Institute, Karlsruhe Institute of Technology, for their support and numerous hints during this study. Financial support for this study was provided to O. Eisen by the Deutsche Forschungsgemeinschaft (DFG) “Emmy Noether” programme grant EI 672/5-1. We thank A. Brisbourne and the anonymous reviewer for their comments which greatly helped to improve the manuscript and H. Gudmundsson for editing of this manuscript. Edited by: H. Gudmundsson