Introduction
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 c axis
than parallel to it, as shown in laboratory studies .
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
. As ice is a non-Newtonian fluid, it is softer
and deforms more easily on the flanks of the ice dome or divide due to the
higher deviatoric stress there compared to the centre of the dome. Thus, the
vertical flow is slower at the dome itself than on the flanks. This leads to
an apparent upwarping of the isochronous layers. The development and
influence of anisotropic fabric on the flow of ice at divides and the effects
on the development of Raymond bumps were investigated by, for instance,
and . At ice divides features like double
bumps and synclines are observed , next to single bumps.
were able to reproduce these double bumps and synclines by including
anisotropic rheology in a full-Stokes model. Hence, they are presently considered
to be direct evidence of the existence of developed anisotropic fabric.
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
. Considerable progress in radar imaging over the last decade
make it now possible to image the very bottom layer of ice sheets
. The radar data show an often fuzzy basal layer, with
a rough upper surface and considerably disturbed coherency of radar return
power. The presence of the basal layer turns out to be widespread, especially
in Antarctica (CReSIS, P. Gogineni, personal communication, 2014). As the basal ice near
the bed is subject to higher stresses and elevated temperatures than the ice
above, it is the region where ice physical properties on the microscale
change most rapidly . These include changes in crystal
orientation fabric (COF) properties and distribution.
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 in discrete depth intervals. From this
we gain information about the local anisotropic conditions at the ice-core
location. Radar data have also been used to analyse the changing COF over
depth . The challenge
in analysing radar data is to distinguish the COF-induced reflections from
the numerous conductivity-induced reflections. This distinction is important
as conductivity-induced layers are isochrones; by following
conductivity-induced reflections in radar data, layers of equal age can be
followed over large distances. Currently, identifying and tracing undisturbed
layering is one of the main methods being used to identify the location of a
site for a potentially 1.5 My old ice core in East Antarctica
.
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 ,
who derived equations for the calculation of seismic velocities for solid
cone and surface cone fabrics. He fitted curves to the slowness surface
(inverse of the phase velocity) calculated from an elasticity tensor measured
by means of ultrasonic sounding. This was applied to data from Dome C,
Antarctica, by . investigated the
anisotropic ice fabric at Byrd Station, Antarctica, for which he used
ultrasonic logging. To determine the anisotropic seismic velocities for
different cone fabrics, he calculated an average from the single crystal
velocity for the encountered directions. This approach was used later by
for analysing the crystal anisotropy from borehole sonic
logging at Dome C, Antarctica.
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
and Greenland . These
reflections were interpreted as arising from an abrupt change in fabric
orientation. However, to analyse the reflection signature and determine the
actual change in COF, we first need an understanding of the reflection
coefficient for changing incoming angles for the transition between different
anisotropic fabrics.
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 c axis distribution from the COF
eigenvalues. We then integrate using a monocrystal elasticity tensor for
these derived distributions to obtain the elasticity tensor for the different
anisotropic fabrics (Sect. ). Based on these derived
elasticity tensors, we calculate seismic velocities and reflection
coefficients for different c axis distributions. As examples, we investigate
the compressional wave velocity variations with increasing angle for
different fabrics and the reflection coefficients for a change from isotropic
to girdle fabric for compressional and shear waves. Further, we analyse the
influence of anisotropy on the reflection signature of the ice–bed interface
and discuss these results in Sect. . This is the first part
of two companion papers. The calculations introduced here will be applied to
ice-core and seismic data from Kohnen Station, Antarctica, in Part II,
.
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 c axis distribution, the glaciological terms, the
common stress regime and the corresponding eigenvalue range. In the second
part the seismic term for the anisotropic regime is given together with the
opening angles derived from the COF eigenvalues to calculate the elasticity
tensor.
Fabrics
Glaciological context
Seismic context
Envelope
Term
Stress regime
Eigenvalues
Term
Opening angle
Isotropic
Uniform
λ1=λ2=λ3=1/3
Isotropic
φ=χ = 90∘
Cone (cluster in mineralogy)
Simple shear
λ1=λ2 λ3≥λ1,λ2
Vertical trans- versely isotropic (VTI)
φ = χ 0∘≤φ≤90∘
Vertical single maximum (VSM)
Simple shear
λ1=λ2=0
λ3=1
Vertical trans-versely isotropic (VTI)
φ = χ = 0∘
Thick girdle
Uniaxial compression, extension
λ2=λ3 λ1=1-2λ2
Horizontal trans-versely isotropic (HTI)
φ = 90∘ 0∘≤χ≤90∘
Partial girdle
Axial compression, extension
λ1=0 0≤λ2≤0.5 λ3=1-λ2
Orthorhombic
χ = 0∘ 0∘≤φ≤90∘
Ice crystal anisotropy
The ice crystal is an anisotropic, hexagonal crystal with the basal plane
perpendicular to the ice crystal's c axis. Due to the existing stresses
within glaciers and ice sheets, these anisotropic ice crystals can be forced
to align in one or several specific directions. In such cases the crystal's
c axis is oriented perpendicular to the main direction of stress
. Depending on the stress regime, different COF distributions
develop. Common stress regimes in glaciers are simple shear and uniaxial
stress (Table ). At ice domes simple shear can be
observed, such that the ice crystals orient towards the vertical; i.e cone
distributions can be found, also called cluster distributions in mineralogy.
At ice divides, with a main direction of extension and compression
perpendicular to that, ice crystals tend to orient in one plane, i.e. in
girdle distributions.
Different fabric distributions were discussed by ,
who classifies eight different fabric groups.
Of these we will use the three most common fabrics observed in glacier ice
in the following analysis of the influence of ice crystal anisotropy on seismic
wave propagation: (i) the cluster (or cone) distribution, (ii) the thick
girdle distribution and (iii) the partial girdle distribution. These
distributions are shown in Table . The sketches (first
row) show the enveloping of the specific c axes' distribution for the different fabrics. We will use the term cone fabric instead of cluster
fabric hereinafter, as it is the more commonly used term in glaciology.
The most extreme forms of anisotropy we can expect in ice are the isotopic
fabric, with a uniform distribution of ice crystals, and the vertical single
maximum (VSM) fabric, where all ice crystals are oriented vertically. Note that
the term “lattice-preferred orientation (LPO)” is used as well in the literature
to refer to the orientation of the crystals , in addition to
COF.
Crystal orientation fabric measurements
The standard method of measuring COF distributions is by analysing thin
sections from ice cores under polarised light using an automatic fabric
analyser . The c axis orientation of each
single crystal is determined and can be given as a unit vector (c).
These orientations can be presented in Schmidt plots, an equal-area
projection of a sphere onto a plane, or as eigenvalues λ1,λ2,λ3 of the weighted orientation tensor:
Aij=W∑l=1n(cicj)l,withi,j=1,2,3.
The number of grains is given by n, and W is a weighting function, with
weighting e.g. by grain number (W=1/n) or by area. The three eigenvalues,
with λ1≤λ2≤λ3 and ∑λi=1,
determine the extension of a rotation ellipsoid. The corresponding
eigenvectors cannot be given when the orientation of the ice core within the
borehole is not measured in geolocated directions. Hence, the direction to
which these eigenvalues apply is often unknown.
Another possibility to describe the anisotropic fabric is to calculate the
spherical aperture from the orientation tensor. Hence, the c axis
distribution is given in the form of one opening angle for the enveloping
cone . However, this limits the analysis of anisotropy
to cone fabrics (Table ).
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 θ, the dashed arrow the phase velocity with
phase angle ϑ for the anisotropic case.
Seismic anisotropy
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
shows the anisotropic wavefront for a P wave travelling in a VSM fabric (red
line) and the spherical wavefront for a P wave in isotropic ice fabric
(dashed black line). For the anisotropic case, group and phase velocity, as
well as group angle θ and phase angle ϑ, are no longer the
same. The group velocity determines the travel time. The phase velocity vector
is normal to the wavefront. Thus, the phase velocity and phase angle
ϑ are needed for the calculation of reflection and transmission
angles as well as reflection coefficients in anisotropic media.
For an anisotropic medium the linear relationship between tensors of stress
σmn and strain τop is described by Hooke's law:
σmn=cmnopτop,
with the elasticity tensor cmnop and m,n,o,p=1,2,3. In the isotropic
case these 81 components of the elasticity tensor can be reduced to the two
well-known Lamé parameters. In the general anisotropic case, symmetry
considerations of the strain and stress tensors apply, as do
thermodynamic considerations . Hence, the general anisotropic
elasticity tensor consists of 21 independent components and is referred to as
triclinic.
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:
ρ∂2um∂t2-cmnop∂2uo∂xn∂xp=0,
with ρ the density of the material, t time, the components um and
uo of the displacement vector u and the different spatial
directions xn, xp. Solving this equation leads to an eigenvalue
problem, the Christoffel equation. For a detailed derivation see,
e.g. .
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 for the elasticity tensor
cmnop→Cij. Therefore, the index combinations of mn and
op are replaced by indices between 1 and 6 (11≡1, 22≡2,
33≡3, 23≡4, 13≡5, 12≡6). Considering only
certain symmetries reduces the unknowns of the elasticity tensor Cij
further. For the analysis of anisotropic ice we consider cone, thick gridle and
partial girdle fabric. The connection between the different fabric types and
symmetry classes, i.e. seismic terminology for this fabric, can be found in
Table . Partial girdle fabric is the fabric with the
lowest symmetry, corresponding to an orthorhombic medium, with nine unknowns:
Cij=C11C12C13000C12C22C23000C13C23C33000000C44000000C55000000C66.
In the case of orthorhombic media three symmetry planes, i.e. orthogonal planes
of mirror symmetry, exist. The number of unknowns can be reduced further to
five unknowns if transversely isotropic media exist, resulting in an
anisotropy with a single axis of rotation symmetry. This is normally
distinguished in vertical transversely isotropic (VTI) and horizontal
transversely isotropic (HTI) media, with a vertical and horizontal axis of
rotation symmetry, respectively. A vertical cone fabric, including VSM
fabric, would be classified as VTI medium, while a thick girdle fabric as given
in Table would be classified as HTI medium. This
distinction is important for the calculation of seismic velocities and
reflection coefficients as the calculation simplifies for wave propagation
within symmetry planes of the anisotropic fabric (Sect. )
Calculation of elasticity tensor from COF eigenvalues
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 c axis
distribution. Thus, we are able to take into account cone as well as girdle
fabric distributions. We distinguish between an opening angle χ in
x1 direction and an opening angle φ in x2 direction in a
coordinate system where the x3 axis is pointing downwards
(Table ). These opening angles will be calculated from the
COF eigenvalues.
The two opening angles determine the kind of fabric
(Table ). If the angles φ and χ are equal,
the c axis distribution is a cone distribution with the cone opening angle
φ=χ; i.e. it is a VTI medium. The two extrema of this distribution
are the uniform distributions, i.e. the isotropic case, and the VSM fabric.
All c axes are oriented vertically in the case of a VSM fabric. The eigenvalues
are λ1=λ2=0 and λ3=1, and the cone opening angle is
0∘. The ice crystals are randomly oriented in the case of isotropic
fabric. The eigenvalues are then λ1=λ2=λ3=1/3, and the
cone opening angle is 90∘. The thick girdle fabric is an HTI media:
the c axes are distributed between two planes with a certain distance, so
that the opening angle φ in x2 direction is 90∘ and χ
in x1 direction gives the thickness of the girdle. The partial girdle
fabric, an orthorhombic medium, is a distribution where all ice crystal c axes
are in one plane, but only within a slice of this plane, so that the opening
angle χ in x1 direction is 0∘ and φ in
x2 direction gives the size of the slice within the plane. A girdle fabric
with χ = 0∘ and φ = 90∘ would correspond to the
eigenvalues λ1 = 0 and λ2 = λ3 = 0.5.
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
(C11=13.93±0.04 GNm-2; C33=15.01±0.05 GNm-2; C55=3.01±0.01 GNm-2;
C12=7.08±0.04 GNm-2; C13=5.77±0.02 GNm-2). The c axis of this ice crystal is oriented
vertically here, parallel to the x3 direction (Table ).
From COF eigenvalues to opening angles
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 suggests
a logarithmic representation of the eigenvalues and classification by a slope
m=ln(λ3/λ2)ln(λ2/λ1).
The fabric is a cone fabric with m>1 and a girdle fabric with m<1.
However, we want to put a stronger tendency towards a classification of the
fabric as cone fabric. In the seismic sense a cone fabric is a VTI medium. It
is easier to calculate velocities and reflection coefficients for VTI media
compared to girdle fabric, i.e. HTI media. Hence, we use a threshold value to
distinguish between cone and girdle fabric. If λ1≤0.1 and
λ2≥0.2, the fabric is classified as girdle fabric; everything
else is classified as cone fabric. Additionally, we set a threshold to
distinguish within the girdle fabric between partial and thick girdle fabric.
If λ1≤0.05, the fabric is classified as partial girdle, and otherwise
as thick girdle fabric. By distinguishing between these fabrics we know that
φ=χ for the cone fabric, φ = 90∘ for the thick
girdle fabric and χ = 0∘ for the partial girdle fabric
(Table ).
Girdle fabrics classified as HTI media are within the
[x2, x3] plane. When the girdle is rotated around the x3 axis, the
rotation is given by the azimuth ψ.
In the next step the remaining, unknown opening angle for the different
fabrics needs to be calculated from the eigenvalues, i.e. φ for the
cone fabric, χ for the thick girdle fabric and φ for the partial
girdle fabric. for instance connects the opening angle
φ of a cone fabric with the eigenvalue λ3 by
λ3=1-2/3sin2φ. To verify this calculation we determined the
eigenvalues for cone angles between 0 and 90∘. In total 10 000
randomly distributed vectors were created, giving a random distribution of
c axes. For each cone angle the vectors within this cone angle were selected.
The eigenvalues for this cone angle were then calculated from these vectors.
The process was repeated 100 times for each cone angle φ. The
calculated λ3(φ) values from the equation given by
differ by up to 15∘ for φ. For a more
precise connection of λ3 and φ than available from the
literature, a 4th-order polynomial was fitted to the λ3–φ
values (Appendix ). The same was done for the calculation of χ
from λ1 for thick girdle fabrics, as well as for the calculation of
φ from λ3 for partial girdle fabrics (Appendix ).
The orientation of the girdle is normally not known. Thus, the azimuth ψ
(Fig. ) of the girdle fabric cannot be determined from
the eigenvalues. This is only possible if the eigenvector belonging to the
eigenvalue λ1, the normal to the plane of the girdle, is known in
geolocated directions. Hence, in the following we normally assume girdle
fabrics to be orientated as HTI medium with the azimuth ψ = 0∘ for
the calculation of the elasticity tensor.
From opening angles to the elasticity tensor
The elasticity tensor of the polycrystal can now be derived using the
measured elasticity tensor for a single ice crystal and the derived angles
χ and φ. For the calculation of the polycrystal elasticity
tensor Cij we follow the idea of . They use the
concept of the and bounds. This concept was
developed to calculate the elasticity tensor of isotropic polycrystals,
containing different crystals. This concept is generalised by
to calculate the elasticity tensor for anisotropic
fabrics.
assumed that the strain on the polycrystal introduces the
same uniform strain in all monocrystals. However,
assumed that the stress on the polycrystal introduces the same uniform stress
in all monocrystals. To calculate the elasticity tensor of the polycrystal
with the assumption, one has to average over the elasticity
tensor Cijm of the monocrystal (superscript m). In the case of the
assumption, the compliance tensor of the polycrystal is
calculated by averaging over the compliance tensor Sijm of the
single crystals. The compliance tensor of a crystal is the inverse of the
elasticity tensor, here given in terms of Hooke's law
(Eq. ):
τmn=smnopσop.
For the inversion of elasticity to compliance tensor and vice versa see
e.g. . The method of and is
an approximation of the elasticity tensor due to violation of local
equilibrium and compatibility conditions across grain boundaries,
respectively. showed that the concepts of
and of give the upper and lower limit for the elastic moduli
of the polycrystal Cij, referred to as Voigt–Reuss bounds,
CijR≤Cij≤CijV,
where the superscripts R and V denote the and
calculation, respectively.
To obtain the elasticity tensor of the anisotropic polycrystal Cij from
the elasticity tensor of the monocrystal Cijm with different
orientations, one has to integrate the elasticity tensor
C̃ijm(ϕ) with a probability density function
F(ϕ) for the different c axes' orientations, where ϕ gives the
minimum (ϕ1) and maximum (ϕ2) extent of the c axes in the plane.
This plane is perpendicular to the corresponding rotation axis, so that the
elasticity tensor C̃ijm(ϕ) is determined from the
monocrystal elasticity tensor Cijm using the rotation matrix
RijC:
C̃ijm(ϕ)=(RijC)TCijmRijC.
The rotation matrices RijC for the different directions in
space are given in Appendix ; (RijC)T is
the transpose of RijC. The same applies for the calculation
of the monocrystal compliance tensor depending on ϕ:
S̃ijm(ϕ)=(RijS)TSijmRSRij,
with the rotation matrix RijS for the compliance tensor
(Appendix ) and its transpose (RijS)T. For
a uniform distribution of the c axis orientations the probability density
function can be given by
F(ϕ)=1ϕ2-ϕ1forϕ1≤ϕ≤ϕ2,=0forϕ2≤ϕ≤π;-π≤ϕ≤ϕ1,
which is symmetric around the main orientation, so that ϕ1=-ϕ0 and
ϕ2=+ϕ0. The elasticity tensor of the anisotropic polycrystal is
then calculated by
Cij=12ϕ0∫-ϕ0+ϕ0C̃ijm(ϕ)dϕ,
and the compliance tensor is calculated by
Sij=12ϕ0∫-ϕ0+ϕ0S̃ijm(ϕ)dϕ.
After considering the orthorhombic symmetry and some rearranging of the
results of Eqs. () and (), the
components of the elasticity tensor and compliance tensor of a polycrystal
can be expressed in compact form. The results are different for c axes'
distributions in the different spatial directions x1, x2 and x3. As
an example, the equations for the elasticity and compliance tensor for a
rotation around the x1 direction are given in Appendix . This
would correspond to a c axis distribution in the [x2, x3] plane. The
equations for rotation around the x2 axis and the x3 axis can equally
be derived from Eqs. () and ().
Steps for calculation of elasticity tensor
(Eq. ) or compliance tensor
(Eq. ) for different fabrics
(Table ).
Step
Rotation axis
Angle
Cone
1
x1
φ=χ
2
x3
90∘
Partial girdle
1
x1
φ
Thick girdle
1
x1
90∘
2
x2
χ
The different rotation directions to calculate the polycrystal elasticity
tensor Cij from a vertically oriented monocrystal elasticity tensor
Cijm for cone, thick girdle and partial girdle fabric are
listed in Table . They are also valid for the
compliance tensor. For the calculation of the elasticity tensor of a partial
girdle (Table ) the elasticity tensor of the monocrystal
Cijm is rotated around the x1 axis with the opening angle
of the partial girdle in x2 direction (φ). The elasticity tensor
is then calculated using Eq. () with ϕ0=φ. For
a thick girdle, φ = 90∘ in order to gain a full girdle in the
[x2, x3] plane in the first step. In a second step this elasticity
tensor obtained for a full girdle is then rotated around the x2 axis with
ϕ0=χ. For cone fabrics with different opening angles the elasticity
tensor of a monocrystal is rotated around the x1 axis
(Eq. ) in a first step using the cone opening angle
(ϕ0=φ=χ); afterwards, the obtained elasticity tensor is
rotated around the x3 axis with ϕ0 = 90∘.
Limitations of the method
developed the approach to calculate the polycrystal
elasticity tensor from the monocrystal elasticity tensors for what they
call S1 (vertical single maximum), S2 (horizontal girdle) and S3 (horizontal
partial girdle) ice for given opening angles. They found that the
Voigt–Reuss bounds for these fabrics are within 4.2 % of each other, and
concluded from this that either calculation, by means of the elasticity
tensor (Eq. ), or compliance tensor
(Eq. ) can be used to calculate the elasticity
tensor of the polycrystal. We use the approach of not
only for the calculation of partial girdle fabrics but also for the
calculation of the polycrystal elasticity tensor of thick girdle and cone
fabrics.
By comparing the individual components of the elasticity tensor derived
following (Eq. ) with those of the
elasticity tensor derived following (Eq.
and taking the inverse of the compliance tensor),
the largest difference of 4.2 % for all the investigated fabrics can be found
for the components C44 (S44) of a partial girdle with an opening
angle of 50 and 90∘. Thus, for all fabrics in this study,
the Voigt–Reuss bounds are within 4.2 % of each other and we follow
in their argumentation that either calculation can be
used. However, when using the calculation, no extra step in the
calculation is needed to invert the compliance tensor. Therefore, for all
further calculations the approach by is used
(Eq. ).
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 ). This classification introduces artificial
discontinuities in the velocity profile over depth, calculated from an ice
core. These discontinuities only reflect the calculation method, not sudden
changes in the prevailing fabric Part II,. This
limitation, introduced by the classification of the different fabric groups,
could be overcome by calculating the opening angels directly from the derived
c axis vectors. Another possibility would be to calculate the elasticity
tensor using the orientation distribution function (ODF), e.g. using the
open-source software METX . The calculation of the elasticity
tensor in this software is likewise based on Voigt–Reuss bounds, as is done
in this study. However, in glaciology the fabric distribution is normally
presented in the compact form of the COF eigenvalues. With the here-presented
framework the information of the eigenvalues can directly be used for the
calculation of the elasticity tensor, without further information. To enable
direct applicability of our method to existing ice-core data sets, we except
the limitations of our approach for the sake of ease of use.
For the calculation of the anisotropic polycrystal from the monocrystal
neither grain size nor grain boundaries are considered.
modelled the number of grains that are necessary to homogenise the elastic
properties of polycrystalline ice and found that at least 230 grains are
needed for girdle fabric (S2 ice). This number of ice crystals should be
reached with seismic waves in ice of around 300 Hz, i.e. a wavelength of more
than 10 m and ice crystals with ≤ 0.1 m diameter on average. Additionally,
computed two cases, with and without grain boundary sliding,
and found a difference of up to 25 % in Young's modulus and the Poisson ratio. In
the absence of grain-boundary sliding the anisotropy mainly defines the elastic
behaviour. Otherwise, grain shape and grain-boundary sliding become important
as well. A certain mistake is, thus, made for the calculation of the
polycrystal by only considering the influence of the anisotropy of the
monocrystal.
The resultant polycrystal elasticity tensors depend of course on the choice
of the monocrystal elasticity tensor. Different authors have measured
and calculated the monocrystal elasticity tensor.
A comparison of the different elasticity tensors used can be found in Part II
. There we investigate results of a vertical seismic
profiling survey in comparison to quantities from measured COF eigenvalues.
We find the best agreement between measured and calculated velocities using
the monocrystal elasticity tensor of for the derivation of
the polycrystal elasticity tensor.
Seismic velocities and reflection coefficients in anisotropic ice
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
e.g.. These solutions are still valid within the symmetry
planes of HTI media. To be able to calculate seismic velocities for the
different fabrics in ice, we will use a calculation of velocities for
orthorhombic media derived by (Sect. ). We
compare our calculated velocities, based on the derived elasticity tensor,
with the well-known velocities for a solid cone that were derived by
(Sect. ).
For the calculation of the reflection coefficient we use exact
as well as approximate
calculations (Sect. ). We show the reflection coefficients for an
abrupt change from isotropic to partial girdle fabric here as an example
(Sect. ). Additionally, we investigate the influence on the
reflection signature of an anisotropic ice mass above the base
(Sect. ).
Velocities in orthorhombic media
For the special case of wave propagation in ice with a developed cone fabric
anisotropy, derived equations of the slowness surface for
P, SV and SH waves. The phase velocities are given by the inverse of the
slowness surface. To calculate the slowness surface over different angles,
first derived the elasticity tensor from single natural
ice crystals by measurements of ultrasonic pulses of 600 kHz. With the
derived equations, velocities for different incoming angles ϑ
dependent upon the cone opening angle φ can be calculated. It is not
possible to calculate velocities for girdle fabrics with this approach.
Using the derived elasticity tensor, we are now able to calculate velocities
for different COF distributions. We use the equations derived by
for the calculation of phase velocities vph
(vp, vsv, vsh) as a function of the phase
angle ϑ for orthorhombic media as given in Appendix
(Eqs. –).
From these phase velocities we have to calculate the group velocities for the
calculation of travel times. The calculation of the group velocity vector
vg can be found, e.g., in and
. If the propagation of the seismic wave is within
symmetry planes of the anisotropic fabric, the group velocity and group angle
can be given in compact form. The group velocity vg is then
calculated from the phase velocity vph by
vg=vph1+1vph∂vph∂ϑ2,
with the group angle θ in the symmetry plane defined by
tanθ=tanϑ+1vph∂vph∂ϑ1-1vph∂vph∂ϑtanϑ.
Outside the symmetry planes of, e.g., HTI media, all components of the group
velocity vector vg have to be considered
(Appendix ).
Phase (dashed lines) and group velocities (solid lines) over the
corresponding phase ϑ and group angle θ for P (red curves),
SH (blue curves) and SV waves (light blue curves) of a VSM fabric. The
SV-wave group velocity shows a triplication for group angles θ between
43 and 47∘.
Figure shows the phase (dashed curves) and group
velocities (solid curves) as a function of the corresponding phase
ϑ and group angle θ of P (red), SV (light blue) and
SH wave (blue) for a VSM fabric. The largest difference between phase and
group velocity can be observed for the SV wave (light blue curves), with a
triplication in the group velocity for group angles of 43–47∘. Here
three different velocities are given for each angle. Due to the small spread
of these velocities, we do not expect that this triplication is of relevance
for applications given the current day accuracy of measurements. The
SV-velocity is largest for 45∘ incoming angle (phase as well as group
angle) with 2180 m s-1, decreasing for 0 and 90∘ to 1810 m s-1.
Variations for the SH wave are rather small, with velocities increasing
between 0 and 90∘ from 1810 to 1930 m s-1, i.e. 6 %. The
P-wave velocity has a minimum at ∼ 51∘ incoming angle:
3770 m s-1. The highest wave speed is observed for waves parallel to the c axis
of an ice crystal (0∘ incoming angle) at 4040 m s-1, and 150 m s-1 (4 %)
slower perpendicular to it.
P-wave phase velocities over phase angle ϑ for different
fabrics. P-wave velocity for (a) different cone opening angles
(φ=χ), (b) partial girdle fabric (χ = 0∘) and (c) thick
girdle fabric (φ = 90∘) within the [x2, x3] plane, (e)
partial girdle fabric (χ = 0∘) and (f) thick girdle fabric
(φ=90∘) within the [x1, x3] plane calculated with
Eq. () given by . (d) shows the P-wave
velocity for different cone opening angles (φ=χ) calculated with
the equation given by . The contour lines give the velocity
differences in percent, in relation to the maximum velocity of the respective
fabric group.
Velocities for anisotropic ice
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 show the P-wave phase velocity for different cone
and girdle fabrics calculated with the equations given in
and the equations derived by for a solid cone. The phase
velocity for the SH and SV wave as well as the corresponding group
velocities can be displayed accordingly . Here, we will limit
our analysis to P waves. However, with the derived elasticity tensor, SH- and
SV-wave velocities can just as well be investigated, and the effect of S-wave
splitting can be analysed.
Figure d shows the velocities calculated from the equations
derived by for a solid cone from the elasticity tensor he
measured at -10 ∘C. These velocities were corrected to
-16 ∘C for better comparison with the
other results, where we use the elasticity tensor of
measured at -16 ∘C. The other subfigures are phase velocities
calculated with Eq. () from an elasticity tensor
derived following the steps in Table with
the elasticity tensor measured by . The top row
(Fig. ) shows velocities for cone fabric (subfigure a: VTI)
as well as partial girdle fabric (b: HTI) and thick girdle fabric (c: HTI) in
the [x2, x3] plane, while the bottom row shows velocities for cone
fabric calculated following (d: VTI) as well as partial
girdle fabric (f: ψ = 90∘) and thick girdle fabric
(e: ψ = 90∘) in the [x1, x3] plane.
The partial girdle (χ = 0∘, Fig. , b, e) with
φ = 90∘ displays the same fabric as the thick girdle
(φ = 90∘, Fig. , c, f) with χ = 0∘.
The same applies to the cone fabric with an opening angle of 90∘
(Fig. , a, d) as well as the thick girdle fabric
(φ = 90∘) with χ = 90∘ (Fig. c,
f), both showing isotropic c axes' distribution. Apart from
's velocities, these velocities for the isotropic
state (Fig. a, c, f) are obviously not isotropic. Slight
variations still exist for these velocities with increasing incoming angle.
This is due to artefacts that seem to appear from the derivation of the
elasticity tensor for the isotropic state using the single crystal elasticity
tensor.
It should also be noted that for a thick girdle with
φ = χ = 90∘ the variations over the incoming angle are just
reversed to those of the cone fabric with opening angle
φ = χ = 90∘. This reflects the difference in the calculation
of the elasticity tensor from cone fabric and girdle fabric. While a girdle
with φ = 90∘ (χ = 0∘) is calculated in the first
step for both fabrics (Table ) by integration
with rotation around the x1 axis, the second step is an integration with
rotation around the x3 axis for the cone fabric and around the x2 axis
for the thick girdle fabric.
The higher velocities calculated with the equations of
(Fig. , d) are due to the difference in the elasticity
tensor, as the elasticity tensor derived by was used for
the calculation in all of the other subfigures (Fig. a–c,
e, f). The calculation exhibits an isotropic state for
φ = χ = 90∘. However, this is only possible as
used fitted curves for the derivation of the slowness
surface.
Reflection coefficients
The calculation of reflection coefficients for different incoming angles is
already rather complicated for layered isotropic media given by the Zoeppritz
equations e.g.. In the case of anisotropic media most of
the studies have been done for VTI media and in
terms of Thomsen parameters . A comprehensive overview of
the different calculations of reflection coefficients for VTI and HTI media
is given by .
In the following, we use equations derived by by means of
perturbation theory for the calculation of englacial reflection horizons.
These equations for general anisotropy were simplified by
for weak contrast interfaces. They are, thus, especially practical for the
reflection coefficients in ice. For the isotropic reference values the
elasticity tensor for isotropic ice can be used and no average needs to be
taken over different materials. The reflection coefficients for the
anisotropic material are then calculated as perturbations of the isotropic
ice fabric. Thus, reflection coefficients for P, SV and SH waves are
obtained. The equations for the calculation of reflection coefficients are
given in Appendix . The Rshsh and Rsvsv
reflection coefficients are restricted to a symmetry plane of the layered
medium. The indices give the polarisation of the incoming and reflected wave;
e.g. Rpp is the reflection coefficient for an incoming P wave,
reflected as P wave, equivalent for Rshsh and Rsvsv.
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
, which were further developed by . Exact
solutions for VTI media are, for example, given by and
.
Reflection coefficients for the boundary between an isotropic
(upper) layer and a partial girdle fabric (lower) layer with different
opening angles φ (χ = 0∘) of the girdle. The reflection
coefficients are calculated with equations given in Sect. for
different incoming phase angles ϑ. The subfigures (a), (b) and (c)
show the reflection coefficients for a girdle orientation (lower layer)
perpendicular to the travel path of the wave (HTI media) for PP, SHSH and
SVSV reflection, respectively. The subfigures (d), (e) and (f) show the
reflection coefficients for a girdle orientation parallel to the travel path
of the wave (azimuth ψ = 90∘) for PP, SHSH and SVSV reflection,
respectively.
Reflection coefficients for anisotropic ice
With the equations given in Appendix
reflection coefficients can be calculated for interfaces between different
fabrics. Figure shows as an example the
Rpp, Rshsh and Rsvsv reflection
coefficient for the transition at a layer interface from an isotropic fabric
to a partial girdle fabric, both for HTI media (ψ = 0∘) and with
an azimuth of ψ = 90∘.
The reflection coefficients are given for angles of incidence between
0 and 60∘. This has two reasons. Firstly, most seismic
surveys do not exceed an incoming angle of 60∘ as this already
corresponds to a large offset compared to the probed depth. Secondly and more
important, the calculation of the reflection coefficients using
Eqs. ()–() is not exact. Instead, the error increases
with increasing incoming angle.
P-wave reflection coefficients for ice–bed interface with different
bed properties as a function of phase angle of incidence ϑ:
basement (black), lithified sediments (red), dilatant sediments (gray) and
water (blue). The solid and dotted lines are the reflection coefficients for
an isotropic ice overburden, the dashed and dashed-dotted lines for the
anisotropic (VSM) overburden. The solid and dashed lines are the reflection
coefficients calculated with exact equations for VTI media given by
and . The dotted and dashed-dotted lines
are approximate calculations following the approach by for the
isotropic case and that of for the anisotropic case,
respectively. Property values for the bed and isotropic ice are taken from
. For the anisotropic ice the elasticity tensor given by
is used.
The largest magnitude of reflection coefficients can be observed for the
SVSV reflection (Fig. ). However, the
reflection coefficients are ≤0.1 for all fabric combinations shown here.
Most significantly, for the PP reflection the reflection coefficients between
different anisotropic fabrics are small. The PP reflection between, for
example, isotropic and VSM-fabric ice
for normal incidence is <0.02. For
comparison the reflection coefficient between isotropic and lithified
sediments (Fig. ) is ∼0.4. Hence, reflection
coefficients at the ice–bed interface are an order of magnitude larger than
reflection coefficients for the transition between different anisotropic
fabrics. To be able to observe englacial seismic reflections, abrupt changes
(i.e. within a wavelength) with significant variations in the orientation of
the ice crystals are needed. Such englacial reflections have been observed in
data from Greenland and Antarctica , and also in the Swiss/Italian Alps . These
reflections can indicate a change in the fabric. However, the investigation
of reflection signatures (amplitude versus offset, AVO) of englacial
reflectors is difficult due to the small reflection coefficients, and the
small range they cover with changing incoming angle.
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
(φ = 0∘, Fig. ) from 0
to 60∘ is between 0.019 and 0.036. To put these values in perspective,
we consider error bars for reflection coefficients as determined for ice–bed
interfaces. It cannot be expected that the error bars for measuring the
reflection coefficient of englacial reflections would be smaller than those
given for the bed reflection coefficients. analysed the
reflection amplitude for the ice–bed interface from a survey near the South
Pole. For the reflection coefficients they derive from the seismic data, they
give error bars of ±0.04, with increasing error bars for decreasing
incoming angles, limited by ±0.2. The change in the reflection
coefficient with offset for englacial reflection that we calculate is smaller
than the given error bars. Thus, it is not possible to derive information
about the anisotropic fabric from englacial reflections using AVO analysis, at
the moment. To be able to derive fabric information from AVO analysis the
error in determining the reflection coefficient from seismic data needs to be
reduced, e.g. better shooting techniques to reduce the signal-to-noise ratio
(SNR) in the data or a better understanding of the source amplitude as well
as the damping of seismic waves in ice.
P-wave velocity, S-wave velocity and density for different bed
scenarios and isotropic ice as given in . These values are
used for the calculation of reflection coefficients given in
Fig. .
Material
vp in m s-1
vs in m s-1
ρ in kg cm-3
Ice
3810
1860
920
Basement
5200
2800
2700
Lithfied sediment
3750
2450
2450
Dilatant sediment
1700
200
1800
Water
1498
0
1000
Reflection coefficients for ice–bed interface
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 e.g. or by AVO analysis
. Figure shows
reflection coefficients for the transition from isotropic and anisotropic
(VSM fabric) ice to different possible bed properties
(Table ). The values for density, P-wave and S-wave
velocity, for the different bed scenarios and the isotropic ice, are taken
from . For the anisotropic VSM fabric the elasticity tensor
of is used.
Exact solutions are calculated using the equations given by
, with corrections by . Their equations
were used to calculate the exact reflection coefficients for the isotropic
ice above the bed (solid lines) and for the anisotropic ice above the bed
(dashed lines) shown in Fig. . The approximate
reflection coefficients for the isotropic ice above the bed (dotted lines)
are calculated using equations given in . The approximate
reflection coefficients for the VSM fabric above the bed (dashed-dotted
lines) are calculated using equations given in .
The differences between the isotropic (solid lines) and anisotropic
reflection coefficients (dashed lines) are small (≤0.04) for the exact
solutions. The approximate calculations fit well to the exact solutions up to
a group angle of about 30∘, with differences of the same order as
isotropic to anisotropic variations. However, differences between exact and
approximate reflection coefficients become large for increasing phase angle
(≥ 30∘). Thus, errors introduced by using approximate
calculations for the reflection coefficients are larger than the effect of
anisotropic ice fabric above the bed.
The observable differences of reflection coefficients for an isotropic and a
VSM-fabric overburden are ≤0.04, i.e. smaller then the smallest error
bars given by (Sect. ). The VSM fabric is the
strongest anisotropy to be expected in ice. If an anisotropic layer exists
above the bed, it yields a different reflection coefficient compared to the
case of the isotropic ice overburden. However, the difference between the
isotropic overburden reflection coefficient and the anisotropic overburden
reflection coefficient is within the range of the error bars given by
. Thus, the anisotropic fabric will not have a measurable
influence on the analysis of the bed properties by means of the AVO method,
given the current degree of data accuracy and SNR.
Conclusions
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
. However, with our method it is now also possible to
calculate velocities for girdle fabrics. Further, we can use the derived
elasticity tensors to investigate the reflections coefficients in anisotropic
ice.
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 c axes' vectors.
This would avoid our classification into cone, partial girdle and thick
girdle fabric. Nevertheless, the approach presented here offers the
opportunity to use the readily available COF data from ice cores and go
towards an investigation of the seismic wave field in ice without the
limitation to velocities only. The inclusion of further properties
influencing the propagation of seismic waves in ice, like density and
temperature, will offer the opportunity to model the complete wave field.
Hence, we are confident that it will become feasible in the future to derive
physical properties of the ice from analyses of the complete observed wave
field by full waveform inversions.