Crystallographic texture (or fabric) evolution with depth along ice cores can be evaluated using borehole sonic logging measurements. These measurements provide the velocities of elastic waves that depend on the ice polycrystal anisotropy, and they can further be related to the ice texture. To do so, elastic velocities need to be inverted from a modeling approach that relate elastic velocities to ice texture. So far, two different approaches can be found. A classical model is based on the effective medium theory; the velocities are derived from elastic wave propagation in a homogeneous medium characterized by an average elasticity tensor. Alternatively, a velocity averaging approach was used in the glaciology community that averages the velocities from a given population of single crystals with different orientations.

In this paper, we show that the velocity averaging method is erroneous in the
present context. This is demonstrated for the case of waves propagating along
the clustering direction of a highly textured polycrystal, characterized by
crystallographic

Because of the weak elastic anisotropy of ice single crystal, the inversion of the measured velocities requires accurate modeling approaches. We demonstrate here that the inversion method based on the effective medium theory provides physically based results and should therefore be favored.

Wave propagation in glaciology is mostly regarded in the context of seismic
waves; see, e.g.,

In a previous paper

This is the goal of the present paper. We will show that the velocity
averaging method is based on an erroneous fundamental assumption. To do so,
we perform a demonstration for the cases of cluster and girdle ice textures
with VTI. It is found that a wave propagating along the vertical axis is
associated with two different shear velocities, which is unphysical since the
polarizations of the shear waves are in the plane of isotropy. In

Considering the low elastic anisotropy of the ice single crystal, the accuracy of the inversion procedure from the measured velocities to ice texture is essential. In particular, a model relying on a rigorous mathematical formalism is required, in order to describe the case of complex textures, beyond the simple case of VTI clusters. With regard to this requirement, we will show that the model based on the effective medium theory appears more reliable.

In this section, we recall classical results (i) on the propagation of elastic waves in a crystal or in a polycrystal characterized by a uniform elasticity tensor and (ii) on the elasticity tensor of polycrystals, considered at the scale of many grains as an equivalent “single” crystal. This allows us to introduce the notations that will be used further in the text and to clarify in a self-consistent way some properties that will be needed.

In the following, we denote

In the previous section, we have considered the uniform elasticity tensor of
a polycrystal. This elasticity tensor is derived from the characteristics of
the grains which compose the polycrystal. For ice, each grain is composed of
the same single crystal, with hexagonal symmetry being characterized by its
elasticity tensor

The anisotropy at the macroscopic scale (at the scale of many grains) results
from the many (or few) possible orientations of the

For the numerical application, we will use the elastic constants as used in

Spherical angles (

First configuration of a VTI girdle with a single zenith
angle

Second configuration of the usual VTI clustered texture.

In this section, we compare the elastic velocities obtained from the velocity
averaging method, as used in

Velocities from the velocity averaging method: first, solve the Christoffel equation for given (

Velocities of the effective medium: first, compute the effective elasticity
tensor using

Two examples of VTI structures will be presented and the propagation along
the vertical direction is considered. The first structure is artificial, with
a

We consider a wave propagating along the vertical axis, thus

In this method, we first derive the velocities in a grain, and this is done
without lack of generality for a wave propagating along

The second step in the velocity averaging method can be applied:

With the probability function given by Eq. (

Next, the velocities of the elastic waves propagating along the

Illustration of the inconsistency of the velocity averaging method
for the two VTI textures shown in Fig.

The first configuration is shown in Fig.

The velocities obtained from the velocity averaging method, Sect.

These velocities are reported in Fig.

This texture is shown in Fig.

The velocities of the effective medium (

Let us now discuss Fig.

Bennett starts with the slowness in a single crystal

An additional angle

It is difficult to anticipate the consequences of such an approach in the case
of other textures, since the weights used to deduce the shear wave velocity
in Eq. (

In this paper, we have proposed a critical analysis of the velocity averaging
method that has been used recently in the post treatment of the wave
velocities deduced from borehole sonic logging measurements along ice cores.
We have illustrated the error made in the fundamental assumption postulated
at the basis of this method. This critical analysis is performed here in the
case of simple VTI textures for which the error leads to an unphysical
result and thus allows for a clear demonstration. For VTI textures, Bennett
circumvented this problem by postulating weighted forms of the shear wave
velocities

The sonic logging measurements are sought as a proxy of the ice polycrystal
anisotropy. Beyond the case of clustered VTI textures, complex textures could
be characterized by such methods (or at least transitions between various
textures with depth). On the one hand, the anisotropy of the ice single
crystal is weak, and thus the anisotropy of ice polycrystals is even weaker.
On the other hand, ice along boreholes is reasonably free of impurities
when compared to most of the materials studied by similar sonic measurements.
The recent sonic logging campaign performed at EPICA Dome C has revealed the
feasibility of such measurements

We are thankfull to A. Diez and an anonymous referee for helping to improve the paper. Maurine Montagnat is supported by the French CNRS (Centre National de la Recherche Scientifique) and the institutes INSIS (Institut des Sciences de l'Ingénierie et des Systemes) and INSU (Institut National des Sciences de l'Univers). Edited by: O. Eisen Reviewed by: A. Diez and one anonymous referee