Ice microstructure and fabric

12 Pages·2007·0.21 MB·English

JournalofGlaciology,Vol.52,No.179,2006 619
Ice microstructure and fabric: an up-to-date approach for
measuring textures
Gae¨l DURAND,1 O. GAGLIARDINI,2 Throstur THORSTEINSSON,3
Anders SVENSSON,1 Sepp KIPFSTUHL,4 Dorthe DAHL-JENSEN1
1Ice and Climate Group, The Niels Bohr Institute, Rockefeller Complex, Juliane Maries Vej 30,
DK-2100 Copenhagen, Denmark
E-mail: [email protected]
2Laboratoire de Glaciologie et Ge´ophysique de l’Environnement (CNRS–UJF), 54 rue Molie`re, BP 96,
38402 Saint-Martin-d’He`res Cedex, France
3Institute of Earth Sciences, Building of Natural Sciences, Askja, Sturlugata 7, IS-101 Reykjav´ık, Iceland
4Alfred Wegener Institute for Polar and Marine Research, Columbusstrasse, D-27568 Bremerhaven, Germany
ABSTRACT. Automaticc-axesanalyzershavebeendevelopedoverthepastfewyears,leadingtoalarge
improvementinthedataavailableforanalysisoficecrystaltexture.Suchanincreaseinthequalityand
quantity of data allows for stricter statistical estimates. The current textural parameters, i.e. fabric
(crystallographic orientations) and microstructure (grain-boundary networks), are presented. These
parametersdefinethestateofthepolycrystalandgiveinformationaboutthedeformationundergoneby
the ice. To reflect the findingsfromautomaticmeasurements, some parameterdefinitionsareupdated
1
andnewparametersareproposed.Moreover,aMATLAB toolboxhasbeendevelopedtoextractallthe
textural parameters. This toolbox, which can be downloaded online, is briefly described.
1. INTRODUCTION It is worth noting that ice flow is strongly dependent on
the texture. Firstly, the deformation of an ice crystal occurs
Ice cores provide a wonderful record to investigate past bydislocationslip,mainlyalongthebasalplane,confirming
climates. Since ice sheets form by successive deposition of the predominant visco-plastic anisotropy of the ice crystal,
snow layers, travelling down to the deepest part of the ice i.e. the resistance to shear on non-basal planes can be 60
sheet is a journey into the past. Ice-core studies provide a times higherthan resistancetoshearalongthebasal planes
large set of climatic parameters, such as past temperature (Duval and others, 1983). Due tothe intrinsic anisotropyof
variationswhichcanbe revealed through isotopicdata, the ice, the c axes rotate during deformation towards compres-
atmospheric composition revealed through gas trapped in sional axes and away from tensional axes (Azuma and
bubbles and the atmospheric circulation inferred from Higashi, 1985; Castelnau and Duval, 1994; Van der Veen
impurity concentrations. Although there is an immense and Whillans, 1994), and complex feedbacks occur
amount of information available, dating is the Achilles’ between flow and fabrics. As an example, under compres-
heel of these archives. For sites with high accumulation sion,whencaxesrotatetowardthecompressionalaxis,the
rates, counting of annual layers enables estimation of the applied stress becomes increasingly perpendicular to the
age (Greenland Icecore Project (GRIP), Greenland Ice basal plane, and the ice becomes harder to compress.
Sheet Project 2 (GISP2) and NorthGRIP cores among Secondly, the rheology may depend on the mean crystal
others; see, e.g., Alleyand others, 1997b). At depthswhere size, as finer grains could explain a third of the shear strain
annual layers are not resolvable, and for sites with low rate enhancement observed in ice deposited during glacial
accumulation rates, flow models have to be used to periods (Cuffey and others, 2000). Understanding the
reconstruct the sinking of the ice layers and then the age reasons for grain-size variations with climatic changes is
canbe estimatedwith depth(DomeC,Vostokand DomeF thus an important task for texture studies (Durand and
(Antarctica) cores among others; see, e.g., Parrenin and others,2006).Finally,recrystallizationprocesseshavetobe
others, 2001). Consequently, understanding the deform- taken into account. As an example, rotation recrystalliza-
ation of the ice is of primary importance for correct tion, also known as polygonization, is characterized by
climatic interpretation. basaldislocationsthatgrouptogetherinwallsperpendicular
From a material science point of view, ice is an to the basal planes and form sub-boundaries. The misor-
assemblage of crystals with a hexagonal crystallographic ientation between two subgrains increases gradually, and
structure.Thecrystalorientationisspecifiedbythedirection grains may split into smaller grains. Such processes
ofitsaandcaxes.Duetothebirefringenceofice,thecaxis complicate the interpretation of texture measurements.
of a single crystal, which is perpendicular to the basal Since the texture records the deformation history of the
planes, can be easily revealed by optical measurements. In ice,anunderstandingofprocessesaffectingthetexturemay
what follows, ‘fabric’refers to the orientations of the c axes lead to a better understanding of ice flow. Moreover, the
of the ice polycrystal, whereas ‘microstructure’refers to the complex feedbacks between fabric development and de-
grain-boundarynetwork.Thetermtextureincludesboththe formation are now incorporated in local flow modelling
fabric and microstructure. Note that this terminology is not (Gagliardini and Meyssonnier, 2000). In such models, both
universallyaccepted,andotherusagesofthesetermscanbe the velocity and the fabric fields are calculated, allowing
found in the literature. comparison with measurements. Such comparisons require
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620 Durandandothers:Icemicrostructureandfabric
some fabric parameters that can be determined both the quality of the thin section: melting more than
theoreticallyandexperimentally.Lastbutnotleast,insome necessary can induce bubble trapping at the glass–ice
cases, the studyof the texture can help to explain observed interface; conversely, too little heat applied to the glass
flow disturbances such as folds (Alley and others, 1997a; platewillcauseaweakbondbetweeniceandglassand
Raynaud and others, 2005). This is crucial for extracting the thin section may be damaged while reducing the
palaeoclimatic records from an ice core. thickness.Thismethodneedspracticebeforesatisfactory
Duetothebirefringenceoficecrystals,icetexturecanbe resultscanbeobtained.Thesamplecanalsobefixedby
revealed by the examination of a thin section under cross- using water droplets on its border as in step 2. This
polarized light. Earlier studies of ice texture were made procedure is easier than that proposed by Langway
manually,whichwastediousandparticularlytime-consum- (1958), and the quality of the final thin section is
ingwork.Muchprogresshasbeenmadeindevelopingice- generally better.
texture instruments, and now automatic measurements of
4. Using a bandsaw, the sample is cut parallel to the glass
texture can be performed. Wilen and others (2003)
plate, leaving 1–2mm of ice adhering to the glass.
presented a review of the automatic ice-fabric analyzers
(AIFAs) used in the glaciological community. In the cited 5. The section is reduced to a thickness of 0.3mm with a
works, the AIFAs were mainly used to extract the grain microtome knife. The thickness is an important par-
orientations of an ice sample, and microstructure was ameter for grain-boundary determination using cross-
considered as a by-product of the measurements. polarized light. Sharp colour transitions between grains
As plentiful and more robust measurements have now arerequiredandinterferencefringes,whichappearifthe
been obtained using an automatic ice-texture analyzer grain boundary is too thick, should be avoided. The
(AITA; since one should also extract the microstructure of optimal thickness corresponds to grain colour in the
a sample using an AIFA, AITA is a more appropriate term), brown–yellow range (Gayand Weiss, 1999).
some of the textural parameters measured so far are, in our
It is obvious that the measurements result from two-
view, now obsolete. Moreover, new parameters can be
dimensional (2-D) cuts of a three-dimensional (3-D)
defined. These parameters better describe the texture and
structure. This point, referred to as the sectioning effect in
shouldbeusedinpreferencetoearlierones.Thispaperaims
what follows, must always be kept in mind, as it influences
toreviewandproposenewmethodstodescribeicetexture.
the interpretation of the measurements made on the
Thepaperisorganizedasfollows:insection2,wediscussin
microstructure (see Underwood (1970) for a review of
detail the operations needed to obtain the texture informa-
stereologicalproblems,andsection3.1.2foradiscussionof
tion from an ice sample. In section 3, we review the
the influence of 2-D cuts on mean grain radius estimation).
parameters used so far to describe the microstructure and
Thec-axismeasurementisnotinfluencedbythis2-Deffect,
propose new definitions. Section 4 deals with fabric
1 assuming a small disorientation within a grain.
parameters. Finally, section 5 briefly presents a MATLAB
tool to determine all these texture parameters. 2.2. Texture measurements
Theclassicalmanualtechniqueformeasuringthefabricofa
2. MEASUREMENTS thinsectionisbasedontheRigsby-stageprocedure(Rigsby,
1951) that was standardized by Langway (1958). Micro-
2.1. Sample preparation
structuralparameters,suchasthemeangrainsize,werealso
Most of the studies of ice texture follow the standard estimatedmanually(seesection3.1).Morerecently,image-
procedure detailed in Langway (1958) to prepare thin analysis procedures have allowed extraction of microstruc-
sections.Here,webrieflyrecallthisprocedureandpropose tureandnewparametersrelatedtotheshapeofgrains(Gay
some improvements: and Weiss, 1999; Svensson and others, 2003).
Today, using an AITA, the whole texture of a polycrystal
1. A vertical (parallel to the core axis) or a horizontal
canbemeasured:fabricandmicrostructuralparameterscan
(perpendiculartothecoreaxis)thicksectionabout1cm
now be obtained together on the same sample. Indeed,
thickiscutfromtheicecore.Thesamplelocationdiffers
Gagliardiniandothers(2004)haveshownthatthemeasured
from core to core, but a large sample area is preferable
cross-sectionalareashouldbeusedasastatisticalweightof
(this point is detailed later).
the polycrystal constituents in order to improve the fabric
2. Onesurfaceofthesamplehastobeflattenedinorderto description. Complete texture measurements are also
obtainaplanesurface.AsproposedbyLangway(1958), pertinent, as mapping of the spatial distribution of c axes
sandpaper can be used, but Thorsteinsson (1996) can reveal additional information about recrystallization
proposedanotherprocedure:Thethicksectionisplaced processes (Alleyand others, 1995; see also section 4.7).
onaglassplate,withafewdropsofwateralongitsside Two major steps have to be carried out to extract the
to glue it to the glass plate. The plate is fixed to a texture from the output of an AITA:
microtomeandisshavedtoproduceasmoothandplane
1. Extraction of grain boundaries. Gay and Weiss (1999)
surface.Thenthesampleisremovedfromtheglassplate
proposed an image analysis algorithm to extract the
by breaking the frozen-water droplets with a cutter. In
microstructure from three images of one thin section
our experience, the second procedure is preferable as it
taken under cross-polarized light. After some filtering to
produces better results.
remove noise, a Sobel filter is used to detect large
3. A glass plate is then frozen or glued onto the smooth discontinuities in intensity, which generally correspond
surface. Langway (1958) proposed the glass plate be to grain boundaries. This method works well for
warmedandthesamplegluedbyrefreezingofwater.The randomlyorientedcrystals,asthecolourvariationsfrom
amount ofheat appliedtotheglassplate willdetermine grain to grain are important (see Fig. 1). For more
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Durandandothers:Icemicrostructureandfabric 621
oriented fabrics, colours between grains are similar and
grain-boundary detection is more difficult and some
manual corrections are needed. Note also that the
resulting microstructure is naturally strongly dependent
on the quality of the thin section. Most of the existing
automaticmachinesproduceimagesofthethinsections
thatcanbeusedtoextractthemicrostructure(Wilenand
others, 2003). An adaptation of the Gay and Weiss
(1999) algorithm can easily be made from the output of
these analyzers.
2. Assigning an orientation for each grain detected in (1).
Fig.1.Imageofathinsectionundercross-polarizedlight(left)and
UsinganAITA,theorientationisknownforeachpixelof the corresponding texture (right). Grain boundaries appear white,
the grain. The initial measurements fora given pixel are andeachgraindefinedwithinthemicrostructurehasanorientation
the co-latitude, (cid:1) , and the longitude, ’ , if this pixel is detailedinthecolourmapgivenbythepolarplotprojectionshown
p p
notlocatedonagrainboundary.Anaveragevalueofthe onthebottomright.
c axis over all pixels inside a grain, as well as the
dispersion, can be estimated within each grain. These
calculations are detailed in section 4.5. Thorsteinssonandothers(1997)estimatedthemeangrain
size alongthe GRIPcore usingthe linear interceptmethod.
Finally, from outputs of the AITA, one obtains the complete
This method consists of counting the number of grain
description of the texture at the grain scale. That is the
boundariesthatinterceptahorizontal(orvertical)line.This
microstructure, which contains all the topological informa-
leadsto theestimate ofa meanintercept length, hLi, which
tion, and the fabric which gives the mean orientation of all
foraregularsurfaceisproportionaltothesquarerootofthe
non-intersecting grains within the microstructure. Such
area of the considered object (Underwood, 1970). This
results allowthe constructionof images suchas that shown
methodsuffersdifferentbiases:(1)ifthegrainshapeevolves
in Figure 1, which contain all the information required to
(andthatisthecase;seeDurandandothers,2004),theratio
calculatetheparametersdescribingthetexture(seesections
betweenthemeanareaandhLiisaffected;(2)twistedgrain
3and4).Itisworthnotingthattheoriginaloutputsfromthe
boundaries can intercept a given line more than once; and
AITA should be conserved, as they certainly contain
(3) the link between the grain-area distribution and the
information at the sub-grain scale that is not taken into
distribution of the length of intercept is not straightforward.
account in this work and may be useful at a later date.
DuvalandLorius(1980)estimatedthemeangrainareaby
counting the number of grains within a defined surface. As
this method takes into account the whole population of
3. MICROSTRUCTURAL PARAMETERS
grains, it is more robust than the method proposed byGow
Manualestimationofthegrainsizeistime-consumingwork
(1969). It also provides a more straightforward estimate of
and several methods have been used so far to make these
the mean grain size than the linear intercept method, but it
measurementspracticable.Asdiscussedinsection2,image-
does not provide a grain-size distribution. Note also that
analysis procedures now provide the ability to extract the
corrections are required in ice with significant porosity.
completemicrostructureinanacceptabletime.Therefore,a
moreaccuratedefinitionofthegrainsizecanbemade,and
3.1.2. Mean grain radius
the procedures also give new information on grain-size
A more complete way to measure a characteristic length of
distributions. Comparisons of the different methods used so
the microstructure would be to calculate the average
far are reviewed in section 3.1. volumetric radius: hR i¼1=N PNg V1=3,where N is
Other studies have focused on the grain shape to obtain V g k¼1 k g
the number of non-intersecting grains within the micro-
information about ice deformation (Azuma and others,
structure and V is the volume of grain k. As mentioned
2000; Durand and others, 2004). As for the measurement k
previously,thevolumeoftheicecrystalcannotbemeasured
of the grain size, several methods have been used so far; a
and 2-D sections have to be made. Then a characteristic
comparison is provided in section 3.2.
length of the microstructure could be expressed by the
3.1. Grain size average radius:
3m.1ic.r1o.sEtrsuticmtuarteion of a characteristic length of the hRi¼ 1 XNg A1=2, ð1Þ
N k
Gow(1969)firstproposedmeasuringthemeangrainsizeof gk¼1
a section by estimating the average area of the 50 largest
where A is the cross-sectional area of grain k and
grains. By definition, this method only measures the k
evolution of the largest part of the grain distribution; thus, Ak ¼Npk(cid:2), where (cid:2) is the actual area of a pixel and Npk is
itisnotrepresentativeofthecompletepopulation.Itisalso the number of pixels that compose grain k. An estimate of
worthnotingthattherepresentativityofthe50largestgrains theerrorinducedbythesectioningeffectcanbemadewith
is affected by a change in the grain size (for a constant the help of a 3-D Potts model. In such a model, 3-D
sampling surface). This leads to a bias in the estimation of microstructureisdividedintosmallvolumeelementsandan
grain-growth rate (Gay and Weiss, 1999). The method was orientation is allocated to each element. Grain growth is
developed in order to perform manual measurements and simulated using a Monte Carlo technique: an element is
should be avoided for automatic measurements. randomly selected and a new orientation is randomly
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622 Durandandothers:Icemicrostructureandfabric
Fig.3.EvolutionofthestandarddeviationofhRioverhR ivsthe
Ref
Fig.2.Normalizeddistributionofthelogarithmofthenormalized
inverseofthesquarerootofthenumberofgrains.
radiusofthegrain(R =hRi)foramicrostructuresampledalongthe
k
EPICADomeC(Antarctica)coreat100.1mdepth(blackline).The
log-normaldistributioncorrespondingtothemeasuredparameters
hiDand(cid:3)Dofthismicrostructureisalsoplotted(dottedline),aswell that these microstructures present enough grains to com-
as the distribution obtained for a 2-D Potts microstructure (grey
pletely describe the grain-size distribution. Their mean
line).Thenumberofgrains(275)andthemeanradius(21.3pixels)
radius was then used as a reference, hR i. For each
of the Potts microstructure are comparable to the natural sample Ref
microstructure, 80 subsampling microstructures were ran-
(454grainsandhRi¼21:6pixels).
domly selected. These sub-microstructures contained be-
tween 3 and 200 grains and their mean radius was
calculated and compared to hRi. As hRi=hR i follows the
Ref
central limit theorem, the standard deviation depends only
attributed. Similar orientations between neighbouring ele-
on the number of grains, following the relation
ments (i.e. same grain) will decrease the energy of the
system,andtheneworientationwillthenbeconserved.The (cid:3)pðhRi=hRRefiÞ¼0:44Ng(cid:3)1=2 (see Fig. 3). Of course, for
Pottsmodelisknowntoproperlyreproducethetopological, measurements on natural ice, hR i is unknown, and we
Ref
kinetic,grain-sizedistributionandmorphologicalfeaturesof make the following approximation:
normal grain growth (Anderson and others, 1989), such as
(cid:3) ðhRiÞ¼(cid:3) ðhRi=hR iÞ(cid:1)hR i
recrystallizationprocessesoccurringintheupperpartofice p p Ref Ref
sheets (Alleyand others, 1986). (cid:2)(cid:3) ðhRi=hR iÞ(cid:1)hRi ð2Þ
p Ref
Following Gagliardini and others (2004), we generated
3-D Potts microstructures containing 400(cid:1)400(cid:1)400 ele- and finally obtain (cid:3)p (cid:2)0:44Ng(cid:3)1=2(cid:1)hRi. An error bar that
ments.Asectionwithinthisvolumecontainsapproximately takes into account the sectioning effect ((cid:3) ) as well as the
s
200 non-intersecting grains. For such a section, both hR i populationeffect((cid:3) )cannowbeattachedtohRiasfollows:
V p
andhRicanbemeasuredandcompared.Fromtheseresults,
we demonstrated that the standard deviation (cid:3) of the hRi (cid:3)ðhRiÞ¼(cid:3)(cid:1)sþ(cid:3)p (cid:2)
s
distribution for a given hRVi can be approximated by (cid:2) 0:02þ0:44N(cid:3)1=2 (cid:1)hRi, ð3Þ
(cid:3) (cid:2)0:02hRi (Durand, 2004). g
s
ThemeangrainsizegenerallyincreaseswithdepthAlley where hRi is calculated using Equation (1).
and others, 1986), and for a constant sampling surface the
number of grains will decrease with increasing depth. The 3.1.3. Distribution parameters
statistics for the calculation of the average grain radius can Under normal grain growth, the normalized grain-size
be drastically affected. As for the error induced by the distribution, R =hRi, remains unchanged and is closely
k
sectioning effect, we estimate the error induced by the approximatedbyalog-normaldistribution(Humphreysand
number of grains used in the calculation of the average Hatherly,1996).Thisobservationdoesnothaveatheoretical
radiususingaPottsmodel.Andersonandothers(1989)have explanationasyet;itis,however,ausefuldescription,asthe
shownthata2-DPottsmodelreproducesthecharacteristics distributioncanbecompletelydefinedwithonlytwo,easily
of a microstructure determined from a section of a 3-D measurable,parameters:themean,hi ¼hlnðR =hRiÞi,and
D k
polycrystal under normal grain growth (and the calculation thestandarddeviation,(cid:3) ¼(cid:3)½lnðR =hRiÞ(cid:4).Changesinthe
D k
time is much smaller for 2-D than for 3-D simulations). As normalized distribution could reveal changes in the
showninFigure2,thenormalizedgrain-size distributionof activated recrystallization processes. Such changes could
a 2-D Potts microstructure is highly comparable with a also provide information on the mechanisms affecting the
natural microstructure sampled along the EPICA (European normalgraingrowth,suchaspinningofgrainboundariesby
ProjectforIceCoringinAntarctica)DomeCcore(75806’S, microparticles which induces a shift of hi and (cid:3) (Riege
D D
123821’E) at 100.1m depth (see also section 3.1.3). and others, 1998; Durand and others, 2006).
We simulated 200 microstructures of 10002 pixels From Potts model results, and following the procedure
containing between 622 and 1808 grains. We supposed detailed in section 3.1.2, we also estimate the intrinsic
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Durandandothers:Icemicrostructureandfabric 623
variabilityofhi and(cid:3) inducedbythesectioningeffectand examined under different scales: a small p explores
D D
the change in the population: details, whereas a larger p improves the statistics.
(cid:1) (cid:2) (cid:3) (cid:3) Compared to the inertia tensor presented in Equa-
(cid:3)ðhi Þ¼ 0:06þ1:83(cid:1)N(cid:3)1=2 (cid:1)(cid:3)hi (cid:3) tion (2), M has the additional advantage that it has a
D (cid:1) g (cid:2) D
physicalsignificanceintermsofmechanicaldeformation
(cid:3)ð(cid:3)DÞ¼ 0:04þ0:95(cid:1)Ng(cid:3)1=2 (cid:1)(cid:3)D: (Aubouyandothers,2003).FromthevariationofMwith
respect to a reference M , a statistical strain tensor, U,
0
which coincides with the classical definition of strain
3.2. Grain shape and deformation of the
(Aubouy and others, 2003), can be defined. As M
microstructure 0
cannot be measured for natural ice, Durand and others
Many different descriptors can be used to determine the
(2004)assumethatthedeformationisisochoric,andthat
shape of grains and quantify the anisotropy of a micro-
the reference state is isotropic, leading to the following
structure (Mecke and Stoyan, 2002). Many have been
expression:
applied by the glaciological community, so comparison 0 (cid:1) (cid:2) 1
(cid:4)
bbeetwtriecekny.dAifmfeorenngttrheesumltsofsrtofmreqduifefenrtelyntucsoedrespacraanmseotmerestiamree:s U ¼14R@log0(cid:4)12 log(cid:1)0(cid:4)2(cid:2)AR(cid:3)1: ð5Þ
(cid:4)
1
1. Comparison between the mean vertical and horizontal
Note that some statistical tests, comparable to those
intercept lengths (Thorsteinsson and others, 1997; Gay
presented in section 3.1.2, have been carried out to
and Weiss, 1999). In addition to the inherent artefacts
estimate the intrinsic variability of U that leads to
induced by the measurement of hLi (see section 3.1.1),
thismethodisnotabletocorrectlyquantifyanisotropyin (cid:3)ðUÞ¼0:35N(cid:3)1=2 (Durand and others, 2004).
a non-horizontal plane. A similar method was used by
Note that Durand and others (2004) called M the texture
Svensson and others (2003), as they measured the
tensor and used different notation. To avoid confusion with
bounding box of each grain and calculated the corres-
our definition of texture (see section 1), M will be denoted
pondingaspectratio,i.e.thewidth-to-heightratioofthe
the microstructure anisotropy tensor.
smallest rectangle with horizontal and vertical sides
whichcompletelyenclosesthegrain.Suchamethodalso
does not fully capture grain elongation that is not 4. FABRIC PARAMETERS
oriented in the horizontal or vertical directions, unless
Thissectionpresentssometoolsthatallowdescriptionofthe
the box orientations are rotated.
fabric measured from a thin section containing N non-
g
2. An inertia tensor for each grain can also be calculated intersectingcrystals.Inthissectioneachindividualcrystalis
(Azuma and others,2000; Wilen and others, 2003).The described by:
eigenvalues of the tensor provide the elongation of the
grain, and the eigenvectors give the directions of 1. Its orientation ck. The orientation of the ice crystal is
elongations. Compared to the previous methods, this described by its c-axis orientation, which is defined by
tensor description has the advantage that the obtained twoangles:theco-latitude(cid:1)k 2½0,(cid:5)=2(cid:4)(ortiltangle)and
measurementsdonotdependontheparticularchoiceof the longitude ’k 2½0,2(cid:5)(cid:4) given in the local reference
axes (e.g. vertical and horizontal). frame, R, whichhas its e(cid:1)3 axis perpendicular to the thin
section. The expression of ck in this reference frame is
3. Durandandothers(2004)proposedanewmethodbased
on the vectors, ‘, that link neighbouring triple-junction ck ¼ðcos’ksin(cid:1)k,sin’ksin(cid:1)k,cos(cid:1)kÞ: ð6Þ
locations (where three grain boundaries meet). This
2. Its volume-weighted fraction, f . Larger crystals should
method, which can suffer from some artefacts in porous k
have more influence on the polycrystal behaviour than
ice (triple junctions may occur in pores), is briefly
smaller ones. As suggested by Gagliardini and others
recalled here. More details and applications can be
(2004),toestimatethevolumeofgraink from2-Dthin-
found in Durand and others (2004). A local average
section measurements, one can assume that it is
microstructure anisotropy tensor, M, can be defined as:
* !+ proportional to its measured cross-sectional area, A .
k
‘2 ‘ ‘
M¼ 1 1 2 Then, the volume-weighted fraction is:
¼ 1 XN‘2‘1‘‘ðjÞ(cid:5)‘22‘‘ðjÞ p fk ¼PNj¼Ag13k=A23j=2: ð7Þ
N
(cid:4)k¼1 (cid:5) In what follows, the fabric of the polycrystal is given in
¼R (cid:4)1 0 R(cid:3)1, ð4Þ termsofthedistributionoftheNg c-axisorientations,ck,
0 (cid:4) and the N weight fractions, f .
2 g k
where ð‘ ,‘ Þ are the coordinates of ‘ in the local
1 2
reference frame, R, (the e(cid:1) axis is perpendicular to the 4.1. Rotation of reference frame and polar
3
representations
thin section). The absolute horizontal orientation is
unknown;e(cid:1) isrelativetothethinsection,whichcannot 4.1.1. Rotation of reference frame
1
beorientedwithrespecttothecoreaxis.Ristherotation Most studies of ice fabrics use equal-area point plots (also
which makes M diagonal and ð(cid:4)1,(cid:4)2Þ are the corres- namedSchmidtpointplots)topresentthedistributionof ck
ponding eigenvalues; h(cid:6)ip denotes the average over N axes in a thin section (e.g. Thorsteinsson and others, 1997;
vectors, up to the pth neighbours. A sample can be see also Fig. 4a). In the literature, the polar plots are
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624 Durandandothers:Icemicrostructureandfabric
underlying large population of crystals. The data are then
^
used to create a robust estimate, GðxÞ, of GðxÞ. With N
g
data points, GðxÞ is represented as a superposition of N
g
simpler normal probability-density distributions, g ðxÞ, for
k
N sub-populations,
g
XNg
GðxÞ¼ f g ðxÞ: ð11Þ
k k
k¼1
GðxÞ is then projected onto an equal-area diagram using
Fig. 4. Point plots (a) and contour plots (b) of the inferred
Equation (10) to produce Gðx,yÞ.
probability-densitydistributionsofthefabricat2999.8mdepthin
the Dome C ice core. The probability-density distribution, G, is For each crystal orientation, ck, the orientation-
contouredatlevelsof(cid:3)¼1=ð6(cid:5)Þ. distribution density of the associated kth sub-population
on a hemisphere centered on axis ck is defined as:
(cid:7) (cid:1) (cid:2) (cid:1) (cid:2)(cid:8)
(cid:7) 1
g ðxÞ¼ exp (cid:3) x(cid:3)ck (cid:6) x(cid:3)ck , ð12Þ
k 2(cid:5)(cid:2)2 2(cid:2)2
represented in a horizontal reference frame, Rh, i.e. the in
where(cid:7) isanormalizingfactortoaccountforthespreadof
situ vertical (the core axis) is at the centre of the polar plot
theGaussianonahemisphere.Inpractice,(cid:7)’1ifthereare
(Langway,1958).Therefore,ifthinsectionsarecutvertically,
ten or more crystals to contour.
thefabricsneedtoberotatedforeasiercomparisonoffabric
Each g ðxÞ has a mean orientation given by ck, and the
diagrams. To obtain a horizontal projection from a vertical k
width of the component normal probability-density distri-
thin section, one needs to rotate ck as follows:
bution is (cid:2)¼(cid:2) þ(cid:2) , where (cid:2) is the spread due to
M N M
’h ¼arctanðtan(cid:1)vsin’vÞ 2½0,2(cid:5)(cid:4) measurementerror,and(cid:2)N representsspreadduetolimited
k k k
sample size, N . Using numerical experiments, it can be
(cid:1)h ¼arccosð(cid:3)sin(cid:1)vcos’vÞ 2½0,(cid:5)=2(cid:4) ð8Þ g
k k k shown(cid:9)pthffiffiaffiffitffiffi the ‘best’ (cid:2) (for 40 or more crystals) is
where ’h and (cid:1)h are the longitude and co-latitude in the (cid:2)¼2 Ng.
k k
horizontal reference frame, and ’v and (cid:1)v are those in the The expected value of GðxÞ for isotropic ice is
vertical frame. In this horizontal rekferencekframe Rh; the e(cid:1)h E ¼1=ð2(cid:5)Þ, and real clusters should exceed the
3 Eþ1=ð3(cid:5)Þ contours, that is E(cid:7)2(cid:3), where (cid:3)¼1=ð6(cid:5)Þ.
axis points vertically upward. If the thin section is cut
ThefunctionG shouldbecontouredat2(cid:3),4(cid:3),...intervals,
horizontally R ¼Rh.
with larger spacing if very strong maxima occur. A
comparison between a classical point plot and a contoured
4.1.2. Equal-area (Schmidt) point plots and contoured
data plot is presented in Figure 4 for a Dome C texture
data plots
sampled at 2999.8mdepth. The contour plotwith N ¼80
The equal-area projection (Kocks, 1998, p.54) is achieved g
illustrates very clearly the strength of the clustering of
by the coordinate transformation:
crystalsnearthevertical.Suchanobservationislessobvious
(cid:1)h in the point plot. More details will be given in a forth-
(cid:6)¼2sin k, ð9Þ
coming paper.
2
and
4.2. Strength of fabric and spherical aperture
x ¼(cid:6)cos’h, y ¼(cid:6)sin’h: ð10Þ
k k k k The strength of fabric (also called degree of orientation or
Thistransformationtakesallpointsintheuppperffiffiffihemisphere strength of orientation) is a one-parameter fabric descriptor
thathasbeenwidelyusedintheglaciologycommunity.This
and projects them within a circle of radius 2 on the x-y
is mainlybecausemost of theobservedice fabrics are one-
plane.Anequal-areapointplotthensimplyputsamarkerat
maximum or girdle-type fabrics and therefore present a
the ðx ,y Þ location of each crystal orientation.
k k
symmetryaxis.
The purpose of presenting c-axis distributions is often to
The definition of the strength of fabric that can be found
learn as much as possible about the fabric in nearby ice,
in the literature (Thorsteinsson and others, 1997; Castelnau
not merely to present a specific dataset (Kamb, 1959).
and others, 1998; Gagliardini and Meyssonnier, 2000) is:
Now that data are collected using automatic methods
(cid:10) (cid:10)
(Wilen, 2000), a collection of several hundred to over a (cid:10)(cid:10)XNg (cid:10)(cid:10)
thousand measured crystal orientations is common. This R ¼2(cid:10) f ck(cid:10)(cid:3)1, ð13Þ
s (cid:10) k (cid:10)
results in the problem of overlapping points when k¼1
presenting the data on point plots. Some authors (Langway, where jj:jj is the norm of a vector. This definition of the
1958; Kamb, 1959) have presented fabric information strength of fabric assumes implicitly that the fabric has a
using contour plots. symmetry axis and that this symmetry axis is the axis e(cid:1)
3
Because any sample is limited in its size, Ng, we can, at perpendicular to the thin section. Therefore, this parameter
best, get an imperfect estimate of the underlying popula- is not objective since it depends on the reference frame
tion. Estimates that are robust are desirable. A new where the sum is performed.
approach, which stems largely from Kamb (1959) is Amorerigorousdefinition,butstillnotobjective,forthe
currently under development and is briefly described here. strength of fabric, R , needs to first calculate the ‘best’
s
It is first assumed that there is an unknown ‘global’ symmetry axis for the fabric. Later we will present two
^
orientation probability-density distribution, GðxÞ, for the different methods to effectively determine this fabric
https://doi.org/10.3189/172756506781828377 Published online by Cambridge University Press
Durandandothers:Icemicrostructureandfabric 625
(cid:1) (cid:1)
symmetry axis, c. Here, assuming that c is known, the that, despite the historical point of view and comparisons
strength-of-fabric parameter is calculated as: with old datasets, there is no longer a reason to use these
(cid:10) (cid:10)
(cid:10)(cid:10)XNg (cid:10)(cid:10) two parameters. As shown below, the second-order orien-
R ¼2(cid:10) f signð(cid:1)c(cid:6)ckÞck(cid:10)(cid:3)1: ð14Þ tationtensorcontainsmuchmoreinformationandshouldbe
s (cid:10) k (cid:10)
preferred.
k¼1
The sign operator allows us to perform the sum within the 4.4. Orientation tensors
hemisphere centred on the fabric symmetry axis, (cid:1)c. Never-
4.4.1. Second-order orientation tensor
theless, with this new definition, the same problem still
As suggested by Woodcock (1977), a good way to
occurs for the grains which have their c axis perpendicular
characterize the essential features of an orientation distri-
to (cid:1)c, for which (cid:1)c(cid:6)ck ¼0 and no sign can be determined.
bution is to use the second-order orientation tensor, að2Þ,
Note that,as mentionedpreviously, Equations(14)and (13)
defined as:
are equivalent only if(cid:1)c¼(cid:1)e .
3
The spherical aperture, (cid:8)s, is directly related to the XNg
strength of fabric by: að2Þ ¼ f ck (cid:5)ck, ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi k
(cid:8) ¼arcsin 1(cid:3)R : ð15Þ k¼1
s s
For isotropic ice Rs ¼0 and (cid:8)s ¼90(cid:8) and for a single- whereck isgivenbyEquation(6)andfk byEquation(7).By
maximum fabric R and (cid:8) ¼0(cid:8). For fabrics with girdle construction, að2Þ is symmetric and there exists a symmetry
s¼1 s
tendency, (cid:3)1(cid:9)R<0 and, due to its definition, (cid:8) is reference frame, Rsym (or principal reference frame), in
s
unusable. which að2Þ is diagonal. Let að2Þ (i = 1,2,3) denote the three
i
correspondingeigenvaluesandoe (i=1,2,3)theassociated
4.3. ‘Best’symmetry axis of a fabric i
eigenvectors, whichcorrespondtothethreebasevectorsof
A first method to calculate the ‘best’ symmetry axis of a Rsym. The eigenvalues of að2Þ can be seen as the lengths of
fabricistousealeast-squaresmethodtominimizethesum the axes of the ellipsoid that best fit the density distribution
ofthesquaredanglebetweenthesymmetryaxisandallthec ofthegrainorientations.Theeigenvectorsthengivetheaxis
axes, ck. That is, we need to find(cid:1)c which minimizes: directions of this ellipsoid.
(cid:3)(cid:1)2 ¼XNg fk(cid:3)2k, ð16Þ relaTthioenst:hree eigenvalues að12Þ, að22Þ and að32Þ follow the
k¼1
where (cid:3)k is the angle between(cid:1)c and ck: að12Þþað22Þþað32Þ ¼1 ð19Þ
cos(cid:3)k ¼j(cid:1)c(cid:6)ckj ð17Þ 0(cid:9)að2Þ (cid:9)að2Þ (cid:9)að2Þ (cid:9)1: ð20Þ
3 2 1
(assuming(cid:1)c is a unit vector).
TheabsolutevalueinEquation(17)istotakeintoaccount For an isotropic fabric, að12Þ ¼að22Þ ¼að32Þ ¼1=3, and when
that each vector ck must be in the same hemisphere as (cid:1)c. thefabricistransverselyisotropic,twooftheeigenvaluesare
The minimum value of (cid:3)(cid:1) deppenffiffiffidffiffisffiffiffiffioffiffiffin the fabric type. For a equal:
tpmyeparexfeimcfatlubymriics-oty(cid:5)tpr=oe2pfi(cid:10)acbf(cid:3)r(cid:1)aicb>rpicpffi(cid:5)ffi(cid:3)ffi(cid:1)ffiffi(cid:5)(cid:3)ffiffiffiffiffiffi¼ffi(cid:3)ffiffiffi2ffiffiffiffiffiffi2ffi>ffiffi.(cid:5)(cid:3)D(cid:1)(cid:3)u(cid:10)e20(cid:2)to,w6th1he(cid:8)eraaenbadssoffoolurrtaaesgviinardlgulleee--, aað2ð122ÞÞ (cid:2)(cid:2)aað3ð222ÞÞ <>11==33 ffoorraasgiinrdgllee-fmabarxiicm: umfabric, ð21Þ
this problem is not continuous and may present some local
According to Woodcock (1977), fabrics that have equal
minima.
Asecondsolutionistodefinethesymmetryaxis,(cid:1)c,asthe girdle and cluster tendencies are such that a1ð2Þ=að22Þ ¼
first eigenvector obtained from the calculation of the að2Þ=að2Þ. Then, the use of the parameter
2 3
eigenvalues of the second-order orientation tensor, as (cid:1) . (cid:2). (cid:1) . (cid:2)
presented below. This tensor is obtained from the direct (cid:9)¼ln að2Þ að2Þ ln að2Þ að2Þ
1 2 2 3
calculation of the average of the dyadic product of the
c axes. By definition, there is no ambiguity from the c-axes gives objective information about the fabric shape: (cid:9)>1
orientations for the calculation of the second-order orien- for single-maximum fabric and (cid:9)<1 for girdle-type
tation tensor. fabric.(cid:1)As s.ugges(cid:2)ted by Woodcock (1977), one should use
Thedifferencein(cid:1)cresultingfromthesetwomethodscan C ¼ln að2Þ að2Þ as a measure of the strength of the
be large, especially when the fabric is not concentrated.As 1 3
an example, for a randomly distributed fabric, the angle preferred orientation. C varies from 0 for randomly distrib-
betweenthe(cid:1)cobtainedbythetwodifferentmethodscanbe uted orientations to infinity when all the orientations are
upto308.Thisisbecauseforaperfectlyrandomlydistributed identical.
fabricthereareaninfinitenumberof‘best’symmetryaxes,(cid:1)c The three eigenvectors, oei, define the ‘best’ material
(everyaxisisasymmetryaxisforsuchafabric). symmetryreferenceframeofthesample.Itisoftenusefulto
In what follows, the ‘best’ symmetry axis is determined calculate the angle between the main symmetry axis of the
usingthesecondmethod.Moreover,sinceboththestrength fabric, oei, and the local vertical:
of fabric, R , and the spherical aperture, (cid:8) , require (cid:1) (cid:2)
knowledge osf the mean symmetry axis of sthe grain (cid:1)¼ arccos oe1(cid:6)eh3 2½0,(cid:5)=2(cid:4), ð22Þ
orientations and, consequently, calculation of the eigen-
vectors of the second-order orientation tensor, we believe where the eh are the basis vectors of Rh.
i
https://doi.org/10.3189/172756506781828377 Published online by Cambridge University Press
626 Durandandothers:Icemicrostructureandfabric
The study of higher-order orientation tensors allows orientation tensors to describe fabrics, as was mentioned
investigation of the gap between the actual fabric and the above, for the strength of fabric, R , or the spherical
s
orthotropicfabricconstructedbytheuseofthesecond-order aperture, (cid:8) .
s
orientation tensor.
4.5. Average and dispersion of the c axes within
4.4.2. Higher-order orientation tensors a grain
By definition, the fourth-order orientation tensor is: UsinganAITA,theinitialmeasurementgivestheorientation
XNg of each pixel in the sample. After extraction of the
að4Þ ¼ f ck (cid:5)ck (cid:5)ck (cid:5)ck: ð23Þ microstructure, one has to determine the mean orientation,
k
k¼1 ck,foreachgrain,asusedintheprevioussections.Asfora
The fourth-order orientation tensor contains more informa- grain, Equation (6) applies for the pixel orientation vector.
tion about the fabric than the second-order orientation Since, similarly to a grain, a pixel can be oriented by
tensor and can reproduce non-orthotropic fabric. If the eithercp or(cid:3)cp,themeanorientationofagrain,ck,cannot
fabric is orthotropic then, by construction, the fourth-order be calculated as the average value over the Nk orientations
p
orientationtensorhasthesameprincipalaxisasthesecond- cp.Asforthewholethinsection,themeanorientationofthe
order one, and only 21 of its 81 components are non-nulls
grain has to be determined as the first eigenvector of the
when expressed in the symmetry reference frame Rsym. second-orderorientation tensorcalculatedonthegrain,i.e.
Moreover, according to Cintra and Tucker (1995) only 3 of for a given grain k, ck ¼oek, where oek is the first
these 21 components are independent. The 21 non-null 1 1
eigenvector of the second-order orientation tensor defined
components are:
by Equation (18) in which N has been replaced by Nk.
g p
að4Þ ,að4Þ ,að4Þ , Contrary to a polycrystalline fabric, adjacent pixel
f1122g f1133g f2233g
orientations on the same grain should be very close except
að4Þ ¼að2Þ(cid:3)að4Þ (cid:3)að4Þ ,
1111 11 1122 1133 when crossing a subgrain. Nevertheless, some artefacts are
að4Þ ¼að2Þ(cid:3)að4Þ (cid:3)að4Þ , still present for some pixels and these pixels need to be
2222 22 2211 2233
excludedfromfurthercalculations.Indeed,foragivenview,
að4Þ ¼að2Þ(cid:3)að4Þ (cid:3)að4Þ , ð24Þ thedeterminationoftheextinctionangleconstrainscp tolie
3333 33 3311 3322
alongtwoorthogonalplanes.TheAITAsrepeatthemeasure-
whereaiijj denotesthes(cid:11)eriesofcomponentsindependento(cid:12)f
ments for three different views that should unambiguously
theorderoftheindices a ¼a ¼a ¼a ¼a ¼a .
iijj ijij ijji jiij jiji jjii determine cp. Unfortunately, for some pixels the extinction
The 60 other components are zero for an orthotropic
positionisnotavailablefor(atleast)oneoftheviewsandthis
fabric.
leads to ambiguities, i.e. some cp lie along plane (see the
Foranaturalnon-orthotropicfabric,onecanusethevalue
polarplotsinFig.5).Alsonotethatthepixelsclosetograin
of these 60 components as an estimate of the gap to
boundariesgenerallygiveanoisydeterminationoftheircp.
orthotropy.These60componentscanbegroupedinthetwo
To remove these artefacts, we use the following pro-
classfamiliesaðii4ijÞandaðii4jkÞ withi 6¼j 6¼k.Theyarecomposed cedure. For grain k, oek is calculated a first time by taking
1
of24and36terms,respectively,butbyconstruction,only6 into account all the Nk pixels of the grain. Then the
p
and3,respectively,ofthesetermsaredifferent.
misorientation angle (cid:3) between oek and cp is calculated
Onecanthendefineafourth-orderestimateofthegapto p 1
foreachpixel.Notethat(cid:3) belongstotherange½0;90(cid:4)ascp
orthotropy of a fabric as: p
(cid:1)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) canalsobeorientedby(cid:3)cp.Distributionsof(cid:3)p areplotted
(cid:4)ð4Þ ¼4 (cid:3)að4Þ (cid:3)þ(cid:3)að4Þ (cid:3)þ(cid:3)að4Þ (cid:3)þ(cid:3)að4Þ (cid:3)þ(cid:3)að4Þ (cid:3) in Figure 5 for grains k = 7 and 9 of a texture sampled at
O 1112 1113 2221 2223 3331
(cid:3)(cid:3) (cid:3)(cid:3)(cid:2) (cid:1)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:2) 1525:8m depth on the EPICA Dome C ice core. If 66% of
þ(cid:3)að4Þ (cid:3) þ12 (cid:3)að4Þ (cid:3)þ(cid:3)að4Þ (cid:3)þ(cid:3)að4Þ (cid:3) : ð25Þ the pixels do not present a (cid:3) value smaller than 108 then
3332 1123 2213 3312 p
the grain is rejected and will not be included in fabric
Since, by definition, all the components of að4Þ are lower
parameter calculations. This is the case for grain k ¼ 9.
than 1, we have (cid:4)ð4Þ (cid:9)16. For a perfect orthotropic fabric However, if the distribution of (cid:3) is well centred around 0
O p
(cid:4)ð4Þ ¼0andlargervaluesof(cid:4)ð4Þindicatethatthecaxesdo (more than 66% of the pixels), oek is recalculated, with the
O O 1
not fulfil the orthotropic symmetry conditions. Calculations pixels for which (cid:3) >10(cid:8) excluded from this new calcula-
p
on140samplesalongtheEPICADomeCcorehaveshowna tion,buttheyarestillincludedintheareacalculation.Note
maximum value for (cid:4)ðO4Þ of 2.11. that most of the pixels presenting (cid:3)p >10(cid:8) are close to the
grain boundaries. This limit of (cid:3) >10(cid:8) takes into account
p
4.4.3. Odd-orders orientation tensor thatwithinagraintherotationrecrystallizationcanreacha
Since a grain can be oriented using either ck or (cid:3)ck, care maximum misorientation of (cid:11)10 to 158 (see section 4.7).
must be taken in using the odd-order orientation tensors to For large grains represented by a significant number of
describe the strength of fabric, R (the sum in the definition pixels, one can attempt to quantify the misorientation
s
ofR correspondstothefirst-orderorientationtensor).Ifone within a grain induced, for example, by rotation
cuts all the grains into two equal parts, one oriented by ck recrystallization. To this aim the eigenvalues of the
andtheotherby(cid:3)ck,itisobviousthatallthecomponentsof second-order orientation tensor are studied. For a grain,
assuming that all the pixel c axes are very close, one
the odd-order orientation tensors are nulls. Even if the odd
orientation tensor is expressedin Rsym, the grains that lie in should expect að2Þ (cid:2)1 and að2Þ (cid:2)að2Þ (cid:2)0. If one assumes
1 2 3
theplane(oe ,oe )canstillbeorientedbyeitherck or(cid:3)ck. that all the pixel c axes are randomly distributed within a
1 2
As a conclusion, one should not use the odd-order cone centred on oek and with a half-angle (cid:3) , then the
1 k
https://doi.org/10.3189/172756506781828377 Published online by Cambridge University Press
Durandandothers:Icemicrostructureandfabric 627
eigenvalues are að2Þ¼ð1þcos(cid:3) þcos2(cid:3) Þ=3 and að2Þ ¼
1 k k 2
að2Þ¼ð1(cid:3)að2ÞÞ=2. As an example, since two new grains are
3 1
created if the rotation recrystallization distortion within the
grain becomes larger than (cid:11)158, one should not expect to
findvaluesofa2 smallerthan0.99.Thestudyofað2Þ andað2Þ
1 2 3
can then show whether the pixel c axes are randomly
distrib-uted (að2Þ ¼að2Þ) or if there exists a preferential
2 3
direction of the misorientation (að2Þ >að2Þ ¼0). In the
2 3
present paper, the large error in the pixel orientation
measurement of our apparatus means we have not been
able to obtain pertinent results.
4.6. Dispersion and error bars associated with the
second-order orientation tensor
The standard deviation, (cid:3) , of að2Þ is made up of (1) the
að2Þ
uncertainties induced by the variability of the measure-
ments, (cid:3)m , added to (2) the uncertainties induced by the
að2Þ
sampling on a limited number of grains, (cid:3)p .
að2Þ
Inordertoestimate(cid:3)m ,thesamethinsectionpresenting
að2Þ
a nearly random fabric was measured ten times. Further
measurements were not feasible, as the sample was
sublimating during the experiment. For each grain present
during the ten successive measurements, the mean c axis
wascalculatedusingtheproceduredescribedinsection4.5.
As a result, the standard deviation of the misorientation for
the ten measurementsof ck wasfound tobe (cid:11)0.68 and not
dependent on the grain orientation itself. Note that, even if
sublimation was occurring during the experiment, no
systematic bias was noticed in the grain c-axis measure-
ments. This last point indicates that small variations of the
sample thickness are not critical for the quality of the
measurements. Then, to estimate (cid:3)m , a random Gaussian
að2Þ
noisewithastandarddeviationof0.68wasaddedtoallthe
Fig. 5. (a) Distribution of the misorientation angle between oek
ck and the parameter að2Þ was calculated. This procedure 1
andallthepixelsofgraink=7inthetexturesampledat1525.8m
was repeated 200 times, allowing calculation of (cid:3)m over
að2Þ on the EPICA Dome C ice core. The corresponding polar plot is
the 200 draws. This method should be repeated for each alsodrawn(left).Asthisgrainfulfilstheconditionforanincluded
sample in order to get an estimation of (cid:3)mað2Þ. However, grain(see text), oek1 is recalculated byexcluding the pixelswith a
experiments carried out on 140 samples have shown that misorientation larger than 108. The corresponding polar plot is
(cid:3)m can be neglected whatever the N , as it is at least one also plotted (right). (b) Distribution of the misorientation angle
að2Þ g
order of magnitude lower than (cid:3)pað2Þ. Nevertheless, for between oek1 and all the pixels of grain k = 9 from the same thin
section and the corresponding polar plot. This grain is rejected
manual measurements, for which the standard deviation
on the estimation of ck is (cid:11)58, (cid:3)p may have to be taken from the analysis.
að2Þ
into account to estimate (cid:3) .
að2Þ
Following the method proposed by Gagliardini and
others (2004), a set of 231 fabrics of 10000 grains each central limit theorem is well respected as maxð(cid:3)p Þ
aii
was numerically generated on a grid of the possible values dependslinearlyonN1=2.Theslopeofthelinearregression,
g
of að12Þ and að22Þ, i.e. on the domain defined by 0<að12Þ <1 (cid:7),changeswiththevalueofað2Þ.Itisworthnotingthatfora
1
and 0<að22Þ <1(cid:3)að12Þ. Then, for a given fabric, we givenvalueofað2Þ,(cid:7)isroughlythesame(crossesandcircles
1
randomly chose N grains within the range [100;1000]
g for að2Þ ¼0:8, stars and squares for að2Þ ¼0:6 in Fig. 6a).
andcalculatedthefabricparameters.Foragivennumberof 1 1
grains,Ng,thisoperationwasreproduced300timesandthe Figure 6b shows the evolution of (cid:7) vs the value of að12Þ for
standard deviation, (cid:3)p , was calculated. This procedure the 231 different fabrics. As we did not want to under-
að2Þ estimate (cid:3)p , we chose to make a second-order polynomial
was repeated for different values of Ng. Figure 6 shows the aii
fit only on the largest values of each bin. From this
results forthe second-orderorientation tensor, að2Þ. Inorder polynomial regression, (cid:7), and thus (cid:3)p can be easily
to work with only one parameter, maxð(cid:3)p Þ is attributed to aii
aii estimated through:
the nine components of tensor að2Þ. Figure 6a shows the (cid:7) (cid:1) (cid:2) (cid:8)
1ev0o0lu<tioNng <of10m0a0xðf(cid:3)ropaimiÞ vas N10g1=0200forgruanidneerdsaminpitliianlgsfawbritihc (cid:3)paii ¼ (cid:3)1:64(cid:1) að12Þ 2þ1:86(cid:1)að12Þ(cid:3)0:14 (cid:1)N1g1=2: ð26Þ
defined by að2Þ ¼0:9, að2Þ ¼að2Þ ¼0:05 (black dots). The Then we have (cid:3)p ¼(cid:3)p ¼(cid:3)p ¼(cid:3)p .
1 2 3 að2Þ að2Þ að2Þ að2Þ
1 2 3 ii
https://doi.org/10.3189/172756506781828377 Published online by Cambridge University Press
628 Durandandothers:Icemicrostructureandfabric
Fig. 7. Distribution of the misorientation between neighbouring
grainpairs(blackline),andtheenvelopeofprobabilitydetermined
after the generation of 200 distributions of the misorientation
between random pairs for a texture sampled at 1525.8m on the
EPICADomeCicecore.Thechangesinthegreyintensityshowthe
(cid:3)(68.3%,darkgrey),2(cid:3) (95.4%,grey)and3(cid:3)(99.7%,lightgrey)
confidencelevels.
account the N (cid:1)ðN (cid:3)1Þ=2 pairs within the sample (or
g g
a random subset). They argued that discrepancies between
D(cid:3)N and D(cid:3)R should highlight the occurrence of
recrystallizationprocesses.Rotationrecrystallization,which
leads to a grain splitting into smaller grains with similar
orientations, should present the highest population of low
angles for D(cid:3)N compared to D(cid:3)R. Migration recrystalliza-
tion produces grains at high angles to their neighbours
(Poirier, 1985) and should also be revealed by comparing
Fig.6.(a)The maxð(cid:3)p ÞinducedbythesamplingofN grainsover D(cid:3)N and D(cid:3)R.
aninitialpopulationoafii10000.Dots:að2Þ¼0:9,að2Þ¼gað2Þ¼0:05; Here we propose a new method for calculating D(cid:3)R:
1 2 3
crosses: að2Þ¼0:8, að2Þ¼0:15, að2Þ¼0:05; stars: að2Þ¼0:6, the fabric is reshuffled while relations between neighbours
1 2 3 1 are kept, i.e. the microstructure is unchanged. Two
að2Þ¼0:35, að2Þ¼0:05. The corresponding linear regressions are
2 3 hundred distributions of the misorientation between neigh-
plotted as solid lines. Circles: að2Þ¼0:8, að2Þ¼0:1, að2Þ¼0:1;
1 2 3 bouring grain pairs are reproduced, leading to the esti-
squares : að12Þ¼0:6, að22Þ¼0:2, að32Þ¼0:2. The corresponding mation of a probability-density envelope. This method has
linear regressions are plotted as dashed lines. (b) Slope of the been applied for a texture sampled at 1525.8m on the
linear regression as presented in (a) vs ðað2ÞÞ for the 231 textures EPICA Dome C ice core (see Fig. 7). Numbers of low-angle
1
tested. The maximum of each bin is marked with a cross, and misorientation between neighbouring pairs (0–108) are
corresponding second-order polynomial fit corresponds to the significantly larger than would be expected for a random
blackcurve.
distribution, indicating that rotation recrystallization is
certainly occurring. This confirms the results of Durand
andothers(2006)whoargued,fromresultsofamodel,that
rotation recrystallization should occur along the EPICA
4.7. Misorientation of neighbouring grain pairs Dome C core.
Several studies propose the investigation of the misorienta- Note that a misorientation is expressed by an angle but
tion between neighbouring grain pairs (Alley and others, also by a rotation axis. The distribution of these axes could
1995; Wheeler and others, 2001). The misorientation also reveal information concerning ice flow (Wheeler and
between two grains is defined as the angle, (cid:3), between others,2001).Toourknowledge,suchadistributionhasnot
the two c axes, and the distribution of (cid:3) is calculated for yet been used in glaciology, though it could be helpful in
all the neighbouring grain pairs within the texture; this learning more about ice deformation.
distribution will be denoted D(cid:3)N. Note that the de-
scription of the whole texture is obviously required to
calculate D(cid:3)N. Due to the evolution of the fabric with 5. THE TEXTURE TOOLBOX
depth, direct comparison of D(cid:3)N between samples was To ease and homogenize texturedata processing within the
1
not possible. Alley and others (1995) proposed comparison glaciologicalcommunity,weproposesomeMATLAB tools
of D(cid:3)N with the distribution of angle misorientation of grouped in a texture toolbox. This toolbox has been
random grain pairs (D(cid:3)R) calculated by taking into designed to (1) completely fulfil the requirements proposed
https://doi.org/10.3189/172756506781828377 Published online by Cambridge University Press