an up to date approach to measure textures - Niels Bohr Institutet

Ice microstructure and fabric: an up to dateapproach to measure texturesGaël Durand 1 , O. Gagliardini 2 , Throstur Thorsteinsson 3 ,Anders Svensson 1 , Josef Kipfstuhl 4 , and Dorthe Dahl-Jensen 1September 21, 20061 Ice and Climate Group, Niels Bohr Institute, Rockefeller Complex, JulianeMaries Vej 30, 2100 Copenhagen Ø, DenmarkE-mail: gd@gfy.ku.dk2 Laboratoire de Glaciologie et Géophysique de l’Environnement, CNRS,UJF-Grenoble I, BP 96, F-38402 Saint-Martin d’Hères Cedex, France3 Institute of Earth Sciences, Building of Natural Sciences, Askja, Sturlugata 7IS-101 Reykjavik, Iceland4 Alfred Wegener Institute, Columbusstrasse, D-27568 Bremerhaven, GermanyAbstractAutomatic c-axes analysers have been developed over the last years,leading to a large improvement of the available data for the ice crystalstexture analysis. Such an increase of the quality and quantity ofthe data allows for stricter statistical estimates. The current texturalparameters, i.e. fabric (crystallographic orientations) and microstructure(grain boundaries network), is presented. These parameters definethe state of the polycrystal and give information about the deformationundergone by the ice. In order to take into account the abilityarising from automatic measurements, some parameter definitions areupdated and new parameters are proposed. Moreover, a Matlab toolboxhas been developed to extract all the textural parameters. Thistoolbox, that can be downloaded online, is briefly described.1

1 INTRODUCTIONIce cores provide a wonderful record to investigate past climates. As ice sheetsform by successive deposition of snow layers, a travel down to the deepestpart of the ice sheet, is a journey into the past. Ice core studies provide alarge set of climatic parameters, such as past temperature variations whichcan be revealed through isotopic data, the atmospheric composition throughgas trapped in bubbles or the atmospheric circulation inferred through impuritiesconcentrations. Apart from these immense amount of informationavailable, the dating is still the achilles heel of these archives. For sites withhigh accumulation rates, the counting of annual layers enables estimation ofthe age (GRIP, GISP and NGRIP cores among other, see e.g. Alley and others(1997b)). At depths where annual layers are not resolvable, or for siteswith low accumulation rates, flow models have to be used to reconstruct thesinking of the ice layers and then estimate the age with depth (Dome C,Vostok, Dome F cores among other, see e.g. Parrenin and others (2001)).Consequently, understanding the deformation of the ice is of primary importancefor correct climatic interpretations.From a material science point of view, ice is an assemblage of crystals withhexagonal crystallographic structure. The crystal orientation is specified bythe direction of its a and c axes. Due to the intrinsic birefringence of ice,the c-axis of a single crystal, which is perpendicular to the basal planescan be easily revealed by optical measurements. In what follows, the fabricrefers to the orientations of the c-axes of the ice polycrystal, whereas themicrostructure refers to the grain boundaries network. The term textureincludes both the fabric and microstructure. Note that this terminology isnot universally accepted, and other usages of these terms can be found inthe literature.It is worth noting that ice flow is strongly dependent of the texture.Firstly, the deformation of an ice crystal occurs by dislocation slip mainlyalong basal plane, conferring an outstanding viscoplastic anisotropy of theice crystal, i.e. the resistance to shear on non basal planes can be 60 timeshigher than resistance to shear along the basal planes (Duval and others,1983). Due to the intrinsic anisotropy of ice, the c-axes rotate during deformationtowards compressional axes and away from tensional axes (Azumaand Higashi, 1985; Van der Veen and Whillans, 1994; Castelnau and Duval,1994), and complex feedbacks occur between flow and fabrics. As an example,under compression, when c-axes rotate toward the compressional axis, the2

applied stress becomes increasingly perpendicular to the basal plane, and theice becomes harder to compress. Secondly, the rheology may depend on thecrystal size, as finer grains could explain a third of the shear strain rate enhancementobserved in ice deposited during glacial periods (Cuffey and others,2000). Understanding the reasons for grain size variations with climaticchanges is thus an important task for texture studies (Durand and others,2006). Finally, recrystallization processes have to be taken into account. Asan example, rotation recrystallization, also known as polygonization, is characterizedby basal dislocations that group together in walls perpendicular tothe basal planes and form sub-boundaries. The misorientation between twosubgrains increases gradually, and grains may split into smaller grains. Suchprocesses complicate the interpretation of texture measurements.Because the texture records the deformation history of the ice, the understandingof processes affecting the texture may lead to a better understandingof ice flow. Moreover, the complex feedbacks between fabric development anddeformation are now incorporated in local flow modelling (Gagliardini andMeyssonnier, 2000). In such models, both the velocity and the fabric fieldsare calculated, allowing comparisons with measurements. Such comparisonsrequire some fabric parameters that can be extracted both theoretically andexperimentally. Last but not least, in some cases, the study of the texturecan be of a great help to explain observed flow disturbances such as folds(Alley and others, 1997a; Raynaud and others, 2005). This is crucial forextracting paleoclimatic records from an ice core.Due to the birefringence of the ice crystal, ice texture can be revealedby the examination of a thin section highlighted under cross-polarized light.Earlier studies of ice texture were made manually, a tedious and particularlytime consuming work. Large progress has been made in developing ice textureinstruments, and now automatic measurements of texture can be performed.Wilen and others (2003) presented a review of the Automatic Ice FabricAnalysers (AIFA) used in the glaciological community. In the cited works,the AIFA were mainly used to extract the grain orientations of an ice sample,whereas microstructure was considered as a by-product of the measurements.Accounting for the plentiful and more robust measurements that followfrom the use of an Automatic Ice Texture Analyser (AITA) 1 , some of thetextural parameters measured so far are, to our point of view, now obsolete.1 Since one should also extract the microstructure of a sample using an AIFA, a moreappropriate terminology should be Automatic Ice Texture Analysers (AITA)3

Moreover, new parameters can be defined. These parameters better describethe texture and should be preferably used. This paper aims to review andpropose new methods to describe ice texture. The paper is organized as follows:in Section 2, we discuss in detail the operations needed to obtain thetexture information from an ice sample. In Section 3, we review the parametersused so far to describe the microstructure and propose new definitions.Section 4 deals with fabric parameters. Finally, Section 5 briefly presentsa matlab tool to determine all these texture parameters, before Section 6concludes.2 MEASUREMENTS2.1 Sample preparationMost of the studies of ice texture follow the standard procedure detailed inLangway (1958) to prepare thin sections. Here, we briefly recall this procedureand propose some improvements:(i) A vertical (parallel to the core axis) or a horizontal (perpendicular to thecore axis) thick section about 1 cm of thickness is cut from the ice core. Thesample location differs from core to core, but a large sample area is preferable(this point is detailed later).(ii) One surface of the sample has to be flattened in order to obtain aplane surface. As proposed by Langway (1958), sandpaper can be used,but Thorsteinsson (1996) proposed another procedure : the thick section isplaced on a glass plate, with a few drops of water along its side to glue it tothe glass plate. The plate is fixed to a microtome and is shaved in order toproduce a smooth and plane surface. Then, the sample is removed from theglass plate by breaking the water frozen droplets with a cutter. From ourexperience, the second procedure is preferable as it produces better results.(iii) A glass plate is then frozen or glued onto the smooth surface. Langway(1958) proposed to warm the glass plate, and to glue the sample by refreezingof water. The amount of heat applied to the glass plate will determine thequality of the thin section: melting more than necessary induce bubbles trappingat the glass-ice interface, conversely, too little heat applied to glass platewill cause a weak bond between ice and glass and the thin section would bedamaged while reducing the thickness. This method needs a lot of practicebefore obtaining satisfactory results. The sample can also be fixed by using4

water droplets on its border as in step (ii). This procedure is easier than theone proposed by Langway (1958), and the quality of the final thin section isgenerally better.(iv) Using a band saw, the sample is cut parallel to the glass plate, leavingbetween 1 and 2 mm thick section of ice adhering to the glass.(v) The section is reduced to a thickness of 0.3 mm with the microtome knife.The thickness is an important parameter for the grain boundary determination.Sharp colour transitions between grains are required and interferencefringes appearing for to thick grain boundaries should be avoided. The optimalthickness corresponds to grain’s colour around a brown-yellow range(Gay and Weiss, 1999).It is obvious that the measurements result from 2D cuts of a 3D structure.This point, referred as the sectioning effect in what follows, has always to bekept in mind as it influences the interpretation of the measurements madeon the microstructure (see Underwood, 1970, for a review of steorologicalproblems and section 3.1 for a discussion on the influence of 2D cuts onmean grain radius estimation). The c-axis measurement is not influenced bythis 2D effect assuming a small disorientation within a grain.2.2 Texture measurementsThe classical manual technique to measure the fabric of a thin section is basedon the Rigsby stage procedure (Rigsby, 1951) that was later standardized byLangway (1958). Microstructural parameters, such as the mean grain size,were also estimated manually (see Section 3.1). More recently, image analysisprocedures have allowed extraction of microstructure and new parametersrelated to the shape of grains (Gay and Weiss, 1999; Svensson and others,2003).Today, using an AITA, the whole texture of a polycrystal can be measured:fabric and microstructural parameters can be now easily obtainedtogether on the same sample. Indeed, Gagliardini and others (2004) haveshown that the measured cross-sectional area should be used as a statisticalweight of the polycrystal constituents in order to improve the fabricdescription. Complete texture measurements are also pertinent, as mappingof the spatial distribution of c-axes can reveal additional information aboutrecrystallization processes (Alley and others (1995), see also Section 4.7).Two major steps have to be carried out to extract the texture from the5

output of an automatic fabric analyser:(i) extraction of grain boundaries. Gay and Weiss (1999) proposed an imageanalysis algorithm to extract the microstructure from three images of onethin section taken under cross-polarized light. After some filtering to removenoise, a Sobel filter is used to detect large discontinuities of intensity, whichgenerally corresponds to grain boundaries. This method works nicely forrandomly oriented crystals, as the colour variations from grain to grain areimportant (see figure 1). For more oriented fabrics, colours between grainsare similar and the grain boundary detection is more difficult so that somemanual corrections are needed. Note also that the resulting microstructure isnaturally strongly dependent on the quality of the thin section. Most of theexisting automatic machines produce images of the thin sections that can beused to extract the microstructure (Wilen and others, 2003). An adaptationof the Gay and Weiss (1999) algorithm can easily be made from the outputof these analysers.(ii) assigning an orientation for each grain detected in (i). Using an AITA,the orientation is known for each pixel of the grain. The initial measurementsfor a given pixel are the colatitude θ p and the longitude ϕ p if this pixel is notlocated on a grain boundary. An average value over all pixels inside a grain,as well as the dispersion, of the c axis can be estimated within each grain.These calculations are detailed in Section 4.5.Finally, from outputs of AITA, one obtains the complete description ofthe texture at the grain scale. That is the microstructure, that contains allthe topological information, and the fabric which give the mean orientationof all non-intersecting grains within the microstructure. Such results allowconstruction of images such as the one shown in Figure 1, which contain allthe information required to calculate the parameters describing the texture(see Sections 3 and 4). It is worth noting that the original outputs fromthe AITA should be conserved, as they certainly contain information at thesub-grain scale that is not taken into account in this work and may be usedagain later.3 MICROSTRUCTURAL PARAMETERSManual estimation of the grain size is a time consuming work and severalmethods have been used so far to make this measurements practicable. Asdiscussed in Section 2, image analysis procedures give now the ability to ex-6

tract the complete microstructure in an acceptable time. Therefore, moreaccurate definition of the grain size can be made, but it also gives new informationon grain size distributions. Comparisons of the different methodsused so far are reviewed in Section 3.1.Other studies have focussed on the grain shape to obtain information onthe ice deformation (Azuma and others, 2000; Durand and others, 2004). Asfor the measurement of the grain size, several methods have been used so farand a comparison is provided in Section 3.2.3.1 Grain size3.1.1 Estimations of a characteristic length of the microstructureGow (1969) first proposed to measure the mean grain size of a section by estimatingthe average area of the 50 largest grains. By definition, this methodonly measures the evolution of the largest part of the grain distribution; thus,it is not representative of the complete population. It is also worth noting,that the representativity of the 50 larger grains is affected by a change in thegrain size (for a constant sampling surface). This leads to a bias in the estimationof a grain growth rate (Gay and Weiss, 1999). This method which hasbeen developed in order to perform manual measurements should be avoidedfor automatic measurements.Thorsteinsson and others (1997) estimated the mean grain size along theGRIP core by using the linear intercept method. This method consists ofcounting the number of grain boundaries that intercept an horizontal (orvertical) line. This leads to the estimation of a mean intercept length 〈L〉,which for a regular surface is proportional to the square root of the area of theconsidered object (Underwood, 1970). This method suffers of different biases:(i) if the grain shape evolves (and that is the case, see Durand and others(2004)), the ratio between the mean area and 〈L〉 is affected, (ii) twistedgrain boundaries can intercepted a given line more than once, and (iii) thelink between the grain area distribution and the distribution of the length ofintercept is not straightforward.Duval and Lorius (1980) estimated the mean grain area by counting thenumber of grains within a defined surface. As this method takes into accountthe whole population of grains, it is more robust than the method proposedby Gow (1969). It also provides a more straightforward estimation of themean grain size compared to the linear intercept method, but it does not7

provide the grain size distribution. Note also, that corrections are requiredin ice with notable porosity.3.1.2 Mean grain radiusA more complete way to measure a characteristic length of the microstructure∑would be to calculate the average volumetric radius: 〈R V 〉 = 1 NgN g k=1 V 1/3k,where N g is the number of non-intersecting grains within the microstructure,and V k is the volume of the grain k. As mentioned previously, the volume ofthe ice crystal cannot be measured so far, and 2D sections have to be made.Then a characteristic length of the microstructure could be expressed by theaverage radius:〈R〉 = 1 N gN∑ gk=1A 1/2k(1)where A k is the cross-sectional area of the grain k and A k = Np k × δ where δis the actual area of a pixel. Npk is then the number of pixels that composethe grain k. An estimation of the error induced by the sectioning effect canbe made by the help of a 3D Potts model. In such a model, 3D microstructureis divided into small volume elements and an orientation is allocated toeach element. Grain growth is simulated using a Monte Carlo technique: anelement is randomly selected, and a new orientation is randomly attributed.Similar orientations between neighbouring elements (i.e. same grain) willdecrease the energy of the system, and the new orientation will then beconserved. Such Potts model is well known to properly reproduce the topological,kinetic, grain size distribution and morphological features of normalgrain growth (Anderson and others, 1989), e.g. recrystallization process occurringin the upper part of ice-sheets (Alley and others, 1986).Following Gagliardini and others (2004), we generated 3D Potts microstructurescontaining 400 × 400 × 400 elements. For a section performedwithin this volume, it contains approximately 200 non-intersecting grains.For such a section, both 〈R V 〉 and 〈R〉 can be measured and compared.From these results, we demonstrated that the standard deviation σ s of the〈R〉’s distribution for a given 〈R V 〉 can be approximated by σ s ≈ 0.02 × 〈R〉(Durand, 2004).As mentioned previously, the mean grain size is generally increasing withdepth, and for a constant sampling surface, the number of grains will decrease8

with increasing depth. The statistics for the calculation of the average grainradius can be drastically affected. As for the error induced by the sectioningeffect, we propose to estimate the error induced by the number of grain usedin the calculation of the average through a Potts model. Anderson and others(1989) have shown that a 2D Potts model reproduces the characteristics of amicrostructure determined from a section of a 3D polycrystal under normalgrain growth (and the calculation time is much smaller for 2D simulationsthan for 3D). As shown on Figure 2, the normalized grain size distribution ofa 2D Potts microstructure is highly comparable with a natural microstructuresampled along the EPICA (European Project for Ice Coring in Antarctica)Dome C core (75 ◦ 06 ′ S; 123 ◦ 21 ′ E) at a depth of 100.1 m (see also Section3.1.3).Then we simulated 200 microstructures of 1000 2 pixels containing between622 and 1808 grains. We supposed that these microstructures presentenough grains to completely describe the grain size distribution. Their meanradius was then used as a reference and noted 〈R Ref 〉. For each microstructure,80 sub-sampling microstructures were randomly selected. These submicrostructurecontained between 3 and 200 grains and their mean radiuswas calculated and compared to 〈R〉. 〈R〉/〈R Ref 〉 follows the central limittheorem and the standard deviation depends only on the grain number followingthe relation σ p (〈R〉/〈R Ref 〉) = 0.44Ng−1/2 (see Figure 3). Of course,for measurements on natural ice, 〈R Ref 〉 is unknown, and we make the followingapproximation:σ p (〈R〉) = σ p (〈R〉/〈R Ref 〉) × 〈R Ref 〉 ≈ σ p (〈R〉/〈R Ref 〉) × 〈R〉 (2)And finally obtain σ p ≈ 0.44Ng −1/2 × 〈R〉. An error bar that takes intoaccount the sectioning effect (σ s ) as well as the population effect (σ p ) cannow be attached to 〈R〉 as follows:σ(〈R〉) = σ s + σ p ≈ ( 0.02 + 0.44N −1/2gwhere 〈R〉 is calculated using Equation 1.3.1.3 Distribution parameters)× 〈R〉 (3)Under normal grain growth, the normalized grain size distribution R k /〈R〉remains unchanged and is closely approximated by a lognormal distribution9

(Humphreys and Hatherly, 1996). This observation does not have a theoreticalexplanation so far; it is however a useful description as the distributioncan be completely defined with only two parameters easily measurable, themean 〈〉 D = 〈ln (R k /〈R〉)〉 and the standard deviation σ D = σ (ln (R k /〈R〉)).Changes in the normalized distribution could reveal changes in the activatedrecrystallization processes. Such changes could also provide informationon the mechanisms affecting the normal grain growth, such as pinning ofgrain boundaries by microparticles that induces a shift of 〈〉 D and σ D (Riegeand others, 1998; Durand and others, 2006).From Potts model results, and following the procedure detailed in thesection 3.1.2, we also estimate the intrinsic variability of 〈〉 D and σ D inducedby the sectioning effect and the change in the population:σ (〈〉 D ) = ( )0.06 + 1.83 × Ng−1/2 × |〈〉D |σ (σ D ) = ( )0.04 + 0.95 × Ng−1/2 × σD3.2 Grain shape and deformation of the microstructureMany different descriptors can be used to measure the shape of grains andquantify the anisotropy of a microstructure (Mecke and Stoyan, 2002). Manyof those have been applied by the glaciological community and thereforeintercomparison between different results from different cores can sometimesbe tricky. Among the most used parameters, we can cite:(i) Comparison between the mean vertical and horizontal intercept lengths(Thorsteinsson and others, 1997; Gay and Weiss, 1999). In addition to theinherent artifacts induced by the measurement of 〈L〉 (see section 3.1.1),this method is not able to correctly quantify anisotropy in a non-horizontalplane. A similar method was used by Svensson and others (2003), as theymeasured the bounding box of each grain and calculated the correspondingaspect ratio, i.e. the width-to-height ratio of the smallest rectangle withhorizontal and vertical sides which completely enclose the crystal. Such amethod also does not fully capture grain elongation that is not oriented inthe horizontal-vertical directions, unless the box orientations are rotated.(ii) An inertia tensor for each grain can also be calculated (Azuma and others,2000; Wilen and others, 2003). The eigenvalues of the tensor provide10

the elongation of the grain, and the eigenvectors give the directions of elongations.Compared to the previous methods, this tensor description has theadvantage that the obtained measurements do not depend on the particularchoice of axes (e.g. vertical and horizontal).(iii) Durand and others (2004) proposed a new method based on the vectors lthat link neighbouring triple junction locations (where three grain boundariesmeet). This method, which can suffer from some artifacts in porous ice(triple junctions may occur in pores), is briefly recalled here. More detailsand applications could be found in the previous publication. A local averagemicrostructure anisotropy tensor M can be defined as:〈( )〉l2M = 1 l 1 l 2l 2 l 1 l 2 = 1 N∑( )λ1 0l(k) ⊗ l(k) = RR −1 (4)2 pN0 λ 2k=1where (l 1 , l 2 ) are the coordinates of l in the local reference frame R (the ē 3axis is perpendicular to the thin section). As far as the absolute horizontalorientation is unknown ē 1 is relative to the thin-section, which cannot beoriented with respect to the core axis. R is the rotation which make diagonalM, and (λ 1 , λ 2 ) are the corresponding eigenvalues; 〈·〉 pdenotes the averageover N vectors, up to the p-th neighbours. A sample can be examined underdifferent scales as a small p explores details, whereas a larger p improvesthe statistics. Compared to the inertia tensor presented in (ii), M has anadditional advantage: it has a physical signification in terms of mechanicaldeformation (Aubouy and others, 2003). From the variation of M withrespect to a reference M 0 , a statistical strain tensor U, which coincideswith the classical definition of strain (Aubouy and others, 2003), can bedefined. As M 0 cannot be measured for natural ice, Durand and others(2004) assume that the deformation is isochoric (incompressibility of ice),and that the reference state is isotropic, leading to the following expression:⎛ ( )⎞ ⎛ ( ) ⎞U = 1 2 R ⎝ log √λ1λ 1λ 20( ) ⎠ R −1 = 1 λ0 log √λ124 R ⎝ log λ 1λ 20( ) ⎠ R −1 .λλ 20 log 2λ 1(5)Also note that some statistical tests, comparable to those presented in Section3.1.2, have been carried out to estimate the intrinsic variability of U thatleads to σ (U) = 0.35N −1/2 (Durand and others, 2004).Note that Durand and others (2004) called M the texture tensor andused different notations. To avoid confusion with the present definition of11

texture (see Introduction), M will be denoted the microstructure anisotropytensor.4 FABRIC PARAMETERSThis section presents some tools that allow description of the fabric measuredfrom a thin section containing N g non-intersecting crystals. In this sectioneach individual crystal is described by:(i) its orientation c k . The orientation of the ice crystal is described by itsc-axis orientation, which is defined by two angles: the colatitude θ k ∈ [0, π/2](or tilt angle) and the longitude ϕ k ∈ [0, 2π] given in the local reference frameR, which has its ē 3 axis perpendicular to the thin section. The expressionof c k in this reference frame isc k = (cos ϕ k sin θ k , sin ϕ k sin θ k , cos θ k ) (6)(ii) its volume weighted fraction f k . Larger crystals should be more influentialfor the polycrystal behaviour than smaller ones. As suggested byGagliardini and others (2004), to estimate the actual volume of grain k from2D thin-section measurements, one can assume that it is proportional to itsmeasured cross-sectional area A k . Then, the volume-weighted fraction is:f k =A 3/2k∑ Ngj=1 A3/2 j(7)In what follows, the fabric of the polycrystal is given in terms of the distributionof the N g c-axis orientations c k and the N g weight fractions f k .4.1 Rotation of reference frame and polar representations4.1.1 Rotation of reference frameMost studies of ice fabrics use the equal-area point plots (also named Schmidtpoint plots) to present the distribution of c k axes in a thin section (e.g.Thorsteinsson and others (1997), see also Figure 4a). In the literature, thepolar plots are represented in a horizontal reference frame R h , i.e. the in-situvertical (the core axis) is in the center of the polar plot (Langway, 1958).12

Therefore, if thin sections are cut vertically, the fabrics need to be rotatedfor easier comparison of fabric diagrams. To obtain a horizontal projectionfrom a vertical thin section, one needs to rotate the c k as follows:ϕ h k = arctan(tan θv k sin ϕv k ) ∈ [0, 2π[θk h = arccos(− sin θk v cos ϕ v k) ∈ [0, π/2](8)where ϕ h k and θh k are the longitude and colatitude in the horizontal referenceframe, respectively. In this horizontal reference frame R h ; the ē h 3 axis pointsvertically upward. If the thin section is cut horizontally R = R h .4.1.2 Equal-Area (Schmidt) point plots and and contoured dataplotsThe equal-area projection (Kocks, 1998, p.54) is achieved by the coordinatetransformationandρ = 2 sin θh k2 , (9)x k = ρ cos ϕ h k, y k = ρ sin ϕ h k. (10)This transformation takes all points in the upper hemisphere and projectsthem within a circle of radius √ 2 on the x-y plane. An equal-area point plotthen simply puts a marker at the (x k , y k ) location of each crystal orientation.The purpose of presenting c-axis distributions is often to learn as much aspossible about the fabric in nearby ice, not merely to present a specific dataset (Kamb, 1959). Now that data are collected using automatic methods(Wilen, 2000), a collection of several hundred to over a thousand measuredcrystal orientations is common. This results in the problem of overlappingpoints when presenting the data on point plots. Some authors (Langway,1958; Kamb, 1959) have presented fabric information using contour plots.Because any sample is limited in its size N g , we can, at best, get animperfect estimate of the underlying population. Estimates that are robustare desirable. Thorsteinsson and others (2006) present a new approach,which stems largely from Kamb (1959). They first recognize that there is anunknown “global” orientation probability-density distribution Ĝ(x) for theunderlying large population of crystals. Then, they use the data to create arobust estimate G(x) of Ĝ(x). With N g data points, G(x) is represented as13

a superposition of N g simpler normal probability-density distributions g k (x)for N g sub-populations,N∑ gG(x) = f k g k (x), (11)k=1G(x) is then projected onto an equal-area diagram using Equation 10 toproduce G(x, y).For each crystal orientation c k , the orientation-distribution density ofthe associated kth sub-population on a hemisphere centered on the axis c kis defined asg k (x) =β [2πΩ exp − 1]2 2Ω (x − 2 ck ) · (x − c k ) , (12)where β is a normalizing factor to account for the spread of the Gaussian ona hemisphere. In practise β ≃ 1 if there are 10, or more, crystals to contour.Each g k (x) has a mean orientation given by the observation c k , andthe width of the component normal probability-density distribution is Ω =Ω M + Ω N , where Ω M is the spread due to measurement error, and Ω N representsspread due to limited sample size N g . Using numerical experimentsThorsteinsson and others (2006) have found that the ”best” Ω (for 40 ormore crystals) is, Ω = 2/ √ N g .The expected value of G(x), for isotropic ice, is E = 1/(2π), and realclusters should exceed the E + 1/(3π) contours, that is E ± 2σ, where σ =1/(6π). The function G should be contoured at 2σ, 4σ, · · · intervals, withlarger spacing if very strong maximums occur. A comparison between aclassical point plot and a contoured data plot is presented in Figure 4 for aDome C texture sampled at 2999.8 m depth. The contour plot with N g = 80illustrates very clearly the strength of the clustering of crystals near vertical.Such an observation is less obvious for the point plot. For further information,refer to the original publication (Thorsteinsson and others, 2006).4.2 Strength of fabric and spherical apertureThe strength of fabric (also called degree of orientation or strength of orientation)is a one-parameter fabric descriptor that has been widely used in theglaciology community. This is mainly because most of the observed ice fabrics14

are one-maximum or girdle type fabrics and therefore present a symmetryaxis.The definition of the strength of fabric that can be found in the literature(Thorsteinsson and others, 1997; Castelnau and others, 1998; Gagliardini andMeyssonnier, 2000) is:Ng∑R s = 2|| f k c k || − 1 (13)k=1where ||.|| is the norm of a vector. This definition of the strength of fabricassumes implicitly that the fabric has a symmetry axis and that this symmetryaxis is the axis ē 3 perpendicular to the thin section. Therefore, thisparameter is not objective since it depends on which reference frame the sumis performed.A more rigourous definition, but still not objective, for the strength of fabricR s needs to first calculate the best symmetry axis for the fabric. Wewill present later two different methods to effectively determine this fabricsymmetry axis ¯c. Here, assuming that ¯c is known, the strength of fabricparameter R s is then calculated as:Ng∑R s = 2|| f k sign(¯c.c k )c k || − 1 (14)k=1The sign operator allows to perform the sum within the hemisphere centeredon the fabric symmetry axis ¯c. Nevertheless, with this new definition, thesame problem still occurs for the grains which have their c-axis perpendicularto ¯c, for which ¯c.c k = 0 and no sign can be determined. Note that asmentioned previously, Equations (14) and (13) are equivalent only if ¯c = ē 3 .The spherical aperture α s is directly related to the strength of fabric by:α s = arcsin √ 1 − R s , (15)For isotropic ice R s = 0 and α s = 90 ◦ and for a single maximum fabricR s = 1 and α s = 0 ◦ . For fabrics with girdle tendency, −1 ≤ R < 0 and dueto its definition α s is no more usable.4.3 Best symmetry axis of a fabricA first method to calculate the best symmetry axis of a fabric is to use aleast square method to minimize the sum of the squared angle between the15

symmetry axis and all the c-axes c k . This can be written as:N∑ gFind ¯c which minimize ¯Θ2 = f k Θ 2 k (16)where Θ k is the angle between ¯c and c k :cos Θ k = |¯c.c k | (assuming ¯c is a unit vector) (17)The absolute value in (17) stands for taking into account that each vectorc k must be in the same hemisphere as ¯c. The minimum value of ¯Θ dependson the fabric type. For a perfectly isotropic fabric ¯Θ = √ π − 2 ≈ 61 ◦ andfor a single maximum type fabric √ π − 2 > ¯Θ ≥ 0 whereas for a girdle typefabric π/2 ≥ ¯Θ > √ π − 2. Due to the absolute value, this problem is notcontinuous and may present some local minima.A second solution is to define the symmetry axis ¯c as the first eigenvectorobtained from the calculation of the eigenvalues of the second-order orientationtensor, as presented below. This tensor is obtained from the directcalculation of the average of the dyadic product of the c-axes. By definition,there is no ambiguity from the c-axes orientations for the calculation of thesecond-order orientation tensor.The difference in ¯c resulting from these two methods can be large, especiallywhen the fabric is not concentrated. As an example, for a randomly distributedfabric, the angle between the ¯c obtained by the two different methodscan be up to 30 ◦ . This is because for a perfectly randomly distributedfabric there are an infinite number of “best” symmetry axes ¯c (every axis isa symmetry axis for such fabric).In what follows, the best symmetry axis is determined using this method.Moreover, since both the strength of fabric R s and the spherical aperture α srequire the knowledge of the mean symmetry axis of the grain orientationsand consequently the calculation of the eigenvectors of the second-order orientationtensor, one can conclude that, despite an historical point of viewand comparisons with old datasets, we believe that there is no more reasonto use these two parameters. As shown below, the second-order orientationtensor contains much more information and should be preferred.k=116

4.4 Orientation tensors4.4.1 Second-order orientation tensorAs suggested by Woodcock (1977), a good way to characterize the essentialfeatures of an orientation distribution is to use the second-order orientationtensor a (2) defined as:N∑ ga (2) = f k c k ⊗ c k , (18)k=1where c k is given by Equation 6 and f k by Equation 7. By construction, a (2)is symmetric and there exists a symmetry reference frame R sym (or principalreference frame), in which a (2) is diagonal. Let a (2)i (i = 1, 2, 3) denote thethree corresponding eigenvalues and o e i (i = 1, 2, 3) the associated eigenvectors,which correspond to the three base vectors of R sym . The eigenvalues ofa (2) can be seen as the lengths of the axes of the ellipsoid that best fit thedensity distribution of the grain orientations. The eigenvectors give then theaxis directions of this ellipsoid.The three eigenvalues a (2)1 , a (2)2 and a (2)3 follow the relations:a (2)1 + a (2)2 + a (2)3 = 1 (19)0 ≤ a (2)3 ≤ a (2)2 ≤ a (2)1 ≤ 1 (20)For an isotropic fabric: a (2)1 = a (2)2 = a (2)3 = 1/3 and when the fabric istransversely isotropic, two of the eigenvalues are equal:{a (2)2 ≈ a (2)3 < 1/3 for a single maximum fabric,a (2)1 ≈ a (2)(21)2 > 1/3 for a girdle fabric.According to Woodcock (1977), fabrics that have equal girdle and clustertendencies are such that a (2)1 /a(2) 2 = a (2)2 /a(2) 3 . Then, the use of the parameterk = ln(a (2)1 /a(2) 2 )/ ln(a(2) 2 /a(2) 3 ) gives objective information about the fabricshape: k > 1 for one maximum fabric and k < 1 for girdle fabric. As suggestedby Woodcock (1977), one should used C = ln(a (2)1 /a(2) 3 ) as a measureof the strength of the preferred orientation. C varies from 0 for randomlydistributed orientations to infinity when all the orientations are identical.The three eigenvectors o e i define the best material symmetry reference frameof the sample. It is often useful to calculate the angle between the main17

symmetry axis of the fabric o e 1 and the local vertical:θ = arccos( o e 1 .e h 3 ) ∈ [0, π/2] (22)where the e h i are the basis vectors of R h .The study of higher order orientation tensors allows investigation of the gapbetween the actual fabric and the orthotropic fabric constructed by the useof the second-order orientation tensor.4.4.2 Higher-orders orientation tensorBy definition, the fourth-order orientation tensor is:N∑ ga (4) = f k c k ⊗ c k ⊗ c k ⊗ c k , (23)k=1The fourth-order orientation tensor contains more information about thefabric as the second-order orientation tensor and can reproduce non-orthotropicfabric. If the fabric is orthotropic, then by construction the fourthorderorientation tensor has the same principal axis than the second-orderone, and only 21 of its 81 components are non-nulls when expressed in thesymmetry reference frame R sym . Moreover, according to Cintra and TuckerIII (1995) only 3 of these 21 components are independents. The 21 non-nullcomponents are:a (4){1122} , a(4) {1133} , a(4) {2233} ,a (4)1111 = a(2) 11 − a(4) 1122 − a(4) 1133 ,a (4)2222 = a(2) 22 − a(4) 2211 − a(4) 2233 ,a (4)3333 = a(2) 33 − a(4) 3311 − a(4)3322 , (24)where a {iijj} denotes the series of components independent of the order of theindices {a iijj = a ijij = a ijji = a jiij = a jiji = a jjii }. The 60 other componentsare zero for an orthotropic fabric.For a natural non-orthotropic fabric, one can use the value of these 60components as an estimate of the gap to orthotropy. These 60 componentscan be grouped in the two class families a (4)iiij with i ≠ j ≠ k. They18and a(4)iijk

are composed of 24 and 36 terms, respectively, but by construction, only 6and 3 of these terms are different, respectively.One can then define a fourth-order estimate of the gap to orthotropy ofa fabric as:∆ (4)O=4(|a(4) 1112| + |a (4)1113| + |a (4)2221| + |a (4)2223| + |a (4)3331| + |a (4)3332|)+ 12(|a (4)1123| + |a (4)2213| + |a (4)3312|)(25)Since by definition all the components of a (4) are lower than one, we have∆ (4)O≤ 16. For a perfect orthotropic fabric ∆(4)O= 0 and larger values of ∆(4)Oindicate that the c-axes don’t fulfill the orthotropic symmetry conditions.Experiences carried out on 140 samples along the EPICA Dome C core haveshown a maximal value for ∆ (4)Oof 2.11.4.4.3 Odd orders orientation tensorSince a grain can either be oriented using c k or −c k , the odd order orientationtensors, as the strength of fabric R s 2 , must be used with care. If one cutsall the grains in two equal parts, one oriented by c k and the other by −c k , itis obvious that all the components of the odd order orientation tensors arenulls. Even if the odd orientation tensor is expressed in R sym , the grainsthat lie in the plane ( o e 1 , o e 2 ) can still be oriented either by c k or −c k .As a conclusion, one should not use the odd order orientation tensors todescribe fabrics, as was already mentioned for the strength of fabric R s andthe spherical aperture α s .4.5 Average and dispersion of the c-axes within a grainUsing an AITA, the initial measurement gives the orientation of each pixel inthe sample. After the extraction of the microstructure, one has to determinefor each grain its mean orientation c k , as used in the previous sections. Ofcourse, as for a grain, the same definition (6) applies for the pixel orientationvector.Since, as a grain, a pixel can either be oriented by c p or −c p , the meanorientation of a grain c k cannot be calculated as the average value over theNpk orientations cp . As for the whole thin section, the mean orientation ofthe grain has to be determined as the first eigenvector of the second-order2 the sum in the definition of R does correspond to the first order orientation tensor19

orientation tensor calculated on the grain, i.e.: for a given grain k, c k = o e k 1where o e k 1 is the first eigenvector of the second-order orientation tensor definedby (18) in which N g has been replaced by Np k .Contrary to a polycrystal fabric, adjacent pixel orientations on the samegrain should be very close except when crossing a subgrain. Nevertheless,some artifacts are still present for some pixels and these pixels need to beexcluded for further calculation. Indeed for a given view, the determinationof the extinction angle constrains c p to lie along two orthogonal planes.The AITAs repeat the measurements for three different views that shouldunambiguously determine c p . Unfortunately for some pixels, the extinctionposition is not available for (at least) one of the views and thus leads to someambiguities, i.e. some c p lie along plane (see the polar plots on Figure 5).Also note that the pixels close to grain boundaries gives generally a noisydetermination of their c p .To remove these artifacts, we use the following procedure. For a grain k,o e k 1 is calculated a first time by taking into account all the Npk pixels of thegrain. Then the misorientation angle Θ p between o e k 1 and cp is calculatedfor each pixel. Note that Θ p belongs to the range [0; 90] as c p can also beoriented by −c p . Distributions of Θ p have been plotted on Figure 5 for thegrain k = 7 (a) and k = 9 (b) of a texture sampled at 1525.8 m on theEPICA Dome C ice core. If 66% of the pixels do not present a Θ p smallerthan 10 ◦ then the grain is rejected and will not be taken into account forfabric parameters calculation. This is the case for the grain k = 9. On theother hand, if the distribution of θ p is well centered around 0 (more than66% of the pixels), o e k 1 is recalculated, but the pixels for which Θ p > 10 ◦ areexcluded for this new calculation, but still accounted in the area calculation.Note that most of the pixels presenting Θ p > 10 ◦ are close to the grainboundaries. This limit of Θ p > 10 ◦ takes into account that within a grain,the rotation recrystallization can reach a maximum misorientation of about10 to 15 ◦ (see Section 4.7).For large grains represented by a significant number of pixels, one canattempt to quantify the misorientation within a grain induced for exampleby the rotation recrystallization. To this aim, a solution is to study theeigenvalues of the second-order orientation tensor. For a grain, assumingthat all the pixel c-axes are very close, one should expect a (2)1 ≈ 1 anda (2)2 ≈ a (2)3 ≈ 0. If one assumes that all the pixel c-axes are randomlydistributed within a cone centered on o e k 1 and with a half angle Θ k , then the20

eigenvalues are a (2)1 = (1+cosΘ k +cos 2 Θ k )/3 and a (2)2 = a (2)3 = (1−a (2)1 )/2. Asan example, since two new grains are created if the rotation recrystallizationdistortion within the grain become larger than approximately 15 ◦ , one shouldnot expect to find value of a 2 1 smaller than 0.99. The study of a(2) 2 and a (2)3 canthen shows whether the pixel c-axes are randomly distributed (a (2)2 = a (2)3 ) orif there exists a preferential direction of the misorientation (a (2)2 > a (2)3 = 0).In the present paper, due to the too large error on the pixel orientationmeasurement of our apparatus, we haven’t been able to obtain pertinentresults.4.6 Dispersion and error bars associated to the secondorderorientation tensorThe standard deviation σ a (2) of a (2) , is the contribution of (i) the uncertaintiesinduced by the variability of the measurements, σ m , added to (ii) thea (2)uncertainties induced by the sampling on a limited number of grains , σ p .a (2)In order to estimate σ m , the same thin-section presenting a nearly randomfabric has been measured 10 times. More measurements were not fea-a (2)sible, as the sample was sublimating during the experiment. For each grainpresent during the 10 successive measurements, its mean c-axis was calculatedusing the procedure described in Section 4.5. As a result, the standarddeviation of the misorientation for the 10 measurements of c k was found tobe approximately 0.6 ◦ and not dependent of the grain orientation itself. Notethat, even if sublimation was occurring during the experiment, no systematicbias was noticed on the grain c-axis measurements. This last point indicatesthat small variations of the sample thickness are not critical for the qualityof the measurements. Then, to estimate σ m , a random Gaussian noise witha (2)a standard deviation of 0.6 ◦ is added to all the c k and the parameter a (2) iscalculated. This procedure is reproduced 200 times allowing calculation ofσ m over the 200 draws. This method should be reproduced for each samplea (2)in order to get an estimation of σ m . However, experiments carried out ona (2)140 samples have shown that σ m can be neglected whatever Na (2) g , as it isat least one order of magnitude lower than σ p . Nevertheless, for manuala (2)measurements, for which the standard deviation on the estimation of c k isabout 5 ◦ , σ p may have to be taken into account to estimate σa (2) a (2).21

Following the method proposed by Gagliardini and others (2004), a set of231 fabrics of 10000 grains each was numerically generated on a grid of thepossible values of a (2)1 and a (2)2 , i.e. on the domain defined by 0 < a (2)1 < 1and 0 < a (2)2 < 1 − a (2)1 . Then, for a given fabric, we randomly choose N ggrains within the range [100; 1000] and calculated the fabric parameters. Fora given number N g of grains, this operation was reproduced 300 times and thestandard deviation of σ p was calculated. All this procedure was reproduceda (2)for different values of N g . Figure 6 shows the results for the second-orderorientation tensor a (2) . In order to work with only one parameter, Max(σa p ii)is attributed to the 9 components of the tensor a (2) . Figure 6a shows theevolution of Max(σa p ii) versus N −1/2Gfor undersamplings with 100 < N g

to compare DθN with the distribution of angle misorientation of randomgrain pairs (DθR) calculated by taking into account the N g ×(N g −1)/2 pairswithin the sample (or a random subset). Alley and others (1995) argued thatdiscrepancies between DθN and DθR should highlight the occurrence of recrystallizationprocesses. Rotation recrystallization, that leads to the split ofa grain into smaller grains with similar orientations, should present a highestpopulation of low angles for DθN compared to DθR. On the other hand, migrationrecrystallization produces grains at high angles to their neighbours(Poirier, 1985) and should also be revealed in the comparison between DθNand DθR.Durand and Svensson (2006) recently proposed a new method to calculateDθR: the fabric is reshuffled whereas relations between neighbours arekept, i.e. the microstructure is unchanged. They reproduce such drawinglots 200 times leading to the estimation of a probability-density envelope.This method has been applied for a texture sampled at 1525.8 m on theEPICA Dome C ice core (see Figure 7). Numbers of low angles misorientationbetween neighbouring pairs (from 0 to 10 ◦ ) are significantly larger thanwhat could be expected for a random distribution, indicating that rotationrecrystallization is certainly occurring. Note that it confirms the results ofDurand and others (2006) who argued, from results of a model, that rotationrecrystallization should occur along the EPICA Dome C core.Note that a misorientation is expressed by an angle but also by a rotationaxis. The distribution of these axes could also reveal information (Wheelerand others, 2001). To our knowledge, such distribution has never been usedin glaciology so far, and could be helpful to learn much about ice deformation.5 THE TEXTURE TOOLBOXTo make easier and homogenize texture data processing within the glaciologicalcommunity, we propose some Matlab tools group in the so-called texturetoolbox. This toolbox have been designed to (i) completely fulfill the requirementsproposed in this work, (ii) easily manage a large amount of data, and(iii) have a user-friendly interface. The texture toolbox is briefly describedhere. More information, and downloading of the texture toolbox are availableonline: http://www.gfy.ku.dk/∼www-glac/crystals/micro main.html.A first part is devoted to the microstructure extraction and the calculationof the mean c-axis of each grain. Automatical detection of grain boundaries23

is available through an adaptation of the algorithm proposed by Gay andWeiss (1999) and manual corrections can be added easily to improve the microstructurequality. A tool has also been developed to check the quality ofthe microstructure by examining the c-axes dispersion within a grain (see Section4.5). Once the microstructure is complete, the average c-axis within eachgrain is calculated and rotated into the horizontal reference frame. Then, allthe information required to describe the texture at the grain scale is recordedin a single file.The second main part helps to manage the textural data. Each texturerecorded by the tool described above can be imported (the depth of eachtexture is then specified). All the up to date parameters described in previoussections are calculated systematically and polar plots prepared following themethod presented in Section 4.1.2 (classical schmidt representation can alsobe chosen). If many textures are imported, the evolution of the differentparameters can be easily plotted (versus depth, or age if a dating file isprovided). All the plots can be exported, as well as the raw data. The listof the studied textures can be saved in a project file, allowing earlier workto be found easily.This is of course an open source code, and we encourage anyone whohas new interesting parameters to add it to the texture toolbox. Such a toolshould help to greatly improve our understanding of ice deformation, andcan be used to share up to date techniques of texture analysis.6 CONCLUSIONBecause measurements with good statistics of an ice thin section can beautomatically done, measuring the texture is no longer an unrealistic task.As a consequence, we believe that some of the parameters used so far tocharacterize the microstructure (mean size of the 50 larger grains, aspectratio) and the fabric (R s , α s ) should not longer be used except for comparisonwith earlier work. More robust definitions can be proposed to better describethe observed texture. It is also worth noting that intrinsic variability of themeasurements was estimated through models and statistical tests.AITA are unfortunately not widespread in the glaciological community sofar. However, most of the recommendations presented here can be followedeven for manual measurements. As an example a (2) can be calculated if onlythe fabric is measured, in this case f k is simply equal to 1/N g .24

It is also very important to note that both microstructure and fabricshould be measured together on the same sample. Indeed, Gagliardini and others(2004) show that grain area should be used as a statistical weight forpolycrystal constituents to improve the description of the observed fabric.Moreover, complete texture measurements allow one to look at relationsbetween neighbouring grains, what could give precious indications on therecrystallization processes occurring (see Section 4.7).7 ACKNOWLEDGEMENTSMain part of this work was inspired by discussion arising during the W orkshopon the Intercomparison of Automatic Ice Fabric Analysers held at the NielsBohr Institute, University of Copenhagen, in January 24-26 2002. We aregrateful to the two anonymous reviewers for their very helpful comments.This work is a contribution to the European Project for Ice Coring in Antarctica(EPICA), a joint European Science Foundation/European Commissionscientific programme, funded by the EU and by national contributions fromBelgium, Denmark, France, Germany, Italy, the Netherlands, Norway, Sweden,Switzerland and the United Kingdom. The main logistic support wasprovided by IPEV and PNRA (at Dome C) and AWI (at Dronning MaudLand). This is EPICA publication no. 156.ReferencesAlley, R. B., J. H. Perepezko and C. R. Bentley. 1986, Grain growth in polarice: I. Theory. J. Glaciol., 32(112), 415–424.Alley, R. B., A. J. Gow and D. A. Meese. 1995, Mapping c-axis fabrics tostudy physical processes in ice. J. Glaciol., 41(137), 1995.Alley, R. B., A. J. Gow, D. A. Meese, J. J. Fitzpatrick, E. D. Waddingtonand J. F. Bolzan. 1997a, Grain-scale processes, folding, and stratigraphicdisturbance in the GISP2 ice core. J. Geophys. Res., 102(C12), 26819–26830.Alley, R. B., C. A. Shuman, D. A. Meese, A. J. Gow, K. C. Taylor, K. M. Cuffey,J. J. Fitzpatrick, P. M. Grootes, G. A. Zielinski, M. Ram, G. Spinelli25

and B. Elder. 1997b, Visual-stratigraphic dating of the GISP2 ice core: Basis,reproductibility and application. J. Geophys. Res., 102(C12), 26367–26381.Anderson, M. P., G. S. Grest and D. J. Srolovitz. 1989, Computer simulationof normal grain growth in three dimensions. Philos. Mag. B., 59(3), 293–329.Aubouy, M., Y. Jiang, J. A. Glazier and F. Graner. 2003, A texture tensorto quantify deformations. Granular Matt., 5, 67–70.Azuma, N. and A. Higashi. 1985, Formation processes of ice fabric patternin ice sheets. Ann. Glaciol., 6, 130–134.Azuma, N., Y. Wang, Y. Yoshida, H. Narita, T. Hondoh, H. Shoji andO. Watanabe. 2000, Crystallographic analysis of the Dome Fuji ice core.In: Hondoh T, editor. Physics of ice records. Sapporo: Hokkaido UniversityPress.Castelnau, O. and P. Duval. 1994, Simulations of anisotropy and fabric developmentin polar ices. Ann. Glaciol., 20, 277–282.Castelnau, O., H. Shoji, A. Mangeney, H. Milsch, P. Duval, A. Miyamoto,K. Kawada and O. Watanabe. 1998, Anisotropic Behavior of GRIP Icesand Flow in Central Greenland. Earth Planet. Sci. Lett., 154(1-4), 307–322.Cintra, J. S. and C. L. Tucker III. 1995, Orthotropic closure approximationsfor flow-induced fiber orientation. J. Rheol., 39(10), 1095–1122.Cuffey, K. M., T. Thorsteinsson and E. D. Waddington. 2000, A renewedargument for crystal size control of ice sheet strain rates. J. Geophys. Res.,105(B12), 26547–26557.Durand, G. 2004, Microstructure, recristallisation et déformation desglaces polaires de la carotte EPICA (Dôme Concordia), Antarctique.Ph.D. thesis, Université Joseph Fourier-Grenoble I, http://www-lgge.ujfgrenoble.fr/publiscience/theses/these-durand.pdf.Durand, G. and A. Svensson. 2006, Neighbouring misorientation and normalgrain growth along the NorthGRIP ice core. Forthcoming paper.26

Durand, G., F. Graner and J. Weiss. 2004, Deformation of grain boundariesin polar ice. Eur. Phys. Lett., 67(6), 1038–1044.Durand, G., J. Weiss, V. Lipenkov, J. M. Barnola, G. Krinner, F. Parrenin,B. Delmonte, C. Ritz, P. Duval, R. Roethsliberger and M. Bigler. 2006,Effect of impurities on grain growth in cold ice sheets. J. Geophys. Res.,111(F01015).Duval, P. and C. Lorius. 1980, Crystal size and Climatic record down to thelast ice age from Antarctic ice. Earth Planet. Sci. Lett., 48, 59–64.Duval, P., M. F. Ashby and I. Anderman. 1983, Rate controlling processesin the creep of polycrystalline ice. J. Phys. Chem., 87, 4066–4074.Gagliardini, O. and J. Meyssonnier. 2000, Simulation of Anisotropic Ice Flowand Fabric Evolution Along the GRIP-GISP2 Flow Line (Central Greenland).Ann. Glaciol., 30, 217–223.Gagliardini, O., G. Durand and Y. Wang. 2004, Grain area as a statisticalweight for polycrystal constituents. J. Glaciol, 50(168).Gay, M. and J. Weiss. 1999, Automatic reconstruction of polycrystalline icemicrostructure from image analysis: application to the EPICA ice core atDome Concordia, Antarctica. J. Glaciol, 45(151), 547–554.Gow, A. J. 1969, On the rates of growth of grains and crystals in south polarfirn. J. Glaciol, 8(53), 241–252.Humphreys, F. J. and M. Hatherly. 1996, Recrystallization and related annealingphenomena. Pergamon-Elsevier Science Ltd.Kamb, W. B. 1959, Ice petrofabric observations from Blue Glacier, Washington,in relation to theory and experiment. Journal of Geophysical Research,64(11), 1891–1909.Kocks, U. F. 1998, The Representation of Orientations and Textures. InTexture and Anisotropy, pages 45–101, Cambridge University Press.Langway, C. C. J. 1958, Ice fabrics and the universal stage. SIPRE TechnicalReport, 62.27

Mecke, K. and D. Stoyan. 2002, Morphology of Condensed Matter. In Physicsand Geometry of Spatially Complex Systems, Lecture Notes in Physics,volume 600, Springer, Heidelberg.Parrenin, F., J. Jouzel, C. Waelbroeck, C. Ritz and J. Barnola. 2001, Datingthe vostok ice core by an inverse method. J. Geophys. Res., 106(D23),31837–31851.Poirier, J. P. 1985, Creep of crystals. Cambridge University Press.Raynaud, D., J.-M. Barnola, R. Souchez, R. Lorrain, J.-R. Petit, P. Duvaland V. Y. Lipenkov. 2005, CO2 and climate: the case of Marine IsotopicStage (MIS) 11. Nature, 436, 39–40.Riege, S. P., C. V. Thompson and H. J. Frost. 1998, The effect of particlepinningon grain size distributions in 2D simulations of grain growth. InGrain growth in polycrystalline Materials III, pages 295–301, The Minerals,Metals & Materials Society, h. weiland and b.l. adams and a.d. rolletedition.Rigsby, G. P. 1951, Crystal fabric studies on Emmons Glacier, Mount Rainer,Washington. J. Geol., 59, 596–598.Svensson, A., K. G. Schmidt, D. Dahl-Jensen, S. J. Johnsen, Y. Wang,J. Kipfstuhl and T. Thorsteinsson. 2003, Properties of ice crystals in North-GRIP late-to middle- Holocene ice. Ann. Glaciol, 37, 113–118.Thorsteinsson, T. 1996, Textures and fabrics in the GRIP ice core, in relationto climate history and ice deformation. Reports on Polar Research, Alfred-Wegener-Insitut f´’ur polar und meeresforschung, Bremerhaven, Germany.Thorsteinsson, T., J. Kipfsthul and H. Miller. 1997, Textures and fabrics inthe GRIP project. J. Geophys. Res., 102(C12), 26583–26599.Thorsteinsson, T., E. D. Waddington and L. wilen. 2006, Extracting fabricinformation from thin-section data. Forthcoming paper.Underwood, E. E. 1970, Quantitative Stereology. Reading, MA, Addison-Wesley Publishing.Van der Veen, C. J. and I. M. Whillans. 1994, Development of fabric in ice.Cold Reg. Sci. Technol., 22(2), 171–195.28

Wheeler, J., D. J. Prior, Z. Jiang, R. Spiess and P. W. Trimby. 2001, Thepetrological significance of misorientations between grains. Contrib MineralPetrol, 141, 109–124.Wilen, L. A. 2000, A new technique for ice-fabric analysis. Journal of Glaciology,46(152), 129–139.Wilen, L. A., C. L. Diprinzio, R. B. Alley and N. Azuma. 2003, Development,principles, and, applications of automated ice fabric analysers. Micro. Res.and tech., 62, 2–18.Woodcock, N. H. 1977, Specification of fabric shapes using an eigenvaluesmethod. Geol. Soc. Am. Bull., 88, 1231–1236.29

Figure 1: Image of a thin section under cross polarized light (on the left)and the corresponding texture (on the right). Grain boundaries appears inwhite, and each grain defined within the microstructure have an orientationdetailed in the colour map given by the polar plot projection on the rightbottom corner30

0.100.08population normalisée0.060.040.020.00-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5ln(/R)Figure 2: Normalized distribution of the logarithm of the normalized radiusof the grain (R k /〈R〉) for a microstructure sampled along the EPICA DomeC core at a depth of 100.1 m (black line). The lognormal distribution correspondingto the measured parameters 〈〉 D and σ D of this microstructureis also plotted (black dotted line), as well as the distribution obtained for a2D Potts microstructure (gray line). The number of grains (275) and themean radius (21.3 pixels) of the Potts microstructure are comparable to thenatural sample (respectively 454 grains and 〈R〉 = 21.6 pixels).31

Figure 3: Evolution of the standard deviation of 〈R〉 over 〈R ref 〉 versus theinverse of the square root of the number of grains.Figure 4: Point plots (a) and contour plots (b) of the inferred probabilitydensitydistributions of the fabric at 2999.8 m depth in the Dome C ice core,Antarctica. The probability-density distribution G is contoured at levels ofσ = 1/(6π).32

Figure 5: (a) Distribution of the misorientation angle between o e k 1 and all thepixels of the grain k = 7 in the texture sampled at 1525.8 m on the EPICADome C ice core. The corresponding polar plot is also drawn (left). As thisgrain fulfil the condition for an available grain (see text) o e k 1 is recalculated byexcluding the pixels with a misorientation larger than 10 ◦ . The correspondingpolar plot is also plotted (right). (b) Distribution of the misorientation anglebetween o e k 331 and all the pixels of the grain number 9 from the same thinsection and the corresponding polar plot. This grain is rejected from theanalysis.

a0.0350.030.025pσaii0.020.0150.010.0050.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1N g−0.5b0.40.350.30.25α0.20.150.10.0500.4 0.5 0.6 (2) 0.7 0.8 0.9 1a 1Figure 6: (a) standard deviation σa p of max(a(2) i ) induced by the sampling ofN g grains over an initial population of 10000. Black dots: a (2)1 = 0.9, a (2)2 =a (2)3 = 0.05 ; black crosses: a (2) 341 = 0.8, a (2)2 = 0.15, a (2)3 = 0.05 ; Black stars:a (2)1 = 0.6, a (2)2 = 0.35, a (2)3 = 0.05 ; the corresponding linear regression areplotted with black lines; circles: a (2)1 = 0.8, a (2)2 = 0.1, a (2)3 = 0.1 ; squares: a (2)1 = 0.6, a (2)2 = 0.2, a (2)3 = 0.2 ; the corresponding linear regression areplotted with black dashed lines. (b) slope of the linear regression as presentedin (a) versus max(a (2)i ) for the 231 textures tested. The maximum of eachbin is marked with a cross and corresponding second order polynomial fitcorreponds to the black line.

250200Number of pairs1501005000 10 20 30 40 50 60 70 80 90Angle (degree)Figure 7: Distribution of the misorientation between neighbouring grain pairs(large black line), and the envelope of probability determined after the generationof 200 distributions of the misorientation between random pairs fora texture sampled at 1525.8 m on the EPICA Dome C ice core. The changesin the gray intensity show the σ (68.3%, dark gray), 2σ (95.4%, gray) and3σ (99.7%, light gray) confidence levels.35