Effects of magnetic interactions in anisotropy of magnetic susceptibility: Models, experiments and implications for igneous rock fabrics quantification

Tectonophysics 418 (2006) 3 – 19 www.elsevier.com/locate/tecto

Effects of magnetic interactions in anisotropy of magnetic susceptibility: Models, experiments and implications for igneous rock fabrics quantification Philippe Gaillot a,1 , Michel de Saint-Blanquat b,⁎, Jean-Luc Bouchez b,2 a

Center for Deep Earth Exploration, Japan Agency for Marine-Earth Science and Technology, Yokohama Institute for Earth Sciences, 3173-15 Showa-machi, Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan b CNRS-LMTG, Université Paul-Sabatier, 14 av. Edouard-Belin, 31400 Toulouse, France Received 3 March 2005; received in revised form 15 July 2005; accepted 5 December 2005

Abstract Besides granites of the ilmenite series, in which the anisotropy of magnetic susceptibility (AMS) is mainly controlled by paramagnetic minerals, the AMS of igneous rocks is commonly interpreted as the result of the shape-preferred orientation of unequant ferromagnetic grains. In a few instances, the anisotropy due to the distribution of ferromagnetic grains, irrespective of their shape, has also been proposed as an important AMS source. Former analytical models that consider infinite geometry of identical and uniformly magnetized and coaxial particles confirm that shape fabric may be overcome by dipolar contributions if neighboring grains are close enough to each other to magnetically interact. On these bases we present and experimentally validate a two-grain macroscopic numerical model in which each grain carries its own magnetic anisotropy, volume, orientation and location in space. Compared with analytical predictions and available experiments, our results allow to list and quantify the factors that affect the effects of magnetic interactions. In particular, we discuss the effects of (i) the infinite geometry used in the analytical models, (ii) the intrinsic shape anisotropy of the grains, (iii) the relative orientation in space of the grains, and (iv) the spatial distribution of grains with a particular focus on the inter-grain distance distribution. Using documented case studies, these findings are summarized and discussed in the framework of the generalized total AMS tensor recently introduced by Cañon-Tapia (Cañon-Tapia, E., 2001. Factors affecting the relative importance of shape and distribution anisotropy in rocks: theory and experiments. Tectonophysics, 340, 117–131.). The most important result of our work is that analytical models far overestimate the role of magnetic interaction in rock fabric quantification. Considering natural rocks as an assemblage of interacting and non-interacting grains, and that the effects of interaction are reduced by (i) the finite geometry of the interacting clusters, (ii) the relative orientation between interacting grains, (iii) their heterogeneity in orientation, shape and bulk susceptibility, and (iv) their inter-distance distribution, we reconcile analytical models and experiments with real case studies that minimize the role of magnetic interaction onto the measured AMS. Limitations of our results are discussed and guidelines are provided for the use of AMS in geological interpretation of igneous rock fabrics where magnetic interactions are likely to occur. © 2006 Elsevier B.V. All rights reserved. Keywords: Anisotropy of magnetic susceptibility; Magnetic interactions; Models; Experiments; Igneous rocks

⁎ Corresponding author. Tel.: +33 561 332 616; fax: +33 561 332 560. E-mail addresses: [email protected] (P. Gaillot), [email protected] (M. de Saint-Blanquat), [email protected] (J.-L. Bouchez). 1 Tel.: +81 45 778 5718; fax: +81 45 778 5704. 2 Tel.: +33 561 332 642; fax: +33 561 332 560. 0040-1951/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.tecto.2005.12.010

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1. Introduction Measurement of the anisotropy of magnetic susceptibility (AMS) in a rock provides a second-order tensor represented by an ellipsoid whose principal axes, Kmax ≥ Kint ≥ Kmin, are determined, in both orientation and magnitude, by the contributions of the individual minerals. A set of mineral grains in a rock generates an AMS if the grains combine into a net statistical alignment of their long axes and/or easy directions of magnetization (Stacey and Banerjee, 1974; O'Reilly, 1984). Due to sensitivity and speed of use, AMS, also called magnetic fabric, became widespread in petrofabric studies both for quantitative and semi-quantitative purposes (Tarling and Hrouda, 1993; Borradaile and Henry, 1997). Hence, fabric study of apparently isotropic rocks and structural mapping of complex igneous bodies (Bouchez, 1997) benefit from the AMS technique. However, due to its integrative nature, use of AMS in structural studies is complicated by (i) the mineralogical control of magnetic anisotropy, (ii) the effects of multiple mineral preferred orientations, (iii) the inverse fabric carried by single domain (SD) magnetite or by some silicates (tourmaline, cordierite), and (iv) the anisotropy of spatial distribution through magnetic interaction, also referred to as textural anisotropy (Fuller, 1963; Wolff et al., 1989). Mineralogical control of magnetic fabric (Rochette and Vialon, 1984; Borradaile, 1987; Rochette, 1987; Borradaile and Sarvas, 1990; Housen and Van der Plujim, 1990; Jackson, 1991), multiple mineral-preferred orientations (Borradaile and Tarling, 1981; Hrouda, 1992; Housen et al., 1993; Callot and Guichet, 2003) and inverse fabrics (Potter and Stephenson, 1988; Rochette, 1988; Borradaile and Puumala, 1989; Rochette et al., 1999) have been extensively studied and will not be further considered in this paper. Instead, we focus on the effects of magnetic interactions on the AMS and their implications for the quantification of igneous rock fabrics. Hargraves et al. (1991) experimentally investigated the case of ferromagnetic rocks (basaltic flows, mafic dikes) in which the AMS can be ascribed either to the shape fabric of anisometric ferromagnetic grains (magnetite), or to the non-uniform distribution of isotropic magnetite grains. They pointed out that even if perfectly magnetically isotropic grains are present in a rock, the rock-AMS will be anisotropic if the magnetic grain distribution is anisotropic and if the grains are close enough to each other to magnetically interact. Magnetic interactions between magnetic particles would then be largely responsible for the measured AMS. The role of

magnetic interactions in AMS was further examined by Stephenson (1994) who developed two simple analytical models of magnetic interactions between magnetically isotropic neighboring particles of infinite linear and planar distributions. The resulting large magnetic anisotropies due to magnetic interactions confirmed the findings of Hargraves et al. (1991). Since rocks do contain magnetic grains that are not isotropic, CañonTapia (1996) generalized the models of Stephenson (1994) by studying the effects of individual particle anisotropy on the AMS of an infinite three-dimensional (3D) array of coaxial ellipsoidal particles. These theoretical efforts were complemented, and partly confirmed, by the experiments of Grégoire et al. (1995) who examined the interactions between two natural magnetite grains. In contrast, detailed investigations of real igneous rocks, namely a granodiorite from Brazil (Archanjo et al., 1995) and a syenite from Madagascar (Grégoire et al., 1998), lead to the conclusion that the measured bulk-rock AMS could be entirely explained by grain-shape anisotropy of magnetite despite the presence of clusters of magnetite grains in these rocks. In both studies, the grain-shape tensor of the magnetite grains, independently determined using image analysis techniques, was shown to be similar to the rock-AMS tensor. It becomes therefore important (i) to reconcile the analytical models and the experiments that predict a major contribution of magnetic interaction, with observations on natural samples that minimize the role of magnetic interaction onto the measured AMS, and (ii) to define a quantitative framework that can help in the interpretation of magnetic fabrics. In this paper we develop a new model that facilitates the quantification of the effects of magnetic interactions onto the resultant AMS and provide new experiments that validate it. A brief presentation of the physical bases and hypotheses of the generalized analytical model of Cañon-Tapia (1996) as well as the experimental limitations motivating the development of a numerical model will be given first. Next, we present a new macroscopic numerical model that considers that each individual magnetic grain is represented by its individual AMS ellipsoid, taking into account its 3D-shape, volume, orientation and location in space. This avoids finer scale complexities of magnetic grains such as size and state of magnetic domains and remanence, and makes easier numerical modeling of the effects of magnetic interactions onto the resultant AMS. For this purpose, we provide new experiments that validate the latter model. By comparing available analytical models (Cañon-Tapia, 1996) with experiments (Grégoire et al., 1995), the effects of magnetic interactions will be estimated in the realistic

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configurations of finite geometry, and non-coaxial grains of various shape. Finally, we will discuss the original case studies of Grégoire et al. (1998) and Hargraves et al. (1991) in the frame of the total AMS tensor recently introduced by Cañon-Tapia (2001). 2. Limitations of available analytical models and experiments: motivation for a numerical model Theoretical models and experiments are complementary in testing the languishing idea that magnetic interactions between small magnetite particles like those found in basalt may be responsible for the measured AMS (Fuller, 1963; Morish and Yu, 1995; Davis and Evans, 1976), an idea revived by Hargraves et al. (1991) and recently debated in Martin-Hernandez et al. (2004). Let us summarize the available theory and experiments, and discuss their limitations. Magnetostatic interaction at a microscopic scale is not straightforward to model although it has been the subject of numerous works motivated by the important technological applications of such magnetic behavior (chemical sensors, thermal switches, magnetic media; see Hubert and Schäfer, 1998 for a review). In the geological context of AMS studies, the basic models of Stephenson (1994) for magnetic lineation and foliation, generalized by Cañon-Tapià (1996) to three-dimensional distributions of anisotropic identical and coaxial particles, constitute the basis of calculation of magnetic interaction effects. In this model, (i) any 3D periodic distribution of particles is approximated by an arrangement in which six nearest neighbors act upon a central reference particle (Fig. 1); (ii) all the particles are

Fig. 2. Dipolar field approximation. (a) General case: dipolar field produced in B by a magnetized body having a magnetic moment (Y m) and located in A. For practical reasons, the dipolar field vector is defined by its radial (hr) and tangential (hθ) components. Special configurations used in the analytical calculation of Stephenson (1994) and Cañon-Tapia (1996) that are of interest for the determination of the Y is parallel to the A–B AMS tensor are shown in (b) and (c). (b) H 0 center-to-center line (θ = 0°), the dipolar field equals hr0 and has the Y ; (c) Y same direction of H H 0 is normal to the A–B center-to-center line 0 (θ = 90°), the dipolar field equals hθ (hθ = hr0/2) and has the opposite Y. direction of H 0

identical and coaxial, and (iii) all particles are modeled by an uniformly magnetized ellipsoid producing an outside field approximated by a dipolar field (Sprowl, 1990) (Fig. 2). By investigating the behavior of the particle assemblage subjected to three mutually perpendicular external fields, Hi (i = 1, 2 and 3), coinciding with the axes of symmetry of the array of particles (xi, xj, xk), the former restrictions allow the analytic calculation of the AMS tensor resulting from this infinite geometry. Following Cañon-Tapià (1996), the bulk susceptibility χi, along the xi direction, is vi ¼

viapp

! v 2 1 1 1   v 2p ri3 rj3 rk3 iapp

ð1Þ

where χi-app and the ri,j,k are, respectively, the apparent susceptibility and inter-particle distance along the i-, j- and k-directions. Using the approximation of the apparent susceptibility as the inverse of N, the demagnetization factor (Stoner, 1945), valid for highly permeable material of low shape anisotropy, expression (1) can be rewritten as: Fig. 1. The three-dimensional array of ellipsoidal particles used in the analytical model of Cañon-Tapia (1996). Relative positions of the four nearest neighbors of a reference particle located in the xi–xj plane. Two other particles are symmetrically disposed on sides of the reference particle along the xk direction. All the particles (prolate or oblate) are identical and coaxial. After Cañon-Tapia (1996).

vi ¼

1 v 2 1 1 Ni    2p ri3 rj3 rk3

!:

ð2Þ

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The 3D periodic structure of the array allows to model any configuration including the planar and linear ones first discussed by Stephenson (1994) by letting one or two rns tend to infinity. To facilitate analytical treatment in this model, all the particles are identical and coaxial and various geometrical constraints (particle center to center line is one of the principal axis of the particles) unlikely to be found in real rocks were introduced. Because, these geometrical constraints constitute a severe limitation to the comparison with experimental results and natural rock studies where alignments of grains are not necessary parallel to one of their principal AMS axes, in the next section we develop a two-grain macroscopic numerical model that allows us to explore more realistic geometrical configurations (different grains and non-coaxial grains). 3. Two-grain numerical model and experimental validation 3.1. Macroscopic numerical model

derive the dipolar field produced on any neighboring grain: mg0 ¼ vg M0g ¼ kijg H0

ð3Þ

where m0g is the magnetization of grain g. Considering that any magnetized particle having a magnetic moment, m0g, produces in each point of the space (r, θ) a dipolar g g field, h, defined by its radial (hr0 ) and tangential (hθ0 ) components hgr0

mg0 cosðhÞ; 2pr3

hgh0 ¼

mg0 sinðhÞ 4pr3

ð4Þ

the effective field, Heff0g, experienced by grain, g, owing to the induced magnetization of its neighbor, is iteratively obtained by summing the adjustment terms necessary to estimate the grain to grain magnetic interaction effects on the local effective field (Fig. 3). X Heff g0 ¼ H0 þ h0 ð1 þ correction termsÞ ð5Þ i

m0g,

In general, the resulting magnetic moment, induced by an external field H0 in a particle g with g volume vg and AMS ellipsoid ki,j , can be used to

In the estimation of the effective field experienced by the reference grain and its effect on its magnetization, the latter correction terms take into account the buckling

Fig. 3. Algorithm of the two-grain numerical model and convergence of the iteration process (see text for details).

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effects of grain A on grain B and of grain B on grain A. Since this interaction process implies correction terms that are a fraction of the previous term multiplied by 1/r3, the number of iterations, i, never exceeds 5 (Fig. 3). The resulting susceptibility (K0A and K0B) of each grain is then given by:   Heff g0 g g Keff 0 ¼ K0 1 þ ð6Þ H0 and, the effective susceptibility of the 2-magnetically interacting grains, K0, is finally simply given by: K0 ¼ K0A þ K0B :

ð7Þ

Inversion of the seven directional measurements using the appropriate design matrix (Borradaile and Stupavsky, 1995) allows to derive the resulting AMS ellipsoid of the 2 grains assemblage. By numerically varying the inter-grain distances, and repeating the previous procedure (7-orientation scheme), we can numerically approximate the effects of magnetic interactions on AMS. 3.2. Experimental validation of the numerical model Our numerical model was tested using the experimental protocol shown in Fig. 4. In this protocol, we used rigid plastic disks keeping grain orientations rigorously constant while their separating distance, r, along the z-axis was decreased by tiny constant increments until contact. Experimentally, these conditions are fulfilled using pure magnetite (Fe3O4) grains (Ward's Natural Science Establishment, Inc.) of about half a millimeter in diameter (d). One grain per disk was glued in a small hole centered in one face of a disk. The disks are marked and have the same diameter as that of the sample holder of the susceptometer (Kappabridge KLY-2; Agico) and calibrated transparent slices of weak paramagnetic susceptibility (Table 1) were used as separators between the disks carrying magnetite grains. The AMS of each grain was measured and served as input to the numerical model.

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Table 1 Magnetic properties of plastic and transparent disks and glass lamellae used in our two-grain experiments

Plastic disk Transparency disk Glass lamella

Kmean (10− 6 SI)

P

0.96 1.05 1.63

1.13 1.04 1.01

The following parameters are displayed in Figs. 5 and 6 as a function of d/r (d: length of grain axis parallel to the center-to-center line z-axis; r: distance between grain centers): (1) principal (maximum, intermediate and minimum) susceptibilities (Kmax, Kint and Kmin) for the experiments (symbols) and their numerical model (line); (2) variation of the mean susceptibility (Kmean); (3) L and F parameters (L = Kmax/Kint; F = Kint/Kmin) parameters, (4) variation of the anisotropy parameter P (P = Kmax/Kmin), and finally (5) orientations of the maximal (square), intermediate (triangle) and minimal (circle) AMS directions in the equal area polar projection. Detailed results of the experiments are also given in Tables 2 and 3. The first experiment (experiment #1, Fig. 5) was made using two similar multidomain pure magnetite grains. Their individual volume, principal AMS tensor intensities and orientations, as well as the raw data of the experiments are given in Table 2. The initial measured and modeled AMS ellipsoids are oblate and subparallel to the grain A1, which is the more anisotropic. The intensity of the maximum axis of the experimental and numerically modeled AMS tensors (Kmax) decreases until d/r ~ 0.7, and then increases until d/r = 1, to reach a value much higher than its initial value (Fig. 5a). The intermediate axis, Kint, regularly decreases from the beginning to the end of the experiment. The minimum axis (Kmin) first increases until d/r ~ 0.8, and then decreases, but its final value is higher than its initial one. All these variations lead to a transitory weak decrease in mean susceptibility (Kmean, Fig. 5b), associated to a significant change in the shape of the AMS ellipsoid, from oblate (F N L) to isotrope (F ~ L at d/r ~ 0.7; Fig. 5c), and consequently to a significant reduction in shape anisotropy (P ~ − 9%; Fig. 5d). The

Fig. 4. Geometrical conventions and procedure for experiments #1 and #2.

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final ellipsoid at d/r = 1 is prolate, having a higher mean susceptibility and a higher anisotropy than the initial one. The final orientation of Kmax is close to the initial orientation of Kmin, i.e. Kmax is subparallel to the long axis of the cluster formed by the two touching grains. A weak discrepancy between the experimental and numerical results is observed for Kint in the 0.8 b d/ r b 1 interval, resulting in a poor modeling of L and F, in this interval and an underestimation of Kmean variation (+ 4% in experimental data, + 1% in model at grain contact; Fig. 5b), while the modeling of P is correct (+ 5% at grain contact; Fig. 5d). In this experiment where grains are almost coaxial, a rotation of the directions of the principal anisotropies (Fig. 5e) and a total amplitude variation of the anisotropy ~ 15% are observed and numerically modeled, at least up to d/ r ~ 0.8. Since the three modeled principal susceptibility axes and their orientations are simultaneously determined by the seven-orientation scheme inversion (Borradaile and Stupavsky, 1995), our confidence in the three axes has to be the same. The observed discrepancy in Kint and consequently in Kmean, L and F may come from the geometrical limitation of our experiments (difference between the shapes of the experimental and numerical grains) and from the sensitivity of our equipment in an interval of d/r values where the numerical Kint/Kmin ratio is lower than 1.02. The second experiment (experiment #2, Fig. 6) was made using two grains having contrasting magnetic properties. Grain A2 has a mean susceptibility ~ 7 times higher than grain B2 (Table 3). The initial (noninteracting) orientation of the ellipsoid is very close to the orientation of the individual ellipsoid of the grain A2. Intensity of Kmax remains stable until d/r = 0.6, then increases significantly, whereas Kint continuously and lightly decreases, and Kmin, first slowly increases then finally decreases for d/r N 0.85 (Fig. 6a). A weak discrepancy remains between the experimental and numerically modeled Kint at d/r N 0.8, leading to the same underestimation of Kmean than for experiment # 1 (experiment #2: +4.5%; model #2: + 2.5 %; Fig. 6b). Due to the size difference of the two grains and to their respective orientation in space, a significant increase in L (L = 1.24 at grain contact) and change in P (+ 10%) is Fig. 5. Experiment # 1: two similar grains. (a) Maximum (square), intermediate (triangle) and minimum (circle) susceptibilities; (b) variation of mean susceptibility (Kmean); (c) L (Kmax/Kint) and F (Kint/Kmin) parameters; (d) variation of anisotropy parameter P; (e) equal-area projection of the principal AMS axes. Orientation of individual AMS axes for grains A and B is shown in the equal-area mesh. Symbols: experimental data; lines: model. Shade reflects magnetic interaction contribution (dark: strong; light: weak).

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observed but is restricted to a very limited interval (d/ r N 0.9; Fig. 6c and d). However, smooth variations in principal anisotropy directions covering the entire d/r interval lead to a fabric different of that of the constituting grains (Fig. 6e), and whose importance depends on the intensity of the magnetic interaction. The total rotation of the ellipsoid principal axes is less than for the experiment #1 (30–40° versus 60–90°). In summary, in spite of the noise potentially attributed to geometrical differences between the experimental and numerical models, and/or to grain internal magnetic behavior (surface/edge effect, remanence…), these new experiments validate both our numerical model, at least for our large multidomain magnetite grains where the contribution of remanence is probably not significant. Whereas the numerical model can be extended to three grains or to small clusters of grains, accurate experiments involving more than two grains are difficult to set up, since they would require to experimentally control six parameters per grain (location and orientation in space). Such experiments have not been conducted since along with the increased experimental uncertainties, and the infinite number of situation to explore, they would not significantly increase our understanding of the magnetic interactions. 4. Compared experimental, analytical, and numerical approaches 4.1. Comparison framework The numerical model developed in this paper as well as particular cases of the generalized analytical (infinite geometry) model of Cañon-Tapià (1996) can more or less closely reproduce the 2-grain experiments of Grégoire et al. (1995). The case study that will be presented will emphasize the differences between the numerical, analytical and experimental approaches by evaluating (1) the uncertainty coming from the positioning of the grains on the experimental results; (2) the effects of the hypotheses used in the analytical and numerical models; and (3) the infinite geometry used in the analytical model on the estimation of the magnetic interaction on AMS. Fig. 6. Experiment # 2: two different grains. (a) Maximum (square), intermediate (triangle) and minimum (circle) susceptibilities; (b) variation of mean susceptibility (Kmean); (c) L (Kmax/Kint) and F (Kint/Kmin) parameters, (d) variation of anisotropy parameter P; (e) equal-area projection of the principal AMS axes. Orientation of individual AMS axes for grains A and B is shown in the equal-area mesh. Symbols: experimental data; lines: model. Shade reflects magnetic interaction contribution (dark: weak; light: strong).

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Table 2 Experiment #1: single grain measurements and experimental results compared to our numerical model Kmax Dec. (°) Single grains A1 12 B1 108

Kint

Kmin

Inc. (°)

Int. (× 10− 6 SI)

Dec. (°)

Inc. (°)

15 6

1073 1059

105 200

10 14

Int. (× 10− 6 SI) 945 967

Dec. (°)

Inc. (°)

229 357

72 74

Int. (× 10− 6 SI) 756 932

Kmean

Anisotropy and shape parameters

Int. (× 10− 6 SI)

L

F

P

T

1.135 1.095

1.250 1.038

1.419 1.136

0.274 − 0.423

925 986

Volume of grain (× 10− 4 cm− 3)

5.01 5.05

Experiment Distance d/r

Dec. (°)

Inc. (°)

Int. (× 10− 6 SI)

349 356 0 351 2 357 353 360 350 350 343 345 340 344 335 336 332 329 329 323 318 310 303 305 299

5 12 11 12 14 14 17 16 19 13 19 18 22 26 29 19 36 33 41 47 42 49 54 55 57

2035 2030 2023 2020 2006 2008 1992 1994 1996 1994 1995 1994 2002 1992 1998 1980 1991 2003 2015 2028 2012 2058 2077 2128 2256

Dec. (°) 80 89 92 84 94 91 87 93 85 82 77 80 76 80 71 68 69 64 67 63 53 49 44 52 48

Kmin

Kmean

Anisotropy and shape parameters

Difference (experience − model)

Inc. (°)

Int. (× 10− 6 SI)

Dec. (°)

Inc. (°)

Int. (× 10− 6 SI)

Int. (× 10− 6 SI)

L

F

P

T

ΔKmean (× 10− 6 SI)

ΔP

12 12 12 14 11 14 13 13 13 10 11 14 13 11 10 8 9 8 10 9 6 8 8 12 12

1994 2004 1991 1975 1974 1977 1961 1967 1946 1957 1942 1957 1930 1931 1921 1935 1925 1917 1907 1901 1915 1892 1897 1876 1877

236 223 229 224 221 224 213 221 207 209 194 206 194 191 178 179 170 166 168 162 149 146 140 149 145

76 73 74 71 72 70 68 69 67 73 68 66 64 62 59 69 53 56 48 42 47 40 34 33 31

1712 1723 1723 1738 1752 1766 1783 1788 1790 1798 1801 1804 1806 1814 1824 1839 1835 1848 1836 1842 1852 1836 1843 1827 1811

1914 1919 1913 1911 1910 1917 1912 1916 1911 1916 1912 1918 1913 1912 1914 1918 1917 1923 1919 1924 1927 1929 1939 1944 1981

1.021 1.013 1.016 1.023 1.016 1.016 1.016 1.014 1.025 1.019 1.027 1.019 1.037 1.032 1.040 1.023 1.034 1.045 1.056 1.067 1.051 1.088 1.095 1.134 1.202

1.165 1.163 1.155 1.137 1.127 1.119 1.099 1.101 1.087 1.088 1.078 1.085 1.069 1.064 1.053 1.052 1.049 1.037 1.039 1.032 1.034 1.030 1.029 1.027 1.036

1.189 1.179 1.174 1.162 1.145 1.137 1.117 1.116 1.115 1.109 1.108 1.106 1.109 1.098 1.096 1.077 1.085 1.084 1.097 1.101 1.087 1.120 1.127 1.165 1.245

0.764 0.840 0.803 0.704 0.762 0.759 0.711 0.752 0.537 0.641 0.473 0.622 0.288 0.328 0.139 0.382 0.182 − 0.096 − 0.181 − 0.349 − 0.187 − 0.476 − 0.517 − 0.655 − 0.678

− 6.620 − 1.053 − 6.604 − 6.730 − 5.976 1.218 − 2.665 1.956 − 2.856 2.391 − 0.926 5.290 − 0.070 − 0.074 2.228 5.963 4.609 10.062 6.411 9.322 11.371 10.778 17.537 19.368 45.144

− 0.007 − 0.014 − 0.012 − 0.010 − 0.016 − 0.016 − 0.026 − 0.023 − 0.015 − 0.020 − 0.015 − 0.012 − 0.006 − 0.007 − 0.001 − 0.018 − 0.006 − 0.006 0.007 0.000 − 0.018 − 0.007 − 0.026 − 0.010 − 0.004

P. Gaillot et al. / Tectonophysics 418 (2006) 3–19

0.197 0.259 0.329 0.424 0.479 0.512 0.551 0.568 0.595 0.600 0.620 0.636 0.648 0.677 0.710 0.724 0.746 0.778 0.786 0.831 0.840 0.880 0.913 0.936 1.000

Kint

Kmax

Table 3 Experiment #2: single grain measurements and experimental results compared to our numerical model Kmax Dec. (°)

Int. (× 10− 6 SI)

Dec. (°)

50 2

1613 230

297 315

Kmin Inc. (°)

4 1

Kmean

Anisotropy and shape parameters

Int. (× 10− 6 SI)

Dec. (°)

Inc. (°)

Int. (× 10− 6 SI)

Int. (× 10− 6 SI)

L

F

P

1489 225

31 66

40 88

1279 199

1460 218

1.083 1.022

1.164 1.131

1.261 1.156

Kmean

Anisotropy and shape parameters

Difference (experience − model) ΔKmean (× 10− 6 SI)

ΔP

2.041 2.278 −2.301 0.360 0.723 4.473 3.406 2.932 3.530 4.778 6.891 12.411 26.857 35.402

−0.004 −0.006 −0.008 −0.006 −0.006 0.001 −0.005 0.002 0.005 0.002 −0.009 0.012 0.030 −0.011

T

0.310 0.696

Volume of grain (× 10− 4 cm− 3)

7.72 1.15

Experiment Distance d/r

0.188 0.316 0.401 0.452 0.554 0.598 0.649 0.679 0.711 0.746 0.781 0.829 0.877 0.932 1.000

Kmax

Kint

Kmin

Dec. (°)

Inc. (°)

Int. (× 10− 6 SI)

Dec. (°)

Inc. (°)

Int. (× 10− 6 SI)

Dec. (°)

Inc. (°)

Int. (× 10− 6 SI)

Int. (× 10− 6 SI)

L

F

P

T

214 219 214 216 221 222 225 232 231 231 235 240 247 261 276

42 42 43 49 47 47 49 50 53 53 55 56 59 61 61

1834 1832 1832 1824 1828 1832 1846 1844 1850 1861 1869 1880 1929 2006 2064

121 124 120 127 127 127 128 133 132 133 134 139 140 144 158

3 6 4 1 4 5 6 8 7 6 8 8 10 14 14

1718 1713 1707 1703 1701 1695 1690 1686 1688 1681 1678 1678 1669 1661 1662

27 27 26 38 33 32 33 37 37 39 39 44 45 47 62

48 48 46 41 43 42 40 39 36 36 33 33 29 24 24

1496 1495 1501 1499 1503 1508 1511 1515 1509 1510 1514 1521 1513 1515 1534

1683 1680 1680 1675 1677 1678 1682 1682 1682 1684 1687 1693 1704 1727 1753

1.068 1.069 1.073 1.071 1.075 1.081 1.092 1.094 1.096 1.107 1.114 1.120 1.156 1.208 1.242

1.149 1.146 1.138 1.136 1.132 1.124 1.119 1.113 1.119 1.114 1.108 1.103 1.103 1.096 1.084

1.226 1.225 1.221 1.217 1.216 1.215 1.222 1.217 1.226 1.232 1.235 1.236 1.275 1.324 1.346

0.359 0.340 0.294 0.301 0.263 0.202 0.119 0.085 0.099 0.030 − 0.025 − 0.074 − 0.194 − 0.344 − 0.459

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Single grains A2 202 B2 225

Kint Inc. (°)

11

12

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Using roughly bipyramidal, suboctahedral natural magnetite grains that are slightly elongated parallel to their [001] axis, magnetic interactions between two grains was investigated by Grégoire et al., (1995) in two particular configurations namely “aligned” and “side-byside” (Fig. 7). Measurements of each grain give susceptibility magnitudes of 2.4 and 3.8 SI, and anisotropies P of 1.3 and 1.1 respectively. In the “aligned” configuration, the grain center-to-center line is parallel to the grain long axes, whereas in the “side-by-side” configuration, this line is perpendicular to their long axes (Fig. 7a). Assuming that the geometry of the experiments is well-constrained (perfect coaxiality of the individual AMS tensors) and using the grains principal susceptibility values and volumes (2.76 · 10− 4 and 6.07· 10− 4 cm3) as inputs in the numerical model, we can numerically reproduce these two experiments. On the other hand, using a particular case of the generalized analytical model of Cañon-Tapià (1996), it is also possible to estimate the effects of magnetic interactions between an infinite chain of prolate particles having a magnetic anisotropy (P ~ 1.2) similar to those of the grains used by Grégoire et al. (1995) by (i) selecting highly permeable prolate ellipsoids having a shape anisotropy of a/b = 1.17, where a and b are the polar and equatorial semi-axes of the ellipsoids (a/b = 1.17 ~ P = 1.2; Stoner, 1945); and by (ii) setting rj = rk→∞ in the general formulation of CañonTapià (1996). For each model (experimental, numerical, analytical) and each configuration (aligned, side-by-side), the particles are moved close together along the center-tocenter line and the AMS tensor is measured (symbols)/ computed (line) at each step. The (Kmax, Kint and Kmin) principal susceptibility values, θχmax (θχint) the angle between the direction of the maximum (intermediate) susceptibility axis and the reference particle long axis (initial Kmax), as well as the variation of the mean susceptibility (Kmean) and anisotropy parameter P are plotted as a function of d/r (Figs. 7 and 8). 4.2. Experimental versus numerical Independently of the contribution of magnetic interactions (d/r ~ 0.1) and the geometrical configuration, the first discrepancy between the experimental and numerical results comes from the non-perfect coaxiality

13

of the grains in the experiments. The influence of the mispositioning is not significant when d/r is small but, as previously stressed by our previous two-grain experiments, the relative orientation of the grains becomes fundamental when their separating distance decreases. Regardless of the configuration (aligned or side-by-side), no significant variation is detected up to d/r b 0.5 (one grain-diameter spacing between grains). Experimental and numerical results show that the normalized susceptibilities and degree of anisotropy vary slowly at first, then vary rapidly (d/r N 0.8) as a function of the distance between grains (Fig. 7). Within the limits of the experimental uncertainties (on spacing between grains particularly in the 0.8–1.0 range, and orientation and location of the grains), results of numerical and experimental approaches agree well except near grain contact (d/r ~ 1). Like in the previous experiments, the discrepancy near grain contact may be explained by geometrical differences between the experimental and numerical models, and/or to grain internal magnetic behavior. In the “aligned” configuration (Fig. 7, left), the rapid increase in Kmax and concomitant decrease both in Kint and Kmin make the orientation of Kmax stable whereas experimental Kint (symbols) fluctuates in the Kint − Kmin plane. No significant change in the resulting AMS tensor orientation is observed. Numerically (line), where individual AMS ellipsoids are coaxial, Kmax and Kint keep strictly their original orientations. The difference in the increase/decrease in Kmax/Kint − Kmin results in a significant increase in Kmean (~ +8%). Hence, in such a configuration, magnetic interaction enhances only the particle shape anisotropy, as attested by the increase in the P parameter (~ + 40%). In the “side-by-side” configuration (Fig. 7, right), Kmin decreases in a manner similar to that of the “aligned” experiment. Kint initially increases until it switches with decreasing Kmax at d/ r ~ 0.6 (Fig. 7c). Kmean and P initially show a weak and concomitant decrease (− 0.25% and −1%, respectively) quickly followed by a noticeable increase up to +7% and + 35%, respectively. In such a configuration, depending on the distance between grains, magnetic interactions can thus slightly decrease the magnetic fabric intensity while either keeping magnetic and preferred shape orientations of the particles alike, or produce an inverse magnetic fabric.

Fig. 7. Numerical model versus experiments. Two-grain experiments in the “aligned” (left, open symbols) and “side-by-side” (right, black symbols) configurations of Grégoire et al. (1995): (a) geometry of the grains and mean orientation of the AMS ellipsoid axes; (b) maximal (square), intermediate (triangle) and minimal (circle) susceptibilities; (c) θKmax (θKint) angle between the direction of the maximum (intermediate) susceptibility axis and the particle long axis; (d) variation of mean susceptibility (Kmean); (e) variation of anisotropy parameter P. Symbols: experimental data; lines: numerical model.

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P. Gaillot et al. / Tectonophysics 418 (2006) 3–19

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Table 4 Effects of magnetic interaction in AMS, models and experiments: a synthesis Chain of Infinite

Coaxial

Identical

2-grain experiments Finite geometry

∼ Coaxial

∼ Identical

Non-coaxial

∼ Identical

Aligned prolate particles (Cañon-Tapia, 1996)

Spherical particles (Stephenson, 1994)

Side-by-side prolate particles (Cañon-Tapia, 1996)

Xmean (%) = +38 P (%) = +187 (d/r)c = 0.5 Fabric reinforcement

Xmean (%) = +20 P (%) = +150 (d/r)c = 0.5 Apparition of a fabric

Xmean (%) = +13.5 P (%) = +93 (d/r)c = 0.5 Angular switch

Aligned configuration (Grégoire et al., 1995)

Side-by-side configuration (Grégoire et al., 1995)

Kmean (%) = +8 P (%) b +40 (d/r)c = 0.5 Fabric reinforcement

Kmean (%) = +8 P (%) b +40 (d/r)c = 0.5 Angular switch

Non-identical

Experiment #1 Kmean (%) b +4 P (%) b +6 (d/r)c = 0.5 Rotation of principal orientations Experiment #2 Kmean (%) b +5 P (%) b +12 (d/r)c = 0.8 Rotation of principal orientations

Expressed in terms of (i) variations of mean susceptibility, (ii) variation of anisotropy parameter, P, (iii) critical d/r ratio where interaction becomes significant, and (iv) fabric orientation change, a complex function of the spatial configuration of the grains, and the parameters of each grain (individual AMS tensors). Prolate particles used in analytical models (Cañon-Tapia, 1996) have a shape aspect ratio a/b = 1.17 corresponding to the mean P (P ∼ 1.2) of the 2 grains assemblage used in the experiment of Grégoire et al. (1995) and our experiment #1.

4.3. Experimental–numerical versus analytical: the effect of infinite geometry Whereas trends in these experimental and numerical changes are in good agreement with analytical predictions, their magnitudes are significantly lower (Fig. 8). The maximum experimental/numerical variations of Kmean and P that reach ~ + 8% and +40% respectively do not reach the +38% (+ 13.5%) and + 187% (+ 93%) predicted for the “aligned” (resp. “side-by-side”) chain of prolate grains (calculated for direct comparison with a/b = 1.17). For a chain of isotropic grains (Stephenson, 1994), these variations would have been even higher, reaching up to + 20% and +150%, respectively (Table 4). In addition, the prediction of the angular switch observed in the analytical “side-by-side” configuration appears later (d/r ~ 0.62; Fig. 8) than in the experiments and numerical simulations (d/r ~ 0.52; Fig. 7).

Since the approximations used in the numerical model are acceptable up to d/r ~ 0.8, and produce only minor discrepancies (~ 10 % on principal susceptibility values) for d/r N 0.8, the major discrepancy between the analytical and the experimental/numerical results is attributed to the difference that exists between the geometries used, namely infinite geometry versus 2grain geometry. Considering an infinite chain of particles, the central reference particle experiences an enhanced field produced by two neighbors which themselves experience an enhanced field of same magnitude. The unity cell of this configuration is thus a reference particle experiencing the enhanced field produced by its two closest neighbors, themselves submitted to the enhancing field of their neighbors. This situation strongly differs from the two-grain experiments, where the unity cell is one grain experiencing the enhanced field produced by another

Fig. 8. Analytical model. Numerical experiments involving an infinite chain of identical and coaxial prolate particles in the “aligned” (left, open symbols) and “side-by-side” (right, black symbols) configurations. (a) Geometry; (b) susceptibilities parallel (χp) and normal (χn) to the particles center-to-center line, and the maximal, intermediate and minimal susceptibilities (χmax: square, χint: triangle, χmin: circle); (c) θχmax (θχint) angle between the direction of the maximum (intermediate) susceptibility axis and the particle long axis; (d) variation of mean susceptibility (χmean); (e) variation of anisotropy parameter P. For comparison experimental/numerical results obtained in the “aligned” and “side-by-side” two-grain experiments of Fig. 7 are displayed using bold lines.

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unique grain. Thus, assuming similar and coaxial grains, the two-grain experimental/numerical approach would be a lower bound, while the infinite geometry model would be an upper bound for clusters of n-magnetically interacting grains. More real configurations involving clusters of non-similar and non-coaxial grains are difficult to address as the number of possible situations to explore is infinite. However, the complementarity of these three approaches allowed us to better understand the role of geometry (separating distance, relative orientations between grains) and of intrinsic grain properties on the effects of magnetic interactions on AMS. 5. Discussion: implications for rock fabric quantification In spite of the approximations used in this work (dipole, no remanence, no surface/edge effect), the contribution of magnetic interaction between any two grains on AMS is well reproduced numerically at a macroscopic scale and is experimentally verified at least up to d/r ~ 0.8. We shall first list and quantify (Table 4) the effects of magnetic interaction between grains as a function of (i) the intrinsic shape anisotropy of the grains, (ii) their relative orientations in space, (iii) their spatial distribution with a particular focus on the intergrain distance distribution, and (iv) the grain size and grain susceptibility distributions. (i) Whereas Stephenson (1994) predicted an increase in χmean and P of 20% and 150% for an infinite chain of isotropic magnetic particles at contact, when applied to an infinite chain of prolate particles (a/b = 1.17), these values become equal to 38% and 187% for the “aligned” configuration and equal to 13.5% and 93% for the “side-byside” configuration. These values strongly decrease down to 8% and 40% for the two-grain experiments of Grégoire et al. (1995). Beside the already discussed reinforcing effect of the infinite geometry, these examples illustrate the fact that the AMS is a complex balance between distribution, orientation and shape anisotropies. However, we can easily state that effects of magnetic interactions are more important when the shape anisotropy and the shape-preferred orientation of the magnetic carriers are weak. (ii) When magnetic interaction becomes important, the relative orientations of the interacting grains are crucial in terms of resulting anisotropy directions, especially if these grains do present a shape

anisotropy. Depending on the grain orientations with respect to the alignment, the direction of maximal anisotropy can be enhanced (“aligned” configuration), inverted (“side-by-side” configuration) or simply “deviated” as shown by our experiments #1 and #2. The non-coaxiality between grains will reduce the effect of magnetic interaction in AMS. This is also observed when comparing the AMS of two grains in contact while being coaxial (observed increases in Kmean and P respectively, at ~+8% and ~+40% in the “aligned” experiments of Grégoire et al., 1995), with the AMS of two grains also in contact but with a different orientation (the observed increases in Kmean and P do not exceed +4% and +6%, respectively, in our experiment #1 which involved non-coaxial grains with anisotropies P = 1.3 and P = 1.1). (iii) Another important factor controlling the intensity of magnetic interaction is the spatial distribution of grains. In this aspect, and because the dipolar field responsible for the enhanced field decreases as 1/r3, the critical parameter to question the presence and effects of magnetic interaction is the distribution of grain-separation distances rather than the “anisotropy of distribution” used by Hargraves et al. (1991). Indeed, as shown in the analytical models, if the grains are close enough to magnetically interact, a uniform spatial distribution of grains can produce a dipolar component that overcomes the classical magnetic fabric due to grain shape preferred orientation. Conversely, an anisotropic distribution of dispersed grains may have no effect (or only a minor effect) on the resulting AMS tensor. Intensity of interaction rapidly decreases from grain-contact (d/r = 1) to a separation-distance roughly equal to one quarter of the mean grain size (d/r = 0.8) and becomes insignificant at a separation distance larger than the mean grain size (d/r = 0.5). (iv) Quantifying the effects of grain-separation distance distribution on the intensity of magnetic interaction becomes complex if the grain size and/or bulk susceptibility distributions are heterogeneous. As attested by our experiment #2, heterogeneity in grain size reduces the effects of magnetic interactions, at least in the 0.5 ≤ d/r ≤ 0.8 interval (compare Figs. 5 and 6). Indeed, even if the enhanced field produced by a large grain may be important, it will have a minor effect on the total AMS contribution if this grain magnetically interacts with a small grain of same permeability (or with a large grain of low permeability). Similarly, the

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enhanced field produced by a small grain is restricted to a small distance range, limiting the effect of its dipolar contribution to a domain of influence not larger than one quarter of its size. In natural igneous rocks, the shape, susceptibility, orientation, and separating distances of grains depend on their crystallization and on their deformation histories. Ferromagnetic grains may be small, equant and arrange into clusters or large, unequant, and dispersed. The recently introduced concept of generalized total AMS tensor, T, (Cañon-Tapia, 2001) helps stress the potential implication of magnetic interactions in AMS study of rocks. Following Cañon-Tapia (2001), the generalized AMS tensor is defined as T ¼ xki I þ ð1  xÞkn N

ð8Þ

where x is the fraction of interacting grains, ki and kn are, respectively, the intrinsic susceptibility (magnitude term) of the interacting (i) and the non-interacting (n) fractions, and I and N the orientation tensors (orientation term) of the two latter fractions. Analyzed with care, this equation may cover any natural situation. As examples, we will discuss real cases well-documented by Grégoire et al. (1998) and Hargraves et al. (1991). In this theoretical framework, and in the light of the previous findings, these two extreme cases provide quantitative illustrations of the implications of magnetic interaction on rock fabric analysis. 5.1. AMS of a syenite from Madagascar Following Archanjo et al. (1995), Grégoire et al. (1998) have combined an AMS study and a numerical image analysis study to establish the AMS significance of a syenite from Madagascar. Selected samples were characterized by a high concentration (3%) of ~ 0.5 mm long magnetite grains, forming observable clusters in thin section. Under the general AMS tensor formulation, the magnetic interaction contribution of T encompasses strong (d/r ≥ 0.8) and weak (0.5 b d/r b 0.8) interactions between grains. T thus, can be rewritten as: T ¼ xw kiw Iw þ xs kis Is þ ð1  xÞkn N

ð9Þ

with xw + xs = x, and where w and s subscripts stand for weak and strong interactions. Based on (i) the previous findings on the role of the separating distance between grains and the role of the relative grain orientations on the effects of magnetic interaction on AMS, and (ii) the characterization (grain shape and spatial distribution) of their samples by image analysis (Grégoire et al., 1998), the following approx-

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imations can be derived. Whereas strong magnetic interactions enhance kis (kis N kn), weak interactions produce no significant change in intrinsic susceptibility of the grains (kiw ~ kn). The image analysis of mutually perpendicular sections (6 × 3) shows that about 25% of the grains magnetically interact (x ~ 0.25). Among these 25%, only 5% interact strongly (xs ~ 0.05 and xw ~ 0.2). In addition, due to scattering in the alignment directions of strongly interacting clusters detected using image analysis techniques, the resulting orientation tensor, Is, is poorly defined (Is b N). In contrast, due to the cooperative effects of “aligned” clusters enhancing the shape fabric, the angular separation between Iw and N is less than 15° (Iw ~ N; Grégoire et al., 1998). Summarizing these observations as kis Nkiw fkn

ð10Þ

xf0:25 with ð1  xÞNxw f0:2Nxs f0:05

ð11Þ

Is bIw fN :

ð12Þ

T can be rewritten as T f0:2kn N þ 0:05kis Is þ 0:75kn N f0:95kn N :

ð13Þ

This demonstrates that during emplacement of this syenitic magma, shape anisotropy of magnetite grains is the main carrier of the AMS despite the occurrence of magnetic interactions affecting a fraction of the grains. 5.2. AMS in the experiments of Hargraves et al. (1991) Using several tablets from a magnetically isotropic dike of dolerite, Hargraves et al. (1991) have investigated the effects of magnetic interaction in igneous rocks. In their experiments, changes in the AMS tensor (magnetically isotropic tablets: knN ~ 0) due to interactions take place only by reducing the distance between the tablets. This procedure restricts the potential magnetic interactions to grains that are located at the surface of the tablets. Statistically, the weakly or strongly interacting grains are more or less aligned along the z-direction, perpendicular to the tablet surface. Accordingly, and conformably to the previous experimental and numerical results, the resulting net maximum anisotropy is parallel to z (Iw ~ Is//z). Hence, independently of the fraction of weakly (xw) and strongly (xs) interacting grains and of their respective

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susceptibilities (kiw and kis), T can be approximated by:

contribution in samples where magnetic interaction is suspected.

T fxw kiw Iw þ xs kis Is fxki I with I==z  direction

Acknowledgments ð14Þ

In such a case, where the isolated tablets are magnetically isotropic, implying that both the shape anisotropy and the shape preferred orientation of the grains are random, the total AMS tensor is, indeed, dominated by magnetic interactions between facing grains. The anisotropy degree, P, of the AMS reaches ~ +120% (Hargraves et al., 1991). Using non-perfectly isotropic synthetic samples made with molds or capillary tubes blended by magnetite and epoxy (Hargraves et al., 1991), the effect of magnetic interaction is reduced (P ~ + 50%) since it is now balanced by the non-null (1 − x)knN term. In summary, when ferromagnetic grains are small, equant and arrange into clusters, a situation which corresponds to the experiments of Hargraves et al. (1991), the AMS is mainly due to distribution even if some isolated particles may contribute differently. In contrast, the AMS of large, unequant and dispersed ferromagnetic grains, similar to the cases discussed by Grégoire et al. (1998) and Archanjo et al. (1995) are shape-related even if magnetic interaction is locally present. Providing quantitative guidelines to estimate the effects of magnetic interaction in AMS, this work naturally leads to the following practical question: “Can we isolate the contribution of magnetic interactions from an AMS study?” The answer is negative, since AMS is an integrative measurement that reduces the information on the intrinsic magnetic anisotropies, orientations and spatial distributions of the n magnetic carriers of a given sample (9n parameters) to an ellipsoid described by only 6 parameters. Inversion from the orientation and magnitude of the AMS ellipsoid is not possible, being too poorly constrained. However, magnetic interactions can be estimated using image analysis techniques of 2D sections on which the magnetic fractions would be previously isolated. Grégoire et al. (1998) used wavelets as adaptative anisotropic directional filters in order to detect clusters of grains of predefined size (Gaillot et al., 1997; Darrozes et al., 1997). Other methods used in seismology to investigate the distribution of inter-object distances such as single-link cluster analysis (Frohlich and Davis, 1990) and pair analysis (Eneva and Pavlis, 1988) can also be used. Combined with the quantitative framework proposed in this paper, image analysis results can allow one to characterize, at least semiquantitatively, the intensity and nature of the dipolar

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