Magnetic fabrics and petrofabrics: their orientation distributions and anisotropies
Introduction
The distribution of orientations in three-dimensions interests several fields of earth science, especially structural geology, petrofabrics and paleomagnetism. In particular, the characterization of a mean orientation and the dispersion about it require some attention, even where the data form a unimodal cluster with circular symmetry. Two clear lines of analysis follow. First, one may deal with axes (non-directed lines) such as fold axes, mineral lineations, petrofabric alignments and magnetic susceptibility axes that may be represented on one hemisphere of a stereogram. Most orientation data in structural geology fall into this category, with the notable exceptions of younging directions of strata, facing directions of folds, vergence directions of tectonic movements and paleocurrent directions. These, like paleomagnetic directions and geomagnetic flux possess a polarity requiring both upper and lower hemispheres for their projection. Fisher et al. (1987) reviewed most aspects of this subject and describe the different statistical treatments required to determine the mean orientations, variances and perform hypothesis tests for sample-distributions, where an appropriate theoretical model may be assumed.
Before we proceed further, we must establish the concept of a homogeneous petrofabric. If the orientation distribution of the property that we consider is the same throughout a sample, it can be said to possess a homogeneous petrofabric. If the property was identically oriented throughout, then the petrofabric would have a saturation alignment and there would be no dispersion of the directions and the mean direction would be defined automatically. The latter situation may be approached where stress-induced nucleation of metamorphic minerals produces a very strong alignment. More generally, the petrofabric is represented by an imperfect alignment but is nevertheless homogeneous, at least at the sample scale. The general problem is, given a suite of such samples from different locations, how do we characterize the petrofabric variation? Whether the different locations are from a single site or outcrop, or from several sites spread over an area, any straightforward statistical treatments require a higher level of fabric homogeneity. The dispersion of directions from one outcrop to another must define an inter-site orientation distribution that is also homogenous. As a shorthand description of such directional variation, Flinn (1965) introduced the concept of a fabric ellipsoid, varying continuously in shape from oblate, for a dispersion of axes in a plane (e.g. schistosity, foliation=S) to prolate for a dispersion of axes about one direction (e.g. a linear fabric=L). Structural geologists use Flinn's L–S scheme in connection with dimensions of aligned (perhaps strained) objects so that these are sometimes referred to as flattened (S) and constricted (L) fabrics, with the gamut of possibilities between (Fig. 1).
The determination of the mean direction and calculation of dispersion generally involves the assumption of some mathematical model for the spherical frequency distribution (Fisher et al., 1987, p. 67 et seq.). Different treatments are required for vectors and axes and they must take account of anisotropic clusters, partial and full girdles to be of value in structural geology, petrofabrics or paleomagnetism. Fisher et al. (1987, p. 84 et seq.) recommend the Kent distribution for vectors and the Watson distribution for axes. Natural orientation distributions are imperfect representations of girdles or clusters, with ragged contours and sub-clusters. These may be artefacts of sampling and limited sample-size, in which case statistics calculated using one of the theoretical models may be appropriate: the low confidence of the resulting mean direction is a meaningful measure of uncertainty. Alternatively, it is common for two geological processes to be compounded so that two petrofabric orientation-distributions are inappropriately merged: the resulting statistics may then be meaningless. Unfortunately, it may not be simple to discriminate between these possibilities from geological data and it is usually impossible from the orientation distribution alone.
The initial and most objective characterization of any orientation distribution is to produce some sort of density plot, familiar to structural geologists as contoured stereograms. Visual inspection alerts one to the possibilities of inter-site heterogeneity, subarea-homogeneity and the validity of any statistics that are calculated. Subsequently, it may be possible to associate the separate, more homogeneous sub-populations with different subareas. In this simple scenario, a homogeneous petrofabric simply shows spatial variation on a regional scale. Alternatively, and more commonly than is sometimes realized, different sub-populations of the orientation distribution may be represented in all subareas, though perhaps not to the same degree. Thus, each part of the area shows a heterogenous petrofabric due to two tectonic events or a tectonic event incompletely overprinting a primary fabric. The separation of the subfabrics from a stereogram of the orientation distribution may then be unwise as it is difficult to separate the blend of directional variation due to differing location and that due to the competing petrofabric contributions.
In structural geology, orientation-distributions normally concern axes or undirected lines. Most natural patterns of interest disperse the directions in stretched clusters or along great circles with overall orthorhombic symmetry, for homogeneous petrofabric domains. Scheidegger (1965) described such ordered angular distributions by matrices whose three eigenvalues represent the intensity of the preferred orientation and whose eigenvectors characterize the orientations of peak, minimum and intermediate concentrations. Woodcock (1977) extended the concept of an orientation tensor to describe an anisotropic distribution of orientations that could range from point-cluster, through partial girdles to great circle girdles. He normalized the Eigenvalues so that their sum (EMAX+EINT+EMIN)=1: point clusters have EMAX>EINT≈EMIN and partial girdles have EMAX≈EINT>EMIN. His orientation tensor quantifies the subjective L–S scheme of Flinn (1965).
Further constraints and correspondingly more information are conveyed by structural elements that involve line-plane pairs, e.g. slickensides in fault planes, fold axes within axial planes, mineral lineations within foliations. Each linear and planar element of the pair cannot be treated separately statistically as they are integral features. If that were done, the angular relationship inherent in each line-plane combination would not be preserved in the angular relationships of their mean directions. A special treatment of such orthogonal structural orientation data avoids these problems (Lisle, 1989) but forms part of the more general case of tensor statistics discussed below (Jelinek, 1978).
Most rocks are anisotropic so that at each point in the material their physical properties such as seismic velocity, electrical conductivity, thermal conductivity or magnetic susceptibility vary, depending on the direction in which they are measured. Second-rank tensors describe the variation at a point by three magnitudes each associated with one of three, mutually orthogonal axes (Nye, 1957, p. 7). Thus, the physical property may usually be represented by a magnitude ellipsoid whose maximum, intermediate and minimum axes correspond to the principal values. To structural geologists, the finite strain ellipsoid is a familiar example (Ramsay, 1967). Here, however, we discuss magnetic susceptibility ellipsoids that describe the variation in magnetic susceptibility caused by low-field induced magnetization (AMS; Hrouda, 1982); or by the ellipsoid for anisotropy of anhysteretic remanent magnetization (AARM; Jackson, 1991).
Reviewed at length elsewhere, AMS is readily measured with great precision for any rock and invariably has a direct correspondence with the orientation distribution of crystals in the rock (Fuller, 1963, Uyeda et al., 1963, Hrouda, 1982, Stephenson et al., 1986, Borradaile, 1987, Rochette et al., 1992, Borradaile and Henry, 1997). A related magnetic anisotropy that isolates the orientation distribution of remanence-bearing grains by their AARM was reviewed by Jackson (1991). AARM isolates the magnetic fabric contribution of remanence-bearing grains that usually occur as oxides or sulphides of iron or manganese. Although these occur as low-abundance accessory minerals, this subfabric may represent a different portion of the crystallization or strain history from the rock-forming minerals that contribute only to AMS. AMS combines fabric contributions from all minerals and it is controlled by their preferred crystallographic orientation-distributions, with the exception of magnetite whose grain-shape dictates its AMS contribution. The eccentricity of the fabric ellipsoid is most effectively described by Jelinek's (1981) Pj parameter (sphere=1; ranging upwards with eccentricity) and its shape by Tj (+1=oblate, −1=prolate). An important feature of magnetic fabrics is the enormous variation in average intensity of the property. Averaging maximum, intermediate and minimum susceptibilities for a sample yields its bulk susceptibility, which, for most rocks, ranges from ∼50×10−6 to ∼10,000×10−6 SI. However, pure quartzites or limestones respond diamagnetically with a feeble bulk susceptibility ∼14×10−6, whereas some iron formations may approach the bulk susceptibility of magnetite (≤3,000,000×10−6).
In a perfectly uniform substance, the AMS ellipsoids from different samples would be parallel and have principal axes of matching dimensions. For such perfectly congruent ellipsoids, any single sample ellipsoid characterizes the distribution: it represents the mean. In reality, the AMS ellipsoids vary in shape and orientation from sample to sample. Consequently, the mean AMS tensor's shape cannot be calculated as a scalar mean from the axial lengths of AMS ellipsoids. The shape of the mean tensor underestimates the anisotropy degree (Pj) and yields a shape parameter |Tj| that is too close to zero (consider Fig. 1) for the individual samples because of their scattered orientations. Furthermore, in the case of magnetic petrofabrics, homogeneity is influenced not just by grain orientation-distribution but also by magnetic mineralogy because different minerals have different bulk susceptibilities, magnetic anisotropies and their proportions vary between samples.
Characterizing the mean orientations of principal directions (kMAX, kINT, kMIN) from numerous samples is more complicated and requires the tensor-statistical approach of Jelinek (1978). Consider the non-systematic scatter of directions produced by the variability of natural processes. For a single, homogeneous petrofabric this is the spherical equivalent of the linear Gaussian distribution of errors of observations. The mean anisotropy has orthorhombic symmetry with principal directions defined as maximum, intermediate and minimum susceptibilities (e.g. Fig. 2a–c). For a well-developed, simple, homogeneous petrofabric this much could be gleaned from traditional density plots (orientation-frequency distributions. e.g. Fig. 4, Fig. 7). The tensor-statistical calculation guarantees the constraint that the means of the principal directions are mutually orthogonal (Jelinek, 1978, pp. 52–53). Incorrectly treating AMS fabric directions as independent lines would yield false, non-orthogonal means for the three principal directions of the mean tensor. Whereas the inaccuracies of that unjustifiable procedure may be barely noticeable in the case of a saturation-alignment petrofabric due to stress-controlled metamorphic crystallization (Fig. 4d), it is normally grossly unacceptable. Thus, tensor-statistics are essential for the characterization of fabric orientation-distributions.
A further consideration in the application of tensor-statistics to magnetic fabrics arises from the enormous variation in bulk susceptibility of rocks and minerals. The orientation of the mean tensor can be distorted by the occurrence of high susceptibility minerals, an effect that can be minimized by normalizing sample-intensities (Fig. 3a and b) as recommended by Jelinek (1978, p. 53), either in a separate subfabric or in a congruent fabric that is less abundant and poorly represented (Fig. 3). Normalizing the samples’ according to their bulk susceptibility overcomes this. Examples below show that normalization is generally preferable but may lose some useful information by suppressing the identification of subfabrics that may be significant (e.g. Fig. 3c–e).
Another advantage of tensor-statistics is that they can reveal the mutual interaction of the directional uncertainty of the mean principal directions (Jelinek, 1978, Lienert, 1991). For example, the confidence cone about the mean maximum principal direction is constrained by the confidence cones about the other two mean principal directions (e.g. Fig. 2a–c). Thus, visual inspection of the elongation of the elliptical confidence cones reveals the shape of the mean tensor (e.g. S>L; cf. Tj>0 for an individual AMS ellipsoid). In this paper, the confidence limits presented confine 95% of possible sample means. The reliability of confidence cones is restricted usually to cases where the apical angle of the cone is ≤25° due to approximations in their calculation (Jelinek, 1978, p. 59). For an adequately sampled, petrofabrically-homogeneous site, the AMS-axes’ confidence cones should retain orthorhombic symmetry on the stereoplot (Fig. 2a–c). This is because those constraints permit zero covariance for possible orientations of the principal axes. Subsequent case studies that demonstrate clear or reasonable orthorhombic symmetry are given in Fig. 4, Fig. 5, Fig. 6, Fig. 9, Fig. 10, Fig. 11, Fig. 12. However, for a biased sample of a homogeneous petrofabric, or a sample-distribution comprising multiple subfabrics with different orientation distributions, it is a different matter. Where the possible orientations of all three principal axes covary, the confidence cones may have sub-orthorhombic symmetry.
It will be shown that the orthorhombic symmetry of confidence-cones may be restored or worsened by normalization of the sample-tensors magnitudes, depending on considerations of petrofabric and magnetic mineralogy. There appear to be two clear geological reasons for confidence limits to have sub-orthorhombic symmetry (e.g. Fig. 2d). First, there may be conflicting subfabrics, e.g. differing degrees of tectonic overprint on a primary fabric or asynchronous, noncoaxial tectonic subfabrics. In other words, the underlying orientation-distribution fails to achieve orthorhombic symmetry. This may even occur in a homogeneous, single-event petrofabric where the sample size is insufficient to faithfully represent orthorhombicity. Second, individual observations may derive from samples with different proportions of high-susceptibility accessory minerals. Accessory minerals may not be abundant enough to define a stable subfabric but their high bulk susceptibility may swamp the AMS contribution of the matrix. Thus, the AMS orientation of the matrix will be skewed towards that of a different or poorly represented accessory-subfabric. Of course, the latter situation is remedied where each sample's tensor is normalized by its bulk susceptibility value although that procedure may also suppress some useful information (Jelinek, 1978, p. 53), as discussed below.
Strictly speaking, markedly non-orthorhombic confidence limits indicate that the sample-distribution is unfavorable for treatment by Jelinek's statistical approach. On the one hand, we may be treating an ideal orthorhombic, unimodal population-distribution that has been inadequately sampled, a common and insurmountable problem in field geology. On the other hand, the underlying population-distribution may be multimodal. Nevertheless, whichever the case, Jelinek's statistics are useful as they alert one to either of the two possibilities. In particular, bimodal distributions are commonly subtle but sensitive to detection by Jelinek's asymmetric confidence cones. Moreover, stereographic inspection of orientation distributions usually provides the first opportunity to detect multiple petrofabric events. Therefore, cautious use of Jelinek's tensor statistics is justified even if the sample-distribution does not rigorously satisfy their precise requirements.
The approach here is to examine seven case studies of magnetic fabrics that summarize the orientation distribution of minerals metamorphic, igneous and sedimentary rocks. Anisotropy of low field susceptibility (AMS) is used in all cases, but the orientation-distribution of magnetite was isolated by anisotropy of anhysteretic remanence (AARM) in three case studies. Each study indicates some strength or pitfall of interpretation of the orientation distribution of a sample of tensors, and each is peculiar to the case in hand. Provided we inspect and respect the idiosyncrasies of each sample, the symmetry and normalization procedures for the sample may reveal useful new information concerning fabric heterogeneity and multiple fabrics as well as the traditional information about the shape and orientation of the mean fabric ellipsoid.
Section snippets
Borrowdale Volcanic Group Slates, English Lake District (Fig. 4: n=74)
This sample of 74 cores from a study by Borradaile and Mothersill (1984) are plotted in a fabric orientation-coordinate system with the prominent mineral lineation of chlorite and stretched volcanic lapilli arbitrarily oriented N–S, and the slaty cleavage horizontal. (In all other case studies, the samples are in geographic coordinates.) The AMS fabric is oblate with a well-defined kMAX lineation and a moderately well-defined oblate symmetry, as shown by the similarly developed elongated
Case studies involving two fabrics determined from AMS and AARM
The preceding examples revealed subtleties in the characterization of a sample of imperfectly aligned AMS ellipsoids. AMS blends the contributions of induced magnetization from all minerals; diamagnetic, paramagnetic and remanence-bearing. Consequently, each individually measured AMS tensor merges several different AMS orientation-distributions: one for each mineral present within the sample! For medium-grained and finer samples there are normally enough grains of each mineral to provide a
Conclusions
Traditionally, petrofabrics, structural directions and magnetic fabrics were evaluated by visual inspection of contoured stereogram density plots. Since magnetic fabrics are represented by second-rank tensors, contours of all three axes of each sample tensor are required to fully describe the orientation distribution. It is wiser always to plot all three axes (e.g. Lienert, 1991) to fully appreciate the orientation-distribution. The clustering or girdling of each of the maximum, intermediate
Acknowledgements
This work was supported by operating and equipment grants to Graham Borradaile from NSERC (Ottawa, Canada). France Lagroix provided excellent research assistance while Sam Spivak and Anne Hammond provided invaluable technical support. I particularly thank Mike Jackson, as well as Frantisek Hrouda, Nigel Woodcock and Karel Schumann for correspondence and constructive reviews. The data used in this paper has been collected over the last decade using Sapphire Instruments equipment for AMS (SI2B
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2020, Computers and GeosciencesCitation Excerpt :This may occasionally lead to a decoupling between indexes derived from orientation tensors and indexes derived from fabric tensors (here K > 1 and T = 0). The sub-fabric effect is attenuated by normalising each individual ellipsoid by the root of the sum of the squared axes length to obtain “unit ellipsoids” (see details in Borradaile, 2001). In such a case, mean principal directions and fabric statistics are ruled by the most populated set of ellipsoids rather than by the largest and most anisotropic ellipsoids (Fig. 7b), which mean principal directions localised at individual axes clustering points, and confidence angles and fabric statistics approaching values expected from the set of 50 small ellipsoids (K = 2.1; LS = 0.50; T = 0.46; P′ = 1.5).