Pure and Applied Geophysics. GRAHAM J. BORRADAILE, 1 MIKE STUPAVSKY, 2 and DAWN-ANN METSARANTA 1

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1 Pure appl. geophys. (2008) Ó Birkhäuser Verlag, Basel, 2008 DOI /s Pure and Applied Geophysics Induced Magnetization of Magnetite-titanomagnetite in Alternating Fields Ranging from 400 A/m to 80,000 A/m; Low-field Susceptibility ( A/m) and Beyond GRAHAM J. BORRADAILE, 1 MIKE STUPAVSKY, 2 and DAWN-ANN METSARANTA 1 Abstract For remanence-bearing minerals (RBM) such as magnetite-titanomagnetite, susceptibility to induced magnetism (M) measured in alternating fields (H AC ) is field-dependent. However, for fields B 400 A/m, measured in an AC induction coil instrument (at 19,100 Hz), susceptibility k 0 = M/H AC is sufficiently linear to provide a reproducible rock (or mineral) magnetic characteristic and its anisotropy may be related to arrangements of minerals in rock, or for single mineral grains to their crystalline or shape anisotropy. For any remanence-bearing mineral at higher fields k HF (¼ M/H AC ) is not constant and the term susceptibility is not normally used. This study bridges the responses between traditional low-field susceptibility measurements and those due to high applied fields, for example when studying hysteresis or saturation magnetization of RBM. Where k HF is measured in alternating fields that peak significantly above 400 A/m the M(H AC ) relation is forced to follow a hysteresis loop in which k HF >k 0 for small H AC and where k HF decreases to zero for very large fields that achieve saturation magnetization. Hysteresis nonlinearity is due to remanence acquired with one field direction requiring a reverse field for its cancellation. We investigate the transition from initial, traditional lowfield susceptibility (k 0 ) measurements at 60 A/m, through 24 different fields from 400 A/m to 40,000 A/m (for very high k 0 to 80,000 A/m). This reveals M(H AC ) dependence beyond from conventional k 0 through the range of hysteresis behavior in fields equal to and exceeding that required to achieve saturation magnetization (M S ). We show k HF increases with peak H AC until the peak field is slightly less than saturation magnetization in natural rock samples rich in magnetite (TM 0 ¼ Fe 3 O 4 ) and TM 60 (Fe 2.4 Ti 0.6 O 4 ). All sample suites predominantly contain multidomain grains with subordinate pseudo-single domain and single-domain grains. k/k 0 increases by B 5% for fields up to 2 ka/m. Above 4 ka/m k/ k 0 increases steeply and peaks, usually between 24 ka/m and 30 ka/m where all grains magnetic moments are activated by H AC since this exceeds the coercive force of most grains. For higher peak H AC, k/k 0 declines sharply as increased H AC values more effectively flip M with each field-direction switch, leading to the low gradient at distal portions of the hysteresis loop. For M 0 -TM 60 bearing rocks, susceptibility peaks for fields *12 ka/m and for magnetite rich rocks up to 24 ka/m. These values are approximately 10% of saturation magnetizations (M S ) reported for the pure minerals from hysteresis DC field measurements. Both the field at peak k/k 0 and the peak k/k 0 value appear to be controlled by the dominant domain structure; multidomain behavior has larger k/k 0 peaks at lower H AC. Stacked k/k 0 versus H AC curves for each sample suite (n = 12 to n = 39) were successfully characterized at the 95% level by a polynomial fit that requires the cubic form k/k 0 = a + bh + ch 2 + dh 3. Thus, for most M-TM bearing rocks, susceptibility and anisotropy of susceptibility (AMS) measurements made on different instruments would be sufficiently precise for most geological applications, if peak alternating fields are B700 A/m. 1 Department of Geology and Physics, Lakehead University, Thunder Bay, P7B 5E1 Ontario, Canada. borradaile@lakeheadu.ca Simmers Avenue, Kingsville, N0P 2G0 Ontario, Canada.

2 G. J. Borradaile et al. Pure appl. geophys., Key words: Field-dependent susceptibility, AC susceptibility, rock magnetism, magnetite, titanomagnetite, ophiolite, Troodos, magnetic susceptibility, magnetic fabric, AMS, hysteresis. 1. Susceptibility at Low Fields and Complications with Remanence-bearing Minerals at Higher Fields For low fields, H, magnetic susceptibility (k) is regarded as a constant relating induced magnetization (M) to(h), linearly as M = kh. This simple relationship is valid for certain silicate, carbonate and other rock-forming minerals that respond solely as diamagnets (e.g., pure quartz and pure calcite, k & SI). It is also true for the rock-forming minerals that respond as paramagnets (e.g., pure clay minerals, micas, amphiboles, pyroxenes, etc., 0 < k B 2000 lsi). A negative diamagnetic background contribution is present for all materials but is usually negligible in comparison to a paramagnetic response. However, pure rock-forming minerals are rare in nature. Their intrinsic lattice susceptibility is normally enhanced by inclusions or exsolutions of higher susceptibility remanence-bearing minerals (RBM), especially iron oxides. The presence of RBMs complicates interpretation since even as accessory grains (B1% by volume) or as inclusions, their high susceptibility (e.g., k > SI) may mask the contribution from rock-forming minerals that interest petrofabric studies and form the bulk of the rock. A low concentration of RBM (*0.01% by volume) introduces a nonlinear M(H), hysteretic response since the acquisition of permanent or remanent magnetism must inevitably saturate at a sufficiently high field. Nonlinearity is noticeable at >300 A/m for low coercivity, multidomainal RBM such as magnetite and pyrrhotite, and they may complicate or thwart studies of magnetic anisotropy (magnetic fabrics). Paradoxically, the magnetic memory of RBM provides the basis for paleomagnetism but it hampers the measurement of susceptibility, especially where small differences along different specimen axes must be determined for anisotropy studies. Any significant amount of magnetic memory inevitably causes nonlinearity of the M(H) relationship. Where RBM are multidomainal (MD) and isotropic, the scalar Rayleigh relationship may satisfy ferrous metals (STRUTT, 1886) and is generally extended to ferromagnetic minerals: M ¼ kh þ ah 2 : ð1þ This is only an approximation when applied to rocks with dispersed grains of the titanomagnetite-series (JACKSON et al., 1998) but satisfactory for fields similar in strength to the geomagnetic field (*80 A/m). There, the term ah 2 is usually insignificant in the context of magnetic anisotropy studies although depends on the bulk susceptibility and the spatial distribution of the RBM (BORRADAILE and JACKSON, 2004). If valid, the linear approximation M = kh has other benefits, apart from yielding reproducible susceptibility values for different field values. In particular, one may substitute M = [k].h, where M and H are vectors and [k] is now a second rank tensor. Thus measuring susceptibility in different directions permits the determination of anisotropy of low field susceptibility

3 Field-dependent Susceptibility (AMS) as the tensor [k] but only if the nonlinear terms are insignificant compared to the linear term (kh). In practice, modern measurements of k and AMS use AC induction coil instruments so that any RBM present are cycled along a nonlinear path. This forms a hysteresis loop, since there is a cause-and-effect lag due to remanence acquired in a field in one direction (+H) requiring a larger reverse field (-H) to overcome the material s coercive force (H C ) (Fig. 1a). When the field oscillates between within a small range ( H MAX B 400 A/m) the Rayleigh relationship describes a minor hysteresis loop, near the origin within the characteristic (largest) loop (Fig. 1a). This small, subdued form of the largest loop is described by (Dr. Mike Jackson, pers.comm, 2004): M ¼ðk 0 þ ah MAX ÞHþ a 2 H MAX H 2 : ð2þ k 0 (low field susceptibility) is the slope near the origin; the sign of the second term ðþþ distinguishes between increasing/decreasing H (Fig. 1a). A reproducible value for its slope, k 0, is only obtainable near the origin. Any exposure to higher fields prevents its remeasurement since the M(H) response then follows a hysteresis loop around the origin. The largest possible of these saturates the RBM producing the characteristic hysteresis loop shown in Figure 1a. This defines saturation magnetization (M S ) in the presence of a field (H), saturation remanence M RS (remaining when there is no applied field) and coercive force H C (the reverse field required to cancel M RS ) in the presence of a field. Another important hysteresis parameter realized from a supplementary experiment and not realized in this diagram, is the coercivity of remanence (H CR ) which is the reverse DC field required to erase M RS. These hysteresis characteristics discriminate very clearly between different RBM, as shown below for the main suite of studied specimens (Fig. 2b). 2. Application and Uses of Low-field Susceptibility and its Anisotropy (AMS) Uses of low-field magnetic susceptibility (k 0 ) and its anisotropy [k], known as AMS, are numerous (HROUDA, 1982, 2002; TARLING and HROUDA, 1993). Susceptibility to the geomagnetic field or to different applied fields during surveying is of considerable commercial value (CLARK, 1983; 1997; CLARK et al., 1988). In the laboratory, comparison of low-and high-field responses aids in discriminating the importance of paramagnetic and diamagnetic material versus RBM (HROUDA and JELINEK, 1990). This is more simply evaluated by comparing the anisotropies (AMS) and orientation-distributions of subsamples with different RBM concentrations, as indicated by their bulk susceptibility (k) (HROUDA, 2002; BORRADAILE and GAUTHIER, 2001, 2003). Independent measurement of the anisotropy of RBM, either using isothermal remanence or anhysteretic remanence is more challenging but where possible it is more successful in isolating susceptibility contributions of ordered phases (RBM) from those of rock-forming paramagnetic and diamagnetic minerals (JACKSON, 1991).

4 G. J. Borradaile et al. Pure appl. geophys., (a) (b) Figure 1 (a) Hysteresis relationship between applied DC field, cycling slowly between reverse (-H MAX ) and forward limits (+H MAX ). Where this achieves saturation of magnetization (plateau M S ), it is the loop that characterizes the remanent magnetization properties of the mineral, giving values for maximum permanent magnetization, the saturation remanence (M RS ) at zero field and the field required to annul it, the coercive force H C. In the broadest sense, susceptibility (k = M/H) may be measured at any field but it is only a reproducible, characteristic value when the material has never been magnetized (or demagnetized). Thus, the low-field susceptibility k 0 should only be measured from the linear part of the initial loop as shown in the box. We explore values of k measured in an AC field induction coil that rapidly cycles (400 Hz) over different field ranges until H MAX exceeds the coercive force value and the magnetization reaches or exceeds M S. (b) This causes the susceptibility ratio k/k 0 to peak near the fields that produce saturation magnetization and to decrease as the magnetization reaches a plateau where k = M/H approaches zero. The example is the mean response of a suite of 29 specimens from dominantly magnetite-bearing ultramafic rocks form the ophiolite mantle sequence in Troodos, Cyprus. It shows the envelope for the data given by the mean curve ± standard error.

5 Field-dependent Susceptibility (a) (b) Figure 2 Hysteresis properties and Curie Temperatures for suites of Troodos ophiolite specimens with decreasing degrees of ocean-floor metamorphism (diabase dikes, gabbro, mantle). (a) Ranges of Curie Temperatures found at different stratigraphic levels through the ophiolite. Magnetite dominates in deeper mantle rocks whereas contributions from titanomagnetite increase at progressively shallower levels where the influence of sea-floor metamorphism increases. (b) In this new plot, the use of three characteristic hysteresis parameters discriminates between the different stratigraphic levels and their magnetic-mineralogy. (c) The conventional Day Plot (DAY et al., 1977) less successfully discriminates between remanence-bearing minerals since it uses the ratios of characteristic hysteresis properties. However, it more clearly differentiates types of domain structures and also mixing trends for different grains-sizes and magnetite-titanomagnetite concentrations (DUNLOP, 2002a, b). Since the availability of personal computers and electronic advances, it has become very simple for nonspecialists to infer mineral orientation distributions, e.g., in studies of structural geology, petrofabrics, magma-flow, depositional alignment, and environmental geology (HROUDA, 1982; BORRADAILE and HENRY, 1997; BORRADAILE and JACKSON, 2004; TARLING and HROUDA, 1993; THOMPSON and OLDFIELD, 1986; EVANS and HELLER, 2003). Interpretation of k, more so for AMS, is rarely straightforward since they depend on the abundances, susceptibilities, orientations and spacing of different minerals in a specimen (BORRADAILE and JACKSON, 2004). These advances and applications by geologists rely on the convenience of susceptibility measurement in a computer-controlled induction-coil instrument using an alternating field (H AC ). However, this introduces instrumental variability that affects measurements. 3. Susceptibility Measurement Conditions, Especially Field-dependence JACKSON et al. (1998) wisely remind us of the importance of measurement conditions. For multidomainal RBM, k may be dependent on the measurement field. Since most instruments use AC fields, k may also show some frequency-dependence (DEARING et al.,

6 G. J. Borradaile et al. Pure appl. geophys., 1996; JACKSON and WORM, 2001) especially for minerals such as pyrrhotite (BORRADAILE et al., 1992). The consideration of measurement field and frequency may become increasingly important as the concentration of RBM increases. Observations of k or AMS are truly unique and universal only where RBM are absent or of low abundance and dispersed. Furthermore, exposure to high fields changes k 0 and AMS noticeably for rocks rich in certain multidomainal accessory RBM (B1% by volume) (POTTER and STEPHENSON, 1990), especially magnetite and pyrrhotite. For reproducible determination of low-field susceptibility in specimens rich in multidomain, low coercivity RBM, it may be advisable to demagnetize carefully first, by a nondestructive but isotropic technique. Low-temperature demagnetization is preferable since thermal demagnetization may cause mineralogical change and static three-axis AF demagnetization may reset k 0 and AMS (POTTER and STEPHENSON, 1990). Tumbling AF demagnetization can never expose every axis through the specimen to every demagnetizing field value and thus risks imposing some anisotropy. In geological applications, tactics must be tailored to suit either paleomagnetic or petrofabric goals and it may not be possible to satisfy both with the same specimen. In Figure1a low-field susceptibility is explained in the context of hysteresis, the fundamental loop-relationship of magnetization and field strength for RBM. It is clear that the only reproducible and linear relation, yielding k, may be obtained where a never-magnetized specimen (M ¼ 0) is exposed to a sufficiently low field such that the linear response M ¼ kh is observed. Convenient and routine susceptibility and AMS studies require alternating fields so that H must vary in sign and the response forms the hysteresis loop. Only when H MAX is small does the loop sufficiently approximate a straight line to yield reproducible k-values. In rock magnetic experiments that characterize RBM and their behavior, the field must be alternated from large positive to large negative values in order to achieve saturation magnetization (M S ) in the presence of the applied field and reveal the full-sized hysteresis loop (Fig. 1a). Of course, this leaves the specimen in a permanently magnetized state with a saturation remanent magnetization (M RS ). Field sensitivity of magnetite-titanomagnetite and pyrrhotite has been addressed by HROUDA et al. (2005), DE WALL (2000), DE WALL and WORM (1993), and WORM (1991). DE WALL measured k at 30 A/m and 300 A/m whereas Jackson et al. measured in the range 0.1 to 2000 A/m. We measured k over the range 400 to 40,000 A/m at 400 Hz and to 80,000 A/m at 800 Hz. Frequency dependence of k introduces issues of complex susceptibility, i.e., resistivity versus true susceptibility, and is discussed elsewhere especially with reference to sulphides minerals (BORRADAILE et al., 1992; JACKSON and WORM, 2001; MULLINS and TITE, 1973; WORM et al., 1993). However, in practical applications, for frequencies <20 khz the out-of-phase resistivity component is sufficiently negligible in comparison with the in-phase susceptibility component and higher frequencies permit greater precision in measuring k. Sapphire instruments susceptibility meters use the operating parameters shown in Table 1. Reconnaissance experiments used a Sapphire Instruments unit with fields up to H AC = 1200 A/m operating at 19.2 khz. JACKSON et al., used a similar instrument but

7 Field-dependent Susceptibility Table 1 Susceptibility instrument specifications (Sapphire Instruments sapphiremagnetics.com) Instrument Coil Frequency (Hz) Peak Field A/m Sensitivity lsi SI k C 10 SI2B 19, k C 3 Lakeshore 7130 JACKSON et al. (1998) 10 10, k*3 SI2 HF1 This paper k C 50 SI2 HF2 This paper ,000 k C 1000 SI2 HF3 This paper ,000 k C 10,000 mostly another (Table 1) with H AC ranging up to 2 ka/m. For pure, synthesized minerals, they found modest susceptibility increases with increasing peak H AC for magnetite (k/k 0 B 1.1) and substantial increases for TM 60 (k/k 0 B 2.0) however these did not peak or plateau. Our new instruments extend the peak field, two prototype instruments. SI2HF1 has a satisfactory sensitivity over 400 A/m B H AC B 40,000 A/m, operating at 400 Hz. For very high susceptibility specimens (k 0 > 0.01 SI) SI2HF2 extends the fieldrange to H AC = 80 ka/m, operating at 800 Hz. These H AC maxima exceed the coercive forces (H C ) and the fields required to achieve saturation magnetization (M S ) for magnetite and titanomagnetite (Table 1). Previous AF demagnetization studies in these rocks show that H C is equivalent to that reported for MD and PSD magnetite (5 15 mt or approximately 4 12 ka/m), (BORRADAILE and GAUTHIER, 2006; BORRADAILE and LAGROIX, 2001; BORRADAILE and LUCAS, 2001; BORRADAILE and MIDDLETON, 2006; MIDDLETON et al., 2004). Those paleomagnetic-rock magnetic studies use a minimum of 15 steps to evaluate coercivity spectra which are insufficiently characteristic or diagnostic to allow a meaningful prediction of response to induced magnetization in high fields. Our magnetite and titanomagnetite bearing rocks show k peaking between H AC * 28 ka/m (TM 60 and H AC * 32 ka/m (TM 0 ). These are below the fields required to achieve saturation magnetization. For a rock, extrinsic magnetic properties such as M s (Table 1) depend on RBM-concentration and are underestimated from rock measurements. Intrinsic properties such as Curie temperatures and coercive force (H c ) are not concentration-dependent and thus are diagnostic of the dominant RBM(s). For the suite of TM 0 -TM 60 rocks from Cyprus, these properties are compared (Fig. 2). As an initial control group, we used 16 fresh diabase specimens with obvious MD- PSD magnetite defined by Curie temperature, Verwey transition temperature and hysteresis characteristics (e.g., BORRADAILE et al., 2004; MIDDLETON et al., 2004); the study then focuses on the effects of varied magnetic mineralogy of Cyprus ophiolite specimens (Figs. 2, and 3). The ophiolite stratigraphy shows progressive changes in relative concentrations of TM 0 -TM 60 and their characteristic magnetic properties (Curie temperatures, H C, H CR, and M RS /M S, Fig. 2) as well as corresponding changes in susceptibility (Fig. 3). The classic sequence preserves ophiolite stratigraphy from lavas and underlying shallow dikes, dominated by TM 60, down through lower dikes that are usually of lower susceptibility. These in turn are underlain by gabbro of relatively low

8 G. J. Borradaile et al. Pure appl. geophys., Figure 3 Stratigraphic profile of the Troodos ophiolite beneath the lava sequence, provided by the CY4 borehole. Variation of field-dependent susceptibility measurements, derived from the graphs of Figure 3, shown with depth (three-point moving average). susceptibility but dominated by magnetite; they escaped the ravages of hydrothermal seafloor metamorphism. At the base of the ophiolite, mantle-sequence ultramafic rocks are also dominated by magnetite, although in some cases this is secondary and fine-grained due to serpentinization. The application involves two sampling strategies; one campaign took 49 samples from a 2,250 m vertical borehole known as CY4 (GIBSON et al., 1989) which provides a continuous section from upper dikes to mantle. The second sampling campaign used surface exposures of the same units; Limmasol Forest dikes (n = 18); East Troodos Basal Group Dikes (n = 35); and Mount Troodos serpentinized Mantle rocks (n = 30). 4. Instrumentation The various Sapphire Instruments magnetic susceptibility meters used in the study determine the magnetic susceptibility of a specimen from the measurement of the coil inductance with the specimen inside the coil and with the specimen removed from the coil. The magnetic susceptibility measuring coil is part of a coil and capacitor resonant circuit that determines the oscillator frequency of a square wave oscillator. Coil inductance is determined by measuring the oscillator frequency. The low power S12, SI2B and SI2HF1 meters use a parallel coil and capacitor resonant circuit. The SI2HFl coil magnetic field is varied between 0.1 to 150 A/m by varying the resistance placed in

9 Field-dependent Susceptibility series with the coil and capacitor resonant circuit. Coil fields from 150 to 1200 A/m are achieved by varying the power supply voltage to the square wave oscillator from ±5 to ±40 volts. The higher-power SI2HF2 and SI2HF3 meters use a series coil and capacitor resonant circuit. The magnetic susceptibility measuring coil is the same coil used in the SI-4 AF demagnetizer used to AF treat rock specimens for paleomagnetic studies. The coil is 10 cm long, 4 cm inner diameter and has 30 layers of 20 gauge copper magnet wire with a coil resistance of 23 ohms. The coil has a single layer secondary winding of 26 gauge magnet wire. The coil is in series with a 0.5 microfarad capacitor array having a 5,000 volt AC rating. The power square wave oscillator is made from four high voltage power field effect transistors (FET) in a bridge configuration. The series coil and capacitor circuit is driven from both ends. The signal from the secondary winding is phase-shifted, amplified and is used to drive the optically isolated FET gates in the required sequence to produce resonant oscillation. The coil magnetic field is varied by varying the power supply voltage to the oscillator from 0.6 to 155 volts. Oscillation starts at 0.25 volts but the coil field is non-linear until the power supply voltage reaches 0.6 volts that produces a 500 A/m coil field. A 12-bit digital to analog converter is used to set the power supply voltage to produce the desired coil magnetic field. The coil fields were calibrated with a Hall Effect probe that is linear to 80,000 A/m. The maximum practical coil field is limited to 80,000 A/m by coil heating. This field requires 2.87 amperes current through the coil generating 190 watts of heat. 5. Measurement Procedure The magnetic susceptibility measurement control program was set up to measure 24 magnetic susceptibility for 24 different peak H AC values ranging from 500 A/m to 80,000 A/m. The specimens were translated into and out of the coil on a one meter long plastic track that isolates the operator from the coil magnetic field. The empty coil air1 (A1) coil inductance is first measured with the specimen 20 cm outside the coil. Specimen (S) coil inductance is measured with the specimen in the center of the coil. Then, a second (air2 = A2) coil inductance is measured with the specimen 20 cm outside the coil. The A2 measurement is used by the program to correct the S measurement for the significant coil inductance drift with temperature during the 20 seconds required for the A1-S-A2 measurement cycle. About 6 minutes are required to measure the 24 susceptibility values at different fields. After about one hour of measurement, the rate of coil heating of about 3C per 6 minutes is balanced by the forced air cooling rate maintained by a fan mounted behind the coil. The surface coil temperature is then about 40 C. Coil inductance exhibits two distinct types of variation with temperature. One type is smooth and continuous variation. The measuring program can correct the measured susceptibility for this type of coil inductance variation with temperature. The second type is erratic and discontinuous with temperature; the measuring program cannot correct the measured susceptibility for these discrete

10 G. J. Borradaile et al. Pure appl. geophys., (a) (b) (c) (d) Figure 4 Field-dependence of susceptibility ratio (k/k 0 ) measured in an SI2 induction coil at 400 Hz for fields ranging from 400 A/m to 40,000 A/m. k is measured in the alternating field for some field range that peaks at H and k 0 is the initial or low-field susceptibility. (a-d)curves for individual specimens with mean and standard-error limit curves for each sample (a-d). All are natural rock specimens with a range of grains sizes (domain-structures) for the remanence-bearing minerals. (a) Fresh diabase dominated by multidomain (MD) magnetite. (b) Ophiolite mantle rocks with MD and pseudo-single domain magnetite. (c) Ophiolite dikes at lower level, near transform fault, MD-PSD magnetite and TM 60. (d) Ophiolite dikes at higher level mainly with TM 60. changes of coil inductance with temperature. This type of drift is evident for specimens having k < 5,000 lsi that show slight erratic changes in k between H steps (Fig. 4). Signal stacking was used on some specimens to improve the measurement precision (Fig. 4).

11 Field-dependent Susceptibility 6. Results 6.1. Control Suite of Diabase with Fresh Multidomain Magnetite A control group of sixteen fresh diabase specimens, known from our earlier studies, provided information from specimens in which MD magnetite is the dominant RBM. Unfortunately, although their concentrations are low, grains of pseudo-single domain (PSD) and perhaps even single domain (SD) magnetite are present (BORRADAILE et al., 2004; MIDDLETON et al., 2004). Whereas they contribute little to the susceptibility, they do skew the coercivity spectrum to values higher than that of MD magnetite and complicate the interpretation. The mean low-field susceptibility of these specimens is 60,000 lsi with a standard deviation of ± 11,000. The susceptibility ratio is k/k 0 where k 0 is the low field value and k is measured at some higher field. It is relatively invariant in fields up to *800 A/m, increasing by <2% (Fig. 4a). Thus, for most rocks, susceptibility and anisotropy of susceptibility measurements made on different instruments with different fields, B700 A/m would be sufficiently consistent for most geological applications. However, beyond 1000 A/m, k/k 0 increases rapidly, by 50% at the average peaking H AC of 29.6 (±2.38) ka/m [± confidence limits are standard errors about the mean value]. Increasing peak values of the alternating field (H AC ) cause progressively more grains of higher coercive force (H C ) to be magnetized, increasing k/k 0. The magnitude of coercive force is related to intrinsic crystallographic properties, increasing with dislocation density and domain-wall immobility. Measurements with peak H AC above those corresponding to that which causes the maximum k/k 0 cause progressively more grains to be saturated until k tends to zero and k/k 0 is independent of peak field. This corresponds to the extremities of the hysteresis loop (Fig. 1a). Excluding any obvious outlying curves, the mean curve has been determined together with its standard error limit curves for each sampled rock suite. The mean ± standard error curves are shown as heavy broken lines (Fig. 4). To characterize the field sensitivity response of k/k 0, a double precision polynomial fit was applied to the mean curve, in all cases optimized using a cubic form: k ¼ a þ bh þ ch 2 þ dh 3 : ð3þ k 0 On each raw data graph (e.g., Fig. 4a), the mean curve and ± standard error curves are shown as dashed lines, excluding the influence of any obvious outlier Table 2. Cubic polynomial fits most satisfactorily describe our k/k 0 dependence on measuring field (H AC ). In all cases these were significant at the 95% level, by consideration of R and sample size, n (BORRADAILE, 2003). Relevant polynomial regression information is available in Table 3. Moreover, the near-unity value of the Pearson correlation coefficients (R C 0.972; Table 3) between corresponding observed and fitted k/k 0 values verifies the validity of the cubic fits as data-descriptors. Recall that R 2 (coefficient of variation) expresses the fraction of the variance of the residuals explained by the fit; in the worst case this was 94.5% and in all other cases better than 97.3% (Table 3).

12 G. J. Borradaile et al. Pure appl. geophys., Table 2 Some data for magnetite-titanomagnetite (DUNLOP and O ZDEMIR, 1997) Approximate H C (coercive force) Approx. M S (saturation magnetization) mt ka/m ka/m Lowest MD All Magnetite * 480 Lowest PSD High PSD All TM 60 * 125 Mid-Upper SD Highest SD Tesla equivalent to (1/4p) A/m 1 mt * ka/m Table 3 Data from observed k/k 0 versus H curve sand polynomial fits to the mean data to 40 ka/m series n H in ka/m at peak k ± St. Err. ka/m Polynomial fit specifications to mean empirical curves: k/k 0 = a + bh + ch 2 + dh 3 observed R 2 a b c d diabase E E E-15 CY4 shallow dikes E E E-14 CY4 deep dikes E E E-14 CY4 gabbro E E E-15 CY4 mantle E E E-15 Limmasol Forest dikes E E E-15 Basal Group Dikes E E E-15 Mt Troodos Mantle E E E-15 to 80 ka/m series CY4 dikes <600 m n/a E E E-15 CY4 mantle >1400 m n/a E E E-15 n = number of specimens, H = AC field in ka/m, k = susceptibility at H, k 0 = susceptibility at low field n/a = not applicable because measurements to 80 ka/m produced many noisy curves; therefore this data is from the polynomial fit to the mean curve R = Pearson s correlation coefficient; R 2 expresses fraction of residual-variance satisfied by the fit More rigorously, although not strictly necessary for highly correlated bivariate data, one may wish to examine the correlation matrix (DAVIS, 2002). For observations of the independent variable x i with variance r 2, first arrange the variables in standardized form: z x ¼ Xn i¼1 ðx i xþ : r x Similarly, the dependent variable y is standardized to z y. Then, using elements of the variance-covariance elements for the matrix correlation matrix, for variables x 1,an x 2 e.g.,