analysis of the field dependence of remanent magnetization curves

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Analysis of the field dependence of remanent magnetization curves

R. Egli

Institute of Geophysics, Eidgenossische Technische Hochschule, Zurich, Zurich, Switzerland

Received 12 June 2002; revised 18 October 2002; accepted 30 October 2002; published 7 February 2003.

[1] A new method to calculate and analyze coercivity distributions of measuredacquisition/demagnetization curves of remanent magnetization is presented. Theacquisition/demagnetization curves are linearized by rescaling both the field and themagnetization axes. An appropriate filtering of the linearized curves efficiently removesmeasurement errors prior to evaluating the coercivity distributions. The filtered coercivitydistributions are modeled using a set of generalized probability density functions in orderto estimate the contributions of different magnetic components. An error estimation iscalculated for these functions with analytical and numerical methods in order to evaluatewhether the model is significantly different from the measured data. Three sedimentsamples from Baldeggersee (Switzerland) and three samples of urban atmospheric

particulate matter (PM) have been analyzed using this method. It is found that thecoercivity distributions of some of the magnetic components show significant andconsistent deviations from a logarithmic Gaussian function. Large deviations are foundalso in the coercivity distributions of theoretical AF demagnetization curves of single-domain and multidomain particles. Constraints in the shape of model functions affect theidentification and quantification of magnetic components from remanent magnetizationcurves and should be avoided as far as possible. The generalized probability densityfunction presented in this paper is suitable for appropriate modeling of Gaussian and alarge number of non-Gaussian coercivity distributions. INDEX TERMS: 1540 Geomagnetismand Paleomagnetism: Rock and mineral magnetism; 1519 Geomagnetism and Paleomagnetism: Magnetic

mineralogy and petrology; 1512 Geomagnetism and Paleomagnetism: Environmental magnetism

Citation: Egli, R., Analysis of the field dependence of remanent magnetization curves, J. Geophys. Res., 108(B2), 2081,

doi:10.1029/2002JB002023, 2003.

1. Introduction

[2] Two of the main tasks of environmental magnetismare the identification and the quantification of differentmagnetic phases in a sample, a procedure usually referredto as the unmixing of magnetic components. Twoapproaches have been developed for this purpose: multi-parameter records [Thompson, 1986; Yu and Oldfield, 1989;Verosub and Roberts, 1995; Geiss and Banerjee, 1997] andanalysis of magnetization curves [Thompson, 1986; Rob-

ertson and France, 1994; Carter-Stiglitz et al., 2001]. Themultiparameter approach is experimentally simple and relieson the measurement of different bulk magnetic propertiessuch as isothermal remanent magnetization (IRM), anhyste-retic remanent magnetization (ARM), susceptibility, andhysteresis parameters. Each parameter is a function of theconcentration of the various magnetic components. Theconcentrations can be estimated if the measured parametersare known individually for each component (forward mod-eling). However, the relation between the physical andchemical properties of the magnetic grains (e.g., composi-tion, grain size, and grain shape) on the one hand and their

magnetic properties on the other is complex and usuallyunknown. Many rock magnetic studies are based on syn-thetic samples, but the magnetic properties of such samplescan differ substantially from their natural counterparts. Onthe other hand, natural magnetic components can rarely bemeasured alone, since natural samples often represent com-plex mixtures of more or less altered magnetic crystals withdifferent origins and histories.

[3] The second approach is based on detailed measure-ment of induced magnetizations (hysteresis loops) or rema-

nent magnetizations (IRM, ARM, and thermoremanentmagnetization (TRM)) in variable magnetizing or demag-netizing fields. The absolute value of the first derivative ofthese curves is proportional to the contribution of allmagnetic grains with a given intrinsic coercivity to the totalmagnetization of the sample and is called the coercivitydistribution. If magnetic interactions between the grains ofdifferent components are negligible, the magnetization of asample is a simple linear combination of the contributionsof each magnetic component (finite mixture model withlinear additivity). The coercivity distribution of each mag-netic component is given by a particular (unknown) func-tion of the magnetizing or demagnetizing field (end-member

distribution), and the measured coercivity distribution is alinear combination of these model functions. If all end-

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2081, doi:10.1029/2002JB002023, 2003

Copyright 2003 by the American Geophysical Union.0148-0227/03/2002JB002023$09.00

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member distributions are compatible with a parameterizedfunction, such model functions can be used to fit themeasured data. If n is the number of components and kthe number of parameters of the model function, there are nkparameters which can be adjusted to obtain a model curvethat best reproduces the measurements. This operation,called component analysis, is performed with nonlinear

fitting algorithms [Heslop et al., 2002]. The coefficient,which multiplies each model function, is a measure of themagnetic contribution of the corresponding component.This approach was first proposed by Robertson and France[1994] by assuming that the IRM acquisition curve of eachmagnetic component can be closely approximated by acumulative logarithmic Gaussian function with three param-eters (amplitude, median destructive field, and dispersionparameter). They also proposed a physical model to explainthis assumption. A first application of component analysiswith IRM acquisition curves was described by Eyre [1996]on Chinese loess samples. The intriguing advantage of thisapproach is that a detailed knowledge of the magnetic

properties of the components is not necessary. The magneticproperties are described by the parameters of the modelfunction used in the component analysis, and the value ofthese parameters is deduced from the shape of the measuredmagnetization curve. Furthermore, the same type of magnet-ization is measured under the same physical conditions forall components, allowing a direct comparison between allmagnetic contributions. This approach, however, is limitedby its extreme sensitivity to measurement errors and to theshape of the function chosen to model the end-memberdistributions. Stockhausen [1998] handled the effect ofmeasurement noise by introducing goodness-of-fit parame-ters to indicate how well a measured curve is fitted by a set ofmodel functions. Kruiver et al. [2002] proposed a statisticaltest to compare different models for the component analysisand to decide the number of end-member distributions thatare necessary to fit the measured data sufficiently well. Theyalso developed an alternative approach to componentanalysis, based on a rescaling of the IRM acquisitioncurve (called linear acquisition plots (LAP)) so that acumulative Gaussian function is transformed into a straightline (called standardized acquisition plot (SAP)). The SAPof a mixture of slightly overlapped components withlogarithmic Gaussian coercivity distributions is character-ized by straight segments separated by inflections.

[4] Another important aspect of component analysis isthe modeling of end-member coercivity distributions. Pre-

vious work has shown that natural and artificial end-member distributions can be approximated with logarithmicGaussian functions [Robertson and France, 1994; Stock-hausen, 1998; Kruiver et al., 2001]. However, this is notnecessarily true for all samples, since many factors,including magnetic interactions, affect the shape ofmagnetization curves. Theoretical AF demagnetizationcurves of noninteracting single-domain particles [Egli and

Lowrie, 2002] and of multidomain particles [Xu andDunlop, 1995] cannot be modeled with logarithmicGaussian functions. This also applies for experimental AFdemagnetization curves of artificial samples of sizedmagnetite [Bailey and Dunlop, 1983; Halgedahl, 1998].In all the cases mentioned above there is only one magneticcomponent, however, deviations from a logarithmic Gaus-

sian function could be interpreted as the result of the sum oftwo components with strongly overlapping coercivities. Thelatter argument is of fundamental importance in theinterpretation of component analysis, since it is directlyrelated to the number of inferred components.

[5] We propose here a new approach to the componentanalysis of acquisition and demagnetization curves. First,

we handle the problem of evaluation and removal ofmeasurement noise without the use of component analysis.In this way, filtered coercivity distributions and confidencemargins can be calculated without any assumptions aboutthe magnetic composition of the sample. We then handle theproblem of component analysis by introducing the use ofgeneralized probability distribution functions (PDFs) tomodel end-member coercivity distributions without anyrestrictive assumptions about their shape. We also obtainan error estimation for the distribution parameters used forthe component analysis. The latter is of fundamentalimportance when end-member coercivity distributions ofdifferent samples are compared. Finally, this approach is

tested on three lake sediment samples.

2. Calculation of Coercivity Distributions

2.1. General Properties of Coercivity Distributions

[6] Coercivity distributions are defined as the absolutevalue of the first derivative of progressive acquisition ordemagnetization curves. We indicate the coercivity distri-bution with fX(H), where the index X indicates the originalacquisition or demagnetization curve used to calculate f, and

H is the magnetic field. Furthermore, fX(H)dH is thecontribution of all coercivities between Hand H+ dHto themagnetization indicated by X. Magnetic interactions andthermal activation effects produce differences between thedifferent kinds of magnetizations (IRM and ARM) and thedifferent kinds of demagnetizations (DC or AF), so that arigorous physical interpretation of fX(H) is almost impos-sible. However, coercivity distributions can supply a lot ofinformation about the carriers of magnetization and help inthe discrimination between different magnetic components.We define here a magnetic component as a set of particleswith identical mineralogy and similar physical properties(e.g., grain size and grain shape, morphology, crystallizationdegree, and concentration of defects): examples are thebacterial magnetosomes [Moskowitz et al., 1988], chemicallygrown fine magnetite in soils [Maher and Taylor, 1988], anddetritial magnetite or hematite from a given host rock. The

coercivity distribution of a component is characterized by asimple-shaped functions (e.g., a lognormal distribution or anegative exponential distribution), whose shape is controlledby the statistical distribution of the magnetic properties of theparticles. Often, coercivity distributions of different compo-nents cover the same range of coercivities (e.g., differentmagnetites) or the contribution of one of them is orders ofmagnitude weaker with respect to the others (e.g., hematitecompared to magnetite). For this reason, the contributions ofdifferent components are difficult to recognize directly fromthe acquisition or demagnetization curves but are evident inthe coercivity distributions.

[7] Coercivity distributions are mathematically describedby PDFs and can be calculated on different field scales forbetter isolation of different components. The shape of a

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coercivity distribution changes according to the field scaleadopted, because the integration over all coercivities corre-sponds to the total magnetization of the sample (normal-ization property). The scale change of a distribution f

generates a new distribution f* defined by the followingtransformation rule:

H* g H* ; f* H* f g1 H* ddH*

g1 H* 1

where H* is the new field scale, g the transformation rulebetween the old and the new scale, expressed by an injectivefunction with inverse g1, and f* the coercivity distributionwith respect to the new scale. For example, the transforma-tion rule from a linear to a logarithmic scale is expressed asfollows:

H* logH; f* H* ln10 10H*f 10H* 2

Another useful transformation is the following:

H* Hp; f* H* H* 1=p1

pf H* 1=p 3

where p is a positive exponent. We will refer to thistransformation as the power transformation. The powertransformation converges for p ! 0 to the logarithmictransformation. If 0 < p < 1, high coercivities are quenchedon the field scale, and distributions with large coercivitiesare enhanced. The same effect is obtained with alogarithmic scale and the opposite effect with p > 1. Theeffect of the field scale transformation on the shape of acoercivity distribution is illustrated in Figure 1.

2.2. Calculation of Coercivity Distributions From aMeasured Remanence Curve

[8] As pointed out in section 2.1, a coercivity distributionis the absolute value of the first derivative of a stepwise

Figure 1. Effect of field rescaling on the shape of a coercivity distribution (sediment sample G010 fromBaldeggersee, Switzerland). Four different field scales were chosen: (a) linear field scale, (b) power fieldscale according to (3) with exponent p = 0.5, (c) power field scale according to (3) with exponent p = 0.2,and (d) logarithmic field scale. Notice how the second peak of the coercivity distribution increases inamplitude when the field scale approaches a logarithmic scale. The thickness of the curve represents theestimated error of the coercivity distribution. The field scale transformation has an effect also on theabsolute error of the coercivity distribution.

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acquisition or demagnetization curve. In terms of Fourieranalysis, the first derivative is equivalent to a high passfilter, whose effect is to enhance small details of the originalcurve. For this reason, any information contained in theoriginal curve will be more evident in the resulting coer-civity distribution. This applies also to the measurementerrors, which are generally small in the measured curve butare enhanced in the resulting derivative. The increase ofsmall measurement errors is the main reason why coercivitydistributions have not been used very often in the interpre-

tation of magnetic measurements, despite the potentialadvantages. Possible sources of measurement errors arediscussed in section 2.5. Depending on the curvature ofthe acquisition/demagnetization curve, a minimum numberof steps are required to reproduce the coercivity distributionfree from aliasing effects. The amplitude of measurementerrors and the number of measured points strongly affect thecalculated coercivity distributions, so that changes in themeasuring procedure can produce apparently differentresults. Figure 2 shows the effects of measurement errorson two coercivity distributions of the same sample meas-ured with different degrees of precision.

[9] Measurement errors are commonly removed by fittingthe measured acquisition/demagnetization curve with anarbitrary number of given model curves (cumulative loga-

rithmic Gaussian distributions). These model curves areidentified with the magnetic signal of individual compo-nents. In this way, the calculation of a coercivity distributionrely on its interpretation in terms of component analysis. Away to calculate coercivity distributions without any furtherinterpretation consists in filtering the measurements or theresulting coercivity distributions in order to remove the

measurement noise. Often, the measured curves are asym-metric and require a different degree of filtering at differentfields. A standard low-pass filter would therefore be ineffi-cient in some regions of the curve, while it would signifi-cantly affect the shape of the curve in others.

[10] In the following, we present a technique that permitsthe removal of the measurement noise homogeneously alongthe entire curve and simultaneously estimates the error of theresulting coercivity distribution. The latter is particularlyuseful to avoid misinterpretations of numerical artifacts. Themethod presented here is based only on the followingassumption: all acquisition/demagnetization curves havetwo regions where the corresponding coercivity distribu-

tion is zero on a logarithmic field scale, one at H! 0 andthe other at H ! 1. In other words, any acquisition/demagnetization curve has two horizontal asymptotes on alogarithmic field scale. The physical meaning of thisassumption for H ! 1 is obvious: all magnetic mineralshave a maximum finite coercivity. For H !0, the physicalexplanation is related to thermal activation processes in SDand PSD particles and with domain wall motions in MDparticles. Measurements with a sufficient number of pointsnear H = 0 are necessary in order to obtain a correctcoercivity distribution for small fields. Appropriate scalingof field and magnetization allows linearization of theacquisition/demagnetization curve. On the linearized curve,each measurement point and the related error have the samerelative importance, so that a simple low-pass filter can beapplied to remove the measurement noise with the sameeffectiveness for all coercivities. An acquisition/demagneti-zation curve M= M(H) can be linearized in a simple way byrescaling the field according to the transformation rule H* =

M(H). However, because of the measurement errors, thefunction M(H) is unknown. A good degree of linearization isreached when a model function M0(H) expressed byanalytical functions is taken as transformation rule insteadof M(H). The choice of the appropriate model function

M0(H) becomes simpler if field and magnetization are bothrescaled. If M* = m(M) and H* = g(H) are the rescalingfunctions for the field and the magnetization, respectively,

then the model function is given by M0(H) = m1

(g(H)). Therelation M*(H*) between scaled field and scaled magnetiza-tion approaches a straight line when the model function

M0(H) approaches the (unknown) noise-free magnetizationcurve. The curve defined by e(H*) = M*(H*) H*represents the deviations of M*(H*) from a perfect linearrelation. We call e(H*) the residual curve. If the modelfunction M0(H) used for the scaling procedure is identicalwith the noise-free magnetization curve, then the residualcurve contains only the measurement errors. In reality, sinceit is impossible to guess the noise-free magnetization curve,the residual curve is a superposition of the nonlinearcomponent of M*(H*) and the measurement errors. Thefundamental advantage of considering the residual curveinstead of the original curve is that the measurement errors

Figure 2. Effect of the measurement precision on theshape and significance of the resulting coercivity distribu-tions. The black line represents the coercivity distribution

calculated from the average of eight demagnetization curvesof the same sample shown in Figure 1 (G010). Thethickness of the line is the estimated error of thedistribution. The gray band represents the coercivitydistribution calculated form only one demagnetization curveof the same sample; its thickness is the corresponding errorestimation. The two distributions are identical within theestimated error, indicating the significance of the errorestimation. The presence of two peaks in the coercivitydistribution is evident already from the result of a singlemeasurement curve. However, a third peak at 105 mT is notsignificant and disappears when more precise measurementsare done. This demonstrates the importance of an appro-priate error estimation for the correct interpretation ofcoercivity distributions.



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are highly enhanced in the residual curve and can behomogeneously removed with a low-pass filter. The choiceof the filter parameters is not critical and has little effect onthe shape of the resulting coercivity distribution. Under idealconditions, e(H*) represents the measurement errors, whichcan be simply removed by fixing e(H*) = 0.

[11] The filtered residual curve can be transformed back

into a magnetization curve as follows:

M H m1 L e g1 H g1 H 4where L(.) is the low-pass filter operator. M(H) is nowsupposedly free of measurement errors.

2.3. Coercivity Distribution Calculator (CODICA): AComputer Program for Coercivity Spectra Calculation

[12] CODICA is a computer program based on the scalingmethod described in section 2.2. It calculates a coercivitydistribution from an acquisition/demagnetization curve andgives an estimation of the maximal error of the calculated

distribution. The latter is important for evaluating the sig-nificance of component analysis on the resulting coercivitydistribution. CODICA is available fromthe author on request.

[13] CODICA runs on a Mathematica interface and usesseveral built-in mathematical routines. The functions of theprogram are discussed step by step in Appendix B. Theresults of the main processing steps of a real measurementare shown in Figure 3. The original demagnetization curveis shown in Figure 3a and is characterized by a typicalheavy-tailed behavior at high fields. A first scale transfor-mation is applied to the field axis in order to approach asymmetric sigmoidal function (Figure 3b). The second scaletransformation is applied to the magnetization axis in order

to linearize the demagnetization curve (Figure 3c). Devia-tions of the linearized demagnetization curve from a best-fitline are plotted in the next step (Figure 3d): the resultingcurve corresponds to the residual curve discussed in section2.2. Further rescaling of the field axis allows to obtain aresidual curve that is almost sinusoidal (Figure 3e). ItsFourier spectrum is concentrated in a narrow band around adominant wavelength, so that a simple low-pass filter easilyremoves the high-frequency measurement noise with littleeffect on the final shape of the filtered demagnetizationcurve. Finally, the filtered residuals (Figure 3f ) are convertedback to the original demagnetization curve by reversing theprevious rescaling steps. The result is a demagnetizationcurve, which is supposed to be free from measurementerrors. A coercivity distribution is obtained from the firstderivative of the filtered demagnetization curve (Figure 1).The error estimation is displayed as an error band on the plot(Figure 1) and in a separated plot as a relative error.

2.4. Testing CODICA

[14] CODICA was tested using a synthetic coercivitydistribution given by f(log H) = N(log H, 24, 0.36) +0.04N(log H, 56, 0.12) + 0.01w(log H), where N(x, m, s) is aGaussian function with median m in mT and dispersionparameter s, and w(x) is a Gaussian white noise withvariance 1 (Figure 4). This coercivity distribution is the sumof two components with overlapping coercivities and

different concentrations, which are similar to those encoun-tered in the natural samples presented later in this paper. The

efficiency of CODICA in removing the measurement errorsis compared with a common low-pass filter. Different cutofffrequencies were chosen, and the mean square differencebetween the filtered and the noise-free distributions wascalculated. The results are given in Figure 5a. Because thedistorting effects introduced by a low-pass filter are mostlyavoided after rescaling the magnetization curve, better results

are obtained with CODICA. The distorting effects introducedby the application of low-pass filters were further tested bycomparing the component analysis of the noise-free and thefiltered coercivity distributions. Changes in the shape of thecoercivity distribution are related to changes in the fittingparameters. The difference between the original parametersof the synthetic coercivity distribution and the best-fitparameters of the filtered distributions are plotted in Figure5b. Optimum removal of the measurement noise can beobtained without significant changes of the fitting para-meters. Consequently, the results of a component analysis arenot affected by the filtering procedure of CODICA.

2.5. Measurement Errors

[15] Measurement errors are the main limiting factor in theinterpretation of finite mixture models. Some knowledgeabout the measurement errors is useful to evaluate thesignificance of a component analysis and to optimize themeasurement procedure. Measurement errors may arise from(1) errors in the magnetization measurement, (2) errors in theapplication of the magnetization/demagnetization field, (3)errors induced by viscosity effects if the time interval betweenthe application of the field and the measurement is not thesame for all steps, and (4) errors induced by mechanicalunblocking of magnetic particles under application of highmagnetic fields on unconsolidated samples. These errorsources generate different noise signals that affect the meas-

urement. Simple error propagation equations can be used toestimate the amplitude of the errors; some results are listed inAppendix A. The effect of the four measurement error sourceson the calculation of a coercivity distribution is simulatedgraphically in Figure 6. Mechanical unblocking effects canaccount for large errors at high fields, which are occasionallyobserved in some unconsolidated samples obtained by press-ing a powder in plastic boxes. Magnetic grains that areelectrostatically attached to larger clay particles are goodcandidates for such undesired effects. Mixing the samplepowder with nonmagnetic wax before pressing it has beenfound to be a good solution to reduce measurement problemsat high fields (S. Spassov, personal communication).

3. Component Analysis With CoercivityDistributions

[16] In section 2, we have shown how measurement noisein acquisition/demagnetization curves can be filtered so thaterrors affecting the calculation of coercivity distributions areminimized. Now we turn to the problem of componentanalysis.

3.1. The Finite Mixture Model

[17] Consider a sample which contains a mixture of ndifferent magnetic components (finite mixture model). Each

component has a probability distribution fi(HjQi) for theintrinsic coercivity, which depends on a set of distribution



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parameters Qi = (qi1, . . ., qil). If the magnetization of eachcomponent adds linearly to the others (linear additivity), thebulk coercivity distribution of the sample is given by:

f H Xn

i1 ciMri fi HjQi 5

where ci and Mri are the concentration and the saturatedmagnetization of the ith component, respectively. The bulkmagnetization is given by the sum of the magnetizations ofeach component. Generally, fi(HjQi) is modeled with alogarithmic Gaussian function [Robertson and France,1994].

Figure 3. Calculation of a coercivity distribution using CODICA. Each plot is the original output of a

program step, as discussed in the text. (a) Original data for the AF demagnetization of an ARM (sampleG010, as in Figures 1 and 2). (b) Demagnetization curve with rescaled field compared with a best-fittingtanh function (solid line). The scaling exponent was p = 0.064. (c) Demagnetization curve with rescaledmagnetization and best-fitting line. (d) Residual curve. (e) The residual curve in (d) was rescaled in orderto approach a sinusoidal curve. (f ) A low-pass filter was applied to the residual curve in order to removethe measurement errors. A back-transformation of the filtered residuals through the steps shown in (d),(c), and (b) and subsequent numerical derivation gives the coercivity distribution plotted in Figure 1.



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[18] In case of interactions, linear additivity no longerholds. The shape of f(H) depends on the magnetizationprocess and may differ for acquisition and demagnetizationcurves [Cisowski, 1981]. Linear additivity is destroyed byinteraction effects which may easily occur in syntheticmixtures [Lees, 1997]. Carter-Stiglitz et al. [2001] avoidedthis problem in their synthetic samples by dispersing

potentially interacting pure components in a diamagneticmatrix before mixing them. Their dispersed pure sampleswere taken as end-member components for their unmixingtests. A particular case is given in samples where eachmagnetic component is formed by clusters of similarparticles. Consequently, strong interactions exist withinbut not between the clust ers, provided the volumeconcentration of the clusters is low enough. In this case,(5) can be rewritten to:

fint H Xni1

ciMri Ci fi * HjQi; Ci 6

where Ci is the volume concentration of the grains of the ith

component within the clusters. Linear additivity is preservedin this case. Equation (6) may apply for the synthetic samplesof Carter-Stiglitz et al. [2001] and in natural samples.

3.2. Generalized Coercivity Distributions

[19] Except for artificial samples, the end-member coer-civity distributions of a mineral mixture are unknown, andmodel functions f(xjQ) with a setQ of parameters are usedinstead. A lognormal function is commonly assumed for

f(xjQ). In this case Q = (H1/2, DP), where H1/2 is the mediandestructive field and DP the dispersion parameter [Robert-

son and France, 1994; Kruiver et al., 2001; Heslop et al.,2002]. On a logarithmic field scale, the lognormal function

coincide with the Gauss distribution. Accordingly, an end-member distribution is forced to be symmetrical about logH1/2 and to have a fixed curvature. Skewed, moresquared or less squared distributions cannot berepresented in this way. Deviations of f(xjQ) from alogarithmic Gaussian function are possible, since therelation between chemical and geometric properties of thegrains on the one hand and magnetic properties on the otherare rather complex and nonlinear. In this paper, we willdemonstrate the existence of consistent and systematicdeviations from the logarithmic Gaussian distribution modelin some natural and artificial samples. These deviations cansignificantly affect the results of unmixing models.

[20] As shown in section 3.1, an end-member coercivity

distribution is conveniently described by a PDF called f(x)in the following. The shape of f(x) is controlled by a set ofdistribution centers mn with related dispersion parameterssn,with n 2 N [Tarantola, 1987]. Special cases are given whenn = 1 (m1 is the median, s1 the mean deviation), n = 2 (m2 isthe mean, s2 the standard deviation), and n ! 1 (m1 is themidrange and s1 the half-range). The parameters H1/2 and

DP used by Robertson and France [1993] correspond to m2and s2 on a logarithmic field scale. The symmetry of a PDFis described by the coefficient of skewness s, where s = s3

3/s23

[Evans et al. , 2000]. Symmetric distributions arecharacterized by s = 0 and mn = m2. The curvature of aPDF is described by the coefficient of excess kurtosis k,

where k = s44

/s24

3 [Evans et al., 2000]. The GaussianPDF is characterized by k = 0.

[21] The description of small deviations from a GaussianPDF involves the use of functions with more than twoindependent parameters. It is of great advantage if suchfunctions maintain the general properties of a Gauss PDF:the nth derivative should exist overR and sn < 1 for allvalues ofn 2 N. Furthermore, the Gaussian PDF should be aparticular case of such functions. A good candidate is the

generalized Gaussian distribution GG [Tarantola, 1987],known also as the general error distribution [Evans et al.,2000]. The Gaussian PDF is a special case of GG distri-butions. Other special cases are the Laplace distribution andthe box distribution. The GGdistribution is symmetric: s = 0.

[22] Skewed distributions can be obtained from a sym-metric PDF through an appropriate variable substitution

x* = g(x, q), where q is a parameter related to the skewnessand g(x, q0) x for a given value q0 ofq. If these conditionsare met, the variable substitution generates a set ofdistributions with parameter q, wherein the original PDFis a special case. A suitable transformation applied to theGG distribution gives the following function:

SGG x; m;s; q;p 1211=ps 1 1=p

qeqx* q1ex*=q

eqx* ex*=q

exp 12

lneqx* ex*=q

2

!p" #

7

with x* = (x m)/s and 0 < jqj 1. We will call this PDF theSkewed Generalized Gauss Distribution (SGG). The GGdistribution is a special case of (7) for q = 1, and the Gaussdistribution is a special case of (7) for q = 1 and p = 2.Approximate relations between the distribution parametersand m2, s2, s and kforp ! 2, q ! 1 are listed in Appendix A.Some examples ofSGGdistributions are shown in Figure 7.The four parameters of a SGGdistribution have a hierarchicstructure: m and s control the most evident properties of thePDF, namely the position along the x axis and the width.On the other hand, q and p influence the symmetry and thecurvature of the PDF.

[23] The need of generalized distribution functions tomodel the coercivity distribution of a magnetic componentis evident on the examples shown in Figures 8 and 9.Figure 8a shows the theoretical AF demagnetization curve ofan ARM for an assemblage of noninteracting, uniaxialsingle-domain magnetite particles with lognormally distrib-uted volumes and microcoercivities [Egli and Lowrie,2002]. Figure 8c shows the theoretical AF demagnetization

curve of an ARM for multidomain particles with a Gaussiandistribution of microcoercivities [Xu and Dunlop, 1995].Both cases can be regarded as a single magnetic component.The related coercivity distributions are not Gaussian on alogarithmic field scale but can be fitted well with a SGGdistribution. Similar coercivity distributions are also foundin natural sediments, as the ODP sample of Figures 8e and8f. The magnetic materials presented in Figure 8 are verydifferent; nevertheless, they have similar coercivity dis-tributions with a negative skewness of 1.3 to 1.7.Similar results are obtained also from AF demagnetizationcurves of SIRM. Both the magnetic interactions and themagnetic viscosity generally increase the initial slope of an

AF demagnetization curve, because they affect mainly lowcoercivity contributions to the total magnetization. As a



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consequence, the related coercivity distributions are left-skewed, and the crossing point between normalizedacquisition and demagnetization curves is


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Bailey and Dunlop [1983] have shown that magnetic grainswith a multidomain-type result of the modified LowrieFuller test [Johnson et al., 1975] have a microcoercivitydistribution which is more convex than the negativeexponential distribution. These coercivity distributions arecharacterized by s < 1. Left-skewed coercivity distribu-tions are also needed to fit AF demagnetization curves of aSIRM in artificial samples of sized magnetite with grainsizes between 0.1 and 100 mm [Bailey and Dunlop, 1983;

Halgedahl, 1998], as shown in Figure 9. The coercivity dis-tributions of all grain sizes have a skewness ofs = 0.93 0.1, which is close to that of a negative exponentialdistribution (s = 0.997). However, the importance ofmagnetic interactions in these samples is not clear.

3.3. Error Estimation

[24] Component analysis can be extremely sensitive tomeasurement errors, especially in case of magnetic compo-nents with highly overlapped coercivity distributions. Thus,some distribution parameter estimates may not be significantat all, even if the quality of the measurement is excellent forthe usual standards in rock magnetism. An error estimation

of each distribution parameter is important to avoid mis-interpretations. This problem was first recognized by Stock-

hausen [1998]: he attempted to evaluate the significance ofhis results by introducing parameters that indicate how wella measured curve is fitted by a set model functions. Kruiveret al. [2002] proposed a statistical test to compare differentmodels for the component analysis. However, theseapproaches are useful to evaluate the overall significanceof component analysis (see section 3.4) but do not provideany information about the single components. The latter isobtained with an error estimation for each distribution

parameter. Error estimates for each parameter are providedin the following for a general PDF f(xjQ).

[25] We assume the measured distribution to be given byyi = f(xi) + dyi, where (xi,yi) is a measurement point and dyithe related measurement error. Several methods can be usedto obtain an unbiased estimation Q ofQ [Stockhausen, 1998;

Kruiver et al., 2001; Heslop et al., 2002]. A best-case errorestimate is obtained by means of the Rao-Cramer-Frechetinequality (RCF). If f(xjq) is a Gauss distribution withvariance s2, measured at regular intervals x of x, and ifdyi = dy is independent of xi, the variance of one unknowndistribution parameterq is given by:

var q 2p1=2sx dy 2R

X@qf xjq

2

f xjq dx 8

Figure 5. Comparison between measurement noise removal with and without the rescaling proceduredescribed in the text. The same coercivity distribution of Figure 4 was calculated and the operation wasrepeated 1000 times with different simulations of the measurement error w(log H). Each of the 1000distributions was filtered in the same way as in Figure 4. In (a), the mean squared difference between thefiltered and the noise-free distributions has been plotted as a function of the normalized cutoff frequencyn of the low-pass filter. Filled symbols refer to the output of CODICA, open symbols to the resultsobtained by filtering the data without the rescaling procedure of CODICA. The normalizing factor for thecutoff frequency is chosen to be identical with the value of the cutoff frequency, which minimizes thesquared residuals. The noise removal is more efficient after the rescaling procedure. In (b), the relativeerror of the best-fit parameters m, m, and s of the two Gaussian distributions are plotted as a function ofthe normalized cutoff frequency of the low-pass filter. For n < 0.7nopt, a low-pass filter inducessignificant distortions in the shape of the coercivity distribution. Open symbols refer to the best-fitparameters of the component defined by m = 0.04, m = 56 mT, and s = 0.12. Solid symbols to the othercomponent. Circles refer to m, squares to m, and triangles to s.



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A proof of equation (8) starting from the standardformulation of the RCF inequality is given in Appendix

C. Equation (8) can be used to estimate the minimum errorsof the parameter estimates of a SGGdistribution with p ! 2

and q ! 1. The results are listed in Appendix A. Moreprecise analytical error estimations, which apply asympto-

tically to all unbiased parameter estimates, are obtained witherror propagation methods, however, only in the limiting

Figure 6. Simulated effect of some measurement error sources on the calculation of a coercivity

distribution with following parameters: m = 1, m = 10 mT, and s = 0.38 according to the calculations ofAppendix A. The two curves of each plot give the upper and lower limit of a coercivity distributioncalculated from a demagnetization curve measured at fixed field intervals of 0.2 on a logarithmic fieldscale (H/H% 0.585). Measurements are performed with (a) an absolute measurement error dM= 0.01,(b) a relative measurement errordM/M= 0.01, (c) an absolute error of 0.2 mT affecting the applied peakfield, and (d) a relative error of 2% affecting the applied peak field. In (e) and (f ), measurement errorsarise from mechanical unblocking of a part e = 0.3% of all magnetic particles in an unconsolidatedsample. In (f ), a magnetically harder component (e.g., hematite) with an unremovable magnetization of5% is added to the soft component.



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case of one component. Then, the variance of an unknowndistribution parameterq is given by:

varq x dy 2R

X@qf xjq 2dx

9

A proof of equation (9) is given in Appendix C. Errors of

the unbiased parameter estimates for one SGG distributionare listed in Appendix A.

3.4. Significance Tests

[26] The finite mixture model of equation (6) has n(l+ 1)independent parameters. If a smaller number of parametersis assumed, the mixture model will not fit well the measureddata. There are two possibilities for increasing the numberof model parameters. The first one consists of adding morecomponents to the model, as discussed in the literature[Robertson and France, 1994: 2 components; Eyre, 1996: 4components; Stockhausen, 1998: 2 components; Kruiver etal., 2001: up to 3 components; Heslop et al., 2002: up to 4components]. The second possibility is presented in this

paper and consists of a better definition of the end-memberPDF. Both strategies can suggest wrong conclusions, asdiscussed in section 4, if the unmixing results are notevaluated critically. The problem of finite mixing models isrelated to the fundamental question of how many para-meters should be used to fit experimental data. The additionof new parameters always improves the goodness of fit of amixture model; however, this improvement is not necessa-rily significant. Kruiver et al. [2001] proposed a combina-tion of statistical tests to determine whether the addition ofextra parameters significantly improves the goodness of fit.They apply an F test and a t test to the squared residuals ofone fit model with respect to another model to decide if the

two models are significantly different. We propose here theuse of a Pearsons c2 goodness of fit test [Cowan, 1998],

which allows us to test if an experimental probabilitydistribution is compatible with a given model distribution

f(xjQ) with l unknown parameters q1. . .ql. According to thistest, the two distributions are incompatible at a confidencelevel a (generally a = 0.05) ifc2(Q) > c2nl1;1a, wherec2nl1;1a is the value of the c

2 distribution with n l 1degrees of freedom, evaluated at 1 a. A coercivitydistribution calculated with the method presented in section 2can be used as reference distribution for the Pearsons c2

goodness of fit test, since this method is not based on finitemixture models.

[27] A statistical test alone is not sufficient to evaluate thesignificance of a mixing model, as demonstrated in section 4.Sometimes, the coercivity distributions of two magneticcomponents are widely overlapped, and an extremely highmeasurement precision is required in order to identify thesecomponents. A stack of six demagnetization curves with 72steps each has been used for the analysis of some samplespresented in this paper. Such a high measurement precisioncannot be used as a standard for systematic investigations.Nevertheless, an integrated approach to this problem is

possible, as shown in section 4 on the example of urbanatmospheric particulate samples. In this case, the componentanalysis of an individual sample was very critical. Theaccurate choice of three samples with extremely differentdegrees of pollution allowed to define the number ofmagnetic components and their magnetic properties. Muchless precision would be required for an extended study ofurban atmospheric particles. The component analysis ofstandard quality measurements would be supported bythe detailed information acquired with the accurate analysisof few reference samples. A similar strategy has beenapplied to the measurement of the lake sediments presentedin section 4. In this case, different sources of magnetic

minerals in the sediments were investigated by accuratemeasurements of each sedimentary unit and of samples

Figure 7. Examples of SGG distributions, given by f(x) = SGG (x, m, s, q, and p). (a) Some particularcases with, m2 = 0, s2 = 1, and q = 1 are plotted. The skewness of all curves is zero. Furthermore, p = 1 fora Laplace distribution, p = 2 for a Gauss distribution, and p ! 1 defines a box distribution. (b) Someleft-skewed SGG distributions with m2 = 0 and s2 = 0.5484 are plotted. Right-skewed distributions withthe same shape can be obtained by changing the sign ofm and q. Demagnetization curves of multidomain

magnetite can be modeled with exponential functions. The corresponding coercivity distribution on alogarithmic field scale can be approximated by a SGGdistribution with s = 0.6656, q = 0.4951, and p =2.3273, plotted in (b). The difference between an exponential PDF and its approximation by a SGGdistribution is smaller than the thickness of the curves.



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Figure 8. Examples of significant deviations of calculated and measured coercivity distributions from alogarithmic Gaussian function. (a) Theoretical AF demagnetization curve of an ARM imparted to a set ofrandom oriented, noninteracting single domain magnetite particles [Egli and Lowrie, 2002]. The particleshave lognormally distributed volumes and microcoercivities. The corresponding coercivity distribution(b) is plotted with points and the solid line is the best-fitting SGG function. (c) Theoretical AFdemagnetization curve of an ARM imparted to a set of multidomain particles with Gaussian distributedmicrocoercivities (redrawn from the study of Xu and Dunlop [1995]). The corresponding coercivitydistribution (d) is plotted with points and the solid line is the best-fitting SGG function. (e) AFdemagnetization curve of an ARM imparted to an ODP sediment sample (leg 145) taken in the NorthPacific (courtesy of M. Fuller). The corresponding coercivity distribution is shown in (f ) together with acomponent analysis performed with SGG functions. The coercivity distribution of the soft component(solid line) clearly differs from a log-Gaussian function.



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from the catchment area. Then, AF demagnetization curveswith 20 steps were measured for the entire sedimentcolumn. The measurements were fitted with the coercivitydistributions of the magnetic components identified in threereference samples. Changes in the amount of biogenicmagnetite during the last 120 years could be reconstructedin this way.

[28] We propose the following set of conditions to applyand test mixing models to a large set of samples:

1. Choice of reference samples. Reference samplescontaining the most varied amounts of the same set ofmagnetic components should be chosen for detailed andprecise measurements. These samples should define themost extreme conditions to be taken into account by themixing model.

2. Statistical tests. The mixing model has to pass astatistical significance test (goodness of fit test) for eachreference sample; in other words it should be compatiblewith the measured data within the experimental errors.

3. Errors. All model parameters should be significant,i.e., they should not be affected by large errors.

4. Consistency. The coercivity distributions of the same

components should be identical within the experimentalerror, or they alternatively should show variations which are

consistent with some physical or chemical changes.Furthermore, variations in the concentration of eachmagnetic component should be explained with the help ofsome independent information (geological setting, chemicaland physical processes).

[29] Point 4 implies some knowledge about the potentialsources of magnetic minerals (e.g., magnetite formation insoils [Maher, 1988], titanomagnetites in volcanic rocks[Worm and Jackson, 1999], biogenic magnetite [Moskowitz

et al., 1993], and maghemite in loess [Eyre, 1996]) andabout the properties of magnetization and demagnetizationcurves [Dunlop, 1981, 1986; Bailey and Dunlop, 1983;

Johnson et al., 1975; Halgedahl, 1998; Cisowski, 1980;Hartstra, 1982; Robertson and France, 1994].

4. Interpretation of Coercivity Distributions byComponent Analysis

4.1. Comparison Between Different PDFs

[30] As discussed above, the results of a componentanalysis depend upon the PDF chosen to model the end-member coercivity distributions and particularly on the

number of parameters assigned to each PDF. In this section,we will compare results obtained with a linear combination

Figure 9. Coercivity distribution parameters m, s, q, and p (see text) for the AF demagnetization of IRMin various synthetic samples of sized magnetite. The magnetic components identified in lake sedimentsand PM10 dust samples are also shown for comparison. Numbers beside each point indicate the grainsize in mm. (a) Scatterplot ofm and s, which are a measure of the median and the width of a coercivitydistribution, respectively. The dashed line indicates the value ofs for a negative exponential distribution(Figure 7). Large grains are characterized by low values ofm and high values ofs. Notice the extremelylow value of s measured for component I2 (probably biogenic magnetite). An inverse correlationbetween m and s is evident. (b) Scatterplot ofq and p, which are related to the skewness and the kurtosisof a coercivity distribution, respectively. The cross point of the dashed lines corresponds to the values ofq and p for a log-Gaussian distribution. All samples of sized magnetite show values of q that aresignificantly different from those of a log-Gaussian distribution. They group around mean values of q %0.46. All parameters of the sized magnetites are intermediate between those of a log-Gaussian

distribution and those of a negative exponential distribution. The coercivity distribution of larger grainsizes approaches an exponential distribution.



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of Gaussian distributions on the one hand and a linearcombination of SGG distributions on the other. Since finitemixture models with non-Gaussian coercivity distributionshave not been reported in the literature, it is not possible todecide from a priori informations which kind of PDF shouldbe used as a basis for a mixture model. From themathematical point of view, all PDFs are equivalent, sincethe goodness of fit, which can be reached with a particularmodel, depends only upon the total number of parametersassumed, regardless of how they are assigned to individual

components. Starting from these considerations, and fromthe fact that coercivity distributions of natural and artificial

samples are nearly log-Gaussian, one could ask if the use ofmore complicated PDFs has any physical meaning. We willhandle this problem in the following.

[31] To better understand the problem, we first illustratethe strong similarities that exist between a SGGdistributionwith p ! 1 and q ! 2 on one hand and a linearcombination of two Gaussian distributions on the other. Weconsider three different situations, which are shown inFigure 10. In the first case, two Gaussian distributions withidentical amplitudes and same s, but slightly different

values of m, are fitted with a SGG distribution with p = 1and q > 2. Ifm/s > 1, where m is the difference between

Figure 10. Three cases where a linear combination of two Gaussian functions is very similar to a SGGdistribution. (a) mN(x,m1, s) + mN(x, m2, s) with jm1 m2j < 2s is compared to a best-fit obtained withSGG(x, m, s0, 1,p), where m = (m1 + m2)/2 and p > 2. (b) m1N(x, m, s1) + m2N(x, m,s2) is compared to abest-fit obtained with SGG(x, m, s, 1,p), where p < 2. In (c), SGG(x, m, s, q,p) is compared withm1N(x, m, s1) + m2N(x, m, s2). Below each plot, the difference between the two functions is plotted in

percent of the maximum value of these functions.



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the means of the two Gauss distributions, the resultingfunction has two local maxima and is evidently bimodal.However, as m/s ! 0, the resulting distribution becomesvery similar to a slightly squared PDF (k< 0). In the secondcase, a SGG distribution with q < 2 is fitted to a linearcombination of two Gaussian functions with the same m butdifferent values ofs. The two distributions converge to the

same function for q ! 2. In the third case, a SGGdistribution with p 6 1 is fitted to a linear combination oftwo different Gaussian functions. Convergence of the twodistributions is obtained for jpj ! 1. In all three cases, theSGG distribution and the combination of two Gaussianfunctions can be very similar. The possibility of distin-guishing two overlapping Gaussian functions from a SGGdistribution depends on the noise level of the data to befitted and can be tested with a Pearsons c2 goodness of fittest. If the test is not passed, the fitting models aremathematically equivalent, but the corresponding interpre-tations are drastically different, since the number of inferredcomponents is not the same. Generally, the use of more

complicated PDFs for the end-member coercivity distribu-tions has the effect of reducing the number of componentsneeded to fit a measurement with a sufficient degree ofprecision. Two components with widely overlappingcoercivity distributions may be modeled with one SGGdistribution, and vice versa, a single component with k6 0or s 6 0 may be modeled with a combination of twoGaussian distributions. In both cases, incorrect interpreta-tion may result. An example is given by the samplesdescribed in section 4.2, which contain magnetic compo-nents whose coercivity distribution are similar to thefunctions plotted in Figures 10b and 10c.

[32] If two overlapping PDFs cannot be resolved at thegiven confidence level, the sum of the estimated contribu-tions may still be significant, despite the fact that theindividual values of the estimates are not significant. Inthis case, the two PDF are conveniently modeled as a singlecomponent, eventually by substituting them with a morecomplex PDF. The use of PDFs with more distributionparameters, instead of a large number of distributions withfewer distribution parameters leads to results of the fittingmodel which are more stable against measurement errors.The stable behavior of a fitting with SGG distributions canbe explained by the fact that small deviations from an idealcoercivity distribution, which arise from measurementerrors, are taken into account by variations in skewnessand kurtosis, rather than by variations in the contributions of

the single components. Obviously, the values obtained forskewness and kurtosis may not be significant at all. Asimilar stability can be obtained with Gaussian functions ifsome of them are grouped as if they were one component.However, it is not always evident which distributions grouptogether, and multiple solutions are often possible, asillustrated by the examples described in the followingsection.

4.2. Examples

[33] In this section, measurements of lake sediments andurban atmospheric particulate matter (PM) are presented.4.2.1. Lake Sediments

[34] Lake sediment samples were taken from Baldeg-gersee, Switzerland. This lake is situated on the Swiss

Plateau at 463 m asl, it has a surface area of 5.2 km2 and amaximum water depth of 66 m. The catchment area (67.8km2) has been used intensively for agriculture since thenineteenth century. The lake was formed more than 15,000years ago after the retreat of the Reuss glacier. Hillsaround the catchment area protect the lake from windsand facilitate oxygen depletion in deep waters. Several

packets of varves indicate these depletion periods duringthe last 6000 years. The last and most severe eutrophica-tion event started in 1885, triggered by the development ofhuman activities in the catchment area. The depth toanoxic water column was 60 m in 1885 and rose to 10 min 1970 [Wehrli et al., 1997]. A 1.2 m long gravity corewas taken in 1999 at the center of the lake and sampledevery centimeter. The samples were immediately freezedried to prevent oxidation and pressed into cylindricalplastic boxes. The core covers the last 200 years ofsedimentation [Wehrli et al., 1997]. Magnetite is the majormagnetization carrier, with small amounts of a high-coercivity material, probably fine-grained hematite. The

analysis of coercivity distributions is used here to separatethe detrital component of the magnetic signal from theauthigenic component (the magnetic particles produced bychemical and biological processes in the lake). In order toidentify the detrital component, a sediment sample from asmall river delta of the lake was taken. Since the catchmentarea is geologically and anthropogenically homogeneousall around the lake, this sample is expected to berepresentative of the detrital input. The sample was sievedin acetone in order to isolate the fraction


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[35] Adequate mixture models are obtained with threeor four SGG functions. Component H has generally lowquality parameter estimates, because the available max-imum field of 300 mT was not sufficient to saturate it. Foreach mixture model, a Pearsons c2 goodness of fit testwas performed. The standard error of the distributionparameters was also estimated using the followingprocedure. An appropriate noise signal, which correspondsto the measurement error estimated by CODICA, was

added to the coercivity distribution by means of a randomnumber generator. The component analysis was then

performed on the resulting distribution, and new valueswere obtained for each distribution parameter. Theprocedure was repeated many times (generally 100) inorder to sample a significant set of estimates of thedistribution parameters, which allow the calculation of thestandard deviation for each parameter. The results aresummarized in Table 2. Each end-member coercivitydistribution can be normalized to have a unit saturationremanence and can be drawn separately, as shown in

Figure 13. In this way, the comparison of end-membercoercivity distributions is facilitated.

Figure 11. Example of integrated approach to the component analysis of coercivity distributions. (a)Coercivity distribution of a PM10 sample (GMA) collected in a green area near the city of Zurich(Switzerland). The thickness of the line is the standard deviation of the estimated error. The coercivity

distribution is fitted with one SGG distribution. The difference between measured and fitted curve isplotted below (thick line) together with the measurement error estimation provided by CODICA (thinline pairs). The smallest error estimation refers to the real measurement of six demagnetization curvesof ARM with 72 steps each. The largest error estimation is calculated for the measurement of onedemagnetization curve with 12 steps. The intermediate error estimation refers to the measurement ofsix demagnetization curves with 12 steps each or one demagnetization curve with 72 steps. Themodeled curve is incompatible with the highest precision measurement. Therefore, two magneticcomponents are used for the component analysis shown in (b). In this case, the coercivity distributionis well fitted within the error of the highest precision measurement. Line pairs represent the upper andlower limit for the coercivity distribution of each component. An alternative approach to high-precisionmeasurements was the investigation of similar samples. In (c) and (d), the coercivity distributions ofother two PM10 dust samples are shown. These samples were collected near a high-traffic road in thecenter of Zurich (WDK) and inside a highway tunnel near Zurich (GUH). The presence of twomagnetic components (called N and A) is evident in these samples. Furthermore, the contribution ofcomponent A to the total ARM is related to the degree of pollution of the area (in increasing order:GMA, WDK, and GUH).



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[36] Models with


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Table 3. Summary of Distribution Parameters of the Magnetic Components Found in PM10 Samples

Parameter GMA WDK GUH Comment

m dm, mA m2/kg 338 2 590 2 316 5 Natural mineral dust (N):Coercivity distributions of GMA and WDK are similar.GUH was collected in a tunnel.

100 (m dm)/Mrs 73 1 41 1 31 1m2 dm2, mT 23.8 0.1 19.6 0.1 15.9 0.3s2 ds2 0.456 0.001 0.438 0.001 0.343 0.01s ds 0.58 0.02 0.65 0.01 1.05 0.03k dk 0.76 0.01 0.928 0.001 1.756 0.02m dm, mA m2/kg 128 2 864 2 709 2 Vehicle pollution dust (A):

All coercivity distributions are similar.100 (m dm)/Mrs 27 1 59 1 69 1m2 dm2, mT 77.3 0.2 74.7 0.2 77.5 0.3s2 ds2 0.245 0.001 0.275 0.001 0.23 0.01s ds 0.20 0.02 0.65 0.01 0.2 0.1k dk 0.19 0. 01 0.785 0.002 0.12 0.01

Figure 12. Finite mixture model for the three sediment samples presented in this paper. The solid linerepresents the coercivity distribution of the sample. The thickness of the line being the standard deviationof the estimated error. The other line pairs represent the upper and lower limit for the coercivity

distributions of each identified component (labeled with D, I, and H). Details of the component analysisfor the three samples are listed in Table 2.



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be representative for the detrital input into the lake.Component I has extremely low values of the dispersionparameter: s < 0.15. In the sample with strongestmagnetization (G044) it is evident that component I iscomposed by at least two subcomponents, which can befound individually in the other samples at lowerconcentration. The magnetization of component I variesfrom 6 mA m2/kg in U03F (2.6% of the bulk magnetiza-tion) to 136 mA m2/kg in G044 (80% of the bulkmagnetization). The coercivity distribution of component Iis comparable to that of samples containing intact cells ofmagnetotactic bacteria [Moskowitz, 1988]. Intact and

broken chains of magnetosomes were observed under theelectron microscope in sample G044. Therefore component

I is identified as magnetite grains of bacterial origin. Themagnetic signal of component I may reflect changes in theproduction rate of biogenic magnetite or a possiblereductive dissolution process of fine magnetite grainsduring eutrophication periods. Component H is badlyresolved, because saturation was not reached in any of thethree samples. The highest contribution of this componentis found in sample U03F. The parameters m and s ofcomponents D, I, and H in all three samples are drawn ina scatterplot in Figure 14. The three components are wellgrouped in three different regions of the plot. Component Iis compatible with the magnetic properties of pure single-

domain magnetite, whereas component D contains a smallbut significant amount of magnetic particles with coerciv-

Figure 13. Coercivity distributions of the magnetic components found in the samples analyzed in thispaper. The coercivity distributions are normalized so that the saturation remanence (area under the curve)

equals to 1 to facilitate the comparison between different samples. (a) Component D of lake sedimentsamples G010, G044, and U03F. The coercivity distribution of this component is identical in all threesamples within the confidence levels given by the measurement errors. This component may representdetrital particles transported toward the center of the lake. (b) Component I2 of lake sediment samplesG010 and G044. The coercivity distribution of this component is identical in both samples within theconfidence levels given by the measurement errors. The relatively high coercivity and the extremelysmall value of s are indicative for intact magnetosomes, either isolated or arranged in chains. (c)Component N of the PM10 samples GMA and WDK. The coercivity distribution of this component isslightly different in the two samples. This component may represent the magnetic minerals contained inthe mineral part of natural dust. (d) Component A of PM10 samples GMA, WDK, and GUH. Thiscomponent may represent the magnetic minerals associated with the pollution products of motorvehicles.



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ities >300 mT, probably representing oxidized magnetite.Component H is associated with a magnetically hard

mineral.4.2.2. Urban Atmospheric PM[37] Urban atmospheric PM is the subject of several

studies because of its negative effects on human health[Harrison and Yin, 2000]. Magnetic properties of urban PMhave been recently investigated by several authors [e.g., Shuet al., 2001; Muxworthy et al., 2002] because of the highconcentration of magnetic minerals in urban pollution. Theidentification of various sources of magnetic particles inurban PM would be of great interest for environmentalstudies. Three samples of urban PM


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ized distributions with five parameters can take intoaccount variations in the symmetry (skewness) and thecurvature (kurtosis). End-member coercivity distributionswith significant and systematic deviations from a logarith-mic Gaussian function are needed to interpret theoreticalAF demagnetization curves of single-domain and multi-domain magnetite, as well as measured demagnetization

curves of synthetic samples containing sized magnetitegrains.

2. The significance of the component analysis ofcoercivity distributions should be evaluated with statisticaltests and with an error estimation of each distributionparameter. Analytical expressions for the error estimationshave been developed.

3. Multiple solutions for the component analysis arepossible. In this case, other informations are needed toidentify the correct solution. A comparison between thecomponent analysis of different sediments, which belong tothe same ecological system, is useful for the identificationof magnetic components.

4. Component analysis is applicable to large sets ofsamples with standard precision measurements, provid-ing the number of magnetic components and theircoercivity distribution parameters is known from detailedmeasurements on a selection of few, adequate referencesamples.

Appendix A: Error Calculations

Appendix B: A Short Guide to CODICA

[41] The elaboration of an acquisition/demagnetizationcurve with CODICA consists in the following steps.

B1. Data Checking

[42] The measurement curve is always displayed as a

demagnetization curve (Figure 3a): this does not affect thecalculation of the coercivity distribution. The user is askedto enter an estimation of the measurement error (if known).This estimate may come from inspection of the measure-ment curve and/or experimental experience with the instru-ments used. A combination of a relative and an absoluteerror is assumed to affect the measurements and the AFfield. Systematic errors, like magnetization offsets andtemperature effects on the sample and on the AF coil donot affect significantly the shape of the measured curve andmay not be included. The calculation of a coercivitydistribution and the related error is independent of theerror estimate entered by the user. This first estimate is

needed by the program in order to display a roughestimation of the confidence limits of the measured curve,which should help the user through the following steps ofthe program.

B2. Scaling the Magnetic Field (Figure 3b)

[43] As discussed in section 2.2, an acquisition/demag-netization curve is supposed to have a sigmoidal shapeon a logarithmic field scale. However, the measuredcurves are not symmetrical. Often, they show a long tailat high fields. In case of lognormal coercivity distribu-tions, the measured curve represented on a logarithmicfield scale becomes symmetric. In all other cases thecurve is asymmetric on both a linear and a logarithmicfield scale. An appropriate scale change which offers aset of intermediate scales between linear and logarithmicis defined by the power function H* = Hp, p being apositive exponent. An appropriate value of p is chosen, sothat the scaled curve reaches maximum symmetry. Thesymmetry of the curve is compared with a referencesigmoidal curve, expressed by an analytical function (a tanhfunction in our program). The scaled curve is thereforerepresented together with the best fitting tanh function. Anautomatic routine optimizes the scaling exponent p so thatthe difference between the original curve and the modelcurve is minimal.

B3. Scaling the Magnetization (Figure 3c)[44] A tanh function was chosen as a reference in order to

scale the field, because of its mathematical simplicity. Itdoes not have a particular meaning and any other similarfunction could be used instead. If the measured curve M(H)coincides with a tanh function, the application of the inversefunction arctanh to the magnetization values generates alinear relation between scaled field and scaled magnetiza-tion. The scale transformation applied to the magnetizationvalues is based on the following model for the relationshipbetween the scaled field H* and the measured magnetiza-tion M(H*) in a demagnetization curve:

M H* Mrs 1 tanh a H* H1=2* M0 B1

Table A1. Effect of Different Measurement Error Sources on the

Calculation of a Coercivity Distributiona

Error Source Magnetization Curve Coercivity Distribution

General,h = H/H

Mi = M (Hi) + MM = total error

fi

Hi1Hi2

Mi1MiHi1Hi

Measurementerror: dM

(M)2 = (dM)

2

+ f2(dH)2(f)

2 = 2h2 (dM)

2 + 2f2 (dH)

2

Applied field error:dH

Absolutemeasurementerror: dM = e

M = e f ffiffiffi2p heRelative

measurementerror: dM/M = e

M/M = e f ffiffiffi2p heMAbsolute

applied fielderror: dH = e

M = ef f=f ffiffiffi2p he=HRelative

applied field

error: dH/H = e

M = efH f=fffiffiffi

2p

he

Mechanicalinstability:dM = eHM

M = eHM f ffiffiffi2p heHMaThe numerical calculation of a coercivity distribution from an

acquisition/demagnetization curve with finite differences is given in thefirst row. In the second row, general equations for the error estimation aregiven by assuming an error dM in the measurement of the magnetizationM and an error dH in the applied field H. In the other rows, errorestimations are given for particular cases where the absolute or the relativeerror of M or H are described by a white noise of amplitude e. The lastrow gives an error estimation in the case where a small part e of allmagnetic particles, which are not magnetically unblocked by the appliedfield, becomes mechanically unstable and rotates under the influence of atorque T/ H. The amount of these particles is assumed to be proportionalto T and thus to H. The corresponding magnetization is then proportional

to HM(H), where M(H) is the magnetization of all magnetically blockedparticles.



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where Mrs has the physical meaning of a saturationremanence (if the measured curve is saturated at the highestfield value), M0 has the physical meaning of a residualmagnetization (M0 = 0 if saturation can be reached), H*1/2 isthe scaled median destructive field, and a is a parameter thatcontrols the steepness of the curve. The following scaletransformation

M* artanh 1 MM0Mrs

B2

generates the linear relation M* = a(H* H*1/2) betweenscaled field and scaled magnetization. The four parameters

Mrs, H*1/2, M0, and a have to be chosen in a way that thescaled magnetization curve as linear as possible. Theprogram optimizes the parameters H*1/2 and a automaticallyusing a LevenbergMarquard algorithm for nonlinearfitting. The parameters M0 and M0 + Mrs represent theasymptotic values of the magnetization curve. Theiroptimization is controlled by the user, since it was foundthat the optimization is very unstable with respect to theseparameters. The scaled curve is represented together with aleast squares linear fitting. Deviation from the least squaresline can be minimized with an appropriate choice of M0and M0 + Mrs. In general, too small values of M0 or too

high values of M0 + Mrs produce a flattening at the rightand left end of the scaled curve, respectively. In contrast,too high values of M0 or too small values of M0 + Mrsproduce a steepening of the scaled curve at the right andleft end of the scaled curve, respectively. Randomdeviations from the least squares line indicate the presenceof measurement noise, systematic smooth deviationsindicate a divergence of the measured curve from (B1).Best results are achieved using samples in which onemagnetic component is dominant or in which differentcomponents have a wide range of overlapping coercivities.In both cases the choice of 5 independent parameters forthe two scaling operations (p, Mrs, H*1/2, M0, and a) is

sufficient to achieve an excellent linear relationshipbetween scaled field and scaled magnetization. In case ofpopulations with drastically different coercivity ranges (i.e.,magnetite and hematite) the scaling method is lesseffective, but in this case the separation of the differentcomponents is also less critical and can be performed evendirectly on the measured curve.

B4. Plotting the Residuals (Figure 3d)

[45] Once the measured curve is scaled with respect tofield and magnetization, the deviation of the scaled curvefrom the least squares line is plotted. We will call this

Table A2. A Summary of Properties of the SGG Distribution and the Relative Error Estimationsa

Distribution Properties Relation With the Distribution Parameters RCF InequalityMinimum c2 Fitting

(One Unknown Parameter)

Optimized c2(q) : n l 1

a confidence limits: [cnl1;a, cnl1;1a]

for n l) 1: n l uaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 n l pGeneral parameter q q ffiffi

2p

p1=4dyffiffiffiffiffiffiffisxp

R11

@f x @q 2 dxf x

1=2 q dyffiffiffiffiffix

p

R11

@f x @q 2 dx

1=2Amplitude m dm ffiffiffi2pp1=4dy ffiffiffiffiffiffiffiffiffisxp dm ffiffiffi2pp1=4dy ffiffiffiffiffiffiffiffiffisxpMean m2 = E(x) m s6 1 0:856k dm2

ffiffiffi2

pp1=4 dy

m

ffiffiffiffiffiffiffiffiffiffiffis3x

pdm2 2p1=4 dym


p

SD: s22 = E[(x m2)2] s2(1 + 1.856k)(1 jsj/3) ds2 p1=4 dym


pds2 2

ffiffiffi2

p3p1=4 dy

m


p

Skewness s: E[(x m2)3]/s23 6 sgn q (1 q)2 (1 + 1.856k) ds 4 2p 1=4dymffiffiffiffiffiffiffiffiffisx

pds 48ffiffiffiffi

13p p1=4 dy

m

ffiffiffiffiffiffiffiffiffisx

p

Kurtosis k: E[(x m2)4]/s22 3 2 p dk 3:204 dymffiffiffiffiffiffiffiffiffisx

pdk 8:243 dy

m


p

aThe following notations are used: n is the number of measurements, l the number of estimated parameters, and ua

the a-quantile of the standardizedGaussian distribution (u0.95 = 1.6). Other notations are explained in the text. The second column gives approximate estimations of some distributionproperties when p ! 1 and q ! 2. The third column gives the minimum error estimations of all distribution parameters, according to the RCF inequality.The last column gives the error estimations of the minimum c2 fitting method, when only one parameter is unknown.

Table A3. Summary of Error Estimations for the Parameters of a SGG Distributiona

DistributionMoments

Minimum c2 Fitting(With l = 1)

Num. Est.(l = 1)

Minimum c2

Fitting (With l = 3)Num. Est.

(l = 3)Num. Est.

(l = 5)

Amplitude m dm ffiffiffi2pp1=4dy ffiffiffiffiffiffiffiffiffisxp 0.993 dm ffiffiffi3pp1=4dy ffiffiffiffiffiffiffiffiffisxp 1.203 1.293Mean m2 dm2 2p1=4 dym


p0.990 dm2 2p1=4 dym


p1.021 1.032

SD s2 ds2 ffiffi

83

qp1=4 dy

m


p0.997 ds2 p1=4 dym


p1.139 2.643

Skewness s ds 48ffiffiffiffi13

p p1=4 dym


p0.883 1.411

Kurtosis k dk 8:243 dym


p0.698 2.032

aThe errors are calculated analytically and numerically for the minimum c2 fitting method. The numerical error estimation is obtained by fitting 1000simulated SGG distributions. These distributions are obtained by adding a Gaussian noise signal of known amplitude to a standardized SGG distribution(m = 1, m = 0, s = 1, q = 1, and p = 2). The numerical estimates are reported as the ratio between the results of the numerical simulation and the analyticalequations reported in the second column. The second and the third columns refer to a minimum c2 fitting with one unknown parameter (m, m, s, q, orp),

the fourth and fifth columns to a minimum c2

fitting with three unknown parameters (m, m, and s), and the last column to a minimum c2

fitting where allparameters are unknown.



23/25

deviation the residuals curve. At this step, measurementerrors are enormously enhanced, as can be seen by compar-ing the residuals with the original measured curve (notshown in Figure 3d). The estimated maximum measurementerror is plotted in the form of a band around the residualcurve. If the error estimation entered by the user wascorrect, the amplitude of the random oscillations of the

residual curve should show the same order of magnitude asthe displayed errors.

B5. Scaling the Residuals (Figure 3e)

[46] Generally, the residuals generate a sinusoidal curve,which is more or less quenched at one end. As in step 2,the field axis can be rescaled with a power transformationin order to approach a quite regular sinusoidal curve, whichlater can be filtered in a more effective way. After this newrescaling step, the residual curve is almost sinusoidal. ItsFourier spectrum is concentrated in a narrow band around adominant wavelength, so that a simple low-pass filterwould easily remove the high-frequency measurement

noise.B6. Filtering the Residuals (Figure 3f)

[47] The residual curve is now ready to be filtered inorder to remove the measurement noise. The filter appliedby the program is a modified Butterworth low-pass filter,defined by:

F n 11 nb0=nb 1=2b B3

where n is the frequency of the spectrum, n0 the so-calledcutoff frequency, and b3 1 the order of the filter. The filterparameters n0 and b are chosen by the user. Details of the

residual curve with an extension smaller than 1/n0 on thefield axis are filtered out. The sharpness of the filter iscontrolled by its order b: b ! 1 gives a cutoff filter. Thefilter parameters should be chosen so that the measurementerror is suppressed without changing the global shape of thecurve. This condition is met by choosing the smallest valueofn0, by which the difference between the filtered and theunfiltered curve attains the same maximal amplitude as theestimated measurement errors. The choice of larger valuesof n0 leads to a coercivity spectrum that fits the measuredcurve better but still contains an unremoved component ofthe measurement errors. The choice of larger values of n0may produce a change in the shape of the curve andsuppress significant details.

B7. Calculating the Filtered Demagnetization Curve

[48] Now, the filtered residuals are converted back to theoriginal curve by applying the steps 25 in reverse order.The result is a demagnetization curve, which is supposed tobe free of measurement errors.

B8. Calculating and Plotting the CoercivityDistribution

[49] The filtered demagnetization curve obtained at point7, the coercivity distribution is calculated as the absolutevalue of the demagnetization curve. The user can choosebetween a linear, a logarithmic and a power field scale. The

maximum amplitude of the error of the coercivity distribu-tion is estimated in the program by comparing the measure-

ment curve with the filtered curve. The error estimation isdisplayed as an error band on the plot (Figure 1).

Appendix C: Error Estimation

[50] A measured distribution is given by yi = f(xi) + dyi,where (xi,yi) is a measurement point and dyi the related

measurement error. In following we assume the measure-ment error to be ergodic (i.e., statistically independent of x)

C1. The RCF Theorem

[51] Letf(xjq) be a PDF with distribution parameterq, andX1, . . ., XN a set of N realizations of the statistic variate X.The variance var q of the parameter q estimated with this setof realizations obey the RCF inequality [Cowan, 1998]:

var q

1NPNi1

f xijq @q lnf xijq 2C1

We use the RCF inequality to calculate var^q when f(xjq) ismeasured directly, instead of X1, . . ., XN. For this propose,

we imagine that f(xjq) is calculated from a set of Nrealizations by counting the numbers Ni of realizationswhich belong to given intervals of amplitude xi around aset of reference points x1, . . ., xn. Consequently, yi = Ni/xiand yi ! f(xijq) for N ! 1, xi ! 0. The probabilitydistribution of Ni is a Poisson distribution with expectedvalue E(Ni) = Nf(xi)xi and variance var(Ni) = E(Ni).Because var(Ni)/E

2(Ni) = vardyi/yi2, we obtain E(Ni) = yi

2/vardyi. Inserting the sum of all Ni into (C1) gives:

var q

1

Pni1

yi

vardyi 2 Pn

i1@qf xijq 2

f xijq

C2

For equally spaced reference points, xi+1 xi = x, and ifx ! 0 the summands in (C2) are conveniently replaced byintegrals:

var q

xR11

f2 xjq vardy x dx

R11

@qf xjq 2f xjq dx

C3

A further simplification is obtained by assuming themeasurement errordyi to be ergodic. Then, vardyi = (dy)

2 and:

var q x dy

2

R11

f2 xjq dx R11

@qf xjq 2f xjq dx

C4

If f(xjq) is a Gauss distribution with variance s2, simplifiesfinally to (8).

C2. Error Estimation With Unbiased FittingMethods

[52] Both the maximum likelihood (ML) and the mini-mum c2 fitting method are asymptotically identical andabsolute efficient for n ! 1, where n is the number ofmeasured points [Cowan, 1998]. We handle therefore onlythe minimum c2 method, which is directly related to the

Pearsons c2

goodness of fit test (see section 2.3). Considera set of N realizations X1, . . ., XN of the statistic variate X,



24/25

and a model distribution f(xjQ) which depends on thedistribution parameters Q = (q1, . . ., qk). The c

2 estimator isgiven by:

c2 Q Xni1

Ni xi f xijQ 2xi f xijQ C5

where Ni is the number of realizations which belong to aninterval of amplitude xi around a given value xi. Theminimum c2 estimate Q is the value of q which minimizesc2(Q). In our case, the individual realizations are unknown,but a measure of f(x, Q) is given. As shown before, eachmeasurementyi off(xi,Q) is related to a numberni = yi

2/vardyi of realizations. In this case, (C5) can be written as:

c2 Q Xmi1

yi f xijQ 2vardyi

C6

If the measurement error is ergodic, var dyi = (dy)2 and (C6)

simplifies to:

c2 Q ffi 1dy 2

Xmi1

yi f xijQ 2 C7

Equation (C7) is proportional to the mean quadratic error,and the c2 fitting method converge to a simple least squaresfitting. It should be noted that this result holds only as far asdy(x) is independent of x, and jdyij ( f(xijQ). If for instancethe relative error dyi/yi is ergodic instead ofdyi, c

2(Q) is nolonger related to the mean quadratic error. The minimizationofc2(Q) is performed by setting:

@c2

Q =@qi 0 ; i 1 . . . k C8The estimate Q is a solution of (C8). The variance Q ofQ isobtained by linearizing (C8) for Q ! Q:

varp JTJ 1 Jy J J1; . . . ; Jn T ; y dy21; . . . ; dy2n

Ji J2i1; . . . ;J2ik

;Jij @f xijQ

@pj

C9

If y is ergodic and the measuring points are equally spaced,(C9) can be conveniently approximated with integrals:

varp x dy 2C1C C1; . . . ;Ck T ;Ci C2i1; . . . ;C2ik

; C11; . . . ;Ckk

Cij R

X

@f xjQ @pi

@f xjQ @pj

dx

C10A particularly simple case of (C10) is (9), where only oneparameter is optimized (Q = q).

Notation

H Magnetic field

M Sample magnetizationm Magnetization of a component

f A general distribution functionq A general distribution parameterq An estimate of q

GG Generalized Gauss distributionSGG Skewed Generalized Gauss distributionm2 Mean value of a distributions2 Standard deviation of a distribution

s Skewness of a distributionk Coefficient of excess kurtosis of a distributionm Distribution parameter for the means Distribution parameter for the standard deviationq Distribution parameter for the skewness

p Distribution parameter for the kurtosis

[53] Acknowledgments. I would like to thank S. Spassov for testingCODICA and suggesting many useful improvements. I thank him and M.Fuller for providing me with some test samples. I am grateful to R. Enkinand M. Jackson for their constructive reviews and W. Lowrie for helpingme with the writing task. This work was supported by the ETH researchproject 0-20556-00.

ReferencesBailey, M. E., and D. J. Dunlop, Alternating field characteristics of pseudo-

single-domain (2 14 mm) and multidomain magnetite, Earth Planet. Sci.Lett., 63, 335352, 1983.

Carter-Stiglitz, B., B. Moskowitz, and M. Jackson, Unmixing magneticassemblages and the magnetic behaviour of bimodal mixtures, J. Geo-phys. Res., 106, 26,397 26,411, 2001.

Cisowski, S., Interacting vs. non-interacting single-domain behaviour innatural and synthetic samples, Phys. Earth Planet. Inter., 26, 5662,1981.

Cowan, G., Statistical Data Analysis, Oxford Univ. Press, New York, 1998.Dunlop, D. J., The rock magnetism of fine particles, Phys. Earth Planet.

Inter., 26, 126, 1981.Dunlop, D. J., Coercive forces and coercivity spectra of submicron magne-

tites, Earth Planet. Sci. Lett., 78, 288295, 1986.Egli, R., and W. Lowrie, The anhystere

analysis of the field dependence of remanent magnetization curves

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