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Geophysical Journal International
, Volume 213 (2) – May 1, 2018

10 pages

/lp/ou_press/quantitative-interpretation-of-the-magnetic-susceptibility-frequency-QTnUzeaztt

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.
- ISSN
- 0956-540X
- eISSN
- 1365-246X
- D.O.I.
- 10.1093/gji/ggy007
- Publisher site
- See Article on Publisher Site

Summary Low-field mass-specific magnetic susceptibility (MS) measurements using multifrequency alternating fields are commonly used to evaluate concentration of ferrimagnetic particles in the transition of superparamagnetic (SP) to stable single domain (SSD). In classical palaeomagnetic analyses, this measurement serves as a preliminary assessment of rock samples providing rapid, non-destructive, economical and easy information of magnetic properties. The SP–SSD transition is relevant in environmental studies because it has been associated with several geological and biogeochemical processes affecting magnetic mineralogy. MS is a complex function of mineral-type and grain-size distribution, as well as measuring parameters such as external field magnitude and frequency. In this work, we propose a new technique to obtain quantitative information on grain-size variations of magnetic particles in the SP–SSD transition by inverting frequency-dependent susceptibility. We introduce a descriptive parameter named as ‘limiting frequency effect’ that provides an accurate estimation of MS loss with frequency. Numerical simulations show the methodology capability in providing data fitting and model parameters in many practical situations. Real-data applications with magnetite nanoparticles and core samples from sediments of Poggio le Guaine section of Umbria–Marche Basin (Italy) provide additional information not clearly recognized when interpreting cruder MS data. Caution is needed when interpreting frequency dependence in terms of single relaxation processes, which are not universally applicable and depend upon the nature of magnetic mineral in the material. Nevertheless, the proposed technique is a promising tool for SP–SSD content analyses. Environmental magnetism, Palaeomagnetism, Rock and mineral magnetism, Inverse theory, Numerical modelling 1 INTRODUCTION The existence of magnetic minerals of grain size in the superparamagnetic (SP) to stable single domain (SSD) transition is usually recognized by the decrease of magnetic susceptibility (MS) with frequency, measured at low external fields. This decrease is quantified by a proxy termed as ‘frequency effect’, usually expressed as per cent of the difference of the MS measured at lower and high frequencies with respect to the value taken with the lower frequency (Dearing et al. 1996; Hrouda 2011). The frequency effect depends on specific frequency setups of the experimental apparatus and usually is used to qualitatively identify samples with SP minerals. Testing frequencies usually span the interval between 1 and 10–20 kHz, since viscous relaxation from minerals at the SP–SSD transition provides a susceptibility loss in this frequency interval. The frequency effect has also been called as percentage loss of susceptibility (Dearing et al. 1996). In classical palaeomagnetic studies, the frequency effect serves as a proxy to screen rock samples able to record stable magnetization along the geological time. Stable magnetizations are recorded in ferrimagnetic minerals when their grain sizes near the critical blocking volume of SSD grains. Due to their stable magnetization, SD particles show minor frequency effect because strongly imprinted magnetization is only deflected under weak external fields (Kodama et al. 2014). However, smaller particles below this critical blocking volume embody SP behaviour making them unable to sustain magnetization at room temperatures. Critical size (diameter) is about 30 nm for spherical particles of magnetite (Shcherbakov & Fabian 2005). SP fractions in soil and sediments are of interest in many environmental studies because such fine-ultrafine fractions have been regarded as indicative of environmental and biogeochemical processes (Maher & Taylor 1988; Dearing et al. 2001; Roberts et al. 2012). MS may be enhanced either by an increase of SP particle contents (e.g. by pedogenic processes) or due to a fining of grain size to pseudo-single domain or multidomain (MD) particles (e.g. by transport processes). Frequency-dependent susceptibility (FDS) has proven to be a useful technique for detecting viscous SP particles near the SP–SSD threshold size (Liu et al. 2012). The grain-size distribution (GSD) of SP particles is important in environmental magnetism because these particles may reflect palaeoenvironmental processes such as precipitation and pedogenesis and environmental contamination by anthropogenic pollutants (Kodama 2013). It is possible to probe the SP particle assemblage and assess its GSD by MS measurements using alternating low fields at different operating frequencies (e.g. Worm 1998; Shcherbakov & Fabian 2005; Egli 2009; Hrouda et al. 2013; Kodama 2013). Therefore, knowledge of grain-size-dependent magnetic characteristics may be relevant for environmental studies. When interpreting MS frequency dependence caused by SP grains, careful must be taken for samples containing titanomagnetites, coarse-grained pyrrhotites or hematite, since these minerals present magnetic field dependence that can in some cases be mistaken for frequency dependence (de Wall 2000). Jackson et al. (1998) reported a strong field dependence of AC susceptibility measurements for synthetic titanomagnetites (Fe3xTixO4) and certain basalts from Hawai and Iceland. The observations on field-dependent susceptibility are important because it can be misinterpreted as resulting from grain size and compositional variations (Vahle & Kontny 2005). The frequency effect is a particular parameter associated with FDS, which can be better observed when taking real (in phase) and imaginary (out of phase) measurements for broad frequency intervals. Shcherbakov & Fabian (2005) developed a technique to determine volume distribution using real and imaginary parts of FDS. Hrouda et al. (2013) use the out-of-phase susceptibility to estimate GSD. In most FDS studies, the out-of-phase susceptibility has been associated with viscous relaxation although other mechanisms may give out-of-phase responses as it is the case of low-field hysteresis and eddy currents in more conductive materials (Jackson 2003–2004). Zero-setting fluctuations also compromise out-of-phase measurements when using common three-frequency susceptibility meters (e.g. AGICO Kappabridge instruments) that operate with the susceptibility bridge method (Hrouda et al. 2015). Most two-frequency susceptibility meters (e.g. Bartington Instruments) based on the oscillating method are more effective in measuring the MS real part (Kodama 2013). Low reliability of out-of-phase data has directed most of the interpretative procedures towards MS phase data, and in particular to the frequency effect. This work develops a quantitative technique to interpret the real part of three-frequency low-field mass-specific MS data, usually interpreted under a qualitative basis. We apply the Debye relaxation model to determine a parameter termed as ‘limiting frequency effect’ (LFE), which is comparable to the experimental frequency effect but not dependent on particular frequency values used in data acquisition. As we show, the LFE can be used to estimate relative grain-size variations of magnetic particles in the SP–SSD transition. Numerical simulations are presented to validate the inversion procedure and real measurements taken with a MFK1-FA AGICO Multi-Function Kappabridge of synthetic magnetite nanoparticles and samples from a geological section illustrate the utility of the proposed technique. 2 THEORY For low-intensity external harmonic fields, the mass-specific MS, χ(ω), can be represented by a Debye relaxation function as \begin{equation}\chi (\omega ) = {\chi _{{\rm{hf}}}} + {\rm{\Delta }}\chi \frac{1}{{1 + i\omega \tau }}\end{equation} (1)for angular frequency ω = 2πf and frequency f [Hz], where τ is relaxation time constant, $$i = \sqrt { - 1} $$, χhf is the FDS high-frequency limit such that χhf = χlf − Δχ, with χlf as its low-frequency limit. Fig. 1 illustrates the parameters in eq. (1) as well as specific frequency values of common three-frequency susceptibility meters. The frequency effect is evaluated by the difference of the susceptibility taken with low and high frequencies, normalized by the susceptibility taken with the higher frequency, expressed in percentage (Hrouda 2011). For specific frequencies, the real part of FDS, χr ≡ χ΄(ω), of χ(ω)can be written as \begin{equation}{\chi _r} = {\chi _{{\rm{hf}}}} + {\rm{\Delta }}\chi \frac{1}{{1 + {{\left( {\omega \tau } \right)}^2}}}.\end{equation} (2) Figure 1. View largeDownload slide Theoretical curve of frequency-dependent magnetic susceptibility. The solid line represents model response as captured in broad-band frequency MS and the symbols represent three-frequency data as s measured with Kappabridge instruments. Model parameters are χhf, Δχ = χlf− χhf and τ. The grey dashed lines indicate the position of χhf and τ in the susceptibility and frequency axis, respectively. The black dashed lines represent the inversion constrains (τmin and τmax in the frequency range). Figure 1. View largeDownload slide Theoretical curve of frequency-dependent magnetic susceptibility. The solid line represents model response as captured in broad-band frequency MS and the symbols represent three-frequency data as s measured with Kappabridge instruments. Model parameters are χhf, Δχ = χlf− χhf and τ. The grey dashed lines indicate the position of χhf and τ in the susceptibility and frequency axis, respectively. The black dashed lines represent the inversion constrains (τmin and τmax in the frequency range). Parameter Δχ is the difference between asymptotic limits of the FDS response. Measured FDS responses are usually smaller than Δχ, due to the limited frequency range of common susceptibility meters. A flat susceptibility response is usually indicative of pure SP samples or pure SSD samples. As discussed by Moskowitz (1985), other mechanisms may contribute to viscous behaviours of magnetic materials such as: thermal fluctuations, after-effects associated with the diffusion of ferrous ions and vacancies, time-dependent changes associated with chemical alteration and eddy currents. MD magnetic particles can also produce a viscous signal through disaccommodation phenomena (e.g. Dunlop & Özdemir 1997; Muxworthy & Williams 2006), originated from the displacement of the domain walls during the application of an external field. As illustrated in Fig. 1, a flat susceptibility response is obtained if viscous components are completely absent (i.e. Δχ = 0), or if the inverse of the viscous time constant 1/τ is far beyond the range of measured frequencies. Accordingly, for low-field susceptibility meters working in the frequency range spanned by the three frequencies shown in Fig. 1, SP-SSD relaxation can be assumed as dominating the FDS response. The procedure developed in this work may in principle, be extended to a broader frequency range, to capture other relaxation times. In this case, eq. (2) can be modified to allow more relaxation times, for example. In fact, the frequency effect serves to screen samples previously to palaeomagnetic studies in which magnetization records (intensity and direction) are targeted. For engineered magnetic materials in the pure SP range (e.g. nanoparticles and ferrofluid suspensions) frequency effect still can be observed as a result of lack of grain-size uniformity and particle coupling in aggregates. As presented in eq. (1), parameter Δχ is positive because in SP–SSD transition χlf > χhf. Assuming that low and high FDS limits are associated with limiting responses in the SP–SSD transition, the low-frequency limit, χhf, is an expression of the susceptibility of SP particles, κSP (Worm 1998; Worm & Jackson 1999; Shcherbakov & Fabian 2005; Hrouda 2011; Kodama 2013; Kodama et al. 2014). This is given by Dunlop & Özdemir (1997) as \begin{equation}{\kappa _{{\rm{SP}}}} = \frac{{{\mu _0}v{M_s}}}{{3{k_B}T}},\end{equation} (3)in which Ms is the spontaneous saturation magnetization of the mineral grain, v is particle volume of an uniform SP fraction (spherical grains assumed), μ0 = 4π × 10− 7 Hm−1 is the vacuum magnetic permeability, kB is the Boltzmann constant (1.38 × 10−23 J K−1), and T the temperature. Under such assumptions, χhf is an expression of the susceptibility κSD of non-interacting SSD (or pseudo SSD) particles (Worm 1998; Worm & Jackson 1999; Shcherbakov & Fabian 2005; Hrouda 2011; Kodama 2013; Kodama et al. 2014). According to Stoner & Wohlfart (1948) theory \begin{equation}{\kappa _{{\rm{SD}}}} = \frac{{2{M_s}}}{{3{H_k}}},\end{equation} (4)where Hk is the particle microscopic coercivity, that is related to the macroscopic coercivity, Hk, by \begin{equation}{H_k} = 2.09{H_c}.\end{equation} (5) As in eq. (1), τ is a relaxation time of non-interacting magnetic SD particles of uniform volume v described by Néel (1949) as \begin{equation}\tau = {\tau _0}\ {\rm{exp}}\left( {\frac{{vK}}{{{k_{\rm{B}}}T}}} \right),\end{equation} (6)where K is the anisotropy energy per grain volume (Jm−3) and τ0 is the electron time constant. The quantity vKexpresses the anisotropy energy that holds the magnetization along a preferential direction (easy axis) against random flipping due to thermal energy kBT. Typical values for τ0 range from 10−10 to 10−8 s (e.g. Dormann et al. 1996; Worm 1998) according to mineral crystalline framework and composition. The magnetic anisotropy (Worm 1998; Worm & Jackson 1999) is \begin{equation}K = \frac{1}{2}{\mu _0}{H_k}{M_s}.\end{equation} (7) For common geological materials quantities Δχ and Δχ/χlf can be regarded as proxies to recognize samples with SSD particles, since they are minor (or zero) when the FDS response (real part) is flat. The ratio Δχ/χlfis comparable with the experimental frequency dependence parameter χfd, which, however, is based on the difference between the susceptibility measured at two frequencies. The two parameters coincide in the limit of broad frequency range. Otherwise, χfd may capture none or only a fraction of the SP susceptibility variation, depending on the time constant of viscous phenomena and the frequency range of measurements. For this reason, Δχ/χlf can be termed as ‘LFE’, to distinguish it from the ‘experimental frequency effect’ that is obtained from given frequencies. Δχ/χlf provides a quantity which can be better evaluated with broad range FDS data. To obtain quantitative estimates from the LFE, let us define a ‘transition parameter’ Ft (subscript t standing for transition), given by \begin{equation}{F_t} = {\left( {1 - \frac{{{\rm{\Delta }}\chi }}{{{\chi _{{\rm{lf}}}}}}} \right)^{ - 1}}.\end{equation} (8) This parameter accounts either for the amplitude of the susceptibility variation with frequency as for the relative variation of the particle grain size in the SP–SSD transition. In terms of susceptibility variation, for pure ensemble of SD particles, using eqs (3) and (4), we obtain κSP = FtκSD. In this case, Ft provides how much the SP content enhances the MS. In terms of volume variation, particle volume, v, can be expressed in terms of a reference volume \begin{equation}{v_c} = \frac{{2{k_{\rm{B}}}T}}{{{\mu _0}{H_k}{M_s}}},\end{equation} (9)for v = Ftvc. Because Δχ/χlf ≥ 0, a transition parameter Ft > 1 means that particle volume v; is greater than reference volume vc; and Ft = 1means that there is no grain-size variation. In this sense, volume vc stands for a minimum volume for the particle assemblage affecting the FDS response. The transition parameter can be regarded then as a quantitative proxy about grain-size coarsening with respect to a reference volume (vc) in the SP–SSD transition. The evaluation of this reference volume from τ requires material properties (Hk and Ms) that are specific to given mineral assemblage. The evaluation of Ft otherwise requires only the LFE parameter, thus meaning that relative grain-size variation can be obtained from LFE despite nothing about its absolute value. The quantity$$\ F_t^{ - 1}$$ otherwise can be regarded as expressing a fining proxy, about how much fine or ultrafine particles are shifted from average particles with SD–SSD response. According to eq. (5), the determination of relaxation time, τ, provides volume estimates through \begin{equation}v\ = \frac{{2{k_{\rm{B}}}T}}{{{\mu _0}{H_k}{M_s}}}\ {\rm{ln}}\left( {\frac{\tau }{{{\tau _0}}}} \right),\end{equation} (10)which as formerly pointed out requires previous knowledge of parameters HkMs and τ0, and then knowledge about mineral type. Parameter Ft, otherwise, does not depend on such specific parameters and as such can be linked to grain-size variation with no assumption on mineral composition. Parameter Ft then can be interpreted in terms of grain-size variations even when the evaluation of the relaxation time is hindered by three-frequency data sets. As discussed next, parameters Δχ and χlf can be better estimated from three-frequency data using specific data inversion procedures providing accurate LFE estimates. We stress that results from the proposed technique rely on assumptions of a single relaxation process, which in many cases is valid for low-field measurements, not enough to overcome higher coercivity of other relaxation processes. These assumptions implicitly require previous knowledge about the nature of magnetic minerals in order to better evaluate LFE results. 2.1 Inversion procedure Assuming frequency effect as caused by a single relaxation time, an inversion procedure can be formulated aiming to determine Debye model parameters χhf, Δχ and τ. Let us consider a data set with n measurements (n = 3 in present applications) of $$\chi _r^0( {{\omega _j}} ) \equiv \chi _j^0$$ with frequencies fj, j = 1,…, n. The unknown model parameters can be determined by minimizing a functional $$Q \equiv Q( {\chi _{1, \ldots ,n}^0,{\chi _{{\rm{hf}}}},{\rm{\Delta }}\chi ,\tau } )$$ such that \begin{eqnarray}Q\ \left( {\chi _{1, \ldots ,n}^0,{\chi _{{\rm{hf}}}},{\rm{\Delta }}\chi ,\tau } \right) \!=\! \mathop \sum \limits_{j = 1}^n {\left[ {\chi _j^0 {-} \chi _j^c\left( {\chi _{1, \ldots ,n}^0,{\chi _{{\rm{hf}}}},{\rm{\Delta }}\chi ,\tau } \right)} \right]^2},\end{eqnarray} (11)in which $$\chi _j^c( {\chi _{1, \ldots ,n}^0,{\chi _{{\rm{hf}}}},{\rm{\Delta }}\chi ,\tau } )$$ is the model response evaluated according to eq. (2). A solution $$( {{{\hat{\chi }}_{{\rm{hf}}}},\widehat {\Delta \chi },\hat{\tau }})$$ minimizing the functional Q can be obtained by solving a non-linear problem in which residuals between measured and evaluated values are minimum, as in eq. (11) according to the Euclidean norm of residuals. The solution of this non-linear problem requires a set of initial solutions, or at least a single initial solution, to initiate a searching procedure intending to find local or global minima. Convergence is achieved when Q(χhf, Δχ, τ) is below a threshold, ε, determined by noise level in data. Flow chart in Fig. 2 illustrates the minimization procedure we adopted, which combines an initial stage with genetic algorithm (Chipperfield & Fleming 1995) to find initial solutions $$( {{{\hat{\chi }}_{{\rm{hf}}}},\widehat {\Delta \chi },\hat{\tau }} )$$ whose outputs are used to feed a Marquadt–Levemberg procedure to improve convergence. The algorithms used for this procedure are ‘ga’ and ‘fmincon’, implemented in the MATLAB programming environment. This combined optimization procedure is repeated N times in order to obtain a set of N alternative solutions, all of them allowing data fitting below ε. This set of solutions is used to evaluate model uncertainty for unknown $$( {{{\hat{\chi }}_{{\rm{hf}}}},\widehat {\Delta \chi },\hat{\tau }} )$$. Figure 2. View largeDownload slide Inversion procedure flowchart. The genetic algorithm is initially applied to obtain an initial solution. Once convergence is achieved, the output solution is used as an initial solution by the constrained non-linear optimization procedure. If the output solution passes the convergence test, it is stored and the procedure starts once again to obtain N alternative solutions to test model uncertainity. A single model solution is obtained by the mean of N alternative solutions. Figure 2. View largeDownload slide Inversion procedure flowchart. The genetic algorithm is initially applied to obtain an initial solution. Once convergence is achieved, the output solution is used as an initial solution by the constrained non-linear optimization procedure. If the output solution passes the convergence test, it is stored and the procedure starts once again to obtain N alternative solutions to test model uncertainity. A single model solution is obtained by the mean of N alternative solutions. Due to poor data coverage of the three-frequency data additional constraints may be required to obtain reliable results. These constraints impose positivity for unknown susceptibilities (0 ≤ χhf ≤ χmax) and that the relaxation time is expressed within the experimental frequency range. This constraint implicitly assumes that frequency range (fmin, fmax) of the experimental procedure is able to capture the FDS response, a well-settled assumption when using the frequency effect to identify SP minerals. The feasibility interval for the relaxation time (τmin ≤ τ ≤ τmax) is given by τmin = 1/2πfmax and τmax = 1/2πfmin. The presence of diamagnetic minerals does not compromise the analysis, because the contribution of diamagnetic minerals to MS would be much smaller than SP MS. Ergo χlf < 0 is indicative of diamagnetic minerals dominance not well suited to be interpreted with Neel's theory. Negative susceptibility however is observed in some carbonaceous sediment with low concentrations of magnetite. Under conditions previously discussed, the unknown Debye parameters can be determined by solving a constrained minimization problem, in which the functional $$Q( {\chi _{1, \ldots ,n}^0,{\chi _{{\rm{hf}}}},\Delta \chi ,\tau } )$$is minimized, subject to 0 ≤ χhf ≤ χmax (χmax being approximately one order of magnitude higher than χ measured at the lowest frequency) and τmin ≤ τ ≤ τmax. This problem was solved according to the algorithm illustrated in Fig. 2. 3 NUMERICAL SIMULATION Synthetic tests were conducted to evaluate the capacity of the inversion procedure in recovering Debye parameters by fitting synthetic data with three-frequency values of the Kapabridge susceptibility meter. As formulated in eq. (11), the Debye model is described by three parameters, therefore requiring at least three MS readings to complete the system of non-linear equations. We simulate model responses using the AGICO Kappabridge frequencies of 976, 3904 and 15616 Hz, a common frequency range regarded as well suited to capture FDS response in SP–SSD transition (Pokorny et al. 2006). Four models are tested to simulate different positions of the unknown relaxation times with respect to the experimental frequency range. These numerical simulations aim to verify the efficiency of the inversion procedure in retrieving the Debye parameters from three-frequency data regarding different models. Parameters χhf = 10− 3 (SI), Δχ = 10− 5(SI) are the same in all models but in model (1) τ = 10 μs, in models (2) and (3) τ is shifted towards χlf (lower frequency), and in model (4) towards χhf (higher frequency). In model (3), τ is beyond the range expected for the SP–SSD transition. Model response for broad frequency response and specific three-frequency Kappabridge synthetic data were evaluated from each model (Fig. 3). Normal, zero-mean random noise of 1 per cent of the amplitude was added synthetic data, which is higher than accepted error levels of 0.1 per cent reported for common MFK1-FA Kappabridge readings. The frequency response evaluated from tested model illustrates the incompleteness of three-frequency data when intending to capture FDS loss. Table insets in Fig. 3 summarize true and estimated model parameters obtained from data inversion. Model uncertainty can be accessed by mapping the misfit functional Q for each model solution. These maps were obtained by varying parameters Δχ and τ nearby the inverted solution, parameter χhf kept as invariant. Q is mapped (evaluated) within the imposed inversion constrains, 10−7 ≤ Δχ (SI) ≤ 10−3 and 10−6 ≤ τ (s) ≤ 10−3. Residual misfits are considered acceptable when Q ≤ 0.01 per cent (white contours in Fig. 3). This misfit level allows a kind of uncertainty assessment for model parameters obtained from data inversion. Figure 3. View largeDownload slide Numerical simulation of synthetic data from models 1 to 4. (a), (c), (e) and (g) Model response (solid lines) and data simulated at Kappabridge frequencies (symbols). The vertical dashed lines indicate the position of the relaxation times in the frequency range. Inset tables summarize Debye parameters applied and estimated in the numerical tests. (b), (d), (f) and (h) Minimizing functional Q for a range of variation of parameters Δχand τ, and a fixed value of χhf (true value). Black and white contours represents misfits of 0.1 and 0.01 per cent, respectively. Figure 3. View largeDownload slide Numerical simulation of synthetic data from models 1 to 4. (a), (c), (e) and (g) Model response (solid lines) and data simulated at Kappabridge frequencies (symbols). The vertical dashed lines indicate the position of the relaxation times in the frequency range. Inset tables summarize Debye parameters applied and estimated in the numerical tests. (b), (d), (f) and (h) Minimizing functional Q for a range of variation of parameters Δχand τ, and a fixed value of χhf (true value). Black and white contours represents misfits of 0.1 and 0.01 per cent, respectively. The numerical simulation shows that the proposed inversion scheme can successfully estimate parameter χhf for the four implemented models. Parameter Δχ is well estimated for models 1–3 but not for model 4 (Fig. 3, inset tables). Parameter τ is more subject to uncertain estimates unless under well-constrained intervals defined by measuring frequencies (976–15616 Hz). Uncertainty in τ, however, does not compromise evaluations of χhf and Δχ and the evaluation of parameters derived from them. Larger relaxation times, such as in models 2 and 3, are not well solved for simulated three-frequency data. In general, however, the simulated frequency range is sufficient to obtain Debye parameters when relaxation times is in the order of 10 μs (models 1 and 4) as it is expected for FDS response from SP–SSD transition. These results outline conditions in which three-frequency data inversion are promising or unviable. To evaluate the inversion stability for the simulated models, a rigorous error analysis was conducted: (1) a set of measurements at the three frequencies was generated by adding random Gaussian errors to the synthetic data (10 per cent for 976 Hz and 5 per cent for 3904 and 15 616 Hz). These measurement errors are much higher than the error of 0.1 per cent reported for MK1-F1 Kappabridge (Hrouda 2011); (2) the inversion procedure was performed on the noisy data simulated in (1). Steps (1) and (2) were repeated 2000 times. The results are presented in Fig. 4. The analysis shows that χhf is accurately estimated for all models. The more sensitive parameters Δχ and τ are correctly estimated for models 1–3, even though a large variation occurs for model 4, compromising the mean and the standard deviation. The results presented in Fig. 4 show that inversion procedure is quite stable, except for very low relaxation times. Instability is observed in specific parameters with lower sensitivity, parameter τ in particular. However, parameter χhf is accurately estimated for all models and therefore, the LFE parameter is less impacted by the uncertainty in Δχ. Figure 4. View largeDownload slide Evaluation of inversion stability for models (a)–(d) 1, (e)–(h) 2, (i)–(l) 3 and (m)–(p) 4. (a), (e), (i) and (m) histogram of inversion residual; (b), (f), (j) and (n) histogram of parameters χhf; (c), (g), (k) and (o) histogram of Δχ and (d), (h), (l) and (p) histogram of τ. The x-axis scales represent the inversion constrains for each parameter. A set of measurements generated by adding random Gaussian errors to the synthetic data and the inversion procedure of the noisy data was performed. This procedure was repeated for 2000 times. Figure 4. View largeDownload slide Evaluation of inversion stability for models (a)–(d) 1, (e)–(h) 2, (i)–(l) 3 and (m)–(p) 4. (a), (e), (i) and (m) histogram of inversion residual; (b), (f), (j) and (n) histogram of parameters χhf; (c), (g), (k) and (o) histogram of Δχ and (d), (h), (l) and (p) histogram of τ. The x-axis scales represent the inversion constrains for each parameter. A set of measurements generated by adding random Gaussian errors to the synthetic data and the inversion procedure of the noisy data was performed. This procedure was repeated for 2000 times. 4 CONTROL EXPERIMENT Fine-grained magnetite samples were synthesized in laboratory by a termo-decomposition process (Fig. 5), in which an iron (III) salt is dissolved in a solvent with high boiling temperature ( > 200 °C) in the presence of a reducer substance, an oxygen donor and stabilizing agents (Gomes da Silva et al. 2011). Using this process, magnetite samples are produced with a controlled particle size of approximately 8 nm (inset in Fig. 5). Different magnetite concentrations (0.2, 1 and 5 per cent by weight) were dispersed in paraffin wax, and resampled for susceptibility measurements. Figure 5. View largeDownload slide Transmission electron microscopy (TEM) from nanomagnetite synthetic samples and grain-size distribution. MS frequency response can be produced by particles interaction in clusters (Courtesy, D. Gomes da Silva, S. H. Toma and Koiti Araki). Figure 5. View largeDownload slide Transmission electron microscopy (TEM) from nanomagnetite synthetic samples and grain-size distribution. MS frequency response can be produced by particles interaction in clusters (Courtesy, D. Gomes da Silva, S. H. Toma and Koiti Araki). The MS measurement procedure consisted of measuring each sample applying an oscillating field at a certain frequency, then changing the operating frequency. This procedure was repeated three times, to check data repeatability. As show in Fig. 6(a), the MS increases with magnetite content, varying from 2 to 100 × 10−5(SI). The measured susceptibilities are well fitted by those calculated from the inversion procedure, as shown by the cross-plot in Fig. 6(b). Figure 6. View largeDownload slide (a) Inversion results for the controlled experiment with nanoparticles of magnetite (mass percentage). Calculated (solid lines) and measured MS data (symbols), and corresponding relaxation times (vertical dashed lines) are presented for each data set. Red, magenta and purple represent magnetite concentrations of 0.2, 1 and 5 per cent, respectively. The minimum and maximum relaxation time constraints are indicated by arrows. (b) Cross-plots of measured and calculated data. Figure 6. View largeDownload slide (a) Inversion results for the controlled experiment with nanoparticles of magnetite (mass percentage). Calculated (solid lines) and measured MS data (symbols), and corresponding relaxation times (vertical dashed lines) are presented for each data set. Red, magenta and purple represent magnetite concentrations of 0.2, 1 and 5 per cent, respectively. The minimum and maximum relaxation time constraints are indicated by arrows. (b) Cross-plots of measured and calculated data. Parameters χhf and χlf estimated from the inversion linearly increase (with same rate of magnetite concentration, Fig. 7a). Although there are only three points for each linear fitting, the correlation coefficient R2 = 1 attests to the linear correlation. LFE also increases with magnetite content (Fig. 7b), but it does not follow the same trend observed for the susceptibility asymptotes χhf and χlf. The inversion procedure was repeated 200 times for each synthetic sample, and the standard deviation of the estimated parameters varied, in respect to the mean value of each sample, less than 0.35 per cent for χhf, less than 7 per cent for Δχ and less than 6 per cent for τ. Small deviations, as observed for the pure SP samples measured in this experiment, show the stability of the procedure when the SP–SSD assumption is valid. Figure 7. View largeDownload slide (a) Debye models parameters (χhf and χlf) and (b) LFE as a function of concentration of magnetite nanoparticles. The dotted, dashed and solid lines represent the linear fit obtained for each parameter as a function of magnetite concentration. Figure 7. View largeDownload slide (a) Debye models parameters (χhf and χlf) and (b) LFE as a function of concentration of magnetite nanoparticles. The dotted, dashed and solid lines represent the linear fit obtained for each parameter as a function of magnetite concentration. Inversion estimated parameters χhf, Δχ and τ, are shown in Table 1 with corresponding transition parameter (Ft) and the particle diameter (d). Particle diameter was calculated using eqs (5)–(8), and assuming τ0 = 10− 9s, Hc = 3.66 × 104A m−1 and Ms = 3.9 × 105 A m−1. The magnetite particles diameters estimated from the relaxation times obtained in the inversion were not accurate due to limited three-frequency band, possibly not wide enough to capture the full response of the magnetite particles viscous relaxation. Table 1. Physical parameters obtained for the controlled experiment. Fe3O4 (per cent) χH (SI) Δχ (SI) Ft τ (μs) d (nm) 0.2 2.8×10–5 ± 2×10–7 1.2×10–6 ± 2.5×10–7 1.042 ± 0.020 50.00 ± 2.89 19.40 ± 0.01 1.0 1.9×10–4 ± 6×10–7 7.3×10–6 ± 5.5×10–7 1.045 ± 0.004 13.24 ± 1.48 18.50 ± 0.38 5.0 9.1×10–4 ± 7×10–7 10.0×10–4 ± 7.3×10–7 1.108 ± 0.023 5.00 ± 0.00 17.90 ± 0.00 Fe3O4 (per cent) χH (SI) Δχ (SI) Ft τ (μs) d (nm) 0.2 2.8×10–5 ± 2×10–7 1.2×10–6 ± 2.5×10–7 1.042 ± 0.020 50.00 ± 2.89 19.40 ± 0.01 1.0 1.9×10–4 ± 6×10–7 7.3×10–6 ± 5.5×10–7 1.045 ± 0.004 13.24 ± 1.48 18.50 ± 0.38 5.0 9.1×10–4 ± 7×10–7 10.0×10–4 ± 7.3×10–7 1.108 ± 0.023 5.00 ± 0.00 17.90 ± 0.00 View Large The magnetite diameter was estimated assuming a few physical (τ0, Hk and Ms) constants that can actually have different values depending on mineral composition and crystalline framework. To analyse uncertainty in estimated parameters due to unknown physical constants, particle diameter was calculated for randomly parameters in intervals 10− 10 ≤ τ0 ≤ 10− 8 s, 2.45 × 104 ≤ Hk ≤ 4.1 × 104 Am−1 and 3.8 × 105 ≤ Ms ≤ 4.8 × 105 Am−1 common ranges expected for magnetite (Dunlop & Carter-Stiglitz 2006). Variations in particle diameter then were found in the 12 ≤ d ≤ 24 nm, which partially overlaps nominal diameter of the magnetite nanoparticles aggregates. For the three samples, the transition parameter Ft presents a subtle variation between 1.042 and 1.108 showing that it is not dependent on concentration of magnetic particles. In addition, since all samples were prepared with same particles, this invariance of this parameter is in agreement that grain-size variation should be the same for all tested samples. Higher Ft estimate for 5 per cent sample can be indicative of larger aggregates, as it may be expected in less-diluted samples. Parameter χhf otherwise is clearly correlated to the concentration of SP particles at least for the tested samples prepared with uniform-grain magnetic particles. 5 CASE STUDY As a first example of the application of the methodology to real data, we analyse the MS measurements of the Poggio le Guaine (PLG) core (Coccioni et al. 2012; Savian et al. 2016). The PLG core was drilled to provide a high-resolution magnetic and palaeoenvironmental record of the upper Barremian, Aptian and Albian stages (Coccioni et al. 2012). The basal section of the PLG core, from 86 to 96 m depth comprises limestones and shales from Maiolica Formation (Fm) and Marne a Fucoidi Formation (Fm). Maiolica Fm is composed mostly of white to light grey limestones and sparse few-centimetre-thick layers of black shales relatively rich in organic matter. Contacts between limestone and black shale are sharp and planar. Marne a Fucoidi Fm comprises dominantly grey to olive-green limestones interbedded with thick layers of black shale. The Selli Level represents a 1.95-m-thick interval of black shale in the PLG core, from 89.24 to 91.19 m depth (Coccioni et al. 2012). In this sector, Savian et al. (2016) identified the magnetochron M0r and constrained it with the available chemostratigraphic and biostratigraphic data. Relevant to this study is the contrast in palaeomagnetic stability between the weakly magnetic white limestones of Maiolica and the stable magnetizations measured in the Marne a Fucoidi marls. MS data were measured on samples collected every 2 cm.The measured susceptibilities are well fitted in the inversion procedure, as shown by the cross-plot in Fig. 8. The linear correlation obtained between measured and calculated MS values present a correlation coefficient R2 = 0.99, showing that the technique provides a successful fit with the measured data. Figure 8. View largeDownload slide Cross-plots of measured and calculated data from the Poggio le Guaine (PLG) core. Figure 8. View largeDownload slide Cross-plots of measured and calculated data from the Poggio le Guaine (PLG) core. The partial PLG section presented in Fig. 9 shows the measured MS, the mean inverted parameters χhf and χlf, as well as parameters LFE and $$F_t^{ - 1}$$ and the mean misfit after 480 inversions, at depths between 86 and 96 m, next to the transition between the two lithostratigraphic formations and the M0r polarity Chron defined by Savian et al. (2016). Measured MS vary from 0.7 × 10−8 to 4.5 × 10−8(SI) and estimated χhf and χlf vary from 0.4 × 10−8 to 5 × 10−8 (SI). LFE ranges from 20 to 70 per cent, while $$F_t^{ - 1}$$varies from 0.2 to 0.8 along the interpreted section. Figure 9. View largeDownload slide Depth section for PLG MS data: (a) interpretation from Savian et al. (2016), (b) measured FDS, (c) inverted parameters χhf and χlf, (d) LFE, (e) Ft−1 and (f) rms error. The grey dashed line indicates the depth of the Maiolica and Marne a Fucoidi formations. Figure 9. View largeDownload slide Depth section for PLG MS data: (a) interpretation from Savian et al. (2016), (b) measured FDS, (c) inverted parameters χhf and χlf, (d) LFE, (e) Ft−1 and (f) rms error. The grey dashed line indicates the depth of the Maiolica and Marne a Fucoidi formations. In general, measured MS values decrease with depth, from ∼ 4 × 10−8(SI) at 86 m to ∼1 × 10−8(SI) at 95 m following the lithological boundary between the white limestones of Maiolica Fm and the greenish marls of Marne a Fucoidi Fm. This trend however, is disconnected in the depth interval ∼89–92 m depth, where MS increases from ∼1 × 10−8 to 3 × 10−8 (SI). This depth interval comprehends most of the Selli Level interval, which is clay-rich layer, and the top of the Maiolica Fm. A second increase from 1 × 10−8 to 3 × 10−8 (SI) is observed within the Maiolica Fm, at approximately 95 m depth, near the M1/M0r polarity boundary, where a centimetric intercalation of black shale is observed. MS measurements at the three frequencies show subtle differences across the intervals described above, but MS asymptotes χhf and χlf estimated in the inversion reveal a more notable frequency effect. Calculated LFE is lower than 30 per cent for depths from 86 m to approximately 93 m (except at the top of the Selli Level, where it reaches 75 per cent), and higher than 30 per cent for greater depths. The higher LFE values are found in the Maiolica Fm, suggesting an increase of SP content in this unit, as one would expect from the lower magnetization and unstable palaeomagnetic behaviour presented by most of its samples (Savian et al. 2016). The parameter $$F_t^{ - 1}$$ ranges from 0.20 to 0.84 in the Marne a Fucoidi Fm (86–92.5 m). However, in this interval, most of the $$F_t^{ - 1}$$ values are higher than 0.50. The highest value of $$F_t^{ - 1}$$ is recognized at the boundary between Maiolica Fm. and Marne a Fucoidi Fm. The low values in $$ F_t^{ - 1}$$ in the Maiolica Fm reveal an increase of the SP content in the sediments, which can be interpreted as a zone of finer grains. 6 CONCLUSIONS We have developed a procedure for quantitative interpretation of the low-field MS frequency effect for measurements taken with at least three frequencies. An interpretation parameter was introduced, related to the frequency effect commonly used in FDS interpretation, named ‘LFE’. The LFE accounts for MS measured at all operating frequencies being therefore a more accurate estimation of MS variation in the frequency range. The transition parameter Ft accounts for relative variation of the particle grain size in the SP–SSD transition and it expresses how much the susceptibility is enhanced by the SP fraction. Numerical simulations show the procedure capability in retrieving modeled parameters and adjusting data, and its limitation regarding the range of variation of relaxation times, imposed by experimental parameters (frequency range and number of measurements). The controlled experiment showed that for the data set used (Kappabridge in-phase MS measured at three frequencies), the procedure can lead to erroneous estimations of particle diameter but provided consistent results recognizing no grain-size variation as prescribed by parameter Ft and LFE analysis.The case study with the previously interpreted PLG section shows that the transition parameter was able to distinguish two geological formations. To further applications of FDS quantitative analysis care must be taken when interpreting LFE or derived Neel's model parameters since other relaxation processes can be observed in the same frequency range. This requires further analysis of magnetic properties and judicious integration of multiple data. The study of broad-band frequency data has been a topic of growing interest. Kodama (2013) developed a methodology based on Neel relaxation theory to reconstruct GSD by using a wide-band frequency spectrum of MS. In principle, the procedure developed in this work can be extended to multifrequencies, possibly with more reliable results due to better data coverage. Acknowledgements This work was partially funded by grants #2016/06114-6 and #2016/17767-0, São Paulo Research Foundation (FAPESP) and by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) post-doctoral grant program. We thank researchers from University of São Paulo involved in the PLG core project (FUSP and Petrobras 2405 project) for sharing data. The authors also thank Ramon Egli an anonymous reviewer for the constructive review and Eduard Petrosvky for handling the review process. 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Geophysical Journal International – Oxford University Press

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