^{1}

^{2}

^{*}

^{1}

^{1}

^{1}

^{2}

Edited by: John William Geissman, University of Texas at Dallas, USA

Reviewed by: Peter Aaron Selkin, University of Washington, Tacoma, USA; Maxwell Christopher Brown, Deutsche GeoForschungsZentrum GFZ, Germany

*Correspondence: Marilyn W. L. Monster

This article was submitted to Geomagnetism and Paleomagnetism, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The multispecimen protocol (MSP) is a method to estimate the Earth's magnetic field's past strength from volcanic rocks or archeological materials. By reducing the amount of heating steps and aligning the specimens parallel to the applied field, thermochemical alteration and multi-domain effects are minimized. We present a new software tool, written for Microsoft Excel 2010 in Visual Basic for Applications (VBA), that evaluates paleointensity data acquired using this protocol. In addition to the three ratios (standard, fraction-corrected, and domain-state-corrected) calculated following Dekkers and Böhnel (

The magnetic field of the Earth is generated in its liquid outer core by magnetohydrodynamic processes. In order to better understand these processes, we need more information on how the field behaved in the past, both in terms of its direction and its intensity. However, while it is fairly straightforward to determine paleodirections from lavas or

For low, Earth-like fields, a thermoremanent magnetization (TRM), as found in lavas or pottery, is proportional to its inducing field (e.g., Muxworthy and McClelland,

An alternative approach is the multispecimen protocol (Dekkers and Böhnel,

Fabian and Leonhardt (

A popular software tool for the analysis of Thellier-style paleointensity experiments is ThellierTool 4.0 (Leonhardt et al.,

In the original multispecimen protocol (Dekkers and Böhnel, _{1}, with _{0} being the NRM). In the DSC protocol (Fabian and Leonhardt, _{2} to _{4}). The five measurements are then:

_{0}: NRM

_{1}: Magnetization after heating and cooling in parallel field

_{2}: Magnetization after heating and cooling in anti-parallel field

_{3}: Magnetization after heating in zero-field and cooling in parallel field

_{4}: Same as _{1} (progressive alteration check)

From these five measurements of the vector remanence, the paleointensity is determined for a lava flow. The original MSP-DB method assumes that multidomain effects are negligible. In this case, if the laboratory field is equal to the paleofield, _{1} should be equal to _{0} (and lower or higher if the laboratory field is lower or higher, respectively, than the paleofield). The _{DB} ratio (Dekkers and Böhnel, _{0} and _{1} are the scalar intensities of the two remanences. The fraction-corrected (MSP-FC) and domain-state corrected (MSP-DSC) ratios (both defined in (Fabian and Leonhardt,

The denominator in both equations is equal to twice the amount of NRM lost. The ratios are therefore normalized to the demagnetized part of the NRM rather than to the complete NRM as is the case for the DB ratio. As the fraction NRM lost is often not the same even for samples within one cooling unit, this fraction correction should reduce the amount of scatter (Fabian and Leonhardt, _{lab} = 0μT, _{1}_{0} is equal to minus the amount of NRM lost and therefore the

The parameter α in the numerator of _{DSC} is used to correct the FC ratio for domains state effects. Fabian and Leonhardt (

Apart from these three ratios, a number of parameters estimating the domain state, progressive alteration, and the total (domain-state-induced and alteration-induced) error can be calculated. Further explanation of these parameters is provided in Fabian and Leonhardt (

The MSP-tool workbook consists of several sheets: “manual”; “input”; “parameters”; “parameters, corrected”; and “list of parameters.” Additionally, several plots (MSP-DB, MSP-FC, and MSP-DSC) are provided that can be exported as PNG or CSV files. File actions and calculations are easily carried out by clicking the corresponding button. The user interface is shown in Figure

The “input” sheet is the start sheet. It provides buttons to import, save, and clear data and to carry out the actual MSP calculations. Furthermore, the values of several parameters, such as the α parameter (Fabian and Leonhardt,

MSP-Tool supports two different input formats. Data can be imported using either the three Cartesian components of the remanence or using the intensity, declination and inclination of the remanence (which are automatically converted to Cartesian components). The order of magnitude for every line of data must be the same; if not, for example for data measured on a JR-6 spinner magnetometer, an additional (sixth) column with the exponent can be added. Since the MSP calculations are inherently relative, the units for _{x}, _{y}, _{z} and the intensity can be chosen freely, as long as they are the same for all five measurement steps.

Clicking the “Calculate and plot” button calculates all parameters in the “input” and “parameters” sheets and plots the three ratios in the “plot” sheets. The VBA tool can process all three MSP protocols and will calculate and plot the relevant ratios and parameters accordingly.

The isolated pTRMs (columns G to I) are calculated by first estimating the vector NRM remaining. The latter is obtained by adding the vectors _{1} and _{2} and dividing the result by 2, c.f. the fraction-corrected MSP-FC ratio (Fabian and Leonhardt,

The vector pTRMs are then:

The scalar magnetic moments _{0} to _{4} (column J) are simply the magnitudes of the vector remanences _{0} to _{4}, whereas the alignment-corrected intensities (column K) are obtained by adding up the isolated NRM remaining and the pTRMs. Of course, if the pTRM is parallel to the NRM, _{i, corr} = _{i}.

Columns L and M show the declination (between 0 and 360°) and inclination (between −90 and +90°) of the _{0} to _{4} steps. All declinations and inclinations are given in specimen coordinates. The declinations and inclinations of the isolated NRM remaining and the isolated pTRMs are used to calculate the parameters Δdec and Δinc (columns N and O), which are a measure of how well the specimens were aligned to the laboratory field. If Δdec and Δinc exceed the AAD, they are shown in red. We chose to use Δdec and Δinc rather than a single angular difference because using these two parameters it is easier to spot systematic alignment errors. Please note that in case of large overprints Δdec and Δinc may be high even if the pTRM was aligned perfectly with the NRM (_{0}).

These calculations all implicitly assume that even if the specimens were not aligned properly, at least they were aligned _{1}) or anti-parallel (_{2}) to the NRM. This is a valid assumption if the positions of the sample holders with respect to the furnace were not changed during the experiment. To minimize orientation issues, it is advisable to process each individual specimen on the same holder throughout the experiment, eliminating the need to orient holders for each step in the MSP-DSC experiment. As the alignment correction does not take into account multidomain effects such as pTRM tails, a proper alignment of the samples during the experiment is still paramount—it is merely a tool to suppress unavoidable small experimental misalignments.

The “m2 factor” (column P) is the normalized dot product of _{0} and _{2}. This parameter equals +1 when _{0} and _{2} are parallel and -1 when they are antiparallel. _{2} is multiplied by this factor to correct for “negative” _{2} intensities. Not correcting for this often results in plots with a large amount of scatter (see Figure

Finally, the angle between the NRM lost and the NRM remaining is calculated (Equation 10). If this angle exceeds the AAD, a warning is shown in column R. A large difference may indicate the presence of an overprint, which would invalidate the result. Alternatively, if the sample's alignment was changed between the first two steps, the NRM remaining cannot be accurately calculated and the angular difference between the NRM remaining and the NRM gained may be anomalously high.

The “parameters” and “parameters, corrected” sheets show a number of parameters that were proposed by Fabian and Leonhardt (_{DS}, the progressive alteration ε_{alt}, single-specimen estimates of the paleointensity _{max} and _{est} and estimates of the alteration error and domain-state error. In the “parameters” sheet these are calculated from the uncorrected remanences _{0} to _{4}, whereas the “parameters, corrected” sheet uses the alignment-corrected remanences.

The progressive alteration is defined in a slightly different way than in Fabian and Leonhardt (_{1}_{4} normalized by _{1}, we use:

This helps to distinguish between measurement noise (which averages to zero) and a systematic (alteration-induced) error. Fabian and Leonhardt's (_{alt}|. For definitions of the parameters _{est}, _{max} and three error estimates Δ_{DSC, alt}, Δ_{DSC, ds} and Δ _{i} see Fabian and Leonhardt (

The plot sheets (DB, FC, DSC, and their alignment-corrected versions) show a plot and a table of the data points as well as the calculated paleointensity and its error bounds and two reliability checks: the average alteration parameter ^{2} and χ^{2} of the linear regression, where χ^{2} is defined as the mean quadratic deviation between the measured value of the DB, FC or DSC ratio (_{i, measured}) and the value of the linear regression at that laboratory field (_{i, expected}):

Clicking “Bootstrap confidence interval” calculates and plots the bootstrap average and the confidence interval. The confidence level may be changed in the “input” sheet; its default value is 95%. The bootstrap function resamples the data set with replacement within their error bounds and calculates a linear fit for each bootstrap cycle. Bootstrap cycles that have a standard deviation of less than 10 μT in their average

The

To test MSP-Tool's ability to accurately determine and correct for alignment problems on real rocks, a complete MSP experiment including preliminary ARM test (de Groot et al.,

Before carrying out the actual MSP experiment, next to the ARM test at the MSP temperature (using 17 field steps of up to 150 mT), some other experiments were conducted to assess the specimens' alteration temperature and domain state. These experiments are described in more detail in the Supplementary Material. The susceptibility-vs.-temperature diagrams for site TD did not show any significant alteration after acquisition of the laboratory full-TRM, whereas site PI showed irreversible behavior at temperatures > 300°C (Figure

After the positive ARM test, the MSP-DSC protocol (Fabian and Leonhardt, _{0}) and the four remanences after in-field heating and/or cooling (_{1} to _{4}) were measured on an AGICO JR-6 spinner magnetometer.

In 90% of cases, Δdec and Δinc as calculated by MSP-Tool were within 15° of the intended misalignment angles (see Table

Looking at the plots (Figure

All plots except the DSC plots for site PI and the uncorrected DSC plot for TD reproduced the “paleofield” within error. Lowering the α factor from 0.5 to 0.2 or even 0.0 (in which case MSP-DSC reduces to MSP-FC) improved the “paleointensity” estimate for the DSC protocol (Figure

^{2}, and χ^{2} as a function of the α parameter^{2} and χ^{2} do not change significantly with varying α. The number of samples used in the calculation of the paleointensity

MSP-Tool offers four reliability criteria: two directional criteria (the overprint check, and Δdec and Δinc), the amount of progressive alteration ε_{alt}, and the intersection with the _{1} and _{2}, leading to an inaccurate estimate of the NRM remaining and NRM lost. In case of consistent alignment (i.e. _{1} and _{2} aligned exactly antiparallel) and high Δdec and Δinc, the corrected plots may be preferred. Finally, both the average progressive alteration _{FC} and _{DSC} should pass through (0, −1) as these ratios are normalized to the amount NRM lost rather than the full NRM. Failure to pass through this point (within 10% and/or within error) may indicate that something other than domain-state-related processes is at work and may be a reason to distrust the obtained paleointensity. It is strongly recommended to conduct the ARM test (de Groot et al.,

In the alignment correction procedure, the scalar intensities _{1} to _{4} are calculated by adding up the isolated NRM remaining and pTRM gained. The full-TRM experiment showed that this correction functions rather well, although it should be noted that multidomain effects such as tails may influence the calculation of the NRM remaining and therefore the calculated pTRMs. It is also important to note that the alignment correction is only accurate if the samples were aligned exactly antiparallel during the first and second heating steps. In order for the correction to work, therefore, it is important not to change the orientation of the specimens in the oven between MSP steps. As the alignment correction does not 100% restore specimens that were misaligned by a large angle, it is still paramount to align the specimens with care.

Fabian and Leonhardt (^{2} value was generally at a maximum and χ^{2} at a minimum for low values of α (α < 0.2), although the differences are small. The “paleointensities” obtained from the uncorrected MSP-DSC plots as a function of α varied between 38.9 (α = 0) and 27.9 μT (α = 1) for site PI and between 41.8 (α = 0) and 31.8 μT (α = 1) for site TD. That PI's result is closer to the expected “paleointensity” for lower values of α may be caused by PI's large progressive alteration. By lowering the α parameter, the influence of the more altered _{3} step is also reduced.

MSP-Tool is an easy-to-use VBA-based tool for analyzing multispecimen experiments. It calculates all ratios and parameters from Dekkers and Böhnel (

The ARM test (de Groot et al., _{1} and an underestimate at _{2}, the MSP experiment may be carried out at these two temperatures, providing upper and lower bounds of the actual paleofield.

Misalignment leads to incorrect estimates of the paleofield as the measured vector remanences are shorter (parallel field steps) or longer (antiparallel field step) than the vector NRM remaining plus the vector pTRM lost. This leads to an underestimate of _{1} and therefore of _{DB}, _{FC}, and _{DSC}, and thus an overestimate of the paleofield. MSP-Tool corrects for this by adding up the two separate components (the NRM remaining and the pTRM gained).

Site PI highlights that a large progressive alteration may lead to underestimates in the DSC protocol. As a successful ARM test implies that no significant alteration occurred after one heating step, the alteration must therefore arise from the multiple heating steps in the MSP experiment. It is recommended to rely on the DB plot in such cases, although it should be recognized that the slope correction and domain state correction cannot be applied.

As the slope of the DB plot depends on the entire NRM rather than the amount of NRM lost, inhomogeneity between specimens may lead to significant scatter. Selecting specimens based on similar amounts of NRM lost may substantially improve the DB plots.

The project was designed by MM in conjunction with LdG and MD. MM wrote the VBA code and carried out the experiments. All three authors participated in the data analysis and contributed to the writing of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research was funded by a grant (project number 822.01.002) from the Earth and Life Science Division (ALW) of the Netherlands Organization for Scientific Research (NWO). MM would like to thank Cor G. Langereis for his advice regarding the bootstrap statistics used in the VBA code.

The Supplementary Material for this article can be found online at: