IAGA paleointensity database: distribution and quality of the data set

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1 Physics of the Earth and Planetary Interiors 147 (2004) IAGA paleointensity database: distribution and quality of the data set Mireille Perrin a,, Elisabeth Schnepp a,b a Laboratoire Tectonophysique, UMR CNRS/UMII 5568, Université Montpellier II CC 49, Montpellier Cedex 5, France b GFZ Potsdam, Sektion 3.3, Telegrafenberg, Potsdam, Germany Received 4 November 2003; received in revised form 27 February 2004; accepted 21 June 2004 Abstract The aim of this paper is to provide details about the design of the paleointensity database and focus on the distribution of the available data set. Version 2003 of the paleointensity database comprises 3128 data from 215 references. Only absolute paleointensity determinations from igneous rocks and baked contacts are considered regardless which method of paleointensity determination was used. The entries in the database correspond to cooling units. All polarities as well as all units are considered (normal, reverse or transitional). The Earth s magnetic field is the only surface trace of the working of the geodynamo and is therefore a very important tool for our understanding of the Earth s deep interior, especially its variation with geologic time. However, to study the long-term variation of the paleomagnetic field, non dipole effects have to be removed. In recent years, the production rate of paleointensity data has strongly increased and the most reliable methods were more frequently used. However, the present data set still has an important temporal bias with 35% of the rocks having ages less than 1 Ma, as well as a strong geographic bias towards the northern hemisphere especially North America and Europe. For most time intervals, secular variation cannot be averaged out, neither temporally nor spatially; therefore interpretation of the averaged field is very limited and is avoided here. For the future, a harmonization or a combination of all IAGA databases would be desirable. Furthermore, the input of raw data at the specimen level would be useful in order to allow reinterpretation of data with more developed and sophisticated methods based on our increasing understanding of rock magnetism. Finally new reliable data are very much needed Elsevier B.V. All rights reserved. Keywords: Database; Paleointensity; Paleomagnetism; Secular variation 1. Introduction Since the IUGG meeting held in Vancouver, August 1987, the Paleomagnetism and Rock Magnetism Work- Corresponding author. addresses: perrin@dstu.univ-montp2.fr (M. Perrin), eschnepp@foni.net (E. Schnepp). ing Groups of the International Association of Geomagnetism and Aeronomy (IAGA) have sponsored the development of global databases covering all aspects of paleomagnetic (e.g. directions, intensities, polarity transitions, secular variation) and rock magnetic data. Paleointensity data from volcanic rocks older than 0.03 Ma had been first compiled by Hidefumi Tanaka and Maseru Kono (Tanaka and Kono, 1994). This /$ see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.pepi

2 256 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) initial paleointensity database (PIDB) contained a total of 1123 flow mean results retrieved from 83 references published up to the end of The structure was quite simple (ASCII files) and therefore very easy to use but information for each result was rather restricted. Mike McElhinny included this initial version in his 1995 package (McElhinny and Lock, 1996). In 1995, responsibility for maintaining and updating the PIDB was passed from H. Tanaka and M. Kono to M. Perrin. The same idea of simplicity was kept for the new PIDB but the design of the database has been changed, information as well as user screens allowing selection and visualization of the data have been added. The contribution of V. Shcherbakov to the 1996 version of the PIDB (1340 cooling units, 92 references) was extremely valuable regarding analysis of Russian data (Perrin and Shcherbakov, 1997). Later E. Schnepp contributed to the 0 5 Ma update and the 1998 version of the PIDB (1692 cooling units, 115 references) was released (Perrin et al., 1998). The first aim of this paper is to describe the design of the database, providing basically of a technical manual of the PIDB asked for by many colleagues. We will then underscore the strengths and weaknesses of the available paleointensity data set contained in the 2003 version (3128 cooling units, 215 references), in relation with global analysis. Finally a possible evolution of the PIDB will be considered. All versions of the PIDB are available as Microsoft Access, EXCEL and ASCII files via ftp://saphir.dstu. univ-montp2.fr/paleointdb/ or through the first author s home page PERSO/perrin/index.html. The last update of the PIDB is also included in the package of IAGA sponsored Paleomagnetic Databases available at the web site of the National Geophysical Data Center noaa.gov/seg/potfld/paleo.html. 2. Data recorded in the paleointensity database All absolute paleointensity determinations obtained from igneous rocks and baked contacts are registered in the database, provided that they have been published in a peer-review journal. Each entry corresponds to a mean result for a given cooling unit. All methods of paleointensity determinations as well as all field configurations are taken into account (normal, reverse, transitional or even unknown polarity). We did not follow the age limit chosen by Tanaka and Kono, and also entered paleointensities derived from volcanic rocks younger than 30 Ky. However, we still did not include data from archeological artifacts because the necessary related information or metadata is quite different (e.g. an archeological context would not fit in the fields describing the rock formation; archeological time scales are regional and not obvious to transfer to an absolute time scale; cooling time and anisotropy corrections often used for archeointensity data from pottery are not used for rocks; and so on). Regional archeomagnetic databases are currently under construction by different authors (e.g. Sternberg et al., 1997; Kovacheva, 1997; Genevey et al., 2003). In the future, it would be desirable to design a common archeomagnetic database, compatible with the paleointensity database, so both data sets could be easily merged for analysis. Alteration of the magnetic properties of the rocks during paleointensity experiments, often preclude definition of a reliable estimate. Methods, which aim at correcting paleointensity results for alteration, have been proposed (e.g. Tanguy, 1975; Burakov and Nachasova, 1985; Walton, 1991; McClelland and Briden, 1996; Valet et al., 1996). However all rock properties, chemical as well as magnetic and/or physical, can be strongly modified during alteration processes and frequently in a largely unknown way. Therefore corrections have to be used with a lot of caution, on a case-to-case basis, and whenever possible we always preferred to include uncorrected data in the database. Some of the older paleointensity papers are now rather difficult to find. A complete collection of all references stated in the database is available in Montpellier and copies can be sent upon request. 3. Design of the paleointensity database Following the option chosen for most IAGA paleomagnetic databases, the new paleointensity database was developed using Microsoft Access. It is organized around two main tables: one for the references REF and another for the data DATA which are related to each other by a one to many relation, through the REFNO field (Table 1). In the rest of the paper, tables from the database will be in bold capital letters while the columns will be in italic.

3 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Table 1 Design of the paleointensity database Information about the sampling site, rock formation, age of the unit, directional analysis, and paleointensity determination are included in the DATA table (Table 1). Convention used for classification of rock formation (columns ROCKGROUP, ROCKTYPE, ROCKFORM) is described in Table 2. When the virtual dipole moment (VDM) is not given in the original paper but directional information is available, we entered in the DATA table a recalculated VDM (VDMc) using the classic formula VDMc = 4 R 3 /2µ 0 F SQR(1 + 3cos 2 Inc) where R is the radius of the earth, µ 0 the permeability of free space, F the paleointensity, and Inc the paleoinclination. A ROKMAG column has been added to list available rock magnetic information such as magnetic mineralogy, Curie temperature, and grain size. In the last column COMMENTS, related information is stated such as an older reference when a new study replaced a previous one, reference for ages when not found in the paleointensity reference, use of Virtual axial dipole moment VADM instead of VDM, and so on. Also when a large unexplained difference is observed between the VDM value stated in the paper and the recalculated VDM, the VDMc value is indicated in the COMMENTS column. Lookup tables (Table 1) were designed either to facilitate the entry and the selection of data (AUTHORS, JOURNALS, CONTINENTS, COU- NTRIES, ROCK TYPES, AGE METHODS, IN- TENSITY METHODS) or to give information (INFORMATION, TIMESCALE, VERSION). Different user interfaces (Table 1, Figs. 1 and 2) have also been prepared to make the database easier to consult, to allow the selection of data, and to choose a preferred output: screen, printer or files (Microsoft Excel, Rich Text Format, or MS-DOS Text). Various criteria can be used to select data (Fig. 2a): age, continent, polarity, intensity method, virtual dipole moment, minimum number of determination per cooling unit and/or maximum standard deviation on the paleointensity estimate (in %). For some cooling units, the

4 258 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Table 2 Conventions used to describe the rock formation Rockgroup Rocktype Rockform Igneous Rocks Volcanic rocks Andesite Porphyrite Baked contact Basalt Chilled margin Dolerite Diabase Dome Rhyolite Dacite Ignimbrite Dyke Trachyte Lava flow Neck Pillow Plug Sill Breccia Pyroclastic Scoria Baked pyroclastic Tuff Plutonic Rocks Anorthosite Dyke Diorite Intrusion Gabbro Norite Sill Granite Granodiorite Vein Granophyre Syenite Metamorphic Rocks Sedimentary Rocks Amphibolite Leptinite Argilite Clay Porcellanite Loess Sandstone Siltstone Baked contact intensity estimate results were produced by a mixture of different paleointensity methods. In the data selection form, the following convention was chosen for paleointensity methods (see Table 3 for the abbreviation of the paleointensity methods): - Microwave (Mpp); - Thellier (T+, T, ST, ST+, TZ, Tv, M); - Shaw (S, ST, ST+, SW); - Van Zijl (Z, TZ, WZ, HeZ); - Wilson (W, SW, WB, WZ); - Others (HeZ, ONR). Also, data for a given author or a given reference can be selected (Fig. 2b). To estimate numerical ages when only geological time estimates were given in the original papers, the Odin geologic timescale (Odin, 1994) was used. This is in contrast to the Global Paleomagnetic Database (McElhinny and Lock, 1996), which used a modified version of the Harland geologic timescale (Harland et al., 1990). As can be seen from Table 4, there are no major differences between both timescales, except for the Proterozoic Paleozoic boundary. For recent ages, we followed the 5th Radiocarbon-Dating Conference (Cambridge, 1962), which fixed ad 1950 as zero-age of the BP-timescale. For the few entries having an age Fig. 1. Paleointensity main data form.

5 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Fig. 2. Example of user interfaces for data selection. younger than ad 1950, an age of zero was entered in order to avoid negative ages. 4. Strengths of the paleointensity data set 4.1. Rate of data acquisition The very first paleointensity paper was published in the late 1940s (Koenigsberger, 1938). Except for one other paper (Nagata, 1943), all other data were published after 1959, when Emile and Odette Thellier published their classical work about the intensity of the earth s magnetic field in historic and geologic time (Thellier and Thellier, 1959). After this break-through paper, the production rate was fairly steady for the following 30 years, with an average of three four publications per year (Fig. 3) corresponding to an average of 50 paleointensity values per year (Fig. 4). One of the strengths of the actual paleointensity database is the

6 260 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Table 3 Abbreviations used in the database for the paleointensity methods Intmtd Paleointensity methods Data number HeZ Petrova (Petrova et al., 1979; PIDB ref 5 8) and van Zij 1 17 M Multivectorial method (Yu and Dunlop, 2002; PIDB ref 200) 2 Mpp Perpendicular microwave technique (Hill and Shaw, 1999; PIDB ref 174) 8 ONR NRM/TRM 77 S Shaw (Shaw, 1974; PIDB ref 158) 321 ST Shaw and Thellier 71 ST+ Shaw and (Thellier with ptrm checks) 37 SW Shaw and Wilson 6 T+ Thellier ((Thellier and Thellier, 1959; PIDB ref 75); (Coe, 1967; PIDB ref 216)) 1479 with ptrm checks (Prévot et al., 1985; PIDB ref 56) T Thellier (or Coe) without ptrm checks 858 Tv Thellier corrected according to Valet (Valet et al., 1996; PIDB ref 142) 41 TZ Thellier and van Zijl 32 W Wilson (Wilson, 1961; PIDB ref 80) 32 WB Wilson Burakov (Burakov, 1978; PIDB ref 215) 28 WZ Wilson and van Zijl 49 Z van Zijl (van Zijl et al., 1962; PIDB ref 79) 70 marked increase in the rate of data acquisition in recent times with up to 20 publications for 1999 (Fig. 3). Furthermore, the number of publications has almost doubled in the past five years, and most importantly, the number of cooling units per publication and the number of estimates per cooling units has also significantly increased (Fig. 4). Nevertheless, the acquisition rate remains desperately slow when taking into account the number of data needed to properly average secular variation, a necessary condition for a meaningful global analysis. This does not reflect a lack of interest of the community but rather can be explained by the difficulties in obtaining reliable paleointensity estimates. Most techniques are very time consuming, the failure rate is often high because of the difficulty of finding natural rocks that behave ideally during paleointensity experiments, and rock magnetic properties are not well Fig. 3. Distribution of the number of publications with publication s year. Fig. 4. Cumulative distributions of the number of publications, number of cooling units and number of paleointensity estimates with publication s year.

7 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Table 4 Comparison between Odin and Harland timescales data now are obtained using more and more reliable techniques as illustrated in Fig. 5. All methods of paleointensity determination were subdivided into three groups (A C), which we regard in order of decreasing reliability. Following arguments developed in Prévot and Perrin (1992), the most reliable procedure (group A) is the Thellier technique, either in its original version (Thellier and Thellier, 1959) or as modified by Coe (Coe, 1967), when it is coupled with sliding ptrm checks (e.g. (Prévot et al., 1985)). The B group of paleointensity techniques includes Thellier determinations without ptrm checks and estimates obtained with the Shaw method, in its original version (Shaw, 1974) as well as with modifications (e.g. (Kono, 1978); (Rolph and Shaw, 1985)). Group C corresponds to all other techniques listed in Table 2. The microwave technique, which is the latest method proposed for paleointensity determination ((Walton, 1991); (Hill and Shaw, 1999)), is for the moment included in Group C because of theoretical uncertainties about the equivalence between TRM and microwave-induced TRM (T M RM). However, new results (e.g. (Shaw et al., 1999); (Hill et al., 2002)) are very promising and, with the addition of ptrm checks, the microwave technique is very likely to become a Group A paleointensity method. Up to 1970, two-thirds of the estimates were obtained with Group C paleointensity methods, three-quarters with Group B between , and more than threequarters with Group A since 1994 (Fig. 5). 5. Weaknesses of the paleointensity data set understood apart from the single domain range (see Perrin (1998) for a review of the principal uncertainties about paleointensity experiments and Dunlop and Ozdemir (1997) for detailed explanations). The renewal of interest for theoretical aspects of thermoremanent magnetization (TRM) acquisition is obvious in the past few years, with a very large number of the paleointensity publications dealing with methodological problems Paleointensity methods Another extremely important step in the production of paleointensity estimates is the fact that most of the 5.1. Geographic distribution The geographical distribution of the paleointensity data is extremely uneven (Fig. 6), especially the latitudinal distribution with 91% of the data obtained from the northern hemisphere, and about 80% from the N latitudinal band. Almost half of the data (45%) is from oceanic plates (Pacific, Atlantic and Indian oceans). However, except for recent ages, the whole of the ocean floor is unsuitable for obtaining absolute paleointensities, because of the low temperature alteration of the titanomagnetites. Accordingly samples must be taken from the small islands and most of the work has been done on the Hawaiian islands, the Society islands, the Reunion island, the Canaries

8 262 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Fig. 5. Distribution of the number of cooling units as a function of the paleointensity methods used for different years of publication. Islands, Japan, and New Zealand, with the largest regional data set being from Iceland on the mid Atlantic ridge. The Eurasian plate accounts for 42% of the total data set, America for 10%, Africa for 2%, and Australia Antarctica for 1%. But even within the best-covered plates, Eurasia and America, the regional distribution is also extremely patchy, because it follows the outcrops of volcanic rocks. For the Eurasian plate, most of the data are clustered around western Europe and south-southeastern Asia. For the American plate, 90% of the results are from the northern part of the continent. For most parts of Africa, Australia-Antarctica, and South America, there are no paleointensity data Temporaldistribution The temporal distribution is once again extremely uneven (Fig. 7), with 96% of the data belonging to the Ma interval. In order to study the characteristic of the temporal distribution in a better manner, the geologic time scale was divided into five time intervals: 0 0.1, 0.1 1, 1 20, , and Ma. We know from archeointensity data for the past 10 ka (e.g. Me Elhinny and Senanayake, 1982), that there is a large variation in dipole moment from less than seven to more than Am 2, mainly due to secular variation. As all entries in the database are instant recordings of the geomagnetic field, secular variation has to be averaged out to study the evolution of the mean paleointensity through geologic time, which required at least a few thousand data points per Ma. The 0 1 Ma interval is the only one with the number of data large enough to expect a correct averaging of secular variation, even if we exclude the highly covered 0 10 ka interval (4300 data per Ma between ka, 534 data per Ma between 0.1 and 1 Ma). These average numbers drastically drop for the 1 20 Ma interval (48 data per Ma) and even more for the Ma interval (3 data per Ma). Between 400 and 3500 Ma, only 121 paleointensity estimates are available (0.04 data per Ma). It is therefore clear that, whatever type of time averaging is used in compilation papers, any mean results older than 5 Ma are very likely to be biased by the uncorrected influence of secular variation and/or local non-dipole anomalies Quality of the data set Finally the quality of the available data set is very uneven. Some older data (about 10%) were published without giving the value of the Virtual Dipole Moment (VDM) or the directional information necessary to recalculate it, which makes them useless for any global analysis. As seen previously the reliability of the paleointensity techniques is very variable. The number of

9 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Fig. 6. Geographic distribution of the paleointensity data set. individual estimates per cooling unit is often very low, and about 40% of the mean results were derived from less than three estimates per cooling unit. The internal consistency between values of the ancient field provided by different specimens from the same cooling unit (Thellier, 1971), is so far one of the best criteria to estimate the reliability of a mean paleointensity value. Fig. 8 reproduces Fig. 7 but with the added constrain of a standard deviation about the mean df 20% (Fig. 8a) and df 10% (Fig. 8b). It can be seen that the number of cooling units is drastically reduced by the application of this single selection criterion, even for very recent periods. The quality of some other very important criteria, such as the technical quality of the paleointensity determination and the quality of the age determination, is also variable. In some cases, a lack of information in the original papers does not allow a precise estimation of the quality of the corresponding estimates. For example, the geographic distribution of the proposed period of low field ( Ma) (Perrin and Shcherbakov, 1997) is particularly biased by the huge amount of data from Armenia and Siberia for

10 264 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) Fig. 7. Age distribution of the global paleointensity data set. Fig. 8. Age distribution of selected paleointensity data sets: (a) df 20%; (b) df 10%.

11 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) which an extremely limited amount of information is available. 6. Future of the paleointensity database The current level of information in the paleointensity database is the cooling unit level. The main goal for the future would be to bring this level to the individual results per specimen with a design, which could be as, is illustrated in Fig. 9. The major advantage of this change would be that the main data set would be the raw data set (NRM left, TRM gained) for each treatment step, together with all information about the method used for the determination (e.g. paleointensity technique, equipment used). From such a set of data, it would be possible to visualize at any time the classic projections (NRM/TRM, equal-area and orthogonal plots). Derived data, which at the specimen level would be the individual directional and intensity results with their respective statistics as proposed by the original authors, would also be included. However, with the raw data available, it would always be possible to reanalyze the data according to our growing understanding of TRM characteristics, MD theory, and so on. This could save a lot of time because, at present, the only option when there is any doubt about a mean data point is to go back in the field and resample. From the proposed set of entries at the specimen level, the usual entries at the cooling unit level could be added, including all common metadata that are already in the database (e.g. location, age, rock formation). Finally, it would be very useful to have an interconnection between the PIDB and other magnetic databases (e.g. archeomagnetism, paleomagnetism, rock magnetism) or non-magnetic databases (e.g. geochronology, petrography). At approximately the same time as we were planning this desirable evolution of the paleointensity database, a similar but much larger initiative was taken by the Magnetics Information Consortium (MagIC). MAGIC evolved from the PMAG Workshop held at Scripps Institution of Oceanography in La Jolla, from March At this workshop it was agreed that there is a critical need to update and integrate existing magnetic database efforts sponsored by IAGA to take advantage of the technological advances provided by modern web-based data handling capabilities. This is definitely the final goal which has to be achieved but already the merging of all magnetic databases would not be that simple. The amount of data, which would have Fig. 9. A possible new design for the paleointensity data base.

12 266 M. Perrin, E. Schnepp / Physics of the Earth and Planetary Interiors 147 (2004) to be processed, is also on a very different scale if all databases are to be brought down from the site level to the specimen level, and the cooperation of all contributors would be needed. The PIDB is probably one of the smallest paleomagnetic database in existence but already the number of entries would be multiplied by a factor five to achieve this. The question remains open regarding whether authors dealing with paleointensity data prefer to keep a separate database, with or without the proposed advancement, or if everybody is ready to exert the efforts to create a large, integrated magnetic database. 7. Conclusion The earth s magnetic field is the only surface trace of the working of the geodynamo and is therefore a very important tool for our understanding of the earth s deep interior, especially its variation with geologic time. Because of the relatively limited number of paleointensity data, their extremely uneven geographic and temporal distribution, and differences in the quality of the data, a proper time-averaging of the paleomagnetic field cannot yet be expected, except maybe for the first Ma. To study the long term variation of the paleomagnetic field through geologic time, non dipole effects have to be removed. The time-averaging of the paleomagnetic field is usually considered to be performed over time intervals on the order of a few tens of ka (e.g. Merrill et al., 1996). Thus, for long term analysis of the dipole field, we would like to have a few thousand data per Ma which, with the current publication rate of 100 data per year, would take a few tens of years per Ma, and the Earth is about 4.5 Ga old! The aim of this rough calculation is only to underscore the real need for new paleointensity determinations and also the reason why no quantitative analysis was attempted in this paper. Several papers dealing with this aspect have been published recently (e.g. Selkin and Tauxe, 2000; Heller et al., 2002; Biggin and Thomas, 2003) with somewhat conflicting conclusions. The characteristics of the actual dataset, with the biases underscored in this paper, lead us to believe that a new quantitative analysis would have been useless without the addition of a significant number of new data. The upkeep of the database would be greatly assisted if every paleointensity paper included all the necessary information required to complete every field of the database. Frequently, even in recent papers, there is a lack of essential information such as the site coordinates or the polarity, and it can be very time consuming or impossible to retrieve this information. Furthermore we would appreciate if all authors could send automatically reprint of their papers, or at least the reference, to M. Perrin. Finally, comments or suggestions on the database and its future would also be very welcome. 8. Acknowledgments We would like to thank Valera Shcherbakov for helping to collect and input data from older papers published in Russia, and Andy Biggin for constructive suggestions on the original manuscript. This work was supported by CNRS, contribution CNRS-INSU xx. During her stay in Montpellier, E.S. was supported by the Deutsche Forschungsgemeinschaft, funds Schn 366/3-1 and -2. References Biggin, A.J., Thomas, D.N., Analysis of long-term variations in the geomagnetic poloidal field intensity and evaluation of their relationship with global geodynamics. Geophys. J. Int. 152 (2), Burakov, K.S., Nachasova, I.E., Correcting for chemical change during heating in archeomagnetic determinations of the ancient geomagnetic field intensity. Izvest. Earth Phys. 21, Coe, R.S., Paleointensities of the Earth s magnetic field determined from Tertiary and Quaternary rocks. J. Geophys. Res. 72, Genevey, A.S., Gallet, Y., Margueron, J.C., Eight thousand years of geomagnetic field intensity variations in the eastern Mediterranean - art. no J. Geophys. Res. Solid Earth, 108(B5): NIL 20-NIL 37. Harland, W.B. et al., A Geologic Time Scale Cambridge, 263 pp. Heller, R., Merrill, R.T., McFadden, P.L., The variation of intensity of earth s magnetic field with time. Phys. Earth Planet. Interiors 131 (3 4), Hill, M.J., Gratton, M.N., Shaw, J., A comparison of thermal and microwave palaeomagnetic techniques using lava containing laboratory-induced remanence. Geophys. J. Int. 151 (1), Hill, M.J., Shaw, J., Palaeointensity results for historic lavas from Mt Etna using microwave demagnetization/remagnetization in a modified Thellier-type experiment. Geophys. J. Int. 139 (2),

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