A new fold test in paleomagnetism (rehabilitation of the mean test)

PHYSICS OF THE SOLID EARTH� English Translation� VOL� �� NO� �� NOVEMBER ��

Russian Edition� APRIL ��

A new fold test in paleomagnetism�rehabilitation of the mean test�

S� V� Shipunov

Geological Institute� Russian Academy of Sciences

Introduction

Recent publications provide evidence of unabatedinterest in the fold test� which �a unique and mostused �eld test� enables dating of natural remanence�NRM� components derived from paleomagnetic anal�ysis �Bazhenov and Shipunov� �� McFadden�

�� Shipunov� �� Watson and Enkin� �� Inaccordance with current ideas� most reliable versionsof the fold test are the test of comparing �CFT� av�erage directions of NRM components before and aftertilt correction for rocks of various occurrence �Bazhenovand Shipunov� ��McElhinny� ��McFadden� ��McFadden and Jones� �� its modi�cation� grouptest� and correlation modi�cations of the fold test �cor�relation fold test �CrFT�� Bazhenov and Shipunov�

�� McFadden� �� Shipunov� �� The CFT and CrFT tests are the most e�cient in de�

tecting bias in NRM magnetization The CFA requiresselection of paleomagnetic samples from di�erent occur�rence sites �for example� form di�erent fold limbs� orsubdivision into groups of di�erent occurrence for lab�oratory studies No such condition is necessary for thecorrelation fold test The mean test �MFT�� one of the oldest modi�cation

of the fold test �McElhinny� �� was recently criticizedand admitted unsuitable for dating magnetization rel�ative to fold age �Bazhenov and Shipunov� �� McFadden� ��McFadden and Jones� �� Shipunov�� When applied� this test allows one only to estab�lish correctly the predominance of prefolding or post�folding component of magnetization The numericalcharacteristics of this predominance remain undeter�mined� and therefore one cannot estimate the possibleuncertainty in determining the angle of true direction ofa speci�c component of magnetization Thus in one ofthe MFT applications �Bazhenov and Shipunov� ��the secondary postfolding component of magnetizationamounted to �� with ratio of concentration parame�ters being Ka�Km � �� Recent wide popularity of the MFT method is ex�

Copyright �� by the American Geophysical Union�

��

plained by the simplicity of its numerical realization�clearness of its formulation� and freedom from any re�strictions on selection of paleomagnetic samples fromvarious outcrops� i e � fold limbs Shipunov �� suggested that if� from practical ex�

perience or other considerations� one could determinethe value of the concentration ratio which would indi�cate the presence of a single component� then a testednull hypothesis can be reformulated to make the meantest to be correct physically The present work aims atrehabilitation of this test through a new approach tothe testing procedure without changing its underlyingprinciple

Formulation of the Test

We will further distinguish between the following twonotations of vector distributions The Greek capital� will be used to denote distributions of prefoldingor postfolding magnetization components and distribu�tion of normals to beds �bed poles� The Latin capi�tal F will denote distributions of magnetization com�ponents in stratigraphic and geographic coordinate sys�tems as derived from paleomagnetic analysis We alsonote the following notation used below� a denotes aprefolding component of magnetization or magnetiza�tion in stratigraphic coordinate system� m denotes apostfolding component of magnetization or magnetiza�tion in geographic system of coordinates� and n denotesdistribution of bed poles In some cases� distributions denoted by di�erent sym�

bols can be the same For instance� this is true for thedistribution ��a� of prefolding single�component mag�netization in a stratigraphic coordinate system F �a��F �a� � ��a�� We also assume that none of the dis�tributions under consideration complies with the Fisherdistribution �Fisher� �� A set of NRM vectors in stratigraphic F �a� and geo�

graphic F �m� coordinate systems is known to depend ina certain manner upon a set of bed poles ��n� in eachsampling site of a paleomagnetic collection under study�Bazhenov and Shipunov� �� Khramov� �� If� forexample� a NRM component studied is a prefolding one�then the spatial position of vectors in geographic coordi�

��

shipunov� new fold test in paleomagnetism ��

nate system F �m� depends solely upon the distributions��a� and ��n� no other factors would a�ect the shapeof the distribution F �m� This dependence can be ex�pressed schematically by the formula

F �m� � ff��a��n�g ��

If� apart from the prefolding magnetization compo�nent� a postfolding one is present� then the distributionF �m� is additionally dependent on the postfolding com�ponent distribution ��m��

F �m� � ff��a��n��m�g ��

Thus using �� and �� the problem can be reducedto establishing whether this additional term connectedwith postfolding remagnetization is present in �� Similar considerations are applicable to the case of

a postfolding NRM component studied when the e�ectof the prefolding component ��a� on the magnetizationvector distribution in stratigraphic coordinate systemshould be established�

F �a� � ff��m��n��a�g ��

As is� dependencies �� cannot be expressedthrough simple formulas for concentration parametersof corresponding distributions of the type �Fisher� ��

�Qn � ��Qm � �� Qa �Qn��Qa � ��

where Q is the square root of a concentration parameterof a corresponding distribution Thus a characteristic feature of the suggested test pro�

cedure is its application to both �stratigraphic and geo�graphic� coordinate systems� as is the case for all correctmodi�cations of the fold test If� when applying the test under assumption of

prefolding single�component NRM magnetization� onewould discover that the dependence does not hold true�then the conclusion may be drawn on the presence ofpostfolding component of magnetization �i e � �� in�cludes the additional term ��m�� Similarly� if the testis applied under the assumption of postfolding single�component NRM magnetization and dependence �� isfound out to hold true only with the additional term��a� being nonzero� this would imply the presence of aprefolding magnetization component

Testing Procedure

We describe the testing procedure in more detail Forde�niteness� we will consider the distribution de�nedby �� and �� i e � half of the test only� assuming acomponent under study to be a prefolding one

We formulate the relevant null hypothesis H� as fol�lows� a vector distribution in a stratigraphic coordinatesystem is that of a prefolding magnetization componentonly� i e � �� holds true The alternative hypothesisH� states that data under consideration do not com�ply with �� and can be described only with the helpof the additional term of �� related with postfoldingremagnetization � For a set of N vectors of NRM magnetization� one

calculates mean directions and concentration parame�ters in stratigraphic and geographic coordinate systems��D� I�K�a and �D� I�K�m� respectively �The mean di�rections are used below only as parameters of numeri�cally modeled Fisher samples �� A sample of N vectors is calculated numerically

M times �say� M � �� with the use of the pseudo�random number generator by the Monte Carlo methodand formulas presented in the works by Shipunov �� The vectors are distributed according to theFisher law with its parameters being the mean direc�tion and concentration value of a set under study inthe stratigraphic coordinate system �D� I�K�a calcu�lated above � Each sample thus obtained is recalculated to the

geographic coordinate system using available rock oc�currence elements ��n� � Average directions �not used� and concentration

parameters Kmi �i � �� M � are calculated to obtainsamples in the geographic coordinate system � Thus the array of concentration parameters sorted

in ascending order in a geographic coordinate system isobtained under the assumption that a vector set stud�ied represents a prefolding NRM component �see thenull hypothesis formulated above�� i e � it characterizesa possible cluster distribution in geographic coordinatesystem provided H� is satis�ed If the concentrationparameter Km in the geographic coordinate system cal�culated under item � for a sample studied does notmatch the model distribution Kmi� then H� should bediscarded The discrepancy is estimated by comparingKm with the ��M th value of the sorted array Kmi If Km � Kmi��M �� then the null hypothesis is dis�carded on the �� level In other words� the probabilitythat a collection tested includes a postfolding magneti�zation component is �� The probability of such in�terpretation being invalid is correspondingly �� Thisprocedure known as hypothesis testing by the Barnardmethod �Maindonald� �� is applicable to paleomag�netic analysis if calculation or tabulation of the distri�bution of criterion statistics is not possible provided thenull hypothesis is true �Shipunov� �� If Km � Kmi��M �� then H� is accepted �there are

no grounds for its discarding� Probability of this infer�ence is inde�nite� since the distributionKmi is unknownin the case of the alternative hypothesis H� being true

�� shipunov� new fold test in paleomagnetism

Table �� Original data of collection �

D I A B

��

Note� The mean direction and concentration parameters in strati

graphic and geographic coordinates are D � �� I � ��K � �� and D � �� I � �� K � �� respectively� D�I� A� and B are declination and inclination of magnetization instratigraphic coordinates� azimuth� and angle of dip of beds� re spectively�

A similar procedure is applied for testing the null hy�pothesis H� �F �m� � ��m�� If two null hypotheses are tested �magnetization is

represented by a single prefolding �� or postfolding ��component�� the following four inferences can be de�rived from application of the proposed fold test � Aprefolding component only is found out � A postfold�ing component only is found out � Both prefoldingand postfolding components are found out � No mag�netization components are found out The latter variantmeans that the test does not permit any de�nite conclu�sions to be made for a given distribution of rock occur�rence elements ��n� and the volume N of paleomagneticcollection


D I A B

� � ��

Note� The mean direction and concentration parameters in strati

graphic and geographic coordinates areD � �� I � �� K � ��and D � �� I � �� K � �� respectively� D� I � A� and B aredeclination and inclination of magnetization in stratigraphic co ordinates� azimuth� and angle of dip of beds� respectively�


D I A B

� � ��

Note� The mean direction and concentrationparameters in strati

graphic and geographic coordinates are D � �� I � ��K � �� and D � �� I � �� K � �� respectively� D�I� A� and B are declinatlion and inclination of magnetization instratigraphic coordinates� azimuth� and angle of dip of beds� re spectively�

Examples of Application of the New Test

The real data of McFadden �� will be takenfor case studies Correlation fold tests proposed byBazhenov and Shipunov �� and McFadden ��give the same results when applied to these data Forinstance� prefolding magnetization is established for col�lection � �Table �� N � �� and postfolding is estab�lished for collection � �Table �� N � �� while magneti�zation of the collection � is either synfolding or containsboth prefolding and postfolding components �Table ��N � �� Original data� declination and inclinationin the ancient coordinate system� azimuth and angle ofbed dips� as well as characteristics of these distributionsin stratigraphic and geographic coordinate systems arepresented in Tables �� and �� respectively Figure � illustrates results of the application of the

proposed new fold test to collection � Histograms aand b of Figure � are obtained from testing the nullhypothesis that the vector distribution of the collectionstudied is that of prefolding one�component magnetiza�tion or� which is the same� that the vector distributionof the collection in geographic coordinate system doesnot depend upon any postfolding component �� ratherthan �� holds true� Figure �a presents the clusteringhistogram of modeled samples in a stratigraphic coor�dinate system� and Figure �b shows the histogram ofconcentration parameters recalculated to the geographiccoordinate system of model distributions The valueKm � �� calculated from real data lies within the rangeof possible values Kmi �about � to �� which means thatno postfolding component has been established


Figure �� Results of the new mean test applied to col�lection � �a� b� Hypothesis H�� the vector distributionin stratigraphic coordinates is that of prefolding one�component magnetization Shown are the histogramsof concentration parameters of model samples in �a�stratigraphic and �b� geographic coordinates �c� d�Hypothesis H�� vector distribution in geographic co�ordinates is that of postfolding one�component magne�tization Shown are the histograms of concentrationparameters of model samples in �d� geographic and �c�stratigraphic coordinates

Figures �c and �d characterize testing of the followingnull hypothesis� the vector distribution of a given collec�tion in geographic coordinates is that of one�componentpostfolding magnetization Figure �d shows the clus�tering histogram of model samples in geographic coor�dinate system� and Figure �c shows the histogram ofconcentration parameters recalculated to stratigraphiccoordinates of the model distributions The value Ka �� calculated from real data is much in excess of pos�sible values Kai �about � to �� which con�rms the pres�ence of a prefolding magnetization component Thus a prefolding component only has been estab�

lished for the collection �� which agrees well with theresults of the correlation fold test �see above� Figures � and � show concentration parameters for

model samples obtained in testing collections � and �these results also agree with the results of correlationfold tests �see above� beginning of the section� Thusapplication of the new test to the data considered leadsto the conclusion that at least for these data� sensitivityof the test is not worse as compared with the correlationfold tests

Figure �� Results of the new mean test applied tocollection � See caption to Figure �

In conclusion� we make two comments � Results of the new mean test do not change when

directions are rejected consecutively one by one from thedata tested and the number of samples in use is reducedto �� and � for collections �� and �� respectively The correlation fold test �Bazhenov and Shipunov� �� Shipunov� �� fails to give adequate results upon

Figure �� Results of the new mean test applied tocollection � See caption to Figure �


Figure �� Results of the new mean test applied to collections �� a� Histogram of ratiosof concentration parameters in geographic coordinates to those in stratigraphic ones plotted fortesting a null hypothesis �vector distribution of the collection under study is that of prefolding one�component magnetization� �b� Histogram of ratios of concentration parameters in stratigraphiccoordinates to those in geographic ones plotted for testing a null hypothesis �vector distributionof the collection under study is that of postfolding one�component magnetization�

a number of samples diminished to �� and �� There�fore the new test is more sensitive and applicable topaleomagnetic collections of lesser volume � Model results obtained by the new mean test en�

able de�ning the range of possible values of the ratioKa�Km that match a null hypothesis tested Thusfor collection �� testing of the �rst null hypothesis H�

�� holds true� gives the range �� for the ra�tio Km�Ka� with the most probable values being �� see Figure � and Table �� The value Km�Ka

calculated for the observed distributions is �� i e � itcoincides with possible values In testing the hypoth�esis H� �a postfolding magnetization component is as�

sumed�� Ka�Km values lie between and � � The ratioKa�Km � �� calculated for the observed distributionsdoes not match possible values� allowing the conclusionto be drawn that is similar to the above one In thiscase a critical value Ka�Km was estimated to be � �at the signi�cance level �� This value being exceededimplies the presence of a prefolding component Testing of data from the other two collections gives

results coinciding with those obtained by the mean testmodi�cation considered above �see Figure � and Ta�bles � and �� Thus one may state that two new modi�cations of

the mean test are proposed in the present work� one


Figure �� Listing of the program implementing the new mean test


compares concentration parameters of model sampleswith those of a real collection studied� and the otherdeals with concentration ratios that are used in the oldmodi�cation of the mean test

Identi�cation of SynfoldingMagnetization

The new mean test can be readily applied for estab�lishing a synfolding magnetization component For thispurpose one should �rst determine the intersection ofsmall circles de�ned by NRM directions for each sam�ple in stratigraphic and geographic coordinates Then�projections of the obtained intersection points onto eachsmall circle should be found� thus specifying a suggesteddistribution of a synfolding magnetization component aswell as its mean direction and concentration parameterused below for modeling The synfolding magnetizationdistribution can be estimated by the distribution ob�tained from directions of intersections of small circleswith great circles that pass through the origin of coor�dinates and are speci�ed by directions of small circleintersections and folding rotation axis The following hypothesis is tested� the distribution

obtained is that of synfolding one�component magne�tization In other words� additional terms that are re�sponsible for prefolding and postfolding components arelacking in the formulas

F �m� � ff��s��n��a��m�g ��

F �a� � ff��s��n��a��m�g ��

where ��s� is the distribution of a synfolding magneti�zation component Then� distributions of possible concentration param�

eters Ka and Km are obtained by modeling M timesnumerically the synfolding component and recalculat�ing results to stratigraphic and geographic coordinates Comparing these distributions with real values we eitheraccept or discard the null hypothesis �see more detaileddescription of the testing procedure above�

Conclusions

� Two new modi�cations of the mean fold test areproposed which are based on clustering examination instratigraphic and geographic coordinate systems anduse numerical modeling by the Monte Carlo method�with the Barnard method applied to testing statisticalhypotheses � The proposed modi�cations of the mean test are

free from restrictions and uncertainties of the old� incor�rect version of the test Resolution of the new tests withrespect to a second magnetization component is higheras compared with correlation fold tests now available Moreover� the volume of paleomagnetic collections re�quired for testing is reduced � The new mean tests are promising for identi�cation

of synfolding magnetization

Appendix

Figure � presents the work version of the programNewFoldTest in Turbo Pascal � implementing the newmodi�cation of the mean test �fold test� The programuses procedures of paleomagnetic analysis �the Paleounit� for various vector operations on a unit sphere andfast sorting procedure QuickSort from the Turbo Pascalpackage Full listings of paleomagnetic analysis proce�dures are presented in the work by Shipunov ��

Acknowledgments� This work has been funded by theRussian Foundation of Fundamental Research �grant ��

References

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Fisher� R� A�� Dispersion on a sphere� Proc� R� Soc� LondonA� ��

Khramov� A� N� �Ed�� Paleomagnetism� �� pp�� Nedra�Leningrad� ��

Maindonald� J�� Numerical Algorithms in Applied Statistics�

�� pp�� Finansy i Statistika� Moscow� ��McElhinny� M� W�� Statistical signi�cance of the fold testin paleomagnetism� Geophys� J� R� Astron� Soc��

McFadden� P� L�� A new fold test for paleomagnetic studies�Geophys� J� Int��

McFadden� P� L�� and D� L� Jones� The fold test in paleomagnetism� Geophys� J� R� Astron� Soc��

Shipunov� S� V�� Foundations of Paleomagnetic Analysis

Theory and Applications� �� pp�� Nauka� Moscow� ��Shipunov� S� V�� The pebble test in paleomagnetism� Izv�

Acad� Sci� Russ� Phys� Solid Earth� no� �� Watson� G� S�� and R� J� Enkin� The fold test in paleomagnetism as a parameter estimation problem� Geophys� Res�Lett��

�Received April ��

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A new fold test in paleomagnetism (rehabilitation of the mean test)