GEOPHYSICAL
RESE•CHLETrERS,
VOL.19,NO.23,PAGES
2353-2356,
DECEMBER
2, 1992
SAMPLINGBIAS IN VGP LONGITUDES
GaryD. Egbert
College
ofOceanography,
Oregon
State
University,
Corvallis
Abstract.
I deriveprobability
densities
for virtualgeomag- VGP statistics
by suchunevensampling
is oftenmentioned
neticpole(VGP)longitudes
for a generalstatistical
modelof
as a potentialuncertainty
in thesestudies[e.g., Valet et al.,
localmagnetic
fieldvariations.
I showthatevenfor very
1992], and some efforts have been made to allow for this
simplestatistically
homogeneous
modelsof secularvaria-
complication
[Constable,1992]. Here I explicitlyaddress
thequestion
of whethera uniformdistribution
of VGP longitudesshouldbe expectedin PSV or reversaldatawhenthe
distribution
of fieldsampling
sitesis non-uniform.
tion,
thedistribution
ofVGPlongitudes
ispeaked
90øaway
fromthe samplinglongitude. Thus, when sitesare distributedunevenly, a non-uniform overall distributionof VGP
longitudes
is to beexpected.
Analysis
of therecent
geomagneticfieldindicates
thatthisbiascanbesignificant,
particu-
Directional Densities and the VGP Transformation
laxlywhenrandomerrorsin paleomagnetic
dataareallowed
for. It is possiblethat someof the deviationsfrom uniformityseenin recentcompilations
of paleomagnetic
reversal
TheVGPtransformation
is a one-to-once
mapping
b' --->
•v
from the local magneticfield direction(b') to the direction
dipolewhichwouldcause
thisfield.It
andsecular
variationdataare a statistical
artifactresulting (•) of a geocentric
haslongbeenappreciated
thatthe scatterof localfielddirectionsmeasuredin a seriesof lava flowsis distorted(bothin
shapeand magnitude)by this transformation[Cox, 1970].
HereI deriveexpressions
for VGP directionaldensitiesfor a
fromthe distortionof the VGP transformation,and the non~
uniformity
of paleomagnetic
samplingsites.
Introduction
general statistical model of variations in local field direc-
Recentlyseveralgroupsof researchers
havenotedsystematic
patternsin virtualgeomagnetic
pole (VGP) longitudes
duringreversals,whichtheyhaveinterpreted
in terms
tions.
ofcore-mantle
dynamics[Tric et al., 1991;Clement,1991].
However,the significanceof theseobservations
is in some
here are invariantunder rotations about the Earth's axis, it
dispute
[Valetet al., 1992;Langeriset al., 1992]. Usingthe
rangeof colatitudes
0o. Two fight handedCartesiancoordi-
highestquality data from a limited number of reversals,
nate systemsare considered. In the "standardcoordinate
Valetet al. [1992] testedthe statisticalsignificance
of these
claims,and concludedthat the evidencefor VGP pathsystematics
is not yet compelling. Subsequently,
Laj etak
[1992]have pointedout that when more sophisticated
methods
of analysisare appliedto a slightlylargerset of
reversaldata, the VGP systematicsbecomestatistically
significant.
To setnotationI beginwith a formaldevelopment
of the
VGP transformation. Since all statistical models considered
sufficesto considerobservations
at longitudeqb=0, for a
system"
•, is aligned
withtherotation
axis,and• pointsin
the directionq)---0.The local coordinatesystemat the samplingsite(to = r(sin0o,0, cos0o)in the standardsystem)is
obtained
by rotatingthe standardcoordinate
framean angie
0o alongthe meridianq)=0so thatthe •,'= •o. I will use
primesto indicatethatthe components
of a vectorshouldbe
interpretedin the local coordinatesystem,and "hats"to
VGPshavealsobeenusedin studies
of pale0-secular
varidenote
unitvectors
(thus• = v,•lvll).Thecomponents
in the
ation(PSV)data. Constable
[ 1992]foundthatthehistogram standardand local coordinatesystemsof a fixed vectorare
of VGP longitudes
computedfor a set of 2,244 PSV data related
viax' = Ux andx = Urx', where
fromthelast5 My hastwo approximately
antipodal
peaks,
nearthelongitudes
whichseemto bepreferred
for thereversalVGPpaths.Herestatistical
testsdemonstrated
unequivocallythatthe VGP longitudes
deviatedsignificantly
froma
cos0o 0 sin0o
U=
0
1
0
.
-sin0o 0 cos0o
uniformdistribution,and Constableconcludedthat the PSV
dataprovideevidence
for persistent
non-zonal
components It isreadilyverifiedthatthemagneticfieldsobserved
at ro
of thegeomagnetic
field. In all of thesestatistical
tests,a
for a dipoleat the originwith momentd' areb'= Vd', where
uniform
distribution
of VGP longitudes
wastakenasthenull
(with all vectorsexpressed
in the local coordinate
system)
hypothesis.
V = c diag[ -1, -1, 2 ]. Here c is a constantwhich,because
ForboththereversalandthePSV data,sampling
sitesare we are ultimatelyinterestedonly in directions,can be set to
veryunevenly
distributed.
Forinstance,
forthePSVdataset
1. Thus,whenexpressed
in the standard
coordinate
system,
used
byConstable
therearetwolargeholes
(75o and45ø
the VGP is
wide)in longitudinal
coverage.
The potentialbiasingof
Copyright
1992by theAmerican
Geophysical
Union.
Paper
number92GL02549
0094-8534/92/92
GL-02549503.00
•(b')= uTv-1b'AIUrV
-• b'11
With thisformulation
VGPsareexpressed
in Cartesian
coordinatesasunit vectorswhichare simplyrelatedto the local
magnetic
fieldvectors.
•In fact,withpaleomagnetic
dataonly
the directional
vectorb'= b'/•]b'[[is generally
available
for
computation
of theVGP. However
2353
2354
Egbert:Sampling
Biasin VGP Longitudes
Cv(fy)
= Cv(b')
P•.((D)
= [2/cD(D-2cos2(D
+ sin2(D)]
-1ß
(2)
sothiscomplication
is only apparent.The statistical
proper-
(5)
Foran isotropic
localfieldcovariance
(%x=Cyyy
=%•),
ties of VGPs are insensitive to when or how magnitude
informationis lost. In the following it will often be con-
PL(q))
issharply
peaked
90ø away
fromthesampling
site,
venientto applythe VGP transformation
(1) directlyto the
The plotteddensityis perhapsunrealistic,
sincethescatter
in
unnormalizedlocalfields;thisis justifiedby (2).
In general
themagnetic
fieldobserved
at•o canbewritten
b' = bAo'+ bsr' = dVU•,+ bsv',
withthenon-uniformity
greatest
atlow-latitudes
(Figure
1).
localfielddirections
is notgenerally
isotropic
[Cox,1970;
Harrison, 1980; Merrill and McElhinny, 1983]. Nonetheless,this exampleillustratesthe centralthrustof this note:
isotropicvariationsin local field directionswill resultin
whered is a scalargiving magnitudeand polarityof the
VGPlongitudes
whichtendto be offset90o fromthesam-
dominant
axialdipoleb•D'. Theremainder
of thefield,bsv'
pling longitude. In the next sectionI will considersome
moreplausiblestatisticalmodelsof secularvariationscatter.
(the "secularvariation"),includesboth dipolewobbleand
thenon-dipolefields.Ourgoalis to derivethestatistical
distribution of VGP directionsfor some simple, but in fact
fairlygeneralstatistical
modelsford andbsv'.
First,! consider
the simplercaseof themarginaldistribution of VGP longitudes.Let Pa be thematfixwhichprojects
ontothe horizontalcomponents
of theVGP. Then,with the
VGP as defined in (1), the VGP longitude q• can be
represented
in Cartesiancoordinates
by the unittwo-vector
,, [cosq)•
Pa•(b')_ Pav(b') vL
First I sketcha derivationof the full VGP directionalden-
sity. Evenwhentheultimateinterest
is in VGP longitudes,
thefull VGPdistribution
mayberequired.Forexample,
this
would be necessaryto compute the distributionof the
weighted average VGP longitude statistic MVL
=tan-l(•vyi/•Vxi) used
byValetetaL[1992]
toaverage
VGP longitudesduringa reversaltransition.
We assume
b' = dVU•,+ bsv',whered is a fixedconstant
andbsv'is Gaussian
with covariance
matrix•;sv. Thenthe
unnormalized
VGPv(b')= d•,+ uTv-• bsv'hasmeand•,and
covariance matrix Y-,v
= UTV-•Esv
V-•U.
Let
u = ZT'Av(b
') = dZT'A•.
+ e. Thentherandom
vectore is
=LsinJ
= IIPa•(b')ii
IIPav(b')ll
I!v,il
In thefollowingI referto •L astheVGPlongitude,
andv• as
Gaussianwith covarianceI, andthe corresponding
unitvec-
the "unnormalizedVGP longitude".Since
tor fi = u/lluilhas an angularGaussiandistribution,
with
parameterss=Z• •]Z• zll and m =llZ• zll. Watson
v• = Pav(b')= PaUrV-•bsv
',
(3)
[1983;sec.3.7] showsthat this distributionis well approxithemarginalVGP longitudedistribution
candependonlyon
matedby a Fisherdistribution
withlocation
parmeter•, and
the statisticalpropertiesof bsv',not on the modelassumed a dispersionparameterk which is a function of m. Watson
for the axial dipolepart of the field. Undermild assumptions [1983, pp 117-118] gives asymptoticforms for small and
concerning
the randomvectorbsv',an exactexpression
for
large m
theprobability
density
of •L canbegiven.
First,assumethatbsv'haszeromeanwith covariance
E[bsv'bsv
'r] = Zsv=
k(m)= 4(2•:)-'Arn m <1
k(m)- m2 m>>l,
and a brief table for intermediate values.
¬^
•A"
Thedensity
oftheVGP• = Zv fi/'IlZi;ullcanbecalculated
(where
Z'sZv
is thelowertriangular
partof theCholesky usinga slightgeneralizationof the transformationapproach
decomposition
ofZsv).Thenthevector
u'= Z?bsv'neces- usedabove[Watson,1983;Eq. 3.6.4]. In termsof theFisher
•A"
sarily has an isotropiccovariance,
with E[u'u'r]=I .
Assume
furtherthattheunitvectors
6'= u?llu'llhavea uniform distribution.
This will holdwhenbsv'hasan ellipsoidaldensitycenteredat theorigin(i.e., whenthejoint den-
sity of u' is radiallysymmetric).A specialcasewhich
satisfies
thisconditionis theGaussian
modelfor bsv'of Con-
density
pF(fi;•,k)andfi = Z7 •/•[Z7 vii,theresultis
pu() = lY-.I- p(u;s,k(m))IIrZTvl1-3"2 (6)
= k(m)exp[k
(re)p][4xsinh
k(m)IZul II•vrz•X½ll3/2]
-z
where
p= rv(b')rZ7
9)-'A
(,rZ7
stableand Parker [ 1988].
The unnormalized
VGP longitudevectorVLgivenin (3)
has2x2 covariance
matrix•L=PtlUTV-I•svV-1UpH
T.
Furthermore,
ZZ v,/llZZ'v11
is
uniformly
distributed
on
-•A
the unit circle,so w = vMIZ vll is a lineartransformation
of a uniformly distributed random direction. But
w/llwll
= cry,
sothetransformation
approach
outlined
by
Watson[1983,pp 109-110]canbeusedto compute
thedensity of the randomlongitudinal
directionvector•, =•.
FromEq. (3.6.7) of Watson,thisis
2.5
2.0
0o
=0ø
,,•1o,..5
.5
.0
whereI ZL I denotes
thedeterminant
of
The density has a particularly simple form when
-lOO
o
lOO
Degrees East of Sampling Site
Zsv= diag[c•xx,
c•w, Cyzz].
Inthiscase,
•;L=C•yy
diag[D
2,1]
Fig. 1. Normalized
VGP longitude
densities,
for anisotropic
vL= (cosq•,sinq)),andsimplifying(4) we find
sharply
peaked
90øawayfromthesampling
site.
of localmagnetic
fieldvariations.
Thedensity
is
where
D2=cos2•)oCYxx/(Iyy
+ sin200cYzz/4(Syy.
By writing distribution
Egbert:
Sampling
BiasinVGPLongitudes
2355
1.4
• 8o=O'
1.2
4
.8
X
I=
,6
I::
•
2.54
.2
.0
-100
00=90'
.4 -
00 = 30"
.2
.6
0
.0
100
I
....
I
I
0
-lOO
Degrees East of Sampling Site
100
Degrees East of Sampling
Site
Fig.
2. Densities
for2 < k"2< 3.5insteps
of0.25forasiteat
Fig.3 Densities
forr 2 = 2.54(thevaluesuggested
by the
30ølatitude.
recentgeomagneticfield, when isotropicrandom and sys-
Eq. (6) is actuallyquite genera/,sinceit can be usedto
tematic
errors
areallowed
for),for fivelatitudes
(0ø,22.5ø,
45ø,67.5øand90ø).
compute
the densityof VGP directions
for anymeanand
covariance
of the local field with a Gaussiandistribution.In
fact,thisdistributional
assumption
is muchstronger
than
required.The resultholds(approximately)
providedthe
Fisherdistribution
is a reasonable
approximation
to the
As thegradientof a randomscalarpotentialarisingfrom
internalsources,
bsv'can be expanded
in sphericalharmonics as
bsv'(r)=-V• • (a/r)l+ialmYlm(•).
(necessarily
isotropic)scatterin fi aboutits mean.When
l=lrn=-I
d=0, pv(fi;•,k)reduces
to a uniform
density,
and(6)
Sincethe covarianceof the random potential is assumed
reduces to
pv(•)= [4•1• I'/•(•rZ•)•/2]-• .
(7)
invariant
under
all
rotations
(i.e.,
Cov[q)(rl),q)(r2)] = Cov[q)(Wrl), q)(Wr2)]for all rotations
Eq.(7) couldbeusedfor modeling
themarginaldistribution W) the randomcoefficientsaim must have zero mean and
covariance
Ealmarm'
=c•tt'•m=' [e.g.,Yaglom,
ofVGPs
during
areversal,
byincluding
random
axialdipole diagonal
variations
in Zsu.
1961]. Furthermore,the covariance matrix of the field componentshasthe form [Constableand Parker, 1988]
Statistical
Modelsfor bsu'
Ebsv'rbsv
'= 23s•= c•2n
diag[!,1, r 2]
Theexample
of Figure1 demonstrated
thatanisotropic
distribution
forbsv'resulted
in a highlynon-isotropic
distribution
of VGPs.However,
Cox[ 1970]haspointed
outthat
VGPsaremorenearlyisotropicthanlocalfielddirections.
Furthermore,
it hasfrequently
beenargued
[e.g.,Harrison,
1980; Merrill and McElhinny, 1983; McFaddenand
McElhinny,
1984]thatsecularvariationnecessarily
should
leadto isotropic
scatter
in VGPs.Thiswouldsuggest
that
thebiasevidentin Figure1 may not be relevantto real
paleomagnetic
VGP distributions.
It is instructiveto consider the statisticalmodel for the
geomagnetic
field of Constableand Parker [1988]. In this
(8)
c•2•
=l---1
X;/(/+1)
c• {Xz
2=Z (/+1)2c•?
r2= c•z2/c•2•
2
'
/=1
Bysetting
rt2 = 2+2//wecanwrite
r2= Zwtz:t
2 wherewt= I (l+l)c•t
2 l (l+l)c•
(9)
areweights
whichsumto one. Thusr 2 is theweighted
average
of •q2= 2+2//. It followsthat2 < k-'2_<4 andthat
r 2 = 4 onlyif ts•= 0 for all 1> ! - i.e.,onlyif thefieldis
alwayspurelydipo!ar.Conversely,
r 2 approaches
2 for
fieldswhichare dominatedby shortwavelengthfeatures.
The densityof VGP longitudesfor this modelis given by
model
bsu'is a sample
fromanhomogeneous
(i.e.,rotationallyinvariant
in a statistical
sense)
random
process
onthe
sphere.
It mightwell be arguedthatthismodelis overly
simplistic.
However,
it hasthemaximum
conceivable
sym-
(5) with D 2 = cos20o+(r2/4)sin20o
' For r 2 = 4, or for
0o= 0øor 180ø(sampling
atthepole),D = i andthedensityisuniform.
Otherwise
k is peaked
90ø awayfromthe
mew amongall modelsdominatedby an axial dipole,and
small
K2 (Figure
2),andatlowlatitudes
(Figure
3).
r 2 canbeestimated
fortherecent
geomagnetic
fieldusing
thusseems
a reasonable
null hypothesis
modelwhenone
seeks
to demonstrate
thatpaleomagnetic
datarequiresome
further
breakdown
ofsymmetry.
TABLE 1. Covariance
ZsuEstimated
fromDGRF-85
Bx
By
Bz
samplingsite. The deviationfrom uniformityis greatestfor
theDGRF models[e.g.,Langel1992]. One way to dothisis
tousetheGauss
coefficients
toestimate
ry?andthenuse(9).
ForDGRF-85thisyieldsanestimate
of r 2 = 3.30. An alternativeapproach,which also allowsa check on the form of
the covariancematrix predictedby the homogeneous
statisti-
ca/model,is to usethe DGRF modelsto directlycompute
Bx
1.10
By
0.31
0.91
Bz
0.18
0.04
3.16
All entries
arenormalized
by theaverage
horizontal
field
variance
({5xx
+ {5yy)
= 3.97x107nT
2.
the averageof Zsvacrossthe globe. The resultsfor DGRF85 (Table1) arereasonably
consistent
with (8), andsuggest
avalue
ofr2= 3.16.Thedeviations
(i.e.,c•xx
,•C•yy;
nonzero
off-diagonalelements)are of questionable
significance,
consideringthat only a singlerealizationof the field is available.
Egbert:
Sampling
BiasinVGPLongitudes
2356
References
Forr 2 = 3.16- 3.3thedeviation
ofVGPlongitude
density
from uniformis _+10%or less(Figure2). However,random
errorsin the paleomagnetic
observations,
whicharelikely to
be moreisotropic[e.g.,Merrill andMcElhinny,1983],can
increasethisbiassignificantly.The summaryof datafrom
thepast5 my givenby McElhinnyandMerrill [ 1975;Table
3] suggests
thatwithin-site
variabilityincreases
thescatter
in
average
fielddirections
for individual
lavaflowsby approximately 10%. Assumingthis represents
isotropicmeasurement error, and that the covarianceof the local field due to
PSVisoftheformZsv= o•diag[1,1,k--2],
thecovariance
of
the
recorded
fields
will
be
approximately
1.2xo•diag[1,
1,r•2l wherer•2 = (r2+.2)A.2is theeffectivevalueofr 2. Fork-'2= 3.16,re2 = 2.80.Allowing
forthe
possibilityof correlatedor systematic
errorswhichdo not
affect within-sitescatter(e.g., tectonicrotationor tilting,
localmagnetic
anomalies,
secondar•
overprinting,
orrock
magneticcomplications),
reduces•c• further. For instance,
assuming
thattheseerrorsarecomparable
to thoseestimated
from the within-site scatter (so that total errors increase
between
flowscatter
by20%)yields
r•2 - (r2+.4)/1.4
= 2.54.
Forthisvalueof r 2 deviations
fromuniformity
reach+_25%
at low latitudes(Figure 3).
The resultsgivenhereare not reallyinconsistent
with the
conclusions
of previous
workers.
Evenwithr 2 =2.5,the
VGPsareindeedmorenearlyisotropicthanlocalfielddirections. However,as theseresultssuggest,
neitherdistribution
is likely to be purelyisotropic.
Conclusions
Clement, B.M., Geographicaldistributionof transitional
VGPs:
evidence
fornon-zonal
equatorial
symmetry
during
the Matuyama-Bmhnes geomagnetic reversal, Earth
planet.Sci.Left., 104, 48-58, 1991.
Constable,
C.G.,Link betweengeomagnetic
reversal
paths
andsecularvariationof the field overthe past5Myr,
Nature, 358, 230-233, 1992.
Constable,
C.G.andR.L. Parker,Statistics
of thegeomagneticsecular
variationfor thepast5m.y.,J. Geophys.
Res.,
93, 11,569-11581, 1988.
Cox, A., Latitude dependenceof the angulardispersion
of
the geomagnetic
field, Geophys.J. R. Astron.Soc.,20,
253-269, 1970.
Harrison, C.G.A., Secular variation and excursionsof the
geomagneticfield,J. Geophys.Res., 85, 3511-3522, 1980.
Laj, C., A. Mazaud, R. Weeks, M. Fuller, and E. HerrerroBervera, GeomagneticReversalPaths,Nature, 359, 111112, 1992.
Langel, R.A., International GeomagneticReferenceField
1991revision,Geophysics,
57, 956-959, 1992.
Langeris,C.G., A.A.M. vanHoof, andP. Rochette,Longitudinal confinementof geomagneticreversal paths as a
possiblesedimentary
artifact,Nature,358, 226-230, 1991.
McFadden,P.L. andM.W. McElhinny,A physicalmodelfor
paleosecular
variation,Geophys.J. R. Astron.Soc.,78,
809-830, 1984.
McElhinny,M.W. and R.T. Merrill, Geomagnetic
secular
variationoverthe last 5my,Rev. Geophys.
SpacePhys.,
13, 687-7 08, 1975.
I have shownthat even for simplesphericallysymmetric
statistical models for secular variation of local magnetic
fields,VGPlongitude
densities
will be peaked
90ø away
from the sampling site. This departurefrom uniformity
should
bemostsevereatlowlatitudes,
andwhenr 2 is small.
Tric, E., C. Laj, C. Jehano,J.-P.Valet, C. Kissel,A. Mazaud,
andS. Iccarino,Highresolution
recordof theupperOlduvai transition
from Po Valley (Italy) sediments:
support
for dipolar transitiongeometry?,Phys. Earth planet.
Inter., 65, 319-336, 1991.
Forplausible
effective
values
ofr 2 (which
allowformeas- Valet, J.-P., P. Tucholka,V. Courtillot,and L. Meynadier,
urementerrors)the bias can be substantial.
A 90ø offsetfromtheconcentration
of sampling
sitesis
Paleomagneticconstraintson the geometry of the
geomagneticfield duringreversals,Nature, 356, 400-407,
very clearin the PSV VGP longitudehistograms
presented
1992.
by Constable[1992], and was alsonotedby Valet et al
Merrill R.T. and M.W. McElhinny,The Earth's Magnetic
[1992] in mean VGP longitudescomputedduringreversals.
Field,AcademicPress,London,401 pp., 1983.
While it is by no meansclearthatthisbiaseffectcanexplain
Watson,G.S., Statisa'cs
on Spheres,JohnWiley andSons,
thesystematics
seenin palcomagnetic
VGPs,thispossibility
New York, 238 pp., 1983.
shouldbe carefullyconsideredbeforemoreexcitingphysical
models are endorsed.
Yaglom,A.M., Secondorderhomogeneous
randomfields,in
Proceedingsof the Forth Berkeley Symposiumon
The modelsgiven here make explicit predictions(e.g.,
concerningthe latitudinal dependence
of the bias) which
couldbe tested. This might help to establishhow important
this effect is. In any event,futureeffortsto rigorouslytest
the statisticalsignificanceof VGP systematics
shouldprobably adoptthe generalapproachelucidated
hereandbegin
from more realistic null hypothesismodels of the actual
magneticfieldvariations,
sothattheycanincorporate
possible biasdueto unevensamplingontheEarth'ssurface.
G.D. Egbert, College of Oceanography,
OregonState
University,Oceanography
Admin.Bldg. 104,Corvallis,OR,
Acknowledgments:
I thank R.T. Merrill and S. Levi for
helpfuldiscussions.
(ReceivedSeptember25, 1992;
acceptedOctober26, 1992.)
Mathematical
StatisticsandProbability,vol. 2, pp. 593622,Universityof CaliforniaPress,Berkeley,1961.
97330-5503.