paleosecular variation and the average geomagnetic field ... · 20 july 2006 q07007,...

Paleosecular variation and the average geomagnetic field at±20! latitude

K. P. Lawrence and C. G. ConstableInstitute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California San Diego,9500 Gillman Drive, Mail Code 0225, La Jolla, California 92093, USA ([email protected])

C. L. JohnsonInstitute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California San Diego,9500 Gillman Drive, Mail Code 0225, La Jolla, California 92093, USA

Also at Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, Canada

[1] We assembled a new paleomagnetic directional data set from lava flows and thin dikes for four regionscentered on ±20! latitude: Hawaii, Mexico, the South Pacific, and Reunion. We investigate geomagneticfield behavior over the past 5 Myr and address whether geographical differences are recorded by our dataset. We include inclination data from other globally distributed sites with the ±20! data to determine thebest fitting time-averaged field (TAF) for a two-parameter longitudinally symmetric (zonal) model. Valuesfor our model parameters, the axial quadrupole and octupole terms, are 4% and 6% of the axial dipole,respectively. Our estimate of the quadrupole term is compatible with most previous studies of deviationsfrom a geocentric axial dipole (GAD) field. Our estimated octupole term is larger than that from normalpolarity continental and igneous rocks, and oceanic sediments, but consistent with that from reversedpolarity continental and igneous rocks. The variance reduction compared with a GAD field is !12%, andthe remaining signal is attributed to paleosecular variation (PSV). We examine PSV at ±20! using virtualgeomagnetic pole (VGP) dispersion and comparisons of directional distributions with simulations fromtwo statistical models. Regionally, the Hawaii and Reunion data sets lack transitional magnetic directionsand have similar inclination anomalies and VGP dispersion. In the Pacific hemisphere, Hawaii has a largeinclination anomaly, and the South Pacific exhibits high PSV. The deviation of the TAF from a GADcontradicts earlier ideas of a ‘‘Pacific dipole window,’’ and the strong regional PSV in the South Pacificcontrasts with the generally low secular variation found on short timescales. The TAF and PSV at Hawaiiand Reunion are distinct from values for the South Pacific and Mexico, demonstrating the need for time-averaged and paleosecular variation models that can describe nonzonal field structures. Investigations ofzonal statistical PSV models reveal that recent models are incompatible with the empirical ±20! directionaldistributions and cannot fit the data by simply adjusting relative variance contributions to the PSV. The±20! latitude data set also suggests less PSV and smaller persistent deviations from a geocentric axialdipole field during the Brunhes.

Components: 13,384 words, 11 figures, 5 tables.

Keywords: paleomagnetic; paleosecular variation; time-averaged field.

Index Terms: 1522 Geomagnetism and Paleomagnetism: Paleomagnetic secular variation.

G3G3GeochemistryGeophysics

GeosystemsPublished by AGU and the Geochemical Society

AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES

GeochemistryGeophysics

Geosystems

Article

Volume 7, Number 7

20 July 2006

Q07007, doi:10.1029/2005GC001181

ISSN: 1525-2027

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Copyright 2006 by the American Geophysical Union 1 of 25

http://dx.doi.org/10.1029/2005GC001181

Received 8 November 2005; Revised 28 March 2006; Accepted 20 April 2006; Published 20 July 2006.

Lawrence, K. P., C. G. Constable, and C. L. Johnson (2006), Paleosecular variation and the average geomagnetic field at ±20!latitude, Geochem. Geophys. Geosyst., 7, Q07007, doi:10.1029/2005GC001181.

1. Introduction

[2] Observations of the magnetic field allow re-mote sensing of activity in Earth’s deep interior,providing indirect information about the dynamicsof the liquid outer core over timescales of centuriesto millions of years. Paleomagnetic studies can beused to identify long term temporal variations ofthe geomagnetic field on local, regional, and globalscales. These time variations are known as paleo-secular variation (PSV). The time-averaged field(TAF) over specific time intervals such as thecurrent or past few polarity chrons is also ofinterest. The TAF may indicate long-term depar-tures from the geocentric axial dipole (GAD), themost basic field model used throughout paleomag-netism. Both paleomagnetic field strength anddirection are used to study TAF and PSV; herewe only consider directional data because of theirgreater abundance and reliability.

[3] Paleodirections used in PSV studies can beobtained from both sedimentary and igneous rocks,but in this work we focus exclusively on the latter.Data from igneous rocks, specifically lava flows orthin dikes that cool rapidly during emplacementprovide a record of the paleofield at a single timeand place that, from a geological perspective, canbe considered instantaneous. Given adequate tem-poral and spatial sampling such flows could pro-vide a global record of the field spanning millionsof years. However, sampling is limited by thegeographical distribution of volcanic sources andthe irregular occurrence of eruptions. Statisticalmethods are used to characterize the TAF andPSV since poorly known age relationships amonglava flows prohibit the construction of an accuratesequence of field measurements through time.

[4] PSV is often described in terms of angulardispersion in local field directions or in virtualgeomagnetic pole (VGP) position about GAD. AsEarth’s field is predominantly dipolar, early studiesof PSV attempted to distinguish contributions toangular dispersion from the dipole and nondipoleparts of the field. Contributions from dipole varia-tions were further separated into temporal varia-tions in the dipole moment, and those from

variations in dipole orientation (dipole wobble)with respect to the Earth’s rotation axis. Over thepast 40 years, many models have been proposed toexplain the observed latitudinal variation in angulardispersion: these have been comprehensivelyreviewed by Merrill et al. [1996]. Most modernPSV models are based on a general approachdeveloped by Constable and Parker [1988] thatprovides a complete statistical description of thegeomagnetic field. In these models the measurablepart of the global geomagnetic field generated inEarth’s outer core is represented by the sphericalharmonic expansion of a Laplacian potential. Thestatistics of the temporal variations in the field aredescribed in terms of statistical distributions of thecoefficients of the individual spherical harmonicterms. In the Constable and Parker [1988] model,the spherical harmonic coefficients are consideredto be samples drawn from a Gaussian statisticalprocess with mean and variance specified accord-ing to spherical harmonic degree and order. Manyvariations to this model have since been proposed[e.g., Constable and Johnson, 1999; Hatakeyamaand Kono, 2002; Hulot and Gallet, 1996; Konoand Hiroi, 1996; Quidelleur and Courtillot, 1996;Tauxe and Kent, 2004], none of which are com-pletely satisfactory since no model explainsall aspects of existing PSV data. Nevertheless,recent work by Hulot and Bouligand [2005]shows that this approach can be useful in discrim-inating among various symmetry properties of thepaleofield.

[5] The nature of non-GAD persistent contribu-tions to the TAF is also debated, and there arenumerous TAF models for the time period 0–5 Ma [Carlut and Courtillot, 1998; Constableand Parker, 1988; Gubbins and Kelly, 1993;Hatakeyama and Kono, 2002; Johnson andConstable, 1995, 1997; Kelly and Gubbins,1997; McElhinny et al., 1996b; Quidelleur et al.,1994; Schneider and Kent, 1990]. Almost the onlycommon feature in these TAF models is a smallaxial quadrupole contribution whose size rangesfrom 2.5 to 8% of the axial dipole term [McElhinny,2004]. The axial quadrupole is the simplest param-eterization (in spherical harmonic models for thefield) that explains observations of low-latitude

GeochemistryGeophysicsGeosystems G3G3 lawrence et al.: paleosecular variation 10.1029/2005GC001181

2 of 25

negative inclination anomalies (observed inclina-tion minus that predicted for a GAD) in the paleo-field. Although several authors propose that allfurther departures from GAD arise from inadequa-cies in quality and spatiotemporal data distributions[Carlut and Courtillot, 1998; McElhinny et al.,1996b], others are of the opinion that these depar-tures reflect real field structure [Gubbins and Kelly,1993; Hatakeyama and Kono, 2002; Johnson andConstable, 1995, 1997; Kelly and Gubbins, 1997],including persistent asymmetries between the Pa-cific and Atlantic hemispheres.

[6] In this study, we compile and examine previ-ously published paleodirections derived from lavaflows emplaced over the past 5 million years atlatitudes close to ±20!. We conduct a comprehen-sive analysis of the average field and secularvariation recorded from 4 regions centered onHawaii, Mexico, Reunion, and in the SouthernPacific Ocean. We discuss how data quality andtransitional field directions affect PSV and TAFestimates, and we investigate the possibility ofequatorial and longitudinal asymmetries in both

PSV and the TAF. Low PSV in the Pacific (some-times misleadingly termed the ‘‘Pacific dipolewindow’’) has been the topic of ongoing debate[Johnson and Constable, 1997, 1998; McElhinnyet al., 1996a; McWilliams et al., 1982; Miki et al.,1998; Shibuya et al., 1995], since it was firstproposed by Doell and Cox [1971] using Hawaiiandata. We revisit this issue, comparing the statisticalproperties of the paleofield at these mid andcircum-Pacific and Indian Ocean locales, and as-sess how well our new data compilation is de-scribed by current [Constable and Johnson, 1999;Tauxe and Kent, 2004] PSV models.

2. Data Set

[7] We have compiled paleomagnetic directionaldata from published volcanic flow studies fromHawaii, Mexico, Fiji, Cook, Society, Reunion, andMauritius Islands. We group data from the Fiji,Cook, and Society Islands (our ‘‘South Pacific’’),and merge the Mauritius data with those fromReunion giving four distinct regions (Figure 1).

Figure 1. Empirically derived data density for study regions. Color scale is logarithmic. Boxes show the studylocations. Blue circles represent studies without tectonic rotation reported; green stars are those reportingpostemplacement tectonic rotation. More information on the data from each site is in Table 1.

GeochemistryGeophysicsGeosystems G3G3 lawrence et al.: paleosecular variation 10.1029/2005GC001181lawrence et al.: paleosecular variation 10.1029/2005GC001181

3 of 25

Sampling latitudes lie within 5! of ±20!. Ourcompilations for Hawaii and Reunion include andextend directional data used by Love and Constable[2003], and our compilation for Mexico is verysimilar to that of Mejia et al. [2005]. We limitour analyses to data spanning the past 5 Myr fortwo reasons: first, the number of data availablediminishes rapidly with increasing age; second, forages less than 5 Myr the plate motion correctionsneeded to obtain accurate paleo-site locations aresmall and reasonably well known. A summary ofthe data that we examined is given in Table 1. Thetotal number of paleodirections is 1125 for Hawaii,567 for Mexico, 267 for Reunion, and 690 for theSouth Pacific.

[8] We assigned numerical ages to all the paleo-magnetic sites, so that we could restrict our anal-yses to the period 0–5 Ma and assess the temporaldistribution within our compilation. Where possi-ble we use site ages determined by radiometricmethods and reported in the original reference.When the flow or the region was dated radiomet-rically, but at a different location from the paleo-magnetic site, we assign the reported flow orregion date to the site in question. When only thegeological epoch was reported (e.g., Pliocene orMiocene) the midpoint of the epoch age range wasused. For epochs spanning the Brunhes-Matuyamatransition, sites with reverse polarity were assignedages greater than 0.78 Ma.

[9] The age distributions for each geographicalregion are shown in Figure 2. The temporal distri-bution for each region is nonuniform. Data setsfrom Reunion, Hawaii, and Mexico are heavilybiased to Quaternary age flows. The !0.5 Ma peakin the Hawaiian data distribution reflects extensivesampling of young flows on (the big Island of)Hawaii. The peak around 3–3.5 Ma corresponds toflows on Oahu, Kauai, and Nihoa Islands [Doell,1972a, 1972b; Herrero-Bervera and Coe, 1999;Laj et al., 1999]. The South Pacific age distributionis strongly influenced by studies focused aroundthe Matuyama/Gauss transition [Duncan, 1975;Yamamoto et al., 2002], in the Late Pliocene(!2.5 Ma) and the Brunhes/Matuyama transitionat !0.8 Ma [Chauvin et al., 1990]. Reunion andMauritius sites are younger than 3.5 Ma andprimarily of Brunhes age.

[10] We present our complete data collection inFigure 3. The left column shows an equal areadisplay of local field direction, D and I, the centercolumn shows the equivalent VGP positions andthe right column shows the D0 and I0 directions

[Hoffman, 1984]. D0 and I0 correspond to local fielddirections that have been rotated so that the centerof the projection is the direction from the predictedGAD field at that site. The use of D0, I0 coordinateshelps to remove the effect of latitudinal differencesamong sites. PSV and TAF studies require paleo-magnetic directions from sites that either have notbeen subject to postemplacement tectonic rotationor can be corrected back to their original positionand orientation. Sites with uncorrected tectonicrotations are likely to exhibit increased scatterand/or bias in paleomagnetic directions. A signif-icant percentage of paleodirections from two of thefour regions investigated here are affected bypostemplacement tectonic rotation (Figure 3, greensymbols). Multiple studies note regional tectonicrotation over the past 5 Myr in Mexico [Maillol etal., 1997; Mejia et al., 2005; Nieto-Obergon et al.,1992; Uribe-Cifuentes and Urrutia-Fucugauchi,1999] and in the South Pacific [Falvey, 1978;Inokuchi et al., 1992; James and Falvey, 1978;Malahoff et al., 1982; Tarling, 1967a; Taylor et al.,2000] and one study notes local tilt in Hawaii[Riley et al., 1999]. Two studies of Mexicanvolcanics indicate regional counterclockwise rota-tion [Ruiz-Martinez et al., 2000; Soler-Arechaldeand Urrutia-Fucugauchi, 2000], while a thirdsupports tilting due to listric faulting [Nieto-Obergon et al., 1992]. There is no obvious patternof overall bias from rotation visible in the paleo-magnetic directions or VGP latitudes for the Mex-ican data. The majority of tectonically rotated sitesin the South Pacific subregion result from counter-clockwise rotation of the Fiji platform. This isapparent in the nearly 20! westerly rotation inmagnetic directions from GAD, and in thecorresponding longitudinal bias in VGPs. For theSouth Pacific compilation we examined studiesfrom a wide longitudinal range to maximize thetemporal distribution of the data. However, all buttwo sites west of 200!E are excluded because ofpostemplacement rotations, so that the South Pa-cific region sampling is centered at 205!E ± 5!Eand is heavily biased toward pre-Brunhes age sites.Few Hawaiian data and no Reunion data areconsidered tectonically rotated. We exclude fromfurther analysis all sites with uncorrected tectonicrotation; see Table 1 for details.

[11] Declination and inclination data from Hawaiicluster around the GAD prediction (red triangle). Anegative mean inclination anomaly is present inboth the normal and reverse polarity data andappears as a northward elongation of the datadistribution in D0, I0 coordinates. Few transitional


4 of 25

Table

1.

DataReferencesa

Reference

Latitude,

!NLongitude,

!E

Nt

ra

Nok

nNu

Polarity

Mean(N

)Mean(R)

Max

Min

Max

Min

NR

DD

DI

DD

DI

Mexico(27Studies)

AlorandUribe[1986]

20.32

19.62

258.58

257.61

13

00

00

13

94

"23.8

"5.7

"6.6

"27.1

Alva-Valdivia

etal.[2001]

18.60

18.23

265.31

264.65

12

01

00

114

7"10.0

"1.7

1.6

"2.9

Bohnel

etal.[1997]

19.33

19.33

260.82

260.82

10

00

01

10

"12.7

2.3

––

Bohnel

etal.[1990]

19.00

19.00

261.00

261.00

11

00

00

––

––

––

Bohnel

andNegendank[1981]

19.93

18.15

263.60

262.38

55

018

00

37

34

3"1.7

"5.6

8.4

"30.5

Bohnel

andMolina-G

arza[2002]

21.14

19.05

263.10

255.50

60

00

06

60

"2.0

"5.0

––

Delgado-G

ranadoset

al.[1995]

20.50

19.04

258.45

256.35

36

011

00

25

17

81.1

"0.5

10.1

0.2

Gonzalezet

al.[1997]

19.82

19.10

260.83

257.78

13

00

00

13

13

04.9

5.9

––

Herrero-Bervera

etal.[1986]

19.00

19.00

261.00

261.00

10

00

00

10

10

0"8.4

"9.4

––

Maillolet

al.[1997]

20.82

20.42

255.28

255.03

16

16

00

00

––

––

––

Mejia

etal.[2005]

22.84

18.98

260.75

258.09

16

01

20

13

211

28.7

"6.3

0.9

2.5

Mooseret

al.[1974]

19.73

19.04

261.40

260.02

187

0119

00

68

52

16

2.9

"7.4

"8.5

"11.5

Mora-Alvarezet

al.[1991]

19.37

19.20

260.76

260.66

50

00

14

22

35.0

"36.9

"46.0

"25.5

Moraleset

al.[2001]

19.32

19.04

260.75

260.38

70

00

07

70

"0.4

"7.8

––

Nieto-O

bergonet

al.[1992]

20.97

20.15

255.58

254.97

13

13

00

00

––

––

––

Osete

etal.[2000]

19.58

19.30

260.68

260.40

30

00

30

27

18

9"9.8

"0.3

"15.0

"6.0

Petronille

etal.[2005]

21.21

21.03

255.64

255.05

17

00

10

16

14

2"1.8

"2.0

"10.1

0.8

Rosas-ElgueraandUrrutia-Fucugauchi[1992]

20.25

20.02

259.66

259.40

14

02

00

12

75

"14.8

"8.3

"30.3

"5.1

Ruiz-M

artinez

etal.[2000]

20.66

19.06

263.58

261.19

25

80

00

17

14

3"0.4

"9.3

"1.1

"31.8

Soler-ArechaldeandUrrutia-Fucugauchi[2000]

20.03

19.84

260.19

259.40

99

00

00

––

––

––

Steele[1971,1985]

19.22

19.10

260.37

260.34

35

00

00

35

34

1"0.8

"1.4

"52.6

"24.8

Uribe-CifuentesandUrrutia-Fucugauchi[1999]

20.46

20.13

258.96

258.75

13

13

00

00

––

––

––

Urrutia-Fucugauchi[1996]

19.34

19.18

260.83

260.78

13

00

00

13

13

0"2.0

"2.2

––

Urrutia-Fucugauchiet

al.[2000]

20.78

20.75

256.66

256.50

60

00

06

51

"1.7

"9.0

0.0

"2.0

Urrutia-Fucugauchiet

al.[1988]

20.78

20.75

256.66

256.50

60

00

06

51

0.7

"13.9

91.1

"15.6

Vlaget

al.[2000]

19.10

19.10

260.50

260.50

10

00

01

10

"21.5

15.6

––

Watkinset

al.[1971]

20.85

20.76

256.66

256.65

70

50

02

20

"14.3

"8.5

––

Mexicototals

22.84

18.15

265.31

254.97

567

60

157

61

343

270

73

After

flow

average

548

60

157

51

325

Hawaii

(23Studies)

Brassartet

al.[1997]

20.21

20.07

204.43

204.10

10

00

00

10

10

0"6.4

"4.5

––

BogueandCoe[1984]

22.16

22.06

200.67

200.26

84

00

00

84

32

53

15.4

4.4

"13.4

"5.6

Bogue[2001]

22.10

22.07

200.26

200.24

50

00

00

50

28

22

"11.3

"10.7

"11.9

"6.8


5 of 25

Reference

Latitude,

!NLongitude,

!E

Nt

ra

Nok

nNu

Polarity

Mean(N

)Mean(R)

Max

Min

Max

Min

NR

DD

DI

DD

DI

Castro

andBrown[1987]

19.30

19.30

204.80

204.10

20

00

02

20

12.3

"3.0

––

Coeet

al.[1978]

19.83

19.05

205.16

204.12

16

00

10

15

15

010.1

"2.3

––

Doell[1969]

19.49

19.49

204.40

204.40

54

00

00

54

54

0"5.2

"11.2

––

Doell[1972a]

22.23

21.90

200.68

200.27

119

00

27

092

48

44

"3.3

"11.6

1.6

"8.4

Doell[1972b]

23.60

21.78

199.95

195.30

39

07

00

32

25

7"2.0

"7.8

10.1

"1.8

Doell[1972c]

21.46

21.26

202.34

202.15

26

00

00

26

26

0"1.3

"4.9

––

DoellandCox[1965]

20.13

18.90

204.95

204.42

148

00

36

1111

111

05.7

"4.8

––

DoellandDalrym

ple

[1973]

21.75

21.34

202.80

201.74

99

01

00

98

33

65

0.7

"3.9

"1.3

"14.1

Herrero-Bervera

andCoe[1999]

21.45

21.40

201.87

201.80

65

02

010

53

27

26

3.3

"25.6

"12.2

"8.0

Herrero-Bervera

etal.[2000]

20.87

20.70

203.11

202.12

80

00

08

08

––

"9.1

4.2

Herrero-Bervera

andValet[2002]

21.42

21.26

202.37

202.13

17

00

00

17

17

0"1.4

"3.4

––

Herrero-Bervera

andValet[2003]

21.38

21.28

202.35

202.05

10

00

00

10

010

––

"3.5

"20.5

Herrero-Bervera

andValet[2005]

21.55

21.55

201.77

201.77

45

00

00

45

1134

3.9

"18.1

13.8

"18.9

Holcombet

al.[1986]

19.71

19.06

205.17

204.16

135

05

00

130

130

02.1

0.0

––

Jurado-Chichayet

al.[1996]

19.00

19.00

204.00

204.30

30

00

03

30

3.9

"16.1

––

Lajet

al.[1999]

21.40

21.40

20.80

201.80

105

00

00

105

105

03.0

"14.8

––

Mankinen

andChampion[1993]

19.71

18.97

204.96

204.38

24

00

00

24

24

02.4

"3.6

––

Riley

etal.[1999]

19.34

19.29

204.70

204.70

51

30

00

021

21

018.0

"13.2

––

Tanaka

andKono[1991]

19.34

19.34

205.16

204.51

70

00

07

70

11.3

"0.3

––

Valetet

al.[1998]

19.73

19.20

204.91

204.40

80

00

08

80

"11.2

"0.2

––

Hawaiiantotals

23.60

18.90

205.17

195.30

1125

30

15

64

111005

737

269

After

flowsare

averaged

1015

30

15

64

11895

Reunion(6

Studies)

Chaumalaun[1968]

"21.0

"21.0

55.5

55.5

97

00

21

53

23

12

114.2

"7.4

172.6

1.7

Chauvinet

al.[1991]

"21.0

"21.0

55.5

55.5

30

00

20

28

28

0356.1

"9.7

––

McD

ougallandChaumalaun[1969]

"20.3

"20.3

57.5

57.5

35

018

28

75

2355.3

"19.2

187.5

"12.3

Raiset

al.[1996]

"21.0

"21.0

55.5

55.5

70

00

50

65

65

0356.4

"3.3

––

Senanayake

etal.[1982]

"20.2

"21.0

57.3

55.5

14

07

70

0–

––

––

–Watkins[1973]

"21.0

"21.0

55.5

55.5

21

00

00

21

19

20.6

"2.8

223

79.3

Reuniontotals

"20.2

"21.0

57.5

55.5

267

025

37

61

144

129

15

South

Pacific(13Studies)

Chauvinet

al.[1990]

"17.67

"17.67

210.33

210.33

123

00

00

123

59

64

188.2

0.3

3.7

30.8

Duncan[1975]

"16.50

"17.67

210.58

208.27

60

00

10

59

40

19

185.6

0.0

359.7

7.6

Table

1.(continued)


6 of 25

directions are recorded, as evidenced by the smallpercentage of low VGP latitudes. The South Pa-cific has a highly scattered distribution, a result thatcould be attributed to preferential sampling oftransitional data in French Polynesia. The scatterappears greater to the south in both VGP and D0, I0.The Mexican sites suggest a negative mean incli-nation anomaly, perhaps of smaller amplitude thanat Hawaii. The VGP distribution appears elongatein the north-south direction, in contrast to the morecircular VGP distribution in the Hawaiian data. Thecorresponding D0, I0 plot shows increased scatter tothe north compared to the south. Reunion andMauritius Islands have an approximately 15! NWtrend in the VGP distribution and 5! NNW trend inthe D0, I0 distribution.

[12] In PSV studies it is common to select data apriori on the basis of various choices about thenumber of data, their quality, and the temporal orspatial distribution. Our strategy is more inclusivethanmost.Wediscarded sites that are older than5Ma,tectonically rotated (green symbols in Figure 3), andfor which one or more of the following are notreported: declination, inclination, or a measure ofwithin-site dispersion. When multiple results arereported from a single lava flow we average them(Tables 2a and 2b). Sites with fewer than 3 samples(brown symbols in Figure 3) are eliminated to reducethe effect of orientation error. Apart from thesecriteria we include all available data in our initialanalyses (including transitional directions), andinvestigate how further data selection criteria affectthe results. Table 1 reports all studies consulted forthe data compilation, along with the numbers of sitesfrom each that are excluded for the reasons listedabove. The final number of normal and reversepolarity data used and the mean declination andinclination are given in Table 1. Note that we donot distinguish transitional data as a distinct state ofthe field. Instead all data with positive (negative)VGP latitudes are considered normal (reversed). Weretain 80% of the original directions from Hawaii,57% from Mexico, 70% from the South Pacific and54% from Reunion.

3. Estimates of TAF and PSV

[13] We use our data set to investigate the TAF andPSV at ±20! latitude. We correct each site for platemotion using the model NUVEL 1-A [DeMets etal., 1994] and our estimate of site age. We convertreverse polarity directions to their normal antipode.This provides sufficient data for statistical ana-lyses, but carries the implicit assumption of

Reference

Latitude,

!NLongitude,

!E

Nt

ra

Nok

nNu

Polarity

Mean(N

)Mean(R)

Max

Min

Max

Min

NR

DD

DI

DD

DI

Falvey

[1978]

"16.00

"16.00

168.00

168.00

14

14

00

00

––

––

––

Inokuchiet

al.[1992]

"17.75

"17.75

178.00

178.00

40

40

00

00

––

––

––

James

andFalvey

[1978]

"18.00

"18.00

178.00

178.00

16

16

00

00

––

––

––

Malahoffet

al.[1982]

"17.37

"18.06

178.46

177.08

38

38

00

00

––

––

––

Morinagaet

al.[1991]

"21.25

"22.50

208.67

200.25

18

05

10

12

111

190.3

3.2

319.1

105.4

Roperch

andDuncan[1990]

"16.00

"16.00

210.00

210.00

132

00

30

129

83

46

219.5

39.4

351.9

7.6

Tarling[1967c]

"21.33

"21.39

200.19

200.16

14

00

90

51

4178.6

"8.7

286

75.2

Tarling[1967a]

"18.00

"18.00

178.00

178.00

23

23

00

00

––

––

––

Tarling[1967b]

"15.35

"17.50

168.57

167.63

19

08

21

88

0–

–0.4

12.4

Tayloret

al.[2000]

"16.87

"19.08

178.60

177.42

44

44

00

00

––

––

––

Yamamoto

etal.[2002]

"16.44

"17.65

210.62

207.74

149

00

30

146

92

54

177.9

"6.1

2.4

1.6

South

Pacifictotals

–15.35

–22.50

210.62

167.63

690

175

13

19

1482

284

198

aNt,totalnumberofpaleodirectionsfrom

unique

sitesreported;r,sitesreported

tobeeffected

bytectonicrotation;a,sitesolder

than

5Myr;Nok,

novalueofkappareported

forsite;n,number

ofsiteswith

less

than

3samplesusedforstatistics;Nu,numberofsitesusedin

thethisanalysis;Polarity

N,number

ofnorm

alpolarity

sitesusedin

thisanalysis;Polarity

R,numberofreverse

polarity

used;meanDD,DI,

themeandeclinationandinclinationanom

aliesofthedatausedforboth

norm

al(N

)andreversed(R)polarity

data.

Table

1.(continued)


7 of 25

symmetry in normal and reverse polarity fields. Wethen calculate summary statistics for the combined±20! data set and for each region, to assess whethereither the TAF or PSV is anomalous in the Pacific,specifically at Hawaii.

[14] The summary statistic used to characterize theTAF is the inclination anomaly (DI), while for PSVwe estimate VGP dispersion (SB). The inclinationanomaly (DI) and the declination anomaly (DD) aredefined as follows:

DI # Iobs " IGAD; $1%

DD # Dobs " DGAD; $2%

where Iobs and Dobs are the observed site inclina-tion and declination, and IGAD and DGAD are thepredictions of the geocentric axial dipole field.Inclination anomaly as defined by equation (1) isappropriate for a single site, however we want DIfor the average direction of multiple sites (DI),which is not simply the average of multiplemeasurements of DI. First, paleomagnetic direc-tions are rotated to D0 and I0 to account for dipolarlatitudinal variations, then the vector average iscalculated and rotated back to inclination which isused to compute the average inclination anomalyof a group of measurements, DI .

[15] A property often used to analyze the PSV isangular dispersion of VGPs, S, defined by

S #

!!!!!!!!!!!!!!!!!!!!!

X

N

i#1

D2i

N " 1

v

u

u

t ; $3%

where each i represents a site, Di is the anglebetween the ith VGP latitude and the spin axis andN is the total number of sites. The total angulardispersion, S, is a combination of between-sitedispersion (SB), attributable to PSV, and within-site dispersion (SW) arising from uncertainty in thepaleomagnetic direction for the site. Since these areindependent we can write S2 = SW

2 + SB2. We are

interested in the between-site dispersion given by

SB #

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

X

N

i#1

D2i

N " 1"

S2wi

NS

" #

v

u

u

t : $4%

NS is the average number of samples per site. Thewithin-site VGP dispersion is approximated by

Sw & 81'!!!

kp : $5%

Most published studies include k, an estimate ofk, and the 95% confidence (a95) circle for all

Figure 2. Age distribution of sites in each geographical region. Bin size is 0.5 Myr. Red represents all sites youngerthan 5 Ma including tectonically rotated sites. Blue represents sites not contaminated by tectonic movement and withmore than two samples.


8 of 25

sites. Where only a95 is reported we calculate kusing

a95 &140'!!!!!!

kNp ; $6%

where N is the number of samples per site.Although this approximation fails when k ( 25[Tauxe et al., 1991] such sites have poorlydefined directions. We will see later that theywould normally be rejected from our analyses.Confidence intervals on SB and DI can becalculated using a bootstrap method [Tauxe,1998].

[16] Before we can make a sensible assessment ofPSV and the TAF, two issues need to be addressed;what are the effects of data quality and of transi-tional sites on estimates of the statistics definedabove? Transitional data are typically excluded onthe grounds that reversals and excursions are notpart of normal PSV. However, this choice clearlyaffects the resulting PSV [e.g., Vandamme, 1994]and TAF estimates. It is also known that dataquality affect estimates of TAF and PSV. Forexample, selection criteria based on k or a95 havebeen used in compiling PSV data sets [Johnsonand Constable, 1995; Lee, 1983; McElhinny et al.,

Figure 3. Equal area plots of (a) declination (D) and inclination (I), (b) virtual geomagnetic pole (VGP), and (c) D0

I0. North is 0! declination. In Figure 3c the paleomagnetic directions are rotated such that the expected direction froma geocentric axial dipole (GAD) field is at the center of the projection and reversed polarity sites are mapped into theirantipodal directions [Hoffman, 1984]. Solid (open) blue circles represent sites with no tectonic rotation and projectedonto upper (lower) hemisphere; solid (open) green squares are upper (lower) hemisphere sites affected by tectonicrotation. Solid (open) brown diamonds are sites with n < 3. The GAD direction is represented by the red triangle.


9 of 25

1996b; Quidelleur et al., 1994]. More recently,Tauxe et al. [2003] concluded that data qualityhad a significant effect on estimates of dispersion.Here we examine how transitional sites and dataquality influence estimates of the TAF and PSV.

3.1. Influence of Data Quality

[17] We use the within-site estimate for Fisheriankappa as a measure of data quality. There are twodistinct estimates of kappa that we will employ inthe rest of our analyses. To investigate the effect of

data quality on estimates of PSV we wish to studyhow an unbiased estimate of SB varies as a functionof k. The statistic

ku #N " 2

N " R$7%

is an unbiased estimate for k [McFadden, 1980].From equation (4) it is apparent that to obtain anunbiased estimate of SB one needs an unbiasedestimate of SW. Since SW

2 varies as 1/k (equation(5)) we need an unbiased estimate for 1/k. We use

Table 2a. Averaged or Duplicated Flows Between Studies

Reference Number of Sites Site Names Exclusion and Inclusion Rationale

MexicoXitle FlowMorales et al. [2001] 1 JM Average all these values.Bohnel et al. [1997] 1 XitleGonzalez et al. [1997] 1 S-9Mooser et al. [1974] 2 SdC14, SdC15Urrutia-Fucugauchi [1996] 13 All

Pelado FlowMorales et al. [2001] 1 JB Use Morales et al. [2001] because k of JB > k of

S-10Gonzalez et al. [1997] 1 S-10Tres CrucesVlag et al. [2000] 1 Mean Use Vlag et al. [2000] because it has greater n than

Gonzalez et al. [1997]Gonzalez et al. [1997] 1 S-6

Hawaii1960 ADTanaka and Kono [1991] 1 1960 AD Use Coe et al. [1978] because Tanaka and Kono

[1991] do not report kCoe et al. [1978] 1 1960 AD1950 ADCastro and Brown [1987] 1 1950 AD Use Castro and Brown [1987] because more

through demagnetization techniques employedCoe et al. [1978] 1 1950 ADHonolulu Volcanic SeriesHerrero-Bervera and Valet [2002] 17 All Use data from Herrero-Bervera and Valet [2002]

because it duplicates Doell [1972c] with modernmeasurement techniques.

Doell [1972c] 26 All

Napali Formation (Kauai)Bogue [2001] 50 All Use the sequence of independent flows [Bogue,

2001, Table 2] consisting of grouped sites from thestudies by Bogue [2001] and Bogue and Coe[1984]. Bogue and Paul [1993] is presented inmore detail by Bogue [2001].

Bogue and Paul [1993] 16 AllBogue and Coe [1984] 44 KT1-32 P1-13

Koolau Series (Oahu)Doell and Dalrymple [1973] 33 A-G Use Herrero-Bervera and Valet [2003] because

they have more consistent results than Doell andDalrymple [1973].

Herrero-Bervera and Valet [2003] 10 All

Waianae Series (Oahu)Herrero-Bervera and Coe [1999] 65 All According to Herrero-Bervera and Valet [2003],

there is no overlap with the Waianae seriesreported by Laj et al. [1999]. Herrero-Bervera andValet [2005] is the only published data from theUpper Mammoth portion of the Waianae series.There could be overlap with Doell and Dalrymple[1973] with many of these studies. However, it isvery difficult to determine decisively, and to avoidmissing data we include all of Doell andDalrymple [1973].

Herrero-Bervera and Valet [2003] 10 AllLaj et al. [1999] 105 AllHerrero-Bervera and Valet [2005] 45 AllDoell and Dalrymple [1973] 46 P-Z


10 of 25

the estimator 1/kf, where McFadden [1980] defineskf as

kf #N " 1

N " R: $8%

kf is the estimate of k typically used in paleomag-netic literature. In cases where sample numbers arelarge, the difference between equations (7) and (8)is minimal. Many of the sites we use in our studyonly have three samples, which could havesignificant impact on the k cut-off criterion usedfor analyzing the effects of data quality. Therefore,when investigating how SB and DI vary as afunction of k, we will use ku, the unbiased bestestimate for k.

[18] The effect of data quality on estimates of SB(PSV) and DI (TAF) is shown in Figure 4. For aspecific value of ku, denoted by ku

cut-off (data qual-ity) we exclude sites with ku less than ku

cut-off and

calculate SB and DI for the remaining sites. Largervalues of ku

cut-off correspond to increasingly strin-gent data quality requirements. A clear choiceconcerning data quality would correspond to avalue of ku

cut-off above which estimates of SB andDI are stable, yet both SB and DI are fairly smoothfunctions of ku

cut-off (Figure 4). SB has an averagevalue that decreases from 24! to 18! as ku

cut-off

increases from 0 to 500, and DI has an averagevalue increasing from "5.1! to "2.5!. The absenceof a stable estimate for SB and DI at the largestvalues of ku

cut-off is a consequence of insufficientdata. As has been noted in many previous studies,at the lowest values of ku (less than !40) estimatesof SB are unreliable.

[19] For kucut-off values between 0 and 200 there is a

range in VGP dispersion of only 4! and a change inDI of !1=2!. For simplicity, we choose to excluderather than downweight low quality data. Onthe basis of Figure 4 we select a value of 100 for

Table 2b. Summary of Duplicated Flow Results

Flow n DD DI ku

MexicoPelado Flow 13 "8 1 269Tres Cruces 9 "21.5 15.3 195.6Xitle: Average flow 18 "2.3 "0.6 146.2

Hawaii1960 AD 8 9.1 "1.2 3421950 AD 127 12.9 "1.6 277Honolulu Volcanic Series 17 "1.4 "3.4 116.5Napali Formation 45 N/A, multiple flows define seriesWaianae Series 271 N/A, multiple flows define seriesKoolau Series 10 N/A, multiple flows define series

Figure 4. Combined data set: (a) VGP dispersion (SB) and (b) mean inclination anomaly (DI ) as functions of kucut-off

(see text for a discussion of ku versus k). Blue dashed line is the average value, the solid red lines represent the 95%confidence interval, and the solid green line is the number of sites as a function of ku

cut-off.


11 of 25

kucut-off, meaning that we exclude all sites with kuless than 100 from further analyses. This retains abalance between number and quality of data and isconsistent with previous studies such as Tauxe etal. [2003].

3.2. Influence of Transitional Data

[20] One of the major problems that plagues pa-leomagnetic field analyses is how to define andtreat transitional data. Typically, transitional dataare defined as measurements with absolute value ofVGP latitude less than 45!, and are excluded fromestimates of PSV and the TAF. The arbitrariness ofthis criterion is highlighted by several studies[Clement, 2000; Coe et al., 2000] showing thatby this definition the apparent length of a reversalvaries substantially with location. Vandamme[1994] proposed a technique that iteratively iden-tifies a critical VGP latitude for transitional data, toproduce an estimate for SB, but this methodassumes that there is a clearly identifiable distinc-tion between normal and transitional data. Weanalyze SB and DI as functions of VGP latitude(lVGP) cut-off, using the data quality criteriondiscussed above (Figure 5). SB changes graduallyas a function of lVGP, and there is no cleardistinction between stable polarity and transitionaldata. SB has a maximum value of 24! when all sitesare included, a value of 15! when lVGP equals 45!and drops off rapidly for lVGP greater than 70!.Estimates of SB (and DI) for lVGP greater than 70!are unreliable, since they will exclude a significantfraction of sites exhibiting typical field variabilityduring a stable polarity chron. As with VGP

dispersion, inclination anomaly is most stable forlVGP between 20! and 65! and has an averagevalue of "4!. Similar analyses and conclusionswere presented by Shibuya et al. [1995] and Tauxeet al. [2003].

[21] The results shown in Figure 5 show no cleardelineation between transitional and nontransitionaldata. Consequently, truncating the data by arbi-trarily defining some directions as ‘‘transitional’’ isill advised. It may be particularly problematicwhen considering smaller spatial or temporal sub-sets of the data, which fail to sample the fieldadequately. If transitional data are to be incorpo-rated in analyses then the number of transitionalsites included must represent the equivalent dura-tion of time the field spent in transition. In thisstudy there are 97 data sites with an absolute valueof lVGP less than 45! which is !8% of the entiredata set (with ku > 100). Over the past 5 Myr thefield has reversed 10 times [Cande and Kent,1995] with an average reversal duration of 6 kyr[Constable, 2003]; one would infer that the fieldspends about 1% of the time in transition. At facevalue this suggests that we have a sampling biastoward transitional data in our data set, but thesituation is more complex. Excursions during theBrunhes are becoming increasingly well docu-mented, and there are probably between 10 and20 of these [Langereis et al., 1997; Lund et al.,1998, 2001]. By their very nature excursions cor-respond to VGP directions that lie far from thegeographic axis. If we estimate the average excur-sion length at half that for a reversal (i.e., 3 kyr) andsuppose that 10 excursions and one transition

Figure 5. Combined data set: (a) VGP dispersion (SB) and (b) mean inclination anomaly (DI ) as functions of VGPlatitude cut-off (lVGP). Blue dashed line is the average value, the solid red lines represent the 95% confidenceinterval, and the solid green line is the number of sites as a function of cut-off. The sites used in calculating SB and DIfor lVGP have values of ku > 100.


12 of 25

occurred since 800 kyr, then the expected percent-age of directions with lVGP < 45! rises to 4.5%.Longer duration excursions or more of them couldplausibly raise the expected percentage of data withlVGP < 45! as high as that found in our data set.Nevertheless, it is clear that temporal sampling isnonuniform among the various regions. In particu-lar the prevalence of data from the Matuyama-Brunhes and Gauss-Matuyama transitions (totaling20% transitional sites) in the South Pacific data setindicate that those transitions are over-represented,while other time intervals that might be expected tocontain excursions are under-represented.

[22] With a more comprehensive and accuratelydated data set, it might be possible to explore theinfluence of temporal sampling, and what happenswith the appropriate representative sampling oftransitions. This is not possible with the currentdata. However, we can compare the Brunhes datawith those from earlier epochs, namely theMatuyama, Gilbert and Gauss chrons (here desig-nated non-Brunhes). The merging of the non-Brunhes data is necessary to provide large enoughdata groups. The dispersions in the Brunhes areconsistently smaller than for non-Brunhes data,except for Mexico, where they are indistinguish-able (Figure 6). The 95% confidence intervals atReunion overlap for the two time periods, but

Hawaii and the combined data set are significantlydifferent. The mean inclination anomalies are alsosmaller during the Brunhes than during non-Brunhes periods. Again, Mexico has equivalentmean values and the 95% confidence intervals forReunion overlap. These results indicate that theTAF is closer to GAD and that there is less SVduring the Brunhes than during than in the polarityintervals prior to the Brunhes. Most importantly,these results seem robust even when consideringpossible bias from the South Pacific. Althoughthere are no data for the Brunhes in our SouthPacific group, data from Easter Island (at 27!S)offer some support for the observation of lowerdispersion during the Brunhes [Brown, 2002; Mikiet al., 1998].

3.3. Regional Variations in Field Behavior

[23] We are now in a position to assess the time-averaged field and paleosecular variation on aregional level. We use DI as a measure of departureof the TAF from GAD and SB as a measure of PSV,and present regional and combined data resultsin Table 3. Nonzero time-averaged declinationanomalies have been used to argue for persistentlongitudinal structure in the TAF [Gubbins andKelly, 1993; Johnson and Constable, 1995, 1997;Kelly and Gubbins, 1997; Love and Constable,

Figure 6. Brunhes (0–0.78 Ma) (blue) versus older (0.78–5.0 Ma) (red) (a) inclination anomaly (DI ) and(b) dispersion (SB) estimates for the combined data set (dot), Hawaii (diamond), Mexico (open square), Reunion(triangle), and the South Pacific (solid square). Data having a kappa < 100 are excluded. The error bars represent 95%confidence intervals. Each point is labeled with the number of data points used to calculate that value.


13 of 25

2003], we focus here on the time-averaged incli-nation anomaly. In contrast to some previousstudies [e.g., Johnson and Constable, 1995],time-averaged declination anomalies for the fourregions studied here are small (typically less than1!). In addition, inclination information is availablefor sediment cores; the same is not true of absolutedeclination information. Although we do not con-sider such data sets in detail here, our results willbe useful for future comparisons of lava flow andsediment data sets. Our null hypothesis is that bothPSV and any departures from GAD in the TAF areaxisymmetric and only exhibit latitudinal signa-tures. Following the results of sections 3.1 and 3.2we reject all data with ku < 100, and calculatestatistics, first for all remaining data, and thenexcluding those with jlVGPj < 45! which we labelas transitional. Reunion and Hawaii have angulardispersions (16.4!) that are smaller than otherregions, but the uncertainties overlap with thosefrom the estimates of dispersion for Mexico. TheSouth Pacific has the largest angular dispersion(34.4 ± 3.0!), and is significantly different fromother study localities. Hawaii has the largest aver-age inclination anomaly ("6.3 ± 1.0!), distinctlylarger than that at Mexico ("0.4 ± 2.9!). However,within uncertainties, DI for Hawaii does not differfrom Reunion or the South Pacific. The combined±20! data set has an average inclination anomaly of"5.1! and an angular dispersion of 22.7!.

[24] We note here that plate motion correctionscontribute only a 0.1! change in estimates ofinclination anomaly for the combined 20! dataset and less than 0.5! change to VGP dispersionestimates. Approximately 25% of our data set isfrom studies published before 1980, and might nothave received adequate laboratory cleaning bymodern standards. If we were to perform the sameanalysis using data only published after 1980 theresults are unchanged within the confidence inter-vals, except for SB, the dispersion for Hawaii andthe combined data set. This is a result of differenttemporal sampling in the post-1980 data, which

sample 70% less of the Matuyama polarity chronthan the entire collection. This results in a de-creased dispersion for Hawaii and therefore thecombined data set.

[25] When transitional data are excluded, there areonly small changes in DI except for the SouthPacific region, which shows a slightly positiveaverage inclination anomaly (0.6 ± 2.2!). Alldispersion estimates decrease as one would expect,and the confidence limits tighten accordingly.Hawaii still has the smallest mean dispersion, andit becomes measurably different from that at bothMexico and the South Pacific, but not Reunion.The South Pacific and the combined data sets havemuch reduced dispersion. This is due to largenumber of transitional directions from the SouthPacific: they make up 20% of that region’s data.The angular dispersion for the combined ±20! dataset is 15.6 ± 0.6! when transitional data areexcluded. This is comparable with estimates madeby McElhinny and McFadden [1997] for globallydistributed data at this latitude.

4. Time-Averaged Field andPaleosecular Variation Models

[26] We turn now to a more detailed evaluation ofthe statistical distributions of our observations andcomparisons with distributions predicted from twocurrent paleosecular variation models. The ob-served D0 and I0 for the combined ±20! directionsare shown in Figure 7a, however the large numberof data conceals details of field behavior. Anempirical probability density plot (Figure 7b) expo-ses more structure in the distribution, which iselongated in the north-south direction, and showsconsiderable scatter. Structure in Figure 7 reflectsboth the time-averaged field and paleosecularvariation.

[27] The density distribution in Figure 7b gives aclear idea of the shape of the data distribution, but

Table 3. Regional Property Results For Sites With ku > 100a

Hawaii Mexico Reunion South Pacific Combined

N 785 136 85 351 1357

DI "6.3 ± 1.0! "0.4 ± 2.9! "5.2 ± 2.4! "3.7 ± 3.4! "5.1 ± 1.0!SB 16.4 ± 1.3! 21.1 ± 4.4! 16.4 ± 3.9! 34.4 ± 3.0! 22.7 ± 1.4!NjVGPj> 45 766 132 84 280 1262

DI jVGPj>45 "6.5 ± 1.2! "2.9 ± 2.1! "4.9 ± 2.3! 0.6 ± 2.2! "4.4 ± 0.8!SB jVGPj > 45 14.6 ± 0.7! 16.2 ± 2.0! 14.4 ± 2.0! 17.7 ± 1.5! 15.6 ± 0.6!

aDI is average inclination anomaly; SB is angular dispersion. jVGPj > 45 indicates statistic was calculated using only sites with jlVGPj > 45!.


14 of 25

it is not suitable for testing whether the data arecompatible with predictions from a particularstatistical model for PSV, or detecting small differ-ences among such models. We use one-dimen-sional distributions of declination and inclinationdata and the 2-sample Kolmogorov-Smirnov (K-S)test [Press et al., 1992] to assess whether theobservations are compatible with the distributionsof simulations of D and I from specific models.We apply the K-S test directly to D and I, sincewe can use our statistical models to predictdirections at the same latitudes and longitudes asour data. The D and I directional distributions ofour data are shown in Figure 8 as both 2-D(Figure 8a) and 1-D (Figure 8b) probability den-sity functions. Figure 8a provides the visual linkbetween the 2-D PDF for D0 and I0 (Figure 7b) andthe 1-D PDFs for D and I (Figures 8b and 8c).The 1-D PDF of declination has a single peak at0!. Inclination shows two peaks, at approximately±40!, representing the normal polarity inclinationof northern and southern hemisphere sites, respec-tively. In Figure 8a these peaks occur in the upperand lower hemispheres, respectively. Recall thatreverse polarity sites have been mapped to theirantipodal directions.

4.1. PSV Model Formulation

[28] We compare observed paleomagnetic direc-tions with simulations from PSV models proposedby Constable and Johnson [1999], CJ98, andTauxe and Kent [2004], TK03. These models arebased on a statistical approach to paleosecularvariation initiated by Constable and Parker[1988], who produced a PSV model that we labelCP88. The geomagnetic field is described by aspherical harmonic expansion in which the scalarmagnetic potential due to an internal field gener-ated at the core is given by

Y r; q;f$ % # aX

1

l#1

X

l

m#0

a

r

$ %l)1gml cosmf) hml sinmf& '

* Pml cos q$ %; $9%

where a is Earth’s radius, r is radius, q is colatitude,f is longitude, l and m are harmonic degree andorder, Pl

m are partially normalized Schmidt func-tions, gl

m and hlm are the Gauss coefficients

representing the field model. Specifically, g10 is

the axial dipole term, while g20 and g3

0 are the axialquadrupole and axial octupole terms, respectively.Constable and Parker [1988] proposed that a giant

Figure 7. (a) Equal area projection of D0, I0 for the combined data set correct for plate motions and selected usingthe following criteria: age <5 Ma, n > 2, ku > 100, and no tectonic rotation postemplacement. (b) Two-dimensionalempirical PDF of the data shown in Figure 7a. Color scale is logarithmic.


15 of 25

Gaussian process can represent PSV. The sphericalharmonic coefficients gl

m and hlm are assumed to be

temporally independent random variables withGaussian distributions having means (m) andvariances (s) described by

Gml ! N mmgl;s

mgl

$ %

$10%

and

Hml ! N mmhl; s

mhl

& '

: $11%

In general, the means and variances can vary withboth spherical harmonic degree and order.Although the specific model parameters used forCP88 are now considered inappropriate to describethe 0–5 Ma PSV, the basic formalism has beenwidely used. CJ98 and TK03 are both zonalmodels, modifications of CP88 proposed to explainlatitudinal variation of SB, but using distinctlydifferent philosophies. CJ98 satisfies SB variationsby imposing increased variance in the l = 2, m = 1terms putting excess energy into PSV at degree 2.TK03 preserves part of Constable and Parker’soriginal conceptual model in the form of a whitespatial power spectrum at the core-mantle bound-ary, by assigning different variances to the dipole

(l " m odd) and nondipole (l " m even) terms. Thiseffectively identifies a different behavior forequatorially symmetric and anti-symmetric con-tributions to the field (for more on this, see Hulotand Bouligand [2005]). Two further characteristicsdifferentiate TK03 and CJ98: TK03 has a lowermean for g1

0 than CJ98 and thus fits recentestimates of the observed mean field strength[Selkin and Tauxe, 2000]; CJ98 has a small averageaxial quadrupole term, whereas in TK03 all theGauss coefficients except g1

0 have zero mean.

4.2. Time-Averaged Field

[29] As noted above the statistical PSV modelsrequire estimates of the time-averaged field (mglmand mhlm). Previous PSV models have used eitherGAD [Tauxe and Kent, 2004] or GAD plus an axialquadrupole contribution [Constable and Johnson,1999; Constable and Parker, 1988]. The averagefield plays an important role when comparing datawith PSV models. Here we investigate a range ofTAF models that are compatible with our observa-tions. We restrict ourselves to zonal models toavoid unnecessary complications, noting that aver-age declination anomalies within each region areless than a degree, while keeping in mind that thedifferences in inclination anomalies may yet de-

Figure 8. (a) Two-dimensional empirical PDF of D, I for the ±20! data set shown in Figure 6. Color scale islogarithmic. One-dimensional empirical PDF of (b) D and (c) I for the same data set. All reverse polarities have beenflipped to their antipodes using VGP latitude as a determinant of polarity.


16 of 25

mand nonzonal structure in the TAF. If we attributeour average inclination anomaly for the combineddata set to a dipole plus quadrupole field the g2

0

term required is approximately 10% of the g10 term.

This is a larger quadrupole contribution to thepaleomagnetic field than previously suggested(see McElhinny [2004] for a review) and is mainlycontrolled by the sizable negative inclinationanomaly at Hawaii. To ensure that our estimateof the TAF is compatible with other high qualitydata sets we compile additional data that allow usto model the time-averaged latitudinal variation ininclination (Figure 9). We then investigate themisfit of models that include g1

0, g20, and g3

0 termsto these data.

[30] The variation of inclination anomaly withlatitude for our data compilation is shown inFigure 9. We supplement our ±20! data set withthe highest quality data present in the global data-base of McElhinny and McFadden [1997] (Demaglevel 4). We also include data collected as part ofthe Time-Averaged Field Investigations (TAFI)project [Johnson et al., 2005]. TAFI data are fromsites in: the Aleutians [Stone and Layer, 2006],British Columbia [Mejia et al., 2002], the SnakeRiver Plain [Tauxe et al., 2004b], San FranciscoVolcanics [Tauxe et al., 2003], Costa Rica (inpreparation), Chile [Brown et al., 2004b; L. L.Brown, personal communication, 2005], Argentina[Brown et al., 2004a; Mejia et al., 2004], Ecuador

[Opdyke et al., 2006], Australia [Opdyke andMusgrave, 2004], Antarctica [Tauxe et al.,2004a], the Azores [Johnson et al., 1998], and theCanaries [Tauxe et al., 2000]. Data from the ±20!data set and the TAFI sites supercede and replacethose fromMcElhinny and McFadden [1997] in ourcompilation. For consistency with the data com-piled by McElhinny and McFadden [1997] weexclude all data with VGP latitudes less than 45!.While this is not a complete global data set, itprovides a reasonable latitudinal data distributionwith which to investigate zonal TAF field models.

[31] We investigate the misfit of TAF models to ourdata for varying contributions from g2

0 and g30,

relative to g10. For given values of G2 = (g2

0/g10)

and G3 = (g30/g1

0) terms we calculate thecorresponding inclination anomalies at our datalocations. We compute the root mean square(RMS) misfit of the predicted to observed inclina-tions using

RMS #

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

1

N

X

N

i#1

Iobs " Ii$ %2v

u

u

t : $12%

Figure 10 shows contours of RMS misfit for TAFfield models where g1

0 is set to "1.0 and the g20 and

g30 contributions vary from 0 to ± 0.5. The best fit

model has G2 = 0.04 and G3 = 0.06 (black dot inFigure 10) and an RMS misfit of 13.8!. The RMS

Figure 9. Inclination anomaly (DI ) as a function of latitude for data from each region of this study, Hawaii (blue),Mexico (red), South Pacific (green), Reunion (purple), and McElhinny and McFadden [1997] and TAFI data sites[Johnson et al., 2005] that passed data selection criterion (black). Models of the TAF for illustrative combinations ofg20 and g3

0 are also shown. Specifically, the best fit model for all data sites is shown in dashed red.


17 of 25

misfit of GAD (black star in Figure 10) is 14.7!.Predicted inclination anomalies for various TAFmodels are also shown in Figure 9. It is clear thatthe latitudinal distribution of data affects estimatesof G2 and G3, for example improved sampling atlow latitudes would better constrain the bounds onG2.

4.3. Tests of PSV Models

[32] We now examine whether existing zonal PSVmodels can be made to fit our new ±20! data setwith an appropriate TAF. We use statistical modelsCJ98 and TK03 and prescribe two distinct TAFs,either GAD or G2 = 0.04 and G3 = 0.06 as derivedin the previous section. We generate data at thesame locations as our paleomagnetic sites. Wesimulate one sample per site and assign within-sitek equivalent to that reported at each site in theobserved data set. We compare our models withthe data using 1-D empirical PDFs and thecorresponding cumulative distribution functions(CDFs) for declination and inclination. We assessthe model fit using the 2-sample K-S test forinclination and a slightly modified K-S test fordeclination, which exploits the Kuiper statistic, amore robust test of data that are circularly distrib-uted [Press et al., 1992].

[33] Comparisons of PSV models with our ±20!data are shown in Figure 11. For all models and

data (black line) the declination PDF is unimodalwhile the inclination PDF is bimodal since we haveboth northern and southern hemisphere sites. Dif-ferences between the observed and predicted incli-nation distributions are seen in both the CDF andPDF whereas declination differences are seen moreeasily in the CDF. CJ98 (blue line) appears topredict the mean declination and inclination well,as we might expect, given that we are using ourbest fit TAF model. But the model fails when itcomes to fitting the shape of the distributions. Wequantify this in Table 4 using the K-S test. In theK-S test the measure of difference between twodistributions is characterized by the statistic d, thatmeasures the maximum distance between sampledistribution functions for two data sets. Smallvalues of d correspond to samples from similardistributions. Associated with the d-statistic is aprobability (significance level) that describes thelikelihood of getting a d value this large or greaterif the underlying distributions are the same. Thesignificance level varies monotonically from 0 to 1,where 1 is a 100% probability. The results of K-Sare given for a number of different models in Table4. Neither of the recent CJ98 nor TK03 models isadequate to model the data, although the nature ofthe mismatch is somewhat different, when ourpreferred TAF model is used. Declination performsworst for CJ98 and inclination for TK03: forTK03, there is a probability of 0.00003 of gettingsuch a large mismatch in inclination distributions ifit were the correct model and 0.02 in declination;CJ98 also performs poorly.

[34] One might wonder whether simple modifica-tions to these models would improve the situation,and we did explore this possibility for TK03 byvarying b, the ratio of standard deviations in theequatorially symmetric to antisymmetric familiesof Gauss coefficients. Tauxe and Kent [2004] chosethis parameter to be 3.8 to satisfy latitudinalvariations in VGP dispersion, but another end-member model is given by the earlier CP88 model(green line, with b = 1.0), which essentially exhib-its no latitudinal variation. We investigated TK03type models with b ranging from 1 to 5 in steps of0.1 and find b = 2.0 has the best fit to our data (redline). It should be noted, however, that even thisbest fit has what would generally be regarded as anunacceptable low probability (p = 0.001) for thedeclination distribution, and it is unlikely to satisfythe global variations in VGP dispersion. Table 4also indicates that CP88 provides an acceptable fitto inclination but not declination. As b is varied thefit to declination is improved at the expense of

Figure 10. Contour plot of RMS misfit of inclinationanomaly using data compiled from our ±20! data set, theTAFI data set [Johnson et al., 2005], and the McElhinnydatabase [McElhinny and McFadden, 1997]. See text forcomplete references and data descriptions. Axes arevalues of Gauss coefficients, g2

0 and g30; g1

0 has been setto "1.0. The dot represents the best fit model of g2

0 ="0.04 and g3

0 = "0.06, and the star is GAD.


18 of 25

inclination. For example, the CP88 model has thehighest inclination significance levels but com-pared to CJ98 and the best fitting model it hasthe lowest declination significance levels. The bestfit model increases the fit to the declination signif-icantly from CP88 but does not increase theinclination fit. Although b = 2.0 provides the bestfit to the new 20! data set, however, none of themodels in this family can be considered adequateto represent the directional data distributions.

5. Conclusions

[35] The results presented in Table 3 are in agree-ment with previous studies. We briefly recap theresults for each of the TAF and PSV and considerwhether these have implications for regional var-iations in the overall geomagnetic field structureand the nature of its secular variation.

[36] When data from all 4 regions are combined theaverage inclination anomaly for the ±20! latitude

band is "5.1 ± 1.0!, or "4.4 ± 0.8! when transi-tional data are excluded. The best fitting 2-param-eter zonal time-averaged magnetic field model to aglobal data set containing our newly compiled±20! data set (but excluding our transitional datafor compatibility with the rest of the global data

Figure 11. Empirical cumulative distribution functions (CDFs) and probability density functions (PDFs) ofobserved ±20! data (black line) and three PSV models. The CJ98 PSV model with a TAF of G2 = 4% and G3 = 6% isshown in blue. The TK03 model with variations in beta from 1.0 (green) and 2.0 (red) are also shown for the sameTAF as used with CJ98.

Table 4. Significance Level Results of K-S Testa

Significance Level TAF Model GAD

CJ98Dec. 1.27E-5 5.71E-6Inc. 0.0541 3.03E-14

CP88Dec. 1.71E-8 6.35E-9Inc. 0.1755 5.86E-15

Best Fit ModelDec. 0.0014 0.0009Inc. 0.1606 1.23E-12

TK03b= 3.8

Dec. 0.0217 0.0389Inc. 2.74E-5 2.96E-14

aModel: G2 = 0.04 and G3 = 0.06.


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set) has G2 = 0.04, and G3 = 0.06. The value forG2 is consistent with previous studies, while G3 iscomparable to that found for reversed polaritycontinental and igneous rocks. It is, however, largerthan that estimated from normal polarity continen-tal and igneous rocks and data from oceanic sedi-ments [McElhinny et al., 1996b]. Creer [1983]proposed that the G3 term is overestimated by afew degrees if averages of unit vectors are used inplace of the full vector field directions (includingfield intensity). However, the estimate of G3 wedetermine is too large to be attributed to this bias.

[37] Within the new data set there are substantialregional differences in inclination anomaly. Hawaiihas the largest DI , and it differs from that found ineither the Mexico or South Pacific regions, but isvery similar to that from Reunion. Thus the Ha-waiian and Reunion data could be explained interms of a TAF consisting of GAD plus an equato-rially symmetric contribution like g2

0. The distinc-tion in DI between Hawaii and Mexico is clearregardless of whether transitional data are included.However, in comparing Hawaii with the SouthPacific we find that when the many transitionaldata from South Pacific are retained, the largeangular dispersion renders the almost 3! differencein average inclination anomaly insignificant at the95% confidence level. This brings up issues abouttemporal and spatial sampling, which we revisitbelow. If transitional data are excluded, Hawaii hasa TAF, which is distinct from both the Mexico andthe South Pacific data sets, raising the possibility ofregional differences around the Pacific. The exclu-sion of transitional data for the South Pacificgenerates a small positive DI for that region,contributing to the relatively large value of G3 inthe best fitting model.

[38] PSV results from our ±20! data set indicatethat Hawaii and Reunion exhibit the lowest VGPdispersion. This result does not depend on theinclusion or exclusion of transitional data: exclu-sion reduces SB by about 2! in each case. BothMexico and the South Pacific have higher VGPdispersion than either Hawaii or Reunion: the 95%confidence intervals for Reunion overlap withthose for Mexico and for the nontransitional SouthPacific data, but Hawaii is significantly differentfrom both Mexico and the South Pacific regardlessof what transitional data are excluded.

[39] Hawaii has long been at the center of debateabout low secular variation (SV) in the Pacificregion. For the modern and historical field previousdiscussions have emphasized hemispherical differ-

ences in SV at the core-mantle boundary [Bloxhamet al., 1989; Gubbins and Gibbons, 2004], whilefor PSV [e.g., Shibuya et al., 1995], Hawaii hasbeen a representative case of low VGP dispersionfor the Pacific hemisphere. Millennial scale field[Korte and Constable, 2005] are based on arche-omagnetic and lake sediment data sets, which seemto exhibit less high frequency content in the Pacificregions. In this study, we examined three distinctregions within the Pacific hemisphere, and finddifferent results in Hawaii compared with bothMexico (on the eastern border of the hemisphere),and the South Pacific, which lies well within theregion identified as anomalous at the core-mantleboundary in historical field models [Jackson et al.,2000; Johnson and Constable, 1998]. The Hawai-ian results are most similar to the Reunion region,which in contrast to the Pacific has high modernSV rates [see, e.g., Constable and Korte, 2006,Figure 1]. If we interpret all the differences in DIand SB as being purely geomagnetic in origin thiswould suggest a more complicated view of geo-magnetic paleosecular variation than has beenconsidered so far.

[40] Before we can accept a geomagnetic origin forthese results we must consider the effects of sam-pling bias. The potential biases come from differingnumbers and quality of data, and variable temporalsampling, in particular serial correlations and theeffects of transitional data. The TAF results arerobust with respect to data quality issues (Figure 4)and number of data. The inclusion or exclusion oftransitional directions (and selection of lVGP cut-off, Figure 5, and Table 3) has minimal effectexcept for the South Pacific. The same cannot besaid for SB, the VGP dispersion due to paleosec-ular variation. From Table 3 we see that thepercentages of transitional data at Hawaii, Mexico,Reunion and the South Pacific are 2.5%, 3%,1%, and 20%, respectively. Hawaii and Reunionhave astonishingly few transitional directions. TheSouth Pacific clearly has over-representation oftransitional data in some time intervals, becauseof the inclusion of studies specifically targetingtwo field transitions, but this may be balanced by alack of data from excursional intervals during theBrunhes, for example. One further point worthnoting is that a larger fraction of the transitionaldata are excluded by the ku criterion than of thestable polarity observations highlighting the diffi-culty of obtaining reliable transitional records at alllocations. The relative reduction in our dispersionestimates corresponds to the fraction of transitionaldata.


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[41] The large DI at Hawaii is unlikely due totemporal sampling bias, because a similar valueis found from three drilled cores on the big islandof Hawaii, that span 0–400 ka in total (summa-rized by Herrero-Bervera and Valet [2003], Holt etal. [1996], Laj et al. [2002], and Teanby et al.[2002]). This strongly suggests that the field atHawaii is indeed unusual over timescales of 0–5 Myr. An immediate consequence of this is thatthe 2-parameter zonal TAF model is too simplistic,a result that has been noted previously [Gubbinsand Kelly, 1993; Johnson and Constable, 1995]. Itis important to remember that a nonzonal TAF fieldmodel may be necessary even if there is no meandeclination anomaly.

[42] An exploration of statistical models for PSV[Constable and Johnson, 1999; Constable andParker, 1988; Tauxe and Kent, 2004] shows thatnone of these are adequate to describe the ±20!data set. This is not a surprise! The prescription ofallowed variance in any of these models is some-what ad hoc and too restrictive. Even after adjust-ing to an appropriate zonal TAF model for our data,we were unable to simultaneously enforce thecorrect variance for the distributions of declinationand inclination (Figure 11 and Table 4). Furtherefforts are clearly necessary to produce a realisticglobal PSV model.

[43] This data compilation and others like it pro-vide new opportunities for assessing TAF and PSV.Large regional data sets can be used to determinethe local TAF departure from GAD, and to studygeographical differences in variance of the mag-netic field. The ultimate goal is to find a statisticaldescription for PSV that correctly predictsthe directional and intensity distributions at anylocation.

Acknowledgments

[44] We thank Lisa Tauxe and Victoria Mejia for usefuldiscussions. We also thank David Gubbins and Jean-PierreValet for thorough reviews. This work was funded by NSFgrant EAR-0337712.

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