journal orf I~n.derr.3d.3ua.3- :-ese-arch
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Closing Pandora's Box: Additional Insights on Inclination Bias Using a
Random Walk Approach
A fundamental working assumption in paleomagnetic studies is that the Earth's magnetic field averages to
a geocentric axial dipole [GAD] when sufficiently sampled. One of the main tools for evaluating the GAD hypothesis
in pre-Cenozoic times is based on the distribution of inclination values. Recent studies of inclination-only data show
a bias towards low inclination and a number of alternative explanations have been offered to explain this bias.
The inclination-only analysis relies on the fact that the planet has been adequately sampled in a spatially and/
or temporally random manner. Inclination-only studies might misrepresent the field because the extant
global paleomagnetic database does not provide an adequate sampling of the field. In this study, we examine
other sources of bias in the database. We find that the apparent contributions of quadrupolar and octupolar
fields may depend upon the inning procedure used. For example, the Cenozoic database can be favorably
compared to GAD when assigned to temporal bins based on geologic periods, but is decidedly non-GAD
when averaged on a finer temporal scale. We also demonstrate that the Paleozoic inclination distribution may
result from a regional sampling bias and we quantitatively assess the probability that the Precambrian
global paleomagnetic dataset sufficiently integrates the time-averaged Earth's magnetic field. Our analysis
suggests that the extant inclination database contains myriad forms of bias and may not represent the
Earth's magnetic field. Unfortunately, the analysis cannot rule out the existence of persistent non-dipolar fields
in geologic time. The global paleomagnetic database does indeed show a rather consistent bias towards
low-inclination values [median inclination is 40oo versus 49oo for the GAD]. Models of the earth's magnetic field
and the thermal evolution of the planet may yield additional clues regarding its GAD or non-GAD nature.
Tests of the geocentric axial dipole [GAD] assumption probing deeper into geologic time have produced
disparate results. Evans  used inclination-only data from paleomagnetic studies and concluded that
the frequency distribution of those data were indistinguishable from that of an expected GAD field.
Subsequent inclination-only studies by Piper and Grant  and Kent and Smethurst  suggested that
there were periods in earth history when the magnetic field differed significantly from GAD. Deviations from the
GAD field for Paleozoic and Mesozoic times were also supported by recent studies by Torsvik and Van der Voo
 and Van der Voo and Torsvik . Hollerbach and Jones  argued that the size of the inner core has
a stabilizing effect on the geodynamo and a smaller inner core might result in persistent higher harmonic fields [e.
g. quadrupolar and octupolar] in the Paleozoic and Precambrian. Bloxham  tested the effects of a smaller
inner core [0.25 present-day] and found that a smaller sized core produced insignificant deviations from the
GAD model. Instead, Bloxham  argued that the large octupolar component inferred from inclination-only
data arises from the periodic effects of lateral heat transfer across the core-mantle boundary. McElhinny
 concludes that the GAD is a good approximation over the last 400 Myr, and we cannot assume the
presence of an axial geocentric octupole term because poor sampling coverage will produce false higher
order components. A number of other, non-geodynamo causes for the observed low-inclination bias have
been proposed and were discussed by Kent and Smethurst .
A successful inclination-only analysis relies either on sufficiently distributed sampling sites or that the sampling
sites become randomized via continental drift. Meert et al.  recently challenged the sensitivity of the
inclination-only method on resolving the GAD field through the use of a random walk model. The random walk
model, assumes a GAD planet and generates inclination data for well-distributed sites on randomly
drifting continents. Meert et al.  concluded that the current paleomagnetic database does not represent
a sufficiently random sample and therefore the non-GAD features observed in previous studies are simply due to
the effects of poor spatial-temporal coverage in the extant database. Here we examine several other flaws
in conducting inclination-only analyses and extend our random walk models to look at very small sample sizes.
Period Bins or Observations x12 x22 Ncrit RMSEA
Cenozoic [0-65 Ma] 253 3.63 9.19 426.3 0.07 NC
Mesozoic [65-250 Ma] 342 7.18 24.50 216.9 0.10 0.28*
Paleozoic [250-550 Ma] 352 32.23 113.48 48.9 0.21 0.11
Precambrian [550-3500 Ma] 531 20.39 108.76 76.6 0.17 0.14
All [0-3500 Ma] 1478 -- 135.75 169.8 0.11 0.16
Phanerozoic 947 -- 54.33 270.8 .091 0.18
Mesozoic+Cenozoic 595 -- 28.72 321 .083 0.28
Mesozoic+Cenozoic-Bloxham 3671 -- 603 95.4 .153 0.28*
X12, c12, as calculated by the original authors; X22, c22 as calculated in this study [note they vary slightly from the numbers
reported in Meert et al., 2003 due to a slightly refined best-fit program] Ncrit=critical N-index or Hoelter Index, RMSEA =
root mean square error of approximation, G2 and G3 are best-fit calculation to the observed binned distribution; NC=not
calculated since the results are indistinguishable from GAD.
* Best fit is significantly different than the observed distribution.
EVALUATION OF PREVIOUS MODELS
Evans  and Kent and Smethurst  used a inning technique to help filter out spatial-temporal biases
in their inclination analysis. Both previous studies used a spatial inning of 10oo x 10oo and temporal bins were
based on geologic periods with the exception that Kent and Smethurst evaluated the entire Precambrian
using 50 Ma intervals. We do not fault the rationale of using spatial-temporal inning; however, we note that
the choice of breakpoints can greatly affect the perceived inclination bias. For example, Kent and Smethurst
 argued that the Cenozoic and Mesozoic inclination distributions were indistinguishable from GAD. Meert et
al.  noted an error in the chi-square statistic [X2] calculation resulting in a Mesozoic distribution that
was significantly different from GAD above the 99% confidence level [see Table 1, X2 critical value 99%
=20.09]. However when the Mesozoic and Cenozoic distributions are added together, the resultant
inclination distribution is also significantly different from GAD [Table 1, figure la,b] with a best fit when
the quadrupolar contribution [G2] is ï¿½ 0.28 and the octupolar contribution [G3] is +0.1425. The best fit for
the Phanerozoic inclination distribution [Table 1, Figure lc,d] differs from GAD with a best fit when G2=ï¿½ 0.18
G Z= SO 0.
Pr cambian e Bstr n a ' n
ï¿½ 3=0 . 1426.
" i U.
3 : t .__ _ _ F_______M__
A i * 42 a Wa
1 0.1=2-1/ 01
Figure 1. [a] Inclination distributions for the Cenozoic and Mesozoic based on the analysis of Kent
SS [ T d i r i i
Phanerozoic based on the analysis of Kent and Smethurst . The dashed line represents the bestrwrozol
fit obtained w n G= a0d G +0.14215.  D ion of 2 a a tat
S t t . t t t ona
G3=+0.1425. [b] Distribution of G2 and G3 values that are statistically indistinguishable from
h 6" \.1 \d 630.. G*A0.144
PrMtiibrian Precambrian _______
0 10 X0 3 l 40 to B Mz V 4 0 .0 42 COa 01 .401 0 0C O i ( PS0 02 0;
lhxmnac InI G2
"I All All
Figure 1. [a] Inclination distributions for the Cenozoic and Mesozoic based on the analysis of Kent
and Smethurst . The dashed line represents the best fit obtained when G2=ï¿½ 0.28 and
G3=+0.1425. [b] Distribution of G2 and G3 values that are statistically indistinguishable from
the observed distribution for the Cenozoic and Mesozoic. [c] Inclination distributions for the
Phanerozoic based on the analysis of Kent and Smethurst . The dashed line represents the best
fit obtained when G2=ï¿½ 0.18 and G3=+0.1425. [d] Distribution of G2 and G3 values that are
statistically indistinguishable from the observed distribution for the Phanerozoic. [e]
Inclination distributions for the Precambrian based on the analysis of Kent and Smethurst .
The dashed line represents the best fit obtained when G2=ï¿½ 0.144 and G3=+0.232. [f] Distribution of
G2 and G3 values that are statistically indistinguishable from the observed distribution for
the Precambrian. [g] Inclination distributions for the entire database based on the analysis of Kent
and Smethurst . The dashed line represents the best fit obtained when G2=ï¿½ 0.18 and
G3=+0.1425. [b] Distribution of G2 and G3 values that are statistically indistinguishable from
the observed distribution for the entire database.
The Precambrian inclination distribution [Table 1, Figure le,f] gives a best fit when G2=ï¿½0.14 and G3=+0.232. If
we combine all the inclination distributions, the resulting best fit is obtained when G2=ï¿½0.18 and G3=+0.14
[Table 1, Figure lg,h].
Bloxham  examined the effects of an intermittent Y20 pattern of lower mantle heat flux variation.
The assumption was that such a pattern would inhibit the emergence of a poloidal field in equatorial regions and
lead to the expression of a predominately octupolar contribution to the magnetic field. He examined
unbinned inclination data for the Cenozoic+Mesozoic, the Paleozoic, and the Precambrian. He concluded that
the Mesozoic+Cenozoic distributions resembled the GAD because 250 Ma is too short a period to adequately
average the Earth's magnetic field and detect these octupolar components. Although a Y20 pattern of lower
mantle convection may result in a predominantly octupolar contribution to the field, we identify several problems
with the analysis of Bloxham . Bloxham  did not apply any statistical tests in an effort to distinguish
if the Mesozoic and Cenozoic distributions were different from GAD. We used the updated global
paleomagnetic database and applied the same selection criteria to the inclination data [n=3671 unbinned values]
and obtained the distribution shown in Figure 2a. This distribution is significantly different from GAD [Table 1]
using all 3 statistical parameters [see Meert et al., 2003] and therefore, if these inclination values faithfully reflect
the magnetic field, then the past 250 Ma also shows significant departures from GAD. Secondly, the Paleozoic
[lasting 293 million years] is only slightly longer, and less well-sampled [see below], than the combined
Mesozoic+Cenozoic [250 Ma]. Lastly, Bloxham  argues that quadrupolar terms average to zero in his
model, yet our analysis of the inclination-only data [see Figure 1] would indicate that most inclination
distributions are best modeled with a nonzero G2 term. Nevertheless, we cannot dismiss this possibility and note
that when we combine all the binned inclination data from the database and plot it as a cumulative frequency
curve [Figure 2b, median inclination 400], a best fit is obtained to the Bloxham  model when the amplitude
is ~17% of the superadiabatic heat flux.
Ceno +Meso i'=3671)
ï¿½ / _GAD
LL 4 --
0 10 20 30 40 So 80 70 So 40
r D-i (b)
Sf t n istri tin i e i i i
Figure 2. [a] Frequency distribution for the unbinned Cenozoic+Mesozoic inclination data from the
2003 global paleomagnetic database. The distribution is significantly different from GAD. [b]
Cumulative frequency of inclination data based on the best fit to the entire binned dataset of Kent
and Smethurst  in comparison to the expected GAD cumulative frequency curve. The best fit
line closely approximates the curve obtained in the Bloxham  model with a Y20 amplitude of
17% of the superadiabatic heat flux.
Kent and Smethurst  demonstrated that the Cenozoic dataset, when binned by geological Period [Neogene
and Paleogene], was indistinguishable from GAD. The reason for the GAD fit is best explained by the even
distribution of sampling sites rather than the effects of randomization via continental drift. We note here that
the similarity to GAD is also due to the inning method applied. Assuming that the GAD-like distribution arises
solely from the even distribution of sites, we should be able to bin the data at a finer temporal scale and obtain
a GAD-like distribution. Figure 3 shows the Cenozoic data binned at 5 Ma and 10 Ma intervals compared to GAD
and the Neogene-Paleogene inning of Kent and Smethurst . The 10 Ma inning produces a total of 519
bins and the resulting distribution is significantly different than GAD above the 99% confidence level
[c2=51.93; Ncrit=154; RMSEA=0.119]. The 5 Ma binned distribution is nearly identical to the 10 Ma
distribution; however the 5 Ma procedure produces 655 spatial-temporal bins and is also significantly different
from GAD at well above the 99% confidence interval.
" - GAD
- K&S (1998)
5- 5 Ma bins Js.-4
_ -10 Ma bins
0 10 20 30 40 50 60 70 80 9C
Figure 3. The frequency distribution of the Cenozoic database compared to the GAD distribution. The
Kent and Smethurst temporal bin [Neogene+Paleogene] produced a frequency that is
indistinguishable from GAD. A finer temporal inning [either 5 Ma or 10 Ma] produces distributions
that are significantly different from GAD.
The Paleozoic inclination distributions of Kent and Smethurst  and Piper and Grant  both showed a
low-latitude bias. Figure 4 [a-c] shows the spatial-temporal distribution of Paleozoic sampling sites. Most of
the sampling sites are from North America and Europe with significantly fewer results from the former
Gondwana elements. This low-inclination bias may have its origins in a strongly non-dipolar field or it may arise
from sampling bias. Meert et al.  argued for the latter explanation and both Bloxham  and Kent
and Smethurst argued for the former explanation. One way to test for sampling bias [in addition to
those conducted by Meert et al., 2003] is to assume that we have faithfully sampled a GAD field in the Paleozoic
and represent the motion of the continental blocks via their apparent polar wander paths [APWP's; see also section
6 below]. We compiled Paleozoic APWP's from the published literature for Siberia, Baltica, Laurentia and
Gondwana [Torsvik et al., 1996; Piper, 1987; Van der Voo, 1993; Smethurst et al., 1998]. These apparent
polar wander paths were then smoothed and divided into 20 Ma segments for the period from 550-250 Ma.
Sampling sites on each of the continents were placed at 5-degree intervals and samples were collected every 20
Ma based on their predicted latitudes from the APWP's. Figure 5a shows the synthetic distribution of inclination
data for Laurentia with a clear low-latitude bias. Figure 5b shows the combined Baltica-Laurentia
distribution compared to the global compilation obtained by Kent and Smethurst . A best fit to the
synthetic distribution is obtained with a pure octupole [G3] contribution of 22.4%. The best fit to the Kent
and Smethurst distribution required a G2=ï¿½0.11 and a G3=+0.28. Figure 5c shows the synthetic
distribution for Gondwana and figure 5d shows the sum of all the synthetic data. Figure 5d indicates that if the
above mentioned continents were well-sampled in the Paleozoic, the resultant inclination distribution would have
a low-inclination bias. Based on this analysis and those conducted by Meert et al.  we conclude that it is
not possible to use inclination-only data in the Paleozoic to distinguish between sampling bias and contributions
from non-dipole fields.
..r ,_ -
Figure 4. [a] Spatial distribution of the Paleozoic database. [b] Temporal distribution of the
Paleozoic paleomagnetic database shown as a cumulative frequency. The median age is 375 Ma and
[c] The spatial-temporal distribution of the Paleozoic database.
__,___ Inclinationr| I Inclination I
r'C ACADD 0
20 Y, 0 `
Figure 5. [a] A synthetic inclination frequency distribution for Laurentian sites based on a
smoothed apparent polar wander path with a sampling frequency of 20 Ma and a spatial inning of
5 degrees [Lau] [b] A synthetic inclination frequency distribution for combined Baltica+Laurentian
[LB] sites based on a smoothed apparent polar wander paths with a sampling frequency of 20 Ma and
a spatial inning of 5 degrees; KS= Phanerozoic data from Kent and Smethurst ; BF= Best Fit
Line. [c] A synthetic inclination frequency distribution for Gondwana sites based on a smoothed
apparent polar wander path with a sampling frequency of 20 Ma and a spatial inning of 5 degrees
and [d] A synthetic inclination frequency distribution for Siberia+Laurentia+Gondwana+Baltica
sites based on a smoothed apparent polar wander path with a sampling frequency of 20 Ma and a
spatial inning of 5 degrees.
The Precambrian inclination-only distribution shows a significant departure from GAD [Kent and Smethurst,
1998; Meert et al., 2003]. We analyzed the 2003 global paleomagnetic database according to the procedures
outlined in Kent and Smethurst  for the interval from 550-4000 Ma. The 1362 values resulted in 549
spatial-temporal bins. The resultant inclination distribution is not radically different from the previous study [fig
6] and shows a bias towards low inclinations. Meert et al.  argued that the low-inclination bias might
arise from incomplete sampling in the Precambrian. Figure 7 [a-c] shows the spatial-temporal bias in the
Precambrian dataset. The study locations are concentrated in Europe and North America and 80% of the data
are younger than 2000 Ma [median <1400 Ma; Fig 7b]. Figure 7d shows the inclination distribution that would
be expected for the present-day locations of these sites and demonstrates that plate motion must play an
important role in producing a random distribution of sampling sites in the Precambrian.
0 10 20 30 40 50 60 70 80 90
Figure 6. Inclination frequency distribution of Precambrian data from the 2003 edition of the
global paleomagnetic database versus the GAD model. The data were binned in 50 Ma temporal
intervals and 10 degree spatial intervals. The total number of bins was 549 [from 1362
individual inclination values].
Meert et al.  argued that a sizeable dataset is necessary to adequately test the GAD hypothesis.
The requirement placed on the inclination-only analysis is that the sampling must guarantee [with 95% confidence
or better] that the field has been adequately sampled. A further condition is that this requirement is met for
whatever temporal period is examined. For example, a small dataset collected for one particular time interval
may result in a distribution that is indistinguishable from GAD. However, additional samples added during the
next time interval may result in a non-GAD distribution. When there is a clear spatial bias to the data we require
a sampling interval guaranteed to faithfully represent the average magnetic field. The Precambrian dataset
sampled less than 5% of the available spatial-temporal bins available making it unlikely to generate the
required random sample. Meert et al.  demonstrated with several examples that such a small dataset
is unlikely to sufficiently test a GAD field, but the argument was not quantified in detail.
Here, we test small sample sizes as follows. The random-walk model was conducted on a GAD planet. Samples
were collected every 50 Ma and plate direction changes were conducted every 75 Ma. Plate velocities ranged from
0 to 8 cm yr-1. We ran the model with an increasing number of sampling sites starting with 11 distributed sites
and ending with 39 distributed sampling sites. The 11 sites produce a total of 550 'bins' and is comparable to
the sample size used in the Kent and Smethurst  evaluation. Each model was run for 2500 Ma and
100 iterations. The program compiled a listing of acceptable representations of the known GAD field using the X2
test, the Ncrit+RMSEA combination or the x2-RMSEA combination [see Meert et al., 2003]. We define a GAD-like
fit as being statistically indistinguishable from the GAD distribution using the critical values outlined in Meert et
, ' I. ln= m
" : - , .
. .V. "
Figure 7. [a] Spatial distribution of the Precambrian database. [b] Temporal distribution of
the Precambrian paleomagnetic database shown as a cumulative frequency. The median age is 1400
Ma and [c] The spatial-temporal distribution of the Precambrian database and [d] expected present-
day inclination values of sampled Precambrian sites showing the sample bias inherent in this dataset.
Small Sample Runs
S x2 RMSEA and Ncrit x2
[ai]1t [end]2 [end]3
Prec11 551 53.4% 5.5% 42.0%
Prec19 970 60.6% 23.8% 47.0%
Prec29 1480 41.8% 62.0% 34.0%
Prec39 2041 22.7% 75.0% 21.0%
1 Uses only the X2 value in the analysis
2 Both the RMSEA and Ncrit values must reach critical levels of significance at the end of each run.
3 Uses only the final X2 value in the analysis
Meert et al.  describes the sensitivity of the X2 test to small and large sample sizes. A small sample size
will almost always be indistinguishable from the expected and large sample sizes will nearly always indicate
a significant difference from the expected distribution. Our first sample run [using 11 distributed sites] showed
that only 53% of the distributions were indistinguishable from GAD using the X2 test [Table 2]. As sample sites
were added to the model, the number of GAD-like distributions generally decreased [Table 2, Figure 8a] with
only 22.7% acceptable fits when there are 39 distributed sites. However, we note an additional complexity
in interpreting these results because the X2 values oscillate over the sampling interval and many of the acceptable
fits are achieved at low-N [see Figure 8b]. Therefore, we also looked at the percentage of GAD-like fits achieved
at the end of the run and found that these also generally decreased with increasing sample sizes [Table 2]. Lastly,
we note that in no case did we achieve the required 95% level using only the X2 test. In contrast, we found that
the number of GAD-like fits based on the RMSEA+Ncrit values increased in dramatic fashion with increasing N [from
a low of 10% to 75% when n=39 sites; Figure 8a]. Although none of these values reached the requisite 95%
level, Meert et al.  showed that when the number of samples is large and the runs are lengthy, the
95% confidence criterion is met.
a, Chi-Square Onl
S5O 700 900 1100 1300 i5s) 1700 190 2100 2300
Number of Total Bins
Figure 8. [a] The percentage of GAD-like distributions based on the X2 values obtained at 50 Ma
intervals compared to the number of binned data [gray solid line; see also Table 2], the percentage
of GAD-like distributions based on the X2 values obtained at the last step of the simulation [2500
Ma; dashed line] and the percentage of GAD-like distributions based on the root mean square error
of approximation and Ncrit indices obtained at the last step of the simulation [2500 Ma; dark line].
[b] Large graph shows the change in the X2 value at each 50 Ma step of one simulation. In this
particular case, the resultant distribution is GAD-like only at the very beginning and very end of the
run. The inset graph is given to demonstrate the variable drift rates generated by the random walk model.
COMPARISON OF NORTH AMERICA'S ACTUAL AND SYNTHETIC DRIFT HISTORY
In an effort to further test the perceived low-inclination bias in the paleomagnetic database, our research included
a comparison of North America's actual paleomagnetic dataset with a synthetic dataset. The synthetic dataset
was generated using a time-averaged APWP [apparent polar wander path] for North America with sampling
locations positioned at 5 degree intervals within the continental boundaries [as above]. Assuming our synthetic
model generated a robust sample, a comparison to the extant dataset [Figure 9] indicates a statistically
significant difference [>99% CI] between the observed and expected results [X =21.842;
Ncrit=198.58; RMSEA=0.106]. This result introduces the possibility of a variable contribution of the higher
harmonic fields throughout at least the last 600 Myr. Alternatively, the difference between these two studies may
be explained by an inadequate sample size of North America in the paleomagnetic database particularly since
both samples show a bias towards low inclinations. Meert et al.  argued for the possibility of an under
sampled North America.
1Niiorlh AImeican IP a
14 , "" ..... ., GA)
- 12 1 ip theSK^ir North
0 20 40 60 80 100
Figure 9. Comparison between a synthetic inclination dataset generated using a uniform distribution
of sites in North America, the GAD field and North American inclination data from the 2003
global paleomagnetic database.
INCLINATION SHALLOWING OF SEDIMENTARY ROCKS
McElhinny  describes inclination shallowing as a compression of the magnetic minerals through compaction
and dewatering of sedimentary rocks. This phenomenon is important in our study of inclination distributions
because, if present, it would cause a bias in the data collected from sedimentary rocks. We compared the
inclination distributions of sedimentary and igneous rocks to test the possibility of inclination shallowing
in sedimentary rocks. Igneous rocks are not subject to inclination shallowing owing to their different mode
of remanence acquisition.
We gathered the inclination data for sedimentary, extrusive igneous, and intrusive igneous rocks and performed
a statistical comparison for each of the three types of rocks [Figure 10]. The comparison because the
sedimentary and extrusive rocks yielded a difference between the two rocks types at a 99% confidence
[X2=55.84; Ncrit=268.3; RMSEA=0.091]. The sedimentary compared to the intrusive igneous rocks also resulted in
a significant difference between the two rock types at a 99% confidence level [X2=150.8; Ncrit=99.95;
RMSEA=0.15]. The difference between these two samples is consistent with inclination shallowing in
sedimentary rocks arising from DRM [Detrital remanent magnetization] processes explained by McElhinny
; however, the test is inconclusive because both intrusive and extrusive igneous rocks also show a
shallow bias. Our paper points to inclination-only studies as a highly unreliable and inconsistent technique to test
the earth's magnetic GAD properties.
18 srrI~ly Rorrk- 0Is Ignr is Rats
0 20 40 60 80 100
Figure 10. Comparison between the inclination dataset from sedimentary, extrusive and intrusive
rocks from the 2003 global paleomagnetic database and GAD.
One of the main tools for evaluating the GAD hypothesis in pre-Cenozoic time is based on the distribution
of inclination values. Recent studies of inclination-only data show a bias towards low inclination, and a number
of alternative explanations were forwarded to explain this bias. One of the assumptions made in the analysis is
that the planet has been adequately sampled in a spatially and temporally random manner. In a recent paper,
Meert et al.  argued that the extant paleomagnetic database is not capable of adequately testing the
GAD hypothesis. Here we have examined other sources of bias in the database. We found that the
apparent contributions of quadrupolar and octupolar fields may depend upon the inning procedure used.
For example, the Cenozoic database can be favorably compared to GAD when assigned to temporal bins based
on geologic periods, but is decidedly non-GAD when averaged on a finer temporal scale. We also demonstrated
that the Paleozoic inclination distribution may result from a regional sampling bias. We also quantitatively assess
the probability that the Precambrian global paleomagnetic dataset might reflect integrated behavior of the
Earth's magnetic field. Although GAD-like fits were obtained with small, randomly distributed sites, the probability
of obtaining a good representation of the field was under 50%.
Unfortunately, our analysis cannot rule out the existence of persistent non-dipolar fields in geologic time. The
global paleomagnetic database does indeed show a rather consistent bias towards low-inclination values
[median inclination is 400 versus 490 for the GAD]. We recognize that there may possibly be many explanations
for these results, but this paper points to an inadequate sampling in the global paleomagnetic database and
possibly shallowing of sedimentary inclinations due to the DRM processes. Models of the earth's magnetic field
and the thermal evolution of the planet may yield additional clues regarding its GAD or non-GAD nature. We
argue that we may be misled by relying on inclination data from an inadequately sampled planet.
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