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Full vector spherical harmonic analysis of the Holocene geomagnetic field

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Full vector spherical harmonic analysis of the Holocene geomagnetic field

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Content FULL VECTOR SPHERICAL HARMONIC ANALYSIS OF THE HOLOCENE
GEOMAGNETIC FIELD
by
Marcia Richardson
A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(EARTH SCIENCES)
May 2014
Copyright 2014 Marcia Richardson
Table of Contents

Abstract iv

Chapter 1 - Introduction 1
1.1 Earth’s Current Magnetic Field 2
1.2 Earth’s Historic Field 4
1.3 Earth’s Field Over the Past 7,000 Years 6
1.4 Rationale 10
1.5 Approach 13

Chapter 2 - PSVMOD2.0 Data and Regional Studies 26
2.1 PSVMOD2.0 Data 26
2.2 Regional Comparison – Non Arctic North America 27
2.2.1 Methodology 28
2.2.2 Assessment of PSV Record Chronologies 30
2.2.3 Regional Comparison of Feature Ages 32
2.2.4 Waveform Analysis 34
2.2.5 Paleointensity Analysis 38
2.2.6 Summary of North America Regional Study 44

Chapter 3 - Modeling 59
3.1 Description of Earth’s magnetic field – the forward problem 59
3.2 Solving the inverse problem 63
3.3 Solving the non-linear problem 67
3.3.1 Formulation of the Jacobian for the non-linear problem 71
3.3.2 Modeling Methodology 74
3.3.3 Model Non-uniqueness 75

Chapter 4 - Model Results 78
4.1 Model Validation 78
4.2 Comparison with GUFM1 78
4.2.1 Comparison with GUFM1 – 1900 AD 80
4.2.2 Comparison with GUFM1 – 1800 AD 89
4.3 Models prior to 1800 AD 97

Chapter 5 – Model Results - Implications for Field Generation 119
5.0.1 High latitude flux lobes – Northern Hemisphere 119
5.0.2 Reverse flux patches near North Geographic Pole 120
5.0.3 High latitude flux lobes – Southern Hemisphere 121
5.0.4 High latitude flux lobes – Hemispheric symmetry 122
ii
5.1 South Atlantic Anomaly 123

Chapter 6 – Spherical Harmonic Analysis of the Solar Magnetic
Field Over the Past Three Solar Cycles 125
6.1 Introduction – A Comparison between the Sun and the Earth 125
6.2 Geomagnetic Polarity Reversals 128
6.2.1 Geomagnetic Excursions 133
6.3 The Solar Magnetic Field 134
6.3.1 Solar Magnetograms 134
6.4 Analysis 136
6.41. Large scale solar field – Spherical harmonic coefficients 136
6.4.2 Ratios of low degree axial coefficients at the photosphere 138
6.4.3 Harmonic contents of the solar magnetic field 139
6.4.4 Preferred reversal paths of the solar field 139
6.4.5 Symmetry families 140
6.4.6 Solar coronal field 140
6.4.7 Current solar cycle 141
6.4.8 Conclusions 141

References 156

Appendix A – Tables of site locations 163
Appendix B – North America Site Locations 166
Appendix C – Expanded Regional Site Maps 168
iii
Abstract
High-quality time-series paleomagnetic measurements have been used to derive
spherical harmonic models of Earth's magnetic eld for the past 2,000 years. A
newly-developed data compilation, PSVMOD2.0 consists of time-series directional
and intensity records that signicantly improve the data quality and global dis-
tribution used to develop previous spherical harmonic models. PSVMOD2.0 con-
sists of 185 paleomagnetic time series records from 85 global sites, including 30
full-vector records (inclination, declination and intensity). It includes data from
additional sites in the Southern Hemisphere and Arctic and includes globally dis-
tributed sediment relative paleointensity records, signicantly improving global
coverage over previous models. PSVMOD2.0 records have been assessed in a series
of 7 regional intercomparison studies, four in the Northern Hemisphere and 3 in
the southern hemisphere. Comparisons on a regional basis have improved the qual-
ity and chronology of the data and allowed investigation of spatial coherence and
the scale length associated with paleomagnetic secular variation (PSV) features.
We have developed a modeling methodology based on nonlinear inversion of the
PSVMOD2.0 directional and intensity records. Models of the geomagnetic eld
in 100-year snapshots have been derived for the past 2,000 with the ultimate goal
of developing models spanning the past 8,000 years. We validate the models and
iv
the methodology by comparing with the GUFM1 historical models during the 400-
year period of overlap. We nd that the spatial distribution of sites and quality
of data are sucient to derive models that agree with GUFM1 in the large-scale
characteristics of the eld. We use the the models derived in this study to down-
ward continue the eld to the core-mantle boundary and examine characteristics
of the large-scale structure of the magnetic eld at the source region. The derived
models are temporally consistent from one epoch to the next and exhibit many of
the expected characteristics of the eld over time (high-latitude
ux lobes, South
Atlantic reverse
ux patch, north pole reverse or null
ux region).
v
Chapter 1
Introduction
Earth's magnetic eld is produced in the outer core by a self-sustaining dynamo
process and is typically described by a global spherical harmonic model. The eld
has been accurately characterized in recent decades by high quality spacecraft
measurements as well as measurements made at globally distributed ground-based
magnetic observatories. To assess the eld prior to the era of space-borne magnetic
measurements we rely on extensive sets of historical observations that have been
compiled starting in the 15th century as part of the global navigation process.
Although uncertainties are signicant and spatial and temporal coverage more
sparse, these records have been used to model the eld over the past four centuries.
Prior to this time we must rely on paleomagnetic records to characterize Earth's
magnetic eld. Paleomagnetic records suer from even greater limitations in
data quality and age control as well as poorer temporal and spatial coverage. In
past decades increasing amounts of data have become available and a number of
global spherical harmonic models have been derived to describe the centennial to
millennial-scale geomagnetic eld [32]. These models are widely used by the scien-
tic community and although they have provided some insight into the variability
of the eld over the past several millennia, we suggest that they may not accurately
capture the details of paleomagnetic eld variability. We have adopted a dierent
approach and have derived spherical harmonic models of the Holocene geomag-
netic eld based on selected high-quality time-series paleomagnetic measurements
of direction and intensity. The new data compilation, referred to as PSVMOD2.0,
1
signicantly improves the data quality and global distribution over previous spher-
ical harmonic models. To further improve the quality and chronology of the data
and assess real characteristics of secular variation, these paleomagnetic time-series
records have been evaluated in individual regional studies.
1.1 Earth's Current Magnetic Field
High quality global measurements of Earth's magnetic eld from space have
been available since the late 1970's. These measurements coupled with obser-
vations from a global network of ground-based observatories have provided an
unprecedented view of Earth's magnetic eld.
Spherical harmonic models based on these data suggest that although about
90% of Earth's eld at the surface can be accounted for by a dipole tilted roughly
11

from the axis of rotation [78], the eld diers markedly from a simple tilted
dipole. Figure 1.1 shows the magnetic eld intensity at the surface (panel a), the
radial component of the eld at the surface (panel b) and the radial eld at the
top of the outer core (panel c) based on the CHAOS model of Olsen et al. [60] for
the epoch 2005. These maps reveal additional complexity beyond a tilted dipole
which would be symmetric about the magnetic axis with maximum intensity at
the poles and minimum at the equator. The major features that are evident in
the surface eld intensity are two strong maxima in the northern hemisphere over
Arctic Canada and Siberia (60,000 nT) and one slightly stronger (66,700 nT)
intensity maxima in the southern hemisphere. The eld intensity low (23,000 nT)
over South America, the well-known South Atlantic anomaly, is a distinct feature.
These features are evident in a map of the radial eld at the surface, panel b, as
well.
2
Based on the assumption that the mantle is a perfect insulator, models of
Earth's eld can be extrapolated to estimate the eld at the top of the source
region, the core-mantle boundary. Here maps of the radial component of the eld,
the fraction that escapes the core and which is responsible for the eld we see at
the Earth's surface, is far less dipolar (panel c, gure 1.1). Not surprisingly, more
small-scale structure is evident here than in analogous maps at the surface because
geometrical attenuation damps higher degree terms with distance from the source.
(The eld produced by terms of the spherical harmonic series vary according to
degree, n with distance from the source as 1=r
(n+2)
). Several remarkable features
show up including a large reverse
ux patch extending from South America to
Africa, presumably the source of the South Atlantic Anomaly observed at Earth's
surface. The two high intensity
ux lobes in the northern hemisphere separated
by roughly 120

are paired with concentrations of
ux at similar longitudes in the
south. (This is dierent from the picture at the surface, with one strong
ux lobe
in the southern hemisphere.) Contrary to expectation for a dipolar eld, a low
intensity
ux patch seen in the polar region is a prominent feature, particularly in
the northern hemisphere.
Earth's magnetic eld changes on time scales of years to centuries, referred
to as secular variation. Signicant changes in Earth's eld have occurred over
the recent era of high quality spacecraft measurements. A comparison of main
eld models over the roughly thirty year interval that these measurements have
been made reveal some notable trends including substantial growth of the South
Atlantic Anomaly accompanied by a sustained decrease in the strength of the dipole
moment which continues to decay at an overall rate of nearly 6% per century.
3
1.2 Earth's Historic Field
Models of the geomagnetic eld over the past four centuries have been developed
based on extensive sets of historical observations. Observations became available
starting in the 15th century when measurements were made as part of the navi-
gation process. A global distribution of observatory data has been available since
the mid-19th century [27]. Earliest measurements were directional only, rst decli-
nation and later inclination. Intensity measurements became available in the mid
19th century when Gauss invented a method of measuring the eld magnitude.
The GUFM1 model of Jackson et al., [25] is the current best representation of
Earth's eld over the past four centuries based on these historical observations.
Snapshots in 100-year intervals of the radial eld at the core-mantle boundary
based on GUFM1 are shown in gure 1.2. Some of the large scale characteristics
seen in Earth's present eld are seen in the historic eld as well. The most distinc-
tive features evident at every epoch are the high latitude
ux lobes over Canada
and Siberia and their counterparts almost symmetrically placed at the same lon-
gitude in the southern hemisphere. These lobes are responsible for the predomi-
nantly axial dipole eld structure observed at the surface. This series of models
suggest that the
ux lobes have been approximately stationary for the past four
centuries, wobbling slightly about a mean position. Persistent reverse-
ux patches
are a prominent feature of the eld at the core surface. The most prominent is a
large-scale patch that extends from southern Africa to South America, presumably
a signature of the South Atlantic Anomaly. Its growth and migration have been
linked to the rapid decay of the axial dipole since the mid 1800s. Reverse-
ux
patches close to the geographic North pole, where the eld should be strongest are
evident throughout most of the past 400 years. A similar, but weaker, feature over
the South pole exists for most epochs as well.
4
The rapid decay in the strength of Earth's dipole moment, almost 6% per cen-
tury, is depicted in the top panel of gure 1.3. The solid red line is the magnitude of
the dipole moment since intensity measurements became available in 1832. Prior to
this time, the value is estimated from the linear extrapolation of 15 nT/year based
on the analysis of Barraclough [3]. The bottom two panels show the contribution
to the axial moment by the normal and reversed magnetic
ux by hemisphere since
1840 and demonstrate that the current dipole decrease is primarily attributable to
an increase in reversed
ux in the southern hemisphere [61].
At Earth's surface westward drift is one of the most striking aspects of secular
variation of the magnetic eld and was rst recognized by Halley three centuries
ago. The westward motion of magnetic features has been estimated at roughly .2

per year. At the core-mantle boundary westward moving eld concentrations are
seen, especially at low latitude and they are especially clear in the Atlantic hemi-
sphere (90

E to 90

W), with eld evolution in the Pacic hemisphere much more
subdued [6]. The Pacic hemisphere is characterized by lower amplitude features
and less systematic secular variation. It has been suggested that this hemispheric
asymmetry could be due to the in
uence of lower mantle inhomogeneities on the
dynamo in the core [14].
Contour plots of GUFM1 in a polar projection show evidence for zero-
ux spots
near the north and south geographic poles (gure 1.4). First noted by Bloxham
[7], zero-
ux patches are opposite that of a dipole eld which would be expected
to be strongest over the poles. Analysis of the radial eld based on GUFM1 with
the axisymmetric eld removed and ltered to remove eld components varying on
timescales longer than 400 years [15] (gure 1.5) shows relatively stable
ux lobes
at high latitudes as well as westward moving features exhibiting wavelike motion,
particularly striking at the equator.
5
Summarizing some of the major features of Earth's eld based on historic model
GUFM1: 1) two relatively stable
ux lobes at high latitudes over arctic Canada and
Siberia 2) rapid decay of the dipole moment, 5-6% per century accompanied by 3)
rapid growth of reverse
ux in the southern hemisphere 4) hemispheric asymmetry
in the style of secular variation with westward wavelike motion in the Atlantic
hemisphere and more subdued secular variation in the Pacic.
1.3 Earth's Field Over the Past 7,000 Years
To understand Earth's eld on timescales longer than the past four centuries,
we must rely on paleomagnetic records, including archeomagnetic and lava
ow
data, and lake and marine sediments. Paleomagnetic records suer from general
weaknesses including lack of global distribution, inherently lower resolution and
greater age uncertainties. Archeomagnetic materials and lava
ow records have
thermoremanent magnetization acquired when the material cools in the presence
of an ambient magnetic eld. This magnetization is acquired over a short period
of time thus represents an instantaneous record of the eld. The disadvantages of
this type of record are that they are scarce before 2000 B.P. and are geographically
biased toward certain areas such as Europe. Sediments acquire their magnetiza-
tion as depositional remanent magnetization (DRM) or post-depositional remanent
magnetization (PDRM), alignment of magnetic grains with the geomagnetic eld in
the water column or the interstitial spaces below the water-sediment interface. The
advantage of using sediment records is that they have the potential of extending
high resolution records further back in time and providing greater global distribu-
tion. The disadvantages include that fact that they are inherently lower resolution
6
and provide a record only of relative eld intensity, thus intensity variations must
be calibrated for use in global modeling [70].
Early archeomagnetic eld models were built as a sequence of snapshots and
restricted to only the rst few degrees of the eld [23]. The rst attempt at deriving
spherical harmonic models of the Holocene geomagnetic eld based on a reasonable
global distribution of directional measurements was made by Constable et al. [13].
In that analysis, 24 distributed times series records of archeomagnetic, lava
ow
and lake sediment data were used to produce models of the geomagnetic eld at
100-year intervals extending from 1000 BC to 1800 AD. These models provided an
unprecedented view of paleomagnetic secular variation (PSV) and demonstrated
a consistent evolution of the eld over the past 3,000 years. The largest scale
features, the Northern hemisphere
ux lobes were clearly resolved and the pattern
of secular variation was characterized.
Since that time, a series of global spherical harmonic models based on ever
increasing amounts of paleomagnetic data have been derived [31],[37],[36],[34]. The
models are referred to as CALSxK.n, (Continuous Archeomagnetic and Lake Sed-
iment models spanning x thousand years and version number n). We will focus
our discussion on CALS7K.2 which spans the past 7,000 years and is widely used
by the scientic community. The model results are typically presented as snap-
shots or animations of contour maps of the eld at the core-mantle boundary
(http://earthref.org/ERDA/431/). Figure 1.6 shows that CALS7K.2 surface con-
tour maps are smooth in comparison with similar maps based on historical data
(gure 1.2). The models resolve the high latitude
ux concentrations in the North-
ern Hemisphere seen in the historic models, although they are highly attenuated
and they hint that a third lobe makes an appearance during some time periods.
There is some evidence for
ux lobes in the southern hemisphere for some epochs
7
as well. The authors claim that during some intervals dipolar structure gives way
to more complex structure (eg., 1900 BC, gure 1.6). Other than the high latitude
northern hemisphere
ux lobes, it is dicult to identify features of the models that
are robust. Animations of those models have given a visual sense of the pattern of
variability of the Holocene eld but it is not clear how much they have improved
our understanding beyond that.
The approach taken in developing the CALSxK.n series of models is to incorpo-
rate all available paleomagnetic data from all sources. CALS7K.2 relies on almost
33,000 measurements of direction and intensity [34]. These data are obtained from
a wide variety of sources, including original literature and existing databases. Sig-
nicant dierences exist between the various data sources, in techniques applied
to obtain and analyze the data, levels of documentation and quality tests applied.
Although the data from which CALS7K.2 is derived include some time series mea-
surements, a large fraction of the data are isolated, spot measurements, and do not
allow for independent assessment of chronology and regional consistency that time
series measurements provide (see chapter 2). In deriving the CALSxK.n series of
models, the authors claim that for a signicant part of the data set it is dicult
to obtain independent, realistic and internally consistent estimates of the uncer-
tainty. They further state that some observations incorporated into their models
are mutually inconsistent. The typical level of variability of measurements used to
derive CALS7K.2 can be seen in a number of globally distributed archeointensity
records (gure 1.7).
Although the data compilation on which CALS7K.2 is based is extensive, the
global distribution of the data is very inhomogeneous in both space and time. The
southern hemisphere is underrepresented, with data concentrated in Europe and
Asia. Only 15% of the data comes from the southern hemisphere, with only one
8
site providing archeomagnetic directional records and two sites providing archeo-
magnetic intensity records. While archeomagnetic data is heavily biased toward
Europe, lake sediment records are more globally distributed, but these records
must be calibrated to provide absolute intensities.
In addition to issues with data quality and global distribution of records, the
approach taken in developing the CALSxK.n models is to seek to obtain a model
that balances tting the model closely against nding a model with minimum
complexity. Unnecessary structure is excluded from the model [63]. Although
this is a reasonable approach and is used by other modelers, coupled with an
extremely noisy and inconsistent data set, we propose that it results in a model
which suppresses real structure and is overly smoothed.
One use of paleomagnetic models is to obtain an estimate of the geomagnetic
dipole moment and its evolution over time. In a comparison of the dipole moment
based on CALS7K.2 with VADMs (Virtual Axial Dipole Moments) from other
studies, Korte and Constable [33] found that CALS7K.2 underestimated the pale-
omagnetic dipole moment by 19% (gure 1.8). VADMs are the equivalent axial
dipole moment calculated from magnetic eld measurements at a particular site.
The eld due to contributions by higher order multipoles may be `aliased' into the
VADM, resulting in a value that is higher than the actual dipole moment deter-
mined from spherical harmonic analysis, so an overestimate might be expected.
They concluded that part of their underestimate is due to contributions from the
non-dipole eld that is included in the calculation of the VADM and part is due
to data quality issues and they further conclude that these issues are aggravated
by geographic biases in sampling. Comparisons with the historical GUFM1 model
for the intervals of time that overlap also suggest that the model underestimates
the dipole moment (gure 1.8). Other, more recent studies also conclude that
9
CALS7K.2 underestimates the dipole moment and suggest that better paleomag-
netic data is needed to improve the models [77],[58].
An accurate global magnetic eld model should predict local directional and
intensity measurements, yet a comparison of CALS7K.2 with regional data records
often indicates a relatively poor t. Figure 1.9 shows several examples of globally
distributed records [37]. Although CALS7K.2 agrees with the measurements for
some records (eg., BIW, declination), for others there is not even agreement in the
general trends.
The authors state, signicant uncertainties in millennial scale geomagnetic eld
reconstructions based on the CALSxK.n series of models remain. Despite these
deciencies, these are the only global spherical harmonic models of the Holocene
geomagnetic eld available for use by the scientic community [34].
1.4 Rationale
The main motivation for developing paleomagnetic eld models is of course, to
extend the time scale over which we can examine the evolution of the Earth's eld.
With accurate models extending over millennia, we can test ideas about the eld
based on present day and historic observations, including the longevity of mag-
netic features and secular variation. Does the dramatic decrease in dipole moment
and accompanied growth of the South Atlantic Anomaly persist in magnetic eld
models on longer timescales? The existence and variability of high latitude
ux
lobes and westward drift of magnetic features can be put into longer-term context
with accurate paleomagnetic eld models.
Models of the Earth's eld can be downward continued to provide a picture
of the large-scale radial eld at the core-mantle boundary under the assumption
10
that the eld sources are internal to the Earth's core and that the mantle is an
insulator [7]. Earth's main eld can serve as a useful probe for investigating the
understanding the physical processes that control the geodynamo, including pat-
terns of convection and the possible in
uence of the lower mantle on dynamo
activity.
A high degree of accuracy is important in models derived from surface mea-
surements since inaccuracies in model parameters are amplied when extrapolated
to the source region at the top of the outer core. Spherical harmonic terms vary
according to (1=r)
(n+2)
, where n is the spherical harmonic degree (see equations
3.5 through 3.7).
Based on the ideas of Lund [70] we argue that if we can characterize the
Holocene eld, we will have a near complete view of the dynamical pattern of
PSV. Power spectra of unit-vector and paleointensity records from the western
North Atlantic for the last 50 thousand years have a cut-o or corner frequency
that limits normal PSV to periods of 10-10,000 years (gure 1.10). Both unit-
vector and intensity exhibit `red' spectra below a frequency of 10
4
/yr (period
of 10,000 years). They interpret the `red' spectral interval to represent the real
dynamical variability of PSV due to ongoing dynamo activity in the Earth's core.
There are no clear spectral peaks in this band because complexities in the dynami-
cal process will smooth out any shorter-term apparent periodicities that appear in
individual records over shorter time intervals. Both spectra have a corner frequency
at lower frequencies where the spectra turn into white (
at) spectra. They inter-
pret that corner frequency to be the limit of the normal dynamical PSV variability;
the lower frequencies in the white spectrum beyond the corner frequency are sim-
ply longer-term trends in PSV due to random-walk of the same PSV dynamical
process.
11
Accurate paleomagnetic models can provide an important point of comparison
with the results of dynamo simulations. Substantial progress has been made in
recent decades in the development of fully three-dimensional, non-linear dynamo
models. These models are important for clarifying the physical processes that
generate the main eld and it is important to compare simulation results with
real characteristics of Earth's eld. Comparisons between dynamo models and
observations are limited in part because the simulations are quite recent but also
the paleomagnetic data and models are not suciently reliable. Simulations are
able to describe some features of the geomagnetic eld quite well, but are lacking
in other aspects. Many dynamo models can replicate observations such as the
fundamental dominance of the axial dipole over geologic time. Others are able to
replicate longitude drift in magnetic features [16].
Accurate paleomagnetic eld models are important for understanding the pro-
duction of cosmogenic isotopes (primarily
10
Be,
14
C and
36
Cl) in Earth's atmo-
sphere and these records are invaluable for understanding the relation between
past climate change, the Earth's magnetic eld, and variations in the solar activ-
ity. Cosmogenic nuclide production rates are in
uenced by the temporal variability
of the solar magnetic eld that modulates the
ux and energy of galactic cosmic
rays that penetrate the heliosphere, as well as the geomagnetic eld, especially
the dipole strength and tilt, but also the more complex aspects of eld varia-
tion as well. Greater solar activity or a stronger geomagnetic dipole increases the
overall shielding of galactic cosmic rays. Variability in radionuclide production at
time scales longer than 3000 years is often attributed to variable geomagnetic eld
intensity and shorter term variations are attributed to modulation by the solar
magnetic eld, although some combination of solar and geomagnetic modulation
could apply to all timescales and one of the largest uncertainties in reconstructions
12
of past solar activity remains accurate reconstructions of the geomagnetic eld [69].
Most attempts to quantify geomagnetic eects on cosmogenic nuclide production
rates have relied on geocentric dipole models however these models can lead to sig-
nicant systematic errors by ignoring non-dipole eects. Recent eorts to model
cosmogenic isotope distribution incorporating the CALS7k.2 model indicate that
the distribution of cosmogenic isotopes is signicantly aected by the non-dipole
eld over the past 7 kyr [41].
1.5 Aproach
We have derived high quality spherical harmonic models of the geomagnetic
eld with an approach that involves several unique aspects: 1) use of globally dis-
tributed, high-quality time series data, as opposed to isolated spot measurements
2) use of a signicant amount of paleointensity data 3) assessment and intercom-
parison of paleomagnetic records on a regional basis.
The newly developed data compilation, PSVMOD2.0, is a family of Holocene
times series records based on a large number of high quality paleomagnetic records
for the past 8000 years. The data set includes 72 sites around the world, which
yield 65 directional PSV time series records and 35 paleointensity time series.
The global distribution is shown in gure 2.1). This data compilation lls in
signicant spatial gaps, including a number of additional sites from the Southern
Hemisphere and Arctic, improving global coverage over previous models that has
been lacking to date. The data are fully described in chapter 2. Twenty-eight
records in PSVMOD2.0 are true full-vector times series with both directional and
paleointensity data. In contrast, the CALS7K.2 model incorporated only a small
amount of paleointensity data.
13
Our use of time series records in 100-year increments, is an improvement over
isolated spot measurements used to derive other models. They provide the advan-
tage of an independent assessment of data quality between successive epochs and
between spatially neighboring records. Spatial correlation permits an indepen-
dent assessment of chronology as well. Error estimates are less certain when using
isolated PSV data. We have been able to assess the accuracy of individual PSV
records, test for systematic age dierences associated with particular records, iden-
tify outlier or poor quality records, and improve the chronology and quality of the
data used.
In each region we have replicated the study of Lund [44] in which a number of
Holocene records from North America were compared to assess regional aspects of
PSV, including evidence for spatial coherence and the scale length, systematic age
dierences associated with particular PSV features indicating eastward or west-
ward drift, distinctive PSV space-time waveform patterns caused by core dynamo
processes. By correlating individual features such as minima or maxima of inclina-
tion or declination, they found that distinctive waveform features could be traced
over scale-lengths that span North America ( 4000 km) without a signicant
change in pattern. We have used a similar approach to improve our understanding
of PSV, in particular the relationship between intensity and directional variability
by comparing PSV on a regional versus global scale and look for persistent pat-
terns of directional and intensity variability. The regional studies are described in
chapter 2.
To derive the eld models, standard, well-established modeling techniques are
used (see, for example [26] and [19]) and are fully described in chapter 3. Global
models generally rely on the standard spherical harmonic representation in terms
of Gauss coecients, g
m
n
and h
m
n
. The inverse problem of nding the coecients
14
is generally solved by nding a model that minimizes the least-squares dierence
between the model predictions and the data. More generally, when directional
measurements, I and D or eld intensity, F, are involved, the relation between the
model parameters and data are nonlinear and the model must be found iteratively.
The methodology for solving the nonlinear problem is also detailed in chapter 3.
Validation of the methodology and approach was done by comparing our
derived eld models with GUFM1 historical models during the period of over-
lap. We compared our newly derived models for epochs 1900 through 1600 AD.
The results of this comparison are detailed in chapter 4. An initial set of models
spanning the past 2,000 years has been derived. These models have been used
to extrapolate the eld to the core-mantle boundary and study characteristics of
the eld at the source region. The model results are described in chapter 4. An
important objective is to examine the eld at the core-mantle boundary for char-
acteristics that have been identied in the eld over the past four hundred years -
static features, drifting features, secular variation, regions of lower than expected
eld strength and suggestions of wave-like features. Inferences and conclusions
regarding eld generation and dynamo process from our models are presented in
chapter 4.
15

( nT)
–1000
0 1000
( μ T)

(nT)
20000 40000 60000
-70000 70000 0
a)
b)
c)
Figure 1.1: Earth's magnetic eld as given by the CHAOS model [60] in 2005.
Panel a shows the surface eld intensity, panel b is the radial eld at the surface,
and panel c is the radial eld at the top of the core.
16
1590
1690
1790
1890
1990
(10
5
nT)
-10 -7.5 -5 -2.5 0 2.5 5 7.5 10
Figure 1.2: Radial eld, Bz at core mantle boundary based on GUFM1 historical
model from 1590-1990.
17
Southern hemisphere
Southern hemisphere
Northern hemisphere
Northern hemisphere
[10
22
Am
2
]
[10
22
Am
2
]
7.8
8.2
8.6
9
9.4
1600 1700 1800 1900 1990
Dipole Moment
(x 10
22 2
)
Year
Am
Figure 1.3: Top panel shows the total dipole moment from 1590-1990 based on
GUFM1 model. Intensity data was not available starting in 1832 (dashed blue
line). The bottom panels show the normal and reversed
ux in the northern and
southern hemisphere.
18
180˚
210˚
240˚
270˚
300˚
330˚
0˚
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60˚
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-
-4
-250
-200
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-50
180˚
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0˚
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0˚
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-
1700
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330˚
0˚
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0˚
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1800
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
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120˚
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300˚
330˚
0˚
30˚
60˚
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120˚
150˚
1900
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
1990
1600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
μΤ
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
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120˚
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330˚
0˚
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330˚
0˚
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210˚
240˚
270˚
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330˚
0˚
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0˚
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330˚
0˚
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60˚
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120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
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180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-
-
-1
1600
1700
1800
1900
1990
Figure 1.4: Contour plots of radial eld at core-mantle boundary in a polar projection based on GUFM1 historic eld
model.The left two panels are centered on the north geographic pole and the right two columns are centered on the
south pole.
19
Longitude (degrees east)
Time (years)
−150 −120 −90 −60 −30 0 30 60 90 120 150
1650
1700
1750
1800
1850
1900
1950
−10
−8
−6
−4
−2
0
2
4
6
8
10
Longitude (degrees east)
−150 −12 0 −9 0 −6 0 −30 0 30 60 90 120 150
1650
1700
1750
1800
1850
1900
1950
−10
−8
−6
−4
−2
0
2
4
6
8
10
Longitude (degrees east)
−150−120 −9 0 −6 0 −3 0 0 30 60 90 120 150
1650
1700
1750
1800
1850
1900
1950
−10
−8
−6
−4
−2
0
2
4
6
8
10
Longitude (degrees east)
−150 −12 0 −9 0 −6 0 −30 0 30 6 0 9 0 120 150
1650
1700
1750
1800
1850
1900
1950
−10
−8
−6
−4
−2
0
2
4
6
8
10
Longitude (degrees east)
−150 −120 −90 −60 −30 0 30 60 90 120 150
1650
1700
1750
1800
1850
1900
1950
−10
−8
−6
−4
−2
0
2
4
6
8
10
Longitude (degrees east)
−15 0 −120 −9 0 −6 0 −3 0 0 30 6 0 90 120 150
1650
1700
1750
1800
1850
1900
1950
−10
−8
−6
−4
−2
0
2
4
6
8
10
B
r
(10
4
nT)
Time (years)
B
r
(10
4
nT)
B
r
(10
4
nT) B
r
(10
4
nT)
B
r
(10
4
nT) B
r
(10
4
nT)
Time (years)
Time (years) Time (years)
Time (years)
~ ~
~ ~
~ ~
60
º
N
40
º
N
20
º
N equator
20
º
S 60
º
S
Figure 1.5: Time-longitude plot of the vertical magnetic eld based on GUFM1
with time-averaged axisymmetric eld removed and high-pass lter with 400 year
cuto period. Data are shown for 20

, 40

, 60

N and S and at the equator.
20
Figure 1.6: Snapshots of the radial component of the magnetic eld at the core-
mantle boundary based on CALS7K.2 paleomagnetic eld model.
21
preparation, 2005) and will be discussed in more
detail there.
7. Conclusions
[45] We have compiled a data set of both direc-
tional and intensity paleomagnetic and archeomag-
netic data from the past 7000 years, consisting of
16,085 values of inclination, 13,080 values of
declination, and 3188 values of intensity, respec-
tively. The data were compiled from existing data-
bases and original literature. Data series have not
been smoothed any more than the final results
given in the original literature. All radiocarbon
ages have been calibrated in a consistent way with
the CALIB program by Stuiver and Reimer [1993],
version 4.3. Ages calibrated by older methods have
been recalibrated wherever original
14
C ages were
available. To maintain independent age scales,
adjustments to timescales based on paleomagnetic
comparisons have been avoided. Special thought
has been given to data uncertainty estimates. Data
and dating uncertainties given in the data sources
were considered and adopted or new estimates
assigned in an attempt to obtain consistent uncer-
tainty estimates for the whole data set.
[46] Data distribution is significantly inhomoge-
neous in time and space, but seems to be sufficient
forglobalmodelingattempts.Inasubsequentpaper
we will use this data set for improving CALS3K.1
anddevelopingaCALS7Kmodelforthepast7000
years [see Korte and Constable, 2005]. As we
might have missed suitable data for this work and
new data might already have become available, we
encourage the reader to let us know, so that we can
updatethedatacompilationandfurtherimproveour
global millennial scale models in the future.
Acknowledgments
[47] We wish to thank all the colleagues who collected
paleomagnetic and archeomagnetic samples/cores, carried
Figure 12. Archeomagnetic intensity data in mT. Red line is model prediction from CALS3K.1 [Korte and
Constable, 2003]; short blue line is from GUFM (1590–1990 [Jackson et al.,2000]).
Geochemistry
Geophysics
Geosystems
G
3
G
3
korte et al.: geomagnetic field models, 1 10.1029/2004GC000800
25 of 32
Figure 1.7: Global distribution of paleomagnetic records used in CALS7K.2.
22
McElhinny and Senanayake
Yang et al.
CALS7K.2
CALS7K.2 V ADMS
Valet, 2008
Figure 1.8: Left panel shows the CALS7K.2 dipole moment (solid line), CALS7k.2 VADMs (circles) and VADMS
from earlier studies, McElhinny and Senanayake (1982) (diamonds) and Yang (2000) (triangles). Right panel shows a
comparison of CALS7K.2 dipole moment with more recent studies of Valet and Hongre. 23
Declination
Inclination
Lake Biwa, Japan
Lake Baikal, Russia
Lake Victoria, Uganda
Vatndalsvatn, Iceland
Figure 1.9: Figure shows a comparison of some individula data records and the model.
24
0.001
0.01
0.1
1
0.01 0.1 1
Normalized Spectral Power
Frequency (cycles per 1000 yrs)
unit-vector
paleointensity
Stacked spectra from three
sediment PSV records from the
western North Atlantic Ocean:
1) JPC-14 (13-65,000 yrs BP)
2) CH88-10P (12-71,000 yrs BP)
3) CH89-9P (12-60,000 yrs BP)

Figure 1.10: Spectral analysis of Pleistocene eld variability for the last 70,000
years in the Western North Atlantic Ocean using a multi-taper method. The (unit-
vector) directional and paleointensity variability are shown separately. Arrows
indicate corner frequencies in their spectra.
25
Chapter 2
PSVMOD2.0 Data and Regional
Studies
2.1 PSVMOD2.0 Data
We are now developing PSVMOD2.0, a family of Holocene PSV time series data
in 100-year intervals for the past 8,000 years. The PSVMOD2.0 database currently
consists of data from 82 globally distributed distinct sites. It is comprised of 74
inclination, 67 declination and 44 paleointensity time-series records, for a total of
185 records. Figure 2.1 is a map showing the locations of the global distribution
of sites. Blue triangles denote directional records and red circles indicate sites
where intensity records were obtained. The seven rectangles indicate the areas
within which regional comparisons of records were made. Tables A.1 through A.4
in the appendix, lists each of the PSVMOD2.0 sites and their location, latitude
and longitude by region. All longitudes are east longitude and southern latitudes
are denoted by an S.
The advantage of time series data is that they provide an independent assess-
ment of data quality between successive epochs and between spatially neighboring
records. Error estimates are less certain when using isolated PSV data as opposed
to spatially adjacent records. The regional correlation of PSV records permits an
independent assessment of chronology as well. Regional comparisons for all seven
26
areas have been made in developing PSVMOD2.0 however the most fully devel-
oped is for North America described in section 2.2. A detailed map of the sites in
the North America regional study is shown in gure 2.2 and detailed maps of the
other regions are shown in appendix C.
2.2 Regional comparison - Non-Arctic North
America
One approach to improving our understanding of PSV, especially the relation-
ship between intensity and directional variability, is to compare PSV on a regional
versus global scale and look for persistent patterns of directional and intensity
variability. Lund [44] applied the regional approach to directional PSV data from
North America and our regional study of North America relies heavily on that
previous analysis. Lund [44] compared 9 unit-vector records (inclination, declina-
tion, but not intensity) from a portion of continental North America (35

- 48

N,
241

- 286

E; a 13

x 45

region) to characterize the regional pattern of Holocene
PSV. Compared to that earlier study we have included a larger number of PSV
records that span a greater portion of North America (17

- 51

N, 205

- 295

E; a 34

x 90

region). We have updated that earlier analysis with 21 directional
PSV records, of which 12 have associated paleointensity records. In this regional
study we limit our analysis here to sites below51

N latitude. A map of the sites
are shown in gure 2.2. Blue triangles denote directional records and red circles
indicate sites where intensity records were obtained. This expanded data set with
full-vector information provides an important new view of geodynamo variability
and energetics during the Holocene for the North America region.
27
2.2.1 Methodology
Details of the North America PSV records including the locations and site
codes are summarized in the appendix B. All of the PSV records, which we
consider, have been carefully developed and dated by the original researchers. Data
measurement/sampling errors have been discussed by the original researchers in all
cases. All radiocarbon dates and chronologies have been corrected using CALIB6.0
[72] and all plotted ages are in Years AD/BC.
Our initial assumption is that there are no systematic errors in the PSV data or
the assigned chronologies. However, meaningful PSV analysis on a regional scale
requires analysis of spatially related records for which data quality, reproducibility,
and chronology have been assessed. We rst correlate individual features of incli-
nation (high/low extremes), declination (east/west extremes), and paleointensity
(high/low extremes) among individual PSV records to test whether the eld vari-
ability is reproducible among neighboring sites and to assess the extent to which
correlatable features can be seen over the region. This comparison gives us an esti-
mate of the quality of individual records, the spatial extent over which correlation
is possible, and identies records or intervals within records, which are anomalous
relative to other records, presumably due to sampling/measurement errors.
Individual PSV features are next tabulated, based on the ages assigned by the
original researchers, to estimate the average feature ages and test for systematic
age dierences between records. Age dierences may be due to non-synchronous
eld variability, such as westward or eastward drift of PSV over the region or due
to systematic errors in dating associated with individual records. In selected cases,
we have identied intervals within individual records where the PSV feature ages
are anomalous by comparison to the other records. We presume the systematic
age osets are due to problems in chronology. In these cases, we have revised
28
the chronologies of individual records by comparison to other neighboring records.
All of the records should then have the comparable data quality and time control
necessary to properly assess the regional pattern of PSV during the Holocene.
Four of the records (HAW, MAM, PSVL, WUS) were derived by stacking indi-
vidual, absolutely-dated paleomagnetic measurements from archeological materi-
als or lava
ows in a small region and normalizing the results to a standard site.
The remaining 17 records were derived from paleomagnetic studies of dated lake
or marine sediments. As previously mentioned, the archeological/lava-
ow PSV
records are considered to be more accurate recorders of eld variability than are the
sediment records; they are also expected to have smaller sampling/measurement
errors than the sediments, under normal study conditions.
Inclination and declination features from the individual PSV records have been
compared and a number of reproducible features have been identied, which can be
correlated over the entire region. Selected PSV records and their labeled features
are shown in gures 2.3 and 2.4. 18 inclination and 18 declination features have
been identied. Several of the features were not identied in the original study by
Lund [44] but some inclination (,
,,) and declination (,,) features have
been identied and used previously for paleomagnetic correlations and dating of
sites in the western USA [40] [5].
12 paleointensity time series have been developed for the North America region.
3 of the time series were derived from absolute paleointensity measurements using
Thellier-type intensity reconstructions [76] and 9 were derived from normaliza-
tion of sediment NRMs by other rock magnetic variables to reconstruct relative
paleointensity records [39][74].
29
Paleointensity features from the individual PSV records have been compared
and a number of reproducible features have been identied, which can be gener-
ally correlated over the entire region. Selected paleointensity records and their
labeled features are shown in gure 2.5. We have identied 11 correlatable pale-
ointensity features. In section 2.2.5 we describe in detail the procedure that uses
the 3 absolute paleointensity records to renormalize the other 9 sediment relative
paleointensity records to units of Virtual Axial Dipole Moment (VADM).
2.2.2 Assessment of PSV Record Chronologies
All of the feature ages identied in the records were averaged to form composite
averages. The inclination, declination, and paleointensity feature ages for each
record were then plotted against the composite averages. Select records are shown
in gure 2.6. We expect there to be a 1:1 correspondence between individual
record feature ages and the composite averages, within the limits of feature age
uncertainties, if there are no signicant, systematic dating errors in the records.
That is the case for sixteen of the PSV records, for example, LSC plotted in
the upper left panel of gure 2.6. The individual inclination, declination, and
paleointensity features for LSC are always within1 of the average ages for the
same features. It is important to note, that we presume (and nd to be true) that
all correlatable inclination, declination, and paleointensity features in one record
have the same age pattern when correlated to any other record in the region.
Five records, however, have intervals where groups of feature ages are signi-
cantly dierent from the composite averages. These records are MAR, BLUI, FIS,
KLM, and SEN. The individual feature ages for these ve records are plotted
against composite averages in gure 2.6. The possible reasons for age dierences
in individual records are either dating errors within the record or local/regional
30
dierences in the timing of real eld variability. The latter could be caused by
drift of the regional geomagnetic eld (eastward/westward, northward/southward)
or by some more general localized anomaly in eld variability. We can assess which
of these two possibilities is likely by comparing the PSV feature ages in each of
the anomalous records with the three or four closest records (sites within a few
hundred km). We assume that all sites less than a few hundred km apart must
have the same record of PSV variability and all PSV features must be synchronous
in age.
KLM was compared with nearby sites (see gure 2.2) LSC, ELK, and PEP to
assess its age dierences. LSC, ELK, and PEP agree with one another and with
the composite averages to within the uncertainty of each feature. Only KLM is in
disagreement. This was noted in the original publication of the KLM results by
Lund and Banerjee [48]. We have corrected the KLM chronology by altering two
linear trends in the relative chronology as noted in the bottom left panel of gure
2.6. This brings back the KLM feature ages to agreement with the composite
averages. SEN (bottom right panel of gure 2.6) was compared to LEB and SAN,
but only SAN overlaps SEN in the interval of SEN anomalous feature ages. SAN
has feature ages consistent with the composite averages (as do 2220 and ECS
further to the east). Therefore, we presume SEN is in error in this time interval
and several linear trends in the relative chronology, as noted in gure 2.6, were
altered to bring the SEN feature ages into agreement with the composite averages.
MAR, BLUI, and FIS are all located in the Pacic-Northwest and all have
intervals of anomalous feature ages. MAR and BLUI have feature ages that are
anomalously young, while FIS has feature ages that are anomalously old. At least
one of these has to be wrong. SAAN, PSVL, and PYR are the three closest
normal records to MAR and BLUI in the west and records farther to the east are
31
mostly normal as well. Given the normal nature of the surrounding records, we
can think of no geomagnetically reasonable way that the MAR and BLUI ages
can be correct. We have therefore corrected the MAR and BLUI chronologies by
altering linear trend intervals, as noted in gures 2.6. FIS is an easier case, in that
its anomalous feature ages are not consistent with any surrounding records and
three of the closest records (SAAN, PSVL, and PYR) are all normal. Therefore,
we have corrected the FIS chronology by altering linear trends as noted in gure
2.6. In the cases of KLM, SEN, MAR, BLUI, and FIS, the feature ages in the
corrected intervals were not used for the nal regional comparison of feature ages
described below.
2.2.3 Regional Comparison of Feature Ages
We next re-averaged all of the individual feature ages after removing the feature
ages in anomalous intervals from the ve records discussed in section 2.2.2. Feature
averages typically had 1 uncertainties of less than 100 years for the last 4000
years and less than 200 years for the older interval. Our rst hypothesis is that
feature ages are not signicantly dierent within our 34

x 90

region, and that
Holocene PSV in the North America region is synchronous without any signicant
evidence for drift (eastward/westward or northward/southward). This assumption
is for the individual correlatable features and must acknowledge that there is some
evidence for slow (millennial-scale) trends in the directional data that dier across
the North America region. For example, there is a notable increasing trend to
inclinations over the last800 years in the east (LSC, SAN, ECS in gure 2.3)
while inclinations in the west show less to no trend (HAW, WAI, PSVL, FIS).
Lund (1996) compared feature ages in a 13

x 45

region by grouping ages in
two ways: sites in the west, central or east, and sites in the north versus south.
32
He saw evidence that sites in the north and central (which contains the most
northerly sites) have younger feature ages than equivalent feature ages at sites to
the south (and both west and east). He interpreted this to indicate a preference for
northward drift between1000 AD and 4000 BC. He found no signicant evidence
for westward or eastward drift.
This analysis has been repeated and is shown in gure 2.7. All individual
feature ages for inclination, declination, and paleointensity are plotted as a function
of east longitude (gure 2.7, top panels) and north latitude (bottom panels). The
dashed lines represent average ages for each feature. Features that are signicantly
dierent from average ages (1) are highlighted in grey. The 1 variation for each
of these features alone, however are larger than the 1 variation for the feature
averages. The most notable example is the HAW paleointensity record (see also
2.5). The HAW intensity features between 0 AD and 2500 BC are all anomalously
young although it should be noted that the data from which the features were
picked are extremely sparse.
Overall, there is no signicant evidence for either westward/eastward or north-
ward/southward drift in the North America region. It is possible that Lund [44]
saw evidence for northward drift in his study because no correction was made to
the chronologies of KLM, SEN, FIS, and BLUI in that study. Those anomalous
chronologies were identied and not used in our analysis. This evidence against any
persistent westward drift of the Holocene geomagnetic eld in the North American
region is consistent with the previous assessment by Lund [47] but directly contra-
dicts the analysis of Yukutake [80], McFadden et al. [50] and Merrill, McElhinny,
and McFadden (gure 4.1 [54]) based on correlating late Holocene paleomagnetic
inclination or declination features alone.
33
2.2.4 Waveform Analysis
Individual PSV features document temporal variability in the geomagnetic-eld
source dynamo process. The regional correlation/synchrony of these individual
features indicates that the dynamo process is coherent over the spatial scale of our
North America region. The fact that the same correlations work for inclination,
declination, and paleointensity features indicates that the regional dynamo process
creates a total vector pattern of variability that is synchronous within the region.
We think that this is characteristic of the geomagnetic eld and that any behavior
of individual scalar PSV features (anywhere in the World) must have comparable
space/time behavior in the other PSV features.
Lund [44] compared the timing of directional PSV features and noted that the
inclination and declination features were largely out of phase, often 90

out of
phase. This out-of-phase relationship produces large open loops in the PSV direc-
tional variability [67][68], often termed circularity [47]. He searched for looping in
the North America region by producing Bauer plots [4] of inclination versus decli-
nation and looking for loops with at least 180

of arc and>4

of radial curvature.
Figure 2.8 is an example from that study.
In this study the preference for an out-of phase relationship between inclina-
tion and declination is illustrated in gure 2.9. The average ages of all declina-
tion (horizontal axis) and inclination (vertical axis) features are identied by lines
extending to the central 1:1 trending line. Large-amplitude features are illustrated
by solid lines; more minor, lower-amplitude features are illustrated by dashed lines.
Wherever solid lines intersect the trend line, an open circle indicates no signicant
overlap in timing between inclination and declination features while a closed cir-
cle indicates synchronous inclination and declination features (to within our limit
of resolution). Only 4 out of 18 times are inclination and declination features
34
synchronous (in phase). All other times, the individual inclination or declination
features are oset (out-of-phase) with respect to one another. This oset charac-
teristic of PSV features naturally leads to open looping of the geomagnetic eld.
Lund [44] identied 6
1
=2 large open loops in directional motion over the last
10,000 years; most were clockwise in their circularity but two were counterclockwise
([44], gure 11). His oldest loop is older than our study time interval, but we have
labeled his youngest 5
1
=2 loops A-F (A being oldest) and searched for these patterns
in our more extended region.
We have grouped our sites into 6 sub-regions, within which we expect circularity
to be the same among all sites. The sub-regions are: 1) St.Lawrence Seaway (2220,
ECS) 2) Pennsylvania/New York (SAN, LEB, SEN) 3) Minnesota (ELK, KLM,
LSC, PEP) 4) Western USA (SAAN, MAR, BLUI, PSVL, FIS, PYR, OWL, ZAC,
WUS) 5) Mesoamerica (MAM only) and 6) Hawaii (HAW, WAI).
Four out of the six sub-regions that are in the northernmost tier (regions 1
- 4, above) all show evidence for an early large clockwise loop, A (5500-7000
BC) followed by a counter-clockwise loop, B (4000-5500 BC). These are followed
by two clockwise loops, C and D, (200 AD-4000 BC). Loop D is followed by a
counter-clockwise loop, E (1500 AD-0 AD). The pattern of the last500 years
is less clear. All sites show a trend in directions associated with a large movement
of the North magnetic pole into northern Canada.
The eastern sites show a half loop with clockwise circularity, F, but the western
sites show no strong sense of circularity. Loops A, B, and C are all1500 years
in duration, but loop D is closer to2400 years in duration. Loops E and F are
again1500 years in duration (for one full loop). There is some limited evidence
for clockwise loop D in the Mesoamerican sub-region, but the MAM data set only
35
extends back2000 years. MAM shows the same pattern in loops E and F seen
at the more northerly sites.
The Hawaii sub-region is signicantly dierent. HAW only goes back3000
years, so WAI presents the only evidence for loops A, B, and C. WAI shows clear
evidence of open looping with loops1000 -1500 years in duration, but the time
of loop B is clockwise, not counterclockwise. There is a large loop between 0 AD
and 2000 BC (time of loop D), but WAI shows clockwise looping while HAW has
counterclockwise looping. At this time, we suspect that the Hawaii sub-region is
behaving dierently from the continental North America sub-regions in circularity,
but more records are needed to better assess these observations. If this is true,
then the Hawaii sub-region may not be part of the North America region for PSV
variability even though we appear to be able to correlate individual PSV features
into that sub-region.
Three aspects of circularity are worth noting. First, the individual open loops,
whether clockwise or counterclockwise, are all associated with several individual
PSV features. This is noted in gure 2.9 where the intervals of individual loops
are identied (A-F) on the 1:1 trend line (white for clockwise circularity and grey
for counterclockwise circularity); each complete loop has at least 5 individual PSV
features associated with it. If we consider individual scalar PSV features to be
simple waveforms, then looping must be considered a complex waveform associated
with a more complex pattern of dynamo source mechanism.
The second feature to note is that the two intervals of counterclockwise cir-
cularity, loops B and E (grey intervals on the 1:1 trend line in gure 2.9), are
associated with the only times of in-phase PSV feature variation (closed circles
in gure 2.9). This suggests that intervals of counterclockwise circularity have
36
some dierent pattern of dynamo activity associated with them than do intervals
of clockwise circularity.
The last feature to note is that the major loops are typically1500 years in
duration (although loop D is closer to2400 years). But, there are also smaller
loops,500 years in duration, that intermittently appear, as well. The complica-
tion is that these shorter duration loops also have smaller sizes and are often not
signicant, given the several degree uncertainty in individual PSV eld measure-
ments. Also, there are times where small loops in dierent records from the same
sub-region show dierent senses of circularity in the same time interval. We think
that the presence of smaller loops,500 years in duration, is real, but we have not
tried to quantify them in any ner detail.
Lund and Banerjee [48] noted that the LSC record appeared to have a par-
ticular type of complex waveform variability characterized by a 90

out-of phase
relationship with declinations slowly moving eastward to a maximum, followed by
fast switch to a westerly maximum coupled with an inclination high at the time of
declination switching. They identied ve of these waveforms at200 BP, 2400
BP, 4800 BP, 7000 BP, and 9500 BP (radiocarbon years) ([44], gure 12). These
appeared to recur every2400 years and be synchronous with all of the large
clockwise loops. We have labeled these ve waveforms (, ,
, and ). The
youngest four are noted in gure 2.9. We have also tentatively identied another
of these waveforms at1500 BC and labeled it by a ? in gure 2.9. With our
revised chronology, waveforms gaps between , ,
, and are not2400 years;
instead they vary from as little as 1600 years to as much as 3100 years. If waveform
? is included then the gaps between them become closer to1800 years (range of
1500-2300 years). Each of these complex waveforms occurs during clockwise cir-
cularity, but it is not certain that we can nd a particular loop to go with each of
37
them. Features , ?, and
are associated with an extended interval of clockwise
looping, but we have really only broken that region into 2 loops, C and D.
2.2.5 Paleointensity Analysis
Twelve of the records from North America region are full-vector records that
contain both directional and paleointensity information. Three of the paleoin-
tensity records were derived from absolute paleointensity measurements using
Thellier-type intensity reconstructions [76]. Another nine paleointensity records
were derived from normalization of sediment NRMs by other rock magnetic vari-
ables to reconstruct relative paleointensity records. Appendix B summarizes the
paleointensity records and key characteristics of the individual records. The
archeological/lava-
ow paleointensity records are considered to be more accurate
recorders of paleointensity variability than are the sediment records; they are also
expected to have smaller sampling/measurement errors than the sediments, under
normal study conditions.
Remanent magnetization carried by detrital grains is thought to be a linear
function of the ambient magnetic eld but is in
uenced by lithological factors,
grain size and concentration of magnetic material in the sediment and properties
of the non-magnetic matrix ([39], [29], [74], [75]). Typically calibration of sediment
intensity records has been done by using nearby paleointensity records or estimates
of global virtual axial dipole moment (VADM) ([29], [12], [64], [8]). An alternate
approach used by Korte and Constable [35] is to calibrate relative paleointensity
records using a global paleomagnetic eld model (CALS7K.2).
In this section we describe a new methodology for renormalizing sediment rela-
tive intensity values to absolute intensity values which we have applied to records
from North America. Paleointensity features from 12 individual PSV records have
38
been compared as part of our regional study and we have identied 11 repro-
ducible features which can be generally correlated over the entire region. Selected
paleointensity records and their labeled features are shown in gure 2.5. Our anal-
ysis of full-vector PSV (inclination, declination, paleointensity) indicates that all
correlatable features within the North America region are synchronous within an
uncertainty of 100 years. We use the paleointensity correlations to assess
the quality of the sediment relative paleointensity records relative to the absolute
paleointensity records and renormalize the sediment relative intensity values to
absolute values (virtual axial dipole moment, VADM).
The sediment relative paleointensity records and archeomagnetic/lava-
ow
absolute paleointensity records were compared by rst measuring the intensity
values for each correlatable feature. All sediment relative paleointensity records
were rst renormalized to a mean value of 1 and the amplitudes of numbered high-
and low-intensity features were tabulated. We did not try to rescale the variance
in each record. We next combined the two absolute paleointensity records from the
Western USA (PSVL, WUS, shown in gure 2.5), calculated their Virtual Axial
Dipole Moments (VADMs) from the local absolute paleointensity values, and tab-
ulated the amplitudes of their numbered high- and low-intensity features. We
then carried out a linear regression between the intensity values and VADMs of
correlatable features (gure 2.11), using two linear ts, one anchored to the origin
(solid lines) and one (unanchored) that
oated within the range of comparable
data (dashed lines).
Six of the sediment records (PEP, LEB, ECS, 2220, PYR, and LSC) had both
unanchored and anchored linear ts with correlation coecients greater than 0.9;
in each case the anchored linear ts were not signicantly dierent from the unan-
chored ts. This is the behavior expected for relative sediment paleointensity
39
records ([24], [28]); sediment remanence should increase linearly with applied eld
over the normal range of geomagnetic eld values. It is not clear, however, that any-
one has previously documented the linearity of relative paleointensity magnitude
versus applied eld (based on archeomagnetic data) before in such a quantitative
manner.
Two of the records (ELK, FIS) had similarly good correlation coecients for
the unanchored linear ts (ELK=0.95, FIS=0.93), but the anchored linear ts
were signicantly dierent from the unanchored ts and had signicantly poorer
correlation coecients. One record (SEN) had a lower unanchored correlation
coecient (SEN=0.78), and an anchored linear t that was signicantly dierent
and even lower in its correlation coecient. We interpret these dierences between
the unanchored and anchored linear ts to result from undiagnosed complications
to the original sediment paleointensity normalization.
It is likely that these three records had environmental changes occur to aect
the relative paleointensity variability. Two possibilities exist. First, changing envi-
ronmental conditions may have altered the relation between remanence acquired
versus the applied eld. Alternatively, there could have been a change in the rock
magnetic parameters used in the sediment normalization due to changing envi-
ronmental conditions. The second possibility has been considered regularly and
rules established to limit development of sediment relative paleointensity records
if rock magnetic variability is too large [29][74]. However, it is not obvious that
signicant environmental changes did occur in these three records to alter the pale-
ointensity acquisition/estimation process. In all three cases, the sediments appear
to be homogeneous with no large changes in apparent environmental conditions
during sediment deposition.
40
LSC, however, provides a means of further assessing this problem. LSC was
studied by Lund [43] and Lund and Banerjee [48]. The published records did
not include a relative paleointensity estimate because, even though the sediments
appeared to be homogeneous over the entire core length, the rock magnetic data
indicated a signicant, but slow variation in NRM and ARM coercivity due to a
subtle but signicant variation in mean magnetic grain size. The unpublished
NRM/ARM ratio for LSC is shown in gure 2.12, (open circles). Lund and
Schwartz [45] renormalized the LSC ARM values by removing a smooth (but non-
linear) trend in ARM coercivity; the renormalized ARM was then used to estimate
relative paleointensity (gure 2.12, closed circles). The linear regression for the
original NRM/ARM ratio for LSC is also shown in 2.11. It has a marked linear,
but unanchored, relationship versus the VADM values with a correlation coecient
of 0.85. We have used that unanchored linear t to renormalize the original LSC
paleointensity values; the result is shown in gure 2.12b (closed circles). We also
show in 2.12b (open circles) the renormalized LSC paleointensity values using the
results of Lund and Schwartz [45]. It is clear that normalizing the original LSC
paleointensity estimates by an unanchored linear t to absolute paleointensity data
produces a result that is not signicantly dierent from one where the original ARM
data were smoothly (and non-linearly) detrended prior to normalization. When
rock magnetic data can be used to improve paleointensity normalization, that is
preferred method for developing a more reliable relative paleointensity record. In
the absence of such rock magnetic data, we think that renormalization using the
unanchored linear t to improve the relative paleointensity record is a better esti-
mate (when the unanchored data have a good least-squares t to absolute data).
We have used this method to improve the paleointensity estimates for ELK, FIS,
and SEN.
41
After renormalizing all records to VADM, we carried out a cross-correlation
among all our paleointensity records. 10 of the 12 records have average cross-
correlations with all other records of 0.54-0.67. Two of the records (HAW, ECS)
have average cross-correlations less than 0.35. We presume that this is because
these two sites, which are at the extreme edges of our North America region,
are at the edge of (or beyond, in the case of HAW?) our region of strong PSV
coherence. We have averaged our 10 VADM paleointensity records with strong
cross-correlation and the results are shown in gure 2.10. The average cross corre-
lation between pairs of these 10 records is 0.65. We note a large-amplitude, long-
duration trend/cycle in the nal composite paleointensity record with an intensity
high2000 years ago and an intensity low7000 years ago. This pattern has been
noted before in the individual studies and is also seen on a global scale. We also
note5 millennial-scale peaks (and troughs) in intensity (features 1, 3, 5, 7, 9 in
gure 2.10 with highs at 1500 AD, 200 BC, 1800 BC, 3400 BC, and 6200 BC.
Renormalization of sediment relative paleointensity records to absolute pale-
ointensity values (usually VADM) is commonly done when new records are devel-
oped. The methods of renormalization normally presume some linear relationship
to another, independent record. The comparison records may be from the same
region (e.g., Europe [73], North America [8]) or from dierent parts of the World
(e.g., Antarctica [79]). The absolute records used for renormalization may be esti-
mates of global dipole moment [49] [59], [30]. One method has been to use the
spherical harmonic model CALS7K.2 from Korte and Constable [37], which can
predict the expected pattern of local eld variability from a global SHA model.
All of these comparisons use the absolute paleointensity estimates to linearly scale
42
the relative paleointensity estimates. This is graphically equivalent to the least-
squares t to intensity peaks and troughs we have used with curve-tting forced
to the origin (anchored).
This study diers in sometimes using least-squares tting that is not forced
to the origin (unanchored). This seems reasonable based on the analysis of LSC,
where we have shown that non-linear correction of the sediment relative paleoin-
tensity record, when rock magnetic complications are known, and linear correla-
tion with an anchored least-squares t produces an almost identical result using
the unanchored least-squares t to the raw data. This approach is corroborated
by the strong cross-correlations between anchored and unanchored paleointensity
records (0.64) in the North America region and among all the anchored records
alone (0.65). (This analysis leaves out the 3 records from the edges of our region,
HAW, ECS, 2220.)
To summarize, we have carried out a renormalization analysis of our 12 pale-
ointensity records in order to more quantitatively compare them and incorporate
them into a full-vector spherical harmonic analysis. We have used least-squares
tting of paleointensity features as a basis for renormalization to absolute values.
Two of the paleointensity records were combined (WUS, PSVL) and used to com-
pare against the 9 sediment relative paleointensity records. Least-squares tting
produced strong ts for 6 of the sediment records (PEP, LEB, ECS, 2220, PYR,
LSC). The anchored ts (forced to the origin) and unanchored ts (not forced
through the origin)were not signicantly dierent and averaged better than 0.9
This is the expected pattern of relative paleointensity variability - that it will scale
linearly with varying absolute geomagnetic eld intensity.
3 of the sediment relative paleointensity records, however, had good (0.9)
to decent(0.78) least-squares ts only for the unanchored tting. The anchored
43
least-squares ts were signicantly poorer. We interpret this to be due to undi-
agnosed rock magnetic complications in these records. Comparison with the LSC
record suggests that these unanchored ts are still useful in paleointensity com-
parisons.
Intensity eld variability was not available to Lund [44] in his PSV analysis
and its addition here provides us with the rst opportunity to compare directional
and paleointensity variability and assess PSV from a full vector perspective. Coun-
terclockwise circularity is associated with the two longest-duration intensity lows
at 1200 AD-0 AD and 4000 BC-5800 BC. All ve millennial-scale intensity highs
appear to be associated with clockwise circularity (although the intensity high at
1500 AD is close to the boundary between clockwise and counterclockwise circu-
larity). Moreover, the ve intensity highs are near (but not exactly in phase) with
the complex waveforms noted by Lund and Banerjee [48] (waveforms , , ?,
,
and in gures 2.9 and 2.10). All of these observations point to intensity variabil-
ity occurring in sync with directional variability in the North America region. It
does not appear, however, that the specic intensity/directional pattern of vari-
ability is simple. Simple models that have been applied before to explain complex
waveforms, such as radial-dipole modeling or dynamo waves ([48], [62], [21]), do
not appear to exactly t the timing of intensity variability associated with these
waveforms.
2.2.6 Summary of North America Regional Study
This regional analysis of PSV in North America signicantly improves the ear-
lier study of Lund [44] by spanning a greater spatial extent and including a larger
number of PSV records, several that are full-vector (inclination, declination, and
paleointensity). The dominant pattern of PSV, characterized by looping implies
44
a spatial region of coherence of at least4000 km with a temporal coherence on
the order of 1000 years. This analysis of Holocene paleomagnetic secular variation
(PSV) records from North America is able to improve our understanding of the
dynamical character of PSV on a regional scale.
21 directional records and 12 paleointensity records (from the same sites) have
been included to provide a full-vector PSV analysis. 18 inclination, 18 declination,
and 11 paleointensity features that are correlatable over the entire North America
region have been identied. The original chronologies used to date individual PSV
features indicate that for 16 out of the 21 sites PSV features are synchronous (to
within100 years) over the entire region. At ve sites, we noted that selected
PSV features dier in age from neighboring sites, and determined that this was
due to dating problems in those 5 records. We corrected those age variances for
our subsequent analysis.
We have tested the idea that individual PSV features may dier in age due
to regional PSV (drift either E/W or N/S) using only the ages of PSV features
from 16 sites where we did not alter chronologies. We see no evidence for persistent
long-term drift (either W/E or N/S) in PSV within the North America region. The
fact that I, D, and paleointensity features all share the same pattern and timing of
variability indicates that the dynamo process which generates the PSV variability
is indeed a full-vector process and coherent on the spatial scale of North America.
This also suggests that PSV variability anywhere in the World must have the
same full-vector variability on a regional scale, with inclination declination, and
paleointensity features all covarying together synchronously.
Lund [44] noted two types of more complex waveforms (given that individual
PSV features are simple waveforms) that are prevalent in the North America PSV
45
- millennial-scale open looping (circularity) of the geomagnetic eld and individ-
ual, distinctive 90 out-of-phase oscillations in I and D. We see clear evidence for
both patterns of waveform variability. North America has been dominated by
open-looping (generally out-of-phase relationship between I and D) with clockwise
circularity for most of the last 8000 years, but there have been two distinct inter-
vals of counterclockwise circularity at 1500 AD-0 AD and 4000 BC-5500 BC. We
see 5
1
=2 loops (A-F) in the last 8000 years, A, C, D, and F are clockwise while
B and D are counterclockwise. The loops normally last1500 years, but loop D
lasts2400 years. We see evidence for smaller, shorter-duration (500 yr) loops,
as well, but we have made no attempt to more closely analyze them. It is impor-
tant to note that the Hawaii records (HAW, WAI) also have millennial-scale open
looping, but the sense of looping is opposite that of the other records. This may
indicate that these records are not part of the North America region in PSV, even
though we can correlate PSV features in these records.
The 90

out-of-phase I and D oscillations, noted by Lund and Banerjee [48]
and identied in Lund [44], are present in all the North America records (except
HAW/WAI?). They noted four distinct oscillations in the last 8000 years, but
we nd evidence for an additional one. The timing between oscillations noted
previously (2400 radiocarbon years) is dierent in the current analysis (1800
year oscillation pattern) due to the use of calibrated age dating.
The composite paleointensity record developed for this study of North America
has two distinctive scales of variability: 1) a long trend/cycle with a broad pale-
ointensity high near 300 BC and a broad paleointensity low near 5000 BC and 2)
a millennial-scale pattern of 5 paleointensity cycles with highs at 1500 AD, 200
BC, 1800 BC, 3400 BC, and 6200 BC. The two intervals of counterclockwise cir-
cularity are associated with the two longest intervals of low paleointensity, while
46
the intervals of clockwise circularity and 5 I/D oscillations are associated with the
5 paleointensity highs. These observations all suggest that the direction of looping
and intensity amplitude dene two dierent regimes of dynamo activity within the
Holocene North America region, counterclockwise circularity and paleointensity
lows versus clockwise circularity, paleointensity highs, and I/D oscillations.
47
-80˚ -80˚
-60˚ -60˚
-40˚ -40˚
-20˚ -20˚
0˚ 0˚
20˚ 20˚
40˚ 40˚
60˚ 60˚
80˚ 80˚
North America
Arctic
Europe/West Asia
Africa
Southwest Pacic
East Asia
South America/Antarctica
7/25/13
directional records
intensity records
Figure 2.1: Global map of PSVMOD2.0 sites. Blue triangles denote sites with directional records and red circles denote
intensity records. 7 areas in which regional comparison were made are indicated.
48
200˚
200˚
210˚
210˚
220˚
220˚
230˚
230˚
240˚
240˚
250˚
250˚
260˚
260˚
270˚
270˚
280˚
280˚
290˚
290˚
300˚
300˚
310˚
310˚
320˚
320˚
20˚ 20˚
30˚ 30˚
40˚ 40˚
50˚ 50˚
60˚ 60˚
BLUI
ECS
ELK
FIS
HAW
LSC
MAM
MAR
OWL
PEP
PSVL
PYR
SAAN
SAN
SEN
WAI
WUS
ZAC
LEB
2220
KLM
Figure 2.2: PSVMOD2.0 North American sites. Blue triangle denotes sites with directional records and red circle
denotes intensity records.
49
2000
0 -2000 -4000 -6000
Years (AD/BC)
SAN LSC PSVL
HAW
WAI ECS FIS
Figure 2
α
β
δ
1
2
3
4
5
6
7
8
9
10
11
12
α
1
2
3
4
5
6
7
8
9
10
11
12
6’ 6’
9’
9’’
9’
9’’
60° 80° 60° 80° 40°
α
β
δ
1
2
3
5
6
7
10
11
γ
γ
60° 80° 40°
α
δ
1
2
3
5
6
7
8
9
10
11
12
6’
9’
9’’
γ
20° 40°
α
β
δ
1
2
3
4
5
6
7
8
9
10
11
6’
9’
9’’
γ
60° 40°
α
β
2
3
4
5
6
7
8
9
10
11/12?
9’
9’’
γ
60° 70° 80°
Figure 2.3: Selected inclination records from the North America region using dating
of the original publications. 18 individual inclination features that are correctable
among all sites are noted in the records.
2000
0 -2000 -4000 -6000
Years (AD/BC)
LSC PSVL ECS FIS
Figure 3
SAN
WAI
HAW
α
β
δ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
α
β
δ
1
2
5
6
7
0° 20° 0° 40°
α
β
δ
1
5
6
7
8
9
10
11
12
13
14
0° 20°
α
β
δ
1
2
4
5
6
7
8
9
10
11
12
13
14
0° 20° 40° −20° −40° 0° 20° −20°
α
β
δ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0° 20°
δ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
5
40° −20°
Declination
Figure 2.4: Selected declination records from the North America region using
dating of the original publications. 18 individual declination features that are
correctable among all sites are noted in the records.
50
2000
0 -2000 -4000 -6000
Years (AD/BC)
1
2
3
4
5
6
7
8
8’
1
2
2’
3
4
5
6
1.0 1.4 0.6
LEB LSC
WUS
PSVL
HAW ECS FIS
2
2’
3
4
5
6
7
8
8’
9
1.0 1.2 0.8 1.0 1.2 1.4 0.8 0.5 1.4
1
2
2’
3
4
5
6
7
8
8’
9
1.0 1.5 2.0
1
2
3
4
5
6
7
8
8’
2’
Renormalized Relative Paleointensty
10 15 5 10 15
Virtual Axial Dipole Moment
1
2
2’
3
4/6
7
8
8’
9
Figure 4
Figure 2.5: Selected paleointensity records from the North America region using
dating of the original publications. 11 individual paleointensity features that are
correctable among all sites are noted in the records.
51
2000
0
-2000
-4000
-6000
2000 0 -2000 -4000 -6000 2000 0 -2000 -4000
2000
0
-2000
-4000
-6000
2000
0
-2000
-4000
-6000
Years (AD/BC) Years (AD/BC)
Years
(AD/BC)
Years
(AD/BC)
Years
(AD/BC)
LSC
Figure 5
MAR
FIS BLUI
KLM SEN
corrected
interval
corrected
interval
corrected
interval
corrected
interval
corrected
interval
Figure 2.6: Plots of inclination, declination and paleointensity feature ages for 6
records plotted versus average feature ages. 1:1 trend lines are indicated by dashed
lines.
52
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
15 20 25 30 35 40 45 50
NorthAm-Paleoint-transpose
YEARS (AD/BC)
NORTH LATITUDE (°)
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
15 20 25 30 35 40 45 50 55
NorthAm-DEC-transpose
YEARS (AD/BC)
NORTH LATITUDE (°)
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
200 220 240 260 280 300
NorthAm-Paleoint-transpose
AGE (AD/BC)
EAST LONGITUDE (°)
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
200 220 240 260 280 300
NorthAm-DEC-transpose
YEARS (AD/BC)
EAST LONGITUDE (°)
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
200 220 240 260 280 300
NorthAm-INC-transpose
YEARS (AD/BC)
EAST LONGITUDE (°)
Inclination Declination Paleointensity
200° 220° 240° 260° 280° 300° 200° 220° 240° 260° 280° 200° 220° 240° 260° 280°
East Longitude
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
15 20 25 30 35 40 45 50 55
NorthAm-INC-transpose
YEARS (AD/BC)
NORTH LATITUDE (°)
East Longitude East Longitude
20° 30° 40° 50° 20° 30° 40° 50° 20° 30° 40° 50°
North Latitude North Latitude North Latitude
2000
0
-2000
-4000
-6000
2000
0
-2000
-4000
-6000
Years
(AD/BC)
Years
(AD/BC)
Figure 6
Figure 2.7: The ages of all PSV features are plotted at top by longitude and at the bottom by latitude. The dashed
lines represent the average age for each feature. Mostly the individual feature ages match the feature average age and
suggest no signicant age oset for feature as a function of latitude or longitude. Grey zones indicate intervals where
individual ages are dierent from their expected average.
53
8020 LUND' HOLOCENE PALEOMAGNETIC SECULAR VARIATION FROM NORTH AMERICA
45 ø FIS
,
fiS o
Barraclough, 1982] suggest that these three waveforms (North
America, Europe, East Asia) are regionally coherent, but
independent of one another. That is, there is no evidence that
one evolved from another by simple westward or eastward drift
[Lund, 1989b]. It is more likely that this waveform pattern
results from a distinctive pattern of outer core fluid flow as part
of the dynamo process and that this fluid flow pattern occurs
intermittently throughout the outer core producing similar PSV
waveforms around the world.
Discussion
Under the "frozen-flux" hypothesis [Roberts and Scott,
1965; Backus, 1968], a number of workers [e.g., Bullard et aL,
so ILSC /
olybp D * - u /
1
-
70ol I I I I I I / 60 ø
o
50øls I I I I I I 65 I I I I
AN
55ø I I I I I
-
ssø ELK
z
D 60-
ß <60 ø
Z --
16
D7 D6
D5 -'
65
70 [1-'i 19 I I I
_ 0 o _5 o
0 ø 5 ø 1 0 ø 1 5 ø 20 ø 25 ø 30 ø
z
o65
z
c70
z
75' --
I10
1
4620 ybp
I I I I I I
_6 ø _4 ø _2 ø 0 ø 2 ø 4 ø 6 ø
I
0 ø 5 ø 10 ø 15 ø 20 ø 25 ø
DECLINATION
Figure 8. Bauer plots of vector variation between 2000 and
6000 years B.P., a period dominated by clockwise circularity.
The sense of circularity associated with large-scale loops is
noted by the arrows. Individual features are labeled for clarity
or comparison. See text for more detailed discussion of the
circularity patterns.
CC1 (features I1-5 and D1-5 in Figure 5) has a distinctive
double easterly swing in declination, a marked low in
inclination, and a phase relationship associated with
counterclockwise looping. This pattern is present in all of the
North American PSV records, although it is not as clearly
observed in LEB due to higher noise levels in the data which
mask some of the waveform detail. A remarkably similar
waveform, but about 1000 years younger, is present in
European archeomagnetic records (Figure 13) and sediment
paleomagnetic records [Turner and Thompson, 1982]. A
similar waveform is also present in east Asian (Figure 13)
archeomagnetic records of the last 1000 years. Analysis of
intervening archeomagnetic records [Lund, 1989b] and
historic patterns of secular variation [e.g., Thompson and
SAN 11o
z 54øI -
I -
62 ø I / 6700 ybp -
66 ø OD12 -
5800 ybp
79100 _6 o _2o 2 ø 6 ø 10 ø
DECLINATION
Figure 9. Bauer plots of vector variation between 6000 and
7000 years B.P., a period dominated by counterclockwise
circularity. The sense of circularity associated with large-scale
loops is noted by the arrows. Individual features are labeled for
clarity or comparison. See text for more detailed discussion of
the circularity patterns.
Figure 2.8: Example of a Bauer plot of inclination versus declination (gure 8 from
the the study of Lund [44]). Figure shows the variation between 2000 and 6000
B.P. The sense of circularity determined in that study is indicated by the arrows.
54
-2000
-1500
-1000
-500
0
500
1000
1500
2000
-2000 -1500 -1000 -500 0 500 1000 1500 2000
8
Inclination Feature Age (Years AD/BC)
-6000
-5500
-5000
-4500
-4000
-3500
-3000
-2500
-2000
-6000 -5500 -5000 -4500 -4000 -3500 -3000 -2500 -2000
1 2 4 5 6 7
13 12 11
2
3
5
1
6
7
9
8
10
11
12
14
9’’
Declination Feature Age (years AD/BC)
Declination Feature Age (years AD/BC)
Inclination Feature Age (Years AD/BC)
Figure 7
F(c) E(cc) D(c)
C(c)
B(cc) A(c)
x
x
x
x
?
α
β
γ
δ
Figure 2.9: Plot of average inclination feature ages (horizontal lines) versus decli-
nation feature ages (vertical lines). The lines extend to the 1:1 trend line. Solid
lines (with numbered PSV features) are large amplitude features; dashed lines
are smaller amplitude features. Solid dots indicate large amplitude inclination
and declination features which meet on the 1:1 trend line indicating they are not
signicantly dierent in age, thus indicate in-phase PSV behavior. Open dots indi-
cate out-of-phase PSV behavior. Associated zones of clockwise circularity (open
boxed intervals) and counterlockwise circularity (grey intervals) and designated
loops (A-F) are noted on the trend line.
55
6
7
8
9
10
11
12
13
14
-6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000
Years (AD/BC)
VADM (x1022 A/m)
Figure 8
α β
γ
δ
A(c) B(cc) C(c) D(c) E(cc) F(c)
1
3
5
7
9
?
Figure 2.10: Composite paleointensity variability for the North America region for
the last 8000 years. 5 distinctive paleointensity highs are numbered. Variability in
looping from 2.9 is shown at the bottom for comparison.
56
0
0.5
1
1.5
0 5 10 15
0
0.5
1
1.5
2
0 5 10 15
0
0.5
1
1.5
0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15
0
0.5
1
1.5
0 5 10 15
0
0.5
1
1.5
0 5 10 15
LSC
renomalized
NRM/ARM
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15
LSC
original
NRM/ARM
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15
ELK
NRM/ARM
2220
NRM/IRM
PYR
NRM/CHI
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15
LEB
NRM/ARM
Relative Paelointensity Relative Paelointensity Relative Paelointensity Relative Paelointensity Relative Paelointensity
SEN
NRM/ARM
FIS
NRM/ARM
PEP
NRM/ARM
0
0.5
1
1.5
0 5 10 15
ECS
NRM/IRM
Absolute Paleointensity (VADM) Absolute Paleointensity (VADM)
Figure 3
Figure 2.11: Least squares ts between individual sediment relative paleointensity
records and the composite absolute paleointensity record of PSVL and WUS. Solid
(dashed) lines indicate anchored (unanchored) ts.
57
6
7
8
9
10
11
12
13
14
-8000 -6000 -4000 -2000 0 2000
VADM
Years AD/BC
0
0.5
1
1.5
2
-8000 -6000 -4000 -2000 0 2000
Relative Paleointensity
Years AD/BC
A
B
Figure 4 Figure 2.12: A) Open circles are the original sediment relative paleointensity record
from LSC [43]. Closed circles are the renormalized relative paleointensity record
of Lund and Schwartz [45] after correcting for a nonlinear rock magnetic trend.
B) Open circles are the renormalized relative paleointensity record from LSC and
closed circles are the renormalized LSC record using unanchored least-squares t-
ting.
58
Chapter 3
Modeling
In this chapter the mathematical methods we have developed to generate mod-
els of the paleomagnetic eld are described. The modeling procedure follows closely
that used to generate models of the historic magnetic eld, GUFM1 [25] and the
CALSxK.n [37] series of paleomagnetic models. The methods are outlined here
and are described in a number of references [26], [19]. First, in section 3.1 the
equations that describe Earth's magnetic eld are presented (the forward prob-
lem). In section 3.2 the methodology for formulating and solving the general
inverse problem is described as well as the specic methodology typically applied
to generate geomagnetic eld models. The problem we are solving is non-linear
because the relationship between the data (declination, inclination and intensity)
is non-linearly related to the model parameters (g
nm
;h
nm
). Section 3.3 describes
the basic methodology we use to solve this non-linear inverse problem.
3.1 Description of Earth's magnetic eld - the
forward problem
A description of Earth's magnetic eld is typically based on the standard spher-
ical harmonic formulation in which the measured magnetospheric magnetic eld
59
can be considered the sum of the internal planetary magnetic eld plus the con-
tribution by external sources. The potential due to these two sources is given
by:
V =a
1
X
n=1

a
r

n+1
T
n
i
+

r
a

n
T
n
e
T
n
i
=
n
X
m=0
P
nm
(cos ) [g
nm
cos(m) +h
nm
sin(m)]
T
n
e
=
n
X
m=0
P
nm
(cos ) [G
nm
cos(m) +H
nm
sin(m)] (3.1)
where a is the planetary radius,P
nm
(cos) are the Schmidt quasi-normalized asso-
ciated Legendre functions of degree n and order m traditionally used in magnetic
eld modeling. g
nm
and h
nm
are the internal spherical harmonic coecients and
G
nm
and H
nm
are the coecients associated with the external terms [26].
In the absence of local currents the magnetic eld can be expressed as the
gradient of a scalar potential B =rV and in a spherical polar coordinate system
(r, ;) the components of the magnetic eld due to both internal and external
sources becomes:
B
r
=
@V
@r
=
1
X
n=1
n
X
m=0
"
(n + 1)

a
r

(n+2)
[g
nm
cos(m) +h
nm
sin(m)]
n

r
a

(n1)
[G
nm
cos(m) +H
nm
sin(m)]
#
P
nm
(cos)
(3.2)
B

=
1
r
@V
@
=
1
X
n=1
n
X
m=0
"

a
r

(n+2)
[g
nm
cos(m) +h
nm
sin(m)]
+

r
a

(n1)
[G
nm
cos(m) +H
nm
sin(m)]
#
dP
nm
(cos)
d
(3.3)
60
B

=
1
r sin
@V
@
=
1
sin
1
X
n=1
n
X
m=0
(
m

a
r

(n+2)
[g
nm
sin(m)h
nm
cos(m)]
+

r
a

(n1)
[G
nm
cos(m) +H
nm
sin(m)]
)
P
nm
(cos)
(3.4)
The equations for the internal eld separately can be written:
B
r
=
@V
@r
=
1
X
n=1
n
X
m=0
"
(n + 1)

a
r

(n+2)
[g
nm
cos(m) +h
nm
sin(m)]
#
P
nm
(cos)
(3.5)
B

=
1
r
@V
@
=
1
X
n=1
n
X
m=0
"

a
r

(n+2)
[g
nm
cos(m) +h
nm
sin(m)]
#
dP
nm
(cos)
d
(3.6)
B

=
1
r sin
@V
@
=
1
sin
1
X
n=1
n
X
m=0
(
m

a
r

(n+2)
[g
nm
sin(m)h
nm
cos(m)]P
nm
(cos)
(3.7)
Models of the historical and paleomagnetic eld typically include only internal
sources. No attempt is made to estimate the eld due to external sources and from
here forward in the description of the modeling external terms will be dropped.
Equations 3.5 through 3.7 can be written in matrix form as:
61
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
B
r1
B
1
B
1
.
.
.
B
ri
B
i
B
i
.
.
.
B
rn
B
n
B
n
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
A
r11
A
r12
A
r1m
A
11
A
12
A
1m
A
11
A
12
A
1m
.
.
.
.
.
.
.
.
.
.
.
.
A
ri1
A
ri2
A
rim
A
i1
A
i2
A
im
A
i1
A
i2
A
im
.
.
.
.
.
.
.
.
.
.
.
.
A
rn1
A
rn2
A
rnm
A
n1
A
n2
A
nm
A
n1
A
n2
A
nm
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A

0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
g
1
g
2
g
n
h
1
.
.
.
h
m
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
(3.8)
where A
l;n;m
are the elements of an n x m matrix (n=number of data points,
m=number of model parameters) that describe how a given data point depends
on the model parameters. The index l refers to the r, , or component of the
eld. A is usually referred to as the design matrix. As an example, referring
back to equation 3.5, the following elements of the array represent the r, , and
components associated with the g
10
term:
A
1;1;1
= 2

a
r

3
P
10
= 2

a
r

3
cos
A
2;1;1
=

a
r

3
@P
10
@
=

a
r

3
sin
A
3;1;1
= 0 (3.9)
62
3.2 Solving the inverse problem
Deriving a global magnetic eld model from observations is a typical geophys-
ical inverse problem. Derivation of the standard least squares solution of a linear
problem can be found in a number of references [19] [26]. The equations describing
this and any general forward problem can be written as d = Am + e where d is
the data vector, A is the n x m design matrix, that depends on the physics of the
problem, m is a vector of model parameters and e is a vector of errors associated
with the measurements. The problem can be restated as:
d
i
=
m
X
j=1
A
ij
m
j
+e
i
(3.10)
which can be rearranged:
e
i
=d
i

m
X
j=1
A
ij
m
j
(3.11)
In the over-determined problem (more equations than unknowns) the least-squares
solution minimizes the sum of the square of the errors:
E
2
=
n
X
i=1
e
i
2
=
n
X
i=1
(d
i

m
X
j=1
A
ij
m
j
)
2
(3.12)
The model parameters are chosen to minimize E by setting
@E
2
@m
k
2
= 0 and with
some rearranging can be written in matrix form as:
A
T
Am =A
T
d (3.13)
63
These are called the normal equations and the matrix A
T
A is the generalized
least-squares inverse. The solution to 3.13 is given by:
m = (A
T
A)
1
A
T
d (3.14)
Typically the measurements are weighted according to their measurement errors.
If we assume that the errors are random with zero mean, the problem can be
written as:
(A
T
C
1
e
A)m =A
T
C
1
e
d (3.15)
where C
e
is the covariance matrix, which for independent random variables is
diagonal with variances on the diagonal.
And the solution can be written:
m = (A
T
C
1
e
A)
1
A
T
C
1
e
d (3.16)
It is well-established that the models are non-unique because of the spatial lim-
itations and errors on the data [63]. However it is possible to demonstrate the
existence of a solution by constructing a model that is compatible with the
data. Innitely many models may satisfy the paleomagnetic measurements. The
approach that has become standard in geomagnetic eld modeling is to choose a
model that has a minimum complexity while still being compatible with the data
[19]. This amounts to minimizing a combination of the error E and a solution N:
T () =E +
2
N (3.17)
where is the trade-o parameter that determines the relative importance of E
and N. It is squared to indicate that it is a positive value. Minimizing T is a
64
= infinity
θ
desired error
desired norm
knee
= 0
norm N
error E
θ
Figure 3.1: Ideal trade o curve showing the mist E versus the model norm N,
a measure of the model complexity. The knee of the curve represents a balance
between tting the model well and obtaining a simple model.
compromise between tting the data better, minimizing E and choosing a simple
model by minimizing N. If =0 the solution is the conventional solution of the
normal equations. A plot of E versus N for dierent values of is referred to as
the trade-o curve.
The minimization can be derived by dierentiation in a manner similar to the
derivation of 3.14 and the equations become:
(A
T
C
1
e
A +
2
W )m =A
T
C
1
e
d (3.18)
and the model parameters are given by:
m = (A
T
C
1
e
A +W )
1
A
T
C
1
e
d (3.19)
It is typical in global geomagnetic modeling to chose a norm based on a plau-
sible physical parameter. The approach, referred to as regularization [63] damps
65
small spatial scales from the model solution. Common choices of smoothing norm
minimize 1) the eld magnitude at the core-mantle boundary, (R
1n
, equation 3.20)
2) the root-mean-square value of the radial eld at the core-mantle boundary (R
2n
,
equation 3.21) or 3) the root-mean-square value of the spatial gradient ofB
r
(R
3n
,
equation 3.22 ).
R
1n
=

r
a
r
c

2(n+2)
(n + 1)
X
m
g
2
nm
+h
2
nm
(3.20)
R
2n
=

r
a
r
c

2(n+2)
(n + 1)
2
2n + 1
X
m
g
2
nm
+h
2
nm
(3.21)
R
3n
=

r
a
r
c

2(n+2)
n(n + 1)
3
2n + 1
X
m
g
2
nm
+h
2
nm
(3.22)
The severity of damping increases from R
1n
to R
3n
. All three smoothing
norms have been investigated during model development. With these denitions,
equation 3.19 can be rewritten as:
m = (A
T
C
1
e
A +R
T
R)
1
A
T
C
1
e
d (3.23)
where is the Lagrange multiplier. In practice the Lagrange multiplier is varied
until we obtain the desired t to the data. A desirable property of regularization
is that the model solution will typically converge before an arbitrary truncation
level is reached. While the data distribution does limit the resolution of small scale
structure, truncation leads to spatial aliasing eects, that is mapping power from
the higher order components into the dipole and quadrupole terms, thus aecting
the reliability of the low-order terms. Early global models based on paleomagnetic
66
data truncated the spherical harmonic series at an arbitrary level, based on the
premise that only low-degree terms could be adequately resolved. Constable et al.
[13] were the rst to expand the spherical harmonic series to higher degrees and
apply regularization techniques in this way. In our model development we choose
the maximum degree and order of the spherical harmonic expansion high enough
so that the roughness of the models is determined by the data and regularization,
not the truncation level. It does not mean however, that we can actually resolve
the lower degree and order coecients with condence.
3.3 Solving the non-linear problem
One further complication arises from the fact that the measured components
of the magnetic eld used as input to the model are declination, inclination and
intensity all of which are non-linearly related to the model parameters. If the
inverse problem was one of solving equations 3.5 through 3.7, the problem would
be a linear one in which the data (B
r
;B

;B

) are linearly related to the model
parameters (g
nm
;h
nm
).
Inclination, declination and intensity are given by the following relationships:
D =tan
1
(Y=X)D=2
I =tan
1
(Z=H)=2I=2
B = (X
2
+Y
2
+Z
2
)
1=2
(3.24)
with X (north), Y (east), and Z (down) in geographic coordinates (gure 3.2). For
67
D
I
X
Y
Z
Geographic
North Magnetic
North
B
H
Figure 3.2: Inclination and declination.
an assumed spherical earth, the relationship between geographic and geocentric
(spherical) coordinates is:
X =B

Y =B

Z =B
r
H = (X
2
+Y
2
)
1=2
(3.25)
H is the horizontal eld. Proper modeling of the geomagnetic eld takes account of
the dierence between geographic coordinates, the coordinate frame in which the
68
measurements are made and geocentric coordinates the reference frame in which
the equations are described. Equations 3.25 hold for a spherical Earth, however,
the Earth is best approximated as a spheroid, and the relation between X,Y,Z in
geographic coordinates and geocentric spherical coordinates is:
X =B

cos( )B
r
sin( ) (3.26)
Y =B

(3.27)
Z =B

sin( )B
r
cos( ) (3.28)
where
sin =sin sin cos cos (3.29)
and is the geographic or geodetic latitude minus the geocentric latitude. Details
of these calculations are given in [26].
Linearization is often used for solving nonlinear inverse problems. The mathe-
matics of the problem are linearized, a likely initial model is chosen and the mist is
calculated. A solution is obtained by proceeding iteratively from the initial model,
seeking a small change in the model that reduces the residuals. The new model is
used as the basis of another small improvement. The procedure is repeated until a
sucient number of models have been searched and a model is found that ts the
data. The general procedure which can be found in various references [19] is as
follows. The general relationship between the data and the model can be written:
d =F (m) +e (3.30)
69
where F denotes a nonlinear function or formula. We initiate the iteration from
some starting modelm
o
and seek a small improvementm
o
by linearized inversion
of the residuals;
d
o
=dF (m
o
) (3.31)
d
o
=A
o
m
o
+e (3.32)
where the derivative is evaluated at m = m
o
. The matrix A
o
is derived from the
nonlinear formula F(m) for each iteration and the zero subscript denotes the rst
iteration. The matrix A is referred to as the Jacobian and its elements are partial
derivatives of the data with respect to the model:
(A
o
)
ij
=
@F
i
@m
j
(3.33)
The linearized equations of condition are solved for m
o
in just the same way as
for the linear inverse problem. The improved solution becomes:
m
1
=m
o
+m
o
(3.34)
If m
1
does not provide a good enough solution the whole procedure is repeated
starting with m
1
instead of m
o
. The general equations for the ith iteration are:
m
i
= (A
T
C
1
e
A +R
T
R)
1
A
T
C
1
e
d
i
(3.35)
where d
i
is the dierence between the data and the model at the ith iteration.
The iteration procedure is repeated until m
i+1
m
i
is less than a prescribed
value, that is when further iterations fail to produce signicant changes in the
70
model. An alternative criteria for convergence is that the improvement in the t
to the data is no longer improved by further iterations. This is preferred over using
the convergence of the model since we have an idea of how small the value should
be based on the estimates of the errors in the data. Furthermore, there is no point
in continuing the iterations when there are no further improvements in the t to
the data.
3.3.1 Formulation of the Jacobian for the non-linear prob-
lem
Solving the nonlinear problem requires constructing the matrix of partial
derivatives that relate the data (declination, inclination and intensity) to the model
parameters. The mathematics are straightforward, but messy and require several
steps that will be described here. Based on equations 3.24 the elements of the
matrix A, which are the partial derivatives of the data with respect to the model
parameters can be obtained using the chain rule for dierentiation for the incli-
nation (I), declination (D), horizontal eld (H) and total eld (F). For example,
since the horizontal eld H is a function of X and Y, the partial dierential can be
written:
@H
@g
i
=
@H
@X
@X
@g
i
+
@H
@Y
@Y
@g
i
(3.36)
@H
@X
=
X
(X
2
+Y
2
)
1=2
=
X
H
(3.37)
@H
@Y
=
Y
(X
2
+Y
2
)
1=2
=
Y
H
(3.38)
Putting this together we get:
@H
@g
i
=
X
H
@X
@g
i
+
Y
H
@Y
@g
i
(3.39)
71
and similarly:
@H
@h
i
=
X
H
@X
@h
i
+
Y
H
@Y
@h
i
(3.40)
The partial derivatives of the declination with respect to the model parameters
can be similarly derived since:
d(tan
1
u)
dx
=
1
1 +u
2
du
dx
(3.41)
and u = (y=x)
du
dx
=
Y
X
2
(3.42)
and
d(tan
1
u)
dx
=
Y
H
2
(3.43)
Similarly:
d(tan
1
u)
dy
=
X
H
2
(3.44)
Putting this together to obtain the partial derivatives gives:
@D
@g
i
=
@D
@X
@X
@g
i
+
@D
@Y
@Y
@g
i
(3.45)
@D
@g
i
=
X
H
2
@Y
@g
i

Y
H
2
@X
@g
i
(3.46)
Derivations of the other partial derivatives can be derived in a similar manner and
will simply be stated here:
@D
@h
i
=
X
H
2
@Y
@h
i

Y
H
2
@X
@h
i
(3.47)
@I
@g
i
=
H
F
@Z
@g
i

XZ
HF
@X
@g
i

YZ
HF
@Y
@g
i
(3.48)
72
@F
@g
i
=
X
F
@X
@g
i
+
Y
F
@Y
@g
i
+
Z
F
@Z
@g
i
(3.49)
with a similar equation for the derivatives with respect to h
i
.
Based on equations 3.26 through 3.28, the relationship between the partial
derivatives of X, Y, and Z with respect to the model parameters and B
r
, B

, and
B

is:
@X
@g
i
=
@B

@g
i
cos
@B
r
@g
i
sin (3.50)
@Y
@g
i
=
@B

@g
i
(3.51)
@Z
@g
i
=
@B

@g
i
sin
@B

@g
i
cos (3.52)
And the relationships between the partial derivatives of the components of the
magnetic eld and the Associated Legendre functions is:
@B
r
@g
i
=

a
r
!
n+2
(n + 1) Pcos m (3.53)
@B

@g
i
=

a
r
!
n+2
@P
@
cos m (3.54)
@B

@g
i
=

a
r
!
n+2
m
sin
P sin m (3.55)
Combining equations 3.50 through 3.52 and 3.26 through 3.28 yields expressions
which can be written in matrix form as:
0
B
B
B
@
@X=@g
i
@Y=@g
i
@Z=@g
i
1
C
C
C
A
=
0
B
B
B
@
sin cos 0
0 1 0
cos sin 0
1
C
C
C
A

a
r

(n+2)
0
B
B
B
@
(n + 1)P cos(m)
@P
@
cos(m)
m
sin
P sin(m)
1
C
C
C
A
(3.56)
73
which can then be inserted into equations 3.46, 3.48 and 3.46 to form the elements
of the Jacobian. There are analogous equations for the partial derivatives with
respect toh
nm
. With each iteration during the inversion process, a new Jacobian,
A is retrieved and an an incremental improvement in the model is obtained as
described in equation 3.35.
3.3.2 Modeling methodology
The elements described so far in this chapter are combined into a methodol-
ogy depicted schematically in gure 3.3. The inversion proceeds iteratively from
a starting model. For the initial epoch in our study 1900 AD, we chose an axial
dipole model withg
10
= -30T. In the iterative modeling procedure, if the solution
converges, then is there is a unique minimum the nal solution should be indepen-
dent of the starting model. At each iteration, to achieve a thorough search of all
parameter space, the minimization procedure involves sweeping through a large
range of values for the Lagrange multiplier, at each iteration, and computing a
new model corresponding to each value of at each iteration. This amounts to
varying the model complexity though a wide range by varying the regularization.
At each iteration the preferred model is the the one corresponding to the minimum
mist. The root mean square mist of each of these models is computed and the
initial model for the next iteration is the model vector with the minimum mist
at the previous iteration.
74
The error estimates for paleomagnetic data in general are not very well-
determined. In an attempt to assign coherent errors estimates to the entire time-
series data set, based on experience with the data described in chapter 2, we have
assigned minimum errors,
i
of 5

in declination, 3

in inclination and 10% for
intensity. To assess the models during the inversion,the root-mean-square mist is
calculated separately for the declination, inclination and intensity and is given by:
rms error =
v
u
u
t
1
N
N
X
i=1

x
i
^ x
i

i

2
(3.57)
For independent errors the square of the resulting error is the sum of the individual
errors squared [32]. The quantity we use to assess the total mist.
It is sometimes the case in the inversion process, that we obtain models for
which the mist is greater than that of the previous iteration. This is described
as an inversion that is not well-behaved described by Parker [63]. In this case
a stabilization procedure is applied in which the new model at the ith iterations
computed using:
m
new
=fm
i
+ (1f)m
i1
(3.58)
where 0 < f < 1
3.3.3 Model Non-uniqueness
The models we develop and any models of the geomagnetic eld are necessarily
non-unique for various reasons. It is well known from Backus and Gilbert [1] that
there are many models of a continuous function, such as Br, that are capable of
tting a nite dataset. This can be described as an under-determined problem
and all problems with more unknowns than data will have an innite number of
75
possible solutions. The inverse problem of nding the model coecients is generally
solved by nding a model minimizing the least-squares dierence between the
model predictions and the data, together with a measure of the eld complexity to
help resolve the issue of nonuniqueness. Errors associated with the data contribute
to the problem of non-uniqueness as well. We do not claim that the models we
present in chapter 4 are unique in any sense, however they represent models that
spatially smooth and t the data adequately.
76
Input Data Vector:
Inclination, Declination,
Intensity
Iterate to nd a solution
Vary the model complexity
END
Stabilize
Find the best
tting model
for that iteration
Guess at an
intial model
n iterations
λ
Is the model
well behaved?
No
Does the
model
t well
enough?
Does the
model
t well
enough?
No
No
vary
Figure 3.3: Flow chart depicting methodology used in modeling process.
77
Chapter 4
Model Results
4.1 Model Validation
Using the methodology described in chapter 3 and depicted schematically in
(gure 3.3) we validate the modeling software and methodology by comparing our
model results with the GUFM1 historical model [27] during the time of overlap.
GUFM1 is described brie
y in section 1.2. It is an historical model of the geo-
magnetic eld that spans the interval 1590 -1990 and is based on a massive data
compilation, more than 36,000 measurements of declination, inclination and inten-
sity. It is thought to be an excellent representation of the secular variation of the
eld over the past 4 centuries and is the standard model used to describe Earth's
eld over this time interval.
4.2 Comparison with GUFM1
In our analysis we generate eld models at regular intervals in 100-year snap-
shots based on the PSVMOD2.0 time-series data. For the most recent epochs,
1900, 1800, 1700 and 1600 AD, we can compare our models with GUFM1. To val-
idate the models, several parameters and graphical representations are required.
The procedure can be outlined as follows: We initiate the inversion by iterating
from a starting axial dipole model withg
10
= -30T. We assess the initial rms error
between the data and the axial dipole model. Although we do not expect the data
78
to t this simple model well, a common measure of the success of the inversion is
to compare the original mist with the nal mist, as a percentage reduction in
the residual [19]. Global maps of the residuals, the dierence between the model
and the data and plots of the distribution of residuals (eg., gure 4.1) provide an
initial insight into the data quality and identication of possible outliers.
As the inversion proceeds, the model and relevant parameters (roughness, mis-
t, Lagrange multiplier) associated with the minimum mist at each step of the
inversion are saved. A well-behaved inversion exhibits a decrease in residuals with
each iteration and an increase in roughness or model complexity. We allow for 20
iterations of the inversion, although in all cases there is no additional improvement
in error after fewer than10 iterations. In our model development we choose a
maximum degree and order of 10 for the spherical harmonic expansion, high enough
so that the roughness of the models is determined by the data and regularization,
not the truncation level, and spatial aliasing eects are not a concern.
In developing a modeling methodology, we have examined all three smoothing
norms, equations 3.20 through 3.22. We examine the power spectra of the preferred
model and chose a model with power in the higher order harmonics that is most
similar to GUFM1. We nd that it is often the case that more severe regularization
(such asR
3n
) is required. We plot the trade-o curve, gure 3.1 and the preferred
model is chosen at the knee of the curve, a compromise between model mist and
complexity. Once the preferred model is selected we examine the distribution and
global maps of residuals, as we did for the starting model. Although we have
developed a methodology for identication of outliers, elimination of outliers has
not yet been incorporated into our modeling strategy.
A visual comparison of contour plots of the radial eld at the core-mantle
boundary between our preferred model and GUFM1 provide a global view of the
79
similarity in the large scale structure of the eld (for example, gures 4.4 and 4.5).
Finally, we can make a direct comparison by examining the actual values of the
spherical harmonic coecients of both models (e.g., table 4.1).
4.2.1 Comparison with GUFM1 - 1900 AD
In this section we describe how we have derived the preferred model for 1900
AD and compare it with GUFM1 for that epoch. The PSVMOD2.0 time series
values of inclination, declination and intensity are chosen to match GUFM1 closely
at 1900 AD, thus while this model serves to validate our software and methodology
and to test whether the global data distribution is sucient to recover a model
similar to GUFM1 it provides no insight related to the data errors. The input data
for 1900 AD consists of 179 values total, 70 inclination values, 65 declination and
44 intensity values. We perform 20 iterations of the inversion procedure starting
from an axial dipole model with g
10
= -30T. One way to compare the quality of
the solution obtained is to examine the improvement in residuals over the starting
model. The root-mean-square mist (equation 3.57) to the initial axial dipole
model for the individual parameters is inclination: 2.99, declination: 5.84 and
intensity 1.56. The total mist, the root sum square of these quantities is 6.78.
Histograms and global maps of the residuals (data minus axial dipole) are shown
in gure 4.1. The mean, median and standard deviation of the distributions are
shown in each panel of the histograms and the red line shows a normal distribution
for a mean and variance equal to that of each parameter. Outliers, greater than
3 are labeled by site name. The sites MUR (270.1

longitude, 81.6

latitude)
and SAW (276.1

longitude, 79.3

latitude) are identied as outliers in declination
compared to an axial dipole model. Histograms and residual maps are used to
80
identify and map outliers and are useful tools to test the goodness of t of the
starting model.
A range of Lagrange multipliers is investigated for each iteration from 1 x 10
11
to 0.9. At each step in the iteration the model for which the total root-mean-
square mist (the sum of the rms mist in declination, inclination and intensity)
is a minimum is saved. Our preferred models are chosen to be those that match
the power spectra of GUFM1 most closely and for this epoch it is obtained with
the most stringent smoothing norm (R
3n
). The behavior of the inversion is shown
in gure 4.2. The total rms mist versus iteration number is shown in the bottom
panel. The model roughness or complexity based on equation 3.22 is shown in
the middle panel and the tradeo curve (model complexity or roughness, versus
error) at the top. As expected for a well-behaved inversion, at each step the error
decreases and the model roughness or complexity increases. The preferred model
is chosen at the knee of the trade-o curve and for this epoch that occurs at the
fth iteration.
The mist between the data and the nal model for the individual parameters
is inclination: 0.115, declination: 0.234, intensity: 0.159 and total 0.508. This
amounts to a 92.5% reduction in the mist compared with the starting model. A
global map of the residuals, gure 4.3 shows how well the preferred model ts the
data. The symbol size as well as color is chosen to match the magnitude of the
residuals, thus the symbols are almost not visible on the global maps. Histograms
corresponding to the dierence between the data and the nal preferred model
show the excellent agreement as well.
A direct comparison between the values of the spherical harmonic coecients
for the two models demonstrate how well we are able to retrieve a model similar to
81
GUFM1. The model coecients through degree 5, for 1900 AD are given in table
4.1.
Finally, a side by side comparison of the model based on PSVMOD2.0 with
GUFM1 reveals the similarity in the large scale structure of the eld. Figures
4.4 and 4.5 are maps of the radial eld downward continued to the core-mantle
boundary for GUFM1 on the left and PSVMOD2.0 on the right. With the global
distribution of data in PSVMOD2.0 we are able to match much of the large-scale
structure seen in GUFM1. Although the structure is more subdued due to the
more limited global distribution of data, the signicant large-scale features of the
eld are similar, including high latitude northern
ux lobes, reverse
ux region in
the South Atlantic, and a weak
ux patch over the north geographic pole.
82
−30 −20 −10 0 10 20 30
0
5
10
15
20
Inclination
mean =3.328
median =1.2045
sd =8.4201
−150 −100 −50 0 50 100
0
10
20
30
40
Declination
mean =2.0865
median =6.4
sd =29.3597
−30 −20 −10 0 10 20 30
0
5
10
Intensity
mean =2.343
median =0.87179
sd =7.8502
MUR, SAW
1900 AD PSVMOD2.0 Axial Dipole
Inclination
2
4
6
8
10
12
14
16
18
20
Declination
20
40
60
80
100
120
Intensity
2
4
6
8
10
12
14
Figure 4.1: Global maps and histogram for 1900 AD showing the residuals (data minus model) of inclination, declination
and intensity for the initial model, an axial dipole with g
10
= -30T. Symbol size as well as color are scaled according
to the magnitude of the parameter.
83
7.5 10
4
8 10
4
8.5 10
4
9 10
4
9.5 10
4
0.26 0.28 0.3 0.32 0.34 0.36 0.38
roughness (µT
2
)
total rms misfit
7.5 10
4
8 10
4
8.5 10
4
9 10
4
9.5 10
4
0 5 10 15 20
roughness (µT
2
)
iteration
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0 5 10 15 20
total rms misfit
iteration
Figure 4.2: Top panel is the trade-o curve of model roughness versus total rms
mist. Middle panel shows increase in model roughness and bottom panel shows
the reduction in root-mean-square error as a function of iteration number for PSV-
MOD2.0 data at 1900 AD epoch.
84
−1 −0.5 0 0.5 1 1.5 2
0
10
20
30
Inclination
mean =0.046076
median =−0.0093951
sd =0.3461
−5 −4 −3 −2 −1 0 1 2 3 4
0
10
20
30
Declination
mean =0.023422
median =0.13458
sd =1.1786
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
5
10
15
20
Intensity
mean =0.028091
median =0.0748
sd =0.52156
1900 AD PSVMOD2.0 Preferred Model
Inclination
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Declination

0.5
1
1.5
2
2.5
3
3.5
4
Intensity
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4.3: Global maps and histograms for 1900 AD showing the residuals (data minus model) of inclination, declination
and intensity for the preferred model.
85
PSVMOD2.0 GUFM1 PSVMOD2.0 GUFM1
g
10
-31.603 -31.492 g
43
-0.206 -0.364
g
11
-2.505 -2.303 g
44
0.448 0.155
h
11
6.549 5.930 h
41
0.624 0.253
g
20
-0.616 -0.702 h
42
0.124 -0.107
g
21
2.945 2.942 h
43
-0.366 -0.174
g
22
1.060 0.935 h
44
0.125 -0.066
h
21
-1.560 -1.011 g
50
-0.003 -0.175
h
22
1.241 1.143 g
51
0.058 0.348
g
30
1.242 1.044 g
52
-0.006 0.275
g
31
-1.29 -1.486 g
53
-0.029 -0.001
g
32
1.185 1.261 g
54
-0.175 -0.063
g
33
0.186 0.588 g
55
-0.043 0.052
h
31
-0.509 -0.377 h
51
-0.455 -0.265
h
32
-0.379 0.0146 h
52
0.158 0.033
h
33
0.926 0.525 h
53
-0.072 -0.007
g
40
0.515 0.874 h
54
-0.312 -0.12
g
41
0.886 0.637 h
55
0.167 0.008
g
42
0.825 0.625
Table 4.1: Spherical harmonic coecients through degree 5 for the preferred model
based on PSVMOD2.0 and GUFM1 for 1900 AD.
86
-600
-600
-400
-400
-200
-200
-200
0
0
0
0
0
200
200
200
200
400
400
400
600
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
-600
-600
-400
-400
-400
-200
-200
-200
-200
-200
0
0
0
200
200
200
400
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
μT
-600
-500
-500
-400
-400
-300
-200
-200
-200
-100
-100
-100
0
0
0
100
100
100
200
200
200
300
300
400
-60 ˚ -60˚
-30 ˚ -30˚
0 ˚ 0˚
30 ˚ 30˚
60 ˚ 60 ˚
-600
-500
-500
-400
-400
-300
-300
-200
-200
-200
-100
-100
0
0
0
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
μT
GUFM1 1900 AD PSVMOD2 1900 AD
Figure 4.4: Contour maps (Hammer projection) of GUFM1 on the left and our preferred model based on PSVMOD2
on the right for 1900 AD. Maps in the top panels are centered on 180

longitude and maps in the bottom panels are
centered on 0

longitude. Units are T.
87
180 ˚
210 ˚
240 ˚
270˚
300 ˚
330˚
0˚
30 ˚
60˚
90 ˚
120 ˚
150 ˚
180 ˚
210 ˚
240 ˚
270˚
300 ˚
330˚
0˚
30 ˚
60˚
90 ˚
120 ˚
150 ˚
-600
-550
-500
-500
-450
-450
-400
-400
-350
-350
-300
-250
-250
-200
-200
-150
-150
-150
-100
-100
-100
-50
0
180 ˚
210˚
240 ˚
270˚
300 ˚
330 ˚
0˚
30 ˚
60˚
90˚
120˚
150 ˚
180 ˚
210˚
240 ˚
270˚
300 ˚
330 ˚
0˚
30 ˚
60˚
90˚
120˚
150 ˚
-200
-150
0
50
100
150
200
200
200
250
250
300
300
350
350
400
400
450
450
500
500
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-700
-600
-600
-500
-500
-400
-400
-400
-400
-300
-300
-300
-200
-200
-100
-100
0
0
100
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-200
-100
0
0
100
100
200
200
200
300
400
500
500
600
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
GUFM1 1900 AD PSVMOD2.0 1900 AD
μΤ
μΤ
Figure 4.5: Contour maps (polar projection) of GUFM1 on the left and our preferred model based on PSVMOD2 on
the right for 1900 AD. Maps in the top panels are centered on the south pole and maps in the bottom panels are
centered on the north pole. Units are T.
88
4.2.2 Comparison with GUFM1 - 1800 AD
We have applied the same analysis described in section 4.2.1 to PSVMOD2.0
time series data at 1800 AD. For this epoch there are 72 inclination, 66 declina-
tion and 43 intensity measurements, a total of 181 measurements. We start by
examining the rms mist (equation 3.57) between the initial axial dipole model
with g
10
= -30T and the data for this epoch. This is the starting model for the
nonlinear inversion and although we do not expect good agreement with the data,
it provides an initial assessment of data quality and a point of comparison with
the nal model. For the individual parameters, the mist in inclination is: 2.38,
declination: 6.22, intensity: 1.32 and total: 9.92. Global maps and histograms
of the data compared to an axial dipole are shown in gure 4.6. The sites high
latitude sites, MUR (270.1

longitude, 81.6

latitude) and SAW (276.1

longitude,
79.3

latitude) are identied as outliers in declination with values greater than 3
compared to an axial dipole model. The red line in each histogram shows a normal
distribution for a mean and variance equal to that of each parameter.
Figure 4.7 shows the behavior of the inversion as the iterations proceed. The
bottom panel shows the increase in model roughness or complexity and the middle
panel shows the reduction in total rms mist with iteration number. The top
panel is the trade-o curve of model complexity versus rms mist. Again the
preferred model is chosen at the fth iteration of the inversion at the knee of the
trade-o curve. We obtained a power spectra most similar to GUFM1 using the
R
2
smoothing norm. R
1
gave rather poor agreement. The preferred solution is
obtained on the fth iteration using the R
2
smoothing norm. The nal mist for
the individual parameters is inclination: 0.18 declination: 0.43 intensity: 0.26 and
total: 0.534.
89
Maps and histograms of the preferred model are shown in gure 4.8. The model
shows excellent agreement with GUFM1 for this epoch. A few sites with some
values greater than 3, BIR and SEN (inclination), NZE and POU (declination)
and LGN (intensity) are identied in the histograms. We make no use of the
identication of outliers but plan to incorporate it into our modeling strategy in
future eorts.
Table 4.2 lists the spherical harmonic coecients through degree 5 for GUFM1
and our preferred model for this epoch. The values agree quite well, especially the
lower degree terms.
90
−20 −15 −10 −5 0 5 10 15 20 25
0
5
10
15
Inclination
mean =2.1046
median =2.1022
sd =6.8614
−150 −100 −50 0 50 100
0
10
20
30
Declination
mean =0.83924
median =3.49
sd =31.3636
−15 −10 −5 0 5 10 15 20 25
0
5
10
15
20
Intensity
mean =5.0094
median =3.592
sd =6.0235
MUR,SAW
Inclination
2
4
6
8
10
12
14
Intensity
2
4
6
8
10
12
14
16
Declination

20
40
60
80
100
120
140
1800 AD PSVMOD2.0 Axial Dipole
Figure 4.6: Global maps and histograms for 1800 AD showing the residuals (data minus model) of inclination, declination
and intensity for the initial model, an axial dipole with g
10
= -30T. Symbol size as well as color are scaled according
to the magnitude of the parameter. Units are degrees for inclination and declination and T for intensity.
91
0.45
0.5
0.55
0.6
0.65
0.7
0.75
1 10
6
1.2 10
6
1.4 10
6
1.6 10
6
1.8 10
6
2 10
6
2.2 10
6
roughness (µT
2
)
total rms misfit
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0 5 10 15 20
total rms misfit
iteration
1 10
6
1.2 10
6
1.4 10
6
1.6 10
6
1.8 10
6
2 10
6
2.2 10
6
0 5 10 15 20
roughness (µT
2
)
iteration
Figure 4.7: Top panel is the trade-o curve of model roughness versus error. Middle
panel shows increase in model roughness and bottom panel shows the reduction in
root-mean-square error as a function of iteration number for PSVMOD2.0 data at
1800 AD epoch.
92
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
10
20
30
Inclination
mean =0.0022775
median =0.020154
sd =0.54333
−10 −8 −6 −4 −2 0 2 4 6
0
10
20
30
Declination
mean =−0.43052
median =−0.38702
sd =2.124
−6 −4 −2 0 2 4 6 8
0
10
20
30
40
Intensity
mean =0.076074
median =−0.020367
sd =1.456
LGN
NZE,POU
BIR,SEN
Inclination
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Declination
1
2
3
4
5
6
7
8
Intensity
1
2
3
4
5
6
1800 AD PSVMOD2.0 Preferred Model
Figure 4.8: Global maps and histograms for 1800 AD showing the residuals (data minus model) of inclination, declination
and intensity for the preferred model. Symbol size as well as color are scaled according to the magnitude of the
parameter.
93
PSVMOD2.0 GUFM1 PSVMOD2.0 GUFM1
g
10
-33.06 -32.82 g
43
0.646 -0.017
g
11
-2.162 -3.144 g
44
0.714 -0.144
h
11
6.620 5.651 h
41
0.069 -0.022
g
20
0.331 0.236 h
42
0.203 -0.502
g
21
1.656 2.510 h
43
-0.794 -0.260
g
22
-0.270 -0.676 h
44
-0.784 -0.209
h
21
-0.837 0.583 g
50
0.218 -0.029
h
22
0.620 1.088 g
51
0.029 0.377
g
30
0.966 0.963 g
52
0.608 0.475
g
31
-0.455 -0.843 g
53
-0.439 -0.053
g
32
1.340 1.281 g
54
-0.463 -0.055
g
33
-0.416 -0.056 g
55
0.021 0.079
h
31
-0.0007 -0.741 h
51
-0.421 -0.230
h
32
-0.160 0.250 h
52
-0.081 -0.027
h
33
0.932 0.988 h
53
0.505 -0.019
g
40
0.399 0.560 h
54
0.109 -0.079
g
41
0.603 0.5991 h
55
-0.325 0.002
g
42
-0.214 0.259
Table 4.2: Spherical harmonic coecients through degree 5 for the preferred model
based on PSVMOD2.0 and GUFM1 for 1800 AD.
94
-600
-500
-500
-400
-400
-300
-300
-300
-300
-200
-200
-200
-200
-100 -100
-100
-100
0
0
0
0
100
100
200
200
300
300
300
400
400
500
600
600
600
700
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
-600
-500
-500
-400 -300
-300
-300
-200
-200 -200
-200
-100
-100
-100
0
0
0
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
-700
-700
-600
-600
-500
-500
-400
-400 -300
-300
-300
-200
-100
-100
-100
L
0
0
0
0
0
100
100
100
100
200
200
200
300
400
500
500
600
700
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
-700
-700
-600
-600
-500
-400
-400
-400
-300
-300
-300
-300
-300
-300
-200 -200
-200
-100
-100
-100
-100
-100
H
H
0
0
0
100
100
100
200
200
200
200
300
300
300
400
500 500
600
700
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
μT
GUFM1 1800 AD
PSVMOD2 1800 AD
μT
Figure 4.9: GUFM1 model for epoch 1800, Hammer projection.
95
180˚
210 ˚
240˚
270˚
300 ˚
330˚
0 ˚
30 ˚
60˚
90˚
120˚
150 ˚
180˚
210 ˚
240˚
270˚
300 ˚
330˚
0 ˚
30 ˚
60˚
90˚
120˚
150 ˚
-700
-700
-600
-600
-500
-500 -400
-400
-400
-400
-300
-300
-300
-200
-200
-200
-200
-100
-100
-100
-100
0
0
180˚
210˚
240 ˚
270˚
300 ˚
330 ˚
0˚
30 ˚
60˚
90 ˚
120 ˚
150 ˚
180˚
210˚
240 ˚
270˚
300 ˚
330 ˚
0˚
30 ˚
60˚
90 ˚
120 ˚
150 ˚
0
0
100
100
100
200
200
300
400
400
500
500
600
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
μT
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-600
-500
-500
-400
-400
-300
-300
-300
-200
-100
-100
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-300
-200
-100
0
0
100
200
200
300
300
400
400
500
500
600
600
600
700
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 1800 AD
GUFM1 1800 AD
Figure 4.10: GUFM1 model for epoch 1800, polar projection
96
4.3 Models prior to 1800 AD
Using the same methodology described in section 4.1, we have derived models
based on the PSVMOD2.0 time series records in 100-year intervals spanning the
past 2,000 years. The steps in the analysis are completely analogous to those
described above. Some of the relevant parameters are summarized in table 4.3.
As expected, in general, the rms mist increase for earlier models when the data
errors are larger.
Contour plots of the radial eld at the core-mantle boundary for the preferred
models for 1600 AD and 1700 AD are shown in gures 4.11 through 4.13 and
the GUFM1 models are shown for comparison. Maps in the top panels (Hammer
projection) are centered on 180

longitude and maps in the bottom panels are
centered on 0

longitude. Polar maps in the top panels are centered on the south
pole and the bottom panel is centered on the north pole. Units are T.
Figures 4.15 through 4.30 show contour plots of the preferred PSVMOD2 model
at each epoch in polar and Hammer projection from 1500 AD through 0 AD/BC.
For these gures, GUFM1 is no longer available for comparison. For all gures,
maps in the top panels (Hammer projection) are centered on 180

longitude and
maps in the bottom panels are centered on 0

longitude. Polar maps in the top
panels are centered on the south pole and the bottom panel is centered on the
north pole. Units are T.
97
Epoch total rms error Smoothing Norm
1900 0.51 R
3
1800 0.53 R
2
1700 1.17 R
3
1600 2.09 R
2
1500 2.12 R
3
1400 2.47 R
3
1300 3.20 R
3
1200 3.76 R
3
1100 2.98 R
3
1000 3.74 R
3
900 2.95 R
3
800 3.21 R
3
700 2.87 R
3
600 3.14 R
3
500 3.12 R
3
400 3.95 R
3
300 4.12 R
3
200 4.64 R
3
100 3.29 R
3
0 3.13 R
3
Table 4.3: Total rms error for preferred PSVMOD2.0 models and smoothing norm
from 0 AD to 1900AD.
98
0˚
-700
-650
-500
-500
-450
-450
-400
-400
-350
-300
-300
-200
-150
-150
-150
-100
-100
-50
-50
0
0
50
50
50
100
100
100
150
150
250
250
300
300
350
400
400
450
500
500
550
600
-60˚ -60˚
-30˚ -30˚
0˚
30˚ 30˚
60˚ 60˚
H
-700
-650
-600
-500
-450
-400
-350
-350
-300
-300
-300
-250
-250
-250
-200
-200
-150
-100
-50
-50
0
0
50
50
50
100
100
100
150
150
200
200
250
250
300
300
350
350
400
400
450
450
0
500
600
650
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
H
-750 -700
-650
-600
-500
-450
-400
-350
-300
-300
-300
-250
-250
-250
-250
-200
-200
-200
-150
-150
-150
-100
0
-50
-50
0
0
50
00
100
150
150
200
200
250
250
250
300
300
350
350
400
400
450
500
-60˚ -60˚
-30˚ -30˚
0˚
30˚ 30˚
60˚ 60˚
-700
-650 -550
-500
-500
-450
-450
-400
-350
-250
-250
-200
-200
-150
-100
-100
-100
-100
-50
-50
-50
-50
0
0
0
0
200
200
400
400
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
GUFM1 1700 AD
PSVMOD2 1700 AD
μT μT
Figure 4.11: Contour maps (Hammer projection) of GUFM1 on the left and our preferred model based on PSVMOD2
on the right for 1700 AD. Maps in the top panels are centered on 180

longitude and maps in the bottom panels are
centered on 0

longitude. Units are T.
99
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-600
-500
-400
-300
-300
-300
-200
-200
-100
-100
0
0 100
200
300
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
0
100
200
200
300
300
400
400
500
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-650
-600
-550
-500
-500
-450
-450
-400
-400
-400
-400
-350
-350
-300
-300
-250
-200
-150
-150
-100
-100
-50
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-50
50
100
100
150
200
250
300 350
400
450
450
450
500
500
550
600
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
GUFM1 1700 AD PSVMOD2 1700 AD
μT
μT
Figure 4.12: Contour maps (polar projection) of GUFM1 on the left and our preferred model based on PSVMOD2
on the right for 1700 AD. Maps in the top panels are centered on the south pole and maps in the bottom panels are
centered on the north pole. Units are T. 100
-550
-550
-500
-500
-450
-450
-450
-400
-400
-400
-350
-300
-300
-300
-250
-250
-200
-200
-150
-150
-150
-100
-50
-50
0
0
50
50
50
100
100
200
200
300
300
300
350
350
350
400
400
400
400
450
450
450
500
500
550
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
L
-600
-550
-550
-500
-400
-350
-300
-300
-300
-250
-250
-200
-200
-150
-100
-100
-100
-50
-50
-50
0
0
0
200
200
400
400
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
-550
-500
-500
-450
-400
-400
-350
-300
-300
-250
-250
-200
-200
-100
-100
-100
-50
-50
0
0
0
50
50
100
100
100
150
150
200
250
250
300
350
400
400
500
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
-550
-500
-500
-4
-450
-400
-300
-250
-200
-200
-150
-150
-100
-100
-100
-50
0
0
50
100
100
150
200
200
250
300
300
350
400
400
0
450
500
500
0
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
GUFM1 1600 AD PSVMOD2 1600 AD
μT μT
Figure 4.13: Contour maps (Hammer projection) of GUFM1 on the left and our preferred model based on PSVMOD2
on the right for 1600 AD. Maps in the top panels are centered on 180

longitude and maps in the bottom panels are
centered on 0

longitude. Units are T.
101
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-600
-500
-500
-400
-400
-300
-300
-200
-200
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
100
200
300
300
400
400
500
500
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-550
-500
-500
-450
-400
-400
-350
-350
-300
-300
-300
-250
-250
-250
-200
-200
-150
-150
-100
-100
-100
-50
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-50
0
50
100
150
200
250
300
350
400
400
500
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
GUFM1 1600 AD
PSVMOD2 1600 AD
μT
μT
Figure 4.14: Contour maps (polar projection) of GUFM1 on the left and our preferred model based on PSVMOD2
on the right for 1600 AD. Maps in the top panels are centered on the south pole and maps in the bottom panels are
centered on the north pole. Units are T.
102
-800
-700
0
-400 -350
-350
-300
-300
-300
-250
-250
-200
-200
-200
-150
-150
-150
-100
-100
0
-50
0
0
0
0
50
50
50
100
100
150
200
250
300
300
350
350
400
400
500
550
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
L
-800
-800
-600 0
0
-350
-350
-300
-300
-250
-250
-200
-150
-150
-150
-100
-100
-50
0
0
0
0
200
200
200
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
μT
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
L
-800
-700
-700
-600
-600
-500
-400
-300
-300
-300
-200
-200
-200
-100
-100
-100
0
0
0
100
200
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-200
200
200
300
300
400
400
500
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 1500 AD
μT
Figure 4.15: Preferred PSVMOD2.0 model 1500 AD, Hammer projection.
103
H
-650
-500
-500 -450
-450
-400 -400
-400
-350
-350
-300
-300
-300
-250
-250
-250
-200
-200
-200
-150
-150
-100
-100
-50
-50
0
0
0
50
50
50
100
100
150
150
200
250
250
300
300
300
350
350
350
350
400
400
450
600
650
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
H H
L
L
L
L
L
L
-700
-650
-500
-450
-450
-400
-400
-400
-400
-350
-350
-300
-250
-250
-250
-200
-150
-150
-150
-100
-100
-100
-100
-50
-50
-50
-50
L
H
0
0
0
200
200
200
400
400
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-500
-400
-400
-400
-300
-300
-300
-200
-200
-200
-200
-100
-100
0
100
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
0
100
200
200
300
300
400
400
400
500
600
700
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 1400 AD
μT
μT
Figure 4.16: Preferred PSVMOD2.0 model 1400 AD.
104
L
L
L
-800
-750
-700
-550
-550
-500
-450
-450
-400
-400
-400
-350
-350
-350
-300
-300
-300
-250
-250
-200
-200
-150
-150
-150
-150
-100
-100
-100
-50
-50
-50
L
0
0
0
50
50
100
100
100
100
150
150
150
150
200
200
200
200
250
250
250
250
300 350
400
450
500
500
-60˚
-60˚
-30˚
-30˚
0˚ 0˚
30˚
30˚
60˚
60˚
L
L
L
L
-550
-500 -450
-450
-400
-400
-350
-350
-300
-300
-300
-250
-250 -200
00
-150
-100
-100
-50
-50
-50
0
0 0
200
200
200
400
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
PSVMOD2 1300 AD
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-500
-500
-400
-400
-400
-300
-300
-200
-200
-100
0 200
200
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
200
200
300
400
500
500
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
μT
μT
Figure 4.17: Preferred PSVMOD2.0 model 1300 AD.
105
L
L
L
-800
-750
-700
-600
-550
-500
-500
-450
-450
-400
-400
-350
-350
-300
-300
-300
-300
-250
-250
-250
-250
-200
-200
-200
-150
-150
-150
-100
-100
-100
-50
-50
-50
L
0
0
0
50
50
50
100
100
150
150
200
200
200
250
250
250
250
300
300
300
350
350
400
400
450
550 600
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
L
L
-600
-550
-500
-450
-450
-400
-400
-350
-300
-300
-250
-250
-200
-200
-150
-150
-150
-100
-100
-100
-50
-50
-50
0
0
0
200
200
200
200
400
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-600
-500
-500
-400
-400
-300
-300
-300
-200
-200
-100
-100
100
200
200
300
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
200
200
300
400
400
600
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 1200 AD
μT
μT
Figure 4.18: Preferred PSVMOD2.0 model 1200 AD.
106
L
L
-750
-700
-650
-550
-550
-500
-500
-450
-450
-450
-400
-400
-350
-300
-300
-250
-250
-250
-200
-200
-200
-200
-150
-150
-150
-150
-100
-100
-100
-100
-50
-50
-50
-50
L
0
0
0
0
50
50
50
50
50
100
100
100
100
100
150
150
150
200
200
250
250
300
300
350
350
400
400
450
450
500
550
600
650
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
-550
-500
-500
-450
-450
-400 -350
-300
-300
-300
-250
-250
-250
-250
-200
-200
-200
-200
-150
-150
-150
-100
-100
-100
-100
-50
-50
-50
0
0
0
200
200
200
400
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700 -600 -500
-500
-500
-400
-400
-300
-200
-200
-100
-100
0
0
100
400
500
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-300
100
100
200 300
400
500
600
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 1100 AD
μT
μT
Figure 4.19: Preferred PSVMOD2.0 model 1100 AD.
107
L
L
L
-550 -500
-500
-450
-450
-450
-400
-400
-400
-350
-350
-350
-300
-300
-300
-250
-250
-250
-200
-200 -200
-200
-200
-150
-150
-150
-150
-150
-150
-100
-100
-100
-100
-50
0
-50
-50
-50 -50
L
0
0
0
0
0
0
50
50
50
50
50
100
100
100
100
150
150
150
200
200
250
250
250
300
300
300
300
350
400
450
450
500
500
500
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
H
L
-550
-550
-500
-500
-500
-450
-450
-400
-400
-400
-350
-350
-350
-300
-300
-300
-250 -200
-200
-150
-150
-150
-150
-100
-100
-100
-100
-50
-50
-50
-50
-50
0
0
0
0
0
200
200
200
400
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
PSVMOD2 1000 AD
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-500
-500
-400
-400
-400
-400
-300
-300
-300
-200
-200
-200
-100
-100
-100
-100
0
0
0
0
100
100
200
200
400
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-200
-200
100
200
300
300
400
500
500
500
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
μT
μT μT
μT
Figure 4.20: Preferred PSVMOD2.0 model 1000 AD.
108
L
-600 -550
-500
-500
-500 -450
-400
-400
-350
-350
-250
-250
-200
-200
-200
-200
-150
-150
-150
-150
-100
-100
-100
-100
-50
-50
-50 -50
0
0
0
0
0
50
50
50
50
50
100
100
100
150
150
200
200
200
250
300
400
450
500
550
-60 ˚ -60 ˚
-30 ˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
-600
-550
-550
-500
-500
-450 -400 -350
-300
-300
-250
-250
-250
-200
-200
-200
-200
-200
-150
-150
-150
-150
-100
-100
-100
-100
-100
-50
-50
-50
-50
-50
-50
L
0
0
0
0
200
200
200
200
400
400
-1000 -800 -600 -400 -200 0 100 300 500 700 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-600
-500
-500
-500
-400
-400
-300
-200
-200
-100
-100
-100
0
0
100
200
200
300
400
500
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
L
L
0
0
100
100
200
300
400
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 900 AD
μT
μT
Figure 4.21: Preferred PSVMOD2.0 model 900 AD.
109
-600
-550
-500
-500
-450
-400
-400
-400
-350
-350
-350
-350
-300
-300
-250
-250
-200
-200
-200
-200
-200
-150
-150
-150
-100
-100
-100
-50
-50
-50
-50
-50
0
0
0
0
0
50
50
50
50
50
50
100
100
100
100
100
150
150
150
150
150
200
200
300
300
350
350
450
500 550
600
650
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
-600
-500
-350
-350
-350
-350
-300
-300
-300
-250
-250
-250
-250
-200
-200
-200
-200
-200
-150
-150
-100
-100
-100
-50
-50
-50
-50
0
0
0
0
0
200
200
200
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-600
-500
-500
-400
-400
-300
-300
-200
-200
-100
-100
0
0
100
100
200
200
300
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-300
0
100
200
200
300
400
500
600
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 800 AD
μT
μT
Figure 4.22: Preferred PSVMOD2.0 model 800 AD.
110
L
-750 -700
-600
-550
-550
-500
-500
-450
-400
-400
-350
-350
-300-250
-250
-200
-200
0
-150
-150
-150
-100
-100
0
-100
-100
-50
-50
-50
-50
-50
0
0
0
0
50
50
50
50
100
100
100
100
150
150
150
200
200
200
250
2 0
250
300
300
350
350
400
550
-60˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
L
L
L
-750
-700
0-600
-600
-600
-550
-500
450
-400
-350
-250
-200
-150
-100
-100
-50
-50
-50
-50
-50
0
0
0
0
200
200
200
200
400
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-600
-600
-600
-500
-500
-400
-400
-300
-300
-200
-200
-200
-100
-100
0
200
300
300
400
400
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-100
0
100
100
200
200
300
400
500
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 700 AD
μT
μT
Figure 4.23: Preferred PSVMOD2.0 model 700 AD.
111
L -650
-650
-600
-600
-600
-550
-550
-500 -450
-450
-400
-400
-400
-350
-350
-300 -250
-250
-200
-200
-200
-200
-150
-150
-150
-150
-150
-150 -100
-100
-100
-100
-50
-50
-50
-50
0
0
0
0
0
50
50
50
50
100
100
100
150
200
250
250
300
300
300
350
350
400
450
500
550
-60˚ -60 ˚
-30 ˚ -30 ˚
0˚ 0˚
30˚ 30˚
60 ˚ 60 ˚
H
-650
-650
-650
-600
-600
-600
-550
-500
-400
-350
-350
-300
-300
-250
-250
-250
-200
-200
-200
-200
-150
-150
-150
-150
-150
-100
-100
-100
-50
-50
-50
-50
0
0
0
0
200
200
200
400
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-600
-600
-600
-500
-500
-400
-400
-300
-200
-200
-200
-100
0
100
200
200
300
300
400
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-200
0
100 200
300
400
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 600 AD
μT
μT
Figure 4.24: Preferred PSVMOD2.0 model 600 AD.
112
-650 -600
-550
-550
-550
-500
-500
-450
-450
-450
-400
-400
-400
-350
-350
-300
-300
-300
-250
-250
-250
-250
-200
-200
-200
-200
-200
-150
-150
-150
-150
-100
-100
-100
-50
-50
-50
-50
-50
L
0
0
0
0
0
0
50
50
50
50
50
50
100
100
100
100
100
150
150
200
200
250
300
350
350
450
500
500
550
600
600
-60 ˚ -60˚
-30˚ -30˚
0˚ 0˚
30˚ 30˚
60˚ 60˚
L
L -650
-650
-600
-550
-550
-500
-450
-400
-300
-300
-250
-250
-250
-250
-200
-200
-200
-200
-200
-150
-150
-150
-100
-100
-50 -50
0
0
0
0
0
200
200
200
400
600
600
-1000 -800 -600 -400 -200 0 100 300 500 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-600
-600
-500
-500
-500
-500
-400
-400
-300
-300
-200
-200
-100
L
0
0
0
100
100
200
200
300
400
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-300
0
100
100
200
300
400
500
600
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 500 AD
μT
μT
Figure 4.25: Preferred PSVMOD2.0 model 500 AD.
113
-750 -700
-700 -650
-600
-600
-400
-400
-350
-350
-350
-300
-300
-250
-250
-200
-150 -150
-100
-100
-100
-100
-50
-50
-50
0
0
0
0
0
50
50
50
50
50
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100
100
100
150
150
200
200
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250
250
300
300
300
400
400
450
500
-60 ˚ -60 ˚
-30˚ -30˚
0 ˚ 0˚
30 ˚ 30 ˚
60˚ 60 ˚
L
-700 -650
-650
-600
-550
-550
-500
-450
-400
-400
-350
-350
-300
-250
-250
-200
-150
-150
-150
-100
-100
-100
-100
-100
-50
-50
0
0
0 0
200
200
200
200
400
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
H
-700
-600
-600
-500
-500
-500
-400
-400
-300
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-200
-100
-100
0
100
200
200
300
400
500
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
100
200
300
300
400
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 400 AD
μT
μT
Figure 4.26: Preferred PSVMOD2.0 model 400 AD.
114
H -850 -800
-750
-650
450
-400 -350
-350
-300
-300
-300
-250
-250 -250
-250
-200
-200
-150
-150
-150
-150
-100
-100
-100
-100
-50
-50
-50
-500
0
0
0
0
50
50
50
50
100
100
150
150
150
150
200
200
200
200
250
300
300
350
350
400
500
550
-60˚ -60˚
-30 ˚ -30 ˚
0 ˚ 0˚
30 ˚ 30 ˚
60 ˚ 60 ˚
H -800 -750
-700
-650
-650
-600
-600
-550
-550
-500
-500 -450 -400
-350
-350
-300
-300
-250
-250
-250
-250
-200
-200
-200 -200
-150
-150
-150
-150
-150
-100
-100
-100
-100
-100
-50
-50
-50
-50
-50
L
0
0 0
0
0
200
200
200
200
200
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-800
-700
-600
-600
-500
-500
-400 -300
-300
-300
-200
-200
-100
-100
-100
0
100
200
200
300
400
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-300
100
200
200
300
300
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 300 AD
μT
μT
Figure 4.27: Preferred PSVMOD2.0 model 300 AD.
115
H
-800-750
-750
-700
-700
-650
-650
-600
-550
-500
-500-450
-400
-400
-350 -300
-300
-250
-250
-250
-200
-200
-200
-200
-150
-150
-150
-150
-150
-150
-100
-100
-100
-100
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-50
-50
-50
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0
0
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50
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50
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150
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250
250
250
300
350
350
400
500
550
600
-60 ˚ -60˚
-30 ˚ -30 ˚
0
˚ 0 ˚
30 ˚ 30 ˚
60 ˚ 60 ˚
H -750 -700
-700
-650
-600
-600
-550 -550
-500
-500
-450
-450
-400
-400
-400 -350
-350
-250
-250
-200
-200
-200
-150
-150
-150
-150
-100
-100
-100
-50
-50
-50
0
0
0
200
200
200
200
400
400
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-700
-700
-600
600
-500
-500
-500
-400
-400
-300
-300
-300
-200
-200
-100
-100
0
0
0
100
200
200
300
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
L
-100
0
100
200
300
500
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 200 AD
μT
μT
Figure 4.28: Preferred PSVMOD2.0 model 200 AD.
116
H
L
-950 -850
-800
-750 -700 -650 -600 -550
-550
-300
-300
-250
-200
-200
-150
-150
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-100
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-50
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250 250
300
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350
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350
400
400
600
-60 ˚ -60 ˚
-30 ˚ -30 ˚
0 ˚ 0˚
30 ˚ 30 ˚
60 ˚ 60 ˚ L
L H
L
L
L
-900 -850
-800
-750
-700
-700
-650
-650
-650
-600
-600
-600
-550
-550 -550
-500
-500
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-400
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-200
-150
-150
-100
-100
-100
-50
-50
-50
H
H
H
H
H
H
H
H
0
0
0
0
0
200
200
200
200
400
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
H
-800 -700
-700
-600
-600
-600
-500
-500
-400
-400
-300
-300
-200
-200
-200
-100
-100
-100
0
0
100
200
300
400
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
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60˚
90˚
120˚
150˚
-100
100
200
300
300
400
600
-1000
-900
-800
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-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 100 AD
μT
μT
Figure 4.29: Preferred PSVMOD2.0 model 100 AD.
117
H
L
-900 -850 -800
-750
-700
-700
-650
-500
-450 -400
-400
-350
-350
-300
-300
-250
-250
-250
-250
-200
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-200
-150
-150
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-100
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-50
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400
450
500
-60 ˚
-60 ˚
-30 ˚
-30 ˚
0˚ 0˚
30 ˚
30˚
60˚
60˚
L
L
-650
-600
-600
-550 -500 -450
-450
-400
-350
-300
-250-200
-200
-200
-150
-150
-150
-100
-100 -100
-100
-100
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-50
-50
-50
0
0
0
200
200
200
400
600
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
L
H
-800
-700
-600
-600
-500
-500
-400
-400
-400
-300
-300
-200
-200
-100
-100
0
0
0
100
100
200
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
180˚
210˚
240˚
270˚
300˚
330˚
0˚
30˚
60˚
90˚
120˚
150˚
-400
-300
-100
0
100
100
200
200
300
400
500
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
PSVMOD2 0
μT
μT
Figure 4.30: Preferred PSVMOD2.0 model 0 AD-BC.
118
Chapter 5
Model Results - Implications for
Field Generation
We have generated spherical harmonic models spanning the past 2,000 years
based on our newly-derived global time series, PSVMOD2.0. Many of the large-
scale characteristics of the current and historical geomagnetic eld (described in
chapter 1) are seen in the models derived in this study. Our models for the
past 400 years compared favorably with the historical model GUFM1. We have
demonstrated that with the limited spatial coverage of PSVMOD2.0, relatively to
GUFM1, we can retrieve models with similar large-scale structure. In general over
the entire 2,000 year interval the models are temporally consistent from one epoch
to the next.
5.0.1 High latitude
ux lobes - Northern Hemisphere
Prominent features of GUFM1 over the past 400 years are two high latitude
ux
lobes in the northern hemisphere over North America and Siberia near longitudes
of270

and 120

. Their location and behavior is most easily tracked in polar
contour maps (gures 4.5, 4.10, 4.12, 4.14). The
ux lobes are thought to re
ect
the pattern of convection in the
uid outer core. It has been proposed by Busse
[10] that in a rapidly rotating annulus, such as the Earth's outer core, convection
takes the form of rolls parallel to the rotation axis and tangent to the inner core
boundary at the equator (gure 5.1). In this pattern of convection,
uid spirals
119
into the convection rolls, concentrating
ux. The convection rolls terminate where
the projection of the inner core boundary along the rotation axis meets the core-
mantle boundary at roughly 70

latitude North and South and dene a region
within known as the tangent cylinder. Latitudes of the observed
ux lobes coincide
with the edge of the tangent cylinder, supporting the idea that they represent the
ends of convection rolls. According to theory, the
ux lobes that are responsible
for the dipolar structure of the eld should be equally spaced in longitude, yet in
current models we see only two, over Siberia and Canada separated by120

. The
third, `missing'
ux lobe should be centered near 0

longitude.
For each of the four epochs during the period of overlap 1600 - 1900 AD,
the location of high latitude
ux lobes in the northern hemisphere based on our
models is consistent with GUFM1. There are two
ux lobes and they are centered
at270

and 120

longitude. In both GUFM1 and PSVMOD2.0 models, the
Siberian
ux lobe appears to shift to eastward to lower longitudes and poleward
at 1600 AD. From 1500 AD through 1000 AD there appear to be two
ux lobes in
the northern hemisphere, one near270

and another close to the geographic pole.
For most earlier epochs before 1000 AD, two
ux lobes can be identied at270

and 120

, although the lobe near 120

is often shifted close to the geographic pole
(for example at 700 AD). A third
ux concentration is evident as well throughout
the earlier time periods, near30

longitude (especially evident in the polar maps)
but it is centered at a latitude much lower than expected for
ux concentrations
associated with convection parallel to the tangent cylinder described above.
5.0.2 Reverse
ux patch near the North Geographic Pole
Another manifestation of the described pattern of convection is a distinctly
dierent convective regime inside the tangent cylinder. Large-scale
ow within
120
the tangent cylinder is thought to be moving, above and below the solid inner
core as an upwelling polar vortex. The presence of reversed
ux patches near
the geographic poles is consistent with a pattern of upwelling, which diuses the

ux, weakens the eld and is responsible for the null patches. Time-series records
from sites at high latitudes, included in PSVMOD2.0 are able to more accurately
characterize the eld in the polar regions.
GUFM1 exhibits reversed
ux patches at the North geographic pole for 1800
and 1900 AD. For 1700 and 1600 AD they are more accurately described as zero-

ux patches. The eld is weak at the geographic pole but not of the opposite sign.
PSVMOD2.0 models during the period of overlap with GUFM1 exhibit weak or
reversed
ux but during some intervals (e.g., 1700 AD) it is not located exactly
at the pole. Reverse or zero-
ux patch cannot be identied in models from 1300
AD through 600 AD although they return in models from 500 - 0 AD. For many
epochs they are apparent close to the geographic pole, consistent with a dierent
pattern of convection within the tangent cylinder.
5.0.3 High latitude
ux lobes - Southern Hemisphere
If the pattern of convection thought to produce
ux concentrations at high
latitudes exists, one would expect to nd the counterpart of northern hemisphere

ux lobes in the southern hemisphere at similar longitudes. Flux concentrations
at high southern latitudes are more dicult to interpret, probably because of the
relative paucity of data in the region. Two
ux lobes can be identied in GUFM1
at longitudes270

and 120

for 1900 and 1800 AD. For 1700 and 1600 AD
there are still two, but they have shifted slightly eastward in longitude. Models
based on PSVMOD2.0 during this interval of overlap with GUFM1 do not agree
particularly well in the location (or number, or existence) of
ux concentrations
121
in the southern hemisphere. The epochs from 1900 AD through 1600 AD can be
described as follows: 1) 1900 - one
ux concentration at200

2) 1800 - three

ux concentrations, but only one at high latitude (270

) 3) 1700 - only one
ux
concentration at60

longitude, in contrast to GUFM1 4) two
ux lobes near
300

and 150

(roughly consistent with GUFM1).
5.0.4 High latitude
ux lobes - Hemispheric symmetry
For most epochs GUFM1 exhibits evidence that the northern hemisphere
ux
lobes have a counterpart at similar, but not exactly the same longitudes in the
southern hemisphere. The discrepancy in longitude may be due to the quality of
the model due to the lack of data in the southern hemisphere. For 1800 - 1500
AD it could be argued that PSVMOD2.0 northern hemisphere
ux lobes have a
counterpart in the southern hemisphere. For earlier intervals the evidence is not
compelling. The concentration of
ux near the southern geographic pole is complex
and dicult to interpret.
Another interesting aspect of
ux lobe behavior based on GUFM1 is the appar-
ent absence of drift in longitude. The theory predicts that the whole pattern should
drift in longitude azimuthally, yet it appears that the pattern has been relatively
stationary over the past four centuries, wobbling slightly about a mean position.
One idea that has been proposed is that mantle convection results in lateral varia-
tions in core-mantle boundary heat
ux which may in
uence where the convection
rolls form. This persistent non-axisymmetric morphology of the geomagnetic eld
is hypothesized to re
ect lower mantle boundary conditions (likely temperature
anomalies) that help organize core convection. A more accurate characterization
of
ux lobe motion over longer timescales provides an important constraint on
conditions in the lower mantle.
122
5.1 South Atlantic Anomaly
Another large-scale feature of the geomagnetic eld of interest is the region on
the core-mantle boundary that includes the weakened and reverse
ux at midlat-
itudes in the southern hemisphere, the source of the South Atlantic Anomaly. In
current models of Earth's eld it appears as an area several thousand kilometers
in diameter with a minimum just o the coast of Brazil. As discussed in chap-
ter 1, the decay of Earth's dipole over the past centuries is thought to be related
to its growth. Contour maps of GUFM1 suggest some movement westward and
growth in areal extent from 1600 - 1900 AD. Models based on PSVMOD2.0 con-
sistently exhibit a reverse
ux area in this region but there is no persistent pattern
of movement of growth over time.
123
Figure 5.1: Pattern of convection in Earth's outer core proposed by Busse [10].
124
Chapter 6
Spherical Harmonic Analysis of
the Solar Magnetic Field Over the
Past Three Solar Cycles
6.1 Introduction - A Comparison between the
Sun and Earth
Understanding Earth's magnetic eld at its source requires extrapolating mod-
els based on surface measurements downward to the top of the liquid outer core
where convective motions sustain the geodynamo. In contrast, the solar magnetic
eld is routinely measured directly at the photosphere, just above its source region.
Although there are many obvious dierences between the magnetic elds of the
Sun and the Earth, it is likely that aspects of their internal dynamos are similar
including the fact that both undergo complete reversals of their large-scale mag-
netic polarity. Because of the absence of high quality data records, there are many
unanswered questions regarding geomagnetic polarity reversals. Yet high quality
measurements of the solar magnetic eld have been made on a daily basis over the
past several decades and the details of the evolution of the solar magnetic eld
can be tracked through three reversals. These measurements of the solar magnetic
eld can be used to inform our understanding of Earth polarity reversals.
125
Despite obvious dierences in the major physical properties of the Sun and
the Earth, including their size, rotation rate, sources of energy and strength and
characteristics of their magnetic elds, there are some important commonalities as
well. For both the Earth, a rapidly rotating planet and the Sun, a slowly rotating
star more than one hundred times its size, it is widely accepted that their magnetic
elds are sustained by dynamo processes occurring deep in the interior [11]. At
Earth, convective motions in the
uid outer core, a spherical shell that comprises
more than a third of the Earth's radius, generate the magnetic eld, whose large-
scale structure at the surface is primarily dipolar. Models of Earth's eld can be
extrapolated downward to the surface of the outer core where the eld is generated.
Here the magnetic eld retains its dipolar nature but is characterized by small-
scale multipolar magnetic structure, including paired, bipolar
ux patches [6]. The
global solar magnetic eld is generated in the convection zone in the outer third of
the Sun. The solar photosphere, its visible surface, is characterized by a complex
small-scale, highly multi-polar magnetic eld, including sunspots, paired bipolar
regions of intense magnetic
ux. The complexity of the photospheric eld pattern
simplies in the overlying corona as higher order multipoles fall o more rapidly
with distance and the eld is primarily dipolar.
Based on characteristics of the magnetic eld and proximity to the source
region, we can identify analogous regions at the Sun and Earth. The eld at the
surface of Earth's outer core, characterized by small-scale magnetic structure is
analogous to the complex eld at the solar photosphere and the eld in the solar
corona, dominated by the dipole and low-order multipoles, is similar to the dipolar
eld measured at Earth's surface.
Another similarity between the Sun and Earth is the complete reversal of the
polarity of their large-scale magnetic elds. For the Sun the dominant magnetic
126
polarity reverses in a regular, periodic manner roughly every eleven years. At
Earth, polarity reversals have occurred intermittently hundreds of times in the
past, the most recent 780,000 years ago [56].
Reversals of the Earth's magnetic eld are poorly understood phenomenon since
our knowledge relies on the fragmentary nature of the geologic record. Reversals
are thought to be short in duration, possibly thousands of years, and because
of this, data from the geomagnetic transitional state is sparse. Paleomagnetic
records that have recorded geomagnetic polarity reversals with good time resolu-
tion through transitions are rare. To understand the morphology of the eld, a
global distribution of records is needed as well. Despite several decades of reversal
studies, ideas about magnetic eld morphology during reversals are still debated.
Yet understanding the nature of geomagnetic polarity reversals is one of the most
fundamental problems in geomagnetism since the temporal evolution of eld geom-
etry during a reversal is intimately related to the process of eld generation in the
Earth's outer core [20].
In contrast, the Sun provides an accessible example of how magnetic elds are
generated in stars. The solar magnetic eld has been monitored by direct mea-
surement for more than thirty-ve years, over four solar cycles and three complete
reversals of the Sun's magnetic polarity. Direct measurements of the photospheric
eld contrast with our knowledge of Earth's eld at the surface of the outer core
which is inferred by downward continuation of measurements at the surface. Based
on the identication of similar morphological regions at the Sun and Earth dened
by similar magnetic eld characteristics and analogous aspects of eld behavior, we
suggest that solar polarity reversals can provide an instructive analogy for geomag-
netic polarity reversals. In this analysis we use the spherical harmonic coecients
determined from routine measurements of the solar magnetic eld to investigate
127
the varying multipolar nature of the eld over the past four solar cycles, with a
particular emphasis on reversals of solar magnetic polarity. How can a comparison
of the varying multipolar nature of the solar magnetic eld with that of Earth
inform our understanding of geomagnetic polarity reversals?
6.2 Geomagnetic Polarity Reversals
We rely on paleomagnetic records of inclination, declination and intensity from
sedimentary and volcanic rocks that have acquired their magnetization during
polarity reversals to reconstruct the record of Earth's eld. Based on analysis of
these records certain characteristics are generally accepted.
There is general consensus that the eld has reversed polarity hundreds of times
in Earth's history. Reversal durations are short (1000's of years) compared with
the stable polarity intervals separating them. The average duration of polarity
intervals varies from a few tens of thousands of years to superchrons which have
lasted millions of years. Paleomagnetic records show that the intensity of the eld
decreases substantially during a reversal, to as low as 10 percent of the intensity
outside the transition.
Other details of the reversal process remain poorly understood. In particular,
a description of directional changes during polarity transitions remains controver-
sial. Because of the importance to dynamo theory, the morphology of the eld
during polarity reversals has been the subject of intense study. Many paleomag-
netic studies use the concept of a virtual geomagnetic pole (VGP) to describe eld
properties. Virtual geomagnetic poles (VGPs) are derived from records of inclina-
tion and declination and denote the position of the magnetic pole of a geocentric
dipole which would have produced the locally observed measurement.
128
In recent decades, as an increasing amount of high-quality paleomagnetic data
have become available and a number of phenomenological models have been pro-
posed to describe the reversal process. Many of these and are summarized by
Merrill and McFadden [56]. Earliest studies suggested that the geomagnetic eld
largely retained its dipolar nature, and reversals occur either by a decay of the
dipole eld and a subsequent build up in the opposite direction or by a rotation
of the dipole without a change in intensity. Subsequent studies found VGP paths
obtained from records at dierent site locations were dierent, suggesting the the
reversal was non-dipolar. Others proposed that the transitional eld is character-
ized by higher order zonal harmonics and that the eld is signicantly asymmetric
with respect to the equator. There is no general consensus on any of these points.
It has been proposed that VGPs form clusters (hang-up points) where the
VGPs lingers for a relatively long period and that the transitional eld seems to
prefer two distinct longitude bands along the Americas and roughly 180

away
along Australia and East Asia, preferred paths (gure 6.1). If this is true, there
are important geomagnetic implications, since preferred longitude bands suggest
some form of lower mantle control during polarity reversals. All of these ideas
about eld morphology during reversals are still strongly debated and there is
little scientic consensus.
Symmetry Families
In the context of dynamo theory, symmetry properties of the magnetic eld
are important and symmetry families have been dened and investigated for the
Earth [71] and other planets [65]. Theorists typically divide the terms in the
spherical harmonic series describing the geodynamo into two families, those that
are antisymmetic with respect to the equator and those that are symmetric. For
129
Figure 6.1: Earth preferred paths.
the anti-symmetric family, referred to as primary or dipolar, (n-m) is an odd integer
and for terms in the symmetric or quadrupole family (n-m) is even (where n and
m are spherical harmonic coecient degree and order, respectively).
The relative contribution of the two symmetry families to the paleomagnetic
eld has been examined [52]. In its usual state, the geomagnetic eld is dominated
by the anti-symmetric, primary family (the axial dipole), however it has been
suggested that the symmetric family might increase substantially during a reversal
and that a polarity transition is characterized by increasing interaction between
the two families [55]. This has been found to occur in the dynamo simulation of a
reversal of Glatzmaier and Roberts [18] and [17].
The idea of symmetry families originated with Bullard and Gellman [9] who
pointed out that any divergence-free vector eld can be separated into a toroidal
part T, and a poloidal part, S . In their formulation, dynamo action is described
as a component of
uid
ow (which they express as s

, for example) acting on an
existing component of the magnetic eld (S

) to produce an alternate component
of the eld (T

). In the Bullard-Gellman analysis, many interactions have no
130
eect. For example, the whole class oft

T

S

interactions are null. This raises the
possibility of the existence of separate dipole and quadrupole dynamo mechanisms.
Roberts and Stix [66] pointed out that in dynamo eld generation by mean eld
electrodynamics in which small-scale turbulent
uctuations of magnetic eld and

uid
ow occur about large scale mean-eld values result in similar interactions.
In a sphere with a mean velocity eld symmetric about the equator and an-eect
with odd parity, physically reasonable for a source arising from the Coriolis force
in a rotating sphere, the magnetic eld solutions separate into two independent
dipole and quadrupole families.
In a study of paleomagnetic data over the last 200 million years, Lee and Lilley
[38] examined the axial dipole, quadrupole and octupole coecients and concluded
that the octupole coecient may be linked more closely to the dipole coecient
than the quadrupole coecient. Figure 6.2 shows their analysis of the geomag-
netic eld over the past 195 million years. The top panel shows the ratio of the
quadrupole to dipole coecient and the bottom panel shows the octupole:dipole
ratio. The octupole:dipole ratio is positive throughout the interval indicating that
the signs of those terms covary. The quadrupole:dipole is as likely to be positive
as negative suggesting these terms do not covary. Lee and Lilley were the rst to
suggest that these two families might be useful in paleomagnetic analysis.
McFadden et al., [51] used the idea of symmetry families to produce a very
simple model for the observed latitudinal variation of VGP scatter. They found
that the dipole and quadrupole families contribute very dierently to the scatter of
VGP with latitude. In a contemporary model of Earth's eld at the time (IGRF65),
terms of the dipole family had a strong, almost linear latitudinal dependent con-
tribution to VGP scatter, whereas the quadrupole contribution varied little with
131
804 S. LEE and F. E. M. LILLEY
(a)
(b)
Fig. 1. Behaviour of the geomagnetic field over the last 195Myr from LEE et al. (1986). (a) The G2
coefficient. (b) The G3 coefficient. The vertical bars represent 95% confidence intervals for the plotted
points. The points are plotted in the middle of the time spans which they represent. These time spans
are: present-5, 5-22.5, 22.5-45, 45-80, 80-110, 110-145, and 145-195Myr.
polarity paleomagnetic data combined, and the work of LEE (1983) found similar
results for normal and reversed polarity data taken separately. As discussed in the
previous section, such consistency must mean that g02 and g03 have generally reversed
together with g01, indicating that on the time-scales of geomagnetic reversals all three
coefficients g1, g2 and g3 are linked. The present speculative interpretation of this
result is that such reversals are essentially magnetic phenomena, with magnetic field
reversing but (mean) velocity field remaining unchanged.
Figure 6.2: Figure showing the ratio of the axial quadrupole to axial dipole coe-
cients, g
20
/g
10
(panel a) and ratio of the axial octupole to axial dipole coecients
g
30
/g
10
(panel b) for the geomagnetic eld at the core mantle boundary over the
last 195 Myr [38].
132
latitude. They suggested that this picture could describe paleosecular variation in
the lava
ow data they analyzed spanning the past 5 million years.
Merrill and McFadden [55] proposed that although the dipole and quadrupole
families are decoupled most of the time, they may be coupled across polarity tran-
sitions and suggested a model in which reversals are initiated when the quadrupole
family couples more strongly with the dipole family. Evidence for decoupling of
eld symmetry families away from polarity transitions was based on a comparison
of the DGRF models of Earth's eld for 1965 and 1985. That analysis suggested
that there is no consistent drift in the dipole family members, but in general
quadrupole family members tend to drift westward at approximately :5

per year.
Evidence for closer coupling of the transitional eld came from analysis which
indicated that the g
10
coecients and g
20
coecients reverse sign together across
polarity transitions [53].
6.2.1 Geomagnetic Excursions
There is also growing evidence in paleomagnetic records for excursions, short-
lived periods characterized by extremely anomalous magnetic eld directions [46].
They seem to occur during times of anomalously low paleointensity and some think
they may be aborted reversals. Excursions are one of the less well-understood
aspects of eld behavior. As with polarity reversals, we have an incomplete pic-
ture of the excursional eld because few excursions have sucient paleomagnetic
records. Examples are shown in gures 6.4 and 6.3.
133
6.3 The Solar Magnetic Field
6.3.1 Solar Magnetograms
Magnetograms are composite images of the strength and direction of the mag-
netic eld over the surface of the Sun. They rely on the principle of Zeeman
splitting of the 5250.2

A Fe I spectral line in proportion to the strength of the
magnetic eld. A detailed description of the measurement technique is provided
by Hoeksema [22]. Magnetograms are obtained by scanning the solar disk and
daily magnetograms of the large-scale photospheric eld have been made at the
Wilcox Solar Observatory (WSO) at Stanford University since May, 1976. They
provide a continuous record of the evolution of the solar magnetic eld that spans
more than four solar cycles and three complete reversals of the solar magnetic
polarity.
An important use of these data is to determine the conguration of the coronal
as well as the heliospheric magnetic eld by means of a Potential Field Source
Surface model 6.5. In the model the coronal eld is approximated as a potential
eld which is assumed to be completely radial at some height (the source surface)
above the photosphere, as suggested by some eclipse pictures. Beyond the source
surface, the model assumes that the eld is simply advected radially outward into
the heliosphere by the solar wind. Using the observed line-of site photospheric
eld as one boundary condition and the requirement of a purely radial eld at the
source surface, a solution to Laplace's equation (r
2
, thusB =rV ) can be found
and the coecients associated with the standard spherical harmonic expansion for
the solar magnetic eld at the photosphere can be derived. Although the model
makes some simplifying assumptions, the accuracy and validity of the model has
been repeatedly conrmed by comparison with spacecraft measurements of the
134
interplanetary magnetic eld. The spherical harmonic coecients to degree 20
describing the solar magnetic eld based on the model for each solar rotation are
available on a continuous basis since measurements began more than thirty-ve
years ago. Coecients to degree 9 for the standard spherical harmonic series
expansion describing the solar magnetic eld were obtained from the online data
archive (http://wso.stanford.edu/). Coecients in units ofTesla (.01 Gauss) are
available on a monthly basis from May, 1976 through August, 2012.
Over time the source surface model calculations have been modied to agree
more closely with measurements in interplanetary space or at Earth. The two
models produced are referred to as the classic and the radial models. The classic
computation locates the source surface at 2.5 solar radii (Rs), assumes that the
photospheric eld has a meridional component and requires a somewhat ad hoc
polar eld correction to more closely match the observations of the interplanetary
magnetic eld structure at Earth. The radial computation assumes the eld in
the photosphere is radial and uses no polar eld correction. Two variants of the
radial model have been produced. One locates the source surface at 2.5 Rs and
the other at 3.25 Rs. The radial model with a lower source surface radius of 2.5 Rs
gives a better overall match to the structure of the inner corona and estimations
of the maximum inclination of the heliospheric current sheet to the ecliptic and
is recommended by Hoeksema (http://wso.stanford.edu/) if only because of its
simplicity. All results presented here use the radial model with the source surface
located at 2.5 Rs.
135
6.4 Analysis
6.4.1 Large scale solar eld - Spherical harmonic coe-
cients
Figure 6.6 shows the behavior of the large-scale solar magnetic eld for more
than three complete solar cycles, starting in May, 1976 based on the (WSO) Source
Surface solar magnetic eld model spherical harmonic coecients. The most recent
data included is from August, 2012. Panel a shows the monthly sunspot numbers
that track the solar cycle, for solar cycles 21 to 24. The sunspot number is produced
by the Solar In
uences Data Analysis Center for the Sunspot Index (SIDC) at the
Royal Observatory of Belgium. These data are obtained from the National Oceanic
and Atmospheric Administration's National Geophysical Data Center website at:
http://www.ngdc.noaa.gov/stp/solar/ssndata.html. This plot is included for ref-
erence in many of the subsequent gures. Panel b shows the axial dipole coecient,
g
10
, panel c is the axial quadrupole g
20
, and panel d the axial octupole, g
30
. The
magnetic polarity reversal is identied as the solar rotation in which g
10
changes
sign, indicated by the vertical blue lines. Solar polarity reversals occur near the
sunspot number maxima. Three complete reversals are shown and the polarity
reversal for the current solar cycle (24) appears to be imminent. The reversal in
solar cycle 21 occurs during Carrington Rotation 1694 in April, 1980. The reversal
during solar cycle 22 is somewhat ambiguous. The sign of g
10
rst changes dur-
ing Carrington Rotation 1823 and oscillates between positive and negative values
for the next half year, (December, 1990 - May, 1991). The reversal of solar cycle
23 occurs during Carrington Rotation 1958, in January, 2000. The time between
reversals ranges between 129 - 135 solar rotations.
136
For each of the three solar polarity reversals, the change in sign of the axial
dipole is accompanied by a change in sign of the octupole coecient, g
30
(panel
c). This spherical harmonic coecient represents magnetic
ux that, like the
dipole eld, is antisymmetric with respect to the solar equator and concentrated
poleward. This gure shows that at the photosphere the magnitude and sign of
the axial dipole and axial octupole vary together and out of phase with the solar
cycle, reaching their maximum values at solar minimum.
Panel d of gure 6.6 tracks the colatitude of the dipole and shows that the
reversal occurs by the magnetic dipole rotating through the solar equatorial plane,
(90

colatitude, indicated by the dashed line). For a signicant fraction of the
solar cycle, the dipole eld is not aligned with either the southern or northern
solar geographic pole. Over a complete solar cycle the dipole colatitude is greater
than 45

away from either solar pole for approximately 17% of the time. For the
geomagnetic eld, excursions are dened as magnetic eld directions whose equiv-
alent virtual geomagnetic pole is between 45

and 135

away from the geographic
pole. By analogy with Earth, using this criteria, the solar magnetic eld would be
considered to be in an excursional state much of the time.
Panels f and g of gure 6.6 show how the magnetic dipole energy is apportioned
over the solar cycle. Panel e shows the magnitude of the equatorial dipole spherical
harmonic coecients (g
2
11
+h
2
11
)
1=2
and panel f shows the magnitude of the axial
dipole coecient. These two parameters vary roughly in anti-phase, as the axial
and total dipole weakens leading to the polarity reversal the magnitude of the
equatorial dipole increases. The polarity reversal occurs when the axial dipole
is at its minimum value. The equatorial dipole remains enhanced through solar
maximum, spiking to a peak value as solar activity decreases during the descending
phase of the solar cycle.
137
The polarity reversal can thus be characterized as a decay in total (and axial)
dipole energy simultaneous with a rotation of the magnetic dipole through the
solar equator to the opposite pole.
6.4.2 Ratios of low degree axial eld coecients at the
photosphere
In gure 6.7 we examine the ratios of the low degree axial spherical harmonic
coecients to the axial dipole coecient. Panels a and b are again the sunspot
number and axial dipole coecient g
10
, for reference. In panel c, the ratio of the
quadrupole to dipole g
20
/g
10
is shown and in panel d the ratio of the octupole to
dipole coecient g
30
/g
10
is shown. The vertical scales are truncated and do not
show the full range of values because the ratio is not meaningful as peak values near
the reversals are simply due to the axial dipole coecient, g
10
approaching zero.
The octupole:dipole ratio is always positive (with the exception of brief intervals
near the reversal), indicating that the signs of those two coecients are always
the same. The quadrupole:dipole ratio is as likely to be positive as negative. The
quadrupole/dipole ratio exhibits an approximately annual periodicity, oscillating
about a mean value of zero. Inspection of the individual coecients (see gure
6.6) suggests this is primarily due to an annual variability in the axial quadrupole
coecient.
Figure 6.7 can be directly compared with the analogous gure for Earth's pale-
omagnetic eld (gure 6.2). From analysis of the paleomagnetic eld, g
10
and g
30
typically have the same sign, whereas like the solar eld g
20
and g
10
are just as
likely to have the same sign as not.
138
6.4.3 Harmonic content of the solar magnetic eld
It is instructive to examine the mean square magnetic eld as a function of
harmonic degree n, originally dened by Lowes [42]:
R
n
= (n + 1)
n
X
m=0

g
2
nm
+h
2
nm

(6.1)
As a function of degree n, R
n
forms the spatial magnetic power spectrum. R
1
represents the dipole power, R
2
the quadrupole power, etc. Figures 6.8 and 6.9
show R
n
for degrees 1 through 9. The dipole power varies out of phase with the
solar cycle as expected, but the variation in R
3
is less apparent. This can be seen
most easily in gures 6.10 and 6.11 which show these same parameters plotted on
the same logarithmic vertical scales. During solar minimum the magnitude of R
3
is comparable to R
1
. The solar reversal occurs when the total dipole power is at
its minimum value and the total octupole power is near a minimum as well.
6.4.4 Preferred reversal paths of the solar eld
The latitude and longitude of the solar magnetic dipole can be calculated
directly from the spherical harmonic coecients and the reversal path examined
for evidence of a preferred path as suggested in some reversal models for Earth.
Figure 6.12 shows the path the solar dipole takes during the reversals of solar
cycles 21 through 23. The circle indicates the starting point and the square the
end point. In all cases, there is a tendency for the reversal path to meander instead
of taking a direct path from pole to pole. The reversal paths of cycles 21 and 22
seem to occupy roughly the same solar hemisphere through the polarity change.
The reversal of cycle 23 does not follow the same pattern. An alternate display of
the same data is shown in gure 6.13 for the three reversals. The data are color
139
coded according to date as the reversal progresses. Depending on when the start
and end times of the reversals are dened, there appears to be a tendency in solar
cycles 21 and 22 for the dipole to return to approximately the same solar longitude.
There is a considerable body of evidence that suggests that certain solar activity
is associated with specic solar longitudes. Analysis of measurements of the solar
wind speed and interplanetary magnetic eld showed evidence for a persistent
dependence on solar longitude and was consistent with a model in which the solar
magnetic dipole returns to the same longitude after each reversal [57]. There are
also hints of preferred longitudes in Sun-like stars. Observation of stellar magnetic
activity (monitored via CA II emission) provides evidence that some stars have
preferred longitudes that persist over several activity cycles [2].
6.4.5 Symmetry families
Figure 6.14 showsR
n
grouped according to the even and odd symmetry families.
For ease of calculation, only terms of the solar eld to degree 6 are included. For
the reversals of cycles 21 and 22 the peak value in the odd family occurs just prior
to the polarity reversal. Both the odd and even families vary with the solar cycle
although the even terms vary more dramatically over the solar cycle than the odd
terms.
6.4.6 Solar coronal eld
As mentioned in section 6.1, as part of our analogy between the solar magnetic
eld and that of the Earth, we have identied analogous regions based on proximity
to the source region where the eld is generated. The photospheric eld investi-
gated thus far, is analogous to the magnetic eld at the core-mantle boundary and
the eld in the solar corona is analogous to the eld measured on the surface of the
140
Earth. We extrapolate the mean square magnetic eld R
n
to the solar corona at
a distance of 2.5 Rs. Since the nth harmonic falls o as r
(n+2)
, the mean square
eld as a function of harmonic degree at the solar corona is:
R
n
(r) =R
n
(a)(a=r)
2l+4
(6.2)
where a refers to the solar photosphere. Figure 6.15 shows R
n
at the solar corona
for degree 1-3. Power in the higher degree terms is attenuated signicantly by
2.5 Rs and is not visible on this vertical scale. At the solar corona, the dipole
dominates the total eld except right at solar maximum.
6.4.7 Current solar cycle
From virtually all the parameters examined, it is clear that the magnitude
of solar magnetic eld has declined dramatically over the past three solar cycles.
Relative to the peak value reached in solar cycle 21, the maximum value of the
axial dipole coecient, g
10
declined by 15.8 percent in cycle 22 and a full 46.6 %
by cycle 23. Similar values for the total dipole power, R
1
are 24% and 61.3%.
The total quadrupole power R
2
declined by 13.5% and 50.9% relative to cycle 21
and octupole power, R
3
declined by 42% and 59.5%. Power in the higher order
harmonics show similar trends.
6.4.8 Conclusions
Based on this analysis, we can make certain inferences about reversals of the
solar magnetic eld. A detailed examination of the morphology of the solar eld
through the reversal shows that the magnetic dipole rotates through the equa-
tor as the reversal occurs. There is a tendency for the magnetic dipole to follow
141
a preferred path in latitude and longitude from one rotation to the next. The
magnitude of the solar dipole decreases and the reversal occurs when it is a mini-
mum. Although the magnitude of the octupole eld g
30
follows a similar pattern
and is a minima at the reversal, there is no convincing evidence that symmetry
families play a strong role. The relative proportion of the symmetry families does
not change leading up to the reversal. Whether the geomagnetic eld behaves in a
similar manner is not certain but understanding reversals of the solar eld provides
a framework for thinking about reversals on Earth.
142
Figure 6.3: Declination, inclination and relative paleointensity for the Laschamp
Excursion [46].
143
Figure 6.4: VGP paths associated with Laschamp Excursion records from JPC14.
A distinctive pattern of two successive clockwise looping is observed in the VGP
path [46].
Figure 6.5: Schematic depicting solar potential eld source surface model used to
derive spherical harmonic coecients.
144
-200
-100
100
200
g
20
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
-300
-100
100
300
g
10
b)
-300
-100
100
300
g
30
d)
0
30
60
90
120
150
dipole
colatitude
e)
0
50
100
150
equatorial
dipole
f)
0
50
100
150
200
1975 1980 1985 1990 1995 2000 2005 2010 2015
|g
10
|
g)
Figure 6.6: Monthly sunspot number (panel a), spherical harmonic dipole coef-
cient, g
10
(panel b), axial ocutpole coecient g
30
, (panel c), dipole colatitude
(panel d), magnitude of equatorial dipole (panel e, see text for description) and
magnitude of the axial dipole coecient (panel f). Units of coecients areTesla.
Solar cycle numbers are shown in the top panel.
145
0
50
100
150
200
sunspot number
a)
-3
-2
-1
0
1
2
3
g
20
/g
10
c)
-4
-2
0
2
4
1975 1980 1985 1990 1995 2000 2005 2010 2015
g
30
/g
10
d)
-300
-200
-100
0
100
200
300
g
10
b)
Figure 6.7: Monthly sunspot number (panel a), spherical harmonic dipole coe-
cient, g
10
(panel b), ratio of the axial quadrupole to axial dipole coecients (g
20
/g
10
(panel c) and ratio of the axial octupole to axial dipole coecients g
30
/g
10
(panel d) for Vertical scales in panels c and d are truncated at low values so that
details of data can be seen.
146
0
5 10
4
1 10
5
1.5 10
5
2 10
5
2.5 10
5
degree 2
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
0
2 10
4
4 10
4
6 10
4
8 10
4
1 10
5
degree 1
b)
0
2 10
5
4 10
5
6 10
5
8 10
5
degree 3
d)
0
5 10
5
1 10
6
1.5 10
6
degree 4
e)
0
5 10
5
1 10
6
1.5 10
6
2 10
6
2.5 10
6
1975 1980 1985 1990 1995 2000 2005 2010 2015
degree 5
f)
Figure 6.8: Mean square eld at the photosphere for harmonic degrees 1 through
5.
147
0
1 10
6
2 10
6
3 10
6
4 10
6
degree 7
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
0
1 10
6
2 10
6
3 10
6
4 10
6
degree 6
b)
0
1 10
6
2 10
6
3 10
6
4 10
6
5 10
6
6 10
6
degree 8
d)
0
2 10
6
4 10
6
6 10
6
8 10
6
1975 1980 1985 1990 1995 2000 2005 2010 2015
degree 9
e)
Figure 6.9: Mean square eld at the photosphere for harmonic degrees 6 through
9.
148
100
1000
10
4
10
5
10
6
degree 2
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
100
1000
10
4
10
5
10
6
degree 1
b)
100
1000
10
4
10
5
10
6
degree 3
d)
100
1000
10
4
10
5
10
6
degree 4
e)
100
1000
10
4
10
5
10
6
1975 1980 1985 1990 1995 2000 2005 2010 2015
degree 5
f)
Figure 6.10: Mean square eld at the photosphere for harmonic degrees 1 through
5, log plot.
149
100
1000
10
4
10
5
10
6
degree 7
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
100
1000
10
4
10
5
10
6
degree 6
b)
100
1000
10
4
10
5
10
6
degree 8
d)
100
1000
10
4
10
5
10
6
1975 1980 1985 1990 1995 2000 2005 2010 2015
degree 9
e)
Figure 6.11: Mean square eld at the photosphere for harmonic degrees 6 through
9, log plot.
150
-80 ˚ -80 ˚
-60 ˚ -60˚
-40 ˚ -40 ˚
-20˚ -20 ˚
0˚ 0˚
20˚ 20 ˚
40 ˚ 40˚
60 ˚ 60 ˚
80˚ 80˚
-80˚ -80˚
-60˚
-60 ˚
-40 ˚ -40˚
-20˚ -20˚
0˚ 0˚
20 ˚ 20 ˚
40˚ 40˚
60 ˚
60 ˚
80˚ 80˚
-80 ˚ -80˚
-60˚ -60˚
-40˚ -40˚
-20˚ -20˚
0˚ 0˚
20˚ 20 ˚
40˚ 40 ˚
60 ˚ 60 ˚
80˚ 80 ˚
Figure 6.12: Solar dipole reversal paths for solar cycles 21-23. Circle marks the
starting point and square is the end point of the reversal.
151
0 50 100 150 200 250 300 350
0
20
40
60
80
100
120
140
160
180
Carrington Rotations 1668−1697

1978.4
1978.6
1978.8
1979
1979.2
1979.4
1979.6
1979.8
1980
1980.2
1980.4
0 50 100 150 200 250 300 350
0
20
40
60
80
100
120
140
160
180
Carrington Rotations 1791−1855

1988
1988.5
1989
1989.5
1990
1990.5
1991
1991.5
1992
0 50 100 150 200 250 300 350
0
20
40
60
80
100
120
140
160
180
Carrington Rotations 1942−1970

1999
1999.2
1999.4
1999.6
1999.8
2000
2000.2
2000.4
2000.6
2000.8
Figure 6.13: Solar dipole reversal paths for solar cycles 21-23 color coded according
to time.
152
8 10
5
2.4 10
6
4 10
6
5.6 10
6
even
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
8 10
5
2.4 10
6
4 10
6
5.6 10
6
odd
b)
0
10
20
30
40
50
1975 1980 1985 1990 1995 2000 2005 2010 2015
odd/even
d)
Figure 6.14: R
n
at the solar photosphere to degree 6, symmetry families are shown.
153
0
100
200
300
400
degree 2
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
0
100
200
300
400
degree 1
b)
0
100
200
300
400
1975 1980 1985 1990 1995 2000 2005 2010 2015
degree 3
d)
Figure 6.15: R
n
at the solar corona. (Attenuated by 2n+4). Same vertical scale
154
0
100
200
300
400
degree 2
c)
0
50
100
150
200
sunspot number
a)
21 22 23 24
0
100
200
300
400
odd
even
degree 1
b)
0
100
200
300
400
degree 3
d)
0
100
200
300
400
1975 1980 1985 1990 1995 2000 2005 2010 2015
e)
degree 1 - 3
Figure 6.16: R
n
at the solar corona. (Attenuated by 2n+4). Same vertical scale.
Odd and even as function of degree through degree 3 and sum of even and odd.
155
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162
Appendix A
Tables of site locations
163
Site code Site location Lat Lon
North America
2220 St. Lawrence Swy 291.4 48.6
BLUI Blue Lake, Idaho 243.2 46.2
ECS St. Lawrence Swy 295.5 48.2
ELK Elk Lake 264.8 47.2
FIS Fish Lake 241.4 42.6
HAW Hawaii 205 19.5
KLM Kylen Lake 268.2 47.3
LEB Lake LeBeouf 280.1 41.9
LSC Lake St. Croix 267.2 45
MAM Mesoamerica 265 17
MAR Mara Lake 241 50.7
OWL Owens Lake 242 36.4
PEP Lake Pepin 268 44.4
PSVL W. USA 241.4 42.6
PYR Pyramid Lake 240.4 40
SAAN Saanich Inlet 236.5 48.6
SAN Sandy Lake 279.9 41.3
SEN Seneca Lake 283 43
WAI Lake Waiau 204.5 19.8
WUS W. USA 250 35
ZAC Zaca Lake 240 34.8
Table A.1: PSVMOD2.0 sites and locations for North America. Longitudes are
east longitude.
164
Site code Site location Lat Lon
Europe
983 Greenland/Iceland 336.4 60.4
2269 N. Iceland 339.5 66.4
2322 E. Greenland 329.5 67.1
ASL Lake Aslikul 54.1 54.2
BIR Lake Birkat Ram 35.7 33.2
BUL Bulgaria 25.0 43.0
CAS Central Asia 58.0 38.0
CAU Caucasus 45.0 40.0
EGY Egypt 33.0 26.0
GEO Georgia 44.0 42.0
GRE Greece 23.5 37.0
HUN Caucasus 45.0 40.0
HZM Lake Holzmaar 6.8 50.1
ICE Lake Vatnsdalsvatn 338.0 67.0
KIN Lake Kinneret 35.7 32.4
LGN Lough Neagh 353.6 54.4
LLO Loch Lomond 356.0 55.0
POH Lake Pohjajarvi 28.0 62.8
SWE Lake Frangsjon 19.7 64.0
TRI Lake Trikhonis 21.5 38.6
UKR Ukraine 30.0 50.0
VUK Lake Vuokonjarvi 29.5 63.4
WUR Western Europe 0.0 50.0
Table A.2: PSVMOD2.0 sites and locations for Europe. Longitudes are east lon-
gitude.
165
Site code Site location Lat Lon
East Asia
1202 East China Sea 122.5 24.8
BAI Lake Baikal 106.1 52.4
BIW Lake Biwa 136.1 35.3
IND India 79 11
JPN Japan 135 35
KOR Korea 127.2 36.5
LAM(PC1341) Lake Lama 90.2 69.5
MD81 Philippine Coast 125.8 6.5
MD77 Indonesia Coast 119.1 1.4
MD62 Indonesia Coast 117.9 4.7 S
MON Mongolia 106 46
PRC NE China 115 35
Arctic
BAF Ban Island Fjords 290.0 69.0
B650 Beaufort Sea 228.4 71.3
B803 Beaufort Sea 224.1 70.6
CH05 Chukchi Sea 201.6 72.7
CH06 Chukchi Sea 203.0 72.7
CH08 Chukchi Sea 203.2 71.6
GFL Grandfather Lake 201.4 59.8
JPC15 Chukchi Sea 206.6 72.0
JPC16 Chukchi Sea 206.6 72.0
GGC19 Chukchi Sea 204.5 72.1
MUR Murray Lake 270.1 81.6
SAW Sawtooth Lake 276.1 79.3
hline
Table A.3: PSVMOD2.0 sites and locations for East Asia and the Arctic. Longi-
tudes are east longitude and sites in the southern hemisphere are indicated by an
S.
166
Site code Site location Lat Lon
South America and Antarctica
1098 Palmer Deep 295.7 64.9 S
1233 Chile Margin 285.5 41.0 S
ESC Lake Escondido 288.4 41.1 S
GER Gerlache-Boyd 298.3 63.1 S
LIS Larsen-A Ice Shelf 299.6 64.8 S
MOR Lake Moreno 288.45 41.1 S
PER Peru Archeomag 285 10 S
POT Laguna Potrok Aike 289.6 52 S
TRE Laguna El Trebol 288.5 41.1 S
WIL Wilkes Land Basin 144.1 64.9 S
Southwest Pacic
AUS SE Australia 145 38 S
EAC Lake Eacham 145.6 17.3 S
KEI Lake Keilembete 142.9 38.2 S
NZE New Zealand 174.9 35.8 S
POU Lake Pounui 175.2 41.3 S
Africa
TUR Lake Turkana 36.6 2.6
Table A.4: PSVMOD2.0 sites and locations for South America and Antarctica,
the Southwest Pacic and Africa. Longitudes are east longitude and sites in the
southern hemisphere are indicated by an S.
167
Appendix B
North America Site Locations
168
Table 1: Summary of (non-Arctic) North American Paleomagnetic Secular Variation Records Used
Directional Data Sets
Site Site Age Range # of Final Sed.
Code Location Lat(°N) Long(°E) (ybp) Type Dates Chron. Rate Qual.
2220 St. Lawrence Swy 48.6 291.4 0-8000 sediment independent ~100 cm/ky A
BLUI Blue Lake, Idaho 46.2 243.2 0-3800 sediment 6 independent ~100 cm/ky A
ECS St. Lawrence Swy 48.2 295.5 0-8000 sediment >10 independent ~100 cm/ky A
ELK Elk Lake 47.2 264.8 0-8000 sediment varved independent ~150 cm/ky A
FIS Fish Lake 42.6 241.4 0-8000 sediment 16+3 ashes independent 100 cm/ky A
HAW Hawaii 19.5 205 0-3000 lava - independent - A
KLM Kylen Lake 47.3 268.2 4200-8000 sediment 5 independent 130 cm/ky A
LEB Lake LeBeouf 41.9 280.1 0-4000 sediment 6 derived 200 cm/ky A
LSC Lake St. Croix 45 267.2 0-8000 sediment 9 independent 200 cm/ky A
MAM Mesoamerica 17 265 0-2000 archeomag - independent - A
MAR Mara Lake 50.7 241 0-4500 sediment 2 ashes independent ~100 cm/ky A
OWL Owens Lake 36.4 242 0-3000 sediment 9 derived 180 cm/ky A
PEP Lake Pepin 44.4 268 0-8000 sediment 1 derived 150 cm/ky B
PSVL W. USA 42.6 241.4 0-8000 lava - independent - A
PYR Pyramid Lake 40 240.4 0-3000 sediment 14 derived 180 cm/ky A
SAAN Saanich Inlet 48.6 236.5 0-8000 sediment Varved independent ~700 cm/ky A
SAN Sandy Lake 41.3 279.9 0-8000 sediment 8 independent ~180 cm/ky A
SEN Seneca Lake 43 286 0-8000 sediment 13 independent ~150 cm/ky A
WAI Lake Waiau 19.8 204.5 0-8000 sediment 5 1ndependent ~40 cm/ky A
WUS W. USA 35 250 0-2000 archeomag - independent - A
ZAC Zaca Lake 34.8 240 0-2800 sediment 12 independent ~300 cm/ky A

Intensity Data Sets
Site Site Age Range # of Final Sed.
Code Location Lat(°N) Long(°E) (ybp) Type Dates Chron. Rate Qual.
2220 St. Lawrence Swy 48.6°N 291.4°E 0-8000 sediment independent ~100 cm/ky A
ECS St. Lawrence Swy 48.2 295.5 0-8000 sediment >10 independent ~100 cm/ky A
ELK Elk Lake 47.2 264.8 0-8000 sediment varved independent ~150 cm/ky A
FIS Fish Lake 42.6 241.4 0-8000 sediment 16+3 ashes independent 100 cm/ky A
HAW Hawaii 19.5 205 0-8000 lava - independent - A
LEB Lake LeBeouf 41.9 280.1 0-4000 sediment 6 derived 200 cm/ky A
LSC Lake St. Croix 45 267.2 0-8000 sediment 9 independent 200 cm/ky A
PEP Lake Pepin 44.4 268 0-8000 sediment 1 derived 150 cm/ky A
PSVL "Lava, W. USA" 42.6 241.4 0-8000 lava - independent - A
PYR Pyramid Lake 40 240.4 0-3000 sediment 14 derived 180 cm/ky A
SEN Seneca Lake 43 286 0-8000 sediment 13 independent ~150 cm/ky A
WUS W. USA 35 250 0-2500 archeomag - independent - A
Figure B.1: Table showing a summary of sites used in North America regional study. 169
Appendix C
Expanded Regional Site Maps
170
320˚
320˚
330˚
330˚
340˚
340˚
350˚
350˚
0˚
0˚
10˚
10˚
20˚
20˚
30˚
30˚
40˚
40˚
50˚
50˚
60˚
60˚
70˚
70˚
20˚ 20˚
30˚ 30˚
40˚ 40˚
50˚ 50˚
60˚ 60˚
70˚ 70˚
983
2269
2322
ASL
BIR
BUL
CAS
CAU
EGY
GEO
GRE
HUN
HZM
ICE
KIN
LGN
LLO
POH
SWE
TRI
UKR
VUK
WUR
Figure C.1: PSVMOD2.0 European sites. Blue triangle denotes sites with directional records and red circle denotes
intensity records.
171
70˚
70˚
80˚
80˚
90˚
90˚
100˚
100˚
110˚
110˚
120˚
120˚
130˚
130˚
140˚
140˚
-10˚ -10˚
0˚ 0˚
10˚ 10˚
20˚ 20˚
30˚ 30˚
40˚ 40˚
50˚ 50˚
60˚ 60˚
70˚ 70˚
1202
BAI
BIW
IND
JPN
KOR
LAM
MD81
MD77
MD62
MON
PRC
Figure C.2: PSVMOD2.0 East Asian sites. Blue triangle denotes sites with direc-
tional records and red circle denotes intensity records.
172
190˚
190˚
200˚
200˚
210˚
210˚
220˚
220˚
230˚
230˚
240˚
240˚
250˚
250˚
260˚
260˚
270˚
270˚
280˚
280˚
290˚
290˚
300˚
300˚
50˚ 50˚
60˚ 60˚
70˚ 70˚
80˚ 80˚
BAF
B650
B803
CH05 CH06
CH08
GFL
JPC15/16 GGC19
MUR
SAW
Figure C.3: PSVMOD2.0 Arctic sites. Blue triangle denotes sites with directional
records and red circle denotes intensity records.
173
250˚
250˚
260˚
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Figure C.4: PSVMOD2.0 South American and Antarctic sites. Blue triangle
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174
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Figure C.5: PSVMOD2.0 Africa site. Blue triangle denotes sites with directional
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Abstract (if available)

Abstract High‐quality time‐series paleomagnetic measurements have been used to derive spherical harmonic models of Earth's magnetic field for the past 2,000 years. A newly‐developed data compilation, PSVMOD2.0 consists of time‐series directional and intensity records that significantly improve the data quality and global distribution used to develop previous spherical harmonic models. PSVMOD2.0 consists of 185 paleomagnetic time series records from 85 global sites, including 30 full‐vector records (inclination, declination and intensity). It includes data from additional sites in the Southern Hemisphere and Arctic and includes globally distributed sediment relative paleointensity records, significantly improving global coverage over previous models. PSVMOD2.0 records have been assessed in a series of 7 regional intercomparison studies, four in the Northern Hemisphere and 3 in the southern hemisphere. Comparisons on a regional basis have improved the quality and chronology of the data and allowed investigation of spatial coherence and the scale length associated with paleomagnetic secular variation (PSV) features. We have developed a modeling methodology based on nonlinear inversion of the PSVMOD2.0 directional and intensity records. Models of the geomagnetic field in 100‐year snapshots have been derived for the past 2,000 with the ultimate goal of developing models spanning the past 8,000 years. We validate the models and the methodology by comparing with the GUFM1 historical models during the 400‐year period of overlap. We find that the spatial distribution of sites and quality of data are sufficient to derive models that agree with GUFM1 in the large‐scale characteristics of the field. We use the the models derived in this study to downward continue the field to the core‐mantle boundary and examine characteristics of the large‐scale structure of the magnetic field at the source region. The derived models are temporally consistent from one epoch to the next and exhibit many of the expected characteristics of the field over time (high‐latitude flux lobes, South Atlantic reverse flux patch, north pole reverse or null flux region).

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Richardson, Marcia (author)

Core Title Full vector spherical harmonic analysis of the Holocene geomagnetic field

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Geological Sciences

Publication Date 04/24/2014

Defense Date 04/24/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,paleomagnetic field

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lund, Steven P. (committee chair), Daeppen, Werner (committee member), Däppen, Werner (committee member), Miller, Meghan S. (committee member), Sammis, Charles G. (committee member)

Creator Email marci_burton@yahoo.com,mburton@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-385115

Unique identifier UC11295746

Identifier etd-Richardson-2413.pdf (filename),usctheses-c3-385115 (legacy record id)

Legacy Identifier etd-Richardson-2413.pdf

Dmrecord 385115

Document Type Dissertation

Format application/pdf (imt)

Rights Richardson, Marcia

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA