chapter3.tex

\chapter{Methods for Producing a Reliable APWP}\label{chap:Reliab}  % chktex 36
\textit{This chapter mainly describes how to generate paleomagnetic APWPs using
168 different methods, and then the application of the new APW path similarity
measuring tool described in the last chapter to find the best APWP generating
method(s). The final results tell us that the ``Age Position Picking (APP)''  % chktex 36
method is better than the traditional ``Age Mean Picking (AMP)'' method for
making a reliable paleomagnetic APWP and weighting does not seem to have much
effect.}
\vfill
\minitoc\newpage

\section{Introduction}

A paleomagnetic pole, also known as paleopole, has many attributes, including
sampling site, number of samples, sample rock characteristics, age determination
and its uncertainty, pole location and its uncertainty etc. An APWP is generated
by combining paleopoles into mean poles, for a particular rigid block over the
desired age range to produce a smoothed path (see detailed definition of APWP in
Section~\ref{sec:applPlateTec}). An APWP reflects cumulative motions of a
continental fragment relative to the Earth's spin axis. So, in order to build an
APWP, an object continent or continental fragment should be picked first. The
general ``boundary'' of this continent or continental fragment should be
determined tectonically. This closed tectonic ``boundary'' is then used to
constrain the paleomagnetic datasets through picking only those with their
sampling sites lying inside the closed ``boundary''. See
Appendix~\ref{appen4chp3} for examples about how the paleopole datasets are
constrained for a particular tectonic plate during a specific time interval.
Then the paleopoles from those sampling sites are statistically combined
together into mean poles when their ages are close. The final products, those
mean poles, are then connected according to their estimated ages to make a
time-series path: a paleomagnetic APWP\@.

So it is important to ask a question like what can be called a reliable
paleomagnetic APWP or how to build a reliable APWP, because, for example (and
also what we want to focus our work on here is), there are many different
perspectives to consider when we combine paleopoles into mean poles. For
instance, how to group paleopoles by age when combining together into a mean
pole, and what statistical method should be used to do this combining? Should
those paleopoles be treated equally in this process of combining, and if not
what factor should be used to weight them, and how? These are the factors that
might affect the end result.

In this chapter, paleopoles' attributes are used to assess their influence on
mean poles (see details in Section~\ref{sec:w}), or to determine if they should
be omitted for producing a `better' mean pole (see details in
Section~\ref{sec:f}). Different key attributes, that can be quantified, or their
combination are considered. In addition, when we are merging paleopoles to
produce a smoothed mean path, we make choices not only about which data are
included or excluded, but how data are combined. Moving averaging is used to
combine data (see details in Section~\ref{sec:p}). With application of two
moving average methods, where each moving window includes data with (1) only
middle point of lower and higher age limits considered as paleopole's age and
(2) whole dating uncertainty considered as paleopole's age range, different
APWPs are produced to see which attribute is more important and also which
moving average method performs better. To answer these questions, we need a
reference path (see details in Section~\ref{sec:refpath}) and a tool to measure
similarity between the paleomagnetic APWPs and the reference path, which was
developed and introduced in Chapter~\ref{chap:Metho}. We have two reference
paths derived from plate kinematics models tied to different absolute reference
frames. These reference models are thought to be more accurate and more precise,
so that if the paleomagnetic APWPs are similar to the reference path, they are
regarded as more `reliable'. The question is which model predicted reference
path is a better reference or is there any difference between these two
reference paths? In this chapter, we will also try to answer this question.

Our results indicate that no single attribute as a filter or correction
universally works better than others for all the continents or continental
fragments. However, the moving average method, that considers the whole range of
paleopoles' temporal uncertainty when calculating mean pole's age is needed
(APP), is clearly a better way of generating reliable APWPs than the one that
only considers the midpoint of temporal uncertainty as a paleopole's age (AMP).
Weighting of individual paleopoles is proved to be unnecessary. The two
reference paths are not significantly different.

\subsection{How An APWP Is Generated and Key Factors}

As mentioned above, the first question to ask is that which tectonic plate the
paleopole we are working on belongs to (see Section~\ref{sec:f1} and details
about how it is solved in Appendix~\ref{appen4chp3}). The fact is that
paleomagnetic data is not just about pole location but integration of multiple
attributes. And that not all data are created equal is directly reflected by
these attributes. So once paleomagnetic data's host tectonic plate is
determined, we encounter another problem: the attributes, for example,
uncertainties of ages and also locations, can vary greatly for different
paleopoles (see details in Sections~\ref{sec:ageu} and~\ref{sec:posu}).
Paleopole's attributes like age and location obviously can be quantified and
then used to weigh paleopoles or to omit some of them through setting up a range
of acceptable values. In addition, the age uncertainty can also be used to
change the way how we do moving averaging when we calculate mean poles through
considering the whole uncertainty range instead of only considering the midpoint
of the uncertainty range in the traditional way (this is how we also tested two
different moving average methods). Besides age and position uncertainties of
paleopoles, the data consistency is also needed to be investigated carefully
(see details in Section~\ref{sec:datcons}), because some paleopoles' polarity
given in the GPMDB 4.6b~\cite[updated in 2016 by the Ivar Giaever Geomagnetic
Laboratory team, in collaboration with Pisarevsky]{M96,P05} could be wrong
although most are correct. Data density is also vital (see details in
Section~\ref{sec:datden}) because paleomagnetism is basically built on
statistics. In addition, when the paleopoles were published (publication year)
is also a key factor that potentially indicates the quality of the paleopoles
(see details in Section~\ref{sec:puby}), simply because in earlier times,
paleomagnetic instrumentation and demagnetisation techniques were not as
advanced as today. In consideration of all these factors, we organise them in
the general processing steps below in order to generate a paleomagnetic APWP\@.

\subsubsection{Picking/Binning Step}

The moving average method is used to combine paleopoles into mean poles. The
moving time window/bin picks a group of paleopoles at each time step (see
description in Section~\ref{sec:p} and Fig.~\ref{fig-nac-maplat}). So the key
factors are:

\paragraph{Width of time window}
(trade off between number of paleopoles and amount of smoothing): Smoothness of
paleomagnetic APWPs generated by moving averaging could change with different
time window lengths and steps. If window length is too wide and step is too
long, final path would be so smoothed that actual details would be missed. If
window length is too narrow and step is too short, final path would be jerky
piecewise so that too much noise would be introduced because there are too few
paleopoles in each window. So a balance needs to be achieved.

\paragraph{Age uncertainty in magnetisation,}
particularly when that uncertainty is larger than the selection window. The
midpoint between the maximum and minimum ages is commonly used as a selection
point (see an example in upper panel of Fig.~\ref{fig-nac-maplat}), but when the
minimum age is constrained by a field test (common when the age range is large;
see Section~\ref{sec:ageu}), the assumption that this is the most likely age is
questionable. As is also discussed in Section~\ref{sec:ageu}, if field test
shows magnetisation acquired prior to, for example, a folding event that is tens
of millions of years after initial rock formation, a passed field test is not
actually very useful. So if age uncertainty is very large, the possibility of
remagnetisation is very high, which means the age is probably closer to the
minimum age instead of the middle age. Since the paleopole's true age could be
anywhere within the age range, an alternative picking method includes a
paleopole if any part of its age range falls within the moving window (see an
example in lower panel of Fig.~\ref{fig-nac-maplat}).

\subsubsection{Filtering Step}

This step filters out bad data that is likely to be unreliable, based on known
information about the paleopoles. These can be subdivided into two groups:

\paragraph{Characteristics that indicate the paleopole is or is not
well-constrained (precise)}
(e.g., has a small $\alpha$95 uncertainty ellipse, large $\kappa$)

\paragraph{Characteristics that indicate the paleopole is or is not reliable
(accurate):}
\begin{itemize}
  \item Lithology, particularly if it increases the risk of inclination
    flattening in sediments (see Section~\ref{sec:datcons}).
  \item Risk of unidentified local rotations in deformed areas (see
    Section~\ref{sec:datcons}).
  \item Publication year \textemdash{} younger studies that use stepwise
    demagnetisation techniques more likely to remove overprints and isolate a
    primary remanence (see Section~\ref{sec:puby}).
  \item Sampling strategy \textemdash{} need sufficient number of contributing
    samples and sites (N and B in Table~\ref{tab-fld}) to have a good chance of
    averaging out secular variation (see Section~\ref{sec:datcons}).
  \item Field tests that constrain the magnetisation age (see
    Section~\ref{sec:ageu}).
\end{itemize}

\subsubsection{Calculation Step}

Normally all paleopoles are treated equally when calculating the \emph{Fisher}
mean~\citep[see how to calculate a \emph{Fisher} mean using the formulas 6.12, 6.13,
6.14 and 6.15 in][chap. 6; note that instead of direction declination and
inclination expected in those formulas, pole longitude and latitude should be
used]{B92}, but as an alternative or supplement to filtering paleopoles can
potentially be weighted according to the factors that potentially influence
their precision and/or accuracy such as the A95 uncertainty or age uncertainty,
and a weighted \emph{Fisher} mean calculated, giving the `better' paleopoles more
influence on the result mean pole. So, for example, each paleopole has got a
weight before calculating for a mean. In this thesis these weights $w_i$
are integrated into the $\sum\limits_{i=1}^N l_i$, $\sum\limits_{i=1}^N m_i$ and
$\sum\limits_{i=1}^N n_i$ of the formulas 6.13 and 6.14 in~\citet[chap.~6]{B92},
where $w$ means weight and $i$ is the same as in those two formulas. Then for a
weighted \emph{Fisher} mean, the two formulas 6.13 and 6.14 in~\citet[chap.~6]{B92}
become
%
\begin{equation}
  l = \frac{\sum\limits_{i=1}^N l_{i}w_i}{R} \quad\quad
  m = \frac{\sum\limits_{i=1}^N m_{i}w_i}{R} \quad\quad
  n = \frac{\sum\limits_{i=1}^N n_{i}w_i}{R}
\end{equation}
%
and
%
\begin{equation}
  R = {\left( \sum\limits_{i=1}^N l_{i}w_i \right)}^2
    + {\left( \sum\limits_{i=1}^N m_{i}w_i \right)}^2
    + {\left( \sum\limits_{i=1}^N n_{i}w_i \right)}^2
\end{equation}
%
Note that the trade-offs and difficulties are similar for filtering, e.g.\
giving poles with low A95 more weight presumes that they are closer to the
actual mean, which is not necessarily the case. Therefore, weighing on different
paleomagnetic factors is hoped to help us have some insight into these complex
issues.

\subsection{Existing Solutions and General Issues}\label{sec:si}

As mentioned in the concluding paragraph of Section~\ref{sec:f2}, if not all
paleopoles are created equal, the question becomes: how should paleopoles of
varying quality be combined? For a certain set of paleopoles, how can we produce
an APWP that is both:
\begin{enumerate}
  \item well constrained: the mean pole for each time step has a low spatial
    uncertainty (small 95\% confidence ellipse), and
  \item accurate: the mean pole position is close to its actual position at each
    time step.
\end{enumerate}

Previous work on this question have focussed on the filtering out so-called
``bad'' data before calculation of the mean pole. Commonly used schemes include:
\begin{enumerate}
  \item~\citet{v90} (V90). V90 includes seven criteria (see the details in the
    concluding paragraph of Section~\ref{sec:f2}). The Q (quality) factor is the
    total number of criteria satisfied (0\textendash7)~\citep{v88}. The Q factor
    is a very straightforward way to get a quantitative reliability score, and
    is widely used for filtering paleopoles prior to calculating
    APWPs~\citep[e.g.][]{T12,Ma16,F19}. Further each paleopole can also be
    conveniently weighted in proportion to its Q factor in the calculations of
    APWPs~\citep{T92}. But at the same time this is a fairly basic filter that
    lumps together criteria that may not be equally important.
  \item~\citet{B02} (BC02). BC02 data quality criteria use only paleopoles with
    $\alpha$95 less than 10\degree\ in the Cenozoic and 15\degree\ in the
    Mesozoic, derived from at least 36 samples from at least 6 sampling sites
    (see also the concluding paragraph of Section~\ref{sec:f2}). While
    straightforward and convenient to apply, these stringent criteria mean some
    useful data may be filtered out and wasted, especially for a period where
    there are only limited number of paleopoles.
  \item~\citet{S05} (SS05). SS05 is similar to, but less stringent than, BC02.
    SS05 uses only paleopoles with $\alpha$95 of $\leq$15\degree\ and an age
    uncertainty of $\leq$40 Myr, derived from at least $\geq$4 sampling sites
    with $\geq$4 samples/site, and at least some sample demagnetisation (See
    also the last paragraph of Section~\ref{subp:ss05}).
\end{enumerate}

Although they are often used, and affect the path and uncertainties of the
resulting APWP, there has been limited study of how effective these filtering
methods are at reconstructing a `true' APWP compared to unfiltered data.
Furthermore, the focus on the filtering stage ignores the possible impact of
binning/windowing (i.e.\ combining paleopoles through moving-averaging) and
weighting. This study more rigorously investigates the effects of the choices
made at every stage on the resulting APWP\@. By focussing on paleomagnetic data
from the last $\sim$120 Myr, where plate motions are independently constrained
by reconstructions of seafloor spreading tied into a hotspot reference frame, we
can also verify which choices actually lead to a better-constrained and accurate
record of past plate motions.


\section{Methods}

\subsection{General Approach}

In this study, we use paleopoles extracted from the GPMDB 4.6b~\citep{M96,P05}
to generate APWPs for the period 120\textendash0 Ma. A range of possible APW
paths for North America, India and Australia can be generated from the extracted
sets of paleopoles using various binning, filtering and weighting methods
(Table.~\ref{tab-pick} and~\ref{tab-weit}). These paths can then be compared to
synthetic APWPs independently generated from an absolute plate motion model. The
three plates chosen have different attributes, both in terms of the input data
set and the nature of the reference APWP\@.

\subsection{Paleomagnetic Data}

\subsubsection{GPMDB Field Codes}

Data analysis includes manipulation of data fields/columns in the GPMDB\@. In
the following content, the codes of the several specific fields used will be
referred to. They are listed in Table~\ref{tab-fld} for easy reference.

\begin{table}[!ht]
  \centering
  \caption{List of the used fields and field codes of the GPMDB.}\label{tab-fld}
  \begin{tabular}{p{0.16\linewidth} p{0.79\linewidth}}
    \toprule
    Field Code & Meaning \\ \midrule
    LOMAGAGE & Lower best estimate of the magnetic age of the magnetisation component \\
    HIMAGAGE & Upper best estimate of the magnetic age of the magnetisation component \\
    B & Number of sites \\
    N & Number of samples \\
    ED95 & Radius of circle of 95\% confidence about mean direction, i.e. $\alpha$95 \\
    EP95 & Radius of circle of 95\% confidence about paleopole position \\
    KD & \emph{Fisher} precision parameter for mean direction \\
    DP & Half-angle of confidence on the pole in the direction of paleomeridian \\
    DM & Half-angle of confidence on the pole perpendicular to paleomeridian \\
    K\_NORM & \emph{Fisher} precision parameter for Normal directions \\
    ROCKNAME & Name of sample rock \\
    ROCKTYPE & Type of sample rock \\
    STATUS & Indicates if results have been superseded \\
    COMMENTS & General information including details of the origin of LOMAGAGE
      and HIMAGAGE\@. If the result is a combined pole this field contains
      information on the data included in the combined result \\
    \bottomrule
  \end{tabular}
\end{table}

\subsubsection{Paleomagnetic Data of Three Representative Continents}

Collections of paleopoles with a minimum age (LOMAGAGE) $\leq$135 Ma for the
North American, the Indian and Australian plates, were extracted from the GPMDB
4.6b. In order to include valid paleopoles from blocks that moved independently
prior to 120 Ma, which therefore should have different assigned plate codes in
the GPMDB, the spatial join technique~\citep{J07} was used to find all sampling
sites (correlating with their derived paleopoles) within the geographic region
that defines the rigid plate within the period of interest (see
Appendix~\ref{appen4chp3} for details):

\begin{figure}[tbp]
  \captionsetup{singlelinecheck=false,justification=raggedright}
  \centering
  \includegraphics[width=.9\textwidth]{../../paper/tex/GeophysJInt/figures/120NAhist.pdf}
  \caption[Distribution of 120\textendash0 Ma North American paleopoles]{(continued on next page)}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[(continued) Distribution of 120\textendash0 Ma North American
    paleopoles]{Temporal distribution of 120\textendash0 Ma North American
    paleopoles in 10/5 Myr window/step length. For distribution a, each bin only
    counts in the midpoints (circles) of pole uncertainty bars (not including
    those right at bin edges); For distribution b, as long as the bar intersects
    with the bin (not including those intersecting only at one of bin edges), it
    is counted in. Of red bars, only two poles, 83\textendash77 and
    80\textendash65 Ma, are from igneous and also sedimentary; only one pole,
    72\textendash40 Ma, is from igneous and also metamorphic. Inside the
    parentheses, except r, i and m, the left are non-redbed sedimentary rocks
    derived (black bars; only two poles, 146\textendash65 and 2\textendash0 Ma
    [RESULTNO 6679 and 1227], are from sedimentary and also
    metamorphic).}\label{fig-120NAhist}
\end{figure}

\paragraph{For North America,}
the search region was defined by the North American craton (NAC), Avalon/Acadia
and Piedmont blocks, as defined by the recently published plate model
of~\citet{Y18}. Following extraction, 58 palepoles from southwestern North
America that have been affected by regional rotations since 36 Ma~\citep{Mc06},
were removed. The final dataset consists of 135 paleopoles
(Fig.~\ref{fig-120NAhist}), with 76 of them (${\sim}56$\%; average age
uncertainty ${\sim}14$ Myr, average EP95 ${\sim}9.3$\degree) sampled from
dominantly igneous sequences, 56 (${\sim}42$\%; average age uncertainty
${\sim}27.5$ Myr, average EP95 ${\sim}10.5$\degree) sampled from mostly
sedimentary sequences, including 6 from redbeds, and 3 (${\sim}2$\%) from
metamorphic sequences. The principal features of the age distribution are a
larger number of young ($<$5 Ma) poles, and relatively fewer poles in the Late
Cretaceous and Miocene.

\begin{figure}[tbp]
  \centering
  \includegraphics[width=.93\textwidth]{../../paper/tex/GeophysJInt/figures/120INhist.pdf}
\end{figure}
\begin{figure}[!ht]
  \addtocounter{figure}{1}
  \ContinuedFloat\caption[Distribution of 120\textendash0 Ma Indian paleopoles]{Temporal
    distribution of 120\textendash0 Ma Indian paleopoles. For red bars, only one
    pole, 67\textendash64 Ma (RESULTNO 8593), is from igneous and also
    sedimentary. See Fig.~\ref{fig-120NAhist} for more information.}\label{fig-120INhist}
\end{figure}

\paragraph{For India,}
the Indian block as defined by~\citet{Y18} was used, but following extraction 31
paleopoles associated with parts of the northern margin that have undergone
regional rotations since the Jurassic~\citep{G15} were removed. The final
dataset consists of 75 paleopoles (Fig.~\ref{fig-120INhist}), with 39 of them
(52\%; average age uncertainty ${\sim}5$ Myr, average EP95 ${\sim}7.7$\degree)
sampled from dominantly igneous sequences and 36 (48\%; average age uncertainty
${\sim}14$ Myr, average EP95 ${\sim}7$\degree) sampled from mostly sedimentary
sequences, including 3 from redbeds. There is a high concentration of poles from
the latest Cretaceous\textendash{}Early Cenozoic (${\sim}70$\textendash60 Ma),
many of which are igneous; in younger and older intervals, there are fewer,
mostly sedimentary poles.

\begin{figure}
  \centering
  \includegraphics[width=.94\textwidth]{../../paper/tex/GeophysJInt/figures/120AUhist.pdf}
\end{figure}
\begin{figure}[!ht]
  \addtocounter{figure}{1}
  \ContinuedFloat\caption[Distribution of 120\textendash0 Ma Australian paleopoles]{Temporal
    distribution of 120\textendash0 Ma Australian paleopoles. For black bars,
    only four poles, 100\textendash80 Ma (RESULTNO 1106), 10\textendash2 Ma
    (RESULTNO 1208), 4\textendash2 Ma (RESULTNO 140) and 1\textendash0 Ma
    (RESULTNO 1963), are from sedimentary and also igneous. For red bars, only
    one pole, 65\textendash25 Ma (RESULTNO 1872), is from igneous and also
    sedimentary rocks, and only one pole, 1\textendash0 Ma (RESULTNO 1147), is
    from igneous and also metamorphic rocks. See Fig.~\ref{fig-120NAhist} for
    more information.}\label{fig-120AUhist}
\end{figure}

\paragraph{For Australia,}
the Australia, Sumba, and Timor blocks as defined by~\citet{Y18} were used, in
combination with data from the Tasmania block younger than ${\sim}100$ Ma (with
a maximum age (HIMAGAGE) $\leq$100 Ma), prior to which it was not fixed with
respect to Australia~\citep{Y18}. The final dataset consists of 99 paleopoles
(Fig.~\ref{fig-120AUhist}), with 61 of them (${\sim}62$\%; average age
uncertainty ${\sim}23.5$ Myr, average EP95 ${\sim}10.9$\degree) sampled from
dominantly igneous sequences, and 38 (${\sim}38$\%; average age uncertainty
${\sim}23$ Myr, average EP95 ${\sim}9.4$\degree) sampled from mostly sedimentary
sequences, including 9 from redbeds. The temporal distribution of poles is
relatively uniform.

\bigskip
Compared with North American (Fig.~\ref{fig-120NAhist}) and Australian
(Fig.~\ref{fig-120AUhist}) paleopoles, Indian paleopoles have a relatively lower
density and a higher prevalence of sedimentary paleopoles, except during the
period of ${\sim}70$\textendash60 Ma (Fig.~\ref{fig-120INhist}). For North
America and India, sedimentary paleopoles have significantly larger average age
uncertainty, but about same for Australia, there is no igneous and sedimentary
${\alpha}95$ difference.

\subsection{APWP Generation}\label{sec:apwpg}

Multiple APWPs were generated using the selected poles for each of the three
plates as follows:

\paragraph{Picking/binning.} A moving average or moving window technique was
used: paleopoles were selected for each APWP time step (initially 5 Myr step
length from 0 to 120 Ma) if their age fell within a window centered on the
current step age. In this study, the width of the moving window was always twice
that of time step (i.e.\ initially 10 Myr), such that each window half-overlaps
with its neighbours.

\paragraph{Filtering.} Poles with characteristics thought to correspond to
poor data quality, or lacking characteristics thought to correspond to good
data quality, were discarded (or in some cases, corrected).

\paragraph{Weighting.} Calculation of a weighted \emph{Fisher} mean~\citep{F53} of
the remaining poles within each window, using weighting functions intended to
increase the influence of higher quality poles relative to lower quality ones.

\bigskip
Twenty-eight different picking and filtering algorithms were tested
(Table~\ref{tab-pick}, referred to hereafter as Pk), in combination with 6
different weighting algorithms (Table~\ref{tab-weit}, referred to hereafter as
Wt), for the three plates. The effects of changing the time step length and
width of the moving window, and the reference path, were also examined.

\subsubsection{Picking/Binning}\label{sec:p}

In studies where the moving window method is used to calculate an
APWP~\citep{T99,T08}, a paleopole is generally considered to fall in the current
window only if the midpoint of its age limits fall within that window. If a
paleopole has a large age uncertainty compared to the size of the moving window,
it will not be included in the moving windows close to the beginning and end of
the age range, which are arguably more likely magnetisation ages than the
midpoint. To investigate this issue, we compare the performance of moving
windows populated using the midpoint picking criterion, referred to hereafter
as ``Age Mean Picking'' (AMP\@; even-numbered algorithms in
Table~\ref{tab-pick}, Fig.~\ref{fig-dif} and subsequent figures), to a less
restrictive picking criterion where a paleopole is included in the current
moving window if any part of its age limits falls within that window, referred
to hereafter as ``Age Position Picking'' (APP\@; odd-numbered algorithms in
Table~\ref{tab-pick}, Fig.~\ref{fig-dif} and subsequent figures). The APP
algorithm will pick more paleopoles in each moving window than the AMP algorithm
(Figs.~\ref{fig-120NAhist}-\ref{fig-nac-maplat}).

If, for example, we have a paleopole with an acquisition age of 10\textendash20
Ma, and we have a 2 Myr moving window with a 1 Myr age step, then it is included
in just the 14\textendash16 Ma window (for the midpoint age of 15 Ma) for
AMP\@. For APP, this paleopole falls in the 9\textendash11, 10\textendash12,
11\textendash13, 12\textendash14 \ldots 17\textendash19, 18\textendash20 and
19\textendash21 Ma windows. Each original paleopole is therefore represented in
the mean poles calculated over its entire possible acquisition age.

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.7\textwidth]{../../paper/tex/GeophysJInt/figures/binning.pdf}
  \caption[How moving average methods work]{An example of 10 Myr moving window
    and 5 Myr step in the two moving average methods, AMP and APP, based on
    paleopoles of the $NAC$. White circles are the midpoints of low and high
    magnetic ages. The vertical axis has no specific meaning here. For example,
    for the window of 15 Ma to 5 Ma (the dashed-line bin), the AMP method
    calculates the \emph{Fisher} mean pole (dark triangle in Fig.~\ref{fig-mhsPred}) of
    only 5 paleopoles, while the APP method calculates the mean pole (dark
    circle in Fig.~\ref{fig-mhsPred}) of 9 paleopoles.}\label{fig-nac-maplat}
\end{figure}

A shorter step and narrower window will potentially increase the clustering of
the selected paleopoles, but will reduce their number. Conversely, a longer step
and wider window will increase the number of paleopoles, but potentially
decrease their clustering. To investigate these trade-offs, every
picking/filtering and weighting method was also used to generate APWPs with a
time step and window doubled to 10 Myr and 20 Myr, respectively. Paths generated
using AMP and APP with no filtering, and every weighting method, with time steps
from 1 Myr to 15 Myr in 1 Myr increments and window widths from 2 Myr to 30 Myr
in 2 Myr increments, were also analysed. In all cases the oldest time step was
120 Ma.

\subsubsection{Filtering}\label{sec:f}

14 different filters or corrections (Table~\ref{tab-pick}) were applied to both
data picked using the AMP moving window method (even numbers) and data picked
using the APP moving window method (odd numbers), resulting in a total of 28
unique picking algorithms. The filters or corrections can be characterised as
follows:

\begin{table}[!ht]
  \centering
  \caption{List of all picking/binning algorithms developed here.}\label{tab-pick}
  \begin{tabular}{@{}ll@{}}
    \toprule
    No. & Picking Algorithm \\ \midrule
    0 & AMP\@: Age Mean Picking, see Section~\ref{sec:apwpg} \\
    1 & APP\@: Age Position Picking \\
    2 & AMP (``$\alpha$95/Age range'' no more than ``15/20'') \\
    3 & APP (``$\alpha$95/Age range'' no more than ``15/20'') \\
    4 & AMP (mainly or only igneous) \\
    5 & APP (mainly or only igneous) \\
    6 & AMP (contain igneous and not necessarily mainly) \\
    7 & APP (contain igneous and not necessarily mainly) \\
    8 & AMP (unflatten sedimentary) \\
    9 & APP (unflatten sedimentary) \\
    10 & AMP (nonredbeds) \\
    11 & APP (nonredbeds) \\
    12 & AMP (unflatten redbeds) \\
    13 & APP (unflatten redbeds) \\
    14 & AMP (published after 1983) \\
    15 & APP (published after 1983) \\
    16 & AMP (published before 1983) \\
    17 & APP (published before 1983) \\
    18 & AMP (exclude commented local rot or secondary print) \\
    19 & APP (exclude commented local rot or secondary print) \\
    20 & AMP (exclude local rot or correct it if suggested) \\
    21 & APP (exclude local rot or correct it if suggested) \\
    22 & AMP (filtered using SS05 palaeomagnetic reliability criteria) \\
    23 & APP (filtered using SS05 palaeomagnetic reliability criteria) \\
    24 & AMP (exclude superseded data already included in other results) \\
    25 & APP (exclude superseded data already included in other results) \\
    26 & AMP (comb of 22 and 24) \\
    27 & APP (comb of 23 and 25) \\ \bottomrule
  \end{tabular}
  \raggedright{\\Notes: SS05,~\citet{S05}}
\end{table}

\paragraph{No modification (Pk 0,1).}

\paragraph{Removal of poles with large spatial and temporal uncertainties
(Pk 2,3).} Paleopoles with both large $\alpha$95 (ED95 $>15\degree$,
following the BC02 threshold for the Mesozoic) and wide acquisition age limits
(difference between HIMAGAGE and LOMAGAGE $>20$ Myr, following the V90 criteria
about age within a half of a geological period; the average of the geological
periods between 120 and 0 Ma [Quaternary, Neogene, Paleogene and Cretaceous] is
about 20 Myr) which are less likely to provide a good estimate of the actual
pole position within any specific age window, were excluded.

\paragraph{Prefer poles from igneous rocks (Pk 4,5, 6,7).} Pk 4,5 removes
paleopoles potentially affected by inclination flattening by selecting only
paleopoles coded as igneous or mostly igneous (ROCKTYPE starting with
``intrusive'' or ``extrusive''). In fact, most of the paleopoles picked by Pk
4,5 are derived from igneous-only rocks. Pk 6,7 select paleopoles coded as
containing igneous (ROCKTYPE containing ``intrusive'' or ``extrusive''); this is
a less strict filter, because the dominant rock type could potentially be
another lithology. Therefore Pk 6 also includes paleopoles from Pk 4, and Pk 7
also includes paleopoles from Pk 5.

\paragraph{Correct sedimentary poles for inclination shallowing (Pk 8,9).}
Rather than excluding paleopoles from sedimentary rocks, paleopoles coded as
sedimentary or redbeds were instead corrected for inclination flattening using
the flattening function $\tan I_o = f \tan I_f$~\citep{K55}, where $I_o$ is the
observed inclination, $I_f$ is the unflattened inclination, and $f$ is the
flattening factor (also known as shallowing coefficient; 1=no flattening,
0=completely flattened). Here $f=0.6$ is used in all cases,
following~\citet{T12}, unless the rock type (ROCKTYPE field in the database) is
not sedimentary dominated but contains sedimentary rocks. In these cases,
$f=0.8$ is used instead, following the minimum anisotropy-of-thermal-remanence
determined f-correction~\citep{D11,Do11}.

\paragraph{Remove redbeds (Pk 10,11) or correct them for inclination
shallowing (Pk 12,13).} Bias toward shallow inclinations is also observed
in paleomagnetic data derived from red-beds~\cite[e.g., in central Asia,
Mediterranean region, North America, etc.]{T04,K04,T07,B10}. This bias can be
addressed by removing the source (Pk 10,11; ROCKTYPE containing redbeds), or
correcting for inclination flattening, setting $f=0.6$ as previously described
(Pk 12,13). In the latter case, the assumption is being made that the
redbeds are carrying a detrital paleomagnetic signal.

\paragraph{Prefer poles from younger (Pk 14,15, 24,25) or older (Pk 16,17)
studies.} Advancements in equipment (e.g., cryogenic magnetometers) and
analytical techniques (e.g., stepwise demagnetisation) mean that more recently
published paleopoles are potentially more reliable than older ones. Pk 14,15
removes any paleopoles published prior to 1983 (YEAR $>$ 1983), the mean
publication date for paleopoles in the GPMDB 4.6b. Pk 24,25 takes a similar but
less aggressive approach by excluding paleopoles that have been superseded (99
paleopoles) by later studies from the same sequence, which are presumed to
represent a more accurate determination of the paleopole position. The
``STATTUS'' field of the GPMDB 4.6b indicates if a paleopole has been
superseded. Conversely, removing paleopoles published after 1983 (Pk 16,17;
YEAR $\leq$ 1983) should have a negative effect.

\paragraph{Exclude suspected local rotations and secondary overprints (Pk
18,19), or correct for them where possible (Pk 20,21).} Secondary remanence
components and local tectonic deformation can both displace the measured pole
position away from its ``true'' position. Such poles can be identified based on
demagnetisation data, or comparison to the pre-existing APWP\@. Pk 18,19
removes paleopoles that were identified as such in the COMMENTS field: 66
paleopoles affected by local rotations are identified by carefully going through
all the COMMENTs containing information about rotation; paleopoles affected by
secondary overprints are extracted with the keyword ``econd'' identified in the
COMMENTS\@. A subset (19) of the 66 paleopoles identified have a suggested
correction associated with them; Pk 20,21 retains these paleopoles after
applying the suggested correction.

\paragraph{SS05 quality criteria (Pk 22,23).}\label{subp:ss05} As with Pk 2,3,
SS05~\citep{S05} removes paleopoles with high spatial ($\alpha$95 $>15\degree$)
and temporal (age range $>$ 40 Myr) uncertainty, but additionally remove
paleopoles where samples had poor sampling coverage (sampling sites' quantity
[B] of $<4$, samples' quantity [N] of $<4$ times of the sites [B]) and were not
subjected to even a blanket demagnetisation treatment (laboratory cleaning
procedure code DEMAGCODE $<2$). Pk 26,27 also use these criteria, but
further excludes superseded data.

\bigskip
Some of the picking (Table~\ref{tab-pick}) and weighting (Table~\ref{tab-weit})
methods developed here are also connected with the V90 Q factor (see
Section~\ref{sec:si}). For example, Pk 2,3 and Wt 2, 4, 5 are related to the V90
criteria 1; Pk 2,3, 22,23, 26,27 and Wt 1, 3 are related to the V90 criteria 2;
Pk 22,23 are related to the V90 criteria 3; The data constraining described in
Appendix~\ref{appen4chp3} is related to the V90 criteria 5; and Pk 18,19 are
related to the V90 criteria 7.

\subsubsection{Weighting}\label{sec:w}

Following filtering, weights were assigned to each of the remaining paleopoles
using one of the following six algorithms (Table~\ref{tab-weit}), prior to
calculation of a weighted \emph{Fisher} mean:
%
\begin{longtable}[h]{@{}c|l@{}}
\caption{List of all weighting algorithms developed here.}\label{tab-weit}
\\\hline\multicolumn{1}{|p{.25in}|}{\textbf{No.}} & \multicolumn{1}{p{5.5in}|}{\textbf{Weighting Algorithm}} \\
\hline\endfirsthead%
\multicolumn{2}{r}{{\bfseries \tablename\ \thetable{} --- continued from previous page}} \\ \hline
\multicolumn{1}{|p{.25in}|}{\textbf{No.}} & \multicolumn{1}{p{5.5in}|}{\textbf{Weighting Algorithm}} \\ \hline
\endhead%
\hline\multicolumn{2}{|r|}{{\bfseries continued on next page}} \\ \hline
\endfoot\hline
\endlastfoot0 & None (No weighting) \\ \hline
1 & Larger numbers of sites (B) \& observations (N), greater $weight$ ($w$):\\
  & \abovedisplayskip=0pt\begin{minipage}{5.5in}\begin{equation*}w=\left\{\begin{array}{ll}
    0.2 & \textrm{, if both B \& N are missing, or B$\leq1$ \& N$\leq1$} \\
    (1-\frac{1}{B})\times0.5 & \textrm{, if N is missing or N$\leq1$, \& B$>$1} \\
    (1-\frac{1}{N})\times0.5 & \textrm{, if B is missing or B$\leq1$, \& N$>$1} \\
    (1-\frac{1}{B})\times(1-\frac{1}{N}) & \textrm{, if B$>$1 \& N$>$1}
\end{array}\right.\end{equation*}\end{minipage} \\ \hline
2 & Lower age uncertainty, greater weight: \\
  & \begin{minipage}{5.5in}age\_range=HIMAGAGE-LOMAGAGE \\
    age\_midpoint = (HIMAGAGE+LOMAGAGE)$\times$0.5 \\
    if age\_midpoint$<$2.58 (Ma; start of the Quaternary, according to GSA Geologic Time Scale), \\
    \vbox{\begin{equation*}w=\left\{\begin{array}{ll}
    1 & \textrm{, if age\_range$\leq$1.29 (from $\frac{2.58\textendash0}{2}$)} \\
    \frac{1.29}{age\_range} & \textrm{, if age\_range$>$1.29}
    \end{array}\right.\end{equation*}} \\
    if 2.58$\leq$age\_midpoint$<$23.03 (Ma; Neogene), \\
    \vbox{\begin{equation*}w=\left\{\begin{array}{ll}
    1 & \textrm{, if age\_range$\leq$10.225 (from $\frac{23.03\textendash2.58}{2}$)} \\
    \frac{10.225}{age\_range} & \textrm{, if age\_range$>$10.225}
    \end{array}\right.\end{equation*}} \\
    if 23.03$\leq$age\_midpoint$<$201.3 (Ma; Paleogene, Cretaceous, Jurassic), \\
    \vbox{\begin{equation*}w=\left\{\begin{array}{ll}
    1 & \textrm{, if age\_range$\leq$15} \\
    \frac{15}{age\_range} & \textrm{, if age\_range$>$15}
    \end{array}\right.\end{equation*}}
    \end{minipage} \\ \hline
3 & Lower $\alpha$95, greater weight: \\
  & \begin{minipage}{5.5in}Positive half Normal distribution with a mean and standard deviation \\
    of 0 and 10, scaled with $10\sqrt{2\pi}$ (to make the peak reach 1) \\
    \vbox{\begin{equation*}w=e^{-\frac{\alpha_{95}^2}{200}},\end{equation*}} \\
    where \\
    \abovedisplayskip=0pt\belowdisplayskip=0pt\vbox{\begin{equation*}\alpha95=\left\{\begin{array}{ll}
    ED95 & \\
    DP & \textrm{, if ED95 is missing} \\
    \frac{140}{\sqrt{KD\times{}N}} & \textrm{, if ED95 \& DP are missing} \\
    \frac{140}{\sqrt{K\_NORM\times{}N}} & \textrm{, if ED95, DP \& KD are missing} \\
    \frac{140}{\sqrt{K\_NORM\times{}B}} & \textrm{, if ED95, DP, KD \& N are missing} \\
    \frac{140}{\sqrt{1.7\times{}B}} & \textrm{, if ED95, DP, KD \& K\_NORM are missing,} \\
    & \textrm{using the lowest KD value ${\sim}1.7$ in GPMDB 4.6b,}
    \end{array}\right.\end{equation*}}
    finally $w$=0 if this $\alpha$95 completely overlaps with another smaller
    $\alpha$95 whose paleopole is exactly derived from the same place and same
    rock.
    \end{minipage} \\ \hline
4 & Age uncertainty Position to bin (more overlap, greater weight): \\
  & \begin{minipage}{5.5in}wha, window high age; wla, window low age \\
    \vbox{\begin{equation*}w=\left\{\begin{array}{ll}
    \frac{wha-LOMAGAGE}{age\_range} & \textrm{, if LOMAGAGE$>$wla \& HIMAGAGE$>$wha} \\
    \frac{HIMAGAGE-wla}{age\_range} & \textrm{, if LOMAGAGE$<$wla \& HIMAGAGE$<$wha} \\
    \frac{wha-wla}{age\_range} & \textrm{, if LOMAGAGE$<$wla \& HIMAGAGE$>$wha} \\
    1 & \textrm{, if LOMAGAGE$>$wla \& HIMAGAGE$<$wha}
    \end{array}\right.\end{equation*}}
    \end{minipage} \\ \hline
5 & Combining 3 and 4: average of the two weights from 3 and 4 \\
\end{longtable}
%
\paragraph{No weighting (Wt 0).} Weighting factor=1 for all paleopoles.

\paragraph{Weighting by sample and site number (Wt 1).} Paleopoles derived
from more individually oriented samples (observations; N) collected from more
sampling levels/sites (B) are more likely to average out secular variation and
accurately sample the GAD field~\citep{v90,B02,T20}, and are given a weighting
closer to 1. Unfortunately, in the GPMDB, a paleopole's B or N is not always
given. As shown in (Table~\ref{tab-weit} no. 1), in such cases the calculated
weights were modified to give lower weights overall whilst still accounting for
partial information, such as N being reported but not B.

For number of sampling sites B$>$1 and number of samples N$\leq$1, there are 8
such paleopoles for 120\textendash0 Ma North America, India 4, and Australia 1.
For B$\leq$1 and N$>$1, there are 20 such paleopoles for 120\textendash0 Ma
North America, India 26, and Australia 22. For B$\leq$1 and N$\leq$1, there are
only 23 such paleopoles for the whole GPMDB 4.6b, including 18 with neither B
value nor N value given; while for 120\textendash0 Ma there is no such paleopole
found in North America, India and Australia.

\paragraph{Weighting by age uncertainty (Wt 2)} Above a maximum age range
that represents a well-constrained age, defined as half of each geological
period in the Phanerozoic Eon (e.g., Quaternary, Neogene; here the geological
period that the middle point of the paleopole's age range falls within is
assigned)~\citep{v90,T20} or 15 Myr (the halves of the Paleogene, Cretaceous,
and Jurassic periods are all at least 20 Myr, which is large for these
relatively young geologic periods), whichever is smaller, paleopoles are given
an increasingly small weight as the age uncertainty (the high magnetic age $-$
the low magnetic age) increases (No. 2 in Table~\ref{tab-weit}).

\paragraph{Weighting by spatial uncertainty (Wt 3).} Paleopoles with a
smaller $\alpha$95 confidence ellipse are given a higher weighting than those
with a larger $\alpha$95, using a Gaussian/Normal distribution centered on 0
with standard deviation of 10. However, a paleopole's $\alpha$95 is not always
given in the database. If $\alpha$95 is not given, DP (the semi-axis of the
confidence ellipse along the great circle path from site to pole) is assigned as
$\alpha$95. If DP is not given either, $\alpha$95 was further approximated by
$\frac{140}{\sqrt{KD\times{}N}}$, where KD is \emph{Fisher} precision parameter for
mean direction if this parameter is given, or \emph{Fisher} precision parameter for
Normal directions (K\_NORM) if only K\_NORM is given when KD is missing. If N is
not given, B is used as N. If even K\_NORM is also missing, the lowest KD value
${\sim}1.7$ in GPMDB 4.6b is used as KD\@. It is also worthwhile to mention that
if specimens, where two paleopoles are derived, are exactly from the same place
and same rock (by checking if ROCKNAME, ROCKTYPE and sampling site are the
same), and one $\alpha$95 is completely inside the other $\alpha$95, a zero is
assigned as the weight of the data with the larger $\alpha$95. In fact, in the
above described procedure A95 (circle of 95\% confidence about mean pole) is a
better alternative instead of $\alpha$95, because A95 is directly reflecting the
spatial uncertainty of the paleopoles. However, most paleopoles' A95s are not
given in GPMDB 4.6b, so $\alpha$95 is used instead since $\alpha$95 is also
indirectly reflecting the quality of the paleopole.

\paragraph{Weighting by degree of overlap between moving window and pole age
uncertainty (Wt 4).} If a large fraction of the age range for an individual
paleopole falls within the current window, it is given a higher weighting than a
pole where the overlap is smaller, because it is more likely to be close to the
true pole position in the window interval. In other words, if window intersects
with part of age range, weight= (intersecting part) / (age range width).

\paragraph{Weighting by both spatial and temporal uncertainty (Wt 5).} This
weighting method is a combination of Wt 3 (but here the standard deviation of
the weighting function is changed to 15, which is also a threshold used
by~\citet{B02} for filtering Mesozoic data through $\alpha$95, to try making a
difference from Wt 3) and Wt 4. It takes the average of sums of the weights
generated by Wt 3 and 4. For weighting by both spatial and temporal uncertainty,
Wt 2 and 4 can also be combined into a new weighting method, which can be a part
of future work.

\subsubsection{Scaling of Weights}

The weights obtained from the six different weighting functions
(Table~\ref{tab-weit}) are then integrated into \emph{Fisher} mean function~\citep{F53}
to calculate a weighted \emph{Fisher} mean. First, weight values are used to scale
Cartesian x, y and z components of each individual paleopole's geographic
coordinate. Then these individual coordinates are combined through \emph{Fisher}
resultant vector $R$ function~\citep[see][chap.~11]{T20}. Therefore the mean
pole location, its spatial uncertainty A95, and \emph{Fisher} precision parameter are
all weighted along with $R$.

Weights are integrated by being multiplied with the variable we would like to do
weighting to. For example, here the weights can be directly multiplied with the
Cartesian x, y and z components of each paleopole. However, this sort of direct
multiplying causes the decreasing of $R$, which further sensitively and
extremely increases $\alpha$95 (Fig.~\ref{fig-alpha95}), especially because $R$
is always less and usually much less than $N$ and $N$ is usually not that high
(more than ${\sim}50$ is rather rare, around 10 averagely). The consequence
would be that all the $\alpha$95 ellipses of mean poles are extremely large in
size and difficult to be spatially differentiated.  % see example Mar 15 2019 in f-i/making_of_reliable_APWPs

\begin{figure*}
  \centering
  \begin{subfigure}{.49\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/alpha95_000082.png}
    \caption{perspective view a}
  \end{subfigure}
  \begin{subfigure}{.49\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/alpha95_000094.png}
    \caption{perspective view b}
  \end{subfigure}
  \caption[Visualization of the equation used to estimate $\alpha$95 from R and
    N]{Visualization of Equation 11.9 of ``Essentials of Paleomagnetism: Fifth
    Web Edition'', illustrating the relationship between the radius of the
    circle of 95\% confidence ($p$=0.05) about the mean, $\alpha$95, resultant
    vector $R$ and number of directions (or paleopoles) $N$. Note that R$<$N and
    N$\geq$2.}\label{fig-alpha95}
\end{figure*}

Therefore, in this thesis the weights are scaled before being multiplied with
the Cartesian x, y and z components by
%
\begin{equation*}
  ScaledW_i=\frac{N \times W_i}{\sum\limits_{i=1}^{N} W_i},
\end{equation*}
%
where $N$ is number of paleopoles for making a mean pole. So the scaled weight
$ScaledW_i$ could be greater than 1 because each weight $W_i$ is actually scaled
through being divided by the mean of the weights ($\frac{\sum\limits_{i=1}^{N}
W_i}{N}$). This scaling does not only keep the effect of weighting but also
avoid dramatically changing $R$ and indirectly and extremely changing
$\alpha$95.

\subsection{Reference Paths}\label{sec:refpath}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=1.01\textwidth]{../../paper/tex/GeophysJInt/figures/fhs_m.pdf}
  \caption[120\textendash0 Ma MHM vs FHM predicted APWP of North America]{MHM
    predicted 120\textendash0 Ma APWP (solid line) for North America through the
    North America\textendash{}Nubia\textendash{}Mantle plate circuit. The FHM
    predicted path (dashed line with shaded uncertainties) is also shown for
    comparison. The age step is 5 Myr. Compared with the 10 Ma paleomagnetic
    mean pole calculated by the AMP method (dark triangle), the coeval mean pole
    derived from the APP method is closer to both FHM and MHM predicted 10 Ma
    poles, which indicates more data diluting the effect of outliers. See also
    the paleopoles that the two mean poles are composed of in
    Fig.~\ref{fig-nac-maplat}.}\label{fig-mhsPred}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=1.01\textwidth]{../../paper/tex/GeophysJInt/figures/fhs501_m.pdf}
  \caption[120\textendash0 Ma MHM vs FHM predicted APWP of India]{MHM predicted
    120\textendash0 Ma APWP (solid line) for India through the
    India\textendash{}Somalia\textendash{}Nubia\textendash{}Mantle plate circuit.
    Its age step is 5 Myr. The dashed line is the FHM predicted path shown for
    comparison. The inset shows paths for fast moving India and also much slower
    moving North America shown in Fig.~\ref{fig-mhsPred}.}\label{fig-mhsPred501}
\end{figure}

A prediction of the expected APWP for any plate can be generated using a plate
kinematic model (e.g.\ the last ${\sim}180$\textendash200 Myr of plate motions
reconstructed from spreading ridges in ocean basins) that is tied in to an
absolute reference frame. Here, we use the rotations of~\citet{O05}, which
describe motion of Nubian Plate relative to the Indo-Atlantic hotspots back to
120 Ma. North America is linked to this absolute frame of reference across the
Mid-Atlantic ridge, using North America-Nubia rotations from chron C1n
(0\textendash0.78 Ma) from~\citet{D10}, to chron C2An (2.7 Ma)
from~\citet{Sh12}, to C5n.1ny (9.74 Ma) from~\citet{M99}, to C5n.2o (10.949 Ma)
from~\citet{G13}, to C6ny (19.05 Ma) from~\citet{M99}, to C6no (20.131 Ma)
from~\citet{G13}, to C34ny (83.5 Ma) from~\citet{M99}, from C34ny to
${\sim}118.1$ Ma from~\citet{S12}, and to closure at C34no (120.6 Ma)
from~\citet{G13}. India is linked via the East African Rift Valley
(Somalia-Nubia rotations from chron C1n (0\textendash0.78 Ma) from~\citet{D17},
to C2A.2no (3.22 Ma) from~\citet{H05}, and to closure at C7.2m (25.01 Ma) and
chron C34 (85 Ma) from~\citet{R16}, and finally extended to 120 Ma because there
was no known relative motion between Somalia and Nubia from 120 Ma to 85 Ma
according to the rotations from~\citet{M16}). Australia is linked via the East
African Rift Valley, then the SW Indian Ridge (E Antarctica-Somalia rotations
from chron C1n (0\textendash0.78 Ma) from~\citet{D17}, to C2A.2no (3.22 Ma)
from~\citet{H05}, to C5n.2no (10.95 Ma) from~\citet{L02}, to C13ny (33.06 Ma)
from~\citet{P08}, to C29no (64.75 Ma) from~\citet{C10}, to C34y (83 Ma)
from~\citet{R16}, to 96 Ma from~\citet{M01}, and to closure at chron M0 (120.6
Ma) from~\citet{M08}), and SE Indian Ridge (Australia-East Antarctica rotations
from chron C1n (0\textendash0.78 Ma) from~\citet{D17}, to C6no (20.13 Ma)
from~\citet{C04}, to C8o (26 Ma) from~\citet{G18}, to C17n.3no (38.11 Ma)
from~\citet{C04}, to C34ny (83.5 Ma) from~\citet{Wh13}, to the Quiet Zone
Boundary (96 Ma) from~\citet{W07}, to full closure at 136 Ma from~\citet{Wh13}).
The geomagnetic polarity timescales of~\citet{C95} for Late Cretaceous and
Cenozoic, and of~\citet{Gr94} for Early to Late Cretaceous time are used to
convert from chron boundaries to absolute ages. A long table listing these
rotation parameters with covariance uncertainties (Table~\ref{tab:rot}) is
included in Appendix~\ref{appen4chp3}, and the calculated rotations for the
North American, Indian and Australian reference APWPs in the hotspot reference
(Figs.~\ref{fig-mhsPred}-\ref{fig-mhsPred801}) are also listed in
Table~\ref{tab:refAPWP}.

\begin{figure}[!ht]
  \centering
  \includegraphics[width=1.01\textwidth]{../../paper/tex/GeophysJInt/figures/mhs801.pdf}
  \caption[120\textendash0 Ma MHM vs FHM predicted APWP of Australia]{MHM
    predicted 120\textendash0 Ma APWP (solid line) for Australia through the
    Australia\textendash{}East
    Antarctica\textendash{}Somalia\textendash{}Nubia\textendash{}Mantle plate
    circuit. Its age step is 5 Myr. The dashed line is the FHM predicted path
    shown for comparison. The inset shows paths for fast moving India shown in
    Fig.~\ref{fig-mhsPred501}, much slower moving North America shown in
    Fig.~\ref{fig-mhsPred}, and also relatively intermediate moving
    Australia.}\label{fig-mhsPred801}
\end{figure}

Where possible, poles which were published uncertainty estimates were used.
Where no uncertainty estimates were available, values of the covariance matrix
were set to an arbitrarily small value (1E–15). Where this occurs, the spatial
uncertainties for the reference APWP are likely underestimated. However, but
because the uncertainties for the Nubia-hotspot rotations are substantially
larger than for rotations derived from fitting magnetic isochrons, the effect is
small.

To reconstruct a reference APWP at the required time steps for comparison with
the paleomagnetic APWPs, rotations and their associated uncertainties were
interpolated between constraining finite rotation poles according to the method
of~\citet{D08}, assuming constant rates.

Neither the hotspot reference frame nor the paleomagnetic reference frame are
truly fixed with respect to the solid Earth. In the former case, hotspots are
not truly stationary in the mantle~\citep{S98}; in the latter, true polar wander
(TPW) may also lead to differential movements of the solid earth with respect to
the spin axis~\citep{E03}. In reality, it is difficult to untangle these
effects. Whilst there is little clear evidence for significant TPW in the past
${\sim}120$ Myr~\citep{C00,R04}, modeling suggests that the effects of hotspot
drift can start to become significant over 80\textendash100 Myr
timescales~\citep{O05}. Because paleomagnetic APWPs have large associated
spatial uncertainties, a synthetic APWP calculated using a fixed-hotspot
reference frame is unlikely to deviate significantly from the `true' APWP, and
most comparison experiments use a fixed-hotspot model (FHM) reference path for
North America (Fig.~\ref{fig-mhsPred}), India (Fig.~\ref{fig-mhsPred501}) and
Australia (Fig.~\ref{fig-mhsPred801}). However, the full set of comparisons for
the 28 picking methods and 6 weighting methods was also run for reference paths
generated using the moving-hotspot model (MHM) rotations of~\citet{O05}, which
incorporate motions of the Indo-Atlantic hotspots relative to the mantle derived
from mantle convection modeling.

When comparing the synthetic APW paths for the three plates (inset,
Fig.~\ref{fig-mhsPred801}), there are clear differences. The predicted mean
north pole for North America at 120 Myr is still at ${\sim}75^{\circ}$N
(Fig.~\ref{fig-mhsPred}), indicating rather slow drift with respect to the spin
axis; this is due to a large component of the North American plate's absolute
motion in the past 120 Myr being to the east. In contrast, the rapid northward
motion of the Indian plate in the same period, particularly prior to its
collision with Asia at ${\sim}50$\textendash55 Ma~\citep{N10}, is reflected by
the 120 Ma predicted mean north pole being located at ${\sim}20^{\circ}$N
(Fig.~\ref{fig-mhsPred501}). Australia represents an intermediate case, with
north westerly plate motion from ${\sim}120$\textendash60 Myr changing to more
rapid northward motion from ${\sim}60$\textendash55 Ma to the present~\citep{W07}.
When comparing the FHM and MHM tracks, significant differences in the oldest
parts (before ${\sim}80$ Ma) are apparent for India and North America.

These differences in the reference path due to different plate kinematics is
another variable that may affect the performance of the different weighting
algorithm for different plates, in addition to the distribution and type of the
contributing mean poles used to generate the paleomagnetic APWPs.

\subsection{Comparison Algorithm}

Comparisons between APWPs generated using different picking and weighting
algorithms and the synthetic reference APWPs were performed using the composite
path difference ($\mathcal{CPD}$) algorithm described in
Chapter~\ref{chap:Metho}, with equal weighting given to the spatial, length and
angular differences (i.e., W$_s$ = W$_l$ = W$_a$ = $\frac{1}{3}$). A lower
$\mathcal{CPD}$ score generally indicates a `better' fit, although a lower score
can also potentially result from comparison to a more poorly constrained path
with large uncertainties, which are less likely to be significantly different.
An additional `Fit Quality' (FQ) metric can help to distinguish such cases, by
assigning the two paths being compared a letter score based on the average size
of their uncertainty ellipses (Chapter~\ref{sec:FQ}). Here, the first letter
refers to the generated APWP, and the second letter refers to the reference
path. In this study, the reference path FQ score is fixed for each plate;
because the spatial uncertainties for paths generated from plate motion models
are small compared to those typical for paleomagnetic data, this reference path
FQ is always rated `A'.

This does not only help find the most similar paleomagnetic APWP (from the best
algorithm) to the reference APWP, but also help further test and demonstrate the
validity of the similarity measuring tool in practise.

\begin{figure}[tbp]
  \vspace*{-.1cm}
  \centering
  \begin{subfigure}{.94\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101_120_0.pdf}
    \caption{minimum 0.00573 (25(0)), maximum 0.08601 (16(2)), mean 0.032403,
      median 0.032395}\label{fig-na-dif} % subcaption
  \end{subfigure}
  \vspace{.1em} % here you can insert horizontal or vertical space
  \begin{subfigure}{.94\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501_120_0.pdf}
    \caption{minimum 0.023 (19(0)), maximum 0.5137 (8(3)), mean 0.1182, median
      0.0835}\label{fig-in-dif} % subcaption
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.94\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801_120_0.pdf}
    \caption{minimum 0.00227 (11(0)), maximum 0.3934 (26(3)), mean 0.08373,
      median 0.05}\label{fig-au-dif} % subcaption
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[$\mathcal{CPD}$ of each plate's paleomagnetic APWPs vs its FHM
    predicted APWP]{Equal-weight composite path difference ($\mathcal{CPD}$)
    values between each continent's paleomagnetic APWPs and its predicted APWP
    from FHM and related plate circuits. The paths are in 10/5 Myr bin/step. The
    $\mathcal{CPD}s$ less than and more than one-standard-deviation interval of
    the whole 168 labeled values (lower and upper 15.866 per cent) are colored
    in green and red. Columns with the same values are connected. The
    percentages of removed paleopoles are derived relative to Pk 1, corrected
    relative to each corresponding picking method (Pk 8,9, 12,13; 1 paleopole
    removed and 1 corrected by Pk 20,21 for India). Fit quality (FQ) for each
    $\mathcal{CPD}$ is greyscale-coded.}\label{fig-dif} % caption for whole figure
\end{figure}

\section{Results}

\subsection{Baseline Results: 10/5 Myr Window/Step, Fixed-hotspot
Reference}\label{sec:base}

Fig.~\ref{fig-dif} shows the $\mathcal{CPD}$ scores for the APWPs generated with
all 28 picking methods (AMP and APP with one of 14 separate filters applied) and
one of 6 weighted mean calculations then applied, compared to the FHM reference
path (squares and dashed line in Figs.~\ref{fig-mhsPred}-\ref{fig-mhsPred801})
for North America (Fig.~\ref{fig-na-dif}), India (Fig.~\ref{fig-in-dif}) and
Australia (Fig.~\ref{fig-au-dif}). The 27 lowest and highest of the 168
scores for each plate (values greater than 1 standard deviation from the mean)
are marked in green and red, respectively. Different combinations of windowing
method, filtering and weighting clearly affect the difference score, with
$\mathcal{CPD}$ values ranging from 0.0023 to 0.5137. The fits for paths with
low difference scores are clearly much better than for those with high ones
(Fig.~\ref{fig-difbw}). From Fig.~\ref{fig-dif}, it is clear that:

\begin{figure}[tbp]
  \centering
  \begin{subfigure}{.42\textwidth} % width of left subfigure
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_10_5_25_0.pdf}
    \caption{N America: min 0.00573 (25(0)), FQ
      B-A}\label{fig-nac-105250}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth} % width of right subfigure
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_10_5_16_2.pdf}
    \caption{N America: max 0.08601 (16(2)), FQ
      B-A}\label{fig-nac-105162}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_10_5_19_0.pdf}
    \caption{India: minimum 0.023 (19(0)), FQ C-A}\label{fig-ind-105190}
  \end{subfigure}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_10_5_8_3.pdf}
    \caption{India: maximum 0.5137 (8(3)), FQ C-A}\label{fig-ind-10583}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_10_5_11_0.pdf}
    \caption{Australia: min 0.00227 (11(0)), FQ C-A}\label{fig-au-105110}
  \end{subfigure}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_10_5_26_3.pdf}
    \caption{Australia: max 0.3934 (26(3)), FQ C-A}\label{fig-au-105263}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[Best and worst $\mathcal{CPD}$s (10/5 Myr window/step)]{Path
    comparisons with best and worst $\mathcal{CPD}$ values shown in
    Fig.~\ref{fig-dif}. The parenthetical remarks are Pk No (Wt No).}\label{fig-difbw}
\end{figure}

\begin{enumerate}
  \item There is much more variation in scores along the horizontal axes than
    the vertical axes (see self-explanatory topography of bands in
    Fig.~\ref{fig-dif}), suggesting that the choice of windowing and filtering
    method (Table~\ref{tab-pick}) has a much greater impact than weighting
    (Table~\ref{tab-weit}).
  \item Many of the highest scores (worst fits) are associated with
    even-numbered picking and filtering methods, i.e., those which use the AMP
    windowing algorithm. Even so, Pk 4 and 6 are among the best methods for
    India.
  \item The magnitude and range of $\mathcal{CPD}$ scores for each of the three
    plates is different, with the North American plate having the lowest
    magnitudes and range (Fig.~\ref{fig-na-dif}), and the Indian plate having
    the highest (Fig.~\ref{fig-in-dif}). The scores for the Australian plate are
    generally closer to the equivalent scores for the North American plate than
    to the scores for the Indian plate (Fig.~\ref{fig-d-dif}).
  \item Although there is some overlap (e.g., Pk 19, 21 [best], and 2, 16
    [worst] for all the three plates or for both India and Australia; 1, 11, 13,
    19, 21, 25 [best], and 2, 14, 16, 22, 26 [worst] for both North America and
    Australia; 5, 7, 19, 21 [best], 2, 16, 18 [worst] for both North America and
    India), the best- and worst-performing picking/filtering and weighting
    algorithms are not exactly the same for each plate.
  \item Relating to FQ ratings: North America's lower $\mathcal{CPD}$ scores are
    also associated with consistently good FQ B-A grades; India's higher
    $\mathcal{CPD}$ scores are also associated with lower FQ (mostly C-A,
    including the no-filter methods, so there are some improvements in FQ with
    some filters), and Australia is a mix of B-A (including the no-filter
    methods) and C-A (which represent a reduction in FQ with some filters). In
    general, lower $\mathcal{CPD}$ scores also appear to be associated with
    better FQ (e.g. B-A rather than C-A).
\end{enumerate}

\subsection{Effects of Windowing Method}

Dividing $\mathcal{CPD}$ scores according to whether the AMP or APP windowing
method was used (Fig.~\ref{fig-difAMPvsAPP}) confirms that whilst the lowest
$\mathcal{CPD}$ scores for paths generated by the AMP windowing algorithm are
close to the lowest scores generated using the APP method, the highest scores
are much higher (Fig.~\ref{fig-boxAMPvsAPP}). The mean of the $\mathcal{CPD}$
scores for APWPs generated using AMP is greater than the maximum APP-derived
score for the Indian and Australian plates
(Fig.~\ref{fig-difAMPvsAPP},~\subref{fig-in-difAMPvsAPP},~\subref{fig-au-difAMPvsAPP}
and Fig.~\ref{fig-boxAMPvsAPP},~\subref{fig-in-boxAMPvsAPP},~\subref{fig-au-boxAMPvsAPP}),
and more than 1 standard deviation greater than the APP mean for North American
APWPs (Figs.~\ref{fig-na-difAMPvsAPP},~\ref{fig-na-boxAMPvsAPP}).

\begin{figure}[tbp]
  \centering
  \begin{subfigure}{1.01\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101_120_0APPvsAMP.pdf}
    \caption{AMP\@: minimum 0.0265 (4(1)), maximum 0.086 (16(2)), mean 0.0451,
      median 0.0457; APP\@: minimum 0.0057 (25(0)), maximum 0.05835 (17(5)),
      mean 0.0197, median 0.01494}\label{fig-na-difAMPvsAPP}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1.01\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501_120_0APPvsAMP.pdf}
    \caption{AMP\@: minimum 0.047 (4(3)), maximum 0.5137 (8(3)), mean 0.1688,
      median 0.1892; APP\@: minimum 0.023 (19(0)), maximum 0.1359 (3(4)), mean
      0.0676, median 0.0602}\label{fig-in-difAMPvsAPP}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1.01\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801_120_0APPvsAMP.pdf}
    \caption{AMP\@: minimum 0.0286 (3(0)), maximum 0.3934 (26(3)), mean 0.12675,
      median 0.0938; APP\@: minimum 0.00227 (11(0)), maximum 0.11445 (3(2)),
      mean 0.0407, median 0.0256}\label{fig-au-difAMPvsAPP}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[$\mathcal{CPD}$ of each plate's paleomagnetic APWPs vs its FHM
    predicted APWP (AMP vs APP)]{Separated results from AMP and APP in
    Fig.~\ref{fig-dif}. For each grid block (left: AMP, right: APP), the
    $\mathcal{CPD}s$ less than one-standard-deviation interval of the whole 84
    values are labeled in green, more than one-standard-deviation interval
    labeled in red.}\label{fig-difAMPvsAPP}
\end{figure}

For each of the 84 possible combinations of filter method and weighting, the
AMP-derived score is typically 3\textendash5 times higher than the equivalent
APP-derived score (Fig.~\ref{fig-boxAMPvsAPP}, insets). APP-generated paths
yield a lower $\mathcal{CPD}$ score than the equivalent AMP-generated path for
82 (97.6\%) of the North America scores, 72 (85.7\%) of the India scores, and 84
(100\%) of the Australia scores. Furthermore, APP-generated paths yield not only
a lower $\mathcal{CPD}$ score but also a same or better FQ than the equivalent
AMP-generated path for 78 (92.9\%) of the North America scores, 67 (79.8\%) of
the India scores, and 76 (90.5\%) of the Australia scores. For the North
American and Indian plates, filtering that prefers igneous paleopoles (Pk 4,5
and 6,7) or removes paleopoles with large various uncertainties (Pk 22,23 and
26,27) is most likely to produce AMP scores close to (less than 1.5 times) or
less than the APP scores
(Fig.~\ref{fig-difAMPvsAPP},~\subref{fig-na-difAMPvsAPP},~\subref{fig-in-difAMPvsAPP}).
In the former case, the scores are comparable and relatively low; in the latter
case, they are comparable but relatively high. For the Australian plate, only
correcting for sedimentary inclination shallowing (Pk 8,9) produces comparable
but moderate scores (Fig.~\ref{fig-au-difAMPvsAPP}).

\begin{figure}[tbp]
  \centering
  \begin{subfigure}{.92\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101_10_5FHM.pdf}
    \caption{}\label{fig-na-boxAMPvsAPP}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501_10_5FHM.pdf}
    \caption{}\label{fig-in-boxAMPvsAPP}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801_10_5FHM.pdf}
    \caption{}\label{fig-au-boxAMPvsAPP}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[Box-and-whisker and cross (inset) plots of
    Fig.~\ref{fig-difAMPvsAPP}]{Box-and-whisker and cross (inset) plots of
    Fig.~\ref{fig-difAMPvsAPP}. The $\mathcal{CPD}$s from same filter and weighting
    method (red and blue dots plotted with box-and-whisker) are connected; some
    special cases where $\mathcal{CPD}$ from AMP lower than from APP are highlighted
    using darker connecting lines. Dot symbols are semi-transparent so a darker
    color indicates a greater number of data at a given
    $\mathcal{CPD}$.}\label{fig-boxAMPvsAPP}
\end{figure}

On the North American plate, the APP-derived $\mathcal{CPD}$ score is most
likely to be significantly better (defined as more than 5 times higher) when
Wt 0 (no weighting) and 1 (by sample and site number) have been
applied (Fig.~\ref{fig-na-difAMPvsAPP}). This pattern is also seen for the
Australian plate, but removal of poles published after 1983 (Pk 16,17) also
results in significantly better performance of the APP method
(Fig.~\ref{fig-au-difAMPvsAPP}). For the Indian plate, the largest difference
occurs when paleopoles with sedimentary inclination shallowing (Pk 8,9) are
corrected, or suspected overprints or local rotations are removed (Pk
18,19, Fig.~\ref{fig-in-difAMPvsAPP}).

\subsection{Effects of Filtering and Weighting}

When $\mathcal{CPD}$ scores are separated by windowing method
(Fig.~\ref{fig-difAMPvsAPP}), the effects of particular filtering and weighting
methods become easier to discern. In general, different filters (rows) produce
larger variations in scores than different weighting methods (columns). With the
exception of AMP-derived paths for India, the $\mathcal{CPD}$ score for paths
with no filtering (Pk 0,1) and no weighting (Wt 0) is lower than the mean scores
for that plate and windowing method. Also with the exception of India, the FQ
for paths with no filtering (Pk 0,1) and no weighting (Wt 0) is better than or
the same as all the other FQ\@. Therefore filtering and weighting at best
slightly improves, and at worst significantly degrades, the APWP fit to the
reference path.

\subsubsection{Filter Aggression}

When considering the effects of filtering, it is important to consider how many
paleopoles within the data set have been removed (related to Pk 2\textendash7,
10,11, and 14\textendash27) or corrected (Pk 8,9 and 12,13): if there is very
little alteration of the data set, little change from no filtering (Pk 0,1)
would be expected. In terms of the numbers of paleopoles affected, the most
consequential filters are:
%
\begin{figure}[tbp]
  \centering
  \begin{subfigure}{.99\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/f501d101.pdf}
    \caption{Differences between Fig.~\ref{fig-in-dif} (India) and
      Fig.~\ref{fig-na-dif} (North America)}\label{fig-i-n-dif}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.99\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/f801d101.pdf}
    \caption{Differences between Fig.~\ref{fig-au-dif} (Australia) and
      Fig.~\ref{fig-na-dif} (North America)}\label{fig-a-n-dif}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.99\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/f501d801.pdf}
    \caption{Differences between Fig.~\ref{fig-in-dif} (India) and
      Fig.~\ref{fig-au-dif} (Australia)}\label{fig-i-a-dif}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[Differences of $\mathcal{CPD}$ of each plate's paleomagnetic APWPs vs
    its FHM predicted APWP]{Differences between different continents' grids in
    Fig.~\ref{fig-dif}. The absolute difference values less than
    1.96-standard-deviation interval of the whole 168 values are labeled in
    green, more than 1.96-standard-deviation interval labeled in
    red.}\label{fig-d-dif}
\end{figure}
%
\begin{enumerate}
  \item removal of sedimentary paleopoles (Pk 4,5 and 6,7), which removes
    ${\sim}40$\textendash50\% of the datasets on all 3 plates, with the highest
    proportion being removed on the Indian plate. Although Pk 4,5 are more
    strict, it does not remove many more paleopoles than Pk 6,7. For North
    America and India, the numbers of filtered paleopoles for Pk 4,5 and 6,7 are
    actually the same.
  \item correction of sedimentary paleopoles for inclination flattening (filter
    in Pk 8,9), which affects 38\textendash48\% of the dataset, with the highest
    affected proportion on the Indian plate.
  \item removal of paleopoles with large temporal and spatial uncertainty (Pk
    2,3, 22,23), particularly for the Australian plate, where the SS05 filtering
    criteria removes ${\sim}70$\% of the paleopoles. Filter in Pk 26,27 combines
    Pk 22,23 and 24,25, but no or very few (in the only case of Australia only 1
    additional paleopole) are actually removed.
  \item filtering based on publication date (Pk 14,15 and 16,17), with the ratio
    of pre/post 1983 poles varying from about 50/50 on the North American plate
    to about 70/30 on the Australian plate.
\end{enumerate}

Conversely, filtering or correction for redbeds (Pk 10,11 and 12,13), local
rotations and overprints (Pk 18,19 and 20,21 [only one paleopole influenced by
local rotation removed, and only one corrected, for just India; ${\sim}2.7$\%,
Fig.~\ref{fig-in-dif}]), or superseded data (Pk 24,25) affected 4\% or less of
the paleopoles on any plate.

Note that there is an important trade-off in that precision of a mean pole is
determined by the number of contributing paleopoles (A95 is proportional to
1/$\sqrt{n}$), so an overly aggressive filter might improve accuracy at the
expense of precision. This is particularly an issue where data density is low
(see Section~\ref{sec:datden}).

\subsubsection{Filter Performance}

Focussing on the filtering and weighting methods with aggressive filtering,
some commonalities in the best- and worst-performing methods can be observed,
although there are usually exceptions for particular plates and/or windowing
methods:
%
\begin{figure}[!ht]
  \captionsetup[subfigure]{singlelinecheck=off,justification=raggedright,aboveskip=-6pt,belowskip=-6pt}
  \centering
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101npoles105w0.pdf}
    \caption{}\label{fig-na-dsw0}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101npoles105w1.pdf}
    \caption{}\label{fig-na-dsw1}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101npoles105w2.pdf}
    \caption{}\label{fig-na-dsw2}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101npoles105w3.pdf}
    \caption{}\label{fig-na-dsw3}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101npoles105w4.pdf}
    \caption{}\label{fig-na-dsw4}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101npoles105w5.pdf}
    \caption{}\label{fig-na-dsw5}
  \end{subfigure}
  \caption[$d_s$ of each pair of poles for North American 10/5 Myr APWPs]{Tested
    spatial difference ($d_s$) values (color shaded) between North American
    paleomagnetic APWPs and its predicted APWP from the FHM and related plate
    circuits. The paths are in 10/5 Myr bin/step. (a) The number labels on the
    grids (including grid heights) are the numbers of paleopoles that are
    contributing to make each mean pole; (b-f) Only different $d_s$ from (a) are
    plotted, and numbers of contributing paleopoles are exactly the same as in
    (a).}\label{fig-nads}
\end{figure}

\begin{figure}[!ht]
  \captionsetup[subfigure]{singlelinecheck=off,justification=raggedright,aboveskip=-6pt,belowskip=-6pt}
  \centering
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nsegs105w0.pdf}
    \caption{}\label{fig-na-dlw0}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nsegs105w1.pdf}
    \caption{}\label{fig-na-dlw1}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nsegs105w2.pdf}
    \caption{}\label{fig-na-dlw2}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nsegs105w3.pdf}
    \caption{}\label{fig-na-dlw3}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nsegs105w4.pdf}
    \caption{}\label{fig-na-dlw4}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nsegs105w5.pdf}
    \caption{}\label{fig-na-dlw5}
  \end{subfigure}
  \caption[$d_l$ of each pair of segments for North American 10/5 Myr
    APWPs]{Tested length difference ($d_l$) values (color shaded) between North
    American paleomagnetic APWPs and its predicted APWP from FHM and related
    plate circuits. The paths are in 10/5 Myr bin/step. (a) The labeled numbers
    on the grids are the averaged numbers of paleopoles that are contributing to
    each segment's two mean poles; (b-f) Only different $d_l$ from (a) are
    plotted, and numbers of contributing paleopoles are exactly the same as in
    (a).}\label{fig-nadl}
\end{figure}
%
\begin{enumerate}
  \item For all three plates, higher $\mathcal{CPD}$ scores are commonly
    associated with aggressively filtering based on $\alpha$95 and age range (Pk
    2,3, 22,23, 26,27), with the exception of AMP-derived paths for India, where
    Pk 22 and 26 produce some of the lowest scores, and also improve FQ\@.
  \item For North America and India, low scores are commonly associated with
    removal of non-igneous poles (Pk 4,5 and 6,7), particularly for AMP-derived
    (Pk 4 and 6) paths (still mostly higher than but very close to APP (Pk 5 and
    7) scores). These filters also improve or sustain FQ when APP replaces AMP,
    except Wt 0,1 reduced FQ to APP's C-A from AMP's B-A for North America. On
    the Australian plate, these filters are less effective, and also reduce
    FQ\@.
  \item For North America and Australia, correction for inclination flattening
    generates $\mathcal{CPD}$ scores very similar to scores with no filtering
    for AMP-derived paths (Pk 8 versus Pk 0; for Australia FQ reduced), and
    increases scores for APP-derived paths (Pk 9 versus Pk 1). In contrast, for
    India generally there is a small decrease in $\mathcal{CPD}$ scores compared
    to no filtering for both AMP- and APP-derived paths.
  \item Many of the highest difference scores for North America and India occur
    when paleopoles published after 1983 are removed (Pk 16,17), whilst removing
    paleopoles published before 1983 (Pk 14,15) generates $\mathcal{CPD}$ scores
    comparable to scores with no filtering (relatively much lower). In contrast,
    for Australia Pk 14,15 produces some of the highest $\mathcal{CPD}$ scores,
    and Pk 16,17 have little effect on the $\mathcal{CPD}$ score, although FQ is
    commonly reduced. It is noteworthy that the number of older studies (65/68;
    Fig.~\ref{fig-au-dif}) is almost 2.5 times of the number of newer studies
    (27/29) for the Australian plate.
\end{enumerate}

\begin{figure}[!ht]
  \captionsetup[subfigure]{singlelinecheck=off,justification=raggedright,aboveskip=-6pt,belowskip=-6pt}
  \centering
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nocs105w0.pdf}
    \caption{}\label{fig-na-daw0}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nocs105w1.pdf}
    \caption{}\label{fig-na-daw1}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nocs105w2.pdf}
    \caption{}\label{fig-na-daw2}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nocs105w3.pdf}
    \caption{}\label{fig-na-daw3}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nocs105w4.pdf}
    \caption{}\label{fig-na-daw4}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.495\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101nocs105w5.pdf}
    \caption{}\label{fig-na-daw5}
  \end{subfigure}
  \caption[$d_a$ of each pair of segment-oreintation-changes for North American
    10/5 Myr APWPs]{Tested angular difference ($d_a$) values (color shaded)
    between North American paleomagnetic APWPs and its predicted APWP from FHM
    and related plate circuits. The paths are in 10/5 Myr bin/step. (a) The
    labeled numbers on the grids are the averaged numbers of paleopoles that are
    contributing to each segment-orientation-change's three mean poles; (b-f)
    Only different $d_a$ from (a) are plotted, and numbers of contributing
    paleopoles are exactly the same as in (a).}\label{fig-nada}
\end{figure}

\begin{figure}
  \centering
  \begin{subfigure}{.95\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101_20_10_120_0.pdf}
    \caption{minimum 0.00991 (15(0)), maximum 0.0654 (14(2)),
      mean 0.0339, median 0.0296371}\label{fig-na-dif2010}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.95\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501_20_10_120_0.pdf}
    \caption{minimum 0.0142822 (6(3)), maximum 0.154278 (16(3)), mean
      0.07384, median 0.072088}\label{fig-in-dif2010}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.95\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801_20_10_120_0.pdf}
    \caption{min 0 (11(0,1),19(3)), max 0.324438 (22(3)),
      mean 0.0682, median 0.036432}\label{fig-au-dif2010}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[$\mathcal{CPD}$ of each plate's paleomagnetic APWPs vs its FHM
    predicted APWP (20/10 Myr window/step)]{As Fig.~\ref{fig-dif}, here the
    paths are generated in 20/10 Myr bin/step. The $\mathcal{CPD}s$ less than
    and more than one-standard-deviation interval of the whole 168 labeled
    values are colored in green and red. Compare the numbers of picked
    paleopoles with those in Fig.~\ref{fig-dif}.}\label{fig-dif2010}
\end{figure}

Whilst it is generally true that methods with low-aggression filters do not
generate scores that differ much from the no-filtering scores, there are some
exceptions:
%
\begin{enumerate}
  \item For North America and Australia, removing superseded paleopoles (Pk
    24,25) produces lower $\mathcal{CPD}$ scores with their FQ unchanged.
  \item Removing paleopoles suspected to be affected by overprints or local
    rotations (Pk 18,19) consistently produces lower $\mathcal{CPD}$ scores with
    their FQ unchanged for Indian APP-derived paths.
\end{enumerate}

\subsubsection{Weighting Performance}

Compared to the variations resulting from different windowing methods and
filters, Fig.~\ref{fig-difAMPvsAPP}, and
Figs.~\ref{fig-nads},~\ref{fig-nadl},~\ref{fig-nada} indicate that the effect of
weighting the data prior to calculating a \emph{Fisher} mean is generally small
or diluted. The individual $d_s$, $d_l$ and $d_a$ difference scores are not
being affected too much for the different applied weighting schemes. Where an
effect can be seen, it is negative, generating larger $\mathcal{CPD}$ scores but
sometimes better FQ (Fig.~\ref{fig-dif}) and sometimes worse FQ
(Fig.~\ref{fig-dif2010},~\subref{fig-na-dif2010},~\subref{fig-in-dif2010}).
%
\begin{enumerate}
  \item For APP-derived paths from the North American and Australian plates, Wt
    0 (no weighting) and 1 (weighting by sample and site number) usually produce
    slightly better $\mathcal{CPD}$ scores with FQ generally unchanged than
    other weighting methods.
  \item Wt 3 (weighting by spatial uncertainty) seems most likely to generate
    much higher $\mathcal{CPD}$ scores, particularly for AMP-derived paths, and
    particular for the Indian plate.
  \item Wt 0, 1 and 5 generally produce lower similarity scores than Wt 2, 3 and
    4.
\end{enumerate}

\begin{figure}[tbp]
  \centering
  \vspace{-.2cm}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_20_10_15_0.pdf}
    \caption{N America: min 0.00991 (15(0)), FQ B-A}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_20_10_14_2.pdf}
    \caption{N America: max 0.0654 (14(2)), FQ B-A}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_20_10_6_3.pdf}
    \caption{India: minimum 0.0143 (6(3)), FQ C-A}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_20_10_16_3.pdf}
    \caption{India: maximum 0.1543 (16(3)), FQ C-A}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_20_10_11_0.pdf}
    \caption{Australia: min 0 (11(0)), FQ B-A}\label{fig-au-2010110}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_20_10_22_3.pdf}
    \caption{Australia: max 0.3244 (22(3)), FQ C-A}\label{fig-au-2010223}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[Best and worst $\mathcal{CPD}$s (20/10 Myr window/step)]{Path
    comparisons with best and worst $\mathcal{CPD}$ values shown in
    Fig.~\ref{fig-dif2010}. The parenthetical remarks are Pk No (Wt
    No).}\label{fig-dif2010bw}
\end{figure}

\subsection{Effects of Window Size}\label{sec:winsiz}

Fig.~\ref{fig-dif2010} shows the $\mathcal{CPD}$ scores and FQ for the APWPs
generated with the 28 picking and 6 weighting methods, compared to the FHM
reference paths, with the picking time window width increased from 10 to 20 Myr,
and the window step increased from 5 to 10 Myr.

\subsubsection{Overall Change}

The overall effect of increased window and step size varies between plates
(Fig.~\ref{fig-f105d2010}). 118/168 (${\sim}70$\%) of equivalent $\mathcal{CPD}$
scores for the North American plate increase, indicating reduced similarity with
respect to the reference path (Fig.~\ref{fig-101f105d2010}). The increased
occurrence of A-A rather than B-A FQ ratings (Fig.~\ref{fig-na-dif2010} versus
Fig.\ref{fig-na-dif}) indicate that in some cases this may be due to a better
constrained path becoming more distinguishable. Decreased scores are confined to
particular, mostly AMP-derived picking methods: Pk 2, 22 and 26 (filtering based
on ${\alpha}95$ and age uncertainties), 4 and 6 (removal of sedimentary
paleopoles), and 16,17 (removal of post–1983 results). It is clearly due to the
corresponding large increase (commonly to two times) of N in each increased
window for these AMP-derived quality filters (even Pk numbers in
Table~\ref{tab:Ws2N}). Although there are also corresponding increases of N in
each increased window for the APP-derived quality filters (odd Pk numbers in
Table~\ref{tab:Ws2N}), they only increase to around 1.4 times 10/5-Myr N.

\begin{table}[!ht]
  \centering
  \caption{Changes of average number of paleopoles per window/mean pole along
    with bin/step from 10/5 Myr to 20/10 Myr.}\label{tab:Ws2N}
  \resizebox{.6\textwidth}{!}{%
    \begin{tabular}{@{}ccccccc@{}}
      \toprule
      \multicolumn{1}{c}{\multirow{3}{*}{Pk}} & \multicolumn{6}{c}{Average Number of Paleopoles per Window} \\ \cmidrule(l){2-7} 
      \multicolumn{1}{c}{} & \multicolumn{2}{c}{N America} & \multicolumn{2}{c}{India} & \multicolumn{2}{c}{Australia} \\ \cmidrule(l){2-7} 
      \multicolumn{1}{c}{} & \multicolumn{1}{c}{10/5} & \multicolumn{1}{c}{20/10} & \multicolumn{1}{c}{10/5} & \multicolumn{1}{c}{20/10} & \multicolumn{1}{c}{10/5} & \multicolumn{1}{c}{20/10} \\ \midrule
      0 & 10 & 18.1 & 6.4 & 11.1 & 6.2 & 12 \\
      1 & 21 & 30.3 & 10.2 & 14.8 & 19.6 & 25.4 \\
      2 & 8.2 & 15 & 5.8 & 9.2 & 4.3 & 7.5 \\
      3 & 10.2 & 17 & 8 & 10.5 & 7.6 & 10.9 \\
      4 & 5.9 & 10.5 & 10.6 & 12.8 & 4.1 & 8 \\
      5 & 10.8 & 15.9 & 8.8 & 11.7 & 11.4 & 14.8 \\
      6 & 5.9 & 10.5 & 10.6 & 12.8 & 4.3 & 8.3 \\
      7 & 10.8 & 15.9 & 8.8 & 11.7 & 11.6 & 15.2 \\
      8 & 10 & 18.1 & 6.4 & 11.1 & 6.2 & 12 \\
      9 & 21 & 30.3 & 10.2 & 14.8 & 19.6 & 25.4 \\
      10 & 9.7 & 17.5 & 6.3 & 10.8 & 5.9 & 11.5 \\
      11 & 19.3 & 28.2 & 9.9 & 14.5 & 18.5 & 24 \\
      12 & 10 & 18.1 & 6.4 & 11.1 & 6.2 & 12 \\
      13 & 21 & 30.3 & 10.2 & 14.8 & 19.6 & 25.4 \\
      14 & 5.2 & 9.5 & 2.9 & 4.7 & 2 & 3.1 \\
      15 & 12.8 & 17.5 & 4.6 & 6.1 & 4.4 & 6 \\
      16 & 6.2 & 9.3 & 5.1 & 7.3 & 5 & 9.7 \\
      17 & 8.2 & 12.8 & 6.5 & 9.2 & 15.2 & 19.4 \\
      18 & 9.9 & 17.8 & 6 & 10.3 & 5.8 & 11.3 \\
      19 & 20.8 & 29.9 & 8.7 & 12.7 & 17.3 & 22.8 \\
      20 & 10 & 18.1 & 6.3 & 10.9 & 6.2 & 12 \\
      21 & 21 & 30.3 & 10 & 14.5 & 19.6 & 25.4 \\
      22 & 6.1 & 11.3 & 6.4 & 7.6 & 2.5 & 3.7 \\
      23 & 8.3 & 13.3 & 5.2 & 6.6 & 5 & 6.5 \\
      24 & 9.9 & 17.8 & 6.4 & 11.1 & 6 & 11.6 \\
      25 & 20.8 & 29.9 & 10.2 & 14.8 & 19.2 & 24.8 \\
      26 & 6.1 & 11.3 & 6.4 & 7.6 & 2.4 & 3.6 \\
      27 & 8.3 & 13.3 & 5.2 & 6.6 & 4.8 & 6.2 \\ \bottomrule
    \end{tabular}%
  }
\end{table}

For the Indian and Australian plates, there is a trend towards decreased
$\mathcal{CPD}$ scores, with 153/168 (${\sim}91$\%) and 104/168 (${\sim}62$\%)
of equivalent $\mathcal{CPD}$ scores being lower than for the 10/5 Myr
window/step, respectively
(Fig.~\ref{fig-f105d2010},~\subref{fig-501f105d2010},~\subref{fig-801f105d2010}).
Where the score increases, the actual difference is close to 0, indicating very
little change, with the exception of Pk 14 (AMP, removal of pre–1983 results) on
the Australian plate. The largest score decreases are associated with Wt 3 (AMP,
${\alpha}95$ and age filtering) on the Indian plate, and Pk 15 (APP, removal of
pre–1983 results) on the Australian plate. Most FQ ratings remain unchanged
(Fig.~\ref{fig-in-dif2010} versus Fig.~\ref{fig-in-dif},
Fig.~\ref{fig-au-dif2010} versus Fig.~\ref{fig-au-dif}): most obviously, ratings
for Pk 5 and 7 (APP, removal of sedimentary poles) on the Indian plate and Pk 14
on the Australian plate change from B-A to C-A.

\begin{figure}[tbp]
  \captionsetup{singlelinecheck=false,justification=raggedright,aboveskip=-6pt,belowskip=-6pt}
  \centering
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101f105d2010.pdf}
    \caption{Generally Bin/Step 10/5 Myr is better
      ($\frac{59}{84}\approx70.24\%$)}\label{fig-101f105d2010}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501f105d2010.pdf}
    \caption{Generally Bin/Step 20/10 Myr is better
      ($\frac{153}{168}\approx91.07\%$)}\label{fig-501f105d2010}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801f105d2010.pdf}
    \caption{Generally 20/10 Myr is better
      ($\frac{13}{21}\approx61.9\%$)}\label{fig-801f105d2010}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[Differences between grids in Fig.~\ref{fig-dif} (10/5 Myr bin/step)
    and Fig.~\ref{fig-dif2010} (20/10 Myr bin/step)]{Differences between grids in Fig.~\ref{fig-dif} (10/5 Myr bin/step)
    and Fig.~\ref{fig-dif2010} (20/10 Myr bin/step). The absolute difference
    values less than 1.96-standard-deviation interval of the whole 168 values
    are labeled in green, more than 1.96-standard-deviation interval labeled in
    red. The strikethrough labels show positive differences.}\label{fig-f105d2010}
\end{figure}

\subsubsection{Relative Performance of Methods}

As for the baseline results (Section~\ref{sec:base}, Fig.~\ref{fig-dif}), the
effects of windowing and filtering, particularly the windowing method (APP
versus AMP), are much more apparent than the effects of weighting. APP-derived
scores still outperform AMP-derived ones, but the overall degree of difference
is reduced for the North American and Indian plates, with percentage of
APP-derived $\mathcal{CPD}$ scores more than 3 times the equivalent AMP-derived
score reducing to ${\sim}26$\% and ${\sim}4$\% (from ${\sim}44$\% and
${\sim}40$\%), respectively. For the Australian plate, this percentage increases
from ${\sim}58$\% to ${\sim}70$\% (Fig.~\ref{fig-dif2010}). AMP-derived
$\mathcal{CPD}$ scores seem more sensitive to change of moving window/step size
than APP-derived ones because both range and mean of AMP-derived scores are
apparently decreased along with the increase of moving window/step size. The
mean AMP-derived still exceeds the max APP-derived for the Indian and Australian
plates though (Fig.~\ref{fig-boxAMPvsAPP2010} versus
Fig.~\ref{fig-boxAMPvsAPP}). In addition, APP-generated paths yield a lower
$\mathcal{CPD}$ score and also a same or better FQ than the equivalent
AMP-generated path for 43 (51.2\%) (reduced from 78 (92.9\%)) of the North
America scores, 52 (61.9\%) (reduced from 67 (79.8\%)) of the India scores, and
83 (98.8\%) (increased from 76 (90.5\%)) of the Australia scores.

\begin{figure}
  \captionsetup[subfigure]{singlelinecheck=off,justification=raggedright,aboveskip=0pt,belowskip=-6pt}
  \centering
  \begin{subfigure}{1\textwidth}
    \caption{}\label{fig-na-boxAMPvsAPP2010}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101_20_10FHM.pdf}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \caption{}\label{fig-in-boxAMPvsAPP2010}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501_20_10FHM.pdf}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \caption{}\label{fig-au-boxAMPvsAPP2010}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801_20_10FHM.pdf}
  \end{subfigure}
  \caption[Box-and-whisker and cross (inset) plots of Fig.~\ref{fig-dif2010}]{Box-and-whisker and cross (inset) plots of Fig.~\ref{fig-dif2010}.
The $\mathcal{CPD}$s from same filter and weighting method (red and blue dots
plotted with box-and-whisker) are connected. Dot symbols are semi-transparent so
a darker color indicates a greater number of data at a given
$\mathcal{CPD}$.}\label{fig-boxAMPvsAPP2010}
\end{figure}

This is largely the result of larger changes in the AMP-derived $\mathcal{CPD}$
scores compared to changes in the APP-derived ones with the wider time window.
The scores for APP-derived methods Pk 1 (no filtering), other non-aggressive
filters (Pk 13, 19, 21, 25), and Pk 9 (correction for inclination flattening),
are particularly stable (Fig.~\ref{fig-f105d2010}) for all three plates.
Filtering according to the SS05 criteria (Pk 23 and 27) also yields comparable
scores for both time windows on the North American and Indian plates.

\begin{landscape}
\begin{table}[tbp]
\centering
\caption{Consistency check on comparisons of picking methods' performance between 20/10 and 10/5 Myr window/step. Notes: E means expected; UE means unexpected.}\label{tab-2010vs105F}
\resizebox{.84\width}{!}{%
\setlength{\tabcolsep}{2pt}
\begin{tabular}{@{}cccllcllcccccl@{}}
\toprule
\multicolumn{2}{c}{Comparisons} & \multicolumn{3}{c}{Consistency of best} & \multicolumn{3}{c}{Consistency of worst} & \multicolumn{6}{c}{If CPD values for 20/10 are lower (Y/N)} \\ \midrule
10/5 & 20/10 & Y/N & \multicolumn{1}{c}{Special case} & Same Pk & Y/N & \multicolumn{1}{c}{Special case} & \multicolumn{1}{c}{Same Pk} & Mean & Median & Max & Min & All & If N, UE case \\ \midrule
\multicolumn{14}{c}{FHM} \\ \midrule
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-na-dif}\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-na-dif2010}\end{tabular}} & \multirow{4}{*}{Y} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}3 more best:\\ Pk 4, 6, 9\\ only for\\ 20/10 (E)\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}1, 5, 7,\\ 11, 13,\\ 15, 19,\\ 21, 25\end{tabular}} & \multirow{4}{*}{N} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}Pk 2, 5, 7, 17, 22,\\ 26 for 10/5 (E);\\ 0, 8, 10, 12, 20,\\ 24 for 20/10 (UE)\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}14,\\ 16,\\ 18\end{tabular}} & \multirow{4}{*}{N} & \multirow{4}{*}{Y} & \multirow{4}{*}{Y} & \multirow{4}{*}{N} & \multirow{4}{*}{N} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}Positive\\ values\\ in\\ Fig.~\ref{fig-101f105d2010}\end{tabular}} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \midrule
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-in-dif}\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-in-dif2010}\end{tabular}} & \multirow{4}{*}{Y} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}4 more best:\\ Pk 9, 21, 22,\\ 26 only for\\ 10/5 (UE)\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}4\textendash7,\\ 19\end{tabular}} & \multirow{4}{*}{N} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}Pk 8 for\\ 10/5 (E);\\ 23, 27 for\\ 20/10 (UE)\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}0, 2,\\ 10, 12,\\ 16, 18,\\ 20, 24\end{tabular}} & \multirow{4}{*}{Y} & \multirow{4}{*}{Y} & \multirow{4}{*}{Y} & \multirow{4}{*}{Y} & \multirow{4}{*}{N} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}Positive\\ values\\ in\\ Fig.~\ref{fig-501f105d2010}\end{tabular}} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \midrule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-au-dif}\end{tabular}} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-au-dif2010}\end{tabular}} & \multirow{3}{*}{Y} & \multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}2 more best:\\ Pk 23, 27 only\\ for 20/10 (E)\end{tabular}} & \multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}1, 11, 13,\\ 17, 19,\\ 21, 25\end{tabular}} & \multirow{3}{*}{Y} & \multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}1 more worst:\\ Pk 4 only for\\ 10/5 (E)\end{tabular}} & \multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}2, 14,\\ 16, 22,\\ 26\end{tabular}} & \multirow{3}{*}{Y} & \multirow{3}{*}{Y} & \multirow{3}{*}{Y} & \multirow{3}{*}{Y} & \multirow{3}{*}{N} & \multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}Positive\\ values in\\ Fig.~\ref{fig-801f105d2010}\end{tabular}} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \midrule
\multicolumn{14}{c}{MHM} \\ \midrule
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-na-difm}\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-na-dif2010m}\end{tabular}} & \multirow{5}{*}{N} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}Pk 1, 9, 11,\\ 13, 19, 21,\\ 25 for 10/5\\ (UE); 22, 26\\ for 20/10 (E)\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}5,\\ 7,\\ 15\end{tabular}} & \multirow{5}{*}{N} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}Pk 16,17\\ only for 10/5\\ (E); 8 only\\ for 20/10\\ (UE)\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}0, 10\\ 12, 14,\\ 18,\\ 20,\\ 24\end{tabular}} & \multirow{5}{*}{N} & \multirow{5}{*}{Y} & \multirow{5}{*}{N} & \multirow{5}{*}{N} & \multirow{5}{*}{N} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}$(0,8,10\textendash12,14,18,20,24,25)(0\textendash5)$\\ $(1,13,19,21)(0,1,5)$  3(0,3,5)\\ $(5,7)(0\textendash2,4,5)$  15(0\textendash4)\\ $(9,23,27)(0,1,3,5)$\\ account for 60.71\%\end{tabular}} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \midrule
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-in-difm}\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-in-dif2010m}\end{tabular}} & \multirow{5}{*}{N} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}Pk 19, 21\\ only for 10/5\\ (UE); 4, 6\\ only for\\ 20/10 (E)\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}5,\\ 7,\\ 22,\\ 26\end{tabular}} & \multirow{5}{*}{Y} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}3 more\\ worst: Pk\\ 8, 14, 20\\ for 10/5\\ (E)\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}0, 2,\\ 10, 12,\\ 16,\\ 18,\\ 24\end{tabular}} & \multirow{5}{*}{Y} & \multirow{5}{*}{Y} & \multirow{5}{*}{Y} & \multirow{5}{*}{Y} & \multirow{5}{*}{N} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}$(1,25)(0,3,5)$  15(3)\\ $(19,23,27)(0\textendash5)$\\ 21(0,1,3\textendash5)\\ $(22,26)(4)$;\\ account for 19.05\%\end{tabular}} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \midrule
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-au-difm}\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fig.\\\ref{fig-au-dif2010m}\end{tabular}} & \multirow{5}{*}{Y} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}3 more best:\\ Pk 15, 23,\\ 27 only\\ for 20/10\\ (E)\end{tabular}} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}1, 11,\\ 13, 17,\\ 19,\\ 21,\\ 25\end{tabular}} & \multirow{5}{*}{Y} & \multirow{5}{*}{\textit{None}} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}2,\\ 14,\\ 16,\\ 22,\\ 26\end{tabular}} & \multirow{5}{*}{Y} & \multirow{5}{*}{Y} & \multirow{5}{*}{Y} & \multirow{5}{*}{N} & \multirow{5}{*}{N} & \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}$(0,20)(5)$  $(5,7)(3)$\\ $(1,11,13,16,18,19,21,25)(0,1,3,5)$\\ $(8,17)(0\textendash3,5)$  9(0,1)  10(1,5)\\ 12(2)  14(2,4,5)  $(22,26)(0\textendash2,4,5)$\\ 24(1,2,5); account for 39.88\%\end{tabular}} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \bottomrule
\end{tabular}%
}
\end{table}
\end{landscape}

As for the 10/5 window, the lowest AMP/APP differences are for Pk 4,5 and 6,7
(igneous poles preferred \textendash{} low scores) and Pk 2,3 and 22,23
(filtering for spatial and temporal uncertainty \textendash{} high scores) for
the North American and Indian plates; and Pk 8,9 (correction for inclination
shallowing \textendash{} moderate scores) for the Australian plates. Likewise,
the highest AMP/APP difference is still associated with Pk 0,1 (no filtering;
low scores) on the North American and Australian plates, and Pk 16,17 on the
Australian plate. The low AMP/APP difference category in the 20/10 window still
remains a similar pattern and shows even lower AMP/APP differences for these
methods. In addition, there are more particular methods that are added to this
category, for example, Pk 14,15 for India (Fig.~\ref{fig-in-dif2010}). This
seems indicating that increased window/step sizes corresponds to lower AMP/APP
differences (later further proved to be true in Fig.~\ref{fig-WinStpVsCPD}).

Overall, the relative performance of the different filtering methods when using
the 20/10 window/step remains similar to that observed for the 10/5 window/step
(Figs.~\ref{fig-dif},~\ref{fig-dif2010}). Some of the lowest scores are still
produced by Pk 1 (APP, no filtering). Of the aggressive filters, removing
sedimentary paleopoles (Pk 4,5 and 6,7) still perform well, selecting only
spatially and temporally well-constrained paleopoles (Pk 2,3 and 22,23) still
perform relatively poorly, and correction for inclination flattening (Pk 8,9)
has little effect.

In addition, using only paleopoles published before 1983 (Pk 16,17) still
produces relatively high $\mathcal{CPD}$ scores for the North American and
Indian plates, although the scores for the AMP-derived Pk 16 are substantially
reduced with the wider time window
(Fig.~\ref{fig-f105d2010},~\subref{fig-101f105d2010},~\subref{fig-501f105d2010}).
Likewise, for Australia, using only paleopoles published after 1983 (Pk 14,15)
still produces higher scores than Pk 16,17: scores for the wider time window are
even higher for the AMP-derived Pk 14, but substantially lower for the
APP-derived Pk 15 (Fig.~\ref{fig-801f105d2010}).

\subsubsection{Other Window Sizes and Steps}

\paragraph{What to expect is}
the $\mathcal{CPD}$ values for larger window/step size should be generally lower
than those for smaller window/step size, which further could result in more best
methods and less worst methods.

\begin{landscape}
\renewcommand*{\arraystretch}{.9}  % arraystretch sets the vertical (row) spacing
\begin{table}[tbp]
\centering
\caption{Equal-weight 120\textendash0 Ma $\mathcal{CPD}s$ for the three
  representative continents' paleomagnetic APWPs compared with their FHM
  predicted APWPs. The best are in dark green and underlined, second best in
  green and third in light green. FQ is shown in cell background color: blue,
  A-A\@; no color, B-A\@; orange, C-A\@; red, D-A.}\label{tab-pk0vs1bs}
\resizebox{.76\width}{!}{%
\setlength{\tabcolsep}{1pt}
\begin{tabular}{@{}lllllllllllll@{}}
\toprule
\multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Window/step\\ size (Myr)\end{tabular}}} & \multicolumn{6}{c}{Pk 0} & \multicolumn{6}{c}{Pk 1} \\ \cmidrule(l){2-13} 
\multicolumn{1}{c}{} & \multicolumn{1}{c}{Wt 0} & \multicolumn{1}{c}{Wt 1} & \multicolumn{1}{c}{Wt 2} & \multicolumn{1}{c}{Wt 3} & \multicolumn{1}{c}{Wt 4} & \multicolumn{1}{c}{Wt 5} & \multicolumn{1}{c}{Wt 0} & \multicolumn{1}{c}{Wt 1} & \multicolumn{1}{c}{Wt 2} & \multicolumn{1}{c}{Wt 3} & \multicolumn{1}{c}{Wt 4} & \multicolumn{1}{c}{Wt 5} \\ \midrule
\multicolumn{13}{c}{North America} \\ \midrule
2/1 & \cellcolor[HTML]{F8A102}0.2864 & \cellcolor[HTML]{F8A102}0.2925 & \cellcolor[HTML]{F8A102}0.28685 & \cellcolor[HTML]{F8A102}0.280944 & \cellcolor[HTML]{F8A102}0.287 & \cellcolor[HTML]{F8A102}0.29324 & {\color[HTML]{34FF34} \textbf{0.005759}} & 0.0091586 & {\color[HTML]{32CB00} \textbf{0.01528}} & {\color[HTML]{009901} {\ul\textbf{0.0090683}}} & 0.03601 & {\color[HTML]{32CB00} \textbf{0.009034}} \\
4/2 & \cellcolor[HTML]{F8A102}0.10584 & \cellcolor[HTML]{F8A102}0.09228 & \cellcolor[HTML]{F8A102}0.10938 & \cellcolor[HTML]{F8A102}0.11301 & 0.1131 & \cellcolor[HTML]{F8A102}0.11263 & {\color[HTML]{009901} {\ul\textbf{0.00525}}} & 0.007665 & 0.01701 & {\color[HTML]{32CB00} \textbf{0.009612}} & 0.03504 & {\color[HTML]{009901} {\ul\textbf{0.007863}}} \\
6/3 & 0.06486 & 0.064078 & 0.077426 & 0.073621 & 0.082867 & 0.075522 & {\color[HTML]{32CB00} \textbf{0.005418}} & {\color[HTML]{34FF34} \textbf{0.007522}} & 0.017936 & {\color[HTML]{34FF34} \textbf{0.0097317}} & {\color[HTML]{34FF34} \textbf{0.01983}} & 0.0095782 \\
8/4 & 0.06934 & 0.07168 & 0.082474 & 0.0707846 & 0.11473 & 0.071686 & 0.006 & {\color[HTML]{32CB00} \textbf{0.006217}} & {\color[HTML]{34FF34} \textbf{0.01556}} & 0.01242 & 0.02163 & 0.011483 \\
10/5 & 0.04412 & 0.04486 & 0.04739 & 0.04212 & 0.04753 & 0.04445 & 0.00591 & {\color[HTML]{009901} {\ul\textbf{0.00616}}} & {\color[HTML]{009901} {\ul\textbf{0.01262}}} & 0.01208 & {\color[HTML]{009901} {\ul\textbf{0.01529}}} & {\color[HTML]{34FF34} \textbf{0.00941}} \\
12/6 & {\color[HTML]{32CB00} \textbf{0.03271}} & {\color[HTML]{32CB00} \textbf{0.02987}} & {\color[HTML]{009901} {\ul\textbf{0.03024}}} & {\color[HTML]{009901} {\ul\textbf{0.0276143}}} & {\color[HTML]{009901} {\ul\textbf{0.03012}}} & {\color[HTML]{009901} {\ul\textbf{0.02988}}} & 0.008736 & 0.00902 & \cellcolor[HTML]{3531FF}0.01715 & 0.015665 & \cellcolor[HTML]{3531FF}{\color[HTML]{32CB00} \textbf{0.01642}} & 0.01453 \\
14/7(119\textendash0) & 0.0367 & 0.03695 & 0.03498 & 0.0347 & {\color[HTML]{34FF34} \textbf{0.0319}} & 0.0352 & 0.0115 & 0.01198 & \cellcolor[HTML]{3531FF}0.023922 & 0.01477 & \cellcolor[HTML]{3531FF}0.02205 & 0.012985 \\
16/8 & 0.052386 & 0.04923 & 0.04964 & 0.04653 & 0.047712 & 0.048523 & 0.01138 & 0.0117 & \cellcolor[HTML]{3531FF}0.02177 & 0.015409 & \cellcolor[HTML]{3531FF}0.023068 & 0.01616 \\
20/10 & \cellcolor[HTML]{3531FF}0.04972 & \cellcolor[HTML]{3531FF}0.05356 & \cellcolor[HTML]{3531FF}0.0562 & \cellcolor[HTML]{3531FF}0.05352 & 0.0542 & \cellcolor[HTML]{3531FF}0.04902 & 0.014 & 0.01728 & \cellcolor[HTML]{3531FF}0.02896 & 0.0181 & \cellcolor[HTML]{3531FF}0.02702 & 0.0185 \\
24/12 & \cellcolor[HTML]{3531FF}{\color[HTML]{009901} {\ul\textbf{0.031642}}} & \cellcolor[HTML]{3531FF}{\color[HTML]{34FF34} \textbf{0.0342}} & \cellcolor[HTML]{3531FF}{\color[HTML]{34FF34} \textbf{0.03273}} & \cellcolor[HTML]{3531FF}{\color[HTML]{34FF34} \textbf{0.03469}} & \cellcolor[HTML]{3531FF}{\color[HTML]{32CB00} \textbf{0.0314765}} & \cellcolor[HTML]{3531FF}{\color[HTML]{32CB00} \textbf{0.031253}} & 0.0164 & 0.016737 & \cellcolor[HTML]{3531FF}0.02521 & \cellcolor[HTML]{3531FF}0.01907 & \cellcolor[HTML]{3531FF}0.02397 & \cellcolor[HTML]{3531FF}0.01924 \\
30/15 & \cellcolor[HTML]{3531FF}{\color[HTML]{34FF34} \textbf{0.0345}} & \cellcolor[HTML]{3531FF}{\color[HTML]{009901} {\ul\textbf{0.0298}}} & \cellcolor[HTML]{3531FF}{\color[HTML]{32CB00} \textbf{0.0317}} & \cellcolor[HTML]{3531FF}{\color[HTML]{32CB00} \textbf{0.0307}} & \cellcolor[HTML]{3531FF}0.03402 & \cellcolor[HTML]{3531FF}{\color[HTML]{34FF34} \textbf{0.0341}} & \cellcolor[HTML]{3531FF}0.0206 & \cellcolor[HTML]{3531FF}0.0211 & \cellcolor[HTML]{3531FF}0.0313 & \cellcolor[HTML]{3531FF}0.0272 & \cellcolor[HTML]{3531FF}0.0293 & \cellcolor[HTML]{3531FF}0.0277 \\ \midrule
\multicolumn{13}{c}{India} \\ \midrule
2/1 & \cellcolor[HTML]{F8A102}0.25838 & \cellcolor[HTML]{F8A102}0.258757 & \cellcolor[HTML]{F8A102}0.258701 & \cellcolor[HTML]{F8A102}0.275184 & \cellcolor[HTML]{F8A102}0.265442 & \cellcolor[HTML]{F8A102}0.258692 & \cellcolor[HTML]{FE0000}0.05446 & \cellcolor[HTML]{FE0000}0.05639 & \cellcolor[HTML]{F8A102}0.06456 & \cellcolor[HTML]{F8A102}0.06075 & \cellcolor[HTML]{F8A102}0.0677596 & \cellcolor[HTML]{F8A102}0.0583844 \\
4/2 & \cellcolor[HTML]{F8A102}0.19345 & \cellcolor[HTML]{F8A102}0.20276 & \cellcolor[HTML]{F8A102}0.19344 & \cellcolor[HTML]{F8A102}0.24597 & \cellcolor[HTML]{F8A102}0.203264 & \cellcolor[HTML]{F8A102}0.19423 & \cellcolor[HTML]{F8A102}0.05588 & \cellcolor[HTML]{F8A102}0.056691 & \cellcolor[HTML]{F8A102}0.066803 & \cellcolor[HTML]{F8A102}0.062283 & \cellcolor[HTML]{F8A102}0.067216 & \cellcolor[HTML]{F8A102}0.060121 \\
6/3 & \cellcolor[HTML]{F8A102}0.17396 & \cellcolor[HTML]{F8A102}0.17476 & \cellcolor[HTML]{F8A102}0.173975 & \cellcolor[HTML]{F8A102}0.229325 & \cellcolor[HTML]{F8A102}0.18514 & \cellcolor[HTML]{F8A102}0.175174 & \cellcolor[HTML]{F8A102}0.057887 & \cellcolor[HTML]{F8A102}0.05882 & \cellcolor[HTML]{F8A102}0.0664754 & \cellcolor[HTML]{F8A102}0.0627534 & \cellcolor[HTML]{F8A102}0.06725 & \cellcolor[HTML]{F8A102}0.06083 \\
8/4 & \cellcolor[HTML]{F8A102}0.15309 & \cellcolor[HTML]{F8A102}0.14966 & \cellcolor[HTML]{F8A102}0.15321 & \cellcolor[HTML]{F8A102}0.17412 & \cellcolor[HTML]{F8A102}0.172154 & \cellcolor[HTML]{F8A102}0.15124 & \cellcolor[HTML]{F8A102}0.05761 & \cellcolor[HTML]{F8A102}0.05848 & \cellcolor[HTML]{F8A102}0.06671 & \cellcolor[HTML]{F8A102}0.06139 & \cellcolor[HTML]{F8A102}0.0669 & \cellcolor[HTML]{F8A102}0.05941 \\
10/5 & \cellcolor[HTML]{F8A102}0.1839 & \cellcolor[HTML]{F8A102}0.1845 & \cellcolor[HTML]{F8A102}0.1909 & \cellcolor[HTML]{F8A102}0.2586 & \cellcolor[HTML]{F8A102}0.1972 & \cellcolor[HTML]{F8A102}0.1852 & \cellcolor[HTML]{F8A102}0.0545 & \cellcolor[HTML]{F8A102}0.0554 & \cellcolor[HTML]{F8A102}0.0662 & \cellcolor[HTML]{F8A102}0.0589 & \cellcolor[HTML]{F8A102}0.066 & \cellcolor[HTML]{F8A102}0.06 \\
12/6 & \cellcolor[HTML]{F8A102}0.108924 & \cellcolor[HTML]{F8A102}0.1035 & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.11205}} & \cellcolor[HTML]{F8A102}0.11831 & \cellcolor[HTML]{F8A102}0.121497 & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.10587}} & \cellcolor[HTML]{F8A102}0.059897 & \cellcolor[HTML]{F8A102}0.06068 & \cellcolor[HTML]{F8A102}0.06993 & \cellcolor[HTML]{F8A102}0.064499 & \cellcolor[HTML]{F8A102}0.06951 & \cellcolor[HTML]{F8A102}0.062945 \\
14/7(119\textendash0) & \cellcolor[HTML]{F8A102}0.112537 & \cellcolor[HTML]{F8A102}0.112885 & \cellcolor[HTML]{F8A102}0.126554 & \cellcolor[HTML]{F8A102}0.139516 & \cellcolor[HTML]{F8A102}0.132359 & \cellcolor[HTML]{F8A102}0.114195 & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.04942}} & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.0502588}} & \cellcolor[HTML]{F8A102}0.0579931 & \cellcolor[HTML]{F8A102}0.060018 & \cellcolor[HTML]{F8A102}0.0654112 & \cellcolor[HTML]{F8A102}0.0582519 \\
16/8 & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.104461}} & \cellcolor[HTML]{F8A102}0.104463 & \cellcolor[HTML]{F8A102}0.121002 & \cellcolor[HTML]{F8A102}0.110942 & \cellcolor[HTML]{F8A102}0.119599 & \cellcolor[HTML]{F8A102}0.118336 & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.051735}} & \cellcolor[HTML]{F8A102}0.052813 & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.055188}} & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.056389}} & \cellcolor[HTML]{F8A102}0.0574883 & \cellcolor[HTML]{F8A102}0.055042 \\
20/10 & \cellcolor[HTML]{F8A102}0.1052 & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.1015}} & \cellcolor[HTML]{F8A102}0.1198 & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.1096}} & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.1174}} & \cellcolor[HTML]{F8A102}0.1166 & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.0492}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.0501}} & \cellcolor[HTML]{F8A102}0.0585 & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.053}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.0536}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.052}}} \\
24/12 & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.053143}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.05356}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.056986}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.05747}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.05558}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.0553047}}} & \cellcolor[HTML]{F8A102}0.051926 & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.045995}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{009901} {\ul\textbf{0.04868}}} & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.0557455}} & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.05698}} & \cellcolor[HTML]{F8A102}{\color[HTML]{34FF34} \textbf{0.05456}} \\
30/15 & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.05617}} & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.0754578}} & {\color[HTML]{32CB00} \textbf{0.0775947}} & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.0575459}} & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.0565421}} & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.056635}} & \cellcolor[HTML]{F8A102}0.0523614 & \cellcolor[HTML]{F8A102}0.0519862 & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.054158}} & \cellcolor[HTML]{F8A102}0.0563985 & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.0555998}} & \cellcolor[HTML]{F8A102}{\color[HTML]{32CB00} \textbf{0.0543501}} \\ \midrule
\multicolumn{13}{c}{Australia} \\ \midrule
2/1 & 0.471554 & 0.529417 & 0.52834 & 0.563622 & 0.628921 & 0.55071 & \cellcolor[HTML]{F8A102}0.00498465 & 0.0047458 & 0.0247455 & 0.0072776 & 0.035936 & {\color[HTML]{34FF34} \textbf{0.0039155}} \\
4/2 & \cellcolor[HTML]{F8A102}0.26822 & 0.29862 & \cellcolor[HTML]{F8A102}0.25881 & 0.276464 & \cellcolor[HTML]{F8A102}0.33741 & \cellcolor[HTML]{F8A102}0.268965 & 0.004543 & 0.004578 & 0.023895 & 0.005417 & 0.032511 & {\color[HTML]{009901} {\ul\textbf{0.0037674}}} \\
6/3 & \cellcolor[HTML]{F8A102}0.090985 & \cellcolor[HTML]{F8A102}0.0947676 & \cellcolor[HTML]{F8A102}0.09448 & \cellcolor[HTML]{F8A102}0.08977 & \cellcolor[HTML]{F8A102}0.10083 & \cellcolor[HTML]{F8A102}0.091376 & {\color[HTML]{32CB00} \textbf{0.0040955}} & 0.004847 & 0.02199 & {\color[HTML]{34FF34} \textbf{0.0046074}} & 0.023489 & 0.0064495 \\
8/4 & 0.0870445 & 0.097201 & 0.057284 & 0.086078 & 0.06245 & 0.04463 & 0.004265 & {\color[HTML]{34FF34} \textbf{0.004191}} & 0.026156 & 0.0067075 & 0.02533 & 0.00757145 \\
10/5 & 0.0315 & 0.039 & {\color[HTML]{34FF34} \textbf{0.0509}} & 0.0305 & 0.0601 & 0.031 & 0.0045 & 0.0048 & 0.0199 & 0.0058 & 0.0288 & 0.0089 \\
12/6 & 0.027348 & {\color[HTML]{34FF34} \textbf{0.026942}} & 0.057592 & 0.026687 & 0.055426 & {\color[HTML]{009901} {\ul\textbf{0.0270495}}} & 0.004341 & 0.0049061 & 0.020529 & 0.007813 & 0.020667 & 0.0054794 \\
14/7(119\textendash0) & 0.02725 & 0.033687 & {\color[HTML]{32CB00} \textbf{0.046242}} & {\color[HTML]{34FF34} \textbf{0.0252737}} & {\color[HTML]{34FF34} \textbf{0.04651}} & {\color[HTML]{32CB00} \textbf{0.027222}} & {\color[HTML]{009901} {\ul\textbf{0.0017226}}} & {\color[HTML]{009901} {\ul\textbf{0.00354029}}} & {\color[HTML]{34FF34} \textbf{0.0119085}} & {\color[HTML]{009901} {\ul\textbf{0.00157378}}} & 0.0121774 & 0.00765843 \\
16/8 & 0.028261 & 0.037993 & 0.0536856 & 0.0278993 & 0.05263 & {\color[HTML]{34FF34} \textbf{0.0282745}} & 0.0050338 & {\color[HTML]{32CB00} \textbf{0.0037223}} & 0.018136 & 0.0047548 & 0.014537 & 0.01061 \\
20/10 & {\color[HTML]{34FF34} \textbf{0.0257}} & 0.0378 & 0.0616 & 0.0254 & 0.0526 & 0.0373 & 0.00544 & 0.012 & 0.023 & 0.0162 & {\color[HTML]{34FF34} \textbf{0.0119}} & 0.0137 \\
24/12 & {\color[HTML]{32CB00} \textbf{0.0219624}} & {\color[HTML]{32CB00} \textbf{0.0214394}} & {\color[HTML]{009901} {\ul\textbf{0.043667}}} & {\color[HTML]{32CB00} \textbf{0.0216832}} & {\color[HTML]{32CB00} \textbf{0.0372685}} & 0.0345906 & 0.0058473 & 0.0063788 & {\color[HTML]{32CB00} \textbf{0.0059937}} & 0.0125858 & {\color[HTML]{009901} {\ul\textbf{0.002742}}} & 0.0051643 \\
30/15 & {\color[HTML]{009901} {\ul\textbf{0.014614}}} & {\color[HTML]{009901} {\ul\textbf{0.0183819}}} & 0.0786527 & {\color[HTML]{009901} {\ul\textbf{0.014254}}} & {\color[HTML]{009901} {\ul\textbf{0.031029}}} & 0.0293416 & {\color[HTML]{34FF34} \textbf{0.00412276}} & 0.00444694 & {\color[HTML]{009901} {\ul\textbf{0.00448627}}} & {\color[HTML]{32CB00} \textbf{0.00397289}} & {\color[HTML]{32CB00} \textbf{0.0054747}} & {\color[HTML]{32CB00} \textbf{0.00387337}} \\ \bottomrule
\end{tabular}%
}
\end{table}
\end{landscape}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=1\textwidth]{../../paper/tex/GeophysJInt/figures/WinStpVsCPD.pdf}
  \caption[Sliding window and step sizes vs $\mathcal{CPD}$]{Plot of the
    equal-weight $\mathcal{CPD}$ scores collected in Table~\ref{tab-pk0vs1bs}.
    Note that here the step size is always half of the sliding window size and
    the reference path is the FHM derived.}\label{fig-WinStpVsCPD}
\end{figure}

\begin{figure}[tbp]
  \captionsetup[subfigure]{aboveskip=2pt,belowskip=-3pt}
  \centering
  \begin{subfigure}{.99\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101_120_0m.pdf}
    \caption{minimum 0.00268588 (5(1)),
    maximum 0.0892467 (16(3)), mean 0.03674, median 0.03177075}\label{fig-na-difm}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.99\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501_120_0m.pdf}
    \caption{minimum 0.0232517 (19(1)), maximum 0.51876 (8(3)),
    mean 0.11364, median 0.076556}\label{fig-in-difm}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.99\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801_120_0m.pdf}
    \caption{minimum 0.00122074 (17(5)), maximum
    0.367952 (26(3)), mean 0.077, median 0.03893}\label{fig-au-difm}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[$\mathcal{CPD}$ of each plate's paleomagnetic APWPs vs
    its MHM predicted APWP]{As Fig.~\ref{fig-dif}, here the reference path is
    predicted from MHM\@. See the numbers of the picked paleopoles for methods
    in Fig.~\ref{fig-dif}.}\label{fig-difm}
\end{figure}

\begin{table}[!ht]
  \centering
  \caption{Average number of paleopoles per window/mean pole for only Pk 0 (AMP
    with no weighting) and Pk 1 (APP with no weighting) from bin/step 2/1 Myr to
    30/15 Myr.}\label{tab:dwsAN}
  \resizebox{.8\textwidth}{!}{%
    \begin{tabular}{@{}cccccccccccc@{}}
      \toprule
      \multirow{2}{*}{Pk} & \multicolumn{11}{c}{Average Number of Paleopoles per \textbf{Window}/Mean pole} \\ \cmidrule(l){2-12} 
       & \textbf{2}/1 & \textbf{4}/2 & \textbf{6}/3 & \textbf{8}/4 & \textbf{10}/5 & \textbf{12}/6 & \textbf{14}/7 & \textbf{16}/8 & \textbf{20}/10 & \textbf{24}/12 & \textbf{30}/15 \\ \midrule
       \multicolumn{12}{c}{N America} \\ \midrule
       \textit{0} & 2.7 & 4.5 & 6.3 & 8.1 & 10 & 12.1 & 13.9 & 14.9 & 18.1 & 22.4 & 27.2 \\
       \textit{1} & 14 & 15.9 & 18 & 19.8 & 21 & 23.5 & 25 & 27.1 & 30.3 & 33.9 & 38.9 \\
       \multicolumn{12}{c}{India} \\ \midrule
       \textit{0} & 2.9 & 3.6 & 4.5 & 5 & 6.4 & 7.7 & 8.2 & 9 & 11.1 & 13.5 & 16.3 \\
       \textit{1} & 6.3 & 7.5 & 8.4 & 9.7 & 10.2 & 11.9 & 12.1 & 14.5 & 14.8 & 16.9 & 19.6 \\
       \multicolumn{12}{c}{Australia} \\ \midrule
       \textit{0} & 2.4 & 3.2 & 4.3 & 5.7 & 6.2 & 7.8 & 9.8 & 11.3 & 12 & 16.3 & 18.3 \\
       \textit{1} & 15.8 & 17.1 & 19 & 19.5 & 19.6 & 22.6 & 24.1 & 24.6 & 25.4 & 30.7 & 32.4 \\ \bottomrule
    \end{tabular}%
    }
\end{table}

\paragraph{The results}
are summarised in Table~\ref{tab-2010vs105F}, Table~\ref{tab-pk0vs1bs} and
Fig.~\ref{fig-WinStpVsCPD}. Based on these results, the effect of doubling the
size of the time window and step is negligible. $\mathcal{CPD}$ scores for APWPs
generated with no filtering (Pk 0,1) over a wider range of time windows and
steps, from 2/1 to 30/15 Myr are shown in Fig.~\ref{fig-WinStpVsCPD} and
Table~\ref{tab-pk0vs1bs}. Scores for APP-derived paths (Pk 1) remain stable over
the whole tested range. AMP-derived paths (Pk 0) have higher scores for windows
narrower than 10 Myr, but are much more stable and converging to APP-derived
paths for windows wider than 10\textendash12 Myr, whilst remaining generally
higher than the equivalent APP scores. The convergence is correlated with the
decrease of differences of average number of paleopoles per window/mean pole
between Pk 0 and Pk 1 when window/step size increases (Table~\ref{tab:dwsAN}).


\subsection{Moving versus Fixed Hotspot Reference}

Fig.~\ref{fig-difm} (10/5 Myr window/step) and Fig.~\ref{fig-dif2010m} (20/10
Myr) show the $\mathcal{CPD}$ scores for the APWPs generated with all 28 picking
and 6 weighting methods, compared to the MHM reference paths (stars and solid
line in Figs.~\ref{fig-mhsPred}-\ref{fig-mhsPred801}).

\begin{figure}[tbp]
  \centering
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_10_5_5_1m.pdf}
    \caption{N America: min 0.002686 (5(1)), FQ C-A}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_10_5_16_3m.pdf}
    \caption{N America: max 0.08925 (16(3)), FQ B-A}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_10_5_19_1m.pdf}
    \caption{India: minimum 0.0233 (19(1)), FQ C-A}
  \end{subfigure}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_10_5_8_3m.pdf}
    \caption{India: maximum 0.5188 (8(3)), FQ C-A}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_10_5_17_5m.pdf}
    \caption{Australia: min 0.00122 (17(5)), FQ B-A}
  \end{subfigure}
  \begin{subfigure}{.42\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_10_5_26_3m.pdf}
    \caption{Australia: max 0.368 (26(3)), FQ C-A}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[Best and worst $\mathcal{CPD}$s (10/5 Myr window/step; MHM)]{Path
    comparisons with best and worst $\mathcal{CPD}$ values shown in
    Fig.~\ref{fig-difm}. The parenthetical remarks are Pk No (Wt
    No).}\label{fig-difbwm}
\end{figure}

The overall patterns and trends described for the FHM reference paths in
Sections~\ref{sec:base}-\ref{sec:winsiz} remain largely unchanged, and the mean,
median and range of $\mathcal{CPD}$ scores are very similar (Fig.~\ref{fig-dif}
versus Fig.~\ref{fig-difm}, and Fig.~\ref{fig-dif2010} versus
Fig.~\ref{fig-dif2010m}). From a direct comparison of scores for the 10/5 Myr
window/step (Fig.~\ref{fig-dmf}), largely positive MHM-FHM differences for North
America (Fig.~\ref{fig-mf101}) indicate most methods generate slightly better
fits to the fixed hotspot reference; largely negative MHM-FHM differences for
India (Fig.~\ref{fig-mf501}) and Australia (Fig.~\ref{fig-mf801}) indicate most
methods generate slightly better fits to the moving hotspot reference. For North
America, large changes seem favored by Wt 3, and small changes seem favored by
Wt 5 or Pk 15 (Fig.~\ref{fig-dmf}). Pk 3 brings minor changes to both North
America and India. Pk 22 and 26 show large changes and Pk 17, 19, 21 and 25 show
minor changes for both India and Australia. However, the absolute differences in
$\mathcal{CPD}$ scores for equivalent methods are all lower than 0.066, and
actually most are less than 0.01.

\begin{figure}[!ht]
  \captionsetup[subfigure]{aboveskip=2pt,belowskip=-3pt}
  \centering
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/101_20_10_120_0m.pdf}
    \caption{(continued on next page) minimum 0.00565784 (5(1)), maximum 0.104618 (14(2)),
      mean 0.043777, median 0.02679955}\label{fig-na-dif2010m}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/501_20_10_120_0m.pdf}
    \caption{minimum 0.0177 (6(3)), maximum 0.15937 (16(1)), mean 0.0745,
      median 0.061346}\label{fig-in-dif2010m}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/801_20_10_120_0m.pdf}
    \caption{minimum 0.00282 (23(4)), maximum 0.31998 (22(3)), mean
      0.062766, median 0.02803}\label{fig-au-dif2010m}
  \end{subfigure}
  \caption[$\mathcal{CPD}$ of each plate's paleomagnetic APWPs vs its MHM
    predicted APWP (20/10 Myr window/step)]{(continued from previous page) As
    Fig.~\ref{fig-dif2010}, here the reference path is predicted from MHM\@. See
    the numbers of picked paleopoles in Fig.~\ref{fig-dif}.}\label{fig-dif2010m}
\end{figure}

\begin{figure}
  \vspace{-1.5cm}
  \centering
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_20_10_5_1m.pdf}
    \caption{N America: min 0.00566 (5(1)), FQ B-A}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_101comb_20_10_14_2m.pdf}
    \caption{N America: max 0.1046 (14(2)), FQ B-A}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_20_10_6_3m.pdf}
    \caption{India: minimum 0.0177 (6(3)), FQ C-A}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_501comb_20_10_16_1m.pdf}
    \caption{India: maximum 0.1594 (16(1)), FQ C-A}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_20_10_23_4m.pdf}
    \caption{Australia: min 0.0028 (23(4)), FQ C-A}
  \end{subfigure}
  \begin{subfigure}{.43\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/ay18_801comb_20_10_22_3m.pdf}
    \caption{Australia: max 0.32 (22(3)), FQ C-A}
  \end{subfigure}
  \caption[Best and worst $\mathcal{CPD}$s (20/10 Myr window/step; MHM)]{Path
    comparisons with best and worst $\mathcal{CPD}$ values shown in
    Fig.~\ref{fig-dif2010m}. The parenthetical remarks are Pk No (Wt
    No).}\label{fig-dif2010bwm}
\end{figure}

APP-derived $\mathcal{CPD}$ values still outperform AMP-derived ones. For 10/5
Myr bin/step, the overall degree of difference is slightly reduced for North
America, India and Australia, with percentage of APP-derived $\mathcal{CPD}$
more than 3 times the equivalent AMP-derived $\mathcal{CPD}$ reducing to about
19.05\%, 38.1\% and 48.81\% respectively (from about 40.5\%, 39.3\% and 50\% of
FHM's). For 20/10 Myr bin/step, the percentage is reducing to about 3.6\% and
46.43\% for India and Australia (from about 3.8\% and 71.4\% of FHM's); whilst
for North America this percentage increases from 28.6\% (FHM) to
${\sim}44.05$\%.

\begin{figure}[tbp]
  \captionsetup[subfigure]{aboveskip=2pt,belowskip=-3pt}
  \centering
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/mf101.pdf}
    \caption{Fig.~\ref{fig-na-difm}-Fig.~\ref{fig-na-dif}; Percentage for
      positive values: ${\sim}66.67$\%}\label{fig-mf101}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/mf501.pdf}
    \caption{Fig.~\ref{fig-in-difm}-Fig.~\ref{fig-in-dif}; Percentage for
      positive values: ${\sim}34.52$\%}\label{fig-mf501}
  \end{subfigure}
  \vspace{.1em}
  \begin{subfigure}{1\textwidth}
    \includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/mf801.pdf}
    \caption{Fig.~\ref{fig-au-difm}-Fig.~\ref{fig-au-dif}; Percentage for
      positive values: ${\sim}19.62$\%}\label{fig-mf801}
  \end{subfigure}
\end{figure}
\begin{figure}[!ht]
  \ContinuedFloat\caption[Differences between results from FHM and MHM]{Differences between
    results from two different reference paths, FHM (Fig.~\ref{fig-dif}) and MHM
    (Fig.~\ref{fig-difm}) derived. The absolute difference values less than
    1.96-standard-deviation interval of the whole 168 values are labeled in
    green, more than 1.96-standard-deviation interval labeled in red. The
    strikethrough labels show positive differences.}\label{fig-dmf}
\end{figure}

This indicates that for comparing with paleomagnetic APWPs choosing fixed- or
moving-hotspot model to generate a reference path does not make much difference.
Therefore, selecting from absolute reference frames, the fixed- or
moving-hotspot model, is not a priority for calculating a reference path.

\begin{landscape}
%\setlength\tabcolsep{3pt}
\begin{table}[tbp]
\centering
\caption{Performance statistics of all the picking and weighting methods for 10/5 and 20/10 Myr window/step path comparisons.}\label{tab-bw}
\resizebox{.92\width}{!}{%
\setlength{\tabcolsep}{2pt}
\begin{tabular}{@{}clllllcccccccc@{}}
\toprule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Hotspot\\ model\\ for ref path\end{tabular}} & \multicolumn{1}{c}{\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}$\mathcal{CPD}$\\ grid\end{tabular}}} & \multicolumn{2}{c}{\multirow{2}{*}{Best}} & \multicolumn{2}{c}{\multirow{2}{*}{Worst}} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\% of\\ APP better\\ than AMP\end{tabular}} & \multicolumn{6}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Count occurrences of each\\ Wt being best for 28 Pks\end{tabular}}} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Pk14,15 (New\\ Studies) better than\\ Pk16,17(Old)\end{tabular}} \\
 & \multicolumn{1}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} &  & \multicolumn{6}{c}{} &  \\ \cmidrule(lr){3-6} \cmidrule(lr){8-13}
 & \multicolumn{1}{c}{} & \multicolumn{1}{c}{Pk} & \multicolumn{1}{c}{Wt} & \multicolumn{1}{c}{Pk} & \multicolumn{1}{c}{Wt} &  & 0 & 1 & 2 & 3 & 4 & 5 &  \\ \midrule
\multirow{12}{*}{FHM} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-na-dif}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}1, 5, 7, 11, 13\\ 15, \textbf{19}, \textbf{21}, 25\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 1,\\ 3, 5\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2, 5, 7, 14, \textbf{16},\\ 17, 18, 22, 26\end{tabular}} & \multirow{24}{*}{0\textendash5} & \multirow{2}{*}{97.6} & \multirow{2}{*}{\textbf{9}} & \multirow{2}{*}{4} & \multirow{2}{*}{4} & \multirow{2}{*}{7} & \multirow{2}{*}{0} & \multirow{2}{*}{5} & \multirow{4}{*}{Y} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-5} \cmidrule(lr){7-13}
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-in-dif}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}4\textendash7, 9, \textbf{19}\\ \textbf{21}, 22, 26\end{tabular}} & \multirow{2}{*}{0\textendash5} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 2, 8, 10, 12,\\ \textbf{16}, 18, 20, 24\end{tabular}} &  & \multirow{2}{*}{85.7} & \multirow{2}{*}{\textbf{15}} & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{6} & \multirow{4}{*}{2} & \multirow{2}{*}{1} &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-5} \cmidrule(lr){7-11} \cmidrule(l){13-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-au-dif}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}1, 11, 13, 17,\\ \textbf{19}, \textbf{21}, 25\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}0,\\ 1,\\ 3,\\ 5\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2, 4, 14,\\ \textbf{16}, 22, 26\end{tabular}} &  & \multirow{2}{*}{100} & \multirow{2}{*}{\textbf{10}} & \multirow{2}{*}{5} & \multirow{2}{*}{1} & \multirow{2}{*}{7} &  & \multirow{2}{*}{3} & \multirow{2}{*}{N} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-3} \cmidrule(lr){5-5} \cmidrule(l){7-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-na-dif2010}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}1, 4\textendash7, 9, 11, 13\\ 15, \textbf{19}, \textbf{21}, 25\end{tabular}} &  & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 8, 10, 12, 14,\\ \textbf{16}, 18, 20, 24\end{tabular}} &  & \multirow{2}{*}{72.6} & \multirow{2}{*}{\textbf{14}} & \multirow{2}{*}{2} & \multirow{6}{*}{0} & \multirow{2}{*}{3} & \multirow{2}{*}{3} & \multirow{2}{*}{6} & \multirow{2}{*}{N, Y} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-5} \cmidrule(lr){7-9} \cmidrule(l){11-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-in-dif2010}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}4\textendash7,\\ \textbf{19}\end{tabular}} & \multirow{2}{*}{0\textendash5} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 2, 10, 12, \textbf{16},\\ 18, 20, 23, 24, 27\end{tabular}} &  & \multirow{2}{*}{69} & \multirow{2}{*}{\textbf{11}} & \multirow{4}{*}{6} &  & \multirow{2}{*}{9} & \multirow{4}{*}{1} & \multirow{2}{*}{1} & \multirow{2}{*}{Y, 4y2n} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-5} \cmidrule(lr){7-8} \cmidrule(lr){11-11} \cmidrule(l){13-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-au-dif2010}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}1, 11, 13, 17, \textbf{19},\\ \textbf{21}, 23, 25, 27\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 1,\\ 3\textendash5\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2, 14, \textbf{16},\\ 22, 26\end{tabular}} &  & \multirow{2}{*}{100} & \multirow{2}{*}{\textit{9}} &  &  & \multirow{2}{*}{\textbf{10}} &  & \multirow{4}{*}{2} & \multirow{2}{*}{N} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(r){1-5} \cmidrule(lr){7-12} \cmidrule(l){14-14} 
\multirow{12}{*}{MHM} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-na-difm}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}1, 5, 7, 9, 11, 13,\\ 15, \textbf{19}, \textbf{21}, 25\end{tabular}} & \multirow{4}{*}{0\textendash5} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 10, 12, 14,\\ \textbf{16}\textendash18, 20, 24\end{tabular}} &  & \multirow{2}{*}{97.6} & \multirow{2}{*}{\textbf{10}} & \multirow{2}{*}{9} & \multirow{2}{*}{3} & \multirow{2}{*}{4} & \multirow{2}{*}{0} &  & \multirow{4}{*}{Y} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-3} \cmidrule(lr){5-5} \cmidrule(lr){7-13}
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-in-difm}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}5, 7, \textbf{19},\\ \textbf{21}, 22, 26\end{tabular}} &  & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 2, 8, 10, 12,\\ 14, \textbf{16}, 18, 20, 24\end{tabular}} &  & \multirow{2}{*}{85.7} & \multirow{2}{*}{\textbf{12}} & \multirow{2}{*}{2} & \multirow{2}{*}{0} & \multirow{2}{*}{7} & \multirow{4}{*}{4} & \multirow{2}{*}{3} &  \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-5} \cmidrule(lr){7-11} \cmidrule(l){13-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-au-difm}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}1, 11, 13, 17,\\ \textbf{19}, \textbf{21}, 25\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0\textendash3,\\ 5\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2, 14, \textbf{16},\\ 22, 26\end{tabular}} &  & \multirow{2}{*}{98.8} & \multirow{2}{*}{\textit{6}} & \multirow{2}{*}{4} & \multirow{2}{*}{2} & \multirow{2}{*}{\textbf{10}} &  & \multirow{2}{*}{2} & \multirow{2}{*}{N} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-5} \cmidrule(l){7-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-na-dif2010m}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}5, 7, 15,\\ 22, 26\end{tabular}} & \multirow{6}{*}{0\textendash5} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 8, 10, 12,\\ 14, 18, 20, 24\end{tabular}} &  & \multirow{2}{*}{76.2} & \multirow{2}{*}{\textbf{10}} & \multirow{2}{*}{8} & \multirow{2}{*}{1} & \multirow{2}{*}{4} & \multirow{2}{*}{1} & \multirow{2}{*}{4} & \multirow{2}{*}{1y5n, Y} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-3} \cmidrule(lr){5-5} \cmidrule(l){7-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-in-dif2010m}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}4\textendash7,\\ 22, 26\end{tabular}} &  & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}0, 2, 10, 12,\\ \textbf{16}, 18, 24\end{tabular}} &  & \multirow{2}{*}{70.2} & \multirow{2}{*}{\textit{6}} & \multirow{2}{*}{\textbf{7}} & \multirow{4}{*}{0} & \multirow{2}{*}{6} & \multirow{2}{*}{3} & \multirow{2}{*}{6} & \multirow{2}{*}{Y, 3y3n} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \cmidrule(lr){2-3} \cmidrule(lr){5-5} \cmidrule(lr){7-9} \cmidrule(l){11-14} 
 & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Fig.\\\ref{fig-au-dif2010m}\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}1, 11, 13, 15, 17,\\ \textbf{19}, \textbf{21}, 23, 25, 27\end{tabular}} &  & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2, 14, \textbf{16},\\ 22, 26\end{tabular}} &  & \multirow{2}{*}{100} & \multirow{2}{*}{\textbf{12}} & \multirow{2}{*}{1} &  & \multirow{2}{*}{10} & \multirow{2}{*}{4} & \multirow{2}{*}{1} & \multirow{2}{*}{N} \\
 &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \bottomrule
\end{tabular}%
}
\end{table}
\end{landscape}

\section{Discussion}

\subsection{Question: Why Do APP Methods Generally Produce Better Similarities
than AMP Methods?}

\subsubsection{Perspective of Conceptual Difference between AMP and APP}

Paleomagnetic (mean) A95 represents precision (how well constrained calculated
poles are), and (mean) coeval poles' GCD represents accuracy (how close
calculated poles are to the reference path). Compared with AMP
(Fig.~\ref{fig-A95mG105F},~\subref{fig-A95GCD105F}
and~\subref{fig-A95SGCD105F}), APP usually improves both and generates paths
with higher accuracy and also higher precision.

\begin{figure}[!ht]
  \captionsetup[subfigure]{singlelinecheck=off,justification=raggedright,aboveskip=-6pt,belowskip=-6pt}
  \centering
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95gcd_10_5F.pdf}\label{fig-A95GCD105F}
  \end{subfigure}
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95Sgcd_10_5F.pdf}\label{fig-A95SGCD105F}
  \end{subfigure}
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95mSld10_5F.pdf}\label{fig-A95mSld105F}
  \end{subfigure}
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95mSad10_5F.pdf}\label{fig-A95mSad105F}
  \end{subfigure}
  \caption[APP spatially better than AMP (arrow)]{Paleomagnetic APWP's mean A95
    versus (a) mean GCD, (b) significant spatial difference $d_s$, (c)
    significant length difference $d_l$, and (d) significant angular difference
    $d_a$ between paleomagnetic APWP and its corresponding FHM-and-plate-circuit
    predicted APWP\@. Arrowtails are the results from AMP, while arrowheads are
    from APP\@. Black color filled arrowheads are the small number of special
    cases of AMP derived equal-weight $\mathcal{CPD}$s better than APP (see
    details in Fig.~\ref{fig-dif} and Table~\ref{tab-bw}).}\label{fig-A95mG105F}
\end{figure}

\begin{figure}[!ht]
  \captionsetup[subfigure]{singlelinecheck=off,justification=raggedright,aboveskip=-6pt,belowskip=-6pt}
  \centering
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95gcdd.pdf}\label{fig-A95GCD105Fd}
  \end{subfigure}
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95Sgcd_10_5Fd.pdf}\label{fig-A95SGCD105Fd}
  \end{subfigure}
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95mSld10_5Fd.pdf}\label{fig-A95mSld105Fd}
  \end{subfigure}
  \begin{subfigure}[htbp]{.49\textwidth}
    \caption{}\includegraphics[width=\textwidth]{../../paper/tex/GeophysJInt/figures/a95mSad10_5Fd.pdf}\label{fig-A95mSad105Fd}
  \end{subfigure}
  \caption[APP spatially better than AMP (dot)]{Differences of APP and AMP
    coordinates shown in Fig.~\ref{fig-A95mG105F}. Crosses locates the small
    minority cases of AMP derived equal-weight $\mathcal{CPD}$s better than APP
    (see details in Fig.~\ref{fig-dif} and
    Table~\ref{tab-bw}).}\label{fig-A95mG105Fd}
\end{figure}

The fact that APP increases the number of contributing paleopoles (N) in each
sliding window (Fig.~\ref{fig-nac-maplat}) could potentially average out some
``bad'' (i.e.\ inaccurate) poles and improve the fit between the paleomagnetic
APWPs and the model-predicted APWPs. The general effects that APP brings include
the decreases in paleomagnetic A95s, or/and distances between compared coeval
poles of paleomagnetic APWP and reference APWP (Fig.~\ref{fig-A95mG105F} and
Fig.~\ref{fig-A95mG105Fd}). However, if the added paleopoles were all or mostly
``bad'', the improvement of fit would not occur. So the improvement of fit is
not only because of the increase in N, but also because the majority of the
additional poles are ``good''. AMP only regards the time uncertainty of each
pole as one midpoint. Then this midpoint is treated as the most likely age of
that mean pole. This is usually incorrect. The age uncertainty of paleopole is
not obtained from a probability density function derived from an observed
frequency distribution. As defined, the time uncertainty's lower (older) limit
is a stratigraphic age, and its upper (younger) limit could be also a
stratigraphic age or be constrained by a tectonic event using the field tests
(e.g.\ fold/tilt test and conglomerate test). So the true age of the paleopole
could be any one that is not older than the lower limit and also not younger
than the upper limit. In other words, the midpoint could be the true age of the
pole, but it is not the most likely age of that pole. If the midpoint is the
most likely age of a pole, AMP should generate a path that is closer to the
reference. However, in most cases APP generates better similarities (See the
high proportions of APP better than AMP in Table~\ref{tab-bw}). Most reasonably,
the midpoint should be regarded as one possibility of all uniformly (not
necessarily normally bell shaped, or U shaped, or left or right skewed)
distributed ages between the two time limits.

So APP remains the effect of a paleopole borne on the mean poles during all the
period of its age uncertainty, and use the increased number of paleopoles (N)
to average out the negative effect of those ``bad'' poles, including the
paleopoles that should not be included at that age for mean pole.

\subsubsection{Perspective of Stability Comparison between AMP and APP}

Fitting curves by moving averaging change with different time window lengths and
time increment lengths (i.e.\ steps) (e.g., the similarity of the pair in
Fig.~\ref{fig-au-2010110} is improved a bit compared to
Fig.~\ref{fig-au-105110}). A balance needs to be made between having windows
that are too wide and steps that are too long, which will smooth the data so
much we miss actual details in the APWP (e.g.\ those 20/10 Myr window/step
paleomagnetic paths in Fig.~\ref{fig-dif2010bw} and even 30/15 Myr window and
step; Table~\ref{tab-pk0vs1bs} and Fig.~\ref{fig-WinStpVsCPD}); and windows that
are too narrow and steps that are too short, which introduces noise by having
too few paleopoles in each window (e.g.\ 2 Myr window 1 Myr step;
Table~\ref{tab-pk0vs1bs} and Fig.~\ref{fig-WinStpVsCPD}). There is a dependence
here on data density: higher density allows smaller windows/steps. A variety of
ways of binning the data (here 30\textendash2 Myr window size and half of the
size as step) are being tested to see which one produces the better and more
appropriately smoothed fit.

\begin{figure}
  \centering
  \includegraphics[width=.95\textwidth]{../../paper/tex/GeophysJInt/figures/draper.pdf}
  \caption[Regression of $\mathcal{CPD}$ on average number of paleopoles per
    window for Wt 0]{Regression of equal-weight $\mathcal{CPD}$ on average
    number of paleopoles per window for Wt 0. Plotted data include all the
    bin/step 10/5 and 20/10 Myr calculations on the 28 picking methods for North
    America, India and Australia. See plots for Wt 1\textendash5 in
    Figs.~\ref{fig-regr1},~\ref{fig-regr2},~\ref{fig-regr3},~\ref{fig-regr4}
    and~\ref{fig-regr5} of Appendix~\ref{appen4chp3}.}\label{fig-regr}
\end{figure}

Note that there are 135, 75 and 99 paleopoles contributing to 120\textendash0 Ma
APWPs for North America, India and Australia respectively. Only for North
America, does the reason of 10/5 Myr unexpectedly generally better than 20/10
could be its relatively larger average number of paleopoles per window, compared
with India and Australia? Since theoretically for each sliding window, the more
``bad'' paleopoles it contains, the worse similarity we should obtain. In
contrast, the less paleopoles the window contains, the weaker the effect of
averaging out ``bad'' poles' influence would be. So is there a threshold average
number of paleopoles per window for making a reliable paleomagnetic APWP\@? For
example, for making a 120\textendash0 Ma APWP, do the regression results
(Fig.~\ref{fig-regr}) indicate the best average number of paleopoles per window
we need should be some value between 19 and 28? APP definitely improves both
average number of paleopoles per window and $\mathcal{CPD}$
(Fig.~\ref{fig-regr}). Here another test is implemented as follows. With the
results from the 10/5 and 20/10 bin/step together, 2/1, 4/2, 6/3, 8/4, 12/6,
14/7, 16/8, 20/10, 24/12 and 30/15 Myr bin/step are also used to generate
paleomagnetic APWPs for North America, India and Australia to see which bin/step
size would make a paleomagnetic APWP closest to the reference path. Will the
similarities they generate be generally worse than those the 10/5 Myr bin/step
generates? Or will they be better first and then worse than those the 10/5 Myr
bin/step generates when the bin/step sizes increase up to 20/10 Myr? For the
best results (Table~\ref{tab-pk0vs1bs}), as expected, AMP needs wider sliding
window and step to get closer to the reference path while APP does not
(Fig.~\ref{fig-WinStpVsCPD}). Even when the best performing sizes of sliding
window and step are assigned for AMP, the results from APP are still much better
than those from AMP\@. Picking methods (directly related to N) are still the key
influence factor of choosing a better sliding window size and step size of
moving averaging, although weighting methods are also important.

\subsubsection{Summary}

\paragraph{If AMP has to be used,} better results can be obtained by using
larger sizes of sliding window and step, commonly more than 24/12 Myr. In
addition, we should be cautious when Wt 3 is used with AMP\@.

\paragraph{APP is still recommended,} not only because the temporal uncertainty
is incorporated into the algorithm but also the results from APP are not as that
sensitive as AMP to the changes of sliding window and step sizes. In fact, for
APP the results from different window and step sizes are much more stable than
those from AMP (Fig.~\ref{fig-WinStpVsCPD}). This means we actually do not need
to worry about what sizes should be chosen for the sliding window and step when
we use APP method.

\subsection{Question: Why Do AMP Methods Sometimes Unexceptionally Produce
Better Similarities than APP Methods?}

Because of smaller number of paleopoles (not necessarily ``bad'') involved in
each sliding window, the mean poles produced by AMP should be relatively far
from its contemporary model-predicted pole. In other words, AMP intends to give
fairly small change in accuracy. This also could potentially bring more
distinguishable $d_s$ for AMP\@ if the corresponding A95 is not large enough.
For example, in Fig.~\ref{fig-na-dif}, there are only two special (of 84 APP
versus AMP comparisons) cases Pk (Wt) 4(3), 6(3) better than 5(3), 7(3)
respectively. Compared with the Pk (Wt) 4(3) APWP, although most of the mean
paleopoles are closer to the FHM predicted APWP and also the number of the
significant pole pairs is two less for the APP derived path (i.e. 5(3)), the
A95s are smaller and most importantly there are one more $d_a$ significant
orientation-change pair and one more $d_l$ significant segment pair
(Table~\ref{tab-w3p4vs5}). If we observe carefully, it is because of the much
smaller 15 Ma A95 for 5(3). A similar phenomenon occurs in the case of 6(3)
versus 7(3): a relatively much smaller paleomagnetic A95 causes more
distinguishable $d_a$ and $d_l$ for the APP results, and they offset the
improvement of spatial similarity $d_s$ APP brings.

For Pk (Wt) 2(0) versus 3(0) for 20/10 Myr window/step for North America, all
their $d_a$ and $d_l$ are indistinguishable. Compared with the results from AMP,
although the coeval pole GCDs are generally unchanged or decreased or even
increased (but not too much) for APP, this spatial improvement is not able to
offset the negative effects of also generally unchanged or decreased or even
increased (but not too much) paleomagnetic A95s, which potentially brings more
statistically distinguishable coeval poles (e.g.\ the 20 Ma and 110 Ma poles for
Pk 3 and Wt 0; Table~\ref{tab-w5p4vs5}). This further causes greater
distinguishable mean $d_s$ from the APP methods. The similar phenomenon occurs
for Pk 2 versus 3 with Wt 2, 3 and 5, Pk 4 versus 5 with Wt 1, 3 and 5, and Pk 6
versus 7 with Wt 1, 3 and 5, and so on (Fig.~\ref{fig-na-dif2010}).

In addition, compared with AMP, APP potentially could generate more mean poles,
because for some sliding window there is no paleopole involved at all for AMP\@
while there are paleopoles involved for APP\@. For APP, the mean poles at all
ages should be composed of more paleopoles than it is for AMP, which should
generally decrease both coeval pole distance and paleomagnetic A95. However,
sometimes a rare case (e.g.\ the 0 Ma comparison shown in
Table~\ref{tab-501w0p22vs23}) happens. It is sometimes the case that an
additional very ``bad'' paleopole gets included by APP and this increases both
coeval pole distance and paleomagnetic A95 even though N increases. Such cases
include Pk 22 versus 23 (actually exactly the same as Pk 26 versus
27) with all the six types of weightings (Fig.~\ref{fig-in-dif}).

So generally as we discussed in the last section APP decreases the distances
between paleomagnetic APWPs and the hotspot and ocean-floor spreading model
predicted APWP, and also the uncertainties of paleomagnetic APWPs. However, as
we described in this section, special cases, like decreased A95 potentially
allows coeval poles to be differentiated if the coeval poles' distance is not
decreased effectively or even increased, or very ``bad'' paleopoles got involved
in some sliding windows, occur. In summary, when the negative effect from these
types of rare cases is beyond the positive effect the generally improved mean
poles contribute, the composite difference score would increase. However, this
phenomenon seldom occurs (Table~\ref{tab-bw}).

\begin{table}[!ht]
\centering
\caption{One example of the Type 1 rare cases where AMP gives better similarity
result than APP does from North America (10/5 Myr window/step). Only
statistically significant values are listed here.}\label{tab-w3p4vs5}
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}llllllllll@{}}
\toprule
\multicolumn{2}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}FHM\\ predicted\end{tabular}}} & \multicolumn{4}{c}{Pk4(Wt3)} & \multicolumn{4}{c}{Pk5(Wt3)} \\ \cmidrule(l){3-10} 
\multicolumn{2}{c}{} & \multicolumn{2}{c}{ds} & \multicolumn{2}{c}{dl} & \multicolumn{2}{c}{ds} & \multicolumn{2}{c}{da} \\ \midrule
Age (Ma) & $\mathbf{dm}/\mathbf{dp}$ (\degree) & Pmag A95 (\degree) & Dist (\degree) & Age (Ma) & Diff (\degree) & Pmag A95 (\degree) & Dist (\degree) & Age (Ma) & Diff (\degree) \\ \midrule
10 & 1.44607/0.793714 & 14.876819 & \textbf{9.937} & 105\textendash110 & 5.91855 & \multicolumn{2}{l}{} & \textit{\textbf{10\textendash15\textendash20}} & \textbf{126.59} \\ \cmidrule(lr){5-6}
15 & 1.2875/0.816514 & \multicolumn{2}{l}{\multirow{2}{*}{}} & \multicolumn{2}{l}{\multirow{11}{*}{}} & 2.0857 & \textbf{11.805} & \multicolumn{2}{l}{} \\ \cmidrule(l){9-10} 
25 & 2.48031/1.10915 & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} & 6.3358 & \textbf{6.873} & \multicolumn{2}{c}{dl} \\ \cmidrule(l){9-10} 
55 & 3.58782/2.14032 & 4.6347 & \textbf{5.372} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} & Age (Ma) & Diff () \\ \cmidrule(l){9-10} 
60 & 4.85938/3.17602 & \multicolumn{2}{l}{\multirow{2}{*}{}} & \multicolumn{2}{l}{} & 6.5922 & 6.215 & \textit{\textbf{10\textendash15}} & \textbf{13.52} \\
65 & 3.68984/2.30014 & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} & 8.6632 & \textbf{7.6} & \textit{\textbf{15\textendash20}} & \textbf{14.68} \\ \cmidrule(l){9-10} 
75 & 2.6435/1.54052 & 9.0812 & \textbf{8.836} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{\multirow{4}{*}{}} & \multicolumn{2}{l}{\multirow{6}{*}{}} \\
100 & 2.8983/2.68346 & 8.892 & \textbf{8.455} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} \\
105 & 2.32328/1.74639 & 5.3 & \textbf{5.03} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} \\
110 & 4.13015/2.25964 & 3.8 & \textbf{9.8064} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} & \multicolumn{2}{l}{} \\
115 & 4.63512/2.58006 & 19.6676 & \textbf{9.3345} & \multicolumn{2}{l}{} & 8.5 & \textbf{11.704} & \multicolumn{2}{l}{} \\
120 & 7.34408/4.06043 & 3.515 & \textbf{17.35} & \multicolumn{2}{l}{} & 7.728 & \textbf{15.258} & \multicolumn{2}{l}{} \\ \cmidrule(r){1-4} \cmidrule(lr){7-8}
\end{tabular}%
}
\end{table}

\begin{table}[!ht]
\centering
\caption{One example of the Type 2 rare cases where AMP gives better similarity
result than APP does from North America (20/10 Myr window/step). Only
statistically significant values are listed here.}\label{tab-w5p4vs5}
\resizebox{.7\textwidth}{!}{%
\begin{tabular}{@{}llllll@{}}
\toprule
\multicolumn{1}{c}{\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Age\\ (Ma)\end{tabular}}} & \multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}FHM\\ predicted\end{tabular}}} & \multicolumn{4}{c}{ds} \\ \cmidrule(l){3-6} 
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Pk2(Wt0)} & \multicolumn{2}{c}{Pk3(Wt0)} \\ \cmidrule(l){2-6} 
\multicolumn{1}{c}{} & $\mathbf{dm}/\mathbf{dp}$ (\degree) & Pmag A95 (\degree) & Dist (\degree) & Pmag A95 (\degree) & Dist (\degree) \\ \midrule
0 & 0 & 3.97 & \textbf{5.714} & 3.97 & \textbf{5.714} \\
10 & 1.44607/0.793714 & 3.879 & \textbf{6.034} & 3.879 & \textbf{6.034} \\
20 & 1.58039/1.10047 & \multicolumn{2}{l}{} & 6.771 & \textbf{6.934} \\
50 & 3.57782/1.61328 & 3.8644 & \textbf{7.304} & 4.03 & \textbf{8.6} \\
60 & 4.85938/3.17602 & 5.716 & \textbf{8.457} & 5.55 & \textbf{7.367} \\
100 & 2.8983/2.68346 & 10.769 & \textbf{7.308} & 10.769 & \textbf{7.308} \\
110 & 4.13015/2.25964 & \multicolumn{2}{l}{} & 3.29 & \textbf{8.311} \\
120 & 7.34408/4.06043 & 3.38 & \textbf{16.41} & 3.083 & \textbf{16.728} \\ \bottomrule
\end{tabular}%
}
\end{table}

\begin{table}[!ht]
  \centering
  \caption{One example of the Type 3 rare cases where AMP gives better
  similarity result than APP does from India (10/5 Myr window/step). Only
  statistically significant values are listed here. Note that for the
  bold-number ages, there is no mean poles at all for the ``Pk 22 (Wt 2)''
  case.}\label{tab-501w0p22vs23}
  \resizebox{\textwidth}{!}{%
    \begin{tabular}{@{}llllllllll@{}}
      \toprule
      \multicolumn{2}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}FHM\\ predicted\end{tabular}}} & \multicolumn{3}{c}{Pk22(Wt 2)} & \multicolumn{5}{c}{Pk23(Wt2)} \\ \cmidrule(l){3-10} 
      \multicolumn{2}{c}{} & \multicolumn{3}{c}{ds} & \multicolumn{3}{c}{ds} & \multicolumn{2}{c}{dl} \\ \midrule
      \multicolumn{1}{c}{Age (Ma)} & $\mathbf{dm}/\mathbf{dp}$ (\degree) & Pmag $\mathbf{dm}/\mathbf{dp}$ (\degree) & Dist (\degree) & N & Pmag $\mathbf{dm}/\mathbf{dp}$ (\degree) & Dist (\degree) & N & Age (Ma) & Diff (\degree) \\ \midrule
      0 & 0 & \textbf{6.28} & \textbf{12.72} & \textit{\textbf{2}} & \textbf{23.54} & \textbf{18.14} & \textit{\textbf{3}} & \textbf{80\textendash85} & 6.286 \\
      10 & 1.12124/0.673225 & 5.4/3.1 & 29.9 & 1 & 5.4/3.1 & 29.9 & 1 & \textbf{110\textendash115} & 16.684 \\ \cmidrule(l){9-10} 
      \textbf{15} & 1.1347/0.8127 & \multicolumn{3}{l}{\multirow{5}{*}{}} & 5.4/3.1 & 28.28 & 1 & \multicolumn{2}{l}{\multirow{13}{*}{}} \\
      \textbf{60} & 4.79687/3.07133 & \multicolumn{3}{l}{} & 8.817 & 8.28 & 20 & \multicolumn{2}{l}{} \\
      \textbf{70} & 4.26508/2.48783 & \multicolumn{3}{l}{} & 3.26 & 4.464 & 20 & \multicolumn{2}{l}{} \\
      \textbf{75} & 2.6777/1.57975 & \multicolumn{3}{l}{} & 5 & 4.477 & 1 & \multicolumn{2}{l}{} \\
      \textbf{80} & 4.20828/2.50294 & \multicolumn{3}{l}{} & 5 & 3.358 & 1 & \multicolumn{2}{l}{} \\
      85 & 2.50744/1.24746 & 5 & 7.632 & 1 & 5 & 7.632 & 1 & \multicolumn{2}{l}{} \\
      \textbf{90} & 3.88998/1.43423 & \multicolumn{3}{l}{\multirow{5}{*}{}} & 5 & 10.884 & 1 & \multicolumn{2}{l}{} \\
      \textbf{95} & 2.23389/1.6247 & \multicolumn{3}{l}{} & 5 & 11.099 & 1 & \multicolumn{2}{l}{} \\
      \textbf{100} & 2.8062/2.59819 & \multicolumn{3}{l}{} & 5 & 11.4155 & 1 & \multicolumn{2}{l}{} \\
      \textbf{105} & 2.32328/1.74639 & \multicolumn{3}{l}{} & 5 & 14.908 & 1 & \multicolumn{2}{l}{} \\
      \textbf{110} & 4.55519/2.49218 & \multicolumn{3}{l}{} & 6.8/4.9 & 13.962 & 1 & \multicolumn{2}{l}{} \\
      115 & 4.63512/2.58006 & 10.73 & 10.508 & 5 & 10.73 & 10.508 & 5 & \multicolumn{2}{l}{} \\
      120 & 6.02639/3.3319 & 10.73 & 10.508 & 5 & 10.73 & 10.508 & 5 & \multicolumn{2}{l}{} \\ \cmidrule(r){1-8}
    \end{tabular}%
  }
\end{table}

Other Type 1 (e.g. Table~\ref{tab-w3p4vs5}) cases: Fig.~\ref{fig-na-dif2010} Pk
(Wt) 2(1) versus 3(1). Fig.~\ref{fig-na-difm} 4(3) versus 5(3), 6(3) versus
7(3).

Other Type 2 (e.g. Table~\ref{tab-w5p4vs5}) cases: Fig.~\ref{fig-na-dif2010} Pk
(Wt) 2(0, 2, 3, 5) versus 3(0, 2, 3, 5), 4(1, 3, 5) versus 5(1, 3, 5), 6(1, 3,
5) versus 7(1, 3, 5). Fig.~\ref{fig-in-dif2010} 4(0\textendash5) versus
5(0\textendash5), 6(0\textendash5) versus 7(0\textendash5), 14(2, 3) versus
15(2, 3), 22(0, 2, 3) versus 23(0, 2, 3), 26(0, 2, 3) versus 27(0, 2, 3).
Fig.~\ref{fig-au-dif2010} 4(2) versus 5(2). Fig.~\ref{fig-au-difm} 8(5) versus
9(5). Fig.~\ref{fig-na-dif2010m} 2(0\textendash5) versus 3(0\textendash5), 4(2)
versus 5(2), 6(2) versus 7(2), 22(0\textendash5) versus 23(0\textendash5),
26(0\textendash5) versus 27(0\textendash5). Fig.~\ref{fig-in-dif2010m}
4(0\textendash5) versus 5(0\textendash5), 6(0\textendash5) versus
7(0\textendash5), 14(3) versus 15(3).

Combined Type 1 and 2 cases: Fig.~\ref{fig-na-dif2010} Pk (Wt) 22(0\textendash5)
versus 23(0\textendash5), 26(0\textendash5) versus 27(0\textendash5).
Fig.~\ref{fig-in-dif2010} Pk (Wt) 22(1, 4, 5) versus 23(1, 4, 5), 26(1, 4, 5)
versus 27(1, 4, 5). Fig.~\ref{fig-in-dif2010m} 22(0\textendash5) versus
23(0\textendash5), 26(0\textendash5) versus 27(0\textendash5).

Other Type 3 (e.g. Table~\ref{tab-501w0p22vs23}) cases: Fig.~\ref{fig-na-difm}
4(2\textendash5) versus 5(2\textendash5). Fig.~\ref{fig-in-difm}
22(0\textendash5) versus 23(0\textendash5), 26(0\textendash5) versus
27(0\textendash5).


\subsection{Question: Why Does Weighting Not Affect Similarity?}

Generally, weighting does not affect the similarity scores dramatically, because
the six results from the six weighting methods are mostly very close to each
other (Fig.~\ref{fig-dif}, Fig.~\ref{fig-dif2010}, Fig.~\ref{fig-difm},
Fig.~\ref{fig-dif2010m}). This closeness is also generally observed in the form
of clusters in Fig.~\ref{fig-w} and Fig.~\ref{fig-wu}. In addition, from the
general statistics of performance of the six weighting methods shown in
Table~\ref{tab-bw}, Wt 0 mostly performs the best or at least the second best,
which means no weighting works better in general.

When the above-mentioned question, about why APP generally produces better fits
than AMP, is tackled, we already find that both accuracy (how closely do the
pole-pairs, segments and angles match) and precision (how large are the
uncertainties on the pairs/segments/angles, i.e.\ how difficult do they
distinguish) can be the factors that finally determine the difference score.
Although another factor, resolution (how many pairs/segments/angles are actually
being compared) can also influence the difference score (e.g.
Table~\ref{tab-pk0vs1bs}), here this factor is not relevant to comparisons
between different weightings for a certain picking method, because the numbers
of picked paleopoles are the same for the six weighting methods.

Therefore, at the very basic level, for example, a lower score is the result of
one of, or combination of:

1. A reduction in the difference scores of significantly different
pairs/segments, straightforwardly interpreted as a better fit (improved
accuracy).

2. Previously significantly different pairs/segments becoming insignificant.
This can occur either because the fit is better (improved accuracy), or because
of an increase in the uncertainty of the mean poles (they become less
distinguishable \textemdash{} decreased precision).

Fig.~\ref{fig-wp} and Fig.~\ref{fig-wpu} show that the proportions of results
from Wt 1\textendash5 to result from Wt 0 are generally in the second quadrant
of the coordinate plane, where the proportional change in difference score is
positive whereas the proportional change in paleomagnetic FQ score is negative.
This means the five weightings (Wt 1\textendash5) do have an effect, not
obviously in accuracy but mainly in improving precision, which could potentially
make more pairs of poles/segments/angle-changes distinguishable. Or even if
accuracy is improved by a small amount, the improved precision tends to cancel
out the effects from the improved accuracy.

There are a few special cases where one or two of the six weighting methods
gives a result with a dramatically worsened difference score, e.g.\ for
weighting method Wt 3 (Fig.~\ref{fig-dif}). From the labeled dots of Wt 3 in
Fig.~\ref{fig-w}, Fig.~\ref{fig-wu}, Fig.~\ref{fig-wp} and Fig.~\ref{fig-wpu},
it appears that Wt 3 is indeed improving precision but not accuracy (at least
not enough to offset the effects from improved precision) so that this precision
improvement worsens the final score. Highly improved precision without
corresponding improved accuracy results in more significant differences in shape
metrics: Wt 3 is exactly performing in this way.

\begin{figure}
  \vspace{-4mm}
  \captionsetup{skip=-6pt}
  \centering
  \includegraphics[width=1\textwidth]{../../paper/tex/GeophysJInt/figures/w.pdf}
  \caption[Paleomagnetic APWP's FQ score vs significant difference score (i.e.
    $\mathcal{CPD}$)]{10/5 Myr bin/step paleomagnetic APWP's FQ score (different
    from FQ, see the definitions of FQ and FQ score in Chapter~\ref{chap:Metho})
    versus $\mathcal{CPD}$ score (reference path: FHM predicted) for the 28
    different picking methods. See the $\mathcal{CPD}$ scores and Ppath-Rpath FQ
    in Fig.~\ref{fig-dif}. Only those results dramatically worsened by Wt 3 are
    labeled.}\label{fig-w}
\end{figure}

\begin{figure}
  \vspace{-4mm}
  \captionsetup{skip=-6pt}
  \centering
  \includegraphics[width=1\textwidth]{../../paper/tex/GeophysJInt/figures/wu.pdf}
  \caption[Paleomagnetic APWP's FQ score vs raw difference score (i.e., no
    statistical testing)]{10/5 Myr bin/step paleomagnetic APWP's FQ score
    (different from FQ, see the definitions of FQ and FQ score in
    Chapter~\ref{chap:Metho}) versus raw difference score
    ($\frac{\Delta_s+\Delta_l+\Delta_a}{3}$; reference path: FHM predicted) for
    the 28 different picking methods. See FQ versus $\mathcal{CPD}$ score in
    Fig.~\ref{fig-w}. Only those results dramatically worsened by Wt 3 are
    labeled.}\label{fig-wu}
\end{figure}

\begin{figure}
  \vspace{-4mm}
  \captionsetup{skip=-10pt}
  \centering
  \includegraphics[width=1\textwidth]{../../paper/tex/GeophysJInt/figures/wp.pdf}
  \caption[Proportional changes of Wt 1\textendash5 to 0: Paleomagnetic APWP's
    FQ score vs $\mathcal{CPD}$]{Proportion of Wt 1\textendash5 to 0:
    Proportional change in 10/5 Myr bin/step paleomagnetic APWP's FQ score
    (different from FQ, see the definitions of FQ and FQ score in
    Chapter~\ref{chap:Metho}) versus proportional change in $\mathcal{CPD}$
    score (reference path: FHM predicted) for the 28 different picking methods.
    See the $\mathcal{CPD}$ scores and Ppath-Rpath FQ in Fig.~\ref{fig-dif}.
    Only those results dramatically worsened by Wt 3 are labeled.}\label{fig-wp}
\end{figure}

\begin{figure}
  \vspace{-4mm}
  \captionsetup{skip=-10pt}
  \centering
  \includegraphics[width=1\textwidth]{../../paper/tex/GeophysJInt/figures/wpu.pdf}
  \caption[Proportional changes of Wt 1\textendash5 to 0: Paleomagnetic APWP's
    FQ score vs raw difference score]{Proportion of Wt 1\textendash5 to 0:
    Proportional change in 10/5 Myr bin/step paleomagnetic APWP's FQ score
    (different from FQ, see the definitions of FQ and FQ score in
    Chapter~\ref{chap:Metho}) versus proportional change in raw difference score
    ($\frac{\Delta_s+\Delta_l+\Delta_a}{3}$; reference path: FHM predicted) for
    the 28 different picking methods. See FQ versus $\mathcal{CPD}$ score in
    Fig.~\ref{fig-wp}. Only those results dramatically worsened by Wt 3 are
    labeled.}\label{fig-wpu}
\end{figure}

In addition, for Wt 3, a very small size of $\alpha$95 (high precision) could be
caused by those sampled directions not covering a long enough period (thought to
be at least about $10^4$ years) to ``average out'' secular variation for giving
a paleopole. That is to say, the smallest $\alpha$95s are being weighted more
than they deserve.

Generally, weighting has some effect because different weighting functions give
obviously different results. However, weighting does not improve fit and
generally no weighting (Wt 0) generates the best fit, although in most cases
weighting does improve precision. Wt 2 or 4 seems to rarely have generated the
best $\mathcal{CPD}$ score (Table~\ref{tab-bw}), compared with other weighting
methods, but they seem performs better with APP in terms of FQ
(Table~\ref{tab-pk0vs1bs}). There is no general pattern about which weighting
(of Wt 1\textendash5) is better or worse. So weighting when calculating mean
poles for a paleomagnetic APWP is not absolutely necessary. However, there are
cases where weighting is better or worse for some specific continent or some
specific picking method. For example, Wt 3 works generally fine with Australian
data (Table~\ref{tab-bw}). However, Wt 3 is not recommended for North America
and India.

\subsection{Question: Why Are The Best and Worst Methods Sometimes Not
Consistent?}

For all three continents, North America, India and Australia, Pk 19 (APP with
local rotation or secondary print excluded) and 21 (APP with local rotation and
secondary print corrected) are consistently the best or at least relatively
better (for example, in Fig.~\ref{fig-in-dif2010}, Pk 21 results are not colored
in green, but ranging from 0.0483\textendash0.0561 which is still much less than
the mean, 0.074 and the median, 0.072); whereas Pk 16 (AMP with publications
before 1983) and 18 (AMP with local rotation or secondary print excluded) are
consistently the worst or at least relatively worse.

Nevertheless, each continent has its own consistently best and worst picking
methods, that are not the best or worst for other continents. For example, Pk 15
(APP with publications after 1983) works well for North America. We know that 70
North American paleopoles (about 53\%) have contributed to Pk 15 for
120\textendash0 Ma. However, only 28 Indian paleopoles (about 38\%) and 29
Australian paleopoles (about 30\%) have contributed to Pk 15. In contrast, Pk 17
(APP with publications before 1983) works well only for Australia. This also
indicates that the number of paleopoles is a key factor that affects the fit
score. The fact that Pk 22 (AMP with SS05 criteria) doesn't work well only for
Australia also reflects the importance of this factor.

In summary, the reason why some best and worst methods are not consistent is
generally that different continents have their unique data sets which can be
affected very differently by a particular feature. In addition, different
continents get their own paleomagnetic studies in varying degrees.


\section{Final Conclusions}

From the perspective of the general similarities between those paleomagnetic
APWPs and the hot spot model and ocean-floor spreading model predicted APWPs,
GAD hypothesis is proved valid for at least the last 120 Myr.

\subsection{Universal Rules for Processing Paleomagnetic Data}
%
\begin{description}
  \item Although filters (all the picking methods where number of paleopoles
    shrinks compared to Pk 0 and Pk 1; see Fig.~\ref{fig-dif}) reduce N and
    would therefore potentially reduce mean pole precision, some filters do
    improve the similarity score. For example, Pk 25 (APP without superseded
    data) always produces better scores than Pk 1 for all the three continents.
    However, Pk 0 and Pk 1 (no filtering and corrections applied) still
    generally perform well.
  \item APP (adding data to a time window with overlapping age selection
    criterion) is better than AMP for making paleomagnetic APWPs, in both
    situations when there are lots of data (APP even better, e.g.\ for North
    America and Australia) and not much data (APP still a better option, e.g.\
    for India; Table~\ref{tab-bw}). APP combined with most filters/corrections
    (Pk 3, 5, 7, 9, 11, \ldots, 27) generally produce worse scores than APP
    without any filter/correction (Pk 1).
  \item In most cases the APP methods produce better $\mathcal{CPD}$ similarity
    scores than the AMP methods (Table~\ref{tab-bw}).
  \item Weighting does not improve the fit but usually slightly improves mean
    pole precision. For many of the methods, no weighting is the best performer
    (Table~\ref{tab-bw}). For example, $\mathcal{CPD}$ score is likely worse for
    the methods of combining Wt 3 and AMP\@.
  \item APP itself helps incorporate temporal uncertainty into the algorithm.
    With the bootstrap test that incorporates spatial uncertainty into the
    $\mathcal{CPD}$ algorithm, both spatial and temporal uncertainties are
    successfully considered in APP methods.
\end{description}

\subsection{Conditional Rules for Processing Paleomagnetic Data}
%
\begin{description}
  \item Pk 16 (AMP with data from old studies) is not recommended for generating
    a paleomagnetic APWP for North America, India and Australia. Pk 18 (AMP
    without data affected by rotations and secondary print) is not recommended
    for generating a paleomagnetic APWP for North America and India.
\end{description}

\subsection{Summary}

According to the results we have from the three continents, North America, India
and Australia, using the similarity measuring tool developed in
Chapter~\ref{chap:Metho}, it is recommended that APP should be used to select
the input paleopoles. According to the paleomagnetic data analysed from the
three continents, the results from any size for sliding window and step are
extremely close to each other when the APP method is used, compared with the
results from the AMP method (Table~\ref{tab-pk0vs1bs} and
Fig.~\ref{fig-WinStpVsCPD}). So any size for binning and stepping is ok when APP
is used. Then filtering is actually not necessary. However, some
filtering/correction methods (e.g.\ Pk 5, 7 (igneous-derived), 11, 13
(nonredbeds or corrected redbeds derived), 19, 21 (non-local-rotation/reprinted
or corrected-rotation derived) and 25 (non-superseded data derived)) are fine
too and will not give worse or even worst results that the other
filtering/correction methods (especially when combined with AMP) could give.
With APP used, weighting is actually not necessary either. If a weighting has to
be used, Wt 1 (related to the number of paleomagnetic sampling sites and
observations) is generally more effective than the other four given weighting
methods (Fig.~\ref{fig-flow}).

If AMP has to be used, a relatively wide sliding window and step are needed.
According to our tests, more than 20/10 Myr is recommended. In addition, AMP
works relatively better with igneous-derived data (i.e.\ Pk 4 and 6), which
indicates that if we have fewer data, these data need to be better in quality
(Fig.~\ref{fig-flow}).

\begin{figure}
  \centering
  \includegraphics[width=1.01\textwidth]{../../paper/tex/GeophysJInt/figures/flowchart.pdf}
  \caption[Flowchart of making a reliable paleomagnetic APWP]{Flowchart for
    recommended procedure of processing paleomagnetic data.}\label{fig-flow}
\end{figure}