prompt
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that case the discrepancy was not improved.
\section{Summary}
The formation energy, $\Delta E$, magnetic, $S_{magn}$, and configuration, $S_{conf}$, entropy contributions
were calculated and analyzed in the full range of the FeV-$\sigma$-phase
existence using the electronic band structure calculations by means of the KKR method.
It was found that $\Delta E$-values strongly depend on the Fe concentration, and their
variation observed for different site
occupancies is characteristic of a given lattice site. The changes also
strongly depend on the number of Fe atoms on the sites. Similar, increasing dependence on the
sample's composition exhibits the $S_{magn}$ term, but for a given composition it weakly depends on
the lattice site. On the other hand, the composition dependence of the $S_{conf}$ term was found to be weak,
although for a given composition, $S_{conf}$, shows well-defined dependence on the lattice site, and for each site,
a rather strong dependence on the number of Fe atoms occupying the site.
The sublattice occupancies were determined for various compositions and
temperatures by minimizing the free energy. The occupancies of A and
D sites were found to be almost independent of $T$ and $x$, while the occupancies of the sites B, C
and D showed a higher sensitivity to $T$ and $x$. The KKR calculations combined
with the analysis of the entropy contributions clearly showed that
Fe atoms preferably occupy A and D sites in the FeV-$\sigma$ phase. This agrees well
with the neutron diffraction measurements, and it confirms the general observation for $\sigma$
in binary alloy systems that the smaller atoms (in this case Fe) tend to mostly reside on the sites
having the shortest distance to the nearest-neighbors (A and D), while the bigger atoms (here V) prefer to occupy the sites
having more fee space to accommodate them (B, C, E).
\begin{acknowledgments}
This work, supported by the European Communities under the contract of
Association between EURATOM and IPPLM, was carried out within the framework
of the European Fusion Development Agreement. It was also supported by the
Ministry of Science and Higher Education, Warsaw (grant No. N N202 228837).
We also acknowledge financial support from the Accelerated Metallurgy Project,
which is co-funded by the European Commission in the 7th Framework Programme
(contract NMP4-LA-2011-263206),
by the European Space Agency and by the individual partner organizations.
\end{acknowledgments}
|
|
\section{Introduction}
\label{sec:introduction}
Turbulence in galaxy clusters can arise from many sources, ranging from mergers and cosmological structure formation \citep{lau09,vazza09,vazza11,zuhone10}, to galactic wakes \citep{kim07,ruszkowski11}, and AGN feedback (\citet{mcnamara07}, and references therein). It is generally expected to be highly subsonic, with Mach numbers ${\cal M}\sim 0.1-0.5$. Turbulence has wide-ranging and pivotal effects on ICM physics. It could dominate metal transport \citep{rebusco05,simionescu08}, accelerate particles, as required in a prominent model of radio halos \citep{brunetti01,brunetti07}, generate and amplify magnetic fields \citep{subramanian06,ryu08,cho09,ruszkowski11a}, and provide pressure support, thus impacting X-ray mass measurements \citep{lau09}, and Sunyaev-Zeldovich (SZ) measurements of the thermal pressure \citep{shaw10,battaglia11,battaglia11a,parrish12}. The unknown level of non-thermal pressure support introduces systematic deviations in the mass calibration of clusters and could strongly affect their use for cosmology. A particularly interesting effect of turbulence is its impact on the thermal state of the gas, potentially allowing it to stave off catastrophic cooling. It can do by dissipation of turbulent motions \citep{churazov04,kunz11}, or turbulent diffusion of heat \citep{cho03,kim03,dennis05}. More subtly, it can do so by affecting magnetic field topology; by randomizing the $B$-field, it can restore thermal conduction to $\sim 1/3$ of the Spitzer rate \citep{ruszkowski10,ruszkowski11,parrish10}. Besides turbulence, a variety of bulk motions such as streaming, shocks, and sloshing have been observed\footnote{While rotation has not been directly seen, it is also expected from cosmological simulations \citep{lau11}. Its effects are generally too small to be detected by the methods discussed in this paper.}. Such (often laminar) gas motions are interesting in their own right. For instance, gas sloshing in the potential well of clusters, which produces observed cold fronts---contact discontinuities between gas of very different entropies---has gleaned information about hydrodynamic instabilities, magnetic fields,
thermal conductivity and viscosity of ICM \citep[and references
therein]{Markevitch2007}.
Current observational constraints on ICM turbulence are fairly weak, and mostly indirect. They come from the analysis of pressure maps \citep{schuecker04}, the lack of detection of resonant-line scattering \citep{churazov04,werner10}, Faraday rotation maps \citep{vogt05,enslin06}, and deviations from hydrostatic equilibrium with thermal pressure alone \citep{Churazov2008,churazov10,Zhang2008}. In general, these studies constrain cluster cores and either place upper bounds on turbulence, or indicate (with large uncertainties) that it could be present with energy densities $\sim 5-30\%$ that of thermal values. The energy density in turbulence is expected to increase strongly with radius \citep{shaw10,battaglia11}, though observational evidence for this is indirect. The most direct means to constrain gas motions is through Doppler broadening of strong emission lines, but this remains undetected with current technology. By examining the widths of emission lines with XMM RGS,
\citet{Sanders2010} found a 90\% upper limit of 274 ${\rm km s^{-1}}$
(13\% of the sound speed) on the turbulent velocity in the inner 30
kpc of Abell 1385; analysis of other systems provides much weaker bounds ($~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} 500 {\rm km s^{-1}}$; \citet{sanders11}).
Numerical simulations provide further insights
\citep[e.g.][]{Lau2009,Vazza2009, Vazza2010}. However, due to limited resolution and frequent exclusion of important physical
ingredients such as AGN jets, magnetic fields, radiative cooling, anisotropic viscosity, it is difficult to draw robust conclusions.
The forthcoming Astro-H mission\footnote{http://astro-h.isas.jaxa.jp/} (launch date 2014)
represents our best hope of gaining a robust understanding of gas motions in
ICM\footnote{Much farther in the future, the ATHENA mission (http://sci.esa.int/ixo) could significantly advance the same goals.}. With unprecedented spectral resolution (FWHM $\sim$ 4-5 eV),
Astro-H could not only measure the widths of emission lines,
therefore constraining the turbulent amplitude, but also probe the line
shapes. Somewhat surprisingly, very few studies have been conducted to extract velocity
information from the shape of emission lines in a realistic observational context. Current work has focused on studying a Gaussian approximation to the line \citep{rebusco08}, and interpreting the radial variation of the line width and line center \citep{zhuravleva12}. For instance, \citet{zhuravleva12} show how the radial variation of line width is related to the structure function of the velocity field, and also how the 3D velocity field can be recovered from the projected velocity field. However, such inferences generally require angular resolution comparable to characteristic scale lengths of the velocity field, and are likely feasible only for one or two very nearby clusters such as Perseus (though such studies do represent a very exciting possibility for ATHENA). At the same time, it has long been apparent that turbulence in clusters leads to significant non-Gaussianity in the line shape---indeed, these were clearly visible in the early simulations of \cite{sunyaev03}. \citet{Inogamov2003} presented a deep and insightful discussion of the origin of line shapes, albeit in an idealized Kolmogorov cascade model for cluster turbulence (for instance, they do not consider the effect of gas sloshing and cold fronts). Heuristically, one can consider non-Gaussianity to arise when the size of the emitting region (heavily weighted toward the center in clusters) is not much larger than the characteristic outer scale of the velocity field. The central limit theorem does not hold as the number of independent emitters is small (and/or in large scale bulk flows, the motion of different emitters is highly correlated). In a series of papers, Lazarian and his
collaborators considered the relationship between the turbulent
spectrum and the spectral line shape in the ISM \citep{Lazarian2000,
Lazarian2006,Chepurnov2006}. However, since they focused on the supersonic and compressible turbulence seen in the ISM---a regime where thermal broadening is negligible and density fluctuations are considerable---the methods they employ are not readily suitable for the mild subsonic turbulence expected in the ICM.
We therefore aim to study how velocity information
can be recovered from the emission line profile in the ICM context, in a realistic observational setting. In particular, we advance the notion that the profile can be separated into different
modes, which have a meaningful physical interpretation. As we will discuss in more detail below, many processes in the
ICM could give rise to velocity fields composed of
distinct components. For instance, the sharp contact discontinuity in velocity in cold fronts will give rise to a bimodal velocity field where one component is significantly offset from another. Another interesting scenario arises if turbulence is not volume-filling (due, for instance, to anistropic stirring by AGN jets). Then spectral lines of different width (with and without turbulent broadening) will be superimposed on one another. When seen in the same field of view (FOV),
these components correspond to different modes in the line
profile, and decomposing the velocity field into dominant modes can
yield valuable quantitative information (for instance, the volume
filling factor). For the upcoming Astro-H mission, mode
separation in the spectrum is necessary and important since the poor
angular resolution make it hard to spatially resolve different
components---indeed, the high spectral resolution of Astro-H is our
best tool for inferring the complex structure of the velocity
field. We use standard mixture modeling techniques and Fisher
matrix/Markov chain Monte Carlo error analysis to quantify how well we
could separate and constrain different
components from a single spectrum, and then establish what we can learn from about the underlying velocity field from such a component separation.
\begin{table}
\caption{Specifications of the Soft X-ray Spectroscopy System
onboard the Astro-H telescope.}
\label{tbl:specs}
\begin{center}
\begin{tabular}{l c}
\hline \hline
Effective area ${\rm cm^2}$ at 6 keV& 225\\
Energy range (keV) & 0.3-12.0 \\
Angular resolution in half power diameter (arcmin) & 1.3\\
Field of view (${\rm arcmin}^2)$ & $3.05\times 3.05$\\
Energy resolution in FWHM (eV) & 5\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\caption{Photon counts $N_p$ in the He-like iron line and physical length
corresponding to angular resolution in HPD (1.3 arcmin) for a few
nearby galaxy clusters. The photons are accumulated from 1 FOV through
the cluster center over $10^6$ seconds. }
\label{tbl:clusters}
\begin{center}
\begin{tabular}{l c c c }
\hline \hline
Cluster Name& Redshift & $d_{1.3}$ (kpc) & $N_p (\times
10^4~{\rm phot})$ \\
\hline
PERSEUS & 0.0183 & 28.81 & 5.8\\
PKS0745 & 0.1028 & 146.76 & 3.8\\
A0478 & 0.0900 & 130.36 & 3.7\\
A2029 & 0.0767 & 112.79 & 3.4\\
A0085 & 0.0556 & 83.78 & 3.1\\
A1795 & 0.0616 & 92.17 & 2.6\\
A0496 & 0.0328 & 50.76 & 1.9\\
A3571 & 0.0397 & 60.94 & 1.7\\
A3112 & 0.0750 & 110.51 & 1.7\\
A2142 & 0.0899 & 130.23 & 1.6\\
2A0335 & 0.0349 & 53.87 & 1.3\\
HYDRA-A & 0.0538 & 81.23 & 1.3\\
A1651 & 0.0860 & 125.13 & 1.1\\
A3526 & 0.0103 & 16.37 & 0.8\\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
Before proceeding to the main discussion, we first list a few
specifications of the Astro-H mission, on which our
discussions are based. Our study mainly takes the advantage of the
high spectral resolution of the Soft X-ray Spectroscopy System (SXS)
onboard the Astro-H telescope. Its properties, taken from the
``Astro-H Quick
Reference''\footnote{http://astro-h.isas.jaxa.jp/doc/ahqr.pdf}, are
given in Table \ref{tbl:specs}.
The energy resolution is 5 eV in FWHM\footnote{It has shown to be even lower--4 eV--in laboratory tests \citep{porter10}.},
corresponding to a standard deviation of 2.12 eV. For comparison, the
thermal broadening of the Fe 6.7 keV line is 2.07 eV for a 5 keV cluster, while
broadening by isotropic Mach number ${\cal M} \sim 0.2$ motions is 2.9 eV. Thus, for the highly subsonic motions in the core with Mach numbers ${\cal M} \sim 0.1-0.3$ generally seen in cosmological simulations, the instrumental, thermal and turbulent contributions to line broadening are all roughly comparable. In contrast to the impressive energy resolution, the angular
resolution of Astro-H is poor: 1.3 arcmin in half power diameter
(HPD).
Therefore, different velocity components are likely to show up in
the same spectrum. Based on these specifications, Table
\ref{tbl:clusters} shows the expected photon counts in the He-like
iron line at 6.7 keV for a few of the brightest nearby clusters ($z \leq 0.1$). The photons are accumulated in one FOV through
the cluster center over $10^6$ seconds; the $\sim$ several x $10^{4}$ photons collected should allow good statistical separation of mixtures if present. The density distributions and
cluster temperatures are taken from \citet{Chen2007}, and metallicity
is assumed to be 0.3 ${\rm Z_{\odot}}$.
Also shown are the physical lengths corresponding to the
angular resolution in HPD, which are
$\sim$100 kpc; comparable to the core size. We therefore do not expect Astro-H to spatially resolve many
structures.
The remainder of the paper is organized as follows. In
\S~\ref{sec:motivations}, we discuss possible scenarios that could
give rise to multiple component spectra, further motivating the current
study. In \S~\ref{sec:methodology}, we develop the methodology to be
used in this paper. In \S~\ref{sec:constraints}, we discuss how
accurately different components could be recovered in idealized toy models, to build our understanding of the applicability and capabilities of the method. In \S~\ref{sec:application}, we
apply our statistical method to realistic simulations of galaxy clusters, where we have full knowledge of the underlying velocity field, and see what information we can recover. In \S~\ref{sec:conclusions}, we
conclude by summarizing the main results.
\section{Motivation}
\label{sec:motivations}
In this section, we motivate the current study by giving examples of very common processes operating in the ICM which could give rise to multi-component velocity fields: bulk motions from mergers and sloshing, and AGN feedback.
\subsection{Bulk Motions}
\label{subsec:example}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-0}{\resizebox{120mm}{!}{\includegraphics{f1.ps}}}
\end{tabular}
\caption{Density map on a slice through the cluster center. The dashed line shows the direction along which the profiles in Fig. \ref{fig:profiles} are
computed, while the perpendicular dotted line--chosen to maximize line of sight velocity shear
--indicates the observation direction for the solid red velocity PDF shown in
Fig. \ref{fig:spectrum}. The dotted red line shows an alternate viewing direction with much less velocity shear; its velocity PDF is given by the thin red line in Fig. \ref{fig:profiles}.}
\label{fig:density}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-0}{\resizebox{90mm}{!}{\includegraphics{f2.ps}}}
\end{tabular}
\caption{Velocity fields on the same slide as in Fig.
\ref{fig:density}, overlaid with density (solid blue curves) and
temperature (dashed red curves) contours. The large purple arrow
indicates the location of the cold front.}
\label{fig:velocity}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{90mm}{!}{\includegraphics{f3.eps}}}
\end{tabular}
\caption{Density (dashed curve), temperature (dotted curve) and
line-of-sight velocity (solid curve in the bottom panel) profiles
along the a direction perpendicular to the cold front, as
indicated in Fig. \ref{fig:density} with a dashed line.
Here, the position of the cold front is given by the
vertical (cyan) line at 85 kpc. }
\label{fig:profiles}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{60mm}{!}{\includegraphics{f4.eps}}}
\end{tabular}
\caption{The thick solid (red) curve is the normalized emission-weighted
velocity PDF from a box centered on the dotted line in Fig.
\ref{fig:density}. The box is 100 kpc long, 100 kpc wide and 1 Mpc
deep. The thick dashed green curve,
is the corresponding profile of the He-like iron line at 6.7 keV
(see top axis for energy scale), including the effects of thermal
broadening, while the thick dot-dashed purple curve also includes
instrumental broadening.
The thin lines show the same curves for
the line of sight given by the dotted red line in
Fig. \ref{fig:density}.
}
\label{fig:spectrum}
\end{figure}
Thus far, most constraints on gas bulk motions comes from observations of sharp density gradients in the plane of the sky. Classic bow shocks have been seen in a handful of violent mergers. Much more common are ``cold fronts'' \citep{markevitch07}: sharp contact discontinuities between gas phases of different entropies, discovered in the last decade thanks to the high-resolution of the {\it Chandra} X-ray telescope. They are seen both in mergers (where the cold gas arises from the surviving cores of infalling subclusters) and relaxed cool core clusters (where they are produced by the displacement and subsequent sloshing of the low-entropy central gas in the gravitational potential well of the cluster). They are remarkably ubiquitous, even in relaxed cool core clusters with no signs of recent mergers, which often exhibit several such cold fronts at different radii from the density peak. For instance, they are seen in more than half of all cool core clusters; given projection effects, most if not all cool core clusters should exhibit such features. Evidently, coherent gas bulk motions are extremely common if not universal\footnote{Indeed, we show here the very first cluster we simulated from random initial conditions, which already exhibited cold front like features.}, and their effects must be taken into account when interpreting Astro-H spectra. Generically, we would expect bulk motions to offset the centroids of emitting regions with significant line-of-sight relative velocity. Cold fronts have been used to probe the amplitude and direction of gas motions in the plane of the sky; combining this with line-of-sight information from the spectrum could prove very powerful indeed.
Our example is taken from an adiabatic numerical simulation from cosmological initial conditions with the
adaptive mesh code Enzo \citep{Bryan1999,Norman1999,O'Shea2004}.
We assume a $\Lambda$CDM cosmology with cosmological parameters
consistent with the seventh year {\it WMAP} results \citep{komatsu11}:
$\Omega_m=0.274$, $\Omega_{\Lambda}=0.726$, $\Omega_b=0.045$,
$h=0.705$, $\sigma_8=0.810$, $n_s=0.96$.
The
simulation has a box size of 64
Mpc, and a root grid of $128^3$. We picked the most massive cluster
($M \sim 2\times 10^{14}~\rm M_{\odot}$) from the fixed-grid initial run, and
re-simulate it with much higher resolution. The highest spatial resolution is
11 kpc in the cluster center.
The cluster has a disturbed morphology, and shows a ``cold
front''-like feature in the core.
Note that our adiabatic simulation necessarily produces a NCC cluster.
The density and velocity fields on a
slice through the cluster center are shown in Fig. \ref{fig:density} and \ref{fig:velocity},
respectively. In the position indicated by the large arrow in Fig.
\ref{fig:velocity}, the density, temperature and velocity all change
rapidly. This is clearly shown in Fig. \ref{fig:profiles}, which shows
density, temperature and velocity profiles along a line perpendicular
to the front (indicated in Fig. \ref{fig:density} with a dashed
line). At $\sim 85$ kpc from the cluster center, the density decreases
while the temperature increases rapidly, as expected in a cold
front (for a shock, the temperature jump would be opposite). Furthermore, the pressure is continuous across the front,
while the tangential velocity changes direction discontinuously across the front---both well-known features of cold fronts \citep{markevitch07}.
For an observation direction along the white dotted line in Fig.
\ref{fig:density}, Fig. \ref{fig:spectrum} shows the
emission-weighted probability distribution function (PDF) of the
line-of-sight velocity. Motivated by Table \ref{tbl:clusters}, we
extract the emission-weighted PDF from an volume with an area of $100\times100~{\rm kpc}^2$ and a
depth of 1 Mpc (this last number represents the line of sight depth, and is chosen for convenience. Our results are insensitive to it as long as it is much larger than the core size, where most of the photons come from).
The PDF clearly shows two peaks, centered at -400 ${\rm km \, s^{-1}}$
and 250 ${\rm km \, s^{-1}}$, corresponding to the gas on different side of the
cold front. After convolution with thermal broadening,
the dashed line shows the profile of the He-like iron line at 6.7 keV,
while the dot-dashed line also includes the instrumental broadening of
Astro-H. They also clearly
show double peak features.
The above case is a somewhat idealized ``best case'' scenario, where we have assumed the
viewing angle to be along the direction of maximum line of sight velocity shear, thus
maximizing the separation between the two peaks in the velocity
PDF. For a more general viewing angle, the separation would not be so
clear, as we show with the thin curves in
Fig. \ref{fig:spectrum}. This is the PDF along the red dotted curve in
\ref{fig:density},
which has very small line-of-sight bulk flow.
There is only one large peak, but with a long tail. From
Fig. \ref{fig:velocity}, we see this long tail comes from the
gas surrounding the cold clump, which has shear
velocities with large components along the LOS.
Therefore the PDF can also be separated into two
components -- a narrow component emitted by the cold clump and a broad
component from the ambient gas. The offset between the components is a measure of the LOS contact discontinuity in the bulk velocity, while smaller scale shear contributes to the width.
Such a decomposition of the line-of-sight velocity, combined with spatially resolved temperature and density information in the plane of the sky from X-ray imaging, could shed more light on the 3D velocity field as well as physical information such as the gas viscosity.
\subsection{Volume-filling Factor of Turbulence}
\label{subsec:others}
The previous section highlighted a situation where strong shear or bulk
motion gives rise to different components with offset centroids
(``separation driven'' case). Another regime where different components
could arise is when the two components have markedly different widths
(``width driven'' case). We saw an example of this at the end of the
previous section: a narrow component due to a cold, kinematically
quiescent clump, and a broader component due to the sheared
surrounding ambient gas. More generally, different widths arise when
turbulence varies spatially. The case when turbulence is only
partially volume-filling is a particularly interesting special
case. Many of the physical effects of turbulence depend not only on
its energy density, but its volume filling fraction $f_{\rm V}$, which
is often implicitly assumed to be unity. For instance, for turbulence
to stave off catastrophic cooling, it must be volume-filling. This is
by no means assured. For instance, analytic models
\citep{subramanian06} of turbulence generation during minor mergers
predict $f_{\rm V}\sim 0.2-0.3$ to be small, but area-filling (i.e.,
the projection of turbulent wakes on the sky cover a large fraction of
the cluster area, $f_{\rm S} \sim O(1)$). Interestingly, cosmological
AMR simulations which use vorticity as a diagnostic for turbulence
find good agreement; $f_{\rm V} ~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} 0.3$ and $f_{\rm S} \sim O(1)$
for all runs \citep{iapichino08}. In our own simulations of stirring
by galaxies \citep{ruszkowski11}, we have seen both high and low
values of $f_{\rm V}$, depending on modeling assumptions. If $g$ modes
are excited by orbiting galaxies (which requires the driving orbital
frequency $\omega$ to be less than the Brunt-V\"ais\"al\"a \, frequency $\omega_{\rm
BV}$--a requirement which depends on both the gravitational
potential and temperature/entropy profile of the gas), then
volume-filling turbulence is excited; otherwise turbulence excited by
dynamical friction is potentially confined to thin ``streaks'' behind
galaxies (see also \citet{balbus90,kim07}). If turbulence is patchy,
we might expect spectral lines to have a narrow thermal Doppler core
(produced in quiescent regions), with turbulently broadened tails. In
the context of our mixture model, measuring the fraction of photons in
the second component might allow a quantitive measure of $f_{\rm V}$.
Yet another context in which strongly spatially varying or partial volume-filling turbulence could result in multiple components in the velocity PDF is AGN feedback, which is ubiquitous in most cool core clusters (\citet{Birzan2004}; for a recent review, see \citet{mcnamara12}). AGN jets are launched over a narrow solid angle and are fundamentally anisotropic; thus, their ability to sustain isotropic heating in the core has often been questioned. Isotropization of the injected energy could arise from weak shocks and sound waves \citep{fabian03}, frequent re-orientation of jets by randomly oriented accretion disks \citep{king07}, jet precession \citep{dunn06,gaspari11}, and cavities being blown about by cluster weather \citep{bruggen05,heinz06,morsony10}. As above, AGN could also excite g-mode oscillations; an intriguing example is a cross-like structure on 100 kpc scales in the ICM surrounding 3C 401 \citep{reynolds05}. A measurement of $f_{\rm V}$ could thus constrain the efficacy of such mechanisms in isotropizing AGN energy deposition throughout the core. The expansion of AGN-driven cavities can also introduce high bulk velocities and shear (corresponding more to the ``separation-drive'' regime); this is potentially directly measurable with ATHENA's excellent angular and spectral resolution \citep{Heinz2010a}, but would require indirect methods such as mixture modeling with the poor angular resolution of Astro-H. In \S \ref{sec:application}, we analyze an AGN feedback simulation kindly provided to us by M. Br\"{u}ggen.
\subsection{Physical Significance of Mixture Model Parameters}
\label{section:physical_significance}
In \S \ref{sec:methodology}, we lay out our methodology for recovering mixture model parameters, and in subsequent sections we describe how accurately these can be constrained. These parameters are the mixture weights $f_{i}$, and means and variances $\mu_{i}, \sigma_{i}^{2}$, of the fitted Gaussians. Given the results of this section, we can tentatively ascribe physical significance to these parameters. The mixture weights $f_{i}$ represent the emission-weighted fraction of the volume in each distinguishable velocity component. The Gaussian means $\mu_{i}$ represent the bulk velocity of a given component. In particular, the difference between the means is a measure of the LOS shear between these components (e.g., as arises at a cold front). Note that this shear due to bulk motions can be considerably larger than the centroid shift due to variance in the mean, induced by turbulent motion with a finite coherence length. The latter is given by $\mu_{i} \sim \sigma_{i}/\
\sqrt{N}$, where $N\sim L_{\rm emit}/l_{\rm v}$ is the average number of eddies pierced by the line of sight, and $L_{\rm emit},l_{\rm v}$ are the size of the emitting region and the coherence length of the velocity field respectively \citep{rebusco08,zhuravleva12}. The variances $\sigma_{i}$ represents turbulent broadening or shear due to the small scale velocity field.
\section{Methodology}
\label{sec:methodology}
We have argued that the X-ray spectrum from galaxy clusters should have multiple distinct components. Uncovering these components is the domain of {\it mixture modeling}, a mature field of statistics with a large body of literature. We will specialize to the case of Gaussian mixture modeling, when Gaussians are used as the set of basis functions for the different components. This is an obvious choice, since thermal and instrumental broadening are both Gaussian, and turbulent
broadening can be well approximated with a Gaussian when the injection scale is much smaller than the size of the emitting regions
(\citet{Inogamov2003}; i.e., once coherent bulk motions have been separated out by classification into different mixtures, the remaining small scale velocity field is well approximated by a Gaussian). It is also by far the best studied case. Mixture modeling has been applied to many problems in astrophysics, such as detecting bimodality in globular cluster metallicities \citep{ashman94, muratov10} linear regression \citep{kelly07}, background-source separation \citep{guglielmetti09}, and detecting variability in time-series \citep{shin09}, though to our knowledge it has not been applied to analyzing spectra. It should be noted that the specialization to Gaussian mixture is not necessarily restrictive; for instance, Gaussian mixtures have been used to model quasar luminosity functions \citep{kelly08}. For us, the fact that Gaussians are a natural basis function allows us to model the spectra compactly with a small number of mixtures, and assign physical meaning to these different components.
Consider a model in which the observations $x_{1},\ldots,x_{n}$ are distributed as a sum of $k$ Gaussian mixtures:
\begin{equation}
f(x|\theta) = \sum^{k}_{j=1} \omega_{i} f_{j}(x|\mu_{j},\sigma^{2}_{j}),
\label{eqn:mixture}
\end{equation}
where $f_{j}(x|\mu_{j},\sigma^{2}_{j})$ are normal densities with
unknown means $\mu_{j}$ and variances $\sigma_{j}^{2}$, and
$\omega_{i}$ are the mixture weights. The parameters which must be
estimated for each mixture are therefore
$\theta_{j}=(\omega_{j},\mu_{j},\sigma_{j}^{2})$, and the function $f$
can be viewed as the probability of drawing a data point with value
$x$ given the model parameters $\theta$. Parameter estimation in this
case suffers from the well-known {\it missing data problem}, in the
sense that the information on which distribution $j$ a data point
$x_{i}$ belongs to has been lost. In addition, the number of mixtures
$k$ may not be a priori known\footnote{In this case, the optimal
number of mixtures can also be estimated from the data, via simple
criteria such as the Bayesian Information Criterion (see equation
\ref{eqn:bic}), or more sophisticated techniques in so-called
Infinite Gaussian Mixture Models. In this paper, we only investigate
separating the two most dominant components of the spectrum, which
have the highest signal-to-noise. The data is generally not of
sufficient quality to allow solving for more than two mixtures
(strong parameter degeneracies develop). Physical interpretation is
also most straightforward for the two dominant mixtures.}. Standard
techniques for overcoming this are a variant of maximum likelihood
techniques known as Expectation Maximization (EM; \citet{dempster77}),
or Maximum a Posterior estimation (MAP; see references in Appendix),
which generally involves Markov Chain Monte Carlo (MCMC) sampling from
the posterior. They are both two-step iterative procedures in which
parameter estimation and data point membership are considered
separately. Since they do not require binning of the data, all
information is preserved. We have experimented extensively with
both. However, due to the large number of data points ($\sim 10^{4}$
photons) in this application, we have found that the much simpler and
faster procedure of fitting to the binned data yields virtually
identical results. In the Appendix, we describe our implementation of
Gibbs sampling MAP and how it compares with the much simpler method we
use in this paper.
Here, we simply bin the data and adopt as our log-likelihood the C-statistic \citep{Cash1979}:
\begin{eqnarray}
-2{\rm ln} {\cal L}(p|d) = -2 \sum_{i=1}^{N_{bin}}n_i {\rm ln} e_i- e_i -
{\rm ln} n_i !
\label{eqn:cashc}
\end{eqnarray}
where ${\cal L}(p|d)$ is the likelihood of the parameters $p$ given the data $d$, $N_{bin}$ is the number of bins, $n_i$ and $e_i$ are the
observed and expected number of counts in the i-th bin; $n_i,e_i$ are obviously functions of the data $d$ and the unknown model parameters $p$ respectively. It assumes that the number of data points in each bins is Poisson distributed (indeed, it is simply the log of the Poisson likelihood). As we describe in the Appendix, maximizing this statistic produces identical results to more rigorous mixture modeling techniques for large number of data points, when the bin size is sufficiently small. Naively, for a large number of data points one might expect $\chi^{2}$ minimization to work equally well. However, in fitting distributions we are sensitive to the wings of the Gaussian basis functions, when the expected number of counts in a bin is small and the data is therefore Poisson rather than Gaussian distributed.
With the likelihood specified in Equation \ref{eqn:cashc}, we sample
from the posterior using Metropolis-Hastings MCMC, adapted from CosmoMC \citep{Lewis2002}. Each run draws $\sim 10^5$ samples. The first
30\% are regarded as burn-in and are ignored in the post-analyses. For all
the runs, we visually exam the trace plots to check for convergence. The MCMC analysis yields the
best-fit MAP parameters as well as the full posterior distribution of
parameters, which allows us to estimate confidence intervals. In all
cases, we use non-informative (uniform) priors; the range of possibilities for the turbulent velocity field is
sufficiently large that only very weak priors are justifiable. The only obvious prior we use is $0 < f_{i} <
1$. Note that there are two identical modes in the likelihood, since
it is invariant under permutation of the mixture indices--the
well-known identifiability or ``label-switching'' problem. Generally, in
a $k$ component mixture, there are $k!$ identical modes in the
likelihood. During the course of a Monte-Carlo simulation, instead of
singling out a single mode of the posterior, the simulation may visit
portions of multiple modes, resulting in a sample mean which in fact
lies in a very low probability region, as well as an unrealistic
probability distribution. We enforce identifiability in a very simple
manner by demanding $\mu_{1} < \mu_{2}$ and hence $s \equiv
\mu_{2}-\mu_{1} >0$. While this is known to sometimes be problematic
\citep{celeux00,jasra05}, in practice it suffices for our simple
models.
For a large number of data points, the distribution of model parameters becomes asymptotically Gaussian, in which case the Fisher matrix can be used to quickly estimate joint parameter uncertainties
(e.g., \citet{Tegmark1997}).
As a consistency check, we therefore also calculate the Fisher matrix whenever the input model is simple enough to
be expressed analytically. It is defined as:
\begin{eqnarray}
F_{ij}=-\left<\frac{\partial^{2}\ln{\cal{L}}}{\partial p_i \partial p_j}\right>,
\label{eqn:fisher}
\end{eqnarray}
where $p_i$ is the i-th model parameter. The best attainable
covariance matrix is simply the inverse of the Fisher matrix,
\begin{eqnarray}
C_{ij}=(F^{-1})_{ij},
\end{eqnarray}
and the marginalized error on an individual parameter $p_i$ is
$\sqrt{(F^{-1})_{ii}}$. Differences
between the MCMC and the Fisher matrix error bars generally indicate the
non-Gaussianity of the likelihood surface (or equivalently, that the log-likelihood cannot be truncated at second order in a Taylor expansion).
\section{Idealized Models}
\label{sec:constraints}
\subsection{Two component Gaussian mixture models: General Results}
\label{subsec:general}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-0}{\resizebox{80mm}{!}{\includegraphics{f5.eps}}}
\end{tabular}
\caption{Constraints on the five model parameters with $N_{d}=10^{4}$ data
points, as a function of $s/(\sigma_{1}+\sigma_{2})$, where $s=\mu_{2}-\mu_{1}$ is the separation between the means. The curves and points
are the results obtained using the
Fisher matrix and MCMC methods, respectively. The
dashed lines and circles [green], dotted
lines and upward triangles [blue], solid
lines [red], dot-dashed lines and downward
triangles [purple] and dot-dot-dashed lines and diamonds [brown] correspond
to $\sigma_{2}=(0.6,0.8,1,1.2,1.4) \sigma_{1}$, respectively.}
\label{fig:separation}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{0}{\resizebox{85mm}{!}{\includegraphics{f6.ps}}}
\end{tabular}
\caption{Error contours for a ``SD'' case ($\sigma_1=1$, $\sigma_2=0.8$,
$s = 2 (\sigma_1+\sigma_2)$): contours depict the
68\%, 95\% confidence levels for the marginalized distribution; the
shadings shows the mean likelihood of the samples; the solid and dashed curves
in the 1-D plots are the fully marginalized posterior and relative
mean likelihood of the samples, respectively.
}
\label{fig:corr_sd}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{0}{\resizebox{85mm}{!}{\includegraphics{f7.ps}}}
\end{tabular}
\caption{Same as Fig. \ref{fig:corr_sd} but for a ``WD'' case:
$\sigma_1=1$, $\sigma_2=0.8$, $s=0.25 (\sigma_1+\sigma_2)$.
Note the increased parameter degeneracies.}
\label{fig:corr_wd}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{60mm}{!}{\includegraphics{f8.eps}}}
\end{tabular}
\caption{Constraints on the model parameters for a 2 Gaussian mixture as a function of the
fractional difference in width, when $s=0.2$ and $\sigma_{1}=1$. As in Fig. \ref{fig:separation},
the lines and points show the results obtained with the Fisher matrix
and MCMC technique, respectively. The solid
lines and squares [red], dashed lines and circles [green], dotted
lines and upward triangles [blue], dot-dashed lines and downward
triangles [purple] and dot-dot-dashed lines and diamonds [brown] are
the constraints on $f_1$, $\mu_1$, $s$, $\sigma_1$ and $\sigma_2$, respectively}
\label{fig:width}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{0}{\resizebox{80mm}{!}{\includegraphics{f9.eps}}}
\end{tabular}
\caption{Constraints on the five free parameters for different values
of $N_d$ and $f_1$. The solid curves and squares are the results for
the fiducial case: $N_d=10^4$, $\sigma_1=1$, $\sigma_2=1$ and
$f_1=0.4$; results for $N_d=10^5$, $f_1=0.3$, and $f_1=0.5$ are
shown with dashed curves and circles, dotted curves and downward
triangles, and dot-dashed curves and upward triangles, respectively.
}
\label{fig:frnp}
\end{figure}
Before focusing on the specific application to galaxy clusters, we first consider a
more general problem: how well two Gaussian profiles can be
separated. As mentioned previously, Astro-H data quality is generally
only sufficient to allow solving for the two most dominant mixtures. A
two mixture component is likely the most common scenario, with the most straightforward physical
interpretation. These
results serve to guide and motivate our later discussions.
Consider therefore the profile:
\begin{eqnarray}
p(x)=\sum_{i=1,2}f_i G(x-\mu_i,\sigma_i),
\label{eqn:profile}
\end{eqnarray}
where $f_i$ is the fraction of each component, while $\mu_i$ and
$\sigma_i$ are the mean and standard deviation of i-th Gaussian
function. Given the constraint $\sum f_i =1$, there are only five model
parameters, which we choose to be: $f_1$, $\mu_1$, $s(\equiv
\mu_2 - \mu_1)$, $\sigma_1$ and $\sigma_2$. Note, $\mu_2$ has been
replaced by $s$ (the separation between the two Gaussians) since
the latter, as we will see more clearly later, usually carries clearer
physical meaning.
The constraints, expressed in term of standard deviations $\Delta$
throughout this paper, are forecasted with both the MCMC and Fisher matrix
methods
(for the MCMC runs, they correspond to $68\%, 95\%$ confidence
intervals for $\Delta,2 \Delta$ respectively, even if the parameter
distribution is non-Gaussian). For each model we create a Monte-Carlo
realization with $N_{d}$ data points and forecast constraints for this
data set. Motivated by Table \ref{tbl:clusters}, we assume $N_d=10^4$.
In general, the standard deviation of the model parameters $\Delta p \propto 1/\sqrt{N_{\rm d}}$, though there are some subtleties---see further discussions below.
The constraints also depend on how much the two components differ; if they are difficult to distinguish, mixture modeling will fail. For Gaussian components, they may
differ in fraction $f_{i}$, mean $\mu_{i}$ or width $\sigma_{i}$.
Here, we shall mostly focus on a situation when the mixing fractions are
comparable: $f_{1}=0.4,f_{2}=0.6$, and focus on how mixture separation
can be driven by differences in mean (``separation dominated'', or SD),
or width (``width drive'', or WD).
In practice, we care mostly about the case when the mixing fractions are roughly comparable, since then the different components are of comparable importance in reconstructing the (emission-weighted) velocity field. As a practical matter, it also becomes increasingly difficult to perform mixture modeling when one component dominates (though see \S \ref{subsec:prospects}).
The results are shown in Fig. \ref{fig:separation} -
\ref{fig:frnp}. Fig. \ref{fig:separation} shows the constraints as a
function of the separation $s=\mu_{2}-\mu_{1}$, normalized by the sum
of the standard deviations: $s/(\sigma_{1}+\sigma_{2})$. Different line
types
and point types indicate different values of $\sigma_2=
(0.6, 0.8, 1, 1.2, 1.4) \, \sigma_1$. Note that all five parameters scale with
$s/(\sigma_{1}+\sigma_{2})$ in the same way, with fractional errors
which are all roughly comparable. We can identify three distinct
regimes:
\begin{itemize}
\item{{\bf Separation Driven} For $s/(\sigma_{1}+\sigma_{2}) ~\rlap{$>$}{\lower 1.0ex\hbox{$\sim$}} 2$,
the fractional errors converge to an asymptotic constant value,
independent of $s/(\sigma_{1}+\sigma_{2})$. In this regime, the
separation is so large that
different components could be viewed as individually constrained
without mixing from other components. Except for $\Delta(s)$ (which
depends on $\Delta (\mu_{2}) \sim \sigma_2/\sqrt{f_i N_d}$), this
asymptotic convergence is also independent of $\sigma_{2}$ (i.e., the
relative widths of the distributions don't matter when the separation
is large).
The asymptotic values for
$\Delta(\mu_i)$, $\Delta(\sigma_i)/\sigma_i$ and $\Delta(f_i)$ are
$\sigma_i/\sqrt{f_i N_d}$, $1/\sqrt{2 f_i N_d}$ and
$\sqrt{{f_i(1-f_i)}/{N_d}}$, respectively; given our $N_{d}=10^{4}$,
this corresponds to $\sim 1\%$ accuracy in parameter constraints. }
\item{{\bf Hybrid} For $0.3 ~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} s/(\sigma_{1}+\sigma_{2}) ~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} 2$,
the separation is comparable to the sum of widths. The mixing
between different components become severe and the quality of
parameter constraints decrease rapidly with decreasing $s$. Since
constraints are increasing driven by data points in the tails of
the respective mixtures (which drive distinguishability), the
effective number of data points $N_{\rm eff} < N_{d}$
falls. Strong parameter degeneracies also develop.}
\item{{\bf Width Driven} When $s/(\sigma_{1}+\sigma_{2}) ~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} 0.3$,
the separation between the distribution becomes negligible, and
component separation is driven almost entirely by differences in
width (note how parameter uncertainties blow up at low $s$ when
$\sigma_{1}=\sigma{2}$). It is driven to an asymptotic value
determined by the effective number of data points in the tails of
the mixtures, $N_{\rm eff}$. }
\end{itemize}
The results
obtained with the Fisher matrix (lines) and MCMC (points) agree well
with each other when the mixtures are easily distinguishable (when $s/(\sigma_{1}+\sigma_{2})$ is large or
$\sigma_2/\sigma_1$ is reasonably far away from 1). Otherwise, discrepancies between these two methods are
clear. These discrepancies are caused by the non-Gaussianity of the
likelihood surfaces and the priors we placed in the MCMC runs.
In this regime, one therefore cannot use the Fisher matrix approximation to the full error distribution.
Fig. \ref{fig:corr_sd} and \ref{fig:corr_wd} show the marginalized likelihood
distributions and error contours for two example
runs. Fig. \ref{fig:corr_sd} is in the SD
regime ($s = 2 (\sigma_1+\sigma_2)$). The likelihood distributions are
very close to Gaussian, explaining the consistency between the Fisher
matrix and MCMC results. The contours allow direct reading of
correlations among parameters. The strongest correlation is between
$\mu_1$ and $s$. As expected, they are negatively correlated, since
$s=\mu_2-\mu_1$ while the $\mu_i$'s are uncorrelated. Fig. \ref{fig:corr_sd} is in the WD
regime ($s = 0.25 (\sigma_1+\sigma_2)$). The likelihood distributions
now deviate from Gaussians, and the correlations among parameters are
much stronger. These are all consistent with the facts that the
constraints are worse (due to larger parameter degeneracies) and the
Fisher matrix results are no long in agreement with MCMC results (due
to non-Gaussianity of the likelihood surface).
In Fig. \ref{fig:width}, we show how the constraints vary with
differences in the Gaussian width in the ``width dominated'' regime. The width of the first component is
fixed to $\sigma_{1}=1$, while the separation $s=0.2$ (note that $s/(\sigma_{1}+\sigma_{2}) \sim 0.2$ is typically the minimal value expected in cluster turbulence when there are no bulk flows, and is due solely to error in the mean; see \S\ref{section:physical_significance}). In general, the
constraints improve as the differences in width
increase, consistent with intuitive expectations. However, the
constraints on $s$ and $\sigma_2$ turn over around $\sigma_2 \sim
2\sigma_1$, beyond which they increase with $\sigma_2$\footnote{The turnover
does not appear in the last panel of Fig. \ref{fig:separation},
because there the y-axis is $\Delta(\sigma_2)/\sigma_2$ rather than $\Delta(\sigma_2)$}.
This can be understood as follows: the error on these quantities receive contributions from confusion error (which dominates at low $\sigma_{2}$) and scaling with $\sigma_{2}$ (since $\Delta(\mu_{i}) \sim \sigma_{i}/\sqrt{f_{i} N_{d}}$ and $\Delta \sigma_{i} \sim \sigma_{i}/\sqrt{2 f_{i} N_{d}}$; this dominates at high $\sigma_{2}$).
On the other hand, the error on the mixing fraction $f_{1}$ scales strongly with the difference in widths,
since it is driven solely by confusion error. However, for other parameters the scaling is significantly weaker. For most cluster scenarios, the width-driven regime gives relative errors of $\Delta p/p \sim 10\%$, which is still small.
Fig. \ref{fig:frnp} shows how the constraints vary with $N_d$ and
$f_1$. The fiducial case (solid curves and squares) is computed
assuming $N_d=10^4$, $\sigma_1=1$, $\sigma_2=0.8$ and $f_1=0.4$,
exactly the same as the dotted curves and upward triangles in
Fig. \ref{fig:separation}. As we increase the $N_d$ by a factor of
10, most constraints are improved by a factor of $\sqrt{10}$,
consistent with our expectation that $\Delta p_{i}/p_{i}
\propto 1/\sqrt{N_{d}}$. This is
despite the fact that only in the asymptotic SD case are relative
errors quantitatively given by the Poisson limit $\Delta p_{i}/p_{i}
\approx 1/\sqrt{(f_{i} N_{d})}$. This is because when mixtures overlap
and are in the hybrid/WD regimes, results are driven by the
distribution tails, where the effective number of data points is still
$N_{\rm eff} \propto N_{\rm d}$.
For $s$ and $\mu_1$, however, MCMC results
show better improvements in the WD regime than factors of
$\sqrt{10}$. This might be due to reduced parameter degeneracies from the
larger number of data points. Varying $f_1$ to 0.3 and 0.5 mildly impacts the
results. As the $f_1$ goes closer to 0.5, constraints improve for most
parameters, except for $f_1$ and $\sigma_2$. Constraints on $f_1$ are
almost unchanged while constraints on $\sigma_2$ are degraded, because
fewer data points are available in the second component to
constrain $\sigma_2$.
Based on Fig. \ref{fig:separation} and \ref{fig:width}, we can already
anticipate the constraints from Astro-H: when there is significant bulk flow and the modes
have a large relative velocity $v_{\rm bulk} > \sigma_{\rm turb}$, parameters can be constrained to $\sim 1\%$ accuracy (SD regime); when
the relative velocity is small but the widths are different by a
reasonable (a few tens of percents) amount, the parameter estimates are accurate at the $\sim 10\%$ level (WD regime). Given the modeling uncertainties in the physical interpretation of these parameter estimates, such accuracy is more than adequate. Next, we will consider two specific
examples of the SD regime and WD regime respectively.
\subsection{Application to Clusters: the Single Line Scenario}
\label{subsec:case1}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{100mm}{!}{\includegraphics{f10.eps}}}
\end{tabular}
\caption{The mock spectra (data points) and best-fit models for the WD
(upper panel) and SD (lower panel) cases in the single line
scenario. The red solid and dot-dashed lines are the input overall spectra and individual components respectively. The green dashed lines are the recovered components. The recovery is remarkably accurate, even when (as in the top panel) the spectra is visually indistinguishable from a single component Gaussian.}
\label{fig:spec_single}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{100mm}{!}{\includegraphics{f11.eps}}}
\end{tabular}
\caption{Same as Fig. \ref{fig:spec_single} but for the entire iron line
complex. The continuum has also been included, assuming a
metallicity of 0.3 ${\rm Z_{\odot}}$.}
\label{fig:spec_multiple}
\end{figure}
\begin{table*}
\caption{Input parameters for the WD case and recovered
best-fit parameters together with their 1-$\sigma$ errors. Also
shown are the predicted uncertainties using the Fisher matrix technique. Note that $v_{\rm pec},v_{\rm rel}$ are line of sight quantities, while $v_{\rm tb,1}, v_{\rm tb,2}$ are 3D velocity dispersions (assuming $v_{\rm 3D}^{2}=3 v_{\rm 1D}^{2}$).}
\label{tbl:para_wd}
\begin{center}
\begin{tabular}{c|l| c c c c c}
\hline \hline
&&$f_1$&$v_{pec}$(km/s)&$v_{rel}$(km/s)&$v_{tb,1}$(km/s)&$v_{tb,2}$(km/s)\\
&Input & 0.4 & 0 & 100 & 150 & 300 \\
\hline
\multirow{2}{*}{Single line}&MCMC & $0.46_{-0.12}^{+0.18}$ & $11.34_{-11.32}^{+11.62}$ & $89.48_{-13.85}^{+28.06}$ & $164.89_{-33.97}^{+33.75}$ & $298.82_{-13.34}^{+11.43}$ \\
&Fisher Matrix & (0.12) & (12.99) & (14.06) & (37.03) & (10.75)\\
\hline
\multirow{2}{*}{Multiple lines}&MCMC & $0.42_{-0.09}^{+0.25}$ & $-14.24_{-9.79}^{+17.15}$ & $130.72_{-11.19}^{+65.05}$ & $145.48_{-33.65}^{+42.59}$ & $291.64_{-34.19}^{+5.40}$ \\
&Fisher Matrix & (0.12) & (11.28) & (17.37) & (36.75) & (10.65)\\
\hline
\multirow{1}{*}{Multiple lines}&MCMC & $0.64_{-0.13}^{+0.22}$ & $5.44_{-9.08}^{+18.10}$ & $147.88_{-34.85}^{+124.25}$ & $180.16_{-21.43}^{+26.58}$ & $308.14_{-83.92}^{+13.44}$ \\
plus continuum&Fisher Matrix & (0.19) & (15.36) & (27.04) & (53.07) & (17.19)\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\caption{Same as Table \ref{tbl:para_wd} but for the SD case.}
\label{tbl:para_sd}
\begin{center}
\begin{tabular}{c|l| c c c c c}
\hline \hline
&&$f_1$&$v_{pec}$(km/s)&$v_{rel}$(km/s)&$v_{tb,1}$(km/s)&$v_{tb,2}$(km/s)\\
&Input & 0.4 & 0 & 500 & 150 & 300 \\
\hline
\multirow{2}{*}{Single line}&MCMC & $0.40_{-0.02}^{+0.01}$ & $2.17_{-6.40}^{+7.28}$ & $493.65_{-4.82}^{+4.54}$ & $152.71_{-5.93}^{+7.62}$ & $310.98_{-5.33}^{+6.65}$ \\
&Fisher Matrix & (0.01) & (6.40) & (4.52) & (11.45) & (9.77)\\
\hline
\multirow{2}{*}{Multiple lines}&MCMC & $0.41_{-0.01}^{+0.01}$ & $2.47_{-4.72}^{+6.18}$ & $505.28_{-4.13}^{+3.96}$ & $148.71_{-9.61}^{+14.19}$ & $294.13_{-9.43}^{+8.84}$ \\
&Fisher Matrix & (0.01) & (5.73) & (4.15) & (12.39) & (9.24)\\
\hline
\multirow{1}{*}{Multiple lines}&MCMC & $0.40_{-0.02}^{+0.02}$ & $-0.01_{-8.19}^{+7.88}$ & $486.43_{-4.98}^{+4.90}$ & $167.92_{-16.89}^{+15.33}$ & $309.75_{-14.03}^{+14.04}$ \\
plus continuum&Fisher Matrix & (0.02) & (6.72) & (4.61) & (15.75) & (12.50)\\
\hline
\end{tabular}
\end{center}
\end{table*}
We begin our discussion of mixture modeling of cluster emission line spectra with the simplest case. For now we ignore
line blending and continuum emission, and only consider one emission
line -- the He-like iron line at 6.7 keV. Again, we assume the PDF is
composed of two Gaussian components. Most of these assumptions will be
relaxed later. We assume the cluster is isothermal with a temperature
of 5 keV. The assumption of an isothermal
distribution is of course somewhat crude for the entire cluster. However,
for nearby clusters, the emission-weighted spectrum is accumulated from a small
area where temperature variations are generally mild ($< 0.5$ keV).
Moreover,
our results are not very sensitive to the
temperature distribution.
We express our results in terms of the bulk peculiar velocity of
the first component ($v_{pec}$), the relative velocity
between the two components ($v_{rel}$), and the 3D turbulent velocity dispersions of each component
($v_{\rm tb,1}$ and $v_{\rm tb,2}$). We assume isotropic turbulence,
so the line of sight velocity dispersion is $v_{\rm
tb,i}/\sqrt{3}$. They are related to the Gaussian PDF via:
\begin{eqnarray}
\mu_1&=&\nu_0+\nu_0\frac{v_{pec}}{c},\\\nonumber
s&=&\nu_0\frac{v_{rel}}{c},\\\nonumber
\sigma_i&=&\sqrt{\sigma_{tb,i}^2+\sigma_{ther}^2+\sigma_{instr}^2},\\\nonumber
\sigma_{tb,i}&=&\nu_0\frac{v_{tb,i}}{\sqrt{3}c},\\\nonumber
\sigma_{ther}&=&\frac{\nu_0}{c}\sqrt{\frac{kT}{Am_p}},\\\nonumber
\label{eqn:profile}
\end{eqnarray}
where $\nu_0$ is the line frequency in the rest frame,
$\sigma_{instr}$ is the standard deviation of instrumental noise
(FWHM/2.35), $A$ is the atomic weight of iron and $m_p$ is the proton mass.
In our WD
example, we assume $(v_{\rm tb,1},v_{\rm tb,2})=(150, 300) \, {\rm km
\, s^{-1}}$, and $v_{\rm rel}=100 \, {\rm km \, s^{-1}}$. For the SD
example, we assume the same $v_{\rm tb,1},v_{\rm tb,2}$, but $v_{\rm
rel}=500 \, {\rm km \, s^{-1}}$. In all cases, we assume the bulk
velocity zero-point $v_{\rm pec}=0 \, {\rm km \,
s^{-1}}$.\footnote{Note that if the redshift of the collisionless
component of the cluster (which does not participate in gas bulk
motions) can be determined to high accuracy by spectroscopy of
numerous galaxies, then $v_{\rm pec}, v_{\rm rel}-v_{\rm pec}$ give the
line of sight bulk velocities with respect to the cluster potential
well. For instance, for nearby clusters where $N_{\rm
gal} \sim 400$ galaxy redshifts have been measured,
the relative error in the center of mass redshift is $\sim 1000 {\rm km \, s^{-1}}/(\sqrt{3}\sqrt{N_{\rm gal}}) \sim 30 {\rm km \, s^{-1}}$.
Otherwise, only $v_{\rm rel}$ (the relative bulk velocity) is of physical significance.}
Sloshing in the cluster potential well generally results bulk motions
with transonic Mach numbers \citep{markevitch07}, so such a value is
realistic for a 5 keV cluster (with sound speed $c_{\rm s} \sim 1000
\, {\rm km \, s^{-1}}$) along an arbitrary line of sight---indeed,
such velocities are found in the simulated cluster in
\S\ref{subsec:example}.
With these assumptions, the widths of the first
and second component, including instrumental, thermal and turbulent broadening, are 3.54 and 4.87 eV; the offsets
between peaks are 1.12 and 11.17 eV for the WD and SD cases, respectively. These parameter choices correspond to $s/(\sigma_{1}+\sigma_{2})=(0.13,1.3)$ respectively, and thus can be compared to expectations from Fig. \ref{fig:separation}. Note that the SD case is not quite in the asymptotic regime $s/(\sigma_{1}+\sigma_{2}) ~\rlap{$>$}{\lower 1.0ex\hbox{$\sim$}} 3$ yet (where the relative errors would be $\sim 1/\sqrt{f_{i} N_{d}} \sim 1\%$), but it is fairly close.
The mock spectra and best-fit models for $10^4$ photons are shown in
Fig. \ref{fig:spec_single}. The best-fit parameters and their
uncertainties are listed in the first row of Table \ref{tbl:para_wd} and
\ref{tbl:para_sd}. In accordance with expectations from \S\ref{subsec:general}, component recovery is remarkably accurate. Even is the WD case, which is visually indistinguishable from a single Gaussian (see top panel of Fig. \ref{fig:spec_single}), the decomposition into the original mixtures is very good, and most velocities are constrained to within $\sim 10-30 \, {\rm km \, s^{-1}}$, which is significantly higher accuracy than needed to model the physical effects of bulk motions and turbulence in the cluster. This showcases the great
potential of high spectral resolution instruments. Of particular interest is the constraint on the mixing fraction, which
is a very good indicator of our ability to separate different
components. A confident detection of multiple components should have
$f_1/\Delta(f_1)$ larger than a few, i.e., the best-fit fraction should be at
least a few $\sigma$ away from ``non-detection'' ($f_1$ or $f_2$ equal
to 0). In the single line scenario, the $1-\sigma$ error of $f_1$ is 0.01 and
0.12 in the SD and WD cases, respectively, consistent with our
expectations from discussions in the previous section. However, a large
fraction of the constraints in the WD case is from the tails, and could
easily be affected by continuum emission (see discussion
below). Also note the general consistency between Fisher matrix and MCMC
techniques, indicating the Gaussian shape of the likelihood
surface for this scenario.
\subsection{The Impact of Multiple Lines and Continuum}
\label{subsec:case2}
In this section, we consider the impact of multiple lines and
continuum emission. Iron lines appear as a line complex between 6.6
and 6.75 keV, and these lines inevitably blend together. Multiple
lines have two competing effects. First, taking all lines into account--all of which have identical mixture decompositions--means more photons, which reduces shot noise in parameter estimates. The photons from the entire line complex is about twice that from the He-like iron line alone. Secondly, as different
lines blend together, information contained in the shape of individual
lines is partly lost due to blending in the line wings. The latter are crucial to driving parameter estimation in the hybrid and WD cases (note however, from Fig. \ref{fig:spec_multiple} that the lowest and highest energy lines in the complex have low/high energy line wings respectively which are unaffected by blending. This is particularly important in the case of the high energy He-like line, which is by far the strongest line in the complex). These two factors have opposite effects on the
constraints. As in the previous sub-section,
we run MCMC chains and Fisher matrices to estimate the constraints.
The properties of the line complex were taken from ATOMDB database\footnote{http://www.atomdb.org/}
(v. 2.0.1). To save computing time, we only included the ten strongest lines
lines. Fisher matrix estimates including more lines show negligible
difference.
The results are listed in the second row of Table \ref{tbl:para_wd} and
\ref{tbl:para_sd}. In the SD case, the constraints estimated using both
MCMC and Fisher matrix techniques are very close to those in the
single line scenario, indicating almost total cancellation between
the effects just mentioned. In the WD case, the constraints from the
Fisher matrix technique are again close to the single line
scenario. However, the results from MCMC runs show asymmetry, and in
general, the constraints are worse than in the single line scenario. Line
blending seems to make the likelihood surface significantly non-Gaussian.
Next, we include the effect of continuum emission. Continuum acts as a source of background noise. Even though we can measure and subtract the continuum, doing so introduces shot noise, particularly in the line wings when Fe line emission and continuum brightness can become comparable, or continuum emission could even dominate. The relative level of
continuum and Fe line emission is controlled by metallicity; larger metallicities imply brighter lines. The mean
metallicity of clusters is typically ${\rm Z} \sim 0.3 \, {\rm Z_{\odot}}$, which we shall assume, though the metallicity in the cluster center is often higher due to contributions from the cD galaxy. We apply our mixture model incorporating both the effects of line blending and continuum; the results are shown in Table \ref{tbl:para_wd} and
\ref{tbl:para_sd}, and in Fig. \ref{fig:spec_multiple} (for
the purpose of clarity, only 1/3 of the data points are shown in this figure).
The results are as one might expect. In the SD case, the constraints are only slightly worsened, since the mixtures
are clearly separated, and almost all the $\sim f_{i} N_{d}$ points in a given mixture can be used for parameter estimates; only a small fraction in the line tails are contaminated by line blending and the continuum. The constraints in the WD case are more badly affected, since the constraints in this case are largely drawn from the tails; here, the
differences between the MCMC and Fisher matrix techniques are also further
enlarged.
The presence of the continuum and line blending limit the
domain of the WD regime, which is no longer strictly independent of
$s/(\sigma_{1}+\sigma_{2})$. For instance, if we assume $v_{\rm rel} =
50 \, {\rm km \, s^{-1}}$ (corresponding to
$s/(\sigma_{1}+\sigma_{2})= 0.067$), the MCMC simulations fail to
converge). They thus limit our ability to constrain components with
small separations, though in practice such small separations should be
rare.
\subsection{Model Selection: When is a Mixture Model fit Justified?}
\label{subsec:prospects}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{60mm}{!}{\includegraphics{f12.eps}}}
\end{tabular}
\caption{Model selection: regions of the $f_1$ - $v_{rel}$ plane where the double component
model is preferred according to the BIC, for $v_{\rm tb,1}=100 \, {\rm km \, s^{-1}}$, $v_{\rm tb,2}=(200,300,400) \, {\rm km \, s^{-1}}$, and $f_{1}=0.4$, $v_{\rm pec}=0 \, {\rm km \, s^{-1}}$. }
\label{fig:select1}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{60mm}{!}{\includegraphics{f13.eps}}}
\end{tabular}
\caption{Model selection: shaded regions shows the regions of $(v_{\rm rel},v_{\rm tb,2})$ parameter space where the double
component model is preferred according to BIC, while the hatched
regions are where mixing fraction is accurately constrained: $\Delta(f_{1}) < (0.2,0.1)$ (cyan and purple hatches respectively). All other parameters are as in Fig. \ref{fig:select1}.}
\label{fig:select2}
\end{figure}
Thus far, we have only considered how accurately mixture model parameters can be constrained. However, this begs the question of whether a mixture model approach is justified at all, particularly when (as in the WD case) the observed emission line is visually indistinguishable from a single Gaussian. Introducing additional parameters will always result in an improved fit, even when these parameters are largely irrelevant and of little physical significance. This is essentially a model selection problem. We use information criteria (e.g., see \citet{liddle04}) which penalize models with more parameters, to identify preferred models. While they have solid underpinnings in statistical theory, fortunately, they have very simple analytic expressions. In this paper, we use the Bayesian Information Criterion (BIC; \citet{Schwarz1978}):
\begin{eqnarray}
BIC\equiv -2 \ln{\cal L}_\mathrm{max}+k\ln N
\label{eqn:bic}
\end{eqnarray}
where ${\cal L}_\mathrm{max}$ is the maximum likelihood achievable by
the model, $k$ is the number of free parameters, and $N$ is the number
of data points; the preferred model is one which minimizes BIC. The BIC comes from the Bayes factor \citep{jeffreys61}, which gives the posterior odds of one model against another. We use it over the closely related Akaike Information Criterion (AIC; \citet{akaike74}), which places a lower penalty on additional model parameters. Thus, we adopt a conservative criterion for preferring mixture models. The absolute value of the BIC has no significance, only the relative value between models. A difference of 2 is regarded as positive evidence, and of 6 or more as strong evidence, to prefer the model with lower BIC \citep{jeffreys61,mukherjee98}. Note that the BIC does not incorporate prior information. This is possible with the more sophisticated notion of Bayesian evidence (e.g., \citet{mackay03}), but involves expensive integrals over likelihood space, and is unnecessary in our case since we adopt uninformative priors.
We aim to distinguish the double component model with $k=5$ (free
parameters: $(v_{\rm pec}, v_{\rm rel}, f_{1}, v_{\rm tb,1}, v_{\rm
tb,2}$), against the single component model with $k=2$ (free
parameters: $\mu, \sigma$). We create simulated data sets which have
two underlying components, and see which regions of parameter space
the BIC will correctly prefer the two component model.
Our simulated line profiles incorporate the additional effects of
thermal and instrumental broadening, continuum, and line blending.
Rather than exploring the full 5 dimensional space, we explore the most interesting subspace to see where model selection is effective. In Fig. \ref{fig:select1} we explore model selection in the $f_1$ - $v_{\rm rel}$ plane, for $v_{\rm tb,2}=(200,300,400) \, {\rm km \, s^{-1}}$, and $v_{\rm pec}=0 \, {\rm km \, s^{-1}}$, $f_{1}=0.4$, $v_{\rm tb,1}=100 \, {\rm km \, s^{-1}}$. The plot shows where the BIC for the double component fit is smaller than that for the single component fit (note that the BIC is obtained by allowing for variation in all fitted parameters; we are just plotting model selection in a subspace).
When $v_{\rm tb,2}=400 \, {\rm km \, s^{-1}}$, all values of $v_{\rm rel}$ and all $0.1 < f_{1} < 0.9$ permit correct selection of the double component model. The result is very similar for $v_{\rm tb,2}=300 \, {\rm km \, s^{-1}}$, but for $v_{\rm tb,2}=200 \, {\rm km \, s^{-1}}$, if both $f_{1}, v_{\rm rel}$ assume low values, the double component model is not preferred. Overall, it is reassuring to see that model selection is not very sensitive to $f_{1}$, since we previously restricted our studies to $f_{1}=0.4$. Thus, even if a smaller fraction of the emission weighted volume has a markedly different velocity structure, it will be detectable in the spectrum. In Fig. \ref{fig:select2}, we show the regions of $(v_{\rm rel},v_{\rm tb,2})$ parameter space where the double
component model is preferred according to BIC. Overall, as expected,
the mixtures can be distinguished if $v_{\rm rel}$ or $v_{\rm tb,2}$
are large; for Astro-H and with the adopted parameters, this is of
order $200 \, {\rm km \, s^{-1}}$. In addition, we show the regions
where the mixing fraction $f_{1}$ is accurately constrained to
$\Delta (f_{1}) < (0.1,0.2)$, since the error on the mixing fraction
should be a good indicator of our ability to distinguish
mixtures. We use the Fisher matrix formalism to calculate these
constraints. The results are qualitatively similar that obtained
with the BIC, though somewhat more restrictive.
\subsection{Non-Gaussian Mixture Components}
\label{subsec: nongaussianity}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{100mm}{!}{\includegraphics{f14.eps}}}
\end{tabular}
\caption{The velocity PDFs in the WD (upper panel) and SD (lower
panel) cases. The solid (red) and dashed (green) curves are the input and
recovered PDFs, respectively. The thick curves are the overall PDFs,
while the thin curves show the individual components. }
\label{fig:nongaus}
\end{figure}
\begin{table*}
\caption{Non-Gaussian mixture components: input parameters (obtained by shifting and rescaling the non-Gaussian mixtures) and recovered
best-fit parameters together with their 1-$\sigma$ errors, for both the WD and SD cases, as in Fig. \ref{fig:nongaus}.
Note that the Fisher matrix results--which require an analytic likelihood--assume Gaussian mixtures, and hence are the same as Tables \ref{tbl:para_wd} and \ref{tbl:para_sd}.}
\label{tbl:nongaus}
\begin{center}
\begin{tabular}{c|l| c c c c c}
\hline \hline
&&$f_1$&$v_{pec}$(km/s)&$v_{rel}$(km/s)&$v_{tb,1}$(km/s)&$v_{tb,2}$(km/s)\\
\hline
\multirow{3}{*}{WD}&Input & 0.4 & 0 & 100 & 150 & 300 \\
&MCMC & $0.17_{-0.01}^{+0.49}$ & $-33.18_{-11.54}^{+37.53}$ & $112.49_{-10.96}^{+89.48}$ & $63.80_{-14.55}^{+138.51}$ & $284.99_{-39.82}^{+4.86}$ \\
&Fisher Matrix & (0.19) & (15.36) & (27.04) & (53.07) & (17.19)\\
\hline
\multirow{3}{*}{SD}&Input & 0.4 & 0 & 500 & 150 & 300 \\
&MCMC & $0.43_{-0.02}^{+0.01}$ & $8.37_{-7.44}^{+5.83}$ & $509.22_{-4.49}^{+4.49}$ & $162.28_{-17.94}^{+12.09}$ & $284.55_{-10.77}^{+13.81}$ \\
&Fisher Matrix & (0.02) & (6.72) & (4.61) & (15.75) & (12.50)\\
\hline
\end{tabular}
\end{center}
\end{table*}
All the preceding discussions are based on the assumption that the
PDFs of individual components are Gaussian, which is not true in general. As we see in
Fig. \ref{fig:spectrum}, individual mixtures show deviations from Gaussianity, i.e. Gaussians are a good but imperfect set of basis functions. In principle, this can be dealt with by fitting higher order mixture models, but in practice the data quality from Astro-H does not allow this; parameter estimation becomes unstable and large degeneracies develop, particularly since the higher order mixtures generally have low mixing fractions $f_{i}$. Unless their velocity means or widths are very different, the physical interpretation of these additional components is also more difficult. Here we construct a simple toy model to isolate the effects of non-Gaussian components. As there are many flavors of non-Gaussianity, the results we show are meant to be illustrative rather than definitive.
To this end, we extract PDFs from a simulated
relaxed cluster, use them as the ``basis'' PDFs of individual components, resize
and combine them to produce a composite PDF, which is in turn used to
generate mock spectra. The ``basis'' PDFs are extracted from a simulation
by \citet{Vazza2010}, which the authors kindly made public;
we sample different PDFs by looking along different lines of sight.
The cluster, labeled as E14, has a mass of ${\rm M} \sim
10^{15}~\rm M_{\odot}$ and experienced its latest major merger at
$z>1$. Due to shot noise which arises from the finite resolution (25
${\rm kpc} \, h^{-1}$) of this simulation--which results in a small
number of cells--we are
forced to extract the emission weighted velocity PDFs from a large volume of
$400\times 400 \times 1000~{\rm kpc}^3$.
The PDFs are
shifted (to match means), linearly rescaled (to match variances) and combined to produce the same WD and SD cases in
\S~\ref{subsec:case2}.\footnote{We emphasize that this procedure is {\it not} meant to simulate what an realistic observation would see, which we treat in \S\ref{sec:application}. It is a toy model in the spirit of the preceding sub-sections, where we use simulations to generate non-Gaussian mixture components.}
We then convolve the composite PDFs with thermal
broadening and instrumental noise for the entire Fe line complex, and
add continuum
to produce mock spectra. Finally, we
fit the mock spectra
to separate and constrain the two components. The results are shown in
Fig. \ref{fig:nongaus} and Table \ref{tbl:nongaus}. In
Fig. \ref{fig:nongaus}, the solid (red) curves are the input PDFs
while the dashed (green) curves are the best-fit model. The thick and
thin curves are the total PDF and individual components,
respectively
(note that because we display the velocity PDF rather than the
spectrum, the multiple lines in the Fe complex, as well as the
continuum and thermal/instrumental broadening, are not shown. However,
all these effects are included in the simulations). In the SD case
(lower panel), the two
components are recovered almost perfectly. In the WD case (upper
panel), however, there are some discrepancies between the input and
output PDFs. The same conclusion can be drawn from Table
\ref{tbl:nongaus}; in the WD case, the best-fit values of $f_1$
and $v_{tb,1}$ are somewhat different from the input values. However,
they are still within the (large) errors. Comparing Table \ref{tbl:nongaus} with Table
\ref{tbl:para_wd} and \ref{tbl:para_sd}, we see that at least in this case,
non-Gaussian components have limited effect on the results. Note the
strong discrepancy between MCMC and Fisher matrix error bars in both
cases, and in particular the strong asymmetry in MCMC errors.
We repeated the same exercise several times with PDFs randomly drawn
along different lines of sight
from the same
simulation. In most attempts, we are able to recover the input
parameter values within the uncertainties. Thus, conclusions
based on Gaussian components are still applicable when the true
PDFs deviate from Gaussianity by a reasonable amount.
Instrumental and thermal broadening, which {\it gaussian}, effectively
smooth out small scale deviations from Gaussianity.
\section{Results from Numerical Simulations}
\label{sec:application}
\subsection{Cold Front Cluster}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{100mm}{!}{\includegraphics{f15.eps}}}
\end{tabular}
\caption{``Cold front'' cluster: the solid (red) curves are the same velocity PDFs as in
Fig. \ref{fig:spectrum}. The dashed (green) curves are the recovered
PDFs from the best-fit models, and the dotted (blue) curves are the
individual components. Numerical values of the fit parameters are in Table \ref{tbl:app1}.}
\label{fig:app1}
\end{figure}
\begin{table*}
\caption{``Cold front'' cluster: best-fit parameters and their uncertainties for the
PDFs in Fig. \ref{fig:app1}, obtained using the Enzo simulation described in \S\ref{subsec:example}. Case 1 and 2 are the top and bottom panels of Fig. \ref{fig:app1} respectively. The ``true values'' are
obtained by fitting the PDF directly,
while the recovered values are obtained from the mock spectrum of
$10^4$ photons, which includes line blending, thermal and instrumental broadening,
and continuum emission.}
\label{tbl:app1}
\begin{center}
\begin{tabular}{c | l | c c c c c}
\hline \hline
&&$f_1$&$v_{pec}$(km/s)&$v_{rel}$(km/s)&$v_{tb,1}$(km/s)&$v_{tb,2}$(km/s)\\
\hline
\multirow{2}{*}{Case 1}&True values & $0.40_{-0.01}^{+0.01}$ & $-359.53_{-1.93}^{+1.73}$ & $517.15_{-2.76}^{+2.24}$ & $155.00_{-2.68}^{+2.49}$ & $269.39_{-3.02}^{+4.04}$ \\
&Recovered& $0.43_{-0.01}^{+0.01}$ & $-348.67_{-6.00}^{+6.34}$ & $522.79_{-4.82}^{+3.31}$ & $165.52_{-16.27}^{+13.16}$ & $247.04_{-9.66}^{+12.70}$ \\
\hline
\multirow{2}{*}{Case 2}&True values& $0.78_{-0.02}^{+0.02}$ & $-40.49_{-1.52}^{+1.70}$ & $142.10_{-6.40}^{+11.28}$ & $128.32_{-2.01}^{+2.17}$ & $220.24_{-6.93}^{+4.22}$ \\
&Recovered& $0.80_{-0.26}^{+0.11}$ & $-37.14_{-13.42}^{+9.09}$ & $136.29_{-60.12}^{+78.13}$ & $131.28_{-38.74}^{+11.02}$ & $237.73_{-53.31}^{+23.30}$ \\
\hline
\end{tabular}
\end{center}
\end{table*}
Finally, we apply our tool to cluster simulations. In the first
example, we attempt to recover the velocity PDFs shown in
Fig. \ref{fig:spectrum}, which derives from a cosmological ENZO
simulation of a cold front cluster. These two cases, which come from
different lines of sight through the same cluster, correspond to
$s/(\sigma_{1}+\sigma_{2}) = {1.42}$ and
$s/(\sigma_{1}+\sigma_{2}) = {0.42}$
respectively, i.e. in the ``separation-driven'' and ``width-driven''
regimes. We first fit the PDFs with a mixture model when no sources of
noise or confusion are present, to derive the ``true'' parameter
values.
We then generate mock spectra by adding thermal and instrumental broadening,
continuum emission and line blending to the PDFs,
and
then apply mixture modeling to the results. The results are given in Fig. \ref{fig:app1} and Table \ref{tbl:app1}.
Overall, the results are very good. The best-fit models
successfully recover the general features of the PDFs, and accurate
parameter estimates with uncertainties which are consistent with our
estimates from the
toy models -- on the order of $\sim 10\%$ for the width-drive case
(case 2) and $\sim 1\%$ for the separation-driven case (case 1). No
systematic biases appear to be present. As we discussed in
\S\ref{section:physical_significance}, these parameters all have
physical significance: $v_{\rm tb,i}$ relates to the turbulent energy
density in each component, $f_{i}$ to the emission weighted volume
fraction of each component, and $v_{\rm rel}$ to the bulk velocity
shear between them. We also applied the single component model to the
same mock spectra, and compared the BIC values. In both cases, the double
component model is preferred (case 1: $BIC_{\rm double}-BIC_{\rm
single}= -1002$; case 2: $BIC_{\rm double}-BIC_{\rm single}= -10$).
\subsection{AGN Feedback Cluster}
\begin{figure}
\begin{tabular}{c}
\rotatebox{0}{\resizebox{85mm}{!}{\includegraphics{f16.eps}}}
\end{tabular}
\caption{Density map and velocity field on the $y-z$ plane. The size
of the figure is 1 Mpc; the bulk motion along the y-direction has
been substracted out.
}
\label{fig:agnmap}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{-90}{\resizebox{60mm}{!}{\includegraphics{f17.eps}}}
\end{tabular}
\caption{``AGN feedback'' cluster: the solid (red) curves are the velocity PDFs from the simulation. The dashed (green) curves are the recovered
PDFs from the best-fit models, and the dotted (blue) curves are the
individual components. Numerical values of fits are in Table \ref{tbl:app2}.}
\label{fig:agn}
\end{figure}
\begin{figure}
\begin{tabular}{c}
\rotatebox{0}{\resizebox{85mm}{!}{\includegraphics{f18.ps}}}
\end{tabular}
\caption{Error contours for the ``AGN feedback'' case: contours depict the
68\%, 95\% confidence levels for the marginalized distribution; the
shadings shows the mean likelihood of the samples; the solid and dashed curves
in the 1-D plots are the fully marginalized posterior and relative
mean likelihood of the samples,
respectively. The stars and vertical lines label the
positions of the true values.}
\label{fig:agn_2D}
\end{figure}
\begin{table*}
\caption{``AGN feedback'' cluster: best-fit parameters and their uncertainties for the
simulated PDF in Fig. \ref{fig:agn}. The ``true values'' are
obtained by fitting the PDF directly,
while the recovered values are obtained from the mock spectrum of
$10^5$ photons, which includes line blending, thermal broadening,
instrument noise, and continuum emission.}
\label{tbl:app2}
\begin{center}
\begin{tabular}{ l | c c c c c}
\hline \hline
&$f_1$&$v_{pec}$(km/s)&$v_{rel}$(km/s)&$v_{tb,1}$(km/s)&$v_{tb,2}$(km/s)\\
\hline
True values & $0.28_{-0.01}^{+0.01}$ & $-5.89_{-0.84}^{+0.78}$ & $2.91_{-1.72}^{+2.16}$ & $44.72_{-1.32}^{+1.83}$ & $240.07_{-1.89}^{+2.84}$ \\
Recovered& $0.30_{-0.01}^{+0.08}$ & $-1.58_{-1.96}^{+2.66}$ & $-4.38_{-4.98}^{+2.94}$ & $23.43_{-10.56}^{+50.80}$ & $246.80_{-3.42}^{+10.89}$ \\
\hline
\end{tabular}
\end{center}
\end{table*}
The second example is a FLASH simulation with static gravity and radiative cooling of a cluster with an AGN in
the center (hereafter denoted as ``AGN feedback''); a simulation
snapshot was kindly provided to us by Marcus Br\"{u}ggen. The
simulated cluster, meant to mimic Hydra A, is described in
\citet{bruggen07} and \citet{simionescu09}; numerous plots of the
velocity field can also be found in \citet{vazza12}. Here, we briefly
summarize some properties.
The box size was 1 Mpc and AMR resolution reached a peak of 0.5 kpc
in the center, and a maximum of (1,4,8) kpc outside (16,100,200) kpc
respectively. A bipolar jet 2 kpc in diameter with power $L_{\rm jet}
= 3 \times 10^{45} \, {\rm erg \, s^{-1}}$ was then introduced; for
the analyzed snapshot the bulk velocity along the jet is $\sim
1500-1800 \, {\rm km s^{-1}}$, and a $M\sim 1.3$ shock has been driven into the
surrounding ICM. The
AGN was also given a bulk velocity of $\sim670 \, {\rm km \, s^{-1}}$ along the direction of
(-1,1,0) relative to the ambient ICM, to mimic the observed offset
between the shock center and the AGN in Hydra A.
In Fig \ref{fig:agnmap}, we show a 1 Mpc size density and velocity field map on the y-z plane through the
center. The size of the figure is
1 Mpc. The large bulk velocity along the x-direction has been
subtracted from the figure.
The outflows from the
AGN stir
the gas in the central region ($\sim 300$ kpc in radius),
while the ambient gas
is left relatively quiescent\footnote{The
small velocity dispersion ($\sim 45 \, {\rm km \, s^{-1}}$) of the quiescent region in this example come from the fact that apart from AGN outburst, it is a relaxed cluster which has not experienced any recent major mergers. Note, however, that the initial conditions come from cosmological GADGET SPH simulations where the small scale gas motions may not have been fully resolved.}. The velocity field is predominantly radial outside 100 kpc (associated with jet expansion and the running shock), while it is close to isotropic within 100 kpc, indicating that instabilities have efficiently isotropized and distributed the jet power.\footnote{Note, however, as also discussed by \citet{vazza12}, that these simulations are purely hydrodynamic, and magnetohydrodynamic (MHD) effects can strongly affect fluid instabilities and energy transfer from AGN bubbles to the ICM \citep{ruszkowski07,dursi08,oneill09}. For instance, 3D MHD simulations of bipolar jets by \citet{oneill10} find to the contrary that jet energy is not efficiently distributed/isotropized, remaining instead near the jet/cocoon boundary.} In Fig. \ref{fig:agn}, we plot the
emission-weighted velocity PDF along the z-direction
inside an area of $1\times1~{\rm Mpc}^2$. The division between turbulent and quiescent gas shows
up in the velocity PDF as a double Gaussian distribution -- a narrow
Gaussian corresponding to the quiescent gas outside the core, and a broad
Gaussian corresponding to the turbulent gas in the center. This is an example of non-volume filling turbulence discussed in \S~\ref{subsec:others}.
Note that we have pessimistically chosen a viewing direction in which
there are no bulk motions (similar to the ``width driven'' case of the
preceding example). For other viewing angles, the jet expansion drives
bulk motions which result in two clear peaks in the spectrum (similar
to the preceding ``separation driven'' case).
The scales in this Hydra A example are so large that in this
particular instance, the velocity structure could be spatially
resolved by Astro-H. However, Hydra A is of course an extremely rare
and energetic outburst; for more typical jet luminosities of $L_{\rm
jet} \sim 10^{44} \, {\rm erg \, s^{-1}}$, the turbulently stirred
region will be at least a factor of $\sim 30^{1/3}\sim 3$ smaller or
$\sim 100$ kpc in size, and hence barely resolved by Astro-H. In this
instance, mixture modeling will still be required to uncover the
filling fraction of turbulence. Also, as previously discussed, MHD
simulations show that motions are not efficiently isotropized and
distributed within the region of influence of the AGN, so in reality
there could be small scale intermittency in turbulence which would be
spatially unresolved, but detectable with mixture modeling.
To approximate such situations, we analyze the spectrum with the
velocity PDF shown in Fig. \ref{fig:agn}, where the effects of line
blending, thermal broadening,
instrument noise, and continuum emission have been included. This
is a clear example of the ``width driven'' scenario, with
$\sigma_{1}/\sigma_{2} = {0.70}$.
We were unable to recover the velocity PDF from the mock spectrum with
$10^4$ photons. The estimated BIC values for the
single and double component models using the ``true values'' indeed
show that the single component model is preferred for $10^4$ photons
($BIC_{\rm double}-BIC_{\rm single}$=17).
However, with $10^5$ photons (which is for instance, possible for
Perseus; see Table \ref{tbl:clusters}), the two components could be
easily separated (in this case, $BIC_{\rm double}-BIC_{\rm single}$=-102). The
results are given in Table
\ref{tbl:app2}. Again, here the ``true values'' are obtained by fitting the
PDF directly, by generating a Monte-Carlo sample of $10^4$
photons. The corresponding 2-D error contours
and marginalized posterior are shown in Fig. \ref{fig:agn_2D}. Note
the firm lower limit of $\sim 30\%$ to the quiescent component; a
clear detection that turbulence is not volume-filling. The velocity
dispersion and hence the energy density in the turbulent component are
also accurately recovered.
\section{Conclusions}
\label{sec:conclusions}
Gas motions can have profound influence on many physical processes in
the ICM, but thus far we have lacked a direct measurement of
turbulence in clusters. Upcoming X-ray missions--in particular
Astro-H--are poised to change that, by directly measuring turbulent
broadening of spectral lines. Thus far, most work has focussed on how
gas motions can alter the mean and width of X-ray emission lines from
galaxy clusters. However, the detailed shape of the line profile has
valuable information beyond these first two moments. Exploiting the
line shape (and thus the high spectral resolution of upcoming missions
such as Astro-H) can in many cases ameliorate poor angular resolution
in inferring the 3D velocity field. The main point of this paper is
that the line-of sight velocity PDF can often be meaningfully
decomposed into multiple distinct and physically significant
components. The
separation is based on deviations of line profiles from a single
Gaussian shape, driven by either the difference in width (``width-driven'', WD) or
mean (``separation-driven'', SD) of the components. Such a mixture
decomposition yields {\it qualitatively} different results from a
single component fit, and the recovered mixture parameters have
physical significance. For instance, bulk flows and sloshing produce
components with offset means, while partial volume-filling turbulence
from AGN or galaxy stirring leads to components with different
widths. The offset between components allows us to measure gas bulk
motions and separate them from small-scale turbulence, while component
fractions and widths constrain the emission weighted volume and
turbulent energy density in each component. With the MCMC algorithm
and Fisher matrix techniques, we evaluate the
prospects of using Gaussian mixture models to separate and constrain different velocity modes in galaxy clusters
from the 6.7 keV Fe line complex. We found that with the $10^4$ photons (which is feasible for the $\sim 14$ nearest clusters; see Table \ref{tbl:clusters}), the components could be
constrained with $\sim 10$\% accuracy in WD cases, and $\sim 1$\%
accuracy in SD cases, in both toy models and simulations of clusters with cold fronts and AGN feedback respectively. Continuum emission degrades the constraints
in WD cases, while it has little impact on the SD cases. On the other hand, line blending appear to have little impact. We generally find
that Astro-H is effective in separating different components when
either the offset between the components or the width of one of the
components is larger than $\sim 200$ km/s. Using PDFs taken from
numerical simulations as ``basis'' functions, we find that reasonable deviations from Gaussianity in the mixture components do not affect our results. We also study error scalings and use information criteria to determine when a mixture model is preferred.
Many extensions of this method are possible. For instance: (i) It
would be interesting to compare the separation between bulk/turbulent
motions obtained from mock X-ray spectra by mixture modeling, with
algorithms for performing this separation for the full 3D velocity
field in numerical simulations (e.g., \citet{vazza12}), to see how
close the correspondence is. (ii) In this study, we have assumed that
due to Astro-H's poor spatial resolution, only line-of-sight
information about the velocity field is possible. In principle, it
should be possible also to obtain information about variation of the
velocity field in the plane of the sky. For nearby clusters such as
Perseus, it should be possible to examine the line shape as a function
of projected radial position to obtain a full 3D reconstruction of the
velocity field (a more detailed implementation of the suggestion by
\citet{zhuravleva12} to study the variation of line center and width
with projected radial position). It would be very interesting to study
the variation of mixture parameters as a function of position in
high-resolution simulations. Even for more distant clusters, a
coarse-grained tiling of the cluster should be possible. (iii) High
resolution X-ray imaging of cold-front clusters yield information
about density/temperature contact discontinuities in the plane of the
sky. This has already been used infer the presence of sloshing and
bulk motions, as well as physical properties of the ICM such as
viscosity and thermal conductivity. Combining information about the
density/temperature contact discontinuity in the plane of the sky with
the line of sight information obtained by mixture modeling could
enhance our understanding of gas sloshing in clusters, and give more
precise constraints on velocities. It would likewise be interesting to
employ mixture modeling on spectra of the violent merger clusters with
classic bow shocks.
More generally, mixture modeling of spectra should prove useful whenever there are good reasons to believe that there are multiple components to the thermal or velocity field, and/or the line profile shows significant deviations from Gaussianity. For instance, it might be fruitful to consider applications to the ISM (e.g., \citet{falgarone04,Lazarian2006}), or Ly$\alpha$ emission from galaxies (e.g., \citet{hansen06,dijkstra12}).
\vspace{-0.5\baselineskip}
\section*{Acknowledgments}
We thank Marcus Br\"{u}ggen for kindly providing a simulation snapshot
of AGN feedback from \citet{bruggen07}, the authors of
\citet{Vazza2010} for making their simulation data publicly available,
and Brendon Kelly, Chris Reynolds, Franco Vazza, Sebastian Heinz and
Fanesca Young for
helpful conversations or correspondence. We acknowledge NSF grant
AST0908480 for support. SPO thanks UCLA and KITP (supported in part by
the National Science Foundation under Grant No. NSF PHY05-51164) for
hospitality. We acknowledge the use of facilities at the Center for
Scientific Computing at the CNSI and MRL (supported by NSF MRSEC
(DMR-1121053) and NSF CNS-0960316).
|
|
\section{Case Study of Narrative Transitions}
In this section, we take clips from one data video as a case to illustrate the effect of transitions in connecting narratives and visualizations.
We highlight characteristic clips of these videos and mark transitions in \textbf{bold} texts.
This video is about global wealth inequality\cite{TheRulesOrg2013}.
It presents a series of data facts about the wealth comparison between the poor and the rich.
Most facts are linked through crafted transitions rather than simply fading in/off.
We introduce five clips (\autoref{fig:case-1}) in detail.
The story starts with a pie chart, which shows the population distribution (\autoref{fig:case-1}(a1)).
The red color represents the richest people, which is only 1\%,
while the pewter represents the other 99\% of the people.
Then, the pie chart \includegraphics[width=8.5em, trim=0 0.45em 0 0]{case-UpdatingContent-1.pdf} to present another data fact (\autoref{fig:case-1}(a2-a5)), where the thin red sector expands, while the pewter part reduces.
By establishing contrasts of wealth and popularity between two groups of people, this transition reveals that the 1\% richest people own much more than the rest.
Within the transition, the color encodes the data category (the richest people and the rest).
The color encoding also preserves the narrative information during the transitions of two pie charts to keep viewers oriented.
The following footage consists of two icons and texts (\autoref{fig:case-1}(b1)).
The scene shows the fact that 3 billion poorest people and 300 richest people own the same wealth.
Afterward, the narrator mentions that the number of people it takes to fill a mid-size commercial aircraft has more wealth than the combined populations of
four countries.
In the clip, an aircraft icon flies through the scene, and the previous pictograph quits the scene under the \textbf{Truck} transition (\autoref{fig:case-1}(b2-b4)).
The icon moves from left to right and guides the viewers' attention to the following world map.
After this contextual data fact, the topic turns from inequality over the population to that over regions (\autoref{fig:case-1}(b5)).
The aircraft in this clip is a \includegraphics[width=5em, trim=0 0.5em 0 0]{case-RSTGuide.pdf} that attracts viewers' attention and shifts the presented topic insensibly.
Then, the world map splits into two parts by two colors (\autoref{fig:case-1}(c1)).
Developed regions and less developed regions are encoded with red and blue, respectively.
These regions first \includegraphics[width=2.4em, trim=0 0.5em 0 0]{case-Splitting.pdf} from the whole world map and then \includegraphics[width=3em, trim=0 0.4em 0 0]{case-Merging.pdf} into two different circular areas (\autoref{fig:case-1}(c2-c4)).
Each circular area represents the total wealth of the merged regions.
The areas consist of a proportional area chart of global wealth measured by poor and rich regions.
The chart then \includegraphics[width=8.5em, trim=0 0.4em 0 0]{case-UpdatingContent-2.pdf} by \includegraphics[width=3.5em, trim=0 0.45em 0 0]{case-Scaling.pdf} to show the comparison of the rich regions and the poor regions over two hundred years (\autoref{fig:case-1}(d1-d5)).
During this process, the layout and the color of the two circles keep constant, whereas the sizes of the two circular areas change according to the data.
The change in sizes creates a prominent contrast on the wealth of the rich and poor regions.
\section{Discussion and Conclusion} \label{section:discussion-conclusion}
In this study, we investigate the taxonomy of transition designs in data videos.
First, we collect the dataset of 284 professional data videos with 3909 clips.
These videos cover various visual styles of visualization contents and diverse transition designs.
Based on a content analysis on these clips, we propose a taxonomy of narrative transitions in data videos.
With regard to the change of visual variables, we conduct a more in-depth analysis of data-driven narrative transitions that preserve narrative meanings of contents, namely, \textit{Preserving Guide} and \textit{Narrative Agent}.
The proposed taxonomy for narrative transitions takes a step in this direction and hopes to encourage future research.
First, following the taxonomy,
evaluations on specific transition designs are needed to assess the effectiveness of engagement and memorability of data videos.
Designers can take transition designs into consideration when building the attention cues of data videos, for example, highlighting data facts or guiding viewer's attention.
Second, our taxonomy provides a new way of inspecting the relationship between narrative transitions and story sequences in data videos.
Narrative transitions can not only highlight visual changes in presentations but also enrich narrations.
We plan to propose a comprehensive model that considers both narrative transitions and story sequences.
Third, our taxonomy can inspire the transition design for other genres of narrative visualizations, for example, animated long-form web articles or data-GIFs\cite{Shu2021}.
Finally, except for narrative, future work can consider other messages that transition can present, for example, visualization rhetoric\cite{Hullman2011} and acquire codes\cite{Byrne2015}.
\section{Design Suggestions} \label{section:discussion-design-suggestions}
Informed by the content analysis and taxonomy, we summarize a series of design suggestions for transition designs in data videos from the following three aspects, \textit{i.e.}, clip forms, content relation, and visualization types.
Given that transitions of the \textit{Refresh}, \textit{Halftime}, \textit{Camera Motion} category are not specialized for any specific content, including clip forms, content relation, and visualization types, we summarized available transitions of \textit{Preserving Guide} and \textit{Narrative Agent} in these three aspects in \autoref{fig:suggestions}.
The table presents available transition choices in different situations but with no effectiveness evaluation yet.
We take typical transitions as examples to explain our design suggestions.
\begin{figure*}[htb]
\centering
\includegraphics[width=\textwidth]{statistics-transitions.pdf}
\caption{
(a) Five transition categories loosely ordered by the relevance of data content between scenes.
(b) Result of our content analysis showing the statistics of visualization-to-visualization, visualization-to-non-visualization, non-visualization-to-visualization transitions in our collected videos.
}
\label{fig:statistics-transitions}
\end{figure*}
\subsection{Clip Form}
A data video has a series of clips, some of which are data clips, while others are not.
These clips have different forms, indicating that they have different styles in presenting video content.
For example, videos such as \textit{We're quitting smoking, so why is big tobacco booming?}\cite{Guardian2019} from \textit{The Guardian}, introduce social topics through an arranged multimedia sequence, which includes expert interviewing scenes and data visualizations.
By contrast, videos, such as \textit{Global Wealth Inequality}~\cite{TheRulesOrg2013} shown in \autoref{fig:case-1}, mainly consist of animated graphics.
Some videos, such as the famous \textit{Hans Rosling's 200 Countries, 200 Years, 4 Minutes -- The Joy of Stats}~\cite{BBC2010}, are a mixture of animated data visualizations and shot scenes of the interpreter.
The transition designs in different clip forms vary.
\textbf{Use shared visual variables to articulate the transitions between visualizations.}
Videos that have transition clips of only visualization contexts take a big proportion of 94\% (267/285) in our data video dataset.
Transitions between these clips are used to transform video content between different charts.
Staged animation in Heer and Robertson's work\cite{Heer2007} is suitable in this scenario.
In our taxonomy, this type of staged animation belongs to \textit{Preserving Guide} and \textit{Narrative Agent}.
Within these two transitions, those preserved visual variables encode the same narrative meanings of charts in the previous and coming scenes, thereby keeping the viewers oriented during the change of video context.
Transition designs between different visualizations can use \textit{RST Guide}, \textit{Staying Guide}, \textit{Updating Content}, etc., according to the forms of visualizations.
Suggestions on transitions between specific visualization types are discussed in detail in \autoref{section:discussion-vistype}.
\textbf{Build the connection between visualization and other video content through transitions.}
The transitions of \textit{Preserving Guide} and \textit{Narrative Agent} can fluently connect visualization contents with others in data videos (see as \autoref{fig:intro-example}(a)).
The transition designs of these clip forms can utilize the similar visual variables between animated icons and items of visualizations (e.g., the circular shape of icons and pie charts and the color of icons and specific encoding in charts).
These same or similar visual variables can create a bridge of non-data-driven animated icons and data-driven visualization items with a seamless meaning transformation between them.
Moreover, texts and numbers can be reused in the coming scenes as titles or labels of charts without an additional set-up animation (\autoref{fig:teaser}(a1-a3)).
\begin{figure*}[htb]
\centering
\includegraphics[width=0.88\textwidth]{suggestions.pdf}
\caption{
Available transitions in clip forms, content relations and visualization types.
}
\label{fig:suggestions}
\end{figure*}
\subsection{Content Relation}
Data videos are author-driven stories\cite{Segel2010}.
These videos have linear storylines presented in an arranged order of scenes.
That is, authors need to carefully consider the relationship between consecutive scenes and use elaborate transitions.
Based on our analysis, the specific transitions can help reveal the following content relations in data videos.
\textbf{Question \& Answer}:
\textbf{Reuse elements in question when answering.}
To lead viewers from the establisher and initial parts of a story to its peak gradually \cite{Amini2015}, designers can use attractive questions as a start and then use visualizations to answer the question by presenting data facts.
In this process, the contents of the questioning scene, such as texts, icons, or even question marks, can be used to create a \textit{Preserving Guide} transition.
For example, texts of the question can be used as an \textit{RST Guide}, and they can move to the visualization as a related annotation of charts.
By doing so, the question and related answers become a fluent sequence without an abrupt change.
\textbf{Whole and Part}:
\textbf{Group items, expand colors, or zoom.}
This content relation means presenting different granularities of data in contexts.
By grouping partial items together or splitting them, a \textit{Merging/Splitting} transition can reveal the affiliation of partial items and the whole entity (\autoref{fig:case-1}(c1-c5), (a1-a5)).
This content relation also includes topics and subtopics.
If related contents of subtopics exist in the scene of the whole topic, then designers can use \textit{Zooming} (\autoref{fig:teaser}(d1-e1)) or \textit{Expanding Guide} (\autoref{fig:teaser}(f1-f3)) to demonstrate the subtopics in detail after the presentation of the whole topic.
Those related contents (e.g., the red point in \autoref{fig:teaser}(d1), the blue sector in \autoref{fig:teaser}(f1)) are the medium of these types of transitions.
With a zoom-in camera motion or a background expanding animation, the meaning of a further introduction of this topic can be illustrated, especially the subtopic that the expanding color or the zooming focus represents.
\textbf{Progress} and \textbf{Supplement}:
\textbf{Present items one by one.}
Telling the story progressively is common in storytelling.
If contents are planned on a canvas layout, then designers can use a \textit{Pedestal/Truck} camera motion to present them individually (\autoref{fig:case-2}(b1-b5)).
If the canvas layout is not created for scenes, then designers can use \textit{Rack Focus} to blur the previous content and present the supplement overlaid on the blurred background (\autoref{fig:teaser}(l1-l3)).
The contents of the previous scene can be an \textit{RST Guide} to conduct a supplement in the coming scenes (\autoref{fig:case-2}(c1-c5, c5-c6))
The \textit{Morphing} transition can also be used in a progress narrative to present a changing process of contents (\autoref{fig:case-3}(c1-c5)).
\textbf{Contrast}:
\textbf{Preserve contexts for comparison.}
Contrasts among different data can play a role in strong arguments for narrative.
A contrast can be gradually created.
Accordingly, visualizations in the previous scenes can be regarded as a whole \textit{RST Guide}.
The whole part translates to a new position but is preserved in the scene while making space for other contents for contrast.
For example, a designer wants to present two pie charts about the sex ratio of two classes for contrast.
After presenting the pie chart of class A, she preserves the chart and lays it on the left side of the screen.
Then, she presents the pie chart of class B on the right side.
Moreover, designers can use \textit{Narrative Agent} transitions (e.g., \textit{Scaling}) to present the contrast of data in different situations by updating related video contents (\autoref{fig:case-1}(d1-d5)).
\subsection{Data Visualization Type} \label{section:discussion-vistype}
Our content analysis demonstrates design tactics for different visualization types.
We summarize these tactics on the basis of our data video dataset and taxonomy, including how to transform \textit{from} and \textit{to} them.
The frequency of transitions \textit{from} and \textit{to} specific visualizations are presented in our supplemental materials.
We first present the general transition designs for visualizations and then specific tactics for different visualization types orderly in the following.
If details in the visualization will be presented, then designers can use \textit{Camera Motions}, such as \textit{Pedestal/Truck} and \textit{Zooming}, to present them for demonstration.
Another way is to use \textit{Updating Content} to present these details in an arranged order.
\textit{Updating Content} can be used for presenting the update of data on the basis of the same form of charts.
Designers can make good use of the same elements in different scenes.
For example, the same color area can be used as an \textit{Expanding Guide}.
The annotation texts or icons can be used as a \textit{RST Guide}, and the same basic geometry shapes, such as circle and rectangle, can be used as \textit{Staying Guide}.
The individual visualization types have different visual components.
Accordingly, the transition designs of the varying visualization types are also distinguished according to these components.
We list the specific tactics for the following visualization types as a supplement for the general tactics mentioned above.
These types of visualization are commonly used in our data video dataset.
\textbf{Line Chart}, \textbf{Scatter Plot}, and \textbf{Bar Chart}:
These three types of charts have many necessary components, for example, axes, legends, label texts, as well as data-encoded lines, points, and bars.
Designers can take these components as a medium for the \textit{RST Guide} transition.
Besides, the axial component can also be a \textit{Staying Guide} when continuously presenting these three types of charts with different contents.
\textbf{Map}:
Maps can present spatial data.
The overview and detailed conditions of maps are essential for presenting data patterns.
\textit{Camera Motion} is a widely used transition to transform between different presenting levels or places on maps.
\textit{Zooming} can indicate the geographical information.
For instance, a particular condition of the city in a country can be shown by zooming from the whole map of the country into the city, and vice versa.
The horizontal or vertical movement of the camera, for example, \textit{Pedestal/Truck}, creates a feeling of space of the scene and reveals the relative localization of different regions.
The shape of a specific region on maps is a good choice for a \textit{RST Guide} if it is also shown in near scenes.
The color of one sub-region can be an \textit{Expanding Guide}, that is, expanded as a background color when presenting detailed data of this subregion.
\textbf{Proportional Area Chart}, \textbf{Pie Chart}, and \textbf{Donut Chart}:
These types of charts present the proportional relation of data.
The \textit{Merging} transition can gather parts that represent different categories of data from the previous scenes and combine them as a whole chart.
The \textit{Splitting} transition is on the opposite.
Varying colors are used to encode different proportional areas in these charts.
These areas can be well used to present the whole and part levels of information as the medium for the \textit{Expanding Guide} (\autoref{fig:case-3}(a1-a5)).
Designers can also use \textit{Scaling} and \textit{Morphing} transitions to present the change of proportional relation of data by revising the size or shape of changing parts.
\textbf{Diagram}:
Diagrams can present pipelines.
In contrast with statistical charts, this type of visualization presents the progress of a topic rather than the data values.
Designers can present each procedure through \textit{Pedestal/Truck} in sequence and use \textit{Zooming} to show the whole diagram at last.
The arrows between procedures can be used as the attention cues of the \textit{RST Guide} transition and link up the whole pipeline.
As for transforming from and to diagrams, icons, and texts in the procedure of a diagram can be used as the \textit{RST Guide}.
\textbf{Pictograph} and \textbf{Number, Icon and Text}:
These types of visualizations mainly consist of animated icons.
Such icons may have their animations or interactions with other elements in the scene.
Transitions from and to these visualizations can make good use of these icons, for example, taking them as the medium of the \textit{RST Guide} transition.
In that way, the rotating, scaling, and translating animations of icons are equal to the transition of scenes, thereby making the transformation seamless in the narrative.
\section{Related Work}
In this section, we introduce related work with regard to animated transitions and data videos.
Animated transitions are commonly used in visualization presentations.
Heer and Robertson\cite{Heer2007} examined the effect of animated transitions between statistical charts and proposed detailed designs concerning the congruence of contexts.
On the basis of an evaluation of those designs, they indicated that some animated transitions could keep viewers oriented.
This notion means that well-designed transitions between visualizations can improve the viewer's perception and cognition.
Researchers have recently conducted more specific work on visualization transition designs for different presentation tasks, such as the analogy between visualizations~\cite{Ruchikachorn2015}, aggregation operations\cite{Kim2019}, and visual grouping\cite{Chalbi2020}.
As for design details, Thompson et al.\cite{Thompson2020} proposed several dimensions of animated data graphics and classified transitions based on these dimensions.
Researchers also used animated transitions for data-driven storytelling.
In Segel and Heer's design space, as a type of visual narrative tactic, \textit{transition guidance} promotes the use of conventional methods to achieve the continuity among different shots to keep viewers oriented when old scenes destruct and new scenes come\cite{Segel2010}.
Animated transitions can be a type of transition guidance when telling stories.
Hullman et al.~\cite{Hullman2013} investigated the arrangements of narrative sequences based on transitions.
Furthermore, Kim et al.~\cite{Kim2017} developed GraphScape, a synthetic model to evaluate the transition cost of visualization sequences.
Researchers also use transitions to make stories cohesive in timelines~\cite{Brehmer2016} and slideshows~\cite{Wang2018}.
Amini et al.~\cite{Amini2017} include \textit{Transition} in the taxonomy of data clip types.
They expanded a few transition designs in their proposed authoring tool, DataClips, from and to pictographs based on Heer's work~\cite{Heer2007}.
However, the discussed transition designs in the previous work have not well covered all transitions that connect contexts in the narrative of data videos.
Data videos have a series of different forms, for example, animated graphics that consist of icons and characters, standard charts, pictographs.
Given that not only visualizations exist in the data videos, but also iconic motion graphics, transition designs between scenes have a large space.
In this paper, we expand the taxonomy to include specific designs and usage scenarios of transition clips.
Additionally, we pay attention to how they cement the context in the linear narrative in data videos.
\section{Taxonomy of Transitions} \label{section:taxonomy}
This section first presents an overview of our proposed taxonomy for narrative transitions in data videos and then introduces data-driven transitions in detail.
\subsection{Overview}
According to the definition of narrative transitions (Sec. 3.1), we examine the designs from two aspects, namely,
data-driven animated transitions (e.g., staged animations~\cite{Heer2007}, \autoref{fig:intro-example}(b), and \autoref{fig:case-1}(a)(c)(d)) and non-data-driven ones (e.g., fade-in/off, \autoref{fig:intro-example}(a), and \autoref{fig:case-1}(b)).
We list five identified transition types in an order of the relevance to data, where the first three pertain to non-data-driven transitions and the last two belong to data-driven ones.
The frequency of each transition type in the dataset (3909 clips) is also reported.
Note that multiple types can be combined together to establish a transition in a clip.
For example, \autoref{fig:case-1}(b) uses a camera motion and a preserving guide in the case.
We provide animated illustrations of these narrative transitions on the website: \url{https://narrativetransitions.github.io/home/}.
\textbf{Refresh} (53.0\%, 2070) means a complete update of the previous scene (\autoref{fig:taxonomy}(a)).
In this type of transitions, no connection exists between the last frame of the previous scene and the first frame of the coming scene.
Usually, this mechanism can be used to present a new topic or an abrupt turn.
We place the combinations of the destruction of previous scenes and creation of coming scenes in this category, for example,~\textit{Hard Cut},~\textit{Fade}, and \textit{Wipe}.
\textbf{Halftime} (2.5\%, 98) adds a new scene between video context (\autoref{fig:taxonomy}(b)).
Such a scene is comparable to a quick half time or the stage curtain between the previous scene and the coming scene.
\textbf{Camera Motion} (14.9\%, 583) updates the scene due to the changing viewpoints or screen focus.
The seven subtypes of camera motion are\cite{Rea2015,Storyblocks2019}: \textit{Pedestal}, \textit{Truck}, \textit{Tilt}, \textit{Pan}, \textit{Dolly}, \textit{Zoom}, and \textit{Rack Focus}.
\textit{Pedestal} means moving the camera vertically.
By contrast, \textit{Truck} means moving the camera horizontally.
\textit{Tilt} and \textit{Pan} also mean moving the camera in the vertical and horizontal direction, respectively.
However, they both require the camera to keep a stable focus during the movement.
\textit{Dolly} means moving the camera forward or backward.
\textit{Zoom} means changing the focal distance.
\textit{Rack Focus} means changing the focus in the scene, for example, bokeh effects.
Camera movements can create a sense of space over scenes, and usually, they are used to present spatial visualizations, such as maps.
The changes in the focus of scenes can highlight the key points of narrations according to the video designer's intention.
\textbf{Preserving Guide} (22.9\%, 894) reuses elements in the previous scene as a visual guide that directs to the next scene (\autoref{fig:taxonomy}(d)).
For example, a preserving guide could be a flying icon (\autoref{fig:case-1}(b)), a colored area, or a stable line, to lead the viewers' attention between two consecutive scenes.
This guide can be used with camera motions to construct fluid transformations (\autoref{fig:case-1}(b)).
\textbf{Narrative Agent} (19.9\%, 777) means substitutes for data during data-driven storytelling (\autoref{fig:taxonomy}(e)).
The transitioned elements could be regarded as agents of data attributes or data values.
This transition illustrates the change of data such as \textit{Scaling} and \textit{Merging}.
Different transition designs are useful to connect diverse narrative states.
For example, in bar charts, data-encoded bars can be used as \textit{Narrative Agents} to present the change of data.
Another example is that \textit{Camera Motion} can be used to transform among different places on maps because this transition can generate a sense of space.
\vspace{-5pt}
\begin{figure}[htb]
\setlength{\abovecaptionskip}{5pt}
\setlength{\belowcaptionskip}{-10pt}
\centering
\includegraphics[width=0.9\columnwidth]{taxonomy-grey-new.pdf}
\caption{
Taxonomy of transition designs in data videos (a-e).
We also present changed and unchanged visual variables of \textit{Preserving Guide} and \textit{Narrative Agent} transitions (f).
}
\label{fig:taxonomy}
\end{figure}
\begin{figure*}[htb]
\setlength{\abovecaptionskip}{3pt}
\setlength{\belowcaptionskip}{-12pt}
\centering
\includegraphics[width=0.85\textwidth]{case-1-new.pdf}
\caption{
Screenshots of the video \textit{Global Wealth Inequality}\cite{TheRulesOrg2013}.
We added green marks on the screenshots to show the animation between two consecutive scenes.
We presented the transition types, visual content and the corresponding transcripts (gray text) under each clip sequence.
}
\label{fig:case-1}
\end{figure*}
\subsection{Data-driven Narrative Transition}
Transitions, such as \textbf{Refresh}, \textbf{Halftime} and \textbf{Camera Motion}, can be used not only in data videos but also in other video narratives.
\textbf{Refresh} is the most popular transitions in the dataset.
This kind of transition has been integrated in existing video authoring tools~\cite{Premiere2020,AfterEffects2020} and can be easily employed in videos.
\textbf{Preserving Guide} and \textbf{Narrative Agent} need craft animations to conduct fluent narrative.
These two transitions are usually used in data videos to preserve data attributes or encoding data-driven insights, which present narrative information of the visualization content of data videos.
We paid special attention to \textbf{Preserving Guide} and \textbf{Narrative Agent} transitions to understand the data-driven design patterns of transitions in data videos.
They make up a large proportion of the studied clips,
especially in the 1644 vis-to-vis transitions, among which,
477 are \textbf{Preserving Guide} (29.0\%) and 617 are \textbf{Narrative Agent} (37.5\%).
Within these transitions, changed and unchanged visual elements both exist.
We focus on the visual contents within clips in terms of Bertin's variables of the image\cite{Bertin1983}.
Those changed variables consist of the animation of transitions, while unchanged ones preserve the narrative information to maintain the congruence of the previous scenes and coming scenes.
The narrative information could be data attributes and values preserved in special visual variables.
The constant variables, which are similar to data agents, help viewers understand what narrative is going on.
We list them in \autoref{fig:taxonomy}(f).
Note that these agent elements are not isolated from each other in practice.
In the following sections, we introduce the narrative transitions of data videos in detail.
Illustrations about each type of transition are presented in \autoref{fig:taxonomy}(d) and (e).
\subsubsection{Preserving Guide}
\includegraphics[width=16em, trim=0 0.45em 0 0]{RSTGuide.pdf}
A single or a group of visual items in the previous scene are kept and reused in the coming scenes.
However, the layout of these visual items, for instance, orientation, size, and position, may change.
These visual elements maintain the narrative information in previous scenes by preserving the same appearance, and at the same time, guide viewers' attention to a new scene by changing the layout.
This transition is particularly useful in setting up a supplement, correlation, or a comparison.
We abbreviate this transition as \textbf{RST Guide}.
\includegraphics[width=12.5em, trim=0 0.45em 0 0]{ESGuide.pdf}
This type means two related situations.
The first situation is to expand the color of a specific item in the previous scene to the background color of the coming scene.
Often, the coming scene introduces details about the item.
The other situation is to shrink the background color of the previous scene to a specific item of the coming scene.
This type of transition is usually used when presenting the relation between the whole and the parts.
Color is an essential visual variable in this type of transition: the same color of adjacent scenes illustrates the same subject.
\includegraphics[width=6.5em, trim=0 0.45em 0 0]{StayingGuide.pdf}
This type of transition means that the layout of the visual items in the previous scene remains the same, and additional items are interacting with the existed items.
The stable visual items are unchanged cue in the scene, and they can be regarded as the basis for the additional items entering, growing, and leaving.
\subsubsection{Narrative Agent}
\includegraphics[width=8em, trim=0 0.45em 0 0]{UpdatingContent.pdf}
This type of transition means updating the visual content without changing the shapes, positions, and colors of the contents in the scenes.
In this transition, the shape, position, and color maintain the narrative information; however, the number of the contents changes to show the differences.
Such a mechanism is used in data videos for highlighting or presenting data sequentially.
\includegraphics[width=3.7em, trim=0 0.45em 0 0]{Scaling.pdf}
These transitions change sizes to show the value change.
Although the size changes, the other visual variables remain the same, thereby making the contents coherent in different scenes.
\includegraphics[width=4.5em, trim=0 0.45em 0 0]{Morphing.pdf}
This type of transition mainly morphs icons or shapes from old ones to new ones.
This transition conveys insights into the transformation between data.
\includegraphics[width=8em, trim=0 0.45em 0 0]{MergingSplitting.pdf}
Merging means combining separate contents into a group, while splitting means separating a group of contents.
This type of transition describes insights into the gathering or scattering.
The whole-part relationship of data can be clearly illustrated by using this type of transition.
\section{Bibliography Instructions}
\section{Content Analysis}
We first state the definition of a transition in data videos in our research scope, followed by the adopted methodology.
\subsection{Definition}
Prior work has defined transitions in narrative visualizations\cite{Hullman2013}.
Hullman et al.~\cite{Hullman2013} consider the change between two independent visual expressions as a transition.
In the previous investigation of data videos, Amini et al. \cite{Amini2017} find that ``most transitions in these videos are a combination of destroy/create clips rather than staged animations described in~\cite{Heer2007}''.
However, after collecting and analyzing 284 data videos, we found many more different transition designs in data videos compared with the previous work.
For example, transitions in \autoref{fig:intro-example} connect the contents by using the shared visual elements of the successive scenes.
The design of using shared visual elements is not strictly the staged animation~\cite{Heer2007}, but they have similarities in reusing visual elements and keeping audience oriented in syntax and in semantic.
In our work, we follow the prior definition of transitions ~\cite{Heer2007,Hullman2013,Amini2017} and expand it in a wider scenario.
First, we identify \textit{clips} in data videos, where a clip is an elemental unit of the data video sequence~\cite{Amini2017}.
Each clip is considered containing two narrative states and one transition between these two states.
A \textit{narrative state} in a clip is defined as an informationally-distinct scene for presenting data facts or other video narrative, following the definition of \textit{narrative visualization state} ~\cite{Hullman2013}.
Therefore, we define the \textit{transition} (also called \textit{narrative transition}) in our paper as the change of two narrative states in a clip, where the narrative state can be
\textit{1)} an animated content without visualizations or data-driven arguments;
or \textit{2)} a visualization content that includes standard charts, pictographs, and other data-driven arguments.
\subsection{Methodology}
We conducted a content analysis~\cite{Krippendorff2018}, which was also used in previous work ~\cite{Segel2010,Hullman2011,Byrne2015} to gain a comprehensive understanding of the transition design in data videos.
First, we collected 284 data videos from reputable sources, such as the leading media outlets, design associations' portfolios, and video sites.
We gathered these videos following the same criteria in Amini's work \cite{Amini2017}.
The videos should convey data-driven arguments and contain related visualizations.
The wide range of topics in the videos includes science, finance, politics, sports, and history.
The video dataset includes animated motion graphics, photography videos, and combinations of them.
We took apart these videos into single clips.
We only focused on the clips that have at least one scene with visualizations or data-driven arguments.
Finally, our dataset contains 3909 clips: 1644 vis-to-vis clips (42.1\%), 1104 others-to-vis clips (28.2\%) and 1161 vis-to-others clips (29.7\%).
To examine the detailed transition designs in clips, we first reviewed selected samples and proposed an initial taxonomy considering the changing visual variables (e.g., position, color, and shape) ~\cite{Bertin1983} during the transition.
We iteratively improved the taxonomy through multiple rounds of discussion among authors and attempts to label sampled transitions.
Ultimately, we achieved an agreement on the final taxonomy (Sec. \ref{section:taxonomy}).
We mainly considered the following questions when analyzing the transition design:
\begin{compactitem}[$\bullet$]
\item What are the narrative states (e.g., visualizations, animated icons) of the transition?
\item What visual variables have changed, and what unchanged during the transition?
\item How are the transitions visually presented?
\item What is the narrative relationship between two states?
\end{compactitem}
Based on the taxonomy, two authors independently coded all the clips and reached an initial consensus on 87.7\% (3430) transitions.
The conflicts (12.3\%, 479) were further resolved through discussion.
Based on the agreement, we conducted a quantitative analysis on the visualization types and transition types in each clip.
The complete results are attached to the supplemental material.
\section{Use Case}
In this section, we demonstrate a use case (\autoref{fig:teaser}) on how to leverage our taxonomy to design a data video.
The video is about COVID-19, which introduces the severe condition of the COVID-19 spreading by presenting a series of data facts with elaborate transitions.
The data is collected from the official notification, and the whole video can be found in the supplementary materials.
First, the video begins with a series of texts that indicate that the video is about the COVID-19 outbreak (\autoref{fig:teaser}(a1)).
With the texts preserving the keywords, a \textit{RST Guide} transition connects the previous scene with a world map to show the data of the total cases in different regions over the world (\autoref{fig:teaser}(a2-a3)).
The color depth on the map represents the severity of the infection in the corresponding area, which becomes deeper over time in China.
A \textit{Zooming} transition brings the viewer into a detailed portrait of the total confirmed case data in China (\autoref{fig:teaser}(b1-b3)).
The number rapidly increases over time.
Then, texts of date and number serve as the medium of the \textit{RST Guide} transition (\autoref{fig:teaser}(c1)).
In combination with the transition, the number seamlessly transforms into one point of the line chart (\autoref{fig:teaser}(c2-d1)).
The line steeply grows, and a \textit{Zooming} transition enlarges the presentation of the highest points of the line chart (\autoref{fig:teaser}(d1-e1)).
Then, the time stops growing, and the \textit{Updating Content} transition is used to show the details of February 12 (\autoref{fig:teaser}(e1-e2)).
The point that represents the total cases on February 12 keeps its circular shape and transforms into a pie chart of the proportion of confirmed cases in Wuhan and other places in China (\autoref{fig:teaser}(e1)).
The proportion of total confirmed cases in Wuhan is over half of the total cases in China.
To show a further detail, an \textit{Expanding Guide} transition expands the color of the sector that represents Wuhan in the pie chart to the background of the scene, thereby indicating that the following scene is about the detailed condition of Wuhan (\autoref{fig:teaser}(f1-f3)).
A group of people icons represents the whole crowd of confirmed cases in Wuhan (\autoref{fig:teaser}(g1)).
The icons consist of a proportional pictograph by \textit{Splitting} into three groups to show the number of existing confirmed, cured, and dead cases (\autoref{fig:teaser}(g1-g3)).
Thereafter, the icons of each group are \textit{Merged} into a proportional area chart that represents the percentage of each group (\autoref{fig:teaser}(h1-i1)).
Then, a \textit{Shrinking Guide} transition transforms the proportional area chart into the previous pie chart (\autoref{fig:teaser}(i1-i2)).
Then a \textit{Zooming} transition guide viewers' sight to the world map and raises the statistical map of total cases over the world (\autoref{fig:teaser}(j1-j2)).
As time goes, by the \textit{Updating Content} transition, the global situation turned terrible (\autoref{fig:teaser}(k1-k3)).
Lastly, a \textit{Rack Focus} transition blurs the world map and shows a proposal that the whole world should unite together to defeat COVID-19 (\autoref{fig:teaser}(l1-l3)).
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