prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What molecule was the focus of the study?

Read the following text and answer briefly.

Context 1: \section{Introduction}

Spectral line surveys have revealed that high-mass star-forming
regions are rich reservoirs of molecules from simple diatomic species
to complex and larger molecules (e g ,
\citealt{schilke1997b,hatchell1998b,comito2005,bisschop2007}) 
However, there have been rarely studies undertaken to investigate the
chemical evolution during massive star formation from the earliest
evolutionary stages, i e , from High-Mass Starless Cores (HMSCs) and
High-Mass Cores with embedded low- to intermediate-mass protostars
destined to become massive stars, via High-Mass Protostellar Objects
(HMPOs) to the final stars that are able to produce Ultracompact H{\sc
  ii} regions (UCH{\sc ii}s, see \citealt{beuther2006b} for a recent
description of the evolutionary sequence)  The first two evolutionary
stages are found within so-called Infrared Dark Clouds (IRDCs)   While
for low-mass stars the chemical evolution from early molecular
freeze-out to more evolved protostellar cores is well studied (e g ,
\citealt{bergin1997,dutrey1997,pavlyuchenkov2006,joergensen2007}),
it is far from clear whether similar evolutionary patterns are present
during massive star formation 

To better understand the chemical evolution of high-mass star-forming
regions we initiated a program to investigate the chemical properties
from IRDCs to UCH{\sc ii}s from an observational and theoretical
perspective  We start with single-dish line surveys toward a large
sample obtaining their basic characteristics, and then perform
detailed studies of selected sources using interferometers on smaller
scales  These observations are accompanied by theoretical modeling of
the chemical processes   Long-term goals are the chemical
characterization of the evolutionary sequence in massive star
formation, the development of chemical clocks, and the identification
of molecules as astrophysical tools to study the physical processes
during different evolutionary stages  Here, we present an initial
study of the reactive radical ethynyl (C$_2$H) combining single-dish
and interferometer observations with chemical modeling   Although
C$_2$H was previously observed in low-mass cores and Photon Dominated
Regions (e g , \citealt{millar1984,jansen1995}), so far it was not
systematically investigated in the framework of high-mass star
formation 

\section{Observations}
\label{obs}

The 21 massive star-forming regions were observed with the Atacama
Pathfinder Experiment (APEX) in the 875\,$\mu$m window in fall 2006 
We observed 1\,GHz from 338 to 339\,GHz and 1\,GHz in the image
sideband from 349 to 350\,GHz   The spectral resolution was
0 1\,km\,s$^{-1}$, but we smoothed the data to
$\sim$0 9\,km\,s$^{-1}$  The average system temperatures were around
200\,K, each source had on-source integration times between 5 and 16
min  The data were converted to main-beam temperatures with forward
and beam efficiencies of 0 97 and 0 73, respectively
\citep{belloche2006}  The average $1\sigma$ rms was 0 4\,K   The main
spectral features of interest are the C$_2$H lines around 349 4\,GHz
with upper level excitation energies $E_u/k$ of 42\,K (line blends of
C$_2$H$(4_{5,5}-3_{4,4})$ \& C$_2$H$(4_{5,4}-3_{4,3})$ at
349 338\,GHz, and C$_2$H$(4_{4,4}-3_{3,3})$ \&
C$_2$H$(4_{4,3}-3_{3,2})$ at 349 399\,GHz)  The beam size was $\sim
18''$ 

The original Submillimeter Array (SMA) C$_2$H data toward the
HMPO\,18089-1732 were first presented in \citet{beuther2005c}  There
we used the compact and extended configurations resulting in good
images for all spectral lines except of C$_2$H  For this project, we
re-worked on these data only using the compact configuration  Because
the C$_2$H emission is distributed on larger scales (see
\S\ref{results}), we were now able to derive a C$_2$H image  The
integration range was from 32 to 35\,km\,s$^{-1}$, and the achieved
$1\sigma$ rms of the C$_2$H image was 450\,mJy\,beam$^{-1}$   For more
details on these observations see \citet{beuther2005c} 

\section{Results}
\label{results}

The sources were selected to cover all evolutionary stages from IRDCs
via HMPOs to UCH{\sc ii}s  We derived our target list from the samples
of \citet{klein2005,fontani2005,hill2005,beltran2006}   Table
\ref{sample} lists the observed sources, their coordinates, distances,
luminosities and a first order classification into the evolutionary
sub-groups IRDCs, HMPOs and UCH{\sc ii}s based on the previously
available data  Although this classification is only based on a
limited set of data, here we are just interested in general
evolutionary trends  Hence, the division into the three main classes
is sufficient 

Figure \ref{spectra} presents sample spectra toward one source of each
evolutionary group  While we see several CH$_3$OH lines as well as
SO$_2$ and H$_2$CS toward some of the HMPOs and UCH{\sc ii}s but not
toward the IRDCs, the surprising result of this comparison is the
presence of the C$_2$H lines around 349 4\,GHz toward all source types
from young IRDCs via the HMPOs to evolved UCH{\sc ii}s   Table
\ref{sample} lists the peak brightness temperatures, the integrated
intensities and the FWHM line-widths of the C$_2$H line blend at
349 399\,GHz  The separation of the two lines of 1 375\,MHz already
corresponds to a line-width of 1 2\,km\,s$^{-1}$  We have three C$_2$H
non-detections (2 IRDCs and 1 HMPO), however, with no clear trend with
respect to the distances or the luminosities (the latter comparison is
only possible for the HMPOs)  While IRDCs are on average colder than
more evolved sources, and have lower brightness temperatures, the
non-detections are more probable due to the relatively low sensitivity
of the short observations (\S\ref{obs})  Hence, the data indicate
that the C$_2$H lines are detected independent of the evolutionary
stage of the sources in contrast to the situation with other
molecules   When comparing the line-widths between the different
sub-groups, one finds only a marginal difference between the IRDCs and
the HMPOs (the average $\Delta v$ of the two groups are 2 8 and
3 1\,km\,s$^{-1}$)   However, the UCH{\sc ii}s exhibit significantly
broader line-widths with an average value of 5 5\,km\,s$^{-1}$ 

Intrigued by this finding, we wanted to understand the C$_2$H spatial
structure during the different evolutionary stages   Therefore, we
went back to a dataset obtained with the Submillimeter Array toward
the hypercompact H{\sc ii} region IRAS\,18089-1732 with a much higher
spatial resolution of $\sim 1''$ \citep{beuther2005c}   Albeit this
hypercompact H{\sc ii} region belongs to the class of HMPOs, it is
already in a relatively evolved stage and has formed a hot core with a
rich molecular spectrum   \citet{beuther2005c} showed the spectral
detection of the C$_2$H lines toward this source, but they did not
present any spatially resolved images  To recover large-scale
structure, we restricted the data to those from the compact SMA
configuration (\S\ref{obs})  With this refinement, we were able to
produce a spatially resolved C$_2$H map of the line blend at
349 338\,GHz with an angular resolution of $2 9''\times 1 4''$
(corresponding to an average linear resolution of 7700\,AU at the
given distance of 3 6\,kpc)  Figure \ref{18089} presents the
integrated C$_2$H emission with a contour overlay of the 860\,$\mu$m
continuum source outlining the position of the massive protostar  In
contrast to almost all other molecular lines that peak along with the
dust continuum \citep{beuther2005c}, the C$_2$H emission surrounds the
continuum peak in a shell-like fashion 

\section{Discussion and Conclusions}

To understand the observations, we conducted a simple chemical
modeling of massive star-forming regions  A 1D cloud model with a mass
of 1200\,M$_\sun$, an outer radius of 0 36\,pc and a power-law density
profile ($\rho\propto r^p$ with $p=-1 5$) is the initially assumed
configuration  Three cases are studied: (1) a cold isothermal cloud
with $T=10$\,K, (2) $T=50$\,K, and (3) a warm model with a temperature
profile $T\propto r^q$ with $q=-0 4$ and a temperature at the outer
radius of 44\,K  The cloud is illuminated by the interstellar UV
radiation field (IRSF, \citealt{draine1978}) and by cosmic ray
particles (CRP)  The ISRF attenuation by single-sized $0 1\mu$m
silicate grains at a given radius is calculated in a plane-parallel
geometry following \citet{vandishoeck1988}  The CRP ionization rate is
assumed to be $1 3\times 10^{-17}$~s$^{-1}$ \citep{spitzer1968}  The
gas-grain chemical model by \citet{vasyunin2008} with the desorption
energies and surface reactions from \citet{garrod2006} is used 
Gas-phase reaction rates are taken from RATE\,06 \citep{woodall2007},
initial abundances, were adopted from the ``low metal'' set of
\citet{lee1998} 

Figure \ref{model} presents the C$_2$H abundances for the three models
at two different time steps: (a) 100\,yr, and (b) in a more evolved
stage after $5\times10^4$\,yr  The C$_2$H abundance is high toward the
core center right from the beginning of the evolution, similar to
previous models (e g , \citealt{millar1985,herbst1986,turner1999}) 
During the evolution, the C$_2$H abundance stays approximately
constant at the outer core edges, whereas it decreases by more than
three orders of magnitude in the center, except for the cold $T=10$~K
model   The C$_2$H abundance profiles for all three models show
similar behavior 

The chemical evolution of ethynyl is determined by relative removal
rates of carbon and oxygen atoms or ions into molecules like CO, OH,
H$_2$O  Light ionized hydrocarbons CH$^+_{\rm n}$ (n=2  5) are quickly
formed by radiative association of C$^+$ with H$_2$ and hydrogen
addition reactions: C$^+$ $\rightarrow$ CH$_2^+$ $\rightarrow$
CH$_3^+$ $\rightarrow$ CH$_5^+$   The protonated methane reacts with
electrons, CO, C, OH, and more complex species at later stage and
forms methane   The CH$_4$ molecules undergo reactive collisions with
C$^+$, producing C$_2$H$_2^+$ and C$_2$H$_3^+$  An alternative way to
produce C$_2$H$_2^+$ is the dissociative recombination of CH$_5^+$
into CH$_3$ followed by reactions with C$^+$   Finally, C$_2$H$_2^+$
and C$_2$H$_3^+$ dissociatively recombine into CH, C$_2$H, and
C$_2$H$_2$  The major removal for C$_2$H is either the direct
neutral-neutral reaction with O that forms CO, or the same reaction
but with heavier carbon chain ions that are formed from C$_2$H by
subsequent insertion of carbon  At later times, depletion and
gas-phase reactions with more complex species may enter into this
cycle   At the cloud edge the interstellar UV radiation
instantaneously dissociates CO despite its self-shielding,
re-enriching the gas with elemental carbon 

The transformation of C$_2$H into CO and other species proceeds
efficiently in dense regions, in particular in the ``warm'' model
where endothermic reactions result in rich molecular complexity of the
gas (see Fig ~\ref{model})   In contrast, in the ``cold'' 10\,K model
gas-grain interactions and surface reactions become important   As a
result, a large fraction of oxygen is locked in water ice that is hard
to desorb ($E_{\rm des} \sim 5500$~K), while half of the elemental
carbon goes to volatile methane ice ($E_{\rm des} \sim 1300$~K)  Upon
CRP heating of dust grains, this leads to much higher gas-phase
abundance of C$_2$H in the cloud core for the cold model compared to
the warm model  The effect is not that strong for less dense regions
at larger radii from the center 

Since the C$_2$H emission is anti-correlated with the dust continuum
emission in the case of IRAS\,18089-1732 (Fig \,\ref{18089}), we do
not have the H$_2$ column densities to quantitatively compare the
abundance profiles of IRAS\,18089-1732 with our model  However, data
and model allow a qualitative comparison of the spatial structures 
Estimating an exact evolutionary time for IRAS\,18089-1732 is hardly
possible, but based on the strong molecular line emission, its high
central gas temperatures and the observed outflow-disk system
\citep{beuther2004a,beuther2004b,beuther2005c}, an approximate age of
$5\times10^4$\,yr appears reasonable   Although dynamical and chemical
times are not necessarily exactly the same, in high-mass star
formation they should not differ to much: Following the models by
\citet{mckee2003} or \citet{krumholz2006b}, the luminosity rises
strongly right from the onset of collapse which can be considered as a
starting point for the chemical evolution  At the same time disks and
outflows evolve, which should hence have similar time-scales   The
diameter of the shell-like C$_2$H structure in IRAS\,18089-1732 is
$\sim 5''$ (Fig \,\ref{18089}), or $\sim$9000\,AU in radius at the
given distance of 3 6\,kpc   This value is well matched by the modeled
region with decreased C$_2$H abundance (Fig \,\ref{model})   Although
in principle optical depths and/or excitation effects could mimic the
C$_2$H morphology, we consider this as unlikely because the other
observed molecules with many different transitions all peak toward the
central submm continuum emission in IRAS\,18089-1732
\citep{beuther2005c}  Since C$_2$H is the only exception in that rich
dataset, chemical effects appear the more plausible explanation 

The fact that we see C$_2$H at the earliest and the later evolutionary
stages can be explained by the reactive nature of C$_2$H: it is
produced quickly early on and gets replenished at the core edges by
the UV photodissociation of CO  The inner ``chemical'' hole observed
toward IRAS\,18089-1732 can be explained by C$_2$H being consumed in
the chemical network forming CO and more complex molecules like larger
carbon-hydrogen complexes and/or depletion 

The data show that C$_2$H is not suited to investigate the central gas
cores in more evolved sources, however, our analysis indicates that
C$_2$H may be a suitable tracer of the earliest stages of (massive)
star formation, like N$_2$H$^+$ or NH$_3$ (e g ,
\citealt{bergin2002,tafalla2004,beuther2005a,pillai2006})  While a
spatial analysis of the line emission will give insights into the
kinematics of the gas and also the evolutionary stage from chemical
models, multiple C$_2$H lines will even allow a temperature
characterization  With its lowest $J=1-0$ transitions around 87\,GHz,
C$_2$H has easily accessible spectral lines in several bands between
the 3\,mm and 850\,$\mu$m   Furthermore, even the 349\,GHz lines
presented here have still relatively low upper level excitation
energies ($E_u/k\sim42$\,K), hence allowing to study cold cores even
at sub-millimeter wavelengths   This prediction can further be proved
via high spectral and spatial resolution observations of different
C$_2$H lines toward young IRDCs 

\acknowledgments{H B  acknowledges financial support
  by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft
  (DFG, grant BE2578)  }



Context 2: \section{Introduction}
The Schwarzschild solution plays a key role in teaching about general relativity: It describes the simplest version of a black hole  By Birkhoff's theorem, it more generally describes the gravitational field around any spherical mass distribution, such as the Sun in our own Solar system  As one of two particularly simple, yet physically relevant examples of a non-trivial metric (the other being the FLRW spacetime of an expanding universe), it is particularly well-suited for teaching about general techniques of ``reading'' and interpreting a spacetime metric 

Consider undergraduate courses where students are introduced to selected concepts and results from general relativity without exposing them to the full mathematical formalism  Such courses have the advantage of introducing students to one of the two great fundamental theories of 20th century physics early on (the other being quantum mechanics); they also profit from subject matter that meets with considerable interest from students \cite{Hartle2006} Using the terminology of Christensen and Moore,\cite{Christensen2012} in the ``calculus only'' approach pioneered by Taylor and Wheeler,\cite{Taylor2001,Taylor2018} spacetime metrics are not derived, but taken as given, and the focus is on learning how to interpret a given spacetime metric  Similar presentations can be found in the first part of the ``physics first'' approach exemplified by Hartle's text book,\cite{Hartle2003} where the concepts of the metric and of geodesics are introduced early on, and their physical consequences explored, while the mathematics necessary for the Einstein equations is only introduced at a later stage  

Whenever the approach involves an exploration of simple metrics such as the Schwarzschild solution, but stops short of the formalism required for the full tensorial form of Einstein's equations, access to a simple derivation of the Schwarzschild solution that does not make use of the advanced formalism can be a considerable advantage 

Simplified derivations of the Schwarzschild solution have a long tradition within general relativity education,\cite{Schiff1960,Harwit1973} although specific simplifications have met with criticism \cite{Rindler1968} This article presents a derivation which requires no deeper knowledge of the formalism of differential geometry beyond an understanding of how to interpret a given spacetime metric $\mathrm{d} s^2$  The derivation avoids the criticism levelled at attempts to derive the Schwarzschild solution from the Einstein equivalence principle in combination with a Newtonian limit,\cite{Gruber1988} relying as it does on a simplified version of the vacuum Einstein equation 

More specifically, I combine the restrictions imposed by the symmetry with the simple form of Einstein's equations formulated by Baez and Bunn \cite{BaezBunn2005} That same strategy was followed by Kassner in 2017,\cite{Kassner2017} but in this text, I use the ``infalling coordinates'' that are commonly associated with the Gullstrand-Painlev\'e form of the Schwarzschild metric,\cite{Martel2001,Visser2005,HamiltonLisle2008} not the more common Schwarzschild coordinates  That choice simplifies the argument even further  In the end, what is required is no more than the solution of an ordinary differential equation for a single function, which yields to standard methods, to obtain the desired result 

\section{Coordinates adapted to spherical symmetry and staticity}
\label{SymmetriesCoordinates}

Assume that the spacetime we are interested in is spherically symmetric and static  In general relativity, a symmetry amounts to the possibility of being able to choose coordinates that are adapted to the symmetry, at least within a restricted sub-region of the spacetime in question  That the spacetime is static is taken to mean that we can introduce a (non-unique) time coordinate ${t}$ so that our description of spacetime geometry does not depend explicitly on ${t}$, and that space and time are completely separate --- in the coordinates adapted to the symmetry, there are no ``mixed terms'' involving $\mathrm{d} {t}$ times the differential of a space coordinate in the metric  If we use ${t}$ to slice our spacetime into three-dimensional hyperplanes, each corresponding to ``space at time ${t}$,'' then each of those 3-spaces has the same spatial geometry  A mixed term would indicate that those slices of space would need to be shifted relative to another in order to identify corresponding points  The mixed term's absence indicates that in adapted coordinates, there is no need for such an extra shift  In those coordinates, we can talk about the 3-spaces as just ``space,'' without the need for specifying which of the slices we are referring to 

In the case of spherical symmetry, we can introduce spherical coordinates that are adapted to the symmetry: a radial coordinate $r$ and the usual angular coordinates $\vartheta,\varphi$, so that the spherical shell at constant $r$ has the total area $4\pi r^2$  In consequence, the part of our metric involving $\mathrm{d}\vartheta$ and $\mathrm{d}\varphi$ will have the standard form
\begin{equation}
r^2(\mathrm{d}\vartheta^2+\sin^2\theta\mathrm{d}\varphi^2) \equiv r^2\mathrm{d}\Omega^2,
\end{equation}
where the right-hand side defines $\mathrm{d}\Omega^2$, the infinitesimal solid angle corresponding to each particular combination of $\mathrm{d}\vartheta$ and $\mathrm{d}\varphi$ 

The radial coordinate slices space into spherical shells, each corresponding to a particular value $r=const $ The rotations around the origin, which are the symmetry transformations of spherical symmetry, map each of those spherical shells onto itself, and they leave all physical quantities that do not explicitly depend on $\vartheta$ or $\varphi$ invariant 

In what follows, we will use the basic structures introduced in this way --- the slices of simultaneous ${t}$, the radial directions within each slice, the angular coordinates spanning the symmetry--adapted spherical shells of area $4\pi r^2$ --- as auxiliary structures for introducing spacetime coordinates  For now, let us write down the shape that our metric has by simple virtue of the spherical symmetry, the requirement that the spacetime be static, and the adapted coordinates, namely
\begin{equation}
\mathrm{d} s^2 = -c^2F(r) \mathrm{d} {t}^2 + G(r) \mathrm{d} r^2 + r^2\:\mathrm{d}\Omega^2  
\label{StaticForm}
\end{equation}
Students familiar with  ``reading'' a spacetime metric will immediately recognize the sign difference between the parts describing space and describing time that is characteristic for spacetime, and the speed of light $c$ that gives us the correct physical dimensions  That there is no explicit dependence on $\varphi$ and $\vartheta$ in the remaining functions $F$ and $G$ is a direct consequence of spherical symmetry  That the factor in front of $\mathrm{d}\Omega^2$ is $r^2$ is a consequence of our coordinate choice, with spherical angular coordinates so that the area of a spherical surface of constant radius $r$ is $4\pi r^2$  That there is no explicit dependence on ${t}$ is one consequence of the spacetime being static; the absence of the mixed term $\mathrm{d} {t}\cdot \mathrm{d} r$ is another  We are left with two unknown functions $F(r)$ and $G(r)$  In the following, let us call ${t}$ and $r$ the {\em static coordinates}  
 
Note that, since $G(r)$ is as yet undefined, we have not yet chosen a specific physical meaning for the length measurements associated with our $r$ coordinate  But because of the $\mathrm{d}\Omega^2$ part, it is clear that whatever choice we make, the locally orthogonal lengths $r\cdot\mathrm{d}\vartheta$ and $r\cdot\sin\vartheta\cdot\mathrm{d}\varphi$ will have the same physical interpretation as for the length measurement corresponding to $\mathrm{d} r$ 

\section{Infalling observer coordinates}
\label{Sec:InfallingObservers}

Now that we know what the radial directions are, at each moment of time ${t}$, we follow Visser\cite{Visser2005} as well as Hamilton and Lisle\cite{HamiltonLisle2008} in defining a family of radially infalling observers   Observers in that family are in free fall along the radial direction, starting out at rest at infinity: In mapping each observer's radial progression in terms of the static coordinate time ${t}$, we adjust initial conditions, specifically: the choice of initial speed at some fixed time ${t}$, in just the right way that the radial coordinate speed goes to zero for each observer in the same way as $r\to\infty $

It is true that talking about ``infalling'' observers already reflects our expectation that our solution should describe the spacetime of a spherically symmetric mass   As we know from the Newtonian limit, such a mass attracts test particles in its vicinity  It should be noted, though, that all our calculations would also be compatible with the limit of no mass being present  In that case, ``infalling'' would be a misnomer, as our family of observers would merely hover in empty space at unchanging positions in $r$  

We can imagine infinitesimal local coordinate systems associated with our observers --- think of the observer mapping out space and time by defining three orthogonal axes, and by measuring time with a co-moving clock  We assume all such little coordinate systems to be non-rotating --- otherwise, we would break spherical symmetry, since rotation would locally pick out a plane of rotation that is distinguishable from the other planes  The radial direction is a natural choice for the first space axis of those little free-falling systems  The other directions, we take to point to observers falling side by side with our coordinate-defining observer --- and to remain pointed at a specific such other observer, once the choice of direction is made 

We assume our infalling observers' clocks to be synchronised at some fixed radius value $r$  By spherical symmetry, those clocks should then be synchronised at {\em all} values of $r$  Anything else would indicate direction-dependent differences for the infalling observers and their clocks, after all  Hence, at any given static time ${t}$, all the infalling observers who are at radius value $r$ show the same proper time $T$ on the ideal clocks travelling along with them  

Once our definition is complete, our static, spherically symmetric spacetime is filled with infalling observers from that family: Whenever we consider an event $\cal E$, there will be an observer from that family passing by at that time, at that location  

Now, consider the coordinate speed of those infalling observers  If we position ourselves at some constant radius value $r$ and watch the falling observers fly by, then we can express both their proper time rate and their coordinate speed in the $r$ direction in terms of $r$ and ${t}$  We can combine the two pieces of information to obtain the rate of change in radial position $r$ with proper time $T$ for those infalling observers  But since the initial conditions for those observers are the same, and since our spacetime is, by assumption, static, the resulting function can only depend on $r$, and not explicitly on ${t}$  Let us rescale that function with the speed of light to make it dimensionless, give it an overall minus sign to make it positive for infalling particles, and call it $\beta(r)$,
\begin{equation}
\beta(r)\equiv -\frac{1}{c}\frac{\mathrm{d} r}{\mathrm{d} T}(r) 
\label{betaDefinition}
\end{equation}

Recall from section \ref{SymmetriesCoordinates} that we also still have the freedom to decide on the physical meaning of $r$  We make the choice of making $\mathrm{d} r$ the physical length measured by one of our infalling observers at the relevant location in spacetime, at constant time $T$  Via our angular coordinates, that implies that length measurements orthogonal to the radial direction, $r\cdot\mathrm{d}\vartheta$ and $r\cdot\sin\vartheta\:\mathrm{d}\varphi$ inherit the same physical interpretation 

As a next step, we transform our metric (\ref{StaticForm}) from the static form into the form appropriate for our coordinate choice $r$ and $T$  We do so by writing the static time coordinate as a function ${t}(T,r)$ in terms of infalling observer time and radius value  In consequence,
\begin{equation}
\mathrm{d} {t} = \frac{\partial{t}}{\partial T}\cdot\mathrm{d} T+ \frac{\partial {t}}{\partial r}\cdot\mathrm{d} r,
\end{equation}
and our new metric now has the form
\begin{align}
 \mathrm{d} s^2 = {} & -c^2 F(r)\left(\frac{\partial t}{\partial T}\right)^2\mathrm{d} T^2 \nonumber \\[0 2em]
 & -2c^2F(r)\left(\frac{\partial t}{\partial T}\right)\left(\frac{\partial t}{\partial r}\right)\mathrm{d} T\:\mathrm{d} r \nonumber \\[0 2em]
 & +\left[G(r)-c^2F(r)\left(\frac{\partial t}{\partial r}\right)^2\right]\mathrm{d} r^2+r^2\:\mathrm{d}\Omega^2 
 \end{align}
At face value, this looks like we are moving the wrong way, away from simplification, since we now have more functions, and they depend on two variables instead of one 

But in fact, this new formulation paves the way for an even simpler form of the metric  Consider a specific event, which happens at given radius value $r$  In a small region around that event, we will introduce a new coordinate $\bar{r}$ to parametrize the radial direction  We want this coordinate to be co-moving with our infalling observers at $r$; each such observer then has a position $\bar{r}=const $ that does not change over time  

Key to our next step is that we {\em know} the metric for the local length and time measurements made by any one of our free-falling observers  By Einstein's equivalence principle, the metric is that of special relativity  Locally, namely whenever tidal effects can be neglected, spacetime geometry for any non-rotating observer in free fall is indistinguishable from Minkowski spacetime as described by a local inertial system 

Since we have chosen both the time coordinate $T$ and the physical meaning of the radial coordinate $r$ so as to conform with the measurements of the local infalling observer, the transformation between $\bar{r}$ and $r$ is particularly simple: It has the form of a Galilei transformation
\begin{equation}
\mathrm{d}\bar{r}= \mathrm{d} r + \beta(r)c\:\mathrm{d} T 
\label{barRshift}
\end{equation}
In that way, as it should be by definition, radial coordinate differences at constant $T$ are the same in both systems, while for an observer at constant $\bar{r},$ with $\mathrm{d} \bar{r}=0$, the relation between $\mathrm{d} r$ and $\mathrm{d} T$ is consistent with the definition of the function $\beta(r)$ in (\ref{betaDefinition}) 

Are you surprised that this is not a Lorentz transformation, as one might expect from special relativity? Don't be  We are not transforming from one local inertial coordinate system to another  The $T$ is already the time coordinate of the infalling observers, so both coordinate systems have the same definition of simultaneity, and time dilation plays no role in this particular transformation  Also, we have chosen $r$ intervals to correspond to length measurements of the infalling observers, so there is no Lorentz contraction, either  It is the consequence of these special choices that gives the relation (\ref{barRshift}) its simple form 

Last but not least, when we analyse specifically an infinitesimal neighbourhood of the point $r,\vartheta,\varphi$, let us make the choice that directly at our point of interest, we make $\bar{r}$ coincide with $r$  Since before, we had only fixed the differential $\mathrm{d} \bar{r}$, we do have the remaining freedom of choosing a constant offset for $\bar{r}$ that yields the desired result 

By Einstein's equivalence principle, the metric in terms of the locally co-moving coordinates $T,\bar{r},\vartheta,\varphi$ is the spherical-coordinate version of the Minkowski metric,
\begin{equation}
\mathrm{d} s^2 = -c^2\mathrm{d} T^2 + \mathrm{d}\bar{r}^2 + \bar{r}^2\mathrm{d}\Omega 
\end{equation}
This version can, of course, be obtained by taking the more familiar Cartesian-coordinate version
\begin{equation}
\mathrm{d} s^2=-c^2\mathrm{d} T^2 + \mathrm{d} X^2 + \mathrm{d} Y^2 + \mathrm{d} Z^2,
\label{CartesianMinkowski}
\end{equation}
applying the definition of Cartesian coordinates $X,Y,Z$ in terms of spherical coordinates $\bar{r},\vartheta,\varphi$
\begin{equation}
x= \bar{r}\:\sin\vartheta\:\cos\varphi, \;\;
y= \bar{r}\:\sin\vartheta\:\sin\varphi, \;\;
z= \bar{r}\:\cos\vartheta,
\end{equation}
to express $\mathrm{d} X, \mathrm{d} Y, \mathrm{d} Z$ in terms of $\mathrm{d} \bar{r}, \mathrm{d}\vartheta, \mathrm{d}\varphi$, and substitute the result into (\ref{CartesianMinkowski}) 

By noting that we have chosen $\bar{r}$ so that, at the specific spacetime event where we are evaluating the metric, $\bar{r}=r$, while, for small radial coordinate shifts around that location, we have the relation (\ref{barRshift}), we can now write down the same metric in the coordinates $T, r, \vartheta,\varphi$, namely as
\begin{equation}
\mathrm{d} s^2 = -c^2\left[
1-\beta(r)^2
\right] \mathrm{d} T^2+2c\beta(r)\mathrm{d} r\:\mathrm{d} T
+\mathrm{d} r^2+r^2\mathrm{d}\Omega^2 
\label{preMetric}
\end{equation}
Since we can repeat that local procedure at any event in our spacetime, this result is our general form of the metric, for all values of $r$  This, then is the promised simplification: By exploiting the symmetries of our solutions as well as the properties of infalling observers, we have reduced our metric to a simple form with no more than one unknown function of one variable, namely $\beta(r)$ 

So far, what I have presented is no more than a long-form version of the initial steps of the derivation given by Visser in his heuristic derivation of the Schwarzschild metric \cite{Visser2005} In the next section, we will deviate from Visser's derivation 

\section{$\beta(r)$ from tidal deformations}
\label{TidalSection}

In the previous section, we had exploited symmetries and Einstein's equivalence principle  In order to determine $\beta(r)$, we need to bring in additional information, namely the Einstein equations, which link the matter content with the geometry of spacetime  For our solution, we only aim to describe the spacetime metric outside whatever spherically-symmetric matter distribution resides in (or around) the center of our spherical symmetry  That amounts to applying the {\em vacuum Einstein equations} 

More specifically, we use a particularly simple and intuitive form of the vacuum Einstein equations, which can be found in a seminal article by Baez and Bunn:\cite{BaezBunn2005} Consider a locally flat free-fall system around a specific event $\cal E$, with a time coordinate $\tau$, local proper time, where the event we are studying corresponds to $\tau=0$  In that system, describe a small sphere of freely floating test particles, which we shall call a {\em test ball}  The particles need to be at rest relative to each other at $\tau=0$  Let the volume of the test ball be $V(\tau)$  Then the vacuum version of Einstein's equations states that
\begin{equation}
\left \frac{\mathrm{d}^2 V}{\mathrm{d}\tau^2}\right|_{\tau=0} = 0 
\label{EinsteinVacuum}
\end{equation}
In words: If there is no matter or energy inside, the volume of such a test ball remains constant in the first order (those were our initial conditions) and the second order (by eq ~[\ref{EinsteinVacuum}])  

If you are familiar with Wheeler's brief summary of Einstein's equations, ``spacetime grips mass, telling it how to move'' and ``mass grips spacetime, telling it how to curve'',\cite{Wheeler1990} you will immediately recognise that this is a specific way for the structure of spacetime telling the test ball particles how to move  The calculation later in this section provides the second part: It will amount to using (\ref{EinsteinVacuum}) to determine the structure of spacetime, namely the still missing function $\beta(r)$, and that is the way for mass, in this case: for the absence of mass, to tell spacetime how to curve 

Note that equation (\ref{EinsteinVacuum}) also holds true in Newtonian gravity  So in a way, this version of Einstein's equation can be seen as a second-order extension of the usual Einstein equivalence principle: Ordinarily, the equivalence principle is a statement about physics in the absence of tidal forces  Equation (\ref{EinsteinVacuum}) adds to this that the lowest-order correction for tidal forces in a freely falling reference frame is that specified by Newtonian gravity  This makes sense, since by going into a free-fall frame, and restricting our attention to a small spacetime region, we have automatically created a weak-gravity situation  In such a situation, tidal corrections are approximately the same as those described by Newton  This argument can serve as a heuristic justification of (\ref{EinsteinVacuum}) 

In 2017, Kassner made use of the Baez-Bunn form of Einstein's vacuum equation to derive the Schwarzschild solution, starting from what we have encountered as the static form of the metric (\ref{StaticForm}) \cite{Kassner2017} We follow the same general recipe, but using the infalling coordinates introduced in section \ref{Sec:InfallingObservers}, which makes our derivation even simpler 

Consider five test particles in a small region of space  Let the motion of each be the same as for the local representative from our coordinate-defining family of infalling observers  We take the central particle $C$ to be at radial coordinate value $r=R$ at the time of the snapshot shown in Fig ~\ref{TestParticlesOutside}  The other four are offset relative to the central particle: As described in the local inertial system that is co-moving with the central particle, one of the particles is shifted by $\Delta l$ upwards in the radial direction, another downward, while two of the particles are offset orthogonally by the same distance 
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0 5\linewidth]{01-free-fall-particles pdf}
\caption{Five test particles in our spherically-symmetric spacetime}
\label{TestParticlesOutside}
\end{center}
\end{figure}
The $\Delta l$ is meant to be infinitesimally small, so while Fig ~\ref{TestParticlesOutside} is of course showing a rather large $\Delta l$ so as to display the geometry of the situation more clearly, we will in the following only keep terms linear in $\Delta l$  

Consider a generic particle, which moves as if it were part of our coordinate-defining family of infalling observers, and which at the time $T_0$ is at $r=r_0$  By a Taylor expansion, that particle's subsequent movement is given by
\begin{equation}
r(T) = r_0 + \frac{\mathrm{d} r}{\mathrm{d} T}(T_0) \cdot \Delta T +\frac12 \frac{\mathrm{d}^2 r}{\mathrm{d} T^2}(T_0) \cdot \Delta T^2
\label{TaylorREvo}
\end{equation}
where $\Delta T\equiv T-T_0$  We know from (\ref{betaDefinition}) that the derivative in the linear term can be expressed in terms of $\beta(r)$; by the same token,
\begin{equation}
\frac{\mathrm{d}^2 r}{\mathrm{d} T^2} = -c\frac{\mathrm{d}\beta}{\mathrm{d} T}=-c\beta' \frac{\mathrm{d} r}{\mathrm{d} T} = c^2\beta\cdot\beta',
\end{equation}
where the prime denotes differentiation of $\beta$ with respect to its argument  Since, in the following, the product of $\beta$ and its first derivative will occur quite often, let us introduce the abbreviation
\begin{equation}
B(r) \equiv \beta(r)\cdot\beta'(r) 
\label{BigBDefinition}
\end{equation}
With these results, can rewrite the Taylor expansion (\ref{TaylorREvo}) as 
\begin{equation}
r(T) = r_0 -c\beta(r_0)\cdot\Delta T + \frac12 c^2B(r_0)\cdot\Delta T^2 
\label{RadialOrbitTime}
\end{equation}
In order to find $r_C(T)$ for our central particle, we simply insert $r_0=R$ into that expression  If, on the other hand, we want to write down the time evolution for particles $U$ and $D$, let us denote it by $r_{U,D}(T)$, we need to evaluate the expression (\ref{RadialOrbitTime}) at the initial location $r_0=R\pm\Delta l$  Since $\Delta l$ is small, we can make a Taylor expansion of $\beta(r)$ and its derivative around $r=R$, and neglect everything beyond the terms linear in $\Delta l$  The result is
\begin{multline}
r_{U,D}(T)=R \pm\Delta l-c\left[
\beta(R)\pm\beta'(R)\Delta l
\right]\Delta T \\[0 2em]
+\frac{c^2}{2}\big[
B(R)\pm B'(R)\Delta l
\big]\Delta T^2
\end{multline}
In consequence, the distance between the upper and lower particle, $d_{\parallel}(T)\equiv r_U(T)-r_D(T),$ changes over time as
\begin{equation}
d_{\parallel}(T) =  2\Delta l\left[
1-c\beta'(R)\Delta T+\frac12c^2 B'(R)\Delta T^2
\right] 
\label{dParallel}
\end{equation}
Next, let us look at how the distance between the particles $L$ and $R$ changes over time  The initial radial coordinate value for each of the particles is
\begin{equation}
r(T_0) = \sqrt{R^2+\Delta l^2}=R\left[1+\frac12\left(\frac{\Delta l}{R}\right)^2\right]\approx R,
\end{equation}
that is, equal to $R,$ as long as we neglect any terms that are higher than linear in $\Delta l$  In consequence, $r_{L,R}(t)$ is the same function as for our central particle, given by eq ~(\ref{RadialOrbitTime}) with $r_0=R$  The transversal (in Fig ~\ref{TestParticlesOutside}: horizontal) distance $d_{\perp}(T)$ between the particles $L$ and $R$ changes in proportion to the radius value,
\begin{align}
d_{\perp}(T) &= 2\Delta l\cdot\frac{r_{L}(T)}{R} \nonumber \\
                 &=2\Delta \left[1-\frac{c\beta(R)}{R}\Delta T+\frac{c^2}{2}\frac{B(R)}{R}\Delta T^2\right] 
                \label{dPerp}
\end{align}
With these preparations, consider the vacuum Einstein equation (\ref{EinsteinVacuum}) for the volume of a test ball  Initially, our particles $C, U, D, L, R$ define a circle, which is deformed to an ellipse  By demanding rotational symmetry around the radial direction, we can construct the associated ellipsoid, which is initially a spherical surface  That ellipsoid has one axis in radial direction, whose length is $d_{\parallel}(T)$, and two axes that are transversal and each have the length  $d_{\perp}(t)$  But that ellipsoid is not quite yet the test ball we need  After all, the particles of the test ball need to be at rest initially, at time $T_0$, in the co-moving system defined by the central particle $C$  Our defining particles are not, as the terms linear in $\Delta T$ in both (\ref{dParallel}) and (\ref{dPerp}) show, where the coefficients of $\Delta T$ correspond to the particles' initial velocities  

In order to define our test ball, we need to consider particles at the same location, undergoing the same acceleration, but which are initially at rest relative to the central particle $C$  

We could go back to the drawing board, back to Fig ~\ref{TestParticlesOutside}, make a more general Ansatz that includes initial velocities which measure the divergence of the motion of our test ball particles from that of the infalling-observer particles, and repeat our calculation while including those additional velocity terms  But there is a short-cut  The only consequence of those additional velocity terms will be to change the terms linear in $\Delta T$ in equations (\ref{dParallel}) and (\ref{dPerp})  And we already know the end result: We will choose the additional terms so as to cancel the terms linear in $\Delta T$ in the current versions of (\ref{dParallel}) and (\ref{dPerp})  But by that reasoning, we can skip the explicit steps in between, and write down the final result right away  The time evolution of the radial-direction diameter of our test ball, let us call it $L_{\parallel}(T)$, must be the same as $d_{\parallel}(T)$, but without the term linear in $\Delta T$  Likewise, the time evolution $L_{\perp}(T)$ of the two transversal diameters must be equal to $d_{\perp}(T)$, but again without the term linear in $\Delta T$  The result is
\begin{align}
L_{\parallel}(T)  &=  2\Delta l \left[1+\frac12c^2B'(R)\Delta T^2\right] \\
L_{\perp}(T) &= 2\Delta l \left[1+\frac{c^2}{2}\frac{B(R)}{R}\Delta T^2\right] 
\end{align}
Thus, our test ball volume is
\begin{align}
V(T) &= \frac{\pi}{6}L_{\parallel}(T) L_{\perp}^2(T) \\
       &= \left \frac{4\pi}{3}\Delta l^3\left[1+{c^2}\left( \frac{B(r)}{r} + \frac{B'(r)}{2}\right)\Delta T^2\right]\right|_{r=R}
\end{align}
For the second time derivative of $V(T)$ to vanish at the time $T=T_0$, we must have
\begin{equation}
\frac{B(r)}{r} + \frac{B'(r)}{2}= 0
\label{VolumeConditionR}
\end{equation}
for all values of $r$  This is readily solved by the standard method of separation of variables: We can rewrite (\ref{VolumeConditionR}) as
\begin{equation}
\frac{\mathrm{d} B}{B} = -2\frac{\mathrm{d} r}{r},
\end{equation}
which is readily integrated to give
\begin{equation}
\ln(B) = -\ln(r^{2}) + const   \;\; \Rightarrow \;\; \ln(Br^2) = C',
\end{equation}
with a constant $C'$, which upon taking the exponential gives us
\begin{equation}
Br^2= C,
\label{BSolution}
\end{equation}
with a constant $C$  Note that the constant $C$ can be negative --- there is no reason the constant $C'$ needs to be real; only our eventual function $B(r)$ needs to be that, and it is clear that (\ref{BSolution}) satisfies the differential equation
(\ref{VolumeConditionR}) for any constant $C$, positive, zero, or negative  By (\ref{BigBDefinition}), the solution (\ref{BSolution}) corresponds to the differential equation
\begin{equation}
\beta(r)\beta'(r) = \frac{C}{r^2}
\end{equation}
for our function $\beta$; with another separation of variables, we can re-write this as 
\begin{equation}
\beta\cdot\mathrm{d}\beta=C\frac{\mathrm{d} r}{r^2} 
\end{equation}
Both sides are readily integrated up; we can solve the result for $\beta(r)$ and obtain
\begin{equation}
\beta(r) = \sqrt{
-\frac{2C}{r} +2D
},
\end{equation}
where $D$ is the second integration constant, and where we have chosen the proper sign, since we know that $\beta(r)>0$  That brings us to the last step: The requirement that, for large values of $r$, the description provided by our solution should correspond to the results from Newtonian gravity  First of all, we note that our initial condition for the infalling observers, which had those observers start out at zero speed at infinity, means that we must choose $D=0$  Then, as we would expect, $\beta(r)$ for large values of $r$ becomes very small, corresponding to small speeds  But at slow speeds, time and length intervals as measured by the infalling observer will become arbitrarily close to time and length intervals as measured by an observer at rest in our static coordinate system at constant $r$, using the static time coordinate ${t}$  As is usual, we identify these coordinates with those of an approximately Newtonian description  In that description, the radial velocity is
\begin{equation}
v(r) = \sqrt{\frac{2GM}{r}},
\end{equation}
which follows directly from energy conservation for the sum of each observer's kinetic and Newtonian-gravitational potential energy  This fixes the remaining integration constant as
\begin{equation}
C = -\frac{GM}{c^2},
\end{equation}
and the final form of our function $\beta(r)$ becomes
\begin{equation}
\beta(r) = \sqrt{\frac{2GM}{rc^2}} 
\end{equation}
Inserting this result in (\ref{preMetric}), we obtain the metric
\begin{equation}
\mathrm{d} s^2 = -c^2\left[
1-\frac{2GM}{rc^2}
\right]\mathrm{d} T^2+2\sqrt{\frac{2GM}{r}}\mathrm{d} r\:\mathrm{d} T+\mathrm{d} r^2+r^2\mathrm{d}\Omega^2 
\label{GPMetric}
\end{equation}
This is known as the Gullstrand-Painlev\'e version of the Schwarzschild metric \cite{Martel2001,Visser2005,HamiltonLisle2008} A  last transformation step brings us back to the traditional Schwarzschild form  Recall our discussion in sec ~\ref{SymmetriesCoordinates}, leading up to the explicitly static form (\ref{StaticForm}) of the metric? The main difference between our current form and the static version is the mixed term containing $\mathrm{d} r\:\mathrm{d} T$ in (\ref{GPMetric})  Everything else already has the required shape  Inserting the Ansatz
\begin{equation}
\mathrm{d} T = \mathrm{d} t + \xi(r) \mathrm{d} r
\end{equation}
into the metric (\ref{GPMetric}), it is straightforward to see that the mixed term vanishes iff our transformation is
\begin{equation}
\mathrm{d} T = \mathrm{d} t +\frac{\sqrt{2GM/r}}{c^2\left(1-\frac{2GM}{rc^2}\right)}\mathrm{d} r 
\label{TtTrafo}
\end{equation}
Substitute this into (\ref{GPMetric}), and the result is the familiar form of the Schwarzschild metric in Schwarzschild's original coordinates $t,r,\vartheta,\varphi$, 
\begin{equation}
\mathrm{d} s^2 = -c^2\left(1-\frac{2GM}{c^2 r}
\right)\mathrm{d} t^2 + \frac{\mathrm{d} r^2}{\left(1-\frac{2GM}{c^2 r}
\right)} + r^2\mathrm{d}\Omega^2 
\end{equation}

\section{Conclusion}
Using coordinates adapted to the symmetries, we were able to write down the spherically symmetric, static spacetime metric  On this basis, and using the family of infalling observers that is characteristic for the Gullstrand-Painlev\'e solution, we wrote down the metric in the form (\ref{preMetric}), with a single unknown function $\beta(r)$  From the simplified form (\ref{EinsteinVacuum}) of the vacuum Einstein equations, as applied to a test ball in free fall alongside one of our family of observers, we were able to determine $\beta(r)$, up to two integration constants  By using the Einstein equation, we escape the restrictions imposed on simplified derivations by Gruber et al \cite{Gruber1988} 

From the initial condition for our infalling observers, as well as from the Newtonian limit at large distances from our center of symmetry, we were able to fix the values of the two intergration constants  Our derivation does not require knowledge of advanced mathematical concepts beyond the ability to properly interpret a given metric line element $\mathrm{d} s^2$  Even our analysis of tidal effects proceeds via a simple second-order Taylor expansion, leading to differential equations for $\beta(r)$ that are readily solved using two applications of the method of separation of variables  

What is new about the derivation presented here is the combination of the Baez-Bunn equations with the infalling coordinates typical for the Gullstrand-Painlev\'e form of the metric --- this combination is what, in the end, makes our derivation particularly simple  In turn, this simplicity is what should make the derivation particularly useful in the context of teaching general relativity in an undergraduate setting 

The derivation proceeds close to the physics, and gives ample opportunity to discuss interesting properties of Einstein's theory of gravity  Students who are presented with this derivation, either as a demonstration or as a (guided) exercise, will come to understand the way that symmetries determine the form of a metric, the deductions that can be made from Einstein's equivalence principle, and last but not least that we need to go beyond the equivalence principle, and consider tidal forces, to completely define our solution 

\section*{Acknowledgements}

I would like to thank Thomas M\"uller for helpful comments on an earlier version of this text 





ground_truth: The focus of the study was on the reactive radical ethynyl (C$_2$H).

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How is the function beta(r) determined in the derivation?

Read the following text and answer briefly.

Context 4: Sowing the Seeds of Future Research: Data Sharing, Citation and Reuse in Archaeobotany
Reading: Sowing the Seeds of Future Research: Data Sharing, Citation and Reuse in Archaeobotany
University of Oxford, GB
Lisa is a post-doctoral research fellow at All Souls College, University of Oxford  Her publications include the co-authored volume The Rural Economy of Roman Britain (Britannia Monographs, 2017)  Her research interests are focussed on agricultural practices in the later prehistoric and Roman period and the utilisation of archaeobotanical data to investigate human-plant relationships 
The practices of data sharing, data citation and data reuse are all crucial aspects of the reproducibility of archaeological research  This article builds on the small number of studies reviewing data sharing and citation practices in archaeology, focussing on the data-rich sub-discipline of archaeobotany  Archaeobotany is a sub-discipline built on the time-intensive collection of data on archaeological plant remains, in order to investigate crop choice, crop husbandry, diet, vegetation and a wide range of other past human-plant relationships  Within archaeobotany, the level and form of data sharing is currently unknown  This article first reviews the form of data shared and the method of data sharing in 239 articles across 16 journals which present primary plant macrofossil studies  Second, it assesses data-citation in meta-analysis studies in 107 articles across 20 journals  Third, it assesses data reuse practices in archaeobotany, before exploring how these research practices can be improved to benefit the rigour and reuse of archaeobotanical research 
Keywords: Archaeobotany, Data reuse, Data sharing, Open science
How to Cite: Lodwick, L , 2019  Sowing the Seeds of Future Research: Data Sharing, Citation and Reuse in Archaeobotany  Open Quaternary, 5(1), p 7  DOI: http://doi org/10 5334/oq 62
Accepted on 29 May 2019 Submitted on 25 Mar 2019
Archaeology is a discipline built on the production and analysis of quantitative data pertaining to past human behaviour  As each archaeological deposit is a unique occurrence, ensuring that the data resulting from excavation and analysis are preserved and accessible is crucially important  Currently, there is a general perception of a low level of data sharing and reuse  Such a low level of data availability would prevent the assessment of research findings and the reuse of data in meta-analysis (Kansa & Kansa 2013; Moore & Richards 2015)  As observed across scientific disciplines, there is a major problem in the reproduction of scientific findings, commonly known as the ‘replication crisis’ (Costello et al  2013)  A range of intersecting debates contribute to this, including access to academic findings (open access), open data, access to software and access to methodologies, which can be broadly grouped as open science practices  Without these, the way that scientific findings can be verified and built upon is impaired  Questions of reproducibility have been raised in recent years in archaeology, with considerations of a range of practices which can improve the reproducibility of findings, and a recent call for the application of open science principles to archaeology (Marwick et al  2017)  Discussion has so far focussed on access to grey literature (Evans 2015), data sharing (Atici et al  2013), data citation practices (Marwick & Pilaar Birch 2018) and computational reproducibility (Marwick 2017), with a focus on lithics, zooarchaeological evidence, and archaeological site reports 
Quantitative assessments of current levels of data sharing, data citation and reuse remain limited in archaeology  The focus of evaluation has been on the uptake of large-scale digital archives for the preservation and dissemination of digital data, such as the Archaeology Data Service (ADS), utilised by developer-led and research projects, and recommended for use by many research funders in the UK (Richards 2002; Wright and Richards 2018)  Much less focus has been paid to the data-sharing practices of individuals or small-groups of university-based researchers who may be disseminating their research largely through journal articles  Recent work on the availability of data on lithics assemblages found a low level of data sharing (Marwick & Pilaar Birch 2018) and there are perceptions of low levels of data reuse (Huggett 2018; Kintigh et al  2018)  Within zooarchaeology numerous studies have explored issues of data sharing and reuse (Kansa & Kansa 2013, 2014), and the sub-discipline is seen as one of the most advanced areas of archaeology in regards to open science (Cooper & Green 2016: 273)  Beyond zooarchaeology, however, explicit discussion has remained limited 
This paper assesses data sharing and reuse practices in archaeology through the case study of archaeobotany – a long established sub-discipline within archaeology which has well-established principles of data recording  Archaeobotany is an interesting case study for data sharing in archaeology as it straddles the division of archaeology between scientific and more traditional techniques  Quantitative data on archaeological plant remains are also of interest to a range of other fields, including ecology, environmental studies, biology and earth sciences  The key issues of data sharing and data reuse (Atici et al  2013) have been touched upon in archaeobotany over the past decade within broader discussions on data quality (Van der Veen, Livarda & Hill 2007; Van der Veen, Hill & Livarda 2013)  These earlier studies focussed on the quality and availability of archaeobotanical data from developer-funded excavations in Britain and Cultural Resource Management in North America (Vanderwarker et al  2016: 156)  However, no discussion of data-sharing and reuse in academic archaeobotany occurred  A recent review of digital methods in archaeobotany is the notable exception, with discussions of the challenges and methods of data sharing (Warinner & d’Alpoim Guedes 2014) 
Currently, we have no evidence for the levels of data sharing and reuse within archaeobotany  This article provides the first quantitative assessment of 1) data publication in recent archaeobotanical journal articles 2) data citation in recent archaeobotanical meta-analysis 3) the reuse of archaeobotanical datasets, in order to assess whether practices need to change and how such changes can take place 
2  Data Publication and Re-use Practices in Archaeobotany
2 1  History of data production and publication
Archaeobotanical data falls within the category of observational data in archaeology (Marwick & Pilaar Birch 2018)  Archaeobotanical data is considered as the quantitative assessment of plant macrofossils present within a sample from a discrete archaeological context, which can include species identification, plant part, levels of identification (cf  – confer or “compares to”), and a range of quantification methods including count, minimum number of individuals, levels of abundance and weight (Popper 1988)  Archaeobotanical data is usually entered into a two-way data table organised by sample number  Alongside the counts of individual taxa, other information is also necessary to interpret archaeobotanical data, including sample volume, flot volume, charcoal volume, flot weight, level of preservation, sample number, context number, feature number, feature type and period  Beyond taxonomic identifications, a range of other types of data are increasingly gathered on individual plant macrofossils (morphometric measurements, isotopic values, aDNA) 
Archaeobotanical training places a strong emphasis on recording data on a sample-by-sample basis (Jacomet & Kreuz 1999: 138–139; Jones & Charles 2009; Pearsall 2016: 97–107)  Time-consuming methodologies utilised in the pursuit of accurate sample-level data recording include sub-sampling and splitting samples into size fractions and counting a statistically useful number of items per sample (Van der Veen & Fieller 1982)  The creation of sample-level data means analysis is often undertaken on the basis of individual samples, for instance the assessment of crop-processing stages and weed ecological evidence for crop husbandry practices  The analysis of sample level data also enables archaeobotanical finds to be integrated alongside contextual evidence from archaeological sites  Requirements for the publication of this data are in place in some archaeological guidelines, for instance current Historic England guidelines for archaeological practice in England (Campbell, Moffett & Straker 2011: 8) 
From the earliest archaeobotanical reports, such as Reid’s work at Roman Silchester, the sample from which plant remains were recovered was noted (Lodwick 2017a), but often results were reported as a list of taxa, or long catalogues of detailed botanical descriptions with seed counts, such as Knörzer’s work at Neuss (Knörzer 1970)  Early systematic archaeobotanical reports displayed data within in-text tables, for example Jones’s work at Ashville (Jones 1978) and the two-way data table has been the standard form of reporting archaeobotanical data ever since  Often data tables are presented within book chapters or appendices, but the financial, space and time constraints of book publishing are limiting  Furthermore, there is the perception that specialist data was not necessary for publication (Barker 2001)  Hence, alternative methods of the dissemination of specialist archaeological data were pursued in the later twentieth century 
From the 1980s, archaeobotanical data tables were often consigned to microfiche following a Council for British Archaeology and Department of Environment report (Moore & Richards 2015: 31), with the example of the excavation of Roman Colchester where the contents of all archaeobotanical samples were available on microfiche (Murphy 1992)  An alternative in the 2000s was providing data tables on CD Rom as seen, for instance, in the CD accompanying the study of a Roman farmstead in the Upper Thames Valley (Robinson 2007) or the One Poultry excavations in London (Hill and Rowsome 2011)  Meanwhile, the inception of the Archaeology Data Service, a digital repository for heritage data, in 1996 meant archaeological datasets were increasingly digitally archived, for instance the data from the Channel Tunnel Rail Link Project (Foreman 2018) or a recent large-scale research excavation at Silchester (University of Reading 2018)  In these cases, archaeobotanical data is available to download as a  csv file 
Whilst the data publication strategy of large excavations was shifting, the availability of data from post-excavation assessment reports has remained challenging  So-called ‘grey literature’ results from the initial evaluation stage of developer-funded investigations and accompanying post-excavation assessment often contain a semi-quantitative evaluation of archaeobotanical samples on a scale of abundance  Whilst paper reports were initially deposited with county Historic Environment Records, a process of digitisation focussing on the Roman period has meant many pdfs are now available through the ADS (Allen et al  2018), whilst born-digital reports are now deposited through OASIS (Online AccesS to the Index of archaeological investigationS), as part of the reporting process (Evans 2015), althought the extent to which specialist appendices are included is variable 
These varying ‘publication’ strategies means archaeobotanical data is often available somewhere for recent developer-funded excavations and large-scale developer-funded excavations, even if much of this data is as a printed table or  pdf file (Evans 2015; Evans and Moore 2014)  However, academic journals are typically perceived as the most high-status publication venue for archaeobotanical data, and a crucial publication venue for academics in order to comply with institutional requirements and the norms of career progression  Aside from the problem of access to pay-walled journals by those without institutional subscriptions to all journals, the publication of primary data alongside research articles faces various problems, from the outright lack of inclusion of data, to problematic curation of supplementary data and a lack of peer review of data (Costello et al  2013; Warinner and d’Alpoim Guedes 2014: 155; Whitlock, 2011)  The extent of these problems for archaeobotany is currently unknown  Given the growth in archaeobotanical data production as methodologies are introduced into many new regions and periods over the last decade, it is vital that we know whether the mass of new data being produced is made available and is being reused 
Recent important advances within archaeobotanical data sharing have focussed on the construction of the ARBODAT database, developed by Angela Kreuz at the Kommission für Archäologische Landesforschung in Hessen  The database is used by a range of researchers in Germany, the Czech Republic, France and England (Kreuz & Schäfer 2002)  Data sharing enabled by the use of this database has facilitated research on Neolithic agriculture in Austria, Bulgaria and Germany (Kreuz et al  2005), and Bronze Age agriculture in Europe (Stika and Heiss 2012)  The use of this database makes data integration between specialists easier due to the shared data structure and metadata description, but often the primary archaeobotanical data is not made publicly available 
2 2  Meta-analysis in archaeobotany
Beyond the need to preserve information, a key reason for the formal sharing of archaeobotanical data is in its reuse to facilitate subsequent research  There has been a long-standing concern within archaeobotany with the need to aggregate datasets and identify temporal and spatial patterns  The palaeobotanist Clement Reid maintained his own database of Quaternary plant records in the late nineteenth century (Reid 1899), which formed the foundation of Godwin’s Quaternary database (Godwin 1975)  Mid-twentieth century studies of prehistoric plant use compiled lists of archaeobotanical materials incorporating full references and the location of the archive (Jessen & Helbaek 1944)  The International Work Group for Palaeoethnobotany was itself founded in 1968 in part with the aim to compile archaeobotanical data, first realised through the publication of Progress in Old World Palaeoethnobotany (Van Zeist, Wasylikowa & Behre 1991), and subsequently through the publication of annual lists of new records of cultivated plants (Kroll 1997) 
To take England as an example, regional reviews produced by state heritage authorities have provided catalogues of archaeobotanical datasets in particular time periods and regions (e g  Murphy 1998)  When one archaeobotanist has undertaken the majority of study within a region, pieces of synthesis within books have provided a relatively comprehensive review, for instance in the Thames Valley, UK (Lambrick & Robinson 2009)  Over the last decade regional synthesis has occurred within several funded reviews which produced catalogues of sites with archaeobotanical data (Lodwick 2014; McKerracher 2018; Parks 2012) and a series of funded projects in France have enabled regional synthesis (Lepetz & Zech-Matterne 2017)  However, many of these reviews are not accompanied by an available underlying database, and draw upon reports which are themselves hard to access 
Through the 1990s and 2000s, a series of databases were constructed in order to collate data from sites in a particular region and facilitate synthetic research  However, these databases have all placed the role of data archiving onto later projects specifically funded to collate data, rather than sourcing datasets at the time of publication  Such a model is unsustainable, and is unlikely to result in all available datasets being compiled  The Archaeobotanical Computer Database (ABCD), published in 1996 in the first issue of Internet Archaeology, contained much of the archaeobotanical data from Britain available at the time of publication, largely at the level of individual samples  The database was compiled between 1989 and 1994 and is still accessible through the accompanying online journal publication (Tomlinson & Hall 1996)  The ABCD made major contributions to recent reviews of the Roman and Medieval periods (Van der Veen, Livarda & Hill 2008; Van der Veen, Hill & Livarda 2013)  However, the database could only be centrally updated, with the online resource remaining a static version, lacking much of the new data produced subsequent to the implementation of PPG16 in 1990  The ADEMNES database, created through a research project undertaken at the Universities of Freiburg and Tübingen, contains data from 533 eastern Mediterranean and Near Eastern sites (Riehl & Kümmel 2005)  Kroll has maintained the Archaeobotanical Literature Database to accompany the Vegetation History and Archaeobotany articles (Kroll 2005) now accessible as a database (Kirleis & Schmültz 2018)  Numerous other databases have collated archaeobotanical studies, including the COMPAG project (Fuller et al  2015), the Cultural Evolution of Neolithic Europe project (Colledge 2016), RADAR in the Netherlands (van Haaster and Brinkkemper 1995), BRAIN Botanical Records of Archaeobotany Italian Network (Mercuri et al  2015) and CZAD – Archaeobotanical database of Czech Republic (CZAD 2019) 
The majority of databases have a restricted regional coverage, whilst research-project driven period-specific databases provide overlapping content  Whilst there are a wide range of archaeobotanical databases available, few contain primary datasets (other than the ABCD) which can be downloaded as  csv files  Data which is most commonly available are bibliographic references per site, with some indications of mode of preservation, quantity of archaeobotanical data, and sometimes taxa present  The databases do not inter-relate to each other, and function primarily as bibliographic sources enabling researchers to find comparative sites or to identify published datasets which need to be re-tabulated prior to meta-analysis  The IWGP website curates a list of resources, but otherwise the resources are often disseminated through the archaeobotany jiscmail list 
Beyond the aim of cataloguing archaeobotanical data within a region and period, meta-analysis is often used in archaeobotany to identify spatial and chronological trends in a range of past human activities, for instance crop choice, crop husbandry practices, plant food consumption, the trade in luxury foods or the use of plants in ritual  Meta-analysis can be undertaken on the basis of simple presence/absence data per site, but in order for such analysis to be rigorous and comparable, sample-level data must be utilised  For instance, sample-level data is required for meta-studies, in order to identify high-quality samples of unmixed crops for weed ecology analysis (Bogaard 2004), to assess the importance of context in the evaluation of wild plant foods (Wallace et al  2019), or to use volumetric measurements as a proxy for scale (Lodwick 2017b)  The reuse of archaeobotanical data also extends to include datasets used as “controls” in commonly used forms of statistical analysis, for instance Jones’s weed data from Amorgos, Greece, which is utilised as a control group in discriminant analysis of crop-processing stage (Jones 1984), and ethnographic observations of crop items in different crop-processing stages (Jones 1990) 
2 3  Open data principles and solutions
Debates over issues of data publication and meta-analysis have been on-going across scientific disciplines over the last decade (Editors 2009), and have been summarised within principles of open science, as recently set out in relation to archaeology (Marwick et al  2017)  Open Data is one of the three core principles for promoting transparency in social science (Miguel et al  2014)  The FAIR principles, developed by representatives from academia, industry, funding agencies, industry and publishers, provide four principles which data sharing should meet for use by both humans and machines – Findability, Accessibility, Interoperability, and Reusability (Wilkinson et al  2016)  A recent report assessing the adoption and impact of FAIR principles across academia in the UK included archaeology as a case study (Allen and Hartland 2018: 46)  It reported how the ADS was often used to archive data, but that “The journal itself provides the “story” about the data, the layer that describes what the data is, how it was collected and what the author thinks it means ” The report also raises the problem that smaller projects may not have the funding to utilise the ADS, meaning that other repositories are utilised  Increasingly, archaeological data is made available through a wide range of data repositories (OSF, Mendeley Data, Zenodo, Open Context), university data repositories (e g  ORA-Data), or social networking sites for academics (Academia edu, ResearchGate)



ground_truth: Using the vacuum Einstein equation and the Baez-Bunn form.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Does DUO contain more instances per image than COCO?

Read the following text and answer briefly.

Context 3: \section{Introduction}
Underwater robot picking is to use the robot to automatically capture sea creatures like holothurian, echinus, scallop, or starfish in an open-sea farm where underwater object detection is the key technology for locating creatures  Until now, the datasets used in this community are released by the Underwater Robot Professional Contest (URPC$\protect\footnote{Underwater Robot Professional Contest: {\bf http://en cnurpc org} }$) beginning from 2017, in which URPC2017 and URPC2018 are most often used for research  Unfortunately, as the information listed in Table \ref{Info}, URPC series datasets do not provide the annotation file of the test set and cannot be downloaded after the contest  
Therefore, researchers \cite{2020arXiv200511552C,2019arXiv191103029L} first have to divide the training data into two subsets, including a new subset of training data and a new subset of testing data, and then train their proposed method and other \emph{SOTA} methods  On the one hand, training other methods results in a significant increase in workload  On the other hand, different researchers divide different datasets in different ways, 
\begin{table}[t]
\renewcommand\tabcolsep{3 5pt}
\caption{Information about all the collected datasets  * denotes the test set's annotations are not available  \emph{3} in Class means three types of creatures are labeled, \emph{i e ,} holothurian, echinus, and scallop  \emph{4} means four types of creatures are labeled (starfish added)  Retention represents the proportion of images that retain after similar images have been removed }
\centering 
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Dataset&Train&Test&Class&Retention&Year \\ 
\hline 
URPC2017&17,655&985*&3&15\%&2017 \\
\hline
URPC2018&2,901&800*&4&99\%&2018 \\
\hline
URPC2019&4,757&1,029*&4&86\%&2019 \\
\hline
URPC2020$_{ZJ}$&5,543&2,000*&4&82\%&2020 \\
\hline
URPC2020$_{DL}$&6,575&2,400*&4&80\%&2020 \\
\hline
UDD&1,827&400&3&84\%&2020 \\
\hline 

\end{tabular}
\label{Info}
\end{table}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=1\linewidth]{example pdf}
\end{center}
   \caption{Examples in DUO, which show a variety of scenarios in underwater environments }
\label{exam}
\end{figure*}
causing there is no unified benchmark to compare the performance of different algorithms 
In terms of the content of the dataset images, there are a large number of similar or duplicate images in the URPC datasets  URPC2017 only retains 15\% images after removing similar images compared to other datasets  Thus the detector trained on URPC2017 is easy to overfit and cannot reflect the real performance 
For other URPC datasets, the latter also includes images from the former, \emph{e g }, URPC2019 adds 2,000 new images compared to URPC2018; compared with URPC2019, URPC2020$_{ZJ}$ adds 800 new images  The URPC2020$_{DL}$ adds 1,000 new images compared to the URPC2020$_{ZJ}$  It is worth mentioning that the annotation of all datasets is incomplete; some datasets lack the starfish labels and it is easy to find error or missing labels  \cite{DBLP:conf/iclr/ZhangBHRV17} pointed out that although the CNN model has a strong fitting ability for any dataset, the existence of dirty data will significantly weaken its robustness 
Therefore, a reasonable dataset (containing a small number of similar images as well as an accurate annotation) and a corresponding recognized benchmark are urgently needed to promote community development 


To address these issues, we introduce a dataset called Detecting Underwater Objects (DUO) by collecting and re-annotating all the available underwater datasets  It contains 7,782 underwater images after deleting overly similar images and has a more accurate annotation with four types of classes (\emph{i e ,} holothurian, echinus, scallop, and starfish)  
Besides, based on the MMDetection$\protect\footnote{MMDetection is an open source object detection toolbox based on PyTorch  {\bf https://github com/open-mmlab/mmdetection}}$ \cite{chen2019mmdetection} framework, we also provide a \emph{SOTA} detector benchmark containing efficiency and accuracy indicators, providing a reference for both academic research and industrial applications  It is worth noting that JETSON AGX XAVIER$\protect\footnote{JETSON AGX XAVIER is an embedded development board produced by NVIDIA which could be deployed in an underwater robot  Please refer {\bf https://developer nvidia com/embedded/jetson-agx-xavier-developer-kit} for more information }$ was used to assess all the detectors in the efficiency test in order to simulate robot-embedded environment  DUO will be released in https://github com/chongweiliu soon 

In summary, the contributions of this paper can be listed as follows 

  $\bullet$ By collecting and re-annotating all relevant datasets, we introduce a dataset called DUO with more reasonable annotations as well as a variety of underwater scenes 

  $\bullet$ We provide a corresponding benchmark of \emph{SOTA} detectors on DUO including efficiency and accuracy indicators which could be a reference for both academic research and industrial applications  


\pagestyle{empty}
\section{Background}
In the year of 2017, underwater object detection for open-sea farming is first proposed in the target recognition track of Underwater Robot Picking Contest 2017$\protect\footnote{From 2020, the name has been changed into Underwater Robot Professional Contest which is also short for URPC }$ (URPC2017) which aims to promote the development of theory, technology, and industry of the underwater agile robot and fill the blank of the grabbing task of the underwater agile robot  The competition sets up a target recognition track, a fixed-point grasping track, and an autonomous grasping track  The target recognition track concentrates on finding the {\bf high accuracy and efficiency} algorithm which could be used in an underwater robot for automatically grasping 

The datasets we used to generate the DUO are listed below  The detailed information has been shown in Table \ref{Info} 

  {\bf URPC2017}: It contains 17,655 images for training and 985 images for testing and the resolution of all the images is 720$\times$405  All the images are taken from 6 videos at an interval of 10 frames  However, all the videos were filmed in an artificial simulated environment and pictures from the same video look almost identical  
  
   {\bf URPC2018}: It contains 2,901 images for training and 800 images for testing and the resolutions of the images are 586$\times$480, 704$\times$576, 720$\times$405, and 1,920$\times$1,080  The test set's annotations are not available  Besides, some images were also collected from an artificial underwater environment 
  
  {\bf URPC2019}: It contains 4,757 images for training and 1029 images for testing and the highest resolution of the images is 3,840$\times$2,160 captured by a GOPro camera  The test set's annotations are also not available and it contains images from the former contests 
  
  {\bf URPC2020$_{ZJ}$}: From 2020, the URPC will be held twice a year  It was held first in Zhanjiang, China, in April and then in Dalian, China, in August  URPC2020$_{ZJ}$ means the dataset released in the first URPC2020 and URPC2020$_{DL}$ means the dataset released in the second URPC2020  This dataset contains 5,543 images for training and 2,000 images for testing and the highest resolution of the images is 3,840$\times$2,160  The test set's annotations are also not available 
  
  {\bf URPC2020$_{DL}$}: This dataset contains 6,575 images for training and 2,400 images for testing and the highest resolution of the images is 3,840$\times$2,160  The test set's annotations are also not available 
  
  {\bf UDD \cite{2020arXiv200301446W}}: This dataset contains 1,827 images for training and 400 images for testing and the highest resolution of the images is 3,840$\times$2,160  All the images are captured by a diver and a robot in a real open-sea farm 

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\linewidth]{pie pdf}
\end{center}
   \caption{The proportion distribution of the objects in DUO }
\label{pie}
\end{figure}



\begin{figure*}
  \centering
  \subfigure[]{\includegraphics[width=3 45in]{imagesize pdf}}
  \subfigure[]{\includegraphics[width=3 45in]{numInstance pdf}}
  \caption{(a) The distribution of instance sizes for DUO; (b) The number of categories per image }
  \label{sum}
\end{figure*}
\section{Proposed Dataset}

\subsection{Image Deduplicating}
As we explained in Section 1, there are a large number of similar or repeated images in the series of URPC datasets  Therefore, it is important to delete duplicate or overly similar images and keep a variety of underwater scenarios when we merge these datasets together  Here we employ the Perceptual Hash algorithm (PHash) to remove those images  PHash has the special property that the hash value is dependent on the image content, and it remains approximately the same if the content is not significantly modified  Thus we can easily distinguish different scenarios and delete duplicate images within one scenario  

After deduplicating, we obtain 7,782 images (6,671 images for training; 1,111 for testing)  The retention rate of the new dataset is 95\%, which means that there are only a few similar images in the new dataset  Figure \ref{exam} shows that our dataset also retains various underwater scenes 

\subsection{Image Re-annotation}
Due to the small size of objects and the blur underwater environment, there are always missing or wrong labels in the existing annotation files  In addition, some test sets' annotation files are not available and some datasets do not have the starfish annotation  In order to address these issues, we follow the next process which combines a CNN model and manual annotation to re-annotate these images  Specifically, we first train a detector (\emph{i e ,} GFL \cite{li2020generalized}) with the originally labeled images  After that, the trained detector predicts all the 7,782 images  We treat the prediction as the groundtruth and use it to train the GFL again  We get the final GFL prediction called {\bf the coarse annotation}  Next, we use manual correction to get the final annotation called {\bf the fine annotation}  Notably, we adopt the COCO \cite{Belongie2014} annotation form as the final format 
\subsection{Dataset Statistics}
{\bf The proportion of classes}: The total number of objects is 74,515  Holothurian, echinus, scallop, and starfish are 7,887, 50,156, 1,924, and 14,548, respectively  Figure \ref{pie} shows the proportion of each creatures where echinus accounts for 67 3\% of the total  The whole data distribution shows an obvious long-tail distribution because the different economic benefits of different seafoods determine the different breed quantities 

{\bf The distribution of instance sizes}: Figure \ref{sum}(a) shows an instance size distribution of DUO  \emph{Percent of image size} represents the ratio of object area to image area, and \emph{Percent of instance} represents the ratio of the corresponding number of objects to the total number of objects  Because of these small creatures and high-resolution images, the vast majority of objects occupy 0 3\% to 1 5\% of the image area 

{\bf The instance number per image}: Figure \ref{sum}(b) illustrates the number of categories per image for DUO  \emph{Number of instances} represents the number of objects one image has, and \emph{ Percentage of images} represents the ratio of the corresponding number of images to the total number of images  Most images contain between 5 and 15 instances, with an average of 9 57 instances per image 

{\bf Summary}:
In general, smaller objects are harder to detect  For PASCAL VOC \cite{Everingham2007The} or COCO \cite{Belongie2014}, roughly 50\% of all objects occupy no more than 10\% of the image itself, and others evenly occupy from 10\% to 100\%  
In the aspect of instances number per image, COCO contains 7 7 instances per image and VOC contains 3  In comparison, DUO has 9 57 instances per image and most instances less than 1 5\% of the image size 
Therefore, DUO contains almost exclusively massive small instances and has the long-tail distribution at the same time, which means it is promising to design a detector to deal with massive small objects and stay high efficiency at the same time for underwater robot picking 

\section{Benchmark}
Because the aim of underwater object detection for robot picking is to find {\bf the high accuracy and efficiency} algorithm, we consider both the accuracy and efficiency evaluations in the benchmark as shown in Table \ref{ben} 

\subsection{Evaluation Metrics}
Here we adopt the standard COCO metrics (mean average precision, \emph{i e ,} mAP) for the accuracy evaluation and also provide the mAP of each class due to the long-tail distribution 

{\bf AP} -- mAP at IoU=0 50:0 05:0 95 

{\bf AP$_{50}$} --  mAP at IoU=0 50 

{\bf AP$_{75}$} --  mAP at IoU=0 75  

{\bf AP$_{S}$} --   {\bf AP} for small objects of area smaller than 32$^{2}$ 

{\bf AP$_{M}$} --   {\bf AP} for objects of area between 32$^{2}$ and 96$^{2}$ 

{\bf AP$_{L}$} --   {\bf AP} for large objects of area bigger than 96$^{2}$ 

{\bf AP$_{Ho}$} --  {\bf AP} in holothurian 

{\bf AP$_{Ec}$} --  {\bf AP} in echinus 

{\bf AP$_{Sc}$} --  {\bf AP} in scallop 

{\bf AP$_{St}$} --  {\bf AP} in starfish 


For the efficiency evaluation, we provide three metrics:

{\bf Param } --  The parameters of a detector 

{\bf FLOPs} --  Floating-point operations per second 

{\bf FPS} --  Frames per second 

Notably, {\bf FLOPs} is calculated under the 512$\times$512 input image size and {\bf FPS} is tested on a JETSON AGX XAVIER under MODE$\_$30W$\_$ALL  

\subsection{Standard Training Configuration}
We follow a widely used open-source toolbox, \emph{i e ,} MMDetection (V2 5 0) to produce up our benchmark  During the training, the standard configurations are as follows:

  $\bullet$ We initialize the backbone models (\emph{e g ,} ResNet50) with pre-trained parameters on ImageNet \cite{Deng2009ImageNet} 

  $\bullet$ We resize each image into 512 $\times$ 512 pixels both in training and testing  Each image is flipped horizontally with 0 5 probability during training 

  $\bullet$ We normalize RGB channels by subtracting 123 675, 116 28, 103 53 and dividing by 58 395, 57 12, 57 375, respectively 

  $\bullet$ SGD method is adopted to optimize the model  The initial learning rate is set to be 0 005 in a single GTX 1080Ti with batchsize 4 and is decreased by 0 1 at the 8th and 11th epoch, respectively  WarmUp \cite{2019arXiv190307071L} is also employed in the first 500 iterations  Totally there are 12 training epochs 

  $\bullet$ Testing time augmentation (\emph{i e ,} flipping test or multi-scale testing) is not employed 



\subsection{Benchmark Analysis}
Table \ref{ben} shows the benchmark for the \emph{SOTA} methods  Multi- and one- stage detectors with three kinds of backbones (\emph{i e ,} ResNet18, 50, 101) give a comprehensive assessment on DUO  We also deploy all the methods to AGX to assess efficiency 

In general, the multi-stage (Cascade R-CNN) detectors have high accuracy and low efficiency, while the one-stage (RetinaNet) detectors have low accuracy and high efficiency  However, due to recent studies \cite{zhang2019bridging} on the allocation of more reasonable positive and negative samples in training, one-stage detectors (ATSS or GFL) can achieve both high accuracy and high efficiency 

\begin{table*}[htbp]
\renewcommand\tabcolsep{3 0pt}

\begin{center}
\caption{Benchmark of \emph{SOTA} detectors (single-model and single-scale results) on DUO  FPS is measured on the same machine with a JETSON AGX XAVIER under the same MMDetection framework, using a batch size of 1 whenever possible  R: ResNet } 
\label{ben}
\begin{tabular}{|l|l|c|c|c|ccc|ccc|cccc|}
\hline
Method&Backbone&Param &FLOPs&FPS&AP&AP$_{50}$&AP$_{75}$&AP$_{S}$&AP$_{M}$&AP$_{L}$&AP$_{Ho}$&AP$_{Ec}$&AP$_{Sc}$&AP$_{St}$ \\ 
\hline 
\emph{multi-stage:} &&&&&&&&&&&&&& \\

\multirow{3}{*}{Faster R-CNN \cite{Ren2015Faster}}
&R-18&28 14M&49 75G&5 7&50 1&72 6&57 8&42 9&51 9&48 7&49 1&60 1&31 6&59 7\\
&R-50&41 14M&63 26G&4 7&54 8&75 9&63 1&53 0&56 2&53 8&55 5&62 4&38 7&62 5\\
&R-101&60 13M&82 74G&3 7&53 8&75 4&61 6&39 0&55 2&52 8&54 3&62 0&38 5&60 4\\
\hline

\multirow{3}{*}{Cascade R-CNN \cite{Cai_2019}}
&R-18&55 93M&77 54G&3 4&52 7&73 4&60 3&\bf 49 0&54 7&50 9&51 4&62 3&34 9&62 3\\
&R-50&68 94M&91 06G&3 0&55 6&75 5&63 8&44 9&57 4&54 4&56 8&63 6&38 7&63 5\\
&R-101&87 93M&110 53G&2 6&56 0&76 1&63 6&51 2&57 5&54 7&56 2&63 9&41 3&62 6\\
\hline

\multirow{3}{*}{Grid R-CNN \cite{lu2019grid}}
&R-18&51 24M&163 15G&3 9&51 9&72 1&59 2&40 4&54 2&50 1&50 7&61 8&33 3&61 9\\
&R-50&64 24M&176 67G&3 4&55 9&75 8&64 3&40 9&57 5&54 8&56 7&62 9&39 5&64 4\\
&R-101&83 24M&196 14G&2 8&55 6&75 6&62 9&45 6&57 1&54 5&55 5&62 9&41 0&62 9\\
\hline

\multirow{3}{*}{RepPoints \cite{yang2019reppoints}}
&R-18&20 11M&\bf 35 60G&5 6&51 7&76 9&57 8&43 8&54 0&49 7&50 8&63 3&33 6&59 2\\
&R-50&36 60M&48 54G&4 8&56 0&80 2&63 1&40 8&58 5&53 7&56 7&65 7&39 3&62 3\\
&R-101&55 60M&68 02G&3 8&55 4&79 0&62 6&42 2&57 3&53 9&56 0&65 8&39 0&60 9\\
\hline 
\hline 
\emph{one-stage:} &&&&&&&&&&&&&& \\
\multirow{3}{*}{RetinaNet \cite{Lin2017Focal}}
&R-18&19 68M&39 68G&7 1&44 7&66 3&50 7&29 3&47 6&42 5&46 9&54 2&23 9&53 8\\
&R-50&36 17M&52 62G&5 9&49 3&70 3&55 4&36 5&51 9&47 6&54 4&56 6&27 8&58 3\\
&R-101&55 16M&72 10G&4 5&50 4&71 7&57 3&34 6&52 8&49 0&54 6&57 0&33 7&56 3\\
\hline 

\multirow{3}{*}{FreeAnchor \cite{2019arXiv190902466Z}}
&R-18&19 68M&39 68G&6 8&49 0&71 9&55 3&38 6&51 7&46 7&47 2&62 8&28 6&57 6\\
&R-50&36 17M&52 62G&5 8&54 4&76 6&62 5&38 1&55 7&53 4&55 3&65 2&35 3&61 8\\
&R-101&55 16M&72 10G&4 4&54 6&76 9&62 9&36 5&56 5&52 9&54 0&65 1&38 4&60 7\\
\hline 

\multirow{3}{*}{FoveaBox \cite{DBLP:journals/corr/abs-1904-03797}}
&R-18&21 20M&44 75G&6 7&51 6&74 9&57 4&40 0&53 6&49 8&51 0&61 9&34 6&59 1\\
&R-50&37 69M&57 69G&5 5&55 3&77 8&62 3&44 7&57 4&53 4&57 9&64 2&36 4&62 8\\
&R-101&56 68M&77 16G&4 2&54 7&77 3&62 3&37 7&57 1&52 4&55 3&63 6&38 9&60 8\\
\hline 

\multirow{3}{*}{PAA \cite{2020arXiv200708103K}}
&R-18&\bf 18 94M&38 84G&3 0&52 6&75 3&58 8&41 3&55 1&50 2&49 9&64 6&35 6&60 5\\
&R-50&31 89M&51 55G&2 9&56 8&79 0&63 8&38 9&58 9&54 9&56 5&66 9&39 9&64 0\\
&R-101&50 89M&71 03G&2 4&56 5&78 5&63 7&40 9&58 7&54 5&55 8&66 5&42 0&61 6\\
\hline 

\multirow{3}{*}{FSAF \cite{zhu2019feature}}
&R-18&19 53M&38 88G&\bf 7 4&49 6&74 3&55 1&43 4&51 8&47 5&45 5&63 5&30 3&58 9\\
&R-50&36 02M&51 82G&6 0&54 9&79 3&62 1&46 2&56 7&53 3&53 7&66 4&36 8&62 5\\
&R-101&55 01M&55 01G&4 5&54 6&78 7&61 9&46 0&57 1&52 2&53 0&66 3&38 2&61 1\\
\hline 

\multirow{3}{*}{FCOS \cite{DBLP:journals/corr/abs-1904-01355}}
&R-18&\bf 18 94M&38 84G&6 5&48 4&72 8&53 7&30 7&50 9&46 3&46 5&61 5&29 1&56 6\\
&R-50&31 84M&50 34G&5 4&53 0&77 1&59 9&39 7&55 6&50 5&52 3&64 5&35 2&60 0\\
&R-101&50 78M&69 81G&4 2&53 2&77 3&60 1&43 4&55 4&51 2&51 7&64 1&38 5&58 5\\
\hline 

\multirow{3}{*}{ATSS \cite{zhang2019bridging}}
&R-18&\bf 18 94M&38 84G&6 0&54 0&76 5&60 9&44 1&56 6&51 4&52 6&65 5&35 8&61 9\\
&R-50&31 89M&51 55G&5 2&58 2&\bf 80 1&66 5&43 9&60 6&55 9&\bf 58 6&67 6&41 8&64 6\\
&R-101&50 89M&71 03G&3 8&57 6&79 4&65 3&46 5&60 3&55 0&57 7&67 2&42 6&62 9\\
\hline 

\multirow{3}{*}{GFL \cite{li2020generalized}}
&R-18&19 09M&39 63G&6 3&54 4&75 5&61 9&35 0&57 1&51 8&51 8&66 9&36 5&62 5\\
&R-50&32 04M&52 35G&5 5&\bf 58 6&79 3&\bf 66 7&46 5&\bf 61 6&55 6&\bf 58 6&\bf 69 1&41 3&\bf 65 3\\
&R-101&51 03M&71 82G&4 1&58 3&79 3&65 5&45 1&60 5&\bf 56 3&57 0&\bf 69 1&\bf 43 0&64 0\\


\hline 
\end{tabular}
\end{center}
\end{table*}
Therefore, in terms of accuracy, the accuracy difference between the multi- and the one- stage methods in AP is not obvious, and the AP$_{S}$ of different methods is always the lowest among the three size AP  For class AP, AP$_{Sc}$ lags significantly behind the other three classes because it has the smallest number of instances  In terms of efficiency, large parameters and FLOPs result in low FPS on AGX, with a maximum FPS of 7 4, which is hardly deployable on underwater robot  Finally, we also found that ResNet101 was not significantly improved over ResNet50, which means that a very deep network may not be useful for detecting small creatures in underwater scenarios  

Consequently, the design of high accuracy and high efficiency detector is still the main direction in this field and there is still large space to improve the performance 
In order to achieve this goal, a shallow backbone with strong multi-scale feature fusion ability can be proposed to extract the discriminant features of small scale aquatic organisms; a specially designed training strategy may overcome the DUO's long-tail distribution, such as a more reasonable positive/negative label sampling mechanism or a class-balanced image allocation strategy within a training batch 

\section{Conclusion}
In this paper, we introduce a dataset (DUO) and a corresponding benchmark to fill in the gaps in the community  DUO contains a variety of underwater scenes and more reasonable annotations  Benchmark includes efficiency and accuracy indicators to conduct a comprehensive evaluation of the \emph{SOTA} decoders  The two contributions could serve as a reference for academic research and industrial applications, as well as promote community development 



ground_truth: Yes, DUO has 9.57 instances per image while COCO contains 7.7.