PhotDM.tex

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\title{IVOA Photometry Data Model}

\ivoagroup{Data Model Working Group}

\author{Jesús Salgado}
\author{Mireille Louys}
\author{Laurent Michel}
\author{Carlos Rodrigo}
\author{Pedro Osuna}
\author{Mark Allen}
\author{Jonathan McDowell}
\author{Deborah Baines}
\author{Jesús Maíz Apellániz}
\author{Evanthia Hatziminaoglou}
\author{Sebastien \mbox{Derriere}}
\author{Gerard Lemson}


\editor{Jesús Salgado}
\editor{Mireille Louys}


\begin{document}

\begin{abstract}
The Photometry Data Model (\textbf{PhotDM}) standard describes photometry
filters, photometric systems, magnitude systems, zero points and their
interrelation with the other IVOA data models through a simple data model.
Particular attention is given necessarily to optical photometry where
specifications of magnitude systems and photometric zero points are required
to convert photometric measurements into physical flux density units.
\end{abstract}

\section*{Acknowledgments}
We acknowledge the support of EuroVO Science Advisory Committee for the review of the
initial versions of PhotDM-1.0 , and the Spanish VO developers who have contributed
to the data model reference implementations.
The design changes and the modeling for the translation to VODML was supported in part by the
ESCAPE project (the European Science Cluster of Astronomy and Particle Physics ESFRI Research Infrastructures)
that has received funding from the European Horizon 2020 research and innovation programme under the Grant Agreement n. 824064.


\pagebreak

\section*{Link to IVOA Architecture}
The figure below shows where the IVOA Photometry Data Model fits within the
IVOA architecture:


%%%%%%%%%%%%%%%%%%%% Figure/Image No: 1 starts here %%%%%%%%%%%%%%%%%%%%

\begin{figure}[H]
\centering

% As of ivoatex 1.2, the architecture diagram is generated by ivoatex in
% SVG; copy ivoatex/archdiag-full.xml to archdiag.xml and throw out
% all lines not relevant to your standard.
% Notes don't generally need this.  If you don't copy archdiag.xml,
% you must remove archdiag.svg from FIGURES in the Makefile.

\includegraphics[width=0.9\textwidth]{role_diagram.pdf}
\caption{PhotDM can be used to enhance and abstract SSAP and TAP access, in particular
to help on the automatic translation of magnitudes to fluxes and to add provenance
metadata to these magnitudes. Also it can be used to generate SEDs using the SpectralDM.
PhotDM-1.1 makes use of UTypes, for backward compatibility to PhotDM-1.0 , and of UCDs as well as VOUnits.
PhotDM can have  serialisations in the VOTable format. This version relies on the VODML modeling principles and
is described in a VODML xml reference document .}
\label{fig:archdiag}
\end{figure}

\section*{Changes from Version 1.0} \label{changesTable}

%%%%%%%%%%%%%%%%%%%% change log starts here %%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
 			\centering
\begin{tabular}{p{3.in}p{1in}p{0.8in}}
\hline
%row no:1
\multicolumn{1}{|p{3.75in}}{\textbf{Change}} &
\multicolumn{1}{|p{0.72in}}{\textbf{Section}} &
\multicolumn{1}{|p{0.9in}|}{\textbf{Date}} \\
\hline

%row no:2
\multicolumn{1}{|p{3.75in}}{First PhotDM 1.1 Latex version} &
\multicolumn{1}{|p{0.72in}}{All} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2021/10/07}} \\
\hline

%row no:3
\multicolumn{1}{|p{3.75in}}{Use the Modelio class diagram
figure} &
\multicolumn{1}{|p{0.72in}}{\ref{datamodel}} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2021/10/25}} \\
\hline

%row no:4
\multicolumn{1}{|p{3.75in}}{Data model summary updates and
correction of UCD tags} &
\multicolumn{1}{|p{0.72in}}{All} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2021/12/21}} \\
\hline

%row no:5
\multicolumn{1}{|p{3.75in}}{Add changes table} &
\multicolumn{1}{|p{0.72in}}{\ref{datamodel}} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2022/02/02}} \\
\hline

%row no:6
\multicolumn{1}{|p{3.75in}}{Add mapping example} &
\multicolumn{1}{|p{0.72in}}{\ref{appendixmapping}} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2022/03/01}} \\
\hline

%row no:7
\multicolumn{1}{|p{3.75in}}{v1.1 Proposed Recommendation} &
\multicolumn{1}{|p{0.72in}}{All} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2022/03/01}} \\
\hline

%row no:8
\multicolumn{1}{|p{3.75in}}{v1.1 PR / first round answer to RFC comments} &
\multicolumn{1}{|p{0.72in}}{All} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2022/05/23}} \\
\hline

%row no:8
\multicolumn{1}{|p{3.75in}}{v1.1 PR / answer to RFC comments from WGs} &
\multicolumn{1}{|p{0.72in}}{All} &
\multicolumn{1}{|p{0.9in}|}{{\fontsize{10pt}{12.0pt}\selectfont
2022/10/05}} \\

\hline
\end{tabular}
\end{table}
\pagebreak

\section{Introduction}
A key role of the VO is to help astronomers find data and to
combine that data in a scientifically meaningful way. A Spectral
Energy Distribution (SED) is an example of combining data whereby
flux density measurements of an astrophysical source at different
spectral energy coordinates (wavelengths/frequencies/energy)
\citep{doi:10.1146/annurev.astro.41.082801.100251,longo,connell,brujine}
are plotted as a
graph of flux density versus a spectral energy coordinate. SEDs that
cover a wide range of the electromagnetic spectrum are particularly
useful for identifying the underlying physical processes operating
in the astrophysical source, and the use of SEDs is becoming more
prevalent as astronomy takes an increasingly multi-wavelength
approach. To combine individual flux density measurements and their
spectral energy coordinates into an SED, these photometric measurements
must be described in sufficient detail to allow for the conversion to
compatible flux density and spectral energy units,  taking into
account the nature of the spectral energy bandpass of the measurements,
as well as the apertures and other details of  the measurements.
This document outlines a photometry data model to describe photometric
measurements in a standard way.

The Photometry Data Model aims to describe the essential elements
of flux density measurements made within all spectral energy domains
across the electromagnetic spectrum. In some domains this is
relatively straight forward, such as in radio astronomy where
measurements are commonly expressed in flux density units, and
where data are readily combined into SEDs. The data model fields
required to describe such a radio flux density measurement includes
a specification of the bandpass, the units of the measurement and
the associated uncertainties. Optical photometry measurements are
however commonly expressed in magnitudes, and a greater level of
description of the magnitude systems and bandpasses are required
to support the conversion of these measurements into flux densities
that could be combined into an SED. As such, much of this document
is necessarily devoted to defining the data model fields required
to describe optical photometry measurements.

Astronomical flux density measurements will often require a
greater level of description than provided by this simple model.
The level of accuracy required depends strongly on the scientific
use of the data. A study of broadband SEDs of active galaxies may
tolerate 20$\%$  uncertainties in the flux density measurements,
and it is usually sufficient in these cases to use average values
for the spectral energy coordinates of the bandpasses. Fitting to
stellar models or science that employs photometric measurements to
derive photometric redshifts requires a much greater level of accuracy.
To manage the different levels of description we take the overall
approach that the Photometry Data Model should include the most
generic elements required to describe photometric measurements,
and that the Photometry Data Model is intended to be used in
coordination with the IVOA Spectrum Data model and the IVOA
Characterization Data Model.

The scientific use case that has guided the choice of the level
of description of the metadata fields in the Photometry Data Model
is the use of the large collections of photometric data that are
published in catalogues (e.g. Vizier, \url{http://vizier.u-strasbg.fr/})
in SEDs. The Photometry Data Model provides the metadata fields for
describing the photometry measurements in catalogues, so that those
data could then be added to, or compared with an SED.

The intended practical use of the Photometry Data Model is that the
metadata fields defined here will be included in the metadata of
catalogues, or of photometry data stored as a pseudo-spectrum. These
data would then be made accessible using Simple Spectral Access Protocol
(SSAP) or Table Access Protocol (TAP) services so that the photometric
measurements can be used and combined in scientific software tools.

The proposed model is based on the description of the photometry
filters, and the description of how\ the\ units of the measurement
are related to flux density. The photometry filter description may
be as simple as specifying a central spectral energy coordinate and
a bandpass width. The more detailed description of optical bandpasses
is supported by allowing for specification of filter transmission
curves, and the   photometric zero points necessary for the conversion
of magnitudes to flux densities.

Information on the properties of filters is not always easily available,
and is sometimes only specified in manuals or in the literature and often
not in digital form. To aid the use of filters information, in particular
as part of the Photometry Data Model metadata fields, we propose a
mechanism for referencing external filter information. Such a
\textit{Filter Profile Service} has been implemented in the Spanish Virtual Observatory
\citep{2012ivoa.rept.1015R}, \citep{2020sea..confE.182R} and exposes this information so software
client applications could discover it.

The following sections of this document summarize some key points
about astronomical photometry (Section 2). The detailed metadata
structure of the data model is presented in Section 3. Section 4
describes use cases in which the model description could be used in
making photometry data available through VO protocols, and, very briefly,
how scientific tools could use this information.


The Photometry Data Model is related to other IVOA Data models (Spectrum DM, Characterization DM,
Observation and Provenance DM),  and is intended to provide photometry
metadata for data that would be accessed via the IVOA Data Access Protocols
such as SSAP (Simple Spectra Access Protocol) or TAP (Table Access Protocol).

As with most of the VO Data Models, the Photometry Data Model makes use of STC, UTypes, Units
and UCDs. PhotDM is serialisable with a VOTable.

\section{Astronomical Photometry}
Astronomical photometry refers to measuring the brightness, flux or
intensity of an astrophysical object. Consider an astronomical source
with a flux density at the observer $F(x)$, where $x$ is a spectral coordinate
(wavelength, frequency or energy).  The photometry measurement will be related
to $<F>$ a flux weighted integral of this flux density over an observed band
with a relative spectral response $T(x)$. The flux weighted integral in its most
simple form is defined as


\begin{equation} \label{eq:1}
\langle F\rangle = \int F(x)T(x)dx
\end{equation}

Calibration of photometric measurements is in general done by comparison to a
reference spectrum that has a known effective flux density $f_0$ at a specific
spectral band.


For this reference spectrum, the flux weighted integral is defined as:

\begin{equation} \label{eq:2}
\langle F_R \rangle= \int F_R (x)T(x)dx
\end{equation}


so that the effective flux density of the source can be evaluated as:

\begin{equation} \label{eq:3}
f = f_0 ( \frac{\langle F\rangle }{\langle F_R \rangle } )
\end{equation}

This represents the most simple and easy to use flux measurement. Flux
measurements expressed in physical flux density units can be easily combined
into SEDs. Many flux measurements published in catalogues of radio sources
for example are simple flux densities of this form.
\par

In optical photometry measurements are often expressed as magnitudes and it
is necessary to define the magnitude system being used, and the zero point
fluxes of the reference spectrum.
\par

Pogson magnitudes are defined as:

\begin{equation} \label{eq:4}
m = -2.5\log_{10} (\langle F\rangle )
\end{equation}

which when compared to a reference spectrum $F_R$ leads to:

\begin{equation} \label{eq:5}
m = m_R -2.5 \log_{10} \left(\langle F\rangle /\langle F_R \rangle \right)
\end{equation}

As explained above, this is equivalent to:

\begin{equation} \label{eq:6}
m = m_R - 2.5 \log_{10} (f/f_0 )
\end{equation}


so that

\begin{equation} \label{eq:7}
f = f_0 10^{-0.4(m - m_{R})}
\end{equation}


Using this expression a measurement in magnitudes can be converted to a
flux density, given the zero point flux of the reference spectrum.  The
magnitude of reference $m_{R}$ and the zero point $f_0$ will be defined
in the document. $m_{R}$ is initially chosen to be zero (or one for
linear photometric systems) in most of the photometric systems although,
continuous recalibration of the photometric system could
produce a deviation of this initial value.
\par

There are a number of magnitude systems that are defined by the reference
spectrum. The three most commonly used magnitude systems are the Vega
magnitude, $AB_{\nu }$ magnitude and $ST_{\lambda }$ magnitude systems.
The Vega magnitude system uses the spectrum of Vega (Alpha Lyrae) as the
reference spectrum $F_R (x)$. The $AB_{\nu }$ magnitude system uses
reference spectrum defined by a constant flux density per unit frequency
($F_{\nu }$) and the
$ST_{\lambda }$ magnitude system uses a reference spectrum of a constant
flux density per unit wavelength ($F_{\lambda }$). The values of $F_{\nu }$
and $F_{\lambda }$ that respectively define the zero points
$m_{AB,\nu } =0$ and
$m_{ST,\lambda } =0$ have been chosen to be the mean flux density of
Vega in the Johnson V band.

\begin{equation} \label{eq:7_0}
m_{AB,\nu } = 0
\end{equation}

\begin{equation} \label{eq:8}
f_\nu = 3.631 \times 10^{-20}\,erg\,cm^{-2}\,s^{-1}\,Hz^{-1}
\end{equation}

\begin{equation} \label{eq:9}
m_{ST,\lambda } = 0
\end{equation}

\begin{equation} \label{eq:10}
f_\lambda = 3.631 \times 10^{-9}\,erg\,cm^{-2}\,s^{-1}\,\angstrom^{-1}
\end{equation}


A convenient graphical representation of these systems is shown in
Figure 2.1 of the Synphot manual (publication date (December 1998)):

\url{http://stsdas.stsci.edu/Files/SynphotManual.pdf}.
\par

For a photometric system that uses Vega magnitudes, the zero point
flux for each filter is the average flux density of Vega over that
bandpass $f_{Vega}$. Some typical values of $f_{Vega}$ are tabulated in
\citep{2001eaa..book.....M} for the Johnson photometric system. Although
the agreed Vega spectrum has changed historically, the commonly referred to
spectrum of Vega in digital form described in \citep{2004AJ....127.3508B}
is available as file at \emph{alpha\_lyr\_stis\_010.fits} at:\par
\url{https://archive.stsci.edu/hlsps/reference-atlases/cdbs/current_calspec/}
\par

In the AB system, the flux density (in units of $erg\, cm^{-2}\, s^{-1}\, Hz^{-1}$)
corresponding to a given magnitude is simply obtained via:

\begin{equation} \label{eq:11}
f_{\nu} = 10^{-0.4(m_{AB}+48.6)}
\end{equation}

And, in the same way, in the ST system, the flux density (in units of $erg\,
cm^{-2}\, s^{-1}\, \angstrom^{-1}$) corresponding to a given magnitude is:

\begin{equation} \label{eq:12}
f_{\lambda} = 10^{-0.4(m_{ST}+21.1)}
\end{equation}

Another magnitude system is the Asinh magnitude system in which magnitudes
are defined as
\begin{equation} \label{eq:13}
m = \frac{2.5}{ln(10)} \left[ sinh^{-1}(\frac{f}{2bf_0}) +ln(b) \right]
\end{equation}


where b is known as the softening parameter. Details of the Asinh magnitude
system and the softening parameters are described in
\url{http://www.sdss.org/DR7/algorithms/photometry.html}
\par

\section{Photometry Data Model} \label{datamodel}
The model shown in Figure~\ref{fig:overview}, organizes the structure and detailed metadata
fields of the Photometry Data Model in a logical manner, and shows the
relationship to other IVOA data models. The metadata fields for each class
specify the essential elements required to describe a photometric measurement.
\par

The main class in this diagram is Photometry Filter. This class contains
all the attributes necessary to describe a filter from the data discovery
point of view.
\par

A Photometric System is a grouping of individual Photometry Filters. This may
represent a particular set of filters that are related in some way.
\par

A magnitude system is characterised for a certain reference spectrum that
will produce a certain zero point for a certain photometry filter. This
reference spectrum could be an ideal one (as in STmag and ABmag systems), a
Vega-like spectrum (as in VegaMag systems) (please notice that different Vega
spectrum versions have been historically used) or any other. In many cases,
the reference spectrum has been calculated as an average of spectra from several
astronomical objects. This would be characterised by a set of Source instances.
\par

A zero point would then be a flux value that can be considered as zero
magnitude, so its value will allow conversions from fluxes to magnitudes
and the other way around. It has associated a photometry filter and it also
depends on the magnitude system (reference spectrum) used to calculate this
magnitude.
\par

There are different types of zero points (Pogson, asinh, linear etc) that will
essentially differ in the way that getFluxFromMagnitude and getMagnitudeFromFlux
operators are implemented plus extra information that could be needed to do
these conversions.
\par

An intermediate class, PhotCal, can be understood as a certain photometry
filter instance, i.e., a certain photometry filter using a certain magnitude
system and linked to a certain zero point class.
%mir as suggested by MCD
PhotDM can interact with Spectral data models (any version) using PhotCal
as a node. It binds the filter and
zeropoint information to the Flux Axis calibration. It can be understood as the
calibration configuration used, bringing together a specific photometry filter
instance with magnitude system and zero point. It leverages the handling of
photometric data through IVOA protocols e.g. SSAP or TAP services.
\par

A Spectrum would have a Characterization Coordsys element that will have
associated a certain PhotCal element in the case of photometry data. Using
this information, magnitudes from different photometric systems could be
compared between them or compared to spectroscopic data expressed in flux.
\par

%%%%%%%%%%%%%%%%%%%% Figure/Image No: 22 starts here %%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
\includegraphics[angle=90,width=5.98in,height=7.19in]{./schema/PhotometryOverviewDiagram_20220520.png}
\caption{Overview class diagram of the Photometry Data Model maintained with Modelio 3.8.
Blue boxes indicate PhotDM pure classes and yellow ones indicate classes that have been reused from other
IVOA data models, connected to them by roles.}
\label{fig:overview}
\end{figure}

\begin{figure}[H]
\includegraphics[angle=0,width=5.98in ]{./schema/BaseDataTypesdiagram.png}
\caption{Attributes in the PhotDM classes belong to the base data types defined in the ivoa: VODML template model which is extended here by two classes : UCD for the UCD semantic tag, and ISOTime for time stamps defined and formatted following the DALI specification \citep{2017ivoa.spec.0517D} .}
\end{figure}

%%%%%%%%%%%%%%%%%%%% Figure/Image No: 22 Ends here %%%%%%%%%%%%%%%%%%%%



 %%%%%%%%%%%%  Starting New Page here %%%%%%%%%%%%%%

\newpage

 %%%%%%%%%%%%  Starting New Page here %%%%%%%%%%%%%%

\newpage
In order to fully describe values of the magnitudes inside photometry point
instances, the former PhotDM-1.0 used  physical quantity classes which used to
aggregate all the basic fields that compose a physical measurement: value,
error, units, etc. However, within the present specification, we will describe
individual attributes of the different quantities separately.
This is due to the fact that the granularity of quantities as defined in the former PhotDM-1.0
did not match the one proposed in the VODML ivoa template. The new UTypes for PhotDM-1.1 are also
generated from these individual attributes to facilitate the use within IVOA Data Access Layer protocols.

\par

\subsection{PhotometricSystem Class}
This class briefly describes the photometric system that contains a set of
photometry filters. Photometry filters can be contained in a certain
photometric system as part of the same observatory/telescope or as part
of a known system.
\par

\subsubsection{PhotometricSystem.description: String}
This String contains a human readable short-text representation of the
photometric system. This will allow client applications to display
textual information to final users.
\par

Examples:

%%%%%%%%%%%%%%%%%%%% Table No: 2 starts here %%%%%%%%%%%%%%%%%%%%

%row no:1
Sloan \par
Johnson
\bigskip

%%%%%%%%%%%%%%%%%%%% Table No: 2 ends here %%%%%%%%%%%%%%%%%%%%


\subsubsection{PhotometricSystem.detectorType: integer}
Detector type associated to this photometric system. Possible
values are:


%%%%%%%%%%%%%%%%%%%% Table No: 3 starts here %%%%%%%%%%%%%%%%%%%%


\begin{table}[ht]
 			\centering
\begin{tabular}{p{2.42in}p{0.8in}p{1.55in}}
\hline
%row no:1
\multicolumn{1}{|p{2.42in}}{Type of detector} &
\multicolumn{1}{|p{0.8in}}{Value} &
\multicolumn{1}{|p{1.55in}|}{Examples} \\
\hline
%row no:2
\multicolumn{1}{|p{2.42in}}{Energy Counter} &
\multicolumn{1}{|p{0.8in}}{0 (default)} &
\multicolumn{1}{|p{1.55in}|}{Energy amplifiers devices} \\
\hline
%row no:3
\multicolumn{1}{|p{2.42in}}{Photon Counter} &
\multicolumn{1}{|p{0.8in}}{1} &
\multicolumn{1}{|p{1.55in}|}{CCDs or photomultipliers} \\
\hline
\end{tabular}
\caption{Detector Types}
 \end{table}


%%%%%%%%%%%%%%%%%%%% Table No: 3 ends here %%%%%%%%%%%%%%%%%%%%

This will be used in order to decide how to calculate the
flux average in, e.g., the synthetic photometry calculations.
% mir MCD Suggestion
At present, this list is exhaustive.
See the description of the photometry filter
transmission curve section \ref{sec:transmissioncurve} to understand how to use this field.
\par

\subsection{PhotometryFilter Class}
This is the main class that describes a photometry filter.
\par

\subsubsection{PhotometryFilter.identifier: String}
This field identifies, in a unique way, within a certain Photometry
Filter Profile service, a filter. Although the main requirement of
this data model field is to be unique within a Filter Profile Service,
the suggested syntax would be:
\par

%%%%%%%%%%%%%%%%%%%% Table No: 4 starts here %%%%%%%%%%%%%%%%%%%%

%row no:1
Facility/Subcategory/Band[/Suffix]
\bigskip

%%%%%%%%%%%%%%%%%%%% Table No: 4 ends here %%%%%%%%%%%%%%%%%%%%

where \textit{Facility} is the telescope, observatory, space mission,
etc that has this filter, \textit{Subcategory} is a meaningful
classification of filters within a facility (usually instrument),
\textit{Band} is the generic name used to describe the wavelength
band used by this filter and \textit{Suffix} is optional metadata added
to the unique identifier string to ensure uniqueness within a Filter
Profile Service.
\par

Example:
\par
%%%%%%%%%%%%%%%%%%%% Table No: 5 starts here %%%%%%%%%%%%%%%%%%%%
SDSS/SDSS.G/G
\bigskip

%%%%%%%%%%%%%%%%%%%% Table No: 5 ends here %%%%%%%%%%%%%%%%%%%%

\subsubsection{PhotometryFilter.fpsIdentifier: String}
IVOA identifier of the filter profile service where this photometry
filter is registered to be used in the discovery of all the relevant
photometry filter properties.
\par

This identifier follows the IVOA syntax defined for IVOA
identifiers \citep{2016ivoa.spec.0523D} which gives a string built up as:
\par

%%%%%%%%%%%%%%%%%%%% Table No: 6 starts here %%%%%%%%%%%%%%%%%%%%
ivo://<ivoa authority id>/<resource key>
\bigskip


%%%%%%%%%%%%%%%%%%%% Table No: 6 ends here %%%%%%%%%%%%%%%%%%%%
Example:
\par

%%%%%%%%%%%%%%%%%%%% Table No: 7 starts here %%%%%%%%%%%%%%%%%%%%
ivo://svo.cab/fps
\bigskip



%%%%%%%%%%%%%%%%%%%% Table No: 7 ends here %%%%%%%%%%%%%%%%%%%%

where svo.cab is the authority id, fps is the resource key of the service.
\par

Whenever the definition of the Filter Profile Service (FPS) is standardised,
the service url of the filter profile service could be obtained
from the registry by requesting the associated information of this
registry resource, e.g., once registered the service URL associated
to this Filter Profile Service would be, e.g.:
\par

%%%%%%%%%%%%%%%%%%%% Table No: 8 starts here %%%%%%%%%%%%%%%%%%%%
http://svo.cab.inta-csic.es/theory/fps/
\bigskip

%%%%%%%%%%%%%%%%%%%% Table No: 8 ends here %%%%%%%%%%%%%%%%%%%%
At this stage, only one filter profile service exists so the service
URL would be the previously mentioned.

\subsubsection{PhotometryFilter.name: String}
This String contains a human readable representation of the filter
name. This will allow client applications to display information
to the final user.
\par

Example:
\par

%%%%%%%%%%%%%%%%%%%% Table No: 9 starts here %%%%%%%%%%%%%%%%%%%%
SDSS.G
\bigskip



%%%%%%%%%%%%%%%%%%%% Table No: 9 ends here %%%%%%%%%%%%%%%%%%%%

\subsubsection{PhotometryFilter.description: String}
This String contains a verbose human readable string description of the
filter. This will allow client applications to display text information
to the final user.
\par

\subsubsection{PhotometryFilter.bandName: String}
This String contains a standard representation of the spectral band
associated to this filter (if any). This information is useful for human
interpretation but it is discouraged to use it for discovery purposes. The
reason is that a filter is not always properly represented by a standard
band so filters could be lost in a query response.
\par

Examples:
\par


%%%%%%%%%%%%%%%%%%%% Table No: 10 starts here %%%%%%%%%%%%%%%%%%%%
U \par B  \par V
\bigskip

%%%%%%%%%%%%%%%%%%%% Table No: 10 ends here %%%%%%%%%%%%%%%%%%%%

Where U,B,V corresponds to ultraviolet, blue and visible respectively.
\par

\subsubsection{PhotometryFilter Time Validity Range}
The following fields will be used to characterize the validity time range of
this specific photometry filter configuration. This is particularly useful
for ground based telescopes where filter, electronics, etc. could easily
change generating versions of the same photometry filter.
Validity time stamps are expressed as ISOTime as specified in DALI \citep{2017ivoa.spec.0517D}
timestamps\par
%%%%%%%%%%%%%%%%%%%% Table No: 11 starts here %%%%%%%%%%%%%%%%%%%%
YYYY-MM-DD[T[hh[:mm[:ss[.s]]]]]
\bigskip
%%%%%%%%%%%%%%%%%%%% Table No: 11 ends here %%%%%%%%%%%%%%%%%%%%

\paragraph{PhotometryFilter.dateValidityFrom: ISOTime}
Start time of the time coverage when this filter configuration is
applicable.

\paragraph{PhotometryFilter.dateValidityTo: ISOTime}
End time of the time coverage when this filter configuration is
applicable.

\subsubsection{PhotometryFilter.transmissionCurve}
\label{sec:transmissioncurve}
Here we consider how wavelengths/frequencies are filtered in the whole
acquisition chain for a calibrated observation stemming from a given
data collection.
\par

This means that within the same data collection most observations will
point to the same PhotometryFilter.transmissionCurve.
\par

The effective transmission curve may be represented as a 2-D graph that
describes the transmission properties of the filter over a wavelength
range defined by the filter bandpass.
\par

It is composed of a spectral coordinate in the x-axis and a scalar in
the y-axis. This effective response curve encloses all the possible
components that modify the energy/photon collection, including detector,
telescope and even atmosphere for transmission curves referenced in
measurements.
Most modern surveys try to reduce everything according to a given airmass
(e.g. 1.3) and this is particularly important for ground-based filters
with $\lambda < 4000 \angstrom $ or  $\lambda > 7000 \angstrom $.
\par

%%%%%%%%%%%%%%%%%%%% Figure/Image No: 3 starts here %%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
	\begin{center}
		\includegraphics[width=4.24in,height=3.12in]{./media/image25.png}
		\caption{Transmission Curve example}
	\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%% Figure/Image No: 3 Ends here %%%%%%%%%%%%%%%%%%%%

\par

This curve can be used, e.g. for the creation of synthetic photometry
\citep{1996BaltA...5..459S,2004A&A...422..205G} from an observational
or a theoretical spectrum by applying it to the spectrum in the filter band-pass.
Taking as input a certain flux, the effective flux as seen using a certain
filter would be, for energy counters \citep{2007ASPC..364..227M}:

\begin{equation} \label{eq:14}
f(\lambda_{eff}) = \frac{\int T(\lambda)F(\lambda)d\lambda}{\int T(\lambda)d\lambda}
\end{equation}

And for photon counters (like CCDs or photomultipliers):

\begin{equation} \label{eq:15}
f(\lambda_{eff}) = \frac{\int T(\lambda)F(\lambda)\lambda d\lambda}{\int T(\lambda)\lambda d\lambda}
\end{equation}

Where $T(\lambda)$ is the transmission curve, $f(\lambda_{eff})$ is the flux of
the spectrum. As the transmission curve is defined only in the filter
band-pass, the limits of the integrals corresponds to the spectral range where
the transmission curve is defined (stored as \textit{PhotometryFilter.bandwidth}
in this data model)
\par

The transmission curve can be closely (although not fully) identified as an array
of points as in a spectrum. There are various ways to provide this information
either directly in an embedded table, or using a reference to a serialised table
file.
\par

Spectral and transmission coordinates can be gathered directly as a table using
 TransmissionPoint attributes (see \ref{serialisationfilter}). Transmission points
of the curve are stored into a simple table using spectrum data fields with their utypes, ucds and units.
\par

\paragraph{PhotometryFilter.transmissionCurve.access}
If the transmission curve is hooked as an external file, we use the \textit{Access} class
defined in the Observation Core Components data model \citep{2017ivoa.spec.0509L} and inherited
from the SSA specification \citep{2012ivoa.spec.0210T}.
\par

\subparagraph{PhotometryFilter.transmissionCurve.access.reference}

The access reference is a URI (typically a URL) which can be used to retrieve the
specific dataset described in a row of the query table response. \par

\subparagraph{PhotometryFilter.transmissionCurve.access.format}
The PhotometryFilter.transmissionCurve.access.format data model field tells the MIME
type of the file pointed to and used to store the curve points. Values for this
string can generally be:\par

%%%%%%%%%%%%%%%%%%%% Table No: 13 starts here %%%%%%%%%%%%%%%%%%%%
application/fits \par
application/x-votable+xml \par
text/csv \par
text/xml
\bigskip


%%%%%%%%%%%%%%%%%%%% Table No: 13 ends here %%%%%%%%%%%%%%%%%%%%

The file content will be a spectrum serialisation with
\textit{PhotometryFilter.transmissionCurve.spectrum.Dataset.DataModel} set to
$``$Spectrum1.1$"$  for instance, and all necessary fields for the spectral and
flux coordinates.
\par

\subparagraph{PhotometryFilter.transmissionCurve.access.size}
Approximate estimated size of the dataset, specified in kilobytes. This would
help the client estimate download times and storage requirements when generating
execution plans. Only an approximate, order of magnitude value is required (a value
rounded up to the nearest hundred kB would be sufficient).\par

\paragraph{PhotometryFilter.transmissionCurve.transmissionPoint}
The transmission curve is a mathematical function that describes the transmission
fraction of a certain filter in a defined spectral range. This function can be discretised
as a set of transmission points and every point will be composed by five attributes:
\par

\begin{itemize}
	\item{spectralValue, type real, with the spectral coordinate (wavelength, energy or frequency) value}
	\item{spectralErrorValue, type real, with the error on the spectral coordinate}
	\item{unit, type Unit, with the unit used for the spectralValue. This is also linked to the type of spectral coordinate used}
	\item{ucd, type UCD, used to indicates the type of spectral coordinate from the ontological point of view}
	\item{One transmissionValue, unitless real, with a value between 0 and 1}
\end{itemize}

One example for a (unit, ucd) combination applied to the PhotometryFilter.transmissionCurve.transmissionPoint could be, for example:
(\texttt{angstrom}, \texttt{em.wl})
indicating that the spectral coordinate is provided as wavelength and in angstroms . 

\par

Although the PhotDM-1.0 enforced the usage of (\texttt{angstrom}, \texttt{em.wl}), other adequate combinations of units and ucds are supported by this data model version 
according to which spectral regime the spectral values apply. This is up to the data provider to use a consistent combination.

The Unit and UCD strings follow specific constraints defined in the IVOA standards and are
implemented using type restrictions on strings in VODML.
The  serialisation of the transmission curve factorises the (unit, ucd) pair for all points of course.
\par

\subsubsection{PhotometryFilter.spectralLocation.value: real}
A spectral coordinate value that can be considered by the data provider as the
most representative for this specific filter band-pass. The selection of this
value should take into account the filter transmission curve profile and in
general should be close to the wavelength mean value, defined
in \citet{1982AJ.....87..670O} as:
\par
\begin{equation} \label{eq:16}
\lambda_{mean} = \frac{\int T(\lambda)\lambda d\lambda}{\int T(\lambda)d\lambda}
\end{equation}

where $\lambda_{mean}$ is the spectral bounds mean value, $T(\lambda)$ is
the transmission curve (see below), $\lambda$ is the wavelength. Please
notice that, since the transmission curve will only be defined in a specific
spectral range, the integrals will also be effectively defined in this
spectral range.
\par

Another convenient definition of an effective wavelength is the
$``$pivot wavelength$"$  defined as follows:

\begin{equation} \label{eq:17}
\lambda_{pivot} = \sqrt{\frac{\int T(\lambda)\lambda d\lambda}{\int T(\lambda)\frac{d\lambda}{\lambda}}}
\end{equation}

It can be proved that the pivot wavelength fulfills the following
relation between the $f_\lambda$ and  $f_\nu $:

\begin{equation} \label{eq:18}
\langle f_\nu \rangle =\langle f_\lambda \rangle \lambda^2_{pivot}/c
\end{equation}

Other definitions for effective wavelengths commonly used in the literature
are source dependent as, e.g., the isophotal wavelength:
\begin{equation} \label{eq:19}
\lambda_{mean} = \frac{\int \lambda F_\lambda(\lambda)T(\lambda)d\lambda}{\int F_\lambda(\lambda)T(\lambda)d\lambda}
\end{equation}

Or the photon distribution based effective wavelength:
\begin{equation} \label{eq:20}
\lambda_{mean} = \frac{\int \lambda^2 F_\lambda(\lambda)T(\lambda)d\lambda}{\int \lambda F_\lambda(\lambda)T(\lambda)d\lambda}
\end{equation}

but these source dependent definitions have two caveats:

\begin{itemize}
	\item{Real spectra do not necessarily satisfy the requirements of the mean value theorem,
	which could produce multiple values for the wavelength.}

	\item{Calculation of these wavelengths implies the knowledge of $F_\lambda $ (usually
	what you want to measure) and it does not look like an intrinsic property of the
	photometry filter.}
\end{itemize}\par

\subsubsection{PhotometryFilter.bandwidth: Bandwidth}
A reference range along the spectral axis coverage of the referenced photometry filter.
\par

The basic elements of this object are described
within the context of a photometry filter as follows.
\par

\paragraph{PhotometryFilter.bandwidth.UCD: String}
Unified Content Description (UCD) string that specifies the nature of the bandwidth object.
\par

\paragraph{PhotometryFilter.bandwidth.unit: IVOA.Unit}
Field that specifies the units of the bandwidth object.
\par

\paragraph{PhotometryFilter.bandwidth.extent: real}
For square filters (100$\%$  between the minimum and maximum wavelength and 0$\%$  otherwise),
the bandwidth could be described as $\lambda_{max} - \lambda_{min}$.
\par

However, for real filters, the bandwidth is not very usable to describe the band-pass of the
filter, but the effective width, that can be described as follows:
\begin{equation} \label{eq:21}
w = \frac{\int T(\lambda)d\lambda}{Max(T(\lambda))}
\end{equation}
where $w$ is the effective width, $T(\lambda)$ is the transmission curve (see below)
and $Max(T(\lambda))$ the maximum value of the transmission curve. As in previous points,
please notice that, since the transmission curve will be only defined in a specific spectral
range, the integrals will also be defined in this spectral range.
\par

\paragraph{PhotometryFilter.bandwidth.start: real}
%mir suggestion from MCD review
Also called $\lambda_{min}$ in the rest of the document, this is a spectral value that better describes
the reasonable minimum value of the spectral range of the filter band-pass. In general,
although this
will not be imposed in order to allow a better description for different types of
transmission curves, this quantity will be close to:
\begin{equation} \label{eq:22}
%mir suggestion from MCD review
\lambda_{min} = \lambda_{mean} - \frac{w}{2}
\end{equation}

In practice, this could be taken as the minimum spectral value of the filter transmission curve.
\par

\paragraph{PhotometryFilter.bandwidth.stop: real}
Also called $\lambda_{max}$ in the rest of the document, this is a spectral value that
better describes the reasonable maximum value of the spectral range of the filter band-pass.
In general,
although this will not be imposed in order to allow a better description for different
types of transmission curves, this quantity will be close to:
\begin{equation} \label{eq:23}
\lambda_{max} = \lambda_{mean} + \frac{w}{2}
\end{equation}

In practice, this could be taken as the maximum spectral value of the filter transmission
curve.\par

\subsection{PhotCal Class}
PhotCal is a class to describe the use of a photometry filter by using a certain magnitude system
configuration. It has associated a certain zero point object.
\par

\subsubsection{PhotCal.identifier: String}
This field identifies, in a unique way, within a certain Photometry Filter Profile
service, a zero point assigned to a filter and a certain photometric system type.
Although the main requirement of the identifier is to be unique within a Filter
Profile Service, the suggested syntax would be:
\par

%%%%%%%%%%%%%%%%%%%% Table No: 15 starts here %%%%%%%%%%%%%%%%%%%%
Facility/Subcategory/Band/Magnitude System Type[/Suffix]
\bigskip


%%%%%%%%%%%%%%%%%%%% Table No: 15 ends here %%%%%%%%%%%%%%%%%%%%
where \textit{Facility} is the telescope, observatory, space mission, etc that
has this filter, \textit{Subcategory} is a meaningful classification of filters
within a facility (usually instrument), \textit{Band} is the generic name used to
describe the spectral band used by this filter, \textit{Magnitude System Type}
makes reference to the type of system as per classification within this document
and \textit{Suffix} is optional metadata added to the unique identifier string to
ensure uniqueness within a Filter Profile Service.
\par

Please notice the suggested syntax of PhotCal unique identifier syntax corresponds
with the Photometry Filter unique identifier concatenated with the photometric
system type.
\par

Example:
\par



%%%%%%%%%%%%%%%%%%%% Table No: 16 starts here %%%%%%%%%%%%%%%%%%%%
SDSS/SDSS.G/G/AB
\bigskip


%%%%%%%%%%%%%%%%%%%% Table No: 16 ends here %%%%%%%%%%%%%%%%%%%%


\subsubsection{PhotCal.zeroPoint: ZeroPoint}
Zero point object associated to this PhotCal instance.
\par

\subsubsection{PhotCal.magnitudeSystem: MagnitudeSystem}
Magnitude system object associated to this phot cal instance.
\par

\subsection{ZeroPoint Class}
This class is used to characterize a zero point flux obtained during the
calibration of a certain photometry filter on a certain photometric system
configuration. This object includes references to the relevant Photometric
System and Photometry Filter objects.
\par

\subsubsection{ZeroPoint.flux.value: real}
Flux of an astronomical object that produces a magnitude of reference
(usually set as zero) for this particular filter and photometric system.
This quantity is necessary to convert to flux a certain magnitude.
\par

For Pogson magnitudes (see section \ref{sect:Pogson}) it will be used in the following way:
\par
\begin{equation} \label{eq:24}
f = f_0 10^{-(m-m_R )/2.5}
\end{equation}

See ZeroPoint.type description for other definitions.
\par

The flux could be expressed as $f_{\lambda}$ or $f_{\nu}$, leaving the
characterization of the type of flux to the units in which this quantity
is expressed.
\par

\subsubsection{ZeroPoint.referenceMagnitude.value: real}
Most of the time, the zero point flux is defined for a magnitude=0 value.
However, to give room to other cases, another reference magnitude value can
be given instead of zero. The use of this reference magnitude is described in
the different getMagnitudeFromFlux() and getFluxFromMagnitude() zero point
extension operations.
\par

Please notice that, by default, reference magnitude will be zero unless
specified otherwise.
\par

The reference magnitude is a dimensionless variable.
\par

\subsubsection{ZeroPoint.referenceMagnitude.error: real}
Total error estimated of the reference magnitude whenever applicable. Reference
Magnitude error is a dimensionless variable.\par

\subsubsection{ZeroPoint.type: integer}
Usual definition of magnitudes, also called Pogson magnitudes, can be improved
for faint sources by replacing the usual logarithm with an inverse hyperbolic
sine function. These kinds of magnitudes are called $``$asinh magnitudes$"$
or $``$luptitudes$"$  \citep{2004A&A...422..205G}.\par

%%%%%%%%%%%%%%%%%%%% Table No: 17 starts here %%%%%%%%%%%%%%%%%%%%


\begin{table}[ht]
 			\centering
\begin{tabular}{p{2.42in}p{0.8in}p{1.55in}}
\hline
%row no:1
\multicolumn{1}{|p{2.42in}}{Zero point type} &
\multicolumn{1}{|p{0.8in}}{Value} &
\multicolumn{1}{|p{1.55in}|}{Description} \\
\hline
%row no:2
\multicolumn{1}{|p{2.42in}}{Pogson} &
\multicolumn{1}{|p{0.8in}}{0 (default)} &
\multicolumn{1}{|p{1.55in}|}{Usual definition} \\
\hline
%row no:3
\multicolumn{1}{|p{2.42in}}{Asinh} &
\multicolumn{1}{|p{0.8in}}{1} &
\multicolumn{1}{|p{1.55in}|}{Used for faint sources, replacing the usual
logarithm with an inverse hyperbolic sine function.} \\
\hline
%row no:4
\multicolumn{1}{|p{2.42in}}{LinearFlux} &
\multicolumn{1}{|p{0.8in}}{2} &
\multicolumn{1}{|p{1.55in}|}{Linear (not logarithmic) magnitudes used in
Radio, Far Infrared, X-Ray spectral } \\
\hline
\end{tabular}
\caption{Types of Zero Points}
 \end{table}


%%%%%%%%%%%%%%%%%%%% Table No: 17 ends here %%%%%%%%%%%%%%%%%%%%

The main difference between the three types of zero points is the
conversion formulae to be used when translating magnitudes into flux and
reverse.\par

In the ZeroPoint class we define two conversion functions;
getMagnitudeFromFlux() and getFluxFromMagnitude() defined as:\par

\begin{itemize}
	\item getMagnitudeFromFlux()\par

\begin{itemize}
	\item{Input Parameters: Flux given in units defined in the
	ZeroPoint.unit data model field.\par}

	\item{Output Result: Corresponding magnitude in double.\par}


\vspace{\baselineskip}

\end{itemize}
	\item  getFluxFromMagnitude()\par

\begin{itemize}
	\item{Input Parameters: Magnitude as real number\par}

	\item{Output Result: Corresponding flux given in units defined in
	the ZeroPoint.unit data model field.}
\end{itemize}
\end{itemize}
\par


\subsection{PogsonZeroPoint Class}
\label{sect:Pogson}
Extension of ZeroPoint to accommodate standard logarithm magnitudes. It
has no supplementary attributes but specific conversion functions.
\par

\subsubsection{PogsonZeroPoint.getFluxFromMagnitude()}
Operator to convert from a flux to a magnitude for Pogson magnitudes. For
Pogson magnitudes, the usual definition should be used:
\par
\begin{equation} \label{eq:25}
f = f_0 10^{-(m-m_R)/2.5}
\end{equation}

Where $f$ is the associated flux, $f_0$ is the flux of reference, $m_R$ is
the magnitude of reference (by default equals to zero) and $m$ is the
observed magnitude.
\par

\subsubsection{PogsonZeroPoint.getMagnitudeFromFlux()}
Operator to convert from a flux to a magnitude for Pogson magnitudes.
For Pogson magnitudes, the usual definition should be used:
\begin{equation} \label{eq:26}
m = m_R - 2.5\log(\frac{f}{f_0 })
\end{equation}

Where $f$ is the associated flux, $f_0$ is the flux of reference,
$m_R$ is the magnitude of reference (by default equals to zero) and
$m$ is the observed magnitude.
\par

\subsection{AsinhZeroPoint Class}
Extension of ZeroPoint  to describe asinh magnitudes, a.k.a. luptitudes.
\par

\subsubsection{AsinhZeroPoint.softeningParameter: real}
Parameter used to correct the calculation of magnitudes for faint
sources. Usually called b. See \citep{1999AJ....118.1406L} for
a formal explanation.
\par

Example:
\par

%%%%%%%%%%%%%%%%%%%% Table No: 18 starts here %%%%%%%%%%%%%%%%%%%%

\begin{table}[ht]
\centering
    \begin{tabular}{|m{2.7cm}|m{8cm}|}
	\hline %inserts double horizontal lines
Band & Softening Parameters (b coefficients) \\
\hline
 $U$ & $1.4 \times 10^{10}$ \\
 $G$ & $0.9 \times 10^{10}$ \\
 $R$ & $1.2 \times 10^{10}$ \\
 $I$ & $1.8 \times 10^{10}$ \\
 $Z$ & $7.4 \times 10^{10}$ \\
\hline
\end{tabular}
\caption{Values used for SDSS DR5 asinh magnitudes}

\end{table}


%%%%%%%%%%%%%%%%%%%% Table No: 18 ends here %%%%%%%%%%%%%%%%%%%%

\subsubsection{AsinhZeroPoint.getFluxFromMagnitude()}
For asinh magnitudes, the operator to be used is:\par
\begin{equation} \label{eq:27}
f = f_0 10^{-(m-m_R )/2.5} \left[ 1-b^2 10^{2(m-m_R )/2.5}\right]
\end{equation}

Where $f$ is the flux of the observed source, $f_0$ is the zero
point flux value, $m$ is the magnitude assigned to this source,
$m_0$ is the reference magnitude (default value to zero unless
specified otherwise) and a new parameter appears, $b$, called the
softening parameter which is referenced in this data model as the
AsihnZeroPoint.softeningParameter.
\par

\subsubsection{AsinhZeroPoint.getMagnitudeFromFlux()}
For asinh magnitudes, the operator to be used is:\par
\begin{equation} \label{eq:28}
m = m_R - \frac{-2.5}{ln(10)}\left[ sinh^{-1}\left (\frac{f}{2bf_0}\right) + ln(b) \right]
\end{equation}

Where $m$ is the magnitude assigned to this source, $m_R$ is the
reference magnitude (default value to zero unless specified otherwise),
$f$ is the flux of the observed source, $f_0$ is the zero point flux value,
and a new parameter appears, $b$ , called the softening parameter,
which is referenced in this data model as the AsihnZeroPoint.softeningParameter.
\par

It can be seen that Pogson and Asinh magnitudes are the same if b=0
although, numerically it is recommended to use different equations
to prevent infinites. See \ref{a.1conversion}
\par

\subsection{LinearFluxZeroPoint Class}
Extension of ZeroPoint  to describe simple linear flux photometry,
commonly used in Radio, Far Infrared and X-ray spectral ranges.
Although not being magnitudes as such, relative linear flux measurements
can be included as a special and trivial case of magnitude.\par

\subsubsection{LinearFluxZeroPoint.getFluxFromMagnitude()}
For Linear Flux measurements, conversion used would be a linear
relation instead of a logarithmic one:
\begin{equation} \label{eq:29}
f = f_0\frac{m}{m_R}
\end{equation}

Where $f$ is the associated flux, $f_0$ is the flux of reference, $m_R$ is
the measurement of reference (default value to one, for this type of zero
points, unless specified otherwise) and $m$ is the relative observed
measurement.
\par

\subsubsection{LinearFluxZeroPoint.getMagnitudeFromFlux()}
For Linear Flux measurements, linear conversion should be used to obtain
the relative observed measurement:
\begin{equation} \label{eq:30}
m = m_R \frac{f}{f_0}
\end{equation}

Where $m$ is the relative observed measurement, $m_R$ is the measurement
of reference (default value to one for this type of zero points unless
specified otherwise), $f$ is the associated flux and $f_0$ is the flux
of reference.\par

\subsection{MagnitudeSystem Class}
The main difference between magnitude systems is the reference spectrum
used to evaluate the magnitudes. In some occasions, the magnitude system
will have a real spectrum of an existing source to calibrate all the magnitudes.
In other occasions, a synthetic spectrum will be used.
\par

\subsubsection{MagnitudeSystem.type: String}
Photometric system type used to calculate the associated zero point.
Some example values are:
\par



%%%%%%%%%%%%%%%%%%%% Table No: 19 starts here %%%%%%%%%%%%%%%%%%%%


\begin{table}[H]
 			\centering
\begin{tabular}{|c|}
\hline
\bf{MagnitudeSystem Type}\\
\hline
VegaMag\\
\hline
ABMag\\
\hline
STMag\\
\hline
\end{tabular}
\caption{Magnitude System Types}
 \end{table}


%%%%%%%%%%%%%%%%%%%% Table No: 19 ends here %%%%%%%%%%%%%%%%%%%%

The list is not exhaustive. The principal difference between these
photometric systems is the reference spectrum used to calculate the
zero point. See section \ref{referenceSpectrum} for a detailed description.
\par

\subsubsection{MagnitudeSystem.referenceSpectrum: URI} \label{referenceSpectrum}
This describes the spectrum of an astronomical object used as
reference to perform photometric calibration.
\par

This points to a Spectrum object as defined in the IVOA spectrum data
model \citep{2011ivoa.spec.1120M}. Instead of having the whole spectrum
attached, we define a link to it as referenceSpectrumURI.
The spectrum will be retrieved by invoking the URI. For example:
\par


\begin{table}[H]
 			\centering
\begin{tabular}{p{4.42in}}
\hline
http://svo2.cab.inta-csic.es/theory/fps/morefiles/vega.dat \\
\hline
\end{tabular}
\caption{Vega spectrum}
 \end{table}

Some typical types of photometric systems are:
\par

\begin{itemize}
	\item{VegaMag: Makes use of Vega ($\alpha $Lyr) as the primary calibrating
	star. PhotometricSystem.referenceSpectrum would be the Vega SED\par}

	\item{ABmag: Makes use of a reference spectrum of constant flux
	density per unit frequency $f_\nu $:}
\end{itemize}
\begin{equation} \label{eq:31}
f_0^{AB} = 3.631 \times 10^{-20} erg\, s^{-1} cm^{-2} Hz^{-1}
\end{equation}

\begin{itemize}
	\item{STmag: Introduced for the HST project, it makes use of a reference spectrum of
	constant flux density per unit of wavelength:
\begin{equation} \label{eq:32}
f_0^{ST} = 3.631 \times 10^{-9} erg\, s^{-1} cm^{-2} \angstrom ^{-1}
\end{equation}}

\end{itemize}
\section{Use Case: Conversion from magnitude to flux, using a Filter Profile Service}
The following fields are the minimal information needed in a DAL service response (SSAP
or TAP) or into a serialisation of the magnitude information in a catalogue in order to
allow the conversion from magnitudes to fluxes if a filter profile service is used:
\par

\begin{itemize}
	\item{It MUST have one field with\\ utype=$"$spec:Spectrum.Data.FluxAxis.value$"$
	and UCD="phot.mag$"$  by measurement that includes the magnitude associated to
	this measurement. \par}

Attributes\ to characterize the error of the measurement like\\
spec:Spectrum.Data.FluxAxis.Accuracy.StatError,\\
spec:Spectrum.Data.FluxAxis.Accuracy.SysError, etc could also be present
in the response.
\par

	\item{It MUST have one field per catalogue or measurement
	with\\ utype=$"$photdm:PhotCal.identifier$"$  that includes the identifier within the
	filter profile service of the filter.}
\end{itemize}\par

The normal workflow used by an application to do the conversion would be:
\par

\begin{itemize}
	\item{Go to the registry to obtain registration details of the Filter Profile service,
	using the IVOA identifier. In particular, the service URL of the service will be used
	to query this service using the uniqueIdentifier.\par}

	\item{Query the Filter Profile Service to obtain basic information of this filter. This
	information would be, at least:\par}

\begin{itemize}
	\item photdm:PhotCal.ZeroPoint.flux.value\par

	\item photdm:PhotCal.ZeroPoint.flux.unitexpression\par

	\item photdm:PhotCal.ZeroPoint.type\par

	\item photdm:PhotometryFilter.spectralLocation.value
\end{itemize}
\end{itemize}\par

And optionally, any other information that could be used for a better use of the selected
data, as, e.g. the Photometry Filter related information.
\par

Please notice that all the information of the Filter Profile Service can be overwritten
either in the DAL service or in the data serialisation. As an example, it could be
decided that the ZeroPoint.flux to be used was not the general one for this filter
within the filter profile service but the night one. In this case, this corrected
value would appear in the DAL response or in the data serialisation so this value,
and not the one on the Filter Profile Service will be used for the conversions.
\par

Flux could be then calculated as (for Pogson magnitudes, i.e. Zeropoint.type=0 and
reference magnitude = 0)
\begin{equation} \label{eq:33}
f = f_0 10^{-m/2.5}
\end{equation}

Where $f_0$ is the ZeroPoint.flux.value, $m$ is the magnitude associated to the
measurement and $f$ is the associated flux. The type of flux ($f_\lambda $ or $f_\nu $)
and the associated units, although they can be indirectly deduced from the field
MagnitudeSystem, will be the same as those for the ZeroPoint.flux.
\par
In case ZeroPoint.type=1 (asinh magnitudes) the value of
AsinhZeroPoint.softeningParameter.value should also be used to modify the conversion
formula to:
\begin{equation} \label{eq:34}
f = f_0 10^{-m/2.5}\left[ 1 - b^2 10^{2m/2.5}\right]
\end{equation}

\begin{appendices}
\section{Conversions}
\subsection{Zero point magnitude and zero point flux} \label{a.1conversion}
In the present model and in order to provide a uniform treatment for all the
different photometric systems, we have used the zero point flux as the quantity
to characterise the photometry filter. The type of flux ($f_\nu $ or $f_\lambda$)
and the units of any converted flux from a magnitude would coincide with the ones
used to express the zero point flux. The zero point flux contains information
about the type of flux that is lost in the zero point magnitude. To show this, we
provide here a couple of conversion examples to illustrate the problem.
\par
The zero point flux can also be interpreted as a magnitude in the following way.
Taking the equation \ref{eq:33} and clearing the magnitude:
\begin{equation} \label{eq:35}
m=-2.5\log_{10}(f/f_0 )=-2.5\log_{10}(f)+2.5\log_{10}(f_0 )=-2.5\log_{10} (f)+m_R
\end{equation}

Where we have defined $m_R$, zero point magnitude, as the magnitude associated to
the zero point flux:
\begin{equation} \label{eq:36}
m_R = 2.5\log_{10} (f_0 )
\end{equation} 
 
For ABmag photometric systems, the magnitudes are usually defined as:
\begin{equation} \label{eq:37}
m_{AB,\nu } = -2.5\log_{10} (f_\nu ) - 48.6
\end{equation}

As per (\ref{eq:32}), the monochromatic $f_\nu $ flux for ABmag magnitudes is:
\begin{equation} \label{eq:38}
f_{R}^{AB}=3.631 \times 10^{-20} erg\, s^{-1} cm^{-2} Hz^{-1}
\end{equation}
so the magnitude of reference is:
\begin{equation} \label{eq:39}
m_R = 2.5\log_{10} (f_{0}^{AB})=2.5 log_{10}(3.631 \times 10^{-20})=-48.6
\end{equation}

Other systems usually define the zero point flux as a $f_\lambda $ flux, as
it is usually done by, e.g., STMag systems. The usual definition of magnitudes for this photometric system is:
\begin{equation} \label{eq:40}
m_{ST,\lambda }=-2.5\log_{10} (f_\lambda )-21.1
\end{equation}

As per (\ref{eq:33}), the monochromatic
$f_\lambda $ flux for these systems is:
\begin{equation} \label{eq:41}
f_0 ^{ST} = 3.631 \times 10^{-9} erg\, s^{-1} cm^{-2} \angstrom ^{-1}
\end{equation}

what is consistent with the usual definition of magnitudes as:
\begin{equation} \label{eq:42}
m_R =2.5\log_{10} (f_0 ^{ST})=2.5\log_{10} (3.631 \times 10^{-9})=-21.1
\end{equation}
\par
Both cases are physically different but the magnitude of reference does not
reflect information by itself this difference. Setting the zero point flux as the
property to characterise the photometric system, the data model contains a 
self-consistent metadata and a more general way to describe all the photometric
systems.

\subsection{Interrelation between Pogson and Asinh magnitudes}
It can be proved that, if b=0, Pogson and Asinh magnitudes are the same:
\begin{equation} \label{eq:43}
\frac{f}{f_0}=10^{-m/2.5}\left[ 1-b^2 10^{2m/2.5}\right] \text{\textbar}_{b=0} =10^{-m/2.5}
\end{equation}
and
\begin{equation} \label{eq:44}
m=\frac{-2.5}{ln(10)}\left[ sinh^{-1} \left(\frac{f}{2bf_0}\right) + ln(b)\right] \text{\textbar}_{b=0} =
\end{equation}
\[
\frac{-2.5}{ln(10)}\left[ ln\left(\frac{f}{2bf_0}\right) + \sqrt{1+\frac{f^2}{4b^2 f_0^2}} + ln(b)\right] \text{\textbar}_{b=0} =
\]
\[
\frac{-2.5}{ln(10)}\left[ ln\left(\frac{f}{2f_0}\right) + \sqrt{b^2+\frac{f^2}{4 f_0^2}}\right] \text{\textbar}_{b=0} =
\]
\[
\frac{-2.5}{ln(10)}\left[ ln(\frac{f}{f_0}) \right] =
\]
\[
-2.5\log{\frac{f}{f_0}}
\]


Although, as can be seen in the previous calculation, Asinh magnitudes are equivalent to Pogson
when b=0, it is recommended to use different implementation conversion codes for both types of magnitudes as a general implementation both for Pogson (b=0) and asinh (b>0) could easily produce numerical infinites during the evaluation.
\par
\section{Data Model VODML documentation}

The VODML description of this model is available at \\ \url{http://ivoa.net/xml/}.

The documentation page generated from this reference XML document is available on the PhotDM-1.1 RFC discussion page : \\
\url{https://wiki.ivoa.net/internal/IVOA/PhotDM11RFC/Photv1_1_PR_20220520.html}

\section{Data Model Summary}
Here is the table summary  of the model, backward compatible to PhotDM-1.0.
It shows the UTypes and UCDs used for each data model element.

%%%%%%%%%%%%%%%%%%%% Table No: 20 starts here %%%%%%%%%%%%%%%%%%%%

\begin{table}[H]
 			\centering
 			\begin{adjustbox}{angle=90}
\begin{tabular}{p{2.5in}|p{1.5in}|p{2in}|p{0.74in}|p{0.35in}}
%row no:1
\multicolumn{5}{p{\dimexpr6.59in+8\tabcolsep\relax}}{\centering
{\fontsize{8pt}{8pt}\selectfont \textbf{General Metadata}}} \\
\hline
%row no:2
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UType}}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UCD 1+}}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Meaning}}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Default value}}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Data type}}} \\
\hline
%row no:3
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont Datamodel.name}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.id }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Data Model Identification }} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont PhotDM-v1.0}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:4
\multicolumn{5}{p{\dimexpr6.59in+8\tabcolsep\relax}}{\centering
{\fontsize{8pt}{8pt}\selectfont \textbf{Photometric System Metadata}}} \\
\hline
%row no:5
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UType}}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UCD 1+}}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Meaning}}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Default value}}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Data type}}} \\
\hline
%row no:6
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometricSystem.description}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.note }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont String representation Photometric System}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:7
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometricSystem.detectorType}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.code }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Type of detector
(e.g energy or photon counter). Possible values defined by enumeration}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont 0\  (Energy Counter)}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont int}} \\
\hline
\end{tabular}
\end{adjustbox}
 \end{table}

%Photometry Filter General Metadata
\newpage
\begin{table}[H]
\centering
\begin{adjustbox}{angle=90}
\begin{tabular}{p{2.5in}|p{1.5in}|p{2in}|p{0.74in}|p{0.35in}}
%row no:8
\multicolumn{5}{p{\dimexpr6.59in+8\tabcolsep\relax}}{\centering
{\fontsize{8pt}{8pt}\selectfont \textbf{Photometry Filter General Metadata}}} \\
\hline
%row no:9
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UType}}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UCD 1+}}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Meaning}}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Default value}}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Data type}}} \\
\hline
%row no:10
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.identifer}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ref.ivoid }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Unique identifer of
filter within a Filter Profile Service (FPS)}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:11
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.fpsIdentifier}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ref.ivoid }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont IVOA identifier of the
Filter Profile Service}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:12
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.name}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.id;instr.filter }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Filter Name in the instrumental }
\par {\fontsize{10pt}{12.0pt}\selectfont configuration\  }} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:13
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.description}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.note }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Text description of the filter band}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
\end{tabular}
\end{adjustbox}
 \end{table}


%%%%%%%%%%%%%%%%%%%% Table No: 20 ends here %%%%%%%%%%%%%%%%%%%%


 %%%%%%%%%%%%  Starting New Page here %%%%%%%%%%%%%%

\newpage


%%%%%%%%%%%%%%%%%%%% Table No: 21 starts here %%%%%%%%%%%%%%%%%%%%
%Photometry Filter Access Metadata
\begin{table}[H]
\centering
\begin{adjustbox}{angle=90}
\begin{tabular}{p{2.5in}|p{1.5in}|p{2in}|p{0.74in}|p{0.35in}}
%row no:1
\multicolumn{5}{p{\dimexpr6.59in+8\tabcolsep\relax}}{\centering
{\fontsize{10pt}{12.0pt}\selectfont \textbf{Transmission Curve Metadata}}} \\
\hline
%row no:2 
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UType}}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UCD 1+}}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Meaning}}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Default value}}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Data type}}} \\
\hline
%row no:3
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.transmissionCurve.\ access.reference}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ref.uri }} &

\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont URI to the
effective transmission curve}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont URI type}} \\
\hline
%row no:4
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.transmissionCurve.\ access.format}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.code}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont File format of the pointed transmission curve}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:5
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.transmissionCurve.\ access.size}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Size in kilobytes of the pointed transmission curve}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont int}} \\
\hline
%row no:6
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photDM.PhotometryFilter.transmissionCurve.\ transmissionPoint.spectralValue}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont em.*}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Spectral value
of one element of the transmission curve representation}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:7
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photDM.PhotometryFilter.transmissionCurve.\ transmissionPoint.spectralErrorValue}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont em.*}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Error of the spectral value
of each element of the transmission curve representation}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:8
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photDM.PhotometryFilter.transmissionCurve.\ transmissionPoint.unit}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Unit of the spectral value
of each element of the transmission curve representation}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:9
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photDM.PhotometryFilter.transmissionCurve.\ transmissionPoint.ucd}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont UCD of the spectral value
of each element of the transmission curve representation}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont UCD}} \\
\hline
%row no:10
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photDM.PhotometryFilter.transmissionCurve.\ transmissionPoint.transmissionValue}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}
\selectfont phys.transmission}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Transmission value of
one element of the transmission curve representation}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline

\end{tabular}
\end{adjustbox}
 \end{table}


%%%%%%%%%%%%%%%%%%%% Table No: 20 ends here %%%%%%%%%%%%%%%%%%%%


 %%%%%%%%%%%%  Starting New Page here %%%%%%%%%%%%%%

\newpage


%%%%%%%%%%%%%%%%%%%% Table No: 21 starts here %%%%%%%%%%%%%%%%%%%%
%Photometry Filter Spectral Axis Coverage
\begin{table}[H]
\centering
\begin{adjustbox}{angle=90}
\begin{tabular}{p{2.5in}|p{1.5in}|p{2in}|p{0.74in}|p{0.35in}}
%row no:1
\multicolumn{5}{p{\dimexpr6.59in+8\tabcolsep\relax}}{\centering
{\fontsize{8pt}{8pt}\selectfont \textbf{Photometry Filter Spectral Axis Coverage}}} \\
\hline
%row no:8
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UType}}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UCD 1+}}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Meaning}}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Default value}}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Data type}}} \\
\hline
%row no:9
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.bandName}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont instr.bandpass }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Generic name for the
filter spectral band}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:10
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.spectralLocation.\ value}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont em.*;meta.main as appropriate}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Reference position along the
spectral axis. Spectral coordinate of the Zero Point }} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:11
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.spectralLocation.\ unitexpression}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.unit }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Unit of the spectral axis used
to characterize it}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont }} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:12
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.spectralLocation.\ UCD}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ucd }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont UCD for the nature of
spectral axis wl, freq, energy}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont em.*\ as appropriate}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:13
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.bandwidth.unit.\ expression}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.unit}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Unit of the spectral extent
used to characterize the bandwidth object}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont }} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:14
\multicolumn{1}{p{2.5in}}{
{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.\ bandwidth.UCD}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ucd }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont UCD for the nature of
spectral bandwidth: wl, freq, energy, wave number, ...}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont em.*\ as appropriate}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:15
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.bandwidth.extent.\ value}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont instr.bandwidth}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Spectral axis extent of the filter}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:16
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.bandwidth.\ start.value}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont em.*;stat.min}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Minimum value of the filter
spectral coverage}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:17
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.bandwidth.\ stop.value}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont em.*;stat.max}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Maximum value of the
filter spectral coverage}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:18
\end{tabular}
\end{adjustbox}
 \end{table}


%%%%%%%%%%%%%%%%%%%% Table No: 20 ends here %%%%%%%%%%%%%%%%%%%%


 %%%%%%%%%%%%  Starting New Page here %%%%%%%%%%%%%%

\newpage


%%%%%%%%%%%%%%%%%%%% Table No: 21 starts here %%%%%%%%%%%%%%%%%%%%
%PhotCal Metadata

\begin{table}[H]
\centering
\begin{adjustbox}{angle=90}
\begin{tabular}{p{2.5in}|p{1.5in}|p{2in}|p{0.74in}|p{0.35in}}
%row no:7
\multicolumn{5}{p{\dimexpr6.59in+8\tabcolsep\relax}}{\centering
{\fontsize{10pt}{12.0pt}\selectfont \textbf{PhotCal Metadata}}} \\
\hline
%row no:19
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UType}}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{UCD 1+}}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Meaning}}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Default value}}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont \textbf{Data type}}} \\
\hline
%row no:20
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotometryFilter.dateValidityFrom}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont time.start}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Time stamp for Start of validity for
this filter in ISOTime format }} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string }} \\
\hline
%row no:21
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotometryFilter.dateValidityTo}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont time.end}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Time stamp for Stop of validity
for this filter in ISOTime format }} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string }} \\
\hline
%row no:24
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotCal.identifier}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ref.ivoid }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Unique identifier of the Photometry }
\par {\fontsize{8pt}{8pt}\selectfont Calibration instance within a FPS}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:25
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotCal.zeroPoint.\-flux.unitexpression}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.unit }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont unit for Zero point flux}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont Jy}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:26
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotCal.zeroPoint.\-flux.UCD}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ucd }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont ucd for Zero point flux}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont phot.flux.density}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:27
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotCal.zeroPoint.\-flux.value}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont phot.flux.density }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont flux value at Zero point associated
to this filter}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:28
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotCal.zeroPoint.\-flux.error}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont stat.error;phot.flux.density}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Error in the flux value at Zero point
associated to this filter}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline
%row no:29
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotCal.zeroPoint.\newline referenceMagnitude.value}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont phot.mag}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Reference magnitude used for zero point}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont 0.0}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real} \par } \\
\hline
%row no:30
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotCal.zeroPoint.\newline referenceMagnitude.error}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont stat.error;phot.mag}} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Error in the reference magnitude
used for zero point}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont 0.0}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real} \par } \\
\hline
%row no:31
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotCal.zeroPoint.type}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.code }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Type of zero point}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont 0}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont int}} \\
\hline
%row no:32
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}\selectfont photdm:PhotCal.magnitudeSystem.type}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.code }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Type of magnitude system}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont VegaMag}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont string}} \\
\hline
%row no:33
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:PhotCal.magnitudeSystem.\-ReferenceSpectrumURI}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont meta.ref.uri }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont Reference SED or spectrum for
this magnitude system}} &
\multicolumn{1}{p{0.74in}}{} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont uri type}} \\
\hline
%row no:34
\multicolumn{1}{p{2.5in}}{{\fontsize{8pt}{8pt}
\selectfont photdm:AsinhZeroPoint.softeningParameter}} &
\multicolumn{1}{p{1.5in}}{{\fontsize{8pt}{8pt}\selectfont phot.calib }} &
\multicolumn{1}{p{2in}}{{\fontsize{8pt}{8pt}\selectfont  Correction parameter
for luptitudes}} &
\multicolumn{1}{p{0.74in}}{{\fontsize{8pt}{8pt}\selectfont 0.0}} &
\multicolumn{1}{p{0.35in}}{{\fontsize{8pt}{8pt}\selectfont real}} \\
\hline

\hline

\end{tabular}
\end{adjustbox}
\end{table}

%%%%%%%%%%%%%%%%%%%% Table No: 21 ends here %%%%%%%%%%%%%%%%%%%%

The UTypes given here are to be considered opaque strings pointing to data model
elements that can be used by consumers of instance documents.

ZeroPoints may belong to one of three categories: Pogson, Asinh or LinearFlux
(leaving room for other future extensions). The treatment of the different
categories ZeroPoints differs from the algorithmic point of view.
However, the data structure only differs in the current DM by the addition
of the softening parameter attached to the Asinh case.

\par

\section{Data Model Serialisations} \label{serialisation}
\subsection{Filter Profile Service Serialisation} \label{serialisationfilter}

\lstset{
    language=xml,
    tabsize=3,
    frame=lines,
    %frame=shadowbox,
    rulesepcolor=\color{gray},
    xleftmargin=20pt,
    framexleftmargin=15pt,
    keywordstyle=\color{black}\bf,
    commentstyle=\color{OliveGreen},
    numberstyle=\tiny\color{codegray},
    stringstyle=\color{red},
    numbers=left,
    numberstyle=\tiny,
    numbersep=2pt,
    breaklines=true,
    showstringspaces=false,
    basicstyle=\footnotesize,
    emph={dmid,dmtype,dmrole,value},emphstyle={\color{magenta}}
    }

The following serialisation is an example of a response of a filter profile
service making use of the PhotDM PhotometryFilter through UTypes:
\par
%%%%%%%%%%%%%%%%%%%% listing phot system 2Mass : starts here %%%%%%%%%%%%%%%%%%%%
\lstinputlisting[language=XML,breaklines, caption={SVO Filter Profile
Service output: Simple VOTable Format}]{./serializations/fpsFilterExample.xml}
%%%%%%%%%%%%%%%%%%%% Table No:  ends here %%%%%%%%%%%%%%%%%%%%

\subsection{Photometric Data in Cone Search and TAP services}
Cone Search and TAP services delivering photometric data 
also provide the photometric filter metadata as well as zero point and other calibration information.

Various ways have been proposed and used using PARAM and GROUP elements for the PhotDM class attributes. 

As an example, the serialisation of the 2MASS catalogue in a cone search service,
could have the following information in the VOTable header:
\par
%%%%%%%%%%%%%%%%%%%% Table No: starts here %%%%%%%%%%%%%%%%%%%%
\lstinputlisting[language=XML,breaklines, caption={VOTable response example
from Cone Search service}]{./serializations/conesearch_ex.xml}
%%%%%%%%%%%%%%%%%%%% Table No:  ends here %%%%%%%%%%%%%%%%%%%%

More details on how to serialise the response are contained in the IVOA Note \citep{derriere} and used in catalog services .

\subsection{Serialisation using VODML model} \label{appendixmapping}
The VODML view of PhotDM also allows the serialisation of PhotDM elements
using mapping techniques into, e.g. VOTable formats. We show some of the possible
serialisations in line with the MIVOT mapping effort of the IVOA DM working group.

\subsubsection{Photometric calibration:  example 1}
This example describes the PhotCal Vega calibration of the 2MASS Ks filter.
\par
%%%%%%%%%%%%%%%%%%%% Table No: starts here %%%%%%%%%%%%%%%%%%%%
\lstinputlisting[language=XML,breaklines, caption={XML mapping block for the
2MASS Ks Filter, based on the VODML representation of PhotDM-1.1 .}]{./serializations/photdm.2MASS.2MASS.Ks.xml}

%%%%%%%%%%%%%%%%%%%% Table No:  ends here %%%%%%%%%%%%%%%%%%%%

 \subsubsection{The full 2MASS photometric system with all filters: example 2}


\lstinputlisting[language=XML,caption={Annotation of a SED data set using a PhotDM-1.1 XML mapping block for the 2MASS Photometric
system.}] {./serializations/exampleXML2MASS.xml}

\end{appendices}

\bibliography{bibliography,ivoatex/ivoabib,ivoatex/docrepo}

\end{document}