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\begin{document}

\title{IRAM Memo 2015-3\\[3\bigskipamount]
  NOEMA sensitivity estimator}%
\author{J. Pety$^{1,2}$, J. Boissier$^{1}$, E. Reynier$^{1}$, S. Bardeau$^{1}$\\[0.5cm]
  1. IRAM (Grenoble) \\
  2. Observatoire de Paris}%
\date{Jul, 22$^{nd}$ 2019\\
  Version 3.1}

\maketitle

\begin{abstract}
  This memo describes the equations used in the NOEMA sensitivity estimator
  available in the \texttt{GILDAS/ASTRO} program.
\end{abstract}

\newpage{}

\tableofcontents{}

\newpage{}

\section{Generalities}

\subsection{The interferometric point source sensitivity}

The point source sensitivity for an interferometric measurement reads
\begin{equation}
  \noise[Jy] = \frac{\JypK\,\Tsys}{\atmeff\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}},
  \label{eq:noise:1}
\end{equation}
where \noise[Jy] is the rms noise flux obtained by integration with an
interferometer of \nant{} identical antenna during the \tint{} integration
time in a frequency resolution \dnu{} with a system temperature given by
\Tsys{}.  \JypK{} is the Jy/K typical conversion factor of the
interferometer. It reads
\begin{equation}
  \JypK = \speceff\,\JyAnt,
\end{equation}
where \speceff{} is the spectrometer efficiency and \JyAnt{} the typical
conversion factor of each interferometer antenna. \JyAnt{} is defined as
\begin{equation}
  \JyAnt = \frac{2\kbolt}{\Area},
\end{equation}
where \kbolt{} is the Boltzman constant, and \Area{} the effective antenna
collecting area~\citep[eq. 3-113 in][]{kra82}.

\JypK{} characterizes the hardware, \ie{} it assumes excellent atmospheric
conditions. The atmospheric decorrelation is taken into account through an
additional efficiency factor, \atmeff{}, that is directly related to the
atmospheric rms phase noise (\pharms) through
\begin{equation}
  \atmeff{} = e^{-\frac{\pharms^2}{2}}.
\end{equation} 

Equation~\ref{eq:noise:1} is true only when the source is unresolved,
\ie{}, there is no effect of beam dilution. In practice this is rarely the
case because the interferometer tries to resolve the source. Thus, this
noise formula should be used with caution when preparing the
observations. In practice, this formula is useful when one wishes to
compare the sensitivity of two different interferometer. Indeed, this point
source sensitivity is independent of the interferometer synthesized beam
that depends on the details of the observations and, in particular, the
interferometer configuration and the completeness of the Earth synthesis.

\subsection{The interferometric extended source sensitivity}

The sensitivity of an interferometer to an extended source reads
\begin{equation}
  \noise[K] = \frac{\beam[prim]^2}{\beam[maj]\,\beam[min]}\,\frac{\Tsys}{\atmeff\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}},
  \label{eq:noise:2}
\end{equation}
where \noise[K] is the rms noise brightness, \beam[prim] the half primary
beam width, and \beam[maj] and \beam[min] the half beamwidth along the
major and minor axes of the synthesized beam.

This formula clearly states that the sensitivity to extended sources
depends on the dilution of the synthesized beam in the primary beam. For a
given interferometer, the primary beamwidth is a fixed quantity while the
synthesized beam is to first order proportional to the longest baseline in
the current interferometer configuration. Hence, doubling the largest
baseline will multiply \noise[K] by a factor $4 (=2^2)$ for the same
integration time or it will multiply the integration time by a factor
$16 (=2^4)$ in order to reach the same sensitivity. This just reflects that
while the interferometer tries to mimic a single-dish antenna of same
diameter as the largest baseline, all the antenna of the interferometer
only fill a fraction of the total collecting area of the single-dish, this
fractions decreasing with a power of two as the baseline linearly
increases.

It is easy to show that \noise[K] and \noise[Jy] are linked through
\begin{equation}
  \noise[K] = \frac{4\,\ln{2}\,\wave^2}{2\pi\,\kbolt\,\beam[maj]\beam[min]} \,\noise[Jy],
\end{equation}
where \wave{} is the observed wavelength. In practice, time/sensitivity
estimator usually computes the relationship between \tint{} and \noise[Jy],
and then the relationship between \noise[K] and \noise[Jy].

\subsection{System temperature}

\begin{figure}[t]
  \centering %
  \includegraphics[width=\linewidth]{astro-noema-time-estimator-tsys-table}
  \caption{\textbf{Top:} Summer (red) and Winter (blue) semester $\Tsys$
    for different precipitable water vapor (PWV) amount and for a source at
    zenith. The numbers indicate PWV values assumed in the
    computation. \textbf{Middle:} Assumed forward effiencies in the
    computation. \textbf{Bottom:} Assumed receiver temperatures in the
    computation.}
  \label{fig:tsys:table}
\end{figure}

\begin{figure}[t]
  \centering %
  \includegraphics[width=\linewidth]{astro-noema-time-estimator-tsys-minmax}
  \caption{\textbf{Top:} Minimum and maximum $\Tsys$ obtained in the
    intermediate frequency bandwidth as a function of the local oscillator
    frequency used in the tuning. \textbf{Middle:} Assumed forward effiencies in the
    computation. \textbf{Bottom:} Assumed receiver temperatures in the
    computation.}
  \label{fig:tsys:minmax}
\end{figure}

\begin{figure}[t]
  \centering %
  \includegraphics[width=\linewidth]{astro-noema-time-estimator-tsys-cont}
  \caption{\textbf{Top:} Averaged ``continuum'' $\Tsys$ as a function of
    the local oscillator frequency used in the tuning. \textbf{Middle:}
    Assumed forward effiencies in the computation. \textbf{Bottom:} Assumed
    receiver temperatures in the computation.}
  \label{fig:tsys:cont}
\end{figure}

The system temperature is a summary of the noise added by the system. This
noise comes from 1) the receiver and the optics, 2) the emission of the
sky, and 3) the emission picked up by the secondary side lobes of the
telescope. It is usual to approximate it (in the \Tas{} scale) with
\begin{equation}
  \Tsys = \frac{\paren{1+\gim}\exp\cbrace{\taus\,\Airmass}}{\Feff}
  \bracket{\Feff\,\Tatm\,\paren{1-\exp\cbrace{-\taus\,\Airmass}}+\paren{1-\Feff}\Tcab+\Trec},
\end{equation}
where \gim{} is the receiver image gain, \Feff{} the telescope forward
efficiency, $\Airmass = 1/\sin(\emr{elevation})$ the airmass, \taus{} the
atmospheric opacity in the signal band, \Tatm{} the mean physical
atmospheric temperature, \Tcab{} the ambient temperature in the receiver
cabine and \Trec{} the noise equivalent temperature of the receiver and the
optics. All those parameters are easily measured, except \taus{}, which
depends on the amount of water vapor in the atmosphere and which is
estimated by complex atmospheric models.

In the ASTRO sensitivity estimator in detailed mode, the system temperature
is computed using an atmospheric model (ATMOSPHERE command).  In proposal
mode, the $\Tsys$ is interpolated in frequency and airmass from tabulated
values (see Fig.~\ref{fig:tsys:table}). The airmass is estimated using the
maximum elevation of a source at the chosen Declination. The values are
different for summer and winter due to the different atmospheric
characteristics. Moreover, the chosen amount of precipitable water vapor
depends on the receiver band (in addition to the season) because the NOEMA
operation team schedule the different receiver bands according to the
actual weather (high frequency bands are scheduled only during the best
weather conditions).

The $\Tsys$ can vary significantly over the large bandwitdh of the 2SB
NOEMA receivers. Figure~\ref{fig:tsys:minmax} shows the minimum and maximum
system temperature inside the IF bandwidth for all possible local
oscillator tunings. As a result, for continuum estimation, a frequency
averaged $\Tsys$ is interpolated from a pre-computed table in PROPOSAL
mode.  The input frequency in that case is the LO frequency of the tuning
(see Fig.~\ref{fig:tsys:cont}).  The averaging is done such as $1/<\Tsys>^2
= 1/N \sum 1/\Tsys^2$.  This is not implemented in DETAILED mode, due to
the prohibitive computing cost of the atmospheric model over the $2 \times
8\,$GHz.

\subsection{The number of polarizations}

All NOEMA antennas are equipped with dual polarization receivers.  They
measure the signal coming from the pointed direction in two perpendicular
polarizations in the same frequency range.  For the current generation of
receiver (2006) and correlators, one or two polarizations are processed by
the correlators, depending on the project settings.  We thus have to
introduce the number of polarizations \npol{}, which can be set to 1 or 2
and insert it in the radiometer equation with
\begin{equation}
  \noise[Jy] = \frac{\JypK\,\Tsys}{\atmeff\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}.
  \label{eq:noise:3}
\end{equation}

\section{Observing mode and elapsed telescope time}

The goal of a time estimator is to find the elapsed telescope time
(\ttel{}) needed to obtain a given rms noise, while a sensitivity estimator
aims at finding the rms noise obtained when observing during \ttel{}.  The
total integration time spent on-source \ton{} is shorter than the elapsed
telescope time due to several factors.
\begin{enumerate}
\item At the beginning of a project a significant time ($\ttune \sim 40$
  minutes) is spent in receiver tuning and calibration observations before
  observing the actual astronomical target. This means that even for a very
  short ON source time, a project cannot be shorter than \ttune{}.
\item After this initial phase, the observing mode does not dedicate 100\%
  of the time to the astronomical target. Part of the time is spent for
  calibration (pointing, focus, atmospheric calibration,...) and to slew
  the telescopes between useful integrations.
\end{enumerate}
As of Gildas Jul17 release, the input time of the ASTRO sensitivity
estimator is telescope time. The actual on source time is then computed
taking into account those two points. 

This computation depends on the observing mode. There are three main
observation kinds.
\begin{description}
\item[Single-source, single-field observations] where the telescope tracks
  a single source during the full integration time. This mode is used when
  the signal-to-noise ratio is the limiting factor.
\item[Track-sharing, single-field observations] where the telescope
  regularly cycles between a few close-by sources. This mode is used when
  the sources are so bright that the limiting factor is the Earth
  synthesis, not the signal-to-noise ratio.
\item[Single-source mosaicking] where the telescope regularly cycles
  between close-by pointings that usually follows a hexagonal compact
  pattern whose side is $\wave/(2\dprim)$, where \dprim{} is the diameter
  of the interferometer antennas. This modes is used to image sources wider
  than the primary beam field of view.
\end{description}
In the following, we will work out the equations needed by the
sensitivity estimator for each of these observing modes.

\subsection{Single-source, single-field observations}
\label{sec:single-field}


That's the simplest case. The point source sensitivity in this case is
\begin{equation}
  \noise[Jy] = \frac{\JypK\,\Tsys}{\atmeff\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
\end{equation}
where \ton{} is the time spent on-source. It is related to the total elapsed telescope time \ttel{} through:
\begin{equation}
%  \ton = \frac{\ttel \,-\,  \ntrack \times  \ttune}{\obseff}
  \ton = \frac{\Delta t_{\emr{obs}}}{\obseff}
  \label{eq:tel-vs-on}
\end{equation}


$\Delta t_{\emr{obs}}$ is the time actually spent in observations
(i.e. after instrument tuning and calibration).  We have $\Delta
t_{\emr{obs}} = \ttel \,-\, n_{\emr{tracks}} \times \ttune$. As
described above \ttune{} is estimated to be 40 min (based on history
of observed projects).  \ntrack{} is the number of tracks to be
observed to reach the requested telescope time.  It is computed as
$\ntrack = \frac{\ttel}{\ttrack}$ where \ttrack is the typical
duration of a track, which depends on the source declination:
    \begin{itemize}
    \item 8 hours for a delination larger than $0\deg$.
    \item 0 hours for a declination lower than $-30\deg$.
    \item A linear interpolation with the declination between these two
      values.
    \end{itemize}



Finally \obseff{} quantifies the efficiency of the observing, not
taking into account the initial calibrations and tuning.  It is
estimated to be about 1.6. Note that the exact value will depend on
several parameters such as the number of calibrators and the distance
between the source and the calibrator(s).


\subsection{Track-sharing, single-field observations}
\label{sec:track-sharing}

In this case, the telescope time is equally divided between the \nsource{}
observed sources. This yields
\begin{equation}
  \noise[Jy] = \frac{\JypK\,\Tsys}{\atmeff\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
\end{equation}
\begin{equation}
  \mbox{with} \quad
  \ton =  \frac{\ttel \,-\,  n_{\emr{tracks}} \times  \ttune}{\obseff \times \nsource}
\end{equation}

Note that it is technically feasible to observe sources in
track-sharing with different integration times. This case is not
implemented yet in the sensitivity estimator and the different
sensitivities should be computed independently.

\subsection{Mosaicking}
\label{sec:mosaicking}
Mosaicking is a particular case of wide-field imaging: The user wishes to
observe a given field of view larger than the primary beam size with a
sensitivity as uniform as possible. The on-source time is thus equally
divided between the independent primary beams in the targeted field
of view. To first order, we thus yield
\begin{equation}
  \noise[Jy] = \frac{\JypK\,\Tsys}{\atmeff\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
  \label{eq:mos}
\end{equation}
\begin{equation}
  \quad \mbox{with} \quad
  \ton =  \frac{\ttel \,-\,  n_{\emr{tracks}} \times  \ttune}{\obseff \times \nbeam}
  \quad \mbox{where} \quad
  \nbeam = \frac{\Amap}{\Abeam},
\end{equation}
where \nbeam{} is the number of independent resolution elements, \Amap{} is
the area of the targeted field of view, and \Abeam is the area of the
resolution element. The \Amap{} area is a user input while the \Abeam{}
area is linked to the telescope full width at half maximum (\beam{}) by
\begin{equation}
  \Abeam = \frac{0.8\,\pi\,\beam[prim]^2}{4\,\ln(2)},
\end{equation}
The 0.8 factor represents the truncation of the beam at 20\% of its
maximum, which is performed during the imaging process.

There are several subtleties in this computation.
\begin{itemize}
\item \Amap{} must be larger than 2 times \Abeam{} (below this we advise to
  use the track sharing mode with two independent fields).
\item The processing (imaging and deconvolution) of a mosaic implies a
  division by the primary beam of the interferometer. As the primary beam
  is to first order a Gaussian decreasing to zero, this implies that the
  noise of the mosaic will vary over the field of view. In particular it
  increases sharply at the edges of the field of view. In other words,
  Eq.~\ref{eq:mos} does not apply to the mosaic edges!
\item The cycling of the pointings of the mosaic should ensure Nyquist
  sampling of the observed field of view. This implies that there is an
  important redundancy between the pointings, contrary to track sharing
  where the sources are supposed to be fully independent on the sky. For
  instance, when mosaicking with a hexagonal compact pattern, each line of
  sight will be observed by 7 contiguous pointings, except at the mosaic
  edges. It can thus been shown that the number of mosaic pointings,
  \npoint, is related to the number of independent elements through
  \begin{equation}
    \npoint = \nbeam \, \paren{\frac{7}{4}}^2,
  \end{equation}
  for a correctly sampled mosaic. Equation~\ref{eq:mos} is only valid
  inside a correctly sampled mosaic.
\item The pointings of a mosaic must be observed in short time cycles to
  ensure that all pointings are observed with similar weather conditions
  and that they share similar $uv$ coverage. This minimizes the
  shift-variant part of the interferometer wide-field imaging
  response. This calls for the shortest possible integration time per
  pointing. However, the interferometer takes time to slew from one
  pointing to the next one without integrating. 
  As a result, the observing efficiency \obseff{} is degraded in the cases of mosaics and we have another
   relationship between the elapsed telescope time and the on-source time as:
  \begin{equation}
    \ton =  \frac{\ttel \,-\,  n_{\emr{tracks}} \times  \ttune}{\obseff \moseff \times \nbeam}
    \quad \mbox{with} \quad
    \moseff = \frac{\tint+\tslew}{\tint},
  \end{equation}
  where \tint{} is the integration time per pointing and \tslew{} is the
  time to slew between two consecutive pointings. Having a large
  integration time per pointing compared to \tslew{} will decrease the
  mosaicking overhead. This requirement is in sharp contrast with the
  previous one, namely the need to homogenize the interferometer wide-field
  response. The best compromise comes from two different considerations.
  \begin{enumerate}
  \item The smallest integration time is set by the acquisition system (for
    instance, the maximum achievable data rate). In pratice, we enforce
    that
    \begin{equation}
      \tintmin = 10\,\mbox{sec}.
    \end{equation}
  \item The distance covered by a visibility in the $uv$-plane during an
    integration should always smaller than the distance associated to
    tolerable aliasing (see Pety and Rodr\'iguez-Fern\'andez 2010 for more
    details).  This can be written as the following condition (Eq.~C.3 in
    this article)
    \begin{equation}
      \frac{\tint}{1 s} << \frac{6900}{\beam[alias] / \beam[syn]},
      \label{eq:uvcond}
    \end{equation}
    where $\beam[alias]$ is the map angular size, and $\beam[syn]$ the
    angular resolution. For a given angular resolution, the interferometer
    minimum integration time corresponds to
    \begin{equation}
      \tintmin \le \frac{1}{\eta}\,\frac{6\,900}{1\,\emr{sec}}\,
                   \sqrt{\frac{\beam[maj]\beam[min]}{\Amap}},
    \end{equation}
    where $\eta$ is a ad-hoc integer set to 5 to ensure the condition
    defined in Eq.~\ref{eq:uvcond}.
  \end{enumerate}
  As the typical slew time between two pointings is $\tslew = 11\,$sec, we
  yield that
  \begin{equation}
    1 \le \moseff \le 2.3
  \end{equation}
\item If the time to cycle all the pointings, \tcycle, is set to 45
  minutes, we yield that the maximum number of pointing per track is
  \begin{equation}
    \npointmax = \frac{\tcycle}{\tintmin+\tslew} = 130.
  \end{equation}
\item Finally, if the PI wishes to observe an area that will require more
  that 130 pointings per independent track, the estimator will ask to
  either increase the requested elapsed telescope time or to decrease the
  requested field-of-view area. The computation is done as follows.
  \begin{enumerate}
  \item The number of track is then computed as described in section~\ref{sec:single-field}.
  \item The number of point per track is then
    $\npointpertrack = \paren{\frac{7}{4}}^2 \, \frac{\nbeam}{\ntrack}.$
    This value must be lower than \npointmax.
  \end{enumerate}
\end{itemize}

In summary, the sensitivity of a Nyquist sampled mosaic is 
\begin{equation}
  \noise[Jy] = \frac{\JypK\,\Tsys}{\atmeff\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
\end{equation}
\begin{equation}
  \mbox{with} \quad
  \ton =  \frac{\ttel \,-\,  n_{\emr{tracks}} \times  \ttune}{\obseff \moseff  \nbeam}
\end{equation}

\begin{thebibliography}{9}
  \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\bibitem[{Kraus (1982)}]{kra82} Kraus, J. D., in McGraw-Hill 1982 (1966),
  Radio Astronomy.
\end{thebibliography}

\end{document}

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