Sensitivity estimation

Step #1: Initialization

Initialize the values of $\ensuremath{\epsilon_\ensuremath{\mathrm{tel}}}$ , $\ensuremath{d_\ensuremath{\mathrm{submap}}}$ , $\ensuremath{d_\ensuremath{\mathrm{scan}}}$ , $\ensuremath{d_\ensuremath{\mathrm{cal}}}$ , $\ensuremath{\eta_\ensuremath{\mathrm{cal}}}$ , $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{pointing}}}$ , $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{focus}}}$ , $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{skydips}}}$ , $\ensuremath{t^\ensuremath{\mathrm{overhead}}_\ensuremath{\mathrm{scan}}}$ , $\ensuremath{t^\ensuremath{\mathrm{overhead}}_\ensuremath{\mathrm{subscan}}}$ , $\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{array}}}$ , $\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{throw}}}$ , $\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}}$ , etc...

Step #2: User input

Get the telescope time ( $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{tel}}}$ ), the number of sources ( $\ensuremath{n_\ensuremath{\mathrm{source}}}$ ) and the elevation ( $\ensuremath{\mathrm{el}}$ ). For mapping, get the map sizes. There are two cases:

EKH restoration

The following checks and computations must be done.

$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{sou}}}$	$\textstyle \le$	$\displaystyle 0.5^\circ,$	(29)
$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{tot}}}$	$\textstyle =$	$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{sou}}},$	(30)
$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{tot}}}$	$\textstyle =$	$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathr... ...throw}}}+ \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}}$	(31)
$\displaystyle \ensuremath{\alpha}$	$\textstyle =$	$\displaystyle \mbox{an empirical increasing function of \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{sou}}}{}.}$	(32)

Shift-and-add restoration

The following checks and computations must be done.

$\ensuremath{\alpha}$ is set to the minimum of the empirical increasing function of $\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{sou}}}$ because this restoration algorithm assumes that the restored sources are almost point-like.
$\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{submap}}}= 0.5\,\... ...\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{edge}}} \right) }$ with $\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{edge}}}= \ensurem... ...{throw}}}+\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}}$ .
If $\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{mos}}}\,\ens... ...}}}\le \ensuremath{\Delta^\ensuremath{\mathrm{2}}_\ensuremath{\mathrm{submap}}}$ , then

$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{tot}}}$ $\textstyle =$ $\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{mos}}},$ (33)

$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{tot}}}$ $\textstyle =$ $\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathr... ...hrow}}}+ \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}},$ (34)

$\displaystyle \ensuremath{n_\ensuremath{\mathrm{source}}}$ $\textstyle =$ $\displaystyle 1.$ (35)
Else

$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{tot}}}$ $\textstyle =$ $\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{submap}}},$ (36)

$\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{tot}}}$ $\textstyle =$ $\displaystyle \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{sub... ...hrow}}}+ \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}},$ (37)

$\displaystyle \ensuremath{n_\ensuremath{\mathrm{source}}}$ $\textstyle =$ $\displaystyle \frac{\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{... ...}}}}{\ensuremath{\Delta^\ensuremath{\mathrm{2}}_\ensuremath{\mathrm{submap}}}}.$ (38)

We note that in this case, $\ensuremath{n_\ensuremath{\mathrm{source}}}$ is not here a real number of source but more a number of submap: It does not need to be an integer.

Step #3: Computation of $\ensuremath{n_\ensuremath{\mathrm{cal}}}$ and $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{source}}}$

$\displaystyle 1.$		$\displaystyle \ensuremath{n_\ensuremath{\mathrm{cal}}}= \ensuremath{\mathrm{cei... ...}}}+\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{cal}}}} \right) },$	(39)
$\displaystyle 2.$		$\displaystyle \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{source}}... ...rm{}}_\ensuremath{\mathrm{cal}}}}{\ensuremath{n_\ensuremath{\mathrm{source}}}}.$	(40)

If $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{source}}}< 0$ , then return the following error message: ``The telescope time must at least be $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{cal}}}/\ensuremath{\epsilon_\ensuremath{\mathrm{tel}}}$ .''

Step #4: Computation of $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{subscan}}}$ , $\ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{max}}}$ and $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{scan}}}$

OnOff

$\displaystyle 1.$		$\displaystyle \mbox{\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{su... ...remath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{max}}}{} are fixed,}$	(41)
$\displaystyle 2.$		$\displaystyle \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{scan}}}=... ...}}) + \ensuremath{t^\ensuremath{\mathrm{overhead}}_\ensuremath{\mathrm{scan}}}.$	(42)

Mapping

$\displaystyle 1.$	$\displaystyle \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{subscan}... ...Vert}}_\ensuremath{\mathrm{tot}}}}{\ensuremath{v_\ensuremath{\mathrm{\Vert}}}},$	(43)
$\displaystyle 2.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{m... ...ath{t^\ensuremath{\mathrm{overhead}}_\ensuremath{\mathrm{subscan}}}} \right) },$	(44)
$\displaystyle 3.$	$\displaystyle \mbox{if} \quad \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensu... ... error message advising to increase \ensuremath{d_\ensuremath{\mathrm{scan}}},}$	(45)
$\displaystyle 4.$	$\displaystyle \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{scan}}}=... ...}}) + \ensuremath{t^\ensuremath{\mathrm{overhead}}_\ensuremath{\mathrm{scan}}}.$	(46)

Step #5: Computation of $\ensuremath{n_\ensuremath{\mathrm{scan}}}$ , $\ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{res}}}$ and $\ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{tot}}}$

$\displaystyle 1.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{scan}}}= \ensuremath{\mathrm{fl... ...}}{\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{scan}}}} \right) },$	(47)
$\displaystyle 2.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{r... ...ath{t^\ensuremath{\mathrm{overhead}}_\ensuremath{\mathrm{subscan}}}} \right) },$	(48)
$\displaystyle 3.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{t... ...max}}}+ \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{res}}}.$	(49)

Step #6: Computation of $\ensuremath{n_\ensuremath{\mathrm{cover}}}$ (Mapping only)

$\displaystyle 1.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{c... ...th{\mathrm{\bot}}_\ensuremath{\mathrm{tot}}}}{\ensuremath{\delta}}+1 \right) },$	(50)
$\displaystyle 2.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{cover}}}= \ensuremath{\mathrm{f... ...remath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{cover}}}} \right) },$	(51)
$\displaystyle 3.$	$\displaystyle \mbox{if} \quad \ensuremath{n_\ensuremath{\mathrm{cover}}}< 1, \q... ...to increase \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{tel}}}{}},$	(52)
$\displaystyle 4.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{t... ...}}}\, \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{cover}}}.$	(53)

Step #7: Computation of actual $\ensuremath{n_\ensuremath{\mathrm{scan}}}$ and $\ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{res}}}$ (Mapping only)

$\displaystyle 1.$		$\displaystyle \ensuremath{n_\ensuremath{\mathrm{scan}}}= \ensuremath{\mathrm{fl... ...suremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{max}}}} \right) },$	(54)
$\displaystyle 2.$		$\displaystyle \ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{r... ...{scan}}}\ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{max}}}.$	(55)

Step #8: Computation of $\ensuremath{\sigma_\ensuremath{\mathrm{}}}$

OnOff

$\displaystyle 1.$		$\displaystyle \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{\ensurem... ...res}}}) \, \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{subscan}}},$	(56)
$\displaystyle 2.$		$\displaystyle \ensuremath{\sigma_\ensuremath{\mathrm{}}}= \frac{\ensuremath{\ma... ...\mathrm{}}_\ensuremath{\mathrm{\ensuremath{\sigma_\ensuremath{\mathrm{}}}}}}}}.$	(57)

Mapping

$\begin{displaymath} \ensuremath{\sigma_\ensuremath{\mathrm{}}}= \ensuremath{\al... ...athrm{\Vert}}}}{\ensuremath{n_\ensuremath{\mathrm{cover}}}}}, \end{displaymath}$

(58)

Step #9: Computation of the actual $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{source}}}$ , $\ensuremath{n_\ensuremath{\mathrm{cal}}}$ and $\ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{tel}}}$

We note that 1) the actual number of scan is $\ensuremath{n_\ensuremath{\mathrm{scan}}}+1$ when $\ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{res}}}\ge 1$ and $\ensuremath{n_\ensuremath{\mathrm{scan}}}$ when $\ensuremath{n_\ensuremath{\mathrm{subscan}}^\ensuremath{\mathrm{res}}}= 0$ and 2) the true elapsed telescope time might be smaller than the user input.

$\displaystyle 1.$	$\displaystyle \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{source}}... ...uremath{t^\ensuremath{\mathrm{overhead}}_\ensuremath{\mathrm{scan}}} \right] },$	(59)
$\displaystyle 2.$	$\displaystyle \ensuremath{n_\ensuremath{\mathrm{cal}}}= \ensuremath{\mathrm{cei... ...uremath{\mathrm{source}}}}{\ensuremath{d_\ensuremath{\mathrm{cal}}}} \right) },$	(60)
$\displaystyle 3.$	$\displaystyle \ensuremath{t^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{tel}}}= ... ...}_\ensuremath{\mathrm{cal}}}}{\ensuremath{\epsilon_\ensuremath{\mathrm{tel}}}}.$	(61)

Time estimation

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Algorithms

Gildas manager 2014-07-01