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\providecommand{\casa}{\mbox{\sc casa}}%
\providecommand{\miriad}{\mbox{\sc miriad}}%
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\providecommand{\greg}{\mbox{\sc greg}}%
\providecommand{\gtvirt}{\mbox{\sc gtvirt}}%
\providecommand{\atm}{\mbox{\sc atm}}%
\providecommand{\telcal}{\mbox{\sc telcal}}%
\providecommand{\astro}{\mbox{\sc astro}}%
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\lhead{\scshape Comparison of ATM versions}
\rhead{\scshape \nouppercase{\rightmark}}   %
\fancyfoot[C]{\bfseries\thepage {}}


%
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%
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%
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%
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%
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\providecommand{\Tamb}[1][]{\ensuremath{T_\ensuremath{\mathrm{amb}}\ifthenelse{\equal{#1}{}}{}{^\ensuremath{\mathrm{#1}}}}}%
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\providecommand{\C}{\ensuremath{C}}%
\providecommand{\cdark}{\ensuremath{c_\ensuremath{\mathrm{dark}}}}%
\providecommand{\el}{\ensuremath{\mathrm{elevation}}}%
\providecommand{\airmass}{\ensuremath{a}}%
\providecommand{\gcal}{\ensuremath{\mathcal{G}}}%
\providecommand{\gim}{\ensuremath{G_\ensuremath{\mathrm{im}}}}%
\providecommand{\gimtrue}{\ensuremath{G_\ensuremath{\mathrm{im}}^\ensuremath{\mathrm{true}}}}%
\providecommand{\gimmeas}{\ensuremath{G_\ensuremath{\mathrm{im}}^\ensuremath{\mathrm{meas}}}}%
\providecommand{\g}{\ensuremath{g}}%
\providecommand{\Feff}{\ensuremath{F_\ensuremath{\mathrm{eff}}}}%
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\providecommand{\Feffmeas}{\ensuremath{F_\ensuremath{\mathrm{eff}}^\ensuremath{\mathrm{meas}}}}%
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\providecommand{\tausig}{\ensuremath{\tau\ifthenelse{\equal{sig}{}}{}{_\ensuremath{\mathrm{sig}}}}}%
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\providecommand{\tauwet}{\ensuremath{\tau\ifthenelse{\equal{wet}{}}{}{_\ensuremath{\mathrm{wet}}}}}%
\providecommand{\tautrue}{\ensuremath{\tau\ifthenelse{\equal{true}{}}{}{_\ensuremath{\mathrm{true}}}}}%
\providecommand{\taumodel}{\ensuremath{\tau\ifthenelse{\equal{mod}{}}{}{_\ensuremath{\mathrm{mod}}}}}%
\providecommand{\water}{\ensuremath{\mathrm{w_{H_2O}}}}%
\providecommand{\awater}{\ensuremath{\mathrm{\alpha_{H_2O}}}} 




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  \ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}= \ensuremath{F_\ensuremath{\mathrm{eff}}}\ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{tot}{}}{}{^\ensuremath{\mathrm{tot}}}} + (1-\ensuremath{F_\ensuremath{\mathrm{eff}}})\ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}.
  
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  \ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{tot}{}}{}{^\ensuremath{\mathrm{tot}}}} = \frac{\ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{sig}{}}{}{^\ensuremath{\mathrm{sig}}}}+\ensuremath{G_\ensuremath{\mathrm{im}}}\,\ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{ima}{}}{}{^\ensuremath{\mathrm{ima}}}}}{1+\ensuremath{G_\ensuremath{\mathrm{im}}}},
  
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  \ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{sig}{}}{}{^\ensuremath{\mathrm{sig}}}} = \ensuremath{T_\ensuremath{\mathrm{atm}}\ifthenelse{\equal{sig}{}}{}{^\ensuremath{\mathrm{sig}}}}\,\ensuremath{\displaystyle\left\{ 1-\exp\ensuremath{\displaystyle\left(  -\ensuremath{a}\ensuremath{\tau\ifthenelse{\equal{sig}{}}{}{_\ensuremath{\mathrm{sig}}}} \right) } \right\}}
  \quad \mbox{and} \quad
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  \frac{\ensuremath{T_\ensuremath{\mathrm{hot}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}-\ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{tot}{}}{}{^\ensuremath{\mathrm{tot}}}}}{\ensuremath{C}_\ensuremath{\mathrm{hot}}-\ensuremath{C}_\ensuremath{\mathrm{sky}}^{tot}} = 
  \frac{\ensuremath{T_\ensuremath{\mathrm{hot}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}-\ensuremath{T_\ensuremath{\mathrm{cold}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}}{\ensuremath{C}_\ensuremath{\mathrm{hot}}-\ensuremath{C}_\ensuremath{\mathrm{cold}}},
  
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  \ensuremath{T_\ensuremath{\mathrm{a}}\ifthenelse{\equal{\star}{}}{}{^\ensuremath{\mathrm{\star}}}}= \ensuremath{T_\ensuremath{\mathrm{cal}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}\,\frac{\ensuremath{C}_\ensuremath{\mathrm{on}}-\ensuremath{C}_\ensuremath{\mathrm{off}}}{\ensuremath{C}_\ensuremath{\mathrm{hot}}-\ensuremath{C}_\ensuremath{\mathrm{off}}},
  
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$\textstyle =$%
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$\displaystyle (1+\ensuremath{G_\ensuremath{\mathrm{im}}})\,\ensuremath{\displaystyle\left[  \ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{sig}{}}{}{^\ensuremath{\mathrm{sig}}}}-\ensuremath{T_\ensuremath{\mathrm{bg}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} \right] }$%
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$\textstyle +$%
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$\displaystyle (1+\ensuremath{G_\ensuremath{\mathrm{im}}})\,\ensuremath{\displaystyle\left[  \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}-\ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{sig}{}}{}{^\ensuremath{\mathrm{sig}}}} \right] }\,\exp(\ensuremath{a}\ensuremath{\tau\ifthenelse{\equal{sig}{}}{}{_\ensuremath{\mathrm{sig}}}})$%
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$\displaystyle \ensuremath{G_\ensuremath{\mathrm{im}}}\,\ensuremath{\displaystyle\left[  \ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{sig}{}}{}{^\ensuremath{\mathrm{sig}}}}-\ensuremath{T_\ensuremath{\mathrm{bg}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} \right] }\,\ensuremath{\displaystyle\left[  \exp\ensuremath{\displaystyle\left\{ \ensuremath{a}(\ensuremath{\tau\ifthenelse{\equal{sig}{}}{}{_\ensuremath{\mathrm{sig}}}}-\ensuremath{\tau\ifthenelse{\equal{ima}{}}{}{_\ensuremath{\mathrm{ima}}}}) \right\}}-1 \right] }$%
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$\displaystyle \frac{1+\ensuremath{G_\ensuremath{\mathrm{im}}}}{\ensuremath{F_\ensuremath{\mathrm{eff}}}}\,\ensuremath{\displaystyle\left[  \ensuremath{T_\ensuremath{\mathrm{hot}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}-\ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} \right] }\,\exp(\ensuremath{a}\ensuremath{\tau\ifthenelse{\equal{sig}{}}{}{_\ensuremath{\mathrm{sig}}}}).$%
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  \ensuremath{T_\ensuremath{\mathrm{cal}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}= (\ensuremath{T_\ensuremath{\mathrm{hot}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}-\ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}})\frac{1+\ensuremath{G_\ensuremath{\mathrm{im}}}}{\ensuremath{F_\ensuremath{\mathrm{eff}}}\,\exp\ensuremath{\displaystyle\left(  -\ensuremath{a}\ensuremath{\tau\ifthenelse{\equal{sig}{}}{}{_\ensuremath{\mathrm{sig}}}} \right) }}.
  
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%
\providecommand{\OO}{\ensuremath{\mathrm{O_2}}}%


%
\providecommand{\OOO}{\ensuremath{\mathrm{O_3}}}%


%
\providecommand{\HOH}{\ensuremath{\mathrm{H_2O}}}%

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$1-\exp(-\ensuremath{a}\tau)$%
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$\ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}\propto \ensuremath{T_\ensuremath{\mathrm{atm}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}\,
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\begin{displaymath}
  
    \frac{\ensuremath{T_\ensuremath{\mathrm{cal}}\ifthenelse{\equal{meas}{}}{}{^\ensuremath{\mathrm{meas}}}}-\ensuremath{T_\ensuremath{\mathrm{cal}}\ifthenelse{\equal{true}{}}{}{^\ensuremath{\mathrm{true}}}}}{\ensuremath{T_\ensuremath{\mathrm{cal}}\ifthenelse{\equal{true}{}}{}{^\ensuremath{\mathrm{true}}}}}
  = \frac{\ensuremath{F_\ensuremath{\mathrm{eff}}^\ensuremath{\mathrm{true}}}(1+\ensuremath{G_\ensuremath{\mathrm{im}}^\ensuremath{\mathrm{meas}}})}{\ensuremath{F_\ensuremath{\mathrm{eff}}^\ensuremath{\mathrm{meas}}}(1+\ensuremath{G_\ensuremath{\mathrm{im}}^\ensuremath{\mathrm{true}}})}
    \exp\ensuremath{\displaystyle\left[  \ensuremath{a}(\ensuremath{\tau\ifthenelse{\equal{mod}{}}{}{_\ensuremath{\mathrm{mod}}}}-\ensuremath{\tau\ifthenelse{\equal{true}{}}{}{_\ensuremath{\mathrm{true}}}}) \right] }-1
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\stepcounter{section}
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\stepcounter{section}
\appendix
\stepcounter{section}
\stepcounter{section}

%
\providecommand{\FigATMvsFreq}[1]{%
  \begin{figure}[htbp]
    \centering
    \includegraphics[width=0.425\textheight,angle=270]{atm-comparison-#1-20deg}
  \end{figure}
  \begin{figure}[htbp]
    \centering
    \includegraphics[width=0.425\textheight,angle=270]{atm-comparison-#1-45deg}
  \end{figure}}%

\stepcounter{section}

\end{document}
