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Low opacity approximation

To outline the relative importance of all parameters, I will now derive an approximate formula for the calibration factor $Tcal$ in the limiting (but frequent) case where the opacities are weak and equal in the image and signal bands.
\begin{displaymath}
T_{sky} = \tau * T_{atm}
\end{displaymath} (25)

where $T_{atm}$ is the physical temperature of the absorbing layers. Thus
\begin{displaymath}
T_{emi} = F_{eff} * \tau * T_{atm} + (1-F_{eff}) * T_{cab}
\end{displaymath} (26)


\begin{displaymath}
\tau = (T_{emi} - (1-F_{eff}) * T_{cab}) / ( F_{eff} * T_{atm} )
\end{displaymath} (27)


\begin{displaymath}
Tcal = C_{eff} * (T_{load} - T_{emi}) (1+Gain\_i) / ((1-\tau) * B_s)
\end{displaymath} (28)

Eliminating $\tau$ gives
\begin{displaymath}
Tcal = C_{eff} \frac{(T_{load}-T_{emi}) * (1+Gain\_i) * T_{a...
...f}}
{B_s * (F_{eff} * (T_{atm}-T_{cab}) + T_{cab} - T_{emi})}
\end{displaymath} (29)

Although $T_{cab}$ is a weighted average of the physical temperature of the cabin and the outside temperature, it is not very different from $T_{load}$. Hence we can simplify :

\begin{displaymath}
Tcal = C_{eff} \frac{F_{eff} * (1+Gain\_i) * T_{atm}}
{B_s * ( 1 - F_{eff} * \frac{T_{cab}-T_{atm}}{T_{cab}-T_{emi}})}
\end{displaymath} (30)

It is important to note that in this formula, $T_{atm}$ is dependent mostly on the outside temperature and pressure (and site altitude of course), and weakly on $\tau$ because $T_{atm}$ is the (physical) temperature of the absorbing layers. $T_{emi}$ is the effective temperature seen by the antenna, and hence small compared to $T_{cab}$. The parameters are $F_{eff}$, $B_{eff}$ and $Gain\_i$. Setting reasonable numbers :
$T_{cab} = 290 K$, $T_{atm} = 240 K$, $T_{emi} = 50 K$, and $C_{eff} =
1$,
yield
\begin{displaymath}
Tcal = \frac{240 * (1+Gain\_i) * F_{eff}}{B_s * (1 - 0.2*F_{eff})}
\end{displaymath} (31)


next up previous contents
Next: SKYDIP effects Up: Critical parameters, biases and Previous: Critical parameters, biases and   Contents
Gildas manager 2018-06-20