8.5.3 General case

The receiver is not purely single-sideband. Let us denote by G^l and G^u the normalized gains in the receiver lower and upper sidebands, G^l + G^u = 1. The atmosphere opacity per km varies with altitude as does the air temperature.

Then, the above expressions of T_sky and T_emi should be explicited for each sideband (j= u or l):

  

$$(1-e^{-{\tau}^j}) T_{atm}^j$$ T_sky^j = (8.36)

$$T_{sky}^l \eta_f G^l + T_{sky}^u \eta_f G^u+ (1-\eta_f) T_{gr}$$ T_emi = (8.37)

The atmospheric transmission model ATM [Cernicharo 1988] allows to calculate iteratively $\tau_\nu$ from a load+ sky measurement. The values of $\tau^l, \tau^u$are calculated for the Standard US atmosphere (parameters are: Winter-, Spring-, or Summer-temperature T, altitude, latitude, water vapor w) by summing up the contributions of O₂ ($A_\nu(T)$), H₂O ($B_\nu(T)$) and O₃ ($C_\nu(T)$) (including rare isotopes and vibrationally excited states):

$$\tau_\nu= A_\nu(T) + w B_\nu(T) + C_\nu(T)$$ (8.38)

Depending on $\nu$, the values of $A_\nu(T), B_\nu(T), C_\nu(T)$ are derived either from a Table or estimated from empirical formulae. A first guess of the amount of precipitable water is made from the ambient temperature, pressure and humidity by using the relations of Sec.8.1.4. Then, the expected T_sky and T_emi are calculated from the two expressions above and T_emi is compared to its value measured from the the observation of the atmosphere and the load (Eq.8.35). The value of w is changed and the calculation of $\tau^j, T_{sky}^j$ and T_emi restarted. Normally, the process converges after 2 to 4 iterations. Once w and T_emi are known, the calibration factor T_cal can be derived

$$T_{cal}^j = ({T_{load} - T_{emi}})e^{\tau^j}$$ (8.39)

and the data calibrated in the T_A^* scale using Eq.8.32.