a. Discussion of required constancy of physical parameters

In a first step we now discuss the constancy of physical parameters within the first two terms of the right-hand side of Eq. (4) during a q measurement. In a second step, an error analysis is presented on the basis of a numerical model.

The spectral correction factors η for the two radiometers (indexes 1 and 2) and valid for blackbody radiation are shown in Fig. 1. Assuming a temperature change of 10 K at the chopper during the q measurement l = (η_2,B,CHO)/(η_1,B,CHO) will change by about 1% since the correction factor of the target radiometer (index 1) increases by 0.5% (curve 3 of Fig. 1), whereas the correction factor of the reference radiometer (index 2) decreases by 0.5% (curve 1 of Fig. 1), going from 300 to 290 K. If the target radiometer is equipped with another type of dome (curve 2, Fig. 1) the change in l becomes negligible since both correction factors change approximately in the same way. This means that depending on the optical materials used in the two radiometers one may have to consider the changes in l. Since the temperature of the chopper is not measured and not accessible from the formalism (B_CHO has been eliminated), assumptions have to be made for the time profile of the chopper temperature to perform a first-order correction.

The changes in η_1,TAR and η_2,B,REF are negligible (<0.05%) since the temperature drifts of the corresponding sources are much smaller (ΔT < 1 K) as compared to the chopper.

The changes in the two temperature correction factors c₁ and c₂ are negligible (<0.05%) since the temperature drift of the detectors is small (<0.2 K) and the correction factors are in the order of 0.2% K⁻¹.

The change of the target radiance E_1,TAR during the measurement of q can be held negligible (<0.1%) by setting the radiometer head on a blackbody source at constant temperature.

The temperature of the reference blackbody T_REF, however, will show a drift since there is some thermal interaction between the cold/hot chopper and the reference blackbody. Measurements show that this temperature drifts up to 1 K during a q measurement, which corresponds to a change of B_REF of 1.3% at 300 K. Therefore, the assumption of thermal steady state during q measurement in Lorenz et al. (1996) is too idealistic and may result in considerable errors of the calibration parameters q and R₁.

In principle a temperature stabilization of the reference blackbody could solve this problem. The additional system increases the costs and is not useful for the purpose of atmospheric flux measurement. These problems can be avoided using the new calibration scheme.

b. Model calculations

To demonstrate the errors mentioned above, a simplified model of the calibration procedure (q measurement and calibration of R₁) has been programmed including the errors introduced to a typical airborne measurement of the downward flux.

The simplifications in the model are as follows. 1) All spectral correction factors are constant (=1), and 2) the temperature correction factors c₁ and c₂ are constant (=1).

As mentioned in section 2a, only the spectral correction factors forming l = (η_2,B,CHO)/(η_1,B,CHO) can show the changes that have to be considered. On the basis of reasonable assumptions (time profile of chopper temperature), these changes can be established with sufficient accuracy if needed, so they are not considered in the model. The changes to the temperature correction factors c₁ and c₂ are negligible, as pointed out above.

Thus the model reflects errors introduced by neglecting temperature drifts of the reference blackbody during q measurement.

The logical structure of the model calculation is demonstrated in the following steps:

1. Model the q measurement

   • Assume a model value of q and R₁;

   • define a series of temperature values for T_REF and T_CHO describing the temperature drifts of the reference source and the chopper as observed during a typical q measurement;

   • calculate the corresponding model radiometer outputs U₁ and U₂ using the model values of q and R₁ and Eq. (1) above and Eq. (5c) of Lorenz et al. (1996),

     

     

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   • determine the measured value of q, q_m from the linear regression of Eq. (3).

2. Model the R₁ calibration according to Eq. (11) of Lorenz et al. (1996):

   

   

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   • Perform the model calibration of R₁ assuming three calibration irradiances E_1,TAR, as used in a typical calibration, and take the corresponding temperatures T_CHO and T_REF, as can be found from experimental data;

   • calculate the corresponding model radiometer outputs U₁ and U₂ using Eqs. (1) and (5) and the model values R₁ and q; and

   • determine the measured value of R₁, R_1m from Eq. (6) using q_m.

3. Model an atmospheric measurement, Eq. (6) of Lorenz et al. (1996):

   

   

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   • Assume a model target irradiance E_1,TAR that is typical for airborne measurements;

   • assume a value for T_REF and a temperature difference between T_REF and T_CHO as a model variable—studying various flight situations, one can find plausible sets for these quantities;

   • calculate the true radiometer outputs U₁ and U₂ for the measurement in the atmosphere using Eqs. (1) and (5) and the model values R₁ and q;

   • determine the measured value of E_1,TAR, E_1,TAR,m with Eq. (7) using R_1m and q_m; and

   • note the difference E_1,TAR,m − E_1,TAR leads to the absolute error shown in the graphs below.

Figures 2 and 3 show the absolute error for atmospheric downward fluxes of 30 and 90 W m⁻² caused by neglecting the changes of T_REF during q measurement as part of the calibration. The two curves in each graph correspond to the two temperatures of the reference source, which have been chosen as being representative for measurements of radiation fluxes at upper-tropospheric levels. The simulations show for this case a positive error of the atmospheric measurement, on the order of 1 to 4 W m⁻², depending on the thermal situation of the instrument. Since the thermal situation of the instrument can rapidly change during the various flight conditions, this error will produce changes on the calculated fluxes that look like an instability of the instrument. Thus, one of the most important advantages of the chopped pyrgeometer, its radiometric stability inherent to the modulation process, is lost to a certain extent.

It must be clearly pointed out that the errors presented here are introduced only by assuming a steady-state condition during q measurement, thus neglecting the drift of the reference temperature that occurs during actual measurement.

In the paper of Lorenz et al. (1996) some results have been presented of the pre-EUCREX Intercomparison Campaign of 1992 where five pyrgeometers (two Eppley pyrgeometers, two Foot pyrgeometers, one chopped pyrgeometer) have been flown simultaneously on three airplanes. Results of a boxlike pattern flown at an altitude of 6.1 km MSL have been analyzed. The results of all five instruments for measurements along the legs and the changes between the legs of the flight box show that the chopped pyrgeometer provides a radiometric stability that is at least as good as that of the Foot pyrgeometers. Another significant finding was the fact that the chopped pyrgeometer yields the smallest flux values of all five instruments throughout the four legs of the box. The difference was about 10 W m⁻² as compared to the average value (∼90 W m⁻²) of the other four instruments. The model calculation presented above would predict a decrease of the measured flux values in the order of 1 W m⁻² [ Fig. 3: (T_CHO − T_REF ∼ 0), T_REF ∼ 230 K] for the chopped pyrgeometer if the new calibration method would have been applied, thus creating a somewhat greater difference to the findings of the other instruments. Since calibration sensitive components (detector, dome, etc.) of the chopped pyrgeometer used for the pre-EUCREX measurements have been changed in the meantime, it is not possible to apply the new calibration to the original instrument.

a. Calibration of the target radiometer

As is obvious from Eq. (8) the target radiometer can be calibrated with a blackbody source if the temperature of the chopper is known during calibration. This can be achieved if the radiometer head is in an isothermal condition during the measurement of a calibration point; then the chopper temperature equals the reference temperature, which is measured by PT100 (accuracy: ±0.15 K, −50°C < T < 50°C). In this case ΔB_T can be calculated since B_CHO and E_1,TAR are known blackbody radiations and η_1,TAR and η_1,B,CHO are spectral correction factors derived for the known temperatures of the radiators.

The procedure to achieve an isothermal radiometer head during target radiometer calibration is as following. In a thermally quiet laboratory the radiometer head (switched off) is put under a protection box for some hours to reach isothermal conditions. After that, the box is removed and the head is switched on and positioned on a blackbody source. These activities can be performed in less than 10 s; thus isothermal conditions are maintained during that time with sufficient accuracy. Immediately after correct position on the blackbody, a signal of the target radiometer can be registered (time constant of the radiometer signals is on the order of 0.1 s) for a few seconds. During that time the signal of the reference radiometer begins to drift away from zero since the temperature of the chopper slowly drifts toward the temperature of the calibration blackbody source, thus creating a radiometric difference ΔB_R [Eq. (9)] between chopper and reference source. Therefore, the signal of the reference radiometer can be used to indicate proper isothermal conditions during the measurement and to establish an error value; this signal can be maintained at less than 10 mV throughout the measurement, corresponding to a ΔT of 0.1 K (300 K blackbody). The whole procedure is repeated for each calibration point.

The graph (Fig. 4) of a target radiometer calibration due to Eq. (8) shows three calibration points that correspond to three temperatures of the calibration blackbody source (280.4, 300.3, and 310.4 K) and the corresponding linear regression. Each of the three calibration points contains n (n ≈ 10) samples (1-s sampling interval) registered during the first n seconds after positioning the radiometer head on the calibration blackbody source; the points are too close together to be resolved in the plot. The linear regression line shows a zero offset of about 4 mV, corresponding to a ΔT of 0.2 K (300-K blackbody), which is consistent with the accuracy of a PT100. This offset can be explained as follows. The radiometric measurement compares the radiation of the reference source with that of the calibration source (both related to the chopper radiation). On the other hand, the calculation of the radiometric terms in Eqs. (8) and (9) is based on the measurement of two independent temperatures (calibration blackbody and reference source). For the purpose of demonstration we assume that the calibration blackbody, the chopper, and the reference source are at the same temperature, that is, isothermal conditions are valid. In this case both radiometer signals will be zero. The temperature measurements at the two sources, however, will show some difference due to systematic errors of the PT100 sensors. Consequently, the radiometric evaluation will predict a radiometric difference ΔB that is somewhat different from zero. Thus the observed offset is due to a more or less unavoidable difference between the measured temperature and the real radiometric temperature of the two sources. If we assume a constant error in the absolute temperature measurement over the range of interest, the slope of the linear regression provides the correct response; the offset is constant and can be omitted in subsequent data evaluation.

Figure 5 shows the deviation of the individual samples of the three calibration points from the regression line, where 1 mV corresponds to a ΔT of 0.05 K (300-K blackbody). Since the sign changes between the sample groups of highest and lowest calibration temperature to the group of the intermediate temperature, the calibration shows a small nonlinearity on the order of 0.1 K (≈0.15% of the 300-K blackbody); this nonlinearity may be due to small errors in spectral correction factors or errors caused by deviations from a flat spectral emissivity over a wide spectral range (1–100 μm) of the black radiating surfaces (reference source and chopper).

b. Calibration of the reference radiometer

The calibration of the reference radiometer is now straightforward because R₁ is already known. One brings the radiometer head to a nonisothermal condition, for example, by warming the chopper with hot air, and puts the radiometer head on a calibration blackbody, which is conveniently at ambient temperature. The chopper temperature will drift toward ambient and create varying output signals for both radiometers. The evaluation of the measurement is as follows. Using Eq. (8) one can calculate the chopper radiance B_CHO for each measurement sample of the target radiometer output U₁ because R₁ and all other quantities are known. The chopper radiation now can be used in Eq. (9) to calculate ΔB_R and perform a linear regression to find R̃₂.

The procedure to calibrate the reference radiometer is somewhat similar to the q measurement discussed above. The important difference, however, is that no assumptions are necessary of a thermal steady state of various radiometer components since all dynamic quantities, including the temperature of the reference source, are known and can be correctly used in the formalism.

Figures 6 and 7 show an example of the reference radiometer calibration. The blackbody calibrator is at ambient temperature in this case. Before starting the measurement, the chopper is warmed to achieve nonzero outputs in both radiometers. During the measurement the signals of the two radiometers are drifting toward zero (Fig. 6) since the chopper slowly drifts to ambient temperature. The plot for calibration (Fig. 7) according to Eq. (9) shows the data points and the linear regression. The measurement starts with a temperature difference between chopper and reference source of about 7 K, which corresponds to the data points at high values of ΔB_R. The data points taken for the calibration cover 140 samples (1-s sampling interval); after that period the further change of the temperature difference (now ≈ 1.3 K) gets slower, thus additional information gain is low. The parameters of the linear regression show a nonzero y intercept of the regression line of 17 mV, which corresponds to a ΔT of 0.15 K of a 300-K blackbody. The nature of this offset is the same as discussed above for the target radiometer.