4.1 Regression using HIRS/3 channels 11 and 12

HIRS/3 channel 11 is centred at a wavelength of 7.3 µm. While the strong water vapour ν₂ vibration–rotation band has its peak line strengths at about the channel 12 wavelength (≈6.5 µm), channel 11 is centred on the long-wave side of this band, off the peak with lower line strengths, and thus channel 11 is characteristic of the water vapour in lower levels than channel 12. In a standard midlatitude summer atmosphere channel 11 peaks at about 5 km altitude (see Fig. 2 of Gierens and Eleftheratos, 2016).

Using the channel spectral response function for channel 11 on NOAA 15, radiative transfer calculations have been performed for the radiosonde profiles used above. Figure 5 shows the resulting brightness temperatures, T_11∕15, plotted against the previously computed ΔT₁₂ for the set of Lindenberg profiles. As expected, T_11∕15 is generally higher than the channel 12 brightness temperatures because it characterises the temperature in the mid-troposphere where the channel 11 weighting function peaks. T_11∕15 ranges from 248 to 268 K for the Lindenberg profiles. Figure 5 also shows a linear correlation between ΔT₁₂ and T_11∕15, although with a large scatter. The linear Pearson correlation coefficient is −0.68. Its square is 0.46, that is, variations of T_11∕15 represent almost half of the variations in ΔT₁₂. The remaining scatter is not surprising given the tremendous variability of relative humidity profiles. One additional piece of information is clearly insufficient to capture all this variability. Nevertheless, the correlation is clearly visible. We have made use of it to construct a correction to the HIRS/3 measured channel 12 brightness temperatures, a correction that leads to a pseudo-channel 12 brightness temperature as if a HIRS/2 instrument had measured it.

Figure 5Scatter plot showing a linear correlation for the difference between channel 12 brightness temperatures (NOAA 15 minus NOAA 14) and the NOAA 15 channel 11 brightness temperature computed using the Lindenberg profiles. The linear Pearson correlation coefficient is −0.68.

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Figure 6(a) Scatter plot showing a linear correlation between a linear superposition of channel 11 and 12 brightness temperatures from NOAA 15 (abscissa, ${\widehat{T}}_{\mathrm{12}/\mathrm{15}}$) with the corresponding channel 12 brightness temperature for the same profile but computed with the NOAA 14 channel response function. Note that the fit line has slope 1.000 and the intercept is close to zero ($\mathrm{2}\times {\mathrm{10}}^{-\mathrm{4}}$). The linear correlation is R=0.986. All data are computed using the Lindenberg profiles. (b) The same data, plotted with the difference in $T_{\mathrm{12}/\mathrm{14}}-{\widehat{T}}_{\mathrm{12}/\mathrm{15}}$ on the y axis.

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For this purpose we try a bilinear regression ² of the following kind:

Here, ${\widehat{T}}_{\mathrm{12}/\mathrm{15}}$ is the desired pseudo-channel 12 brightness temperature that is equivalent to a HIRS/2 measurement. In other words it is the T_12∕15 that would have been measured by a HIRS/2 instrument. For the calculation of ${\widehat{T}}_{\mathrm{12}/\mathrm{15}}$ only the nadir brightness temperatures have been retained as it seems that the off-nadir directions do not yield differing information. The two data vectors containing the brightness temperatures of channels 11 and 12 are linearly correlated with R=0.71, but they point in different directions; that is, they are not co-linear. Regression thus yields a unique result, namely

Figure 7(a) Test of the superposition method using radiosonde profiles from the two GRUAN stations Sodankylä, Finland, and Manus, Papua New Guinea. The diagonal line (y=x) is included to check the result: it is not a fit. (b) The same data, plotted with the difference in $T_{\mathrm{12}/\mathrm{14}}-{\widehat{T}}_{\mathrm{12}/\mathrm{15}}$ on the y axis.

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The 1 σ uncertainty estimates of the parameters b,c are both ±0.01. The corresponding data pairs are shown in Fig. 6. The slope and intercept of the regression (black line in panel a) are 1.000 and $\mathrm{2}\times {\mathrm{10}}^{-\mathrm{4}}$, respectively, and the linear correlation between the linear superposition of channel 11 and 12 brightness temperatures and that of the pseudo-channel 12 brightness temperature is 0.986. Panel (b) shows the same data but with the difference between regressand and regressor on the y axis. Maximum deviations from the zero line are about +3 and −2 K. Mean and standard deviation of the residuals are 0.0±0.6 K.

4.2 Test with independent radiosonde profiles

Using the linear superposition of channel 11 and 12 brightness temperatures for the considered atmospheric profiles from the two GRUAN stations, Sodankylä and Manus, leads to the data pairs shown in Fig. 7. The black diagonal line in this figure is not the result of a best fit or a regression, but is y=x, plotted to guide the eye in checking the result. The residual means ($T_{\mathrm{12}/\mathrm{14}}-{\widehat{T}}_{\mathrm{12}/\mathrm{15}}$) and their standard deviations are 0.3±1.3 K for Sodankylä and $-\mathrm{0.4}\pm \mathrm{1.3}$ K for Manus. Again, we see that the regression using just one additional piece of information is not able to provide a complete correction with an average residual of zero. At least these residuals are much smaller than the original differences between T_12∕14 and T_12∕15 shown in Fig. 2. Obviously, the superposition methods works well for these data, representing a polar and an equatorial atmosphere.

5.1 Superposition of weighting functions

The superposition of channels 11 and 12 is equivalent to a superposition of their weighting functions. Fig. 8 gives an example. The weighting functions are generic functions, as in Gierens and Eleftheratos (2016), assuming a water vapour scale height of 2 km and peak altitudes of 8.5 km for ch. 12/15 (red curve), 7.5 km for ch. 12/14 (black) and 5 km for ch. 11/15 (blue). The black curve with circles represents the superposition of channels 11 and 12 on NOAA 15 with the weights b and c derived above. The superposition curve (its upper tail, its peak and about half of its lower tail) is between the corresponding channel 12 weighting functions. We note here that the superposition weighting function has some weight at lower altitudes where both channel 12 weighting functions are already very low. Overall, we see that the superposition method eventually brings the pseudo-channel 12 brightness temperature of NOAA 15 closer to the level of the corresponding channel 12 brightness temperature of NOAA 14.

Figure 8 (and actual weighting functions shown in the Supplement, Fig. S1) shows that there is some possibility that channel 11 sees the ground when the atmosphere is quite dry. In such cases, which might occur at high latitudes, the superposition will not work. High brightness temperatures in both channels 10 and 11 could indicate such an event. Indeed, the (high-latitude) Sodankylä data show larger scatter in Fig. 7 than the (equatorial) data from Manus, which might result from unwanted ground influence at the high-latitude station.

An interesting alternative interpretation of the coefficients resulting from the bilinear regression may derive from the following consideration: it is possible to rewrite Eq. (1) as a weighted mean of three temperatures:

From this interpretation and Eq. (2) follows $a^{\prime }=-\mathrm{0.14655}$ and it turns out that T₀=241.6 K, which is remarkably close to 240 K, the T₀ used as a reference in the retrieval schemes developed by Soden and Bretherton (1993), Stephens et al. (1996) and Jackson and Bates (2001). At the altitude where the channel 12 weighting function peaks the temperature is, on average, close to T₀. The remarkable fact is that the regression results just in this T₀ for the constant part and not anything else – a finding that could not be expected a priori.

Figure 8Examples of weighting functions for channels 11 and 12 on NOAA 15 (blue and red), their superposition (black with circles) and channel 12 on NOAA 14 (black).

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5.2 Application to real data

For the same set of 1004 days of common operation of NOAA 14 and NOAA 15 as used in GE17, we have compared the channel 12 brightness temperatures and daily averages on a 2.5^∘ × 2.5^∘ grid in the northern midlatitudes, 30 to 70^∘ N. Differing from the previous paper, we use the original non-intercalibrated brightness temperatures. For NOAA 15 we compute the linear superposition derived above, that is, ${\widehat{T}}_{\mathrm{12},\mathrm{15}}$, while for NOAA 14 we use T_12,14. The 2-D histogram of these data pairs is shown in Fig. 9. It is remarkable how similar this histogram is to a corresponding one shown as Fig. 2 in GE17, which displays the intercalibrated data. The ordinary least squares linear fit through the new data pairs (solid line) has the equation:

with a slightly smaller slope and a slightly larger intercept than in GE17 using the intercalibrated data pairs (0.8290 and 41.63, respectively). These coefficients have been determined using a non-linear least-squares Marquardt–Levenberg algorithm (Press et al., 1989, chap. 14.4). The slope has a 1σ uncertainty of 0.001. Because the linear least squares fit suffers from regression dilution when errors of the regressor (${\widehat{T}}_{\mathrm{12},\mathrm{15}}$) are not taken into account, we also compute the bivariate regression (dash-dotted), which has the equation:

and this has a slightly smaller slope and larger intercept than the corresponding fit through the intercalibrated data (which has 0.994 and 2.007, respectively). As such the quoted uncertainty of the slope coefficient (0.001) refers only to the ordinary least squares fit. The difference between the slope coefficients of the ordinary and bivariate regression is considerably larger than the error estimate given above (i.e. 0.8025 vs. 0.9256). If we take this difference as a measure of uncertainty of the slope parameter, then the differences between the present parameters and those in GE17 (i.e. 0.8025 vs. 0.8290 for the ordinary least squares and 0.9256 vs. 0.994 for the bivariate regression) are relatively small. In this sense we may state that this comparison remarkably shows that two essentially different methods used to treat the HIRS 2 to HIRS 3 transition lead to very similar results.

Figure 92-D histogram of brightness temperatures, displaying ${\widehat{T}}_{\mathrm{12}/\mathrm{15}}$ on the abscissa and T_12∕14 on the ordinate axes. The data are from 1004 common days of operation of NOAA 14 and NOAA 15. The dashed diagonal line represents x=y, the solid line is the best fit according to an ordinary least squares regression and the dashed–dotted line is the bivariate regression line.

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In pursuit of the goal to study changes in upper-tropospheric humidity with respect to ice (UTHi) we applied the retrieval formula of Jackson and Bates (2001) to ${\widehat{T}}_{\mathrm{12},\mathrm{15}}$ and to T_12,14 of the common 1004 days. A density plot of the corresponding data pairs of UTHi is displayed in Fig. 10. Obviously the result is not satisfying; the plot closely resembles the corresponding scatter of data pairs produced from the intercalibrated data (Shi and Bates, 2011) that are shown in Fig. 1 of GE17. Unfortunately the superposition method does not solve the problem of a considerable overestimation of the number of supersaturation events recorded with HIRS 3 and 4 instruments and it seems that the pseudo-channel 12 data have to be treated with the cdf-matching technique developed by GE17 in the same way as the intercalibrated data. This is beyond the scope of the present paper.

Figure 10Heat map displaying UTHi computed using ${\widehat{T}}_{\mathrm{12}/\mathrm{15}}$ on the ordinate against values computed from the original T_12∕14 on the abscissa. Obviously the problem concerning the excess of supersaturation cases in the NOAA 15 data remains even with this new kind of data treatment. The colour scale shows the number of events in each 1 % × 1 % pixel.

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The new method is an independent approach for an intercalibrated HIRS channel 12 data set, based on results of radiative transfer calculations, classification of profile characteristics and a superposition with information delivered by channel 11. The intercalibration of Shi and Bates (2011) is instead based on pixelwise direct corrections, where the brightness temperature-dependent corrections are determined from regressions of the first kind between subsequent satellite pairs. As Figs. 9 and 10 show, both methods seem to produce very similar results. The statistically based method of Shi and Bates (2011) is thus supported by an independent method, and results obtained from data intercalibrated with either method should be more trustworthy. We thus consider our question from the beginning to be answered positively: whether combining HIRS 2 and HIRS 3 data into a single time series is justified. Although both methods produce similar results, as we see in Fig. 10, neither method of intercalibration solves the problems with the discrepancy in the range of high UTHi values which results from a corresponding discrepancy at the low tail of channel 12 brightness temperatures (see GE17). It is probable that this problem does not originate from the intercalibration procedure, since for the radiative transfer calculation it makes no difference whether the humidity profile contains a very humid upper troposphere with supersaturated layers or not. In each case it provides the corresponding brightness temperature. It is more probable that the problem with the lower tail of the T₁₂ distribution comes from the retrieval method, which is based on linearisations around certain “tangential points”, thermodynamic properties typical of the upper troposphere (e.g. the T₀=240 K mentioned above) and that this linear approach is not completely sufficient in cases in which actual properties are too far away from the tangential points.