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4. Extinction coefficient behaviour with wavelength

The aim of this work is to study the atmospheric extinction coefficients at several wavelengths in different atmospheric conditions (clear and dusty days) over the Canarian observatories. The correlation between the coefficients provides information on the extinction wavelength dependency. If we represent the tex2html_wrap_inline1311 versus tex2html_wrap_inline1313 slopes equal to unity imply a grey behaviour. Figure 2 (click here) shows the extinction coefficients at several wavelengths versus the one at 680 nm. Looking at this figure some features are immediately apparent. The functions tex2html_wrap_inline1311=f(K680) are straight lines but at low values of K680 the slope changes. This is clearly seen mainly at 450 nm and 500 nm. To verify this and calculate the value at which the slope changes we do the following. For each one of the extinction coefficients pairs in Fig. 2 (click here), we fit 9 straight lines covering different but overlapping regions on the x-axis. For 450 nm versus 680 nm, only 5 straight lines were fitted due to the lower number of points, mainly at high extinction values. Table 2 (click here) shows the intervals and the number of points used in the fits. The fitting procedure is a standard least-squares method with errors in both coordinates, which we obtain from the computation of extinction coefficient.

Figure 2: Extinction coefficients (mag airmass-1) for all the wavelengths used in this work versus the one at the common channel to all the observations (680 nm). Notice the slope changes for extinction coefficients smaller that 0.1 mag airmass-1

Figure 3 (click here) shows the tex2html_wrap_inline1307 values with the straight lines fitted to the different overlapping regions. The slopes obtained in these regions are plotted in the insets. The ordinates of the insets correspond to the slopes and the abscissae represent the regions at which these slopes are obtained. A point with abscissa of 0.3 means that the slope has been obtained from points with abscissae between 0.3 and 0.8 mag airmass-1; a point with abscissa 0.2 has been obtained from 0.2 to 0.8 mag airmass-1, and so on (see Table 2 (click here)). Therefore there are 9 points (determination of the slopes) in the insets, except for K450 (Fig. 3 (click here)a) in which there are only 5.

Figure 3: Extinction coefficients (mag airmass-1) of Fig. 2 (click here) plotted separately and fitted to straight lines for several overlapping regions values of K680 as explained in the text (see Table 2 (click here)). In the insets are shown the slopes of the fitted lines (see text)


Region K680 No. of points for No. of points for No. of points for No. of points for
(mag airmass-1) tex2html_wrap_inline1297 = 450 nm tex2html_wrap_inline1297 = 500 nm tex2html_wrap_inline1297 = 770 nm tex2html_wrap_inline1297 = 870 nm
0.3 - 0.8 - 45 22 41
0.2 - 0.8 - 76 32 70
0.1 - 0.8 - 144 63 123
0.09 - 0.8 - 158 68 131
0.08 - 0.8 8 175 71 144
0.07 - 0.8 12 213 84 165
0.06 - 0.8 20 392 121 250
0.05 - 0.8 69 667 277 488
0.00 - 0.8 87 713 343 533
Table 2: Number of points (N) used in each of the linear fits to the lines K=a+bK680 on several and overlapping regions values of K680 (see text)


The slopes of the straight lines are close to 1 for the high extinction values, but points with low extinction values have higher slopes. In addition the increase of the slopes is smaller as we go to higher wavelengths and slopes are always close to one for K870 versus K680.

The dependence on wavelength of the atmospheric aerosol extinction coefficient for visible light is in good agreement with the Ångstrom formula (see for example Cachorro et al. 1989):
where tex2html_wrap_inline1379 is the turbidity coefficient and tex2html_wrap_inline1381 is the Ångstrom exponent. This formula is usually written as:
with tex2html_wrap_inline1381 being the slope of a straight line. From the logarithm scale, it can be seen that the extinction coefficient varies less between the two longest wavelengths of the interval than between the two smallest. This explains why the slopes for 870 and 770 nm are much smaller than the slopes at 450 and 550 nm in Fig. 3 (click here). On the other hand, the fact that the slopes are equal to 1 for only the higher extinction values indicates from (3) that tex2html_wrap_inline1385, which is a value of tex2html_wrap_inline1381 related to aerosol particles of the dust size. When other points of lower tex2html_wrap_inline1307 are introduced into the fits, tex2html_wrap_inline1381 needs to increase to lead higher values of the slopes and, tex2html_wrap_inline1381 increases as aerosol sizes decrease. Therefore, this different behaviour of the slopes is produced by the effect of Saharan dust.

The first evidence of this effect is based on the long experience of our observers. In all our observing campaigns, observers fill in a detailed daily running logbook in which all the observing conditions are recorded. In particular, the quality of the day is classified as coronal, pure, diffuse and absorbent (dusty). When there is a considerable amount of dust in the sky, it is easy to evaluate its quality as the inhabitants of the islands know very well. When the sky is blue and clear is also easy to classify it as a coronal or pure day (in this work termed "clear" days). The classification problem arises when the day is somewhat diffuse and it is not possible to know if it is produced by water vapour, aerosols, etc. A detailed comparison of notes with their respective extinction coefficients shows that on clear days (annotated as coronal-pure) the extinction values at 680 nm are never higher than about 0.07-0.09 mag airmass-1, while on dusty days (diffuse-absorbent) they are always higher. The classification of the days based on our observers' skills, is absolutely valid for clear and dusty skies, but ambiguous for the extinction coefficient range between about 0.07-0.1 mag /airmass at 680 nm. To estimate the extinction coefficient at which the effect of the dust starts to appear, we have performed a more objective calculation.

On dusty days, the standard deviation (sigma) of the daily linear fit increases (see Fig. 1 (click here)b). Using the sigma of the fits, it is possible to see the effects of the dust in more detail. In Fig. 4 (click here) we show K680 divided by the standard deviation of the Bouguer law fit (sigma), versus K680. As is clearly seen in this figure, at low K680, the K680/sigma ratio is concentrated in a narrow vertical band, whereas for high extinction values it is not (low broad horizontal). The first band corresponds to clear days with low extinction and low sigma; of course, the quality of clear days is not always the same and this produces the large spread of points on the y-axis. The second band corresponds to dusty days with high extinction values but also with high dispersion. Our problem is to find the K680 value threshold that can separate both. The inset of Fig. 4 (click here) shows the region corresponding to this "transition'' band where it seems that the separation between the two bands could be around 0.075 mag airmass-1 (at 680 nm) in agreement with the experience of our observers. The exact value K680 at which the slope change occurs can be somewhat affected by the selection of the days in the analysis. If all the days had been used, very probably this point would have been ill defined as the amount of "intermediate'' K values would have increased. However this would only affect the precision at which this value is known.

Figure 4: Extinction coefficient at 680 nm divided by its sigma value obtained from the Bouguer law fit versus extinction at 680 nm. The two bands (narrow vertical and spread horizontal) corresponds to clear and dusty days, respectively. In the inset, the interval corresponding to the "transition'' region between the two bands

Therefore, we choose this threshold value as a border point at which the effects of the dust begin to be significant. Probably with this classification, a small amount of points could be swapped between the two bands. This value of 0.075 mag airmass-1 (at 680 nm) really represents an inferior limit for the selection of clear days and it can be increased at least up to 0.09 mag airmass-1 (at 680 nm; a test made changing the value of the borderline between 0.07-0.09 mag airmass-1 shows that the result does not change significantly). On this basis, we divide each of the four extinction curves of Fig. 3 (click here) in two: extinction values lower and higher that 0.075 mag airmass-1 (at 680 nm). Then straight lines are fitted to each of the 8 curves as explained before. Figure 5 (click here) shows this analysis and Table 3 (click here) the slopes obtained in the fits. From Fig. 5 (click here) and Table 3 (click here) it seems to be clear that the Saharan dust at Teide Observatory has a grey behaviour in the wavelength used in this work (this result was first proposed by Whittet et al. 1987 for Roque de los Muchachos Observatory, La Palma, Canary Islands).

As we mentioned at the end of Sect. 3, we will discuss now the effect of neglecting the temporal variation of the extinction as function of time. Computing the extinction coefficients in the morning and in the afternoon instead of, for example, every hour, we are obtaining an average of the extinction during that half of the day. If we compute the extinction every hour we would get 4 or 5 values around the value obtained using the half day (this have been checked for several days). As the days have been separated into clear and dusty days with no atmospheric mixing conditions, the temporal values of the extinction would be on the straight lines of Fig. 5 (click here) (left panel) for clear days and in Fig. 5 (click here) (right panel) for dusty days. In conclusion, if we had obtained the extinction values on a shorter time basis, one hour for example, we would have 4 or 5 times more points in the figures but our results would be the same.


tex2html_wrap_inline1297(nm) Clear days (a) Dusty days (b)
450 2.132 (+/-) 0.105 1.028 (+/-) 0.042
500 1.333 (+/-) 0.025 1.011 (+/-) 0.015
770 1.159 (+/-) 0.033 0.999 (+/-) 0.009
870 1.026 (+/-) 0.029 1.009 (+/-) 0.012
Table 3: Slopes of the lines tex2html_wrap_inline1429 corresponding to Figs. 5 (click here)a and 5b) (see text)


Figure 5: Extinction coefficients together with the best linear fits, once the days have been separated in clear days (extinction at 680 nm < 0.075 mag airmass-1) and dusty days (extinction at 680 nm > 0.075 mag airmass-1). See Table 3 (click here)

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