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 versus 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
=*f*(*K*_{680}) are straight lines but at low values of
*K*_{680} 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 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 *K*_{450} (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
*K*_{680} as explained in the text (see Table 2 (click here)). In the
insets are shown the slopes of the fitted
lines (see text)

Region K_{680} | No. of points for | No. of points for | No. of points for | No. of points for |

(mag airmass^{-1}) | = 450 nm | = 500 nm | = 770 nm | = 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 |

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 *K*_{870} versus
*K*_{680}.

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 is the turbidity coefficient and is the
Ångstrom exponent. This formula is usually written as:

with 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 , which is a value of related to aerosol particles of the dust size. When other points
of lower are introduced into the fits,
needs to increase to lead higher values of the slopes and,
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 *K*_{680} divided by the standard deviation of the Bouguer law fit
(sigma), versus *K*_{680}. As is clearly seen in this figure, at low
*K*_{680}, the *K*_{680}/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 *K*_{680} 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 *K*_{680} 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.

(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 |

**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)