## I I

0 1000 2000 3000 4000

Altitude in m

Fig. 6.21. Phase fluctuation as function of the altitude between two points 100-step 500-1600 m apart, from bottom to top.

For the subject of this chapter it is of more interest to look at the amplitude fluctuations, because these are directly related to the fluctuations in the self-emission of the atmosphere by virtue of Kirchhoffs law:

where h and a are the coefficient of emission and absorption, respectively; B(v,T) is

Planck's radiation law (or its Rayleigh-Jeans approximation, where appropriate). The antenna measures fluctuations in power, which are proportional to the square of the amplitude fluctuations. Using Eq. (6.29) with Eq. (6.32) and integrating along L to infinity, we obtain for the mean square logarithmic fluctuation in atmospheric brightness temperature ATB :

The atmospheric average brightness temperature must be calculated for a standard model of the atmosphere. Atmospheric physicists have made an extensive effort to establish such a standard model. For wavelengths in the centimeter and millimeter range the model of Liebe (1989) is widely accepted. Note that the brightness temperature, and hence the fluctuating component of atmospheric emission and absorption is strongly determined by water vapour. This is the main reason why antennas for work in the mm-wavelength region are located at high sites in dry areas of the world. Also, it should be mentioned that the above discussion assumes a clear atmosphere without influence of clouds and frontal activity. The turbulence discussed here occurs in the quiet atmosphere because of small fluctuations in temperature, pressure and humidity. In Fig. 6.22 we show the relative power fluctuations as function of wavelength as computed from the above equations [Mat.6.4]. We see a rather strong increase with decreasing wavelength, which is caused mainly by the increased opacity of the atmospheric water vapour. It should be borne in mind that this curve is valid for the frequency regions outside the pressure broadened absorption lines (see Fig. 6.19), where the above approach breaks down. It is also clear that the relative fluctuation level decreases slowly with increasing altitude, about a factor two for 5000 m. We should however remember that the actual atmospheric brightness temperature also

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