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largest attenuation, and only the lowest part of the atmosphere – in literature called skin
depth – contributes to the signal detected by radiometer. The frequencies away from the
peak, are less attenuated and radiation from higher layers in the atmosphere will contribute
only to the deteriorating of the measured signal’s noise ratio.
AskneandWestwater(1986);Troitsky(1986)describedamulti-frequency(alsocalledmulti
channel) passive microwave radiometer, which gave the possibility to measure temperature
profiles in the troposphere (up to 10 km) but their lower troposphere’s layers accuracy is
not good. A more simple technique for the microwave remote sensing of the boundary layer
temperature is based on measuring of the brightness temperature of the atmosphere proper
in the center of the oxygen absorption band. Troitsky et al. (1993) and Kadygrov and Pick
(1998) described an angular scanning single-channel microwave radiometer centered proper
on molecular oxygen band at 60 GHz. For our application we will consider this second as case
study.
As shown in Figure 180 at frequencies ν = 60 Ghz we can suppose (with a good accuracy)
an absorption coefficient independent with altitude h or αν(h) = αν(0) =constant and a skin
depth around at h = 300 m. If we suppose that the skin depth equal to the boundary layer’s
height Hb at the zenith by integrating Eq. (339) follows:
obtaining
and also:
τ(Hb,θ) = 1
cosθ
Hb
0
αν(z)dz = |αν(0)|(Hb −0)
cosθ
|αν(0)| = |αν(h)| 1
300 m
= |αν(0)300 m|
cos0
1, (343)
(344)
H(θ) ≤ Hb = cosθ
cosθ300 m (345)
|αν(0)|
0 ≤ H(θ) ≤ 300 m. (346)
Thus, the remote temperature sensing is conducted by measurements of the brightness
temperature at the different elevation angles θ = 0–90◦ . In this case the depth of contributing
radiation layer is a range from 0–300 m (Troitsky et al., 1993) in the hypothesis that the
layers of an atmosphere which are above than 300 m will not influence the measure of T ν0
b (θ).
More in general we can rewrite the Eqs. (340) and (341) of brightness temperature as
function of the elevation angle θ and expressed by a defined integral function of T(z) and
Eq. (342) weighting function calculated at the frequency ν0
T ν0
b (θ) =
H
0
T(z)W ν0 (z,θ)dz. (347)
This is a Fredholm integral of the first kind and the superior limit of integration is finite
and coincident with the limit of atmosphere’s altitude sensed generally not more than Hb = 2
km. In this hypothesis we are supposing the layers of an atmosphere which are higher than 2
km do not influence T ν0
b (θ) measurements.
The previous integral function Eq. (347) may be solved for an unknown temperature profile
T(z), given a set of measured radiometer brightness temperature data Tb(θ) at different
elevation angles θ.
One of the first inversion alghorithm used was a variation of the Twomey-Tikhonov retrieval
algorithm (Tikhonov and Arsenin, 1977) in form of generalized variation (Troitsky
et al., 1993). Inversion techniques for upward-looking radiometers are generally based on the
temperature climatology at the site that is typically derived from in situ radiosonde measurements.
The inversion method use an initial-guess profile, usually derived from radiosonde
observations or tethered balloon, and use temperature brightness measurements to correct
this initial guess.
At this purpose instead of T(z) the deviation from the restriction function can be minimized
on the manifold of positively determined function (a class of normalized function).
268 DTU Wind Energy-E-Report-0029(EN)