Publishers version - DTU Orbit

Figure 175: (a) Left: Conventions assumed into the integration of the RTE. (b) Right: Conventions
assumed into the integration of the RTE
value Iν(0):
Iν(0) = Iν(s0)exp(−τν(s0))+ 2kν2
c 2
s0
0
αν(s)T(s)exp(−τν(s)ds)ds. (336)
Substituting the equivalent formulation of Iν(s) expressed in Eq. (332) we can rewrite Eq.
(336) as a function of the brightness temperature Tb(s):
Tb(0) = Tb(s0)exp(−τν(s0))+
s0
0
T(s)αν(s)exp(−τν(s))ds, (337)
where Tb(s0)exp(−τν(s0)) is the brightness temperature contribute to the atmosphere from
cosmic background sources attenuated by the optical depth τν(s0) existing between ground
level and the point s0.
Referring to polar axis system of the Figure 175(b), operating the substitution ds =
dz/(cosθ), where θ is angle off zenith axis of the radiometer’s beam (for zenith θ = 0◦ )
and assuming z as the atmosphere layer’s level, the measured ground level brightness temperature
for an upward looking radiometer expressed as function of frequency ν, and elevation
angle θ it will become
Tb(0,ν,θ) Tb(ν,θ) = Tb(∞)exp(−τν(∞))+ 1
cosθ
with the optical depth at z-layer’s quote :
τν(z,θ) = 1
cosθ
z
0
∞
0
T(z)αν(z)exp(−τν(z))dz
(338)
αν(z ′ )dz ′ . (339)
AssumingthecosmicbackgroundcontributeTb(∞)exp(−τ(∞))asnegligiblefortheeffect
of the opacity of the entire atmosphere, we can rewrite Eq. (338) at last as:
∞
1
Tb(ν,θ) = T(z)
0 cosθ αν(z)exp(−τν(z,θ))
dz (340)
or:
Tb(ν,θ) =
∞
0
T(z)W(ν,z,θ)dz, (341)
in which the ground level measured brightness temperatures Tb(ν,θ) is expressed as a convolution
integral involving a temperature weighting function
W(ν,z,θ) 1
cosθ αν(z)exp(−τν(z,θ)) (342)
(also defined kernel function) and the thermodynamic temperature profile T(z). From a
physical point of view the brightness temperature Tb(ν,θ) of Eq. (341) can be considered
a “weighted” average over the thermodynamic temperature of the atmosphere along the
integration path
numerically is the integral sum of the elementary emission term T(z)dz from each volume’s element,
264 DTU Wind Energy-E-Report-0029(EN)