Global Positioning System Reference
In-Depth Information
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temperature
T
cosmic
, which results from the residual cosmic radiation of outer space
that is left from the Big Bang. Thus
∞
T
cosmic
e
−τ
(
∞
)
e
−τ
(s)
ds
T
b
=
+
T(s)
α
(6.47)
0
∞
τ
(
∞
)
=
α
(s) ds
(6.48)
0
T
cosmic
=
2
.
7
K
(6.49)
The brightness temperature (6.47) depends on the atmospheric profiles of physical
temperature
T
and absorption
. For the atmosphere the latter is a function of pres-
sure, temperature, and humidity. Equation (6.47) represents the forward problem, i.e.,
given temperature and absorption profiles along the path one can compute brightness
temperature. The inverse solution of (6.47) is of much practical interest. It potentially
allows the determination of atmospheric properties such as
T
and
α
[20
Lin
—
2.9
——
Nor
PgE
α
, as well as their
spatial distribution from brightness temperature measurements.
Consider the following special cases. Assume that the temperature
T
is constant.
Neglecting the cosmic term, using
d
τ = α
ds
, the radiative transfer equation (6.47)
becomes
T
τ
(a)
0
T
1
e
−τ
(a)
e
−τ
d
T
b
=
τ =
−
(6.50)
[20
Fo
r a large optical depth
τ
(a)
1weget
T
b
=
T
and the radiometer acts like a
th
ermometer. For a small optical path
τ
(a)
1weget
T
b
=
T
τ
(a)
. If the temperature
is
known, then
(a)
can be determined. If we also know the absorption properties of
th
e constituencies, it might be possible to estimate the concentration of a particular
co
nstituent of the atmosphere.
For the sake of clarity, we reiterate that (6.44) defines the Rayleigh-Jeans bright-
ne
ss temperature. The thermodynamic brightness temperature is defined as the tem-
pe
rature of a blackbody radiator that produces the same intensity as the source
be
ing observed. The latter definition refers to the physical temperature, whereas the
Ra
yleigh-Jeans definition directly relates to the radiated intensity. The difference be-
tw
een both definitions can be traced back to the approximation implied in (6.43). A
gr
aphical representation of the differences is found in Janssen (1993, p. 10).
τ
6.
3.2 Absorption Line Profiles
Microwave radiometers measure the brightness temperature. In ground-based ra-
diometry, the relevant molecules are water vapor, diatomic oxygen (O
2
), and liquid
water. Mathematical models have been developed for the absorption. For isolated
molecules, the quantum mechanic transitions occur at well-defined resonance fre-
quencies (line spectrum). Collision with other molecules broadens these spectral
