Global Positioning System Reference
In-Depth Information
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
With stated assumptions, Equation (6.38) becomes a standard differential equation
with all terms depending only on the intensity along the path of propagation. The
solution can be written as
s
0
I(f,s
0
)e
−τ
(s
0
)
B(f, T ) e
−τ
(s)
I(f,
0
)
=
+
α
ds
(6.41)
0
s
0
α
(s
)ds
τ
(s)
=
(6.42)
Equation (6.41) is called the radiative transfer equation.
I(f,
0
)
is the intensity at the
measurement location
s
=
0, and
I(f,s
0
)
is the intensity at some boundary location
s
=
s
0
. The symbol
τ
(s)
denotes the optical depth or the opacity.
[20
kT
, as is the case for microwaves and longer waves, the denominator
in (6.40) can be expanded in terms of
hf / k T
. After truncating the expansion, the
Planck function becomes the Rayleigh-Jeans approximation
If
hf
Lin
—
5.5
——
No
PgE
2
f
2
kT
c
2
2
kT
λ
B(
λ
,T)
≈
=
(6.43)
2
Th
e symbol
denotes the wavelength. Expression (6.43) expresses a linear relation-
sh
ip between Planck function and temperature
T
. For a given opacity (6.42) the in-
ten
sity (6.41) is proportional to the temperature of the field of view of the radiometer
an
tenna given (6.43).
The Rayleigh-Jeans brightness temperature
T
b
(f )
is defined by
λ
[20
2
2
k
I(f)
≡
λ
T
b
(f )
(6.44)
T
b
(f )
is measured in degrees Kelvin; it is a simple function of the intensity of the
ra
diation at the measurement location. If we declare the space beyond the boundary
s
0
as the background space, we can write the Rayleigh-Jeans background brightness
te
mperature as
2
2
k
I(f,s
0
)
≡
λ
T
b
0
(f )
(6.45)
=
Using definitions (6.44) and (6.45), the approximation (6.43), and
T
T
b
, the
radiative transfer equation (6.41) becomes
s
0
T
b
0
e
−τ
(s
0
)
e
−τ
(s)
ds
T
b
=
+
T(s)
α
(6.46)
0
This is Chandrasekhar's equation of radiative transfer as used in microwave remote
sensing. For ground-based GPS applications, the sensor (radiometer) is on the ground
(
s
=
0) and senses all the way to
s
=∞
.
T
b
0
becomes the cosmic background
