340
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
the Rayleigh return, possible shift of the laser line, and uncertainties in its spectral
purity and center wavelength.
14.2.3.1 Absorption Cross-Section Errors
A general expression of the absorption cross section at a wavenumber ν is given by
σ ν
ε ν ν
( )
( , ) (
, , )
=
−
S T
p T
i
Λ
0
,
(14.8)
where ν 0 is the center wavenumber of an absorption line, S(T,ε) is the line strength
of the transition at temperature T and initial state energy ε, Λ(ν – ν 0 , p i , T) is the line
shape function, and p i is the partial pressure. For water vapor, these two functions are
S T
S T T
k T T
B
( , )
( / ) exp
/
ε
ε
=
−
−
0 0
3 2
0
1 1
(14.9)
Λ
Λ
(
, , )
(
, , )
ln /
Re[ ( )]
ν ν
ν ν
π
−
≈
−
=
⋅
−
0
0
1
2
2
p T
p T
f
w z
i
V
i
D
(14.10)
and
z
if
f
C
D
=
−
+
(
)
/ ln
ν ν 0
2
2
,
where S 0 is the absorption line strength under standard conditions, k B is the Boltzmann
constant, and Λ V is the Voigt absorption line function, which is a good approximation
for the actual line shape. Re indicates the real part of the complex function, and i = 1 .
The parameters f D and f C of the Voigt function describe the linewidths (full width
half maximum [FWHM]) of Doppler and collision broadening, respectively. Both
of them are pressure and temperature dependent. For a gas mixture, the effective
collision broadened width is given by
f
f p T
f p T
p
p
T
T
C
C i
i
i
C i
i
C i
=
=
∑ ,
,
( , )
( , )
,
2
0 0
0
0
η
∑
2
i
,
(14.11)
where f C,i (p 0 ,T 0 ) is the collision broadened width at standard conditions, and η C,i
is the temperature exponent of collision broadening for a single component of the
mixture. The linewidth of the Doppler broadened spectrum is
f
k T
mc
D
B
=
⋅
8
2
2
0
ln ν .
(14.12)
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
the Rayleigh return, possible shift of the laser line, and uncertainties in its spectral
purity and center wavelength.
14.2.3.1 Absorption Cross-Section Errors
A general expression of the absorption cross section at a wavenumber ν is given by
σ ν
ε ν ν
( )
( , ) (
, , )
=
−
S T
p T
i
Λ
0
,
(14.8)
where ν 0 is the center wavenumber of an absorption line, S(T,ε) is the line strength
of the transition at temperature T and initial state energy ε, Λ(ν – ν 0 , p i , T) is the line
shape function, and p i is the partial pressure. For water vapor, these two functions are
S T
S T T
k T T
B
( , )
( / ) exp
/
ε
ε
=
−
−
0 0
3 2
0
1 1
(14.9)
Λ
Λ
(
, , )
(
, , )
ln /
Re[ ( )]
ν ν
ν ν
π
−
≈
−
=
⋅
−
0
0
1
2
2
p T
p T
f
w z
i
V
i
D
(14.10)
and
z
if
f
C
D
=
−
+
(
)
/ ln
ν ν 0
2
2
,
where S 0 is the absorption line strength under standard conditions, k B is the Boltzmann
constant, and Λ V is the Voigt absorption line function, which is a good approximation
for the actual line shape. Re indicates the real part of the complex function, and i = 1 .
The parameters f D and f C of the Voigt function describe the linewidths (full width
half maximum [FWHM]) of Doppler and collision broadening, respectively. Both
of them are pressure and temperature dependent. For a gas mixture, the effective
collision broadened width is given by
f
f p T
f p T
p
p
T
T
C
C i
i
i
C i
i
C i
=
=
∑ ,
,
( , )
( , )
,
2
0 0
0
0
η
∑
2
i
,
(14.11)
where f C,i (p 0 ,T 0 ) is the collision broadened width at standard conditions, and η C,i
is the temperature exponent of collision broadening for a single component of the
mixture. The linewidth of the Doppler broadened spectrum is
f
k T
mc
D
B
=
⋅
8
2
2
0
ln ν .
(14.12)