Priciples of Active and Passive Remote Measurements ...
115
Then, we define an equivalent homogeneous atmosphere of constant pressure Pe, constant temperature Te and amount of absorber U e , that gives the same equivalent width.
{ [l-exp{- {n(z)S(Te)gv(Pe,Te)dz}]dv=W
Jf1V
}z
(5.82)
With the very common two-parameter Curtis-Godson approximation, the equivalent temperature Te is set arbitrarily and pe and U e are tuned so as to satisfy the equivalence in the two
limit regimes of weak and strong absorption.
For weak absorption, if we develop the exponential and keeping in mind the normalization of
gv, a first condition is obtained as
1 n(z)S(T(z))dz = 1 n(z)S(Te)dz
(5.83)
For strong absorption, if Vie consider again that the absorption at the center of the lines is
saturated and that 1 - expO only differs from 1 far from the center, we can write
=1 [- {_I n (z)S(T(z))a L(p(z),T(z)d }]d
W
1 exp
( ) 2
Z
V
llV
z
7r V - Vo
(5.84)
and similarly for Woo Since the denominator is independent of altitude, it can be put out of
the integration over z, the condition of equivalence in strong regime is
(5.85)
From the first condition (5.83), the reference temperature Te is arbitrarily fixed and the equivalent absorber amount U e is
(
S(T(z))
U e = Jz n(z) S(Te) dz
(5.86)
and the strong absorption condition gives
Ue~ = (n(z) aL(p(z), T(z)) S(T(z)) dz
a LO
Jz
a LO
S(Te)
(5.87)
A further simplification consists in omitting the influence of temperature variations, the first
condition is then always satisfied and, if we write the variation of a LO with altitude, the second
condition is
1 n(z)p(z)dz =Pe 1 n(z)dz
(5.88)
c) Random band models
So far, we have introduced the concept of band model for a single isolated line. The absorption
bands (rotation or vibration-rotation bands) correspond to the superposition of many spectral
lines. The averaged transmittance over the spectral interval D.V is then expressed by
TD.v(Z,Z',flo) = ~j Tv(Z,Z',flo)dv = ~ { exp{- t kv(z")p(z")dz"}
D.V D.V
D.V J D.V
Jz'
(5.89)
with
kv(z") = " S;
a;(z")
7' 7r[(v- vo;)2 + aHz")]
(5.90)
The band model conceptually the simplest consists of describing the spectrum step by step
in changing v and in recalculating kv at each step. Since many lines can have, at a given
115
Then, we define an equivalent homogeneous atmosphere of constant pressure Pe, constant temperature Te and amount of absorber U e , that gives the same equivalent width.
{ [l-exp{- {n(z)S(Te)gv(Pe,Te)dz}]dv=W
Jf1V
}z
(5.82)
With the very common two-parameter Curtis-Godson approximation, the equivalent temperature Te is set arbitrarily and pe and U e are tuned so as to satisfy the equivalence in the two
limit regimes of weak and strong absorption.
For weak absorption, if we develop the exponential and keeping in mind the normalization of
gv, a first condition is obtained as
1 n(z)S(T(z))dz = 1 n(z)S(Te)dz
(5.83)
For strong absorption, if Vie consider again that the absorption at the center of the lines is
saturated and that 1 - expO only differs from 1 far from the center, we can write
=1 [- {_I n (z)S(T(z))a L(p(z),T(z)d }]d
W
1 exp
( ) 2
Z
V
llV
z
7r V - Vo
(5.84)
and similarly for Woo Since the denominator is independent of altitude, it can be put out of
the integration over z, the condition of equivalence in strong regime is
(5.85)
From the first condition (5.83), the reference temperature Te is arbitrarily fixed and the equivalent absorber amount U e is
(
S(T(z))
U e = Jz n(z) S(Te) dz
(5.86)
and the strong absorption condition gives
Ue~ = (n(z) aL(p(z), T(z)) S(T(z)) dz
a LO
Jz
a LO
S(Te)
(5.87)
A further simplification consists in omitting the influence of temperature variations, the first
condition is then always satisfied and, if we write the variation of a LO with altitude, the second
condition is
1 n(z)p(z)dz =Pe 1 n(z)dz
(5.88)
c) Random band models
So far, we have introduced the concept of band model for a single isolated line. The absorption
bands (rotation or vibration-rotation bands) correspond to the superposition of many spectral
lines. The averaged transmittance over the spectral interval D.V is then expressed by
TD.v(Z,Z',flo) = ~j Tv(Z,Z',flo)dv = ~ { exp{- t kv(z")p(z")dz"}
D.V D.V
D.V J D.V
Jz'
(5.89)
with
kv(z") = " S;
a;(z")
7' 7r[(v- vo;)2 + aHz")]
(5.90)
The band model conceptually the simplest consists of describing the spectrum step by step
in changing v and in recalculating kv at each step. Since many lines can have, at a given