Geoscience Reference
In-Depth Information
and the following is obtained for the derivative with respect to the coefficient
of the molecular scattering:
∂σ
P
,
m
(
P
)
∂
L
i
(
P
)
∂σ
P
,
m
(
P
i
)
∂σ
z
,
m
(
P
i
)
∂σ
z
,
m
(
P
i
)
Q
H
2
O
(
P
i
)
=
.
(5.33)
∂
Q
H
2
O
(
P
i
)
The derivatives depending on volume coefficient of the molecular scattering
have been calculated above, and the absorption cross-section for H
2
Oiscom-
puted with (5.8).
Theexpressionforthederivativeofthemolecularscatteringvolumecoeffi-
cient is obtained as follows:
∂σ
z
,
m
(
P
i
)
∂
Q
H
2
O
(
P
i
)
=
∂σ
z
,
m
(
P
i
)
∂
∂
m
P
w
,
(5.34)
∂
∂
∂
m
P
w
Q
H
2
O
(
P
i
)
where:
∂σ
z
,
m
(
P
i
)
∂
4
m
m
2
−1
=
σ
z
,
m
(
P
i
)
m
∂
λ
−2
m
∂
P
w
=
10
−6
0.0624 − 0.00068
1 + 0.003661
T
(
P
i
)
∂
P
w
Q
H
2
O
(
P
i
)
=
0.7501
P
i
.
∂
Derivative with respect to volume coefficient of the aerosol absorption.
The
volume extinction coefficient only depends on volume coefficient of the aerosol
absorption that directly yields:
∂
∆τ
(
P
1
,
P
2
))
∂κ
z
,
a
(
P
i
,
=
∂
∆τ
(
P
1
,
P
2
))
∂α
P
(
P
i
)
∂α
P
(
P
i
)
∂α
z
(
P
i
)
∂α
z
(
P
i
)
(
(
λ
∂κ
z
,
a
(
P
i
)
L
j
(
) ,
(5.35)
λ
j
)
∂α
z
(
P
i
)
|
∂κ
z
,
a
(
P
i
))
=
where
1 with taking into account (1.23) and (1.24).
Derivative with respect to volume coefficient of the aerosol scattering.
The
volume coefficients of the absorption and scattering and the phase function
of the aerosol scattering as per (5.9) depend on the volume coefficient of the
aerosol scattering. Therefore, we obtain:
(
∂
∆τ
(
P
1
,
P
2
))
∂σ
z
,
a
(
P
i
,
=
∂
∆τ
(
P
1
,
P
2
))
∂α
P
(
P
i
)
∂α
P
(
P
i
)
∂α
z
(
P
i
)
∂α
z
(
P
i
)
∂σ
z
,
a
(
P
i
)
L
j
(
(
(
λ
) ,
(5.36)
λ
j
)
∂α
z
(
P
i
)
|∂σ
z
,
a
(
P
i
)
=
where
1 with taking into account (1.23) and (1.24).
Then we can write:
∂σ
P
,
a
(
P
)
∂σ
z
,
a
(
P
i
,
∂σ
P
,
a
(
P
i
)
∂σ
z
,
a
(
P
i
,
λ
j
)
=
L
i
(
P
)
λ
)
L
j
(
) .
(5.37)
λ
Search WWH ::
Custom Search