Geoscience Reference
In-Depth Information
σ
a
(
P
i
,
λ
j
). The concentrations of the atmo-
spheric gases will be expressed through the volume-mixing ratio that gives the
simple relation for their counting concentrations:
λ
j
),
κ
a
(
P
i
,
λ
j
), and surface albedo
A
(
P
i
Q
(
P
i
)
kT
(
P
i
)
=
n
(
P
i
)
.
(5.26)
λ
j
to be specified in a general
form for the present, their concrete magnitudes will be obtained on the basis
of the derivative analysis (Sect. 4.4). Note that in practice to simplify the
derivatives computing (and to prevent the errors while programming) the
derivatives are to be written as a chain of the simplest formulas using the
rule of the composite function differentiation. It is also useful even if the
substituting of the derivatives to the general formulas causes the simplification
of the expressions. The other approach effectively simplifying the calculations
is application of the expression of the derivative of the product through the
logarithmic derivatives.
As intermediate values in the grids
P
i
and
Let the sets of altitude levels
P
i
and wavelengths
λ
j
are computed with the linear
interpolation according to the following:
F
(
u
i
)
u
i
+1
−
u
u
−
u
i
u
i
+1
−
u
i
=
F
(
u
)
u
i
+1
−
u
i
+
F
(
u
i
+1
)
,
∂
|∂
the derivative of function
F
(
u
i
) is obtained as the following process:
After determining number
n
from condition
u
n
≤
F
(
u
)
u
≤
u
n
+1
the following
equalities are correct:
∂
F
(
u
)
F
(
u
i
)
=
0for
i<n
or
i>n
+1;
∂
∂
F
(
u
)
u
i
+1
−
u
u
i
+1
−
u
i
F
(
u
i
)
=
=
for
i
n
,
∂
∂
F
(
u
)
u
−
u
i
u
i
+1
−
u
i
F
(
u
i
)
=
=
for
i
n
+1.
∂
∂
|∂
Asthederivativedependsontheargumentonly,specifyitas
F
(
u
)
F
(
u
i
)
≡
L
i
(
u
). Then the derivative with respect to the surface albedo is written as:
∂
=
λ
λ
j
)
(
A
)
L
j
(
).
∂
A
(
∆τ
(
P
1
P
2
), as per (2.1)-(2.4), is the quadratic function
of volume extinction coefficient
Thephotonfreepath
α
(
P
i
). Hence the following algorithm is elab-
∂|∂α
∆τ
(
P
1
,
P
2
)), where inequity
P
2
<P
1
orated for computing derivative
(
P
i
)(
is assumed for the definiteness:
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