Determination of Parameters of the Atmosphere and the Surface in a Clear Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 184

1. Finding numbers n 1 and n 2 from conditions P n 1 ≥ P 1 ≥ P n 1 +1 , P n 2 ≥

P n 2 +1
2. Then three cases are considered depending on the magnitude of differ-
ence n 2 − n 1 : n 2 >n 1 +1

P 2 ≥

∂

∆τ ( P 1 , P 2 )) =

( P i ) (

0for i<n 1 or i>n 2 +1;

∂α

∂ ( P 1 − P i +1 ) 2

( P i − P i +1 )

1
2

∆τ ( P 1 , P 2 )) = =

( P i ) (

, r i n 1 ;

∂α

∂ ( P 1 − P i ) 2

P 1 − P i − 1
2 P i −1 − P i + 1

∆τ ( P 1 , P 2 )) =

( P i ) ( 2 ( P i − P i +1 ), ∂α

=

for i n 2 + 1 ; (5.27)

∂ 1

2 ( P i −1 − P i +1 ), for n 1 +2

∆τ ( P 1 , P 2 )) =

( P i ) (

≤ i ≤ n 2 −1;

∂α

∂

( P i − P 2 ) 2

P i − P 2 − 1
2 P i − P i +1 + 1

∆τ ( P 1 , P 2 ))

= =

( P i ) ( 2 ( P i −1 − P i ), for i

n 2 ;

∂α

∂

( P i −1 − P 2 ) 2
P i −1 − P i

1
2

∆τ ( P 1 , P 2 )) = =

( P i ) (

, r i n 2 +1.

∂α

= n 2 n 1 +1.Thiscasediffersfromthelatterbythederivativebeingequalto:

∂ ( P 1 − P i ) 2 ( P i − P 2 ) 2

P i − P i +1

P 1 − P 2 − 1
2 P i −1 − P i − 1

∆τ ( P 1 , P 2 )) =

( P i ) ( ∂α 2 (5.28)

= = for i n 1 +1 n 2

=

n 2 n 1 :

∂

∆τ ( P 1 , P 2 )) =

( P i ) (

0for i<n 1 or i>n 1 +1;

∂α

∂ ( P 1 − P i +1 ) 2 −( P i − P 2 ) 2

P i − P i +1

P i − P 2 + 1
2

∆τ ( P 1 , P 2 )) =

( P i ) (

,

∂α

(5.29)

= = for i n 1 n 2 ,

∂

( P i −1 − P 2 ) 2 −( P 1 − P i ) 2
P i −1 − P i

P 1 − P i + 1
2

∆τ ( P 1 , P 2 )) =

( P i ) (

,

∂α

= = for i n 1 +1 n 2 +1.

Note that, the volume extinction coefficient in the described algorithm is
applied after recalculating per the pressure unit α P ( P i ), while it has been