Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 194

Relationship ( 2.13 ) makes it possible using T(h) and ʳ i (h)to nd T bi (direct

problem) and by T bi to calculate T(h) and ʳ i T(h) (inverse problem). The proposed

method to solve these problems is based on the use of T(h) and

ʳ i T(h) and their
derivatives at the point h = 0 with some additional limitations resulting from the
transformations made below. Write the relationship ( 2.13 ) in the form:

Z 1

T b k ¼ j k T ð h Þ dF k ð h Þ; ð 2 : 14 Þ

0

where

2 3

Z

h

4 5

F k ð h Þ ¼exp c k ð x Þ dx

0

ʻ is the wavelength.

Integrating ( 2.14 ) by parts, we obtain:

2 3

Z 1

4 5 ¼ j k T 0 þ J 1

T 0 ð h Þ F k ð h Þ dh

T b k ¼ j k T 0 þ ½ ð 2 : 15 Þ

0

Let J 1 = 0, then T b ʻ = ʺ ʻ T 0 .IfdT(h)/dh ≡ 0, then J 1 = 0. Therefore nd

conditions under which J 1 = 0. Let the following relationships be valid:

T ð h Þ ¼ X

n

B k h k c k ð h Þ c k [ 0 ; ð 2 : 16 Þ

k¼0

It follows from ( 2.16 ) that if the coef cients B k meet the condition:

X

n

B k
c k
k

! ¼ 0 ; ð 2 :

17 Þ

k

k¼1

then J 1 =0.
Integrating ( 2.15 ) by parts, we obtain:

2
4

3
5

Z 1

T 0
c k ð 0 Þ þ

T 00 ð h Þc k ð h Þc 0 k ð h Þ T 0 ð h Þ
c

T b k ¼ j k T 0 þ F k ð h Þ dh ð 2 : 18 Þ

2

k ð h Þ

0