Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 196

for

X

n

c N 1 k
k

B k k ð k 1 Þð k N þ 1 Þ k ! ¼ 0 ; ð n N Þ

[

k¼N

Consider the case when

c k ð h Þ ¼exp c k fg

From ( 2.20 ) we obtain:

T ð h Þ ¼ X

N 1

B k exp kc k h ½

k¼0

Now we move on to the inverse problem. Let measurements be made at
wavelengths ʻ 1 , … , ʻ N . Solve the equation:

P 0 N 1 ð h Þ ¼c c k ð h Þ

and nd

Z

P k 1 ð h Þ ¼ c k ð h Þ P k ð h Þ dh ; ð k ¼ 1 ; ...; N Þ

cient on the
frequency and depth of the layer be divided by the product of the known function

Let ʳ ʻ (h)= ˆ ( ʻ ) ˈ (h), that is, the dependence of the absorption coef

ˆ ( ʻ )

and an unknown function ) is set from the empirical

table and its analytical approximation can be constructed. Then, excluding from
( 2.20 ) the second right-hand term (it is equal to zero), we write the relationship:

ˈ (h). As a rule, the function ˆ ( ʻ

h i

T b k ¼ j k X

N

ð N k 1 Þ 0

T ð k 1 Þ ð 0 Þ F k uðkÞ; w

ð 0 Þ; ...; w ð 0 Þ; wð 0 Þ ð 2 : 21 Þ

k¼1

and the solution of the inverse problem is reduced to the solution of a system of
algebraic equations with regard to functions:

T 0 ð 0 Þ; ...; T ð N 1 Þ ð 0 Þ; wð 0 Þ; ...; w

ð N 1 Þ ð 0 Þ

T ð 0 Þ;

Consider the case ˈ (h) = const, that is, ʳ ʻ = ˆ ( ʻ ) > 0. Equation ( 2.21 )is

re-written:

T b k ¼ j k X

N

T ð k 1 Þ ð 0 Þ=u

ð k 1 Þ ðkÞ

ð 2 : 22 Þ

k¼1