The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 138

observational and calculation results but the one, maximum over all points
i = 1,..., N , the path described below will become impassable.

The distance between the observational and calculated values of R

≡

ρ X )) is called adiscrepancy . Thus, finally it is possible to de-

fine the formulated problem as the revealing of the values of the vector X
components through the known observational vector Y corresponding to the
minimum of discrepancy:

( Y , G ( X , U \

N

1
N

Y

= =

( y i − y i ) 2 ,

R G ( X , U \ X ). (4.6)

= i 1

The problem formulated in this manner constitutes the matter of aleast-
squares technique (LST) , proposed by CF Gauss. The following section contains
the consequent elucidating of the LST, its specifics and modification.

4.2
The Least-Square Technique for Inverse Problem Solution

Write the solution of the direct problem explicitly through the vector compo-
nents of the observations and initial parameters:

= = y i g i ( x 1 ,..., x K ), i 1,..., N , (4.7)

where g i (...)arecertainfunctionswherethecomponentsofvector U X are

included, however we will not write them further in the explicit relations.
Substituting (4.7) to the expression for discrepancy (4.6) and considering the
square of discrepancy R 2 as a function of variables x k , k

\

= 1,..., K to obtain its

extremums we derive the following equation system:

∂ R 2

∂

x k =

0,

i.e.thesameindetail:

N

( y i − g i ( x 1 ,..., x K )) ∂

g i ( x 1 ,..., x K )
∂

= = 0, k 1,..., K . (4.8)

x k

= i 1

Inthecommoncaseofnonlinearfunctions g i the direct obtaining of the
solutions of system (4.8) and their analysis for theminimumof discrepancy are
rather complicated. Thus, to beginwith, consider the case of linear functions g i ,
which could be further generalized to the nonlinear dependence. Besides the
problems of obtaining the parameters of the linear dependence with LST often
appear, for example these very problems have been solvedduring the secondary
processing of the airborne irradiance data (Sect. 3.2).