The Problem of Retrieving Atmospheric Parameters from Radiative Observations - Short-Wave Solar Radiation in the Earth's Atmosphere - page 147

the reselection of all possible variants of the parameter values if their quantity
is small, however with a large quantity of parameter values it is unreal. Then
the penalty function method is appropriate, as the constraint equations have
the form x k ( x k −1) = 0.

Lastly, the case where we impose inequality on the desired parameters is
considered. They can be written without the generality loss as:
ψ j ( x 1 ,..., x K ) ≥ 0, j = 1,..., J . (4.34)

But conditions (4.34) could be approximately reducedtotheabove-considered
constraint conditions (4.26) if the differentiable functions are used instead ψ j .

These functions are close to zero, when the inequality is valid and they are
big enough when the inequality is not valid. The following condition could be
used as the simplest constraint satisfying the desired properties:

= ψ j ( x 1 ,..., x k )) , = c j exp(− h j j 1,..., J . (4.35)

Constants h j havethesamemeaningastheabove-consideredonesinrelations
(4.26) because the matrix of the partial derivatives from condition (4.35) is
equal to:
∂

ψ j ∂ψ j

c j

x k =

− h j c j .

∂ ∂

x k

Thus, formally, for coordination with the above-derived equations, the follow-
ing should be set everywhere:

ψ j ∂ψ j = h j = = H ≡ ( h jl ), h jj , h j , l = 0, and ( C n ) jk − c j .

j

∂ x k

The selection of values h j is arbitrary to a certain degree, the larger they are
the more exact the inequalities are, and the stronger the exponent nonlinearity
is. The method of penalty functions is obviously converted to the case when
constraints (4.26) and conditions (4.34) are imposed on the parameters.

4.3
Accounting for Measurement Uncertainties and Regularization
of the Solution

We will begin to consider the impact of the observational uncertainties on the
inverse problem solution from an elementary but rather important relation.
Let some parameters X be expressed linearly through observational results Y :

= X AY + A 0 , (4.36)

where A and A 0 are the specified matrix and vector. Note that all relations
obtained in the previous sectionfinally have just a similar view, though vector X
is treated here in a wider sense: as some value linearly dependent on Y .