variability of the temperature at different levels. Coefficients of correlation between
temperature deviation values at different levels are at other diagonals of the matrix.
They express possible links between temperature deviations at different altitudes.
The ill-posed inverse problem is characterized by extremely strong observational errors influence on the desired solution. Thus let us introduce the error’s
matrix that describes observational errors and is defined by satellite instrument
quality.
S TT ¼
s
2
1
s 12 s 13 s 1n
s 21 s
2
2
s 2n
::: ::: ::: :::
s n1
s
2
nn
Mean square deviations of intensity measurements at corresponded spectral
channels are at the principal diagonal of the error matrix. Coefficients of correlation
between errors at different channels are at other diagonals.
The following is necessary for obtaining the correct solution of the problem
of remote temperature sounding:
1. the desired vector ~ f is to the best of its ability obey the Eq. 9.1;
2. the vector ~ f corresponds natural temperature variability – statistical ensemble
K TT .
The method of statistical regularization is used to satisfy these two conditions.
Then the regularized solution takes the form:
^
~ f ¼ A
T
S
À1 A þ K
À1
TT Þ
À1 A
T
S
À1 ~ f
(9.10)
This solution (9.10) is the result of an entire mathematical course: methods of
solving ill-posed problems.
It is necessary to point out that the vector
^
~ f is not rigorous and exact solution
of the Eq. 9.1 and when substituting to the equation, it provides only approximate
equality. But it is clear that the equality basically could not be rigorous because of
observational errors.
f % A
^
~ f
(9.11)
However, values of the vector
^
~ f are physically justified though do not provide
the rigorous equality (9.1). The abundance of a priori temperature profiles (rich
statistics) is required for the physical justification.
It’s not always possible to have rich statistics, e.g. over oceans, in atmospheres
of other planets. Then the method of maximum smoothness is applied.
It is a priori known that temperature (and any other physical value) profile in the
atmosphere have to be smooth and escapes leaps because atmospheric properties
vary smoothly. The property of smoothness may be formulated as temperature
values at two neighbor levels weakly differ. The mathematical formulation of
92
9 The Thermal Remote Sounding of the Atmosphere
temperature deviation values at different levels are at other diagonals of the matrix.
They express possible links between temperature deviations at different altitudes.
The ill-posed inverse problem is characterized by extremely strong observational errors influence on the desired solution. Thus let us introduce the error’s
matrix that describes observational errors and is defined by satellite instrument
quality.
S TT ¼
s
2
1
s 12 s 13 s 1n
s 21 s
2
2
s 2n
::: ::: ::: :::
s n1
s
2
nn
Mean square deviations of intensity measurements at corresponded spectral
channels are at the principal diagonal of the error matrix. Coefficients of correlation
between errors at different channels are at other diagonals.
The following is necessary for obtaining the correct solution of the problem
of remote temperature sounding:
1. the desired vector ~ f is to the best of its ability obey the Eq. 9.1;
2. the vector ~ f corresponds natural temperature variability – statistical ensemble
K TT .
The method of statistical regularization is used to satisfy these two conditions.
Then the regularized solution takes the form:
^
~ f ¼ A
T
S
À1 A þ K
À1
TT Þ
À1 A
T
S
À1 ~ f
(9.10)
This solution (9.10) is the result of an entire mathematical course: methods of
solving ill-posed problems.
It is necessary to point out that the vector
^
~ f is not rigorous and exact solution
of the Eq. 9.1 and when substituting to the equation, it provides only approximate
equality. But it is clear that the equality basically could not be rigorous because of
observational errors.
f % A
^
~ f
(9.11)
However, values of the vector
^
~ f are physically justified though do not provide
the rigorous equality (9.1). The abundance of a priori temperature profiles (rich
statistics) is required for the physical justification.
It’s not always possible to have rich statistics, e.g. over oceans, in atmospheres
of other planets. Then the method of maximum smoothness is applied.
It is a priori known that temperature (and any other physical value) profile in the
atmosphere have to be smooth and escapes leaps because atmospheric properties
vary smoothly. The property of smoothness may be formulated as temperature
values at two neighbor levels weakly differ. The mathematical formulation of
92
9 The Thermal Remote Sounding of the Atmosphere