The result is closer to the true profile: average deviation is about 3
C.
(c) variant: Diagonal elements of the temperature correlation matrix K TT are
equal to 10, i.e. result could considerably differ from the mean profile.
Values of diagonal elements of the errors matrix S are equal to 10 that points
big observational errors. The result coincides with the mean profile because
observational data are suspicious.
(d) variant: Allowing that desired result strongly differ from mean profile (K TT
¼ 20) and assuming that the high exactness of observations (S ¼ 0.0000001)
we obtain the result close to the formal, rigorous solution (9.6) and unreliable
physically impossible variations of the temperature (Æ3,000 K) at different
altitudes demonstrates main feature of the ill-posed problem.
One should inquire into the exactness of the instrument, which is put on a
satellite board for adequate specifying the matrix S. The instrument on a satellite
board degrades and the exactness deteriorates. Thus the matrix S is to be
corrected while satellite is on the orbit.
Obtained here results demonstrate a strong deviation from the real temperature profile at the tropopause in all cases of initial parameters, where the
temperature varies dramatically. In our case only one value is assumed for all
altitudes in spite of the set of values defining the matrices K TT or S. For better
temperature retrieval at the tropopause more spectral channels are provided. The
real exactness of the temperature remote retrieval is reached at about 0.5–0.7
C.
It is important to choose a successful alternative between variants: smoothness
of profile and closeness to the mean profile and very strong intrusion of the mean
profile and loss of the real information. It is clear that a rich volume of the a
priori information is necessary (a lot of correlation matrix for any season and any
geographical site) for a reliable temperature retrieval.
2. The method of maximum smoothness.
The program “RADNIMBUS.exe” (in directory Lab4) accomplishes the auxiliary calculation for the taken temperature profile. Then the program
“DZ_T_FU1.exe” solves the inverse problem using the method of maximum
smoothness. Result is saved in the file “fu.dat”. The ranges of the regularization
coefficient a variation is proposed as 0.00001–0.1. Values of all elements of the
matrix S are s ij ¼ 1 corresponds to rough observation with significant errors;
s ij ¼ 0.1, corresponds to more exact observation.
The result of the program operation leads to the optimal value of the regularization coefficient a optimal ¼ 0.005, and mean square deviation 5.1
. The next result
presents three profiles: true, mean and retrieved. The error appears 8
and it is too
big.
The following iteration is close to the regularization coefficient value a ¼ 0.005
with the range of variation 0.001–0.01. Values of matrix S elements s ij ¼ 0.1 give
the optimal value of the regularization coefficient: a optimal ¼ 0.001; and the mean
square deviation 3.1
. Thus this retrieved profile is better compatible with the true
one.
9.7 Practice 8
95
C.
(c) variant: Diagonal elements of the temperature correlation matrix K TT are
equal to 10, i.e. result could considerably differ from the mean profile.
Values of diagonal elements of the errors matrix S are equal to 10 that points
big observational errors. The result coincides with the mean profile because
observational data are suspicious.
(d) variant: Allowing that desired result strongly differ from mean profile (K TT
¼ 20) and assuming that the high exactness of observations (S ¼ 0.0000001)
we obtain the result close to the formal, rigorous solution (9.6) and unreliable
physically impossible variations of the temperature (Æ3,000 K) at different
altitudes demonstrates main feature of the ill-posed problem.
One should inquire into the exactness of the instrument, which is put on a
satellite board for adequate specifying the matrix S. The instrument on a satellite
board degrades and the exactness deteriorates. Thus the matrix S is to be
corrected while satellite is on the orbit.
Obtained here results demonstrate a strong deviation from the real temperature profile at the tropopause in all cases of initial parameters, where the
temperature varies dramatically. In our case only one value is assumed for all
altitudes in spite of the set of values defining the matrices K TT or S. For better
temperature retrieval at the tropopause more spectral channels are provided. The
real exactness of the temperature remote retrieval is reached at about 0.5–0.7
C.
It is important to choose a successful alternative between variants: smoothness
of profile and closeness to the mean profile and very strong intrusion of the mean
profile and loss of the real information. It is clear that a rich volume of the a
priori information is necessary (a lot of correlation matrix for any season and any
geographical site) for a reliable temperature retrieval.
2. The method of maximum smoothness.
The program “RADNIMBUS.exe” (in directory Lab4) accomplishes the auxiliary calculation for the taken temperature profile. Then the program
“DZ_T_FU1.exe” solves the inverse problem using the method of maximum
smoothness. Result is saved in the file “fu.dat”. The ranges of the regularization
coefficient a variation is proposed as 0.00001–0.1. Values of all elements of the
matrix S are s ij ¼ 1 corresponds to rough observation with significant errors;
s ij ¼ 0.1, corresponds to more exact observation.
The result of the program operation leads to the optimal value of the regularization coefficient a optimal ¼ 0.005, and mean square deviation 5.1
. The next result
presents three profiles: true, mean and retrieved. The error appears 8
and it is too
big.
The following iteration is close to the regularization coefficient value a ¼ 0.005
with the range of variation 0.001–0.01. Values of matrix S elements s ij ¼ 0.1 give
the optimal value of the regularization coefficient: a optimal ¼ 0.001; and the mean
square deviation 3.1
. Thus this retrieved profile is better compatible with the true
one.
9.7 Practice 8
95