The aim of this project is to understand the advantages and pitfalls of an optimal estimation inversion retrieval scheme as described by Rodgers (1976). As said by Clive Rodgers himself:

The problem that will be examined here is as follows: given a measurement or series of measurements of thermal radiation emitted by the atmosphere, the intensity and spectral distribution of which depend on the state of the atmosphere in a known way, deduce the best estimate of the state of the atmosphere.
There are two distinct aspects to this problem which are not always clearly separated; they may be described as the 'inverse' problem and the 'estimation' problem. The inverse problem is the matter of inverting a known equation which expresses radiation as a function of the atmospheric state, so as to express atmopsheric state in terms of radiation. This is usually an 'ill-posed' problem; i.e., it has no mathematically unique solution.
We therefore have an estimation problem, that is, to find the appropriate criteria which determine the best solution from all the possible ones which are consistent which the observations.

A simple retrieval scheme will be build and used to investigate the effect of the different parameters that determine the outcome of the final retrieval.

1. Use Equations (7.38) and (7.39) from Stephens (1994) to calculate 10 equally spaced temperature weighting functions K between 6 and 24 km. Use an atmospheric scale height H of 7 km.

2. Use equation (12) from Rodgers (1976) to calculate synthetic radiances y from one of the available temperature profiles.

3. Use one of the available temperature profiles or just an isothermal profile for the a priori temperature profile x_a.

4. Set up a measurement covariance matrix S_y where only the diagonal elements are used. Use a standard deviation of 5 %.

5. Set up an a priori covariance matrix S_a according to Equation (16) of Rodgers (1990). Use a length scale l of 3 km and a standard deviation of 50 K.

6. Use Equation (28) of Rodgers (1990) to calculate the contribution functions D_y.

7. Use Equation (8) of Rodgers (1990) to calculate the averaging kernel A.

8. Use Equations (21) and (22) of Rodgers (1976) to calculate the retrieved temperature profile and its covariance matrix S_x.

9. Calculate a retrieval error profile from the retrieval covariance matrix derived in th last step.

10. Plot the weighting functions, contributions functions, averaging kernel, original, a priori and retrieved temperature profile, and retrieval error profile.

• Add some random noise to your synthetic measurements and look how the retrieval is affected.

• Vary the measurement and a priori covariance matrices and look how the retrieval is affected.

• Use different temperature profiles for the original and a priori profiles and look how the retrieval is affected.

• McClatchey Tropical Profile

• McClatchey Mid-Latitude Summer Profile

• McClatchey Mid-Latitude Winter Profile

• This is a file with Fortran routines to do all matrix calculations you need in this project.

• In IDL it is straightforward to do the matrix calculations. An example is the following: Dy=Sa##transpose(K)##invert(K##Sa##transpose(K)+Sy)

• Rodgers, C.D., Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation. Reviews of Geophysics and Space Sciences, 14, 609 - 624, 1976.

• Rodgers, C.D., Characterization and error analysis of profiles retrieved from remote sounding measurements. Journal of Geophysical Research, 95, 5587 - 5595, 1990.

• Stephens, G.L., Remote Sensing of the Lower Atmopshere. Oxford University Press, 1994.

• To make life a little easier it is assumed in the above that the Planck function is linear in temperature (as is the case for microwave observations) and that there is no surface contribution to the observed radiances. Therefore, the radiances can be calculated as a matrix multiplication between the weighting functions and the temperature profile: y = K x.

• The inversion of the a priori matrix specified above is usually highly unstable. Try to avoid any direct inversion of this matrix in your calculations. Using the above mentioned equations will do the job.