AT 652 Atmospheric Remote Sensing
Temperature Retrieval

The aim of this project is to understand the advantages and pitfalls of an optimal estimation inversion retrieval scheme as described by Rodgers (1976,2000). 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.

A full treatment on optimal estimation is beyond the scope of this course, but students would benefit from reading or skimming Daniel Jacob's introductory notes on the subject.


Procedure and Tasks:

  1. Use Equations (7.38) and (7.39) from Remote Sensing of the Lower Atmosphere: An Introduction (1994) to calculate 10 equally spaced temperature weighting functions W between 6 and 24 km. Use an atmospheric scale height H (for pressure) of 7 km. These weighting functions, while idealized, capture the general behavior of weighting functions calculated more exactly for realistic gas absorption cross sections.

  2. Calculate synthetic radiances y from the tropical temperature profile using the linear forward model y = Wx where x represents the true temperature profile. Do not add instrument noise (yet).

  3. You will now attempt to retrieve the temperature profile from the synthetic radiances.

    • Use an isothermal temperature profile at 250 K as the a priori temperature profile xa.
    • Create a covariance matrix Sy representing uncorrlateded noise among the channels with a magnitude of 0.25K.
    • Create apriori covariance matrix Sa representing the uncertainty in the prior temperature profile. Assume 1 sigma errors of +/- 50 K with a decorrelation length of 6 km (eqn 16).
    • Using the synthetic measurements y together with the weighting functions W, the prior temperature profile, and the prior and measurement error covariance matrices, use (equations (21) and (22) ) to calculate the optimal (retrieved) temperature profile along with the a posteriori covariance matrix.
  4. Plot the weighting functions, the true temperature profile, the a priori temperature profile, and the retrived temperature profile. Also plot the error in the temperature profile (retrieved-true) as well as the a posteriori error profile (the square root of the diagonal elements of the posterior error covariance matrix). How does the retrieved profile compare to the true profile? How does it compare to the a priori temperature profile?

  5. Verify the results by:
    a. Showing that the retrieved profile leads to the observed (synthetic) radiances.

    b. The Weighting Function matrix W tells us how the radiances y respond to changes in the true temperature profile x. That is, Δy = WΔx. Determine the changes in the radiances for a change (e.g., 10 K) in the temperature profile at 0, 20, and 50 km (separately). How is the a posteriori error profile related to the sensitivity of the radiances to changes in the temperature profile? (Hint: You do not need to re-run the retrieval to answer this.)


  6. a. Next, add random noise (such that values are normally distributed about zero with a standard deviation of 5 Ky) to your synthetic radiances, and re-run the retrieval. Be sure to tell the retrieval program that the assumed error is now 5 Ky (this goes into vector "y_err" and specifically represents the assumed standard deviation of the error of each channel), and discuss how (and at which altitudes) the retrieval is most affected.

    b. What happens if you tell the retrieval program that the assumed error in y is still very low, like 0.25K, even though it isn't? That is, use the same noisy values for the measured radiances y as in step 6a above, but change y_err back to the small value assumed in step 4. Why do you think this happened?

  7. Complete steps 2-6 once again (you may exclude steps 5 and 6b this time), this time using either the mid-latitude winter temperature profile or mid-latitude summer profile to compute y (your choice) instead of the tropical temperature profile. Use a different a priori profile as well (your choice as well on the type of profile to use).



Files you will need:


Literature:


Remarks:




Last Modified: 10/2/2013