Parameters to assess the impact of regularisation on remotely sensed atmospheric profiles.

The atmosphere of Mars is the second most sounded atmosphere after the one of Earth. The wealth of datasets is now important enough to make direct comparisons of retrieved atmospheric parameters. Those comparisons can reveal biases that can arise from natural differences due to time and location, but also due to instrumental differences, or due to the different forward models used for the retrievals. Associated with the latter, “regularisation” must not be overlooked.

Deriving localised atmospheric parameters from remote sensing is often an ill-posed problem. In practice, ill-posed problems lead to a substantial increase in the level of noise when passing from the measurements to the retrieved values. Several ways to regularise ill-posed problems have been proposed. Amongst them, the most common ones are the Tikhonov method (TM)^1,2, and the optimal estimation method (OEM)³. TM is mainly used to smooth the solution while the OEM relies on an “a priori” on the parameter to be retrieved and a covariance matrix on this “a priori”. The latter serves to limit the allowed discrepancy between the “a priori” and the solution, in addition to smoothing the solution. One needs to be careful that the covariance matrix is often itself singular. Nowadays, the most cited method is OEM, but from a mathematical viewpoint, TM and OEM, are very similar.

The “a priori” and its covariance matrix are crucial information for understanding the “regularisation” applied to the solution. Another important piece of information reflecting the “regularisation” is provided by the averaging kernel matrix. The latter equals the identity matrix if no regularisation is applied, while the kernels resemble bell-curved functions with full-width at half maximum (FWHM) increasing as the amount of regularisation increases.

Most of the comparisons between datasets are possible thanks to the now common practice of providing the data described within a publication. Comparisons are crucial to the validation of results. For the Earth's atmosphere, no comparison is possible without taking into account the averaging kernels^4,5. The full averaging kernel matrix size equals the square of the size of the solution, and it is thus not always possible to provide the whole matrix. We propose to provide three parameters that permit the reconstruction of the averaging kernels: their FWHMs, the values of the peaks, and the area of each averaging kernel.

This talk aims to recall all the assertions above in particular for limb or occultation sounding and to show the importance of providing the averaging kernels associated with the retrieved profiles, as it permits assessing the reliability of a retrieved parameter. The examples in this presentation will be kept very general so that the concepts described here can be used for any other remotely sensed dataset.

References:

• Doicu, A., Trautmann, T. & Schreier, F. Numerical Regularization for Atmospheric Inverse Problems. Numerical Regularization for Atmospheric Inverse Problems (Springer Berlin Heidelberg, 2010). doi:10.1007/978-3-642-05439-6.
• Tikhonov, A. N. & Arsenin, V. Y. Solutions of Ill-Posed Problems. (V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977).
• Rodgers, C. D. Inverse Methods for Atmospheric Sounding. vol. 2 (WORLD SCIENTIFIC, 2000).
• Rodgers, C. D. & Connor, B. J. Intercomparison of remote sounding instruments. J. Geophys. Res. Atmospheres 108, (2003).
• Von Clarmann, T. et al. Overview: Estimating and reporting uncertainties in remotely sensed atmospheric composition and temperature. Atmospheric Meas. Tech. 13, 4393–4436 (2020).