Insights into Tikhonov regularization: application to trace gas column retrieval and the efficient calculation of total column averaging kernels

Abstract

Abstract. Insights are given into Tikhonov regularization and its application to the retrieval of vertical column densities of atmospheric trace gases from remote sensing measurements. The study builds upon the equivalence of the leastsquares profile-scaling approach and Tikhonov regularization method of the first kind with an infinite regularization strength. Here, the vertical profile is expressed relative to a reference profile. On the basis of this, we propose a new algorithm as an extension of the least-squares profile scaling which permits the calculation of total column averaging kernels on arbitrary vertical grids using an analytic expression. Moreover, we discuss the effective null space of the retrieval, which comprises those parts of a vertical trace gas distribution which cannot be inferred from the measurements. Numerically the algorithm can be implemented in a robust and efficient manner. In particular for operational data processing with challenging demands on processing time, the proposed inversion method in combination with highly efficient forward models is an asset. For demonstration purposes, we apply the algorithm to CO column retrieval from simulated measurements in the 2.3 µm spectral region and to O 3 column retrieval from the UV. These represent ideal measurements of a series of spaceborne spectrometers such as SCIA-MACHY, TROPOMI, GOME, and GOME-2. For both spectral ranges, we consider clear-sky and cloudy scenes where clouds are modelled as an elevated Lambertian surface. Here, the smoothing error for the clear-sky and cloudy atmosphere is significant and reaches several percent, depending on the reference profile which is used for scaling. This underlines the importance of the column averaging kernel for a proper interpretation of retrieved column densities. Furthermore, we show that the smoothing due to regularization can be underestimated by calculating the column averaging kernel on a too coarse vertical grid. For both retrievals, this effect becomes negligible for a vertical grid with 20-40 equally thick layers between 0 and 50 km.

Carbon monoxide column retrieval for clear-sky and cloudy atmospheres: a full-mission data set from SCIAMACHY 2.3 µm reflectance measurements

“…This approach has been applied to SCIA-MACHY data before by Gloudemans et al (2008) and Borsdorff et al (2016), and its regularisation type is discussed in detail by Borsdorff et al (2014). Here, we summarise the aspects of the retrieval that are needed for the later discussion.…”

“…Here G is the gain matrix of the inversion and its definition is given by Borsdorff et al (2014). Therewith, we have defined all diagnostic tools for our retrieval.…”

“…Therewith, we have defined all diagnostic tools for our retrieval. A detailed overview of the profile-scaling approach is given in Borsdorff et al (2014).…”

“…The column averaging kernel represents a weighted integration of the true vertical profile, taking into account the particular retrieval sensitivity. The differences between the true column, c true = Cρ true and c eff = a col ρ true , cannot be inferred from the measurement and this is known as the null-space or smoothing error of the retrieval (Borsdorff et al, 2014;Rodgers, 2000),…”

“…Borsdorff et al (2014) presented a numerically efficient algorithm for calculating a col . The column averaging kernel represents a weighted integration of the true vertical profile, taking into account the particular retrieval sensitivity.…”