Technical Note: Variance-covariance matrix and averaging kernels for the Levenberg-Marquardt solution of the retrieval of atmospheric vertical profiles 1Istituto di Fisica Applicata "Nello Carrara" (IFAC) del Consiglio Nazionale delle Ricerche (CNR), Firenze, Italy
2Dipartimento di Fisica, Università di Bologna, Bologna, Italy
Received: 01 October 2009 – Published in Atmos. Chem. Phys. Discuss.: 30 November 2009 Abstract. The variance-covariance matrix (VCM) and the averaging kernel matrix (AKM)
are widely used tools to characterize atmospheric vertical profiles retrieved
from remote sensing measurements. Accurate estimation of these quantities is
essential for both the evaluation of the quality of the retrieved profiles
and for the correct use of the profiles themselves in subsequent applications
such as data comparison, data assimilation and data fusion. We propose a new
method to estimate the VCM and AKM of vertical profiles retrieved using the
Levenberg-Marquardt iterative technique. We apply the new method to the
inversion of simulated limb emission measurements. Then we compare the
obtained VCM and AKM with those resulting from other methods already
published in the literature and with accurate estimates derived using
statistical and numerical estimators. The proposed method accounts for all
the iterations done in the inversion and provides the most accurate VCM and
AKM. Furthermore, it correctly estimates
the VCM and the AKM also if the retrieval iterations are stopped
when a physically meaningful convergence criterion is fulfilled, i.e. before achievement of the numerical convergence at machine precision.
The method can be easily implemented in any Levenberg-Marquardt
iterative retrieval scheme, either constrained or unconstrained, without
significant computational overhead.
Revised: 24 March 2010 – Accepted: 25 March 2010 – Published: 31 March 2010
Citation: Ceccherini, S. and Ridolfi, M.: Technical Note: Variance-covariance matrix and averaging kernels for the Levenberg-Marquardt solution of the retrieval of atmospheric vertical profiles, Atmos. Chem. Phys., 10, 3131-3139, doi:10.5194/acp-10-3131-2010, 2010.