Statistical bias and variance for the regularized inverse problem: application to space-based atmospheric CO2 retrievals

Remote sensing of the atmosphere is typically achieved through measurements that are high-resolution radiance spectra. In this article, our goal is to characterize the first-moment and second-moment properties of the errors obtained when solving the regularized inverse problem associated with space-based atmospheric CO2 retrievals, specifically for the dry air mole fraction of CO2 in a column of the atmosphere. The problem of estimating (or retrieving) state variables is usually ill posed, leading to a solution based on regularization that is often called Optimal Estimation (OE). The difference between the estimated state and the true state is defined to be the retrieval error; error analysis for OE uses a linear approximation to the forward model, resulting in a calculation where the first moment of the retrieval error (the bias) is identically zero. This is inherently unrealistic and not seen in real or simulated retrievals. Nonzero bias is expected since the forward model of radiative transfer is strongly nonlinear in the atmospheric state. In this article, we extend and improve OE's error analysis based on a first-order, multivariate Taylor series expansion, by inducing the second-order terms in the expansion. Specifically, we approximate the bias through the second derivative of the forward model, which results in a formula involving the Hessian array. We propose a stable estimate of it, from which we obtain a second-order expression for the bias and the mean square prediction error of the retrieval.