Cressie, N; Wang, R; Smyth, M; and Miller, C E., Statistical bias and variance for the regularized-inverse problem: application to Space-based atmospheric CO2 retrievals, National Institute for Applied Statistics Research Australia, University of Wollongong, Working Paper 04-16, 2016, 22.
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 CO 2 retrievals, specifically for the dry air mole fraction 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. Non-zero 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 mean squared prediction error of the retrieval.