Sensitivity of Optimal Estimation Satellite Retrievals to Misspecification of the Prior Mean and Covariance, with Application to OCO-2 Retrievals

Optimal Estimation (OE) is a popular algorithm for remote sensing retrievals, partly due to its explicit parameterization of the sources of error and the ability to propagate them into estimates of retrieval uncertainty. These properties require specification of the prior distribution of the state vector. In many remote sensing applications, the true priors are multivariate and hard to characterize properly. Instead, priors are often constructed based on subject-matter expertise, existing empirical knowledge, and a need for computational expediency, resulting in a working prior. This paper explores the retrieval bias and the inaccuracy in retrieval uncertainty caused by explicitly separating the true prior (the probability distribution of the underlying state) from the working prior (the probability distribution used within the OE algorithm), with an application to Orbiting Carbon Observatory-2 (OCO-2) retrievals. We find that, in general, misspecifying the mean in the working prior will lead to biased retrievals, and misspecifying the covariance in the working prior will lead to inaccurate estimates of the retrieval uncertainty, though their effects vary depending on the state-space signal-to-noise ratio of the observing instrument. Our results point towards some attractive properties of a class of uninformative priors that is implicit for least-squares retrievals. Furthermore, our derivations provide a theoretical basis, and an understanding of the trade-offs involved, for the practice of inflating a working-prior covariance in order to reduce the prior's impact on a retrieval (e.g., for OCO-2 retrievals). Finally, our results also lead to practical recommendations for specifying the prior mean and the prior covariance in OE.