National Institute for Applied Statistics Research Australia University of Wollongong, Australia Working Paper 03 -19

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 specication 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 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). We
nd 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 inac-
curate estimates of the retrieval uncertainty, though their eects 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. Further, our derivations provide a theoretical basis, and
an understanding of the trade-os involved, for the popular OE practice of inflating a
working prior covariance in order to reduce the prior's impact on a retrieval. Finally,
our results also lead to practical recommendations for specifying the prior mean and
covariance in OE.