The role of observation and background errors for reconstructing localized features from non-local observations

Most data assimilation (DA) methods define the analysis state (i.e., the optimal state for initializing a numerical model) through a quadratic cost function which penalizes both the differences to a model prior (called background state) and the distance to the observations. This paper studies the impact of observation and background error characteristics on the ability to reconstruct spatially localized features with such methods. While the density of the data employed in the DA process gives an upper limit for the spatial reconstruction, this limit can generally only be achieved if the observations are sufficiently precise. This work discusses how finite observation errors (for given background error statistics) degrade the spatial resolution of the analysis state. For this it expands the cost function minimum into a weighted sum over pseudo inverse (PI) solutions each of which corresponds to a different subset of the available observations (i.e., only a subset of the observations is considered for each of these terms, respectively). Observation errors occur only in the weighting factors of this expansion and therefore determine the extent to which observational information is included in the analysis state. More precisely, the weighting factors of the different PIs can be written in terms of normalized observation errors and the determinant of a correlation matrix which characterizes the overlap of the corresponding observation operators. The presented mathematical results are illustrated with a simple model problem which explicitly shows how the reconstruction of a localized feature depends on observation errors as well as the observation operators’ overlap. ndings of this work generally demonstrate that large observation errors do not only decrease the overall weight which the respective observations obtain in the DA process, they especially reduce the DA systems capability to obtain spatially localized information. Small observation errors are particularly important when processing strongly non-local observations as they are typically obtained from passive remote sensing measurements. These have the potential to smear out signals from localized sources over large regions in model space. Generally, observation errors have to be smaller the more the respective observation operators overlap.