PROM: a method for estimating error covariance functions for variational data assimilation schemes in ocean models 

Abstract

<p>Variational data assimilation methods are an invaluable tool for operational ocean models. These methods require the knowledge of the spatial covariances of the background errors (differences between the numerical model and the true values) and the observation errors (differences between true and measured values). In practice, the true covariances are never known exactly, so the error covariance matrices containing values of the covariance functions at specific locations must be estimated approximately. The Hollingsworth and Lönnberg method (H-L), based on the analysis of innovations (differences between observations and the model), is a widely used method to compute the covariance matrices. It requires to combine into bins the products of innovation data separated by similar distances, compute covariances in each bin and then to find a best fit covariance function. While being a helpful tool, the H-L method has its limitations. We have developed a new mathematical method for computing the background and observation error covariance functions and therefore the error covariance matrices. The method (PROM) uses functional analysis to project the innovation products into a subspace of parametrised functions and then to find the element of this subspace that closest approximate the data. This allows to overcome some shortcomings of the H-L method like the assumption of statistical isotropy, allowing to compute 2D and 3D covariance functions. It also eliminates intermediate steps used in the H-L method such as binning the innovations, and the computation of innovation covariances for each bin, before the best-fit curve can be found. We show that the new method works in situations where the standard H-L method experiences difficulties, especially when observations are scarce. It gives a better estimate than the H-L in a synthetic idealised case where the true covariance function is known. We also show that the new method allows to use the separable convolution mathematical algorithm leading to a dramatic increase of the computational speed, up to an order of magnitude.</p>