Modelling spatially correlated observation errors in variational data assimilation using a diffusion operator on an unstructured mesh

Abstract

We propose a method for representing spatially correlated observation errors in variational data assimilation. The method is based on the numerical solution of a diffusion equation, a technique commonly used for representing spatially correlated background errors. The discretization of the pseudo‐time derivative of the diffusion equation is done implicitly using a backward Euler scheme. The solution of the resulting elliptic equation can be interpreted as a correlation operator whose kernel is a correlation function from the Matérn family. In order to account for the possibly heterogeneous distribution of observations, a spatial discretization technique based on the finite element method (FEM) is chosen where the observation locations are used to define the nodes of an unstructured mesh on which the diffusion equation is solved. By construction, the method leads to a convenient operator for the inverse of the observation‐error correlation matrix, which is an important requirement when applying it with standard minimization algorithms in variational data assimilation. Previous studies have shown that spatially correlated observation errors can also be accounted for by assimilating the observations together with their directional derivatives up to arbitrary order. In the continuous framework, we show that the two approaches are formally equivalent for certain parameter specifications. The FEM provides an appropriate framework for evaluating the derivatives numerically, especially when the observations are heterogeneously distributed. Numerical experiments are performed using a realistic data distribution from the Spinning Enhanced Visible and InfraRed Imager (SEVIRI). Correlations obtained with the FEM‐discretized diffusion operator are compared with those obtained using the analytical Matérn correlation model. The method is shown to produce an accurate representation of the target Matérn function in regions where the data are densely distributed. The presence of large gaps in the data distribution degrades the quality of the mesh and leads to numerical errors in the representation of the Matérn function. Strategies to improve the accuracy of the method in the presence of such gaps are discussed.

Assimilation of wide-swath altimetry observations to correct large-scale river routing model parameters

“…Moreover, we omit error correlations along the swath so that observation errors follow a white noise model. Accounting for spatially-correlated observation errors is an active research area in the field of data assimilation that is beyond the scope of demonstrating the feasibility of assimilating SWOT-type data (Guillet et al, 2018). Finally, in the framework of OSSE, observed and simulated water depths have the same scale as the ISBA-CTRIP model is used to generate both.…”

The impact of using reconditioned correlated observation‐error covariance matrices in the Met Office 1D‐Var system

“…Although much of the initial use of the diagnostic to estimate observation errors focussed on interchannel correlations, this has been extended to spatial correlations (Waller et al ., ; ; ; Cordoba et al ., ) and temporal correlations (Bennitt et al ., ). In addition to the direct use of estimated covariance matrices in data assimilation schemes, information obtained using the DBCP diagnostic has been used to model error covariance functions and operators (e.g., Stewart et al ., ; Michel, ; Simonin et al ., ; Guillet et al ., ). Theoretical work has also studied how well the diagnostic is expected to perform depending on the accuracy of the initial choice of background and OEC matrices for the single step (Waller et al ., ) and the iterative form of the diagnostic (Ménard, ; Bathmann, ).…”

Wide-Swath Altimetric Satellite Data Assimilation With Correlated-Error Reduction

“…For computational reasons, it is common practice in operational DA systems dealing with large observation datasets to make the assumption of uncorrelated observation errors, i.e., to assume the observation covariance matrix diagonal (Liu and Rabier, 2002;Oke et al, 2008;Janjić et al, 2018;Guillet et al, 2019). Indeed, the computational cost of the filters formulated in square root form (e.g., the ensemble transform Kalman filter) becomes linear in the number of observations for a diagonal observation covariance matrix (Brankart et al, 2009).…”

“…In the present study, we focus on the assimilation of SWOT data for an operational context, i.e., under the assumption of a diagonal observation error covariance matrix. Over the years, several techniques have been proposed to reduce the effect of neglecting the observationerror covariances, for instance, by inflating the observation error variances or by parameterizing the error covariances with a diffusion operator (Stewart et al, 2008(Stewart et al, , 2013Brankart et al, 2009;Miyoshi et al, 2013;Waller et al, 2014;Ruggiero et al, 2016;Guillet et al, 2019). None of these techniques are equipped to deal with non-local error correlations (i.e., correlations that do not decrease with the distance).…”