In order to create the cost function for variational data assimilation one needs to compute the background error covariance matrix, and this in turn requires information on the true state of the physical variables being modelled. This introduces a large difficulty which is often overcome by estimating the background error and then using this behaviour of the error to model the entire covariance during assimilation. A common approach for the estimation is the Hollingsworth and Lonnberg method, whereby the model forecasts subtracted from the observational data, known as the innovations, are manipulated using assumptions of correlation. This method requires the spatial binning of the innovation data and a curve fitting scheme that is applied to these bins. During my research I have produced a new method based on the general idea of analysis of innovations similar to the Hollingsworth and Lonnberg. The new method is referred to as a binless analysis of innovations this is because I have removed the need for spatial binning. Instead the innovations are used at their exact latitude and longitude, with the error covariance model and form functions. The method then minimizes a norm of differences to find a best approximation to the background error covariance. This is done using an interpretation of the minimization for the norm which involves inner products and produces a solvable series of equations. The accuracy of our methods has been compared against an implementation of the Hollingsworth and L\"onnberg method, and then assessed in multiple ways; first using qualitative analysis of the background standard deviation and length-scale ratio, secondly by the assessment of assimilated forecasts, and finally by reducing the observations used in the methods to represent the affect of application to sparsely observed systems. In all cases the binless analysis of innovations is similar or better then the results from currently used methods. The positives of our method can be summarised by being able to produce an error estimate with competitive accuracy, without the use of spatial binning and requiring less free parameters leading to a small increase to the range of applications. The additional range of applications comes from the binless methods innate flexibility towards other covariance modelling scheme, as there is no spatial restrictions we are not required to use the isotropic assumption. We have been able to use the binless analysis of innovations method for a simple anisotropic application but for any operational case this would require more research and a larger set of observations. We have included our anisotropic research to demonstrate the potential use for multi-dimensional error covariance models.

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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.