Adaptive Tuning of Numerical Weather Prediction Models: Part I: Randomized Gcv and Related Methods in Three and Four Dimensional Data Assimilation

We consider the problem of ingesting scattered, noisy, heterogeneous data into an imperfect, evolving mathematical model for the purpose of prediction of a dynamical system. In particular we consider models which perform a three or four dimensional data assimilation using variational/statistical methods. This paper is written with global scale medium range numerical weather prediction models in mind, but the results are applicable more generally to computer models of dynamical systems which ingest scattered, noisy data. In variational data assimilation, optimal ingestion of the observational data, and optimal use of prior physical and statistical information involves the choice of numerous weighting, smoothing and tuning parameters which control the ltering and merging of divers sources of information. Generally these weights must be obtained from a partial and imperfect understanding of various sources of errors, and are frequently chosen by a combination of historical information, physical reasoning and trial and error. It is our purpose to show how the methods of generalized cross validation (GCV) and unbiased risk(UBR), and two extensions of GCV and UBR, namely Partial GCV (PGCV) and Partial UBR (PUBR) can be used to estimate certain of the most important tuning, weighting and smoothing parameters, in the context of extremely large operational models (106 degrees of freedom). For comparison we review how maximum likelihood (ML) and generalized maximum likelihood(GML) can be used for some of the same purposes when the model is correctly speci ed. A theoretical proof of the properties of PUBR is given. In order to be useful in operational numerical weather prediction, the estimates must have the potential for operational implementation in the context of the iterative solution of extremely large variational problems. We carry out two `toy' simulations which suggest how this may be done, via the technique of randomized trace estimation. In many examples of approximate solution of extremely large variational problems via iterative methods, the number of iterations can also play a role as a `tuning' parameter. In the second `toy' simulation it is demonstrated how UBR and GCV can also be used to decide when to stop the iteration. Following these demonstrations, which are carried out in variational problems without time dependency, we describe how the methods may be implemented in several four dimensional contexts that appear in the current literature. In this paper, we explicitly only describe the choice of parameters which play the role of signal-to-noise ratios, and number of iterations. Other estimable parameters are speci cally described only in passing. In the second paper in this series, we develop a general theory of parameter estimability, which will provide a general yardstick for selecting parameters which can be estimated by these methods. In the third paper we give a new GCV-like method for the estimation of relative accuracy of heterogeneous sources of data.