A Quantile-Conserving Ensemble Filter Framework. Part I: Updating an Observed Variable

Abstract A general framework for deterministic univariate ensemble filtering is presented. The framework fits a continuous prior probability density function (PDF) to the prior ensemble. A functional representation for the observation likelihood is combined with the prior PDF to get a continuous analysis (posterior) PDF. Cumulative distribution functions for the prior and analysis are also required. The key innovation is that an analysis ensemble is computed so that the quantile of each ensemble member is the same as its prior quantile. Many choices for the prior PDF family and the likelihood function are described. A choice of normal prior with normal likelihood is equivalent to the ensemble adjustment Kalman filter. Some other choices for the prior include gamma, inverse gamma, beta, beta prime, lognormal, and exponential distributions. Both prior distributions and likelihoods can be defined over a set of intervals giving additional flexibility that can be used to implement methods like a Huber likelihood for observations with occasional outliers. Priors and likelihoods can also be defined as sums of distributions allowing choices like bivariate normals or kernel filters. Empirical distributions, for instance piecewise linear approximations to arbitrary PDFs and functions can be used. Another empirical choice leads to the rank histogram filter. Results here are univariate and can be used to compute increments for observed variables or marginal distributions for any variable for a reanalysis. Linear regression of increments can be used to update state variables in a serial filter to build a comprehensive data assimilation system. Part 2 will discuss other methods for extending the framework to multivariate data assimilation. Significance Statement Data assimilation is used to combine information from model forecasts with subsequent observations to obtain better estimates of the current state of the atmosphere or other parts of the Earth system. Ensemble data assimilation uses a number of forecasts to get more information about uncertainty. A new method allows much more flexibility in the assumptions that must be made when doing ensemble data assimilation. As an example, the method can be better for quantities that are bounded like the amount of an atmospheric trace pollutant.