Ultra Rapid Data Assimilation Based on Ensemble Filters

The goal of this work is to analyse and study an ultra-rapid data assimilation (URDA) method for adapting a given ensemble forecast for some particular variable of a dynamical system to given observation data which become available after the standard data assimilation and forecasting steps. Initial ideas have been suggested and tested by Etherthon 2006 and Madaus and Hakim 2015 in the framework of numerical weather prediction. The methods are, however, much more universally applicable to general non-linear dynamical systems as they arise in neuroscience, biology and medicine as well as numerical weather prediction. Here we provide a full analysis in the linear case, we formulate and analyse an ultra-rapid ensemble smoother and test the ideas on the Lorentz 63 dynamical system. In particular, we study the assimilation and preemptive forecasting step of an ultra-rapid data assimilation in comparison to a full ensemble data assimilation step as calculated by an ensemble Kalman square root filter. We show that for linear systems and observation operators, the ultra-rapid assimilation and forecasting is equivalent to a full ensemble Kalman filter step. For non-linear systems this is no longer the case. However, we show that we obtain good results even when rather strong nonlinearities are part of the time interval [t0, tn] under consideration. Then, an ultra-rapid ensemble Kalman smoother is formulated and numerically tested. We show that when the numerical model under consideration is different from the true model, used to generate the nature run and observations, errors in the correlations will also lead to errors in the smoother analysis. The numerical study is based on the popular Lorenz 1963 model system used in geophysics and life sciences. We investigate both the situation where the full system forecast is calculated and the situation important to practical applications where we study reduced data, when only one or two variables are known to the URDA scheme