Ensemble based methods for leapfrog integration in the simplified parameterizations, primitive‐equation dynamics model

Abstract

This paper presents efficient and practical implementations of sequential dataassimilation methods for the Simplified Parameterizations, primitive-EquationDYnamics (SPEEDY) Model, a well-known numerical model, into the dataassimilation community for climate prediction. In the SPEEDY model, the timeevolution of dynamics is performed via the second-order Leapfrog integrationscheme; this time integrator relies on two steps: the position and the velocity. Thecomputational implementation of SPEEDY blends the time integrator and thespatial discretization of dynamics to accelerate algebraic computations. Thus,there is no access to the right-hand side function of the ordinary differentialequations governing the time evolution of model dynamics. Consequently, theSPEEDY model is often treated as a black box wherein positions and velocitieswork as inputs and outputs. Since observations in operational data assimilationonly match position states, we can exploit augmented vector states to propagateanalysis innovations frompositions to velocities. For this purpose, we formulatethree variants of ensemble-based filters and perform numerical experiments toassess their accuracies.We consider two scenarios for the experiments: an idealcase wherein positions and velocities can be observed and a more realistic onewherein measurements are only accessible for position states. Besides, we discussthe effects of the ensemble size on the accuracies of our formulations and,even more, the typical case in which velocities are not updated across assimilationsteps. The results reveal that all filter formulations? accuracies remainthe same in terms of Root-Mean-Square-Error by neglecting observations fromvelocities (a realistic scenario) even for cases wherein the number of measurementsdecreases to 6% of model components. Furthermore, for all discussedfilter implementations, the propagation of analysis increments from positionto velocities improves up to 100% the performance of filter implementationswherein velocities are not updated, a typical operational scenario.