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

This paper presents efficient and practical implementations of sequential data assimilation methods for the Simplified Parameterizations, primitive‐Equation DYnamics (SPEEDY) Model, a well‐known numerical model, into the data assimilation community for climate prediction. In the SPEEDY model, the time evolution of dynamics is performed via the second‐order Leapfrog integration scheme; this time integrator relies on two steps: the position and the velocity. The computational implementation of SPEEDY blends the time integrator and the spatial discretization of dynamics to accelerate algebraic computations. Thus, there is no access to the right‐hand side function of the ordinary differential equations governing the time evolution of model dynamics. Consequently, the SPEEDY model is often treated as a black box wherein positions and velocities work as inputs and outputs. Since observations in operational data assimilation only match position states, we can exploit augmented vector states to propagate analysis innovations from positions to velocities. For this purpose, we formulate three variants of ensemble‐based filters and perform numerical experiments to assess their accuracies. We consider two scenarios for the experiments: an ideal case wherein positions and velocities can be observed and a more realistic one wherein measurements are only accessible for position states. Besides, we discuss the effects of the ensemble size on the accuracies of our formulations and, even more, the typical case in which velocities are not updated across assimilation steps. The results reveal that all filter formulations' accuracies remain the same in terms of Root‐Mean‐Square‐Error by neglecting observations from velocities (a realistic scenario) even for cases wherein the number of measurements decreases to 6% of model components. Furthermore, for all discussed filter implementations, the propagation of analysis increments from position to velocities improves up to 100% the performance of filter implementations wherein velocities are not updated, a typical operational scenario. This article develops efficient ensemble‐based methods to propagate analysis innovations from position states to velocities. This is relevant for numerical models with mixed codes (model dynamics and numerical integrators). The proposed method is deployed in three well‐known ensemble formulations: the ensemble Kalman filter (EnKF) via localization matrix, the EnKF via a modified Cholesky decompositio