Cycling the representer algorithm for variational data assimilation with the Lorenz attractor

Realistic dynamic systems are often strongly nonlinear, particularly those for the ocean and atmosphere. Applying variational data assimilation to these systems requires a tangent linearization of the nonlinear dynamics about a background state for the cost function minimization. The tangent linearization may be accurate for limited time scales. Here it is proposed that linearized assimilation systems may be accurate if the assimilation time period is less than the tangent linear accuracy time limit. In this paper, the cycling representer method is used to test this assumption with the Lorenz attractor. The outer loops usually required to accommodate the linear assimilation for a nonlinear problem may be dropped beyond the early cycles once the solution (and forecast used as the background in the tangent linearization) is sufficiently accurate. The combination of cycling the representer method and limiting the number of outer loops significantly lowers the cost of the overall assimilation problem. In addition, this study shows that weak constraint assimilation corrects tangent linear model inaccuracies and allows extension of the limited assimilation period. Hence, the weak constraint outperforms the strong constraint method. Assimilated solution accuracy at the first cycle end is computed as a function of the initial condition error, model parameter perturbation magnitude, and outer loops. Results indicate that at least five outer loops are needed to achieve solution accuracy in the first cycle for the selected error range. In addition, this study clearly shows that one outer loop in the first cycle does not preclude accuracy convergence in future cycles.