We proposed retrospective optimal interpolation (ROI) that iteratively assimilates an observation set at a post analysis time into a prior analysis at the analysis time. When considering the levels of approximations achieved in ROI and fixed-interval Kalman filter (FIKS) to obtain the variance-minimum solution, we found that ROI should be more accurate than FIKS. We confirmed our suppositions with a numerical experiment using the three-variable Lorenz model. Even though four-dimensional variational assimilation (4D-Var) fails to find the global minimum because a cost function has multiple minima, ROI finds successfully the global minimum of the cost function. We proved that ROI is equivalent to the quasi-static variational assimilation (QSVA), which was devised to overcome the multiple minima problem. However, ROI's ability of overcoming the multiple minima problem depends on the length of analysis window.

An adjoint model is required to implement QSVA. By exploiting the perturbation method, we can implement ROI without using an adjoint model. For the cost-effective implementation of ROI, we developed the reduced-rank formulation of ROI based on the accuracy-saturation property. From the Lorenz 40-variable model experiment, we confirmed the reduced-rank ROI becomes clearly cost-effective as analysis window expands in its length.