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In the numerical prediction of weather or climate events, the uncertainty of the initial values and/or prediction models can bring the forecast result’s uncertainty. Due to the absence of true states, studies on this problem mainly focus on the three subproblems of predictability, i.e., the lower bound of the maximum predictable time, the upper bound of the prediction error, and the lower bound of the maximum allowable initial error. Aimed at the problem of the lower bound estimation of the maximum allowable initial error, this study first illustrates the shortcoming of the existing estimation, and then presents a new estimation based on the initial observation precision and proves it theoretically. Furthermore, the new lower bound estimations of both the two-dimensional ikeda model and lorenz96 model are obtained by using the cnop (conditional nonlinear optimal perturbation) method and a pso (particle swarm optimization) algorithm, and the estimated precisions are also analyzed. Besides, the estimations yielded by the existing and new formulas are compared; the results show that the estimations produced by the existing formula are often incorrect.