Determination of the Backward Predictability Limit and Its Relationship with the Forward Predictability Limit

Abstract

In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space. The forward predictability mainly focuses on the forward evolution of initial errors superposed on the initial state over time, while the backward predictability is mainly concerned with when the given state can be predicted before this state happens. From the results, there is a negative correlation between the local forward and backward predictability limits. That is, the forward predictability limits are higher when the backward predictability limits are lower, and vice versa. We also find that the sum of forward and backward predictability limits of each state tends to fluctuate around the average value of sums of the forward and backward predictability limits of sufficient states. Furthermore, the average value is constant when the states are sufficient. For different chaotic systems, the average value is dependent on the chaotic systems and more complex chaotic systems get a lower average value. For a single chaotic system, the average value depends on the magnitude of initial perturbations. The average values decrease as the magnitudes of initial perturbations increase.

摘要

在研究工作中, 提出了向前与向后可预报性两类可预报性. 然后利用非线性局部Lyapunov指数(NLLE)方法定量估计了Lorenz63和Lorenz96模型中相空间状态点的局部向前与向后可预报期限. 向前可预报性主要关注与叠加在初始状态上初始误差随时间的向前演变, 而向后可预报性则主要关注给定状态在它发生之前何时被预测出来. 研究结果表明, 向前与向后可预报期限具有负相关关系. 也就是说, 当向前可预报期限比较大时, 向后可预报期限比较小, 反之亦然. 我们还发现每一个状态点的向前与向后可预报期限之和均在足够多的状态的两类可预报期限之和的平均值附近振荡. 此外, 当状态点的数目足够多时, 此平均值为常数. 对于不同的混沌系统, 此平均值的大小取决于混沌系统. 更加复杂的混沌系统拥有较低的平均值. 对于单个混沌系统, 此平均值依赖于初始误差量级的大小. 平均值随之初始误差量级的增大而减小.