Comparisons of Two Ensemble Mean Methods in Measuring the Average Error Growth and the Predictability

In this paper, taking the Lorenz system as an example, we compare the influences of the arithmetic mean and the geometric mean on measuring the global and local average error growth. The results show that the geometric mean error (GME) has a smoother growth than the arithmetic mean error (AME) for the global average error growth, and the GME is directly related to the maximal Lyapunov exponent, but the AME is not, as already noted by Krishnamurthy in 1993. Besides these, the GME is shown to be more appropriate than the AME in measuring the mean error growth in terms of the probability distribution of errors. The physical meanings of the saturation levels of the AME and the GME are also shown to be different. However, there is no obvious difference between the local average error growth with the arithmetic mean and the geometric mean, indicating that the choices of the AME or the GME have no influence on the measure of local average predictability.