We thank Prof. Shin Sang-IK for stimulating discussions on the tau test. We also thank three anonymous reviewers for helpful and constructive comments. This work is supported by NSFC40830106, 2012CB955201, GYHY200906016 and NSF.
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Unless otherwise specified, this paper will be confined to the discussion of linear atmospheric responses.
For the real world, the validity of the linearization depends on the time scale. At very short time scale (e.g., subdaily to daily), atmospheric process can be strongly nonlinear. The linear dynamics becomes more valid for a properly long time average, because the averaged high-frequency chaotic nonlinear dynamics tends to become stochastic noise according to the central limit theorem (Gardiner 1997). Thus, for averages of different time scales, the representation of each term could be different, including the noise term. In our idealized model study of this section and in the next section, however, we will ignore the issues related to the nonlinearity. Instead, we will assume the coupled climate process is determined by a purely linear system (1.1) such that we can focus on the comparison of sampling errors in different assessment methods.
The terms “forcing” and “response” here often refer to the slow SST forcing and the rapid atmospheric feedback response to SST, respectively, which are in the convention of ocean–atmosphere interaction studies. They are different from the convention in linear stochastic dynamics such as LIM and FDT, where “forcing” and “response” usually refer to the stochastic noise forcing and the stochastic variables associated with the deterministic terms, respectively.
For models of more grid points, however, a direct estimation of the feedback matrix is subject to larger sampling errors due to the correlation of SST variability among neighboring points (LWL08). Now, the base of the SST forcing needs to be selected carefully. A convenient choice is the leading EOFs (Wen et al. 2010; Fan et al. 2011).
This has been demonstrated in the coupled model using experiments with the ocean forcing
The results are similar for optimal LIM and FDT estimations in 2-day data and weekly data.
This form of the tau test is suggested by Dr. Sang-Ik Shin.