Climate Change 2001:
Working Group I: The Scientific Basis
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12.2.4 Some Important Statistical Considerations

Most recent studies (Hegerl et al., 1996, 1997, 2000, 2001; North and Stevens, 1998; Allen and Tett, 1999; Tett et al., 1999, 2000; Berliner et al., 2000; North and Wu, 2001; Stott et al., 2001) have used a regression approach in which it is assumed that observations can be represented as a linear combination of candidate signals plus noise (see Appendices 12.1 and 12.2). Other approaches, such as pattern correlation (Santer et al., 1995, 1996a; see also Appendix 12.3), complement the regression approach, being particularly valuable in cases where model-simulated response patterns are particularly uncertain. In all cases, the signal patterns are obtained from climate models. In the regression approach, the unknown signal amplitudes are estimated from observations. The uncertainty of these estimates that is caused by natural variability in the observations is expressed with confidence intervals. Detection of an individual signal is achieved when the confidence interval for its amplitude does not include zero. Overall detection (that some climate change has taken place) is achieved when the joint confidence interval on the signals considered does not encompass the origin.

Attribution and consistency
Detecting that some climate change has taken place does not immediately imply that we know the cause of the detected change. The practical approach to attribution that has been taken by climatologists includes a demand for consistency between the signal amplitudes projected by climate models and estimated from observations (Hasselmann, 1997). Consequently, several studies, including Hegerl et al. (1997, 2000) and Tett et al. (1999, 2000) have performed an "attribution" consistency test that is designed to detect inconsistency between observed and model projected signal amplitudes. This test is a useful adjunct to detection because it provides an objective means of identifying model-simulated signal amplitudes that are significantly different from those estimated from observations. However, the test does not give the final word on attribution because it is designed to identify evidence of inconsistency rather than evidence for consistency between modelled and observed estimates of signal strength. A further refinement (e.g., Stott et al., 2001) is to consider the full range of signals believed, on physical grounds, to be likely to have had a significant impact on recent climate change and to identify those subsets of these signals that are consistent with recent observations. If all these subsets contain an anthropogenic component, for example, then at least part of the observed change can be attributed to anthropogenic influence. Levine and Berliner (1999) point out that a test that searches for consistency is available (Brown et al., 1995), but it has not yet been used in attribution studies. Bayesian statisticians approach the problem more directly by estimating the posterior probability that the signal amplitudes projected by climate models are close to those in the observed climate. Berliner et al. (2000) provides a demonstration.

The use of climate models to estimate natural internal variability
Climate models play a critical role in these studies because they provide estimates of natural internal variability as well as the signals. In most studies an estimate of natural internal variability is needed to optimise the search for the signal and this is usually obtained from a long control simulation. In addition, a separate estimate of natural variability is required to determine the uncertainty of the amplitude estimates. Unfortunately, the short instrumental record gives only uncertain estimates of variability on the 30 to 50 year time-scales that are important for detection and attribution and palaeo-data presently lacks the necessary spatial coverage (see Section 12.2.2). Thus a second control integration is generally used to estimate the uncertainty of the amplitude estimates that arises from natural climate variability (e.g., Hegerl et al., 1996; Tett et al., 1999).

Temporal and spatial scales used in detection studies
While a growing number of long control simulations are becoming available, there remain limitations on the spatial scales that can be included in global scale detection and attribution studies. Present day control simulations, which range from 300 to about 2,000 years in length, are not long enough to simultaneously estimate internal variability on the 30 to 50 year time-scale over a broad range of spatial scales. Consequently, detection and attribution studies are conducted in a reduced space that includes only large spatial scales. This space is selected so that it represents the signals well and allows reliable estimation of internal variability on the scales retained (see Appendix 12.4). Recently, the scale selection process has been augmented with a statistical procedure that checks for consistency between model simulated and observed variability on the scales that are retained (Allen and Tett, 1999).

Fixed and temporally-varying response patterns
Detection and attribution studies performed up to the SAR used fixed signal patterns that did not evolve with time. These studies were hampered because the mean large-scale response of climate to different types of anomalous forcing tends to be similar (e.g., Mitchell et al., 1995a; Reader and Boer, 1998; see also Figure 12.3). Recent studies have been able to distinguish more clearly between signals from anthropogenic and other sources by including information from climate models about their temporal evolution. Tett et al. (1999, 2000) and Stott et al. (2001) in related studies have used a space-time approach in which the signal pattern evolves on the decadal time-scale over a 50-year period. North and Wu (2001) also use a space-time approach. North and Stevens (1998) used a related space-frequency approach (see Appendix 12.2).

Allowance for noise in signal patterns
Most studies have assumed that signal patterns are noise free. This is a reasonable assumption for fixed pattern studies (see Appendix 12.2) but space-time estimates of the 20th century climate change obtained from small ensembles of forced climate simulations are contaminated by the model's internal variability. Allen and Tett (1999) point out that noise in the signal patterns will tend to make the standard detection algorithm (e.g., Hasselmann, 1993, 1997) somewhat conservative. Methods for accommodating this source of noise have been available for more than a century (Adcock, 1878; see also Ripley and Thompson, 1987). Allen and Stott (2000) recently applied such a method and found that, while the question of which signals could be detected was generally unaffected, the estimated amplitude of individual signals was sensitive to this modification of the procedure. Another source of uncertainty concerns differences in signal patterns between different models. Recent studies (Allen et al., 2000a,b; Barnett et al., 2000; Hegerl et al., 2000) consider the sensitivity of detection and attribution results to these differences.



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