Estimation of the signal amplitudes, as well as the detection and attribution consistency tests on the amplitudes, requires an estimate of the covariance matrix Cuu of the residual noise field. However, as y typically represents climate variation on time-scales similar to the length of the observed instrumental record, it is difficult to estimate the covariance matrix reliably. Thus the covariance matrix is often estimated from a long control simulation. Even so, the number of independent realisations of u that are available from a typical 1,000 to 2,000-year control simulation is substantially smaller than the dimension of the field, and thus it is not possible to estimate the full covariance matrix. The solution is to replace the full fields y, g1,...,gm and u with vectors of dimension k, where m<k<<n, containing indices of their projections onto the dominant patterns of variability f1,...,fk of u. These patterns are usually taken to be the k highest variance EOFs of a control run (North and Stevens, 1998; Allen and Tett, 1999; Tett et al., 1999) or a forced simulation (Hegerl et al., 1996, 1997; Schnur, 2001). Stott and Tett (1998) showed with a "perfect model" study that climate change in surface air temperature can only be detected at very large spatial scales. Thus Tett et al. (1999) reduce the spatial resolution to a few spherical harmonics prior to EOF truncation. Kim et al. (1996) and Zwiers and Shen (1997) examine the sampling properties of spherical harmonic coefficients when they are estimated from sparse observing networks.
An important decision, therefore, is the choice of k. A key consideration in the choice is that the variability of the residuals should be consistent with the variability of the control simulation in the dimensions that are retained. Allen and Tett (1999) describe a simple test on the residuals that makes this consistency check. Rejection implies that the model-simulated variability is significantly different from that of the residuals. This may happen when the number of retained dimensions, k, is too large because higher order EOFs may contain unrealistically low variance due to sampling deficiencies or scales that are not well represented. In this situation, the use of a smaller value of k can still provide consistent results: there is no need to require that model-simulated variability is perfect on all spatio-temporal scales for it to be adequate on the very large scales used for detection and attribution studies. However, failing the residual check of Allen and Tett (1999) could also indicate that the model does not have the correct timing or pattern of response (in which case the residuals will contain forced variability that is not present in the control regardless of the choice of k) or that the model does not simulate the correct amount of internal variability, even at the largest scales represented by the low order EOFs. In this case, there is no satisfactory choice of k. Previous authors (e.g., Hegerl et al., 1996, 1997; Stevens and North, 1996; North and Stevens, 1998) have made this choice subjectively. Nonetheless, experience in recent studies (Tett et al. 1999; Hegerl et al. 2000, 2001; Stott et al., 2001) indicates that their choices were appropriate.
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