# Appendix 9.A: Methods Used to Detect Externally Forced Signals

This appendix very briefly reviews the statistical methods that have been used in most recent detection and attribution work. Standard ‘frequentist’ methods (methods based on the relative frequency concept of probability) are most often used, but there is also increasing use of Bayesian methods of statistical inference. The following sections briefly describe the optimal fingerprinting technique followed by a short discussion on the differences between the standard and Bayesian approaches to statistical inferences that are relevant to detection and attribution.

## 9.A.1 Optimal Fingerprinting

Optimal fingerprinting is generalised multivariate regression adapted to the detection of climate change and the attribution of change to externally forced climate change signals (Hasselmann, 1979, 1997; Allen and Tett, 1999). The regression model has the form y = Xa +u, where vector y is a filtered version of the observed record, matrix X contains the estimated response patterns to the external forcings (signals) that are under investigation, a is a vector of scaling factors that adjusts the amplitudes of those patterns and u represents internal climate variability. Vector u is usually assumed to be a Gaussian random vector with covariance matrix C. Vector a is estimated with â = (XTC^{-1}X)^{-1}XTC^{-1}y, which is equivalent to (TX)^{-1}XTỹ, where matrix X and vector ỹ represent the signal patterns and observations after normalisation by the climate’s internal variability. This normalisation, standard in linear regression, is used in most detection and attribution approaches to improve the signal-to-noise ratio (see, e.g., Hasselmann, 1979; Allen and Tett, 1999; Mitchell et al., 2001).

The matrix X typically contains signals that are estimated with either an AOGCM, an AGCM (see Sexton et al., 2001, 2003) or a simplified climate model such as an EBM. Because AOGCMs simulate natural internal variability as well as the response to specified anomalous external forcing, AOGCM-simulated climate signals are typically estimated by averaging across an ensemble of simulations (for a discussion of optimal ensemble size and composition, see Sexton et al., 2003). If an observed response is to be attributed to anthropogenic influence, X should at a minimum contain separate natural and anthropogenic responses. In order to relax the assumption that the relative magnitudes of the responses to individual forcings are correctly simulated, X may contain separate responses to all the main forcings, including greenhouse gases, sulphate aerosol, solar irradiance changes and volcanic aerosol. The vector a accounts for possible errors in the amplitude of the external forcing and the amplitude of the climate model’s response by scaling the signal patterns to best match the observations.

Fitting the regression model requires an estimate of the covariance matrix C (i.e., the internal variability), which is usually obtained from unforced variation simulated by AOGCMs (e.g., from long control simulations) because the instrumental record is too short to provide a reliable estimate and may be affected by external forcing. Atmosphere-Ocean General Circulation Models may not simulate natural internal climate variability accurately, particularly at small spatial scales, and thus a residual consistency test (Allen and Tett, 1999) is typically used to assess the model-simulated variability at the scales that are retained in the analysis. To avoid bias (Hegerl et al., 1996, 1997), uncertainty in the estimate of the vector of scaling factors a is usually assessed with a second, statistically independent estimate of the covariance matrix C which is ordinarily obtained from an additional, independent sample of simulated unforced variation.

Signal estimates are obtained by averaging across an ensemble of forced climate change simulations, but contain remnants of the climate’s natural internal variability because the ensembles are finite. When ensembles are small or signals weak, these remnants may bias ordinary least-squares estimates of a downward. This is avoided by estimating a with the total least-squares algorithm (Allen and Stott 2003).