Optimal Fingerprints for the Detection of Time-dependent Climate Change

An optimal linear filter (fingerprint) is derived for the detection of a given time-dependent, multivariate climate change signal in the presence of natural climate variability noise. Application of the fingerprint to the observed (or model simulated) climate data yields a climate change detection variable (detector) with maximal signal-to-noise ratio. The optimal fingerprint is given by the product of the assumed signal pattern and the inverse of the climate variability covariance matrix. The data can consist of any, not necessarily dynamically complete, climate dataset for which estimates of the natural variability covariance matrix exist. The single-pattern analysis readily generalizes to the multipattern case of a climate change signal lying in a prescribed (in practice relatively low dimensional) signal pattern space: the single-pattern result is simply applied separately to each individual base pattern spanning the signal pattern space. Multipattern detection methods can be applied either to test the statistical significance of individual components of a predicted multicomponent climate change response, using separate single-pattern detection tests, or to determine the statistical significance of the complete signal, using a multivariate test. Both detection modes make use of the same set of detectors. The difference in direction of the assumed signal pattern and computed optimal fingerprint vector allows alternative interpretations of the estimated signal associated with the set of optimal detectors. The present analysis yields an estimated signal lying in the assumed signal space, whereas an earlier analysis of the time-independent detection problem by Hasselmann yielded an estimated signal in the computed fingerprint space. The different interpretations can be explained by different choices of the metric used to relate the signal space to the fingerprint space (inverse covariance matrix versus standard Euclidean metric, respectively). Two simple natural variability models are considered: a space-time separability model, and an expansion in terms of P0Ps (principal oscillation patterns). For each model the application of the optimal fingerprint method is illustrated by an example.