Appendix 12.2: Three Approaches to Optimal Detection
Optimal detection studies come in several variants depending upon how the time
evolution of signal amplitude and structure is treated.
Fixed pattern studies (Hegerl et al., 1996, 1997, 2000a; Berliner et
al., 2000; Schnur, 2001) assume that the spatial structure of the signals does
not change during the epoch covered by the instrumental record. This type of
study searches for evidence that the amplitudes of fixed anthropogenic signals
are increasing with time. The observed field y=y(t) that
appears on the left hand side of equation (A12.1.1)
is typically a field of 30 to 50year moving window trends computed from annual
mean observations. The regression equation (A12.1.1)
is solved repeatedly with a fixed signal matrix G as the moving 30 to
50year window is stepped through the available record.
Studies with timevarying patterns allow the shape of the signals, as
well as their amplitudes, to evolve with time. Such studies come in two flavours.
The spacetime approach uses enlarged signal vectors that consist of
a sequence of spatial patterns representing the evolution of the signal through
a short epoch. For example, Tett et al. (1999) use signal vectors composed of
five spatial patterns representing a sequence of decadal means. The enlarged
signal matrix G=G(t) evolves with time as the 5decade
window is moved one decade at a time. The observations are defined similarly
as extended vectors containing a sequence of observed decadal mean temperature
patterns. As with the fixed pattern approach, a separate model is fitted for
each 5decade window so that the evolution of the signal amplitudes can be studied.
The spacefrequency approach (North et al., 1995) uses annual mean signal
patterns that evolve throughout the analysis period. A Fourier transform is
used to map the temporal variation of each signal into the frequency domain.
Only the lowfrequency Fourier coefficients representing decadalscale variability
are retained and gathered into a signal vector. The observations are similarly
transformed. The selection of timescales that is effected by retaining only
certain Fourier coefficients is a form of dimension reduction (see Dimension
Reduction, Appendix 12.4) in the time domain. This is
coupled with spatial dimension reduction that must also be performed. The result
approximates the dimension reduction that is obtained by projecting observations
in space and time on low order spacetime EOFs (North et al., 1995). A further
variation on this theme is obtained by increasing the time resolution of the
signals and the data by using monthly rather than annual means. Climate statistics,
including means, variances and covariances, have annual cycles at this time
resolution, and thus dimension reduction must be performed with cyclostationary
spacetime EOFs (Kim and Wu, 2000).
Given the same amount of data to estimate covariance matrices, the spacetime
and spacefrequency approaches will sacrifice spatial resolution for temporal
resolution.
