Our goal is to develop a filter (a weighted averaging process) that can be applied to the historical data stream in space and time so that the signal or "forced part" of the stream is optimally enhanced by comparison with the natural variability or "noise." By a filter we mean a linear operation on the data stream such as a weighted integral over the globe and over the past. This space- and time-dependent weighting function essentially constitutes a filter. This function must take place despite an imperfect observing system and an imperfect knowledge of the signal. To pose the problem formally, we seek a filter that minimizes the mean squared error between our estimator for the signal and the true signal. Implicit is the assumption that the data stream Tdata(r,t) can be written as a sum of signal and noise parts,
where TS(r,t) is the forced signal and TN(r,t) is the natural variability.
This assumption has been shown to be adequate for a general-circulation-model (GCM) simulation of the surface temperature field for an idealized planet (North et al., 1992). We conjecture that it holds for the more general questions of interest, but it is particularly problematic for studies of the world ocean. The mathematical expression for a filtered data stream, is
where D is the spatial domain and G(r,t;r',t') is the filter kernel.
Consider a planet that experiences periodic (forced) solar luminosity. Simulations of climate indicate that the resulting spectrum of the global average temperature contains a simple red-noise continuum with a sharp peak at the forcing frequency (North et al., 1992), provided the thermal response amplitude is no more than a few degrees. If we know the imposed forcing and response frequency from theory, we can filter the data stream so that only a narrow band of frequencies around the peak is retained. The longer the time series of data, the narrower can be the filter.
Much more can be done if the temporal profile (waveform) being detected is known precisely. In the case of the oscillating solar luminosity, for example, the phase of the imposed sinusoid is known as well as the frequency. Since the noise is independently distributed between sine and cosine components, we can further select within the frequency band to eliminate phases that are unlike the signal. For example, if the signal is all sine component, we can eliminate the cosine component with a simple projection filter. Since the random-noise component for a stationary process distributes variance equally between the sine and cosine components, we can reduce the variance of the noise by a further factor of two, or reduce the noise standard deviation by the square root of two. In this way we can hope to increase the signal-to-noise ratio considerably more than can be accomplished with a simple frequency band-pass filter.
In fact, there is even more enhancement potential at our disposal. If the forced response has a characteristic geographical signature at the responding frequency, we can use our filter to select only geographically similar patterns. This procedure can be employed to eliminate any patterns occurring in the natural variability that are dissimilar (orthogonal) to the signal pattern. It follows that very faint signals can be seen if we know enough about them a priori and if enough of the noise is orthogonal to the signal. Our experimentation with stochastic energy-balance models suggests that at least another factor of two in the signal-to-noise ratio can be gained from this extension to pattern recognition.
While the sinusoidal example is interesting, and potentially useful for the important sunspot-response detection problem (it is important mainly because it may tell us something about climate sensitivity at the decadal-scale), it is more idealized (being nearly a sharp tone) than the one we face in the greenhouse problem, which is spread over a broad continuum of frequencies.
In contrast to the periodically forced system, the greenhouse problem presents a single ramp-like waveform with a characteristic geographical response pattern (land-surface temperatures tend to lead slightly over ocean-surface temperatures). Hence, the response will be composed of a large number of sinusoidal frequencies in the Fourier sense. Since the pattern of response of the surface-temperature field depends strongly on the forcing frequency, we will find it natural to use frequency-dependent empirical orthogonal functions (fdEOFs) as an expansion basis (see, e.g., North, 1984). We stress that this is not just a convenient choice, but one that arises naturally from the formalism as the optimum for such purposes.
In this section we formulate the mean square error (MSE) for the random component of the error made with our estimator of the signal for a given realization. Unfortunately, in the climate problem we have only one realization to work with. In this paper we dispense with mathematical detail, referring the reader to the paper by North et al. (1995) for detailed derivations. The error in question is the difference for an individual realization between the estimator of the signal (filtered data stream) and the actual signal, whose shape in space and time must be known a priori. We form the square of this error and average it over an ensemble of realizations of the same waveform embedded in natural