Detection of Forced Climate Signals. Part 1: Filter Theory

This paper considers the construction of a linear smoothing filter for estimation of the forced part of a change in a climatological field such as the surface temperature. The filter is optimal in the sense that it suppresses the natural variability or “noise” relative to the forced part or “signal” to the maximum extent possible. The technique is adapted from standard signal processing theory. The present treatment takes into account the spatial as well as the temporal variability of both the signal and the noise. In this paper we take the signal's waveform in space-time to be a given deterministic field in space and lime. Formulation of the expression for the minimum mean-squared error for the problem together with a no-bias constraint leads to an integral equation whose solution is the filter. The problem can be solved analytically in terms of the space-time empirical orthogonal function basis set and its eigenvalue spectrum for the natural fluctuations and the projection amplitudes of the signal onto these eigenfunctions. The optimal filter does not depend on the strength of the assumed waveform used in its construction. A lesser mean-square error in estimating the signal occurs when the space-time spectral characteristics of the signal and the noise are highly dissimilar; for example, if the signal is concentrated in a very narrow spectral band and the noise in a very broad band. A few pedagogical exercises suggest that these techniques might be useful in practical situations.