truncated is an unbiased suboptimal approximation to the optimal filter.
Each term in the sum for the signal-to-noise ratio squared is the ratio of the square of the signal projected onto mode n divided by the fdEOF eigenvalue corresponding to that mode (7). Hence, each term can be interpreted as the ratio of the square of the signal amplitude to the noise variance for that mode. If the signal is confined to only a few fdEOF modes, the sum will terminate quickly. If it is broad-band, the sum will have many terms, each one of which is positive. A highly desirable situation is one in which the signal projections are large, for modes with small eigenvalues. Clearly, it is important to estimate the rough size of the signal-to-noise calculation before performing any data manipulation or extensive model simulations, since detection of the forced signal may be impossibly difficult even with an optimal filter.
Energy-balance models (EBMs) have proven to be able to simulate the geographical distribution of the seasonal cycle of surface temperature (North et al., 1983; Hyde et al., 1989). When EBMs are forced with white noise (in space and time), they have also been shown to reproduce the geographical distribution of the variance of natural fluctuations in frequency bands from two months to a few years (Kim and North, 1991). Even the geographic distributions of spatial correlation characteristics from EBMs are in good agreement with observational data for these same frequency bands. The only new parameter in the noise-forced simulations is the strength of the noise forcing. Even this cancels out in the computation of correlations. Such a forced model enables us to compute the fdEOFs exactly. We dub them frequency-dependent theoretical orthogonal functions (fdTOFs), since they come from a theoretical model rather than from data. When we computed these fdTOFs and compared them to those computed from 40 years of surface-temperature data, they were in reasonable agreement (Kim and North, 1993).
We have extended the noise-forced EBMs (nfEBMs) to include a deep ocean similar to that used in a number of studies by the IPCC (IPCC, 1992; see also Hoffert et al., 1980). The model consists of land and ocean distributed on the globe's surface. The ocean consists of a mixed layer of 75 m depth atop a deep ocean, which transports heat by way of uniform upwelling and uniform thermal diffusion. Such an ocean model develops a thermocline at a depth of about 1 km for a reasonable choice of the two new parameters (Kim et al., 1992). The new coupled model still reproduces a good seasonal cycle and second-moment variance and covariance characteristics. Its lower-frequency behavior is thought to be somewhat improved over that of the earlier mixed-layer-only version (Kim and North, 1992).
The new coupled (linear) model is capable of producing fdTOFs for the natural-variability and forced-climate signals. The static sensitivity of the model is somewhat low as climate models go: 2°C for a doubling of carbon dioxide and 1.2°C for a 1 percent increase in solar constant. These last quantities do not depend on the ocean, since they are static (for transition from equilibrium to equilibrium) characteristics. The transient behavior of the model is fairly well understood. For a ramp in greenhouse forcing starting in the year 1750, the model produces about a 0.5°C increase over the last century. One important signature or fingerprint of this type of forcing is that the mid-Asia minus mid-Pacific differences over the last century are about 0.13°C. Hence, in this class of models there is a potentially significant land-sea contrast in the warming that can be exploited for pattern recognition. In greenhouse warming, land leads ocean by a potentially detectable amount. If warming is caused by internal variability of the oceans, we would expect ocean to be leading.
Next we consider a few explicit numerical examples in which the signals and noise are computed from nfEBMs (e.g., Kim and North, 1992; Kim et al., 1992).
We have measurements of the solar constant taken from several independent satellites over the last 12 years. There