1. Introduction

There are a number of different approaches presently being taken toward solving the problem of detecting forced climate change (Hasselmann 1979, 1993, 1997; Bell 1982, 1986; North et al. 1995; North and Kim 1995; Santer et al. 1995; Santer et al. 1996; Hegerl et al. 1996; Hegerl et al. 1997; Stevens and North 1996). The mathematical basis of these approaches share some commonality (Hegerl and North 1997), but the methods by which these techniques are applied and the data they are applied to differ widely among the various research groups. In the event that these different research groups conclude that evidence of climate change has been detected, or not detected, the credibility of their arguments will be enhanced by the use of different approaches to the problem. In fact, the acceptance of such a claim will only come gradually as a diversity of evidence accumulates.

Reluctance to accept the optimal signal detection theory framework stems largely from its mathematical complexity and resulting opaqueness to the research community. For an assertion about anthropogenically induced climate change to be convincing, it must be understandable by scientists in neighboring disciplines. The works of N. Wiener and later treatises on space–time signal processing are neither familiar nor are they easy reading for most physical scientists. One of our goals in this paper is to rederive the optimal detection procedure in a simple framework that will make it more accessible and intuitive. As so often happens after a laborious and rigorous derivation or proof, one finds a simple geometrical interpretation of the process that is more intuitive and leads to a greater understanding. This is the case here, as will be seen in later sections.

The most recent development in the climate signal detection problem is the inclusion of several possible sources of climate forcing. It has become apparent that in the case of anthropogenic climate forcing, one must include both the forcings due to greenhouse gases and to anthropogenic sulfate aerosols (Santer et al. 1995; Santer et al. 1996; Hegerl et al. 1997). Recent research using the GFDL climate models has found that the surface temperature response to these two climate forcings is linearly additive in both space and time (Haywood et al. 1997; Ramaswamy and Chen 1997). There are at least two other significant climate forcings to be considered (IPCC 1996). These are the naturally occuring climate forcings due to volcanic stratospheric aerosols and solar variability (10–11-yr solar cycle). We consider these four climate forcings to be deterministic because we have proxy information about the sources of these forcings and so can include them in our ensemble of simulations.

The signal detection problem is made more difficult by the inclusion of multiple deterministic signals amid a background of natural variability. To detect the climate signal-of-interest, a filter must be constructed, which will not only separate out the signal-of-interest from the background noise but also be able to discriminate the signal-of-interest from the other climate signals present. All of the climate signals considered here have some overlap in their space–time patterns or “fingerprints.”Any filter that allows the signal-of-interest to be passed through will also allow some component of the other signals to be passed through. This additional contaminating signal will be mistakenly identified as part of the signal that is being searched for. The approach we use to avoid this problem is to project out the component of the signal-of-interest, which is perpendicular to the signal of the other three combined climate forcings. This perpendicular component of the signal-of-interest is then used in the construction of the filter. The disadvantage of this approach is that the perpendicular component of the signal will be smaller than the total signal. This will result in some decrease in the signal-to-noise ratio and hence raise the threshold of detection.

As with all procedures of this kind, models play an important role. A model is used to generate a space–time waveform for each of the climate signals or combination of climate signals. Additional models are also used to calculate the optimal weights used in the construction of the optimal filter. The use of four different climate models allows us to make separate estimations of the signal strength. The different estimates are fairly consistent with one another indicating that the procedure is robust with respect to choice of climate model.

A byproduct of our procedure is the theoretical signal-to-noise ratio γ, which allows an assessment to be made of the performance of the procedure without use of the actual observed data. The square of the signal-to-noise ratio γ² can be written as a sum over spectral contributions from different space–time modes in the problem. By looking at the composition of this distribution, we can assess where we are gaining the most performance and where there is negligible contribution.

The main findings are that the volcanic aerosol and greenhouse gas signals have large γ and are highly significant, the anthropogenic aerosol signal has a small γbut is also very significant, and the solar variability signal has both a small γ and very low significance. The total signal obtained when all four climate forcings are combined is so significant (≈99.9% confidence) as to virtually rule out the possibility of its occurence arising from the noise of natural variability.

2. Combining two or more estimators

Consider estimating the temperature of a reservoir by using two measuring devices. Let the individual estimators be of the form T̂_i = T + ϵ_i, where T is the true temperature of the reservoir and ϵ_i is the random error in an individual estimate. The errors are assumed to have a Gaussian distribution with a mean of zero and to be independent (uncorrelated). These properties of the errors allow us to write 〈ϵ_i〉 = 0 and 〈ϵ_iϵ_j〉 = σ²_iδ_ij, where 〈··〉 represents the ensemble average, and σ²_i are the error variances of the individual measuring devices. The individual estimator T̂_i is unbiased, since 〈T̂_i〉 = 〈T + ϵ_i〉 = T + 〈ϵ_i〉 = T.

The optimal weight W, which will minimize the mse, can be found from the above equation by setting the derivative ∂ε²/∂W = 0 and solving for W:

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Substituting this result into (3) produces the optimal value of the mse:

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We define

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so that ε²_opt = 1/η². By substituting (4) into (1) we get the important result

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These results can be generalized for K independent unbiased estimators as

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and

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Equation (8) shows that adding another unbiased measuring device always decreases the mse no matter how poor its quality (how large its error variance). This simple derivation of optimal weighting of independent estimators is familiar to many atmospheric scientists. The result is very intuitive. We simply weight each estimator inversely according to its individual error variance.

The derivation presented above required that the individual errors be uncorrelated with one another. If this were not so, the coordinate axes could simply be rotated to the principal axes of the error or noise covariance matrix. Then the entire formalism goes through as before except in the rotated coordinate system. In climatology, this is the tranformation to the Empirical Orthogonal Function (EOF) basis set.

In two dimensions, this is easily spelled out explicitly. Let the covariance matrix of the noise be given by C_ij= 〈ϵ_iϵ_j〉. Taking the noise to be distributed bivariate normally, the contours of equal probability of occurrence of pairs of values of (ϵ₁, ϵ₂) are given by (Thiébaux 1994)

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which is an ellipse in the (T₁, T₂) plane, and where ρis the correlation coefficient (= 〈ϵ₁ϵ₂〉σ⁻¹₁σ⁻¹₂). In the two-dimensional case, we can find an angle through which to rotate the coordinate axes so that the principal axes of the ellipse will coincide with the new coordinate axes (T^′₁, T^′₂):

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In the new coordinate system, T^′₁ and T^′₂ are uncorrelated. In the case of K dimensions, the figure is an ellipsoid in the K dimensional space, and a simple length preserving rotation can also be used to find the appropriate coordinate system.

3. The fingerprint and optimal estimators

a. The state space

Consider a K-dimensional vector space, which is spanned by a set of orthonormal basis vectors {e₁, e₂, . . . , e_K}. In this state space, the basis vectors lie along the principal axes of a K-dimensional ellipsoid, which defines the Gaussian probability density function. In other words, the noise covariance matrix is diagonal in this state space and the basis vectors are the orthonormal eigenvectors of the covariance matrix. Any Gaussian random variable can be represented by its statistically independent components, which lie along the direction of the basis vectors.

In this state space we can define a noise vector as

Nn₁e₁n₂e₂n_Ke_K(12)

where the n₁, n₂, . . . , n_K are the random noise components, which will differ from one realization to another. Similarly, we can define a signal vector as

Ss₁e₁s₂e₂s_Ke_K(13)

The unit vector in the direction of the signal is written as

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where the magnitude of the signal vector is given by

Ss²₁s²₂s²_K^1/2(15)

Since this is a vector space, the usual vector operations are defined. We also assume that the data can be written as a linear sum of the signal and noise. We can then write a vector for the data:

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As an example of the vector dot product, we form e₁·D:

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which is an unbiased estimator of s₁.

For the climate signal problem, we need to define what we mean by the state space. Consider three stations whose temperature readings are T₁, T₂, and T₃. These readings form the components of a three-dimensional vector, and they define the state of that array of stations at a particular time. The variations will consist of a sum of forced and free contributions. In general, the natural variability part of the temperatures of the three will be correlated so that it will be necessary to rotate the coordinate system to the EOF (principal component) basis to use optimal estimation methods efficiently. We could consider the evolution of this three-vector in time. Its tip will move in the space due to the influence of climate noise and deterministic forcings. In our treatment, we go a step farther and use a spectral representation in time. Since the noise is assumed stationary in time, it is appropriate to use a Fourier basis set. In this case, for each Fourier frequency component, there will be a set of spatial EOFs. For example, the three stations may have different spatial patterns (correlation-free linear combinations) for each frequency. The state space for the climate problem then will consist of the Fourier components (sine and cosine) with each having a string of spatial EOFs attached. In the climate problem, which we will take up later in this paper, we will choose a frequency band containing eight discrete frequencies and 36 spatial boxes. This gives a total of 8 × 2 × 36 = 576 EOF components. In the Fourier representation, the state vector does not move in time but is stationary in an expanded space whose dimension depends on the frequency band chosen.

b. Fingerprint estimator of S

A common problem in climatology is that we know the shape of the signal waveform but not its amplitude. In other words, we know the direction of the signal vector (indicated by the unit vector e_S) and want to find its magnitude S. An unbiased estimator of S is the scalar product e_S·D. We show this for the two-dimensional case:

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since 〈d_k〉 = 〈s_k + n_k〉 = s_k and where the subscript FP indicates “fingerprint.” In other words, we project the data vector D along the direction of the signal vector S. This fingerprint estimator has an mse given by

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The above fingerprint estimator (e_S·D) can also be written as

Ŝ_FPe_Se₁e₁De_Se₂e₂D(21)

This can be generalized to K dimensions and rewritten as

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The fingerprint method is very easy to implement and has attracted some users. On the other hand, it does not take advantage of the fact that the individual variances in the mse might be quite different. Hence, we might want to weigh the information from the component directions differently to take advantage of the fact that one component might be more accurately estimated than another. The way to do this is presented in the next subsection.

c. Optimal estimator of S

Consider that we have a signal vector S and a data vector D, both with K components. For the kth component of S we can write

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For a two-dimensional example of this, refer to Fig. 2. Similarly, we can write for the kth component of D

e_kDd_ks_kn_k(25)

The kth unbiased estimator of S can then be written as

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since

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We have K statistically independent unbiased estimators of S, one for each component direction. The error variance of the kth unbiased estimator can be shown to be

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where 〈n²_k〉 = σ²_k is the noise variance in the kth component. We want to linearly combine these K estimators to give the optimal unbiased estimate of S. This is the same problem as trying to estimate the temperature of the reservoir in the previous section. In analogy with (9) we can write

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where the W_k are the optimal weights given by

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and the estimators Ŝ_k are given by (26). Hence, the optimal estimator of S can be written as

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Equation (31) can be rewritten as

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where Γ is defined as the optimal filter. In the last expression for Ŝ_opt, we show the data vector D factored out to emphasize that the procedure is a projection operation on the data vector D; hence, the term filter.Compare this equation with (22) for the fingerprint estimator.

To emphasize the connection between this derivation and the derivation presented in North et al. (1995), and that appears in Stevens and North (1996), we substitute (23) into (31) and (32), and (25) into (31) and after some rearranging get

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where α is the scaling factor (Stevens and North 1996), and γ² is the square of the theoretical signal-to-noise ratio:

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The scaling factor α is the ratio of the estimated magnitude of the signal in the data to the magnitude of the model signal. Figure 3 is a schematic diagram, which summarizes these ideas. In Stevens and North (1996), the summation over k in (34) and (35) refers to summation over the eigenmodes, while the noise variance σ²_k is replaced by the eigenvalue λ_k corresponding to the kth eigenmode.

Let us recall our assumptions. First and foremost, we assumed the linear additivity of the signal and the noise. We have used the principal component directions to formulate the problem from the beginning. That is, we chose the coordinate axes to be the principal axes of the covariance ellipsoid of the noise vector. We had to assume knowledge of the direction of the signal waveform. Our job is to estimate the strength of the signal in the data given this information and one realization of the data. The quantity γ = Sη is a good a priori measure of the quality of the procedure, since it is the theoretical signal-to-noise ratio. We can use γ² as computed with models to tell us which vector components are most important in the estimation problem without invoking the observed data. This is very important, since we can use our climate models to condition our choice of the subspace within which we can make a reliable estimation of the signal strength without involving the data itself (which might be considered cheating).

4. Interference by other signals

Table 2 lists the separation angles between the possible pairs of signals produced by the four climate forcings used in this study. Also listed are the angles between the individual signals and the signal resulting from the combination of the other three climate forcings. For example, the signal VAG is the signal that resultswhen the forcings due to volcanic aerosols, greenhouse gases, and anthropogenic aerosols are combined. As can be seen in Table 2, the angle between the solar signal S and the signal VAG is 80.9°. Thus, we might expect some contamination from the signal VAG when looking for S in the data. In Table 3, the magnitudes of the four signal vectors are listed along with the magnitude of the signal resulting from combining all four climate forcings. Notice that both V and G are much larger than S.The magnitude of SVGA is smaller than that of G because the forcings V and A act to cool the surface of the earth, while G acts to warm it. Thus, the vector sum S + V + G + A involves some cancellation of components.

5. Errors in the optimal filter

As can be seen from (32) and (33), the construction of the optimal filter Γ requires climate models to generate a signal (for e_S) and the background climate noise (for σ²_k). Since climate models are not perfect representations of reality, they will introduce errors into the detection problem. We now examine some possible errors in the construction of the optimal filter.

All filters have an inherent type of error variance that we call the filter “pass-thru” error. This error arises from the fact that some component of the background noise will always be passed through the filter because by chance it looks like the signal. This error has a theoretical value σ²_pass = 1/γ². The larger the γ of the model signal used to construct the filter, the smaller the pass-thru error. This type of error will exist no matter how closely the models can approximate reality.

Consider the error variance involved in the use of imperfect models in constructing the filter. The first type of error results from using an incorrect fingerprint. In the present context, this means the unit vector e_S has the wrong direction in the state space. An equivalent statement is that the direction cosines e_S·e_k are incorrect. The single constraint on these direction cosines is that their squares must add up to unity. An incorrect fingerprint can lead to a bias in the estimation of the signal strength. We call this type of error the “model bias”error, or σ²_bias.

Of course there are other sources of error that are difficult or impossible to estimate so this estimate of the total error should be considered as the minimum error estimate.

6. Application to 100 yr of temperature data

Stevens and North (1996) have analyzed 100 yr (1894–1993) of surface temperature data based upon 36 10° × 10° boxes distributed over the earth as shown in Fig. 4. The data is from the Jones dataset of monthly temperature anomalies (Jones and Briffa 1992). Each box results from averaging the data of four smaller 5° × 5° boxes. The boxes were chosen based on the continuity, representativeness, and distribution of the data. Each box has 1200 months of continuous data. In our analysis, we excluded the winter half-year of data in the extratropics and kept all 12 months of data in the Tropics as explained in Stevens and North (1996). In future work we intend to keep all months using a cyclostationary EOF basis set (Kim et al. 1996). The retained data form a 36-component multivariate discrete time series. In the absence of external forcing, the time series should be stationary as we have found using 1000-yr time series (control runs) from several coupled ocean–atmosphere GCMs.

By analyzing these long control runs, we were able to calculate the spatial EOFs for each Fourier frequency component (actually, we took the spatial factors to be the same over the narrow frequency band employed in the previous study as well as here). Since our original interest was in detecting the solar signal amid climate noise and other deterministic signals, we restrict our investigation to the “solar band,” which includes only the eight frequencies from 0.06 to 0.13 yr⁻¹. The corresponding timescales range from 16.67 to 7.69 yr. The band chosen here is purposely designed to exclude as much ENSO activity as possible (freq >0.13 yr⁻¹) because the climate models being used do not simulate ENSO very faithfully, if at all. In our previous study (Stevens and North 1996), we retained the high-frequency 0.14 yr⁻¹ harmonic. An additional advantage offered by the low-frequency cutoff is that the temporal eigenfunctions for the 100-yr segment are closer to sines and cosines if the frequency is high compared to the (segment length)⁻¹ (see the appendix of North and Kim 1995). See Fig. 5 for a summary of these considerations. A final important argument for using only the narrow band of frequencies in our study is that present coupled ocean–atmosphere GCMs may not be able to produce the low-frequency spectrum of fluctuations with any reliability. In the band of interest here, even a mixed-layer model appears to be adequate. Fluctuations in this band from the coupled GFDL model appear to be very similar to those in the corresponding mixed-layer model (Manabe and Stouffer 1996).

We compute the signals S, V, G, and A using an energy balance model described in Stevens (1997). Stevens (1997) also describes how the the individual climate forcings are specified in both time and space for use in the EBM. Figure 6 shows the global annual means of the climate forcings used in the EBM. Each frequency in the solar band has a real (cosine) and imaginary (sine) phase component. Combining the 36 spatial components (boxes) with the eight frequencies and the two phases yields a state space of 576 dimensions (36 × 8 × 2). Thus, the signal vectors S, V, G, and A each have 576 components in the state space.

7. Results

Our main goal is to find the optimal estimates of the strengths (magnitudes) of the signals S, V, A, and G. What we actually find is the ratio of the estimated strength of a signal in the data to the strength of the signal produced by the EBM (the scaling factor α). Figure 7 shows the Fourier spectral power (sums of sine squared and cosine squared components at each discrete frequency) for each of the four climate forcings in the frequency band ranging from 0.06 to 0.13 yr⁻¹. Note the steady decrease in power with increasing frequency in the greenhouse gas and the anthropogenic aerosol spectra, indicative of the Fourier transform of a ramp forcing. Of course, these vectors are nearly oppositely directed in the 576-dimensional space (see Table 2). Note also that the units along the vertical axes differ by a factor of 10. Another interesting fact is the amount of power at frequencies of 0.10 yr⁻¹ and 0.11 yr⁻¹ in the forcing spectrum of the volcanic aerosols. We assert that this comes about from the chance spacings of volcanic erruptions over the last 100 yr (Fig. 6). In the combined forcing spectrum, the largest peak is at 0.10 yr⁻¹; this is due to the volcanic aerosols, not to the solar variability, which is smaller by a factor of 10. This is indicative of the potential danger of not considering interfering deterministic signals. In Fig. 8 are shown the responses of the EBM to the individual climate forcings at a frequency of 0.10 yr⁻¹. The (real part)² refers to the real part (cosine) of the discrete fourier transform (DFT), and (imaginary part)² refers to the imaginary (sine) part of the DFT. This phase information is included in the optimal filter and helps to discriminate one signal from another. Notice the large peaks in the response to the volcanic aerosol forcing and the fact that the greenhouse response is predominately in the imaginary part and fairly uniform over the 36 boxes.

In the following sections, we present the results for the component of the signal-of-interest, which is perpendicular to the signal produced by the sum of the other three climate forcings. In this way we form an estimate of the strength of each individual signal without interference from the other signals. In Tables 4 through 8 we show the results for α and γ for each of four climate models used to calculate the optimal weights. EBM refers to a 10000-yr control run from our noise-forced EBM; GFDL (Geophysical Fluid Dynamics Laboratory)/ml refers to a 1000-yr control run from the GFDL mixed-layer GCM; GFDL/c refers to a 1000-yr control run from the GFDL coupled ocean–atmosphere GCM; MPI refers to a 1000-yr control run from the Max Planck Institute coupled ECHAM1/LSG GCM. In the tables, the column labled “Region” refers to the use of all 36 boxes (Global) or a subset of the 36 boxes (Fig. 4). “Tropics” indicates that the 20 boxes in the Tropics (30°N–30°S) were used in the calculations. “Extratropics” indicates that the 16 boxes outside of the Tropics were used. “N. Hem.” indicates that the 24 boxes in the Northern Hemisphere were used, and similarly for S. Hem., W Hem., and E. Hem. (see Stevens and North 1996). In each of these cases, we computed a new set of optimal weights.

Figure 9 shows the global values for the scaling factor α for the four models as well as the mean value of αcalculated from all the regions and models. The error bars represent the error estimate of ±σ_total from (40). If there were no signals present in the data then we would expect 〈α〉 = 0.0. In Fig. 10, we compare results for αcalculated using the EBM global optimal weights in the optimal filters. The results for α from the first column of Fig. 9 (EBM) are shown in Fig. 10 with horizontal error bars for each signal. The histograms of α values in Fig. 10 were calculated using 100 control runs of climate noise generated by the EBM as data input to each optimal filter. Since these control runs contain no signal we should expect a Gaussian distribution of αabout a mean of zero. Notice that the sharpness of the distributions increases with increasing γ, as they should. Also notice that the α values for the anthropogenic aerosol, volcanic aerosol, greenhouse gas, and combined forcing signals are well outside the distributions of the noise.

Finally, the results are summarized in Table 9, which shows the mean values of α and γ when averaged over all regions and models. The error in the estimate of α_meanis also given as the standard deviation of α_mean from a value of zero. This standard deviation can be represented as a confidence level in the estimate of α_mean, since we expect α to be a positive number and so can use a one-tailed test of the hypothesis that the signal is in the observed data.

f. Decomposition of γ²

The above results for the total deterministic signal raises a question about where the contributions to γ²are coming from. Is the large γ due to just the global mean? Consider the formula for the square of the signal-to-noise ratio:

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where T_n is the projection of the signal onto the nth EOF and λ_n is the eigenvalue corresponding to that EOF. The index n is really composed of two parts, the frequency component with real and imaginary phases and the spatial EOF index. We sum the two phase contributions and then display in a three-dimensional bar graph the contributions from the various terms in the last equation. Figure 11 show this decomposition of the total deterministic signal. Note that contributions are spread broadly across many spatial EOF modes. The main point of this analysis is that many EOF components contribute to the γ²; it is not simply the global average (first EOF).

8. Conclusions

In this paper we presented a derivation of the optimal filter, which is equivalent to earlier formulations, but conceptually much simpler because of its relation to the simple optimal weighting of independent estimators and the use of vector notation. A scheme was also developed for estimating individual signal strengths when they are embedded in a mixture of deterministic signals. The method was applied to a series of signals, which might be expected to appear in the observed data.

An important finding of this analysis is the occurrence in the climate-forcing spectrum of strong peaks at the frequencies of 0.10 and 0.11 yr⁻¹ due to the climatically significant volcanic eruptions over the last 100 yr (Fig. 7). In addition, the greenhouse gas forcing has significant power over the entire solar band of frequencies (0.06 to 0.13 yr⁻¹). Either of these two climate forcings is much larger than the solar-cycle forcing over the solar band.

The wider implication of these findings is that all four deterministic climate forcings need to be included in any signal detection effort. If in searching for one such signal the other signals are treated as simply part of the background climate noise, serious contamination of the results may occur. Since all of the signals have some commonality in their climatic space–time patterns, a detection method must be used that isolates the signal-of-interest from the influence of the other climate signals.

Using our method to isolate the individual climate signals, we cannot conclude with much confidence that the solar-cycle signal has been detected in the observed data. This does not rule out its detection in other datasets. However, we conclude that the climate signals due to volcanic aerosols, greenhouse gases, and anthropogenic aerosols have been detected in the observed data with a very high level of confidence (≥99.97%). It is important to note that even if our estimate of the total error variance given in (40) is underestimated by a factor of 2, the confidence level for the detection of the greenhouse gas signal in the observed data is only reduced to 99.93%. Finally, we conclude that the signal due to the combination of the four climate forcings considered in this study is present in the observed data at a confidence level of 99.9%. It should be pointed out that this detection method cannot distinguish between two climate forcings, which have the same space–time response pattern. Thus, any climate forcing that is similar to the greenhouse gas forcing over the past 100 yr would be mistakenly attributed to greenhouse gas forcing.