Climate Dynamics

, Volume 24, Issue 1, pp 45–55

Optimal filtering for Bayesian detection and attribution of climate change

Authors

    • Max Planck Institute for Meteorology
  • Kl. Hasselmann
    • Max Planck Institute for Meteorology
Article

DOI: 10.1007/s00382-004-0456-3

Cite this article as:
Schnur, R. & Hasselmann, K. Clim Dyn (2005) 24: 45. doi:10.1007/s00382-004-0456-3

Abstract

In the conventional approach to the detection of an anthropogenic or other externally forced climate change signal, optimal filters (fingerprints) are used to maximize the ratio of the observed climate change signal to the natural variability noise. If detection is successful, attribution of the observed climate change to the hypothesized forcing mechanism is carried out in a second step by comparing the observed and predicted climate change signals. In contrast, the Bayesian approach to detection and attribution makes no distinction between detection and attribution. The purpose of filtering in this case is to maximize the impact of the evidence, the observed climate change, on the prior probability that the hypothesis of an anthropogenic origin of the observed signal is true. Whereas in the conventional approach model uncertainties have no direct impact on the definition of the optimal detection fingerprint, in optimal Bayesian filtering they play a central role. The number of patterns retained is governed by the magnitude of the predicted signal relative to the model uncertainties, defined in a pattern space normalized by the natural climate variability. Although this results in some reduction of the original phase space, this is not the primary objective of Bayesian filtering, in contrast to the conventional approach, in which dimensional reduction is a necessary prerequisite for enhancing the signal-to-noise ratio. The Bayesian filtering method is illustrated for two anthropogenic forcing hypotheses: greenhouse gases alone, and a combination of greenhouse gases plus sulfate aerosols. The hypotheses are tested against 31-year trends for near-surface temperature, summer and winter diurnal temperature range, and precipitation. Between six and thirteen response patterns can be retained, as compared with the one or two response patterns normally used in the conventional approach. Strong evidence is found for the detection of an anthropogenic climate change in temperature, with some preference given to the combined forcing hypothesis. Detection of recent anthropogenic trends in diurnal temperature range and precipitation is not successful, but there remains strong net evidence for anthropogenic climate change if all data are considered jointly.

Introduction

The degree to which the human-induced climate change predicted by models has been detected and verified today by observations remains a topic of debate. The cautious statement in the Second Assessment Report (Houghton et al. 1996) of the Intergovernmental Panel on Climate Change (IPCC) that “the balance of evidence indicates a discernible human influence on climate” has been reaffirmed and strengthened in the Third Assessment Report (Houghton et al. 2001, see also status reports of the International ad hoc Detection and Attribution Group, Barnett et al. 1999, and IDAG 2004). The global-scale climate change observed in recent decades is indeed difficult to attribute to natural causes. But it is less clear whether climate changes observed on the regional scale, such as rainfall patterns, droughts, storms and other extreme events, which are likely to have the strongest impact on human living conditions, can be attributed to human activity. For these data, as well as for a number of global climate change indices, the time series are too short to establish reliable natural variability statistics, which are needed for conventional statistical detection and attribution tests. Therefore, natural variability is usually estimated from long model control simulations but this approach faces the problem of considerable differences in the simulated variability of different climate models (see Barnett 1999). These differences will be even larger for many non-temperature variables or indices. The errors of the climate models used to predict the climate change signals, against which the observed climate change data are compared, are similarly difficult to estimate. For these reasons, only a subset of the available climate change data, primarily surface temperature and zonally averaged vertical temperature data, have been applied so far for conventional statistical detection and attribution analyses (Hegerl et al. 1997; Santer et al. 1996; Tett et al. 1999; Stott et al. 2001).

To overcome these statistical data base limitations, Bayesian techniques have been proposed (Varis and Kuikka 1997, Hasselmann 1998, Risbey et al. 2000). By introducing subjectively defined probabilities, the Bayesian approach not only extends the permissible data base, but also provides a scientific foundation for investigating the implications of different subjective estimates of the errors of models and data, or of different prior beliefs in the validity of competing climate change forcing hypotheses (see Earman 1992).

As preparation for a detailed Bayesian detection and attribution analysis applied later to a more comprehensive climate change data set, we consider in this study an important technical aspect of the Bayesian approach: how should one weight the data in order to maximize the impact of the data on the probability that a given climate change forcing hypothesis is true?

In the conventional statistical approach to the detection and attribution problem, the data are projected onto a suitably chosen low-dimensional sub-space in order to enhance the signal-to-noise ratio (S/N). Without a reduction in the dimensionality of the climate phase space, the signal is normally swamped by the natural variability noise. The patterns used for the projection (known as fingerprints) can be chosen such that S/N is maximized (Hasselmann 1979, 1997). However, the optimal fingerprint method of the conventional detection and attribution approach cannot be immediately carried over to the Bayesian case, as here the detection and attribution problem is formulated quite differently.

In the conventional approach, the hypothesis of a climate change signal induced by human activity (or some other given external forcing mechanism) is compared against the null hypothesis that the observed climate change is due to natural climate variability. One estimates first the probability p0 that a climate change signal that is as least as large as the observed signal could have occurred within a given observation period through natural climate variability. If p0 is small, a signal is said to have been detected at a significance level of 1−p0 (or, in an alternative terminology, p0). Once detection has been established, one investigates further whether the climate change signal inferred from the observations can be attributed to the postulated forcing mechanism, i.e. whether the predicted and inferred signals are mutually consistent within the estimated error bounds of the observations and the model prediction. The errors of the predicted signal are relevant only for attribution; for the detection test, their only impact is a minor degradation of the detection skill through a distortion of the computed fingerprint pattern relative to the true optimal pattern.

In the Bayesian approach, in contrast, there exists no formal separation between detection and attribution, and the uncertainties of the predicted signal and the natural climate variability play an equally important role for both questions. The observed climate change is treated as new evidence that modifies an initial, subjectively defined probability of the validity of a given hypothesis regarding the cause of the climate change. The evidence changes the initial probability prior to knowledge of the evidence (the “prior”) into a posterior probability (the “posterior”). The modification depends not only on the prior, but also on the ratio of the likelihood of making the relevant observations for the case that the forcing hypothesis is true or, alternatively, false (i.e. that the observations are due to natural climate variability or some other postulated forcing mechanism). The conceptual framework of the Bayesian approach therefore differs fundamentally from the conventional approach, and we shall indeed show that the solution for the optimal filtering or weighting of the observational data is quite different for the two cases.

In the following section we review briefly the basic relations required for the Bayesian analysis. In Sect. 3 and 4 we derive then the optimal data filtering results for the Bayesian case, while Sect. 5 illustrates their application for the example of global mean surface temperatures, summer and winter diurnal temperature ranges and precipitation. The principal conclusions of our analysis are summarized in the last section.

Bayes factors

We consider an ensemble of climate change data consisting of sets of individual indices and/or fields of space-time dependent variables. Following the compressed notation of Hasselmann (1998), we denote the complete set of space-time discretized climate change variables as the vector v. Thus the index i of the vector component vi runs over all climate change variables, as well as over all discretized space and time co-ordinates of the variables, which can be defined at a set of stations or gridpoints.

The covariance matrix of the time-lagged, spatially dependent second moments of the natural variability of the variables is denoted by
$$ C_{ij} = \left\langle {\tilde v_i \tilde v_j } \right\rangle , $$
(1)
where \({\tilde v_i }\) represents the natural climate variability, 〈...〉 the ensemble average, and we assume that all variables vi are defined relative to the ensemble mean, so that \(\left\langle {\tilde v_i } \right\rangle = 0.\)

Although we focus in the following on the set of climate change trajectories in which the time variable is absorbed in the comprehensive vector index i, all relations derived in the following apply formally also to the case that v is regarded as a set of time-dependent variables, v=v(t), where v(t) can consist either of the original data or a derived data set obtained from these by some time-dependent filtering operation (for example, running trends defined over some time interval). The following relations apply then at a given instant in time.

In addition to the natural variability \(\mathbf{\tilde v}\) we consider also the model-related errors v* in the predicted climate change signal vp,
$$ {\mathbf{v}}^p = {\mathbf{v}}^s + {\mathbf{v}}^* $$
(2)
where vs is the (unknown) true climate change signal due to some hypothesized forcing mechanism. The model prediction errors v* can be characterized by their first and second moments. We may set the first moments equal to zero, 〈v*〉=0, since a known bias in the model prediction can be corrected. The second moments are described by the model error covariance matrix
$$ C_{ij}^* = \left\langle {v_i^* v_j^* } \right\rangle . $$
(3)
This is difficult to estimate reliably using conventional statistics and must therefore be based to a large part on subjective Bayesian estimation methods (see discussion in Sect. 4).
The impact of observing a climate change signal v0, the “evidence” e, given a predicted signal vs for some anthropogenic (or other external-forcing) climate change hypothesis H, is given by Bayes’ celebrated theorem:
$$ p(h|e) = \frac{{p(e|h)}} {{p(e)}}p(h), $$
(4)
where p(h) denotes the prior probability that the hypothesis is true, H=h= ”true”, p(h|e) the posterior probability that the hypothesis is true (given the evidence e), and p(e) is the likelihood of obtaining the evidence (making the observations), independent of whether the hypothesis H is true, H=h, or false, \(H = \bar h = {\text{``false''}}.\) Equation (4) may be rewritten, in the notation of Hasselmann (1998), as
$$ \begin{aligned} c = & c_0 l/\{ c_0 l + (1 - c_0 )\hat l\} \\ = & (1 + \beta \hat l/l)^{ - 1} \\ \end{aligned} $$
(5)
where
$$ \beta = \frac{{1 - c_0 }} {{c_0 }}, $$
(6)
c (for “credibility”) =p(h|e), c0=p(h), and the net probability p(e) has been decomposed into the two conditional likelihoods l=p(e|h) and \(\hat l = p(e|\bar h)\) of obtaining the evidence for the case that the hypothesis is true or false, respectively, using the relation
$$ p(e) = p(e|h)p(h) + p(e|\bar h)p(\bar h) = lc_0 + \hat l(1 - c_0 ). $$
(7)
Thus, the relevant parameter determining the impact of the evidence e on the conversion of the prior c0 into the posterior c is the likelihood ratio \(\hat l/l.\)
Instead of considering the likelihood ratio for the case that the hypothesis H is true, H=h, or false, \(H = \bar h\) (i.e., that the complementary hypothesis \({\bar H}\) is true), we can consider generally the impact of the evidence on two arbitrary hypotheses H1, H2 (for example, the hypotheses that an observed climate change can be explained by greenhouse gas emissions alone, H1=h1 or, alternatively, by a combination of greenhouse gas emissions and aerosols, H2=h2, see Hegerl et al. 1997). From Eq. (4) we obtain
$$ \frac{{p(h_1 |e)}} {{p(h_2 |e)}} = \frac{{p(e|h_1 )p(h_1 )}} {{p(e|h_2 )p(h_2 )}}. $$
(8)
Thus the ratio of the posterior probabilities of the two hypotheses is given by the ratio p(h1)/p(h2) of the priors multiplied by the likelihood ratio, or Bayes factor (see Kass and Raftery 1994)
$$ B = \frac{{l_1 }} {{l_2 }} = \frac{{p(e|h_1 )}} {{p(e|h_2 )}}. $$
(9)
The fact that the likelihoods enter in Bayes’ theorem, Eqs. (4, 5) or the derived relation (8), only in the form of ratios provides a conceptually consistent, local description of the impact of the observations. As only probability ratios are involved, one need consider only probability densities defined at the single point in phase space at which the observations have actually been made. In contrast, in the conventional detection method, only a single hypothesis, the null hypothesis, is tested, and one cannot therefore form ratios of probability densities. Since the probability of observing a climate change signal in the infinitesmal phase space element corresponding to the observed climate change at a given point in phase space small is infinitesmally small, the value of the probability density \({\bar l}\) at that point alone is not a useful quantity. To obtain a meaningful finite probability measure, one needs to integrate the probability of observing a signal over some finite phase space region not directly related to the point in phase space at which the observations were made.

We note also that in Bayes’ approach one can consider the impact of the evidence e on the credibility of several alternative anthropogenic climate change hypotheses hj simultaneously (see Smith et al. 2003 for another application of Bayes Factors to different climate change hypotheses). As discussed in the following, the relevant likelihood densities lj=p(e|hj) can be computed in a relatively large common climate change phase space, without restriction, as in the conventional approach (see Hasselmann 1997; Hegerl et al. 1997), to a strongly reduced sub-space spanned only by the signal patterns of the hypotheses being tested.

Optimal filtering in the Bayesian framework

Since the Bayesian detection and attribution analysis can be carried out, in principal, in the full observational climate-change phase space v, one may question whether data filtering or weighting techniques as in the conventional approach are needed. The optimal fingerprint technique in the conventional approach serves two purposes: it enhances the signal-to-noise ratio by reducing the dimension of the space in which the detection test is carried out, and it maximizes S/N by choosing a suitable filter that suppresses the noise relative to the signal.

The first motivation no longer holds for the Bayesian case. As pointed out, in the conventional analysis, the computation of the detection significance level requires integrating the natural-variability probability density over some finite phase space region, specifically over all climate change values greater than the observed climate change signal (“greater” being defined with respect to a metric determined by the inverse covariance matrix of the natural variability). The integral increases rapidly with the number of dimensions of the phase space. It is smallest for a one-dimensional phase space obtained by projecting the full phase space onto a single detection variable (in the optimal case, using a fingerprint pattern derived through a suitable rotation of the predicted signal pattern). Thus the significance level for detection can be significantly enhanced by decreasing the number of phase space dimensions. In the Bayesian case, in contrast, the impact of the “evidence” e depends only on the ratio of likelihood densities, so that the dependence on the phase-space dimension cancels.

However, the second motivation, the maximization of the signal-to-noise ratio, remains. For example, in the case that a single external-forcing climate change hypothesis H is to be tested against the null hypothesis \(\bar H,\) one would like to maximize the Bayes factor \(B = l/\ifmmode\expandafter\hat\else\expandafter\^\fi{l} = p(e|h)/p(e|\ifmmode\expandafter\bar\else\expandafter\=\fi{h}).\) We show below that this can be achieved through a suitable filtering operation. This leads also to dimensional reduction, but the reduction is in general less severe than in the conventional case. In contrast to the conventional optimal fingerprint technique, the filter depends not only on the natural variability covariance spectrum but, as to be expected from Eq. (5), also on the covariance matrix Cij* of the model-prediction errors.

The dimensional reduction follows from the application of a selection/rejection criterion to the pattern components in a transformed phase space. The transformation is composed of three successive transformations:

  1. 1.

    A rotation c=Uv to empirical orthogonal functions (EOF co-ordinates) c defined with respect to the natural variability noise. This diagonalizes the covariance matrix C of the natural variability, \(C \to E = (\hat \sigma _1^2 ,\hat \sigma _2^2 , \ldots ,\hat \sigma _n^2 ).\) In practice, this transformation will normally be combined with a truncation: one retains only those EOFs that can be distinguished from white noise in accordance with the criterions, for example, of North et al. (1982) or Preisendorfer and Overland (1982). Thus one works in a reduced computational space that is typically of the order of 10 to 20.

     
  2. 2.
    A rescaling of the EOF co-ordinates c through a diagonal transformation x=Sc, in components, \(x_i = c_i /\hat \sigma _i ,\) such that the natural variability covariance matrix becomes the unit matrix. The probability distribution of the natural climate variability, which we assume to be Gaussian, is thereby transformed into the normalized isotropic form
    $$ \hat p(x) = (2\pi )^{ - n/2} {\text{exp}}\left\{ { - \frac{1} {2}\sum\limits_{i = 1}^n {x_i^2 } } \right\} $$
    (10)
     
  3. 3.

    Finally, the co-ordinates are rotated again, yielding variables z=V(x), such that the covariance matrix Cij* of the model errors is diagonalized.

     
The net transformation T=VSU leads to EOF co-ordinates z defined with respect to the model error covariance matrix C* The model error probability distribution, which we assume again to be Gaussian, is accordingly given by
$$ p({\mathbf{z}}) = (2\pi )^{ - n/2} \left\{ {\prod\nolimits_{i = 1}^n {\sigma _i } } \right\}^{ - 1} {\text{exp}}\left[ { - \frac{1} {2}\sum\limits_{i = 1}^n {\frac{{(z_i - z_i^p )^2 }} {{\sigma _i^2 }}} } \right]. $$
(11)
The predicted signal in the coordinate system z is given by zp=V(xp).
The normalized isotropic form (10) of the probability distribution of the natural variability remains invariant under a rotation and thus retains the same form with respect to the final coordinate system z:
$$ \hat p({\mathbf{z}}) = (2\pi )^{ - n/2} {\text{exp}}\left\{ { - \frac{1} {2}\sum\limits_{i = 1}^n {z_i^2 } } \right\} $$
(12)
We filter now the phase space such that the Bayes factor \(l/\hat l,\) which compares the anthropogenic climate change hypothesis against the Null hypothesis, or, equivalently \(\ln (l/\hat l),\) is maximized. The maximization is undertaken for the ideal case that the predicted climate change signal is the same as the true climate change signal, and both in turn are equal to the observed signal: vp = vs = v0, or zp=zs=z0.

If the hypothesis of an externally forced climate change signal is true, the likelihood l that the observed climate change signal lies in an infinitesimal phase space element Δz at the point z = zp is p(zpz. If the complementary hypothesis that the observed climate change signal is due to natural climate variability is true, the likelihood \({\hat l}\) that the observed climate change signal v lies in the transformed infinitesimal phase space element Δz at the point z=zp is \(\hat p({\mathbf{z}}^p )\Delta {\mathbf{z}}.\)

Thus, for the assumed situation for which we wish to maximize the logarithm of the Bayes ratio, we have
$$ \ln (l/\hat l) = \ln l - \ln \hat l = \ln p({\mathbf{z}}^p ) - \ln \hat p({\mathbf{z}}^p ) = \sum\limits_{i = 1}^n {\left( {\frac{{(z_i^p )^2 }} {2} - \ln \sigma _i } \right)} . $$
(13)
The maximum ratio \(l/\hat l\) is accordingly obtained by retaining only those pattern components in the transformed phase space z for which
$$ (z_i^p )^2 > 2\ln \sigma _i . $$
(14)
The Bayesian detection and attribution analysis is carried out in this reduced phase space. The number of dimensions retained in the reduced space is seen to depend both on the signal-to-noise ratio of the relevant pattern component, expressed by the left hand side of the inequality (14), and on the rms error of the predicted signal pattern, given by the right hand side. For large signal-to-noise ratios and small prediction errors, the dimension of the reduced space remains rather large. In practice, the dimension is then limited by the number of EOFs that can be reliably determined from a finite data sample (see Preisendorfer and Overland 1982; North et al. 1982; Hegerl et al. 1997; Allen and Tett 1999).
The present result can be compared with the conventional detection approach. In this case one considers only the natural climate variability noise, and carries out only the first two transformations T′=SU leading to the normalized isotropic Gaussian distribution, Eq. (10), for the variable x. If only a single externally forced climate change hypothesis is tested, the maximal signal-to-noise ratio for the case that the observed climate change signal x0 is the same as the predicted signal xp is obtained by projection onto a single detection variable d, using a single optimized fingerprint pattern. From isotropic symmetry (see Hasselmann 1997) it follows that the projection must be given by d=xpTx0, or expressed in EOF co-ordinates,
$$ d = \sum\limits_{i = 1}^n {c_i^p c_i^p /\hat \sigma _i^2 } . $$
(15)
Thus all EOFs are linearly superimposed, after weighting with the inverse of their associated variances, to construct a single fingerprint pattern. In contrast to the Bayesian case, the uncertainty of the model predictions does not enter in the analysis, and only a single projection pattern is finally used.

We note that in both the Bayesian and the conventional case, the filtering is carried out using only prior information on the estimated natural variability, the predicted signal and, in the Bayesian case, the signal uncertainty. No reference is made to the evidence, the observed climate change, which would introduce a statistical bias.

Estimation of Cij*

The model error covariance matrix Cij* consists generally of three contributions: sampling errors due to the internal variability of the model, inherent model errors and errors arising from inter-model differences in the applied forcings. For a realistic model, the internal model climate variability is similar to the real internal climate variability. It can be estimated from the computed model variability of a long control run and/or from a number of integrations of a given climate change scenario using perturbed initial conditions (see Cubasch et al. 1994; Tett et al. 1999). The inherent model errors and forcing-related errors can be judged by intercomparisons of model runs using different models (although systematic errors common to all models cannot be detected in this manner). Since the number of model runs will normally be limited, and systematic model errors can be assessed only subjectively from general model performance, the prescription of the model error covariance matrix C* will necessarily be somewhat subjective. Nevertheless, assuming that the spread in published scenario simulations using different models provides some measure of model and forcing errors (i.e. neglecting an unknown common model error), one can estimate C* directly as the sum of the three contributions as follows:

From m scenario computations, we obtain m estimates va, a = 1,...,m of the climate change signal. Different Monte Carlo simulations with a single model can be counted either separately or averaged into a single run. In the first case, the relative contribution of individual model errors is underestimated, while the second case underestimates the relative contribution of the natural variability. A detailed discussion requires a separate treatment of both contributions, but this will not be pursued here. Some degree of arbitrariness in the weighting, equal or differentiated, of the errors of different models is inevitable, as different modellers will invariably have different assessments of model errors. Indeed, this is one of the motivations for applying Bayesian statistics.

We define the predicted signal as the mean of the individual model predictions:
$$ {\mathbf{v}}^p = \frac{1} {m}\sum\limits_{a = 1}^m {{\mathbf{v}}^a } . $$
(16)
The deviation of an individual model prediction from the mean is then given by
$$ {\mathbf{v}}^{a^* } = {\mathbf{v}}^a - {\mathbf{v}}^p . $$
(17)
To estimate C*, we assume that the full climate-change phase space has been reduced, as discussed above, to a smaller computational space spanned by the first n EOFs pi, i = 1,...,n, typically 10 to 20, of the natural climate variability. When represented in this reduced space, the deviations of the individual predicted climate change signals from the mean are given by
$$ {\mathbf{v}}^{a*} = \sum\limits_{i = 1}^n {c_i^a {\mathbf{p}}^i + \varepsilon ^a } $$
(18)
where εa is the residual error of the truncated expansion, which we ignore.
The matrix C*=Cij* can then be estimated in this truncated EOF space, treating the model runs as independent realizations of a statistical ensemble, as
$$ C_{ij}^* = \frac{1} {m}\sum\limits_{a = 1}^m {c_i^a c_j^a } $$
(19)
(or by the same expression with the factor 1/m replaced by 1/(m−1) to remove the bias incurred by the finite number of realizations).

This straightforward approach will normally run into the problem, however, that the number of independent model experiments is too small to span the selected EOF space. Thus, the estimate Eq. (19) of the model error covariance matrix Cij* will generally be degenerate and will need to be augmented by subjective assessments of the general structure of Cij*. These can be inferred from estimates of the model natural variability derived from long control runs, or from plausible assessments of model errors based on the models’ general performance in reproducing the present climate.

Application

As a simple demonstration and test exercise, we have applied the Bayesian detection and attribution framework as developed to the annual means of near-surface temperature (TAS) and precipitation (PREC) and the summer and winter means of diurnal temperature range (DTR-JJA and DTR-DJF). These variables are sufficiently well covered by both observations and climate model ensemble simulations to permit the straightforward application to the algorithm outlined in Sect. 2 and 3. The Bayesian analysis will be compared with results from the conventional detection method.

We consider three hypotheses:
  • H1 the observed climate change is due to changes in the greenhouse gas (GHG) forcing

  • H2 the observed climate change is due to a combination of greenhouse gas plus sulfate aerosol (GS) forcings

  • H0 the observed climate change can be explained by natural variability alone

Only the direct effect of sulfate aerosols is considered in H2, since there exist only a few model integrations that include both direct and indirect effects. Note that our external forcing hypotheses are not complimentary, in contrast to the conventional detection case, in which one tests always a given hypothesis (externally forced climate change) against the complementary null-hypothesis (natural variability). For illustration we have limited the analysis to a few selected examples, excluding a number of further climate change hypotheses discussed in the literature, such as the volcanic eruption or solar forcing hypotheses.

Although we have termed the hypotheses H1 and H2 for brevity as anthropogenic forcing hypotheses, we note that they represent in fact a superposition of anthropogenic forcing and natural variability. The effect of natural variability has been automatically incorporated in the estimation of the likelihood of observing the predicted anthropogenic climate change, which is determined both by the model uncertainty and the natural climate variability that is superimposed on the anthropogenic signal.

For each of the two anthropogenic hypotheses H1 and H2, a number of (ensemble) simulations from different models were used to estimate the predicted climate change signals vp and the model-error covariance matrix C*. The model data were obtained from the IPCC Data Distribution Centre 1. Details of the models used and the number of ensemble integrations available for each variable and forcing scenario are given in Table 1.
Table 1

Models and ensemble integrations used to estimate the predicted climate change signal and the model error covariance matrix. Ensemble sizes are given for near-surface temperature (TAS), precipitation (PREC) and diurnal temperature range (DTR), for the greenhouse gas-only scenario (GHG) and the greenhouse gas plus sulfate aerosol scenario (GS)

Model

Center

Size of ensemble

 

 

TAS

PREC

DTR

GHG

GS

GHG

GS

GHG

GS

ECHAM3/LSG

MPIfM, Germany

4

2

1

2

1

2

ECHAM4/OPYC3

MPIfM, Germany

1

1

1

1

1

1

HADCM2

Hadley Centre, UK

4

4

4

4

4

4

HADCM3

Hadley Centre, UK

1

0

1

0

0

0

GFDL-R15a

GFDL, USA

2

1

2

1

0

0

CSIRO-Mk2

CSIRO, Australia

1

1

1

1

1

1

CGCM1

CCCma, Canada

1

3

1

3

1

3

Total

14

12

11

12

8

11

Natural climate variability \(\tilde {{\mathbf{v}}}\) was estimated from the ECHAM3/LSG 2000-year control run. Observed annual (temperature and precipitation) and seasonal (DTR) averages were derived from monthly gridded fields of near-surface temperature (Jones et al. 1999), precipitation (Dai et al. 1987) and DTR (Easterling et al. 1997). All model data were interpolated to the resolution of the observations (5°× 5° for surface temperature, 2.5°× 2.5° for precipitation and DTR) and subjected to an observational mask accounting for missing values in the observations. Note that, in this application, we treat each of the climate variables individually rather than jointly. Rather than using the full space-time fields of recent climate evolution (see Allen and Tett 1999) the temporal change is represented by the latest 31-year trends of each variable, as in Hegerl et al. (1996, 1997). Thus, all data v (observations, natural variability and scenario runs) that enter the detection algorithm represent spatial patterns of the trends for the latest available 31-year observational period (first row in Table 2). As pattern basis we used the first 20 EOFs of the overlapping 31-year trend patterns of the long ECHAM3/LSG control run. We then used the “rule of thumb” of North et al. (1982) to determine the number n of EOFs used in the transformations described in Sect. 3. The second row of Table 2 lists the number of EOFs used for each variable, i.e. the dimension of the phase space in which the likelihoods are computed for the optimization procedure. The results are relatively insensitive to the choice of n.
Table 2

Application of the optimal filtering algorithm for Bayesian detection and attribution to the annual means of surface temperature (TAS) and precipitation (PREC), and winter (DJF)/summer (JJA) means of diurnal temperature range (DTR). See text for details

 

TAS

DTR DJF

DTR JJA

PREC

Net

Latest observed 31-year trend covers

1967-97

1964-94

1964-94

1965-95

Number n of EOFs retained from control run

14

16

11

16

(a) Greenhouse gas forcing vs. Natural

Number of patterns selected by Bayesian criterion

13

9*

6*

9

Bayes factor (B10)

>100

0.36

0.11

0.6

>100

log 10(B10)

>2

−0.44

−0.96

−0.22

>2

(b) Greenhouse gas plus sulfate forcing vs. Natural

Number of patterns selected by Bayesian criterion

13

9*

6

9

Bayes factor B20

>100

0.52

0.05

0.46

>100

log 10(B20)

>2

−0.28

−1.33

−0.33

>2

(c) Greenhouse+sulfate hypothesis vs. Greenhouse hypothesis

Bayes factor B21

2.1

1.44

0.43

0.77

1.001

log 10(B21)

0.3

0.16

−0.37

−0.11

0.0004

To estimate the model error covariance matrix Cij*, we averaged first the Monte Carlo realizations of individual model ensembles and then regarded this set of averaged realizations as the basic model ensemble from which Cij* was computed in accordance with the relations given in Sect. 4. As pointed out, this captures the inter-model differences, but underestimates the contribution from the natural model variability. However, as the number of Monte Carlo simulations available for any given model was generally too small for a reliable direct estimation of the uncertainty of the model predictions arising from the model’s natural variability, we estimated this contribution indirectly by assuming that the natural variability of each model could be represented by the natural climate variability \(\tilde v,\) as determined from the long ECHAM3/LSG 2000-year control run. Although there is considerable disparity between the surface temperature variability of control runs of different climate models (see Barnett 1999), the control runs of the models considered here are in broad terms mutually consistent and reproduce the observed natural climate variability reasonably well, considering uncertainty in both the model simulations and the observations. The variability of the ECHAM3/LSG control run appears to somewhat underestimate natural variability in surface temperature (see Stouffer et al. 2000), this should be kept in mind when interpreting the following results. A more complete evaluation of the variability in control runs of different models, including the variability for precipitation and DTR for which no systematic model intercomparisons are available, is beyond the scope of this study.

In x-space, i.e. after diagonalization and normalization with respect to the natural climate variability \(\mathbf{\tilde v}\) the contribution from the models’ natural variability to Cij* is then given by diag(K,...,K), where \(K = \frac{1} {k}\sum\nolimits_{i = 1}^k {\frac{1} {{k_i }}} ,\)k is the number of model ensembles used and the factors 1/ki represent the effect of averaging ki realizations of model i in computing the covariance of the ensemble average due to natural variability. For consistency, we used the same model error covariance matrix for testing both the GHG and the GS hypothesis, computing Cij* from the pooled ensembles of GHG and GS experiments. This ensures that, after the transformation T has been applied, all likelihoods can be computed in the same phase space allowing a direct comparison of hypotheses H1and H2 below. The results of the optimal Bayesian filtering algorithm are summarized in Table 2. Sections (A) and (B) show the number of patterns that pass the optimal selection criterion of Eq. (14) for the two anthropogenic forcing hypotheses H1 and H2, and the associated Bayes factors obtained by computing the likelihoods in the reduced phase space spanned by these patterns only. The number of patterns retained is quite high, considerably higher than the one or two patterns typically used in conventional detection and attribution analyses. However, the phase-space dimension is nevertheless reduced compared with the dimension n of the original space spanned by the control-run EOFs (Fig. 1).
Fig. 1

Number of control-run EOFs spanning the base space in which the analysis is performed (left bars) and number of optimal patterns for the Bayesian and conventional detection and attribution analysis, for near-surface temperature (TAS), winter (DJF) and summer (JJA) diurnal temperature range (DTR), and precipitation (PREC).

Kass and Raftery (1994) have proposed guidelines for the translation of Bayes factors, i.e. the formal computation of the impact of evidence on the odds of one hypothesis over the other, into qualitative “evidence levels”. They suggest the four evidence levels listed in Table 3.
Table 3

Interpretation of Bayes factors (after Kass and Raftery 1994)

log 10(Bk0)

Bk0

Evidence in favour of Hk over H0

0 to 1/2

1 to 3.2

Weak (“not worth more than a bare mention”)

1/2 to 1

3.2 to 10

Substantial

1 to 2

10 to 100

Strong

>2

>100

Decisive

Following these guidelines, a Bayes factor larger than 100, for example, indicates that there is decisive evidence in favor of one hypothesis over the other. We have used a threshold value of 100 for the Bayes factors to avoid spuriously high likelihood ratios, which are unreliable. The n-dimensional probability densities become very small, and the likelihood ratios correspondingly large, if the observation vector lies well outside the confidence ellipsoids of model errors or natural variability, particularly for high phase-space dimensions. Such high likelihood ratios are critically dependent on the Gaussian hypothesis, which is questionable on the limbs of the probability distributions.

The results show that both the GHG and the GS temperature signal can be detected with decisive evidence against natural variability. However, neither of the anthropogenic forcings can be detected in the DTR or precipitation signals. These results are consistent with the findings from an application of the conventional fingerprint detection technique to the ECHAM4/OPYC3 signals and ECHAM3/LSG natural variability (Schnur 2004, Detection of Climate Change Using Advanced Climate Diagnostics: Seasonal and Diurnal Cycle, submitted to Meteorol. Z.). While the temperature signal was found to be highly significant for both forcings, the precipitation and summer DTR signals could not be detected. In the case of winter DTR, the signal passed the conventional detection test for both forcings, but failed the subsequent attribution tests. This is also in accordance with the Bayesian result, in which detection and attribution is combined in a single analysis.

The last column in Table 2 shows the net results for all four variables jointly, obtained by multiplying the individual likelihood ratios (Bayes factors) under the assumption that the different likelihoods are independent. This overestimates the net impact of different observational data, but provides nevertheless a qualitative indication of the advantage of combining different sources of data in Bayesian detection and attribution studies. A comprehensive treatment would need to consider the correlations between different data types for model errors and natural variability in a joint analysis of all variables. Despite the low likelihood ratios for DTR and precipitation, there is still seen to be strong net evidence for detection of the GHG and GS forcings.

The ratio of the Bayes factors B20 and B10 yields also the Bayes factor B21 describing the impact of the evidence on the odds of hypothesis H2 over H1 (see Eq. 9). Thus, B21 represents the impact of the observations on the attribution of the observed climate change to the two alternative anthropogenic forcing mechanisms. In computing B21 as the ratio of B20 and B10, a technical inconsistency can arise if different optimal patterns were selected in the computation of B20 and B10. In this case, the likelihood for the H0 hypothesis, which formally cancels out in the ratio, is computed in two different phase spaces. To avoid this inconsistency, we used only those optimal patterns that fulfilled the Bayesian selection criterion (Eq. 14) in the computations of both B20 and B10, so that the results of sections (A), (B) and (C) in Table 2 are based on the same phase space. This resulted in the omission of one optimal pattern for the cases marked with a star in Table 2.

Section (C) of Table 2 indicates that, for temperature, the likelihood for the GS hypothesis is about twice as high as the likelihood for the GHG hypothesis. This is consistent with the conventional findings, most of which support higher attribution levels for the GS than the GHG hypothesis (Hegerl et al. 1997; Barnett et al. 1999; IDAG 2004). However, a Bayes factor B21 of 2.1 represents only very weak evidence (Table 3). Also, for the net result for all observations, the likelihood ratio for both anthropogenic hypotheses is almost exactly unity. Compared with the conventional analysis, the explicit inclusion of model uncertainties in the Bayesian analysis tends to downgrade the higher attribution level for the combined greenhouse-gas-plus-aerosol forcing relative to greenhouse-gas-only forcing. This is consistent with the considerable scatter found between different models in the attribution significance level for the combined forcing using the conventional approach (Barnett et al. 1999). The scatter arises, in addition to uncertainties in the forcing itself, from the large uncertainties in the model response to sulfate aerosol forcing, an effect which is automatically included in the Bayesian analysis and reduces the impact of the sulfate forcing. For DTR and precipitation, B21 is close to unity and provides no evidence in favour of either of the two hypotheses. This is also consistent with the conventional analysis.

Bayes factors summarize the impact of the evidence on the prior odds of one hypothesis over another. For equal priors, the Bayes factors are equal to the posterior odds. For a complete description, however, one needs to consider generally the impact of the evidence on the full range of prior beliefs with respect to the different hypotheses. Figure 2 illustrates as example three cases representing different prior expert opinions on the relative probabilities of the natural variability and two anthropogenic forcing hypotheses GHG and GS. The first case, “uninformed” priors, assigns equal probability to each hypothesis. The remaining two cases represent “extreme” priors corresponding to a “climate change skeptic” and a “climate change advocate”. The climate change skeptic believes that, with 90% probability, the observed climate change can be explained by natural variability, while the climate change advocate assigns only a 10% probability to this explanation. Both experts are undecided between the two anthropogenic forcing hypotheses GHG and GS, assigning to these equal probabilities of 5% (skeptic) and 45% (advocate). For all three experts, the probabilities of the three hypotheses sum to unity, implying that alternative explanations of climate change are not considered. The degree to which these prior beliefs are changed by the observed trends in the different climate variables is indicated in Fig. 2.
Fig. 2

Posterior probabilities for three cases of prior belief about climate change: “uninformed priors”, “climate change skeptic” and “climate change advocate”. The cases are shown as sets of stacked bars where each bar shows the probabilities for the three hypotheses that the observed climate change can be explained by natural variability, by the greenhouse gas forcing alone (GHG) and by the superposition of greenhouse gas and sulfate aerosol forcing (GS). The prior probabilities are shown in each case by the leftmost bar, followed by the posterior probabilities for temperature (TAS), winter (DJF) and summer (JJA) diurnal temperature range (DTR), precipitation (PREC) and the net result (NET) when all variables are considered simultaneously.

The left group of columns in Fig. 2 shows the results for the uninformed case of equal priors for all three hypotheses. The posterior probabilities in this case are the same as the Bayes factors, which were already listed in Table 2. The near surface temperature data provide decisive evidence for the detection of anthropogenic climate change: the posterior probability for the natural variability hypothesis is negligible. The posterior probability ratio for the GS and GHG hypotheses is about 2 to 1. In the case of diurnal temperature range and precipitation (middle columns in the left group), the observations have the opposite impact: the posterior probability for the natural hypothesis is enhanced relative to the original value of 1/3. However, the net result considering all evidence (right column) still yields decisive evidence for an anthropogenic origin of the observed climate change, with approximately equal probabilities for the GHG and GS forcing hypotheses.

The overall assessment remains unchanged if we start from the prior probabilities of either the “skeptic” or the “advocate” (middle and right groups of columns). The evidence for an anthropogenic origin of the observed temperature change is so large that the collective outcome for all variables is independent of the prior probabilities assigned to the three hypotheses. The climate change advocate would also find sufficient evidence in the winter DTR and precipitation data to favor the anthropogenic forcing hypotheses over natural variability for these data alone (in accordance with the conventional finding that the observed trends for these variables lie on the margin of detectability). However, the “skeptic” favors the hypothesis of natural variablility for these variables. For the still weaker summer DTR signal, the natural variability hypothesis is favored by both the “skeptic” and the “advocate”.

Summary and conclusions

The Bayesian approach to the detection and attribution of anthropogenic climate change differs from the conventional analysis in four important aspects: (1) it allows the incorporation of climate change indices for which the statistical data base is insufficient to reliably support conventional signal-to-noise analysis methods; (2) detection and attribution are not regarded separately, but are treated together as coupled aspects of a single problem; (3) natural climate variability and model uncertainties are similarly combined in a single joint analysis, and (4) the analysis depends only on local probability densities, defined at the specific point in phase space at which the climate change signal has actually been observed. In this work we have focused on the latter three aspects, in preparation for a more extensive later analysis addressing the first aspect.

The limitation to local probability densities is a consequence of Bayes theorem, which can be expressed in terms of Bayes factors. These describe the impact of the observational evidence for two competing hypotheses in terms of the ratio of the local likelihoods for obtaining the observations for each individual hypothesis. Since the dimension of the phase space cancels in the likelihood ratio, there is no need for dimensional reduction, in contrast to the conventional case, in which the signal is normally lost in the noise unless the phase space is drastically reduced to one or two fingerprint patterns.

It can nevertheless be advantageous to reduce the dimension of the phase space dimension also in the Bayesian case in order to enhance the impact of the observations on a given hypothesis. This is achieved by removing patterns that reduce rather than enhance the Bayes factor for the hypothesis, relative to a competing hypothesis such as natural variability. The filter must, of course, be defined without knowledge of the observations. Expressed in terms of a suitably normalized EOF pattern space, the relevant pattern selection criterion is simply that the square of the predicted signal amplitude must be at least twice as large as the logarithm of the standard deviation of the model uncertainty.

As illustration, we applied the Bayesian filtering technique to a small set of climate variables, which had also been subjected to a conventional detection and attribution analysis: near-surface temperature (TAS), winter and summer diurnal temperature range (DTR), and precipitation (PREC). Two forcing mechanisms were considered: greenhouse gas alone (GHG), and greenhouse gas plus aerosols (GS). The estimates of natural climate variability were derived from the Hamburg ECHAM3/LSG simulations. Estimates of the predicted signals and the model uncertainties required for the Bayesian analysis were obtained from sets of ensemble integrations using seven different models. The number of patterns retained in the Bayesian analysis ranged from six (summer DTR) to 13 (TAS), as opposed to two in the conventional analysis.

Using the conventional approach, a climate change signal exceeding the natural variability noise level could be detected for temperature and winter DTR, but not for summer DTR and precipitation. For temperature, both anthropogenic forcing mechanisms were consistent with the observed signal, whereas for winter DTR, the attribution results were dependend on the model control run used for the estimation of natural variability: the GS signal was marginally consistent with observations for ECHAM4/OPYC3 variability, but neither of the anthropogenic forcings was consistent for the case that ECHAM3/LSG natural variability was used.

The Bayesian analysis yielded qualitatively similar results. The observed near-surface temperature change provided decisive evidence in favour of anthropogenic forcing versus natural climate variability, with a weak preference for GS forcing over GHG forcing. However, none of the remaining data enhanced the probabilities for the anthropogenic forcing hypotheses. This applied also to the winter DTR trends, which could be detected and at least marginally attributed to GS in the conventional analysis. The difference can presumably be attributed to the larger uncertainties of the model predictions, in particular for DTR and precipitation, which are not included formally in the conventional approach, but enter explicitly in the Bayesian analysis. In general, the weighting of the information on sulfate aerosol forcing is downgraded in the Bayesian analysis relative to the conventional approach through the inclusion of the model error structure. Thus, the large differences in the impact of sulfate forcing found for different models in conventional analyses (see Barnett et al. 1999) are automatically included and quantified in the Bayesian approach. Despite the negative impact of the observations of diurnal temperature range and precipitation on the priors for the anthropogenic forcing hypotheses, the net impact of all variables together is dominated by the near surface temperature and decisively support the anthropogenic climate change hypotheses, with approximately equal probabilities for greenhouse gas forcing with and without sulfate aerosols.

It is planned to apply the techniques developed and illustrated here to a comprehensive set of climate change indices that have been discussed in the literature (cf. Houghton et al. 2001) as possible evidence of anthropogenic climate change. This requires assessments of the natural variability and anticipated climate change signals for all data, including the signal uncertainties and model error structure. Due to the lack of adequate statistical information for many of these indices, these assessments will need to be augmented to a large part by subjective expert judgements.

Footnotes
1

see IPCC Data Distribution Centre, http://ipcc-ddc.cru.uea.ac.uk/ for further data, model descriptions and references

 

Acknowledgements

This work was supported by the National Oceanic and Atmospheric Administration (NOAA) Climate Change Data and Detection program and the US Department of Energy, Office of Energy Research, as part of the International ad hoc Detection Group effort, and by the European Commission QUARCC project, ENV4-96-0250.

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© Springer-Verlag 2004