Quantifying anthropogenic influence on recent near-surface temperature change

Abstract

We assess the extent to which observed large-scale changes in near-surface temperatures over the latter half of the twentieth century can be attributed to anthropogenic climate change as simulated by a range of climate models. The hypothesis that observed changes are entirely due to internal climate variability is rejected at a high confidence level independent of the climate model used to simulate either the anthropogenic signal or the internal variability. Where the relevant simulations are available, we also consider the alternative hypothesis that observed changes are due entirely to natural external influences, including solar variability and explosive volcanic activity. We allow for the possibility that feedback processes, other than those simulated by the models considered, may be amplifying the observed response to these natural influences by an unknown amount. Even allowing for this possibility, the hypothesis of no anthropogenic influence can be rejected at the 5% level in almost all cases. The influence of anthropogenic greenhouse gases emerges as a substantial contributor to recent observed climate change, with the estimated trend attributable to greenhouse forcing similar in magnitude to the total observed warming over the 20th century. Much greater uncertainty remains in the response to other external influences on climate, particularly the response to anthropogenic sulphate aerosols and to solar and volcanic forcing. Our results remain dependent on model-simulated signal patterns and internal variability, and would benefit considerably from a wider range of simulations, particularly of the responses to natural external forcing.

Appendix total least squares estimation

The estimation procedure used in this study is based on the assumption that the observed climate record, as characterised by the n-element vector y (n = 125 in our example), can be represented as a linear superposition of m model-simulated response-patterns, x _i , denoted as the columns of the m × n matrix X. Ideally, large ensemble simulations should be used to minimise the influence of sampling noise on the model-simulated signals (i.e., to pin down exactly what we are looking for in the observations). In practice, to compare results from models given a wide range of available ensemble-sizes, we need to take sampling variability explicitly into account, giving the detection model:

$$ {\bf y} - \user2{\upsilon}_0 =\sum_{i=1}^{m} ({\bf x}_i - \user2{\upsilon}_i)\beta^{\rm true}_i $$

(5)

$$= ({\bf X} - \varvec{\Upsilon}_X)\varvec{\beta},$$

(6)

in which $\user2{\upsilon}_0$ is the sampling noise in the observations and $\user2{\upsilon}_i$ (or the i ^th column of $\varvec{\Upsilon}_X$) is the noise in the ith model-simulated responses to external forcing.

We assume a linear relationship between the underlying deterministic (noise-free) observed and modelled responses, ${\bf y} - \user2{\upsilon}_0$ and ${\bf x}_i - \user2{\upsilon}_i$, given by the unknown scaling factors, $\beta^{\rm true}_i$. The physical interpretation of these scaling factors is as follows: $\beta^{\rm true}_i$ represents the amount by which we have to scale the $i{\rm th}$ model-simulated response to reproduce the observations, assuming the shape of the response is accurately simulated (apart from sampling error due to the finite size of the ensemble used to generate it) and only the amplitude is unknown.

Assuming both $\user2{\upsilon}_0$ and the $\user2{\upsilon}_i$ are independent and Gaussian distributed, the probability of observing this particular combination of climate observations y and model simulations X for a given set of scaling factors $\varvec{\beta}$—known as the “likelihood” of (y, X) given $\varvec{\beta}$ and the detection model (6)—is given by:

$$ P({\bf y, X} | {\varvec{\beta}}) = \mbox{constant} \times e^{- r^2},$$

(7)

where r ² is the sum of the noise in both y and X implied by that value of $\varvec{\beta}$, weighted by their respective inverse covariance matrices, assuming these are known. That is,

$$ r^2 = \user2{\upsilon}_0^T {\bf C}_0^{-1} \user2{\upsilon}_0 + \sum_{i=1}^{m} \user2{\upsilon}_i^T {\bf C}_i^{-1} \user2{\upsilon}_i ,$$

(8)

where C ₀ and C _i are the expected n ×  n covariance matrix of the noise in the climate observations y and the ith model-simulated response-pattern, x _i , respectively.

What we ultimately want is the distribution of $\varvec{\beta}$ given this set of climate observations and model simulations, or $P(\varvec{\beta} | {\bf y, X})$. This also depends on our prior knowledge, requiring an application of Bayes theorem:

$$ P(\varvec{\beta} | {\bf y, X}) = \frac{P({\bf y, X} | \varvec{\beta}) P(\varvec{\beta})}{P({\bf y, X})}.$$

(9)

In this equation, $P(\varvec{\beta})$ represents our prior knowledge or assumptions about the range of possible values for the $\beta^{\rm true}_i$: that is, how probable we consider each value of $\beta^{\rm true}_i$ before these observations are taken into account. P(y, X) is the probability of obtaining this particular set of observations and model simulations, given those constraints, irrespective of the true value of $\varvec{\beta}$: it acts as a normalisation, ensuring that all probability density functions integrate to unity. In the cases considered here, we will assume both $P(\varvec{\beta})$ and P(y, X) are uniform distributions, giving the “standard” or “classical” detection model used, for example, by Hegerl et al. (1997), and Tett et al. (1999). It is, however, important to recognise that there is no fundamental distinction between the “classical” and “Bayesian” approaches. Classical optimal fingerprinting is simply a special case of a Bayesian analysis in which uniform priors are assumed: that is, in which we assume no prior knowledge whatsoever of the parameters we are trying to estimate and the observations we are likely to obtain.

If we assume that the noise in both y and X is Gaussian and dominated by internal climate variability as simulated by the A-OGCM, then we can make ${\bf C}_i = {\bf C}_0 = {\bf C}_{\rm N}$, the covariance between elements of y-like vectors drawn from an unforced control simulation, simply by scaling the ith response-pattern by $\sqrt{\ell_i}$, the number of simulations in the ith ensemble. We also introduce a so-called “pre-whitening operator”, P, defined such that

$$ {\bf P} {\bf C}_{\rm N} {\bf P}^T = {\bf I}^{(\kappa)} ,$$

(10)

where ${\bf I}^{(\kappa)}$ is a unit matrix of rank κ. The pre-whitening operator, P, will be model-dependent: there is no universal operator to remove correlations in variability simulated by any model. The so-called “rank of the detection space”, κ, need not be the same as the number of observations n—if it is less than n, P not only acts to make the noise equal-variance and uncorrelated in all the elements of the pre-whitened observations and model-simulated response-patterns, but it can also be used to filter out any component of the signal that correlates with modes of variability that are poorly simulated by the A-OGCM. This dual role of the pre-whitening step in climate change detection is discussed at length in Allen and Tett (1999).

The simplest, although by no means unique, way of defining P is to decompose a sequence of “climate noise” realisations from a control integration of the A-OGCM into a set of spatio-temporal patterns, or extended-EOFs (Weare and Nastrom 1982) and associated principal components, using a singlar value decomposition. The κ rows of P are then defined as the κ highest-ranked E-EOFs weighted by their respective inverse singular values. With this choice of weighting, if we were to take another realisation of climate noise, from a different control integration for example, and pre-multiply it by P, we would expect approximately the same noise variance in each of the κ rows of the resulting pre-whitened noise vector, and for the noise to be uncorrelated between rows.

Representing the pre-whitened model-simulated response-patterns, P X as the first m columns of the κ ×  (m + 1) matrix Z, and the pre-whitened observations, P y, as the m + 1 column, the detection model (6) can be re-written (Allen and Stott 2003)

$$ ({\bf Z} -\varvec{\Upsilon}_Z){\bf v} = {\bf 0} , $$

(11)

where $\varvec{\beta}$ has been encoded into the vector v (see below). Equation (11) states that a linear relationship exists between the noise-free observations and the noise-free model-simulated response-patterns or, equivalently, that a vector exists which is orthogonal to every row of ${\bf Z} -\varvec{\Upsilon}_Z$. The pre-whitened noise contamination, $\varvec{\Upsilon}_Z$ is unknown, but an unbiased estimate of the orientation of the vector v can be obtained from the eigenvector of Z ^T Z with the smallest eigenvalue, as was first observed by Adcock (1878). This estimate, ${\bf \tilde{v}}$, also maximises the likelihood function in Eq. (7) by minimising the total (perpendicular) squared distance of the observation-model points (defined by the κ rows of Z) from the m-dimensional plane, orthogonal to ${\bf \tilde{v}}$, which represents the best-fit solution: hence the name Total Least Squares solution (van Huffel and Vanderwaal 1994).

Best-fit scaling factors on the individual model-simulated response-patterns, as required to reproduce the observations in model (6) may be obtained from the ratios,

$$ \beta_i = - \sqrt{\ell_i} \frac{\tilde{v}_i}{\tilde{v}_{m+1}},$$

(12)

recalling that the ith column of Z has been scaled by $\sqrt{\ell_i}$ to equalise the noise variance in all observables. Best-fit reconstructions of the noise-reduced observations and model-simulated response-patterns are obtained by extracting the component of Z orthogonal to v til, thus:

$${\bf \tilde{Z}} = {\bf Z} - {\bf Z} {\bf \tilde{v}} {\bf \tilde{v}}^T.$$

(13)

Confidence intervals on the elements of ${\bf \tilde{v}}$ are obtained using the formulae for uncertainties in an eigen-decomposition given in North et al. (1982), and Allen and Smith (1996), using an independent estimate of C _N to re-estimate the eigenvalues of Z ^T Z to avoid the problem of “artificial skill” noted by Bell (1986)—see Allen and Stott (2003), for details.

Twentieth-century warming trends “attributable” to specific external influences under the detection model (6) are obtained as follows: the columns of ${\bf \tilde{Z}}$ in Eq. (13) represent our best estimate of the noise-free (and hence unobservable) modelled and observed climate change respectively. A range of other reconstructions are also consistent with these observations and this model simulation at a given confidence level, found in a similar manner to the uncertainty range in ${\bf \tilde{v}}$. We compute linear trends in global mean temperature in these “possible” reconstructions by extracting the first spherical harmonic for each of the five decades in the observation vector and regressing them onto a vector which is zero in the 1946–1956 decade, increasing linearly thereafter: this provides the best estimate of the linear trend over the 1906–1996 period in data that has been expressed as anomalies about the 1906–1996 climatology. For the sake of brevity, we refer to this as the “attributable trend over the 20th century”.