9march2010© The Royal Statistical Society and Crown copyright 2010

Bohr famously remarked: “prediction is very diffi cult, es-pecially about the future”. However, questions do not go away just because they are diffi cult: the problems posed by climate change are very real and require answers today. To address these problems requires a combina-tion of skills and expertise from many disciplines, and statistics has a key role to play. Th is article should give a fl avour of some of the statistical issues involved. It is necessarily personal and selective: but climate science provides a rich and important fi eld of application for the statistical community.

What do we mean by climate?

Climate is a complicated concept. To use another oft-repeated quote, “climate is what you expect, weather is what you get”2. In layman’s terms, you could say the “climate” of a particular location represents some kind of average value of the “weather” experienced there. For example, the Copenhagen accord reaffi rms an increase of 2°C in global mean temperature as a threshold for dan-gerous climate change. However, this is not quite enough.

Copenhagen 1

Climate changemaking certain what the uncertainties are

Introduction

At the end of 2009, world leaders returned from the Copenhagen climate change summit. Th e outcome was seen by many as disappointing; but the mere occurrence of the summit testifi es to widespread recognition that climate change is one of the most important issues facing humanity. Just a few of the less obvious potential impacts are on water availability, fl ooding, and health; the malaria mosquito may extend its range, and more frequent sum-mer heatwaves may kill. Authorities in every area need to make their plans; and those plans require the best available information about the climate of the next 10, 20 or even 100 years. Many countries are now develop-ing national strategies to meet this challenge. In the UK, for example, Department for the Environment, Food and Rural Aff airs and other government departments have funded the development of the UKCP09 climate projec-tions for the next century, which are intended as a tool for anybody concerned with climate change adaptation1.

Of course, it is far from trivial to produce credible projections of climate over the next century. As Niels

Questions do not go away just because they are diffi cult. The problems posed by climate change are real

At the Copenhagen conference our leaders failed to agree on a course of firm action against climate change. We present three articles on the post-Copenhagen world. In the first of them, Richard Chandler, Jonathan Rougier and Mat Collins describe statistical work that still needs to be done.

10 march2010

Th e “danger” in “dangerous” arises not only from average conditions, but also from changes in variability, and in particular in extremes such as fl oods, storms and droughts. Arguably a more relevant defi nition is that climate is “the distri-bution of weather”, and a thorough assessment of the impact of climate change must examine changes in this distribution. We speak of global warming, but specifi cally, one might be interest-ed in the distribution of many other quantities as well as surface temperature. Precipitation is one: in many parts of the world survival itself depends on whether there is local rainfall or drought. Other weather-related quantities such as features of the growing season are also im-portant. We refer to all of these quantities as prognostic variables.

How are climate projections made?

Politicians desire certainty, and planners need all the certainty they can get. But clearly, all fu-ture values of prognostic variables are currently uncertain. In principle, one might attempt to determine them by constructing an accurate simulator of the climate system, incorporating all available historical information (although historical values are themselves uncertain, due to measurement errors and data sparsity). How-ever, the climate system is so complex that any simulator necessarily represents a simplifi cation of reality. Modelling even something as funda-mental as the behaviour of a cloud is currently beyond our computing power. Climate science therefore has to simplify. Diff erent approaches to this simplifi cation correspond broadly to dif-ferent allocations of computing resources. Some concentrate their computing power on simulator evaluations, some on the number of processes they include, and some on the spatial and tem-poral resolution of their models. Two extreme cases are energy balance models (EBMs) and general circulation models (GCMs).

Energy Balance models simplify complex processes, down to such levels as “change in stored energy = input energy – output energy”. Th ey use simple equations and have minimal resolution. As a result they are simple enough to be implemented using a calculator or a spreadsheet. Despite their simplicity they have played a crucial role in our understanding of the gross features of climate, notably global and zonally-averaged mean temperatures, and the role of the greenhouse eff ect. However, they are largely phenomenological, and can give only limited guidance about future climate. For example, a warmer climate implies more atmospheric water vapour, and an important role for clouds. But clouds have a complicated

eff ect on temperature, both refl ecting visible light back into space, and absorbing and re-radiating infrared light from the surface of the Earth, depending on their location, shape, and composition. An energy balance model has no room for such subtleties.

At the other end of the scale, GCMs at-tempt to represent the physics of climate proc-esses, and can determine the net outcome of competing eff ects (see Box 1). Crucially for ad-aptation studies, they also have the potential to provide climate information at regional scales. However, they are expensive to create and eval-uate: they are usually run on supercomputers, and take about a month to produce a 100-year simulation. As computers have become more

powerful, the tendency has been for GCMs to share the extra power between including extra processes and increasing the resolution, so that the speed of evaluation has remained roughly constant.

Most climate simulators used today are deterministic and based on physical principles. Statistical and stochastic (that is, probability-based) modelling has hitherto been little used, except in applications involving projections over short (e.g. seasonal) time horizons. For longer horizons, the reliance on physical prin-ciples is crucial for constraining the interac-tions between the millions of variables in the climate system. Nonetheless, there is currently interest in incorporating stochastic elements

Box 1. General circulation models (GCMs)

In very general terms, the dynamics of the climate system can be described using the single equation dy/dt = f(y,x). Here, y(t) represents the values of all prognostic variables at time t and x(t) is a collection of forcing variables, which are considered as external to the system: these might include solar intensity, volcanic emissions, and atmospheric concentrations of particulates and gases such as CO2. The equation may be paraphrased as “the future is entirely determined by the current state of the system and by the forcing”. A GCM attempts to solve this equation for a given initial confi guration y(t0) and forcing scenario {x(t): t > t0}. Typically, t0 is chosen to be a time in the past at which the climate system was relatively stable, such as 1850 (“pre-industrial”). As with all models, however, many approximations are involved. For example, it is impossible to enumerate completely the state of the system at any time, so GCMs actually work with a subset of (typically millions of) elements of y(t). Even here, it is not possible to represent all of the interrelationships between these elements. Moreover, the model equations cannot be solved analytically and numerical methods must be used. This usually involves some kind of discretisation in both time and space: Figure 1 illustrates the spatial discretisation used in two GCMs developed at the Met Offi ce Hadley Centre in the UK. HadCM3 has for some years been one of the world’s leading GCMs; notice, however, the coarse spatial resolution of the grid here (Ireland, for example, consists of just two grid boxes). HadGEM1 is a newer GCM with enhanced resolution, but it is nonetheless clear that the output on a grid can only approximate the spatial variation in y(t). Despite this, GCMs provide credible simulations of many quantities at global and regional scales, and much of our detailed understanding of the climate system derives from this type of simulator.

Figure 1. Schematic illustration of Met Offi ce Hadley Centre climate models

11march2010

into climate simulators. Ultimately, this will allow a better treatment of their limitations, by attributing uncertainty to particular aspects of the physical modelling.

Quantifying uncertainty

Th e simplifi cations and approximations in-volved in any climate simulator rule out a perfect prediction of the prognostic variables

we described earlier; they are thus a source of uncertainty in future projections. We refer to this as structural uncertainty. It is important to recognise two other sets of uncertain quanti-ties, which are specifi c to the individual simu-lators or simulations which use them.

1. Earth system quantities that describe what the Earth is like at a selected moment. Th ey are involved in inferring the prognostic variables, and are treated as simulator inputs. Th ey come in three types: (i) prescribed values for Earth system components treated as fi xed in the simulator, such as topography and vegetation type; (ii) historical values of forcing variables (see Box 1); and (iii) future values of the same forcing variables, described in terms of emissions scenarios (see Box 2). Additionally, dynamical simulations require initialisation values for the prognostic variables.

2. Th e “correct” values of parameters inside the simulator, which are imperfectly known or which do not have an analogue in the climate system. GCMs have hundreds of such parameters, many designed to represent sub-grid-scale processes such as convection within clouds. We write “correct” in quotes because the notion of a correct value, though useful and widespread, is not easy to operationalise in the context of an imperfect simulator.

We refer to uncertainty about the simulator inputs as input uncertainty, and uncertainty

about parameter values as parametric uncer-tainty. Figure 2 is a directed acyclic graph showing the relationships between the various uncertain quantities, along with the role of climate system observations.

Uncertainties neglected

Until recently, standard practice in climate science has been to neglect all three sources of uncertainty. Structural uncertainty has of-ten been treated as negligible; historical input values were replaced by a point estimate; and parameter values were derived from a combi-nation of physical reasoning and a very lim-ited amount of tuning. Th is does not refl ect myopia on the part of climate modellers, who are all too aware of uncertainty about histori-cal inputs and simulator parameters; they are

also aware that their simulators have substan-tial structural failings. It simply refl ects the expense of evaluating a GCM. By and large, climate modellers have chosen to incorporate as much physics as possible, to obtain what they consider the best possible point estimate of future climate. But this has been at the ex-pense of quantifying uncertainty. Ideally we would have multiple simulator evaluations over diff erent values for both the historical inputs and the simulator parameters. Th ere has been some recent progress in this direc-tion: for example, the primary source of information for UKCP09 is an ensemble of simulations from the HadCM3 General Circulation model (see Box 1) in which key parameters have been varied within ranges of uncertainty determined by expert judgement. Th e design of such ensembles poses some interesting challenges and is an area in which the application of statistical principles could undoubtedly lead to major improvements in current practice.

Another route is to use the distribu-tion from a collection of diff erent simulators, termed a multi-model ensemble. For example, in 2007 the International Panel on Climate Change Fourth Assessment Report (AR4) reported results from 23 GCMs4. Th ese simu-lators have diff erent structural assumptions, and therefore produce diff erent evaluations for the same emissions scenario. It is tempting to treat the spread of values as a measure of uncertainty, and this often happens implicitly when the evaluations are displayed together to produce a “cone” expanding out into the future as in Figure 3. Th is cone underestimates un-certainty, however, since no account is taken of parametric and input uncertainty for the

Necessary simplifi cations bring uncertainty, but other

sources of uncertainty also exist

Box 2. Emissions scenarios

The values of forcing variables for climate projections are usually obtained from “emissions sce-narios”. For several years, the scenarios introduced by the Intergovernmental Panel on Climate Change (IPCC) in 2000 have been used routinely in climate research. They tell different stories about the future, in terms of their implications for atmospheric concentrations of greenhouse gases3. The A1 family of scenarios, for example, supposes a future world of rapid and convergent economic growth, global population that peaks in mid-century and declines thereafter, and the introduction of new and more effi cient technologies. Families B1 and A2 correspond to patterns of development that may be expected to lead respectively to lower and higher concentrations than A1. Interestingly, the scenarios were never developed to span possible futures, and they were explicitly not assigned probabilities.

The IPCC scenarios were put together in the early 90s and have now outlived their useful-ness, and a new set of “Representative Concentration Pathways” is under development. Rather than being led by storylines, these are deliberately designed to span a wide range of outcomes and to be clearly distinguishable from one another. In this sense, they conform much more to what mathematicians would recognise as a “basis” for future emissions, in which any particular future can be thought of as some linear combination of pathways.

Figure 2. Directed acyclic graph showing the relationships between prognostic variables, observations, simulator inputs and simulator parameters. In statistical terms, the marginal distributions of inputs and parameters represent input and parametric uncertainty, respectively. The conditional distribution of prognostic variables given inputs and parameters represents the simulator’s structural uncertainty. The conditional distribution of observations given prognostic variables represents measurement uncertainty

Inputs

Parameters

Prognosticvariables

Observations

Simulator

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individual simulators, or of common sources of error such as modelling concepts or code that is shared between simulators. A further complication is that multi-model ensembles cannot be interpreted probabilistically in any conventional sense, because the available simulations constitute an “ensemble of op-portunity”5 rather than being sampled from a well-defi ned population. Th e interpretation of collections of evaluations from individual and multiple GCMs is an active research area for statisticians working in climate science.

Getting involved

Th ere is now a huge amount of interest in rep-resenting and quantifying uncertainty about future climate, and statisticians have an im-portant role to play. We have highlighted some specifi c issues here, but there are many oth-ers – for example, how to infer the changing distribution of weather from a limited number of simulations, and how to “downscale” GCM output to the fi ner resolution often required for impacts studies. Th ere is increasing recog-nition that progress requires multidisciplinary

collaboration, and opportunities for this are becoming more frequent. Within the Royal Statistical Society, the Environmental Statis-tics Section routinely organises climate-related meetings6.

In 2010, two further events in the UK should be of interest to statisticians wishing to get involved in this area. One is the 11th In-ternational Meeting on Statistical Climatology (IMSC), to be held in Edinburgh in July7. Th is provides a forum for interaction between the at-mospheric and statistical science communities. Th e other is a residential research programme at the Newton Institute in Cambridge, entitled “Mathematical and Statistical Approaches to Climate Modelling and Prediction”8. Th e programme runs from August to December and aims to bring together mathematicians, statisticians and climate scientists to work spe-cifi cally on (a) the incorporation of stochastic components into climate simulators and (b) probabilistic climate prediction. Of course, as with all applications, there is a steep learning curve. However, the problems are fascinating, complex and topical; and the potential rewards are well worth the eff ort.

References1. Available from http://ukcp09.

defra.gov.uk/ 2. Heinlein, R. (1973) Time Enough for

Love. New York: Putnam.3. See http://www.grida.no/cli-

mate/ipcc/emission/4. See http://www.ipcc.ch/5. Tebaldi, C. and Knutti, R. (2007) Th e

use of the multi-model ensemble in probabilistic climate projections. Philosophical Transactions of the Royal Society, Series A, 365, 2053–2075.

6. For details, see the Section website (http://www.rss.org.uk/ environmental), the meetings pages of RSS News or join the ENV-STAT mailing list at http://www.jiscmail.ac.uk/fi les/envstat/index.html.

7. See http://cccma.seos.uvic.ca/imsc/11imsc.shtml

8. See http://www.newton.ac.uk/programmes/CLP/index.html

Richard E. Chandler is at the Department of Statistical Science, University College London. Jonathan Rougier is at the Department of Mathematics, Bristol Univer-sity. Mat Collins is based at the Met Offi ce Hadley Centre. All three are involved in the Edinburgh and Cambridge events.

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Figure 3. Projections of annually averaged global mean temperature for the 21st century, derived from 23 different GCMs as used in the IPCC AR4 (data obtained from the KNMI Climate Explorer at http://climexp.knmi.nl/). Projections are shown as changes from the 1980–99 mean. The fi rst panel shows all of the available simulations from the A1B, A2 and B1 emissions scenarios (see Box 2). The remaining panels show the simulations from each scenario individually, with colours corresponding to individual GCMs. Not all GCMs were run for all scenarios. Conversely, some GCMs were run more than once per scenario with different initial confi gurations of the prognostic variables