Uncertainty in Climate Predictions

Douglas Nychka,1National Center for Atmospheric Research (NCAR),Juan M. Restrepo, University of Arizonaand Claudia Tebaldi, Climate Central.

Carbon dioxide (CO2) and other greenhousegases, released into the atmosphere from humanactivities, have altered Earth’s natural climate.Naturally produced greenhouse gases have a dif-ferent chemical signature from the man-madevariety. The increasing concentration of CO2 inthe atmosphere, correlated to human industrialactivities, has been shown through modelingstudies and a host of empirical evidence to causea large fraction of the warming trend recordedin the last decades. Moreover, larger and morerapid future changes are expected in the absenceof significant mitigation.

The uncertainty of climate modeling: all thedetailed topography and heterogeneous physicalprocesses in this aerial image from Colorado isreduced to a single grid point in the typical reso-lution (e.g. 100km×100km) for a global climatemodel. (NCAR Digital Library)

Understanding the interactions among differ-ent components of the Earth’s physical systemand the influence of human activities on the en-vironment, has progressed immensely, but it isfar from complete. The qualitative assessment ofglobal warming needs to be buttressed by morequantitative and precise estimates of differentimpacts of climate change on natural, social andeconomic systems.

A dramatic cumulonimbus cloud with a rainshaft. This thunder storm in Colorado is“weather”. However, based on the sparse veg-etation one can infer that the “climate” for thisarea involves limited rainfall. (NCAR DigitalLibrary)

Quantitative assessments of the impacts of fu-ture climate change are not straightforward andrequire new tools in mathematics and statis-tics providing not only estimates on climateimpacts but also companion measures of uncer-tainty. The Intergovernmental Panel on ClimateChange (IPCC) 2007 report highlights uncer-tainty quantification as one of the most pressingresearch issues in determining our overall abilityto propose how human activities can be mod-ified to mitigate our effects on natural climateand how we can adapt to unavoidable changes.

Climate and weatherClimate refers to the distribution of the state ofthe atmosphere, oceans, and biosphere at timescales of decades, centuries, millennia. It canbe viewed as a forced/diffusive system, withvery complex and long-range interactions be-tween the oceans, the land mass, and the atmo-sphere. Examples of climatic events are the ”IceAges”, droughts and the current global warming.Weather, on the other hand, refers to the state

1Web links in bluePhoto Credits: E. Nychka and NCAR Digital Library

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of the above coupled systems at much shorterspatio-temporal scales. Its behavior can be verysensitive to initial conditions and so is often dif-ficult to predict. Examples of weather events aremore familiar to us: storm systems, hurricanesor tornados. The distribution of weather eventsover a long period of time and at a given locationis the climate.

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Atmospheric concentrations of Carbon Dioxideover the last 10,000 years (large panel) and since1750 (inset panel) showing the rapid increase dueto increased consumption of fossil fuels. (SeeIPCC Fourth Assessment Report)

Projections of future climate are based on theoutput of atmosphere/ocean general circulationmodels and are used to simulate conditions in thefuture based on projected levels of greenhousegases. These models are physically based, com-puter codes that couple the dynamics among theocean, the atmosphere, sea ice and land alongwith biogeochemical processes that affect con-centrations of CO2. It may come as a surprisethat the essential physics for these models israrely debated, drawing on classical fluid dynam-ics and thermodynamics. However, due to thefinite resolution of the models the exact physicalprocesses must be approximated. Due to theirsize and complexity, most climate system mod-els can not achieve a resolution much smallerthan regions of 100 kilometers square. The re-sult is that the model must account for physicalprocesses that are not resolved, i.e. explicitlysimulated, such as cloud formation and com-plex topography. These unresolved processesstill have a strong influence on climate at largescales. The technique of representing unresolvedprocesses is termed a parameterization and is an

active area of climate research. Not surprisingly,parameterizations are also a key factor in modeluncertainty.

Snapshot of a high resolution (∼40km) simula-tion of the global atmosphere using the NCARCommunity Atmosphere Model (CAM) anima-tion

Dynamical systemsTo an applied mathematician, a climate modelcan be viewed as a discrete dynamical systemwith some additional variables entering as exter-nal forces. The state of the atmosphere, oceanand other components at a time t are repre-sented by a large state vector, say xt, of threedimensional fields and the external “forces”, col-lected together as another vector yt, includingthe changes in CO2 over time. A mapping Ftakes the state from time t to t + 1. It uses boththe current state of the climate system, xt andthe values of the external forces, yt to producea dynamical result for the next time step. Con-cisely, xt+1 = F (xt,yt). Here the individualtime steps of the system can be interpreted asgenerating weather while long term averages orlong term probability distributions of the statevector reflect the climate of the system. Sup-pose that a more accurate physical descriptionof xt+1 involves a more detailed or higher res-olution state, say x∗

t , and accordingly for someG, xt+1 = G(x∗

t ,yt) gives a better forecast att + 1. The parameterization problem from thegeosciences is to express this dynamical rela-tionship in terms of the xt vector. This processis also known as closure and presents interest-ing new mathematical problems for geophysicalsystems.

At the same time as climate models have becomeever more complex, efforts are made to pro-duce simpler dynamical systems that are more

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amenable to mathematical analysis. For exam-ple, energy balance models are used to gain anunderstanding of the basic dynamics climate sci-entists and mathematicians are also involved inproposing ways to capture important unresolvedphysics in a consistent way: up-scaling and clo-sure techniques are used to produce coarse mod-els that can convey the unresolved physics in aconsistent way. Recently, stochastic techniquesare being applied to capture (or close) unresolvedor poorly understood phenomena in such a waythat the models have a statistical behavior thatis consistent with observations.

Elevations for the Western US at 4km and128km resolution. The coarser resolution shownis a typical scale for climate model experimentsand the finer scale is at the limits of historicalobservational climate data sets.

Numerical simulationsA numerical experiment with a climate modelinvolves specifying the initial state, x1, and theforcing series yt and then stepping the systemforward many time steps. The intriguing aspectof climate modeling is that although the modelis formulated as a differential or short term map-ping, the climate produced by the model is thelong term distribution (including averages) ofthe state vector. The scientific challenge in

climate modeling is to represent the physicalprocesses of the atmosphere and ocean at timescales on the order of minutes but produce re-alistic simulations of the Earths climate whenvariables, such as surface temperature or rain-fall, are averaged over 20 to 30 years. Typicallya state-of-the-art climate model when run on asuper computer can simulate several “model”years for each “wall clock” day, thus runs of amodel extending 100 years or more are limitedby available computer resources. The computa-tional operations increase roughly by a factor of16 when the grid size is cut in half and this limitplaces practical constraints on any significant in-crease in climate model resolution. For example,moving from a 128km grid to a cloud resolvingscale of 4km would increase the amount of com-putations by a factor of more than 106.

Climate model uncertaintiesThere exist large sources of uncertainty in con-structing and applying climate models. Thesecan be classified into initial conditions, boundaryconditions, parameterizations and model struc-ture. Initial conditions are often dismissed asa source of uncertainty in climate projectionssince the results from model runs are usually av-eraged over decades. However, there still may besome uncertainty introduced for shorter climatepredictions due to the initial state of the deeperlayers of the oceans and the slow time scale partsof the carbon cycle.

Cumulus clouds over the South Pacific, (NCARDigital Library)

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Uncertain forcings Boundary condition uncer-tainty arises because the model experiments,which are otherwise self-contained, rely on inputs(or forcings) that represent external influences.Natural external influences are the solar cycleor volcano eruptions. Man-made influences forclimate change studies are primarily greenhousegas emissions, whose future behavior depend onunpredictable socio-economic and technologicalfactors. Given the uncertainty for future hu-man activities, simulations of future climate aretermed projections rather than predictions todenote their dependence on a particular scenarioof human activity.

Subgrid scale uncertainty Parameter and struc-tural uncertainties are intrinsic to numerical cli-mate models. Because of limited computingpower, climate processes and their interactionscan be represented in computer models only upto a certain spatial and temporal scales (the twobeing naturally linked to each other).

Observed mean winter surface temperatures andthose simulated from the NCAR Community Cli-mate System Model (CCSM). These data are

available as part of CMIP3.The influence of processes existing at finer scalesthan the model grid cells depend on the tech-nique of parametrization or mathematical clo-sure described above.

Climate model bias: the CCSM simulated cli-mate minus the observations.

For example, the percentage of cloud cover andthe type of clouds within a model grid cell areresponsible for feedbacks on the large scale quan-tities, determining how much heat penetrates thelower layers of the atmosphere and how much isreflected back to space. However, the formationof clouds is an extremely complex process thatcannot be measured empirically as a functionof other large scale observable (and/or repre-sentable in a model) quantities. Thus the choiceof the parameters controlling the formation ofclouds within the model cells has to be dictatedby a mixture of expert knowledge and goodness-of-fit arguments.

Parameter uncertainties have started to be sys-tematically explored and quantified by so-calledperturbed physics ensembles (PPE), sets of sim-ulations with a single model but different choicesfor various parameters. The design of theseexperiments and their analysis provide fertileground for the application of statistical model-ing. Because climate model runs are computa-tionally intensive, part of the challenge is to varyseveral model parameters simultaneously usinga limited set of model runs. (See climatepredic-tion.net for a large and particularly innovativePPE based on individuals running parts of an

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experiment on personal computers.) Bayesianstatistical approaches are often preferred becausethey can also incorporate expert judgment in theprior formulation.

Uncertainty due to model construction Struc-tural uncertainty is introduced by scientificchoices of model design and development. Thereare sources of variation across model outputthat cannot be captured by parameter perturba-tions, and can introduce a larger degree of inter-model variability than any PPE experiment.The model output collected under the ClimateModel Intercomparison Project (CMIP) 3 hasprovided the basis for all the future projectionsfeatured in the last IPCC report. Formal sta-tistical analysis of model output, either comingfrom multi-model ensembles such as CMIP3 orfrom PPE experiments is crucial to quantifyingthe uncertainties caused by models approxima-tions. Unfortunately, the most popular use ofthese ensembles is to compute simple descriptivestatistics: model means, medians and ranges.These may not be appropriate because the givenensemble may not be representative of the super-population of all possible models. For example,models that are currently derived from differentclimate groups may share a common origin andso inherit subtle structural similarities. Build-ing mathematical tools that can recognize thesedependencies among models is a difficult butchallenging problem. Closely related to this is-sue is that some models may be “better” thanothers in simulating climate leading to methodsfor weighting models differently when synthesiz-ing their future projections.

A cluster analysis of 19 climate models based ontheir spatial biases in simulating winter (DJF)and summer (JJA) mean surface temperature.Points joined by dashed lines are models from thesame group (e.g. 7,8,9 are from NASA/GoddardInstitute for Space Studies.) The five circledpoints are a “space-filling” subset of the 19 mod-els. (Courtesy of M. Jun, Texas A&M)

Data assimilationThe blending of data and models has been usedsince the early days of weather forecasting. Inthe last 25 years estimation and statistical meth-ods have been introduced to weather predictionproducing significantly better forecasts by han-dling the inherent error of models and data in amore consistent way.

Modeled vs. observed cloud changesJuly 2007 minus July 2006

CAM Total Cloud Changes MODIS Terra Cloud Changes

Sea Ice Area

Fraction Changes

Unlike CAM, MODIS shows variability in the cloud response over open water.

Modeled vs. observed cloud changesJuly 2007 minus July 2006

CAM Total Cloud Changes MODIS Terra Cloud Changes

Sea Ice Area

Fraction Changes

Unlike CAM, MODIS shows variability in the cloud response over open water.

The difference in cloud cover over the Earth froma North pole projection between July 2007 andJuly 2006. Between these two years there wasa large decrease in the sea ice extent. There isa corresponding change in the cloud cover overopen ocean in the observations (MODIS) but themodel (CAM) results do not produce this changein clouds and suggest some problems with cloudformation over polar oceans. To make this com-parison, observational data must be assimilitatedinto CAM to match the model state to a partic-ular period of observation. (Research and figure(Jen Kay, NCAR)

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These methods, known collectively as data as-similation, are especially useful when the datais sparse, both in space and in time. A newarea in climate model development is diagnos-ing problems in their formulation based on howwell they predict weather. This is in contrast tomore traditional methods where long term aver-ages or other statistics from a model simulationare compared to the statistics of observations.Specifically, F is used to make a short term pre-diction and the results are compared to whatwas actually observed. The differences can thenbe used to check or modify parameterizations orother components of the model. Another strat-egy is to assimilate observations with a modelover a period of time and then compare themodel state to other atmospheric measurementswithheld from the assimilation process. Thefigure given above considering cloud cover overopen ocean and its relationship to sea ice is anexample of this method.

The blurring of the distinction between climatemodels and weather forecasting models is a newopportunity for theory on the relationship be-tween short term behavior of F and its longterm average properties. Climate and weatherare inherently nonlinear, and as a result alsohave non-Gaussian distributions. Assimilation

of data and models with strong nonlinearitiesand non-Gaussianity pose special mathemat-ical challenges as they differ from the stan-dard linear formulation of the Kalman Filter.Nonlinear/non-Gaussian methods of data assim-ilation are intended to provide better forecaststhat track complex geophysical processes, suchas cloud cover or sea ice extent. These can im-prove the initial conditions and boundary condi-tions used as a starting point in climate modelsimulations and aid in model development.

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Predictions of a particle trajectory in anonlinear/non-Gaussian test problem. Bench-mark particle filter (blue) and the Diffusion Ker-nel Filter (blue circles) and the traditional dataassimilation method, the extended Kalman Filter(red). True path in black. (Research and figure,P. Krause and J. M. Restrepo, U. Arizona)

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