School of Mathematics, Environmental Modeling Research Group, University of South Australia, Adelaide, Australia

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1 Uncertainty analysis of an integrated greenhouse effect model R. Zapert," J.A. Filar* & P.S. Gaertner* " World Resources Institute, 2050 Project, Washington DC, School of Mathematics, Environmental Modeling Research Group, University of South Australia, Adelaide, Australia Abstract This paper investigates the use of a stochastic dynamical system approach to assessing uncertainty of integrated global climate change models. It uses the formulation of controlled stochastic dynamical systems applied to integrated global change models: dx(t) = [F(X(t))+U(t)]dt + BdW(t), X(to) = C, (0) where X(t) is a state of the system, U(t) is anthropogenic forcing, BW(t) is white noise random excitation, and C a random initial state. The uncertainty is estimated from the distribution of process X(t). The application of this approach is illustrated with the implementation of an uncertainty analysis module of the IMAGE 1.0 (Integrated Model to Assess the Greenhouse Effect, see Rotmans [7]) greenhouse effect model within the WIM (Windows for IMAGE, see Filar et al. [4]) visualization environment. WIM provides a framework for linking global climate models such as IMAGE with regional models of the impacts of climate change and forecast analysis tools such as uncertainty model. Introduction The issues surrounding the anticipated impacts of human activities on the environment, particularly those of the enhanced greenhouse effect, will constitute significant amounts of scientific research in the coming years. The understanding of both the strengths and limitations of the tools used to assess the global change is crucial to the policy maker's ability to recommend appropriate actions. Computer models or simulation models simulate the dynamics of climate system by means of solutions of differential equations.

2 4 Computer Techniques in Environmental Studies These models constitute the majority of the tools used to forecast climatic change predictions. It is important to notice that climate forecasts generated by most models consist of only deterministic equations. Models produce forecasts without any indication of what the confidence interval, or uncertainty range, of the predicted value might be. Yet, modeling of climate change includes significant uncertainties. Deterafebtie r«r*east» F*r*ta*i* with Uat$r*al*$y Eitimfttes Figure 1: Deterministic climate forecast and forecast \vith uncertainty estimates. Atmospheric carbon dioxide concentration. Busmnes as Usual (Ball) and Accelerated Policies (AP) scenarios. Systematic analysis of the uncertainties of even simple climate models requires substantially more computing power than the model itself. Therefore, for the purpose of uncertainty analysis our attention is focused on the simplest climate models described in the next section. Integrated Assessment Models (lams) Practical considerations such as policy analysis demands directed climate modellers' attention towards more flexible models designed to handle simplified and aggregated data. Such integrated models incorporate all major components of global climate change (e.g. the carbon cycle, atmospheric chemistry, radiative forcing, and socio - economic impact modules). The main advantage of the integrated models of climate change is their speed allowing to test several environmental scenarios (e.g. greenhouse gas emission scenarios). However, they have significant limitations resulting from the simplified geography and parametrization of climate processes. Integrated models rely on the climate characteristics derived from the C-GCMs simulations. The Dutch IMAGE LO model was selected as a working example of an integrated global change model. It was developed at the RIVM by Rotmans, see Rotmans [6]. It is a coarse model based on the globally averaged variables.

3 Computer Techniques in Environmental Studies 5 The main components of IMAGE include the carbon cycle model of Goudriaan and Ketner and the ocean carbon model of Wigley and Schlesinger. IMAGE 1.0 became a precursor of more advanced environmental models, two of which are described here. The first, IMAGE 2.0, (see Alcamo f 1]) is an intermediate complexity grid - based model. The climate model within IMAGE 2.0 is part of the larger structure simulating the climate - biosphere dynamics. The second, TARGETS, see Rotmans [6] is an integrated model of global change it includes a climate model as well as population, health, and economy sub-models. For the purpose of the uncertainty analysis the IMAGE 1.0 has been reformulated as a controlled dynamical system: dx(t)/dt = F(X(t)) + U(t), X(to) = C, te[to,t], to=1990, T=2100 (1) whose state X(t) is a 155 components vector representing the most important climate variables (e.g. greenhouse gas concentrations or ocean temperature, see Braddock et al., [2] and Zapert [8]). This formulation allows one to investigate the stability, equilibrium properties, and the uncertainty of the climate system. Under the assumption that the missing sink of carbon has a linear capacity the system (1) is stable at an equilibrium point. Moreover, for given constant emission levels of greenhouse gases the steady state of (1) can be computed numerically, see Braddock et al. [2] and Zapert [8] using fixed point iteration method. Uncertainty of Forecasts of Integrated Models The problem of assessing uncertainties of climate forecasts, here represented by X(t) as in (1), can be approached by modeling the uncertainties as stochastic effects. We represent these by an additional diffusion term BW(t) in the equation (1) and a randomized initial state C. The uncertainty, defined as the 95% confidence interval, is estimated from the distribution of a stochastic process X(t). The distributions of initial state C and the additive noise process BW(t) were chosen to be compatible with the uncertainty ranges given in the literature. The main sources of uncertainty are the measurement of the initial condition, approximation of model parameters, and the random effects such as volcanic activity. El Nino cycle, and the variable cloud cover, see Zapert [8]. Stochastic version of the system equation (1), that is equation (0), can be interpreted as the vector Ito stochastic differential equation. The uncertainties of model forecasts at given times were estimated by computing the confidence intervals of X(t) or alternatively by approximating the first two moments of

4 6 Computer Techniques in Environmental Studies X(t). Demand for computing power (recall that X(t) has dimension 155) together with the fact that the explicit solutions of equation (0) are not in general available suggested the simplified approach via system linearization, see Filar & Zapert [5] and Zapert [8]. If the system dynamics is linear, that is if F(X(t)) = AX(t) in the equation (0), then the distribution of the state vector X(t) can be found explicitly. The system (0) thus becomes: dx(t)/dt = AX(t) + U(t) + BW(t), X(to) = C. (2) Under the standard assumptions about BW(t) and C, vector X(t) for any time t is normally distributed. Moreover, its covariance matrix V(t) = E[X(t)X(t)^] satisfies the following linear matrix differential equation: dv(t)/dt = AV(t) + V(t)A* + B, V(to) = E[C(5]. (3) The last equation provides an effective tool for computing the uncertainty of (2). It was found in the numerical experiments that the linearized model (2) accurately approximates its nonlinear version (0) for the period The uncertainty estimates computed using both versions agree. Moreover, the uncertainty depends very little on the implemented environmental scenario represented by the forcing term U(t). Hence, it only needs to be computed once for the entire range of forcing terms. This important property enabled the implementation of the uncertainty module within the WIM system described in the next section. WIM not only replicates the deterministic IMAGE'S forecast but also computes the associated confidence intervals for the most significant climate variables. In addition, the user can interactively change the magnitude of initial state C uncertainty and the magnitude of the random effects modelled by the diffusion matrix B. Windows for IMAGE (WIM) WIM is a software system designed to provide a framework for combining global climate models such as IMAGE 1.0 with local or regional models of the impacts of the climate change such as SEAL - modeling the impact of the sea level rise on the Louisiana coastline, see Curiel et al., [3]. WIM is based on a graphical user interface that allows users to set up simulations (by creating user defined emission scenarios), analyze and transfer results into other software products in an interactive manner within the MS Windows 3.1 environment. WIM is also a visualization tool enabling users ranging from ones with almost no computer or climate change knowledge to those of advanced knowledge to gain new insights about aspects of climate change. Features in WIM, other than IMAGE, include uncertainty analysis module, a data fitting module, report generation, and full user and scientific documentation. An example of the forecast generated by IMAGE and WIM is

5 Computer Techniques in Environmental Studies 7 presented in Figure 2. It shows different factors contributing to the global sea level rise according to the RIVM Business as Usual scenario. Components of Sea Level Rise zoza 2040 ZOGd 1. Drop In Sea Level due to Antarctic accumulator; [m. - RlXftf Business 2. Sea Level Rise due to Glaciers Melting [meters] - RIMVri Business as 3. Sea Level Rise due to Greenland Melting [meters] - Rfvtvl Business a: 4. Sea Level Rise due to Therms! E«panslcn [meters] - RIVM Business i S. Natural Trend In Sea Level Rise tfnetera] Figure 2: WIM-IMAGE. Sea Level Rise forecast - RIVM Business as Usual scenario. WIM is equipped with a comprehensive on-line documentation system written in hyper-text. It consists of two parts: the user guide and the scientific manual. The scientific manual explains in detail the assumptions, methodology, variables and equations of the underlying models. It also provides a full set of references to the relevant scientific literature. The manual can be regarded not only as the reference source but also as the learning tool in the area of climate change. Moreover, it can be argued that such a comprehensive documentation schould help in avoiding the misuse of the climate change model. The part of the scientific guide devoted to the IMAGE 1.0 model was reprinted with the permition of J. Rotmans and Kluwer Academic Publishers. Figure 3 presents an example of WIM's integrated environment including elements of scientific manual. In addition to the IMAGE 1.0 and SEAL models WIM includes two forecast analysis modules: the Uncertainty Analysis module and Data Fitting module. These modules enhance the WIMs capability as a analysis tool. The uncertainty module in the WIM package is based on the technique of estimating the uncertainty of the model forecasts via successive linearization of the system equations as described in the previous section. This module enables a WIM user to compute uncertainty ranges for the generated predictions for the period Figure 4 displays the uncertainty estimates of the atmospheric carbon dioxide concentrations for two environmental scenarios, e.g. Business as Usual and Accelerated Polices (Forced Trends), representing the two extreme approches to environmental

6 8 Computer Techniques in Environmental Studies protection during next 100 years. Confidence intervals were calculated using initial and system noise values of 1.78 (0.5%) and (0.2%) respectively. Note the scenario overlap in the period Figure 3: Methane module: Diagram of methane cycle and the forecasts of the methane concentration for standard emission scenarios. The data fitting module is designed to approximate the generated forecasts with three types of equations: linear, polynomial, and exponential. It has a facility for segmentating the time interval into four disjoint sub-intevals. The program fits functions of the same type to the forecast in different time periods. It allows the user to compare the behavior of climate variables in different time periods in terms of such quantities as annal change rate or a growth rate. Statistics describing the accuracy of approximations are displayed together with the forecasts and fitting curves. The features of this module are shown in Figure 5 using the forecast of the carbon dioxide concentration for the RIVM Business as Usual scenario with fifty year segmentation's of the time scale. The right hand side of the figure presents the averaged annual change rate (or slope) along with the correlation coefficients for each approximating line.

7 Computer Techniques in Environmental Studies 9 Figure 4: WIM - Uncertainty Analysis Module. Concentration of atmospheric CO2 with uncertainty estimates for Busimies as Usual and Forced I rends emission scenarios. Figure 5: WIM - Data Fit module. Averaged annual change rates of atmospheric Carbon Dioxide concentration.

8 10 Computer Techniques in Environmental Studies Conclusions The main objective of the present paper was to apply the techniques of both deterministic and stochastic dynamical systems to integrated climate models. It was accomplished by formulating IMAGE 1.0 as the system (1) and (0). The uncertainties associated with the model forecasts were computed by either solving Ito equations directly or by computing second moments of the solution process of the linearized system in (2). This approach to the uncertainty of the climate change forecasts exhibits the models inherent capacity to propagate errors. An estimate of the rate of growth of uncertainties of predictions may serve as an important suplementary criterion of the validity of the climate models. The second objective of the paper was to demonstrate the integration and visualization techniques and interactive use of the climate models in the WIM framework. Such an approach links global and regional models. Moreover, by visualizing the calculated forecasts using WIM's graphical user interface it allows for more comprehensive understanding of the underlying processes and easier policy analysis. WIM software is already being used at the US Environmental Protection Agency, World Resources Institute, RIVM, Greenpeace International, and numerous academic institutions as a suplemental policy and climate change analysis tool. Reference [1] Alcamo J,, (ed.), 'IMAGE 2.0: Integrated Modeling of Global Climate Change\ Kluwer, Dordrecht, The Netherlands, [2] Braddock R.D., Filar J.A., Zapert R., Rotmans J., & den Elzen M.G., * Mathematical formulation of the IMAGE greenhouse effect model', Accepted for publication in Applied Mathematical Modelling, [3] Curiel I.J., Filar J.A., & Zapert R., 'Relative Contributios of the Enhanced Greenhouse Effect on the Coastal Changes in Louisiana', Technical report Series 2, University of South Australia, School of Mathematics, Adelaide, Australia, [4] Filar J.A., Gaertner P.S., & Zapert R., 'Windows for IMAGE - User Guide\ University of South Australia, School of Mathematics, Adelaide, [5] Filar J.A. & Zapert R., 'Uncertainty Analysis of a Greenhouse Effect Model', In: C.Carraro and A. Haurie, (eds.), 'Operations Research and Environmental Management', [6] Rotmans J., "IMAGE: An Integrated Model to Assess the Greenhouse Effect", Kluwer, Dordrecht, The Netherlands, [7] Rotmans J., (ed.), 'Global Change and Sustainable Development", RIVM Report no , Bilthoven, The Netherlands, [8] Zapert R., 'Uncertainty Analysis of Enhanced Greenhouse Effect Models', PhD Thesis, University of Maryland at Baltimore County, Baltimore, 1994.