The Decoupled Direct Method for Sensitivity Analysis in a Three-Dimensional Air Quality Model s Implementation, Accuracy, and Efficiency ALAN M. DUNKER,* ,† GREG YARWOOD, ‡ JEROME P. ORTMANN, † AND GARY M. WILSON ‡ Chemical and Environmental Sciences Laboratory, General Motors Research and Developm ent Center, Warren, Michigan 48090-9055, and ENVIRON International Corp., Novato, California 94945-5010 The decoupled direct method (DDM) has been implemented in a three-dimensional (3D) air quality model in order to calculate first-order sensitivities with respect to emissions and initial and boundary concentrations. This required deriving new equations for the sensitivities from the equations of the hybrid chemistry solver and the nonlinear advection algorithm in the model. The sensitivities for the chemistry and advection steps were tested in box-model and rotating- hill simulations, respectively. The complete model was then applied to an ozone episode of the Lake Michigan region during July 7-13, 1995. The DDM was found to be highly accurate for calculating the sensitivity of the 3D model. The sensitivities obtained by perturbing the inputs (brute-force method) converged toward the DDM sensitivities, as the brute-force perturbations became small. Ozone changes predicted with the DDM sensitivities were also compared to actual changes obtained from simulations with reduced inputs. For 40% reductions in volatile organic compound and/or NO x emissions,the predicted changes correlate highly with the actual changes and are directionally correct for nearly all grid cells in the modeling domain. However, the magnitude of the predicted changes is 10-20% smaller than the actual changes on average. Agreement between predicted and actual ozone changes is better for 40% reductions in initial or boundary concentrations. Calculating one sensitivity by the DDM is up to 2.5 times faster than calculating the concentrations alone. Introduction Sensitivity coefficients measure how the concentrations predicted by an air quality model depend on input data and modelparameters.As such,theyhave a varietyofuses.They can provide directional, semiquantitative, or quantitative estimates of the effects of emission changes (1, 2). When combined with estimates of uncertainties in model inputs, sensitivities provide estimates of uncertainties in the pre- dicted concentrations (3).An incrementalreactivityfactor is the sensitivity of the ozone concentration to emissions of individual volatile organic compounds (VOCs) (4, 5). Sen- sitivities can be used to estimate rate constants from experimentalmeasurements(6)or emission inventoriesfrom ambientmeasurements(7).Further,sensitivities can be used to develop a simplified representation of a model over a specific range of input variables and parameters (8). First-order sensitivities describe the linear response of the modelto a change in input parameters,and higher-order sensitivities describe the quadratic,cubic,and higher power responses. Because the number ofhigher-order sensitivities increases rapidlywith the power ofthe response it is generally practicalto calculate onlythe first-order sensitivities,though second-order sensitivities have sometimes been determined (8, 9). The major limitation of first-order sensitivities is that these describe the model response over a limited range of the input parameters. However, first-order sensitivities are stillvaluable for improvingthe efficiencyofa globalsensitivity analysis byselectingthe parameters for study(10)orguiding the Monte Carlo algorithm (11). Furthermore, the results of global sensitivity analyses have often shown that the model response to parameter changes is relatively linear (e.g., the Monte Carlo samples are analyzed by linear regression) (10, 11). Varying the input parameters one by one in separate model simulations and evaluating the change in predicted concentrations is the simplest approach to obtaining sen- sitivities (usually termed the brute-force method [BFM]). More sophisticated methods have been developed and applied to air quality models in an attempt to improve the accuracy, efficiency, and/or convenience of calculating sensitivities.The major approaches are the decoupled direct method (DDM)(12, 13),the Green’s function method (9, 14), automatic differentiation in FORTRAN (ADIFOR) (15-17), and the adjoint method (18). Avariation of the DDM, called DDM-3D, has also been developed recently (19). Of these methods, it appears that the DDM, DDM-3D, and adjoint methods have been applied to three-dimensional (3D) air quality models. The DDM was originally developed for a first-generation model of ozone formation (12). This paper describes a new implementation of the method for a current model, the Comprehensive Air Quality Model with extensions (CAMx) (20, 21). Our focus is on obtaining first-order sensitivities with respect to initial concentrations, boundary concentra- tions, and emissions. However, one can also obtain sensi- tivities with respect to any other input parameters using the DDM. We provide substantial flexibility in defining the sensitivities, e.g., we allow sensitivities to different input chemicalspeciesand to variationsin the spatialand temporal form ofthe inputs.We give details ofthe implementation for the chemistryand advection algorithmsbecause these require the key, new additions to CAMx. Results of separate, stand- alone tests for these algorithms are presented next. We also applied the complete modelto an ozone episode in the Lake Michigan region on July 7-13, 1995. The accuracy of the DDM sensitivities in this application is investigated by comparison to results from the BFM.In addition,we examine how well the first-order sensitivities describe the model response to different input perturbations. The CAMx with the DDM is publicly available (22). CAMx also includes a technique for apportioning ozone concentra- tions to emissions and initial and boundary concentrations of VOC and NOx. In a related paper (23), we compare these ozone source contributions with the corresponding ozone sensitivities obtained by the DDM. *Correspondingauthor phone: (586)986-1625;fax: (586)986-1910; e-mail: alan.m.dunker@gm.com. † General Motors Research and Development Center. ‡ ENVIRON International Corp. Environ. Sci. Technol. 2002, 36, 2965-2976 10.1021/es0112691 CCC: $22.00  2002 American Chemical Society VOL. 36, NO. 13, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 2965 Published on Web 06/04/2002