A version of the direct method for calculating first-order sensitivity coefficients is extended to nonlinear, time-dependent models defined by stiff differential equations. In this approach the auxiliary equations for the sensitivity coefficients are solved separately from the model equations. Accuracy and stability are maintained by using exactly the same time steps and numerical approximations in calculating the sensitivities as are used in calculating the model solution. The decoupling procedure also greatly increases the efficiency of the method by taking advantage of the fact that the auxiliary equations for different sensitivity coefficients are quite similar. The decoupled direct method is applied to stiff chemical mechanisms for the oxidation of hydrocarbons in the atmosphere, the pyrolysis of ethane, and the oxidation of formaldehyde in the presence of carbon monoxide. Sensitivity coefficients are also calculated for the three mechanisms by a method employing Green's function and by actually varying the input parameters. Based on these results, the decoupled direct method has advantages in simplicity, stability, accuracy, efficiency, storage requirements, and program size over other methods, including those using Green's function. Specifically, the decoupled direct method is as much as a factor of 6 more efficient than a recent version of the Green's function method. Extensions of the decoupled direct method are also discussed.

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The ozone source apportionment technology (OSAT) estimates the contributions of different sources to ozone concentrations using a set of tracers for NOx, total VOCs, and ozone and an indicator that ascribes instantaneous ozone production to NOx or VOCs. These source contributions were compared to first-order sensitivities obtained by the decoupled direct method (DDM) for a three-dimensional simulation of an ozone episode in the Lake Michigan region. The cut-point for the OSAT indicator between VOC- and NOx-sensitive ozone production agrees well with the DDM sensitivities to VOC and NOx. In a ranking of the most important contributors to ozone concentrations >80 ppb, the OSAT and DDM results agreed on four of the top five contributors on average. The spatial distributions of the sensitivities and source contributions are similar, and the OSAT and DDM results for ozone >80 ppb correlate well. However, the source contributions ascribe substantially less relative importance to anthropogenic emissions and greater relative importance to the boundary concentrations than do the sensitivities. In regions where NOx inhibits ozone formation and the sensitivity is negative, the source contribution is small and positive. For the same subdivision of the emissions, the OSAT is 14 times faster than the DDM, but the DDM has greater flexibility in defining which emissions to include and generates results for species other than ozone. The first-order sensitivities explain, on average, 70% of the ozone concentrations.

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 NOx 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.