Adjoint Sensitivity Analysis for a Three-Dimensional Photochemical Model: Implementation and Method Comparison PHILIP T. MARTIEN* AND ROBERT A. HARLEY Department of Civil and Environmental Engineering, University of California, Berkeley, CA, USA 94720-1710 DAN G. CACUCI Department of Nuclear Engineering, University of California, Berkeley, California 94720-1730 and Institute for Nuclear Technology and Reactor Safety, University of Karlsruhe, 76131 Karlsruhe, Germany Photochemical air pollution forms when emissions of nitrogen oxides (NO x ) and volatile organic compounds (VOC) react in the atmosphere in the presence of sunlight. The goal of applying three-dimensional photochemical air quality models is usually to conduct sensitivity analysis: for example, to predict changes in an ozone response due to changes in NO x and VOC emissions or other model data. Forward sensitivity analysis methods are best suited to investigating sensitivities of many model responses to changes in a few inputs or parameters. Here we develop a continuous adjoint model and demonstrate an adjoint sensitivity analysis procedure that is well-suited to the complementary case of determining sensitivity of a small number of model responses to many parameters. Sensitivities generated using the adjoint method agree with those generated using other methods. Compared to the forward method, the adjoint method had large disk storage requirements but was more efficient in terms of computer processor time for receptor-based investigations focused on a single response at a specified site and time. The adjoint method also generates sensitivity apportionment fields, which reveal when and where model data are important to the target response. Introduction Near the earth’s surface, air pollutants such as ozone and particulate matter (PM) pose concerns for human health, natural ecosystems, agricultural crop yields, and the pres- ervation of building materials (1). Ozone and secondary PM form by complex nonlinear chemical processes that take place in the atmosphere. As a consequence, these air pollution problems have proved to be difficult to understand and control. Three-dimensional air quality models (AQMs) have been developed to represent the relevant coupled processes of emissions, advection, diffusion, photochemical reactions, and surface deposition (2). Such models require thousands of inputs and model parameters, and it is impossible to know a priori exactly which among the thousands of inputs are most important. Sensitivity analysis addresses the question of how the various model responses, quantities of interest derived from the model output, change with perturbations to the model data, the inputs and parameters that define the model. Sensitivity analysis produces first-order derivatives, or sensitivity coefficients, that can be used to rank the impor- tance of model data to a defined response. A number of studies have applied 3-D photochemical models to rank the reactivity of volatile organic compounds (VOC) (3-5). Other studies have investigated the sensitivity of key responses to perturbations in model parameters, such as reaction rate coefficients, to improve our understanding of the chemical mechanisms in AQMs (6). If sensitivities to all data items are available, then from estimates of the data variances and covariances one can estimate the uncertainty of a given response (7). A complete uncertainty analysis based on the use of sensitivity coefficients has not been made for a three- dimensional AQM. Previous work on estimating uncertainty in air quality models has applied simplified models and used statistical methods to generate uncertainties (8). Other studies have conducted an uncertainty analysis examining a limited set of inputs (9, 10) or using surrogates to approximate the propagation of errors across the modeling domain (11). A computationally efficient method for the case of many parameters and relativity few responses is the adjoint sensitivity analysis procedure (ASAP) (7, 12). Like the Green function (13-15), the adjoint function, also called the importance function (16), is independent of perturbations in the model data and perturbations in the resultant concen- trations. Unlike the Green function, the adjoint function depends on the specific model response selected, and thus must be recalculated for each target response. The com- pensation for this dependence on the response, however, is that the adjoint function is an N-vector instead of an N × N matrix like the Green function kernel. This greatly reduces the necessary computer memory and storage requirements of the method. Adjoint models have found wide application in the atmospheric sciences (17), particularly within meteorological models for variational data assimilation (18-21). Cho et al. (22) used the adjoint method for calculating sensitivities in an atmospheric chemistry model. Variational data assimila- tion has also been used to adjust emissions to minimize differences between predicted concentrations and measure- ments (23, 24). An adjoint model was applied to a simplified multi-box AQM for generating sensitivities (25). Sirkes and Tziperman (26) discussed the distinction between discrete and continuous formulations of adjoint models and showed that discrete models, if not properly formulated, can display numerical instabilities. Schmidt and Martin (27) used an adjoint method in a three-dimensional model to assess the importance of long-range transport of ozone and its precur- sors to the Paris region. Hess and Vukic ´evic ´(28) used a coarse- grid chemical transport model of the free troposphere and the adjoint to the model to examine the transport of a pollutant plume between Asia and Hawaii. Sandu et al. (29, 30) produced and demonstrated automatic code-generation software to facilitate direct and adjoint sensitivity analysis studies of chemical kinetic systems. Recently, Sandu et al. (31) applied these computational tools in a three-dimensional air quality model to conduct sensitivity analysis and data assimilation for an air pollution study in East Asia. The objective of this research is to implement a continuous adjoint sensitivity method in a 3-D photochemical AQM and compare the resource requirements and products obtained from the adjoint procedure to that of other sensitivity * Corresponding author e-mail: ptmartien@baaqmd.gov. Environ. Sci. Technol. 2006, 40, 2663-2670 10.1021/es0510257 CCC: $33.50  2006 American Chemical Society VOL. 40, NO. 8, 2006 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 2663 Published on Web 03/17/2006