Atmosphere 2011, 2, 407-425; doi:10.3390/atmos2030407 

atmosphere

ISSN 2073-4433 

www.mdpi.com/journal/atmosphere 

Review 

Air Quality Response Modeling for Decision Support 

Daniel S. Cohan 1,* and Sergey L. Napelenok 2

1

 Department of Civil and Environmental Engineering, Rice University, 6100 Main Street MS 519, 

Houston, TX 77005, USA 

2

 U.S. Environmental Protection Agency, Research Triangle Park, NC 27711, USA; 

E-Mail: napelenok.sergey@epa.gov 

* Author to whom correspondence should be addressed; E-Mail: cohan@rice.edu; 

Tel.: +1-713-348-5129; Fax: +1-713-348-5268.

Received: 20 June 2011; in revised form: 10 August 2011 / Accepted: 17 August 2011 / 

Published: 26 August 2011 

Abstract: Air quality management relies on photochemical models to predict the responses 

of pollutant concentrations to changes in emissions. Such modeling is especially important 

for secondary pollutants such as ozone and fine particulate matter which vary nonlinearly 

with changes in emissions. Numerous techniques for probing pollutant-emission 

relationships within photochemical models have been developed and deployed for a variety 

of decision support applications. However, atmospheric response modeling remains 

complicated by the challenge of validating sensitivity results against observable data. This 

manuscript reviews the state of the science of atmospheric response modeling as well as 

efforts to characterize the accuracy and uncertainty of sensitivity results. 

Keywords: sensitivity analysis; source apportionment; instrumented models; air quality 

modeling; review 

1. Introduction 

At their essence, photochemical models are tools for estimating the pollutant concentrations that 

would result from given emissions, meteorology, and other specified conditions. Those simulated 

concentrations are critical to forecasting future pollution events, evaluating scientific understanding of 

emission inventories and atmospheric processes, assessing relationships between source emissions and 

OPEN ACCESS
Atmosphere 2011, 2 408

receptor concentrations, and reconstructing atmospheric conditions beyond the limits of available 

observations. The accuracy of simulated concentrations relative to ambient observations is a primary 

measure of a photochemical model’s performance [1].

However, for policy applications, it is often the responses of concentrations to emission changes, 

rather than the concentrations themselves, that most critically inform decision making. For example, 

modeling to demonstrate future attainment of U.S. ambient air quality standards typically compares 

concentrations under base and proposed future emissions to determine the sufficiency of an attainment 

plan [2]. Component measures for such plans can be prioritized based on sensitivity modeling of 

pollutant responsiveness to various emitters [3]. Other applications seek to apportion the origin of 

pollutant concentrations among precursor sources (source apportionment), or to quantify the benefits 

of a control policy. In each case, some form of atmospheric response modeling must be used to 

characterize the responsiveness of concentrations to emissions. 

Atmospheric response modeling is especially important for secondary pollutants which form from 

nonlinear interactions of precursor compounds. Secondary pollutants such as tropospheric ozone and 

(secondary) particulate matter (PM) have been among the leading targets of air pollution control 

efforts in recent decades due to their impacts on human health and the natural environment [4,5]. The 

nonlinear formation of these pollutants [6] necessitates considering spatially and temporally variable

meteorological and chemical conditions, typically via three-dimensional photochemical model 

simulations, to predict concentration responses to emission changes [7]. Other forcings on ambient 

pollutant concentrations may also be of interest in policy applications, such as responses due to climate 

change and changes in land use. 

A major challenge in atmospheric response modeling is evaluating the accuracy of sensitivity 

results. While simulated concentrations can be evaluated against observations from satellites, aircraft, 

and ground-based monitors, the sensitivities of concentrations to emissions are not directly observed.

Dynamic evaluation seeks to evaluate concentration-emission relationships within air quality models 

by considering periods of substantial emission changes [1]. For example, studies may compare 

simulated and observed changes in pollutant concentrations between weekdays and weekends [8,9], or 

across multiple years in which substantial emission changes have been documented [10,11]. Still, 

evaluation of simulated sensitivities is less direct than for concentrations, and is complicated by 

uncertainties in simulated emissions changes and meteorological variability. 

Here, various techniques that have been developed to probe the responses of pollutant concentrations 

to emissions perturbations in photochemical models, and their use to inform policy applications such 

as control strategy development and source apportionment are reviewed. Efforts to characterize the 

accuracy and uncertainty of concentration-emission sensitivity estimates are also reviewed. 

2. Response Modeling Methods 

A wide array of response modeling techniques has been applied to probe relationships between 

concentrations and emissions in photochemical models. Several of the most widely used methods are 

described here: brute-force, decoupled direct method, adjoint, and source apportionment. 

 
Atmosphere 2011, 2 409

2.1. Brute-Force Method 

The simplest and most widely used method for characterizing concentration response to emissions 

in a photochemical model is simply to rerun it with alternate emission rates. This “brute-force” (finite 

difference) method computes differences between concentrations in simulations with base and 

perturbed rates of emissions or other parameters, typically with other conditions held fixed unless 

interactions with climate change are being investigated. A key appeal of brute-force is its ready 

application in any photochemical model, without the substantial programming required to implement 

the other response modeling techniques. 

In some brute-force applications, modelers may construct alternate emission inventories to represent 

specific scenarios of interest to policy development or scientific investigation. For example, scenarios 

may represent emission levels under control strategies [12], or under projected climate change [13,14]. 

Other studies consider hypothetical percentage changes in emissions by species, source sector, or 

source region [15]. In those cases, it is typically assumed that the rate of concentration change per 

emission change can be linearly interpolated or extrapolated to predict the impacts of emission changes 

smaller or larger than the level explicitly modeled. More formally, suppose concentrations Ci(x,t) are 

simulated under both base (E x i,0 ( ,t)

%

) and perturbed (E x i f , ( ,t)

%

) levels of the emission component of 

interest, where Ef

%

 differs from E0

%

 by fraction ∆f: 

( ) 0

1 f E E ≡ + ∆f

% % (1)

C C E 0 0 ≡ ( )

%

(2)

C C E f f ≡ ( )

%

(3)

Notations for species (i = 1,...,N), place (x = (x, y, z)), and time (t) are dropped for convenience. 

The brute-force method would then approximate concentrations Cε

 under any other emission level Eε

%

by linearly scaling the difference in concentrations between the perturbed and base simulations: 

( ) 0

1 ε E E ≡ + ∆ε

% % (4)

0 0 ( ) f

f

ε

∆ε

≈ + −

∆

C C C C (5)

For pollutants with a nonlinear response to emission rates, the accuracy of Equation 5 will tend to 

degrade as the fractional perturbation of interest, ∆, differs more sharply from the explicitly modeled 

perturbation, ∆f. Specifically, if concentrations exhibit a saturating (concave down) response to 

precursor emissions as is typically observed for ozone, Equation 5 will tend to overestimate the benefit 

of emission reductions smaller than ∆f and underestimate the benefit of larger emission reductions, 

assuming an emission reduction (∆f < 0) was simulated in the perturbation case. However, the 

accuracy of Equation 5 is rarely examined explicitly. Selection of perturbation levels relevant to the

policy or scientific questions at hand will tend to enhance the accuracy and usefulness of brute-force

results. The accuracy of brute-force sensitivities can also be improved by applying the central 

difference method shown in Equation 6: 
Atmosphere 2011, 2 410

0 ( ) 2f f fε

ε

−

∆

≈ + −

∆

C C C C (6)

where C− f are the concentrations under emission level E− f

%

. However, this approach doubles the 

needed number of perturbation simulations, and does not address inaccuracies arising from 

extrapolations to input perturbations substantially larger than ∆f. Furthermore, the brute-force method 

has been shown to be susceptible to numerical instability due to model discontinuities and roundoff 

errors [16–18]. 

Another complication of nonlinear concentration-emission response is that the impacts of 

perturbations in multiple emission components will not be strictly additive. One approach to represent

the joint impacts of simultaneous emission perturbations is response surface modeling [19,20]. 

Brute-force modeling is conducted for a matrix of perturbation scenarios, each representing different 

levels of perturbations in the emission components. Statistical analysis of the results is then applied to 

create a response surface that estimates concentrations for any combination of perturbations in the 

emission components [19,20]. Response surface modeling was recently applied to characterize the 

nonlinear response of ozone to precursor emissions in China [21]. Response surfaces also have been 

validated to be highly accurate in “metamodeling” PM concentrations in an underlying model [20]. 

However, as is true for brute-force methods generally, response surfaces require a computationally 

burdensome number of model simulations when many input perturbations are of interest. 

2.2. Decoupled Direct Method 

Various approaches have attempted to avert the need for differencing numerous brute-force 

simulations by computing pollutant sensitivities directly within the photochemical model. Several 

approaches, including the coupled direct method and Green’s function method have been found to 

be unstable or impractical for use in comprehensive photochemical models (see Yang et al. [22] 

and references therein). However, the decoupled direct method (DDM) [23] and adjoint sensitivity 

analysis [24] have been shown to stably and efficiently compute sensitivity relationships and thus have 

been more widely implemented in three-dimensional models [16,17,22,25–28]. 

DDM operates by tracking sensitivity coefficients of all concentrations to specified model inputs or 

parameters, utilizing equations derived from those representing concentrations in the underlying model

to evolve sensitivities. The atmospheric diffusion equation (ADE) representing the transport and 

reactions of chemical compounds in photochemical models can be written in simplified form as: 

( ) ( )

t

∂

= −∇ + ∇ ∇ + + +

∂

C

uC K C R E L (7)

where u(x,t) is the wind field, K(x,t) is the turbulent diffusivity tensor, Ri(x,t) are the net rates of 

chemical production, Ei(x,t) are the emission rates, and the ellipsis denote other processes represented 

by the model. The ADE is solved by operator splitting and is subject to initial and boundary conditions 

described elsewhere [29]. 

Suppose that we are interested in how concentrations vary with perturbations in targeted model 

inputs or parameters Pj(x,t), which may be perturbed from their unperturbed values Pj

 by scaling 

factors , analogously to Equation 4: 
Atmosphere 2011, 2 411

p P P j j j j j = = + ∆ ε ε (1 ) (8)

Since our interest here is in sensitivities to emissions, note that each parameter pj is equivalent to an 

emission component E%

 in the notation of the previous section. 

Semi-normalized sensitivity coefficients ( )

( )

1

,

,

i j S x t of concentration response to pj

 are developed 

by scaling non-normalized local sensitivity coefficients (∂C/∂Pj) by the unperturbed value Pj of the 

parameter: 

( )

( )

1

j j j

j j j j

P P

p ε P ε

∂ ∂ ∂

= = =

∂ ∂ ∂

C C C S (9)

This semi-normalization simplifies calculations by placing sensitivity coefficients into the same units 

as concentrations. For the case of sensitivity to emission rates, DDM computes the evolution of (1) S j by 

substituting Equation 9 into Equation 7, yielding the equation below [22,27]: 

( )

( ) ( ) ( ) ( ) ( )

1

j 1 1 1

j j i j j t

∂

= −∇ + ∇ ∇ + + +

∂

S

uS K S FS E% L (10)

where F is the Jacobian matrix ( / F R C ik i k = ∂ ∂ ) and Ej

%

 is the base level of the emissions inventory 

component represented by sensitivity parameter pj. Alternate terms for representing sensitivities to

parameters other than emissions, as well as initial and boundary conditions for Equation 10, are 

described in other references [22,26]. DDM is a decoupled approach in that for each process 

sensitivities are updated separately from, and after, concentrations. 

Slightly different approaches have been taken to implement Equation 10 in air quality models. 

The original DDM formulation of Dunker [23,30] applies the same algorithms and internal time steps 

for chemistry in Equations 6 and 9, whereas the “DDM-3D” variation of Yang et al. [22] factorizes the 

Jacobian only once per advection time step. The former approach maximizes the accuracy and 

consistency of sensitivity results, whereas the latter enhances computational efficiency. Nevertheless, 

successful implementations of both DDM [26,31] and DDM-3D [18,22,25,27,31,32] in state-of-the-science 

photochemical models have been validated to yield computational savings and consistent results 

relative to brute-force. 

A key distinction between DDM and brute-force is that DDM predicts local sensitivity coefficients 

j

ε

∂

∂

C

, representing responsiveness to infinitesimal changes in a parameter, whereas brute-force predicts 

responses to finite changes, 

ε

∆

∆

C

. Just as the accuracy of brute-force results may degrade as 

interpolated or extrapolated beyond the explicitly modeled perturbation level, linear scaling of local

first-order DDM sensitivities may poorly represent the nonlinear impacts of large perturbations. Thus,

for highly nonlinear relationships, first-order DDM results can be applied reliably only to characterize 

local responsiveness or the impacts of small perturbations. 

Hakami et al. [27,33] showed that the applicability of DDM can be greatly expanded by 

incorporating second-order sensitivity coefficients. Their High-order Decoupled Direct Method 

(HDDM) extended the DDM-3D algorithms to compute second-order or even higher-order sensitivity 
Atmosphere 2011, 2 412

coefficients. Second-order self-sensitivity coefficients ( )

2

2

j j , 2

j

ε

∂

=

∂

C

S represent the curvature of concentration 

response to a single parameter, whereas cross-sensitivity coefficients ( )

2

2

j k,

j k ε ε

∂

=

∂ ∂

C

S characterize how 

sensitivity of concentrations to one parameter changes as a second parameter is varied. 

Validation tests confirmed the consistency of HDDM and brute-force finite differencing in 

estimating second-order local sensitivities [27,33]. More significantly, Hakami et al. showed that 

Taylor series expansions incorporating first- and second-order HDDM coefficients via the equation 

below can reliably predict ozone response to simultaneous large percentage (e.g., 50%) changes in 

multiple parameters [27,33]: 

( ) ( ) ( ) ( ) ( )

2 2

1 1 2 2 2

, 0 ,

2 2 j k

j k

ε ε j j k k j k j k j k

ε ε

ε ε ε ε

∆ ∆

C C S S S S S ≈ + ∆ + ∆ + + + ∆ ∆ (11)

where 

,

j k

Cε ε denotes concentrations when parameters Pj and Pk are perturbed by fractions j ∆ε and 

k ∆ε respectively. Cohan et al. [25] showed that Equation 11 could even be extended with reasonable 

accuracy to predict the ozone impacts of zeroing-out individual emission sources (i.e., 1 j ∆ = − ε ), 

providing results closer to source apportionment. While studies applying Equation 11 with HDDM for 

ozone have proliferated (e.g., [34–36]), no published study has yet extended DDM to second-order for 

PM, although such an extension was recently described in a conference presentation [37]. 

The majority of DDM applications have focused on the response of pollutant concentrations to 

emission changes. These include quantifications and optimizations of emission control strategy impacts, 

in some cases linked with consideration of control costs [38], health effects [39], or climate change [40]. 

Hakami et al. [33] showed that HDDM coefficients could be applied to generate isopleths of ozone 

response to simultaneous changes in VOC and NOx emissions. DDM-computed sensitivities to 

emissions have also been applied through a variety of approaches to conduct inverse modeling of 

emission inventories [41–43]. Sensitivities to reaction rate constants have been applied to assess key

reactions and uncertainties in the chemical mechanisms that drive air quality models [44,45]. All of 

these applications, though achievable using brute-force methods, benefitted from the ability of DDM to

efficiently compute sensitivities to numerous input parameters. 

2.3. Adjoint 

Brute-force and DDM each compute the sensitivities of all model outputs throughout the domain to 

specified model inputs and parameters. Thus, they are well-suited for characterizing how 

concentrations or deposition amounts everywhere are impacted by a limited number of changes in 

emissions or other parameters of interest. However, for some applications, it may be desired to probe 

how specific model outputs are influenced by numerous model parameters. For example, one may seek 

to probe how pollutant concentrations or deposition rates at a single monitor are influenced by many 

model parameters, or how a spatially aggregated metric of pollutant conditions is influenced by 

numerous emitters. As the number of emitters or input parameters of interest grows, it would quickly 

become cumbersome to compute forward sensitivities to each parameter. 
Atmosphere 2011, 2 413

For cases such as these, an adjoint of a model provides an efficient method for calculating 

sensitivity of a few model outputs to numerous model parameters. Adjoints have been implemented to 

some extent in several modern regional and global photochemical models including CMAQ [16], 

CHIMERE [46], STEM [47], GEOS-Chem [17], and others. Adjoints of atmospheric models have 

been used in a variety of applications including attainment studies of ozone in specific airsheds as well 

as for ozone and PM2.5 on continental scales [28,48,49]; inverse modeling of black carbon based on 

ground and ship measurements [50]; long-range transport of ozone [51] and black carbon [52]; and 

data assimilation of ozone and NOx measurements [47]. 

Mathematical description of an adjoint is simplified by introducing the forward tangent linear 

model (TLM), which is analogous to the formulation of DDM (Equation 10). Both DDM and TLM 

refer to the forward method for calculating sensitivity because the perturbation is carried forward 

through the various model processes and in time. 

The ADE in operator form would be: 

C M p = ( ) (12)

where the model output vector C is a function of the model input parameter vector p, and M is the 

operator matrix representing the combination of all processes in the model. 

Analogously, TLM (Equation 10) in operator form can be written as: 

δ δ C L p = ( ) (13)

where L is the Jacobian of the model operator. Furthermore, specific adjoint applications require the 

definition of a scalar function, J, based on model outputs as: 

J J = (C) (14)

In a simple application of calculating a sensitivity field of species i, J = ci, whereas in a more 

complex scenario of inverse modeling, J can be a cost function dependent on the difference between 

modeled and observed values, the difference between initial and perturbed parameters, the observation 

error covariance matrix, and possibly other terms [17]. The goal of the adjoint would then be to 

efficiently determine ∂J/∂p, or ∂ci/∂p in the simplest case. 

Similar to DDM sensitivities (Equation 9), it is possible to define the adjoint variable  as: 

i

i

J

p

λ

∂

=

∂

(15)

Then the adjoint can be defined by applying Lagrange multipliers to Equation 10 and integrating by 

parts by taking into account its initial and boundary conditions [47] as: 

( ) ( ) i T

i i i i t

λ

λ λ λ φ

∂

= −∇ − ∇ ∇ − −

∂

u K F (16)

where φi is the forcing term dependent on specific application and the definition of J. 

In practice, adjoint functions are applied to each model process in sequence, including advection, 

diffusion, chemistry, and aerosol processes. Two methods exist for adjoint implementation. If the 

underlying functions describing a particular process are differentiable and can be used directly, then a 

continuous adjoint for that process is possible. A continuous adjoint for a particular process is 
Atmosphere 2011, 2 414

constructed by differentiating the underlying mathematical equations first and then discretizing the 

adjoint equations of solution. On the other hand, if such calculations are too cumbersome or impossible 

due to the complexity of the process, then the process can be numerically discretized first and later 

differentiated to obtain a discrete adjoint. The advantage of the continuous adjoint is that it obtains an 

exact solution that may be more useful in interpreting the results, because it represents a physical 

time-dependent sensitivity behavior and explicitly assures numerical stability [53]. In contrast, a 

discrete adjoint may exhibit instability and yield a solution inconsistent with the forward model [54]. 

However, the attraction of a discrete adjoint is that it can often be derived using automatic 

differentiation tools that process the forward model code directly. Furthermore, discrete adjoint results 

may be more easily validated, because the resulting gradients do not differ from the actual gradient of 

the numerical cost function as can be the case with continuous adjoints [17]. In practice, both 

approaches are frequently used for an adjoint implementation within a particular forward model that is

solved using operator splitting, where one method may be used for transport processes and another 

for chemistry. 

Hakami et al. [48] demonstrate a potential regulatory application of adjoint modeling to calculate 

sensitivities of a nationwide U.S. ozone national ambient air quality standard (NAAQS) to precursor 

emissions of NOx and VOCs. The sensitivities are presented varying in space and time to provide a 

fully resolved source/receptor relationship for modeled ozone nonattainment. Source categories and 

locations with the greatest impact are quickly identified as potential targets of control strategies. 

2.4. Source Apportionment

Source apportionment seeks to quantify the contributions of various emission sources from specific 

geographic areas or emissions sources to pollutant levels at particular locations. While source 

apportionment in photochemical models may be conducted by a brute-force method zeroing out 

sources one by one, this may become computationally prohibitive if many emitters are of interest. 

Thus, a host of tools has been implemented to enable more efficient probing of source apportionment 

relationships in air quality models. Here, we focus solely on model-based source apportionment 

techniques, not on the wide array of observation-based source apportionment techniques such as 

chemical mass balance and positive matrix factorization [55] that characterize source contributions 

based on ambient data. 

Apportioned fractions of species are typically tracked separately from base concentrations with the 

specific intent to not alter base model predictions. Source apportionment methods are typically less 

computationally costly than zero-out brute-force simulations, but can be limiting in that they only track 

a subset of all possible source and species combinations. Typically, end-point pollutants include either 

ozone or various chemical components of particulate matter, but rarely in the same model. These methods 

consider secondary formation from precursor emissions of nitrogen oxides (NOx = NO and NO2), 

volatile organic compounds (VOCs), sulfur dioxide (SO2), and ammonia (NH3), in addition to primary 

emissions of the tracked species. Usually, boundary and initial conditions are also tracked separately. 

Various approaches have been taken in recent years to develop source apportionment methods. 

These methods are generally designed to provide information of pollutant formation from predefined 

geographic regions and/or chemical precursors. One approach for resolving emissions source 
Atmosphere 2011, 2 415

attribution for PM species is to model the particles as an explicit external mixture while preserving 

source information [56]. In this method, particulate matter precursors from each source are injected 

into the model separately and their identity is preserved as they undergo the various physical and 

chemical transformations. Modifications to the underlying chemical mechanism and the accompanying 

solver are required to accommodate precursor species duplication. A similar approach has been 

demonstrated for resolving ozone formation from tagged VOC sources [57]. Again, modifications to 

the chemistry module are required, but in this case, tracking contributions of different emissions 

sources to concentrations of various VOCs, alkoxy radical (RO), peroxy radical (RO2), and 

hydroperoxyl radical (HO2) is considered. These species dictate the conversion rate of NO to NO2, 

which in turn is used as the indicator for ozone formation. 

Source apportionment implementation for PM and ozone is complicated by the interaction of 

second-generation products from explicitly tagged precursors with one another. Such cases are 

typically handled on a reaction by reaction basis and the products are attributed to sources according to 

the linkage established through their molecular structure. 

Source oriented methods are computationally demanding, because they greatly increase not only the 

number of species transported by the model, but also the computational burden involved in solving a 

larger number of potentially stiff differential equations governing gas and aerosol chemical 

interactions. However, preserving emissions information through the model helps to overcome one of 

the main challenges of source apportionment, which is accounting for the highly nonlinear and time 

variant nature of secondary pollutant formation. Computationally lighter versions of this method track

only a subset of species. For example, the carbon apportionment model has been used as a diagnostic 

tool to track sources of chemically inert particulate organic and elemental carbon [58,59]. 

Other source apportionment techniques rely on simplifying assumptions to reduce the size of the 

numerical problem. For example, the ozone source apportionment technology (OSAT) [60] 

implemented within the Comprehensive Air quality Model with extensions (CAMx) [61] groups 

emissions of NOx and VOCs from each tracked source together with the corresponding ozone formed 

as the result of these emissions into families, and tracks these families through the various processes in 

the air quality model. OSAT adds the flexibility of apportioning ozone to NOx emissions in addition to 

VOCs, and does so by determining NOx- or VOC-limiting ozone formation regimes on the basis of the 

ratio of the production of peroxide (H2O2) to the production of nitric acid (HNO3) [62]. At ratios 

above 0.35, the ozone formed is assigned to emissions of NOx, and at ratios below 0.35 the ozone 

formed is assigned to emissions of VOCs. This method still accounts for different VOC reactivity as 

do the explicit tracking methods, but may have some limitations during instances where ozone is 

titrated through reaction with NO near large emissions sources of NOx, since instantaneous negative 

apportionment is not possible. 

As a variation on OSAT, the anthropogenic precursor culpability assessment (APCA) [61] has been 

implemented in the CAMx model. APCA functions similarly to OSAT, but reallocates the ozone 

production that was attributed to non-controllable sources to the controllable precursor that 

participated in its formation. For example, in a situation where OSAT would attribute ozone 

production to biogenic emissions of VOCs, APCA would reassign the attribution to anthropogenic 

NOx even under VOC-limiting conditions. As such, this method biases towards assigning more ozone 

formation to NOx emissions, in order not to assign culpability to presumably non-controllable biogenics. 
Atmosphere 2011, 2 416

Also stemming from the development of OSAT in CAMx is the particulate matter source 

apportionment technology (PSAT) and its complement, online particulate source apportionment 

(OPSA) [63]. Both of these methods track primary and secondary PM components by making the 

simplifying assumption that each secondary component links directly to one specific precursor. For 

example, NOx emissions lead directly to formation of particulate nitrate. Secondary effects, such as the 

nonlinear chemical and thermodynamic interactions between species are considered through the 

behavior of the bulk, non-apportioned species based only on the equilibrium assumption. Both PSAT 

and OPSA start with the gas-phase OSAT implementation to which PM relevant reactions are added. 

PSAT differs from OPSA in that it apportions the transport of the tagged species based on fluxes of the 

bulk in an offline way, instead of transporting the tagged species themselves as in OPSA. This 

introduces some error to the system, since the PSAT attributes the fluxes linearly to save on 

computational time spent on transport routines. 

The CMAQ model also has implemented methodology for particulate source apportionment in the 

way of tagged species source apportionment (TSSA), which tracks precursors to aerosol sulfate, 

nitrate, ammonium, elemental carbon, secondary organic aerosol, and other aerosol species [64]. 

Similar to the PSAT implementation in CAMx [63], this method also applies information from bulk 

species for most model transformations to the tagged species. However, gaseous chemistry is described 

more explicitly by modifying the underlying chemical mechanism and adding chemical tracers 

relevant for aerosol formation. TSSA also employs mass normalization routines that ensure mass 

conservation, as do OSAT, APCA, and PSAT. This is done by introducing an “other” source category 

automatically as a tracer in addition to user specified geographical and chemical precursor sources. 

An important consideration for interpreting results of any air quality source apportionment method 

is that there is no unique true answer to attributing pollutant concentrations to precursor sources. The 

contributions of various sources may not be strictly additive, since photochemical regime and pollutant 

responsiveness may change as emission rates vary. The emission source assigned the greatest zero-out 

source contribution may not necessarily yield the most benefit from its incremental control, and its 

contribution may vary as other sources are controlled. However, source apportionment results still 

provide a wealth of information for the purpose of control strategy development or culpability assessment. 

Other drawbacks of source apportionment techniques come from the fact that these methods try to 

strike a balance between computational efficiency and explicit treatment of all science processes done

by the base model. Ozone source apportionment techniques frequently make use of various 

assumptions to calculate whether or not the ozone producing regime is either NOx- or VOC-limited. 

Apportionment of inorganic particulate species is frequently done by ignoring or simplifying 

thermodynamic interactions between PM species. Most of the described apportionment techniques 

assume that secondary PM components are linked directly to one precursor species. For example, 

secondary sulfate is traced back only to SOx emissions. Therefore, these methods are unable to capture 

indirect effects such as the replacement of ammonium sulfate with ammonium nitrate that sometimes 

accompanies reductions of SO2 emissions [65], or the influences of atmospheric oxidants and their 

precursors on secondary organic aerosol formation [66]. 

Similar to the other response models discussed, source apportionment techniques aim to generate 

additional information about one or more modeling episodes that could be used to test changes to the 

forcings behind the model resolved state of air quality without the need for additional simulation. One 
Atmosphere 2011, 2 417

such application is detailed by Baker and Foley [67]. In their approach, the CAMx model with PSAT is 

used to estimate annual sulfate and nitrate formation from emissions of precursors from 99 power 

plants in the eastern United States. Emissions from these plants were manually perturbed, and the 

apportionment model repeated to arrive at brute-force sensitivity of source apportionment. Finally, a 

nonlinear regression model was used to describe the relationship of downwind particulate matter 

concentrations to emissions strength and distance from source. This hybrid formulation of two 

response techniques (brute-force and PSAT) together with a statistical model allows for quick 

approximation for impacts of varying emissions at particular facilities or siting a new facility in the 

modeling domain. 

3. Accuracy and Uncertainty of Atmospheric Response Estimates 

Given the critical role of atmospheric response modeling in informing air quality decision-making, 

the accuracy and uncertainty of response estimates are important to examine. All of the instrumented 

modeling techniques above can be validated against the forward model response to ensure that they 

faithfully represent pollutant-emission response within the underlying model. Most validations of 

DDM have shown it to be highly accurate in capturing pollutant response to small-scale emission 

changes. The extent local sensitivity coefficients can predict large-scale response depends on the 

nonlinearity of pollutant formation under the conditions at hand; in some situations, first-order 

sensitivities may be sufficient to predict response for up to ±30% changes in emissions [31,68], 

whereas Taylor expansions including second-order HDDM coefficients can allow accurate representation 

of ozone response to domain-wide emission reduction of at least 50% [25,33]. 

Such validations assess whether a response modeling technique is reliably capturing relationships 

within the underlying model, but do not address the accuracy of the model itself in representing actual 

atmospheric response to emission changes. Simulated pollutant concentrations are routinely evaluated 

against observations [1], but pollutant-emission sensitivities are not directly measured. While 

consistency of simulated and observed concentrations lends confidence to model performance, it does 

not guarantee the accuracy of sensitivity and source apportionment results. Different models or 

alternate settings of a single model can achieve similar performance for pollutant concentrations 

despite yielding markedly different predictions of pollutant-emission response [44,69,70]. Thus, there

is growing interest in the use of observational data to infer pollutant responsiveness to emission 

changes and thereby ground truth modeled sensitivity results. The approaches described in the 

following three paragraphs—dynamic evaluation, observational indicator ratios, and receptor-based 

source apportionment—provide informative though imperfect gauges of response modeling accuracy, 

as each of these methods entails its own set of assumptions and simplifications. 

Dynamic evaluation assesses the ability of models to represent changes in pollutant concentrations 

resulting from changes in emissions or other conditions [1]. For example, differences in pollutant 

concentrations on weekdays and weekends can be used to assess their responsiveness to weekly 

patterns in emission rates [8,9,71]. Other dynamic evaluation studies have considered how pollutant 

levels have responded to long-term trends in emission rates, such as those resulting from major 

emission control policies [10,11,72]. In each case, a key challenge is to accurately characterize the 

changes in emission rates and meteorological conditions across the periods of interest, so that the 
Atmosphere 2011, 2 418

pollutant response can be effectively isolated. However, since characterizations of emission trends and 

meteorology are inevitably imperfect, dynamic evaluation cannot definitively determine 

whether mispredicted trends in concentrations arise from those imperfections or from errors in a 

model’s responsiveness. 

Another check on modeled sensitivities is provided by observational indicator ratios, which 

consider ambient measurements of various species to infer how ozone or secondary PM was formed. 

For example, ratios of species concentrations such as O3/NOy (NOy = NOx and its oxidation products) 

and H2O2/HNO3 have been found to be effective indicators of whether ozone formed under NOx- or 

VOC-limited conditions [62,73]. More quantitative estimates of ozone responsiveness to NOx

emissions can be obtained from ozone production efficiency metrics, which quantify the number of 

ozone molecules produced per molecule of NOx consumed [74]. For PM, the observed ratio between 

free ammonia and total nitrate concentrations may indicate whether formation of wintertime nitrate is 

limited primarily by NOx or ammonia emissions [75]. Such indicator ratios already have a role in some 

source apportionment methods, but could be used further for evaluation purposes. 

For source apportionments, various receptor-based methods such as positive matrix factorization 

and chemical mass balance have been developed for apportioning PM based on observations of its 

constituent compounds [76–78]. Some studies have attempted to compare PM source apportionments 

derived from receptor-based and emission-based models [79], or to develop hybrid approaches 

combining the two methods [80]. Whereas source signatures can be inferred from speciated PM data, 

such approaches are, of course, inapplicable to ozone and other individual gases. 

Apart from comparisons with observational data to gauge accuracy of response modeling, a separate 

line of efforts has sought to characterize the uncertainty in response modeling that arises due to 

uncertain model inputs and formulations. Parametric uncertainties result from uncertain model input 

data and parameters, whereas structural uncertainties result from imperfect numerical formulations of 

physical and chemical processes [81]. Most studies conducted numerous brute-force runs with Monte 

Carlo sampled inputs to characterize changes in pollutant-emission sensitivities resulting from changes 

in uncertain inputs [82–84]. More recent studies have introduced methods utilizing HDDM and 

adjoints for more efficient uncertainty characterization [85–88]. In either case, Bayesian approaches 

could be applied to use observational data to assess the relative likelihoods of various plausible 

simulations and thereby refine uncertainty estimates [89,90]. 

4. Discussion 

The use of response modeling methods in air quality management has increased over recent years 

partly due to significant improvements to the computational capacity of management agencies, and 

partly due to the need to find more creative control strategies in response to lower air quality standards. 

While new probing tools continue to be introduced and refined, a growing number of publications 

apply existing techniques to new air quality challenges. 

As applications of response modeling proliferate through a wider community of users, care must be 

taken to ensure that the methods are applied and interpreted in an effective manner. The use of 

brute-force methods tends to be straightforward. More complex methods such as DDM or adjoint 

modeling typically require more extensive validation to ensure that results are consistent with the 
Atmosphere 2011, 2 419

forward model, both to protect against possible coding errors as well as to check the method’s 

performance in a new photochemical regime. For all methods, proper interpretation of results is crucial. 

For example, extrapolations of local sensitivity coefficients or interpolations of large-scale impacts can 

introduce error when response is highly nonlinear. There may also be important interactions between 

responses not captured by sensitivities to individual parameters. Response surface modeling and 

high-order sensitivity coefficients can each provide accurate results across a range of emission 

perturbations, but can be more complex to apply than other methods. 

The results of multiple response modeling methods can be highly complementary. For example, 

source apportionment computes the impacts of entire emission sources, whereas DDM characterizes 

sensitivities to incremental perturbations. Nevertheless, most studies apply only a single response 

modeling method, despite a handful of studies that have rigorously compared multiple methods [91,92]. 

Zhang et al. [92] showed that first-order DDM effectively simulated ozone response to NOx and VOC 

emission changes of less than 40%; OSAT was better able to simulate source apportionment 

(i.e., 100% emission reductions), but its non-negativity failed to represent VOC-limited regions where 

reductions in NOx may lead to increases in ozone concentrations. Koo et al. [91] compared the results 

of brute-force, PSAT, and DDM characterizations of particulate matter sensitivity and source 

apportionment. They found that first-order DDM could simulate responses of inorganic secondary 

aerosols to emissions perturbations of 20% or more, and of organic and primary aerosols to up to 100%.

PSAT yielded similar results to zero-out source apportionment in most cases, but did not account 

for indirect effects such as when removal of an SO2 emitter frees up oxidants to react with SO2 from 

other sources. 

The response modeling methods reviewed here characterize the responses of pollutants to emission 

changes, but do not necessarily explain the physical or chemical causes of those responses. Other 

techniques such as process analysis [93] characterize the contributions of various physical and 

chemical processes to pollutant concentrations and formation and destruction rates. Process analysis 

has been applied most frequently to quantify the processes influencing tropospheric ozone [94–96]. 

Additional insights into the causes of atmospheric response can be gleaned by substituting alternate 

mechanisms or parameterizations in air quality models. Ultimately, response probing methods 

can perform only as well as the underlying model, so ongoing improvements to the physical and 

chemical representations in atmospheric models, and the scientific understanding that underpins them, 

remain paramount.

Acknowledgements 

The work of Cohan was supported by National Science Foundation CAREER Award Grant 087386. 

Although this manuscript has been reviewed and approved for publication, it does not necessarily 

reflect the policy or views of the U.S. Environmental Protection Agency. The authors would like to 

thank Kirk Baker for helpful input on the manuscript. 
Atmosphere 2011, 2 420

References 

1. Dennis, R.; Fox, T.; Fuentes, M.; Gilliland, A.; Hanna, S.; Hogrefe, C.; Irwin, J.; Rao, S.T.; 

Scheffe, R.; Schere, K.; et al. A framework for evaluating regional-scale numerical photochemical 

modeling systems. Environ. Fluid Mech. 2010, 10, 471–489. 

2. US-EPA. Guidance on the use of models and other analyses for demonstrating attainment of air 

quality goals for ozone, PM2.5, and regional haze. EPA-454/B-07-002. April 2007. 

3. Cohan, D.S.; Boylan, J.W.; Marmur, A.; Khan, M.N. An integrated framework for multipollutant 

air quality management and its application in Georgia. Environmen. Manag. 2007, 40, 

545–554. 

4. Brunekreef, B.; Holgate, S.T. Air pollution and health. Lancet 2002, 360, 1233–1242. 

5. Fuhrer, J. Ozone impacts on vegetation. Ozone Sc. Eng. 2002, 24, 69–74. 

6. Lin, X.; Trainer, M.; Liu, S.C. On the nonlinearity of the tropospheric ozone production. 

J. Geophys. Res. 1988. 93,15879–15888. 

7. Russell, A.G. Regional photochemical air quality modeling: Model formulations, history, and 

state of the science. Annu. Rev. Energy Environ. 1997, 22, 537–588. 

8. Altshuler, S.L.; Arcado, T.D.; Lawson, D.R. Weekday vs. Weekend ambient ozone 

concentrations—Discussion and hypotheses with focus on Northern California. J. Air Waste 

Manag. Assoc. 1995, 45, 967–972. 

9. Pierce, T.; Hogrefe, C.; Rao, S.T.; Porter, P.S.; Ku, J.Y. Dynamic evaluation of a regional air 

quality model: Assessing the emissions-induced weekly ozone cycle. Atmos. Environ. 2010, 44, 

3583–3596. 

10. Gilliland, A.B.; Hogrefe, C.; Pinder, R.W.; Godowitch, J.M.; Foley, K.L.; Rao, S.T. Dynamic 

evaluation of regional air quality models: Assessing changes in O-3 stemming from changes in 

emissions and meteorology. Atmos. Environ. 2008, 42, 5110–5123. 

11. Hogrefe, C.; Civerolo, K.L.; Hao, W.; Ku, J.Y.; Zalewsky, E.E.; Sistla, G. Rethinking the 

assessment of photochemical modeling systems in air quality planning applications. J. Air Waste 

Manag. Assoc. 2008, 58, 1086–1099. 

12. Roth, P.M.; Reynolds, S.D.; Tesche, T.W. Air quality modeling and decisions for ozone reduction 

strategies. J. Air Waste Manag. Assoc. 2005, 55, 1558–1573. 

13. Tagaris, E.; Manomaiphiboon, K.; Liao, K.J.; Leung, L.R.; Woo, J.H.; He, S.; Amar, P.; Russell, 

A.G. Impacts of global climate change and emissions on regional ozone and fine particulate 

matter concentrations over the United States. J. Geophys. Res. Atmos. 2007, 112, 

doi: 10.1029/2006jd008262. 

14. Tao, Z.N.; Williams, A.; Huang, H.C.; Caughey, M.; Liang, X.Z. Sensitivity of U.S. surface 

ozone to future emissions and climate changes. Geophys. Res. Lett. 2007, 34, 

doi:10.1029/2007GL029455. 

15. Roselle, S.J.; Schere, K.L. Modeled response of photochemical oxidants to systematic reductions 

in anthropogenic volatile organic-compound and nox emissions. J. Geophys. Res. Atmos. 1995, 

100, 22929–22941. 

16. Hakami, A.; Henze, D.K.; Seinfeld, J.H.; Singh, K.; Sandu, A.; Kim, S.T.; Byun, D.W.; Li, Q.B. 

The adjoint of CMAQ. Environ. Sci. Technol. 2007, 41, 7807–7817. 
Atmosphere 2011, 2 421

17. Henze, D.K.; Hakami, A.; Seinfeld, J.H. Development of the adjoint of GEOS-Chem. Atmos. 

Chem. Phys. 2007, 7, 2413–2433. 

18. Napelenok, S.L.; Cohan, D.S.; Hu, Y.T.; Russell, A.G. Decoupled direct 3D sensitivity analysis 

for particulate matter (DDM-3D/PM). Atmos. Environ. 2006, 40, 6112–6121. 

19. Hill, J.; Polasky, S.; Nelson, E.; Tilman, D.; Huo, H.; Ludwig, L.; Neumann, J.; Zheng, H.C.; 

Bonta, D. Climate change and health costs of air emissions from biofuels and gasoline. Proc. 

Natl. Acad. Sci. USA 2009, 106, 2077–2082. 

20. US-EPA. Technical Support Document for the proposed PM NAAQS Rule: Response Surface 

Modeling. February 2006. 

21. Xing, J.; Wang, S.X.; Jang, C.; Zhu, Y.; Hao, J.M. Nonlinear response of ozone to precursor 

emission changes in China: A modeling study using response surface methodology. Atmos. Chem. 

Phys. 2011, 11, 5027–5044. 

22. Yang, Y.J.; Wilkinson, J.G.; Russell, A.G. Fast, direct sensitivity analysis of multidimensional 

photochemical models. Environ. Sci. Technol. 1997, 31, 2859–2868. 

23. Dunker, A.M. The decoupled direct method for calculating sensitivity coefficients in chemical 

kinetics. J. Chem. Phys. 1984, 81, 2385–2393. 

24. Elbern, H.; Schmidt, H.; Ebel, A. Variational data assimilation for tropospheric chemistry 

modeling. J. Geophys. Res. Atmos. 1997, 102, 15967–15985. 

25. Cohan, D.S.; Hakami, A.; Hu, Y.T.; Russell, A.G. Nonlinear response of ozone to emissions: 

Source apportionment and sensitivity analysis. Environ. Sci. Technol. 2005, 39, 6739–6748. 

26. Dunker, A.M.; Yarwood, G.; Ortmann, J.P.; Wilson, G.M. The decoupled direct method for 

sensitivity analysis in a three-dimensional air quality model—Implementation, accuracy, and 

efficiency. Environ. Sci. Technol. 2002, 36, 2965–2976. 

27. Hakami, A.; Odman, M.T.; Russell, A.G. High-order, direct sensitivity analysis of 

multidimensional air quality models. Environ. Sci. Technol. 2003, 37, 2442–2452. 

28. Martien, P.T.; Harley, R.A. Adjoint sensitivity analysis for a three-dimensional photochemical 

model: Application to Southern California. Environ. Sci. Technol. 2006, 40, 4200–4210. 

29. Mcrae, G.J.; Goodin, W.R.; Seinfeld, J.H. Development of a 2nd-generation mathematical model 

for urban air pollution. 1. Model formulation. Atmos. Environ. 1982, 16, 679–696. 

30. Dunker, A.M. Efficient calculation of sensitivity coefficients for complex atmospheric models. 

Atmos. Environ. Part A 1981, 15, 1155–1161. 

31. Koo, B.; Dunker, A.M.; Yarwood, G. Implementing the decoupled direct method for sensitivity 

analysis in a particulate matter air quality model. Environ. Sci. Technol. 2007, 41, 2847–2854. 

32. Napelenok, S.L.; Cohan, D.S.; Odman, M.T.; Tonse, S. Extension and evaluation of sensitivity 

analysis capabilities in a photochemical model. Environ. Model. Softw. 2008, 23, 994–999. 

33. Hakami, A.; Odman, M.T.; Russell, A.G. Nonlinearity in atmospheric response: A direct 

sensitivity analysis approach. J. Geophys. Res. Atmos. 2004, 109, doi: 10.1029/2003JD004502. 

34. Jin, L.; Tonse, S.; Cohan, D.S.; Mao, X.L.; Harley, R.A.; Brown, N.J. Sensitivity analysis of 

ozone formation and transport for a central California air pollution episode. Environ. Sci. Technol.

2008, 42, 3683–3689. 

35. Kim, S.; Cohan, D.S.; Byun, D.W. Contributions of inter- and intra-state emissions to ozone over 

Dallas-Fort Worth, Texas. Civ. Eng. Environ. Syst. 2009, 26, 103–116. 
Atmosphere 2011, 2 422

36. Xiao, X.; Cohan, D.S.; Byun, D.W.; Ngan, F. Highly nonlinear ozone formation in the 

Houston region and implications for emission controls. J. Geophys. Res. Atmos. 2010, 115, 

doi: 10.1029/2010JD014435. 

37. Zhang, W.; Russell, A.G. The Decoupled Direct Method for Higher-Order Sensitivity Analysis 

for Particulate Matter in Multidimensional Air Quality Models. In Proceedings of the American 

Meteorological Society Annual Meeting, Atlanta, GA, USA, January 2010; pp. 17–21. 

38. Cohan, D.S.; Tian, D.; Hu, Y.T.; Russell, A.G. Control strategy optimization for attainment and 

exposure mitigation: Case study for ozone in Macon, Georgia. Environ. Manag. 2006, 38, 451–462. 

39. Tagaris, E.; Liao, K.J.; Delucia, A.J.; Deck, L.; Amar, P.; Russell, A.G. Sensitivity of Air 

Pollution-Induced Premature Mortality to Precursor Emissions under the Influence of Climate 

Change. Int. J. Environ. Res. Public Health 2010, 7, 2222–2237. 

40. Liao, K.J.; Tagaris, E.; Manomaiphiboon, K.; Napelenok, S.L.; Woo, J.H.; He, S.; Amar, P.; 

Russell, A.G. Sensitivities of ozone and fine particulate matter formation to emissions under the 

impact of potential future climate change. Environ. Sci. Technol. 2007, 41, 8355–8361. 

41. Hu, Y.T.; Odman, M.T.; Russell, A.G. Top-down analysis of the elemental carbon emissions 

inventory in the United States by inverse modeling using Community Multiscale Air Quality 

model with decoupled direct method (CMAQ-DDM). J. Geophys. Res. Atmos. 2009, 114, 

doi: 10.1029/2009JD011987. 

42. Mendoza-Dominguez, A.; Russell, A.G. Iterative inverse modeling and direct sensitivity analysis 

of a photochemical air quality model. Environ. Sci. Technol. 2000, 34, 4974–4981. 

43. Napelenok, S.L.; Pinder, R.W.; Gilliland, A.B; Martin, R.V. A method for evaluating spatiallyresolved nox emissions using Kalman filter inversion, direct sensitivities, and space-based NO2

observations. Atmos. Chem. Phys. 2008, 8, 5603–5614. 

44. Cohan, D.S.; Koo, B.; Yarwood, G. Influence of uncertain reaction rates on ozone sensitivity to 

emissions. Atmos. Environ. 2010, 44, 3101–3109. 

45. Gao, D.; Stockwell, W.R.; Milford, J.B. Global uncertainty analysis of a regional-scale gas-phase 

chemical mechanism. J. Geophys. Res. 1996, 101, 9107–9119. 

46. Menut, L.; Vautard, R.; Beekmann, M.; Honore, C. Sensitivity of photochemical pollution 

using the adjoint of a simplified chemistry-transport model. J. Geophys. Res. Atmos. 2000, 105, 

15379–15402. 

47. Sandu, A.; Daescu, D.N.; Carmichael, G.R.; Chai, T.F. Adjoint sensitivity analysis of regional air

quality models. J. Comput. Phys. 2005, 204, 222–252. 

48. Hakami, A.; Seinfeld, J.H.; Chai, T.F.; Tang, Y.H.; Carmichael, G.R.; Sandu, A. Adjoint 

sensitivity analysis of ozone nonattainment over the continental United States. Environ. Sci. 

Technol. 2006, 40, 3855–3864. 

49. Henze, D.K.; Seinfeld, J.H.; Shindell, D.T. Inverse modeling and mapping US air quality 

influences of inorganic PM2.5 precursor emissions using the adjoint of GEOS-Chem. Atmos. 

Chem. Phys. 2009, 9, 5877–5903. 

50. Hakami, A.; Henze, D.K.; Seinfeld, J.H.; Chai, T.; Tang, Y.; Carmichael, G.R.; Sandu, A. Adjoint 

inverse modeling of black carbon during the Asian Pacific Regional Aerosol Characterization 

Experiment. J. Geophys. Res. Atmos. 2005, 110, doi: 10.1029/2004JD005671. 
Atmosphere 2011, 2 423

51. Zhang, L.; Jacob, D.J.; Kopacz, M.; Henze, D.K.; Singh, K.; Jaffe, D.A. Intercontinental source 

attribution of ozone pollution at western US sites using an adjoint method. Geophys. Res. Lett.

2009, 36, doi: 10.1029/2009GL037950. 

52. Kopacz, M.; Mauzerall, D.L.; Wang, J.; Leibensperger, E.M.; Henze, D.K.; Singh, K. Origin and 

radiative forcing of black carbon transported to the Himalayas and Tibetan Plateau. Atmos. Chem. 

Phys. 2011, 11, 2837–2852. 

53. Sirkes, Z.; Tziperman, E. Finite difference of adjoint or adjoint of finite difference? Mon. Weather 

Rev. 1997, 49, 5–40. 

54. Sandu, A.; Liu, Z. On the properties of discrete adjoints of numerical methods for the advection 

equation. Int. J. Numer. Methods Fluid. 2008, 56, 769–803. 

55. Hopke, P.K. Receptor Modeling in Environmental Chemistry; John Wiley: New York, NY, 

USA, 1985. 

56. Mysliwiec, M.J.; Kleeman, M.J. Source apportionment of secondary airborne particulate matter in 

a polluted atmosphere. Environ. Sci. Technol. 2002, 36, 5376–5384. 

57. Ying, Q.; Krishnan, A. Source contributions of volatile organic compounds to ozone formation in 

southeast Texas. J. Geophys. Res. Atmos. 2010, 115, doi: 10.1029/2010JD013931. 

58. Bhave, P.V.; Pouliot, G.A.; Zheng, M. Diagnostic model evaluation for carbonaceous PM2.5 using 

organic markers measured in the southeastern US. Environ. Sci. Technol. 2007, 41, 1577–1583. 

59. Samaali, M.; Bouchet, V.S.; Moran, M.D.; Sassi, M. Application of a tagged-species method to 

source apportionment of primary PM(2.5) components in a regional air quality model. Atmos. 

Environ. 2011, 45, 3835–3847. 

60. Dunker, A.M.; Yarwood, G.; Ortmann, J.P.; Wilson, G.M. Comparison of source apportionment 

and source sensitivity of ozone in a three-dimensional air quality model. Environ. Sci. Technol.

2002, 36, 2953–2964. 

61. ENVIRON International Corporation. User’s Guide—Comprehensive Air Quality Model with 

Extensions Version 4.50; Novato, CA, USA, 2008. 

62. Sillman, S.; He, D.Y. Some theoretical results concerning O-3-nox-VOC chemistry and 

nox-VOC indicators. J. Geophys. Res. Atmos. 2002, 107, doi: 10.1029/2001JD001123. 

63. Wagstrom, K.M.; Pandis, S.N.; Yarwood, G.; Wilson, G.M.; Morris, R.E. Development and 

application of a computationally efficient particulate matter apportionment algorithm in a threedimensional chemical transport model. Atmos. Environ. 2008, 42, 5650–5659. 

64. Wang, Z.S.; Chien, C.J.; Tonnesen, G.S. Development of a tagged species source apportionment 

algorithm to characterize three-dimensional transport and transformation of precursors and 

secondary pollutants. J. Geophys. Res. Atmos. 2009, 114, doi: 10.1029/2008JD010846. 

65. Dennis, R.L.; Bhave, P.V.; Pinder, R.W. Observable indicators of the sensitivity of PM2.5 nitrate 

to emission reductions—Part II: Sensitivity to errors in total ammonia and total nitrate of the 

CMAQ-predicted non-linear effect of SO2 emission reductions. Atmos. Environ. 2008, 42, 

1287–1300. 

66. Carlton, A.G.; Pinder, R.W.; Bhave, P.V.; Pouliot, G.A. To What Extent Can Biogenic SOA be 

Controlled? Environ. Sci. Technol. 2010, 44, 3376–3380. 

67. Baker, K.R.; Foley, K.M. A nonlinear regression model estimating single source concentrations of 

primary and secondarily formed PM2.5. Atmos. Environ. 2011, 45, 3758–3767. 
Atmosphere 2011, 2 424

68. Zavala, M.; Lei, W.; Molina, M.J.; Molina, L.T. Modeled and observed ozone sensitivity to 

mobile-source emissions in Mexico City. Atmos. Chem. Phys. 2009, 9, 39–55. 

69. Wu, S.L.; Mickley, L.J.; Jacob, D.J.; Logan, J.A.; Yantosca, R.M.; Rind, D. Why are there large 

differences between models in global budgets of tropospheric ozone? J. Geophys. Res. Atmos.

2007, 112, doi: 10.1029/2006JD007801. 

70. Thunis, P.; Rouil, L.; Cuvelier, C.; Stern, R.; Kerschbaumer, A.; Bessagnet, B.; Schaap, M.; 

Builtjes, P.; Tarrason, L.; Douros, J.; et al. Analysis of model responses to emission-reduction 

scenarios within the citydelta project. Atmos. Environ. 2007, 41, 208–220. 

71. Marr, L.C.; Harley, R.A. Modeling the effect of weekday-weekend differences in motor vehicle 

emissions on photochemical air pollution in central California. Environ. Sci. Technol. 2002, 36, 

4099–4106. 

72. Napelenok, S.L.; Foley, K.M.; Kang, D.W.; Mathur, R.; Pierce, T.; Rao, S.T. Dynamic evaluation 

of regional air quality model’s response to emission reductions in the presence of uncertain 

emission inventories. Atmos. Environ. 2011, 45, 4091–4098. 

73. Sillman, S. The use of noy, H2O2, and HNO3 as indicators for ozone-nox-hydrocarbon sensitivity 

in urban locations. J. Geophys. Res. Atmos. 1995, 100, 14175–14188. 

74. Liu, S.C.; Trainer, M.; Fehsenfeld, F.C.; Parrish, D.D.; Williams, E.J.; Fahey, D.W.; Hubler, G.; 

Murphy, P.C. Ozone production in the rural troposphere and the implications for regional and 

global ozone distributions. J. Geophys. Res. Atmos. 1987, 92, 4191–4207. 

75. Pinder, R.W.; Dennis, R.L.; Bhave, P.V. Observable indicators of the sensitivity of PM2.5 

nitrate to emission reductions—Part I: Derivation of the adjusted gas ratio and applicability at 

regulatory-relevant time scales. Atmos. Environ. 2008, 42, 1275–1286. 

76. Chow, J.C.; Watson, J.G. Review of PM2.5 and PM10 apportionment for fossil fuel combustion 

and other sources by the chemical mass balance receptor model. Energy Fuels 2002, 16, 222–260. 

77. Reff, A.; Eberly, S.I.; Bhave, P.V. Receptor modeling of ambient particulate matter data using 

positive matrix factorization: Review of existing methods. J. Air Waste Manag. Assoc. 2007, 57, 

146–154. 

78. Schauer, J.J.; Rogge, W.F.; Hildemann, L.M.; Mazurek, M.A.; Cass, G.R. Source apportionment 

of airborne particulate matter using organic compounds as tracers. Atmos. Environ. 1996, 30, 

3837–3855. 

79. Marmur, A.; Park, S.K.; Mulholland, J.A.; Tolbert, P.E.; Russell, A.G. Source apportionment of 

PM2.5 in the southeastern United States using receptor and emissions-based models: Conceptual 

differences and implications for time-series health studies. Atmos. Environ. 2006, 40, 2533–2551. 

80. Lee, D.; Balachandran, S.; Pachon, J.; Shankaran, R.; Lee, S.; Mulholland, J.A.; Russell, A.G. 

Ensemble-trained PM2.5 source apportionment approach for health studies. Environ. Sci. Technol.

2009, 43, 7023–7031. 

81. Fine, J.; Vuilleumier, L.; Reynolds, S.; Roth, P.; Brown, N. Evaluating uncertainties in regional 

photochemical air quality modeling. Annu. Rev. Environ. Resour. 2003, 28, 59–106. 

82. Bergin, M.S.; Noblet, G.S.; Petrini, K.; Dhieux, J.R.; Milford, J.B.; Harley, R.A. Formal 

uncertainty analysis of a Lagrangian photochemical air pollution model. Environ. Sci. Technol.

1999, 33, 1116–1126. 
Atmosphere 2011, 2 425

83. Deguillaume, L.; Beekman, M.; Derognat, C. Uncertainty evaluation of ozone production and its 

sensitivity to emission changes over the Ile-de-France region during summer periods. J. Geophys. 

Res. 2008, 113, doi: 10.1029/2007JD009081. 

84. Hanna, S.R.; Lu, Z.; Frey, H.C.; Wheeler, N.; Vukovich, J.; Arunachalam, S.; Fernau, M.; 

Hansen, D.A. Uncertainties in predicted ozone concentrations due to input uncertainties for the 

UAM-V photochemical grid model applied to the July 1995 OTAG domain. Atmos. Environ.

2001, 35, 891–903. 

85. Digar, A.; Cohan, D.S. Efficient Characterization of Pollutant-Emission Response under 

Parametric Uncertainty. Environ. Sci. Technol. 2010, 44, 6724–6730. 

86. Pinder, R.W.; Gilliam, R.C.; Appel, K.W.; Napelenok, S.L.; Foley, K.M.; Gilliland, A.B. 

Efficient probabilistic estimates of surface ozone concentration using an ensemble of model 

configurations and direct sensitivity calculations. Environ. Sci. Technol. 2009, 43, 2388–2393. 

87. Tian, D.; Bergin, M.S.; Cohan, D.S.; Napelenok, S.L.; Hu, Y.; Chang, M.E.; Russell, A.G. 

Uncertainty analysis of ozone formation and response to emission controls using higher-order 

sensitivities. J. Air Waste Manag. Assoc. 2010, 60, 797–804. 

88. Sandu, A.; Zhang, L. Discrete second order adjoints in atmospheric chemical transport modeling. 

J. Comput. Phys. 2008, 227, 5949–5983. 

89. Bergin, M.S.; Milford, J.B. Application of Bayesian Monte Carlo analysis to a Lagrangian 

photochemical air quality model. Atmos. Environ. 2000, 34, 781–792. 

90. Deguillaume, L.; Beekman, M.; Menut, L. Bayesian Monte Carlo analysis applied to 

regional-scale inverse emission modeling for reactive trace gases. J. Geophys. Res. 2007, 112, 

doi: 10.1029/2006JD007518. 

91. Koo, B.; Wilson, G.M.; Morris, R.E.; Dunker, A.M.; Yarwood, G. Comparison of Source 

Apportionment and Sensitivity Analysis in a Particulate Matter Air Quality Model. Environ. Sci. 

Technol. 2009. 43, 6669–6675. 

92. Zhang, Y.; Vijayaraghavan, K.; Seigneur, C. Evaluation of three probing techniques 

in a three-dimensional air quality model. J. Geophys. Res. Atmos. 2005, 110, 

doi: 10.1029/2004JD005248. 

93. Jeffries, H.E.; Tonnesen, S. A comparison of 2 photochemical-reaction mechanisms using 

mass-balance and process analysis. Atmos. Environ. 1994, 28, 2991–3003. 

94. Jang, J.C.C.; Jeffries, H.E.; Tonnesen, S. Sensitivity of ozone to model grid resolution. 2. Detailed 

process analysis for ozone chemistry. Atmos. Environ. 1995, 29, 3101–3114. 

95. Jiang, G.F.; Lamb, B.; Westberg, H. Using back trajectories and process analysis to investigate 

photochemical ozone production in the Puget Sound region. Atmos. Environ. 2003, 37, 1489–1502. 

96. Henderson, B.H.; Jeffries, H.E.; Kim, B.U.; Vizuete, W.G. The influence of model resolution on 

ozone in industrial volatile organic compound plumes. J. Air Waste Manag. Assoc. 2010, 60, 

1105–1117. 

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