A new, path-integral method is presented for apportioning the concentrations
of pollutants predicted by a photochemical model to emissions from different
sources. A novel feature of the method is that it can apportion the
difference in a species concentration between two simulations. For example,
the anthropogenic ozone increment, which is the difference between a
simulation with all emissions present and another simulation with only the
background (e.g., biogenic) emissions included, can be allocated to the
anthropogenic emission sources. The method is based on an existing, exact
mathematical equation. This equation is applied to relate the concentration
difference between simulations to line or path integrals of first-order
sensitivity coefficients. The sensitivities describe the effects of changing
the emissions and are accurately calculated by the decoupled direct method.
The path represents a continuous variation of emissions between the two
simulations, and each path can be viewed as a separate emission-control
strategy. The method does not require auxiliary assumptions, e.g., whether
ozone formation is limited by the availability of volatile organic compounds
(VOCs) or nitrogen oxides (NO

The goal of source apportionment is to determine, quantitatively, how much different emission sources contribute to a given pollutant concentration. Source apportionment is thus a useful tool in developing efficient strategies to meet air quality standards by identifying the most important sources. If emissions are involved in only linear processes between where they are emitted and where they impact a receptor location, the concentration of the pollutant at the receptor is the sum of independent contributions from the individual emission sources. For example, one can define a tracer for each source of primary, unreactive particulate matter (PM) in an air quality model such that the sum of the tracer concentrations is the total primary PM concentration and the tracer concentrations form the source apportionment. However, if a secondary pollutant is formed by nonlinear chemical reactions, source apportionment is more complicated and, indeed, there is no unique apportionment.

Reflecting this non-uniqueness, a number of approaches have been developed
for source apportionment of secondary pollutants. The simplest approach is
source removal or the brute force method. Simulations with and without a
particular source are compared, and the changes in predicted concentrations
are assigned to emissions from that source (Marmur et al., 2006; Tong and
Mauzerall, 2008; Wang et al., 2009; Zhang et al., 2014). A related approach
is the factor-separation method, which for

Another approach involves the use of reactive tracers for individual chemical species, sources, and/or geographic regions (Yarwood et al., 1996; Dunker et al., 2002b; Mysliwiec and Kleeman, 2002; Wagstrom et al., 2008; Wang et al., 2009; Grewe et al., 2010; Butler et al., 2011; Emmons et al., 2012; Kwok et al., 2013). However, various chemical assumptions (beyond those in the chemical mechanism) are usually applied to track production of the secondary pollutant in nonlinear reactions. In addition, the source contributions are often restricted to being positive even if some primary pollutants can inhibit formation of the secondary pollutant. An exception is possible if tracers are assigned to all the chemical species and the model has an appropriate form (Grewe, 2013). Then, chemical assumptions external to the model are unnecessary, and the source contributions need not be positive.

Assignment methods trace through all the reaction pathways from products
back to parent reactants (Bowman and Seinfeld, 1994; Bowman, 2005). These
methods also require extra chemical assumptions for reactions in which a
product results from multiple reactants. Lastly, local sensitivity
coefficients have been used to apportion ozone (O

This work presents a new approach for source apportionment called the path-integral method (PIM). The PIM provides a new, direct mathematical connection between sensitivity analysis and source apportionment and a connection between source apportionment and emission-control strategies. Also, the PIM does not require additional chemical assumptions beyond those in the model itself. An important advantage of the PIM is its ability to allocate to sources a concentration increment, i.e., the difference between two simulations (base and background cases). If the anthropogenic increment is allocated to sources, the PIM requires that the base-case concentration minus the sum of the anthropogenic source contributions equals the background concentration. Other methods do not have this requirement, and thus may ascribe too much or too little importance to the anthropogenic sources. The PIM does require more computational effort than some other source apportionment methods because first-order sensitivities must be calculated at several levels of anthropogenic emissions.

The PIM is applied here to allocate the anthropogenic increments of O

The PIM is based on an exact mathematical equation that is in itself not new.
In particular, the equation is routinely used in thermodynamics (Sect. 2.3).
However, the application of the equation to atmospheric modeling is new. The
equation is the generalization to multiple variables of a familiar
relationship for a single variable, namely that the integral of the
derivative of a function (

For this work, the equation (Kaplan, 1959) takes the form

Although the focus here is on emissions, Eq. (1) can also include parameters
that scale the initial and boundary concentrations. Furthermore, if the
background case has all emissions and initial and boundary concentrations set
to zero, then

The contribution of source

The source contributions depend on the path

Because the sensitivities are integrated over the path

The path

The simplest and shortest integration path, termed the diagonal path, is
defined by

Three possible integration paths when the concentration difference
between the base (point B) and background (point b) cases is allocated to two
sources with emissions scaled by

The Gaussian numerical integration formulas have maximum precision (Isaacson
and Keller, 1966). This means that for a given number of points at which the
integrand is evaluated,

One special case is successive zero-out (SZO) of the sources. In SZO, the
emissions from one source are reduced to zero while leaving all other
emissions unchanged, then the emissions from a second source are reduced to
zero, etc. until the background case is reached. This is a path along the
edges of a hypercube in

Another special case involves expanding the sensitivities in a Taylor series
in the

The dependence of the source contributions on path has an analogy in
thermodynamics. For example, in the case of a single-component gas, the
energy

Time-dependent inputs were developed for CAMx, v6.00, configured with
two cells in a vertical column. The lower cell varied diurnally in height
from 100

Summary of daily emission rates used in the base-case simulation.

The emissions were developed from the national totals in the 2008 US National
Emission Inventory, version 3 (US EPA, 2013b) with several adjustments.
Emissions from wildfires and prescribed fires were excluded because these
vary greatly from year to year and were unusually high in 2008. Also, to
represent summer conditions, emissions from residential wood combustion were
excluded. Further, emissions of NO from lightning were added (Koo et al.,
2010). The emissions were segregated into biogenic (plus lightning) emissions
and five major source categories of anthropogenic emissions: fuel combustion,
industrial sources, on-road vehicles, non-road vehicles, and other emissions.
Vegetation and soil emissions and their speciation are from BEIS3.14 (Pierce
et al., 1998). Anthropogenic emissions of volatile organic compounds (VOCs)
from a major source category were allocated to CB6 species using speciation
profiles from SPECIATE 4.3 for one or two sub-categories of sources
comprising a significant fraction of the VOC emissions (Simon et al., 2010;
US EPA, 2013a). The annual emissions of VOC species, NO

The model and inputs are not intended to be a detailed representation of a specific urban area but rather to provide a useful platform for testing the PIM, specifically different integration formulas and the dependence of the source contributions on paths.

The concentrations of O

Results from the two-cell model simulations. Ozone and formaldehyde concentrations for the base case and the background case and the difference between them (anthropogenic increment).

The O

Dependence of the integrands for allocating O

The Gauss–Legendre formula was tested for accuracy using different numbers
of integration points and different integration variables. One set of tests,
labeled GLns, used the distance

The sum of the source contributions on the three paths was compared to the
anthropogenic concentration increment (right- vs. left-hand sides of Eq. 1)
to determine the accuracy of the formulas. Table 2 gives the mean absolute
error and mean bias of the formulas for O

Average error and bias for different numerical integration
formulas. The sum of the source contributions calculated using the formula
is compared to the anthropogenic increment of O

Table 2 also shows that the accuracy of a formula is lower for the VOCF path
than the other paths when using the same number of points. This difference
can be understood by examining the integrand in Eq. (4). Figure 3 displays
the integrands for allocating

Contributions of sources and VOC, NO

Overall, the GL3r formula for the Diag path and the GL4s formula for the
other paths give quite accurate results and were used to calculate the source
apportionments in Sect. 4.2. Figure S2 gives a comparison of the sum of the
source contributions vs.

Figure 4 presents the apportionment of

The PIM can separate the contributions of all emission species. Figure 4
shows that the CO contributions from on-road and non-road vehicles are not
negligible compared to the VOC contributions of these sources. For on-road
vehicles, the CO contributions are generally 20–45 % of the VOC
contributions, and for non-road vehicles, 10–30 %. HONO emissions are
assigned only to on-road and non-road vehicles and are small (0.8 % of
NO

Figure 5 displays the source contributions to

Apportionment of the anthropogenic O

The different results for the VOCF path can be explained by the fact that the
NO

The source contributions to

Figure S3 contains the apportionment of

As shown in Sect. 4, the PIM can allocate the difference in concentration
between two simulations to emission sources. Consequently, the PIM requires
that the base-case concentration minus the sum of the anthropogenic source
contributions (difference

Another advantage is that the PIM is based on an exact mathematical
relationship that is independent of the chemistry or model and does not
require added relationships or approximations. The PIM allows source
contributions to be either positive or negative. If the secondary pollutant
formation is inhibited by emissions of some species, source, or geographic
area, the sensitivity to these emissions will be negative for at least some
values of the scaling parameter

Once a model has been modified to calculate the first-order sensitivities,
the PIM requires only very simple post-processing of model results,
specifically, calculating a linear combination of sensitivities from
different simulations. This can be readily done with existing
post-processing packages such as the Package for Visualization of
Environmental data (PAVE) or the Visualization Environment for Rich Data
Interpretation (VERDI) (University of North Carolina, 2004, 2014). The PIM is not
focused on just one species, e.g., O

In principle, there is an infinite number of source apportionments available from the PIM. However, each source apportionment is linked to an emission-control strategy. If a control strategy is defined along with the timing of the controls, the number of source apportionments is reduced to just one.

The major disadvantage of the PIM is that it requires more computational effort than other methods because the sensitivities must be determined at several emission levels between the base and background simulations. This disadvantage is mitigated, to some degree, because the additional simulations provide information on how concentrations and sensitivities will change along the emission-control path.

The PIM has been applied in this work to a simplified configuration of CAMx that includes the nonlinear chemistry but not transport or dispersion. However, transport and dispersion do not involve nonlinear interactions among the species. Because the nonlinear dependence of the sensitivities on the integration variable (Fig. 3) is driven by the nonlinear chemistry and a full 3-D configuration should not have any other sources of nonlinearity, the number of integration points required for PIM for a 3-D configuration should be similar to the number required for the simplified configuration (three or four) (Dunker et al., 2015).

Application of the PIM to the special case involving the Taylor series
expansion, input data and emissions for the model simulations, accuracy in
allocating

Edited by: V. Grewe