This work reviews the existing methodologies for source apportionment and sensitivity analysis to identify key differences and stress their implicit limitations. The emphasis is laid on the differences between source “impacts” (sensitivity analysis) and “contributions” (source apportionment) obtained by using four different methodologies: brute-force top-down, brute-force bottom-up, tagged species and decoupled direct method (DDM). A simple theoretical example to compare these approaches is used highlighting differences and potential implications for policy. When the relationships between concentration and emissions are linear, impacts and contributions are equivalent concepts. In this case, source apportionment and sensitivity analysis may be used indifferently for both air quality planning purposes and quantifying source contributions.

However, this study demonstrates that when the relationship between emissions and concentrations is nonlinear, sensitivity approaches are not suitable to retrieve source contributions and source apportionment methods are not appropriate to evaluate the impact of abatement strategies. A quantification of the potential nonlinearities should therefore be the first step prior to source apportionment or planning applications, to prevent any limitations in their use. When nonlinearity is mild, these limitations may, however, be acceptable in the context of the other uncertainties inherent to complex models.

Moreover, when using sensitivity analysis for planning, it is important to note that, under nonlinear circumstances, the calculated impacts will only provide information for the exact conditions (e.g. emission reduction share) that are simulated.

When pollutant concentrations exceed the thresholds set in the legislation,
competent authorities must take actions to abate pollution. Those abatement
strategies consist in reducing the precursor's emission of the different
activity sector to reduce pollutant concentrations but they are challenging
to design because of the complex relationships that link emissions and
pollutants. Indeed, the concentration of a pollutant at a given location
generally results from direct emissions and from interactions in the
atmosphere among different emission precursors, emitted by a variety of
sources. For example, particulate matter (denoted here as PM) results from
the interaction and combination of five different precursors (PPM, NO

Two different approaches are currently used to support air quality decision
makers: source apportionment and sensitivity analysis.

Source apportionment quantifies the

Sensitivity analysis estimates the

The main objective of this work is to review the existing methodologies, identify key differences and stress their implicit limitations. We particularly focus on the differences between concentration “impacts” (sensitivity) and “contributions” (source apportionment) obtained with different methodologies. We make use of a simple theoretical example to compare the approaches, highlight differences and potential implications in terms of policy. In the following sections, we analyse first how these methodologies work in a simple linear case before generalising it to more complex nonlinear situations.

Let us consider

On the other hand, the sensitivity of the concentration to a change of a
given emission source can be quantified via partial derivatives. For
Eq. (1) this gives

The consequences of a linear relationship between concentration and emission
sources are twofold:

All higher-order derivatives (order 2 and beyond) are null, including those involving two or more emission sources (crossed derivatives), as the impact of a change in one emission source is independent from all others.

The first-order partial derivatives are constant and can therefore be calculated with finite differencing, between any couple of emission levels, for example a base case (denoted BC) and a background (denoted as 0).

Equation (2) can directly be used for source apportionment purpose, with

In the next sections, we will explore how this simple conclusion changes when nonlinear relationships are considered. In particular, we will assess which implications (and limitations) these nonlinearities have in terms of source apportionment and sensitivity analysis.

The “brute-force” method consists in estimating the concentration change by performing and subtracting two simulations, one with and the second without a specific emission source to be analysed (Blanchard, 1999; Yarwood et al., 2004).

In nonlinear situations, the concentration change resulting from a set of emission sources is no longer equivalent to the sum of the concentration changes resulting from emission sources changed individually. In the following, we refer to the work of Stein and Alpert (1993) who proposed an approach to decompose an overall impact into single (one emission source only) and combined (multiple emission sources) impacts.

We consider here three precursor's emissions

It is clear from Eq. (3) that the impact of a simultaneous change of two
emission sources is not equivalent to the sum of the individual impacts, as
highlighted by the additional term

In a top-down formulation, the highest emission level is chosen as
reference. The Stein–Alpert formulation for three precursors can then be
expressed similarly to the bottom-up formulation as

Equation (2) shows that, when the relationship between concentration and
several emission sources is linear, the contribution of a specific source
(source apportionment) can be computed as the impact on concentration
obtained by a full reduction of this source (sensitivity). Moreover, the sum
of the impacts on concentration obtained by reduction of the single sources
(

Unlike the Stein–Alpert methodology, the tagged species methodology is designed for source apportionment purposes. This methodology tags each precursor and quantifies its contribution (in terms of mass) to the pollutant concentration.

Tagged algorithms are implemented in several chemical transport model systems (Yarwood et al., 2004; Wagstrom et al., 2008; ENVIRON, 2014; Bhave et al., 2007; Wang et al., 2009; Kranenburg et al., 2013).

In tagging approaches, the effect of the full reduction of all sources is
directly expressed as the sum of the source contributions:

Tagging methodologies split the interaction terms into fractions and
attribute these fractions to the source contributions, on the basis of mass
weighting factors:

On the other hand, the strength of this method is that it allows for a direct comparison of the source contributions with measurements (or measurement-based methods like receptor models).

Note that similar tagging methods are also used in the frame of climate–chemical studies at the global scale (e.g. Horowitz and Jacob, 1999; Lelieveld and Dentener, 2000; Meijer et al., 2000; Grewe, 2004; Gromov et al., 2010; Butler et al., 2011; Emmons et al., 2012; Grewe et al., 2012, 2017).

The decoupled direct method (DDM) is designed to calculate directly
sensitivities to emission changes (Dunker et al., 1984,
2002). It aims to compute the first-order derivatives (which correspond to
the potencies mentioned in Sect. 2):

In the linear case, the first-order derivatives are constant and the first-order approximation of the Taylor formula gives the exact expression of the impact on concentration of an emission change between H and L. When the emission-concentration relationship is nonlinear, the first derivatives are not constants. The first-order Taylor formula cannot take into account the nonlinear effects. It is a linear approximation based on derivatives computed at a given emission reference level (level H in our example). The estimation of the impact on concentration of an emission change between H and L is accurate enough if level L is close enough to level H.

HDDM is another method (Hakami et al., 2003) which aims to increase the accuracy of the DDM method by computing second-order derivatives.

DDM (and HDDM) gives similar information to the Stein–Alpert formulation applied with the brute-force top-down approach (because the reference level is H). For the same reason as for the Stein–Alpert approach, these two methods are suitable for source apportionment purpose only if the relation between concentration and emission is close to linearity.

DDM (and HDDM) approximates the impact on concentration from an emission change between the two levels H and L, using derivatives computed at level H. This impact is accurate enough if level L is close enough to the reference level H.

Dunker (2015) showed how to use first-order sensitivity to determine source contributions between two model cases – e.g. to apportion the difference between the current atmosphere (and natural conditions) to specific human activities. Along the same lines, Simon et al. (2013) used first-order sensitivity to construct emission response surfaces. To cope with potential nonlinearities and the need to compute higher-order derivatives, a powerful alternative is to compute first-order sensitivities at several emission levels.

In this section, examples are designed to illustrate the differences in
terms of contribution and impact estimates when the approaches discussed
previously are used. In these examples, we focus on the formation of
PM in the atmosphere and only consider three formation
processes: direct emissions (primary PM denoted as PPM), formation through
reactions with nitrogen oxides (

We also limit our examples to emissions from three activity sectors. The
residential sector (R) only emits PPM and

In this first example, the quantity of precursors (in terms of mass) is
large enough to feed all reactions. The agricultural sector emits 150

Example of PPM,

Let us first calculate the PM concentration produced with and without each of
the sources:

No source:

One source only:

Two sources:

Three sources:

The contribution of each activity sector is calculated as the concentration
change resulting from a 100

In a bottom-up approach, the concentration associated with the lowest
emission level is considered as the reference. Concentration impacts are
then computed by the difference between any situation (e.g. one, two or
three sources present) and this reference:

With one source:

With two sources:

With three sources:

In a BF-TD approach, the higher emission level (base case,

With one source:

When all emissions from one sector are reduced
(e.g. residential), the other two sector remain active (agricultural and
industry). In this case, the top-down impact is the difference between the
base case concentration and the concentration resulting from the
agricultural and industrial emissions only. A similar reasoning can be made
for all sectors:

With two sources:

The top-down impact due to a full reduction
of two sectors (e.g. residential and agriculture) is similarly computed as
the difference between the base case concentration and the concentration
resulting from the remaining sector (industry):

With three sources:

The impact resulting from the simultaneous reduction of all three sources is similar in the top-down and bottom-up approaches:

Schematic representation of the allocation of PM to its sources in the non-limited example. The expected total PM is displayed in the grey bar on the left.

Compared to brute force, the tagged species approach calculates the share of
each source to the overall concentration change. These shares are referred
to as contributions and have the main property that the sum of the
individual contributions is equal to the overall concentration impact, by
definition, i.e.

In our example, 50 mol of

The contribution attributed to

Note that a decomposition of the nonlinear interaction terms obtained in the top-down or bottom-up approach (using the above stoichiometric ratios) would lead to similar results as for the tagged approach. These results are graphically represented in Fig. 2 (second column).

In this methodology, delta concentrations and interaction terms are
estimated with first-order partial derivatives computed from the highest
emission level (base case in our example). Being a sensitivity approach
using level H as reference, DDM shows clear analogies with the BF-TD:

The first-order derivatives are evaluated using finite differencing between
the BC and a level characterised by emissions that are 10

The concentration changes resulting from a 100

In the linear case (second paragraph) we have seen that a single source
contribution can be computed as the impact resulting from a full reduction
of this source. However, source contributions and concentration impacts
should not be confused as they are different in most situations. The example
presented in this paragraph illustrates this clearly for a nonlinear
system. Indeed the contributions of the single sources computed by the
tagged species approach (

Figure 3 shows that the impact on concentration is proportional to the
emission reduction indicating that the relationship between emission and
concentration changes is linear. However, this example also illustrates the
fact that linearity encompasses two aspects: (1) the interaction terms are
zero (

Evolution of the concentration changes resulting from different percentage of source abatement (top-down approach) for the three sectors (residential, agricultural and industrial).

This example is similar to the previous one, except that the emissions of

When all sources release emissions, the 100 mol of

Note that when the agricultural source is active with only one of the two
other sources (residential or industrial), the

Example with three sources in an ammonia-limited regime. The mass emitted by each source is expressed in moles.

The PM concentrations obtained when one or two sources are active are
similar to the previous example:

The BF-BU approach computes all concentration impacts from the background
concentration (

The top-down approach uses the base case (

The contribution of each source is computed similarly to the non-limited
regime. The production of 33.3 mol of

As shown below, DDM only considers first derivatives, which are not suitable
to estimate higher-order interaction terms. The calculation of the first-order derivatives in this example gives

Schematic representation of the allocation of PM to its sources in the ammonia-limited example. The expected total PM is displayed in the grey bar on the left.

The main difference with respect to the non-limited regime is the appearance of a triple interaction term that will also lead to differences between the BF-TD and the DDM approaches, given the fact that the latter only accounts for first-order terms.

In comparison to the non-limited regime, the calculation of the
concentration impacts resulting from different percentage of emission
reduction shows nonlinear trends (Fig. 6). A discontinuity appears at
50

Evolution of the concentration changes resulting from different percentage of source abatement (top-down approach) for the three sectors (residential, agricultural and industrial).

The methodologies presented in this section aim at decomposing the impact of
an ensemble of sources into different terms attributed to each of the individual
sources. The terms computed by methodologies designed for source
apportionment (like TAG) are named source contributions. Their sum is always
equal to the combined impact of all sources. On the other hand, the terms
computed by sensitivity analysis represent the emission change of each
individual source and their sum is equal to the combined impact of all
sources only if the relationship between emissions and concentrations is
linear (see Sect. 2). Grewe at al. (2010) and Grewe (2013), who used simple
differential equations to reproduce the ozone tropospheric chemistry, also
highlighted this point in their work. In nonlinear situations, the source
contributions computed for source apportionment and the source impacts
computed for sensitivity analysis are different (see Fig. 5, where column 2
shows different results than column 3 or 4). Nonlinearity also implies that
the calculation of the source impacts depends on the bounds used to
estimate the concentration changes (denoted “H” and “L” in Sect. 4).
The BF-BU and BF-TD approaches (columns 3 and 4 in Fig. 5) give different results
because they are not using the same reference level (“L” for the BU and
“H” for the TD as defined in Sect. 4). Moreover, the results depend from
the percentage of emission changes applied to calculate the source impacts
as demonstrated by the different source impacts computed with the BF-TD for
100 and 25

In synthesis, the second example illustrates that all the methodologies tested to find source contributions and source impacts give different results when the relationship emissions–concentrations is nonlinear. A quantification of the potential nonlinearities should therefore be the first step prior to source apportionment or planning applications, to prevent any limitations in their use. When nonlinearity is mild, these limitations may, however, be acceptable in the context of the other uncertainties inherent to complex models.

In this work, we compared source apportionment and sensitivity approaches and investigated their domain of application. While sensitivity analysis refers to impacts to characterise the concentration change resulting from a given emission change, source apportionment aims to quantify contributions by attributing a fraction of the pollutant concentration to each emission source. In the case of linear (or close to linear) relationships between concentration and emissions, impacts and contributions are equivalent (or close to) concepts. Source apportionment may then be used for air quality planning purposes and, vice versa, sensitivity analysis may be used for quantifying sources contributions.

In many cases, however, linearity is not a valid assumption. In such cases, sensitivity approaches cannot be used to retrieve source apportionment information, unless nonlinear interaction terms are explicitly accounted for. On the other hand, source apportionment approaches (e.g. tagged species approach) intrinsically account for these nonlinear interactions into their source contributions. But because it mixes interaction terms and impacts, which may react in opposite directions, source apportionment should not be used to evaluate the impact of abatement strategies.

Even when using sensitivity analysis for planning, it is important to note
that, under nonlinear conditions, the calculated impacts will only provide
information for the exact conditions that are considered. Impacts for an
emission reduction of 50

Fortunately, not all cases are so complex as to require the full quantification of all nonlinear interaction terms. Thunis et al. (2015) showed that for yearly average relationships between emission and concentration changes, linearity is a realistic assumption, implying the possible use of source apportionment and sensitivity analysis for both purposes. Some integrated assessment tools (e.g. GAINS, SHERPA) take advantage of this assumption to retrieve source apportionment information from calculated chemistry transport model sensitivities. Although nonlinearities are important for short-term time averages (e.g. daily means, episodes), they are likely not associated with every process. For instance, nonlinear interactions are expected to be more relevant for secondary pollutants, especially under limited regimes. The challenge consists, therefore, in screening the system for significant nonlinearities and accounting for them by calculating explicitly the relevant nonlinear interaction terms.

One main strength of source apportionment approaches is to provide contribution estimates that can be cross-validated with source apportionment derived from measurements (i.e. receptor modelling; for a detailed description, see e.g. Belis et al., 2013). This step is crucial for the evaluation of chemistry transport models.

No specific code is attached to this work as all presented examples can easily be replicated.

The authors declare that they have no conflict of interest.