GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-10-4245-2017Source apportionment and sensitivity analysis: two methodologies with two different purposesClappierAlainBelisClaudio A.PernigottiDeniseThunisPhilippephilippe.thunis@ec.europa.euUniversité de Strasbourg, Laboratoire Image Ville Environnement, Strasbourg, FranceEuropean Commission, Joint Research Centre, Ispra, ItalyPhilippe Thunis (philippe.thunis@ec.europa.eu)24November20171011424542564July201712July20173October20176October2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://gmd.copernicus.org/articles/10/4245/2017/gmd-10-4245-2017.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/10/4245/2017/gmd-10-4245-2017.pdf
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.
Introduction
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, NOx,
SO2, NH3 and VOC), which can be emitted by different activity
macro-sectors (e.g. residential, transport, industrial and agriculture; Seinfeld and Pandis, 2016).
Two different approaches are currently used to support air quality decision
makers: source apportionment and sensitivity analysis.
Source apportionment quantifies the contribution of an
emission source (or precursor) to the concentration of one pollutant at one
given location.
Sensitivity analysis estimates the impact on pollutant
concentration that results from a change of one or more emission sources.
In practice, source apportionment is often used for planning purposes. It is
indeed intuitive to use source apportionment to detect the activity sectors
that need to be tackled in priority in an air quality plan. On the other
hand, sensitivity analysis is often used as an approach to derive source
contributions, e.g. SHERPA (Thunis et al., 2016), FASST (Crippa et al., 2017) and
GAINS (Kiesewetter et al., 2015).
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.
Linear simplification and implications
Let us consider C a pollutant concentration at one location that is a function
of three variables (E1, E2 and E3), i.e. the emissions of
three precursors or sources within a given domain: C=CE1,E2,E3. For a linear relationship between the function C and
the three variables E1, E2 and E3, we can write
CE1,E2,E3=C0,0,0+P1E1+P2E2+P3E3,
where P1, P2 and P3 are three constant coefficients.
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
∂C∂E1=P1;∂C∂E2=P2;∂C∂E3=P3.
In Clappier et al. (2017) the coefficients (P1, P2 and P3)
are referred to as “potencies” – the authors used this concept to analyse the model response to emission changes in different
European countries.
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).
The potency equations then read as
P1=ΔC0BC1E1BC;P2=ΔC0BC2E2BC;P3=ΔC0BC3E3BC
with
ΔC0BC1=CE1BC,0,0-C0,0,0,ΔC0BC2=C0,E2BC,0-C0,0,0,ΔC0BC3=C0,0,E3BC-C0,0,0.
Together with “potencies”, Clappier et al. (2017) also introduce the
concept of “potential”, defined as the concentration change resulting from
a total reduction of the emissions (from BC to 0). The “potential” can be
calculated via relation (1) applied between the BC and background
levels as
ΔC0BC=ΔC0BC1+ΔC0BC2+ΔC0BC3,
where ΔC0BC=CE1BC,E2BC,E3BC-C0,0,0.
Equation (2) can directly be used for source apportionment purpose, with
ΔC0BC1 the concentration change resulting from a total
reduction of the emission source (or precursor) E1, reflecting the
contribution of E1 to the BC concentration. Similarly, ΔC0BC2 and ΔC0BC3 are the contributions of
E2 and E3. Equation (2) shows that, in the linear case, the
concentration change resulting from a simultaneous reduction of all emission
sources (ΔC0BC) is equal to the sum of the emission source
contributions.
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.
Brute-force method
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.
Bottom-up formulation
We consider here three precursor's emissions E1, E2 and E3, which
are changing from a low (denoted as “L”) to a high level (denoted as
“H”). In a bottom-up approach, the low emission level is chosen as the
reference. With these definitions and notation, the impact on concentration
resulting from a change of one only of the three precursor's emissions can
be written as follows:
ΔCL¯H1=CE1H,E2L,E3L-CE1L,E2L,E3LΔCL¯H2=CE1L,E2H,E3L-CE1L,E2L,E3LΔCL¯H3=CE1L,E2L,E3H-CE1L,E2L,E3L,
while the impact on concentration resulting from the simultaneous changes of
two or three precursor's emissions would be written as
ΔCL¯H1,H2=CE1H,E2H,E3L-CE1L,E2L,E3LΔCL¯H1,H3=CE1H,E2L,E3H-CE1l,E2L,E3LΔCL¯H2,H3=CE1L,E2H,E3H-CE1L,E2L,E3LΔCL¯H¯=CE1H,E2H,E3H-CE1L,E2L,E3L.
Using a similar notation, the decomposition of Stein and Alpert (1993)
applied to two variables (E1 and E2) would read as
ΔCL¯H1,H2=ΔCL¯H1+ΔCL¯H2+C^int,
where ΔCL¯H1 and ΔCL¯H2 are
the impacts induced by the change in emission sources E1 and E2
taken independently, and ΔCL¯H1,H2 is the impact
induced from E1 and E2 taken simultaneously.
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 C^int. This term, which
quantifies the interaction between the two emission sources, can be
calculated using Eq. (3) as
C^int=C^L¯H1,H2=ΔCL¯H1,H2-ΔCL¯H1-ΔCL¯H2.
The Stein–Alpert formulation can similarly be applied with three emission
sources:
ΔCL¯H¯=ΔCL¯H1+ΔCL¯H2+ΔCL¯H3+C^int,
where ΔCL¯H1, ΔCL¯H2 and
ΔCL¯H3 are the impact on concentration resulting
from single emission changes in the sources and
C^int=C^L¯H1,H2+C^L¯H1,H3+C^L¯H2,H3+C^L¯H1,H2,H3,
where C^L¯H1,H2,
C^L¯H1,H3 and C^L¯H2,H3
are the double interaction terms that can be further decomposed via Eq. (4). C^L¯H1,H2,H3 is the triple interaction
term (between E1, E2, E3), which can be decomposed by combining
Eqs. (5) and (6) as
C^L¯H1,H2,H3=ΔCL¯H¯-ΔCL¯H1-ΔCL¯H2-ΔCL¯H3-ΔCL¯H1,H2-ΔCL¯H1,H3-ΔCL¯H2,H3.
Top-down formulation
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
ΔCL¯H¯=ΔCL1H¯+ΔCL2H¯+ΔCL3H¯+C^int,
where ΔCL1H¯, ΔCL2H¯ and
ΔCL3H¯ are the impacts on concentration induced by
reducing E1, E2 and E3 independently, whereas C^int is
the interaction term, which itself can be decomposed into a series of double
interactions and a triple interaction term:
C^int=C^L1,L2H¯+C^L1,L3H¯+C^L2,L3H¯+C^L1,L2,L3H¯.
It is important to stress that the top-down single impacts are not
equivalent to their bottom-up counterparts. The relation between these
bottom-up and top-down impacts can be expressed as (here for the case of
E3)
ΔCL3H¯=CE1H,E2H,E3H-CE1H,E2H,E3L,ΔCL3H¯=CE1H,E2H,E3H-CE1L,E2L,E3L-CE1H,E2H,E3L-CE1L,E2L,E3L,ΔCL3H¯=ΔCL¯H¯-ΔCL¯H1H2.
Using Eqs. (3)–(6), Eq. (9) can be re-expressed as
ΔCL3H¯=ΔCL¯H3+C^L¯H1,H3+C^L¯H2,H3+C^L¯H1,H2,H3.
In other words, the top-down impact on concentration of an emission source
(obtained by switching off the emission source while all others remain
unchanged) is not equivalent to its bottom-up counterpart (obtained by
switching on the emission source while all others are switched off).
Equation (10) indeed clearly shows that additional interaction terms need to
be considered. The implications resulting from these differences are
highlighted in Sect. 5, in which some theoretical examples are described.
Source apportionment and sensitivity analysisTagged species techniques
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
(ΔC0BC1+ΔC0BC2+ΔC0BC3)
is equivalent to the impact on concentration resulting from a simultaneous
abatement of all sources (ΔC0BC). In such a case, the
concentration impacts are equal to source contributions and source
apportionment and sensitivity analysis lead to similar results. This is not
the case, however, when the relationship between concentrations and
emissions is nonlinear. In their approach, Stein and Alpert express the
difference between the impact caused by a simultaneous abatement and the sum
of the impacts caused by individual abatement as interactions between the
different sources. The Stein–Alpert formulation applied between the
BC and background levels is very close to Eq. (2) but with an
additional term that accounts for interactions:
ΔC0BC=ΔC0BC1+ΔC0BC2+ΔC0BC3+C^int.
Because the interaction terms cannot be attributed to a single emission
source as they represent the interaction between two or more emission
sources, the Stein–Alpert methodology does not allow one to identify the
full contribution of each individual source. It cannot therefore be used for
source apportionment purpose, unless the interaction terms are negligible as
in the linear case.
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:
ΔC0BC=δC1+δC2+δC3,
where δC1, δC2 and δC3 are the contributions
of sources E1, E2 and E3 resulting from the tagged species
approach resolution.
Tagging methodologies split the interaction terms into fractions and
attribute these fractions to the source contributions, on the basis of mass
weighting factors:
δC1=ΔC0BC1+αC^int.
Because the tagged species approach mixes interaction terms and single
concentration impacts into sources contributions, it is not suitable to
estimate the effect of emission reduction when nonlinearities are present
(Burr and Zhang, 2011a, b). Indeed, these two types of terms may react in
very different ways to emission reductions. This fact is detailed in the
examples provided below.
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).
DDM
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):
∂C∂E1;∂C∂E2;∂C∂E3.
The Taylor formula is applied at first order to calculate the concentration
change between two emission levels (denoted H and L):
ΔCL¯H¯=ΔE1∂C∂E1H+ΔE2∂C∂E2H+ΔE3∂C∂E3H
with ΔE1=E1H-E1L, ΔE2=E2H-E2L,
ΔE3=E3H-E3L.
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.
Example
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 (NO2) and ammonia, (NH3) and
formation through reactions with sulfur oxide (SO2) and NH3:
PPM→PM2NO2+H2O+12O2→2HNO32NH3+2HNO3→2NH4NO3SO2+H2O+12O2→H2SO42NH3+H2SO4→(NH4)2SO4.
These reactions pathways are summarised by the following system of
reactions:
PPM→PMPPMNO2+NH3→PMNH4NO3SO2+2NH3→PM(NH4)2SO4.
This system is further simplified by assuming that all reactions
have comparable kinetics (reaction speed) and have reached their
equilibrium. From these three reactions, 1 PM mole can be produced by 1 PPM
mole, by the combination of 1 NH3 and 1 NO2 moles or by the
combination of 1 SO2 and 2 NH3 moles.
We also limit our examples to emissions from three activity sectors. The
residential sector (R) only emits PPM and NO2, the agricultural sector
(A) only emits NH3 and the industrial sector (I) only emits PPM and
SO2 (Fig. 1). We assume for convenience that no background pollution
is present (i.e. there is no PM when all emissions are zero). Two situations
are considered: a “non-limited regime” where the NH3 quantity is
sufficient to react with all moles of NO2 and SO2 and a “limited
regime” where the NH3 quantity is not sufficient to react with
all moles of NO2 and SO2.
Non-limited regime
In this first example, the quantity of precursors (in terms of mass) is
large enough to feed all reactions. The agricultural sector emits 150
NH3 moles, which can react with 50 NO2 moles emitted by the
residential sector and 50 SO2 moles emitted by industrial sector.
One hundred
PPM moles are emitted by the residential sector as well by the industrial
sector (Fig. 1).
Example of PPM, NO2, SO2 and NH3 emissions released
by three activity sectors: residential (R), agricultural (A) and industrial
(I). For convenience, no units are associated with emissions and
concentrations.
Let us first calculate the PM concentration produced with and without each of
the sources:
No source:
C0 is the PM concentration obtained when all
emissions are set to zero. Since we assumed a zero background pollution,
C0=0.
One source only:
CR (resp. CA and CI) is the PM
concentration reached when only the residential (resp. agricultural and
industrial) sector releases emissions:
CR=100 produced by PPM emissions (NO2 emissions do not produce PM as
no NH3 is available).
CA=0 because NO2 and SO2 are not available to react with
NH3.
CI=100 produced by the PPM emissions (SO2 emissions do not produce
PM as no NH3 is available).
Two sources:
CRACRI, and CAI are the
concentrations obtained when two (out of three) activity sectors release
their emissions simultaneously (the RA subscripts correspond to residential
and agriculture, RI to residential and industrial, AI to agriculture and
industrial):
CRA=150:100 produced by PPM emissions from the residential
sector and 50 produced by the 50 NO2 released by the residential sector
reacting with the 50 NH3 emitted by agriculture (100 NH3 moles
remain unused).
CRI=200:100 produced by PPM emissions from the residential
sector and 100 produced by PPM emissions from the industrial sector.
CAI=150:100 produced by PPM industrial emissions and 50 from
the combination of 50 SO2 (industry) and 100 NH3 (agriculture).
Three sources:
CRAI is the concentrations obtained when
all emissions are released simultaneously.
CRAI=300:200 from PPM (residential and industry), 50 from reaction
between NO2 and NH3 and 50 from reaction between SO2 and
NH3.
Brute-force bottom-up (BF-BU) method
The contribution of each activity sector is calculated as the concentration
change resulting from a 100 % emission increase from the lowest emission
level (previously denoted “L” or background) to the highest level (denoted
as “H”; the BC CRAI obtained with all emissions).
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:ΔCRBU=CR-C0=100ΔCABU=CA-C0=0ΔCIBU=CI-C0=100.
With two sources:ΔCRABU=CRA-C0=150ΔCRIBU=CRI-C0=200ΔCAIBU=CAI-C0=150.
With three sources:
ΔCRAI=CRAI-C0=300.
To calculate the interaction terms, we use the Stein–Alpert formulation
using Eqs. (5) and (6):
ΔCRAI=ΔCRBU+ΔCABU+ΔCIBU+C^RABU+C^RIBU+C^AIBU+C^RAIBU,
from which the interaction terms are obtained by application of Eqs. (4) and (6):
C^RABU=ΔCRABU-ΔCRBU-ΔCABU=50C^RIBU=ΔCRIBU-ΔCRBU-ΔCIBU=0C^AIBU=ΔCAIBU-ΔCABU-ΔCIBU=50C^RAIBU=ΔCRAI-ΔCRBU-ΔCABU-ΔCIBU-C^RABU-C^RIBU-C^AIBU=0.
As can be seen from this example, the system behaves nonlinearly and the
interaction terms (e.g. C^RABU) are non-zero. Moreover, the sum
of the individual impacts (ΔCRBU+ΔCABU+ΔCIBU=200) underestimates the
overall impact (ΔCRAI=300). These results are
graphically represented in Fig. 2 (third column).
Brute-force top-down (BF-TD) method
In a BF-TD approach, the higher emission level (base case, CRAI) is the
reference and the impact of each activity sector is calculated as the
concentration change resulting from a 100 % emission decrease (of one, two
or three sources) from this reference to the background level:
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:ΔCRTD=CRAI-CAI=150ΔCATD=CRAI-CRI=100ΔCITD=CRAI-CRA=150.
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):ΔCRATD=CRAI-CI=200ΔCRITD=CRAI-CA=300ΔCAITD=CRAI-CR=200.
With three sources:
The impact resulting from the simultaneous
reduction of all three sources is similar in the top-down and bottom-up
approaches:
ΔCRAI=CRAI-C0=300.
The interaction terms can be obtained in a similar way to the bottom-up
approach by using the Stein–Alpert formulation for ΔCRAI:
ΔCRAI=ΔCRTD+ΔCATD+ΔCITD+C^RATD+C^RITD+C^AITD+C^RAITD.
The interaction terms are given by
C^RATD=ΔCRATD-ΔCRTD-ΔCATD=-50C^RITD=ΔCRITD-ΔCRTD-ΔCITD=0C^AITD=ΔCAITD-ΔCATD-ΔCITD=-50C^RAITD=ΔCRAI-ΔCRTD-ΔCATD-ΔCITD-C^RATD-C^RITD-C^AITD=0.
With this approach, a nonlinear behaviour is also observed and interaction
terms are non-zero. It is also interesting to note that the triple
interaction term (C^RAITD) is null. The sum of the individual
impacts (ΔCRTD+ΔCATD+ΔCITD=400) overestimates the
overall impact (ΔCRAI=300). We further discuss
these aspects at the end of this section. These results are graphically
represented in Fig. 2 (fourth and fifth columns).
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.
Tagged species approach
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.
ΔCRAI=δCRTAG+δCATAG+δCITAG.
The sector contributions are computed by tracking the mass of their emitted
species contributing to PM formation (in our example: PMPPM, PMNH4NO3 and
PM(NH4)2SO4):
PMPPM is formed from PPM. The 100 mol
from the residential sector lead to 100 mol of PM. The same applies to the
100 mol from industry.
PMNH4NO3 is formed by combination of NH3 and NO2. The share between
these two contributions is obtained by application of stoichiometric molar
mass ratios:
a1=NO3mNO3m+NH4m=0.78.
In our example, 50 mol of PMNH4NO3 are formed by
combination of NO2 (50 mol) from the residential sector and NH3
(50 mol) from agriculture. The contribution attributed to NO2 is
50×a1, whereas the contribution attributed to NH3 is
50×(1-a1).
PM(NH4)2SO4 is
formed by combination of NH3 and SO2. The following stoichiometric
mass ratio is used:
a2=SO4mSO4m+2NH4m=0.73.
The contribution attributed to SO2 is 50×a2, whereas the
contribution attributed to NH3 is 50×(1-a2).
The contribution of each sector is then obtained as the sum of their
precursor contribution shares as follows:
δCRTAG=100+50×a1=138.7δCATAG=50×1-a1+50×1-a2=24.9δCITAG=100+50×a2=136.4.
By definition the sum of the contributions (δCRTAG+δCATAG+δCITAG=300) is exactly equal to the overall concentration
impact (ΔCRAI=300).
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).
DDM
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:
∂C∂αRTD=150,∂C∂αATD=100,∂C∂αITD=150,
where αR, αA and αI are percentage
emission changes from the BC for the residential, agricultural and
industrial sectors.
The first-order derivatives are evaluated using finite differencing between
the BC and a level characterised by emissions that are 10 % lower
for each activity sector.
The concentration changes resulting from a 100 % emission reduction
(i.e. between the BC and the zero emission case) can be estimated by
setting αR, αA and αI to unity:
ΔCRHDDM=∂C∂αRTD=150ΔCAHDDM=∂C∂αATD=100ΔCIHDDM=∂C∂αITD=150.
We see from this last example that both the total PM and the contribution of
the sources are then comparable with those obtained by the BF-TD method.
Their interpretation is similar (Fig. 2, sixth column). In their work, Koo
et al. (2009) present a detailed comparison between a DDM and a tagged
species approach in a 3-D PM model and show which sensitivities are similar
to apportionment, and which are not.
Comparative overview
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 (δCRTAG=138; δCATAG=24; δCITAG=136) differ from the
concentration impacts resulting from a total abatement of these single
sources computed by the BF-TD (ΔCRTD=150;
ΔCATD=100; ΔCITD=150) method.
Moreover, the sum of the concentration impacts obtained with either the BF-TD or BF-BU approach for single sources does not equal the total concentration
impact (ΔCRAI=300). This is also valid for any selection
of two sectors (ΔCRTD+ΔCATD=250≠ΔCRATD=200). Note that similarly
to BF-TD, the concentration impacts computed as increases from the
background (BF-BU) show the same behaviour (ΔCRBU+ΔCABU=100≠ΔCRABU=150).
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 (C^int=0) and (2) the ratios between concentration change and
emission changes (ΔC/ΔE) remain constant,
regardless of the calculation bounds (denoted “H” and “L” in Sect. 4).
In the current example the ratios ΔC/ΔE are
constant (linear trend of ΔC in Fig. 3) but the
relationship between concentration and emission is not linear because of the
non-zero interaction terms (not shown) (C^RATD=-50 and
C^AITD=-50). However, even with zero interaction terms, we can
still observe a nonlinear behaviour with the emission reduction percentage.
The evaluation of linearity therefore requires two tests: one to quantify
the interaction terms and the second to assess the deviation from a linear
trend with respect to the emission reduction percentage.
Evolution of the concentration changes resulting from different
percentage of source abatement (top-down approach) for the three sectors
(residential, agricultural and industrial).
Limited regime
This example is similar to the previous one, except that the emissions of
NH3 are reduced from 150 to 100 mol.
When all sources release emissions, the 100 mol of NH3 are split
into 100/3 = 33.3 mol which react with NO2 to form 33.3 mol of
PMNH4NO3 and 100×2/3=66.6 mol which react with SO2 to
give 33.3 mol of PM(NH4)2SO4.
Because the mass of NH3 is not enough to react with all the NO2
and SO2 mass, 16.7 mol of NO2 and 16.7 mol of SO2 remain
unused (Fig. 4).
Note that when the agricultural source is active with only one of the two
other sources (residential or industrial), the NH3 100 mol are then
sufficient to consume all the NO2 or SO2 and lead to 50 mol of
PM in either case.
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:
C0=0;CR=100;CA=0;CI=100CRA=150;CRI=200;CAI=150.
But the result differ when all sources are active: CRAI=266.6 (200 from
PPM (residential industry), 33.3 from reaction between NO2 and NH3
and 33.3 from reaction between SO2 and NH3).
Bottom-up brute-force method (BF-BU)
The BF-BU approach computes all concentration impacts from the background
concentration (C0). The Stein–Alpert terms are similar to the
non-limited case, except for ΔCRAI and
C^RAI:
ΔCRBU=100,C^RABU=50,ΔCABU=0,C^RIBU=0,ΔCIBU=100,C^AIBU=50,ΔCRAI=266.6,C^RAI=-33.3.
The limiting effect of NH3 appears only in the negative triple
interaction term (C^RAI). These results are graphically
represented in Fig. 5 (third column).
Top-down brute-force method (BF-TD)
The top-down approach uses the base case (CRAI) concentration as
reference to compute the concentration impact. In this case, all
Stein–Alpert terms are different from the non-limited regime:
ΔCRTD=116.6,C^RATD=-16.6,ΔCATD=66.6,C^RITD=33.3,ΔCITD=116.6,C^AITD=-16.6,ΔCRAI=266.6,C^RAI=-33.3
These results are graphically represented in Fig. 5 (fourth and fifth
columns).
Tagged approach
The contribution of each source is computed similarly to the non-limited
regime. The production of 33.3 mol of PMNH4NO3 and 33.3 mol
of PM(NH4)2SO4 are
split among the different sectors using the stoichiometric coefficients
a1 and a2:
δCRTAG=100+33.3⋅a1=125.8,δCATAG=33.3⋅1-a1+33.3⋅1-a2=16.6δCITAG=100+33.3⋅a2=124.2.
These results are graphically represented in Fig. 5 (second column).
DDM
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
ΔCRHDDM=∂C∂αRTD=111.5,ΔCAHDDM=∂C∂αATD=66.7,ΔCRHDDM=∂C∂αITD=88.1.
These results are graphically represented in Fig. 5 (sixth column).
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.
Comparative overview
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 % reduction for the abatement of industrial emissions. This
discontinuity corresponds to a change of chemical regime. Below the 50 %
reduction level, the quantity of NH3 is not sufficient to feed the
reactions with NO2 and SO2 (with no SO2 reduction, 50 mol
of NO2 and 50 mol of SO2 would require 150 mol of NH3 but
only 100 are available) while beyond this 50 % reduction level the
quantity of NH3 is then enough to feed the reactions with NO2 and
SO2 (with 50 %SO2 reduction, 50 mol of NO2 and 25 mol
of SO2 requires 100 mol of NH3).
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 % emission reductions (columns 4 and 5 in Fig. 5). We
expect that lower percentage emission reductions generate less nonlinearity
and lead to a better agreement between the BF-TD and the DDM method (columns 5 and 6 in Fig. 5).
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.
Conclusions
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 % are only valid for exactly that percentage of
reduction, and extrapolation to air quality planning with any other emission
reduction levels would be inappropriate, unless additional scenarios are
tested. Along the same line of reasoning, the importance of the nonlinear
interaction terms (among precursors) should be quantified as well when
assessing the impact of more sources or precursors. Finally, these
nonlinear interaction terms are in most cases not constant with the
emission reduction intensities, which imposes the further need to
quantify them for different levels of emission reduction. Calculating
sensitivities and interactions at various level of emission reductions seems
the only alternative when nonlinearities are important. In this respect,
new approaches like the path-integral methodology proposed by Grewe et al. (2012) might represent a powerful approach.
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.
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