Introduction
Atmospheric chemistry transport models (ACTMs) provide scientific support
for policy development. It is therefore important to have a quantitative
understanding of the levels of uncertainty associated with model outputs
(AQEG, 2015; Frost et al., 2013;
Rypdal and Winiwarter, 2001). Sensitivity and uncertainty analyses are both
used in this regard. Uncertainty analysis is applied to quantify the propagation of uncertainties of single or multiple inputs through to a model output,
whilst sensitivity analysis is used to investigate input–output
relationships and to apportion the variation in model output to the
different inputs. However, due to the complexity of ACTMs, the relationship
between model inputs and outputs is not analytically tractable, so both
quantities must be estimated by sampling model inputs according to an
experimental design and undertaking multiple model simulations
(Dean et al., 2015;
Norton, 2015; Saltelli et al., 2000; Saltelli and Annoni, 2010).
Typically, model assessment studies focus on uncertainties in the model
parameter values
(Derwent, 1987; Konda
et al., 2010; De Simone et al., 2014) and model-specific structure
(Simpson et al., 2003; Thompson and Selin,
2012). However, for ACTMs the uncertainty in the model input emissions data
could be dominating; for example, previous dispersion model uncertainty
studies identified input emissions as a primary source of uncertainty in
model outputs
(Bergin
et al., 1999; Hanna et al., 2007; Sax and Isakov, 2003). It is also the case
that a major role of ACTMs is to estimate the impact of potential future
changes in emissions on atmospheric composition
(Boldo
et al., 2011; Crippa et al., 2016; Heal et al., 2013; Vieno et al., 2016;
Xing et al., 2011; Zhang et al., 2010).
Thus, the focus of this study is to demonstrate a systematic approach for
quantifying model output sensitivity and uncertainty as a function of the
variation in model input emissions. We used the Fine Resolution Atmospheric
Multi-pollutant Exchange (FRAME) model as a case study. FRAME is a
Lagrangian model that, at a 5 km × 5 km horizontal
resolution over the UK, outputs annual average surface concentrations of sulfur
dioxide (SO2), nitrogen oxides (NOx), ammonia (NH3), nitric
acid (HNO3), particulate ammonium (NH4+), sulfate
(SO42-), and nitrate (NO3-), together with dry and wet
deposition of oxidised sulfur (SOy), oxidised nitrogen (NOy), and
reduced nitrogen (NHx)
(Dore et
al., 2012; Matejko et al., 2009; Singles et al., 1998). The model is
extensively used to provide policy support including generation of
source–receptor matrices for the UK Integrated Assessment Model (UKIAM) and
the estimation of critical load exceedances
(Matejko et al., 2009;
Oxley et al., 2013). Source–receptor matrices link concentration and
deposition with individual emission sources and are used to automate
procedures to estimate the impact of future emission reduction scenarios.
Integrated assessment modelling incorporates technical emissions abatement
costs with cost–benefit analysis and source–receptor data to indicate
cost-effective solutions to protect natural ecosystems from acidic and
nitrogen deposition above defined critical thresholds and to protect human
health from particulate concentrations
(Oxley et al., 2003, 2013).
FRAME uses emissions input data from the UK National Atmospheric Emissions
Inventory (NAEI; http://naei.beis.gov.uk/, last access: 30 October 2016), which are compiled following the international Guidelines for
Reporting Emissions and Projections Data under the Convention on Long-range
Transboundary Air Pollution (United Nations Economic Commission for Europe,
2015). We used the uncertainties published by the NAEI in the Informative
Inventory Report (Misra et al., 2015) as the foundation of the uncertainty
propagation for the FRAME concentration and deposition outputs with respect
to UK emissions of SO2, NOx, and NH3. The uncertainty ranges
for different pollutants reported by the NAEI are estimated using a Monte
Carlo technique which corresponds to the IPCC Tier 2 approach (IPCC, 2006).
In this approach, uncertainty ranges for each source for both emission factor
and activity statistics are associated with a probability distribution and
further used as inputs in a stochastic simulation which calculates output
distributions of total UK emissions for each pollutant. The uncertainties are
expressed as plus or minus half the confidence interval width relative to the
estimated emissions value.
Previously, local one-at-a-time (OAT) sensitivity analysis has been used to
investigate ACTM sensitivity because it is less computationally demanding
than global sensitivity analysis that requires a large number of
simultaneous perturbations of all inputs of interest. However, there are
significant disadvantages associated with OAT analysis: the interactions
between the input parameters and non-linearities in the model response
cannot be identified; additionally, as the number of input parameters
increases, the fraction of parameter space investigated tends to 0
(Jimenez and Landgrebe, 1998; Saltelli
and Annoni, 2010). Therefore, local OAT sensitivity analysis is only
applicable when the effects of the different inputs are all independent of
each other and model response is linear for the range of investigated
inputs. Many previous publications that include ACTM sensitivity analysis
use the OAT approach but fail to acknowledge its limitations
(Appel
et al., 2007; Borge et al., 2008; Capaldo and Pandis, 1997; Labrador et al.,
2005; Makar et al., 2009).
Hence, this study focuses on demonstrating the use of global methods capable
of revealing non-linearity in the model response and the presence of
interactions between inputs in addition to revealing the spatial pattern of
the model response to changes in the input emissions. Global sensitivity and
uncertainty analyses have been applied in many earth science fields such as
hydrological modelling
(Shin et al.,
2013; Yatheendradas et al., 2008), ecological modelling
(Lagerwall
et al., 2014; Makler-Pick et al., 2011; Song et al., 2012), and atmospheric
aerosol modelling
(Carslaw
et al., 2013; Chen et al., 2013; Lee et al., 2011). Increasing computational
resource means this approach is now starting to be applied to ACTMs
(Christian et al., 2017).
In a global sensitivity analysis a sample space is created for all inputs
under investigation from which a set of combinations of model inputs for
different model runs are chosen. The sampling design for model inputs for
uncertainty and sensitivity analysis must balance the needs of covering the
full multidimensional input parameter space at sufficient density to allow
the characterisation of any non-linearities and interactions in the model
response with a small enough number of samples for the total number of model
runs to remain computationally tractable. Simple random sampling is
conceptually the simplest sampling technique but has low efficiency
compared to other sampling approaches and tends to lead to clusters and gaps
in coverage of the input space (Saltelli et
al., 2008). Likewise, full or fractional factorial designs
(Box and Hunter, 1961) do not allow an effective exploration of
the whole input space because for more than a few levels of each input, the
number of model runs becomes very large. Quasi-random sampling, of which the
Sobol' sequence (Sobol', 1967, 1976;
Sobol' and Levitan, 1999) is a popular choice for variance-based sensitivity
analysis, may not work well when the number of sampling points is small
(Saltelli et al., 2008). Therefore, in this
work, Latin hypercube sampling (LHS) (McKay et al., 1979), which is a
stratified space-filling sampling technique, was used. Advances have been
made to optimise the space-filling properties of LHS including maximin
sampling
(Johnson
et al., 1990; Morris and Mitchell, 1995) and integrated mean squared-error
minimisation (Park, 1994).
In summary, this work demonstrates the application of global uncertainty and
sensitivity analysis to an ACTM using the FRAME model as an example.
Methods
Model description
The FRAME model is a Lagrangian model that calculates annual average surface
concentrations of SO2, NOx, NH3, and HNO3, particulate
NH4+, SO42-, and NO3-, and dry and wet deposition of
SOy, NOy, and NHx at 5 km × 5 km horizontal
resolution over the UK (Dore et al., 2012; Fournier et al., 2002; Matejko et
al., 2009; Singles et al., 1998). This spatial resolution corresponds to
> 10 000 model grid squares over the UK land area. The air column
contains 33 vertical layers of varying thickness from 1 m at the surface to
100 m at the top of the mixing layer. The vertical diffusion between layers
is calculated using K-theory. The
air columns move from the boundary of the domain along straight-line
trajectories with varying starting angles at a 1∘ resolution. The
trajectories are defined by an annual wind rose and annually averaged wind
speed generated for the year 2012 from the output of the Weather Research and
Forecast model (www.wrf-model.org, last access: 1 November 2017) (Skamarock et al., 2008) version 3.7.1. The model was run at a 5 km
resolution over the UK with boundary and initial conditions initialised by
the National Centers for Environmental Prediction Final Global Forecast
System (NCEP-GFS-FNL) data
(https://rda.ucar.edu/datasets/ds083.2/, 30 October 2016).
Gridded emissions of SO2, NOx, and NH3 are obtained from the
UK NAEI (http://naei.beis.gov.uk/, 15 October 2016) at a 1 km × 1 km spatial resolution (maps are shown in
Fig. S1 in the Supplement). Input emissions of SO2 and NOx are
split into three categories: UK area, point source, and shipping emissions.
FRAME treats SO2 emissions as 95 % SO2 and 5 % H2SO4,
and NOx emissions as 95 % NO and 5 % NO2. For NH3 emissions, there are only UK area and point source categories. The NH3 emissions
from livestock are distributed spatially according to Hellsten et al. (2008).
All emissions are injected into the air column at different heights according
to the classification of emission sources.
The chemical scheme is described in
Fournier et al. (2004) and
includes gaseous- and aqueous-phase oxidation reactions and conversion of the
gases NH3, SO2, and NOx to particulate matter
(NH4+, NO3-, SO42-). NH4NO3 is
formed by the equilibrium reaction between HNO3 and NH3 and
nitrate aerosol also arises by the deposition of HNO3 onto sea salt or
large particles. H2SO4 reacts with NH3 to form
(NH4)2SO4. The aqueous-phase reactions include the oxidation
of S(IV) by O3 and the metal catalysed reaction with O2. Modelled
dry deposition is land-cover dependent and calculated using a canopy
resistance model. Wet deposition is calculated using scavenging coefficients, and it is driven by rainfall, which is modelled using a constant drizzle
approach based on the measured spatial distribution of annual average
rainfall data with the assumption of an enhanced washout rate over elevated
areas.
A detailed evaluation of model outputs with annually averaged measurements
of pollutant concentrations in air and precipitation concentrations is
discussed elsewhere (Dore et al., 2015).
In this study, all model runs were performed using emissions and meteorology
data for the year 2012 and FRAME model version 9.15.0.
Sensitivity and uncertainty analysis
For both sensitivity and uncertainty analyses a Latin hypercube sampling
design was chosen as it is superior to quasi-random sampling for small
numbers of samples (Saltelli et al., 2008). A
uniform LHS design was created using the R package “lhs” (Carnell,
2016), with the sample optimised by maximising the mean distance between the
design points. The LHS design was created for the scaling coefficients
applied to the model input emissions of UK SO2, NOx, and NH3
and not for the actual values of the input emissions. This means that
emissions from all sources of a particular pollutant were varied by the same
fraction across all grid squares in a particular model run.
For the sensitivity analysis a uniform LHS sample of size n=100 within a
range of ±40 % relative to the baseline for each of the three
input variables was created. This range was chosen to test the overall model
response to changes in emissions (for example to identify non-linearities)
as it encompasses the range of variations in input emissions used for future
scenario simulations with the FRAME model, as well as incorporating emission
reductions applied for the generation of source–receptor relationships for
integrated assessment modelling.
Regression coefficients (RCs) were used as the measure of the sensitivity of
the model response, derived as follows. For each model grid cell and for
each model output variable a multiple linear regression (Eq. 1) was
performed using the data from the n=100 model runs. To obtain the RCs
(bi in Eq. 1), the model inputs Xi and outputs Y were substituted by
corresponding values of fractional change relative to the baseline value.
This simplifies the interpretation of the resulting RCs. An RC represents the
relative effect of changing input Xi on the output Y, e.g. RC = 0.5
signifies a 15 % reduction in the output variable value if an input is
reduced by 30 %. The coefficients of determination (R2) were
evaluated for each fitted model (for every grid cell) to identify if a
significant level of non-linearity in the input–output relationship was
present.
Y=b0+∑i=13biXi
For the uncertainty propagation, the input sampling space was constrained to
the specific uncertainty ranges assigned to the emissions of SO2, NOx, and NH3 in the UK Informative Inventory Report
(Misra et al., 2015) with a new LHS sample n=100.
These uncertainty ranges are derived following published guidelines on
quantifying uncertainties in emissions estimates (IPCC, 2006;
Pulles and Kuenen, 2016). According to the guidelines, uncertainties are
expressed as lower and upper limits of the 95 % confidence interval as a
percentage of the central estimate. The assigned emissions uncertainties
have ±4, ±10, and ±20 % ranges for
SO2, NOx, and NH3 respectively. The probability distributions
were not specified; therefore, it was chosen to use uniform distributions for
the variable ranges from which the LHS sample was created. It is also
acknowledged that a number of other aspects of emissions uncertainty are not
included. For example, the FRAME model cannot capture uncertainty in
assigned seasonal and diurnal cycles in emissions. Uncertainties in the
spatial distributions or in the height of elevated emissions are also not
included.
Box plots of the values of RCs across all
UK land-based model grid squares. Boxes demarcate the median and lower/upper
quartiles of the distributions; whiskers extend to 1.5 times the
interquartile range.
Spatial distributions (at the 5 km × 5 km model grid
resolution) of RCs for particulate NH4+, SO42-, and
NO3- as a function of variation in input emissions of SO2, NOx, and NH3. The model input emissions for which the RC quantifies
the output variable sensitivity is given in brackets in each panel.
The uncertainty values for each grid square were calculated as a half of the
95 % confidence interval relative to the mean value of the output as
recommended in the EMEP/EEA and IPCC Guidebooks (IPCC, 2006;
Pulles and Kuenen, 2016). Relative uncertainty values are presented here.
To assess the contribution of uncertainties in the emissions of SO2,
NOx, and NH3 to the overall output uncertainty, standardised regression
coefficients (SRCs) were calculated as shown in Eq. (2). A multiple linear
regression was performed using the data from the 100 model simulations for
the case of constrained input sampling space. The SRCs (βi in Eq. 2)
were calculated by multiplying the RC by the ratio between the standard
deviations of the input σi and output σY. (σY
is the same for all the βi values for a given output
variable.)
βi=biσiσY
The squared value of SRC (Eq. 3) for linear additive models is equal to the
ratio of variance of the mean of Y when one input variable is fixed, VXi(EX∼i(Y|Xi)), to the unconditional variance of
Y, V(Y) (Saltelli et al., 2008). Thus, SRC
squared represents the fractional contribution of the uncertainties in the
model inputs to the overall uncertainty in the output. For the case of
non-linear models, variance decomposition methods are described in more
detail elsewhere
(Homma
and Saltelli, 1996; Saltelli, 2002; Saltelli et al., 2010; Sobol', 1993). In
the case where a large number of model simulations is not possible, an
emulator-based approach can be used for the uncertainty and sensitivity
analysis
(Blatman
and Sudret, 2010; Lee et al., 2011; Shahsavani and Grimvall, 2011; Storlie
and Helton, 2008).
βi2=VXi(EX∼i(Y|Xi))V(Y)
Results and discussion
Global sensitivity analysis
Figure 1 summarises the distributions of the RC
global sensitivity measure across all model grid cells. RCs show the
sensitivity of each model output variable to the three input emissions
variables (SO2, NOx, and NH3) and can be interpreted as a
magnitude of the response of an output to the unit change in a particular
input when all other inputs are allowed to vary. The magnitude of the RCs
provides useful information not only about the effect of the change in a
particular input on a model output but also allows input sensitivity
ranking to be determined because all inputs were assigned the same range of
variation (±40 %). In the case where the ranges for inputs
differ, SRCs are used to obtain the
input importance ranking instead.
Figure 1 shows (i) that model outputs have varying sensitivities, (ii) that model outputs have varying relative rankings in their sensitivities to SO2, NOx, and NH3 emissions, and (iii) that these output sensitivities to the
emissions also vary spatially across the model grids, as shown by the
spreads in individual box plots. The annual average concentrations of
particulate NH4+, NO3-, and SO42- and annual
dry and wet deposition of SOy for the baseline model run are presented
in Supplement Fig. S2. The actual spatial distributions of
the RCs from Fig. 1 are illustrated in Fig. 2 for the example output
variables of particulate NH4+, NO3-, and
SO42-. Figure 3 shows the equivalent for the example output
variables of dry and wet deposition of SOy. These five output variables
were chosen to illustrate the spatial distribution of uncertainty and
sensitivity metrics. Figures S3 and S4 in Supplement show the
spatial distribution of RCs for other FRAME outputs displayed in Fig. 1.
RC is a first-order sensitivity measure, and it quantifies the average
response of model output to varying a model input Xi when all inputs are
allowed to vary. In this study no second- or higher-order interaction terms
were quantified as their contribution was assumed to be negligible. This was
concluded from the values of the coefficients of determination (R2)
obtained from multiple linear regressions performed; for most output
variables, values of R2 were on average > 0.98, with the exception of
a slightly lower value for HNO3 (R2>0.96). Hence, less than
2 % (4 % for HNO3) of variance in the output could not be explained
by the linear combination of inputs. This finding allows us to conclude that
the FRAME model response is in fact fairly linear within the ±40 %
emission perturbation range investigated. The absence of any substantial
deviations from linearity in the model response and the absence of second- or
higher-order interactions between input variables indicate that the current
use of the FRAME model to produce source–receptor matrices for the use in the
UK Integrated Assessment Model is not subject to undue error from varying
emissions one at a time. Without conducting the global sensitivity analysis, it is not possible to predict a priori for a given model output variable
either the relative sensitivities to the different input factors, such as
emissions, or the spatial variation in these sensitivities that are
illustrated in Figs. 1, 2, and 3.
Spatial distributions (at the 5 km × 5 km model grid
resolution) of RCs of dry (d) and wet (w) deposition of SOy as a
function of variation in input emissions of SO2, NOx, and NH3.
The model input emissions for which the RC quantifies the output variable
sensitivity is given in the brackets in each panel.
With respect to findings from this FRAME model sensitivity analysis for
particulate inorganic components in the UK context, Fig. 1 shows that the
modelled surface concentrations of particulate NH4+ are sensitive
to changes in emissions of all three pollutants, being similarly sensitive
(on average) to emissions of NH3 and SO2 and slightly less
sensitive to emissions of NOx. The sensitivities of NH4+ to
SO2, NOx, and NH3 emission changes were found to vary
substantially around the UK (top row of Fig. 2). The sensitivity of
NH4+ to SO2 emissions is generally lowest in south-east
England, and rises on moving north and west across the UK. Reductions in
emissions are always associated with reductions in NH4+. The broad
geographical pattern of relative sensitivity across the UK of NH4+
to NH3 emissions is approximately the reverse of that to SO2
emissions although with substantial spatial heterogeneity as well. Figure 2
shows that there are instances in north-west Scotland of negative RCs for
the sensitivity of NH4+ to NOx emissions, i.e. areas where
NH4+ increases when NOx emissions are decreased.
Figure 1 similarly shows that surface concentrations of particulate
SO42- are sensitive to changes in emissions of all three of
SO2, NOx, and NH3 (most sensitive to SO2 emissions) but
with a universally negative sensitivity (albeit relatively weak) to NOx
emissions; i.e. particulate SO42- concentrations increase
everywhere by approximately 3 % if NOx emissions are reduced by
40 % (lower row of Fig. 2). This is due to competition between
HNO3 and H2SO4 to react with NH3 and form particles; i.e. reducing NOx emissions means NH3 is more readily available to
react with H2SO4. The positive values of RCs of SO42- to
SO2 emissions are geographically fairly uniform (somewhat lower
sensitivity in the eastern UK), but the relative sensitivity to NH3
emissions is more heterogeneous and greater in the east.
The sensitivity of particulate NO3- concentrations to the
emissions is more straightforward than for particulate NH4+ and
SO42, being dominated by its positive sensitivity to NOx
emissions, weakly sensitive to NH3 emissions, and essentially not
sensitive at all to SO2 emissions (Fig. 1 and middle row of Fig. 2). The sensitivity to NOx emissions is almost unity, such that for
example a 30 % reduction in NOx emissions results in almost the same
30 % reduction in surface NO3-. The spatial distribution of RCs
that represent the sensitivity of NO3- concentrations to NOx (and
NH3) emissions is also geographically more homogenous across the UK
than the sensitivities of NH4+ and SO42- concentrations
(middle row of Fig. 2).
The concentrations of the three inorganic particulate matter components are
determined by the reactions that lead to the formation of
(NH4)2SO4 and NH4NO3. The formation of the former is
irreversible whilst the latter exists in reversible equilibrium with
gas-phase NH3 and HNO3. Changes in emissions of NH3 have an
impact on the formation of both (NH4)2SO4 and NH4NO3
very quickly, and therefore close to the source of the NH3 emissions,
because it reacts directly as NH3. In contrast, the influence of changes
in SO2 and NOx emissions is not so localised. Before they
influence the formation of (NH4)2SO4 and NH4NO3, these gases must be oxidised in the atmosphere to H2SO4 and
HNO3, during which time the air is undergoing transport. The spatial
pattern of the sensitivities of (NH4)2SO4 and
NH4NO3 formation to changes in the UK precursor emissions is
therefore the outcome of many interacting factors: (i) the magnitude of
background import of precursors from outside the UK which could explain the lower sensitivity of inorganic particulate matter components to SO2
emissions in south-east England, (ii) the magnitude and spatial pattern of
the UK precursors, (iii) the time taken for chemical oxidation in relation to
atmospheric transport of air masses, and (iv) the varying dry and wet
deposition spatial patterns that remove from the atmosphere both the
precursor gases and particulate products.
In summary, the broad patterns of the sensitivity results in Figs. 1, 2, and 3 can be explained as follows. The surface concentrations of the
directly emitted pollutants NH3, NOx, and SO2 are
predominantly sensitive only to their respective emissions (Fig. 1). This
is also the case for the deposition of oxidised S and of oxidised and
reduced N. Dry deposition is dominated by the gas-phase components, so the
variations in the dry deposition of NHx and SOy are dominated by
the variations in the emissions of NH3 and SOx respectively, with
the RC values being close to 1. For the dry deposition of NOy, both
NO2 and its oxidation product HNO3 are important. This is
illustrated by the weaker response of dry NOy deposition to changes in
NOx emissions. Wet deposition is a more complex process as this is
dominated by washout of the particles which are the product of chemical
reactions in the atmosphere. This explains lower values of RC for wet
compared to dry deposition.
The considerably more ubiquitous sources of NOx emissions compared with
SO2 emissions means that atmospheric concentrations of gaseous oxidised
N are generally higher than for oxidised S, so the former usually has a greater
influence on NH3 chemistry. Therefore, particulate NO3- is
predominantly controlled by NOx emissions, and changes in SO2
emissions have very little effect on particulate NO3-. However,
because lower NOx emissions lead to lower NH4NO3 formation,
more NH3 is available, which means lower NOx emissions lead to
greater (NH4)2SO4 formation. This explains the inverse
correlation between surface concentrations of SO42- and NOx
emissions. On the other hand, changes in NH3 emissions impact on both
NO3- and SO42- concentrations, both in a positive
direction of association but with a magnitude sensitive to the relative
amounts of the reacting species present, which in turn depends both on the
magnitudes and distances of local sources and on long-range transport.
Likewise, the sensitivity of NH4+ concentrations varies with all
three sets of precursor emissions and with geographical location. The same
is the case for concentrations of HNO3. This is why, aside from some
broad expectations, it is not easily possible to predict the spatial
patterns of the sensitivities of ACTM model output to changes in emissions, and a formal sensitivity analysis is needed.
Uncertainty propagation
The global uncertainty propagation approach for FRAME output variables was
based on the assigned uncertainties in the estimates of the total UK
emissions of SO2 (±4 %), NOx (±10 %), and
NH3 (±20 %) (Misra et al., 2015). As
explained in the “Methods” section, the uncertainties in the input emissions were
assigned uniform distributions, and no uncertainties in either the spatial
or temporal aspects of the emissions are included. No substantial difference
in the resulting model output uncertainty ranges was observed when the
probability distributions of the input emissions were changed to normal. The
distributions of the relative uncertainties across all model grid cells for
each output are shown in Fig. 4. Example maps of the spatial distributions
of the relative uncertainties from Fig. 4 for surface concentrations of
particulate NH4+, NO3-, and SO42- and for dry
and wet deposition of SOy are shown in Fig. 5. Equivalent maps for
the relative uncertainties of the other FRAME output variables are shown in
Supplement Fig. S5.
Distributions of relative uncertainty values calculated for all
FRAME model outputs across all model grid squares given the following input
uncertainty ranges: ±4, ±10, and ±20 % in emissions of SO2, NOx, and NH3 respectively. Boxes demarcate the median and lower and
upper quartiles of the distributions; whiskers extend to 1.5 times the
interquartile range.
Spatial distributions (at the 5 km × 5 km model grid
resolution) of the relative uncertainties in surface concentrations of
particulate NH4+, SO42-, NO3-, and dry and wet
deposition of SOy for uncertainties of ±4, ±10, ±20 % in
emissions of SO2, NOx, and NH3 respectively. The uncertainty
values are represented as a range of ± the baseline value and represent
the 95 % confidence interval.
Figure 4 shows that the surface concentration of NH3 is the most
uncertain output (model grid median uncertainty 19.8 %). This is because
the variation in NH3 surface concentrations is almost entirely driven
by variation in NH3 input emissions (Fig. 1), and this is the most
uncertain input in the presented analysis. The uncertainty in modelled dry
deposition of NHx likewise closely matches the assigned uncertainty in
NH3 emissions (median = 18.8 %). The uncertainty in wet deposition
of NHx is somewhat less than uncertainty in dry deposition (median = 13.4 %) because wet deposition of NHx includes some dissolved
(NH4)2SO4 component which is also sensitive to other
precursor emissions whose uncertainty is estimated to be smaller than for
NH3. Surface concentrations of SO2 and the dry and wet depositions
of SOy have the least uncertainty (medians of 6.0, 4.8, and 3.2 %) for the similar reason that these model outputs are predominantly
sensitive to SO2 emissions (Fig. 1), which has the smallest of the
input uncertainties (±4 %).
Relative uncertainties of particulate SO42- (median = 6.4 %),
NO3- (median = 8.6 %), and NH4+ (median = 7.5 %)
are fairly similar (Fig. 4) even though there are substantial differences in
the assigned uncertainties for emissions of SO2, NOx, and NH3.
The explanation is that particulate matter (PM) components are sensitive to all three inputs (for NO3-, two out of
three inputs) (Fig. 1). There is also wide spatial variation in the
uncertainties of these PM components (Figs. 4 and 5). The relative
uncertainty values in the surface concentration of HNO3 show the largest
variability out of all output variables. This can be explained by the fact
that the concentration of this species is impacted directly by both gas- and
particle-phase processes. The spatial pattern of the relative uncertainty
values does not correlate either with the spatial pattern of emissions or
rainfall, which demonstrates again that the uncertainties of many model
outputs cannot be readily predicted because of the complexity of the
atmospheric processes underpinning them and consequently that formal
uncertainty analysis needs to be applied.
Uncertainty apportionment
Estimated uncertainty of the model output given the uncertainties in model
input emissions is presented in Figs. 4 and 5, but it is also of interest
to know how each of the inputs contributes to the overall uncertainty
individually. This was estimated by calculating squared SRCs (Eq. 3). As an example, Fig. 6 illustrates
the spatial distributions of the fractional contributions of the SO2, NOx, and NH3 emission uncertainties to the overall uncertainties in
surface concentrations of particulate NH4+, NO3- and
SO42- for the assigned uncertainties in the input emissions,
whilst Fig. 7 illustrates a similar theory for the dry and wet deposition of
SOy. The equivalent maps for the other model output variables are
presented in Supplement Figs. S6 and S7.
Figure 6 shows that across nearly all of the UK, uncertainty in concentrations
of particulate NH4+ is mainly driven by the uncertainty in
NH3 emissions. Uncertainty in NOx emissions contributes some
uncertainty to NH4+ concentrations, whilst the uncertainty in
SO2 emissions makes almost no contribution. Northern Ireland is an
exception; here, uncertainties in NOx emissions contribute the most to
the uncertainties in NH4+ concentrations and perturbations in
NH3 emissions have less impact. Concentrations of NH3 in Northern
Ireland are some of the highest anywhere in the UK, whilst NOx
emissions are not high; this means that NH3 will be in excess, so the
formation of NH4NO3 will be largely controlled by HNO3
through NOx emissions. The major contribution to uncertainty in
particulate NO3- derives from uncertainty in NOx emissions
(Fig. 6). However, in the east of Scotland, uncertainty in NH3
emissions contributes up to 78 % of the total uncertainty. There is no
contribution from SO2 emissions uncertainty. An important feature of
the lower panels of Fig. 6 is that by far the major contributor to
uncertainty in particulate SO42- concentrations is the uncertainty
assigned to the NH3 emissions, not the uncertainty in the direct
precursor SO2 emissions. This is because the formation of
(NH4)2SO4 is irreversibly dependent on gaseous NH3 and
emissions of NH3 are much more uncertain than SO2 emissions.
Figure 7 shows the spatial distribution of the squared SRC values for dry
and wet SOy deposition; for these output variables, uncertainty in
NOx does not make any contribution to uncertainty in either case. In
contrast to the situation for particulate SO42- concentrations
shown in Fig. 6, Fig. 7 shows that uncertainty in dry and wet deposition
of SOy is mainly driven by the uncertainty in the SO2 emissions.
Additionally, uncertainty in NH3emissions contributes to the total
uncertainty in dry and wet SOy deposition. The contribution to
uncertainty in wet deposition is higher due to wet deposition being
dominated by the washout of the particles, which include products of the
reactions of NH3 with oxidation products of SOx.
Spatial distributions (at the 5 km × 5 km model grid
resolution) of the squared SRC values which represent the fractional
contribution of the uncertainty in the input emissions given in brackets to
the overall uncertainty in the surface concentrations of particulate
NH4+, SO42-, and NO3-. The uncertainties in the input
emissions are ±4, ±10, and ±20 % for SO2, NOx, and NH3 respectively.
Spatial distributions (at the 5 km × 5 km model grid
resolution) of the squared SRC values which represent the fractional
contribution of the uncertainty in the input emissions given in brackets to
the overall uncertainty in the dry and wet deposition of SOy. The
uncertainties in the input emissions are ±4, ±10, and ±20 % for
SO2, NOx, and NH3 respectively.
Conclusions
We have applied global sensitivity analysis to determine the response of
concentration and deposition output variables of the FRAME atmospheric
chemistry transport model to perturbations of UK emissions of SO2, NOx, and NH3. The benefit of using systematic global sensitivity
analysis is that all dimensions of variable input space are investigated
simultaneously, which is important when the response to a large number of
variables is of interest so inferences can be drawn without assumptions
about the model structure. For complex models such as ACTMs, for which
input–output mapping is not analytically tractable, it is not possible to
predict output sensitivities to multiple input perturbations without
conducting a global sensitivity analysis. Local one-at-a-time sensitivity
analysis is often applied without acknowledging the shortcomings associated
with it.
In this study no substantial deviations from linearity or the presence of
interactions between the model input variables were identified for the FRAME
model in response to input emission perturbations within a ±40 %
range; hence, regression coefficients obtained from multiple linear
regression were chosen as a sensitivity measure. This was not predictable
from a local one-at-a-time sensitivity analysis.
Whilst the sensitivity of surface concentrations of the primary precursor gases
SO2, NOx, and NH3 (and of the deposition of S and N) was dominated
by the emissions of the respective pollutant, the sensitivities of secondary
species such as HNO3 and particulate SO42-, NO3-, and NH4+ to pollutant emissions were more nuanced and
geographically variable. The dry deposition of S and N showed a stronger
response to changes in the emissions of the respective pollutant compared to
wet deposition.
A global uncertainty analysis approach was used to estimate uncertainty
ranges for all FRAME model output variables from the uncertainties assigned
to the UK emissions of SO2, NOx, and NH3 (±4, ±10, and ±20 % respectively) by the UK National
Atmospheric Emissions Inventory. The spatial distribution of the relative
uncertainty was affected by both the sensitivity of the model output to
variations in the inputs and the magnitude of this variation (i.e. the input
uncertainty range); NH3 was the most uncertain input, and as a result
the output variables sensitive to NH3 showed the highest levels of
relative uncertainty in the areas most sensitive to this input. The
uncertainty in the surface concentrations of NH3 and NOx and the
depositions of NHx and NOy was shown to be due to uncertainty in a
single precursor input variable, NH3, and NOx respectively. In
contrast, the concentration of SO2 and deposition of SOy was
affected by uncertainties in both SO2 and NH3 emissions. Likewise,
the relative uncertainties in the modelled surface concentrations of each of
the secondary pollutant variables (NH4+, NO3-,
SO42-, and HNO3) were affected by the uncertainty range of at
least two input variables.
This work has demonstrated a methodology for conducting global sensitivity
and uncertainty analysis for ACTMs. Although, for the FRAME model used here,
the response to emission perturbations was found to be substantially linear
in the investigated input range, the complexity of chemical and physical
processes included in ACTMs means that the input–output relationships, in
particular their spatial patterns, cannot be predicted without conducting a
global sensitivity analysis. The benefit of using global approaches is that
all dimensions of input variable space are investigated simultaneously, so
model input–output relationships can be quantified without the need to make
strong prior assumptions about the model response to perturbations in the
inputs of interest.