Atmospheric chemistry transport models (ACTMs) are widely
used to underpin policy decisions associated with the impact of potential
changes in emissions on future pollutant concentrations and deposition. It
is therefore essential to have a quantitative understanding of the
uncertainty in model output arising from uncertainties in the input
pollutant emissions. ACTMs incorporate complex and non-linear descriptions
of chemical and physical processes which means that interactions and
non-linearities in input–output relationships may not be revealed through
the local one-at-a-time sensitivity analysis typically used. The aim of this
work is to demonstrate a global sensitivity and uncertainty analysis
approach for an ACTM, using as an example the FRAME model, which is
extensively employed in the UK to generate source–receptor matrices for the
UK Integrated Assessment Model and to estimate critical load exceedances. An
optimised Latin hypercube sampling design was used to construct model runs
within

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

FRAME uses emissions input data from the UK National Atmospheric Emissions
Inventory (NAEI;

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.

The FRAME model is a Lagrangian model that calculates annual average surface
concentrations of SO

Gridded emissions of SO

The chemical scheme is described in
Fournier et al. (2004) and
includes gaseous- and aqueous-phase oxidation reactions and conversion of the
gases NH

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.

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 SO

For the sensitivity analysis a uniform LHS sample of size

For the uncertainty propagation, the input sampling space was constrained to
the specific uncertainty ranges assigned to the emissions of SO

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

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 SO

The squared value of SRC (Eq. 3) for linear additive models is equal to the
ratio of variance of the mean of

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 (SO

Figure 1 shows (i) that model outputs have varying sensitivities, (ii) that model outputs have varying relative rankings in their sensitivities to SO

RC is a first-order sensitivity measure, and it quantifies the average
response of model output to varying a model input

Spatial distributions (at the 5 km

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 NH

Figure 1 similarly shows that surface concentrations of particulate
SO

The sensitivity of particulate NO

The concentrations of the three inorganic particulate matter components are
determined by the reactions that lead to the formation of
(NH

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 NH

The considerably more ubiquitous sources of NO

The global uncertainty propagation approach for FRAME output variables was
based on the assigned uncertainties in the estimates of the total UK
emissions of SO

Distributions of relative uncertainty values calculated for all
FRAME model outputs across all model grid squares given the following input
uncertainty ranges:

Spatial distributions (at the 5 km

Figure 4 shows that the surface concentration of NH

Relative uncertainties of particulate SO

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 SO

Figure 6 shows that across nearly all of the UK, uncertainty in concentrations
of particulate NH

Figure 7 shows the spatial distribution of the squared SRC values for dry
and wet SO

Spatial distributions (at the 5 km

Spatial distributions (at the 5 km

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 SO

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

Whilst the sensitivity of surface concentrations of the primary precursor gases
SO

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 SO

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.

The FRAME model code is not available in the public domain
as the model is the intellectual property of the Centre for Ecology &
Hydrology and is only made available to students and researchers who are
collaborating directly with CEH staff. However, all the following output data
are available at

The authors declare that they have no conflict of interest.

Ksenia Aleksankina acknowledges studentship funding from the University of Edinburgh and the NERC Centre for Ecology & Hydrology (NERC CEH project number NEC05006). The CEH funding was provided by the Department for Environment Food & Rural Affairs, contract AQ0947, Support for National Air Pollution Control Strategies 2013-2015 (SNAPS). Edited by: Fiona O'Connor Reviewed by: two anonymous referees