Airborne aerosols are a fascinating component of the Earth's atmosphere. They come in a bewildering variety of shapes, sizes and compositions. More importantly, they can affect the radiative budget and energy and hydrological balances of the atmosphere . Dust aerosols may transport nutrients for the biosphere over long distances and air pollution aerosol can pose health hazards for humans . Aerosols have also been suggested as disease vectors . For a recent review of some of these aspects, see .

Models provide powerful tools to explore the role of aerosols, but require evaluations against observations in order to quantify their skill and detect possible model errors. AEROCOM is an international community of scientists (http://aerocom.met.no) involved in evaluating global aerosol models , but model evaluations are also routinely performed by individual research groups around the world. It is therefore surprising that evaluation strategies themselves have received relatively little scrutiny.

Due to constraints on computational resources, global aerosol–climate models are currently run at spatial resolutions of ∼200 km. This of course limits their ability to resolve fine-scale structure which will affect the comparison of global model data with observations; models and observations represent averages over different spatial areas. Satellite remote sensing observations are made for nominal pixels of 10 km as for MODIS (MODerate resolution Imaging Spectroradiometer) or 17 km as for MISR (Multi-angle Imaging SpectroRadiometer) or 3 km as for SEVIRI (Spinning Enhanced Visible and InfraRed Imager). Ground stations from AErosol RObotic NETwork (AERONET) can be estimated to sample no more than 5 km horizontally away from the site. In situ measurements cover even less of the atmosphere surrounding them; yet, observed aerosol fields are known to exhibit variations over relatively short distances of 10 to 100 km . Note that the spatial resolution of global models also impacts global model data due to the non-linear nature of many physical and chemical processes ; but that is not the topic of this paper.

Recently, the disparity of spatial scales between global models and observations has attracted some attention. Using satellite-retrieved solar surface radiation estimates to assess spatial representativeness, estimated differences of 1–2 % and 2–3 %, respectively, in 5-year seasonal means between either 1∘ × 1∘ or 3∘ × 3∘ areas and point measurements. and showed that the standard deviation, skewness and kurtosis of climate data (e.g. temperature) can be significantly different between point values and gridded values (in their analysis means were identical by construction).

We use high-resolution model simulations (with a 10 km grid spacing) to simulate both perfect global model data and perfect observations. These data are considered perfect in the sense that they are both derived from the same high-resolution simulation that we treat as the truth. In fact, the only difference between the global model data and observations is the area over which the high-resolution simulation is averaged (see Sect. ). No measurement errors are added to the observations. The high-resolution simulations allow us to build up statistics of the difference between observations and model data, under a large variety of scenarios. In particular, we consider different observables like AOT (aerosol optical thickness), PM2.5, number densities and CCN (cloud condensation nuclei) for different regions on the globe. We also evaluate a variety of averaging and interpolation strategies designed to bring model data and observations closer together. These high-resolution model simulations provide us with a toy model of what happens when global model data are evaluated with observations, ignoring both model and observation errors.

Since we simulate global model data as an average over the high-resolution data, a very relevant question is the following: what average is appropriate? This question is closely tied to the question of what the grid-point value of a global model represents and will be addressed later.

Section  introduces the three different models and six different regions for which we have high-resolution simulations. We also explain how the simulated fields were turned into observables. Section  describes in more detail how both global model data and observations are generated from the high-resolution simulations. In particular, Sect.  discusses various interpretations that may be given to a global model's grid-point value. Section  then introduces the concept of spatial sampling as a source of error through some examples. More substantive statistics can be found in Sects. , , , and . An evaluation of several strategies to reduce spatial sampling differences is given in Sect. . A preliminary analysis of the consequences of spatial sampling for the use of flight campaign data can be found in Sect. . The paper concludes with a summary (Sect. ).

Note that Sect.  contains some general guidelines to interpreting many of the figures and statistics that appear in this paper.

Three different regional models were used to create high-resolution simulated fields (10 km, 1 h) of common observables: AOT, extinction, PM2.5, black carbon mass concentration, number densities and CCN. Figure  shows the simulation regions, and Table  summarises the most important information on these simulations.

WRF-Chem (Weather Research and Forecasting model coupled with Chemistry) was run for three regions using the MADE/SORGAM aerosol module , and one region using the GOCART bulk aerosol scheme. The meteorology was nudged to NCEP-FNL operational analysis data. The western Europe (W. Europe) and Oklahoma runs used emission scenarios (TNO MEGAPOLI-2005 or US National Emissions Inventory NEI-2005) with imposed 24 h cycles for the anthropogenic emissions. These regions are characterised by fairly localised spatially fixed sources. The Congo experiment used daily biomass burning emissions derived from MODIS fire counts and is characterised by highly localised sources that differ in location from day to day. The MADE/SORGAM module assumes that the aerosol exists in three modes (Aitken, accumulation and coarse) of varying species mixtures (sulfate, nitrate, organic and black carbon, sea salt and dust). MADE/SORGAM explicitly treats nitrates and SOA (secondary organic aerosol).

An expanded version of EMEP/MSC-W that includes calculations of aerosol bulk optical properties (based on work by , and ) was run at a 0.1∘ × 0.1∘ grid, using ECMWF-IFS meteorology for 2008 and TNO-INERIS emissions for 2009 for Europe. Emissions of black carbon were derived from the emissions of primary PM2.5, using EMEP standard split-factors (per country and sector). Monthly, day-of-week and hourly temporal profiles were applied to the annual emissions. The EMEP chemical scheme includes approximately 160 reactions. The aerosols are represented as bulk mass distributed between a fine fraction (including sulfate, nitrate, ammonium, organic and black carbon sea salt and dust) and a coarse fraction (nitrate, sea salt and dust). Ammonium nitrate is calculated with the equilibrium model MARS, and the formation of coarse nitrate from nitric acid depends on the relative humidity. SOA is calculated using the volatility basis set approach. For all details see and references therein.

NICAM-SPRINTARS (see , and references therein) was run in global mode with a stretched grid that had a resolution of 11 km over a part of Honshu (the largest island of Japan). Its meteorology was nudged to NCEP-FNL reanalysis data. SPRINTARS uses a bulk mass scheme with individual modes for sulfate, organic carbon, black carbon and bins for sea salt and dust. Two different organic/black carbon mixtures are also represented by individual modes. Anthropogenic emissions of black carbon and the SO4 precursor gas SO2 had a prescribed diurnal cycle. SOA were treated in the simple manner of scaling aerosol emissions. Nitrate aerosols were ignored in this SPRINTARS simulation.

Both EMEP and SPRINTARS do not calculate number densities as prognostic variables (SPRINTARS can diagnose them from assumed size distributions) and consequently did not provide those fields for our analysis. Both EMEP and SPRINTARS data were regridded from their original model grids to regular grids with 10 km spacings.

This section briefly describes the main methodology used in this paper. Using the high-resolution simulated fields, we have generated both perfect observations and perfect global model data. The high-resolution field v has a regular rectilinear horizontal grid (10×10 km), and a regular temporal spacing (1 h). Only the vertical spacing is non-regular and differs among the models. The field v can be thought of as three- or four-dimensional data cubes vxyt or vxyzt where x=1…nx and y=1…ny are indices to the horizontal coordinates, z=1…nz is an index to the vertical coordinate and t=1…nt is an index to the time coordinate. In the following, the z coordinate is ignored for brevity's sake. A single perfect observation Oxyt at time t and location x,y is simulated by Oxyt=vxyt.

A perfect global model grid point's value Mxyt can be simulated by averaging vxyt over a global model grid-box area 2Δx+1×2Δy+1 in the high-resolution field: Mxyt=∑i=-ΔxΔx∑j=-ΔyΔywijvx+i;y+j;t, where Δx and Δy represent the longitudinal and latitudinal half-sizes of a grid box, as measured in the coordinate indices. Here w is a normalised weighting function (to be defined later). Note that perfect model data can only be calculated on an inner domain of the high-resolution region of 1+Δx≤x≤nx-Δx;1+Δy≤y≤ny-Δy.

In the case that the location of the observation and the grid point coincide, an instantaneous spatial sampling error can now be defined as ϵxyt=Oxyt-Mxyt, where we use the perfect model value as a reference, since it is the model value that we want to evaluate in actual comparisons of observational and model data. It is straightforward to define a relative sampling error for time-averaged data by εxyt=∑k=t-Δtk=t+ΔtOxyt-Mxyt/∑k=t-Δtk=t+ΔtMxyt, where 2Δt+1 is an arbitrary averaging interval. Using the global model value (instead of the observation) as reference prevents denominators from becoming zero.

A subset of the data cube of our regional simulations is used to build up error statistics. In addition to the limitation imposed by the Eq. () (already discussed), the outer 50 km of the simulated region was excluded from our analysis to reduce boundary effects. Similarly, the first 2 days of each simulation were used as a spin-up and excluded from analysis. At various points in our analysis, we have studied the sensitivity of our results to these choices but found no significant impact.

In Fig. , we show instantaneous simulated AOT and surface black carbon concentration after 10 days in the WRF-Chem W. Europe run. By comparing the field in a small 10×10 km box to the average of a large 210×210 km box surrounding it (approximate size of present-day global model grid box), we assess spatial sampling errors. The centre of the large box we refer to as grid point (of the global model). By moving these two boxes together throughout the region, we can build up statistics of spatial sampling errors (also shown in Fig. ). These errors can reach ∼100 % and form coherent patterns several global model grid boxes large. Time series of the global model and observed values at a single location are shown in Fig. . In the case of AOT, we see that the perfect observation can both over- and underestimate the perfect model value with variations on a timescale of a day or so. The black carbon time series, on the other hand, shows systematic underestimation by the perfect observation over long periods for most of the month (note that events of overestimation also occur but on smaller timescales). Although these time series vary a lot throughout the region, this example is nevertheless typical. A video of the W-Europe simulation of AOT and surface black carbon mass concentrations can be found at http://dx.doi.org/10.5446/18550.

Since these spatial sampling errors are substantial, it makes sense to try and reduce them by temporally averaging the data. In Fig. , we show monthly averaged simulated AOT and surface black carbon concentration from the same run. The spatial sampling errors in monthly averaged observations are also shown in Fig. . They are smaller than the errors for instantaneous fields but are still quite substantial (up to ∼20 % for AOT and ∼65 % for black carbon). Note also that the error patterns have become larger and more coherent. As a matter of fact, notice how closely the patterns in sampling errors for black carbon agree with its emission sources, except that sampling errors are negative (and quite large) where concentrations are quite low. When defining areas downstream from sources where the aerosol is supposedly well-mixed spatially, it is important to consider the grid-box size of the model which is evaluated as much as the length scales involved in the actual aerosol processes.

The effectiveness of temporal averaging is shown in Fig. , where the spatial sampling errors are shown as a function of averaging period. Temporal averaging does decrease spatial sampling errors but not as fast as one would expect if instantaneous sampling errors behaved like independent Gaussian noise. This is understandable because the persistence of emission sources and flow patterns in the atmosphere create temporal correlations in the fields of a few hours to a few days. Note that AOT is more strongly (beneficially) affected by temporal averaging than surface black carbon concentrations.

Before studying these spatial sampling errors in more detail, we consider how (dis)similar they are among different models. The Europe region simulated by EMEP encompasses the W. Europe region simulated by WRF-Chem MADE and so these two models allow ready intercomparison for May 2008; see Fig. . We see that both instantaneous and monthly errors as predicted by WRF-Chem and EMEP are of similar magnitude although WRF-Chem generally produces larger errors. Error magnitudes for different observables behave similarly among WRF-Chem and EMEP: monthly errors for AOT and surface black carbon are the smallest and largest errors, respectively. EMEP monthly error maps (see Fig. ) also look similar to WRF-Chem results (Fig. ), especially for black carbon surface concentrations.

It would be interesting to understand the reason for the differences. From global model studies in the context of AEROCOM (e.g. ), we know that such attribution is difficult and here we limit ourselves to pointing out some obvious differences between WRF-Chem and EMEP. First, there are differences in emission inventories and sea-salt emission parametrisations. Second, although both models were nudged to reanalysis data, transport will be different due to different dynamical cores and vertical resolution (WRF-Chem uses twice the vertical resolution as EMEP). For similar reasons wet and dry deposition are different. Both models also use a very different aerosol scheme (bulk mass or two-moment scheme). All of this will affect aerosol lifetimes, which in turn will affect the spatio-temporal variability of aerosol.

It should also be pointed out that EMEP shows quite a bit of month-to-month variation: e.g. January 2008 errors for AOT and March 2008 errors for surface black carbon concentration are markedly bigger than those estimated for May.

The most important point here is that both models suggest spatial sampling errors of similar magnitude with similar spatial patterns.

Figure  shows relative spatial sampling errors (either instantaneous or monthly) for all observables and the three WRF-Chem MADE regions (see also Table  and Fig. ). Instantaneous RMS errors are large: from 20 % up to 160 % depending on observable and region (the RMS errors are calculated over a single region for the full month, see Table ). There are clear and (mostly) systematic differences among the three regions in that W. Europe shows the largest errors and Congo the smallest. This may be related to the overall wind flow; Congo shows the most laminar flow (and hence most coherent aerosol plumes), while W. Europe shows a very turbulent flow (we do not wish to discount other effects like the spatio-temporal distribution of sources but a full explanation is beyond this paper's scope). Two observables (black carbon concentrations near 2 km a.g.l. for all three regions and surface CCN at S=0.02 % in W. Europe) show errors down to -100 %. In the case of black carbon, this is due to narrow black carbon plumes travelling through an otherwise pristine air layer; the observation often sees the pristine air but the model always includes contributions from the plume. In the case of CCN, the background CCN at S=0.02 % is very low, especially close to sources where many small particles are emitted. However, once in a while a plume of larger particles travels over, giving rise to much larger CCN at low supersaturation S=0.02 %.

The monthly errors can be reduced quite a bit compared to the instantaneous errors. For many observables, RMS errors are 5–15 %, although for observables like surface black carbon concentrations and N10 it can be 30–50 and 30–80 %, respectively, with individual errors reaching over 100 %. Congo represents quite a different situation from the other two regions: the reduction due to averaging is much less, and in the case of surface N10 there is actually a slight increase in errors. An important difference between W. Europe and Oklahoma on the one hand and Congo on the other is that the first have mostly fixed aerosol sources with a prescribed diurnal cycle. The latter has emission sources (fires) in different locations from day to day.

Figure  shows relative spatial sampling errors for the other three regions, all simulated by models with bulk mass schemes for aerosol. In general, spatial sampling errors appear to be a bit smaller than in Fig. , but note the exception of extinction near 2 km a.g.l. Monthly sampling errors over ocean are very low, due to spatial correlations in the near-surface wind-speeds that cause sea-salt aerosol emission. However, large errors are found for extinction over ocean near 2 km a.g.l., that seem partly due to isolated plumes of sea salt but mostly due to a broken cloud field that rains out sea salt locally. Both instantaneous and monthly errors over Japan become larger if only observations over the land area are considered. The Japan region includes parts of the Japan Sea and the North Pacific Ocean that account for more than 80 % of the simulated area. Additionally, the Japan simulation, like the Congo simulation, shows rather laminar flow from mesoscale to synoptic scale. Finally, simple statistics like in Fig.  cannot convey that over an extended region like Europe there are areas with systematically small or large sampling errors due to source locations and orography (see also Figs.  and ).

In the case of actual observations, there may be quite a bit of intermittency in their temporal sampling, suggesting that the spatial sampling decreases we have shown here for monthly averages represent a best-case scenario. For a study of errors due to temporal sampling we refer to .

The vertical distribution of spatial sampling errors can be very different depending on observable and region. Figures  and  show the instantaneous and monthly relative spatial sampling error profiles for extinction, N10 and black carbon concentrations.

We see that although errors are typically largest at and near the surface, this does not preclude large errors higher up in the atmosphere. The instantaneous errors for black carbon concentrations actually show largest errors from 2 to 7 km a.g.l. This is due to black carbon plumes in a relatively pristine background, which also explains why the error distribution is so clearly skewed to negative values (observation sees the pristine background while the model also includes plumes). Black carbon's only source is surface emission, but both extinction and N10 also have sources throughout the troposphere (nucleation, condensation and in-cloud production of sulfate) which likely explains the difference between these observables.

For the monthly errors, most profiles show secondary maxima in sampling errors well above the surface.

We have analysed the sampling errors at their original model levels, which for these simulations occur at fairly constant altitude above ground. Note that the errors estimated in this subsection do not take into account that a global model's grid box may have a vertical extent larger than that of our regional simulations. Taking this into account would only increase the estimated errors. The profiles of spatial sampling errors for the bulk mass simulations are rather constant and therefore not discussed here.

So far we have considered observations at the exact grid point of a global model's grid box which is a useful starting point but also quite unrealistic. For a sample of randomly distributed observations in a 210 by 210 km grid box, only 2 % will be within 10 km of the grid point and 50 % will be more than 84 km away from it. By considering observations located throughout the grid box, and not just its centre, it is possible to show how monthly sampling errors increase with distance of the observation to the grid point; see Fig. . As a matter of fact, 50 % of possible AOT observations have errors at least twice as large as found for an observation at the grid point. Observations in the very corners of the grid box exhibit errors 3 times as large. The increase of sampling errors with distance to the grid point for surface black carbon concentrations is not as large but still significant.

That sampling errors increase with distance may be surprising but can be explained. The evolution of aerosol across a global model grid box may differ quite a bit due to differences in sources, flow and deposition (especially wet). Nevertheless, as is well known from observations, aerosol exhibits correlations over several tens of kilometres and our high-resolution simulations are no different. Hence, an observation at the centre of a grid box will correlate strongly with a large part of that grid box, while an observation in the upper-right corner will only correlate strongly with (part of) the upper-right quadrant of that grid box but less so with the lower-left quadrant. It is important to realise that aerosol in individual 10×10 km2 boxes cannot be considered as independent and identically distributed (i.i.d.) random variables. If aerosol behaved like i.i.d. random variables, sampling errors would not increase with distance.

Figure  shows box-and-whisker plots of monthly sampling errors for several observables, either at the grid point, or at a distance of 70 or 100 km, for the W. Europe region. Similar results can be shown for Oklahoma and Congo, where the relative increase with distance is often (but not always) larger. For all three regions and all observables, the increase for Δq82 at 70 km is between 1.2 and 2.3 × and the increase at 100 km is between 1.4 and 3.4 ×. Instantaneous spatial sampling errors increase less fast with distance but still significantly; typical increases for Δq82 at 70 km is 1.3 for AOT and 1.2 for surface black carbon concentration (i.e. monthly averaging is more beneficial for an observation at the grid point than one at 70 km distance).

As discussed before (Sect. ), the meaning of a global model's grid-point value is not obvious. So far we have assumed that the grid-point value is the unweighted average of the high-resolution field over the global model's grid box (i.e. a constant weighting function w). Here, we explore how the sampling errors depend on different weighting functions. Figure  shows how a constant, linear or Dirac-δ weighting function affects sampling errors as a function of distance to the grid point. For the Dirac-δ weighting function, sampling errors are equal to zero at a distance of zero: the global model's value is equal to the observation (since both are perfect). However, as distance increases, so will the spatial sampling errors. Actually, for distances larger than ∼30 km, the three very different weighting functions give rather similar sampling errors (but notice that more localised weighting functions yield larger errors as expected). Since for randomly distributed observations, only ∼6 % would be closer than 30 km to the grid point, we feel it is justified to conclude that the shape of the weighting function only has a small impact on statistics of spatial sampling errors. The spatio-temporal variation of the field is far more important.

The typical sampling errors when the observation is at the model grid point are lower than those for an observation offset from the grid point. It seems unlikely that we can devise strategies to reduce centre-of-grid-box errors, other than temporal averaging (see Sect. ) or further averaging global model data (and their associated observations) over multiple grid boxes. However, the sampling errors for observations offset from a grid point might be reduced by proper screening, interpolation within the model grid or considering multiple observations at the same time.

Unlike satellite or ground-site observations, measurements taken during a flight campaign cannot be properly averaged over time (at least on timescales from days to months and longer). To simulate the (nearly) instantaneous measurements during horizontal legs of flight campaigns, we use narrow tracks: 10 km wide and 210 km long, centred on the grid point and running in either east–west or north–south direction. Profiles of spatial sampling errors for such flight campaign data can be seen in Fig. . Compared to instantaneous point observations (also shown), the flight campaign observations are less affected by spatial sampling issues because they sample a larger part of the grid box. Even so, significant instantaneous RMS errors exist, varying between 10 and 41 % for extinction, 10 and 46 % for N10 and 21 and 100 % for black carbon concentrations at different altitudes and for different regions (these errors are for a best-case scenario: a grid-box long flight path centred on the grid point). For Congo, spatial sampling errors can be quite different depending on whether the flight path runs north–south or east–west around 6 km a.g.l. Prevailing wind flows are east–west, resulting in similarly orientated plumes. If the flight track observations are within and along such a plume, spatial sampling errors will be large and positively biased. If the flight track observations are across such a plume, errors will be smaller and (over a large domain) unbiased.

The Congo results highlight a particular issue with flight campaign data: if the flight tracks have deliberately been chosen to follow observed aerosol plumes, perfect observations will overestimate perfect model values by significant amounts.

Almost vertical legs of flight campaigns should experience errors like those discussed for point observations, Sect. . Notice that we do not consider the vertical extent of a global model's grid box in our analysis.

The spatial resolutions of current global aerosol models and the observations used to evaluate them are very different. Model grid-point values are representative of areas of ∼ 200 × 200 km2 but individual observations seldom see more than ∼10×10 km2 of the atmosphere. This difference in the field of view should affect the evaluation of models with observations but has received little attention in the literature. We believe our paper is the first systematic and qualitative study of the differences between a perfect model and perfect observations due to spatial sampling.

Using high-resolution simulations for six different regions by two different regional models and one global model, we show that spatial sampling errors can be substantial across a range of observables (AOT, extinction, PM2.5, black carbon concentrations, number concentrations and CCN). These spatial sampling errors fluctuate in time and space, depending on emission sources, grid locations, weather and aerosol processes. Ultimately, they constitute noise that will be present in any model evaluation and that can not be eliminated entirely unless model grid sizes become smaller than observational fields of view.

Assuming observations that do not coincide with the global model's grid point but are offset by 80 km (54 % of randomly located observations in a 210×210 km grid box will be further away), the following statistics are offered. For instantaneous data, RMS spatial sampling errors (defined as observation minus global model value) are larger than 30 %, typically between 40 and 80 % and may go up to 160 % (depending on observable and region). These errors are typically positively skewed and highly non-Gaussian. For monthly data, RMS sampling errors are larger than 10 %, typically between 10 and 40 % and may go up to 75 % (depending on observable and region).

This noise can however be reduced; we have explored the impact of spatial or temporal averaging of data as well as selection of observations based on distance to a grid point or interpolation of model data to the location of an observation. Our study suggests that while increased model resolution will of course be beneficial, resolutions will need to be 4 times higher (50×50 km2 grid-box area) before spatial sampling errors become significantly smaller. In the mean time, we recommend that both model data and observations are spatio-temporally averaged to ensure best agreement. Here the model data must first be spatially interpolated to and temporally collocated with the observation. Optimal averaging procedures will depend on the spatio-temporal sampling of the observations, the characteristics of the observable and the requirements of the scientific community; so we offer no single prescription although the results in this paper provide some guidelines. Optimal strategies for evaluating models with observations need to receive more attention from researchers.

Our results suggest that caution is needed when using in situ measurements in global model evaluation. These measurements consistently led to larger spatial sampling errors than remote sensing measurements like AOT. For instance, monthly surface black carbon concentrations and number densities for our simulations have RMS spatial sampling errors of at least 30 and up to 80 %. Best-case scenarios for flight campaign data still allowed spatial sampling errors of 100 % and typically the observation would underestimate the model.

Regarding the large sampling errors in case of black carbon, other species (e.g. sulfate, sea salt) were not explicitly analysed in this paper but show different results (not shown). Sulfate errors tend to be rather small, probably due to the multitude of sources and relatively long lifetimes. Sea salt, on the other hand, shows large and systematic monthly sampling errors along coastlines (unsurprisingly). Given the size of our global model's grid box, these errors extend quite far into land or over sea. The important point here is that sampling errors for species mass concentrations can be very different dependent on species, and hence have a big impact on the evaluation of a model's particle speciation.

It is likely that the spatial sampling errors estimated in this paper are underestimates. First, showed that model spatial variability over 75 km increased significantly (by 60 to 100 %) when model resolution changed from 15 to 3 km. Our current high-resolution simulations have resolutions of 10 km. Second, our high-resolution simulations do not resolve fine-scale structure below 10 km while many in situ measurements actually have fields of view on the order of millimetres to centimetres (e.g. particle inlets). Third, our models are more limited in the spatio-temporal variation of their emission sources than reality due to assumed and constant diurnal patterns in anthropogenic emissions. Finally, even high-resolution models will have to take a broad view of aerosol and describe average properties (e.g. mass and/or number densities) instead of modelling individual aerosols in all their variety.

On the other hand, it is possible that in areas far away from sources (e.g. the free troposphere over the remote ocean) aerosol has mixed sufficiently to strongly reduce spatial sampling errors (e.g. HIPPO measurements over the Pacific; see also ). Our simulations do not really allow us to explore this scenario.

In the interest of comparing likes to likes, this paper does not consider that real observations may have very intermittent temporal sampling, nor does it consider the impact that precipitation may have on spatio-temporal variability of aerosol for example. These issues are the subject of further investigation.