This analysis uses ground-based observations from AERONET, together with satellite retrievals from the Multiangle Imaging SpectroRadiometer (MISR) and MODIS instruments, and model simulations from the Goddard Earth Observing System (GEOS) Version 5 Nature Run (G5NR). All of these have different spatiotemporal sampling techniques and associated uncertainties in their estimates of AOD. Considering a diverse set of data sources such as this provides a more comprehensive picture of the frequency distributions of AOD than would be obtained from only a single data type. It allows the strengths of individual techniques to be used while helping to avoid erroneous conclusions stemming from limitations of individual techniques. The data sources are described below.

present a method for testing whether a sample of data is consistent with draws from a normally distributed population. Their derivation included some empirical comparisons to other tests, and they found it to have some advantages over those techniques. performed Monte Carlo simulations of various distributions to assess eight different normality tests and found that the Shapiro–Wilk (SW) test has the greatest statistical power in most circumstances. The SW test computes the squared discrepancy between the quantiles of the sample with those expected from random samples from a normally distributed population. Implementations are available in many software packages and languages. The test statistic W for a sample x is defined as 7W=∑i=1Naix(i)2∑i=1Nxi-x¯2, where x(i) indicates the ith smallest sample member (known as the ith order statistic of x), and ai values are weighting coefficients calculated from the expected values (m) and covariance (V) of order statistics from normally distributed data: 8(a1,…,aN)=mTV-1(mTV-1V-1m)1/2. The test requires N≥3, and W can take values between 0 and 1. (their Table 1), later corrected in , provide V for N≤20; provide approximate calculations for larger sample sizes, and provide approximate m and V up to N=5000. Relevant sample sizes in the present study are up to several hundred points. As the normal distribution is symmetrical about its mean, the coefficients ai are symmetrical; e.g. a=(-0.643,-0.281,-0.088,0.088,0.281,0.643) for N=6. The coefficients are larger for the outer elements of ai, corresponding to the tails of the data sample (i.e. the outer order statistics x(i)). The numerator of Eq. () thus represents a tail-weighted squared sum, while the denominator represents a sum of squared deviations from the sample mean x¯. For normally distributed samples these increase around the same rate such that W is close to 1; for non-normal data the denominator increases more rapidly such that W becomes closer to 0.

provides a normalisation for W in order to estimate a p value for the result, i.e. the probability that a W score at least as extreme would be observed under the null hypothesis of the sample being drawn from a normally distributed population. The extension for large N and normalisation are used here. Note that the equivalent test for log-normality is simply W calculated using the logarithm of the data, i.e. here log⁡10τ rather than τ. A high p value indicates consistency with draws from a normal (or log-normal) distribution. One important point to note is that this test only evaluates the degree of departure from normality: it does not test the importance of that departure. As with any test like this, the power (i.e. efficacy at detecting a given departure) is a function of sample size. Thus for large sample sizes it is easier to detect a discrepancy from normally distributed data even if the discrepancy is trivial. Both of these points should be kept in mind when interpreting the results.

The SW test is employed here as follows. First, spatial distributions of AOD are assessed by aggregating the MISR, MODIS, and G5NR data from their native spatial resolutions to 1∘ (as this is a common spatial scale for L3 AOD products and model output) and applying the SW test without any aggregation in time. The resulting aggregates thus have a daily time step for MISR and MODIS (considering all orbits from a given calendar day) and 30 min for G5NR. Next, temporal variations in AOD within a day are assessed by aggregating the AERONET and (previously spatially aggregated) G5NR data on a daily basis and applying the SW test to each site and grid cell. Aggregating G5NR first in space and then in time is more similar to the way polar-orbiting L3 aggregates sample the global aerosol system (as each L2 product is essentially a near-instantaneous snapshot), although the results are not significantly different if G5NR is analysed first in time and then in space. Finally, the resulting daily aggregates from all data sets are aggregated to monthly and seasonal time steps and the SW test applied to each site and grid cell. Seasons are defined as December–January–February (DJF), March–April–May (MAM), June–July–August (JJA), and September–October–November (SON). Note that the monthly and seasonal calculations use daily τ¯l as a basis, although results differ negligibly if τ¯n is used instead.

In each case, at least three data points are required for an aggregate to be considered valid; this is the minimum required for the SW test calculation and also the minimum number of observations for AERONET or MODIS standard processing to report a daily average value and the minimum number of days for MODIS products to report a monthly average value. The SW p value is computed for distributions of τ and log⁡10τ, with resulting p values denoted by p(τ) and p(log⁡10τ), respectively, and the results fall into one of four possible categories.

|τ¯l-τ¯n|≤τt. In this case the choice of normal or log-normal summary statistics may be considered unimportant, as the resulting arithmetic and geometric averages are similar. The threshold τt is taken as 0.01, which is the typical uncertainty in AERONET midvisible AOD , and thus represents a reasonable lower bound on achievable uncertainty in average AOD from models and observations at the present time. It is also similar to the thresholds for AOD accuracy over land (±0.016) and ocean (±0.011) estimated by to be necessary to be able to constrain the aerosol direct radiative effect to ±0.5 W m-2.

The results will be interpreted in terms of relative frequencies of these four categories, as it is important to realise that the idiosyncrasies in real-world data complicate the estimation and calculation of p values. For example, the ideal case of independent random samples from the true population cannot be achieved due to correlated errors in observations or simulations and non-random sampling in space and/or time. SW (or other tests) cannot say whether or not the data are normally or log-normally distributed for any given instance but instead only help say the extent to which the two distributions are reasonable, useful approximations on the whole. In cases of small sample sizes the statistical power of the test may remain small; if the test results for a given area are essentially noise, then similar frequencies of normality and log-normality might be expected. The best that can be done is to keep in mind the limitations of the data, and the statistical tests, in the interpretation of the analysis.

Figures  and respectively show the categorisation results for temporal (from AERONET and G5NR) and spatial (from MISR, MODIS, and G5NR) frequency distributions of AOD on daily scales. As a summary, Table  shows the global mean fractions of data in each category; note that these are the mean of each valid AERONET site and grid cell (i.e. all sufficiently sampled areas are treated equally). Sites and grid cells require at least 50 valid days with data to be included in these statistics. As the spatial sampling between the data sets is quite different (Figs.  and ), the results from the different data sets are not expected to match, but reading the table from left to right gives a sense for how the categorisations change on the different scales assessed. The following general conclusions can be drawn about variability relevant to daily aggregation.

Patterns shown between Figs.  and are similar; i.e. daily AOD frequency distributions tend to have similar shapes whether for temporal aggregation over a day (as from AERONET or model output) or spatial aggregation on scales of 1∘ (as from polar-orbiting satellites). This establishes that it is reasonable to aggregate spatial and temporal data on a daily basis in a similar way.

Figure  provides brief examples of AOD distributions falling into three of the SW test categorisations for AERONET AOD data collected within a single day. As will be shown later, differences between normal and log-normal distributions become more pronounced at longer timescales than in these examples. The case for Midway Island (in the Pacific Ocean), a location dominated by low-AOD maritime conditions , shows a case where the arithmetic- and geometric-mean AOD are both around 0.055, and thus the choice of summary statistic is likely unimportant for most applications (although note that p(log⁡10τ)>p(τ), indicating greater consistency with a log-normal distribution). The case for Moscow (Russia), taken from a period of extreme wildfires during summer 2010, is characterised by intense smoke . Here, the data are more consistent with a normal distribution than a log-normal one, and |τ¯l-τ¯n|=0.05. This example (N=21) illustrates some difficulties in purely visual inference about distribution shape when histograms are sparse; the median number of points contributing to a single day of AERONET data globally was 25. In this case the data are more consistent with a normal distribution due to a closer match toward the tails of the distribution, but the data are consistent with both normal and log-normal distributions under most relevant significance levels. The final case is from Pontianak (Borneo, Indonesia) during an intense period of biomass burning in 2015, an event analysed in detail by . For this date, |τ¯l-τ¯n|=0.08, and the data show greater consistency with log-normality due to the skew of the distribution.

Days where neither normal nor log-normal distributions provide a good fit to AOD observations are commonly those where multiple regimes are present within a grid cell or during a day. Figure  presents a case study from the AERONET site in Essex, Maryland, USA, on 5 July 2011. This day was previously analysed by as a case of rapid AOD enhancement following the development of a cumulus cloud field just after 17:00 UTC (universal time coordinated), near solar noon (Fig. a). The Sun photometer was operated with a 3 min sampling cadence, with 132 points in total throughout the day. For the day as a whole, p(τ)≪pt and p(log⁡10τ)≪pt; i.e. the data are inconsistent with both normal and log-normal distributions, revealed by the bimodality of the histogram in Fig. b. However, for the 111 points before 17:00 UTC p(τ)=0.002 and p(log⁡10τ)=0.009, and for the 14 points after 18:00 UTC, p(τ)=0.38 and p(log⁡10τ)=0.54, in both cases indicating stronger evidence for log-normality than normality. By combining various data sets and lines of evidence, attribute enhancements like these to a combination of humidification and new particle formation rather than cloud contamination in the direct-Sun data, so there is physical reasoning for this bimodality. In situations like this the multimodal-fitting approach of would give a more complete representation of the aerosol field than presenting single-distribution summary statistics. Distribution-agnostic metrics (such as reporting various percentiles of the AOD) are an alternative option.

Note also that the near-universal choice of aggregating daily on a UTC calendar day basis, rather than in terms of local solar time (LST), can further complicate matters for locations far from the meridian. For example, AERONET sites in eastern Asia, Australasia, and the western Americas often contain data from midnight to mid-morning UTC, with a long gap, and then from late evening to midnight UTC. The break in the middle is due to local nighttime, during which no data are collected; i.e. observations from a single UTC day can contain data from 2 local days. If something happens during this gap to affect the AOD distribution, which is often the case due to the diurnal variations or meteorology, this will naturally increase the chances of multimodality. Thus, something as basic as the definition of the day to aggregate to can affect the inferred AOD distribution shape. This could be contributing to some of the cases where neither distribution fits (Figs.  and ) in these parts of the world. This affects all the data sets. This issue has also been explored by , examining the correlation between hourly and daily modelled AOD fields for two different definitions of a day (UTC vs. LST).

While similar, the patterns in Figs.  and are not identical between data sets. G5NR is the only data set which enables both spatial and temporal aggregation on a daily basis. Here, both aggregates show, for example, small differences between τ¯l and τ¯n over much of the global ocean and a higher frequency of large differences over southern and eastern Asia. However, the spatial aggregates also show areas of large difference (fit well by neither normal nor log-normal distributions) for grid cells with strong elevation variations, such as along the edges of the Himalayas or Andes, while the temporal aggregates do not. If the bulk of the aerosol here is low-lying, then this naturally leads to another case of multiple populations within a grid cell. This is not seen to the same extent in the satellite retrievals here, although they are known to undersample (due to misinterpreting spatial heterogeneity of the scene for cloud cover) and sometimes have retrieval artefacts which could distort the distributions . In these cases moving to a finer spatial scale might be useful to provide summary metrics for these populations separately; i.e. 1∘ might be too coarse. The aforementioned retrieval artefacts might also explain some of the discrepancies between MODIS and other results in other mountainous areas such as western North America, Europe, and the Horn of Africa.

G5NR temporal aggregates also show increased incidence of log-normality and of neither distribution fitting well in the Southern Ocean, while G5NR spatial aggregates do not; this implies diurnal cycles which affect the aerosol field here coherently on scales larger than 1∘. A similar feature, with 10 %–20 % occurrence of normally distributed AOD in the Southern Ocean, is seen in the MODIS results. MODIS retrievals are known to report higher AOD here than other data sets (including active sensors, Sun photometry, and data sets with stricter cloud screening); attributed much of this to a combination of cloud contamination and retrieval assumptions of surface wind speed (which affect surface brightness). This latter factor was addressed in more recent MODIS data versions compared to those used by , although the enhanced AOD remains, implying that cloud contamination is still a factor. A similar enhancement was seen in older version of the MISR data product but largely removed in the latest version used here . This implies that the occasional normality seen in MODIS daily AOD in the Southern Ocean is likely to be an artefact of biases in the AOD retrievals. MODIS and MISR also report normal or log-normal AOD distributions each up to about 30 % of the time over various North African and Central Asian deserts, while G5NR does not. Unfortunately, the remoteness of many of these areas means that AERONET has few sites in them. It is therefore hard to resolve the reasons for differences between the various data sets.

AERONET also provides some opportunities to study the spatial distribution of AOD on horizontal scales of tens of kilometres to around 100 km, similar to L3 and global-climate-model resolution. These are mostly in so-called Distributed Regional Aerosol Gridded Observation Networks (DRAGONs) of up to several dozen sites, as detailed by , deployed during intensive operating periods (IOPs) of field campaigns. Some DRAGON deployments have been in areas also containing several long-term AERONET sites (e.g. those around Washington, DC, USA), enabling spatial characterisation (to a lesser extent) outside these IOPs. Furthermore, a few areas have had three or more AERONET sites deployed simultaneously within ∼100 km of each other; often (but not always) this overlap was temporary as one site replaced another. Table  shows the categorisation resulting from applying the SW and AOD difference threshold tests on daily geometric-mean AOD for each of these field campaign deployments or groups of spatially clustered AERONET sites. More details of the DRAGON deployments are available in , and the AERONET web page (https://aeronet.gsfc.nasa.gov, last access: 14 November 2019) provides additional background information and the locations of other clustered sites. Categorisation results are broadly in line with Figs.  and , and Table , in that typically the most common finding is that the difference between daily arithmetic- and geometric-mean AOD is smaller than 0.01. For the 10 field campaign deployment regions listed in Table , log-normality is more commonly observed than normality in 6 of them for the days when the resulting difference in AOD is at least 0.01. For the seven clusters of sites outside of field campaigns (which have fewer, i.e. three to four sites total), log-normality is more common in five. While this is consistent with the picture from the larger-scale analysis, it is also important to recall that these deployments are typically short in time (often weeks to months) and tend to be around major metropolitan areas. As a result the frequencies in Table  might not be extensible to longer time periods at these locations or other environments.

Maps of categorisation of monthly and seasonal AOD aggregates, in both cases from daily AOD, are shown in Figs.  and , respectively. Global-average fractions are again shown in Table . Monthly satellite and AERONET composites require at least 16 contributing months to be considered valid, and seasonal composites require at least eight contributing seasons; for the satellites, using 5 years of data, a maximum of 60 months or 15 seasons are possible. Increasing these thresholds removes some shorter-term AERONET sites and satellite retrievals at high latitudes and some tropical locations, where retrieval coverage is limited. As only 1 year of G5NR data are used, the monthly analysis is performed but seasonal analysis is not. Moving from daily to monthly aggregates in Table , the overall tendency is for AOD differences to become larger (i.e. the fraction within the category |τ¯l-τ¯n|≤0.01 decreases), and the distributions increasingly favour log-normality over normality. Going from monthly to seasonal, the trend is more pronounced both in the absolute fraction of data (Table ) and in the spatial distributions (Fig. ). As in the daily data, some features are broadly consistent between the data sets.

Unlike daily aggregation, for monthly or seasonal aggregation the difference between arithmetic and geometric means is frequently more than 0.01. Thus, monthly and seasonal aggregates are more sensitive to the choice of averaging method. This implies generally larger variability on timescales of months and seasons than of spatial variability within a day, which is consistent with previous work e.g. that has established that temporal colocation on daily, rather than monthly, timescales is important in reducing sampling-related differences between AOD data sets.

The previous portion of the analysis focuses mostly on the occurrence and distinguishability of normal and log-normal distributions for AOD; also relevant are the magnitudes of the differences introduced into the data sets by the choice of averaging method and summary statistic. Figure  shows the difference between geometric- and arithmetic-mean AOD (τ¯n-τ¯l) binned as a function of arithmetic-mean AOD, on daily and monthly timescales, from the four data sets. For the daily plots the G5NR temporal aggregation is shown, although results are similar for the spatial aggregation. Also shown in both panels is the Global Climate Observing System (GCOS) goal uncertainty in AOD for an aerosol climate data record (CDR), which is the greater of 0.03 or 10 % of the AOD . AOD differences approaching or exceeding this level imply that the aggregation method alone causes a sensitivity of similar magnitude to the total desired uncertainty and therefore that if data are to be aggregated, then the choice of an appropriate technique is crucial.

It is important to realise that the AOD difference τ¯l-τ¯n is always zero or negative, as geometric means are always smaller than or equal to arithmetic means . This means that the offsets will always be systematic. For daily data (left panel of Fig. ), the median offset, and its dependence on AOD, are reasonably consistent between all four data sets. The central 68 % of the observed offsets are also somewhat smaller than the GCOS uncertainty requirement and below a total AOD of around 0.6, generally smaller than 0.02.

Even a small offset in reported AOD, if systematic, can have important implications for calculations of climate forcing. This is particularly true for aerosol–cloud interactions, as these are very sensitive to both the anthropogenic perturbation and the natural background state assumed. For example, using perturbed parameter simulations to global climate models, estimated that 45 % of the uncertainty in the global mean forcing due to the cloud albedo effect of aerosols was related to uncertainties in the natural background aerosol burden, compared to 34 % for anthropogenic emissions. Others, including and , have similarly found large dependence of forcing depending on the choice of background. Where AOD is low, such as over much of the global ocean, a small absolute AOD change can be a large relative perturbation. Although the limitations of satellite retrievals for some of these applications are well-known e.g., the same argument may apply if forcing parametrisations are developed from model simulations aggregated in certain ways. As a result, even differences smaller than the GCOS goal uncertainty, such as the daily differences in Fig. , may be significant for these purposes. To paint a more complete picture of the forcing, ideally the overall shape of the distribution, rather than a single number, should be considered; this has also been found to be important for parametrisations related to the cloud radiative effect and rainfall .

In contrast to the daily results, and even in low–moderate AOD loadings around 0.3, for monthly aggregates (right panel of Fig. ) the difference is often similar to or larger than this GCOS uncertainty. This means that the choice of arithmetic- or geometric-mean AOD as a summary metric in itself can and often does introduce systematic offsets in reported monthly AOD of a similar size to the goal total uncertainty for an AOD CDR. While this is not strictly an error (in the mathematical sense), if the analyst is using the arithmetic mean as a summary of log-normally distributed data, then, as stated earlier, the inferred typical (common) level of AOD will be biased. Inferences made may be misleading, and less complete, than those if the shape and width of the distribution were considered explicitly. As with the daily data, the magnitude of the offset is AOD-dependent; the magnitude is, however, less consistent than for the daily results, with the median offset being largest for AERONET. This might in part reflect known tendencies for a high bias in low-AOD conditions and/or low bias in high-AOD conditions in these satellite products , meaning that the difference τ¯l-τ¯n is dampened due to diminished spatial and temporal variability within and between days. The asymmetry of the variabilities (vertical lines) in both panels of Fig.  indicates a dependence of the difference on the specific local conditions, i.e. the factors affecting the width of the distribution.

As Fig.  established that on monthly timescales sensitivities to averaging method often exceed GCOS goal CDR uncertainties, and Fig.  maps how frequently such exceedances occur. The behaviour for seasonal aggregates (not shown) is more pronounced than that of monthly and shows similar spatial features. As seen in earlier parts of this study, the four data sets give broadly consistent spatial patterns but differences in magnitude. Specifically, this is seen most frequently (30 %–90 %, dependent on grid cell and data set) in eastern Asia and Saharan outflow regions, which is unfortunate because these are important and frequently studied components of the global aerosol system. Exceedance of GCOS thresholds in 10 %–40 % of months is also seen fairly consistently across much of eastern North America and Eurasia, South America, southeastern Asia, and southern Africa. This is most common during the summer months (former two cases) and local biomass-burning seasons (other cases), when AOD levels are generally higher. GCOS threshold exceedance is infrequent (observed <10 % of the time) over the remote open ocean in any of the data sets, although they may be slightly elevated for oceanic regions downwind of continental aerosol sources. In all of these regions, the monthly data show higher consistency with log-normality more often than they do than normality (Fig. ), particularly for the AERONET record, which has the most reliable AOD. Therefore, the locations where the difference between τ¯n and τ¯l is largest are also generally those where the data support log-normal summary statistics the most.

A potential counterexample to the need to account for distribution shape is the case of particulate matter (PM) estimation from AOD retrievals, in which case it might be more sensible to report arithmetic-mean AOD than the geometric-mean AOD, even if the underlying distribution is log-normal. This is because the arithmetic mean is directly proportional to the total, while the geometric mean requires knowledge of distribution width as well to return total mass, and in PM studies it is often the total mass which is of interest. However, in practical terms, PM forecasts and nowcasts and daily exposure estimates typically use the finest resolution data available rather than aggregates . Still, this may be relevant for long-term studies looking at total PM exposure levels, which are often finely spatially resolved but are aggregated temporally .

The issue may also be less crucial in analyses where the purpose is to assess the offsets between two data sets (e.g. difference between L3 composites), as opposed to the geophysical fields themselves, as differences between the arithmetic means of two data sets and geometric means of the same two data sets are likely to be of the same sign. The magnitudes will, however, differ depending on the tails of the distributions. As indicated earlier, for AOD, sampling differences between data sets can often be a large determinant of observed offsets . Still, examining multiple differences (e.g. differences between arithmetic means, between geometric means, and median differences) can be informative with respect to how much the distributions are influenced by outliers and as a general indicator of skew. Examples include Fig. 2 of or Fig. 11 of .

As the differences between arithmetic and geometric mean are larger for higher-AOD regions (Figs.  and ), choice of summary statistic could also influence the calculation of AOD trends. Specifically, as τ¯l<τ¯n by an increasing amount as AOD increases, smaller magnitudes of calculated trends would be expected (as the maxima are dampened to a higher degree than the minima). Multiple studies over the past few decades have looked at AOD trends globally and regionally, whether over oceans only e.g. or both oceans and land e.g.. While data sources, periods of analysis, and analysis techniques differ, as do quantitative results, several features tend to be consistently reported.

AOD over the global ocean, and over many ocean basins, has not changed very much.

Using three long-term AERONET sites (one for each of the above features), Table  provides decadal AOD trends calculated using geometric-mean AOD τ¯l and arithmetic-mean AOD τ¯n as a basis. These sites were used in some of the above studies to complement satellite retrieval and model-simulation analyses; in all cases, those studies used arithmetic-mean AOD. Ascension Island is in the South Atlantic Ocean, where reported AOD trends are typically small, and presently has data available from 1998 to 2016. The Goddard Space Flight Center (GSFC) in Maryland, USA, a region of decreasing AOD, has data from 1993 onwards and is one of the longest-running AERONET sites; Solar Village (operated from 1999 to 2013) was at a solar-power farm northwest of Riyadh, Saudi Arabia, and near the maximum of AOD trends reported in previous studies. At these sites, on monthly timescales, the data fell more often into SW category 4 (AOD difference is larger than 0.01, and data are most consistent with draws from a log-normal distribution) 40 %, 61 %, and 70 % of the time for Ascension Island, GSFC, and Cabo Verde, respectively, than category 3 (more consistent with draws from a normal distribution). In most cases the bulk of the remainder of months fell into category 1 (AOD difference smaller than 0.01). On a seasonal basis this preference for log-normality over normality is even more pronounced, corresponding to 68 %, 79 %, and 82 % of the time to category 4, respectively.

The purpose here is not to perform an exhaustive global trend analysis but to assess quantitatively the implications of log-normally distributed AOD on some well-reported features of global aerosol trends. Prior studies typically calculated trends based on deseasonalised monthly mean AOD time series and calculating a linear least-squares regression fit. Deseasonalisation was achieved either by subtracting the mean AOD annual cycle over the time period or, as in and by using harmonic regression to model the annual cycle. took a somewhat different approach by analysing the temporal variability in principal components of monthly AOD fields rather than the AOD fields themselves. In some of these analyses, seasonal trends were calculated by averaging the monthly data within each season, although in this case it is arguably more reasonable to use seasonal aggregates from daily data as a basis. The motivation for considering seasonal trends is that some aerosol features, and their variability, are characteristic of particular seasons (e.g. Ascension Island and GSFC sample transported smoke through summer and autumn but seldom during other seasons; at Solar Village dust storms are most frequent and intense in spring and summer).

Here, linear trends are calculated using both monthly and seasonal aggregates for both normal (i.e. arithmetic mean) and log-normal (i.e.  geometric mean) aggregates, both calculated from daily AOD (τ¯n or τ¯l, respectively). Monthly trends are calculated using the monthly AOD time series, after subtraction of the mean seasonal cycle, as in previous studies. Seasonal trends do not require a deseasonalisation step. The data are fit using linear least-squares regression, with the weights equal to the standard error on the estimated monthly (or seasonal) AOD. For the log-normal averaging this is strictly asymmetric, although it is approximated as symmetric in this case, which has negligible influence on the results. The lower limit for these standard errors is taken as 0.01, corresponding to the AERONET AOD uncertainty. As this is largely dominated by calibration uncertainty , it is not significantly reduced by averaging and can therefore be considered systematic over a single (roughly year-long) deployment but closer to random over a multi-year time series. The standard error on the annual cycle is added in quadrature to the estimated uncertainty in the monthly time series to account for the uncertainty in the deseasonalisation step. Following , the lag-1 autocorrelation is estimated and used to adjust the uncertainty estimates. Furthermore, the χ2 statistics on the fits, which have an expected value of n-2 (where n is the record length and two parameters are fit in the regression), were in most cases somewhat in excess of this (Table ). This implies that the standard errors are not a complete representation of the uncertainty in the time series data and/or that a linear model does not fully describe the variation. Thus, in addition to the autocorrelation correction, the trend uncertainty estimates in Table  are also scaled by χ2/(n-2).

At each of the three sites, the decadal AOD trends are qualitatively the same whether calculated using arithmetic- or geometric-mean AOD time series as a basis. However, as expected, trends using geometric-mean AOD are smaller in magnitude (i.e. increases and declines in AOD are less pronounced). The decrease in magnitude is often of the order of 10 %–30 %, which is typically within the 1σ uncertainty estimates on the calculated AOD trend. The implication of this is that, as a result of assuming an underlying normal distribution, prior studies may be qualitatively correct on the sign of AOD trends but quantitatively have a tendency to overestimate their magnitude (again using the prior definition of “typical” or “common” aerosol loading). For log-normally distributed data, trends in geometric quantities are more straightforward to interpret.

Note that during presentations and reviews of this work, a question arose as to whether, moving from point trends to larger regional-scale trends, the central limit theorem (CLT) would mean that arithmetic and geometric trends would converge. The CLT does not imply that expanding the region (adding more data) means that the AOD would become closer to a normal distribution; this is a common misconception of the CLT. Rather, the uncertainties on estimates of the summary statistic (whether arithmetic or geometric means) will behave approximately according to normal statistics, even if the underlying AOD distribution is not normal. It does not mean that the underlying quantity becomes closer to normally distributed; this misconception of the CLT is discussed in Sect. 3.1.3 of the review by . The answer therefore depends on the form of distributions in space and time across the region (as well as, potentially, the order in which the data are averaged). As the evidence from data sets in this analysis (Figs. , , , and ) indicates that on larger scales AOD data become increasingly often more consistent with the log-normal than the normal distribution, and this was also found by prior spatial analyses , it seems likely that regional-scale (rather than point) AOD trend estimates would show similar, or even accentuated, differences between trends in arithmetic- and geometric-mean trends.

Widely used spatiotemporal aggregates of aerosol data from surface observations, satellite retrievals, and model simulations typically consist of arithmetic means and standard deviations of finer-resolution data. These statistics are most meaningful for normally distributed data, while previous work has indicated that AOD is often distributed close to log-normally on large scales. While one can transform between normal and log-normal summary statistics e.g. Table 1 of, in practice it is typical for data users to (implicitly or explicitly) assume normality in their analyses without considering the underlying distribution shape. This can influence the inferences drawn from the data. This study has illustrated the use of Shapiro–Wilk tests as a comparative tool to assess whether quantities such as AOD are more consistent with draws from normally or log-normally distributed populations. Data from ground-based observations (AERONET), satellite retrievals (MISR and MODIS), and model simulations (G5NR), despite their varying sampling and error characteristics, provide broadly consistent results.

As timescales increase from days to months to seasons, data become increasingly more consistent with log-normal than normal distributions, and the differences between arithmetic- and geometric-mean AOD become larger; assuming normality systematically overstates both the typical level of AOD and its variability. In low-AOD regions such as the open ocean and mountains, often the AOD difference is small enough (<0.01) to be unimportant for many applications especially on daily timescales. However, in continental outflow regions and near source regions over land, and on monthly or seasonal timescales, the difference is frequently larger than the GCOS goal uncertainty in a climate data record (the larger of 0.03 or 10 %). As a result of this, estimated trends in geometric-mean AOD are smaller in magnitude than (although consistent in sign with) those in arithmetic-mean AOD.

As noted earlier, using the arithmetic mean and standard deviation to summarise log-normal data is not “wrong” in a mathematical sense, as one can transform between the two. The danger is in not explicitly considering the underlying distribution when drawing an inference, as the result may be misleading or, at a minimum, less of a full picture than could otherwise be obtained.

The main recommendations from this study for future missions and reprocessing of current data sets and simulations are as follows.

The frequency distribution of a geophysical quantity should be analysed in order to assess how best to aggregate it. This analysis should be done at the spatial and temporal scale or scales of interest for the aggregation because distributions are scale-dependent. The Shapiro–Wilk technique is a powerful tool to assess discrepancies from a normal or log-normal distribution and should be further combined with desired performance thresholds to assess whether discrepancies are scientifically relevant for a given quantity.

The analysis presented here refers to AOD, but the methodology is general. provide goal uncertainties for many geophysical CDRs, which may be helpful for assessing the importance of averaging method in different disciplines. Overall the log-normal distribution seems to be a better reference for AOD aggregates than the normal distribution, on spatial scales of single locations or 1∘ and temporal scales from days to seasons.

It is important to bear in mind that these simple distribution forms are just approximations for the true underlying distribution of a geophysical quantity, and the relevant problem is in identifying one which is a sufficiently accurate representation for a given task. Normal and log-normal distributions are mathematically convenient and represent many data sets reasonably well, which is a motivating factor for considering these two both historically and in the present work. Sometimes multiple distribution forms are suitable: this analysis has shown that often in low-AOD conditions the choice of normal or log-normal representation may not matter for many purposes. Furthermore, while not analysed here, dependent on the choice of parameters, gamma distributions often used to describe cloud particle size distributions, e.g. can be numerically similar to log-normal ones. Indeed, both gamma and log-normal distributions can provide good summaries of rain rates in some circumstances . Sometimes multiple modes are required, and sometimes neither distribution is a suitable approximation.

If only a few distributions or points need to be summarised, then it is of course preferable to show the actual distributions and/or an informative summary which is agnostic to any particular distribution shape, such as a box-and-whisker plot. However for many larger-scale analyses, aggregated outputs from observations and model simulations are likely to remain the format of the choice for many data users due to their convenience and significantly lower computational and storage requirements than full-resolution (e.g. L2) data. While these unavoidably lead to a loss of information, it is important that users consciously consider the underlying distributions that the data sets are drawn from when utilising these summary statistics in research. The above recommendations will result in more statistically and scientifically meaningful data sets and decrease potential systematic biases which can lead to erroneous qualitative and quantitative interpretation about the state of the Earth system.