The first step in our methodology is to divide the world into its major source regions. Most dust is emitted from the so-called “dust belt” of northern Africa, the Middle East, central Asia, and the Chinese and Mongolian deserts (Prospero et al., 2002). In addition, dust is emitted in smaller quantities from Australia, southern Africa, and North and South America. Correspondingly, we divided the world into nine source regions that together account for the overwhelming majority (>99 %) of desert dust emissions simulated in models (Fig. 2a). Our analysis includes both natural and anthropogenic (land-use) emissions of dust in those source regions because our analysis is based on observations that by nature integrate both (but see further discussion in Sect. 5.1). However, our analysis explicitly does not include high-latitude dust sources, which produce dust through different mechanisms and with different properties than desert dust, yet likely dominate the dust loading for some high-latitude regions (Prospero et al., 2012; Bullard et al., 2016; Tobo et al., 2019; Bachelder et al., 2020). The nine source regions partially follow the definition in Mahowald (2007), with the main difference that we divided the North African source region, which accounts for approximately half of global dust emissions (Wu et al., 2020), into western North Africa, eastern North Africa, and the Sahel. Similar dust source regions were also used in more recent studies (Ginoux et al., 2012; Di Biagio et al., 2017).

We use an ensemble of global chemical transport and climate models (see Table 1) to obtain simulations of the emission, transport, and deposition of dust from each of the nine source regions. Specifically, we use simulations from the Community Earth System Model (CESM; Hurrell et al., 2013; Scanza et al., 2018), IMPACT (Ito et al., 2020), ModelE2.1 (Miller et al., 2006; Kelley et al., 2020), GEOS/GOCART (Rienecker et al., 2008; Colarco et al., 2010), MONARCH (Pérez et al., 2011; Badia et al., 2017; Klose et al., 2021), and INCA/IPSL-CM6 (Boucher et al., 2020). These six models were forced with three different reanalysis meteorology datasets (Table 1), which helped sample the uncertainty due to the exact reanalysis meteorology used that past work indicates is substantial (Largeron et al., 2015; Smith et al., 2017; Evan, 2018). Most of the six models were run for the years 2004–2008 or a subset thereof to coincide with the analysis period of regional DAOD in Ridley et al. (2016), which provided most of observational DAOD constraints used in this study (see Table 1). Sensitivity tests indicated that using different years from each simulation resulted in differences of less than 10 % in the inverse model results. Each model either ran a separate simulation for each source region or used “tagged” dust tracers from each source region. The exact setup of each model is described in the Supplement.

Our inverse model uses several results derived from model simulations (Fig. 1). First, for each model we obtained the normalized seasonally averaged column loading l̃r,s,kθ,ϕ, which is the spatial distribution of a unit (1 Tg) of loading originating from source region r for season s and particle size bin k. As such, the units of this field are per square meter (Tg m-2 loading per Tg of loading from source r), and we show annual averages of the normalized bulk dust loading for each model and source region in Fig. S1. Additionally, we obtained the normalized 3D concentration (C̃r,k,sθ,ϕ,P; m-3) and the 2D dust emission (F̃r,k,sθ,ϕ; m-2 yr-1) and (dry and wet) deposition fluxes (D̃r,k,sθ,ϕ; m-2 yr-1) that are associated with a unit of global dust loading for each source region, season, and particle size bin. All model fields were regridded using a modified Akima cubic Hermite interpolation (Akima, 1970) to a common resolution of 1.9∘ latitude by 2.5∘ longitude with 48 vertical levels (see Adebiyi et al., 2020, for further details). As explained further below, since our inverse model only uses normalized model fields per particle size, our results are independent of model tuning of global dust emissions or the simulated relative contributions of the major source regions defined here (Fig. 1). Our results are also not affected by model errors in representing dust mass extinction efficiency or the emitted dust size distribution.

We restricted our analysis to dust with a diameter D≤Dmax= 20 µm because there are insufficient measurements to constrain the abundance of coarser dust particles in the atmosphere (Adebiyi and Kok, 2020). Note, however, that the few measurements that have been made of dust with D>20 µm suggest that it is abundant over and near source regions such as North Africa and accounts for a non-negligible fraction of shortwave and longwave extinction (Ryder et al., 2019). As such, more measurements of “super-coarse” (D>10 µm) and “giant” (D>62.5 µm) dust are needed, which would allow the analysis presented here to be extended to larger particle sizes in the future. Since some of the models in our ensemble do not account for dust with D up to 20 µm, we use the procedure in Adebiyi et al. (2020; see their Sect. 2.3.1) to extend these models to 20 µm. Specifically, we use the normalized 12–20 µm particle size bin simulated by the GEOS/GOCART model to estimate what CESM and GISS ModelE2.1 would have simulated for an additional particle size bin extending to 20 µm (see additional details in the Supplement). We chose this bin specifically from the GEOS/GOCART model because it shows the best agreement against the observational constraint on regional DAOD (Fig. 3).

We next implemented an inverse model to determine the optimal bulk dust loading that must be generated by each source region to produce the best match against constraints on regional DAOD. This inverse model thus requires the spatial pattern of DAOD produced per unit bulk dust loading from each source region, which is the Jacobian matrix of DAOD with respect to dust loading. We obtained this DAOD produced per unit (1 Tg) of bulk dust loading by combining the simulated distributions of a unit of size-resolved dust loading (l̃r,s,kθϕ) with constraints on the globally averaged dust size distribution and extinction efficiency (Kok et al., 2017; Adebiyi and Kok, 2020). The calculations of the Jacobian matrix (this section) and the optimal bulk loading per source region (next section) are performed iteratively because each source region's fractional contribution to global dust loading affects the agreement against the constraint on the globally averaged dust size distribution.

The DAOD produced per unit of bulk dust loading originating from source region r in season s is (Kok et al., 2017) 1Jr,sθ,ϕ=∂τ˘r,sθ,ϕ∂L˘r,s=∑k=1Nbinsϵ‾kf˘r,s,kl̃r,s,kθ,ϕ where L˘r,s is the globally integrated bulk dust loading generated by source region r in season s, τ˘r,sθ,ϕ is the spatial distribution of DAOD due to dust from source region r in season s, Jr,s is the Jacobian matrix (Tg-1) of τ˘r,s with respect to L˘r,s, Nbins is the number of particle size bins in a global model simulation (or derived from the simulated modes), ϵ‾k is the size-dependent mass extinction efficiency (m2 g-1) of particle size bin k defined further below, l̃r,s,kθ,ϕ (m-2) is the simulated seasonally averaged spatial distribution of a unit of dust loading from source region r and particle bin k, and f˘r,s,k (unitless) is the fractional contribution of dust loading in size bin k to the seasonally averaged global dust loading generated by source region r (i.e., ∑kf˘r,s,k=1). As such, Eq. (1) obtains the DAOD produced per unit of dust loading from a given source region and season by adding up the normalized spatial distributions of the loading from each particle size bin, in proportion to each bin's contribution to the globally integrated loading produced by the source region, and then multiplying the size-resolved loading by the mass extinction efficiency (MEE) to obtain the DAOD.

To obtain the Jacobian matrix in Eq. (1) we need to obtain f˘r,s,k, each particle bin's fractional contribution to the globally integrated dust loading generated by source region r in season s. Because models as a group underestimate the mass of particles with larger diameters (D>∼ 5 µm; Kok et al., 2017), we adjust the model size distribution to match a constraint on the globally averaged dust size distribution derived from a combination of observations and models (Adebiyi and Kok, 2020). This procedure retains regional differences in the atmospheric dust size distribution that models simulated for the different source regions, while forcing the globally averaged dust size distribution that results from the summed contributions from all source regions to match the constraint on the globally averaged dust size distribution. That is, 2f˘r,s,k=αkf̃r,s,k∑k=1Nbinsαkf̃r,s,k, where f̃r,s,k is the modeled mass fraction per particle size bin for a given source region r and season s, and αk is the global correction factor for particle size bin k, which is different for each model. We obtained αk by setting the fraction of atmospheric dust in particle size bin k, summed over all source regions and seasons, equal to the constraint on the fractional contribution of particle size bin k to the global dust loading from Adebiyi and Kok (2020). That is, 3αk=∫Dk-Dk+dV‾atm(D)dDdD∑rNsreg∑s=1Nsf̃r,s,kL˘r,s/∑r=1Nsreg∑sNsL˘r,s, where Nsreg=9 is the number of source regions (Fig. 2a) and dV‾atm(D)dD is a realization of the size-normalized (that is, ∫0DmaxdV‾atmdDdD=1, where Dmax=20 µm) globally averaged volume size distribution from Adebiyi and Kok (2020), which was obtained by combining dozens of in situ measurements of dust size distributions with an ensemble of climate model simulations. Further, Dk- and Dk+ are respectively the lower and upper diameter limits of particle size bin k, and L˘r,s is the globally integrated and seasonally averaged bulk dust loading per source region (as obtained from the analysis below). As such, the denominator in Eq. (3) denotes the simulated globally averaged mass fraction, whereas the numerator denotes the globally averaged mass fraction in particle size bin k as constrained from in situ measurements and model simulations by Adebiyi and Kok (2020).

The final ingredient needed to use Eq. (1) to obtain the DAOD produced by a unit (1 Tg) of bulk dust loading from a given source region and season is the MEE (ϵ‾k). We do not use each model's assumed MEE because these tend to be substantially biased compared to measurements (Adebiyi et al., 2020). This bias is largely due to a neglect or underestimation of the asphericity of dust (Huang et al., 2020), which increases the surface-to-volume ratio and thereby enhances the MEE by ∼ 40 % (Kok et al., 2017). We thus follow Kok et al. (2017) in obtaining the MEE from constraints on the dust size distribution and the extinction efficiency of randomly oriented (Ginoux, 2003; Bagheri and Bonadonna, 2016) aspherical dust. That is, 4ϵ‾k=32ρ‾d∫Dk-Dk+Q‾ext(D)DdV‾atm(D)dDdD∫Dk-Dk+dV‾atm(D)dDdD, where Q‾ext(D) is a realization of the globally averaged size-resolved extinction efficiency from the analysis of Kok et al. (2017), which is defined as the extinction cross section divided by the projected area of a sphere with diameter D (πD2/4). The term dV‾atm(D)dD inside the integrals approximates the sub-bin distribution in particle size bin k as the globally averaged dust volume size distribution. Further, ρ‾d= (2.5 ± 0.2) × 103 kg m-3 is the globally averaged density of dust aerosols (Fratini et al., 2007; Reid et al., 2008; Kaaden et al., 2009; Sow et al., 2009). This observationally constrained density of dust is lower than the 2600 to 2650 kg m-3 used in many models (Tegen et al., 2002; Ginoux et al., 2004), most likely because dust aerosols are aggregates with void space that lowers their density below that of individual mineral particles.

The above procedure combined model simulations of the 2D spatial variability of size-resolved dust loading with constraints on dust size distribution and MEE. This procedure yielded the spatial distribution of DAOD that is produced by a unit (1 Tg) of dust loading from a given source region and season. Next, we use an inverse modeling approach to determine how many teragrams (Tg) of loading are needed from each source region to produce optimal agreement against constraints on the seasonal DAOD over areas proximal to major dust source regions.

We use joint observational–modeling constraints on regional DAOD at 550 nm from Ridley et al. (2016). This study used three different satellite AOD retrievals – from the Multi-angle Imaging Radiometer (MISR) and the Moderate Resolution Imaging Spectroradiometer (MODIS) on board the Terra and Aqua satellites – and bias-corrected those satellite data using more accurate ground-based aerosol optical depth measurements from AERONET. Ridley et al. (2016) then used an ensemble of global model simulations to obtain the fraction of AOD that is due to dust in 15 regions for which AOD is dominated by dust. Ridley et al. (2016) thus leveraged the strengths of these different tools by combining the accuracy of ground-based measurements with the global coverage of satellite retrievals and the ability of models to distinguish between different aerosol species. Furthermore, by averaging the resulting DAOD over large areas and long time periods (2004–2008 for each season), this study minimized representation errors that can affect model comparisons to data (Schutgens et al., 2017). An additional strength of the Ridley et al. (2016) analysis is that it transparently propagates a range of uncertainties that are both observationally and modeling based and which we in turn propagate into our own analysis (see Sect. 2.5). We also consider the Ridley et al. (2016) dataset more accurate than aerosol reanalysis products that assimilate similar AOD observations. This is because the Ridley et al. (2016) product includes a transparent quantification of errors that we propagated into the representation of the global dust cycle here and because the partitioning of assimilated AOD into different aerosol species in reanalysis products depends on the underlying aerosol models and is thus susceptible to the large biases in the prognostic aerosol schemes of these models (e.g., Adebiyi et al., 2020; Gliß et al., 2021). Nonetheless, the Ridley et al. (2016) data are subject to some important limitations discussed further in Sect. 5.1.

Although we consider the Ridley et al. (2016) constraints on DAOD to be more accurate than constraints from individual satellite products, AERONET data, or aerosol reanalysis products, this study's results for the Southern Hemisphere (SH) are susceptible to substantial biases. This is because dust makes up a substantially lower fraction of total AOD in the SH than for the main Northern Hemisphere (NH) source regions (e.g., Fig. S2 in Kok et al., 2014a). Therefore, we did not use the Ridley et al. (2016) results for the SH and instead used the seasonally averaged DAOD estimated by Adebiyi et al. (2020) over the three SH regions. These DAOD constraints are based on an ensemble of four aerosol reanalysis products, namely the Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA-2; Gelaro et al., 2017), the Navy Aerosol Analysis and Prediction System (NAAPS; Lynch et al., 2016), the Japanese Reanalysis for Aerosol (JRAero; Yumimoto et al., 2017), and the Copernicus Atmosphere Monitoring Service (CAMS) interim Reanalysis (CAMSiRA; Flemming et al., 2017). The resulting regional DAOD product also includes an error estimation based partially on the spread in DAOD in the four reanalysis products. In addition, we added a region over North America, for which Ridley et al. (2016) did not obtain results and for which we also use the reanalysis-based results of Adebiyi et al. (2020). In total, we thus have constraints with error estimates on the seasonal and area-averaged DAOD over 15 regions (see Fig. 2b and Table 2).

We then used an inverse modeling approach to determine the optimal combination of dust loadings from the nine source regions (denoted with subscript r) that minimizes the disagreement against the DAOD constraint of these 15 observed regions (denoted with subscript p) for each season. We thus need to account for the contribution of each of the nine source regions (Fig. 2a) to the DAOD in each of these 15 observed regions. The seasonally averaged DAOD over the observed region p is 5τ‾sp=∑r=1NsregJr,spL˘r,s, where τ‾sp is the DAOD averaged over observed region p and season s, and Jr,sp (Tg-1) is the Jacobian matrix of τ˘r,sp with respect to L˘r,s, where τ˘r,sp denotes the area-averaged and seasonally averaged DAOD over observed region p that is produced by dust from source region r. The Jacobian matrix Jr,sp is the area-weighted DAOD over observed region p that is produced per unit of bulk dust loading originating from source region r in season s. We obtain Jr,sp by integrating Eq. (1) over Ap, the area of the observed region p (Table 2): 6Jr,sp=∂τ˘r,sp∂L˘r,s=∫Ap∑k=1Nbinsϵ‾kf˘r,kl̃r,s,kθ,ϕdA∫ApdA.

The seasonally averaged globally integrated dust loading generated by each source region (L˘r,s) is thus determined from the number of units of dust loading from each source region r that results in the best agreement against the constraint on DAOD (τ‾sp) over the 15 observed regions. Equation (5) thus represents a system of equations for each simulation in our global model ensemble, which we can write in explicit matrix form for clarity: 7τ‾s1τ‾s2⋯τ‾sNτ,reg=L˘1,sL˘2,s⋯L˘Nsreg,sJ1,s1J1,s2⋯J1,sNτ,regJ2,s1J2,s2⋯J2,sNτ,reg⋮⋮⋱⋮JNsreg,s1JNsreg,s2⋯JNsreg,sNτ,reg.

We used Eq. (7) to obtain the seasonally averaged global dust loading generated by each source region. Specifically, for each season s we used the simplex search optimization method (Lagarias et al., 1998) to determine the nine values of L˘r,s that minimize the cost function of the summed squared deviation (χτ2) between the 15 DAOD constraints and the corresponding regional DAOD calculated from Eq. (7). That is (e.g., Cakmur et al., 2006), 8χτ2=∑p=1Nτ,reg∑r=1NsregL˘r,sJr,sp-τ‾sp2, where Nτ,reg=15 and Nsreg=9. Because the variables in Eqs. (1)–(8) are interdependent, we iterated these equations until convergence was achieved.

After constraining the seasonal dust loading L˘r,s generated by each source region, we now obtain the 2D DAOD and the size-resolved dust loading, emission and deposition fluxes, and 3D concentration. We do so by using the fact that other dust cycle components (DAOD, concentration, deposition) scale linearly with dust loading because our model simulations are driven by reanalysis products (Table 1) such that dust does not impact the meteorology. Each dust field can therefore be obtained by multiplying the simulated normalized dust field (e.g., seasonal dust concentration per unit of dust loading) by the number of units of dust loading per source region and season (L˘r,s).

The 2D DAOD is then 9τ˘sθ,ϕ=∑r=1NsregL˘r,sJr,sθ,ϕ.

The size-resolved and bulk dust loadings are respectively 10l˘s,kθ,ϕ=∑r=1Nsregf˘r,kL˘r,sl̃r,s,kθ,ϕ,and11l˘sθ,ϕ=∑k=1Nbins∑r=1Nsregf˘r,kL˘r,sl̃r,s,kθ,ϕ.

Similarly, the 3D size-resolved and bulk concentrations produced by each source region are 12C˘s,kθ,ϕ,P=∑r=1Nsregf˘r,kL˘r,sC̃r,s,kθ,ϕ,P,and13C˘sθ,ϕ,P=∑k=1Nbins∑rNsregf˘r,kL˘r,sC̃r,s,kθ,ϕ,P, where P is the vertical pressure level. And the size-resolved and bulk emission fluxes are 14F˘s,kθ,ϕ=∑r=1Nsregf˘r,kL˘r,sF̃r,s,kθ,ϕ,and15F˘sθ,ϕ=∑k=1Nbins∑rNsregf˘r,kL˘r,sF̃r,s,kθ,ϕ.

Finally, the size-resolved and bulk deposition fluxes are 16D˘s,kθ,ϕ=∑r=1Nsregf˘r,kL˘r,sD̃r,s,kθ,ϕ,and17D˘sθ,ϕ=∑k=1Nbins∑r=1Nsregf˘r,kL˘r,sD̃r,s,kθ,ϕ.

See the Glossary for further descriptions of each variable. In our companion paper (Kok et al., 2021a), we further partition these fields into the originating source region.

The results represented by Eqs. (9)–(17) require realizations of the various inputs (Fig. 1), which include both model fields and constraints on dust properties and abundance. Because each of these inputs is uncertain and as such is represented by a probability distribution, we obtained two products that sample different aspects of this uncertainty of the inputs, namely “improved model” results and “inverse model” results.

First, we obtained improved model results by sampling over different realizations of observational constraints on dust properties and abundance but using the output of only a single model. That is, we solved Eqs. (1)–(17) a large number of times (100; limited by computational resources), and for each iteration we drew a random realization of each of the observational constraints but used simulation results from a single model. This procedure thus includes a random drawing of realizations of the globally averaged dust size distribution (dV‾atm(D)dD), the extinction efficiency (Q‾ext(D)), the particle density (ρ‾d), and the observed regional DAOD (τ‾sp). As such, the improved model results represent output from a single model (see Table 1) for which DAOD is calculated from loading using the observational constraint on extinction efficiency (Eq. 4) and for which the contributions from different source regions and particle bins are added in such a way to simultaneously match observational constraints on the dust size distribution (Eq. 2) and DAOD (Eq. 8).

Second, we obtained our main product, namely the inverse model product that represents the optimal representation of the global dust cycle. We obtained this product by similarly sampling over different realizations of the input fields, but now including a random drawing of one of the six global model simulations in each of the bootstrap iterations. This additional step propagates uncertainty in model predictions of the normalized size-resolved dust loading, concentration, and deposition fields into our results (Eqs. 9–17). Because different models use different particle size bins (Table 1), we convert the size-resolved results from each bootstrap iteration to common particles size bins of 0.2–0.5, 0.5–1, 1–2.5, 2.5–5, 5–10, and 10–20 µm. We do so by assuming that sub-bin distributions follow the constraint on the globally averaged dust loading (Fig. 1). This assumption will introduce some further error in size-resolved results. For both the inverse model and improved model products, we retained only those bootstrap iterations that produced a root mean square error of less than 0.05 relative to the DAOD constraints; this quality control retained approximately three-quarters of the iterations.

In drawing the realizations of seasonally averaged observed DAOD (τ‾sp), we need to account for correlations of errors between different seasons and regions. Specifically, some of the errors in the calculation of the DAOD in Ridley et al. (2016) and Adebiyi et al. (2020) are systematic, such as errors in satellite retrieval algorithms and systematic model errors in simulations of (dust and non-dust) aerosols. These errors are thus at least partially correlated between seasons and regions, although we cannot establish the exact degree of correlation. We can thus roughly divide the errors into three different categories: errors that are completely random between seasons and regions, systematic errors that are correlated between different seasons for the same region, and systematic errors that are correlated across regions for a given season. The sum of the squared contributions of these three errors equals the square of the total error σ‾sp reported in Table 2. Since we cannot determine what the relative contribution of each of these three types of errors is, we assume that the contribution of each of these three errors is equal. Although the uncertainty in our results as quantified from the bootstrap procedure increases if a larger fraction of the DAOD error is assumed to be systematic, the median results presented in Sect. 4 are not sensitive to the partitioning of this error. The details of the mathematical treatment for calculating these errors are provided in the Supplement.

The bootstrap procedure used in the inverse model product propagates all the quantified random and systematic errors present in the inputs. Nonetheless, it cannot account for systematic biases in these inputs, such as the tendency of models to underestimate coarse dust lifetime (Ansmann et al., 2017; van der Does et al., 2018; Adebiyi et al., 2020). As such, the obtained uncertainty ranges should be interpreted as a lower bound on the actual uncertainty.

We use two sets of independent measurements to evaluate the ability of the inverse model to reproduce the global dust cycle. The first dataset is a compilation of dust surface concentration measurements. Of the 27 total stations in this compilation, 22 are measurements of the bulk dust surface concentration taken in the North Atlantic from the Atmosphere–Ocean Chemistry Experiment (AEROCE; Arimoto et al., 1995) and taken in the Pacific Ocean from the sea–air exchange program (SEAREX; Prospero et al., 1989) for observation periods noted in Table 2 of Wu et al. (2020). These data were obtained by drawing large volumes of air through a filter. To reduce the effects of anthropogenic aerosols, measurements were only taken when the wind was onshore and in excess of 1 m s-1 (Prospero et al., 1989). The mineral dust fraction of the collected particulates was determined either by burning the sample and assuming the ash residue to represent the mineral dust fraction or from their Al content (assumed to be 8 % for mineral dust, corresponding to the Al abundance in Earth's crust) (Prospero, 1999). Note that since these measurements were taken during the period 1981–2000, the dust surface concentration “climatology” obtained from these measurements is for a different time period than that of the model simulations used in the inverse model (Table 1).

Since most of the AEROCE and SEAREX stations are located far downwind of source regions, we also added a dataset of dust surface concentration from the Sahelian Dust Transect that was deployed in 2006 as part of the African Monsoon Multidisciplinary Analysis (AMMA; Lebel et al., 2010; Marticorena et al., 2010). This dataset contains measurements over 5–10 years of the surface concentration of aerosols with an aerodynamic diameter ≤10 µm (PM10,aer) at four stations in the western Sahel (M'Bour, Bambey, Cinzana, and Banizoumbou; see http://www.lisa.u-pec.fr/SDT/, last access: 13 May 2020). As with the AEROCE and SEAREX datasets, only measurements were used for which the wind direction was predominantly coming from dust-dominated regions. As such, these measurements have at least two systematic errors: (i) the AMMA data reported the concentration of all particulate matter, so taking these measurements as being of dust concentration overestimates the true dust concentration, and (ii) measurements taken when wind was not coming from a dust-dominated region were omitted, which could also cause an overestimation of the dust concentration. To mitigate the effect of this second error, we only use seasonally averaged dust concentrations for which >70 % of data was retained. This resulted in the omission of the winter and spring seasons at the Bambey station.

Following Huneeus et al. (2011) and Wu et al. (2020), we additionally added surface concentration measurements of PM10,aer dust from a long-term (May 1995–December 1996) filter-based deployment in Jabiru, northern Australia (Vanderzalm et al., 2003). However, unlike Huneeus et al. (2011) and Wu et al. (2020), we do not use data obtained in Rokumechi (Zimbabwe), which used a similar methodology, because most of the dust at this southern African site originated locally from within and near the national park where the station was located (p. 2649 in Nyanganyura et al., 2007).

To use the measurements of PM10,aer dust in Jabiru and the Sahel, we obtained the PM10,aer dust concentration for those models with size-resolved surface concentrations, namely the inverse model and each model in our ensemble. We did so by first obtaining the geometric diameter that corresponds to an aerodynamic diameter of 10 µm, which is DPM10,aer=caer×10µm=6.8µm. This uses the conversion factor caer=0.68 from Huang et al. (2021), who accounted for the effects of particle shape (Huang et al., 2020) and density to link the aerodynamic and geometric diameters. For each model, we then summed the contributions from particle bins with diameters smaller than DPM10,aer and used a correction factor cPM10,aer for particle size bins that straddle DPM10,aer. This correction factor uses the result from Adebiyi and Kok (2020) that the globally averaged dust size distribution (dV‾atm(D)dlnD) is approximately constant in the range of 5–20 µm such that the fractional contribution to the PM10,aer concentration of a bin that straddles DPM10,aer can be approximated as 18cPM10,aer=ln⁡DPM10,aer/Dk-ln⁡Dk+/Dk-, where Dk- and Dk+ are respectively the lower and upper limits of the particle size bin that straddles the 10 µm aerodynamic diameter (D=6.8 µm).

The second independent dataset that we used to evaluate the inverse model results is a compilation (110 stations) of the deposition flux of dust with a geometric diameter ≤10 µm (PM10) from Albani et al. (2014). This study merged data from previous datasets (Ginoux et al., 2001; Tegen et al., 2002; Lawrence and Neff, 2009; Mahowald et al., 2009) and adjusted these data to cover the 0.1–10 µm geometric diameter range. We obtained the PM10 deposition flux for the inverse model, the MERRA-2 data, and for each model in our ensemble following the approach above for the PM10,aer concentration data. Note that we cannot correct the concentration and deposition flux of the AeroCom Phase I models (next section) to the PM10,aer and PM10 size ranges because of a lack of size-resolved simulation data. We thus used the bulk concentration and deposition fluxes as many of these models simulated the PM10 size range (see Table 3 in Huneeus et al., 2011).

To assess the consistency of the inverse model results with both the independent datasets, we calculated the error-weighted mean square difference between the inverse model results and the observations. This statistic is known as the reduced chi-squared statistic and equals (Bevington and Robinson, 2003) 19χν2=∑iNiMi-Oi2σi2+σm2, where the index i sums over the Ni measurements in the dataset, Oi is the ith measurement in the dataset, Mi is the inverse model result for the location and season of the ith measurement (if applicable), σm is the calculated error in the inverse model result from the bootstrap procedure (see Sect. 2.5), and σi is the error in the measurement. For a model that matches measurements within the experimental error, χν2≈1 (Bevington and Robinson, 2003). Values of χν2 that are ≪1 indicate an overestimate of model or experimental error, whereas values of χν2≫1 indicate either an underestimate of errors or substantial biases in the model or experimental data.

We estimated the experimental errors in the surface concentration measurements by propagating the standard error in monthly averaged surface concentration measurements into seasonal and annual averages. Note that these errors do not include representation errors, which could be important (Schutgens et al., 2017). The errors in deposition data are more difficult to estimate, as these are not usually reported and because deposition fluxes can show large spatial and temporal variability (Avila et al., 1997), leading to larger representation errors. We estimated the relative error in deposition data measurements from the spread in measurements at similar locations. For the cluster of data in southern Europe (eastern Spain, southern France, northern Italy; e.g., Avila et al., 1997; Bonnet and Guieu, 2006), the standard deviation is about an order of magnitude, and for clusters of data north of Cape Verde (e.g., Jickells et al., 1996; Bory and Newton, 2000) and northwest of Tenerife (e.g., Honjo and Manganini, 1993; Kuss and Kremling, 1999), the standard deviation is about a quarter of an order of magnitude. We therefore take the relative error in deposition data as half an order of magnitude. This error is large compared to the inverse model error of approximately a quarter of an order of magnitude for deposition fluxes in the NH.

In order to compare the inverse model's representation of the global dust cycle against climate and chemical transport model simulations, we used the results of an ensemble of simulations for which the prognostic dust cycles were analyzed in detail, namely the AeroCom Phase I simulations of the dust cycle in the year 2000 (Huneeus et al., 2011). As such, the AeroCom simulations were obtained for a year closer to the time period in which most concentration and deposition measurements were taken (see above). We do not use newer AeroCom Phase II and Phase III simulations because only the dust component of Phase I models has been analyzed in detail. We furthermore do not use recently analyzed dust cycle results from CMIP5 models (Pu and Ginoux, 2018; Wu et al., 2020) because less than half of CMIP5 models with prognostic dust cycles reported total deposition fluxes, which are needed for the analyses against measurements (see previous section). In addition, many CMIP5 models did not include a prognostic dust cycle and instead read in pre-calculated dust emissions (Lamarque et al., 2010). But note that CMIP5 model errors against measurements are similar to those for AeroCom models and those for our model ensemble (e.g., compare Figs. 8 and 9 in Wu et al., 2020, against Figs. S9, S10, S12, and S13).

We analyzed the AeroCom Phase I model results to obtain the seasonally and annually averaged DAOD at 550 nm, the dust surface concentration, and the annually averaged total (wet and dry) deposition fluxes for comparisons against measurements and the inverse model results. We also obtained the globally integrated annually averaged dust emission flux, dust loading, and DAOD. We obtained these variables for each of the 13 AeroCom simulations available from the online AeroCom database (see https://aerocom.met.no/, last access: 11 December 2020; this repository does not contain the 14th model simulation analyzed in Huneeus et al., 2011, from the ECMWF model, which is thus omitted here).

We also analyzed the MERRA-2 dust product (Gelaro et al., 2017) in order to compare the inverse model's representation of the global dust cycle against a leading aerosol reanalysis product. We obtained the same variables from the MERRA-2 data as from the AeroCom data, except that we analyzed the MERRA-2 data for the years 2004–2008 to coincide with the regional DAOD constraints (Table 2).

We quantified the agreement of the various models against measurements using Taylor diagrams (Taylor, 2001) and by the correlation coefficients, bias, and root mean square errors (RMSEs). Because the surface concentration and deposition flux measurements span several orders of magnitude, their RMSEs are calculated in log space. We furthermore quantified overall model agreement against measurements by calculating the normalized error Φm against the available data for each hemisphere: 20ΦmNH=13Sτ,mNH∑n=1NmodelSτ,nNHNmodel+Sconc,mNH∑n=1NmodelSconc,nNHNmodel+Sdep,mNH∑n=1NmodelSdep,nNHNmodel,21ΦmSH=12Sconc,mSH∑n=1NmodelSconc,nSHNmodel+Sdep,mSH∑n=1NmodelSdep,nSHNmodel, where n and m index the different models, which include the inverse model, MERRA-2, the six model ensemble members, and the 13 AeroCom models such that Nmodel=21. Further, S denotes the RMSE of a model simulation with the DAOD (subscript τ), surface concentration (subscript conc), and deposition flux (subscript dep) datasets on the annual timescale. These data are split into datasets for the Northern Hemisphere (superscript NH) and Southern Hemisphere (superscript SH). For the SH, there are no accurate observational constraints on DAOD available (see Sect. 2.3), so we calculate the error relative to only the surface concentration and deposition flux datasets. Note that Φm is defined such that Φm=1 implies that a model is average among the 21 models in reproducing the global dust cycle. The lower Φm is, the more accurately it reproduces measurements and observations of the various aspects of the global dust cycle.

To verify the viability of our methodology, we first compare the inverse model's DAOD against the observationally constrained seasonal DAOD of 15 regions (Table 2). As is expected from the inverse modeling methodology, the error is substantially reduced compared to the unmodified ensemble of simulations for all seasons (Fig. 3a–d). This decrease in error is particularly pronounced over North Africa, which we characterized using three different source regions (western North Africa, eastern North Africa, and the Sahel; Fig. 2a) and which shows a decrease in the RMSE of a factor of approximately 3 to 5 depending on the season. Note that the DAOD in the mid-Atlantic region is nonetheless systematically underestimated by both the models in our ensemble and the inverse model. This is a common problem in models that is likely in part due to overly fast removal in models (Ridley et al., 2012; Yu et al., 2019). The RMSE over the relatively minor dust source regions of North America, Australia, South America, and southern Africa is similarly reduced by about a factor of 5. For the East Asia and Middle East–central Asia regions, the decrease in RMSE is about a factor of 1.5 to 2. This relatively smaller decrease in the RMSE likely occurs because we used only one source region each for both these relatively extensive source regions. Consequently, our procedure is unable to eliminate some biases of the model ensemble in these regions, such as an underestimation of DAOD in the Thar desert, which could be due to model underestimations of emissions in this region (Shindell et al., 2013). Future work could thus improve upon our results by using more source regions to better constrain the contributions of the Middle East and Asian source regions to the global dust cycle.

Overall, our procedure achieves a substantial reduction of the total DAOD error summed over the 15 regions, reducing the RMSE by over a factor of 2 from 0.092 to 0.041. This reduction in error is expected, as our methodology minimized the error against these regional DAOD data. Moreover, we find that the reduced chi-squared statistic, which is of order 1 for a model that captures observations within the uncertainties (Bevington and Robinson, 2003), is indeed less than 1 for all seasons except boreal spring. This implies that our methodology results are in good agreement with the observational DAOD constraints. Further, the ability of the inverse model to reproduce the spatial pattern of DAOD on both seasonal (Fig. 3e) and annual (Fig. 3f) timescales is substantially improved relative to both the six models in the model ensemble and the AeroCom Phase I models, and it is similar to that of the MERRA-2 dust product. This is noteworthy as many of the satellite and ground-based AOD observations upon which the observational DAOD is based have been used to inform the dust schemes in the ensemble models (Cakmur et al., 2006; Kok et al., 2014a) and have been assimilated by the MERRA-2 dust product (Buchard et al., 2017; Gelaro et al., 2017; Randles et al., 2017).

We present inverse model results for the dust emission rate, DAOD, column-integrated dust loading, dust surface concentration, and dust deposition flux (Table 3, Fig. 4) and compare these inverse model results against independent measurements in Sect. 4.3. We also provide median estimates with the uncertainty of the main size-resolved properties of the global dust cycle (Fig. 5).

Our results indicate that the global emission rate and loading of dust with a geometric diameter D≤20 µm (PM20) are larger than most models account for. AeroCom models reported an ensemble median global dust emission rate of 1.6 × 103 Tg yr-1 (1 standard error range: 1.0–3.2 × 103 Tg yr-1), and CMIP5 models reported a value of 2.7 (1.7–3.7) × 103 Tg yr-1; both these ensembles included a mix of models simulating dust up to diameters of 10 µm or more (see Fig. S7). Our results indicate that the global emission rate of PM20 dust is 4.6 (3.4–9.1) × 103 Tg yr-1. There are two reasons for this larger global dust emission rate. First, our methodology accounts for dust up to a geometric diameter of 20 µm, which is a larger size range than accounted for in many AeroCom and CMIP5 models (Huneeus et al., 2011; Wu et al., 2020; Fig. S7) and thus results in a larger bulk dust emission flux. Accounting for this larger size range is desirable because observations indicate that ∼ 30 % of PM20 dust loading consists of super-coarse dust (D>10 µm) (Ryder et al., 2019; Adebiyi and Kok, 2020; Fig. 5b). Because super-coarse dust has a shorter lifetime (1.0 (0.4–1.8) d; Fig. 5d) than finer dust, we find that super-coarse dust accounts for ∼ 65 % of the total PM20 dust emission flux, which corresponds to 2.9 (1.8–6.5) × 103 Tg yr-1 (Fig. 5a). This ∼ 65 % relative contribution of the 10 ≤D≤20 µm size range is substantially larger than that inferred from size-resolved measurements of the emitted dust flux (Huang et al., 2021). In order to match in situ atmospheric dust size distributions, current models thus need to emit more super-coarse dust than determined from measurements of the emitted dust flux, which further supports the inference from multiple previous investigations that super-coarse dust deposits too quickly in atmospheric models (Maring et al., 2003; Ansmann et al., 2017; Weinzierl et al., 2017; van der Does et al., 2018). The small mid-visible (550 nm) MEE of super-coarse dust (0.13 (0.12–0.15) m2 g-1; Fig. 5e) causes it to account for only a small fraction (7.2 (5.7–9.3) %) of the total shortwave (SW) DAOD of 0.028 (0.024–0.030) (Fig. 5f and Table 3). However, dust with 10≤D≤20 µm is nonetheless radiatively important because it accounts for a larger fraction of dust absorption of SW radiation (Tegen and Lacis, 1996; Samset et al., 2018) and because it produces ∼ 20 % of the global dust longwave (LW) DAOD of 0.014 ± 0.003 (Fig. 5h).

The second reason that PM20 emission fluxes are larger than accounted for in most models is that observations have shown that many models have a bias towards fine dust (Kok, 2011b; Ansmann et al., 2017; Adebiyi and Kok, 2020). Indeed, models that do include dust up to a 20 µm geometric diameter tend to underestimate the global PM20 dust emission rate relative to our results (Fig. S7). Because coarse dust has a shorter lifetime and a lower MEE (Fig. 5e, f), correcting this fine dust bias requires a substantially larger total emission flux to match DAOD constraints. Many of the models in our ensemble partially addressed the fine bias by using the brittle fragmentation theory parameterization for the emitted dust flux, which is substantially coarser than other emitted dust size distributions (Kok, 2011b). This causes our model ensemble to show a larger emission flux (3.5 (2.7–5.2) × 103 Tg yr-1) than AeroCom models (1.6 (1.0–3.2) × 103 Tg yr-1), although this increase is also due to these more recent models simulating dust out to larger particle diameters (Fig. S7). More recent work has used dozens of in situ measurements to show that the fine dust bias in models is even more substantial than previously reported, specifically that the atmospheric loading of coarse dust with D>5 µm is several times greater than accounted for in most models (Adebiyi and Kok, 2020). Generating this even greater loading of coarse dust thus requires a correspondingly larger emission flux (Table 3; Fig. 5a). Emission fluxes would be even larger if the maximum size range was extended further to include dust with D>20 µm, which measurements indicate is abundant close to source regions and might be important for interactions with longwave radiation (Ryder et al., 2013, 2019; Fig. 5g, h). As previously reported by Adebiyi and Kok (2020), accounting for the substantial atmospheric loading of coarse dust with 5≤D≤20 µm also drives a larger total dust loading, increasing from 20 (12–24) Tg obtained by AeroCom models and 17 (14–36) Tg obtained by CMIP5 models to 26 (22–30) Tg obtained here (Table 3). Since models indicate that the atmospheric loading of non-dust aerosols is around 10 Tg (Textor et al., 2006; Gliß et al., 2021), dust is likely by far the most dominant aerosol species by mass, accounting for approximately three-quarters of the atmosphere's total particulate matter loading.

The constraints on the global dust cycle obtained here are strongest on the DAOD because our inverse model minimizes error with respect to observed regional DAOD (Sect. 4.1). The inverse model then relies on observational constraints on the globally averaged dust size distribution and extinction efficiency to link the DAOD to loading per source region (Sect. 2.2 and 2.3), which adds further uncertainty to our inverse model results. Constraints on dust emission and deposition fluxes are still more uncertain because these further depend on results from the ensemble of models, such as the spatial pattern of emission within individual source regions, transport, and the size-resolved dust lifetime. The lifetime of coarse dust shows especially large variability between models, which substantially adds to the uncertainty in PM20 emission and deposition fluxes because coarse dust dominates these fluxes (Fig. 5a, b). Consequently, the relative uncertainties in global emission and deposition fluxes are several times larger than the relative uncertainty in DAOD (Table 3).

After obtaining inverse model results for key aspects of the global dust cycle, we next evaluate the accuracy of this representation of the global dust cycle using independent measurements of dust surface concentration and dust deposition fluxes (see Sect. 3.1). We divide these results into comparisons for the NH (Sect. 4.3.1) and the SH (Sect. 4.3.2). We do this because we have observationally informed constraints on DAOD for 11 NH regions and therefore expect the inverse model results to show relatively good agreement against independent measurements in the NH. In contrast, we do not have observationally constrained DAOD for the SH; instead, the inverse model used an ensemble of reanalysis products, whose ensemble members might have similar biases as they assimilate similar remote sensing datasets. As such, we expect the inverse model results to show less agreement against independent measurements in the SH.

Our results are subject to a few important limitations. First, although our methodology integrates observational constraints, it still relies on global model simulations to compute a number of key variables, including the spatial pattern and timing of dust emissions within each source region, the vertical distribution of dust, and the deposition flux of dust. All three of these processes are known to be subject to important model errors (e.g., Ginoux, 2003; Huneeus et al., 2011; Kim et al., 2014; Kok et al., 2014a; Evan, 2018; O'Sullivan et al., 2020). As discussed in Sect. 1, accurately simulating the magnitude and spatiotemporal variability of dust emissions represents a fundamental challenge for models. To mitigate this problem, many models prescribe prolific dust sources where geomorphologic processes concentrate fine soil particles as a result of fluvial erosion (Ginoux et al., 2001; Prospero et al., 2002; Tegen et al., 2002; Zender et al., 2003; Koven and Fung, 2008). However, these representations are highly uncertain, as indicated by large differences in the spatial patterns of emissions (Cakmur et al., 2006; Kok et al., 2014a; Wu et al., 2020). In addition to these challenges with simulating dust emissions, many models also underestimate the height at which dust is transported (Yu et al., 2010; Johnson et al., 2012; Kim et al., 2014). Furthermore, excessive diffusion of coarse dust due to numerical sedimentation schemes causes additional problems in many models (Ginoux, 2003; Eastham and Jacob, 2017; Zhuang et al., 2018) and might be partially responsible for a general underestimation of long-range transport of coarse dust relative to measurements and satellite observations (Maring et al., 2003; Ridley et al., 2014; Ansmann et al., 2017; Gasteiger et al., 2017; van der Does et al., 2018; Yu et al., 2019). Because of these various uncertainties in model representations of dust processes, our constraints on dust AOD and loading are the strongest, and constraints on dust emission, deposition, and 3D concentration have greater uncertainty (Table 3). Furthermore, although uncertainties in the products obtained here include the error due to the spread in the results of the models in our ensemble, they do not account for systematic biases between the model ensemble and the real world, which might be substantial in light of the problems in model simulations highlighted above. In addition, some of the other inputs to our methodology, such as the globally averaged dust size distribution (Adebiyi and Kok, 2020), would also be affected by possible biases in model results, such as in deposition. One consequence of our incomplete understanding of dust processes is that observational constraints will remain valuable even as model resolution is increased.

A second limitation of our methodology is that the quality of the inverse model depends on the accuracy of the observational constraints on the globally averaged dust size distribution (Adebiyi and Kok, 2020), extinction efficiency (Kok et al., 2017), and the regional DAOD constraints obtained in Ridley et al. (2016) and Adebiyi et al. (2020). As such, the results presented here are subject to the limitations of those studies. These limitations are described in detail in the corresponding papers and include possible biases due to errors in the dust extinction efficiency due to the assumed tri-axial ellipsoid shape being an imperfect approximation of the highly heterogeneous shape and roughness of real dust particles (Lindqvist et al., 2014; Kok et al., 2017), errors in the remotely sensed optical depth retrieval algorithms for aspherical dust particles (Hsu et al., 2004; Kalashnikova et al., 2005; Dubovik et al., 2006), errors in the cloud-screening algorithms used in satellite and ground-based remote sensing products, errors due to a scarcity of AERONET “ground-truth” data in dust-dominated regions, and systematic differences between clear-sky and all-sky AOD, although studies indicate that such a systematic difference is small for dusty regions (Kim et al., 2014; Ridley et al., 2016; Adebiyi and Kok, 2020). The uncertainty due to many (not all) of these errors has been quantified in the relevant papers, and these errors have thus been propagated into the results in the present study. An additional key limitation is that the Ridley et al. (2016) DAOD constraint uses model simulations of the AOD due to other aerosol species to separate dust AOD from non-dust AOD in dusty regions. As such, consistent biases in model simulations of non-dust AOD would have affected the inferred dust AOD. For instance, a systematic underestimation of biomass burning AOD across models (Reddington et al., 2016; van der Werf et al., 2017) would cause the underestimated biomass burning AOD to instead be assigned to dust, thereby causing an overestimate of dust AOD. This source of error might be particularly important in regions with substantial non-dust aerosol loadings, such as in much of Asia and in the Sahel during the biomass burning season (Yu et al., 2019). Furthermore, the regional DAOD constraints from Adebiyi et al. (2020) for the lesser source regions of Australia, North America, South America, and southern Africa are based on an ensemble of aerosol reanalysis products. These products assimilate remotely sensed AOD and partly rely on prognostic aerosol models to partition this AOD to the different aerosol species (e.g., Randles et al., 2017). Considering the large uncertainties in dust models (Huneeus et al., 2011; Checa-Garcia et al., 2020; Wu et al., 2020), these products could thus be substantially biased in regions for which dust does not dominate AOD.

Another limitation of our results is that the representation of the modern-day global dust cycle is based mostly on model data and regional DAOD constraints for the period 2004–2008. As such, changes in the dust cycle before or after that period are not reflected in our results. For instance, satellite measurements have shown an increase in dust loading in the Middle East (Hsu et al., 2012; Kumar et al., 2019). Further, we assume that dust contributes to loading and deposition in the same season that it is emitted, which is not always true and could generate small inconsistencies. We also use observational constraints on DAOD only at the mid-visible range (550 nm), which is most sensitive to dust with a diameter of ∼ 1–5 µm (Fig. 5f). Dust particles outside this size range are thus partially constrained by correcting model simulations to match the globally averaged dust size distribution inferred in Adebiyi and Kok (2020) and might thus have larger errors than dust with diameters around 1–5 µm. Another important limitation is that many of the models in our ensemble do not explicitly account for anthropogenic (e.g., land-use) sources of dust emission, which might account for ∼ 10 %–25 % of present-climate dust emissions (Mahowald et al., 2004; Tegen et al., 2004; Ginoux et al., 2012). However, the observationally constrained DAOD used here to scale dust emissions and loading does not distinguish between natural and anthropogenic dust and thus inherently includes both. Nonetheless, the omission of land-use dust emissions from many of the models in our ensemble could produce important errors close to anthropogenic dust sources, which might account for a substantial fraction of total emissions in Asia, Australia, southern Africa, and the Americas (Ginoux et al., 2012). Another limitation is that our approach neglects feedback between dust and meteorology, which could be important for certain regions or seasons (Cakmur et al., 2004; Miller et al., 2004; Pérez et al., 2006; Ahn et al., 2007; Heinold et al., 2007; Colarco et al., 2014; Randles et al., 2017).

Finally, the conclusion that our methodology yields an improved representation of the global dust cycle depends on the quality of the independent data used to evaluate the inverse model results. However, these data have a few limitations. First, some of the measurements might have large, unquantified errors. This appears to be the case especially for deposition flux measurements, which show a much larger spread than surface concentration measurements, even for proximal locations. Second, the concentration and deposition data used to evaluate the inverse model results do not coincide in time with the simulations, which could affect the comparisons (see Sect. 3 and further discussions in, e.g., Huneeus et al., 2011). Finally, some aspects of our representation of the global dust cycle were not explicitly tested against measurements. Future work could further investigate the accuracy of the inverse model results through comparisons against additional data, such as visibility data (Mahowald et al., 2007; Shao et al., 2013), dust vertical profile data (Yu et al., 2010; Kim et al., 2014), and remote sensing retrievals of the Ångström exponent (Huneeus et al., 2011). In addition to these limitations with the data, it is also possible that the inverse model better reproduces independent measurements because of canceling errors, for instance between model underestimates in long-range transport of coarse dust and overestimates in emissions from source regions closer to observational sites.

The results in Figs. 6–11 show that our methodology of integrating observational constraints on dust properties and abundance reduces model errors in simulating the global dust cycle. This finding is particularly clear from the results of the six improved models. Each of these models shows a substantial reduction of model error against measurements and observations of the NH dust cycle (Figs. 7d–f, 8a–d), with the average reduction of the errors in improved models equaling ∼ 35 % (Fig. 8e). These findings suggest several ways in which the representation of the global dust cycle can be improved in global and regional models.

First, our results indicate that it is critical for models to account for the substantial asphericity of dust aerosols (Okada et al., 2001; Huang et al., 2020). Dust asphericity enhances the MEE by ∼ 40 % because aspherical dust particles extinguish more radiation than volume-equivalent spherical particles (Kalashnikova and Sokolik, 2004; Potenza et al., 2016; Kok et al., 2017). As such, not accounting for dust asphericity causes an overestimation of the dust loading needed to match DAOD constraints by ∼ 40 % and can thus produce a corresponding bias against concentration and deposition flux measurements. This is illustrated by the MERRA-2 results, which are in good agreement with DAOD constraints (Figs. 7d, e, 8e) but overestimate NH dust deposition flux measurements by ∼ 25 % and surface concentration measurements by ∼ 50 % (Figs. 8a, b and S9 and S10). MERRA-2 uses dust optics from Colarco et al. (2014) based on spheroids, which underestimate dust asphericity (Huang et al., 2020) and yielded a ∼ 25 % enhancement of dust extinction. Accounting for the full extinction enhancement of ∼ 40 % due to dust asphericity would thus reduce the biases of the MERRA-2 dust product against surface concentration and deposition flux measurements. Since most current models either do not account for dust asphericity or substantially underestimate its effect on extinction efficiency (Huang et al., 2020), we recommend that models account for the full enhancement of extinction by dust asphericity, for instance by implementing the constraints on the extinction efficiency of aspherical dust from Kok et al. (2017).

Second, models can be improved by correcting the current substantial underestimation of coarse dust loading. In this study, we integrated a joint observational–modeling constraint on the globally averaged dust size distribution in order to account for the ∼ 17 Tg of coarse dust (5≤D≤20 µm) that observations indicate is present in the atmosphere (Ryder et al., 2019; Adebiyi and Kok, 2020). The finding that our methodology almost eliminates biases against NH measurements (Figs. 6–8) suggests that this constraint on the globally averaged dust size distribution is relatively accurate. This further supports the conclusion from several studies that models substantially underestimate coarse dust loading (Ansmann et al., 2017; van der Does et al., 2018; Ryder et al., 2019; Adebiyi and Kok, 2020; Gliß et al., 2021). Models can thus be improved by eliminating the current underestimation of coarse dust. This could be done either by adjusting the size distribution of emitted dust aerosols such that the size-resolved global dust loading matches the constraints on the globally averaged size distribution (Adebiyi and Kok, 2020) or, preferably, by improving the relevant model physics. Specifically, recent studies indicate that the underestimation of coarse dust is due to both an underestimation of the emission of coarse dust (Huang et al., 2021) and an underestimation of the lifetime of the emitted coarse dust (Maring et al., 2003; Weinzierl et al., 2017). Measurements of the emitted dust size distribution show a much larger flux of dust with D≥10 µm than current parameterizations, including brittle fragmentation theory (Kok, 2011b, a), account for (Huang et al., 2021). The fact that models need to use a fractional contribution of emitted super-coarse dust (10≤D≤20 µm) that is even larger than found by measurements (Fig. 5a; Sow et al., 2009; Rosenberg et al., 2014; Huang et al., 2019, 2021) suggests that models underestimate the lifetime of (super-)coarse dust. This finding further supports the inference from several lines of evidence that models underestimate the lifetime of (super-)coarse dust (Maring et al., 2003; Weinzierl et al., 2017; van der Does et al., 2018). As such, models require improved parameterizations of both the emitted dust size distribution and dry deposition processes to properly account for the abundance of (super-)coarse dust in our atmosphere. Improved parameterizations of the emitted dust size distribution that better account for the large contribution of (super-)coarse dust are under development (Huang et al., 2021). To improve size-resolved dry deposition, we recommend that models account for the ∼ 20 % slowing of the gravitational settling speed due to dust asphericity (Huang et al., 2020). Further improvements in dust deposition parameterizations are likely needed, including accounting for the strong enhancement of upward vertical transport of emitted (super-)coarse dust by topography (Rosenberg et al., 2014; Heisel et al., 2021) and possible reductions of gravitational settling due to electrification and turbulence in dust layers (Ulanowski et al., 2007; Gasteiger et al., 2017; van der Does et al., 2018).

Finally, our results indicate that model accuracy can be substantially improved by correcting biases in the dust loading generated by each main source region (Figs. 3, 8e). These biases could be reduced in two ways. First, models could emulate the procedure developed here and scale the emission of dust from each region to match the observed regional DAOD obtained in Ridley et al. (2016). A second approach would be to scale the simulated (size-resolved) emissions or loading per source region and season to that obtained in our companion paper (Kok et al., 2021a). These improvements would be most effective for simulations of the present-day dust cycle by regional and global models, as well as short-range, medium-range, and seasonal forecasts of dustiness by numerical weather models. Ultimately, parameterizations of dust emissions should be improved to eliminate the need for adjustment of model simulations in this manner. This is critical because without identifying and correcting the problematic model physics, we cannot know how these processes change with climate, for example under global warming or over glacial cycles. Together with uncertainties due to future land-use changes, this problem limits the ability of models to predict future changes in the global dust cycle and its effect on climate and the Earth system (Evan et al., 2016; Kok et al., 2018).

Although we found that the integration of observational constraints on dust properties and abundance is effective in reducing model errors in the representation of the NH dust cycle, we found only slight improvements for the SH dust cycle (Fig. 11e). There are two likely reasons for this finding. First, whereas the inverse model is informed by accurate observational constraints on regional DAOD in the NH, such constraints are less accurate for the less dusty SH (Ridley et al., 2016). And second, the dust cycle simulations used in our ensemble are less accurate for the SH dust cycle than for the NH dust cycle, as indicated by substantially larger root mean square errors relative to measurements for the SH (Fig. 11c, d) than for the NH (Fig. 8c, d). These larger model errors for the SH likely occur because a large fraction of SH dust emissions originates from regions containing sparse vegetation (Ito and Kok, 2017), the effects of which on dust emission are difficult for models to represent accurately (King et al., 2005; Okin, 2008). Additionally, there are fewer data available in the SH from ground-based measurements such as dust surface concentration measurements. And whereas many measurements close to dust source regions are available for the NH, most measurements for the SH are at sites remote from the main dust source regions (Fig. 2c, d), where they are less effective at constraining the main features of the SH dust cycle. There are also fewer satellite retrievals available to constrain simulations of the SH dust cycle. For instance, dust sources such as Patagonia are shrouded by clouds for a larger fraction of the year than most NH sources (Ginoux et al., 2012), which limits constraints on dust emissions and DAOD from satellite retrievals (Gasso and Stein, 2007). Additionally, the errors in satellite retrievals tend to be larger for the SH than for the NH because the relative error decreases with AOD (Kahn et al., 2005; Remer et al., 2005). Considering the important role that the SH dust cycle plays in biogeochemistry, the carbon cycle, and the climate system (Lambert et al., 2008; Hamilton et al., 2020), our results underscore a critical need for more observations to constrain the SH dust cycle.

In addition to identifying mechanisms to improve individual model simulations, this study obtained an improved representation of the global dust cycle that can be used to improve our understanding and quantification of the impact of dust on the Earth system. This addition to the DustCOMM dataset (Adebiyi et al., 2020) contains dust loading, DAOD, (surface) concentration, and (wet and dry) deposition flux fields that are resolved by space, particle size, and season (data are available at https://dustcomm.atmos.ucla.edu/data/K21a/, last access: 12 May 2021). Our results in Sect. 4.3 indicate that this dataset is more accurate than both a large number of climate and chemical transport model simulations and the MERRA-2 dust product. Moreover, whereas MERRA-2 is internally inconsistent because dust loading is adjusted after emission by assimilating AOD measurements (Randles et al., 2017; Wu et al., 2020), our method for integrating observational constraints yields a self-consistent representation of the global dust cycle. Our companion article (Kok et al., 2021a) will supplement this dataset by partitioning all these fields by the originating source region. This dataset representing the seasonally resolved and size-resolved global dust cycle can be used to more accurately quantify dust impacts on the Earth system, such as on climate, weather, the hydrological cycle, biogeochemistry, and human health.

Our dataset of an improved representation of the global dust cycle has an additional strength that amplifies its use: our dataset quantifies and propagates a range of observational and modeling uncertainties (see Sect. 2.5). Comparisons against independent datasets indicate that the propagated error is realistic for the NH and might slightly underestimate the true errors in the SH (Figs. 7 and 10). The availability of realistic errors allows for the propagation of uncertainty into dust impacts constrained using our dataset, such as in the quantification of direct radiative effects and indirect cloud and biogeochemistry effects (Mahowald, 2011). With a few exceptions (Kok et al., 2017; Regayre et al., 2018; Di Biagio et al., 2020), the quantification of the uncertainty of (dust) aerosol direct and indirect radiative effects is uncommon yet critical to robustly constraining (dust) aerosol impacts on the Earth system (Carslaw et al., 2010; Mahowald et al., 2011b). Moreover, the quantification of uncertainties in aerosol effects in both the present-day and pre-industrial climates is crucial to constraining climate sensitivity (Carslaw et al., 2013, 2018).

A second strength of our dataset representing the global dust cycle is that it uses an analytical framework that could be improved and expanded. The framework could be improved by using more accurate observational constraints of dust properties and dust abundance as inputs (see Fig. 1), for instance from several recent DAOD climatologies (Pu and Ginoux, 2018; Voss and Evan, 2020; Gkikas et al., 2021), or by adding additional types of observational constraints, such as on the dust vertical profile (Song et al., 2021). The framework could be expanded by adding calculations of additional dust properties and impacts, such as dust mineralogy and radiative effects. The framework could also be expanded to cover different time periods than the 2004–2008 time period we used here or to constrain the historical variability of the global dust cycle, for instance using time-resolved DAOD climatologies (Voss and Evan, 2020; Gkikas et al., 2021; Song et al., 2021). As such, our approach has the potential to continually improve the representation of the global dust cycle and its impacts on the Earth system.