The description of particle properties is well-established and can be found in textbooks with varying levels of detail. Thus, we can restrict ourselves to a brief summary of those properties that are of special relevance for MOPSMAP.

The microphysical properties of an aerosol particle are described by its shape, size, and chemical composition.

Atmospheric aerosols might be spherical in shape but many types consist of nonspherical particles, often with a large variety of different shapes. Mineral dust e.g., and volcanic ash aerosols e.g., are important examples of the latter, but, for example, pollen, dry sea salt or soot particles are also usually nonspherical. A quite common approach to consider the particle shape is the approximation using spheroids or distributions of spheroids . Spheroids originate from the rotation of ellipses about one of their axes. Only one parameter is required for the shape description. use the “axial ratio” ϵm, which is the ratio between the length of the axis perpendicular to the rotational axis and the length of the rotational axis. By contrast, use the “axis ratio” ϵd, defined as the inverse of ϵm. Spheroids with ϵm<1, ϵd>1 are called prolate (elongated) whereas spheroids with ϵm>1, ϵd<1 are oblate (flat) spheroids. The aspect ratio ϵ′ is the ratio between the longest and the shortest axis, i.e., ϵ′=1ϵm=ϵd in the case of prolate spheroids and ϵ′=ϵm=1ϵd in the case of oblate spheroids. Spheroids with ϵ′=1 are spheres.

The size of a particle is commonly described by its radius or diameter. While this is unambiguous in the case of spheres, more detailed specifications are necessary for any kind of nonspherical particles. Often the size of an equivalent sphere is used for the description of the nonspherical particle size: the volume-equivalent radius rv of a particle with known volume V (containing the particle mass, i.e., without cavities) is rv=3V4π3, whereas the cross-section-equivalent radius rc of a particle with known orientation-averaged geometric cross section Cgeo is rc=Cgeoπ. In the case of spheroids, rc is equal to the radius of a sphere having the same surface area (as used by ). For the conversion between rv and rc, the radius conversion factor ξvc=rvrc=3π4VCgeo3/23 is used . ξvc is equal to 1 in the case of spheres and decreases with increasing deviation from a spherical shape. Another definition of size is given by the radius of a sphere that has the same ratio between volume and geometric cross section as the particle rvcr=3V4Cgeo=ξvc3rc. This definition corresponds to the case “VSEQU” presented by , to the “effective radius” in Eq. (5) of , and is more sensitive to non-sphericity than rv or rc. For example, a particle with rc=1 µm and ξvc=0.9 implies that rv=0.9 µm and rvcr=0.729 µm.

For setting up a data set of optical properties for different wavelengths, it is highly beneficial to make use of the size parameter x=2πrλ. The size parameter x describes the particle size relative to the wavelength λ. The advantage of using x is that optical properties (qext, ω0, and F, as defined below) at a given wavelength are fully determined by its shape, refractive index m, and x. Equivalent size parameters xv, xc, and xvcr are calculated from the equivalent radii, analogously to Eq. ().

The chemical composition of a particle determines its complex wavelength-dependent refractive index m. The imaginary part mi is relevant for the absorption of light inside the particle, whereby an imaginary part of zero corresponds to non-absorbing particles.

The optical properties of a nonspherical particle depend on the orientation of the particle relative to the incident light. In our data set we assume that particles are oriented randomly; thus, the optical properties are stored as orientation averages .

The orientation-averaged optical properties at a given wavelength are fully described by the extinction cross section Cext, the single-scattering albedo ω0 and the scattering matrix F(θ), where θ is the angle by which the incoming light is deflected during the scattering process (“scattering angle”). The extinction cross section Cext can be normalized by the orientation-averaged geometric cross section Cgeo of the particle giving the extinction efficiency qext=CextCgeo=Cextπrc2. The single-scattering albedo ω0 is given by ω0=CscaCext, where Csca is the scattering cross section.

For the scattering matrix F of randomly oriented particles, we use the notation of , i.e., F(θ)=a1(θ)b1(θ)00b1(θ)a2(θ)0000a3(θ)b2(θ)00-b2(θ)a4(θ) with six independent matrix elements. The scattering matrix describes the transformation of the incoming Stokes vector Iinc to the scattered Stokes vector Isca: Isca(θ)=Csca4πR2F(θ)Iinc, where the Stokes vectors have the shape I=IQUV and R is the distance of the observer from the particle. The Stokes vectors I describe the polarization state of light, with the first element I describing its total intensity. Thus, F is relevant for the polarization of the scattered light, and its first element a1, which is known as the phase function, is important for the angular intensity distribution of the scattered light. The phase function is normalized such that ∫0∘180∘a1(θ)⋅sin⁡θ⋅dθ=2.

For many applications it is useful to expand the elements of the scattering matrix using generalized spherical functions . The scattering matrix elements at any scattering angle θ are then determined by a series of θ-independent expansion coefficients α1l, α2l, α3l, α4l, β1l, and β2l, with index l from 0 to lmax⁡, see Eqs. (11)–(16) in . lmax⁡ depends on the required numerical accuracy as well as on the scattering matrix itself. For example, in the case of strong forward scattering peaks (typically occurring at large x), lmax⁡ needs to be larger than in the case of more flat phase functions, to get the same accuracy.

The asymmetry parameter g is an integral property of the phase function: g=12∫0∘180∘cos⁡θ⋅a1(θ)⋅sin⁡θ⋅dθ. g is the average cosine of the scattering angle of the scattered light and is calculated from the expansion coefficients by g=α11/3.

Depending on the particle type, different approaches are available for calculating particle optical properties. For the creation of the MOPSMAP optical data set, we use the well-known Mie theory in the case of spherical particles, which is a numerically exact approach over a very broad range of sizes. For spheroids we use the T-matrix method (TMM), which is a numerically exact method but limited with respect to maximum particle size. For larger spheroids not covered by TMM, we apply the improved geometric optics method (IGOM). For irregularly shaped particles the discrete dipole approximation (DDA) is applied.

Using the codes with the settings described above, a data set of modeled optical properties of single particles in random orientation was created. For spheres, we stored averages over narrow size bins as described above instead of single particle properties. An overview over the wide range of sizes, shapes, and refractive indices of the particles in the data set is given in Tables  and . For each combination of refractive index and shape a separate netCDF file was created, e.g., “spheroid_0.500_1.5200_0.008600.nc” for spheroids with ϵm=0.5 (prolate with ϵ′=2.0) and m=1.52+0.0086i. Each file contains the optical properties on a grid of size parameters. The complete data set requires about 42 gigabytes of storage capacity.

For spheres and spheroids the minimum size parameter is set to 10-6, and the maximum size parameter is set to x≈1005 to cover, e.g., rc=80 µm at λ=500 nm. The size increment is 1 % (i.e., xi+1/xi=1.01) in the case of spheres, 5 % in the case of TMM spheroids, and 10 % for IGOM spheroids. In the case of spheroids having refractive indices most relevant for atmospheric studies, the TMM is applied up to (or close to) the largest possible size parameter with the approach described in Sect. . The maximum size parameter of the TMM calculations is reduced for less relevant refractive indices. An overview is given in Sect. S2 of the Supplement and a detailed list of the maximum size parameters for all m and ϵm combinations can be downloaded from . The maximum size parameter for TMM is in the range 5<x<125, strongly depending on m and particle shape, and determines the lowest size parameter at which IGOM may be applied. The first IGOM size parameter is between 0 and 10 % larger than the maximum TMM size parameter. The TMM and IGOM results for spheroids are merged into a single netCDF file covering the complete size range from x=10-6 to x≈1005, which is sufficient for most applications. For example, for prolate spheroids with ϵ′=1.8 and m=1.56+0i, the size range from x=10-6 to x=88.22 is covered by TMM; IGOM starts at x=89.54. The transition from TMM to IGOM for several scattering angles is demonstrated in Sect. S3 of the Supplement. Since IGOM is an approximation, unrealistic jumps of optical properties may occur at the transition. For typical mineral dust ensembles in the visible spectrum, particles in the IGOM range contribute less than 10 % to the total extinction. IGOM was not applied to mr<1.04; thus, the size parameter range is limited to the TMM range for these refractive indices. A step of 0.04 was selected for the mr grid in the most relevant range (from 1.00 to 1.68) and a wider mr step elsewhere. The development of the data set started with mi=0.0043, and beginning from this value, mi was increased and decreased in steps of a factor 2. Below mi=0.001 and above mi=0.1, the step width is a factor of 2.

The optical data for the irregularly shaped particles (Table ) are limited to xv≤30.2 because of the huge computation requirements for optical modeling of large particles. Nonetheless, the most important range for many applications is covered; e.g., at λ=1064 nm particles up to rv=5.1 µm can be modeled. The m grid for the irregularly shaped particles is limited to the most relevant range for desert dust in the visible spectrum, and the mi step is set to a factor of 2. The quantification of the conversion factor ξvc of the six irregular shapes requires the determination of their orientation-averaged geometric cross sections, which is done numerically.

The optical properties stored for each particle are the extinction efficiency qext, the scattering efficiency qsca, and the expansion coefficients α1l, α2l, α3l, α4l, β1l, and β2l of the scattering matrix. The ADDA and the IGOM code provide the angular-resolved scattering matrix elements, which we converted to the expansion coefficients stored in the data set following the method described by and . We optimized the expansion coefficients for accurate scattering matrices at 180∘, which is probably the most error sensitive angle. As a by-product, lidar applications will certainly benefit from this optimization.

In the case of asymmetric shapes in random orientation, the scattering matrix has 10 independent elements as discussed by . By using only six elements of F (Eq. ) in our data set, we implicitly assume that each irregular model particle (shapes A–F) occurs as often as its mirror particle, which is formed by mirroring at a plane .

Figure  shows an example from the MOPSMAP optical data set. The refractive index is set to m=1.56+0.00215i, which is representative of desert dust particles at visible wavelengths. The properties of spherical particles are shown in blue, whereas the properties of prolate spheroids with ϵ′=1.4 and 3.0 are shown in orange and green, respectively. Red and violet lines denote irregularly shaped particles D and F, respectively. Figure a shows the extinction efficiency qext as a function of cross-section-equivalent size parameter xc. The general shape of the qext(xc) curve is similar for the different shapes; nonetheless, with increasing deviation from a spherical shape, the amplitudes of the oscillations of qext(xc) become smaller and a shift in the maximum qext towards larger xc is found. Figure b shows the single-scattering albedo ω0 for the same particles as Fig. a. For particle sizes comparable to the wavelength, ω0 reaches maxima with values of about 0.991, almost independent of particle shape. ω0 approaches a value of about 0.551 at xc≈1000 for spheres and spheroids. Fig. c shows the asymmetry parameter g. When the particle size becomes comparable to the wavelength, g increases and oscillates as a function of xc, with the strongest oscillations occurring in the case of spheres. There is some shape dependence of g for xc>5; in particular, the aggregate shape results in systematically smaller g than the other shapes for xc>10. The transition from the numerically exact TMM to the IGOM approximation occurs at xc≈125 for ϵ′=1.4 (orange line) and at xc≈27 for ϵ′=3.0 (green line) and is quite smooth.

Usually aerosol particles occur as ensembles of particles of different size, refractive index, and/or shape. The different particles contribute to the optical properties of the ensemble. Assuming that the distance between the particles is large enough for interaction of light with each particle to occur without influence from any other particle “independent scattering”;, the contribution of each particle can be added as described below.

In MOPSMAP particle ensembles are composed of one or more independent modes (the terms “mode” and “component” are often used synonymously in the literature). Each mode in MOPSMAP is characterized by particle size, shape, and refractive index, whereby each property can be described as a fixed value or as a distribution (see below). As these parameters do not necessarily correspond to the grid points of the MOPSMAP data set, for each mode (and each wavelength), decomposition into contributions from the different available m and shapes of the data set needs to be performed.

For a mode containing spheroids, in the most simple but probably most frequently used case of fixed values of mr, mi, and ϵm, linear interpolation in the three-dimensional (mr, mi, ϵm) space of the MOPSMAP data set is performed; i.e., eight grid points contribute to the result, with each grid point weighted according to the normalized distance from the parameters of the mode. For each dimension, the contributing grid points are the nearest grid point smaller or larger than the value of the mode; e.g., for the real part of the refractive index mr mr,i≤mr<mr,i+1. The weight of the grid points mr,i and mr,i+1 is wmr,i=mr,i+1-mrmr,i+1-mr,i,wmr,i+1=mr-mr,imr,i+1-mr,i. Finally the weights for each of the eight contributions are calculated as the products of the weights determined for each dimension. An example is shown in Sect. S4 of the Supplement. The error in the interpolation of the user-specified values between the grid points of the data set is discussed in Sect. 

Under other conditions more or less than eight contributions have to be considered. In the case of spheres or a single irregular shape, an interpolation in the shape dimension is not necessary, so that four contributions are sufficient. In the case of a spheroid aspect ratio distribution, contributions from all required ϵm grid points are considered and weighted according to the given distribution. In the case of a mode containing a non-absorbing fraction (see below), an additional mi grid point, mi=0, may be required. Furthermore, because of the limited size range of irregularly shaped particles in the data set, a special treatment can be applied: a MOPSMAP option is available which substitutes irregularly shaped particles above a selected size parameter with other particle shapes, spherical or nonspherical, as selected by the user. As a consequence, the particle shape of that mode becomes size- and wavelength-dependent and the number of different contributions increases. The total number of contributions for an ensemble, denoted as J in the following, varies because the number of modes is not fixed and, as just discussed, the number of contributions from each mode depends on the characteristics of each mode. This underlines the flexibility of MOPSMAP.

The optical properties of the particle ensemble are calculated for each wavelength by summation over extensive properties of all particles described by the J contributions. This approach corresponds to the so-called external mixing of particles. Each contribution has a size distribution nj(r), i.e., a particle number concentration per particle radius interval from r to r+dr, in the range from rmin⁡,j to rmax⁡,j, which is obtained by multiplying the user-defined size distribution of the mode with the weights obtained during the decomposition. The extinction coefficient αext and the scattering coefficient αsca are calculated by αext=∑j=1J∫rmin⁡,jrmax⁡,jCext,j(r)⋅nj(r)⋅dr, αsca=∑j=1J∫rmin⁡,jrmax⁡,jCsca,j(r)⋅nj(r)⋅dr. The expansion coefficients need to be weighted with Csca,j(r); for example, α1l of a particle ensemble is calculated by α1l=1αsca⋅∑j=1J∫rmin⁡,jrmax⁡,jα1,jl(r)⋅Csca,j(r)⋅nj(r)⋅dr. For the integration of extensive properties over the size distribution, we apply the trapezoidal rule, which assumes linearity between the r grid points.

The size distribution n(r)=dNdr for each mode can be specified in various ways. The MOPSMAP user can either specify a single size, apply size distribution tables in ASCII format, or apply a size distribution parameterization. The following parameterizations are available:

n(r)=12πN0ln⁡σ1rexp⁡-12ln⁡r-ln⁡rmodln⁡σ2   –   log-normal distribution;

The particle shape can be specified independently for each mode and is, within each mode, independent of size and refractive index. In the case of spheroids, either a fixed aspect ratio ϵ′ or an aspect ratio distribution is used. The latter can be given as a table in an ASCII file or it can be parameterized by a modified lognormal distribution n(ϵ′)=dNN0⋅dϵ′=12πσar(ϵ′-1)exp⁡-12ln⁡ϵ′-1-ln⁡ϵ0′-1σar2 with parameters ϵ0′ for the location of the maximum of n(ϵ′) and σar for the width of the distribution.

The refractive index of each mode can either be wavelength-independent or specified as a function of wavelength in an ASCII file. In addition, it is possible to specify for each mode a non-absorbing fraction X. If X>0, the mode is divided, for all sizes and shapes, into a non-absorbing (mi,1=0, relative abundance X) and an absorbing fraction (mi,2=mi/(1-X), relative abundance 1-X). As a consequence, the average mi over all particles of the mode remains equal to the mi as specified by the user. This non-absorbing fraction approach can be used as a parameterization of the refractive index variability within desert dust ensembles as described by and below in Sect. .

For the hygroscopic particle growth the following parameterization rwet(RH)rdry=1+κ⋅RH1-RH13 is implemented in MOPSMAP, where RH is the relative humidity and κ the hygroscopic growth parameter of the particles of each mode. This equation describes the ratio between the size of the particle at a given RH and the size of the particle in a dry environment (RH=0 %). The parameterization implies that this ratio is independent of size; thus, for example in the case of a lognormal size distribution, rmin⁡, rmax⁡, and rmod are multiplied with this ratio, whereas the relative width σ of the distribution is not modified. This is the usual approach though modal representations of aerosol size distributions may also predict higher moments , and thus σ can be a prognostic variable as well. The refractive index is modified by the water taken up following the volume weighting rule. Both RH and κ can be chosen by the user. This parameterization is valid for particles with r>40 nm, where the Kelvin effect can be neglected . It is worth noting that this parameterization differs from the relative humidity dependence implemented in OPAC, which was adapted from .

As output of MOPSMAP the following properties of aerosol ensemble are available. Redundant properties, such as lidar-related properties, are available to facilitate the use of the results:

extinction coefficient αext (m-1)

Scattering matrix elements and the quantities derived from them are calculated from the expansion coefficients. Wavelength-independent properties reff, N, a, v, and M are calculated for each wavelength to demonstrate the numerical accuracy of the integration.

The results are available in ASCII and in netCDF format. The format of the program output is described in the user manual. The netCDF output files can be read by the radiative transfer model uvspec, which is included in libRadtran .

Due to the limited size resolution in the data set and required interpolations between refractive index and aspect ratio grid points, deviations from exact model calculations for specific microphysical properties occur. As examples, Fig.  illustrates deviations introduced for single particle properties, whereas Table  shows deviations for particle ensembles.

In Fig. a and c effects of the limited size resolution on the extinction efficiency qext and the asymmetry parameter g are shown for non-absorbing spheres and spheroids with m=1.52+0i. In particular for spheres with x>10, deviations for single particles can be considerable because of small-scale features that are not resolved in the data set. In the case of spheres these features are implicitly considered in the data set by storing the average over 1000 sizes within each size bin as described above. In the case of spheroids, the data set contains properties calculated for single sizes which may not be fully representative of close-by sizes. However, since the small-scale features are much weaker for spheroids than for spheres, the average deviation for spheroids is much smaller than for spheres.

In Fig. b and d effects due to the required interpolation between the refractive index grid points are illustrated for spheres with m=1.54+0.005i. While the red lines show the properties calculated from the data set, the black lines show Mie calculations done explicitly for m=1.54+0.005i with the same size grid as used in the data set. The comparison illustrates that MOPSMAP calculates optical properties on average correctly, but some smaller-scale features are lost: for example, the extinction efficiency qext(x) in the size parameter range from 20 to 40 is dampened compared to the Mie calculation for m=1.54+0.005i because of the interference of the qext(x) curves for mr=1.52 and mr=1.56 (see gray lines in Fig. b; note that curves for different mi lie almost on top of each other).

For other size ranges, refractive indices, and optical quantities, the effects on the single particle properties are in principle similar but they may vary in magnitude.

Table  investigates the sampling and interpolation errors for a mono-modal lognormal size distribution with a typical width of σ=2.0. The effective radius is reff=1.44 µm, which is a typical value for transported desert aerosol. Sizes up to rmax⁡=4 µm, which corresponds to size parameter xc=40 at λ=628.32 nm, are considered. The left half of Table  compares optical properties calculated from the MOPSMAP data set (columns “data set”) with properties calculated using a high size resolution (columns “highres”), the same resolutions as displayed in Fig. a. For spheres, the results are equal up to at least the fourth digit. In the case of prolate spheroids with ϵ′=2.0, deviations are found for the fourth digit of αext and g. For the lidar-related quantities S and δl, the differences are larger with the relative deviation of δl being 2.6 %. These differences are caused by the high sensitivity of lidar-related quantities, and it is expected that deviations become smaller when shape distributions or wider size distributions are applied.

The right half of Table  demonstrates the effect of the m interpolation for an exemplary m=1.54+0.005i. MOPSMAP calculations (columns “data set”) are compared to results obtained using explicitly this refractive index in the Mie and TMM calculations. While the effect of the m interpolation is very small for αext, g, and δl, it is slightly larger for ω0 and S. The maximum relative effect is found for the lidar ratio S of spheres with a deviation of 1.7 %.

These comparisons demonstrate that deviations found for single particles are largely smoothed out in the case of particle ensembles due to the averaging over a large number of different particles. Only for a few special atmospheric applications, for example, the modeling of a rainbow, the limited resolution of the data set may still lead to a considerable error.

The first example of applications deals with hygroscopic growth. If aerosol particles are hygroscopic, their microphysical and optical properties change with RH. Fig.  shows how optical properties of the 10 OPAC aerosol types , which contain up to four components, some of which are hygroscopic, change with RH. These calculations were performed using the MOPSMAP web interface, where the OPAC aerosol types are available as predefined ensembles and the relative humidity can be chosen by the user. MOPSMAP considers the hygroscopic effect by application of the κ parameterization (Eq. ), which differs from the RH dependency implemented in OPAC.

The upper row of Fig.  shows the normalized extinction coefficient of the different types (indicated by color) at three wavelengths λ (each in a subplot) calculated for RH values of 0, 50, 70, 80, and 90 %. The extinction at all λ is normalized to the extinction at RH =0 % and λ=532 nm. As a consequence, the differences between the columns illustrate the wavelength dependency of the extinction, whereas changes with RH illustrate the hygroscopic effects. For example, for the desert aerosol type (orange color), the wavelength dependency is low, which is related to the large size of the dominant mineral particles, and the hygroscopic effect is relatively weak because mineral particles are hygrophobic. By contrast, for maritime (bluish colors) and antarctic types (purple color), the wavelength dependence is stronger and the hygroscopic effect is strong because of the domination by highly hygroscopic sulfate and sea salt particles. For the continental as well as the urban and arctic types, the wavelength dependence is even stronger and the hygroscopic effect weaker, which may be explained by strong contributions from the soot and water-soluble components which contain quite small particles with κ values significantly smaller than the κ values of sea salt particles e.g.,.

The single-scattering albedo ω0 is shown in the second row of Fig. . ω0 varies strongly with aerosol type, with the highest values of almost 1.0 for the antarctic, maritime clean, and maritime tropical aerosol types. Since water is almost non-absorbing at the considered wavelengths, the water uptake hardly changes ω0 if ω0 is already close to 1.0. The single-scattering albedo of the desert type is much lower, but it is also virtually independent on the RH as this aerosol type does not take up much water. For the other types, an increase in RH results in an increase in ω0.

The extinction-to-mass conversion factor η, which is plotted in the third row of Fig. , is necessary to calculate mass concentrations from extinction coefficient measurements or mass loadings from AOD measurements. An important parameter for η is the particle size e.g., with the consequence that the desert aerosol type, which contains the highest fraction of coarse particles of the considered types, shows the highest η values. Again, the wavelength dependency is significant for the other aerosol types so that the η values at λ=1064 nm (right column) are significantly larger than at λ=532 nm (middle column). The dependence of η on RH is significantly weaker than the dependence of the extinction on RH (upper row), which may be explained by the increase in mass with increasing RH compensating for the increase in extinction.

The bottom row of Fig.  illustrates the mass-to-backscatter conversion factor Z as a function of RH. Z is useful, for example, for comparisons of vertical profiles simulated with aerosol transport models to profiles measured with lidar or ceilometer. The multiplication of simulated aerosol mass concentration M with Z provides simulated β profiles which can be compared with the measurements. The figure shows that there is considerable spread between the different aerosol types, in particular at short wavelengths. RH only has strong effects on the maritime and arctic aerosol types.

Currently the hygroscopic growth of different aerosol components is not ultimately understood, and different κ-values are discussed. With MOPSMAP their influence on the optical properties can easily be determined and used in validation studies.

Aerosol transport models in combination with the optical properties of the aerosol allow one to model the radiative effect of the aerosol. The aerosol is typically modeled in terms of mass concentrations for a limited number of aerosol types divided over a few size bins (sectional aerosol model) or a few modes (modal aerosol models). Thus, realistic optical properties for each size bin of each aerosol type are required for modeling the radiative effects e.g.,.

In this example, we calculated the optical properties of dust at λ=500 nm for the five size bins of the COSMO-MUSCAT model . The size bins are determined by the radius limits 0.1, 0.3, 0.9, 2.6, 8, and 24 µm. We assumed constant dv/dlnr within each bin. Each bin was modeled through the expert mode of the MOPSMAP web interface. The refractive index is m=1.53+0.0078i, which is equal to the value given for the mineral components in OPAC. We considered two cases for the particle shape: on the one hand, spherical particles and, on the other hand, prolate spheroids with the aspect ratio distribution given by . For the latter case we assumed volume-equivalent sizes to keep the particle mass constant.

The calculated phase functions are presented in Fig. , where each size bin is represented by an individual color. The difference between both lines of the same color represents the shape effect. For size bin 1 (0.1 µm <r<0.3 µm, black lines), the difference is small, whereas for all other bins the shape effect is larger. The strongest effects are found for θ>100∘ with differences of up to a factor of 4 between the particle shapes. These angular ranges can be important, for example, for the backscattering of sunlight into space and thus for the aerosol radiative effect. The very strong effect at θ=180∘ is relevant for any lidar application, e.g, the intercomparison of modeled and measured attenuated backscatter profiles .

Calculated parameters relevant for radiative transfer and remote sensing are given in Table . The shape effect on the single-scattering albedo ω0 and the asymmetry parameter g is small except for size bin 2 where g is significantly larger for the spheroids than for the spheres. The extinction-to-mass conversion factor η is systematically smaller for spheroids than for spheres in bins 2–5 because the geometric cross section of the spheroids is ≈5.5% larger than the cross section of the volume-equivalent spheres. The mass-to-backscatter conversion factor Z of the spheroids is lower than the Z of spheres for most size bins, with maximum differences being larger than a factor of 2.

Many in situ measurement setups are limited with respect to the maximum particle size they are able to sample, e.g., because of losses at the inlet or the tubing. In this example, we illustrate the effect of the cutoff for the desert aerosol type from OPAC at RH =0 % .

Figure  illustrates various aerosol properties as a function of the cutoff radius rmax⁡. Fig. a shows properties that are normalized by the values found at rmax⁡=60 µm (where 99.988 % of the total particle cross section is covered, referring to rmax⁡=∞). The PM10 mass, i.e., the mass in the particles with diameter smaller than 10 µm (rmax⁡=5 µm), and the PM2.5 mass (rmax⁡=1.25 µm) are standard parameters to quantify pollution e.g.,. In our example, PM10 and PM2.5 contain only 59.5 and 21.6 % of the total particle mass, respectively. However, PM10 and PM2.5 measurement setups cover 94.4 and 69.0 % of the total geometric cross section, respectively. The single-scattering albedo in the case of PM2.5 is about 0.035–0.071 higher than for the total aerosol, whereas the asymmetry parameter is reduced by about 0.02–0.04. As a further example, if the cutoff is rmax⁡=10 µm, 97.8 % of the total cross section and 75.6 % of the mass are covered; the single-scattering albedo and the asymmetry parameter deviate from the total aerosol by less than 0.008.

This example shows that consideration of maximum size is essential when derived optical properties or mass concentrations are interpreted, and results can be severely misleading if the cutoff radius is not considered. These effects can be easily quantified with MOPSMAP and its web interface.

This example demonstrates how the selection of the size equivalence in the case of nonspherical particles affects various ensemble properties. In MOPSMAP the size-related parameters are either interpreted as rc (default) or as rv or rvcr (see Sect. ) according to the choice of the user. Each size equivalence can be transformed into another by Eqs. () and (). For example, if “volume cross section ratio equivalent” has been chosen in the web interface, and “0.5” for rmod, this would be equivalent to setting 0.5⋅ξvc-3 for rmod when the default “cross section equivalent” is kept (ξvc depending on shape).

To further elucidate the role of the different representations of radii, the same parameters of a lognormal size distribution are applied to the different size interpretations. For this purpose, the parameters are set to rmod=0.5 µm and σ=2 with rmin⁡=0.001 µm, rmax⁡=1.75 µm (reff=0.98 µm), and N0=103.66 cm-3, which results in a concentration of N=100 cm-3 in the range from rmin⁡ to rmax⁡. The effect of the three alternative interpretations on particle size is demonstrated in Fig.  for irregular shape D having ξvc=0.8708. All three size distributions (curves of different color) are plotted in terms of dN/drc(rc) (black axes). For comparison, axes for dN/drv(rv) (red axes) and dN/drvcr(rvcr) (green axes) are also shown. Using these axes, the size distribution curves can be interpreted in terms of the various size equivalences. The comparison between the size distributions clearly shows a shift towards larger sizes when rvcr or rv instead of rc is assumed. For example, assuming rvcr for the lognormal size distribution (green curve) describes the same ensemble as using rmod=ξvc-3⋅0.5 µm =0.8708-3⋅0.5 µm =0.757 µm (see Eq. ) and rmax⁡=0.8708-3⋅1.75 µm =2.65 µm when assuming rc as particle size.

Since the size distributions depend on the selected size equivalence various (optical) properties of the ensemble are also different; a quantification has been provided by MOPSMAP (Table ). The particle mass density is set to 2600 kgm-3, the refractive index is m=1.54+0.005i and the wavelength is λ=532 nm. The first column of Table  shows the optical properties of spherical particles. In the subsequent columns, all particles are assumed to be aggregate particles (shape D) with the same rc (second column, corresponding to the black curve in Fig. ), the same rv (third column, red curve), and the same rvcr (last column, green curve) as the spheres in the first column.

The results are consistent with the increase in particle size from assuming rc over rv to rvcr (see cross section density a, mass concentration M, and also Fig. ). The extinction coefficient αext and the forward volume scattering ã1(0∘) of the nonspherical particles best agree with the spherical counterparts if cross section equivalence is assumed. These properties are known to be sensitive to the particle cross section for particles larger than the wavelength. The absorption is in first approximation proportional to the particle volume if absorption is weak. As a consequence, for the single-scattering albedo ω0, both cross section and volume are relevant and dependencies are more complicated than for αext. The single-scattering albedo ω0 of shape D decreases in Table  from left to right due to the strong increase in particle volume. The selection of the size equivalence has a small effect on the asymmetry parameter g, the backward phase function a1(180∘), the lidar ratio S, and the linear depolarization ratio δl.

These results highlight the importance of a thoughtful selection of the size equivalence. The most appropriate size equivalence certainly depends on the concept of how the size distribution is measured. For example, if scattering by coarse dust particles is measured and the size is inverted assuming spherical particles, assuming cross-section equivalence in subsequent applications with nonspherical particles seems natural as scattering mainly depends on the particle cross section. MOPSMAP and its web interface provides the flexibility to investigate this topic theoretically.

In general, the knowledge on microphysical properties is limited; thus, they are subject to uncertainties. If these uncertainties can be quantified, it is consistent to also quantify the corresponding uncertainties of the optical properties.

In this regard, the sensitivity of a calculated optical property ζ to changes in a microphysical property ψ is an important aspect that can be expressed by the first partial derivative ∂ζ/∂ψ. The Jacobian matrix J is the M×N matrix containing all first partial derivatives for M optical properties and N microphysical properties. The elements of J of an aerosol ensemble can be numerically calculated by perturbing the microphysical properties of the ensemble. For demonstration in the following example we perturb ψ with a factor of 0.99 and 1.01 to numerically calculate the first partial derivatives. A sample script for the calculation of J is provided together with MOPSMAP.

Table  shows an example of J for the optical properties ζ∈{ω0,g,S} and the microphysical properties ψ∈{mr,mi,ϵ′}. J was calculated for a simplified dust ensemble described by one lognormal size mode with rmod=0.1 µm, σ=2.6, rmin⁡=0.001 µm, rmax⁡=20 µm, a refractive index m=1.53+0.0063i, and prolate spheroids with ϵ′=2.0. The wavelength is set to λ=532 nm. This results in ω0=0.9020, g=0.7319, and S=69.95 sr. These properties are most sensitive to mi, which can be clearly seen from Table . For example, a change in mi by 0.001 would result in a change in ω0 of 0.011. An increase in ϵ′ or mi increases g and S, whereas an increase in mr reduces their values. The sensitivity to perturbations of the microphysical properties is particularly strong for the lidar ratio S, which can be seen by comparing S=69.95 sr of the ensemble with the partial derivatives. We emphasize that the accuracy of J is limited by the sampling in the MOPSMAP data set (see also Sect. ); for example, partial derivatives ∂ζ/∂mr are constant between the mr grid points of the data set.

The Jacobian matrix J is valid for a certain set of microphysical properties values and, as mentioned, J can be used to quantify the uncertainty of the calculated properties for a given microphysical uncertainty. However, when uncertainties in the microphysical properties become larger, J may change significantly within the uncertainty range of ψ and other approaches may be required to estimate the uncertainty in the calculated optical properties. A simple approach applicable to this problem is the Monte Carlo method e.g.,. Repeated calculations with microphysical properties randomly chosen within the uncertainty range are performed. The uncertainty of the calculated quantities is determined by the statistics over the different sampled ensembles. In general, the computation time is longer than using J and is proportional to the number of calculated ensembles. Due to the statistical nature of the Monte Carlo method, the final results get more precise with increasing number of sampled ensembles. A script for the Monte Carlo uncertainty propagation is provided together with MOPSMAP. For example, based on the ensemble described above, sampling within the uncertainty ranges rmod=0.1±0.01 µm, σ=2.6±0.1, mr=1.53±0.03, mr=0.0063±0.002, and ϵ′=2.0±0.5 results in the ranges 0.85<ω0<0.94, 0.68<g<0.78, and 29 sr <S< 103 sr.

Mineral dust aerosols are ensembles of different minerals with different refractive indices. Usually the variability in the refractive index of the particles within a dust aerosol ensemble is neglected when modeling its optical properties. In this example, we compare optical properties calculated using the full measured variability in the imaginary part of the refractive index mi to properties calculated with the common assumption of all particles in an ensemble having an average mi. Furthermore, a parameterization of the variability is considered.

We use the desert aerosol type of OPAC . Prolate spheroids with the aspect ratio distribution of are assumed for the mineral components and spherical particles for the WASO component (RH =0 %). The real part of the refractive index is mr=1.53 for all particles. The wavelength in this example is set to λ=355 nm, which is a wavelength where absorption by iron oxide is strong. Because of the variable iron oxide content of individual particles, the variability in mi is large at this wavelength. Consequently, a significant influence on optical properties can be expected. In this example we consider three cases of imaginary part variability: first, we apply the size-resolved distribution of the imaginary part of the refractive index for Saharan dust as derived from mineralogical analysis . Second, we assume the average imaginary part for all particles (it is 0.0175, which is close to 0.0166 given for the mineral components in OPAC at λ=355 nm). Finally, we parameterize the mi distribution with the non-absorbing fraction approach as introduced in Sect. . In this case, we set X=0.5, resulting in 50 % of the mineral particles having mi=0, whereas the other 50 % of the particles have mi=0.0349.

Figure  shows the volume scattering function for the three cases. This figure shows that the sensitivity of the forward scattering to the mi distribution is negligible whereas the sensitivity increases with increasing scattering angle θ. For backward scattering, the difference between the measured mi distribution (red line) and using the average mi (black line) is more than a factor of 2. The parameterization assuming X=0.5 (thick blue line) is in much better agreement with the measured case. The root-mean-square relative deviation between the volume scattering function for the measured distribution and for the average mi is 30 %, whereas it is only 4 % for the parameterization. For comparison two additional X values, i.e., X=0.25 (thin dashed blue line) as well as X=0.75 (thin solid blue line), are also shown in Fig. , but their deviation is larger than for the parameterization with X=0.5. The extinction coefficient αext only changes by less than 0.03 % between the three representations of mi. For ω0 we obtain 0.852 using the measured mi distribution, whereas ω0=0.741 when using the average mi and ω0=0.834 using the parameterization with X=0.5. For the asymmetry parameter g, we obtain 0.744, 0.789, and 0.749 for the measured, averaged, and parameterized cases, respectively. For the lidar ratio S, values of 41, 78, and 42 sr are calculated for the three cases, whereas for the linear depolarization ratio δl values of 0.241, 0.212, and 0.220 are obtained.

These results emphasize that it is important to consider the nonuniform distribution of the absorptive components in the desert dust ensembles for optical modeling of such aerosols at short wavelengths. We have shown in this example that optical properties of Saharan dust can be well simulated with X=0.5. Whether this conclusion holds for other cases of desert dust can easily be investigated by means of MOPSMAP when measurements of mi distributions of further dust types are available.

Integrating nephelometers aim to measure in situ the total scattering coefficient αscatrue of aerosol particles by detecting all scattered light. The angular sensitivities of real nephelometers, however, deviate from the ideal sensitivity, which is the sine of scattering angle θ. For example, nearly forward or nearly backward scattered light does not reach the detectors because of the instrument geometry . This has to be considered during the evaluation of measurements and can be done by applying a truncation correction factor Cts=αscatrue/αscameas to the measured scattering coefficients αscameas. Cts can be calculated theoretically using optical modeling if aerosol microphysical properties and the angular sensitivity of the instrument are known. Some nephelometers not only measure the total scattering coefficient but also the hemispheric backscattering coefficient, which is the scattering integrated from θ=90 to 180∘. For the hemispheric backscattering coefficient, a correction factor also needs to be applied to correct the measured hemispheric backscattering coefficient affected by the nonideal instrument sensitivity. This correction factor Cbs is defined analogously to Cts as the ratio between the true coefficient and the measured one. Note that this hemispheric backscattering coefficient is defined differently from β, which is measured by lidars and used elsewhere in this paper.

Figure  shows modeled correction factors for the total (Fig. a) and the backscatter (Fig. b) channel of an Aurora 3000 nephelometer. The angular sensitivity of the instrument is taken from . For the following sensitivity study the mineral dust refractive index from OPAC , the parameterized mi distribution with X=0.5 (as shown in Sect. ), a lognormal size mode with σ=1.6 and a maximum radius of rmax⁡=5 µm (corresponding to a PM10 inlet) is assumed. The mode radius rmod is varied from 0.01 to 1 µm (horizontal axis) and two cases for the particle shape, i.e., spherical particles (solid lines) and cross-section-equivalent prolate spheroids with the ϵ′ distribution from (dashed lines), are considered. The colors denote the three operating wavelengths of the instrument (450, 525, and 635 nm). The figure shows that the total scattering correction factor Cts mainly depends on particle size. In the case of large particles (rmod=1 µm), the nephelometer underestimates total scattering by a factor of ≈2 if the truncation error is not corrected. Shape only has a small effect on forward scattering; thus, its influence on the correction of the truncation error is less than 3 % (compare dashed and solid lines of the same color). The maximum shape effect on Cbs is 7 %, i.e., indicating that assuming spherical particles for the truncation correction may result in an overestimation of the hemispheric backscattering coefficient.

The correction factors might be recalculated for example when new data on the refractive index or particle shape become available. This example highlights the potential of MOPSMAP as a useful tool for the characterization of optical in situ instruments. In addition, it could be used for the interpretation of angular measurements, for example, as performed with a polar photometer by .

present a data set comprising shape–size distributions of ashes from nine different volcanoes as well as wavelength-dependent refractive indices for five different ash types. The particles were collected between 5 and 265 km from the volcanoes. While refractive indices can also be expected to be valid at larger distances from the volcanoes, the effective radii in the range from 9.5 to 21 µm are probably not realistic for long-range-transported ash. Based on this data set, which is available in the supporting information of , we calculate optical properties of these volcanic ashes with MOPSMAP. Each single particle is modeled as a prolate spheroid with the given size and aspect ratio, as well as with the refractive index given for the type of ash the volcano emits. In addition, we assume a non-absorbing fraction of X=0.5 (as used in Sect. ). The application of this non-absorbing fraction approach seems reasonable when taking into account the variability in the transparency of the particles shown in Fig. 5 of . Due to the data set limits of MOPSMAP, particles with r>47.5 µm are modeled as r=47.5 µm and aspect ratios >5 are set to 5. For each volcano, less than 0.5 % of the particles was affected by these modifications.

Figure  shows the single-scattering albedo ω0 and the asymmetry parameter g for the nine ashes as a function of wavelength between 300 and 1500 nm. Differences of ω0 are up to about 0.12 with ash from Chaitén (Chile) and Mt. Kelud (Indonesia) being the least and most absorbing species, respectively. ω0 is correlated with the ash type, which is mainly a result of the significant variability in mi (see Fig. 16b of ). For all ashes, ω0 increases slightly with wavelength, typically by about 0.05 over the wavelength range shown. The variability in g is less than 0.05, and for all ashes the changes with wavelength are weak with values of less than 0.02. The mass-to-backscatter conversion factor Z varies between 1.16 and 3.38×10-3 m2sr-1g-1 for the nine ashes. The extinction-to-mass conversion factor η at λ=550 nm ranges from 14.8 to 33.0 gm-2 which is considerably higher than known for typical aerosols (e.g., Fig. ) or volcanic ash transported over continental scales (e.g., η between 1.10 and 1.88 gm-2 found by ). In particular the different values of η clearly demonstrate that optical properties of volcanic ash layers drastically change with the distance from the eruption due to changing microphysics.

This example suggests that it is worthwhile considering the specific microphysical properties of each volcano. However, for realistic MOPSMAP calculations valid in the long-range regime, size distributions different from the ones used in this example must certainly be applied whereas the refractive indices are more likely representative.