Introduction
Aerosol particles in the Earth's atmosphere are important in various ways,
for example because of their interaction with electromagnetic radiation and
their effect on cloud properties. Consequently aerosol particles are relevant
for weather and climate. The temporal and spatial variability in their
abundance as well as the variability in their properties is significant which
poses huge challenges in quantifying their effects. This includes the need to
establish extended networks of observations using instruments such as
photometers , lidars , or ceilometers
and the development of models to predict the influence
of particles on the state of the atmosphere; see, e.g., .
Aerosol properties and distributions are often quantified by ground-based and
spaceborne optical remote sensing and by optical in situ measurements. These
measurements are indirect with respect to microphysical properties (e.g.,
particle size) because they measure optical quantities and require the
application of inversion techniques to retrieve microphysical properties.
Precise knowledge on the link between microphysical and optical properties is
needed for the inversion. This link is provided by optical modeling, i.e., the
optical properties of particles are calculated based on their microphysical
properties. Optical modeling is required also for other applications, e.g.,
for radiative transfer, numerical weather prediction, and climate modeling.
As optical modeling can be very time-consuming, it is often inevitable to
pre-calculate optical properties of particles and store them in a lookup
table, which is then accessed by the inversion procedures or subsequent
models.
In our contribution we describe the MOPSMAP (Modeled optical properties
of ensembles of aerosol particles) package, which consists of a data
set of pre-calculated optical properties of single aerosol particles, a
Fortran program which calculates the properties of user-defined aerosol
ensembles from this data set, and a user-friendly web interface for online
calculations. Figure illustrates the overall scheme of the
package, including the optical modeling codes (green box) needed once to
prepare the underlying data set. MOPSMAP is either provided via an
interactive web interface at https://mopsmap.net or via download as an
offline application. The former is possible as MOPSMAP is computational very
efficient. Compared to other data sets with predefined aerosol components,
such as OPAC , compared to existing online Mie tools such as
the one provided by , and compared to GUI tools such as
MiePlot , MOPSMAP is more flexible with respect to the
characteristics of the aerosol ensembles. Moreover, our data set considers
not only spherical particles but also spheroids and a small set of
irregularly shaped dust particles. The output includes
ASCII tables for further evaluation, netCDF files for direct application in the
radiative transfer model uvspec , and plots, e.g., for
educational purposes.
In Sect. 2, after defining aerosol properties, we describe how existing
optical modeling codes were applied (green box in Fig. ) to
create the optical data set of single particles (yellow box). Subsequently,
in Sect. 3, we describe the Fortran program (orange box) that uses this data set to calculate optical
properties of user-defined particle ensembles. The web interface for online
application of the MOPSMAP package is introduced in Sect. 4. To demonstrate
the potential of MOPSMAP, several applications are discussed in Sect. 5
before we sum up our paper and give an outlook.
Background and the MOPSMAP data set
The optical properties of a particle with known microphysical properties are
calculated by optical modeling. For the creation of the basic data set of
MOPSMAP, optical modeling of single particles has been performed. In this
section, we first define microphysical and optical properties of single
particles and then describe how we created the data set using existing
optical modeling codes.
We emphasize that the data set is, in principle, applicable to the complete
electromagnetic spectrum; however, we use, for simplicity, the term “light”
and consequently “optics” instead of more general terms.
Definition of particle properties
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)⋅sinθ⋅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∘cosθ⋅a1(θ)⋅sinθ⋅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.
Optical modeling of single particles
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.
Mie theory
We use the Mie code developed by for optical modeling of
spherical particles. In contrast to the nonspherical particle types
described below, we do not store the optical properties of single particles
(in a strict sense) because the properties of spheres can be strongly
size-dependent, which would require a very high size resolution of our data
set e.g.,. Instead, we store data averaged over very
narrow size bins, allowing us to use a lower size resolution resulting in a
smaller storage footprint of the data set. For each size parameter
grid point x, we actually consider a size parameter bin covering the range from
x/1.01 to x⋅1.01 and apply the Mie code for 1000
logarithmically equidistant sizes within that bin before these results are
averaged and stored.
T-matrix method (TMM)
We use the extended precision version of the code described by
for modeling optical properties of spheroids. To improve the
coverage of the particle spectrum (x, ϵm, and m), internal
parameter values of the TMM code, which primarily determine the limits of the
convergence procedures, were increased (NPN1 = 290; NPNG1 = 870; NPN4 =
260) as discussed by . Though, in general, the TMM provides
exact solutions for scattering problems, nonphysical results might be
obtained due to numerical problems. To reduce the probability of nonphysical
results and to increase the accuracy of the results, the parameter DDELT,
i.e., the absolute accuracy of computing the expansion coefficients, was set
to 10-6 (default 10-3). In non-converging cases, which occurred near the upper limit of the covered size range, the requirements were relaxed
to DDELT = 10-3. Cases that did not converge even with the relaxed
DDELT were not included in the data set. Nevertheless, some nonphysical
results were obtained by this approach, for example, ω0>1, or
outliers of otherwise smooth ω0(x) or g(x) curves. Thus, for
plausibility checks for each particle shape and refractive index, single-scattering albedos ω0 and asymmetry parameters g were plotted over
size parameter x and outliers were recalculated with slightly modified size
parameters. Recalculations with nonphysical results were not included in the
data set, which reduces the upper limit of the covered size range for that
particular particle shape and refractive index.
Improved geometric optics method (IGOM)
Optical properties of large spheroids were calculated with the improved
geometric optics method (IGOM) code provided by and . In
general, this approximation is most accurate if the particle and its
structures are large compared to the wavelength. In addition to reflection,
refraction, and diffraction by the particle, which are considered by
classical geometric optics codes, IGOM also considers the so-called edge
effect contribution to the extinction efficiency qext
. Classical geometric optics results in qext=2,
whereas qext is variable in the case of IGOM. The default settings of
the code were used. The minimum size parameter was selected depending on the
maximum size calculated with TMM.
Discrete dipole approximation code ADDA
Natural nonspherical aerosol particles, such as desert dust particles,
comprise practically an infinite number of particle shapes; thus, it is
impossible to cover the full range of shapes in aerosol models. Moreover, the
shape of each individual particle is never known under realistic atmospheric
conditions. Consequently, typical irregularities such as flat surfaces,
deformations or aggregation of particles can be considered only in an
approximating way. To enable the user of MOPSMAP to investigate the effects
of such irregularities the properties of six exemplary irregular particle
shapes, as introduced by , are provided. The geometric
shapes were constructed using the object modeling language Hyperfun
. The first three shapes are prolate spheroids with varying
aspect ratios (A: ϵ′=1.4; B: ϵ′=1.8; C: ϵ′=2.4)
and surface deformations according to . Shape D is an
aggregate composed of 10 overlapping oblate and prolate spheroids; surface
deformations were applied as for shapes A–C. Shape E and F are edged
particles with flat surfaces and a varying aspect ratio.
The optical properties were calculated with the discrete dipole approximation
code ADDA . A large number of particle orientations needs
to be considered for the determination of orientation-averaged properties.
ADDA provides an optional built-in orientation averaging scheme in which the
calculations for the required number of orientations is done within a single
run. An individual ADDA run using this scheme requires approximately the time
for one orientation multiplied with the number of orientations (typically a
few hundred), which can result in computation times of several weeks for
large x. Because of the long computation times we split them up and
performed independent ADDA runs for each orientation. The
orientation-averaged properties are calculated in a subsequent step using the
ADDA results for the individual orientations (see below).
Microphysics of spheres and spheroids considered in the MOPSMAP data set.
Method
Mie
TMM
IGOM
Particle shape
spheres
oblate and prolate spheroids
ϵ′=1.2, 1.4, ..., 3.0, 3.4, 3.8, ..., 5.0
Size parameter
10-6<xc<1005
10-6<xc<(5-125)
(5-125)<xc<1005
xi+1xi=1.01
xi+1xi=1.05
xi+1xi=1.10
size bins
single size
single size
mr
(0.1, 0.2, ..., 0.9, 1.0)*, 1.04, 1.08, ..., 1.68, 1.76, ..., 2.0, 2.2, ..., 3.0
mi
0, 0.0005375, 0.001075, 0.0015203, 0.00215,
0.0030406, 0.0043, 0.0060811, 0.0086, 0.0121622,
0.0172, 0.0243245, 0.0344, 0.0486490, 0.0688,
0.0972979, 0.1376, 0.2752, 0.5504, 1.1008, 2.2016
* IGOM was not applied to m≤1.0.
Microphysics of irregularly shaped particles considered in the MOPSMAP data set.
Particle shape
Shapes A–F, Fig. 1 of
Size parameter
10-3<xv<30.2; xi+1xi≈1.10; single size
mr
1.48, 1.52, 1.56, 1.60
mi
0, 0.00215, 0.0043, 0.0086, 0.0172, 0.0344, 0.0688
The computational demand of DDA calculations increases strongly with size
parameter x, typically with about x5 to x6. Thus, when aiming for
large x, which is required for mineral dust in the visible wavelength
range, it is necessary to find code parameters and an orientation averaging
approach that provide a compromise between computation speed and accuracy.
The ADDA code mainly allows the following code parameters to be optimized:
DDA formulation
stopping criterion of the iterative solver
number of dipoles per wavelength.
We estimate the accuracy of the ADDA results by comparing
orientation-averaged qext, qsca, a1(0∘),
a1(180∘), and a2(180∘)/a1(180∘) with results
obtained using more strict calculation parameters. Accuracy tests are
performed for shapes B and C, for size parameters xv=10.0, 12.0,
14.4, 17.3, 19.0, and 20.8, and for refractive index m=1.52+0.0043i; i.e., 12 single particle cases are considered in total. By comparing the different
DDA formulations available in ADDA, it was found that the filtered
coupled-dipole technique (ADDA command line parameter “-pol fcd -int fcd”),
as introduced by and applied by , offers
the best compromise between computation speed and accuracy of modeled optical
properties. Using a stopping criterion for the iterative solver of 10-4
instead of 10-3 only has negligible influence on optical properties
(<0.1 %) but requires approximately 30 % more computation time; thus,
we used 10-3 for the ADDA calculations to create our data set. The
extinction efficiency qext and the scattering efficiency
qsca change in all cases by less than 0.3 % if a grid density
of 16 dipoles per wavelength is used instead of 11. The maximum relative
changes due to the change in dipole density are 0.2 % for a1(0∘),
1.7 % for a1(180∘), and 1.9 % for
a2(180∘)/a1(180∘). Because of the large difference in
computation time, which is about a factor of 3–4, and the low loss in accuracy,
about 11 dipoles per wavelength were selected for the MOPSMAP data set. For
xv<10 we use the same dipole set as for xv=10 so
that the number of dipoles per wavelength increases with decreasing
xv, being about 110/xv.
The particle orientation is specified by three Euler angles
(αe, βe, γe) as described
by and basically a step size of 15∘ is applied for
βe and γe resulting in 206 independent ADDA
runs for each irregular particle. The orientation sampling and averaging is
described in detail in Sect. S1.1 of the Supplement.
To test the accuracy of the selected orientation averaging scheme,
orientation-averaged optical properties for shapes B, C, D, and F were
compared to results using a much smaller step of 5∘ for
βe and γe. These calculations consider about
12 times more orientations than the calculations used for MOPSMAP. Details
are presented in Sect. S1.2 of the Supplement. Maximum deviations of less
than 1 % are found for qext, qsca, and
a1(0∘). For backscatter properties, a1(180∘) and
a2(180∘)/a1(180∘), typical deviations are of the order of a
few percent (max. 14 %). Moreover, in Sect. S1.3 of the Supplement, the
selected orientation averaging scheme is applied to spheroids, and their
optical properties are compared to reference TMM results. These deviations
are comparable to those given in Sect. S1.2.
In summary, ADDA with the filtered coupled-dipole technique, at least 11
dipoles per wavelength and a stopping criterion for the iterative solver of
10-3 was used for optical modeling of the irregularly shaped particles
in our data set together with the orientation averaging scheme combining 206
ADDA runs. Tests demonstrate that the modeling accuracy is mainly determined
by the applied orientation averaging scheme.
Optical data set
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
.
Optical properties of single particles (or narrow size bins in the case
of spheres) with fixed refractive index m=1.56+0.00215i as a function of size
parameter. The different colors denote different particle shapes. Panel (a) shows the extinction efficiency qext,
panel (b) the single-scattering albedo ω0, and panel (c) the asymmetry parameter g.
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.
MOPSMAP Fortran program
In this section the basic characteristics of the MOPSMAP Fortran program to
calculate optical properties of particle ensembles are described. Besides a
modern Fortran compiler, e.g., gfortran 6 or above, the netCDF Fortran
development source code is required to build the executable. The computation
time and memory requirements depend on the ensemble complexity and the number
of wavelengths but in general are low for state-of-the-art personal
computers. The Fortran code and the data set are available for download from
, and a web interface (see Sect. 4) provides online
access to most of the functionality of the Fortran program without the
requirement of downloading the code and the data set.
Simplified flow chart of the MOPSMAP Fortran program.
Within each MOPSMAP run the optical properties of a specific user-defined
ensemble are calculated at a user-defined wavelength grid. The ensemble
microphysics and the wavelength grid are defined in an input file. The
details about the options available for the input file are described in a
user manual which is provided together with the code.
Figure shows a flow chart of the MOPSMAP Fortran
program. The program is initialized by reading the input file and a data set
index. The latter contains information on the refractive index and shape grid
and the size parameter ranges covered by the data set. Then, all information
required for the optical modeling is initialized, for example the set of
wavelengths, the refractive indices as a function of wavelength, shape
distributions, and the effect of the hygroscopic growth, before the optical
calculations are performed for each wavelength, as described in the
following.
Calculation of optical properties of particle ensembles
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)=12πN0lnσ1rexp-12lnr-lnrmodlnσ2 – log-normal
distribution;
n(r)=Arαexp-Brγ – modified gamma distribution,
;
n(r)=Aexp-Br – exponential distribution, α=0, γ=1;
n(r)=Arα – power law distribution, Junge distribution, B=0,
;
n(r)=Arαexp-Br – gamma distribution, γ=1,
.
rmod is the mode radius, σ a dimensionless parameter for
the relative width of the distribution, and N0 the total number density
(in the range from rmin=0 to rmax=∞) of the lognormal
distribution. For the subsequent size distributions, parameters A,
α, B, and γ are positive and A controls the scaling of
total number density whereas α, B, and γ are relevant for the
shape of the size distributions. The exponential distribution, power law
distribution, and the gamma distribution are a subset of the modified gamma
distribution with the specific parameter values as given above see
also.
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-12lnϵ′-1-lnϵ0′-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 .
Output of Fortran program
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)
single-scattering albedo ω0
asymmetry parameter g
effective radius reff=∫r3n(r)dr∫r2n(r)dr (µm) (referring to rc, rv, or rvcr as selected by the user)
number density N (m-3) (number of particles per atmospheric volume)
cross section density a (m-1) (particle cross section per atmospheric volume)
volume density v (particle volume per atmospheric volume)
mass concentration M (gm-3) (particle mass per atmospheric volume)
expansion coefficients (α1l to β2l) for elements of scattering matrix
scattering matrix elements (a1 to b2) at user defined angle grid
volume scattering function ã1=αext⋅ω04π⋅a1 (m-1sr-1) at user defined angle grid
backscatter coefficient β=αext⋅ω04π⋅a1(180∘) (m-1sr-1)
lidar ratio S=4πω0a1(180∘) (sr)
linear depolarization ratio δl=a1(180∘)-a2(180∘)a1(180∘)+a2(180∘)
Ångström exponents AEζ=-logζ(λ1)ζ(λ2)logλ1λ2 for ζ∈{αext,αsca,αabs,β}
extinction-to-mass conversion factor η=Mαext (gm-2)
mass-to-backscatter conversion factor Z=βM
(m2sr-1g-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 .
Interpolation and sampling error
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.
Examples illustrating the effect of the limited size resolution of
the MOPSMAP data set (a, c) and the effect of the interpolation
between the refractive index grid points of the data set (b, d);
extinction efficiencies qext (a, b) and asymmetry parameters
g (c, d) as functions of the size parameter from x=0 to x=40 are compared;
in (a) and (c) the high size-resolution calculations (black lines) were performed
with linear x steps of 0.002 in the case of spheres and 0.01 in the case of spheroids;
in (b) and (d) the red lines show properties calculated with MOPSMAP
for m=1.54+0.005i by interpolation between refractive indices included in the
data set (i.e., between m=1.52+0.0043i, m=1.52+0.0060811i,
m=1.56+0.0043i, and m=1.56+0.0060811i, for which the properties are shown
as thin gray lines), and for comparison, the black lines show the properties
calculated by Mie theory explicitly for m=1.54+0.005i using the same
x grid as used by the data set.
Optical properties calculated for a lognormal mode with rmod=0.5 µm,
σ=2.0, rmin=0.001 µm, and rmax=4 µm at λ=628.32 nm.
Two cases of particle shapes are considered: spheres and prolate spheroids with ϵ′=2.0.
The columns “data set” contain values calculated using MOPSMAP with the data set described in Sect. 2.3.
For comparison, the same properties are calculated in the columns “highres” using a high size resolution
and in the columns “explicit” using Mie theory or TMM explicitly at m=1.54+0.005i.
Size sampling example
m-interpolation example
for m=1.52+0i
for m=1.54+0.005i
Spheres
Spheroids
Spheres
Spheroids
Data set
Highres
Data set
Highres
Data set
Explicit
Data set
Explicit
αext (km-1)
4.808
4.808
4.863
4.861
4.793
4.793
4.844
4.846
ω0
1.0000
1.0000
1.0000
1.0000
0.8845
0.8840
0.8892
0.8886
g
0.7045
0.7045
0.7018
0.7021
0.7331
0.7332
0.7382
0.7380
S (sr)
10.52
10.52
42.75
42.30
13.13
13.36
58.25
58.78
δl
0.0000
0.0000
0.3063
0.2986
0.0000
0.0000
0.2502
0.2502
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.
MOPSMAP web interface
A web interface is provided as part of MOPSMAP at https://mopsmap.net.
It was designed to be intuitive for expert and nonexpert users, e.g., for the
demonstration of sensitivities of optical properties on microphysical
properties in the framework of lectures, but also for a lot of scientific
problems as outlined in the following section. The web interface is written
in PHP and uses the SQLite library. After the registration as a user, online
calculations of optical properties of a large range of particle ensembles can
be performed. Input and output can be defined by the user; for nonexpert
users, a lot of default ensembles representative of specific climatological
conditions are already available. The input parameters primarily include the
microphysical properties of the particles. The particles' microphysics are
described by up to four components (each described by an individual lognormal
size distribution), the wavelength dependence of the refractive index and the
shape. Any lognormal size distribution can be used; to facilitate the usage
(e.g., for nonexpert users), the aerosol components from the OPAC data set
, e.g., “mineral coarse mode”, “water-soluble”, or
“soot”, are already included. The same is true for the 10 “aerosol types”
defined in OPAC, e.g., “continental clean”, “urban” or “maritime
polluted”, consisting of a combination of components. Calculations can be
made for a single wavelength, for wavelength ranges or a prescribed
wavelength set (e.g., for a typical aerosol lidar or a AERONET sun
photometer). Moreover, users can define their own wavelength sets, e.g., for a specific radiometer. The
relative humidity is selected by the user and it is effective for all
hygroscopic components according to Eq. (). The hygroscopic
growth of the OPAC components in MOPSMAP differs from the original OPAC
version ; it follows the κ parameterization with the
values proposed by . In the “expert user mode” the
flexibility is further increased: the number of components can be larger than
four, and the size distribution can be given as discrete values on a
user-defined size grid.
Properties of OPAC aerosol types as a function of relative humidity RH
calculated with the κ parameterization implemented in MOPSMAP (Eq. ).
The different colors denote the 10 different OPAC aerosol types as indicated in the legends.
The columns denote different wavelengths λ as indicated above the upper row.
The upper row shows the extinction coefficient normalized to the extinction coefficient
of the same aerosol type at RH =0 % and λ=532 nm.
The single-scattering albedo ω0, the extinction-to-mass conversion
factor η, and the mass-to-backscatter conversion factor Z are
plotted in the subsequent rows.
The output comprises the complete set of optical properties as described in
Sect. . It can be downloaded for further applications and
includes ASCII tables as well as a netCDF file that can be used for radiative
transfer calculations with uvspec of the widely used libRadtran package
. To provide an immediate overview over the results, the most
important parameters, such as extinction coefficient (αext),
single-scattering albedo (ω0), asymmetry parameter (g),
Ångström exponent (AE), or lidar ratio (S), are displayed as tables
when the calculations have been completed. In addition plots of the results
as a function of wavelength and scattering angle are shown as selected by the
user.
All results are stored in the user's personal folder so that all calculations can be reproduced.
Furthermore, all calculations can also easily be rerun with a slightly modified input parameter set.
Applications
In this section a selection of examples is presented to demonstrate the wide
range of applications of MOPSMAP. Many of them can be performed by using the
web interface. Some examples need a local version of MOPSMAP alongside with
scripts that repeatedly call the Fortran program. These scripts are written
in Python and can be downloaded from as part of the
MOPSMAP package.
It is worth mentioning that numerous studies demonstrate the need for optical
modeling of aerosol ensembles, thus illustrating the range of possible
applications of MOPSMAP. Moreover, optical modeling is essential for many
different related modeling activities. It is required, for example, for
closure experiments consistency checks between different measurement
methods involving an aerosol model,
e.g.,,
radiative transfer studies e.g.,, the inversion of
remote-sensing measurements
e.g.,, the inversion of in situ
data e.g.,, aerosol
layer visibility simulations e.g.,, dynamic aerosol
transport models e.g.,, aerosol
characterization e.g.,, and
solar energy e.g.,.
Effect of hygroscopicity
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.
Optical properties for sectional aerosol models
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.,.
Optical properties at λ=500 nm of the five COSMO-MUSCAT
dust size bins. Two cases for the particle shape are considered:
spheres/prolate spheroids. For details, see text.
Bin 1
Bin 2
Bin 3
Bin 4
Bin 5
ω0
0.9632/0.9628
0.9216/0.9264
0.7903/0.7934
0.6450/0.6485
0.5561/0.5601
g
0.6567/0.6585
0.6866/0.7111
0.8088/0.8109
0.8998/0.9017
0.9442/0.9419
η (gm-2)
0.2905/0.3000
0.5594/0.5236
2.230/2.071
6.989/6.633
22.09/20.90
Z (m2sr-1g-1)
4.234×10-2/
1.185×10-1/
1.403×10-2/
1.204×10-3/
8.225×10-5/
3.981×10-2
5.421×10-2
8.901×10-3
7.457×10-4
8.651×10-5
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
.
Phase functions at λ=500 nm of the five COSMO-MUSCAT dust
size bins (different colors) assuming spherical particles (solid lines)
and prolate spheroids (dashed lines). For details, see text.
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.
Effect of cutoff at maximum size
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 % .
Properties of one-modal size distribution at λ=532 nm
consisting of spheres or aggregate particles (shape D,
ξvc=0.8708;
Fig. 1 of ) assuming different size equivalences. For details, see text.
Properties
Spheres
Aggregate particles
Using r
Using rc
Using rv
Using rvcr
αext (km-1)
0.350
0.347
0.449
0.750
ω0
0.897
0.922
0.910
0.883
g
0.722
0.679
0.680
0.689
a1(0∘)
100
97.4
128
222
ã1(0∘) (km-1sr-1)
2.51
2.48
4.17
11.7
a1(180∘)
1.21
0.405
0.420
0.432
S (sr)
11.6
33.6
32.8
33.0
δl
0.000
0.450
0.454
0.454
Cross section density a (km-1)
0.141
0.141
0.186
0.323
Mass concentration M (µgm-3)
482
318
481
1103
Optical and microphysical properties of the OPAC desert aerosol
type as a function of cutoff radius rmax. Panel (a) shows the
normalized extinction coefficient αext at three wavelengths,
the normalized cross section density a, and the normalized mass concentration M.
Normalization to values calculated for rmax=60 µm.
The single-scattering albedo ω0 at the same wavelengths is plotted
in (b), and the asymmetry parameter g in (c).
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.
Effect of the selection of size equivalence of nonspherical particles
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.
Lognormal size distributions (SD) with same rmod,
σ, N0, and rmax assuming different size equivalences for
aggregate particles (shape D, ξvc=0.8708) as applied in Table .
The size distributions are plotted in terms of cross-section-equivalent
sizes (i.e., dN/drc(rc) referring to black
axes and grid). For comparison axes valid for the other size interpretations
are also plotted in red and green, which allows each size distribution to be
interpreted in terms of each size equivalence.
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.
Uncertainty estimation of calculated optical properties
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.
Elements of the Jacobian matrix, i.e., first partial derivatives,
of a dust-like ensemble (see text for details).
∂ω0
∂g
∂S
∂mr
-0.037
-0.428
-360 sr
∂mi
-11.0
+3.69
+2839 sr
∂ϵ′
+0.010
+0.058
+48.3 sr
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.
Effect of refractive index variability
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.
Volume scattering function of dust at λ=355 nm
(arbitrary scale) using either the mi distribution (red) measured by
, the average mi of these measurements (black),
or applying the non-absorbing fraction parameterization with different
X (blue).
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.
Effect of particle shape on the nephelometer truncation error
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.
Modeled correction factors Cts for total scattering
(a) and Cbs for hemispheric backscattering
(b) of an Aurora 3000 nephelometer as a function of particle size. For details, see text.
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
.
Optical properties of ash from different volcanoes close to the source
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.
Modeled wavelength-dependent optical properties for
ashes from different volcanoes. More details on the ash samples are given in Table 1 of . The colors indicate ash type:
basalt is dark blue, basaltic andesite is light blue, andesite is green,
dacite is orange, and rhyolite is red (see Fig. 7 of for reference).
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.
Conclusions
Radiative properties of atmospheric aerosols are relevant for a wide range of
meteorological applications, in particular for radiative transfer
calculations and remote-sensing and in situ techniques. Optical properties
strongly depend on the microphysical properties of the particles – size,
refractive index and shape – properties that are highly variable under ambient
conditions. As a consequence, the application of mean properties could be
questionable. However, the determination of optical properties of specific
aerosol ensembles can be quite time-consuming, in particular when
nonspherical particles shall be considered.
For this purpose we have developed the MOPSMAP package that provides the full
set of optical properties of arbitrary, randomly oriented aerosol ensembles:
single particles of the ensemble can be spherical or spheroidal with size
parameters up to x≈1000. Moreover, a small set of irregular particles
is considered. The refractive index can be 0.1≤mr≤3.0 and 0≤mi≤2.2. The size distribution of the ensemble can either be
parameterized as a lognormal distribution, as a (modified) gamma distribution, or
freely chosen according to individual data. MOPSMAP includes a web interface
for online calculations at https://mopsmap.net, offering the most
frequently used options; for advanced applications or large sets of
computations, the full package is freely available for download. Key
applications of MOPSMAP are expected to be the evaluation of radiometer
measurements in the UV, VIS and near-infrared spectral range or aerosol lidar
measurements. They can help to improve the inversion of such measurements for
aerosol characterization. Furthermore, MOPSMAP can be used to refine optical
properties of aerosols in radiative transfer models or in numerical weather
prediction and chemistry transport models.
The details of the concept underlying MOPSMAP are discussed in this paper.
Several examples are presented to illustrate the potential of the package,
including an example to calculate optical properties for sectional aerosol
models and an example illustrating the effect of maximum size cutoff that
occurs in the inlet system of in situ instruments. In another example,
conversion factors between the backscatter coefficient (available from
lidar/ceilometer measurements or from numerical forecast models) and the mass
concentration of volcanic ashes have been calculated. These conversion
factors are relevant to estimate flight safety after volcanic eruptions and
vary by about a factor of 3 between the nine ashes under investigation.
The concept of MOPSMAP allows continuous upgrades to further extend the range
of applications. For example, the resolution of the refractive index grid could be
increased, new versions of underlying scattering codes could be applied when
available, larger size parameters could be considered, e.g., using DDA for
m close to 1 , and new sets of irregular particles could
be implemented, e.g., those presented by . However such
extensions can be quite time-consuming, so that extensions are expected to be
limited. Moreover, conceptional upgrades will be investigated without knowing
yet whether they can be included in the web interface. Here, a trade-off
between scientific complexity and user-friendliness must be found. Whereas
internal mixing in the case of homogeneous particles is already covered in the
present version, the implementation of a core-shell particle model can be
discussed. Finally, we want to emphasize that the feedback from the users will
help us to set up a priority list of further actions.