Distributions of ellipsoids are often used to simulate the optical properties of non-ellipsoidal atmospheric particles, such as dust. In this work, the applicability of ellipsoids for retrieving the refractive index of dust-like target model particles from scattering data is investigated. This is a pure modeling study, in which stereogrammetrically retrieved model dust shapes are used as targets. The primary objective is to study whether the refractive index of these target particles can be inverted from their scattering matrices using ellipsoidal model particles. To achieve this, first scattering matrices for the target model particles with known refractive indices are computed. First, a non-negative least squares fitting is performed, individually for each scattering matrix element, for 46 differently shaped ellipsoids by using different assumed refractive indices. Then, the fitting error is evaluated to establish whether the ellipsoid ensemble best matches the target scattering matrix elements when the correct refractive index is assumed. Second, we test whether the ellipsoids best match the target data with the correct refractive index, when a predefined (uniform) shape distribution for ellipsoids is assumed, instead of optimizing the shape distribution separately for each tested refractive index. The results show not only that for both of these approaches using ellipsoids with the true refractive index produces good results but also that for each scattering matrix element even better results are acquired by using wrong refractive indices. In addition, the best agreement is obtained for different scattering matrix elements using different refractive indices. The findings imply that retrieval of refractive index of non-ellipsoidal particles whose single-scattering properties have been modeled with ellipsoids may not be reliable. Furthermore, it is demonstrated that the differences in single-scattering albedo and asymmetry parameter between the best-match ellipsoid ensemble and the target particles may give rise to major differences in simulated aerosol radiative effects.

Mineral dust particles are abundant constituents of the Earth's
atmosphere

Dust particles are irregularly shaped and often inhomogeneous, making
accurate computations of their single-scattering properties a challenge

One such simple model geometry is that of ellipsoids. As shown by, e.g.,

Since the third axis of ellipsoids provides an even broader base for fitting than spheroids, they are likely to be able to mimic scattering by a wide variety of different target particles. This great flexibility is, however, potentially also a great risk in remote sensing applications, as it may allow good fits to be obtained with measurements based on wrong parameters. Here, we will investigate this issue with regards to the refractive index. To this end, we will use target data comprising single and ensemble-averaged scattering matrices computed for model particles whose shapes have been derived from real, individual dust particles through stereogrammetry.

Our chosen approach is thus to use computed, synthetic data to test the inversion rather than using real measurements. This approach offers several benefits: mainly the ability to acquire the full scattering matrix at all scattering angles in addition to perfect knowledge of and freedom to adjust the size, shape and composition of the target particle. Moreover, unknown measurement errors are replaced by quantifiable and somewhat controllable simulation uncertainties. Therefore, we strongly believe that a pure modeling study such as this is a highly useful approach for testing retrieval algorithms and simplified model shapes or other parameterizations.

In what follows, two types of analyses are carried out for the scattering
matrix elements. First, we will seek shape distributions for ellipsoids that
mimic the target data as faithfully as possible. Second, for comparison, we
will perform forward modeling and adapt a pre-defined uniform shape
distribution of ellipsoids. In both cases, the analyses are carried out for a
variety of refractive indices. The purpose is to find out how well ellipsoids
can match the target data and whether the best matches are obtained with the
correct refractive index. As the refractive index is wavelength dependent,
the refractive index retrieval cannot apply multiple wavelengths for
additional information without assuming some kind of relationship for the
refractive index at different wavelengths. Therefore, we perform the analysis
only at a single wavelength. The methodology adapted and data used are
presented in Sect.

The interaction of incident radiation with a particle can be
characterized by the scattering equation. One common formulation is
with the Stokes vector

In general, the scattering matrix has 16 elements. However, when

the particles are randomly oriented and

the particles are mirror symmetric, or particles and their mirror particles are present in equal numbers,

In this work, we consider the phase function formulation of the
scattering matrix elements. The scattering matrix is thus normalized
such that

The scattering matrix, like all dimensionless single-scattering
properties, is subject to the scale invariance rule, stating that
these properties depend only on the complex refractive index

In order to evaluate the retrieval results, we need to know the actual
refractive indices of our target model particles. It is also desirable that
the target particles and their scattering properties are representative of
real particles. One option would be to use measured scattering properties,
but then the refractive index would be uncertain. We therefore choose to use
synthetic data, computed using shapes derived from real dust particles by
stereogrammetry. Stereogrammetry is a method for acquiring a
three-dimensional structure of a particle by taking a pair of stereo images
with a scanning electron microscope. The target particle is tilted between
images to change the perspective. By matching known points between the images
from different perspectives, the structure of one half of the particle can be
determined, and a scaled mirroring technique is applied to produce the other
half. The stereogrammetric method is described in detail by

In addition to the original stereogrammetric particles, we discuss results
based on their artificially roughened variations. The surfaces of the
particles were modified using a Monte Carlo ray collision system that creates
several small mounds and craters at the surface, therefore reducing the
artificial surface smoothness caused by the stereogrammetric method while
keeping the overall particle shapes and volumes nearly intact. The roughening
method used is described in more detail by

In principle, the roughened particles may represent the real physical targets of the stereogrammetry study more than the original stereogrammetric shapes due to the fact that the stereogrammetry method can not recreate the fine surface roughness of the physical particles. However, the roughening is based on arbitrarily chosen parameters that have not been related in any way with the (possible) roughness characteristics of the target shapes considered here, or any other dust particles. Therefore, we consider the original unroughened particles as the primary target and use the roughened versions primarily to study the sensitivity of the results to particles' surface roughness. In particular, if moderate changes in surface roughness significantly alter the results of the refractive index retrieval, it can be said that the retrieval algorithm is too sensitive, or the impact of roughness on scattering dominates that of the refractive index.

For the scattering calculations we used version 1.2 MPI of ADDA

Renders of the DDA representations of the four target particles. The particles are depicted here with one-eighth of the dipole resolution compared to the calculations.

The scattering of each target shape was averaged over 8192 random
orientations for all size parameters. Figure

ADDA was run on the Finnish Meteorological Institute Cray XC30 supercomputer
Voima, using 64 computer cores per simulation. Additionally, 10 concurrent
simulations were run in parallel to reduce the total run time. With this
setup, the total amount of CPU time used was approximately 46 000 h.
The resulting scattering matrix elements of the target particles are shown in
Fig.

The six independent non-zero scattering matrix elements of the target particles integrated over the size distribution.

The scattering matrices for ellipsoids are taken from the database by

The particle size distribution for both the ellipsoid and the target
data is a lognormal distribution with

We investigate how well scattering by ellipsoid ensembles can match
scattering matrix elements of target particles. Specifically, the
scattering matrix elements of the individual ellipsoids form a basis,
and we seek the linear combination of the shapes that minimizes the
squared difference to the target data. We want the weights of the
individual particles in the ensemble to have properties of
probabilities, and that imposes two requirements, described by
Eqs. (

Due to the requirements in Eqs. (

The best-fit ensemble will be

Instead of fitting

For

The final normalization for

This ensures that the weights are positive and properly normalized.

In Sect.

In the case of a fixed, uniform shape distribution, instead of using

Number of ellipsoids employed in the fits, and the scattering errors. The numbers shown are means across all 40 refractive indices and four individual particles plus the three-particle ensemble. An ellipsoid is counted as part of the ensemble when its relative weight is at least 0.1 %.

We investigate the validity of the ellipsoid ensemble assumption by
fitting scattering matrix elements of a set of ellipsoids to those of
target particles, as described in Sect.

Additionally, Table

Scattering error

Scattering error

Figures

We show the scattering errors only for the ensemble of the three original unroughened stereogrammetric particles Cal, Agg and Dol. Sil was excluded from the ensemble because of its extreme axis ratio, which was not covered by the ellipsoids used in the study. The figures for individual particles are not shown, because in most cases the plots for the individual particles match those of the ensemble relatively well. If there are discrepancies, they are noted in the text. Similarly, the results of the roughened particles are not shown but are described in the text whenever noteworthy.

Complex refractive index errors for the fitted shape distribution. The values in bold correspond to retrieved refractive indices at the boundary of our refractive index domain and therefore could be even more erroneous in reality.

The results for individual particles are shown in
Tables

Complex refractive index errors for the uniform shape distribution.

It is important to note that the similarities in the

Figure

The good-fit band for

For

Compared to the previous elements, the behavior of

Considering the impact of surface roughness on the retrievals,

In addition to allowing the ellipsoid shape distribution to vary while
searching for the best-fitting

The minimum scattering matrix element errors are found at the maximum real
part and the minimum imaginary part of the refractive index for

For the ensemble

The

The

For

As in the case for the fitted shape distribution,

From Figs.

When performing the analysis for particles with added artificial
surface roughness, the results usually change only slightly (not
shown). However, sometimes the results change dramatically,
particularly for

Illustration of the effect of roughening on

Due to the large variability in the optima location for different
elements and particles, performing the retrieval using a combination
of different elements can yield a wide variety of results, depending
on the scaling and weighing of the error functions and individual
scattering matrix elements. Most notably, in some cases almost any
refractive index could be retrieved, were the weights or error
definitions selected accordingly, because the individual

Illustration of two combined

Target scattering matrix elements for

Figure

It is important to consider various uncertainty sources for the retrieval.
Because the discussion on the accuracy of the ellipsoid database is outside
of the focus of this work, the main concerns are the DDA accuracy and the
retrieval process reliability. DDA accuracy is discussed in detail in
Sect.

Based on these results, it seems that ellipsoids are not reliable for solving the inverse problem of retrieving the refractive index from scattering matrix data of irregular non-ellipsoidal particles, especially when using only individual elements. The retrieval results may be good in some cases for specific combinations of elements but that appears to depend strongly on the details of the combination, and any combination that works in one case might not work in another. Small ensembles of ellipsoids, like those used here, do seem to provide good fits with the correct refractive index, but even better ones with erroneous refractive index. In fact, the good quality of the fits may actually give a misleading impression of the validity of ellipsoid fitting. However, it needs to be emphasized that the ensemble used in this study is a small one, based on just three particles, without any abundance-dependent weighing. Ensembles containing a larger number of different particles, such is the case in the atmosphere, might yield different results.

Clearly, one can not assume that matching optical parameters between two scatterers imply matching physical parameters. While this might be true for isolated cases, it does not hold in the general case. This also opens up questions about solving the inverse problem. Even if a set of model particles is able to replicate some optical data with freely adjustable weights, thus formally solving the forward problem for each parameter individually, can we trust that the physical parameters are also close? This would be a requirement for said model particles to be used in retrieval, but based on this study, it seems like this assumption does not hold, especially if the model shapes do not even closely match the target shape.

In addition to seeing the retrieval errors of refractive indices, it
is interesting to know how the inaccurate retrievals of refractive index propagate into
higher-level applications, such as radiative transfer simulations. To
investigate this, we calculated the single-scattering albedo
(

Variability of single-scattering albedo and asymmetry parameter when the retrieval is based on fitting the ellipsoid shape distribution to individual scattering matrix elements of the particle ensemble. The true values of these parameters for the particle ensemble are also shown for reference.

We next consider the aerosol radiative effects on the top-of-the-atmosphere (TOA) and surface (SFC) net fluxes and atmospheric
absorption (ATM), normalized by the downwelling solar flux

Here

Two cases are contrasted: the REF case, for which

Figure

Due to the larger

The larger

For

Overall, this example suggests that errors in refractive index arising
from inaccurate shape assumptions in the retrieval scheme may result
in very substantial errors in the single-scattering parameters
(especially

In this work we investigated the reliability of the ellipsoid ensemble fitting for retrieving refractive indices of non-ellipsoidal model particles, with shapes retrieved from real dust particles via stereogrammetry. While it is known that ellipsoid ensembles can replicate the scattering of non-ellipsoidal particles closely, it is not known whether such ensembles are linked to the microphysical properties of the target particles. That is, if an ellipsoid shape ensemble of a given refractive index fit the scattering data of a particle extremely closely, does it guarantee that the particle has the same refractive index? This is the implicit assumption that is made in various retrieval processes, but the validity has not been investigated thoroughly before.

This question was studied with a two-step process. First we performed fitting of the scattering matrix elements of ellipsoid ensembles of various refractive indices. Second, we investigated the relationships of the scattering errors of the best-fit ensembles and the deviation of the refractive index of this best-fit ensemble from the true refractive index of the target particle, which was known. As target particles we used individual stereogrammetric particles as well as a small ensemble of them. In addition to having the ellipsoid shape distribution as a free parameter, we investigated the scattering matrix element differences between the target particles and a uniform distribution of ellipsoid shapes.

Based on our results, ellipsoid fitting is not a reliable method for
retrieving the true refractive index of non-ellipsoidal irregular particles,
despite producing good fits to the scattering matrix elements. The retrieval
based on error minimization found the true refractive index for only three
cases out of 120 shown in Tables

When using the modified model particles with added surface roughness,
the retrieval results are usually not affected much. Incidentally,
for most scattering matrix elements
the scattering errors increase, suggesting that ellipsoids do a poorer
job at mimicking scattering by dust particles with added surface
roughness. The retrieval of

Overall, it seems that the refractive index ranges selected were not completely sufficient to find the actual best-fit values, because most retrieved refractive indices were on edge of our complex refractive index space. However, the purpose of this study was not to find the refractive indices with the absolutely best match but rather to investigate whether the refractive index can be retrieved from the angular dependence of scattering from irregular dust particles using simplified model particles.

The analysis results clearly show that the retrieval of

When considering the practical implications of our findings, we must emphasize that few actual retrieval methods are based on an approach adapted here. Additionally, many applications use either spheroids or spheres instead of ellipsoids and have different limitations and error sources than those of ellipsoids. Different instruments employ different types of measurement data, for example, and thus have different vulnerabilities to the inherent biases imposed by the ellipsoid model. Also, we only considered cases with individual particles or a very small ensemble of three particles. Additionally, our target particles may not scatter light like real dust particles even though their shapes are directly derived from those of real dust particles. Therefore, this study should not be taken as a proof that dust refractive index retrieval using ellipsoids does not work. Rather, this study should be considered a cautionary tale that hopefully encourages retrieval teams to test their algorithm with sufficiently realistic reference data, yet we need to emphasize that our retrieval tests were conducted under ideal conditions. We did not have any measurement errors, other external contributions to the “measured” radiation, and we automatically employed the correct size distribution. We note that size and refractive index often have similar effects on scattering, so retrieval of both the size and refractive index may give rise to even larger retrieval errors due to error compensation.

Based on our findings, it would be interesting to carry out similar
investigation employing more complex model shapes for the retrieval.
Unfortunately, the computational burden of such an investigation would be
tremendous. One possible method to facilitate such a study is the shape
matrix method by

The authors wish to thank Lei Bi, Maxim Yurkin, David Crisp and an
anonymous referee for their helpful comments in improving the manuscript.
This research has been funded, in part, by the Academy of Finland
(grant 255 718) and the Finnish Funding Agency for Technology and
Innovation (Tekes; grant 3155/31/2009). Maxim Yurkin is acknowledged
for making his ADDA code publicly available
(