Single scattering by randomly oriented Saharan sand, silt, and clay particles is studied at 441.6 nm wavelength. Numerical simulations using the ray optics approximation and the Lorenz-Mie theory are compared with laboratory measurements of scattering matrix elements reported in the literature. The ray optics approximation is modified with ad hoc simple schemes of Lambertian surface elements and internal screens to study the effect of small-scale surface roughness and internal structures, respectively. Two different Lambertian reflection/refraction matrices with varying depolarization characteristics are applied. Model particle shapes are based on a tentative shape analysis of real Saharan particles. It is found that the traditional ray optics approximation agrees well with measurements only if unrealistically spiky particle shapes and an imaginary part of the refractive index (Im(m)) that is rather small compared with the typical values given in the literature are used. When the Lambertian schemes are applied, the agreement with measurements clearly improves. More importantly, good agreement can then be achieved using realistic particle shapes; this too requires a rather small Im(m). If Im(m) values corresponding to typical values in the literature are used, good fits can be achieved, but unrealistically spiky particle shapes have to be used. Our findings also indicate that sophisticated single-scattering modeling is important even below the ray optics domain. Finally, we demonstrate the importance of sophisticated single-scattering modeling for radiative flux and radiance calculations.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Saharan sand grains are irregularly shaped particles with a rounded shape and a surface covered with structures in many scales. They are also internally inhomogeneous. The importance of these details for light scattering depends on the particle size relative to the wavelength of light considered. The larger the particle, the more important the particle geometry is for scattering. For example, comparison of simulations using spherical and spheroidal shapes clearly shows that simplifications in shape can result in large errors in remote sensing of atmospheric mineral aerosols [e.g., Mishchenko et al., 1995, 1997; Pilinis and Li, 1998]. How well the detailed shape and inhomogeneity can be taken into consideration in light-scattering modeling depends on the model used; the choice of usable methods depends mostly on the particle size relative to the wavelength of light considered.
 For atmospheric mineral particles with a radius over a micron, for which we can expect the shape and structure to be most important at visible wavelengths, there are very few applicable light-scattering methods in which detailed particle geometry can be taken into account [see, e.g., Mishchenko et al., 2000c]. Most analytical methods are restricted to quite simple particle geometries. Numerical volume-integral methods, otherwise quite flexible, are presently limited to rather small particles due to the computational burden. The surface-integral methods can handle only rather simple and large-scale nonsphericities, and they generally cannot handle internal inhomogeneity. When the scale of inhomogeneity is significantly larger than the wavelength of light considered, inhomogeneity cannot be even indirectly incorporated by applying so-called effective-medium approximations [e.g., Chýlek et al., 2000]. It is indeed difficult to accurately model single scattering by large, irregular, inhomogeneous particles.
 If the particles are sufficiently large, however, the so-called ray optics approximation (ROA) can be used. In the ROA it is assumed that the curvature of the particle surface is much larger than the wavelength of the incident radiation everywhere on the particle and the surface can thus be considered locally a plane. In addition, it is assumed that the phase differences between internal and external fields across the surface irregularities are sufficiently large to suppress the interference effects associated with the irregularities [Muinonen et al., 1997]. In practice, ROA appears to be valid for smaller nonspherical than spherical particles, even though a sphere is a shape maximizing the surface curvature. This is because, for spherical or other highly symmetric particles, effects due to interference are strong and they are not included in the ROA. If particles are absorbing, then the usability of the ROA for small particles improves further [see, e.g., Mishchenko et al., 2000c, and references therein]. There is not yet a method available to study the lower limit of particle sizes that the ROA method can handle accurately if the shapes are truly irregular. The ROA can handle realistic mineral-particle shapes; it can also, in principle, handle the inhomogeneity. While atmospheric mineral particles are mostly too small for the ROA to be valid under normal conditions at visible wavelengths, there are occasions when most scattering comes from particles that are in the ray optics domain: for example, a sand storm [d'Almeida, 1987; Ichoku et al., 1999].
 In the geometric optics part of the ROA, the planar surface is thought to reflect incident light specularly. If a particle surface is smooth, such as with liquid droplets, this approach works well. If, on the other hand, there is small-scale roughness on the surface, each plane element can be thought to scatter light rather diffusely. In such conditions, the ROA is in principle not valid. It can be argued, however, that such conditions can be dealt with by modifying the reflection law.
 In the present work, we study the possibility of incorporating the small-scale surface roughness and internal inhomogeneity into ROA and estimate its importance in the case of large Saharan dust particles. Such a study is highly relevant for the atmospheres of Earth and Mars, for example, as mineral dust is the most predominant aerosol class observed from space for both atmospheres, and can significantly affect the radiative properties of these atmospheres [e.g., World Climate Programme, 1983; Kieffer et al., 1992; Husar et al., 1997]. We expect our results to be applicable also for other particles with similar refractive index, shape, and size, as well as for modeling of surface albedos in sand-like regolith. In addition to the radiative energy budget, aerosol particles are important in many remote sensing applications, as their presence affects the measurements of other atmospheric constituents or meteorological quantities.
 Our aim is realistic particle shape modeling, so we apply a statistical shape model, a so-called Gaussian random sphere geometry [e.g., Muinonen, 2000b], with statistical parameters derived from a shape analysis. The details of the shape model and the shape analysis are given in section 2.1. We use a traditional ray optics approximation and modify it for diffusely reflecting surface elements and internal structures. We note that it is not a new idea to include inhomogeneity into a ROA model; for example, Macke et al. [1996a] used spherical inclusions to mimic the effect of soot particles and air bubbles inside ice crystals in a similar fashion, and C.-Labonnote et al.  applied this method to interpret ADEOS-POLDER intensity and polarization measurements over ice clouds. The method has also been applied to planetary regolith particles [see Macke, 2000, and references therein]. The surface roughness has also been incorporated previously in the ROA models and used for ice crystals [see, e.g., Macke et al., 1996b; Yang and Liou, 1998]. However, the handling of roughness in these papers is different from ours: we also try to account for effects arising from roughness in a scale too small to be taken explicitly into account by ray tracing. Our model bears also some resemblance to the scalar, semi-empirical scattering theory by Pollack and Cuzzi . Our ROA model and the key characteristics of single scattering are considered in section 2.2. In section 2.3, we describe a radiative transfer model used to study the radiative properties of our model particles. We compare the ray optics results with those obtained using the Lorenz-Mie theory, and with laboratory measurements of Saharan particles. The measurements by Volten et al. , reviewed in section 3, are especially valuable for us as they include all the independent nonzero scattering matrix elements. Unfortunately, other information about measured scattering matrices of natural mineral aerosol particles is sparse. The results of the single scattering simulations and the radiative transfer simulations are given in section 4. Further discussion of results and their physical interpretation is given in section 5. Finally, the conclusions are drawn in section 6.
2. Theoretical Aspects
2.1. Particle Shape
 Natural mineral particles are irregularly shaped, with the shape varying from particle to particle. For such particles, a statistical shape model is clearly needed. A Gaussian random sphere geometry is thus applied [see, e.g., Muinonen et al., 1996; Muinonen, 2000b], with necessary statistical parameters derived from a shape analysis of a sample of real Saharan particles.
 The shape of a Gaussian random sphere is given by a radius vector
where ϑ and φ are the spherical coordinates, a the mean radius, σ the relative standard deviation of the radius vector, s the so-called logradius, and er a unit vector pointing outward in a radial direction. The logradius is given as a real-valued series of spherical harmonics Ylm, l and m being the degree and the order of the expansion. In principle, the expansion is infinite, but in practice only terms up to lmax are included; the proper value of lmax depends on the covariance function Σs of the logradius, as do the weights slm. Σs is related to the covariance function of radius a2Σr by
where γ is the angular distance between two directions (ϑ1, φ1) and (ϑ2, φ2), and Pl's are the Legendre polynomials with weights cl. The correlation functions of radius and logradius are Cr = Σr/σ2 and Cs = Σs/ln(σ2 + 1). Gaussian random spheres are generated by randomizing the spherical harmonics weight coefficients slm which have zero means and variances proportional to the cl's of the corresponding degree l in equation (4); for details, see, e.g., Muinonen et al.  and Muinonen [2000b]. The Gaussian random sphere statistics are isotropic, so the generated shapes are randomly oriented.
 In order to obtain the covariance function for the Gaussian random sphere model, we carried out a tentative shape analysis for a sample of Saharan particles. For the shape analysis, the sample was first sieved to separate subsamples based on their size. Then, about a hundred particles from each subsample were photographed with a microscope equipped with a CCD camera to achieve a sufficient statistical accuracy. In order to capture the large-scale shape of the particles, a Fourier series was fitted to each particle silhouette, using the center of mass of the silhouette as the origin. The corresponding covariance function for each subsample was then derived from the ensembles of Fourier series by assuming that the covariance function for the silhouettes is a valid approximation for that of random intersections of particles; the Cs for the Gaussian random sphere model was computed from the retrieved covariance function. Similarly, we approximated a2σ2 by the variance of the radius vectors of the silhouettes. A more sophisticated method for the shape analysis is explained by Lamberg et al. .
 Our shape analysis showed that the low-degree Legendre coefficients of the correlation function follow the power law cl ∝ l−ν with ν = 4. Intriguingly, such power-law coefficients are in agreement with those found for the shape of asteroids [Muinonen and Lagerros, 1998] and even for the shape of terrestrial planets [Kaula, 1968]. Using modern fractal analysis [e.g., Peitgen and Saupe, 1988], such a correlation function would indicate nonfractal, two-dimensional overall shapes for Saharan particles [Muinonen, 2002b]. A more detailed shape analysis for Saharan particles using the methods by Lamberg et al.  and Muinonen and Lagerros  is left for the future. As for the higher-degree Legendre coefficients, they may well reveal fractal characteristics for some finite range of degrees. For σ, a value about 0.2 was derived. We also fitted an ellipse to each silhouette and computed the axis ratio of particles from the axes of the ellipse; it was found to be about 1.5 for our sample particles.
 As pointed out above, the inverse proportionality of cl's to the fourth power of l appears to be quite universal for many natural irregular shapes. We thus decided to use this kind of Cs for our model particles. We set cl ∝ l−4 for l ≥ 2, and use a single parameter lmin, which indicates the first nonzero cl, to define the shape of the function. The minimum value for lmin is 2. We call this kind of Cs a power law correlation function to distinguish it from the modified Gaussian correlation function used by, e.g., Volten et al. . Example images of particle shapes generated using the power law correlation function with varying lmin are shown in Figure 1. The shape with lmin = 2 is a realistic shape, whereas the others have additional lower spherical harmonics suppressed.
 Even in the subsample of smallest particles analyzed, the particles were typically larger than about 10 μm in radius, whereas the atmospheric Saharan particles radiatively most important are smaller than that [e.g., Tegen and Lacis, 1996]. Considering the generality of our approach as mentioned above, and the similarity of shapes in the subsamples with different mean size, we assume that our shape analysis is valid also for smaller Saharan particles. In fact, we propose our power law correlation function with ν = 4.0 to be used as a first approximation for any natural mineral particles of unknown shape. The uncertainties involved in the shape analysis are discussed in section 5.
2.2. Single-Scattering Model
 The scattering properties of a single particle are described by its scattering and absorption cross sections σsca and σabs, respectively, and a 4 × 4 scattering phase matrix P connecting the Stokes vectors of incident and scattered light. The phase matrix element P11, the so-called phase function, describes the dependence of scattering on the scattering angle θs for unpolarized incident light. How these single-scattering properties are computed with our ROA model is explained thoroughly by Muinonen [2000b].
 In many applications, the phase function is described by a single number, the so-called asymmetry parameter g, representing the forward-weightiness of scattering. The absorptivity is often characterized by the so-called single-scattering albedo ϖ. These quantities can be given in the ROA by
where qsca is the scattering efficiency, and superscripts G and D stand for the geometric optics and diffraction parts, respectively. Another important parameter for light scattering is the so-called size parameter
where λ is the wavelength. The basic characteristics of scattering do not depend on the particle size but, instead, the size parameter.
 For the geometric optics solution, we use a Monte Carlo ray-tracing model that is a discretized version of the model explained thoroughly, e.g., by Muinonen et al. . The discretization means that the scatterer is described with a wire frame of triangles instead of a continuous function; the procedure is explained by Muinonen [2000a]. The biggest advantage of this is that a wire frame can have any shape, while a continuous function can only represent star-like shapes. The possibility of missing sub-step features of the shape in ray tracing is also eliminated, as the crossings with the particle surface can be analytically solved. Whether the discretized version is faster or slower than the nondiscretized version depends on several factors, the most important being the number of triangle rows in an octant (ntr) used to describe the scatterer. One clear disadvantage of the discretization is that a very large number of triangles is needed to accurately represent a shape. While the high accuracy of shape is not necessary when ensemble-averaged scattering by irregular shapes is considered, it is necessary in case of, e.g., spheres. Increasing ntr slows down simulations greatly. The effect of discretization on scattering is further considered in section 4.1.1.
 In addition, we have modified our ROA model to incorporate the internal inhomogeneity and the small-scale surface roughness into it. Our general assumption is that the surface roughness in scales too small to be taken explicitly into account in the ROA model affects scattering by increasing its diffuse and depolarizing nature. Thus we devised an ad hoc scheme using depolarizing Lambertian surface elements to take this into account, so that in effect, our model particles are partially “white particles” described by van de Hulst . In the scheme, a ray intersecting a particle surface undergoes either a Fresnelian or a Lambertian reflection and refraction, with the probability depending on the surface area fraction fex of Lambertian surface elements. Internal rays cannot undergo total reflection from a Lambertian surface element. Similarly to the surface roughness, we have a Lambertian scheme for the internal structure, in which an internal ray has a possibility to hit randomly oriented Lambertian screens, the probability depending on the free path length δin (in meters) inside the particle (probability = 1 − exp(−path length/δin)). As δin characterizes a material, not a particle, it is independent of particle size. Both the Lambertian surface elements and internal screens also have a given plane albedo, αex and αin, respectively. Thus we have two Lambertian parameters for both schemes in our model.
 Two alternatives were considered for defining the Lambertian reflection. The two alternative vector laws are ad hoc simple generalizations of the scalar Lambertian scattering law (as described by, e.g., Muinonen et al. ). When incident on a Lambertian surface element, an arbitrarily polarized ray is either linearly (in vector law a) or both linearly and circularly fully depolarized (law b), i.e., the Mueller matrix of the incident ray [see, e.g., Muinonen et al., 1996] is multiplied by the Lambertian reflection/transmission matrix [Hansen and Travis, 1974]. The reflection/transmission matrices have either the form diag(1,0,0,1) (in vector law a) or diag(1,0,0,0) (law b), with all nondiagonal elements equal to zero. Then, the resulting reflected/transmitted ray is scattered based on the scalar Lambertian scattering law, i.e., in a direction , where angles θ′ and ϕ′ specify zenith and azimuthal angles with respect to the scattering element, respectively, and ξ is a random number with a uniform distribution in the interval from 0 to 1 [Muinonen et al., 2001]. This procedure applies both for reflected and refracted (transmitted) rays. Both vector laws satisfy the symmetry relationships for phase matrices given by Hansen and Travis , but only the vector law a results in scattering phase matrices that satisfy the backscattering symmetry relation P44(π) = P11(π) − 2P22(π) [see, e.g., Mishchenko et al., 2000a]. Lambertian idealization is, however, widely used and is generally thought to approximate diffusely scattering surface well. Alternative means of modeling non-Fresnelian surface are discussed in section 5.
 Diffraction is solved in the Kirchhoff approximation, assuming the scatterers are spheres with an equivalent cross-sectional surface area. Taking the true shape into account would be much slower, and would provide little benefit.
 Both our geometric optics and diffraction models have been modified to take into account the particle size distribution. In the geometric optics model, this is accomplished by randomly selecting the particle mean radius for each incoming ray using a probability density function derived from the appropriate size distribution, and then weighting the incoming rays by the squares of corresponding particle mean radii. This approach is similar to that given by Nousiainen , but simpler in the sense that here the shape is not a function of size. In the diffraction model the particle sizes are similarly randomized.
where Ni is the total number of particles, σi the geometric standard deviation, Ri the geometric mean radius, and Ni(a) the number density of particles smaller than a in the corresponding modes. The size distribution for each mode, ni(a), can be considered to represent, for example, a distribution for a given mineral composition [d'Almeida, 1987; d'Almeida et al., 1991]. Here, we assume that the particles in different modes are statistically identical except for the size.
 A size distribution can be characterized by the effective radius aeff and the effective variance νeff, defined by Hansen and Travis  as
where 〈S〉 is the ensemble-averaged geometric cross-sectional area. These quantities, representing the first and second moments of the distribution, have been defined under the assumption that scattering is proportional to the cross-sectional surface area. When the assumption is valid, the quantities are usually sufficient to characterize the scattering by a distribution, regardless of the actual shape of the distribution. In the ray optics domain, the assumption is valid as long as the scattering efficiency is constant within the integration range. For absorbing particles, this is generally not the case. Thus aeff and νeff may not be as useful as expected for absorbing particles. In addition, aeff is less representative for scattering when νeff is large. An effective size parameter xeff can be defined by replacing a with aeff in equation (6).
2.3. Radiative Transfer Model
 In order to illustrate the effect of different particle characteristics and the resulting different scattering properties on radiative fluxes and radiances, selected test cases were performed using a one-dimensional (i.e., plane-parallel) Monte Carlo radiative transfer model. The model divides the atmosphere into a set of homogeneous layers (only two layers were needed in the present tests). The optical properties of each layer are defined by the optical thickness τ, the single-scattering albedo ϖ, and the phase function P11. Polarization is not accounted for. To speed up the computations, the integral of the phase function I(θ) = ∫0θP11(θ′)cos θ′dθ′ (normalized to 1 for θ = 180°) is precomputed and inverted to obtain the scattering angle θs = I−1(ξ) corresponding to a random number ξ ∈ [0,1]. Radiances are calculated by tabulating the angular distribution of photons reaching the top-of-the-atmosphere (TOA) or the surface at 1 degree resolution. The correct performance of the code was validated by comparisons with the DISORT algorithm of Stamnes et al.  for smooth phase functions that can be represented exactly using a Legendre series.
3. Measurement Data
 Our main comparison data are the light scattering measurements carried out for Saharan particles by Volten et al. . The scattering data that are relevant to this work include all the independent nonzero scattering matrix elements at λ = 441.6 nm for the scattering angles from 5° to 170° with 5° angular resolution. The scattering matrix F given by Volten et al. is a phase matrix except for an unknown normalization coefficient. Thus, when comparing measured F matrices with simulated P matrices, the normalized matrix elements Fij/F11 and Pij/P11 are readily comparable; as done by Volten et al., F11 and P11 are compared by setting them to unity at θs = 30°.
 The measured sample is the same that we have used for our shape analysis. The sample has been collected from the surface and sieved mechanically to remove the largest particles, as a sufficient amount of airborne material was not available. This means that the size distribution of the sample is not fully representative of the airborne desert aerosol in any specific meteorological condition. The particle radii in the sample vary from about 80 nm to 180 μm, so it includes sand, silt, and clay particles, and a few particles, representing 0.07% of the total surface area of the sample, exceed the upper size limit of a = 100 μm for atmospheric dust particles given by d'Almeida . The smallest particles are clearly too small to satisfy the validity criteria of the ROA at λ = 441.6 nm.
 The size distribution for the sample is given by Volten et al. . Here we have fitted a trimodal lognormal size distribution to it, with special emphasis on conserving aeff and νeff. The fit resulted in the following size distribution parameters: N1 = 36.0, N2 = 6.7 · 10−3, and N3 = 3.15 · 10−5 (per unit volume); σ1 = 3.5, σ2 = 2.4, and σ3 = 1.4 (dimensionless); R1 = 0.017, R2 = 1.35, and R3 = 33.0 (μm). With an integration range for equations (9) and (10) from 80 nm to 180 μm, we obtain aeff = 8.2 μm and νeff = 3.9, consistent with the original distribution. The fit and the measured size distribution are shown in Figure 2. Clearly, the trimodal distribution fits very well with the measured distribution, with slight deviations around 30 μm radius particles. Unfortunately, the error margins of the size distribution due to the particle sizing are not known.
 Typical features of the scattering matrix measured by Volten et al.  for Saharan particles can be seen, e.g., in Figure 3. The most characteristic features are: (1) flat side-scattering and the absence of an increase in scattering towards the back-scattering direction in the phase function; (2) the degree of linear polarization for unpolarized incident light (−P12/P11; hereafter linear polarization) is mostly weakly positive, with a maximum at side-scattering angles; (3) the depolarization ratio (1 − P22/P11) deviates markedly from zero, indicating the particles are clearly nonspherical. These features are consistent with the measurements by Jaggard et al.  for soil dust, and with ray optics simulations by Muinonen et al.  for Gaussian random spheres. The single exception is that the ROA simulations cannot reproduce the measured weak negative linear polarization near the backscattering direction, but this feature is likely to be caused by phase effects that are not included in ROA (for a possible explanation, see, e.g., Muinonen [2002a]). Otherwise the results are in agreement with measurements.
 Due to the large number of different simulations reported in this study, we have divided the results into several subsections. The physical interpretation of the results is presented in section 5.
4.1. Single-Scattering Simulations
4.1.1. Model Parameters
 Before the actual simulations, the ROA model was tested using the nondiscretized version of the model as a benchmark. The necessary resolution in the triangulation, described by the number of triangle rows in an octant (ntr), was tested with Lambertian schemes disabled. It was found that the ntr value needed for an accurate solution depends strongly on the particle shape. As ntr is also a critical parameter for the duration of computations, an optimal value needed to be found. For strongly irregular shapes, the results appeared to converge with as low a value as ntr = 15, while for a spherical test case even a value of 50 was not sufficient to reproduce the rainbow peaks accurately. We decided to set ntr = 25 which is quite sufficient for the irregular shapes used here. The Lambertian scheme was tested by comparing the results with available analytical solutions [e.g., Muinonen et al., 2001]. The model was found to work properly. For the parameter lmax we chose the value of 25, as it guaranteed a sufficient accuracy for the Legendre expansion of the correlation function in all our cases.
 With these settings, we were able to use 1000 rays per shape and 1000 shapes per simulation, totaling a million rays per simulation. The angular resolution of scattering was set to 3°, a compromise between precision and statistical noise both for the geometric optics and the diffraction parts. For radiative transfer simulations, a resolution of 2° was used for the geometric optics part, but a resolution of 0.0032° was used for the diffraction part within the first forward-diffraction peak. This approach increased the resolution of ROA in the forward-scattering angles, where most of the scattered energy is found.
 The model simulations were carried out using λ = 441.6 nm, corresponding to the shorter wavelength used by Volten et al.  in their laboratory measurements. This gave us a larger size parameter and thus better validity of the ROA model. The (relative) refractive index m of Saharan particles corresponding to λ = 441.6 nm was interpolated from the table given by d'Almeida et al.  for desert aerosols, resulting in a value of m = 1.53 + i0.01. This is consistent with the OPAC database [Hess et al., 1998] and was used as a first-guess estimate. Because Im(m) of a mineral is not a constant but varies even orders of magnitude due to changing amounts of absorbing trace constituents, and because it is difficult to measure, we considered Im(m) a free parameter. On the other hand, we fixed Re(m) to 1.53 in all systematic simulations, as it is easier and more reliable to measure than Im(m), varies relatively little in natural silicates and, according to test simulations, affects the results relatively little for the parameters used here. The birefringence was assumed to be insignificant for scattering; this is supported by the scattering measurements by Volten et al. .
 All of our simulations with a size distribution were based on the fitted trimodal size distribution given in section 3. Considering the validity of the ROA model, the whole size distribution could not be used in the ROA simulations. Thus we had to truncate the size distribution. We chose to include only particles larger than a = 2 μm in the truncated distribution. Thus essentially all the particles in mode 3 are included, but only about 10% and 90% of the total surface area in modes 1 and 2, respectively. At λ = 441.6 nm, a = 2 μm corresponds to the size parameter x = 28 which is rather small for the ROA even with irregular shapes. Yet the truncation caused two considerable problems. First, only 37% of the total cross-sectional surface area of the original distribution is included in the truncated distribution, indicating that less than half of the total scattering is accounted for in the simulation. Second, aeff and νeff are changed significantly: after the truncation they are 21.2 μm and 0.95, respectively. These problems must be kept in mind with the ROA simulations. Nevertheless, we are able to fit the ROA simulations very well with measurements (which include also the small particles). The implications of this are discussed in section 5.
 It should be noted that a change in aeff alters xeff correspondingly. As absorption and scattering depend on the quantity Im(m)x instead of Im(m) in ROA, a truncation can indirectly bias Im(m). Test simulations with ROA showed that, in the first approximation, the effect of truncation on xeff can be compensated for by modifying Im(m) accordingly, although this compensation does not account for the change in νeff or take to consideration the fact that the particles truncated are not in the ray optics domain. Thus, as the truncation here increases xeff by a factor of about 2.5, the values of Im(m) used in the ROA simulations can be thought to correspond to a “true” physical value that is about 2.5 times the value used. For example, the first-guess value of Im(m) = 0.01 should be given to the ROA model as 0.004, which we then call “truncation-compensated Im(m).” We emphasize that this compensation is strictly confined to the interpretation of the model parameter Im(m), and that the values of Im(m) given are always values used in the model(s).
 In an attempt to improve the modeling of real Saharan particles, we carried out a large number of different simulations using different parameter combinations and models. We started by comparing the Lorenz-Mie theory with the measurements. Then we tried a traditional ROA model using the truncated size distribution, and studied the possibility of computing the truncated small-particle part using the Lorenz-Mie theory. After that we concentrated on the ROA model alone, investigating the parameters that would give us best agreement. This was done both with and without the Lambertian schemes. The parameters used in different ROA simulations are given in Table 1. The results are shown in section 4.1.2. It should be noted that our “best-fits” are not optimal fits in a rigorous sense. First, the choice of the best-fits was based on subjective judgment rather than on the use of any metrics. Second, the large number of free parameters forced us to use rather coarse resolution in a systematic search of best fits (typically 3 different values for each free parameter). The results of this search were then used to look for even better fits by hand-picking promising parameter combinations.
Table 1. Parameters Used in Different ROA Simulationsa
Simulation Set is the number the simulations are referred with in the text, Var means that the quantity was varied, a dash indicates that the quantity was not used, and SD stands for the truncated size distribution.
4.1.2. Single-Scattering Results
 The comparison of the Lorenz-Mie theory and the measurements by Volten et al.  is shown in Figure 3. Obviously, the simulations and the measurements are not in agreement. The same kind of disagreement can be seen in the study by Jaggard et al.  for soil dust. To demonstrate that the differences are not due to an uncertainty in Im(m), we have included a curve with Im(m) decreased by an order of magnitude. The largest differences are in P22/P11, which is traditionally connected to the particle nonsphericity: minimum values measured are about 0.2, while the Lorenz-Mie theory gives a value exactly one. There are also large differences in other scattering matrix elements. Interestingly, certain element ratios appear to have even a different sign: for example −P12/P11.
 On the other hand, it appears that the ROA itself can be fitted to the measured scattering matrices of irregular mineral particles quite well. For example, Volten et al.  were able to fit their measurements with ROA apparently in a way superior to any previously published study. Good fits required unrealistically spiky particle shapes (lmin ≈ 10), but it was not necessary to take into account the size distribution. We could not improve significantly the fits obtained by Volten et al. by using a size distribution and varying Im(m), σ, and lmin systematically (simulation set 1). We also attempted to compensate for the absence of small particles in the truncated size distribution by computing their scattering using the Lorenz-Mie theory and adding that to the ROA results, but it turned out that this approach only made the agreement worse.
 Before we applied the Lambertian schemes in the fitting, their effect on scattering was tested systematically (simulation set 2). For simplicity, we used single-size simulations with parameters compatible with those used in the best fit of Volten et al.  for the average mineral aerosol. The Im(m) = 0.00213 used corresponds to the Im(m)x = 0.1 adopted by Volten et al. .
 The dependence of scattering on the Lambertian parameters is shown in Figures 4–6which indicate that the effects of the Lambertian parameters on the individual scattering matrix elements are systematic. Also, they appear to have a rather similar impact on scattering. The parameter αin, however, appears to have mostly a weak effect in the cases studied, so we have not included a figure illustrating its effect. Naturally αin is more important for small than for large δin; it also seems that increasing Im(m) increases its importance. The choice of the Lambertian vector law affects practically only the element P44/P11.
 The main effects of the Lambertian parameters for each scattering matrix element can be summarized as follows:
An increase in fex and αex, and a decrease in δin all act to increase P11 in the backscattering direction and decrease it at some angles in the forward-scattering hemisphere, effectively decreasing the asymmetry parameter g.
An increase in fex and a decrease in δin both act to decrease the maximum in −P12/P11 and move it toward the forward-scattering direction. A decrease in αex decreases −P12/P11 slightly and moves the maximum toward the backscattering direction.
An increase in fex and a decrease in δin both act to decrease P22/P11 closer to zero, whereas an increase in αex increases it in the forward hemisphere and decreases it in the backward.
An increase in fex and a decrease in δin both act to decrease P33/P11 at scattering angles smaller than about 120° and increase it at angles larger than that. An increase in αex increases P33/P11 in the forward hemisphere.
An increase in fex and a decrease in δin both act to decrease the side-scattering maximum of −P34/P11. The effect of αex is very weak.
For the vector law a, an increase in fex and a decrease in δin both act to increase P44/P11 outside the forward scattering directions, while an increase in αex acts to increase P44/P11 in the backward hemisphere and decrease it at angles around 50°. For vector law b, an increase in fex and a decrease in δin both act to decrease P44/P11, except close to the backscattering direction. An increase in αex increases P44/P11 in the forward hemisphere.
 The Lambertian schemes also affect absorption, but the effect is not dependent on the vector law used. The Lambertian surface elements are not themselves absorbing, but they affect the amount of energy refracting into the absorbing particle, as well as the ray paths inside the particle. Similarly, Lambertian internal screens affect the ray paths inside the particle. For example, for the parameters used in simulation set 2, we obtain ϖ = 0.816 when Lambertian schemes are not used. When Lambertian surface elements are applied with fex = 0.5 and αex = 0.5, we obtain ϖ = 0.852. The actual effect depends on the contrast between Lambertian and non-Lambertian surface elements, and their area fractions. The effect of Lambertian internal screens is rather weak, although it naturally depends on the parameters used. With the values used here, internal screens hardly affect ϖ.
 When we applied the Lambertian schemes in order to improve the agreement with the measurements, we ran two sets of simulations (simulation sets 3 and 4). The first set (3), in which particle shapes (lmin) were varied, was to study if good agreement can be achieved using the truncation-compensated first-guess Im(m) = 0.004, while the second set (4) was to study how good fits can be achieved using realistic shapes. In the latter set, the emphasis was on Im(m) values as close to the first-guess value as possible. Both sets revealed that the Lambertian schemes have the desired effect on scattering and we can improve our fits significantly. More importantly, we could get good fits using realistic particle shapes (simulation set 4). This was not possible without the Lambertian schemes (simulation set 1) and without decreasing Im(m) below the first-guess value. Good fits for the whole matrix required, however, that we use the vector law b in the Lambertian scheme. If only linear polarization was depolarized (vector law a), P44/P11 values were generally too large. The best-fit cases for simulation sets 3 and 4, along with simulation set 1, are shown in Figure 7. The best-fit parameters can be found in Table 2. Obviously the Lambertian schemes lead to much improved agreement with measurements: the elements P22/P11, −P34/P11, and P44/P11 (with the vector law b) improve most. Because only the P44/P11-element is significantly affected by the choice of the vector law applied and only the vector law b provided us with good fits, rest of the results correspond to the vector law b.
Table 2. Best-Fit Parameters for Simulations Shown in Figure 7a
Simulation Set is the number the simulations are referred with in the text, and SD stands for the truncated size distribution.
 Test simulations proved that we can also get good fits by using the uncompensated Im(m) = 0.01 with the truncated size distribution; this requires larger lmin (≈10) and thus very spiky shape, and lower αex. Also, it turned out that good fits can be achieved using realistic particle shapes with Im(m) values even orders of magnitude smaller than the best-fit value of 0.001. On the other hand, for values larger than Im(m) = 0.001, good fits are difficult if not impossible to achieve without increasing lmin and thus giving up the realistic particle shapes.
 Finally, we studied the size dependence of scattering in the ROA using single-size simulations. The Lambertian schemes were excluded. It turned out that for moderately nonspherical particles with the first-guess Im(m) = 0.01, the particle shape is important only for particles with a ≲ 5 μm (see section 5 for explanation). This result can be scaled for other values of Im(m) by keeping the term Im(m)x constant.
4.2. Radiative Transfer Simulations
 As shown, the scattering matrix measurements by Volten et al.  did not allow us to constrain our single-scattering model unambiguously. Rather, different simulations resulted in different “best-fits,” leading to differences in the calculated radiative fluxes and radiances. These differences are demonstrated here. In addition, a comparison is made with results based on the Lorenz-Mie theory, which is by far the most common approach for computing aerosol single-scattering properties.
 The three best-fits chosen for the comparison are denoted as follows: ROA-0.001 stands for the ray optics approximation with Im(m) = 0.001 (simulation set 1 in Table 2), Lamb-0.001 means ray optics approximation augmented by the Lambertian schemes with Im(m) = 0.001 (simulation set 4), and Lamb-0.004 is the same for Im(m) = 0.004 (simulation set 3). The Lorenz-Mie calculations are denoted as Mie-0.001 and Mie-0.004 for Im(m) = 0.001 and Im(m) = 0.004, respectively. All calculations, also those based on the Lorenz-Mie theory, assume the truncated size distribution used above, and the wavelength is fixed at 441.6 nm. Table 3 summarizes ϖ and g in all these cases, and the corresponding phase functions P11 are displayed in Figure 8a.
Table 3. Single-Scattering Properties for the Radiative Transfer Testsa
The ϖ stands for the single-scattering albedo, and g stands for the asymmetry parameter. The truncated size distribution is assumed. See text for the abbreviations.
 The dust aerosols were assumed to be well-mixed in the layer 700–1013 hPa (about the lowest 3 km) of a tropical model atmosphere. Calculations were made for dust optical thicknesses of τ = 0.4 and τ = 2. Since our size distribution is more compatible with a heavy sand storm than background conditions, we mainly consider the results for τ = 2 below. In addition to dust, the simulations included molecular Rayleigh scattering (total column optical thickness 0.233) and very weak O3 absorption (column optical thickness 0.0008, based on Freidenreich and Ramaswamy ). The surface albedo (i.e., the ratio of the upwelling flux to the downwelling flux) was fixed at 0.3 (Lambertian surface). Three values of solar zenith angle were considered (θ0 = 0°, θ0 = 50°, and θ0 = 80°). All the results shown are from simulations with 108 photons.
Table 4 displays for the τ = 2 cases the values of the TOA albedo (R = /), the normalized downward flux at the surface (SFC) (T = /), and the normalized atmospheric absorption (A = [(F↓ − F↑)TOA − (F↓ − F↑)SFC]/). For comparison, the results for a dust-free atmosphere (τ = 0; NOAER) are also given. The following points can be made:
Overall, the effect of the dust layer is dominated by absorption. The absorptance values (A) are very high particularly in the θ0 = 50° case (55–67%). Especially as regards the absorption (but also R and T), the largest contribution to the differences between the results for different cases comes from the choice of Im(m). Again, we wish to emphasize that the scattering matrix measurements by Volten et al.  did not allow us to fix Im(m) unambiguously.
In all cases, the presence of the dust layer decreases the TOA albedo (R) substantially. The largest albedo values are obtained for Lamb-0.001 and the smallest for Mie-0.004, with absolute differences of ΔR ≈ 0.08 between these two. The relative differences in the albedo of the dust layer itself were quite large. Neglecting all other factors, the albedo ranged in the θ0 = 0° case from 0.013 for Mie-0.004 to 0.032 for Mie-0.001 and to 0.088 for Lamb-0.001. For θ0 = 80°, the respective values were 0.083, 0.185 and 0.222.
The normalized downward flux at the surface (T) is largest for Mie-0.001 and generally smallest for Lamb-0.004, with an absolute difference ΔT = 0.118 for θ0 = 0°. The directly shape and structure-related differences between Mie-0.001 and Lamb-0.001 also reach ΔT = 0.054. The relative differences in T are largest for θ0 = 80°, T being ≈40% smaller for Mie-0.004 and Lamb-0.004 than for Mie-0.001.
Table 4. Radiative Flux Results
The R stands for the top-of-the-atmosphere (TOA) albedo, and T and A are the surface downward flux and atmospheric absorption, respectively, both normalized by the TOA downward flux (). An aerosol optical thickness τ = 2 is assumed, except that τ = 0 in the NOAER case. See text for more explanation.
TOA Albedo (R)
Normalized Downward Flux at the Surface (T)
Normalized Atmospheric Absorption (A)
Figures 8b and 8c display, as another example of the effect of the alternative treatments of dust single-scattering properties, the TOA upward and surface downward radiances for θ0 = 0° (τ = 2). In this case it is sufficient to consider zonally averaged radiances, for which the statistical rms error is below 1% for zenith angles between 2 and 85 degrees. The primary qualitative differences lie between the results based on the Lorenz-Mie theory and those for the ray optics approximations (either with or without the Lambertian elements). In particular, the Mie-0.001 upward radiances show two peaks at zenith angles of θ = 3° and θ = 17°, which correspond to local maxima in P11 near the backward-scattering direction (at 177° and 163°) in Figure 8a. This suggests that singly-scattered radiation contributes substantially to the upward radiances in this case. This feature is weaker in the more strongly absorbing Mie-0.004 case and is absent when nonspherical particles are assumed. For ROA-0.001, Lamb-0.001, and Lamb-0.004, the radiance distribution is basically flat, with larger values near the horizon associated with molecular scattering above the dust layer. The Lamb-0.001 upward radiances are consistently slightly larger than those for ROA-0.001, particularly at relatively small zenith angles.
 In Figure 8c, the downward radiances are distinctly larger in the Lorenz-Mie cases than when nonspherical particles are assumed at small and intermediate zenith angles. However, the opposite is true at zenith angles larger than about 70° (50°) for Im(m) = 0.001 (for Im(m) = 0.004).
 The results for τ = 0.4 are qualitatively similar to those for τ = 2, but the dust radiative effects are naturally smaller (by ≈60–70% for θ0 = 0° and θ0 = 50°, and by ≈50% for θ0 = 80°). The TOA albedo values are by 0.05–0.10 below the dust-free case, and the absorptance values range from 0.14 to 0.25. The absolute differences between different single-scattering approximations are likewise smaller. For example, the normalized downward flux at the surface (T) for ROA-0.001 and Lamb-0.001 is ≈0.016 smaller than that for Mie-0.001 both for θ0 = 0° and θ0 = 50°.
 Comparison of the Lorenz-Mie simulations with the light-scattering measurements by Volten et al.  confirms that real Saharan particles cannot be considered homogeneous, isotropic spheres. Considering the weight of particles smaller than 2 μm to scattering, it is obvious that even particles smaller than 2 μm scatter light significantly differently than the Lorenz-Mie theory indicates. This discrepancy is so significant that if we try to improve our ROA simulation results by taking the truncated part of the size distribution into account by using the Lorenz-Mie theory, the agreement with measurements becomes worse. The measured clearly nonzero depolarization ratio implies that at least the particle shape, and probably also the small-scale surface structure and the internal inhomogeneity are quite important for these small particles also. Uncertainty of Im(m) is not sufficient to explain the discrepancy. We concur with the previous studies [e.g., Mishchenko et al., 1995, 1997; Pilinis and Li, 1998] that the Lorenz-Mie theory does not describe scattering by real Saharan particles accurately.
 The light scattering simulations carried out by Volten et al. , on the other hand, show that the ROA simulations can reproduce the measured scattering quite well, even without using size distributions; this implies that inversion of a Saharan particle size distribution from scattering data might be difficult. Contrary to our expectations, the incorporation of a size distribution to simulations does not improve the agreement significantly. It indicates, however, that the values given for Im(m) by d'Almeida et al.  might be too large. The results by Dubovik et al.  also support this view. In addition, our simulations confirm that unrealistically spiky shapes have to be used to get adequate fits with Fresnelian ROA. While such shapes are not realistic, spikiness can actually mimic diffuse surface scattering. Narrow spikes can also result in scattering similar to that from smaller or less-absorbing particles. Thus spikes with varying size can mimic the effect of a distribution of Im(m). Simulations with realistic particle shapes result in too strong linear polarization and too weak depolarization.
 Due to strong absorption inside the particles, strong linear polarization is clearly connected to the dominance of specular surface scattering. This and the weakness of measured linear polarization led us to believe it would be highly important to incorporate the effect of the small-scale surface roughness. Thus we applied the Lambertian surface elements to mimic the diffuse, depolarizing surface reflection. We also applied the internal Lambertian screens to incorporate the effect of internal inhomogeneity. While both the Lambertian schemes introduce two additional parameters, the ROA model does not become correspondingly more flexible. As the tests (simulation set 2) indicated, the parameter αin has mostly a weak effect, and the three remaining parameters have rather similar effects on scattering. This is not surprising, because all these parameters control the amount of diffuse scattering. Thus changes in different parameters can partially cancel each other, especially so if lmin is also a free parameter. This, unfortunately, makes it more difficult to find the best possible fit, and can also complicate the application of this approach to other problems. On the whole, the surface Lambertian scheme is clearly more important than the internal Lambertian scheme for good fits: the surface Lambertian scheme alone allows good fits with realistic shapes, while the internal Lambertian scheme does not.
 The ROA simulations with proper Lambertian parameters result in significantly improved agreement with the measurements, with the exception of P44/P11 for which good agreement is possible only by using the vector law b. More importantly, the introduction of Lambertian schemes allows us to get good fits by using realistic particle shapes. This requires, however, that we decrease Im(m) to about 0.001 from the first-guess value of 0.004. The inability to simulate the scattering from a whole size distribution accurately prevents us from drawing any decisive conclusions about the true imaginary index of refraction for the measured Saharan particles. By using unrealistically spiky shapes (Figure 1c), good fits can also be achieved by using larger values of Im(m), at least up to 0.01.
 The problems in backscattering direction requires a further discussion. The vector law b which is otherwise consistent with measurements, does not satisfy the condition P44(π) = P11(π) − 2P22(π) for backscattering. Intriguingly, for this condition to hold for the measured matrix, there must be a backscattering spike either in P22/P11 or P44/P11 element, or both. Such a spike might result, e.g., from coherent backscattering [Muinonen, 2002a]. Unfortunately, the measurements do not cover these angles. As neither Fresnelian ROA nor our Lambertian modification include phase effects, we cannot model coherence effects with them. Better means of modeling diffusely scattering surface are currently under investigation. It could be based on empirical bidirectional reflectance distribution matrix, or a radiative transfer modeling within each surface element. The Lambertian approximation appears, however, a reasonable and efficient first approximation.
 Even though the introduction of Lambertian schemes allows good fits with measurements, there are two points to be made. First, even though the results improve, good fits require that we relax our constraints either on Im(m) or the particle shape somewhat. There are several possible reasons for this: (1) Our first guess may not be correct, i.e., the values of Im(m) given in literature do not apply to the Saharan particles measured by Volten et al. . (2) In reality, Im(m) varies from particle to particle. As the absorptivity depends weaker than linearly on Im(m), the use of average Im(m) leads to an enhanced absorption. Here we confined ourselves to using a fixed value, as we have no data about the distribution of Im(m) for Saharan particles and we wanted to limit the number of free parameters. (3) It is possible that Im(m) varies within particles; absorbing components, especially the iron oxide, can be mostly constrained to the particle surface. (4) It may be a side effect of the absence of small particles in the ROA simulations. (5) Real Saharan particles have sharp edges that are not reproduced in our randomly generated shapes: spikes may be mimicking these features. (6) It is possible that our Lambertian schemes do not result in right kind of scattering. (7) It is possible that our shape analysis has not adequately captured all the factors affecting scattering. The assumption that the statistics of silhouettes can be used as statistics for intersections probably leads to slightly overly rounded shapes. In addition, the silhouettes are probably not in perfect random orientation [see, e.g., Jaggard et al., 1981]. It is also possible that the power law correlation function is not correct for the high-degree cl's that contribute to the surface roughness; our shape analysis is constrained to low degrees defining the large-scale shape. Nevertheless, it has been shown that small variations in shape statistics are unimportant for ensemble-averaged scattering properties [Nousiainen and Muinonen, 2002].
 Second, good agreement with the measurements does not prove that our model particles scatter light realistically. The most important reason for this is the obvious one: the absence of small particles in the simulations. However, the fact that we are able to get good fits with measurements even though the simulations do not include small (truncated) particles implies one of the following: (1) The size distribution used exaggerates the number of independently scattering small particles (due to an erroneous distribution or an aggregation of small particles during the scattering measurement). While agglomeration was not detected in the inspections during the scattering measurements, it is possible that the agglomeration of the smallest particles could go unnoticed (H. Volten, personal communication, 2001). (2) The Lambertian schemes somehow take correctly into account scattering by the small truncated particles, or a compensation of errors occurs. (3) A large part of the small truncated particles scatter light similarly to the particles included in the simulation.
 It is quite possible that each of these plays a role. The last explanation, however, seems most plausible, as the measured scattering matrices of Loess, Red Clay and Green Clay [Volten et al., 2001; Muñoz et al., 2001], which all consist of irregular mineral particles with significantly smaller size, show very similar scattering matrix with that of the Saharan sample. Indeed, this seems the most natural explanation for the question why we can simultaneously fit well all scattering matrix elements with our modified ROA method using the truncated size distribution. Additional information can be attained when scattering by small (truncated) particles is solved accurately using a suitable scattering method for nonspherical particles. This is the topic of our future research.
 One potential method for handling the truncated part of the size distribution is the so-called T-matrix method, which would probably perform better than the Lorenz-Mie theory as it can handle nonspherical shapes. We did not test the T-matrix method ourselves, but the published T-matrix phase functions do appear to have increased scattering toward the backward direction similar to the Lorenz-Mie theory [see, e.g., West et al., 1997; Mishchenko et al., 1997, 2000b], indicating that it might have similar problems with fitting to the measurements. Mishchenko and Travis  have studied the linear polarization for desert dust aerosols using the T-matrix method, but their largest size parameters are smaller than the smallest ones used here. Unfortunately, we are unable to estimate the performance of the T-matrix method with respect to other nonzero scattering matrix elements, as we could not find any published results for those. If the T-matrix method is applied for the truncated part of the size distribution, it is probably best to do it using axially symmetric Gaussian random spheres instead of spheroids [e.g., Muinonen, 2000a]. It is questionable if the T-matrix method would work well for larger particles: our results indicate that the surface structure is very important for scattering properties of large particles, and the T-matrix method cannot handle rough surface. Other methods applicable for small particles include the second-order perturbation series approximation for Gaussian random spheres [Muinonen, 2000b; Nousiainen et al., 2001], and so-called volume-integral methods which can handle complex geometries. For example, a finite difference time domain method (FDTD) can be used up to x = 20 with ensemble averaging (P. Yang, personal communication, 2001).
 The finding that particles larger than about a = 5 μm with Im(m) = 0.01 result in similar scattering regardless of the exact shape is due to the fact that large particles absorb most of the light refracted inside them. Thus most scattered light originates from the surface reflection, and the specular surface reflection for an ensemble of convex, randomly oriented particles is independent of shape [van de Hulst, 1981]. Even for such particles, however, it matters whether the surface reflection is diffuse or specular. Were the large Saharan particles specularly reflecting, the Lorenz-Mie theory could perform very well for them.
 We expect the detailed modeling of single scattering to be most important for Saharan particles with mean radii in a range from 1 to 10 μm. For particles smaller than that, the small-scale surface structure and the internal inhomogeneity are not expected to be critical factors, and particles larger than that are usually too sparse to make much of an impact. The lower boundary is an educated guess, but the comparison of measurements with the Lorenz-Mie theory clearly indicates that at least the particle shape, and probably also the inhomogeneity and the small-scale surface roughness should be taken into consideration well below the ray optics domain. This makes this work relevant also for mineral particles in the Martian atmosphere. While the composition of Saharan particles is not an ideal analogy for Martian particles, the particle shapes and surface structures can be quite similar. A random shape approach is especially relevant, because exact shapes of Martian particles are not known. The effective radius of airborne Martian mineral particles is about 2 μm [Kieffer et al., 1992; Ockert-Bell et al., 1997], so the ROA approach may not be the best method to be applied for Martian particles. Nevertheless, scattering measurements for Red Clay with aeff = 1.5 by Volten et al.  and Muñoz et al.  show a very similar scattering pattern with that for our Saharan sample.
 Atmospheric radiative transfer simulations require P11 or g as a parameter. We wish to point out that the values of g in Table 3 vary somewhat even for the cases that are fitted to measurements. For the fitting, we have been forced to normalize P11 and this is a risky procedure: the normalization can force differently sloping phase functions to a seemingly good agreement with each other, thus biasing g. The fact that we can get all the phase matrix elements in good agreement with the measurements speaks for the validity of our modeling approach, but it does not guarantee unbiased results.
 The radiative transfer simulations show that absorption is dominant over scattering in the simulation dust layer, particularly so when θ0 = 50°. This is due to the fact that for θ0 = 0°, the path length for the direct solar beam through the dust layer is shorter, and for θ0 = 80°, Rayleigh scattering reduces more effectively the amount of solar radiation reaching the dust layer. Because of the strong Rayleigh scattering, especially in the θ0 = 80° case, the TOA albedo is clearly less sensitive to the single-scattering properties of dust particles than the reflectance of the dust layer is itself. Because molecular scattering becomes weaker and Im(m) for desert aerosols decreases with increasing wavelength [d'Almeida et al., 1991; Dubovik et al., 2002], we speculate that the assumptions about particle shape and structure are likely to be more important for the TOA albedo at larger wavelengths.
 Finally, it may be noted that the above result regarding dust aerosols decreasing the TOA albedo substantially is in contrast to the findings by Hsu et al. . These authors concluded, based on ERBE and TOMS satellite data, that Saharan dust aerosols have a very weak and noisy effect on the broadband solar TOA albedo over land areas. A similar result emerges from the calculations by von Hoyningen-Huene et al. , who fitted their aerosol model using spectral and angular solar radiation measurements made in Senegal. Most other computational studies [e.g., Carlson and Benjamin, 1980; Ackerman and Chung, 1992; Claquin et al., 1998] suggest that dust aerosols decrease the TOA albedo over deserts at least somewhat. However, Ackerman and Chung  found little difference in ERBE shortwave fluxes between clear and dust-laden regions of the Arabian peninsula. There are at least two reasons leading to the discrepancy between our results and observations.
 First, our simulations do not (and are not intended to) represent the most typical cases in the atmosphere. Ordinarily, the size of desert dust particles is significantly smaller. Using Lorenz-Mie theory and spectral sky-radiance measurements, [Dubovik et al., 2002] retrieved median volume radii of only ≈2 μm for coarse mode desert dust particles. In sand storms, however, the typical particle size can be even larger than in our calculations, at least near the surface: the sandstorm size distribution of d'Almeida  has an effective radius aeff = 26 μm. It appears probable, though, that the largest (i.e., heaviest) particles are confined to the lowest parts of the sand storm, so that the upper parts are characterized by smaller particles. For a given particle shape, smaller particles have larger ϖ and smaller g and therefore they reflect solar radiation more efficiently.
 Second, our computations concern the single wavelength λ = 441.6 nm. At this wavelength, both x and Im(m) of desert dust particles are larger than those at longer wavelengths in the shortwave region [d'Almeida et al., 1991; Dubovik et al., 2002]. Therefore the reflectivity of the dust layer is very likely smaller at λ = 441.6 nm than in the shortwave region on average.
 Light-scattering simulations carried out in this work indicate that the small-scale surface roughness of real Saharan particles should be taken into account when their light-scattering properties are being simulated at visible wavelengths. We expect this to hold also for other similar mineral particles and at near-IR and UV wavelengths. When the particle inhomogeneity is taken into account as well, the agreement between measurements and simulations improves further. The surface roughness appears to be more important, which is likely partially due to the large contribution of surface reflection to the light scattered by absorbing particles.
 The importance of the small-scale surface roughness and internal structures, taken into consideration in our ROA model using Lambertian schemes, depends somewhat on our assumptions about Im(m) of Saharan particles. If the truncation-compensated first-guess value 0.004 is valid or the correct value is higher than that, then they are of the utmost importance. Without (partially) diffuse surface scattering, scattered light is too strongly polarized to be in agreement with the measurements. Even with Lambertian schemes, good agreement requires unrealistically spiky model particle shapes. On the other hand, if the truncation-compensated Im(m) is about 0.001 or smaller, it is possible to obtain good agreement with measurements using realistic particle shapes, as long as at least the surface Lambertian scheme is applied. Rather good fits can also be achieved without using Lambertian schemes, but this requires very spiky shapes.
 The implications of the results for radiative transfer calculations are illustrated in selected test cases. These tests highlight the role of Im(m), but they also show that the results based on the Lorenz-Mie theory can differ significantly from the ray optics results, not only for radiances but also for radiative fluxes. Thus the Lorenz-Mie theory should only be used with considerable caution in radiative transfer applications involving large desert-dust particles.
 The above conclusions depend on the validity of following two assumptions: (1) the exclusion of small particles from simulations does not introduce significant qualitative changes in results; (2) our ad hoc modeling of small-scale surface roughness and internal inhomogeneity is adequate. The former is considered a major assumption, but as shown in section 5, there are good reasons to believe that it can be made. Its quantitative evaluation requires a proper handling of scattering by small nonspherical dust which is carried out in the future. The latter assumption cannot be quantitatively evaluated either, but the Lambertian schemes are intuitively reasonable and perform well: For the first time, good agreement can be achieved between measurements and simulations using realistic particle shapes, and most importantly, this agreement includes all the independent scattering matrix elements. Also, the parameters for the best fits are reasonable: fex is 0.25 for both cases and the value of αex is lower for more absorbing cases. Thus we feel confident that our approach is a valid first approximation to model scattering by natural mineral particles with large size parameter.
 Proper modeling of scattering by mineral aerosol particles is of great importance, as they form the most prominent aerosol class in the Earth's atmosphere. Here we have touched the issue by considering scattering by Saharan particles with large size parameter, but the picture is not the whole until small particles are also modeled properly. Unfortunately, the measurements by Volten et al.  represent a size (parameter) distribution in which a substantial part of scattering comes from both particles below and within the ROA size range. While this complicates the interpretation of the parameters of our modified ROA model, it would also be a problem for fitting the parameters of the scattering model used below the ROA size range. Fitting the free parameters of both models simultaneously would be very complicated. Thus we will concentrate on the small-particle scattering in a different paper and look into possibilities to get measurement data more suitable for such a study. We emphasize that the small-particle scattering dominates in typical weather conditions and its accurate handling is therefore of utmost importance.
 Additional measurements could also help us retrieving more information about the values of the Lambertian parameters relevant for Saharan particles with large size parameter, and perhaps also for the mean value and the distribution of Im(m). It would also be very useful to get scattering matrix measurements very close to the backscattering direction. The ROA model itself can also be further improved. The diffuse scattering should be modeled by physically rigorous means, at least for benchmark purposes. The variability of Im(m) in natural minerals should be taken into account by using a proper distribution for the model Im(m). In addition, the relation between Im(m), particle size, and αex should be established. Thus additional laboratory measurements are important also for the model development.
 We thank Hester Volten from the Free University of Amsterdam for providing us with the data of light scattering measurements of Saharan particles and the corresponding size distribution. We also thank Martti Lehtinen from the University of Helsinki for mineralogical discussions and lending us a microscope for the shape analysis, Juha-Pekka Lunkka from the University of Helsinki for providing us with the sample of Saharan particles, and Kari Lumme from the University of Helsinki for his general contribution in the initial states of this project. Last but not least, we thank the anonymous proofreaders and the referees for their valuable comments.