A case study for feldspar aerosols is conducted to assess the errors introduced by simple model particles in radiance and flux simulations. The spectral radiance field and net flux are computed for a realistic phase function of feldspar aerosols measured in the laboratory at 633 nm. Results are compared to computations with spherical and spheroidal model particles. It is found that the use of spherical model particles introduces large spectral radiance errors at top of atmosphere (TOA) between −6 and 31%. Using a new shape parameterization of spheroids reduces the error range to −1 to 6%. Spherical model particles yield an absolute TOA spectral net flux error of −6.1 mW m−2 nm−1. An equiprobable shape distribution of spheroids results in only minor improvements, but the new shape parameterization yields an error of only −0.8 mW m−2 nm−1. A variation of the refractive index m reveals that the resulting changes in the TOA spectral net flux are slightly smaller than the error caused by assuming the particles to be spherical. However, the uncertainty of m is commonly considered the major error source in aerosol radiative forcing simulations, whereas the use of spherical model particles is often not seriously questioned. This study implies that this notion needs to be reconsidered. Should the relative spectral net flux errors be representative for the entire spectrum, then the use of spherical model particles may be among the major error sources in broadband flux simulations. The new spheroidal shape parameterization can, however, substantially improve the results.
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 Modeling the impact of mineral dust aerosols on radiance is of interest for interpreting remote sensing observations of aerosols, which can provide a better quantification of the spatial and temporal variability of desert dust aerosols in the atmosphere and hence of the impact of dust-like aerosols on climate. Although spherical model particles are still used operationally to a large extent for interpreting optical remote sensing observations of aerosols [Nakajima et al., 1996; Kaufman et al., 1997; Tanré et al., 1997; Torres et al., 1998; Dubovik and King, 2000], it is now widely accepted that they are unsuitable for dust aerosols. Considerable efforts have been expended into quantifying the error in modeling optical properties of nonspherical particles by using spheres. Several studies indicate that model particles as simple as spheroids can reproduce the optical properties of realistic aerosol particles significantly better than spheres [Mishchenko et al., 1997; Kahn et al., 1997; Kahnert et al., 2002a, 2002b; Nousiainen and Vermeulen, 2003; Kahnert, 2004]. Thus remote sensing retrieval algorithms have been developed that are based on using spheroids rather than spheres as model particles [Dubovik et al., 2002]. However, it is still unclear what shape parameterization one should use for spheroidal model particles. Because of a lack of more detailed information, one usually employs an equiprobable shape distribution, but little is known about the appropriateness and the limitations of this approach. Aerosol optical thickness retrievals have recently been reported on the basis of using nonspherical phase functions [von Hoyningen-Huene et al., 2003; Kokhanovski et al., 2004].
 Modeling the impact of mineral dust aerosols on radiative net flux is of particular interest in climate research, since mineral dust aerosols can have a strong direct climate forcing effect in arid regions [Harshvardhan and Cess, 1978; Carlson and Benjamin, 1980; Haywood et al., 1999, 2001]. Increases in mineral aerosol loads due to anthropogenically caused desertification can, on a local scale, even be the strongest anthropogenic climate forcing mechanism in arid regions [Myhre and Stordal, 2001]. Simulations of aerosol climate forcing rates are almost exclusively based on the use of spherical model particles [Myhre and Stordal, 2001; Tegen et al., 1996; Sokolik et al., 1998], and it is often taken for granted that the errors due to the use of Lorentz-Mie theory are negligible in view of the many other error sources affecting aerosol forcing rate simulations. One of those error sources is the uncertainty in the refractive index, which is related to a lack of information about the exact chemical composition of the aerosols and the internal and external mixing of different chemical constituents [Sokolik et al., 2001]. Other error sources are related to incomplete knowledge about the size distribution of aerosol particles and their horizontal and vertical distributions in the atmosphere. A recent study [Myhre and Stordal, 2001] investigated the sensitivity of climate forcing rates of desert dust aerosols to changes in the physical and chemical properties of aerosols and to their vertical distribution in the atmosphere. The authors concluded that the uncertainty of the refractive index and, to a lesser extent, the size distribution are the most important error sources in aerosol forcing rate simulations. Throughout this study, spherical model particles were used for computing aerosol optical properties. In fact, indications have been found that size-shape distributions of spheroids and equivalent size distributions of spheres have comparable single-scattering albedos and asymmetry parameters [Mishchenko et al., 1995], thus suggesting that spheres may indeed be suitable for climate simulations. However, a recent modeling investigation [Kahnert and Kylling, 2004] based on using more complex model particles, more variable shape distributions, and detailed radiative transfer simulations arrived at different conclusions. It was found that spheres may cause errors in spectral net flux simulations that are roughly on the same order of magnitude as those errors caused by the uncertainty in the refractive index, which implies that the use of Lorentz-Mie theory may be among the major error sources in climate forcing simulations. In another investigation [von Hoyningen-Huene and Posse, 1997], combined spectral aerosol optical thickness and sky brightness observations were inverted by using spherical and spheroidal particles. It was found that the use of spherical model particles can lead to large errors for retrieving climate relevant quantities from the observations.
 To the best of our knowledge, no indications for the inappropriateness of spherical model particles for flux simulations have yet been found by using directly measured (rather than retrieved or simulated) single-scattering optical properties of aerosols. This is the approach that is taken in this paper, building on a previous study [Nousiainen et al., 2005], in which single-scattering modeling of irregularly shaped feldspar particles using simplified particle geometries was investigated. In the present article, we present detailed radiative transfer simulations based on using a realistic phase function of feldspar aerosols measured in the laboratory [Volten et al., 2001]. The radiance and net flux results of this simulation serve as a reference case for comparison with results obtained by using spherical and spheroidal model particles. We investigate two different shape parameterizations of spheroids, namely, the commonly used equiprobable parameterization and a parameterization that puts more weight on more aspherical spheroids and little weight on mildly aspherical spheroids. The suitability of using a simple Henyey-Greenstein phase function for radiance and flux simulations is also investigated. The radiance and flux errors are quantified, and the net flux errors are compared to the error due to the uncertainty in the refractive index of the aerosols. In Section 2, the reference case and the electromagnetic scattering computations for computing the optical properties of the model particles are described. In Section 3, the approach for performing the radiative transfer simulations is explained, and results are presented for the spectral radiance computations and spectral net flux computations. Concluding remarks are given in Section 4.
2. Single-Scattering Optical Properties
 The necessary input parameters to radiative transfer simulations are the single-scattering optical properties of the particles, in particular the single-scattering albedo ω and the phase function p(Θ), where Θ denotes the scattering angle in the scattering plane. In this study, a laboratory-measured phase function of feldspar aerosols at 633 nm is used as a reference case. The measured phase function only covers scattering angles from 5° to 173°, and the single-scattering albedo has not been measured in a laboratory at all. To make the measurements useful to us, p(Θ) needs to be extrapolated to the full 180° span, and ω needs to be found. To this end, light-scattering simulations are used in the following two-step procedure.
Liu et al.  suggested to combine the measured phase function with the Lorentz-Mie solution for spherical particles of an equivalent size distribution in the forward scattering direction. This approach works well since the forward peak of the phase function is strongly size sensitive but little sensitive to shape. In the backscattering direction, one can perform an extrapolation of the measurements. It is noted that the backscattering direction does not affect the normalization significantly. Subsequently the phase function can be normalized.
 In the second step a postprocessing of the normalized phase function is performed, in which the phase function is fitted with a suitable ensemble of spheroids. This step builds on an earlier study [Nousiainen et al., 2005], in which we have obtained a best fit pfit(Θ) of the measured feldspar phase function by optimizing a size-shape distribution of spheroidal model particles. The single-scattering computations for the spheroids were performed with Mishchenko's T matrix code. Following Nousiainen et al.  the refractive index m is assumed to be 1.5 + 0.001i. This choice is based on the information available from Klein and Hurlbut  and Egan and Hilgeman . Further, following Nousiainen and Vermeulen , the size distribution n(r) of the feldspar particles was assumed to be lognormal, i.e.
with a mean radius R = 0.167 μm and geometric standard deviation σ = 2.32. This size distribution was used in the single-scattering computations for the spheroids. The spheroids' shape distribution was optimized with an automatic fitting routine [Kahnert, 2004] by minimizing the least squares error between the measurement points with given measurement errors and the ensemble-averaged phase function of the size-shape distribution of spheroids. Unlike the fitting procedure described by Kahnert  we fitted the phase function p(Θ) rather than its logarithm log p(Θ). The result was in excellent agreement with the measurement points over the entire range of measured scattering angles.
 This best fit phase function based on a distribution of spheroids is used to extrapolate the measured phase function in the scattering-angle intervals Θ ∈ [0°,5°) and Θ ∈ (173°,180°] to obtain the actual reference phase function. We normalize the resulting phase function pref such that
The normalized measurement points pref are represented by the circles in Figure 1. pref is decomposed into its Legendre coefficients, which serve as input to the radiative transfer solver.
 The single-scattering albedo is taken directly from the best fit simulation. Since ω is usually rather insensitive to particle shape [Mishchenko et al., 1995; Kahnert, 2004], it is reasonable to assume that ω obtained from the best fit simulation is a very good approximation for the actual (unknown) ω of the feldspar sample (provided that the refractive index is known with sufficient accuracy).
 In practical applications, especially when only fluxes are needed, such as in climate studies, one often uses the asymmetry parameter g in conjunction with the Henyey-Greenstein parameterization of the phase function rather than the exact phase function. The asymmetry parameter for the reference case gref is simply computed from the reference phase function according to
The results for ωref and gref are shown in Table 1.
Table 1. Scalar Optical Properties for the Reference Case and the Ensembles of Model Particles
Spheroids, n = 0
Spheroids, n = 3
 To study the suitability of simple model particles for radiance and flux simulations, the aerosol phase function is simulated by using simple model particles such as spheres and spheroids. For the spherical model particles, Lorentz-Mie computations were carried out for the size distribution given in equation (1) and for m = 1.5 + 0.001i. This yields the phase function, single-scattering albedo, and asymmetry parameter for the spherical model particles. The phase function of the spheres is represented by the solid line in Figure 1. The single-scattering albedo and the asymmetry parameter are given in Table 1.
 Unlike spheres, spheroidal model shapes require defining a shape distribution. As mentioned above, we have already determined a shape distribution that yields an optimum fit of the measured feldspar phase function, which served us to extrapolate the measurement points and to obtain a good estimate of ωref. However, one usually does not have sufficient information to retrieve a best fit shape distribution from observations. It would therefore be unrealistic to use the optimized shape distribution as a test case in our study. In applications one would have to make an a priori assumption about the shape distribution. The most commonly used ansatz is to assume an equiprobable distribution h0(ξ) of shape parameters ξ within an interval [ξi, ξf] given by
where a denotes the diameter of the spheroid along its main symmetry axis, and b denotes its maximum diameter perpendicular to the main symmetry axis, i.e., along any of its dihedral symmetry axes. We performed single-scattering computations for spheroids by using the size distribution given in equation (1), by using m = 1.5 + 0.001i, and by assuming the shape distribution given in equation (4). For all sizes the shape parameter (aspect ratio) was varied between ξ = −1.6 (a/b = 0.385) and ξ = 1.6 (a/b = 2.6). The results for the single-scattering optical properties are shown in Figure 1 (dashed line) and in Table 1.
Nousiainen et al.  tested different shape distribution parameterizations using the same feldspar sample as a reference. These parameterizations have a form
where the normalization constant Nn is determined such that the integral over hn(ξ) yields unity. Clearly, the equiprobable shape distribution in equation (4) is contained in equation (6) as the special case n = 0. It was found [Nousiainen et al., 2005] that the parameter choice n = 3 gives the best representation of the feldspar phase function. In the present study, we therefore test, in addition to the equiprobable ansatz, the spheroidal shape parameterization given in equation (6) with n = 3. For brevity, we will henceforth refer to this as the “∣ξ3∣” shape distribution. The single-scattering optical properties for this ensemble of spheroids are presented in Figure 1 (dash-dotted line) and Table 1.
 We note that the single-scattering computations for spheroids can become numerically unstable for elongated spheroids with large size parameters. Because of these technical limitations we had to restrict our investigation to a wavelength of 633 nm, even though phase function measurements exist [Volten et al., 2001] for feldspar particles at 442 nm.
3. Radiative Transfer Simulations
 To assess the errors in radiance and flux simulations due to representing aerosols by simple model particles we have to run radiative transfer simulations. The single-scattering optical properties for the reference case and for the model particles are used as input to the radiative transfer model. As a radiative transfer driver we use uvspec [Kylling et al., 1998; Mayer et al., 1997] which is part of the libRadtran package (http://www.libradtran.org). The driver sets up the input parameters for the radiative transfer solver based on the atmospheric profile and the surface properties. DISORT [Stamnes et al., 1988] is used as a radiative transfer solver to compute radiance and flux from the radiative transfer equation. Molecular scattering is accounted for by assuming a standard tropical atmosphere with 49 layers [Anderson et al., 1986]. All radiative transfer simulations are carried out at the wavelength of 633 nm. The aerosol optical depth profile used is shown in Table 2. The profile is based on aerosol scattering coefficient measurements under typical background situations [Myhre et al., 2003] and was also used by Kahnert and Kylling . A “typical background situation” refers to dust concentrations in or near desert areas under calm wind conditions, i.e., in the absence of dust storm events. The cumulative optical depth of our aerosol profile is 0.5, corresponding well with what could be expected from typical number densities and size distributions under background situations as reported by d'Almeida . A surface albedo of 0.1 is used as a typical sea surface albedo, as most remote sensing of aerosols involve methods that work best over ocean surfaces. The climatic impact of mineral dust is largest over oceans due to their high single-scattering albedo in combination with the low ocean surface albedo. It is also noted that the outflow regions of the Sahara and the Gobi deserts lie over the ocean.
Table 2. Vertical Profile of the Optical Depth of Dust-like Aerosolsa
Extinction Optical Depth
The same phase function and the same single-scattering albedo were used for all layers.
 A summary of the reference case and the model shape ensembles tested in the radiance and flux computations are provided in Table 3. Model 1 is based on using spherical model particles. Model 2 employs an equiprobable shape distribution of spheroids, whereas model 3 employs a ∣ξ3∣ shape distribution of spheroids. The reference case and models 1–3 are based on computations with the full phase functions. The last four cases are based on using the Henyey-Greenstein parameterization of the phase function along with the asymmetry parameter obtained from the reference case and from models 1–3, respectively. The Henyey-Greenstein approach is commonly taken in climate simulations, since it is computationally much less demanding. The same extinction optical depth profile is used in each case (see Table 2), so each case differs from the others only in the phase function and in the single-scattering albedo. Both p(Θ) and ω are assumed to be constant with altitude in each case. In the Henyey-Greenstein test cases the asymmetry parameter g is assumed to be constant with altitude.
Table 3. Cases Considered in the Intercomparison of Computed Radiances and Fluxes
measured phase function of feldspar aerosols
spherical particles with equivalent size distribution, m = 1.5 + 0.001i
equiprobable shape distribution of spheroids, equivalent sizes, m = 1.5 + 0.001i
∣ξ∣3 shape distribution of spheroids, equivalent sizes, m = 1.5 + 0.001i
Henyey-Greenstein phase function with the exact g value from the reference case
as Henyey-Greenstein ref but with g from model 1
as Henyey-Greenstein ref but with g from model 2
as Henyey-Greenstein ref but with g from model 3
3.1. Angular Distribution of Radiance Field
 The spectral radiance in the reference case where the Sun is at (θ0, ϕ0) = (20°, 0°) is presented in Figure 2. Figure 2 (right) shows a polar plot of the downwelling spectral radiance Iλ,aerosol−(θ, ϕ) at the bottom of the atmosphere (BOA) as a function of the zenith angle θ and the azimuth angle ϕ. One can clearly see a bright region around the direction of incoming sunlight, which is due to the pronounced and relatively broad forward peak in the phase function of the feldspar aerosols (compare Figure 1). Figure 2 (left) presents the upwelling spectral radiance Iλ,aerosol+(θ, ϕ) at the top of the atmosphere (TOA) as a function of the nadir angle θ and the azimuth angle ϕ.
Figure 3 shows the relative differences of the downwelling BOA spectral radiance between the reference case and an equivalent size distribution of spheres (Figure 3, top left), a spheroidal size-shape distribution with equivalent sizes and with an equiprobable shape distribution ∝ ∣ξ∣0 (Figure 3, top right), a corresponding ensemble of spheroids with a ∣ξ∣3 shape distribution (Figure 3, bottom left), and for a Henyey-Greenstein phase function using the exact value of the asymmetry parameter gref and of the single-scattering albedo ωref from the reference case (Figure 3, bottom left, denoted by “Henyey-Greenstein ref” in Tables 3–6). Table 4 shows the maximum and minimum values of the errors δIλ− = (Iλ,model− − Iλ,aerosol−)/Iλ,aerosol− 100%.
Table 4. Maximum and Minimum Percent Errors δIλ,min− and δIλ,max− Between the Downwelling Spectral Radiance for the Various Cases of Particle Models and the Reference Case at BOA
Table 5. Maximum and Minimum Relative Differences δImin+ and δImax+ Between the Upwelling Spectral Radiance for the Various Cases of Particle Models and the Reference Case at TOA
Table 6. Yearly Averaged Spectral Net Flux ΔFλ at TOA and Absolute Difference Between the Various Models and the Reference Casea
 Clearly, the Henyey-Greenstein phase function yields the largest positive and negative errors. The forward peak of the phase function is underestimated by the Henyey-Greenstein parameterization, which results in large negative errors down to −59% around the solar zenith angle. At angles farther away from the forward scattering direction, the Henyey-Greenstein phase function overestimates the reference phase function of the feldspar aerosols, resulting in large positive errors in the spectral net flux simulations up to almost 16%. These results are by no means surprising. The Henyey-Greenstein parameterization is mostly used in flux simulations. The idea is that the positive and negative errors cancel when integrating over all solid angles, provided that one uses the correct asymmetry parameter. Thus one only expects this parameterization to yield a correct angular average of the scattered radiative energy, but not a correct angular distribution thereof. To what extent this parameterization really can produce accurate flux results will be investigated in section 3.2.
 The spherical model particles yield an error range between −12 and 8%. The ∣ξ∣0 shape distribution of spheroids yields errors between −6 and 3%, and the ∣ξ∣3 shape distribution of spheroids deviates from the reference results by −5 to 7%. Thus the error range observed for spheres is larger than that observed for the spheroids by roughly a factor of 2.
Figure 4 shows the corresponding relative errors for the upwelling spectral radiance at the TOA. Maximum and minimum relative errors are given in Table 5. The Henyey-Greenstein results differ from the reference case by between −10 and 2%, which is considerably less than the corresponding range at BOA. Spheres yield by far the largest error range at the TOA. The large errors around the solar nadir direction of up to 31% cover a relatively broad range of nadir angles. They are caused by the strongly enhanced backscattering that one always observes for spherical model particles but rarely for real nonspherical aerosols (compare Figure 1). At nadir directions farther away from the solar nadir angle negative errors down to −6% cover a large band of solid angles; this is caused by the notoriously low side scattering of spherical model particles that is highly untypical for nonspherical aerosols. Again, this can clearly be seen in Figure 1. The ∣ξ∣0 (equiprobable) shape distribution of spheroids causes errors in a range between −2 and 11%, constituting a reduction of error by a factor of 2 as compared to spheres. The ∣ξ∣3 shape distribution of spheroids causes errors that lie in the range between −1 and 6%, a factor of 2 smaller than the corresponding error range of the ∣ξ∣0 shape distribution, and a factor of 4 smaller than the error range of spheres.
3.2. Net Flux at TOA
 As a measure for the direct radiative climate forcing effect of feldspar aerosols we consider the spectral net flux ΔFλ = Fλ− − Fλ+ at TOA. Fλ− denotes the downwelling spectral flux at TOA, which is simply the incoming solar spectral flux. At the wavelength λ = 633 nm, the average solar input is = Fλ− = 1695.7080 mW m−2 nm−1. This value has to be corrected for each day of the year due to the eccentricity of the Earth's orbit. This correction is done automatically by uvspec. Fλ+ represents the upwelling hemispherical spectral flux at TOA, which is obtained by integrating the upwelling spectral radiance at TOA over all solid angles of the upper hemisphere.
 For a geographical latitude of 20°, spectral net flux computations were carried out for all cases listed in Table 3. For simplicity, constant dust concentration and clear-sky conditions are assumed. To compute a yearly averaged spectral net flux, we first computed 24-hour spectral net flux averages for each day of the year by performing a 24-hour time integration over all solar zenith angles for each day, where the solar input was corrected for the varying Earth-Sun distance. The 24-hour averages were subsequently time-integrated over all days of the year. The daily average computations can be simplified by integrating only from sunrise to noon, multiplying the result by 2 (to account for the second half of the day), and dividing by the total length of the day. The yearly average is computed with sufficient precision by only computing spectral net fluxes for every other day, since the variation of the daily averaged spectral net flux is rather smooth.
Table 6 presents the flux computation results for the reference case and the various model particles. The second column shows the yearly averaged spectral net fluxes, and the third column shows the absolute differences between the different models and the reference case. It is important to note that even though the relative differences between the reference case and the different models may seem small, the absolute difference can be quite substantial. Also, it is, after all, the absolute value computed for the net flux that is vital for assessing the climate forcing effect of aerosols. Thus it is also the absolute value of the flux errors that needs to be compared with flux errors due to other uncertainties.
 We clearly see that spherical model particles (model 1) cause the largest yearly averaged spectral net flux errors (−6.1 mW m−2 nm−1). Interestingly enough, the use of an equiprobable shape distribution of spheroids (model 2) still yields an error of −4.7 mW m−2 nm−1, improving the results obtained with spheres only marginally. On the other hand, using a ∣ξ∣3 shape distribution of spheroids (model 3) yields an average net flux error of only −0.8 mW m−2 nm−1, which is a substantial improvement over the results obtained with spheres.
 Actual climate models usually employ simple radiative transfer models with simple parameterizations for the phase function, such as the Henyey-Greenstein parameterization. The last four rows in Table 6 show the results we obtain in the spectral net flux simulations when using a Henyey-Greenstein phase function rather than the exact phase function for each particle type. When using the exact value for the asymmetry parameter from the reference case (g = 0.729) in the Henyey-Greenstein parameterization (denoted by “Henyey-Greenstein ref” in Tables 3–6), the resulting averaged spectral net flux deviates from the reference case by only −0.2 mW m−2nm−1. This confirms that the Henyey-Greenstein phase function is in principle accurate enough for net flux computations. The reason is that this parameterization is based on the asymmetry parameter, which expresses how much radiation is scattered in the forward directions in relation to how much is scattered in the backward directions. Thus when integrating the spectral radiance computed with the Henyey-Greenstein phase function over all downwelling and upwelling directions the errors in the angular distribution of the spectral radiance largely cancel out.
 However, in a realistic situation we do not know the exact value of the asymmetry parameter. Often one simulates the asymmetry parameter by using Lorentz-Mie theory. To assess the error one can expect from this approach, we used the results we obtained for spherical particles (g = 0.697, model 1), and performed radiative transfer simulations with the Henyey-Greenstein parameterization (Henyey-Greenstein 1). As can be seen in Table 6, this results in an error for the spectral net flux of −6.1 mW m−2 nm−1. When the asymmetry parameter computed for an equiprobable shape distribution of spheroids (g = 0.705, model 2) is used in the Henyey-Greenstein parameterization (Henyey-Greenstein 2), the resulting net flux error of −4.8 mW m−2 nm−1 is only somewhat smaller than that obtained with spheres. The reason is that the equiprobable shape distribution of spheroids yields an asymmetry parameter that is clearly closer to that computed with spheres than that obtained for the feldspar particles. In this context it is interesting to note that Mishchenko et al.  studied the asymmetry parameter and single-scattering albedo of a size distribution of spheres and an equivalent size distribution of spheroids with an equiprobable shape distribution, and found that the difference in ω and g between these two particle models is rather small. The results of our study confirm those findings, and we are now able to go one step further and to assess to what extent an equiprobable shape distribution of spheroids is suitable to reproduce the spectral net flux in a realistic situation involving natural aerosol particles. We can conclude that the Henyey-Greenstein radiative transfer simulations for a ∣ξ∣0 shape distribution of spheroids yields a relatively large error that is only slightly smaller than that obtained with spheres.
 On the other hand, our new ∣ξ∣3 shape distribution of spheroids (model 3) yields an asymmetry parameter of g = 0.725, which is close to the exact value of the reference case. When using this g value in the Henyey-Greenstein parameterization (Henyey-Greenstein 3), the resulting spectral net flux differs from that obtained for the reference case by only −1.0 mW m−2 nm−1, which is close to the error we obtain when using the exact phase function for model 3. This result constitutes a major improvement over the use of spheres and equiprobable shape distributions of spheroids.
 The results of this investigation are based on considering only one wavelength, which makes it difficult to assess what errors one can expect in broadband net flux simulations. Unfortunately, we do not know the variation of the feldspar phase function, the single-scattering albedo, or the feldspar refractive index with wavelength. In the absence of such information we can, at best, obtain a rough zeroth-order guess of the broadband net flux errors by assuming that the relative error in relation to the solar spectral flux is constant for each model particle over the entire spectral range. This estimate may actually be rather accurate, since errors from using simplified shapes in scattering simulations are likely to be larger for shorter wavelengths (and hence larger size parameters) and smaller for longer wavelengths (smaller size parameters). Since 40% of the solar energy lies at wavelengths shorter than 633 nm and 60% at longer wavelengths, the net result may actually be quite close to our zeroth-order guess. Using the solar broadband flux of 1373 W m−2, this would yield an estimate for the broadband flux error of −4.9 W m−2 (model 1), −3.8 W m−2 (model 2), −0.6 W m−2 (model 3), −0.2 W m−2 (Henyey-Greenstein ref), −4.9 W m−2 (Henyey-Greenstein 1), −3.9 W m−2 (Henyey-Greenstein 2), and −0.8 W m−2 (Henyey-Greenstein 3). Thus the new shape parameterization (model 3) reduces the error by almost an order of magnitude compared to the error based on using spheres (model 1). We reiterate that these figures are based on assuming cloud-free and background conditions throughout the year, and that they constitute local and not global estimates. We also emphasize that these figures contain a large degree of uncertainty, but they do illustrate how serious the broadband flux errors may be due to using spherical model particles or even an equiprobable shape distribution of spheroids. They also illustrate that spheroidal model particles with the new ∣ξ∣3 shape distribution are likely to yield significantly improved results.
 Considering the difficulties in assessing the broadband net flux errors more accurately, we shall take a different and more systematic approach to put our spectral net flux error estimates into perspective. As mentioned earlier, the uncertainty in the refractive index m is commonly considered the major error source in broadband net flux simulations and thus in quantifying the direct climate forcing effect of dust aerosols [Myhre and Stordal, 2001]. Indeed, for the measured feldspar sample we do not know the exact value of m. We merely know that the real part of the refractive index Re(m) is expected to be between 1.5 and 1.6 [Klein and Hurlbut, 1993], whereas the imaginary part Im(m) is probably lower than 0.001 [Egan and Hilgeman, 1979]. Both the phase function and the single-scattering albedo are sensitive to changes in m. We have used a refractive index of m = 1.5 + 0.001i in all our computations so far. To obtain an estimate of how much the spectral net flux results vary because of the uncertainty in the imaginary part Im(m), we run Lorentz-Mie computations for m = 1.5 + 0i (model 1a), use the resulting phase function and single-scattering albedo as input to the radiative transfer model, perform the averaging over all times of the day and all days of the year, and compare the computed averaged spectral net flux to that obtained by the Lorentz-Mie calculations based on using the reference refractive index (model 1). Likewise, we assess the spectral net flux error due to the uncertainty in Re(m) by performing corresponding Lorentz-Mie computations and radiative transfer simulations for a refractive index of m = 1.6 + 0.001i.
 The results are presented in Table 7. The uncertainty in Im(m) (model 1a) leads to an error in the averaged spectral net flux of −5.8 mW m−2 nm−1. This error is comparable to, and even slightly smaller than the error we obtain from comparing model 1 or Henyey-Greenstein 1 to the reference case of feldspar aerosols (see Table 6). This strongly indicates that using spherical model particles in net flux simulations can lead to errors that are on the same order of magnitude as the error due to the uncertainty in the imaginary part of the refractive index. On the other hand the uncertainty in Re(m) (model 1b) results in an averaged spectral net flux that deviates by only −0.6 mW m−2 nm−1 from the results obtained with the same spherical model particle (model 1) and the reference refractive index. Thus the uncertainty in the real part of the refractive index seems to result in net flux errors that are negligible in comparison to the errors due to using inappropriate model particles, such as spheres. These results agree well with those reported by Kahnert and Kylling .
Table 7. Comparison of Yearly Averaged Spectral Net Fluxes ΔFλ at TOA Computed With Spherical Model Particles and Different Refractive Indices
Net Flux ΔFλ, mW m−2 nm−1
Absolute Net Flux Error Compared to Model 1, mW m−2 nm−1
Spheres, m = 1.5 + 0.001i (model 1)
Spheres, m = 1.5 + 0i (model 1a)
Spheres, m = 1.6 + 0.001i (model 1b)
 It is noted that the error related to the model particle shape in the single-scattering albedo ω has a negligible effect on the net flux errors. One can see in Table 1 that the use of spherical model particles results in an ω error of only 0.2%. In a similar study based on the same optical depth profile as the one in Table 2 [Kahnert and Kylling, 2004] it was found that an ω error of 1% makes a contribution to the net flux error that is a factor of 6 smaller than the error related to the misrepresentation of the phase function.
4. Discussion and Conclusions
 We have investigated the errors in spectral radiance and spectral flux simulations due to representing natural feldspar aerosols by simplified model particles, namely spheres and spheroids. The radiance simulations showed that the use of spheres causes considerably larger errors than the use of spheroids. This effect is particularly pronounced at TOA, which is most relevant for satellite remote sensing, and which also provides the basis for TOA net flux computations. The limitations of spherical model particles for radiance simulations come as no surprise, since the phase function of spheres strongly deviates from that of the reference case (see Figure 1), and since single-scattering strongly dominates at the low optical depths encountered under background situations. However, we also obtained an interesting new result with potentially high relevance for the development of future remote sensing retrieval algorithms based on spheroidal particles. By comparing two different parameterizations of the spheroidal shape distribution, we found that the ∣ξ∣3 shape distribution yields quite accurate results with errors that are reduced by roughly a factor of 2 as compared to the commonly used ∣ξ∣0 (equiprobable) shape distribution of spheroids. Simulations based on spherical model particles were found to give TOA spectral radiance results with an error range that is larger by a factor of 4 than those results obtained with the ∣ξ∣3 spheroidal shape distribution.
 The superior performance of the ∣ξ∣3 shape distribution can be attributed to the fact that it places more weight on more aspherical spheroids and little weight on mildly aspherical particles. Since the former have phase functions more closely resembling those of realistic nonspherical aerosol particles than the latter, this results in a better representation of the aerosol phase function and the asymmetry parameter by the model as compared to the equiprobable ansatz. Laboratory measurements at 442 nm and 633 nm showed that different aerosol samples have qualitatively similar phase functions [Muñoz et al., 2000, 2001; Volten et al., 2001; Muñoz et al., 2002]. Hence the good performance of the ∣ξ∣3 spheroidal shape model is most likely not coincidental. This model will also work well for ensembles of nonspherical aerosols other than feldspar particles, although it will be technically more difficult to perform electromagnetic scattering simulations for samples with larger particle sizes.
 By far the most important result of this study is the observation that the use of spherical model particles proves highly inadequate for simulating spectral net fluxes and thus for computing the direct climate forcing effect of feldspar aerosols. As computations of climate forcing rates of aerosols are almost exclusively based on the use of Lorentz-Mie theory, the implications of this finding may be far reaching and can hardly be overemphasized. The inappropriateness of Lorentz-Mie theory for net flux simulations has already been observed by Kahnert and Kylling , which was a pure modeling study. To the best of our knowledge, our results confirm for the first time the findings of Kahnert and Kylling  by using a directly measured aerosol phase function as a reference case.
 Still, more studies are needed to further substantiate this finding. In particular, there is not enough information to accurately estimate the broadband net flux error. If the relative errors we observed should turn out to be representative for the whole spectral range, then the local broadband net flux error due to using spherical model particles could be on the order of −5 W m−2. Hence the effect is potentially quite significant and calls for further studies.
 To assess the relative importance of the net flux error estimates we compared flux simulations based on Lorentz-Mie computations performed with different refractive indices. We found that the spectral net flux error due to the uncertainty in the refractive index is comparable to, and even slightly smaller than the error due to using spherical model particles. Again, this conclusion agrees well with those recently reported [Kahnert and Kylling, 2004]. As the uncertainty in the refractive index is considered to be the main source of error in assessing the direct climate forcing effect of dust aerosols [Myhre and Stordal, 2001], we can conclude that the use of Lorentz-Mie theory may be among the major error sources in climate forcing simulations of dust aerosols, even though this error source is usually highly underrated and often even completely disregarded. Large efforts are targeted at obtaining better estimates of the refractive index of dust aerosols. When more information in this regard becomes available, the errors related to the use of spherical model particles will gain even more relative importance.
 Further, we found that representing the feldspar aerosols by an equiprobable shape distribution of spheroids yields spectral net flux errors that are almost as large as those obtained by using spheres. However, by using a ∣ξ∣3 spheroidal shape distribution the spectral net flux errors are reduced by almost an order of magnitude. Again, this can be explained by the fact that the ∣ξ∣3 shape distribution selects out those particles that are more likely to give a good representation of the phase function (and thus the asymmetry parameter) of realistic nonspherical particles. Thus the ∣ξ∣3 spheroidal shape distribution is likely to yield results in both remote sensing retrieval algorithms and in climate forcing simulations that are superior to those obtained with an equiprobable spheroidal shape distribution. The use of this shape distribution involves electromagnetic scattering computations for exactly the same types of spheroids as the use of an equiprobable shape distribution, we simply apply different weights in the ensemble average over particle shapes. Thus the new shape parameterization requires essentially no extra work as compared to the equiprobable ensemble ansatz.
 We also observed that it is not necessary to use the exact phase function in net flux simulations. One can obtain accurate results by using the popular Henyey-Greenstein parameterization, provided that the asymmetry parameter used is accurate. Again, the use of Lorentz-Mie simulations for computing the asymmetry parameter results in a high net flux error. Little improvement can be achieved by using an equiprobable shape distribution of spheroids. However, using a ∣ξ∣3 spheroidal shape distribution yields considerably more accurate results. Our recommendation for net flux computations would therefore be to simulate the single-scattering albedo and the asymmetry parameter with a ∣ξ∣3 spheroidal shape distribution and to use the result in a radiative transfer model in conjunction with the Henyey-Greenstein parameterization of the phase function.
 We believe that the net flux errors obtained here constitute a rather conservative estimate. The main reason is that we performed the annual net flux average by only varying the solar input according to the day of the year, while assuming a constant aerosol size distribution typical for background situations over the entire year. Thus we completely disregarded the frequent occurrence of dust storm events that would shift the size distribution toward considerably larger particles. However, for larger size parameters the misrepresentation of the true phase function and the asymmetry parameter by the spherical model particles can be expected to be even more severe, which is likely to yield even higher errors in the spectral net flux simulations as those estimated in our study.
 We make a final comment on the representativeness of our results. Considering the similarity of phase functions of different aerosol types of comparable sizes it is likely that the misrepresentation of the aerosol phase function by spheres will lead to similar errors when conducting net flux simulations for other nonspherical aerosol types. We also note that the conclusions reached by Kahnert and Kylling , although based on more strongly absorbing particles and even smaller sizes than in this study, agree well with our results. Thus it is quite likely that our findings are by no means limited to the special case of feldspar aerosols. However, the findings of this study should be further validated, in particular by conducting similar investigations at other wavelengths. Currently, only phase functions measured at 633 nm and at 442 nm are readily available (http://www.astro.uva.nl/scatter). Thus more data on aerosol phase functions at different wavelengths are needed. Investigations at shorter wavelengths will, due to the larger size parameters, also place higher demands on electromagnetic scattering codes.
 Hester Volten is gratefully acknowledged for making her aerosol phase function data available. Funding for this work was partially provided by the Norwegian Research Council within the MACESIZ project.