Retrieval of aerosol single-scattering properties from diffuse and direct irradiances: Numerical studies

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi:10.1029/2007jd009239, 2008 Retrieval of aerosol single-scattering properties from diffuse and direct irradiances: Numerical studies Rei Kudo, 1 Akihiro Uchiyama, 1 Akihiro Yamazaki, 1 Eriko Kobayashi, 1 and Tomoaki Nishizawa 2 Received 30 July 2007; revised 29 November 2007; accepted 5 February 2008; published 13 May 2008. [1] Two different methods have been developed to retrieve aerosol single-scattering albedo w 0 (l) and asymmetry factor g(l) from the direct and diffuse spectral irradiances measured at the surface. One is the direct method, which assumes the Henyey-Greenstein phase function and retrieves w 0 (l) and g(l) directly from the irradiances. The other is the indirect method, which calculates optimum w 0 (l) and g(l) after roughly retrieving the size distribution and refractive index from the irradiances. Their retrieval accuracies were evaluated from intensive sensitivity tests for water-soluble, dust, and biomass-burning models. The retrieval accuracies of w 0 (l) and g(l) were 0.01 to 0.06 and 0.02 to 0.13 for the direct method, and 0.00 to 0.03 and 0.00 to 0.06 for the indirect method. Citation: Kudo, R., A. Uchiyama, A. Yamazaki, E. Kobayashi, and T. Nishizawa (2008), Retrieval of aerosol single-scattering properties from diffuse and direct irradiances: Numerical studies, J. Geophys. Res., 113,, doi:10.1029/2007jd009239. 1. Introduction [2] Aerosol particles have a major impact on the radiative balance of the Earth-atmosphere system. However, significant uncertainties remain regarding the magnitude and the sign of aerosol radiative forcing, due to incomplete knowledge of aerosol optical properties and their strong temporal and spatial variations [e.g., Haywood et al., 1997; Ramanathan et al., 2001; Yu et al., 2004]. Although ground-based aerosol remote sensing does not provide global coverage, it does continuously provide the aerosol optical properties in the total atmospheric column [Dubovik et al., 2002]. Therefore various retrieval methods of the optical properties from ground-based solar radiation measurements have been studied. [3] Modeling the effect of aerosol optical properties on atmospheric radiation requires the following aerosol singlescattering properties: aerosol optical thickness (loading), phase function (angular dependence of light scattering), and single-scattering albedo (ratio of scattering to scattering + absorption). Single-scattering albedo influences the sign (cooling/heating) of aerosol radiative forcing, while optical thickness and the asymmetry of the phase function (asymmetry factor) influence its magnitude [Hansen et al., 1997]. [4] Recently, some retrieval methods from the combined measurement of the direct irradiance and the angular distribution of sky radiances have been developed. The methods of Nakajima et al. [1996], Dubovik and King [2000], and Dubovik et al. [2000] retrieve the detailed size distribution 1 Meteorological Research Institute, Japan Meteorological Agency, Tsukuba-shi, Ibaraki-ken, Japan. 2 National Institute for Environmental Studies, Tsukuba-shi, Ibaraki-ken, Japan. Copyright 2008 by the American Geophysical Union. 0148-0227/08/2007JD009239 of the fine and coarse aerosol particles and the refractive index at some specific wavelengths. Size distribution and refractive index are important parameters of aerosol particles to calculate the aerosol single-scattering properties. However, Yang and Gordon [1998] neglected the size distribution and refractive index of aerosol particles, and they retrieved optical thickness, phase function, and singlescattering albedo directly from the direct irradiance and angular distribution of sky radiances. Their direct retrieval is not influenced by the shape of aerosol particles. [5] The combined measurements of the direct and diffuse spectral irradiances have been also used to retrieve aerosol optical properties. Using the dependency of the irradiance ratio on solar zenith angle, King and Herman [1979] and King [1979] retrieved the imaginary part of the refractive index and the surface albedo from the ratio of the diffuse irradiance to the direct irradiance. The method of Kassianov et al. [2005] retrieves the mean particle radius, total number of particles for an assumed size distribution, and the imaginary part of the refractive index from the direct and diffuse spectral irradiances at six wavelengths measured by a Multi Filter Rotating Shadow-band Radiometer (MFRSR). [6] The purpose of this work is to develop methods to retrieve aerosol single-scattering properties from direct and diffuse spectral irradiances. Since the diffuse spectral irradiance does not contain the angular distribution of light scattering, it is difficult to exactly retrieve complicated parameters such as size distribution and phase function. Therefore we focused on the retrieval of the aerosol singlescattering albedo and asymmetry factor, and we developed two different methods. One method (the direct method) assumes the Henyey-Greenstein phase function and retrieves single-scattering albedo and asymmetry factor directly from the direct and diffuse spectral irradiances. The other method (the indirect method) calculates the 1of18

optimum single-scattering albedo and asymmetry factor, after roughly retrieving the size distribution and the refractive index from the direct and diffuse spectral irradiances. The difference between the two methods is that the indirect method uses the size distribution and refractive index and the direct method does not. These methods were designed for the direct and diffuse spectral irradiances at seven wavelengths (l = 340, 380, 400, 500, 675, 870, and 1020 nm) by the direct and diffuse spectral radiometers of the Meteorological Research Institute (MRI), Japan. [7] Section 2 describes the detailed techniques of the two methods. Section 3 discusses numerical sensitivity tests for simulation data, performed to evaluate the retrieval accuracies of the two methods. Section 4 summarizes the results. 2. Method 2.1. Direct Method [8] The direct method directly retrieves aerosol singlescattering albedo w 0 (l) and asymmetry factor g(l) from the direct and diffuse spectral irradiances. This direct retrieval method neglects the shape, size distribution, and refractive index of aerosol particles. Furthermore, because size distribution is neglected, the direct method can retrieve w 0 (l) and g(l) from the measurements at one wavelength. 2.1.1. Forward Modeling [9] The measurements by the direct and diffuse radiometers of MRI are the direct and diffuse spectral irradiances at seven wavelengths (340, 380, 400, 500, 675, 870, and 1020 nm). The measured spectral irradiances are used as ratios of the diffuse irradiances to the direct irradiances: F obs ðlþ ¼ Fobs dif ðlþ F obs ðlþ ; where F obs dif (l) and F obs dir (l) are the diffuse and direct spectral irradiances at wavelength l. We constructed the forward model to calculate the irradiance ratios from w 0 (l) and g(l). [10] For the radiative transfer computations in the atmosphere, we employed an improved version of the radiative computation scheme originally developed by Asano and Shiobara [1989] and Nishizawa et al. [2004]. The solar irradiances at seven wavelengths were computed by the doubling and adding method [Lacis and Hansen, 1974]. The model atmosphere was assumed to be plane-parallel and was divided into 23 layers from the surface up to the altitude of 50 km. The seven wavelengths (l = 340, 380, 400, 500, 675, 870, and 1020 nm) were selected to avoid strong gaseous absorption, but slight ozone absorption remained. We used the total ozone amount of climatological data. The vertical ozone profile was approximated by a formula from Green [1964]. The spectral optical thickness due to molecular scattering was taken from the expression of Fröhlich and Shaw [1980] with the depolarization correction by Young [1981]. The ground surface was assumed to be Lambertian. Surface albedo values were given a priori. However, since the seasonal variation of surface albedo could be a source of retrieval errors for long-term measurements, we tested influences of surface albedo errors on the retrievals (section 3). dir ð1þ [11] The aerosol optical thickness, single-scattering albedo, and phase function were required in the radiative transfer computation. The calculated irradiance ratio was described schematically by: F cal ðlþ ¼ F cal w 0 ðlþ; PðQ; lþ; t obs ð Þ ; ð2þ AOT l where Q represents scattering angles, P(Q, l) is the phase obs function, and t AOT (l) is the aerosol optical thickness. obs Aerosol optical thickness t AOT (l) was obtained from obs (l) by the following equation: F dir Fdir obs ðlþ ¼ F 0 m exp m t obs ð Þþt O3ðlÞþt MOL ðlþ ; ð3þ AOT l where F 0 is the exoatmospheric flux, m is the air mass, t O3 (l) is the optical thickness of ozone, and t MOL (l) isthe optical thickness of molecular scattering. For the direct retrieval of g(l), P(Q, l) was approximated by the Henyey- Greenstein function [Henyey and Greenstein, 1941]: PðQ; l 1 gðlþ 2 Þ ¼ 3=2 : ð4þ 1 þ gðlþ 2 2gðlÞcos Q The Henyey-Greenstein phase function can replace the more realistic Mie phase function in most irradiance calculations, with no more than a few percent errors; it is widely used to characterize aerosol or cloud droplet scattering [Hansen, 1969; Boucher, 1998]. Using the Henyey-Greenstein phase function, (2) was rewritten as F cal ðlþ ¼ F cal w 0 ðlþ; gðlþ; t obs ð Þ ; ð5þ AOT l The vertical distribution of aerosol properties can be highly variable, but no information is available about the aerosol vertical profile from irradiances measured at the surface. Therefore the vertical profiles of aerosol optical thickness, single-scattering albedo, and phase function were assumed to be homogeneous from the surface to 3 km altitude. [12] The dependency of the irradiance ratio F cal (l) on w 0 (l), g(l) and air mass are illustrated in Figure 1, where F cal (l) is calculated for l = 500 nm, t AOT (500 nm) = 0.5, air mass 1.0 and 2.0. Here, F cal (l) increases as w 0 (l) and g(l) increase, so there are no unique solutions of w 0 (l) and g(l) estimated from the irradiance ratio. However, the dependency of F cal (l) onw 0 (l) and g(l) at air mass 1.0 is different from the dependency at air mass 2.0. The direct method estimates unique solutions of w 0 (l) and g(l) from this difference. That is, the direct method assumes the stability of w 0 (l) and g(l) for a certain time and retrieves w 0 (l) and g(l) from two irradiance ratios measured at different air masses. The stability of t obs AOT (l) is not assumed, and the values obtained by (3) at each air mass are used. Since the assumption of long-time stability is not appropriate for the real atmosphere, the difference between two air masses should be small. We describe a degree of the difference between two air masses in section 3.2.1 (1). 2.1.2. Inversion Scheme [13] Our inversion scheme is based on the method of maximum likelihood (MML), one of the strategic principles of statistical estimation. MML searches the parameters for 2of18

obs m. The irradiance ratios calculated from a and t AOT forward model (5) are expressed as: Fa; t obs T AOT ¼ ð F cal w 0 ðl i Þ; gðl i Þ; t obs;m1 AOT ð F cal w 0 ðl i Þ; gðl i Þ; t obs;m2 ð AOT l i l i Þ Þ by the [15] The PDF of the measured irradiance ratios F obs is written in the form: P F Fa; t obs AOT jf obs / exp ½ yf ðaþš ¼ exp 1 Fa; tobs AOT F obs ð 2 T sf ð8þ Þ 1 Fa; t obs AOT F obs ; ð9þ Figure 1. Dependency of the irradiance ratio on singlescattering albedo, asymmetry factor, and air mass. The irradiance ratios were calculated by the forward model of the direct method for wavelength 500 nm, optical thickness 0.5. The solid lines denote air mass 1.0, and the long dashed lines denote for air mass 2.0. the best fit of all data, considering the probability density functions (PDFs) of the measurements or a priori data. The best fitted parameters correspond to the maximum of PDF. In the direct method, three PDFs of the measured irradiance ratios and two a priori constraints were used. Two constraints were introduced to eliminate unrealistic oscillations in the retrievals. One was a priori distribution of the retrieved parameters. The other was a priori smoothness of spectral dependences of the retrieved parameters. 2.1.2.1. PDF of Measured Irradiance Ratios [14] The PDF of the measured irradiance ratios at seven wavelengths and at two different air masses is assumed the Gaussian distribution function. The vector forms of the retrieved parameters and two optical thicknesses obtained from two direct irradiances at different air masses are written as: and t obst AOT ¼ a T ¼ ð w 0 ðl i Þ gðl i Þ Þ; ð6þ tobs;m1 AOT l i ð Þ t obs;m2 ð AOT l i Þ ; i ¼ 1; ; 7; where l i is seven wavelengths, m 1 and m 2 are different air masses, and t obs,m AOT (l) is aerosol optical thickness at air mass ð7þ where s F is the covariance matrix of the vector F obs. Here, s F is assumed diagonal because we are not aware of any clear correlation between random errors in measurements at different wavelengths and different air masses. Table 1 presents the standard deviation of F obs used in the sensitivity tests of section 3. Figure 2 is an example of P F (F(a, t obs AOT ) F obs ) in two-dimensional space. In Figure 2, P F (F(a, t obs AOT ) F obs ) was calculated for the irradiance ratios, which were simulated with the Henyey-Greenstein phase function for a wavelength of 500 nm, optical thickness of 0.5, and air masses of 1.0 and 2.0. The best-estimated a corresponds to the maximum value of P F (F(a, t obs AOT ) F obs ). 2.1.2.2. PDF of A Priori Distribution Constraint [16] A priori distributions of retrieved parameters prevent the parameters from converging to the solutions beyond a realistic range. We limited the minimum and maximum of a by the Gaussian distribution function. The minimum and maximum values were taken from the values of the aerosol model in OPAC [Hess et al., 1998]. Table 2 presents the minimum and maximum values used in this study. The PDF of a priori distribution is written as P a ðþ/exp a ½y a ðþ a Š ¼ exp 1 2 ða hai ð Þ 1 ða haiþ ; ÞT s a ð10þ where hai is the medium values of the minimum and maximum; s a is the covariance matrix of the a priori distribution and is assumed diagonal. We considered the interval [a max, a min ] as 68% confidence level [hai + e a, hai e a ](e a ; standard deviation), and defined hai and e a as hai ¼ a max þ a min =2; ð11þ Table 1. Standard Deviations in the Direct Method Fitted Data Standard Deviation Measured irradiance ratio F obs (l) 5% A priori distribution constraint w 0 (l) 0.35, 0.36, 0.36, 0.38, 0.41, 0.44, 0.45 a g(l) 0.25, 0.26, 0.27, 0.28, 0.28, 0.30, 0.32 a A priori smoothness constraint w 0 (l) 0.07 g(l) 0.03 a Values are for seven wavelengths: 340, 380, 400, 500, 675, 870, and 1020 nm. 3of18

2.1.2.4. Optimization [19] According to MML, the best-estimated a corresponds to the maximum of the following joint PDF: PðÞ¼P a F Fa; ð AOT ÞjF obs Pa ðaþp s ðaþ ¼ expf½y F ðaþþy a ðaþþy s ðaþšg: ð16þ [20] The maximum P (a) corresponds to the minimum of the following evaluation function: yðþ¼ a y F ðþþ a y a ðþþ a y s ðþ: a ð17þ [21] The optimum a makes this evaluation function y(a) minimum. We employed the Newton-Gauss method to search for the best estimation. This method searches for the solutions iteratively using the following equation: Figure 2. PDF of measured irradiance ratios for the direct method. PDFs were calculated for irradiance ratios simulated with the Henyey-Greenstein phase function for wavelength 500 nm, for optical thickness 0.5, and for air mass 1.0 and 2.0. e a ¼ a max a min =2: ð12þ The values of e a are given in Table 1. 2.1.2.3. PDF of A Priori Smoothness Constraint [17] For a priori smoothness of the spectral dependencies, we assumed that the second derivatives of a were near zero. Seven wavelengths were used in this study, but the spectral intervals of the seven wavelengths were not constant. Therefore we interpolated a to ^a with the constant spectral interval, 0.05 mm. a T ¼ ð w 0 ðl i Þ gðl i ÞÞ! ^a T ¼ ^w 0 l j ^g lj ; i ¼ 1; ; 7; j ¼ 1; ; 17: ð13þ [18] The second derivative of ^a is written in the following form: f s ðþ a T ¼ ^w 0 l j ^g l j 2^g ljþ1 2^w0 l jþ1 þ ^w0 ðl jþ2 þ ^g ljþ2 Þ: ð14þ The PDF of f s (a) is written as P s ðþ/exp a ½y s ðþ a Š ¼ exp 1 ð 2 f sðþ0 a Þ T ðs s Þ 1 ðf s ðþ0 a Þ ; ð15þ where s s is the covariance matrix and is assumed diagonal. The values of s s were defined as the maximum values in the second derivatives calculated from OPAC data. The standard deviations are listed in Table 1. a next ¼ a cur Ha ð cur Þ 1 ryða cur Þ; ð18þ where a cur is the parameters of the current step, a next is the parameters of the next step, and H is Hessian matrix H ij = @ 2 y/@a i @a j. Here, H contains the second derivatives of the irradiance ratios, @ 2 F(a, t obs AOT )/@a i @a j, and computation of @ 2 F(a, t obs AOT )/@a i @a j is time-consuming. Therefore we ignored @ 2 F(a, t obs AOT )/@a i @a j in the computation of H [Press et al., 1992]. For the conduction of H 1, we employed singular value decomposition (SVD) [Press et al., 1992]. [22] A priori distribution constraint is helpful for searching the realistic estimation. However, if the constraint is too strong, a converges to the solutions near the maximum value of P a (a). For relaxation of the constraint, we introduced the following weight function: We rewrote (17) as: gðþ¼ a 1 ð if y FðÞ1 a Þ y F ðaþ ðif y F ðaþ < 1Þ: yðaþ ¼ y F ðaþþgðaþy a ðaþþy s ðaþ: ð19þ ð20þ [23] Here, y a (a) is always less than 1.0 in the a priori limited region, and always y F (a) > g (a) y a (a). If a approaches the best estimation, y F (a) approaches zero. The term g(a) y a (a) also approaches zero, and the constraint is weakened. If a is far away from the best estimation, the term g(a)y a (a) becomes large and the constraint becomes strong. 2.2. Indirect Method [24] The indirect method calculates optimum w 0 (l) and g(l) after roughly retrieving the size distribution dv(r)/dln r(r is particle radius) and refractive index (real part n(l) and Table 2. Maximum and Minimum of w 0 and g in OPAC Wavelength, nm 340 380 400 500 675 870 1020 w 0 maximum 1.000 1.000 1.000 1.000 1.000 1.000 1.000 minimum 0.295 0.277 0.267 0.226 0.169 0.120 0.094 g maximum 0.947 0.942 0.940 0.915 0.870 0.865 0.869 minimum 0.428 0.406 0.396 0.353 0.298 0.251 0.222 4of18

Table 3. Standard Deviations in the Indirect Method a Fitted Data Standard Deviation Measured irradiance ratio ln(f obs (l)) 0.05 Measured optical thickness ln(t AOT (l)) 0.01/t(500 nm) A priori smoothness constraint ln(dv(r)/d ln r) 1.2 ln(n(l)) 0.2 ln(k(l)) 1.25 Constraint of step correction Dln(dV(r)/d ln r) 2.5 Dln(n(l)) 0.05 Dln(k(l)) 1 a All values are taken from Dubovik and King [2000]. imaginary part k(l)) from the irradiance ratios F obs (l) and optical thickness t obs AOT (l) at seven wavelengths (l = 340, 380, 400, 500, 675, 870, and 1020 nm). This method does not assume the Henyey-Greenstein phase function and the stability of w 0 (l) and g(l) for a certain time. However, the indirect method requires t obs AOT (l) at more than two wavelengths to retrieve dv(r)/dln r. 2.2.1. Forward Modeling [25] The indirect method first retrieves n(l), k(l), and dv(r)/dln r. The retrieved dv(r)/dln r are 22 bins with 0.05 mm r 15 mm and equal step D(ln r) =lnr j +1 ln r j = const. We constructed the forward model to calculated irradiance ratios and optical thickness from n(l), k(l) and dv(r)/dln r. The radiative transfer scheme is the same as that used with the direct method. However, optical thickness cal t AOT (l), single-scattering albedo w 0 (l), and phase function P(Q, l) are calculated from n(l), k(l) and dv(r)/dln r as: Z lnrmax t cal AOT ðlþ ¼ ln r min C ext ðnðlþ; kðlþ; rþ dvðþ r d ln r; d ln r ð21þ t cal AOT ðlþ ¼ tcal AOT nðlþ; kðlþ; dv r j d ln r ; j ¼ 1; ; 22; ð26þ where j is 22 grid points of the size distribution. 2.2.2. Inversion Scheme [26] The inversion scheme of the indirect method is also based on MML. The indirect method uses PDF of the measured irradiance ratio and optical thickness, and PDF of a priori smoothness constraints of the retrieved parameters. The two PDFs are assumed to have Gaussian distributions. 2.2.2.1. PDF of Measured Irradiance Ratios and Optical Thickness [27] The measured irradiance ratios, optical thickness, and retrieved parameters are positive, and so we transformed the measurements and the retrieved parameters to the logarithms. This transformation is a way to avoid retrieval of negative values for fundamentally positive parameters [Dubovik and King, 2000]. The vectors of the measurements and retrieved parameters with logarithmic transformation are written as: F obst ¼ ln t obs ð Þ ln F obs ðl i Þ ; AOT l i ð27þ a T ¼ ln dv r j lnðnðl i ÞÞ lnðkðl i ÞÞ ; d ln r i ¼ 1; ; 7; j ¼ 1; ; 22: ð28þ Z ln rmax t cal AOT;absp ðlþ ¼ ln r min Z ln rmax PðQ; lþ ¼ ln r min C abs ðnðlþ; kðlþ; rþ dvðþ r d ln r; d ln r ð22þ C sca ðq; nðlþ; kðlþ; rþ dvðþ r d ln r; d ln r ð23þ [28] The vector of irradiance ratios and optical thickness calculated by the forward model is Fa ð Þ T ¼ ln t cal AOTð n ð l iþ; kðl i Þ; dv r j d ln r ln F cal nðl i Þ; kðl i Þ; dv r j d ln r : ð29þ w 0 ðlþ ¼ tcal AOT ðlþtcal AOT;absp l t cal AOT l ð Þ ð Þ ; ð24þ cal where t AOT,absp (l) is the optical thickness of aerosol absorption; C ext (n(l), k(l), C absp (n(l), k(l), r) and C sca (Q, n(l), k(l), r) are extinction, absorption, and scattering cross sections of aerosol particles. These cross sections are calculated by the Mie theory. Mie scattering calculation is time-consuming; therefore, C ext (n(l), k(l), r), C absp (n(l), k(l), r), and C sca (Q, n(l), k(l), r) are interpolated from look-up tables over all possible n(l) and k(l). The irradiance ratio and optical thickness determined by the forward model of the indirect method are written as: F cal ðlþ ¼ F cal nðlþ; kðlþ; dv r j d ln r ; ð25þ The PDF of F obs is defined as: P F Fa ð ÞjF obs / exp ½ yf ðþ a Š ¼ exp 1 2 Fa T ðþfobs ð sf Þ 1 Fa ð ÞF obs ; ð30þ where s F is covariance matrix and is assumed diagonal. The values of s F are the measurement accuracies of the irradiance ratios and optical thickness. The standard deviations of the irradiance ratios and optical thickness are presented in Table 3. 2.2.2.2. PDF of A Priori Smoothness Constraint [29] In the indirect method, a priori smoothness constraint of a is considered. We assumed that the second derivatives of ln [dv(r)/dlnr] approach zero and the first derivatives of 5of18

Table 4. Aerosol Models Adapted for Numerical Sensitivity Tests a Aerosol Type s 1 s 2 r V1 r V2 C V1 /C V2 n(l) k(l) t(500) Water-soluble 0.6 0.6 0.118 1.17 2 1.45 0.0035 0.05 0.20 0.50 1.00 Dust 1 0.6 0.8 0.1 3.4 0.066 1.53 0.008 0.50 1.00 Dust 2 0.6 0.6 0.1 1.17 0.066 1.53 0.008 0.50 Biomass-burning 0.4 0.6 0.132 4.5 4 1.52 0.025 0.50 1.00 a s: standard deviation. r V : median radius [mm]. C V : volume concentration for the size distribution. n: real part of refractive index. k: imaginary part of refractive index. t(500): aerosol optical thickness at wavelength 500 nm. All values are taken from Dubovik et al. [2000]. ln[n(l)] and ln [k(l)] approach zero. These derivatives are defined as: f s ðþ a T ¼ ln dv r j 2ln dv r jþ1 d ln r d ln r lnðnðlþþ lnðnðl iþ1þþ lnðl i Þlnðl iþ1 Þ lnðkðl i ÞÞlnðkðl iþ1 ÞÞ Þ; lnðl i Þlnðl iþ1 Þ i ¼ 1; ; 6; j ¼ 1; ; 21: þ ln dv rjþ2 d ln r ð31þ [30] The PDF of the smoothness constraint is given by: P s ðf s ðaþþ / exp½y s ðþ a Š ¼ exp 1 ð 2 f sðaþ0þ T ðs s Þ 1 ðf s ðaþ0þ ð32þ where s s is the covariance matrix and is assumed diagonal. The standard deviations of the smoothness constraint are given in Table 3. These values were taken from Dubovik and King [2000]. 2.2.2.3. Optimization [31] The joint PDF P(a) and the evaluation function y(a) of the indirect method are written as: PðÞ¼P a F Fa ð ÞjF obs Ps ðþ¼exp a f½y F ðaþþy s ðaþšg; ð33þ yðþ¼y a F ðaþþy s ðaþ: ð34þ [32] The Newton-Gauss method was also used to minimize y(a). Here, we employed the convergence improvement used by Dubovik and King [2000] for a stable retrieval. According to Dubovik and King [2000], the convergence improvement produces monotonic convergence of the iterations to the minimum of y(a) in a manner similar to that of the Levengberg-Marquardt method. This method limits the length of the step correction Da = a next a cur in the iterative process by assuming the PDF of Da. The PDF and the evaluation function of Da are defined as: P a ðdaþ / exp 1 Da 0 2 ð y a ðdaþ ¼ 1 Da 0 2 ð ð Þ 1 ðda 0Þ ; ð35þ ÞT s Da ð Þ 1 ðda 0Þ; ð36þ ÞT s Da where s Da is the covariance matrix of Da and is assumed diagonal. The standard deviations of Da are presented in Table 3. These values are taken from Dubovik and King [2000]. Furthermore, y a (Da) is multiplied by the weight function, gðþ¼ a 2ðy F ðaþþ y s ðþ a Þ= N f N a ; ð37þ where N f is the number of all the fitted measurements and a priori data, and N a is the number of retrieved parameters. The value of g(a) decreases with a decrease of y F (a). The length of Da is strongly limited far from the best estimation, but the limitation is weakened near the best estimation. Figure 3. RMSEs of w 0 (500 nm) and g(500 nm) retrieved for seven combinations of two air masses. Each RMSEs were calculated from w 0 (l) and g (l) retrieved for all aerosol models of Table 4. 6of18

Figure 4. Convergence of the dust 1 model (t(500 nm) = 0.5) by the direct method. (a) Evaluation function in the iteration process. (b) Irradiance ratio at air mass 2.0. (c) Irradiance ratio at air mass 2.1. Adding the term of the convergence improvement, (34) is rewritten as: yðþ¼y a F ðþþg a ðþy a a ðþþy a s ðþ: a ð38þ [33] The optimum w 0 (l) and g(l) were computed from a obtained by minimization of (38). Then w 0 (l) was calculated with equations (21) (24). We calculated g(l) from gðlþ ¼ 1 2 Z 1 1 Pðcos QÞcos Qd cos Q ð39þ 3. Numerical Sensitivity Test [34] The numerical sensitivity tests for the simulated direct and diffuse irradiances were performed to evaluate the retrieval accuracies of the direct and indirect methods. 3.1. Simulation Data [35] The direct and diffuse irradiances were simulated for four aerosol models: water-soluble, dust 1, dust 2 and biomass-burning aerosols. These aerosol models were taken from Dubovik et al. [2000]. According to Dubovik et al. [2000], the particle volume size distribution is modeled by the bimodal lognormal size distribution as: dvðþ r d ln r ¼ X2 i¼1 " C v;i pffiffiffiffiffiffiffiffiffi exp 1 # ln r ln r 2 v;i ; ð40þ 2ps i 2 s i where index i denotes fine (i = 1) and coarse (i = 2) modes of aerosol particles, C V,i denotes the volume concentration for the size distribution, r V,i denotes the median radius, and s i denotes the standard deviation. Table 4 presents the parameters of the size distribution and the refractive index 7of18

Figure 5. Single-scattering albedo retrieved by the direct method for air mass combination 2.0 and 2.1: (a) Error free case. (b) Surface albedo error case. (c) Diffuse irradiance error case. (d) Optical thickness error case. (e) Single-scattering albedo error case. (f) Asymmetry factor error case. 8of18

Figure 6. Same as Figure 5 but retrievals of asymmetry factor. 9of18

Figure 7. Open circles indicate comparison of the diffuse irradiances with the Henyey-Greenstein and with the Mie phase functions. Closed circles indicate relative errors of the diffuse irradiances with the Henyey-Greenstein phase function and of the diffuse irradiances with the Mie phase function. These diffuse irradiances were simulated for all aerosol models of Table 4, for seven wavelengths, and for air masses of 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0. for all aerosol models, as well as the values of optical thickness. Clear and hazy conditions were considered for the water-soluble model. Dust and biomass-burning aerosols are usually a consequence of such phenomena as dust storm or extensive fires; correspondingly, these aerosols are often characterized by relatively high optical thickness. The direct and diffuse irradiances at seven wavelengths (340, 380, 400, 500, 675, 870, and 1020 nm) were simulated for all aerosol models (Table 4). The values of surface albedo were taken from the reflectance of a green vegetation model of Dubovik et al. [2000]. The spectral values were 0.03, 0.03, 0.03, 0.06, 0.06, 0.2, and 0.2 at each of the seven wavelengths. The irradiances at six air masses (1.0, 2.0, 3.0, 4.0, 5.0, and 6.0) were simulated to test the influence of retrieval accuracy on air mass. 3.2. Results [36] Numerical sensitivity tests of the direct and indirect methods for simulated irradiances with and without errors were conducted. Three patterns of errors were given, considering the retrievals for the real atmosphere: +50% bias errors to surface albedo, ±5% random errors to diffuse irradiance, and ±0.02 random errors to optical thickness. The random errors for diffuse irradiance and optical thickness were given as instrumental accuracies. The bias errors for surface albedo were given, considering analysis for long-term measurements. [37] The direct method assumes the stability of w 0 (l) and g(l) for a certain time and uses two measurements at different air masses. For only the direct method, two more patterns of random errors were conducted to test influences of the time variation of w 0 (l) and g(l). These two patterns were ±0.05 errors for w 0 (l) and about ±0.07 errors for g(l). The errors for g(l) were given by adding ±30% errors to P(Q, l). 3.2.1. Direct Method 3.2.1.1. The Magnitude of the Difference Between Two Air Masses [38] The direct method requires two irradiance ratios at different air masses. It is necessary to determine the magnitude of the difference between two air masses before the numerical sensitivity tests. We determined the magnitude from the retrieval tests for seven combinations of two air masses. The seven combinations were (1.5 and 1.51), (1.5 and 1.55), (1.5 and 1.6), (1.5 and 1.7), (1.5 and 1.8), (1.5 and 1.9), and (1.5 and 2.0): the differences between two air masses were 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5. Figure 3 is Root Mean Square Errors (RMSEs) calculated from w 0 (l) and g(l) retrieved for all aerosol models of Table 4. RMSEs except for RMSE of (1.5 and 1.51) increased with an increase of air mass difference. The forward model of the direct method uses Henyey-Greenstein phase function. The Henyey-Greenstein phase function failed to reproduce the backscattering peak and tended to overestimate the Mie phase function at side scattering angles [Boucher, 1998]. The side and back patterns of phase function influence the diffuse irradiance at large air mass. Therefore RMSEs increased with an increase of the air mass difference. RMSEs of (1.5 and 1.55) and (1.5 and 1.6) are almost same. Therefore we applied the magnitude of the difference 0.1 to the all numerical sensitivity tests. The air mass difference 0.1 corresponds to the stable condition in about one hour at our observation site. 3.2.1.2. Results of Error-Free Cases [39] Figure 4 illustrates how the iteration converged. This convergence is for t (500 nm) = 0.5 of the dust 1 model. The evaluation function converged sharply at the first iteration. The estimated irradiance ratios at air masses 2.0 and 2.1 agreed well with the truth data. The results for other models also converged before the tenth iteration, and the differences between the estimated and true irradiance ratios were small. [40] Figures 5a and 6a present the retrieval results of the direct method for error-free cases. The retrieval errors of w 0 (l) and g(l) for the water-soluble and biomass-burning models were less than 0.03 and 0.11. Those for the dust 1 and 2 models were less than 0.07 and 0.16. Retrieval errors for the dust models were large. In the error-free cases, only the Henyey-Greenstein phase function caused the retrieval Table 5. RMSEs of Retrievals by the Direct Method a Water-Soluble t(500) < 0.5 t(500) 0.5 Dust 1 Dust 2 Biomass-Burning w 0 no error 0.01:0.01:0.01 0.01:0.01:0.01 0.03:0.04:0.04 0.01:0.02:0.03 0.01:0.01:0.02 error 0.05:0.05:0.03 0.02:0.03:0.02 0.03:0.04:0.06 0.02:0.03:0.04 0.01:0.02:0.04 g no error 0.05:0.03:0.03 0.06:0.03:0.02 0.11:0.10:0.10 0.03:0.03:0.08 0.05:0.04:0.06 error 0.04:0.06:0.05 0.04:0.05:0.03 0.08:0.08:0.12 0.03:0.03:0.08 0.04:0.06:0.13 a Left: ultraviolet region (<400 nm). Middle: visible region (400 to 740 nm). Right: near-infrared region (<740 nm). 10 of 18

Table 6. RMSEs of Retrievals for Various Air Mass Combinations by the Direct Method a Combination of Air Masses 1.0, 1.1 2.0, 2.1 3.0, 3.1 4.0, 4.1 5.0, 5.1 6.0, 6.1 RMSE w 0 0.013 0.022 0.019 0.018 0.016 0.017 g 0.039 0.061 0.051 0.066 0.068 0.077 a RMSEs were calculated from retrievals for all aerosol models in Table 4. errors. The Henyey-Greenstein phase function approximated the downward diffuse irradiances with good accuracies [Hansen, 1969; Boucher, 1998]. We evaluated the differences between the Henyey-Greenstein and Mie phase functions. Figure 7 presents the results of the comparison. The diffuse irradiances were calculated for all aerosol models of Table 4, for seven wavelengths, and for six air masses. The relative errors of the Henyey-Greenstein phase function relative to the Mie phase function were less than 4%. However, the magnitudes of retrieval errors of w 0 (l) and g(l) exceeded 4%. Retrieval errors for the dust models were especially large. Since the Henyey-Greenstein phase function is very smooth and has lower and broader forward peak than the Mie phase function, the function is suitable for small and high absorbing particle but not for large particle with sharp forward peak [Wiscombe, 1977]. Hence the retrieval errors of the dust models were large. Other analytic phase functions are more suitable for aerosol particle than Figure 8. Convergence of the dust 1 model (t(500 nm) = 0.5) by the indirect method. (a) Evaluation function in the iteration process. (b) Irradiance ratio at air mass 2.0. (c) Optical thickness. 11 of 18

Figure 9. Single-scattering albedo retrieved by the indirect method for air mass 2.0. (a) Error-free case. (b) Surface albedo error case. (c) Diffuse flux error case. (d) Optical thickness error case. the Henyey-Greenstein phase function. Examples include the double Henyey-Greenstein phase function [Irvine, 1965] and the modified Henyey-Greenstein phase function [Conette and Shanks, 1992]. The double Henyey-Greenstein phase function is suitable for large particle but needs three parameters. The modified Henyey-Greenstein phase function is suitable for Rayleigh scattering, but not for large particle. We tested the modified Henyey-Greenstein phase function, but the results were almost the same as those of the Henyey-Greenstein phase function. 3.2.1.3. Results of Random-Error Cases [41] Figures 5b and 6b present the results for simulated data with surface albedo errors. The retrieval results of g(l) in the water-soluble model were larger than those of errorfree cases. The given bias errors of the surface albedo increase the diffuse irradiance, and the magnitudes of the diffuse irradiances are the largest in the water-soluble aerosol models. Therefore these results caused the overestimation of g(l). In the results of the other aerosol models, the variations from the error-free cases were slightly large at long wavelengths for both w 0 (l) and g(l). The values of surface albedo given in the sensitivity tests were large for long wavelengths, and the bias errors became large; therefore, retrieval errors of w 0 (l) and g(l) at long wavelengths were larger than those of error-free cases. The land surface albedo in the near-infrared region is larger than in the visible region and has more uncertainties [Li et al., 2002]. When retrieving for long-term measurements, it is important to assume the seasonal variations of the surface albedo. [42] Figures 5c and 6c present the results for diffuse irradiance error cases. Compared to the results of w 0 (l) for the error-free cases, retrieval errors of w 0 (l) for the optically thin cases of the water-soluble model increased. When there are few aerosol particles in the atmosphere, the irradiance ratios become insensitive to the aerosol optical properties. Therefore retrieval errors in optically thin cases 12 of 18

Figure 10. Same as Figure 9 but retrievals of asymmetry factor. exceeded those in optically thick cases. The retrieval results of g(l) in (c) of Figure 6 were almost the same for all models. The same results of g(l) were also observed in Figures 6e and 6f. We describe these results later in this section. [43] Figures 5d and 6d plot the results for optical thickness error cases. Retrieval results of w 0 (l) and g(l) were almost the same as those for the error-free case, indicating that the measurement noise of optical thickness was low and did not strongly influence retrievals. [44] Figures 5e and 6e depict the results considering time variations of w 0 (l). The retrieval errors of w 0 (l) were less than 1.0 for all the aerosol models. Considering ±0.05 errors added to simulated data, retrieved w 0 (l) were like averaged values for time variations of aerosols. However, the retrieval results of g(l) were almost the same for all models. [45] Figures 5f and 6f present the retrieval results with errors for the stability of g(l). The retrieval errors of w 0 (l) were almost the same as those for error-free cases, indicating that time variations of g(l) did not influence the retrievals of w 0 (l). Retrieved g(l) were almost the same values for all models. [46] The results of g(l) in Figures 6c, 6e and 6f were almost the same. A common feature in these retrievals was that the evaluation function of the measurements y F (a) was larger than those for error-free cases. According to equations (18) and (19), when y F (a) was large, the a priori distribution constraint became strong. Consequently, the retrieved and g(l) converged according to the a priori distribution. [47] Table 5 presents RMSEs for retrievals by the direct method. RMSEs for error-free cases were calculated from the results in (a) of Figures 5 and 6. RMSEs for error cases 13 of 18

Figure 11. Size distribution retrieved by the indirect method for air mass 2.0, and for the error-free case. were calculated from results in (b) to (f) of Figures 5 and 6. RMSEs for w 0 (l) and g(l) were 0.01 to 0.06 and 0.02 to 0.13. The direct method accurately retrieved w 0 (l) but not g(l), especially for the dust models. In almost all cases, RMSEs of the ultraviolet and visible regions were smaller than those of the near-infrared region. 3.2.1.4. Dependency of Retrieval Accuracy on Air Mass [48] Table 6 demonstrates the influence of air mass on retrieval errors. RMSEs were calculated from retrieval results for seven wavelengths and for all aerosol models of Table 4. The RMSE of w 0 (l) was small and was not dependent on air mass. The RMSE of g(l) for the smallest air mass (a combination of 1.0 and 1.1) was the smallest, but RMSEs for other air masses were twice as large. The side and back patterns of phase function influenced the diffuse irradiance for large air masses. For both fine and coarse particles, the Henyey-Greenstein phase function failed to reproduce the backscattering peak and tended to overestimate the Mie phase function at side scattering angles [Boucher, 1998]. Consequently, retrieval errors for the large air mass were large. 3.2.2. Indirect Method 3.2.2.1. Results of Error-Free Cases [49] Figure 8 presents an example of convergence by the indirect method. This convergence was for t(500 nm) ofthe dust 1 model. The evaluation function converged at the 20th iteration. The estimated irradiance ratios and optical thickness agreed well with the true values. The results for other models also converged before the 30th iteration, and the differences between the estimation and truth data were small. [50] Figures 9a and 10a plot the results of w 0 (l) and g(l) for the error-free cases. The retrievals of w 0 (l) and g(l) agreed well with the truth data. However, the retrievals of 14 of 18

Figure 12. Same as Figure 8 but retrievals of the real part of the refractive index. g(l) were slightly smaller than the truth for almost all aerosol models. The underestimation of g(l) are noticeable in the results of water-soluble t(500 nm) = 0.5 and dust 1 models. [51] Optimum w 0 (l) and g(l) were calculated from retrieved size distribution dv(r)/dlnr, and refractive indices n(l) and k(l). Retrieval errors of w 0 (l) and g(l) were caused by retrieval errors of dv(r)/dln r, n(l), and k(l). Figures 11, 12, and 13 plot the retrieval results of dv(r)/dln r, n(l), and k(l). The retrievals of dv(r)/dln r for all aerosol models were overestimated at r < 0.2 mm and underestimated at r > 1 mm. According to Yamamoto and Tanaka [1969] and King et al. [1978], dv(r)/dln r is dependent on optical thickness, and dv(r)/dlnr retrieved from optical thickness at 0.34 to 1.02 mm are limited to between 0.1 mm and 4.0 mm. Since the given optical thickness were less sensitive to coarse particles, dv(r)/dlnr at r > 1 mm were underestimated. The dv(r)/dlnr at r < 0.2 mm were overestimated to adjust the calculated optical thickness to the given optical thickness. The retrieval errors of n(l) were between 0.0 and 0.15 in Figure 12. The retrieval errors of k(l) except dust 1 were between 0 and 50% in Figure 13. However, the retrieval errors of k(l) for dust 1 were 50 to 150%. Accuracies of n(l) and k(l) retrieved from AERONET measurements were between 0.025 and 0.04, and between 30% and 100% [Dubovik et al., 2000]. The retrieval errors of n(l) in this study were larger than those of AERONET measurements, but the retrieval accuracies of k(l) except dust 1 were comparable with those of AERONET measurements. The indirect method did not retrieve n(l) accurately. Originally, the irradiance ratio is sensitive to k(l) but is not sensitive to n(l). [52] According to Hansen and Travis [1974], w 0 (l) is dependent on k(l) and dv(r)/dlnr. dv(r)/dln r at 0.2 < r < 15 of 18

Figure 13. Same as Figure 8 but retrievals of the imaginary part of the refractive index. 1.0 mm were retrieved well, but dv(r)/d ln r at r <0.2mm and r >1.0mm were not retrieved well. These errors may cause the retrieval errors of w 0 (l), but w 0 (l) were retrieved accurately. The indirect method searches the optimum value of k(l) for the given irradiance ratios. As a result, single scattering albedo was retrieved well, although dv(r)/dlnr was not retrieved well. This is noticeable in the results of dust 1 model which has the largest dv(r)/dln r at r >1.0mm in all models. This caused large retrieval errors of k(l) for dust 1 model. On the other hand, g(l) is dependent on n(l) and dv(r)/dln r. The underestimation of g(l) in the results of water-soluble t(500 nm) = 0.5 and dust 1 models were caused by the overestimation of n(l) and the underestimation of dv(r)/dlnr at r >1mm. Table 7. RMSEs of Retrievals by the Indirect Method a Water-Soluble t(500) < 0.5 t(500) 0.5 Dust 1 Dust 2 Biomass-Burning w 0 no error 0.00:0.01:0.00 0.01:0.01:0.02 0.01:0.02:0.02 0.00:0.00:0.00 0.02:0.02:0.01 error 0.01:0.01:0.02 0.01:0.01:0.01 0.01:0.02:0.03 0.02:0.02:0.02 0.02:0.02:0.02 g no error 0.05:0.02:0.01 0.06:0.04:0.03 0.03:0.05:0.05 0.00:0.01:0.00 0.02:0.02:0.03 error 0.05:0.03:0.03 0.06:0.04:0.03 0.04:0.05:0.05 0.01:0.01:0.01 0.03:0.02:0.03 a Left: ultra violet region (<400 nm). Middle: visible region (400 to 740 nm). Right: near-infrared region (<740 nm). 16 of 18

Table 8. RMSEs of Retrievals for Various Air Masses by the Indirect Method a Air Mass 1.0 2.0 3.0 4.0 5.0 6.0 RMSE w 0 0.011 0.013 0.014 0.017 0.017 0.012 g 0.030 0.035 0.050 0.058 0.061 0.052 a RMSEs were calculated from retrievals for all aerosol models in Table 4. 3.2.2.2. Results of Random-Error Cases [53] Figures 9b 9d and 10b 10d reveal the retrieval results of w 0 (l) and g(l) with random errors. For all the results, the retrieval errors of w 0 (l) and g(l) were small and almost the same as those of error-free cases. The indirect method was not influenced by the magnitude of the random errors given in these sensitivity tests. However, the retrieval errors of w 0 (l) and g(l) were large for the optically thin case of the water-soluble model. In the optically thin case, it was difficult to retrieve aerosol optical properties because the contribution of the aerosol optical properties to the irradiances was small. Large retrieval errors of g(l) were also observed in the near-infrared region of the biomassburning model. Since the volumes of coarse particles in this model were very small, optical thickness at the near-infrared region was very small. [54] Table 7 presents RMSEs for retrievals by the indirect method. RMSEs for error-free cases were calculated from the results in (a) of Figures 9 and 10. RMSEs for error cases were calculated from the results in (b) to (d) of Figures 9 and 10. RMSEs for w 0 (l) and g(l) were 0.00 to 0.03 and 0.00 to 0.06. These values were smaller than those of the direct method. The indirect method accurately retrieved both w 0 (l) and g(l). 3.2.2.3. Dependency of Retrieval Accuracy on Air Mass [55] Table 8 reveals the dependency of retrieval error on air mass. RMSEs were calculated from retrieval results for seven wavelengths and for all aerosol models of Table 4. RMSEs of g(l) increased slightly with an increase of the air mass. RMSEs of w 0 (l) were almost same for difference of the air mass. The indirect method was not influenced by the difference of the air mass strongly. 4. Summary [56] We have developed two methods to retrieve w 0 (l) and g(l) from direct and diffuse spectral irradiances. This work refers to these methods as the direct method and the indirect method. The direct method neglects the shape, size distribution, and refractive index of aerosol particles and retrieves w 0 (l) and g(l) directly from two irradiance ratios measured at different air masses. This method can retrieve w 0 (l) and g(l) from the measurements at one wavelength because the size distribution is neglected. In the retrieval, the aerosol phase function is approximated by the Henyey-Greenstein function, and the stability of w 0 (l) and g(l) for a certain time is assumed. In contrast, the indirect method calculates optimum w 0 (l) and g(l), after roughly retrieving dv(r)/dlnr, n(l), and k(l) from the measured irradiance ratios and optical thickness. This method does not need the assumption of the Henyey- Greenstein phase function and the stability of w 0 (l) and g(l), but it does require the optical thickness measured at some wavelengths to retrieve the size distribution. Both the direct and indirect methods retrieve w 0 (l) and g(l) simultaneously on the basis of MML. [57] Sensitivity tests for simulated irradiances with and without errors were conducted to evaluate the retrieval accuracies of the two methods. The simulated irradiances were calculated for water-soluble, dust 1, dust 2, and biomass-burning models. Considering the application to the real atmosphere, bias and random errors were added to surface albedo, diffuse irradiances, and optical thickness. For only the direct method, random errors for the assumption of the stability of w 0 (l) and g(l) were tested. RMSEs for retrieved w 0 (l) and g(l) were calculated from the results of these sensitivity tests. RMSEs of the direct method were 0.01 to 0.06 for w 0 (l) and 0.02 to 0.13 for g(l). The direct method accurately retrieved w 0 (l); however, retrieval results of g(l) were inadequate, especially for the dust models, because the Henyey-Greenstein function is not appropriate for the dust model. A more suitable approximated phase function is required to retrieve g(l) accurately. RMSEs of the indirect method were 0.00 to 0.03 for w 0 (l) and 0.00 to 0.06 for g(l). The indirect method accurately retrieved both w 0 (l) and g(l); however, the underestimation of g(l) were observed. These were caused by the underestimation of dv(r)/dln r for the coarse particles (r > 1 mm) and the overestimation of n(l). The retrieval accuracies of w 0 (l) by other studies were 0.03 to 0.07 [retrieval from AERONET measurements, Dubovik et al., 2000], and 0.1 to 4.3% [retrieval from MFRSR measurements, Kassianov et al., 2005]. Our retrieval accuracies of w 0 (l) by both the direct method and indirect method were comparable with these results. The retrieval accuracies of g(l) by other study were 0.1 to 4.4% [retrieval from MFRSR measurements, Kassianov et al., 2005]. The retrieval accuracy of g(l) by the indirect method was also comparable with this study, but that by the direct method was worse. [58] For future work, we plan to apply our methods to the data measured with the direct and diffuse spectral radiometers at MRI for several years. Additionally, we believe the method is applicable to the direct and diffuse broadband irradiances, since the direct method can retrieve aerosol single-scattering properties from the diffuse and direct irradiances at one wavelength. Broadband radiation measurements have been conducted worldwide and for a longterm period. References Asano, S., and M. Shiobara (1989), Aircraft measurements of the radiative effects of tropospheric aerosols: l. Observational results of the radiation budget, J. Meteorol. Soc. Jpn., 67, 847 861. Boucher, O. (1998), On aerosol direct shortwave forcing and the heyneygreenstein phase function, J. Atmos. Sci., 55, 128 134. Cornette, W. M., and J. G. Shanks (1992), Physically reasonable analytic expression for the single-scattering phase function, Appl. Opt., 31, 3152 3160. Dubovik, O., and M. D. King (2000), A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements, J. Geophys. Res., 105(D16), 20,673 20,696. Dubovik, O., A. Smirnov, B. N. Holben, M. D. King, Y. J. Kaufman, T. F. Eck, and I. Slutsker (2000), Accuracy assessments of aerosol optical properties retrieved from Aerosol Robotic Network (AERONET) Sun and sky radiance measurements, J. Geophys. Res., 105(D8), 9791 9806. Dubovik, O., B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, and I. Slutsker (2002), Variability of absorption and optical 17 of 18