Estimate of satellite-derived cloud optical thickness and effective radius errors and their effect on computed domain-averaged irradiances

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jd006668, 2006 Estimate of satellite-derived cloud optical thickness and effective radius errors and their effect on computed domain-averaged irradiances Seiji Kato, 1 Laura M. Hinkelman, 2 and Anning Cheng 1,3 Received 14 September 2005; revised 26 January 2006; accepted 16 May 2006; published 1 September [1] The process of retrieving cloud optical thickness and effective radius from radiances measured by satellite instruments is simulated to determine the error in both the retrieved properties and in the irradiances computed with them. The radiances at 0.64 mm and 3.7 mm are computed for three cloud fields (stratus, stratocumulus, and cumulus) generated by large eddy simulation models. When overcast pixels are assumed and the horizontal flux is neglected in the retrieval process, the error in the domain-averaged retrieved optical thickness from nadir is 1% to 32% (1% to 27%) and the error in the retrieved effective radius is 0% to 67% (0% to 63%) for the solar zenith angle of 30 (50 ). Using the radiance averaged over a 1 km size pixel also introduces error in the optical thickness because of the nonlinear relation between the reflected radiance and optical thickness. Both optical thickness and effective radius errors increase with increasing horizontal inhomogeneity. When the 0.64 mm albedo is computed with the independent column approximation using retrieved properties from nadir (oblique) view for a solar zenith angle of 50, the error is 0.3% to 14% ( 5% to 30%) relative to the albedo from 3-D radiative transfer computations with the true cloud properties. The albedo error occurs even though the radiance at one angle is forced to agree because a plane parallel cloud with a single value of optical thickness and effective radius cannot consistently match the radiance angular distribution. In addition, the error in the retrieved cloud properties contributes to the albedo error. When albedos computed with cloud properties derived from nadir and oblique views are averaged, the albedo error can partially cancel. The absolute error in the narrowband 0.64 mm (3.7 mm) albedo averaged over a 1 1 domain is less than 1.5% (0.6%), 5.0% (4.1%), and 7.1% (11%) in order of increasing inhomogeneity, when albedos computed with cloud properties derived from viewing zenith angles between 0 and 60 are averaged and when the solar zenith angle is between 10 and 50. When the solar zenith angle is 70, the error increases to up to +24% (+37%) for all three scenes. Citation: Kato, S., L. M. Hinkelman, and A. Cheng (2006), Estimate of satellite-derived cloud optical thickness and effective radius errors and their effect on computed domain-averaged irradiances, J. Geophys. Res., 111,, doi: /2005jd Introduction 1 Center for Atmospheric Sciences, Hampton University, Hampton, Virginia, USA. 2 National Institute of Aerospace, Hampton, Virginia, USA. 3 Now at Analytical Services & Materials, Inc., Hampton, Virginia, USA. Copyright 2006 by the American Geophysical Union /06/2005JD [2] Cloud optical thickness and volume-averaged cloud particle size can be estimated from narrowband radiances measured from satellites. A basic assumption of such retrievals is that the radiance at visible wavelengths is more sensitive to cloud optical thickness than cloud particle size, while the radiance at near-infrared wavelengths is sensitive to both particle size and optical thickness [Nakajima and King, 1990]. At visible wavelengths, absorption by cloud particles is negligible so that the single scattering albedo and phase function are weak functions of particle size provided that this size varies within a realistic range. Therefore the reflectance of clouds at visible wavelengths is predominately a function of the optical thickness, which can thus be estimated by observing reflectance at a visible wavelength. Cloud particles absorb near-infrared radiation; absorption by an optically thick cloud layer is approximately proportional to the square root of cloud particle size [Twomey and Bohren, 1980]. Even if clouds are not optically thick, particle size can be determined from a near-infrared radiance if the optical thickness is given and the particle size is not too small. Therefore the cloud optical thickness and particle size can, in principle, be estimated using visible and near-infrared radiances measured by satellitebased instruments [Nakajima and King, 1990; Han et al., 1994; Minnis et al., 1998]. 1of16

2 [3] The above argument is based on one-dimensional radiative transfer theory that neglects horizontal variations of cloud properties within and outside of an instrument field of view. Because clouds are horizontally inhomogeneous, assuming a horizontally uniform cloud and neglecting horizontal fluxes in deriving cloud properties from radiance measurements introduce errors. The global radiation budget is sometimes computed with satellite retrieved cloud properties [e.g., Zhang et al., 1995; Rossow and Zhang, 1995; Charlock et al., 2001]. In the CERES project [Wielicki et al., 1996], for example, irradiances at the surface and three other pressure levels are computed using satellite retrieved cloud properties constrained by top-of-atmosphere irradiances derived from CERES radiance measurements. In order to understand the error in the computed irradiances, we need to answer the following questions: How large is the error in retrieved properties, and how much does this error affect the irradiances subsequently computed with a 1-D radiative transfer model? Unless 3-D effects are parameterized [e.g., Iwabuchi and Hayasaka, 2002; Wyser et al., 2002], implementing a 3-D radiative transfer algorithm in the cloud property retrieval is not practical because it is computationally far too expensive and because the 3-D cloud fields in the actual retrieval process are not known. We can, however, use a 3-D radiative transfer model to understand the errors causedbyapplying1-dradiativetransfertheorytothe retrieval process. [4] Earlier studies have assessed cloud property retrieval errors caused by using 1-D radiative transfer theory. Loeb et al. [1998] show that when cloud optical thickness is derived from forward observing direction (the instrument looks toward the Sun) and when variations of the cloud top height exist, the derived optical thickness tends to be smaller than the true value. Vàrnai and Marshak [2001] estimate the uncertainty in the retrieved optical thickness due to neglecting horizontal flux in the retrieval algorithm. They show that the uncertainty increases with increasing retrieved optical thickness and solar zenith angle when the optical thickness is retrieved from nadir. Coakley et al. [2005] show that assuming an overcast imager pixel causes the cloud fraction and effective radius to be overestimated and the optical thickness of clouds to be underestimated when the pixel is actually only partially filled by clouds. Chambers et al. [2001] estimate the error in the 0.83 mm irradiance when retrieved optical thicknesses from 2-D trade cumulus, broken stratocumulus, and overcast stratus cloud fields are used in a 1-D radiative transfer model by comparing these values with the irradiances derived from angular distribution models. Their result indicates that the error ranges from 11% to 60% depending on viewing angle. [5] Even though some recent studies address the error in the retrieved effective radius [e.g., Cornet et al., 2005)], the error in retrieved particle size caused by applying 1-D radiative transfer theory, hence neglecting horizontal fluxes and cloud inhomogeneity, has not been studied in detail. The error in irradiances computed from cloud optical thickness and effective radius retrieved using 1-D radiative transfer theory is also poorly understood. The purposes of this paper are (1) to investigate the effect of applying 1-D radiative transfer theory in retrieving the cloud optical thickness and effective radius from radiances measurements in two narrowband wavelengths and (2) to estimate the effect of the error in irradiance computations when the retrieved optical thickness and particle size are used. 2. Cloud Property Retrieval Simulation [6] A typical cloud property retrieval process is simulated using liquid water content fields generated by large eddy simulation (LES) models [Stevens et al., 1999]. An advantage of this approach compared to using observed cloud fields is that the true cloud properties are known, so that the retrieval error can be precisely determined. We use three scenes covering the range from uniform to broken cloud fields. The first and second scenes are from different time steps of the same marine stratocumulus cloud simulation (the Atlantic Stratocumulus Transition Experiment, ASTEX [Bretherton et al., 1999]). During this interval, the cloud changes from a relatively horizontally uniform cloud deck (Figure 1a, hereinafter ASTEX-St) to a stratocumulus cloud (Figure 1b, hereinafter ASTEX-Sc). The third scene is a broken boundary layer cloud field (Figure 1c, Atlantic Trade Experiment [Stevens et al., 2001], hereinafter ATEX-Cu). The horizontal (vertical) resolutions of these liquid water content fields are 50 m (25 m) for ASTEX-St and ASTEX-Sc and 125 m (20 to 40 m) for ATEX-Cu and the domain sizes are approximately 3 km 3kmand8km 8 km, respectively. We assume that the cloud particles follow a lognormal distribution with a mode radius of 10mm and standard deviation of 1.42, which corresponds to an effective radius of 13.6 mm, for all three scenes. The extinction cross section, single scattering albedo, and phase function are then computed using Mie theory. The extinction coefficient b for each grid point is determined as b ¼ 3LQ e 4r e r w ; where L is the liquid water content, Q e is the extinction efficiency (the extinction cross section divided by the geometrical cross section of particles), r e is the effective radius, and r w is the density of liquid water. Note that although the cloud particle number concentration tends to be constant with height for boundary layer water clouds, it varies proportionally to the liquid water content in the clouds used for this simulation. This does not introduce any unrealistic results since we are only interested in deviations from the specified values rather than absolute results. [7] Table 1 lists the properties specified for the three cloud fields used in this study. The domain-averaged optical thickness is defined as t ¼ 1 N where N is the total number of columns in the domain. The domain averaged optical thicknesses are 4.6, 3.8, and 6.3 for ASTEX-St, ASTEX-Sc, and ATEX-Cu, respectively. The optical thickness distributions are further characterized by the shape parameter n defined as X N i¼1 t i ; Pðt;t; nþ ¼ 1 n nt n 1 e nt=t ; Gn ð Þ t ð1þ ð2þ ð3þ 2of16

3 Figure 1. Optical thickness, log 10 of the extinction coefficient, and optical thickness distribution for the LES cloud fields, where the extinction coefficients have been computed from liquid water content using equation (1). (a) ASTEX-St. (b) ASTEX-Sc. (c) ATEX-Cu. The mean optical thicknesses shown in the figures are computed using only cloudy columns, so they are different from the values shown in Table 1. The locations of the x-z section shown in the middle plots are indicated by the horizontal lines in the plots in the left-hand column. The curves shown in the plots in the right-hand column are fitted gamma distributions. assuming a gamma distribution. Here t is taken only over the cloudy columns. Horizontal inhomogeneity decreases with n and the albedo is not sensitive to n when it is greater than about 10 [e.g., Barker et al., 1996; Kato et al., 2005]. The maximum likelihood method of Greenwood and Durand [1960] is used to estimate n. Table 1 also includes the standard deviations of cloud top height s z and optical thickness s t, which are also computed using only the cloudy columns. Cloud properties shown in Table 1 are used as the true values when the error in retrieved values are computed in this study. [8] We simulate a typical cloud optical thickness and effective radius retrieval process using radiances measured at two wavelengths. The measured radiances at two monochromatic wavelengths (0.64 mm and 3.7 mm in this study) are computed from the full-resolution cloud fields using the Spherical Harmonics Discrete Ordinate Method (SHDOM [Evans, 1998]) in 3-D mode. Molecular scattering Table 1. LES Model Generated Cloud Properties ASTEX-St ASTEX-Sc ATEX-Cu Mean optical thickness a exp(log mean optical thickness) a Standard deviation optical thickness Shape parameter, n Cloud fraction Effective radius, mm Mean cloud top height, m Standard deviation cloud top height m Mean cloud base height m Cloud field tilt (x direction), b deg Cloud field tilt (y direction), b deg a The domain average t (both mean and log mean) includes both clear and cloudy portions but other properties are averaged over cloudy regions only. b Cloud field tilt is computed by the method explained by Hinkelman et al. [2005]. A positive (negative) angle indicates that clouds tilt toward the positive (negative) x or y direction with height. 3of16

4 Table 2. Domain-Averaged Albedo and Retrieved Cloud Properties at Nadir View Solar Zenith Angle, deg Optical Thickness Shape Parameter Mode Radius, a mm Albedo (0.64 mm) Albedo (3.7 mm) ASTEX-St Truth Retrieved Truth Retrieved ASTEX-Sc Truth Retrieved Truth Retrieved ATEX-Cu Truth Retrieved Truth Retrieved a Effective radius is 1.36 times larger than the mode radius. is included but gaseous absorption is neglected. The surface albedo is set to zero and cyclic boundary conditions are assumed. Narrowband reflectances (p times the upward radiance divided by the downward irradiance) at the height of the highest cloud top in each domain are then calculated from these results. The reflectances are computed at 13 combinations of angles composed of the instrument viewing zenith angles q =0, 30, and 60 and relative azimuth angles f =0,30,60, 120, 150, and 180, where f =0is when the instrument views toward the Sun (forward observing direction). The solar zenith angles used for the following analysis are 30 and 50. Direct solar radiation travels from left to right (positive x direction) in Figure 1 for all scenes. [9] Look-up tables for the cloud property retrievals are prepared using a 1-D radiative transfer model. In the first table, the 0.64 mm reflectance (hereinafter termed the 0.64 mm 1-D reflectance ) is given as a function of optical thickness, viewing zenith angle, relative azimuth angle, and solar zenith angle with the effective radius fixed at 13.6 mm. The other look-up table contains the 3.7 mm wavelength reflectance (hereinafter termed the 3.7 mm 1-D reflectance ) as a function of optical thickness and effective radius as well as viewing zenith angle, relative azimuth angle, and solar zenith angle. The optical thickness increment in both tables is 0.2 in logarithmic space. Four effective radii, 6.8, 13.6, 20.4 and 27.2 mm are used to compute the 3.7 mm 1-D reflectances. To avoid errors caused by differences in model assumptions, the look-up tables are built using SHDOM in 1-D mode. [10] To simulate typical satellite-based cloud retrievals, such as those performed using Moderate Resolution Imaging Spectroradiometer (MODIS [King et al., 1992]) data from the Terra and Aqua satellites, we first average the reflectances obtained from the full-resolution 3-D computations to 1 km resolution using a 1 km 1 km moving window. (These low-resolution values will be referred to as 1 km resolution 3-D reflectances. ) Pixels are identified as cloudy if the 1 km resolution 3-D reflectance at 0.64 mm is greater than a threshold of All cloudy pixels are considered overcast. The optical thickness for each cloudy pixel is retrieved by logarithmically interpolating between optical thickness values so that the tabulated 0.64 mm 1-D reflectance matches the measured 0.64 mm 1 km resolution 3-D reflectance. Using the retrieved optical thickness, the cloud particle effective radius is then retrieved by linearly interpolating the square root of effective radii to match the tabulated 3.7 mm 1-D reflectance with the measured 3.7 mm 1 km resolution 3-D reflectance. The resulting values are evaluated by comparison to the true values from the LES fields averaged up to 1 km resolution. [11] In order to assess the effect of cloud property retrieval errors on subsequent 1-D irradiance computations, we compute narrowband irradiances at 0.64 mm and 3.7 mm using the independent column approximation along with the retrieved cloud fractions, optical thicknesses, and effective radii. For consistency, we employ SHDOM in 1-D mode for these calculations. The error in the retrieved irradiances is evaluated by comparison to values from 3-D SHDOM radiative transfer computations performed on the original LES fields and averaged up to 1 km resolution. Because cloud droplets do not absorb at 0.64 mm, we examine only the albedo at this wavelength but evaluate the errors in the albedo, transmittance, and absorptance at 3.7 mm. 3. Results 3.1. Accuracy of Retrieved Domain-Averaged Cloud Properties [12] Cloud property retrieval results for nadir viewing angle are summarized in Table 2. While the retrieved cloud optical thickness averaged over all pixels is high for the overcast cloud (ASTEX-St), it is low for the nonovercast clouds (ASTEX-Sc and ATEX-Cu). Using the shape parameter n as a measure of horizontal inhomogeneity, the magnitude of the relative optical thickness error, defined as (retrieved - truth)/truth, increases as cloud horizontal inhomogeneity increases (i.e., n decreases). For example, the magnitude of the relative error in the domain-averaged optical thickness retrieved from nadir view with a solar zenith angle of 50 is 1%, 8%, and 27%, respectively, for ASTEX-St, ASTEX-Sc, and ATEX-Cu. (Note that the truth is defined as the linear average of the optical thickness including clear and cloudy columns. Note also that the domain-averaged retrieved value is computed using all possible 1 km 1 km pixels.) The magnitude of the average error in the effective radius retrieved at nadir view 4of16

5 Figure 2. Histograms of 1-km pixels sorted by true cloud fraction over a pixel and the error in the retrieved optical thickness or effective radius for nadir view retrievals. The solar zenith angle is 50. The line and shaded contours indicate the number of pixels in each bin. also increases with increasing cloud inhomogeneity. Compared to the true value of 13.6 mm, the retrieved domainaveraged values at a solar zenith angle of 50 are 13.6, 16.3, and 22.2 mm, respectively, for the stratus, stratocumulus, and cumulus scenes. [13] Much of the observed error in the retrieved optical depth and particle size values at nadir view for ATEX-Cu can be ascribed to cloud brokenness at scales smaller than the 1 km pixels used in the retrievals. When the 0.64 mm reflectance threshold is applied, every 1 km 1 km pixel in each scene is considered to be cloudy. This means that, on average, pixel cloudiness is overestimated in the stratocumulus and cumulus scenes (ASTEX-Sc and ATEX-Cu), for which the true cloud fraction is less than 1. (See cloud properties in Table 1.) In the extreme case of the ATEX-Cu scene, for which the retrieval errors are the largest, approximately 90% of the 1 km nadir view pixels are only partially filled by clouds. When overcast cloud is assumed for a partially filled pixel, a smaller optical thickness is needed to match the observed 0.64 mm reflectance in the retrieval [e.g., Oreopoulos and Davies, 1998]. The underestimation of the optical thickness leads to overestimation of the particle size in the second phase of the retrieval. Figure 2 shows 2-D histograms of 1 km size pixels sorted by true cloud fraction within a pixel and the error in the retrieved cloud optical thickness and effective radius from the pixel. Errors in both retrieved properties are seen to increase as true pixel cloud fraction decreases. Nevertheless, a significant error exists even for truly overcast pixels in ASTEX-Sc and ATEX-Cu. [14] Inaccurate treatment of cloud inhomogeneity in the radiative transfer computations contributes to retrieval errors for both partly cloudy and overcast pixels. A wellknown aspect of this mistreatment is the averaging of nonlinear reflectance function values within a pixel, which causes the retrieved optical thickness to be small [e.g., Zuidema and Evans, 1998; Oreopoulos and Davies, 1998)]. Error also results because the retrieval algorithm 5of16

6 Figure 3. Error in the domain-averaged (a) optical thickness (t) and (b) effective radius (r e ) retrieved from the 1 km resolution reflectance computed with the independent column approximation relative to the true value. Difference between the (c) t and (d) r e retrieved from the 1 km resolution 3-D reflectance and those retrieved from the 1 km resolution reflectance computed with the independent column approximation normalized by the true value. The relative azimuth angle is 30 and 150 for the forward (positive viewing zenith angle) and backward (negative viewing zenith angle) observing directions, respectively. The solar zenith angle is 50. is unable to account for the radiative transport that occurs between neighboring pixels with different scattering coefficients in the true 3-D processes. To estimate the contribution of neglecting horizontal fluxes to retrieved property errors, the retrieval process is repeated for reflectances computed using the independent column approximation (ICA [Cahalan et al., 1994]). As for the 1 km resolution 3-D reflectances, the ICA reflectances are computed for the full-resolution cloud fields and then averaged over all possible 1 km pixels. Figure 3 shows the difference of retrieved optical thickness and effective radius relative to the true value (Figures 3a and 3b). It also shows the difference of the retrieved value from 1 km 3-D reflectance and from 1 km ICA reflectance divided by the true value (Figure 3c and 3d). Therefore the difference shown in 3a and 3b is due to averaging of nonlinear reflectance function values within a pixel and the difference shown in 3c and 3d is due to neglecting horizontal fluxes. (The sum of these two errors is the error shown in Figure 4.) Figure 3 shows that both averaging nonlinear function over a pixel and neglecting horizontal flux contribute the retrieved property errors. Because inhomogeneity within and outside a pixel is determined by cloud fields and both exist together, these two factors contribute the error when they are neglected. [15] As shown in Figure 4, the error in the average retrieved optical thickness exhibits a strong dependence on viewing zenith angle; the absolute value of the relative difference between the retrieved and true values increases with viewing zenith angle, especially in the forward observing direction. This viewing zenith angle dependence also increases with increasing horizontal inhomogeneity. Loeb and Coakley [1998] show that the retrieved optical thickness from overcast pixels decreases with viewing zenith angle in the forward observing direction but is not 6of16

7 Figure 4. Relative difference between the domain-averaged retrieved and true values of optical thickness and effective radius for ASTEX-St, ASTEX-Sc, and ATEX-Cu scenes as a function of instrument viewing zenith angle used for the retrieval. Positive and negative viewing zenith angles indicate the forward and backward observing directions, respectively. The solar zenith angle is 50, and the cloud properties are retrieved from 1 km resolution radiances. Each line indicates a different relative azimuth angle. as sensitive to the viewing zenith angle in the backward observing direction for almost all solar zenith angles. Their resolution changes from 4 to 13 km depending on viewing zenith angle but it is constant in this study. Our result shows that the retrieved optical thickness decreases with viewing zenith angle in the backward observing direction but that the angular dependence is less pronounced compared with that in the forward direction. The error in the retrieved domainaveraged effective radius also shows a strong viewing zenith angle dependence (Figure 4); the relative difference is greatest when the effective radius is retrieved from oblique forward observing directions Effect of Retrieval Errors on Computed Irradiances [16] Figure 5 shows the relative error in the domainaveraged albedo as a function of the viewing zenith angle used in the cloud property retrievals. For all scenes, the 0.64 mm domain-averaged albedo for a solar zenith angle of 30 is smaller than the corresponding 3-D albedo when cloud optical thickness retrieved from the nadir view is used (Table 2). This indicates that, for the solar zenith angle of 30, the effect of the error in the retrieved optical thickness is larger than the effect of the error in the retrieved cloud fraction. When the solar zenith angle is 50, the 0.64 mm domain-averaged albedo from nadir retrievals exceeds the 3-D albedo for ASTEX-Sc and ATEX-Cu scenes. The reason for this albedo error increase with solar zenith angle is discussed later in this section. The shape of the albedo error curve as a function of viewing zenith angle is similar to the shape of the retrieved optical thickness error. However, the sign of the error for ASTEX-Sc and ATEX-Cu scenes changes depending on the viewing zenith angle when the solar zenith angle is 50 (Figure 5) as opposed to the optical thickness error, which is negative for all viewing zenith angles (Figure 2). While the magnitude of the relative errors in retrieved optical thickness and albedo are similar for the ASTEX-St scene, the relative albedo error is less than the relative retrieved optical thickness error for almost all angles for the other two scenes. This indicates that the sensitivity of albedo to changes in radiance is smaller than the sensitivity of retrieved optical thickness to changes in radiance [e.g., Zuidema and Evans, 1998]. This is partly because the retrieved optical thickness error and cloud fraction error partially cancel in albedo computations. [17] For all three scenes, the 3.7 mm domain-averaged albedos are smaller for a solar zenith angle of 30 (Table 2) and larger for a solar zenith angle of 50 than the corresponding 3-D albedos when retrieved cloud optical 7of16

8 Figure 5. Relative error in the 0.64 mm albedo computed with retrieved cloud optical thickness as a function of the viewing angle from which optical thickness is retrieved. Positive and negative viewing zenith angles indicate the forward and backward observing directions, respectively. The solar zenith angle is (top) 30 and (bottom) 50, and the cloud properties are retrieved from 1 km resolution 3-D radiance. Different lines indicate different relative azimuth angles of the radiance used for the retrieval. The error bars are estimated from 2-D cloud fields with SHDOM by the method described in Appendix A. thickness and effective radius from the nadir view are used (Figure 6). The transmittance computed with retrieved cloud properties is larger for ASTEX-St. Because the retrieved cloud fraction is 1 for ASTEX-SC and ATEX-Cu, which leads to underestimation of the transmittance of the direct irradiance, the computed transmittance (direct plus diffuse) of ASTEX-Sc and ATEX-Cu is smaller than the true values. Although the error bars for the ATEX-Cu scene results at 3.7 mm, shown in Figure 6, are large, it is clear that the large error in the retrieve transmittance is caused by the severe overestimation of the cloud fraction in this case. We keep the ATEX-Cu 3.7 mm results despite the size of their estimated error since they make physical sense when compared to the results of the other scenes in this study. [18] Marshak et al. [1998] show that absorption computed with the independent column approximation for a cloud layer is less than the absorption from a 3-D computation for a range of solar zenith angles and fractal cloud scenes. This is because horizontal fluxes increase photon path lengths. Therefore we expect the absorption computed using the retrieved properties and independent column approximation to be low, if the retrieval error is small. Our results for the ASTEX-St scene indicate that cloud absorption computed with retrieved cloud properties is, in fact, lower than the true value. However, for the two nonovercast scenes (ASTEX-Sc and ATEX-Cu), the absorption is overestimated when retrieved cloud properties are used. Errors in both retrieved cloud fraction and effective radius contribute to this overestimation. For example, using the true value of the effective radius of 13.6 mm instead of the retrieved value from nadir (the average retrieved value is 20% larger) and using the retrieved optical thickness and cloud fraction from nadir reduces the absorptance error from 18% to 10% for the ASTEX-Sc scene. (The reduction is approximately equivalent to the square root of the ratio of retrieved and true effective radii.) Using both the true effective radius and cloud fraction further reduces the error to 8%. For the ATEX-Cu scene, which has a larger absolute absorptance error than the ASTEX-Sc, the cloud fraction error dominates in the error; using the true cloud fraction and nadir retrieved cloud optical thickness and effective radius reduces the absorptance error from 35% to less than 1%. Therefore the magnitude of the contribution of the cloud fraction and effective radius errors to the absorptance error depends on the cloud field. In addition, even though the absorption estimated with the independent column approximation is usually less than that from the 3-D calculation, a study by Hinkelman [2003] suggests that the sign of the absorption error depends on cloud field geometry even if the true cloud properties are used; when clouds tilt toward the direct 8of16

9 Figure 6. Relative error in the 3.7 mm albedo, transmittance, and absorptance computed with retrieved cloud optical thickness as a function of the viewing angle from which optical thicknesses are retrieved. Positive and negative viewing zenith angles indicate the forward and backward observing directions, respectively. The solar zenith angle is 50, and the cloud properties are retrieved from 1 km resolution 3-D radiance. Different lines indicate different relative azimuth angles of the radiance used for the retrieval. The error bars are estimated from 2-D cloud fields with SHDOM by the method described in Appendix A. solar radiation (as in ASTEX-St and ASTEX-Sc, Table 1), the independent column approximation can overestimate absorption. [19] Albedos computed from retrieved cloud properties are sometimes averaged over a larger grid box (e.g., [Zhang et al., 1995; Rossow and Zhang, 1995], 1 1 [Young et al., 1998]). While the viewing zenith angle for a specific region is fixed for a geostationary satellite, over time, the same region is viewed from a range of viewing zenith angles by instruments on a Sun-synchronous satellite such as Terra or Aqua. Because the albedo error changes sign depending on viewing zenith angle, as shown in Figures 5 and 6, the errors partially cancel when albedos computed with cloud properties derived from a range of viewing angles are averaged. A typical viewing zenith angle distribution over a 1 1 region derived for CERES instruments operated in cross-track mode on Terra by collecting data over a month is shown in Figure 7. (This is close to the distribution of MODIS viewing zenith angles since MODIS also scans cross-track on the same platforms.) A1 by 1 area is normally viewed from one or two Terra orbits per day. In particular, nadir views, which require the satellite to pass directly over the region, are clustered on a few days each month. Consequently, when the monthly mean viewing zenith angle over a 1 1 area is derived from equally weighted daily mean values, the relative weight of the nadir view is reduced. We use the viewing zenith angle frequency distribution shown in Figure 7 to 9of16

10 Figure 7. Distribution of daily averaged viewing zenith angles derived from a month of CERES cross-track data over a 1 by 1 region. The relative azimuth angle is distributed in a narrow range because the instrument is on a Sun-synchronous satellite. average the errors in the albedos computed from retrieved cloud properties in order to estimate the error in the albedo averaged over a 1 1 region and over a month due to neglecting horizontal cloud inhomogeneity in both the cloud property retrieval algorithm and albedo computations. [20] Figure 8 shows the solar zenith angle-dependent albedo error relative to the 3-D albedo when the albedo is weighted by the viewing zenith angle frequency distribution shown in Figure 7. The magnitude of the 0.64 mm albedo error for ASTEX-St and ASTEX-Sc is less than 1.5% and 5.0% when the solar zenith angle is less than or equal to 50. Similarly, the magnitude of the 3.7 mm albedo error for ASTEX-St and ASTEX-Sc is less than 0.6% and 4.1% when the solar zenith angle is less than or equal to 50. The error is larger for ATEX-Cu for solar zenith angles of 50 or 70, but comparable to the other two scenes when the solar zenith angle is 10 or 30. The error in the albedos computed using the independent column approximation and true cloud properties, also shown in Figure 8, is comparable in magnitude. Note that the domain-averaged ICA albedo for the true cloud properties shown in Figure 8 was computed by a Monte Carlo model because it computes domain-averaged values more effectively and accurately than SHDOM does (F. Evans, personal communication, 2005). Figure 8. Relative albedo error for a 1 1 region as a function of solar zenith angle (open circles and squares). The albedo error is defined as (the albedo computed with retrieved cloud properties - 3-D albedo)/3-d albedo. The error is averaged using the viewing zenith angle distribution function shown in Figure 7. The error bars are estimated using the method described in Appendix A. The dotted lines with solid circles indicate the relative error defined as (ICA albedo - 3-D albedo)/3-d albedo where the ICA albedo for this error estimate is computed with the true cloud properties by a Monte Carlo model. 10 of 16

11 Figure 9. (a) Relative difference between the ICA and 3-D albedos at 0.64 mm as a function of solar zenith angle. Here the retrieved optical thickness from nadir view is used in the ICA computations. The error bars indicate the relative RMS error of the ICA albedo. (b) Relative difference between the ICA and 3-D nadir reflectances, both computed with the true cloud optical thicknesses. (c) Relative difference between the domain-averaged retrieved and true optical thicknesses. [21] In all three cloud scenes, the error in both the 0.64 and 3.7 mm albedos increases with solar zenith angle (Figures 8 and 9a). To investigate these increases, the nadir view 0.64 mm reflectance difference (ICA - 3-D) is plotted as a function of solar zenith angle (Figure 9b). Figure 9b shows that the nadir 3-D reflectance is larger than the reflectance computed with independent column approximation for all three scenes when the solar zenith angle is 70. This nadir view radiance difference at large solar zenith angles agrees with the results of Loeb et al. [1997]. As a consequence, the retrieved optical thickness from nadir view and the albedo computed with it increase with solar zenith angle (Figures 9a and 9c), which also agrees with the results of Zuidema and Evans [1998]. Note that, because the retrieved cloud fraction is larger than the true cloud fraction, the domain averaged albedo computed with retrieved properties can be larger than the 3-D albedo even though the retrieved domain-averaged optical thickness is smaller than the true value (e.g., ATEX-Cu). [22] Although we only evaluate the albedo error for two monochromatic wavelengths, the error at these two wavelengths provides an approximate upper bound of the error for broadband albedo calculations because the effect of the horizontal flux is the largest at wavelengths where absorption is negligible (e.g., 0.64 mm). Because near-ir optical thickness is the visible optical thickness multiplied by the ratio of the extinction cross sections (the near-ir extinction cross section divided by the visible extinction cross section), it is affected by the error in both the retrieved optical thickness and effective radius. Therefore we have evaluated the error at near-ir wavelengths to assess the error in both the retrieved optical thickness and effective radius. Including gaseous absorption is expected to decrease overall horizontal inhomogeneity because absorbing gases are more horizontally uniform than cloud properties. Therefore neglecting gaseous absorption as done in this study also provides an upper bound of the error caused by the net horizontal flux at near-ir wavelengths. 4. Discussion [23] The fact that the retrieved cloud optical thickness varies with viewing angle indicates that the radiance reflected by a plane parallel cloud cannot consistently characterize the radiance angular dependence from 3-D cloud fields with one value of the optical thickness. Similarly, a plane parallel cloud with one combination of optical thickness and effective radius cannot characterize the radiance angular distribution at near-ir wavelengths. Because the irradiance is the radiance integrated over a hemisphere, the irradiance computed with retrieved cloud properties is in general different than the irradiance from the 3-D radiative transfer calculation even though the radiance at one angle is forced to agree. [24] To understand the error in the retrieved properties and the impact on the albedo computed with them, we therefore need to understand the angular dependence of the radiance. Figure 10 shows the error in the reflectance computed with the independent column approximation (r ICA r 3D ), computed with a plane parallel horizontally uniform cloud (r pp r 3D ) and computed with flat boundary clouds (r 3DFB r 3D ). All three differences are from the 3-D reflectance computed for the original cloud fields. The flat boundary clouds are idealized scenes with the same liquid water paths in each column as in the original scenes but constant cloud top and base heights (i.e., flat boundaries). In each scene, the cloud top (base) height is selected to match the highest (lowest) cloud top (base) in the corresponding original domain. The optical thickness of the plane parallel cloud is computed by logarithmically averaging the true cloud optical thickness over the domain. The radiance of the partially cloudy scene (ASTEX-Sc) plane parallel cloud is derived by averaging the clear-sky and cloudy-sky radiances weighted by the fractional area they cover. The (r pp r 3D ) 11 of 16

12 Figure 10. Absolute differences between domain-averaged reflectances computed using various cloud structure and radiative transfer approximations. Solid line indicates independent column approximation (r ICA ) and full 3-D calculations (r 3D ), dash-dotted line indicates 3-D calculation using constant cloud boundary heights (r 3DFB ) and full 3-D calculation (r 3D ), and dashed line indicates 1-D computation for plane parallel cloud (r pp ) and full 3-D computation (r 3D ). The true optical thickness values for the ASTEX-Sc scene are employed for the 3-D and ICA computations. For the computation with the plane parallel cloud, the optical thickness is derived by logarithmically averaging the true values. The clear-sky and cloudy-sky reflectances are then combined using their fractional area coverage as weights. concave lines indicate that the oblique view difference minus nadir view difference is positive. Therefore the radiance from a plane parallel cloud is less isotropic than that from a 3-D cloud especially when the solar zenith angle is 50. Because the line is also concave, the radiance from flat boundary clouds is also less isotropic than the radiance from clouds with variable boundaries, which indicates that cloud boundary variability makes the radiance more isotropic. This agrees with the results by Loeb et al. [1998]. [25] The reflectance computed with the independent column approximation is close to that from 3-D computation. The 0.64 mm reflectance computed using the independent column approximation and the true cloud properties is approximately 1 to 2% larger than r 3D for ASTEX-Sc when the solar zenith angle is 30 and 50. This positive error is partly responsible for the low optical thickness value retrieved from the scene. Similarly, the 3.7 mm reflectance computed using the independent column approximation and the true cloud properties is up to 2% larger for ASTEX-Sc when the solar zenith angle is 30 and 50, which causes the retrieved effective radius from the scene to be high. Note that the ICA computed radiance is less isotropic if retrieved optical thickness is used because the domain-averaged retrieved optical thickness is smaller than the true value (Figure 2) and the radiance reflected by thinner clouds over a dark surface is less isotropic [Loeb et al., 2000]. [26] To illustrate why the radiance angular distributions from 3-D and ICA computations differ, Figure 11 shows a vertical cross section of the ASTEX-Sc cloud field along with the difference between the 3-D (with variable cloud top) and ICA reflectances. The 3-D and ICA reflectances are computed with true cloud properties with cloud field resolutions and are averaged to a 1 km resolution. The variation of cloud top height allows an instrument to view 12 of 16

13 Figure 11. (a) Difference between the ICA and 3-D 0.64 mm reflectance (r ICA r 3D ), (b) corresponding absolute error in the retrieved optical thicknesses, (c) difference between the ICA and 3-D 3.7 mm reflectances, and (d) corresponding absolute error in the retrieved effective radii. (e) Section of the ASTEX-Sc cloud field used for the retrieval shown with contours of log 10 of the extinction coefficient. All radiances are averaged to a 1 km resolution. The solar zenith angle is 50, and direct solar radiation travels from upper left to lower right. deeper parts of clouds that are less illuminated by the direct solar radiation so that they appear to be darker when the radiance is observed at an oblique view. A cloud top viewed from nadir view may or may not be illuminated by direct solar radiation depending on the cloud top structure and solar zenith angle, so that darkening is less pronounced compared with that from an oblique view. In addition, a part of clouds illuminated by other cloud parts might be apparent from nadir view. Therefore variable cloud top structure makes the radiance angular distribution more isotropic (r 3D versus r 3DFB in Figure 9). However, the effect of the horizontal flux due to internal scattering coefficient variability appears to counteract the effect of the horizontal flux on the angular distribution of the reflectance due to cloud boundary variability because r ICA r 3D is smaller than r 3DFB r 3D (Figure 10); the error with no significant viewing zenith angle dependence shown in Figure 3c also supports this. [27] Because the reflectance change with respect to the optical thickness change is angle-dependent, forcing the reflectance to agree with the 3-D 1 km resolution reflectance at one angle is not sufficient for an agreement of the radiance at other angles. Although in order for the albedo computed with retrieved properties to be correct requires only that the reflectance error integrates to zero over the hemisphere, which is less restrictive than requiring the reflectance to be correct at all angles, it is unlikely that the error from all angles will exactly cancel out when the radiance is matched at only one angle. [28] When the albedo is computed from the retrieved optical thickness and effective radius, the relative albedo difference is either nearly equivalent (ASTEX-St) or smaller than (ASTEX-Sc and ATEX-Cu) the relative difference in the retrieved properties, while the shape of the relative difference as a function of viewing angle is similar to the shape of the optical thickness relative difference. When the albedo derived from optical thickness from nadir and oblique views is averaged, the error is further reduced because the error in nadir and oblique views partially cancel. When the solar zenith angle is large (e.g., 70 ), the error in the albedo computed with optical thickness derived from the nadir view is dominated. The nadir view retrieval error is caused by the large difference between ICA and 3-D reflectances. Because the optical thickness derived from nadir increases with solar zenith angle, the relative error in the domain-averaged albedo also increases with solar zenith angle. This strong solar zenith angle-dependent albedo error in all three scenes at solar zenith angle greater than 50 causes a bias error in radiation budget estimates at high latitudes when cloud properties derived from a satellite are used for albedo computations and when latitude and solar zenith angle are correlated; zonally averaged albedos computed from retrieved optical thickness might be affected by the systematic bias error correlated with latitude. [29] The result of this study also suggests a bias error in the retrieved effective radius. A survey of recent in situ cloud particle size measurements indicates that the average effective radius of marine stratiform clouds is 9.6 ± 2.4 mm [Miles et al., 2000]. Effective radii of 15mm or larger, however, are often retrieved from satellite measurements. Even though particle size tends to increase with height and the effective radius from in situ measurements might not represent the global variation, a question arises as to what causes these large retrieved effective radii that are seldom found in in situ measurements. The increase retrieved particle size because of applying an algorithm based on 1-D radiative transfer theory to highly inhomogeneous clouds, in part perhaps, contributes to the large values seen in retrieved effective radii. Therefore an analysis of retrieved effective radii from satellites needs to include only those 13 of 16

14 Figure 12. Relative 0.64 mm domain-averaged albedo error and relative RMS difference as a function of pixel size of the instrument. The relative albedo error is defined as the domain averaged albedo computed with retrieved cloud properties minus the 3-D albedo divided by the 3-D albedo. The relative RMS difference is defined as the RMS albedo difference computed column by column divided by the 3-D domain-averaged albedo. The solid and dashed lines are for the solar zenith angles of 30 and 50, respectively. The relative azimuth angles used for the retrieval are 30 and 150, and the distribution shown in Figure 7 is used for averaging over viewing zenith angles. from uniform clouds to avoid the error in the effective radii affecting the analysis. [30] We used 1 km pixels in this study but the retrieved cloud fraction and optical thickness depend on the pixel size [Davis et al., 1997]. In order to determine an optimum pixel size for albedo computations to obtain a small albedo error and to detect a possible large albedo error from highly inhomogeneous clouds, we simulate the retrieval process changing the size over which the radiance is averaged. Figure 12 shows the error of the domain-averaged albedo and root mean square (RMS) error relative to the domainaveraged 3-D albedo as a function of pixel size. The domain-averaged albedo computed with retrieved cloud properties is further averaged using viewing zenith angle distribution function described earlier. The RMS error is computed by comparing the albedo computed with retrieved cloud properties with the 3-D albedo from each column. The RMS error is also averaged using the viewing zenith angle distribution function. The relative error in the domainaveraged albedo for almost all scenes decreases as the pixel size increases from a few hundred meters to a few kilometers when the solar zenith angle is 30 and 50. The relative albedo error for the ATEX-Cu case increases as the pixel size increases from 1 km when the solar zenith angle is 50. The RMS error decreases as pixel size increases especially when the pixel size is greater than about 1 km. This suggests that about 1 to 3 km is an optimum pixel size for cloud property retrievals if those are used to compute the albedo. Even though the domain size of these three cloud scenes is relatively small compared with the domain size typically used for a radiation budget estimate, which possibly underestimates the variability of the optical thickness within the domain, our result indicates that the shape parameter derived from 1 km size pixels can distinguish these three cloud scenes. Our result also shows that the albedo error increases with decreasing the shape parameter. Therefore the shape parameters computed from the optical thickness derived from 1 km pixels can be used as an indicator of possibly large albedo errors. 5. Conclusions [31] We used three cloud fields generated by large eddy simulation models to investigate the error in cloud optical thickness and effective radius retrieved using a typical twowavelength algorithm based on 1-D radiative transfer theory as well as the error in the irradiances computed from these retrieved properties. The cloud fields used in this study include one overcast and two nonovercast scenes but all scenes are considered overcast by the retrieval. For the overcast scene, the retrieved optical thickness is larger (smaller) than the true value when it is retrieved from nadir (oblique views). The domain-averaged retrieved optical thickness is smaller than the true value when pixels are assumed to be overcast even though the true cloud fraction is less than 1. In this study, the retrieved optical thickness error is found to increase as horizontal inhomogeneity expressed by the retrieved shape parameter increases; for nadir view retrievals at the solar zenith angle of 50, the relative error in the domain-averaged retrieved optical thickness is 1% for a scene with the retrieved shape parameter of 60 and increases to 27% for a scene with the retrieved shape parameter of 0.9. The domain-averaged retrieved effective radius is larger than the true value for all three scenes. When the solar zenith angle is 50, the relative error in the domain-averaged effective radius is negligible for a scene with the retrieved shape parameter of 60 and the relative error increases to 63% for a scene with the retrieved shape parameter of 0.9. The observed errors in the retrieved optical thickness and effective radius are caused by neglecting horizontal fluxes, assuming uniform cloud over a pixel, and using the radiance averaged over a pixel for the retrieval. 14 of 16