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## Abstract

Histograms of nearest neighbor spacings of fair weather cumulus at 15 locations Over the world's oceans are presented based on the analysis of high resolution LANDSAT 3 Multispectral Scanner images for amounts of cloud cover ranging from 0.6% to 37.6%. These histograms are found to be essentially the same at all locations analysed, similarly to our previous findings on the size distributions and the fractal dimensions of the perimeters for this cloud type.

The nearest neighbor spacings are linearly dependent on the effective cloud radii, with a proportionality factor ranging from five to twenty. The histograms peak at about 0.5 km. Nearest-neighbor spacings smaller than about a kilometer, associated with cumulus clouds with an effective radius less than a few hundred meters, have a distribution of cloud centers that is almost independently distributed in the horizontal plane and show a tendency for the formation of clumps. Larger spacings of up to thirty kilometers occur and are associated with the larger clouds. These latter spacings are not independent.

## Abstract

Histograms of nearest neighbor spacings of fair weather cumulus at 15 locations Over the world's oceans are presented based on the analysis of high resolution LANDSAT 3 Multispectral Scanner images for amounts of cloud cover ranging from 0.6% to 37.6%. These histograms are found to be essentially the same at all locations analysed, similarly to our previous findings on the size distributions and the fractal dimensions of the perimeters for this cloud type.

The nearest neighbor spacings are linearly dependent on the effective cloud radii, with a proportionality factor ranging from five to twenty. The histograms peak at about 0.5 km. Nearest-neighbor spacings smaller than about a kilometer, associated with cumulus clouds with an effective radius less than a few hundred meters, have a distribution of cloud centers that is almost independently distributed in the horizontal plane and show a tendency for the formation of clumps. Larger spacings of up to thirty kilometers occur and are associated with the larger clouds. These latter spacings are not independent.

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## Abstract

Two full months (July 2003 and January 2004) of Moderate Resolution Imaging Spectroradiometer (MODIS) Atmosphere Level-3 data from the *Terra* and *Aqua* satellites are analyzed in order to characterize the horizontal variability of vertically integrated cloud optical thickness (“cloud inhomogeneity”) at global scales. The monthly climatology of cloud inhomogeneity is expressed in terms of standard parameters, initially calculated for each day of the month at spatial scales of 1° × 1° and subsequently averaged at monthly, zonal, and global scales. Geographical, diurnal, and seasonal changes of inhomogeneity parameters are examined separately for liquid and ice phases and separately over land and ocean. It is found that cloud inhomogeneity is overall weaker in summer than in winter. For liquid clouds, it is also consistently weaker for local morning than local afternoon and over land than ocean. Cloud inhomogeneity is comparable for liquid and ice clouds on a global scale, but with stronger spatial and temporal variations for the ice phase, and exhibits an average tendency to be weaker for near-overcast or overcast grid points of both phases. Depending on cloud phase, hemisphere, surface type, season, and time of day, hemispheric means of the inhomogeneity parameter *ν* (roughly the square of the ratio of optical thickness mean to standard deviation) have a wide range of ∼1.7 to 4, while for the inhomogeneity parameter *χ* (the ratio of the logarithmic to linear mean) the range is from ∼0.65 to 0.8. The results demonstrate that the MODIS Level-3 dataset is suitable for studying various aspects of cloud inhomogeneity and may prove invaluable for validating future cloud schemes in large-scale models capable of predicting subgrid variability.

## Abstract

Two full months (July 2003 and January 2004) of Moderate Resolution Imaging Spectroradiometer (MODIS) Atmosphere Level-3 data from the *Terra* and *Aqua* satellites are analyzed in order to characterize the horizontal variability of vertically integrated cloud optical thickness (“cloud inhomogeneity”) at global scales. The monthly climatology of cloud inhomogeneity is expressed in terms of standard parameters, initially calculated for each day of the month at spatial scales of 1° × 1° and subsequently averaged at monthly, zonal, and global scales. Geographical, diurnal, and seasonal changes of inhomogeneity parameters are examined separately for liquid and ice phases and separately over land and ocean. It is found that cloud inhomogeneity is overall weaker in summer than in winter. For liquid clouds, it is also consistently weaker for local morning than local afternoon and over land than ocean. Cloud inhomogeneity is comparable for liquid and ice clouds on a global scale, but with stronger spatial and temporal variations for the ice phase, and exhibits an average tendency to be weaker for near-overcast or overcast grid points of both phases. Depending on cloud phase, hemisphere, surface type, season, and time of day, hemispheric means of the inhomogeneity parameter *ν* (roughly the square of the ratio of optical thickness mean to standard deviation) have a wide range of ∼1.7 to 4, while for the inhomogeneity parameter *χ* (the ratio of the logarithmic to linear mean) the range is from ∼0.65 to 0.8. The results demonstrate that the MODIS Level-3 dataset is suitable for studying various aspects of cloud inhomogeneity and may prove invaluable for validating future cloud schemes in large-scale models capable of predicting subgrid variability.

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## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

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## Abstract

This paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feed-back, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a “slope-stability” theorem; i.e., if the local slope of the steady-state icelinc latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed in the text.

## Abstract

This paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feed-back, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a “slope-stability” theorem; i.e., if the local slope of the steady-state icelinc latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed in the text.

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## Abstract

Landsat Multispectral Scanner (MSS) and Thematic Mapper (TM) data, with 80 and 30 m spatial resolution, respectively, have been employed to study the spatial structure of boundary-layer and intertropical convergence zone (ITCZ) clouds. The probability distributions of cloud area and cloud perimeters are found to approximately follow a power-law, with a different power (i.e., fractal dimension) for each cloud type. They are better approximated by a double power-law behavior, indicating a change in the fractal dimension at a characteristic size which depends upon cloud type. The fractal dimension also changes with threshold. The more intense cloud areas are found to have a higher perimeter fractal dimension, perhaps indicative of the increased turbulence at cloud top. A detailed picture of the inhomogeneous spatial structure of various cloud types will contribute to a better understanding of basic cloud processes, and also has implications for the remote sensing of clouds, for their effects on remote sensing of other parameters, and for the parameterization of clouds in general circulation models, all of which rely upon plane-parallel radiative transfer algorithms.

## Abstract

Landsat Multispectral Scanner (MSS) and Thematic Mapper (TM) data, with 80 and 30 m spatial resolution, respectively, have been employed to study the spatial structure of boundary-layer and intertropical convergence zone (ITCZ) clouds. The probability distributions of cloud area and cloud perimeters are found to approximately follow a power-law, with a different power (i.e., fractal dimension) for each cloud type. They are better approximated by a double power-law behavior, indicating a change in the fractal dimension at a characteristic size which depends upon cloud type. The fractal dimension also changes with threshold. The more intense cloud areas are found to have a higher perimeter fractal dimension, perhaps indicative of the increased turbulence at cloud top. A detailed picture of the inhomogeneous spatial structure of various cloud types will contribute to a better understanding of basic cloud processes, and also has implications for the remote sensing of clouds, for their effects on remote sensing of other parameters, and for the parameterization of clouds in general circulation models, all of which rely upon plane-parallel radiative transfer algorithms.

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## Abstract

The interannual variability (IAV) in monthly averaged outgoing infrared radiation (IR, from the NOAA polar orbiting satellites) is observed to be larger during summer than during winter over the north Pacific Ocean. A statistical analysis of the daily observations shows the daily variance to be similar during both seasons while the autocorrelation function is quite different. This leads to a seasonal difference in estimates of the climatic noise level, i.e., the variances expected in summer and winter monthly averages due to the number of effectively independent samples in each average. Because of a less vigorous tropospheric circulation, monthly means of IR during summer are affected by the passage of fewer synoptic-scale disturbances and their attendant cloudiness. Fewer independent samples imply a larger variance in the time averages. While the observed IAV is less in winter, the ratio of the observed IAV to the climatic noise level is larger, suggesting that signals of climatic variability in outgoing IR may be more readily diagnosed during winter in this region. The climatic noise level in monthly averaged IR and cloudiness is also estimated for two other climatic regimes—the quiescent subtropical north Pacific and the ITCZ in the western Pacific.

## Abstract

The interannual variability (IAV) in monthly averaged outgoing infrared radiation (IR, from the NOAA polar orbiting satellites) is observed to be larger during summer than during winter over the north Pacific Ocean. A statistical analysis of the daily observations shows the daily variance to be similar during both seasons while the autocorrelation function is quite different. This leads to a seasonal difference in estimates of the climatic noise level, i.e., the variances expected in summer and winter monthly averages due to the number of effectively independent samples in each average. Because of a less vigorous tropospheric circulation, monthly means of IR during summer are affected by the passage of fewer synoptic-scale disturbances and their attendant cloudiness. Fewer independent samples imply a larger variance in the time averages. While the observed IAV is less in winter, the ratio of the observed IAV to the climatic noise level is larger, suggesting that signals of climatic variability in outgoing IR may be more readily diagnosed during winter in this region. The climatic noise level in monthly averaged IR and cloudiness is also estimated for two other climatic regimes—the quiescent subtropical north Pacific and the ITCZ in the western Pacific.

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## Abstract

The authors present the global plane-parallel shortwave albedo bias of liquid clouds for two months, July 2003 and January 2004. The cloud optical properties necessary to perform the bias calculations come from the operational Moderate Resolution Imaging Spectroradiometer (MODIS) *Terra* and MODIS *Aqua* level-3 datasets. These data, along with ancillary surface albedo and atmospheric information consistent with the MODIS retrievals, are inserted into a broadband shortwave radiative transfer model to calculate the fluxes at the atmospheric column boundaries. The plane-parallel homogeneous (PPH) calculations are based on the mean cloud properties, while independent column approximation (ICA) calculations are based either on 1D histograms of optical thickness or joint 2D histograms of optical thickness and effective radius. The (positive) PPH albedo bias is simply the difference between PPH and ICA albedo calculations. Two types of biases are therefore examined: 1) the bias due to the horizontal inhomogeneity of optical thickness alone (the effective radius is set to the grid mean value) and 2) the bias due to simultaneous variations of optical thickness and effective radius as derived from their joint histograms. The authors find that the global bias of albedo (liquid cloud portion of the grid boxes only) is ∼+0.03, which corresponds to roughly 8% of the global liquid cloud albedo and is only modestly sensitive to the inclusion of horizontal effective radius variability and time of day, but depends strongly on season and latitude. This albedo bias translates to ∼3–3.5 W m^{−2} of bias (stronger negative values) in the diurnally averaged global shortwave cloud radiative forcing, assuming homogeneous conditions for the fraction of the grid box not covered by liquid clouds; zonal values can be as high as 8 W m^{−2}. Finally, the (positive) broadband atmospheric absorptance bias is about an order of magnitude smaller than the albedo bias. The substantial magnitude of the PPH bias underlines the importance of predicting subgrid variability in GCMs and accounting for its effects on cloud–radiation interactions.

## Abstract

The authors present the global plane-parallel shortwave albedo bias of liquid clouds for two months, July 2003 and January 2004. The cloud optical properties necessary to perform the bias calculations come from the operational Moderate Resolution Imaging Spectroradiometer (MODIS) *Terra* and MODIS *Aqua* level-3 datasets. These data, along with ancillary surface albedo and atmospheric information consistent with the MODIS retrievals, are inserted into a broadband shortwave radiative transfer model to calculate the fluxes at the atmospheric column boundaries. The plane-parallel homogeneous (PPH) calculations are based on the mean cloud properties, while independent column approximation (ICA) calculations are based either on 1D histograms of optical thickness or joint 2D histograms of optical thickness and effective radius. The (positive) PPH albedo bias is simply the difference between PPH and ICA albedo calculations. Two types of biases are therefore examined: 1) the bias due to the horizontal inhomogeneity of optical thickness alone (the effective radius is set to the grid mean value) and 2) the bias due to simultaneous variations of optical thickness and effective radius as derived from their joint histograms. The authors find that the global bias of albedo (liquid cloud portion of the grid boxes only) is ∼+0.03, which corresponds to roughly 8% of the global liquid cloud albedo and is only modestly sensitive to the inclusion of horizontal effective radius variability and time of day, but depends strongly on season and latitude. This albedo bias translates to ∼3–3.5 W m^{−2} of bias (stronger negative values) in the diurnally averaged global shortwave cloud radiative forcing, assuming homogeneous conditions for the fraction of the grid box not covered by liquid clouds; zonal values can be as high as 8 W m^{−2}. Finally, the (positive) broadband atmospheric absorptance bias is about an order of magnitude smaller than the albedo bias. The substantial magnitude of the PPH bias underlines the importance of predicting subgrid variability in GCMs and accounting for its effects on cloud–radiation interactions.

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## Abstract

The authors propose a new cloud property retrieval technique that accounts for cloud side illumination and shadowing effects present at high solar zenith angles. The technique uses the normalized difference of nadir reflectivities (NDNR) at a conservative and an absorbing (with respect to liquid water) wavelength. It can be further combined with the inverse nonlocal independent pixel approximation (NIPA) of that corrects for radiative smoothing, thus providing a retrieval framework where all 3D cloud effects can potentially be accounted for. The effectiveness of the new technique is demonstrated using Monte Carlo simulations. Real-world application is shown to be feasible using Thematic Mapper (TM) radiance observations from *Landsat-5* over the Southern Great Plains (SGP) site of the Atmospheric Radiation Measurement (ARM) Program. For the moderately oblique (45°) solar zenith angle of the available Landsat scene, NDNR gives similar regional statistics and histograms when compared with standard independent pixel approximation (IPA), but significant differences at the pixel level. Inverse NIPA is also applied for the first time on observed high-resolution radiances of overcast Landsat subscenes. The dependence of the NIPA-retrieved cloud fields on the parameters of the method is illustrated and practical issues related to the optimal choice of these parameters are discussed.

It is natural to compare novel cloud retrieval techniques with standard IPA retrievals. IPA is useful in revealing the inadequacy of plane parallel theory in certain situations and in demonstrating sensitivities to parameter choices, parameterizations, and assumptions. For example, it is found that IPA has problems in matching modeled and observed band-7 (2.2 *μ*m) reflectance values for ∼6% of the pixels, most of which are at cloud edges. For simultaneous cloud optical depth–droplet effective radius retrievals (where a conservative and an absorptive TM band are needed), it is found that the band-4 (0.83 *μ*m)–band-7 pair was the most well behaved, having less saturation, smaller changes in nominal calibration, and better overall consistency with modeled values than other bands. Mean values of optical depth, effective radius, and liquid water path (LWP) for typical IPA retrievals using this pair are *τ* = 22, *r*
_{
e
} = 11 *μ*m, and LWP = 157 g m^{−2}, respectively. Inclusion of aerosol scattering above clouds results in ∼8% decrease in mean cloud optical depth for an aerosol optical depth of 0.2. Degradation of instrument resolution up to ∼2 km has small effects on the optical property means and histograms, suggesting small actual cloud variability at these scales and/or radiative smoothing. Comparisons with surface instruments (microwave radiometer, pyranometer, and pyrgeometer) verify the statisitical adequacy of the IPA retrievals. Last, cloud fractions derived with a simple threshold method are compared with those from an automated procedure called Automatic Cloud Cover Assessment now in operational use for *Landsat-7.* For the northernmost 2000 scanlines of the scene, the cloud fraction *A*
_{
c
} is 0.585 from thresholding, as compared with *A*
_{
c
} = 0.563 for the automated procedure, and the full scene values are *A*
_{
c
} = 0.870 and *A*
_{
c
} = 0.865, respectively. This suggests that the *Landsat-7* automated procedure will likely give reliable scene-averaged cloud fractions for moderately thick clouds over continental U.S. scenes similar to SGP.

## Abstract

The authors propose a new cloud property retrieval technique that accounts for cloud side illumination and shadowing effects present at high solar zenith angles. The technique uses the normalized difference of nadir reflectivities (NDNR) at a conservative and an absorbing (with respect to liquid water) wavelength. It can be further combined with the inverse nonlocal independent pixel approximation (NIPA) of that corrects for radiative smoothing, thus providing a retrieval framework where all 3D cloud effects can potentially be accounted for. The effectiveness of the new technique is demonstrated using Monte Carlo simulations. Real-world application is shown to be feasible using Thematic Mapper (TM) radiance observations from *Landsat-5* over the Southern Great Plains (SGP) site of the Atmospheric Radiation Measurement (ARM) Program. For the moderately oblique (45°) solar zenith angle of the available Landsat scene, NDNR gives similar regional statistics and histograms when compared with standard independent pixel approximation (IPA), but significant differences at the pixel level. Inverse NIPA is also applied for the first time on observed high-resolution radiances of overcast Landsat subscenes. The dependence of the NIPA-retrieved cloud fields on the parameters of the method is illustrated and practical issues related to the optimal choice of these parameters are discussed.

It is natural to compare novel cloud retrieval techniques with standard IPA retrievals. IPA is useful in revealing the inadequacy of plane parallel theory in certain situations and in demonstrating sensitivities to parameter choices, parameterizations, and assumptions. For example, it is found that IPA has problems in matching modeled and observed band-7 (2.2 *μ*m) reflectance values for ∼6% of the pixels, most of which are at cloud edges. For simultaneous cloud optical depth–droplet effective radius retrievals (where a conservative and an absorptive TM band are needed), it is found that the band-4 (0.83 *μ*m)–band-7 pair was the most well behaved, having less saturation, smaller changes in nominal calibration, and better overall consistency with modeled values than other bands. Mean values of optical depth, effective radius, and liquid water path (LWP) for typical IPA retrievals using this pair are *τ* = 22, *r*
_{
e
} = 11 *μ*m, and LWP = 157 g m^{−2}, respectively. Inclusion of aerosol scattering above clouds results in ∼8% decrease in mean cloud optical depth for an aerosol optical depth of 0.2. Degradation of instrument resolution up to ∼2 km has small effects on the optical property means and histograms, suggesting small actual cloud variability at these scales and/or radiative smoothing. Comparisons with surface instruments (microwave radiometer, pyranometer, and pyrgeometer) verify the statisitical adequacy of the IPA retrievals. Last, cloud fractions derived with a simple threshold method are compared with those from an automated procedure called Automatic Cloud Cover Assessment now in operational use for *Landsat-7.* For the northernmost 2000 scanlines of the scene, the cloud fraction *A*
_{
c
} is 0.585 from thresholding, as compared with *A*
_{
c
} = 0.563 for the automated procedure, and the full scene values are *A*
_{
c
} = 0.870 and *A*
_{
c
} = 0.865, respectively. This suggests that the *Landsat-7* automated procedure will likely give reliable scene-averaged cloud fractions for moderately thick clouds over continental U.S. scenes similar to SGP.

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## Abstract

Conventional wisdom is that lidar pulses do not significantly penetrate clouds having an optical thickness exceeding about *τ* = 2, and that no returns are detectible from more than a shallow skin depth. Yet optically thicker clouds of *τ* ≫ 2 reflect a larger fraction of visible photons and account for much of the earth’s global average albedo. As cloud-layer thickness grows, an increasing fraction of reflected photons are scattered multiple times within the cloud and return from a diffuse concentric halo that grows around the incident pulse, increasing in horizontal area with layer physical thickness. The reflected halo is largely undetected by narrow field-of-view (FOV) receivers commonly used in lidar applications. Cloud Thickness from Offbeam Returns (THOR) is an airborne wide-angle detection system with multiple FOVs, capable of observing the diffuse halo as a wide-angle signal, from which the physical thickness of optically thick clouds can be retrieved. This paper describes the THOR system, demonstrates that the halo signal is stronger for thicker clouds, and presents a validation of physical thickness retrievals for clouds having *τ* > 20, from NASA’s P-3B flights over the Department of Energy’s Atmospheric Radiation Measurement Southern Great Plains site, using the lidar, radar, and other ancillary ground-based data.

## Abstract

Conventional wisdom is that lidar pulses do not significantly penetrate clouds having an optical thickness exceeding about *τ* = 2, and that no returns are detectible from more than a shallow skin depth. Yet optically thicker clouds of *τ* ≫ 2 reflect a larger fraction of visible photons and account for much of the earth’s global average albedo. As cloud-layer thickness grows, an increasing fraction of reflected photons are scattered multiple times within the cloud and return from a diffuse concentric halo that grows around the incident pulse, increasing in horizontal area with layer physical thickness. The reflected halo is largely undetected by narrow field-of-view (FOV) receivers commonly used in lidar applications. Cloud Thickness from Offbeam Returns (THOR) is an airborne wide-angle detection system with multiple FOVs, capable of observing the diffuse halo as a wide-angle signal, from which the physical thickness of optically thick clouds can be retrieved. This paper describes the THOR system, demonstrates that the halo signal is stronger for thicker clouds, and presents a validation of physical thickness retrievals for clouds having *τ* > 20, from NASA’s P-3B flights over the Department of Energy’s Atmospheric Radiation Measurement Southern Great Plains site, using the lidar, radar, and other ancillary ground-based data.

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## Abstract

Monte Carlo radiative transfer methods are employed here to estimate the plane-parallel albedo bias for marine stratocumulus clouds. This is the bias in estimates of the mesoscale-average albedo, which arises from the assumption that cloud liquid water is uniformly distributed. The authors compare such estimates with those based on a more realistic distribution generated from a fractal model of marine stratocumulus clouds belonging to the class of “bounded cascade” models. In this model the cloud top and base are fixed, so that all variations in cloud shape are ignored. The model generates random variations in liquid water along a single horizontal direction, forming fractal cloud streets while conserving the total liquid water in the cloud field. The model reproduces the mean, variance, and skewness of the vertically integrated cloud liquid water, as well as its observed wavenumber spectrum, which is approximately a power law. The Monte Carlo method keeps track of the three-dimensional paths solar photons take through the cloud field, using a vectorized implementation of a direct technique. The simplifications in the cloud field studied here allow the computations to be accelerated. The Monte Carlo results are compared to those of the independent pixel approximation, which neglects net horizontal photon transport. Differences between the Monte Carlo and independent pixel estimates of the mesoscale-average albedo are on the order of 1% for conservative scattering, while the plane-parallel bias itself is an order of magnitude larger. As cloud absorption increases, the independent pixel approximation agrees even more closely with the Monte Carlo estimates. This result holds for a wide range of sun angles and aspect ratios. Thus, horizontal photon transport can be safely neglected in estimates of the area-average flux for such cloud models. This result relies on the rapid falloff of the wavenumber spectrum of stratocumulus, which ensures that the smaller-scale variability, where the radiative transfer is more three-dimensional, contributes less to the plane-parallel albedo bias than the larger scales, which are more variable. The lack of significant three-dimensional effects also relies on the assumption of a relatively simple geometry. Even with these assumptions, the independent pixel approximation is accurate only for fluxes averaged over large horizontal areas, many photon mean free paths in diameter, and not for local radiance values, which depend strongly on the interaction between neighboring cloud elements.

## Abstract

Monte Carlo radiative transfer methods are employed here to estimate the plane-parallel albedo bias for marine stratocumulus clouds. This is the bias in estimates of the mesoscale-average albedo, which arises from the assumption that cloud liquid water is uniformly distributed. The authors compare such estimates with those based on a more realistic distribution generated from a fractal model of marine stratocumulus clouds belonging to the class of “bounded cascade” models. In this model the cloud top and base are fixed, so that all variations in cloud shape are ignored. The model generates random variations in liquid water along a single horizontal direction, forming fractal cloud streets while conserving the total liquid water in the cloud field. The model reproduces the mean, variance, and skewness of the vertically integrated cloud liquid water, as well as its observed wavenumber spectrum, which is approximately a power law. The Monte Carlo method keeps track of the three-dimensional paths solar photons take through the cloud field, using a vectorized implementation of a direct technique. The simplifications in the cloud field studied here allow the computations to be accelerated. The Monte Carlo results are compared to those of the independent pixel approximation, which neglects net horizontal photon transport. Differences between the Monte Carlo and independent pixel estimates of the mesoscale-average albedo are on the order of 1% for conservative scattering, while the plane-parallel bias itself is an order of magnitude larger. As cloud absorption increases, the independent pixel approximation agrees even more closely with the Monte Carlo estimates. This result holds for a wide range of sun angles and aspect ratios. Thus, horizontal photon transport can be safely neglected in estimates of the area-average flux for such cloud models. This result relies on the rapid falloff of the wavenumber spectrum of stratocumulus, which ensures that the smaller-scale variability, where the radiative transfer is more three-dimensional, contributes less to the plane-parallel albedo bias than the larger scales, which are more variable. The lack of significant three-dimensional effects also relies on the assumption of a relatively simple geometry. Even with these assumptions, the independent pixel approximation is accurate only for fluxes averaged over large horizontal areas, many photon mean free paths in diameter, and not for local radiance values, which depend strongly on the interaction between neighboring cloud elements.