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## Abstract

When cloud properties are retrieved from satellite observations, current calculations apply one-dimensional (1D) theory to the three-dimensional (3D) world: they consider only vertical processes and ignore horizontal interactions. This paper proposes a novel approach that estimates 3D effects in cloud optical thickness retrievals. The proposed method combines visible and thermal infrared images to see whether 3D radiative effects make clouds appear asymmetric—that is, whether cloud surfaces tilted toward the sun are systematically brighter than surfaces tilted away from it. The observed asymmetries are then used to estimate 3D effects for 1-km-size pixels as well as 50-km-size areas. Initial results obtained for Moderate-Resolution Imaging Spectroradiometer (MODIS) images reveal that 3D effects cause abundant uncertainties in the 1-km-resolution 1D retrievals. Averaging over 50 km by 50 km areas greatly reduces the errors but does not remove them completely. Conservative estimates show that the mean optical thickness values are biased by more than 10% in 10% of the areas, and the errors in the areas' standard deviation values are more than 10% in about 20% of areas.

## Abstract

When cloud properties are retrieved from satellite observations, current calculations apply one-dimensional (1D) theory to the three-dimensional (3D) world: they consider only vertical processes and ignore horizontal interactions. This paper proposes a novel approach that estimates 3D effects in cloud optical thickness retrievals. The proposed method combines visible and thermal infrared images to see whether 3D radiative effects make clouds appear asymmetric—that is, whether cloud surfaces tilted toward the sun are systematically brighter than surfaces tilted away from it. The observed asymmetries are then used to estimate 3D effects for 1-km-size pixels as well as 50-km-size areas. Initial results obtained for Moderate-Resolution Imaging Spectroradiometer (MODIS) images reveal that 3D effects cause abundant uncertainties in the 1-km-resolution 1D retrievals. Averaging over 50 km by 50 km areas greatly reduces the errors but does not remove them completely. Conservative estimates show that the mean optical thickness values are biased by more than 10% in 10% of the areas, and the errors in the areas' standard deviation values are more than 10% in about 20% of areas.

## Abstract

This paper presents a simple yet general approach to estimate the uncertainties that arise in satellite retrievals of cloud optical depth when the retrievals use one-dimensional radiative transfer theory for heterogeneous clouds that have variations in all three dimensions. For the first time, preliminary error bounds are set to estimate the uncertainty of cloud optical depth retrievals. These estimates can help us better understand the nature of uncertainties that three-dimensional effects can introduce into retrievals of this important product of the Moderate Resolution Imaging Spectroradiometer instrument. The probability distribution of resulting retrieval errors is examined through theoretical simulations of shortwave cloud reflection for a set of cloud fields that represent the variability of stratocumulus clouds. The results are used to illustrate how retrieval uncertainties change with observable and known parameters, such as solar elevation or cloud brightness. Furthermore, the results indicate that a tendency observed in an earlier study—clouds appearing thicker for oblique sun—is indeed caused by three-dimensional radiative effects.

## Abstract

This paper presents a simple yet general approach to estimate the uncertainties that arise in satellite retrievals of cloud optical depth when the retrievals use one-dimensional radiative transfer theory for heterogeneous clouds that have variations in all three dimensions. For the first time, preliminary error bounds are set to estimate the uncertainty of cloud optical depth retrievals. These estimates can help us better understand the nature of uncertainties that three-dimensional effects can introduce into retrievals of this important product of the Moderate Resolution Imaging Spectroradiometer instrument. The probability distribution of resulting retrieval errors is examined through theoretical simulations of shortwave cloud reflection for a set of cloud fields that represent the variability of stratocumulus clouds. The results are used to illustrate how retrieval uncertainties change with observable and known parameters, such as solar elevation or cloud brightness. Furthermore, the results indicate that a tendency observed in an earlier study—clouds appearing thicker for oblique sun—is indeed caused by three-dimensional radiative effects.

## Abstract

A method for inferring cloud optical depth *τ* is introduced and assessed using simulated surface radiometric measurements produced by a Monte Carlo algorithm acting on fields of broken, single-layer, boundary layer clouds derived from Landsat imagery. The method utilizes a 1D radiative transfer model and time series of zenith radiances and irradiances measured at two wavelengths, *λ*
_{1} and *λ*
_{2}, from a single site with surface albedos *α*_{λ1}*α*_{λ2}*τ*′ are obtained through cloud-base reflectances that are approximated by differencing spectral radiances and estimating upwelling fluxes at cloud base. When initialized with suitable values of *α*_{λ1}*α*_{λ2}*h,* this method performs well at all solar zenith angles. Relative mean bias errors for *τ*′ are typically less than 5% for these cases. Relative variances for *τ*′ for given values of inherent *τ* are almost independent of inherent *τ* and are <50%. Errors due to neglect of net horizontal transport in clouds yield slight, but systematic, overestimates for *τ* ≲ 5 and underestimates for larger *τ.* Frequency distributions and power spectra for retrieved and inherent *τ* are often in excellent agreement. Estimates of *τ* depend weakly on errors in *h,* especially when *h* is overestimated. Also, they are almost insensitive to errors in surface albedo when *α*_{λ1}*α*_{λ2}*τ,* particularly large *τ.* In contrast, the conventional method of using only surface irradiance yields almost entirely invalid results when clouds are broken.

Though results are shown only for surfaces resembling green vegetation (i.e., *α*_{λ1}*α*_{λ2}*α*_{λ1}*α*_{λ2}*τ* for broken clouds above many surface types.

## Abstract

A method for inferring cloud optical depth *τ* is introduced and assessed using simulated surface radiometric measurements produced by a Monte Carlo algorithm acting on fields of broken, single-layer, boundary layer clouds derived from Landsat imagery. The method utilizes a 1D radiative transfer model and time series of zenith radiances and irradiances measured at two wavelengths, *λ*
_{1} and *λ*
_{2}, from a single site with surface albedos *α*_{λ1}*α*_{λ2}*τ*′ are obtained through cloud-base reflectances that are approximated by differencing spectral radiances and estimating upwelling fluxes at cloud base. When initialized with suitable values of *α*_{λ1}*α*_{λ2}*h,* this method performs well at all solar zenith angles. Relative mean bias errors for *τ*′ are typically less than 5% for these cases. Relative variances for *τ*′ for given values of inherent *τ* are almost independent of inherent *τ* and are <50%. Errors due to neglect of net horizontal transport in clouds yield slight, but systematic, overestimates for *τ* ≲ 5 and underestimates for larger *τ.* Frequency distributions and power spectra for retrieved and inherent *τ* are often in excellent agreement. Estimates of *τ* depend weakly on errors in *h,* especially when *h* is overestimated. Also, they are almost insensitive to errors in surface albedo when *α*_{λ1}*α*_{λ2}*τ,* particularly large *τ.* In contrast, the conventional method of using only surface irradiance yields almost entirely invalid results when clouds are broken.

Though results are shown only for surfaces resembling green vegetation (i.e., *α*_{λ1}*α*_{λ2}*α*_{λ1}*α*_{λ2}*τ* for broken clouds above many surface types.

## Abstract

In the fourth part of our “Cellular Statistical Models of Broken Cloud Fields” series we use the binary Markov processes framework for quantitative investigation of the effects of low resolution of idealized satellite observations on the statistics of the retrieved cloud masks. We assume that the cloud fields are Markovian and are characterized by the “actual” cloud fraction (CF) and scale length. We use two different models of observations: a simple discrete-point sampling and a more realistic “pixel” protocol. The latter is characterized by a state attribution function (SAF), which has the meaning of the probability that the pixel with a certain CF is declared cloudy in the observed cloud mask. The stochasticity of the SAF means that the cloud–clear attribution is not ideal and can be affected by external or unknown factors. We show that the observed cloud masks can be accurately described as Markov chains of pixels and use the master matrix formalism (introduced in Part III of the series) for analytical computation of their parameters: the “observed” CF and scale length. This procedure allows us to establish a quantitative relationship (which is pixel-size dependent) between the actual and the observed cloud-field statistics. The feasibility of restoring the former from the latter is considered. The adequacy of our analytical approach to idealized observations is evaluated using numerical simulations. Comparison of the observed parameters of the simulated datasets with their theoretical expectations showed an agreement within 0.005 for the CF, while for the scale length it is within 1% in the sampling case and within 4% in the pixel case.

## Abstract

In the fourth part of our “Cellular Statistical Models of Broken Cloud Fields” series we use the binary Markov processes framework for quantitative investigation of the effects of low resolution of idealized satellite observations on the statistics of the retrieved cloud masks. We assume that the cloud fields are Markovian and are characterized by the “actual” cloud fraction (CF) and scale length. We use two different models of observations: a simple discrete-point sampling and a more realistic “pixel” protocol. The latter is characterized by a state attribution function (SAF), which has the meaning of the probability that the pixel with a certain CF is declared cloudy in the observed cloud mask. The stochasticity of the SAF means that the cloud–clear attribution is not ideal and can be affected by external or unknown factors. We show that the observed cloud masks can be accurately described as Markov chains of pixels and use the master matrix formalism (introduced in Part III of the series) for analytical computation of their parameters: the “observed” CF and scale length. This procedure allows us to establish a quantitative relationship (which is pixel-size dependent) between the actual and the observed cloud-field statistics. The feasibility of restoring the former from the latter is considered. The adequacy of our analytical approach to idealized observations is evaluated using numerical simulations. Comparison of the observed parameters of the simulated datasets with their theoretical expectations showed an agreement within 0.005 for the CF, while for the scale length it is within 1% in the sampling case and within 4% in the pixel case.

## Abstract

In the third part of the “Cellular Statistical Models of Broken Cloud Fields” series the cloud statistics formalism developed in the first two parts is interpreted in terms of the theory of Markov processes. The master matrix introduced in this study is a unifying generalization of both the cloud fraction probability distribution function (PDF) and the Markovian transition probability matrix. To illustrate the new concept, the master matrix is used for computation of the moments of the cloud fraction PDF—in particular, the variance—which until now has not been analytically derived in the framework of the authors’ previous work. This paper also serves as a bridge to the proposed future studies of the effects of sampling and averaging on satellite-based cloud masks.

## Abstract

In the third part of the “Cellular Statistical Models of Broken Cloud Fields” series the cloud statistics formalism developed in the first two parts is interpreted in terms of the theory of Markov processes. The master matrix introduced in this study is a unifying generalization of both the cloud fraction probability distribution function (PDF) and the Markovian transition probability matrix. To illustrate the new concept, the master matrix is used for computation of the moments of the cloud fraction PDF—in particular, the variance—which until now has not been analytically derived in the framework of the authors’ previous work. This paper also serves as a bridge to the proposed future studies of the effects of sampling and averaging on satellite-based cloud masks.

## Abstract

Here, previous work using photon diffusion theory to describe radiative transfer through dense plane-parallel clouds at nonabsorbing wavelengths is extended. The focus is on the scaling of space- and time-domain moments for transmitted light with respect to cloud thickness *H* and optical depth *τ*; and the new results are as follows: accurate prefactors for asymptotic scaling, preasymptotic correction terms in closed form, 3D effects for internal variability in *τ,* and the rms transit time or pathlength. Mean pathlength is ∝*H* for dimensional reasons and, from random-walk theory, we already know that it is also ∝(1 – *g*)*τ* for large enough *τ* (*g* being the asymmetry factor). Here, it is shown that the prefactor is precisely 1/2 and that corrections are significant for (1 – *g*)*τ* < 10, which includes most actual boundary layer clouds. It is also shown that rms pathlength is not much larger than the mean for transmittance (its prefactor is *H* as *τ* increases; it is also shown that the transmitted spot shape has a flat top and an exponential tail. Because all preasymptotic corrections are computed here, the diffusion results are accurate when compared to Monte Carlo counterparts for *τ* ≥ 5, whereas the classic scaling relations apply only for *τ* ≥ 70, assuming *g* = 0.85. The temporal quantities shed light on observed absorption properties and optical lightning waveforms. The spatial quantity controls the three-dimensional radiative smoothing process in transmission, which was recently observed in spectral analyses of time series of zenith radiance at 725 nm. Opportunities in ground-based cloud remote sensing using the new developments are described and illustrated with simulations of 3D solar radiative transfer in realistic models of stratocumulus. Finally, since this analytical diffusion study applies only to weakly variable stratus layers, extensions to more complex cloud systems using anomalous diffusion theory are discussed.

## Abstract

Here, previous work using photon diffusion theory to describe radiative transfer through dense plane-parallel clouds at nonabsorbing wavelengths is extended. The focus is on the scaling of space- and time-domain moments for transmitted light with respect to cloud thickness *H* and optical depth *τ*; and the new results are as follows: accurate prefactors for asymptotic scaling, preasymptotic correction terms in closed form, 3D effects for internal variability in *τ,* and the rms transit time or pathlength. Mean pathlength is ∝*H* for dimensional reasons and, from random-walk theory, we already know that it is also ∝(1 – *g*)*τ* for large enough *τ* (*g* being the asymmetry factor). Here, it is shown that the prefactor is precisely 1/2 and that corrections are significant for (1 – *g*)*τ* < 10, which includes most actual boundary layer clouds. It is also shown that rms pathlength is not much larger than the mean for transmittance (its prefactor is *H* as *τ* increases; it is also shown that the transmitted spot shape has a flat top and an exponential tail. Because all preasymptotic corrections are computed here, the diffusion results are accurate when compared to Monte Carlo counterparts for *τ* ≥ 5, whereas the classic scaling relations apply only for *τ* ≥ 70, assuming *g* = 0.85. The temporal quantities shed light on observed absorption properties and optical lightning waveforms. The spatial quantity controls the three-dimensional radiative smoothing process in transmission, which was recently observed in spectral analyses of time series of zenith radiance at 725 nm. Opportunities in ground-based cloud remote sensing using the new developments are described and illustrated with simulations of 3D solar radiative transfer in realistic models of stratocumulus. Finally, since this analytical diffusion study applies only to weakly variable stratus layers, extensions to more complex cloud systems using anomalous diffusion theory are discussed.

## Abstract

A simple and fast algorithm for generating two correlated stochastic two-dimensional (2D) cloud fields is described. The algorithm is illustrated with two broken cumulus cloud fields: cloud optical depth and cloud-top height retrieved from the Moderate Resolution Imaging Spectroradiometer (MODIS). Only two 2D fields are required as an input. The algorithm output is statistical realizations of these two fields with approximately the same correlation and joint distribution functions as the original ones. The major assumption of the algorithm is statistical isotropy of the fields. In contrast to fractals and the Fourier filtering methods frequently used for stochastic cloud modeling, the proposed method is based on spectral models of homogeneous random fields. To retain the same probability density function as the (first) original field, the method of inverse distribution function is used. When the spatial distribution of the first field has been generated, a realization of the correlated second field is simulated using a conditional distribution matrix. This paper serves as a theoretical justification of the publicly available software “Simulation of a two-component cloud field,” which has been recently released. Although 2D rather than full 3D, the stochastic realizations of two correlated cloud fields that mimic statistics of given fields have proven to be very useful to study 3D radiative transfer features of broken cumulus clouds for a better understanding of shortwave radiation and the interpretation of remote sensing retrievals.

## Abstract

A simple and fast algorithm for generating two correlated stochastic two-dimensional (2D) cloud fields is described. The algorithm is illustrated with two broken cumulus cloud fields: cloud optical depth and cloud-top height retrieved from the Moderate Resolution Imaging Spectroradiometer (MODIS). Only two 2D fields are required as an input. The algorithm output is statistical realizations of these two fields with approximately the same correlation and joint distribution functions as the original ones. The major assumption of the algorithm is statistical isotropy of the fields. In contrast to fractals and the Fourier filtering methods frequently used for stochastic cloud modeling, the proposed method is based on spectral models of homogeneous random fields. To retain the same probability density function as the (first) original field, the method of inverse distribution function is used. When the spatial distribution of the first field has been generated, a realization of the correlated second field is simulated using a conditional distribution matrix. This paper serves as a theoretical justification of the publicly available software “Simulation of a two-component cloud field,” which has been recently released. Although 2D rather than full 3D, the stochastic realizations of two correlated cloud fields that mimic statistics of given fields have proven to be very useful to study 3D radiative transfer features of broken cumulus clouds for a better understanding of shortwave radiation and the interpretation of remote sensing retrievals.

## Abstract

The Multiangle Imaging Spectroradiometer (MISR) views the earth with nine cameras, ranging from a 70° zenith angle viewing forward through nadir to 70° viewing aft. MISR does not have an operational cloud optical depth retrieval algorithm, but previous research has hinted that solar reflection measured in multiple directions might improve cloud optical depth retrievals. This study explores the optical depth information content of MISR’s multiple angles using a retrieval simulation approach. Hundreds of realistic boundary-layer cloud fields are generated with large-eddy simulation (LES) models for stratocumulus, small trade cumulus, and land surface–forced fair-weather cumulus. Reflectances in MISR directions are computed with three-dimensional radiative transfer from the LES cloud fields over an ocean surface and averaged to MISR resolution and sampled at MISR 275-m pixel spacing. Neural networks are trained to retrieve the mean and standard deviation of optical depth over different size pixel patches from the mean and standard deviation of simulated MISR reflectances. Various configurations of MISR cameras are input to the retrieval, and the rms retrieval errors are compared. For 5 × 5 pixel patches the already low mean optical depth retrieval error for stratocumulus decreases 41% and 23% (for 25° and 45° solar zenith angles, respectively) from using only the nadir camera to using seven MISR cameras. For cumulus, however, the much higher normalized optical depth retrieval error only decreases around 14%. These small improvements suggest that measurements of solar reflection in multiple directions do not contribute substantially to more accurate optical depth retrievals for small cumulus clouds. The 3D statistical retrievals, however, even with only the nadir camera, are much more accurate for small cumulus than standard nadir plane-parallel retrievals; therefore, this approach may be worth pursuing.

## Abstract

The Multiangle Imaging Spectroradiometer (MISR) views the earth with nine cameras, ranging from a 70° zenith angle viewing forward through nadir to 70° viewing aft. MISR does not have an operational cloud optical depth retrieval algorithm, but previous research has hinted that solar reflection measured in multiple directions might improve cloud optical depth retrievals. This study explores the optical depth information content of MISR’s multiple angles using a retrieval simulation approach. Hundreds of realistic boundary-layer cloud fields are generated with large-eddy simulation (LES) models for stratocumulus, small trade cumulus, and land surface–forced fair-weather cumulus. Reflectances in MISR directions are computed with three-dimensional radiative transfer from the LES cloud fields over an ocean surface and averaged to MISR resolution and sampled at MISR 275-m pixel spacing. Neural networks are trained to retrieve the mean and standard deviation of optical depth over different size pixel patches from the mean and standard deviation of simulated MISR reflectances. Various configurations of MISR cameras are input to the retrieval, and the rms retrieval errors are compared. For 5 × 5 pixel patches the already low mean optical depth retrieval error for stratocumulus decreases 41% and 23% (for 25° and 45° solar zenith angles, respectively) from using only the nadir camera to using seven MISR cameras. For cumulus, however, the much higher normalized optical depth retrieval error only decreases around 14%. These small improvements suggest that measurements of solar reflection in multiple directions do not contribute substantially to more accurate optical depth retrievals for small cumulus clouds. The 3D statistical retrievals, however, even with only the nadir camera, are much more accurate for small cumulus than standard nadir plane-parallel retrievals; therefore, this approach may be worth pursuing.

## Abstract

In this paper, the effect of cloud structure on column absorption by water vapor is investigated. Radiative fluxes above and below horizontally inhomogeneous liquid water clouds are computed using an efficient Monte Carlo technique, the independent pixel approximation, and plane-parallel theory. Cloud inhomogeneity is simulated by two related fractal models that use bounded cascades for the horizontal distribution of optical depth. The first (“clumpy”) model has constant cloud top and base, hence a constant geometrical thickness but varying extinction; the second (“bumpy”) model has constant extinction and cloud base, hence variable cloud-top and geometrical thickness. The spectral range between 0.9 and 1.0 *μ*m (with strong water vapor absorption and negligible cloud liquid water absorption) is selected for a detailed study, not only of domain-averaged quantities, but also radiation fields. Column-absorption fields are calculated as the difference between the two net fluxes above and below clouds. It is shown that 1) redistribution of cloud liquid water decreases column absorption, that is, plane-parallel absorption is larger than the independent pixel approximation one by 1%–3%; 2) 3D radiative effects enhance column absorption by about 0.6% for the clumpy model and 2% for the bumpy model, that is, Monte Carlo absorption is larger than independent pixel approximation absorption—this effect is most pronounced for the bumpy cloud model at solar zenith angle ≈45°; and 3) plane-parallel absorption is larger than 3D Monte Carlo absorption for high solar elevations and nearly equal to it for low solar elevations. Thus, for extended clouds of thickness 1–2 km or less, in an important water vapor absorption band (0.94 *μ*m), the authors do not find a significant enhancement of cloud absorption due to horizontal inhomogeneity.

## Abstract

In this paper, the effect of cloud structure on column absorption by water vapor is investigated. Radiative fluxes above and below horizontally inhomogeneous liquid water clouds are computed using an efficient Monte Carlo technique, the independent pixel approximation, and plane-parallel theory. Cloud inhomogeneity is simulated by two related fractal models that use bounded cascades for the horizontal distribution of optical depth. The first (“clumpy”) model has constant cloud top and base, hence a constant geometrical thickness but varying extinction; the second (“bumpy”) model has constant extinction and cloud base, hence variable cloud-top and geometrical thickness. The spectral range between 0.9 and 1.0 *μ*m (with strong water vapor absorption and negligible cloud liquid water absorption) is selected for a detailed study, not only of domain-averaged quantities, but also radiation fields. Column-absorption fields are calculated as the difference between the two net fluxes above and below clouds. It is shown that 1) redistribution of cloud liquid water decreases column absorption, that is, plane-parallel absorption is larger than the independent pixel approximation one by 1%–3%; 2) 3D radiative effects enhance column absorption by about 0.6% for the clumpy model and 2% for the bumpy model, that is, Monte Carlo absorption is larger than independent pixel approximation absorption—this effect is most pronounced for the bumpy cloud model at solar zenith angle ≈45°; and 3) plane-parallel absorption is larger than 3D Monte Carlo absorption for high solar elevations and nearly equal to it for low solar elevations. Thus, for extended clouds of thickness 1–2 km or less, in an important water vapor absorption band (0.94 *μ*m), the authors do not find a significant enhancement of cloud absorption due to horizontal inhomogeneity.

## Abstract

A new analytical statistical model describing the structure of broken cloud fields is presented. It depends on two parameters (cell size and occupancy probability) and provides chord distributions of clouds and gaps between them by length, as well as the cloud fraction distribution. This approach is based on the assumption that the structure of a cloud field is determined by a semiregular grid of cells (an abstraction of the atmospheric convective cells), which are filled with cloud with some probability. First, a simple discrete model is introduced, where clouds and gaps can occupy an integer number of cells, and then a continuous analog is developed, allowing for arbitrary cloud and gap sizes. The influence of a finite sample size on the retrieved statistics is also described.

## Abstract

A new analytical statistical model describing the structure of broken cloud fields is presented. It depends on two parameters (cell size and occupancy probability) and provides chord distributions of clouds and gaps between them by length, as well as the cloud fraction distribution. This approach is based on the assumption that the structure of a cloud field is determined by a semiregular grid of cells (an abstraction of the atmospheric convective cells), which are filled with cloud with some probability. First, a simple discrete model is introduced, where clouds and gaps can occupy an integer number of cells, and then a continuous analog is developed, allowing for arbitrary cloud and gap sizes. The influence of a finite sample size on the retrieved statistics is also described.