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

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

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 _{λ 1} < α _{λ 2} . Assuming that clouds transport radiation in accordance with 1D theory and have spectrally invariant optical properties, inferred optical depths 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 _{λ 1} , α _{λ 2} , and cloud-base altitude 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 _{λ 1} is underestimated and α _{λ 2} overestimated. Reversing the sign of these errors leads to overestimation of 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 _{λ 1} ≪ α _{λ 2} ), the performance of this method depends little on the values of α _{λ 1} , and α _{λ 2} . Thus, if radiometric data have sufficient signal-to-noise ratios and suitable wavelengths can be found, this method should yield reliable estimates of 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

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 ≈ 0.59); this proves that, in sharp contrast with reflection, pathlength distributions are quite narrow in transmission. If the light originates from a steady point source on a cloud boundary, a fuzzy spot is observed on the opposite boundary. This problem is formally mapped to the pulsed source problem, and it is shown that the rms radius of this spot slowly approaches 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

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

This study investigates the internal structure of marine stratocumulus (Sc) using the spatial fluctuations of liquid water content (LWC) measured along horizontal flights off the coast of southern California during the First ISCCP Regional Experiment (FIRE) in summer of 1987. The results of FIRE 87 data analyses are compared to similar ones for marine Sc probed during the Atlantic Stratocumulus Transition Experiment (ASTEX) in summer 1992 near the Azores. In this first of two parts, the authors use spectral analysis to determine the main scale-invariant regimes, defined by the ranges of scales where wavenumber spectra follow power laws; from there, they discuss stationary issues. Although crucial for obtaining meaningful spatial statistics (e.g., in climate diagnostics), the importance of establishing stationarity—statistical invariance under translation—is often overlooked. The sequel uses multifractal analysis techniques and addresses intermittency issues. By improving our understanding of both nonstationarity and intermittency in atmospheric data, we are in a better position to formulate successful sampling strategies.

Comparing the spectral responses of different instruments to natural LWC variability, the authors find scale breaks (characteristic scales separating two distinct power law regimes) that are spurious, being traceable to well-documented idiosyncrasies of the Johnson–Williams probe and forward scattering spectrometer probes. In data from the King probe, the authors find no such artifacts; all spectra are of the scale-invariant form k ^−β with exponents β in the range 1.1–1.7, depending on the flight. Using the whole FIRE 87 King LWC database, the authors find power-law behavior with β = 1.56 ± 0.06 from 20 m to 20 km. From a spectral vantage point, the ASTEX cloud system behaves statistically like a scaled-up version of FIRE 87: a similar exponent β = 1.43 ± 0.08 is obtained, but the scaling range is shifted to [60 m, 60 km], possibly due to the 2–3 times greater boundary layer thickness.

Finally, the authors reassess the usefulness of spectral analysis:

1. • Its main shortcoming is ambiguity: very different looking stochastic processes can yield similar, even identical, spectra. This problem impedes accurate modeling of the LWC data and, ultimately, is why multifractal methods are required.

2. • Its main asset is applicability in stationary and nonstationary situations alike and, in conjunction with scaling, it can be used to detect nonstationary behavior in data.

Having β > 1, LWC fields in marine Sc are nonstationary within the scaling range and stationary only at larger scales. Nonstationarity implies long-range correlations, and we demonstrate the damage these cause when tying to estimate means and standard deviations with limited amounts of LWC data.

Abstract

Several studies have uncovered a break in the scaling properties of Landsat cloud scenes at nonabsorbing wavelengths. For scales greater than 200–400 m, the wavenumber spectrum is approximately power law in k ^−5/3, but from there down to the smallest observable scales (50–100 m) follows another k ^−β law with β > 3. This implies very smooth radiance fields. The authors reexamine the empirical evidence for this scale break and explain it using fractal cloud models, Monte Carlo simulations, and a Green function approach to multiple scattering theory. In particular, the authors define the “radiative smoothing scale” and relate it to the characteristic scale of horizontal photon transport. The scale break was originally thought to occur at a scale commensurate with either the geometrical thickness Δ _z of the cloud, or with the “transport” mean free path l _t = [(1 − g)σ]⁻¹, which incorporates the effect of forward scattering (σ is extinction and g the asymmetry factor of the phase function). The smoothing scale is found to be approximately l _tΔ _z at cloud top; this is the prediction of diffusion theory which applies when (1 − g)τ = Δ _z  /l _t ≳ 1 (τ is optical thickness). Since the scale break is a tangible effect of net horizontal radiative fluxes excited by the fluctuations of τ, the smoothing scale sets an absolute lower bound on the range where one can neglect these fluxes and use plane-parallel theory locally, even for stratiform clouds. In particular, this constrains the retrieval of cloud properties from remotely sensed data. Finally, the characterization of horizontal photon transport suggests a new lidar technique for joint measurements of optical and geometrical thicknesses at about 0.5-km resolution.

Abstract

This is the second of two papers analyzing the internal liquid water content (LWC) structure of marine stratocumulus (Sc) based on observations taken during the First ICCP (International Commission on Cloud Physics) Regional Experiment (FIRE) 1987 and Atlantic Stratocumulus Transition Experiment (ASTEX) 1992 field programs. Part I examined wavenumber spectra and the three-decade scale range (tens of meters to tens of kilometers) over which scale invariance holds; the inability of spectral analysis to distinguish between different random processes was also underscored. This indetermination is removed in this part by applying multifractal analysis techniques to the LWC fields, leading to a characterization of the role of intermittency in marine Sc.

Two multiscaling statistics are computed and associated nonincreasing hierarchies of exponents are obtained: structure functions and H(q), singular measures and D(q). The real variable q is the order of a statistical moment (e.g., q = 1.0 yields a mean); D(q) quantifies intermittency, H(q) nonstationarity. Being derived from the slopes of lines on log(statistic) versus log(scale) plots, these exponents are only defined when those lines are reasonably straight and where this happens defines the scale-invariant range. Being nonconstant, the derived H(q) and D(q) indicate multifractality rather than monofractality of LWC fields.

Two exponents can serve as first-order measures of nonstationarity and intermittency: H ₁ = H(1) and C ₁ = 1 − D(1). For the ensemble average of all FIRE and all ASTEX data, the authors find the two corresponding points in the (H ₁, C ₁) plane to be close: (0.28, 0.10) for FIRE and (0.29, 0.08) for ASTEX. This indicates that the dynamics determining the internal structure of marine Sc depend little on the local climatology. In contrast, the scatter of spatial averages for the individual flight around the ensemble average illustrates ergodicity violation. Finally, neither multiplicative cascades (with H ₁ = 0) nor additive Gaussian models such as fractional Brownian motions (with C ₁ = 0) adequately reproduce the LWC fluctuations in marine Sc.