Warren J. Wiscombe

Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media

We summarize an advanced, thoroughly documented, and quite general purpose discrete ordinate algorithm for time-independent transfer calculations in vertically inhomogeneous, nonisothermal, plane-parallel media. Atmospheric applications ranging from the UV to the radar region of the electromagnetic spectrum are possible. The physical processes included are thermal emission, scattering, absorption, and bidirectional reflection and emission at the lower boundary. The medium may be forced at the top boundary by parallel or diffuse radiation and by internal and boundary thermal sources as well. We provide a brief account of the theoretical basis as well as a discussion of the numerical implementation of the theory. The recent advances made by ourselves and our collaborators-advances in both formulation and numerical solution-are all incorporated in the algorithm. Prominent among these advances are the complete conquest of two illconditioning problems which afflicted all previous discrete ordinate implementations: (1) the computation of eigenvalues and eigenvectors and (2) the inversion of the matrix determining the constants of integration. Copies of the FORTRAN program on microcomputer diskettes are available for interested users.

A Model for the Spectral Albedo of Snow. I: Pure Snow

Abstract We present a method for calculating the spectral albedo of snow which can be used at any wavelength in the solar spectrum and which accounts for diffusely or directly incident radiation at any zenith angle. For deep snow, the model contains only one adjustable parameter, an effective grain size, which is close to observed grain sizes. A second parameter, the liquid-equivalent depth, is required only for relatively thin snow. In order for the model to make realistic predictions, it must account for the extreme anisotropy of scattering by snow particles. This is done by using the “delta-Eddington” approximation for multiple scattering, together with Mie theory for single scattering. The spectral albedo from 0.3 to 5 μm wavelength is examined as a function of the effective grain size, the solar zenith angle, the snowpack thickness, and the ratio of diffuse to direct solar incidence. The decrease in albedo due to snow aging can be mimicked by reasonable increases in grain size (50–100 μm for new snow...

A Model for the Spectral Albedo of Snow. II: Snow Containing Atmospheric Aerosols

Abstract Small highly absorbing particles, present in concentrations of only 1 part per million by weight (ppmw) or less, can lower snow albedo in the visible by 5–15% from the high values (96–99%) predicted for pure snow in Part I. These particles have, however, no effect on snow albedo beyond 0.9 μm wavelength where ice itself becomes a strong absorber. Thus we have an attractive explanation for the discrepancy between theory and observation described in Part I, a discrepancy which seemingly cannot be resolved on the basis of near-field scattering and nonsphericity effects. Desert dust and carbon soot are the most likely contaminants. But careful measurements of spectral snow albedo in the Arctic and Antarctic paint to a “grey” absorber, one whose imaginary refractive index is nearly constant across the visible spectrum. Thus carbon soot, rather than the red iron oxide normally present in desert dust, is strongly indicated at these sites. Soot particles of radius 0.1 μm, in concentrations of only 0.3 pp...

The albedo of fractal stratocumulus clouds

Abstract An increase in the planetary albedo of the earth-atmosphere system by only 10% can decrease the equilibrium surface temperature to that of the last ice age. Nevertheless, albedo biases of 10% or greater would be introduced into large regions of current climate models if clouds were given their observed liquid water amounts, because of the treatment of clouds as plane parallel. Past work has addressed the effect of cloud shape on albedo; here the focus is on the within-cloud variability of the vertically integrated liquid water. The main result is an estimate of the “plane-parallel albedo bias” using the “independent pixel approximation,” which ignores net horizontal photon transport, from a simple fractal model of marine stratocumulus clouds that ignores the cloud shape. The use of the independent pixel approximation in this context will be justified in a separate Monte Carlo study. The focus on marine stratocumulus clouds is due to their important role in cloud radiative forcing and also that, o...

Multifractal characterizations of nonstationarity and intermittency in geophysical fields: Observed, retrieved, or simulated

Geophysical data rarely show any smoothness at any scale, and this often makes comparison with theoretical model output difficult. However, highly fluctuating signals and fractal structures are typical of open dissipative systems with nonlinear dynamics, the focus of most geophysical research. High levels of variability are excited over a large range of scales by the combined actions of external forcing and internal instability. At very small scales we expect geophysical fields to be smooth, but these are rarely resolved with available instrumentation or simulation tools; nondifferentiable and even discontinuous models are therefore in order. We need methods of statistically analyzing geophysical data, whether measured in situ, remotely sensed or even generated by a computer model, that are adapted to these characteristics. An important preliminary task is to define statistically stationary features in generally nonstationary signals. We first discuss a simple criterion for stationarity in finite data streams that exhibit power law energy spectra and then, guided by developments in turbulence studies, we advocate the use of two ways of analyzing the scale dependence of statistical information: singular measures and qth order structure functions. In nonstationary situations, the approach based on singular measures seeks power law behavior in integrals over all possible scales of a nonnegative stationary field derived from the data, leading to a characterization of the intermittency in this (gradient-related) field. In contrast, the approach based on structure functions uses the signal itself, seeking power laws for the statistical moments of absolute increments over arbitrarily large scales, leading to a characterization of the prevailing nonstationarity in both quantitative and qualitative terms. We explain graphically, step by step, both multifractal statistics which are largely complementary to each other. The geometrical manifestations of nonstationarity and intermittency, “roughness” and “sparseness”, respectively, are illustrated and the associated analytical (differentiability and continuity) properties are discussed. As an example, the two techniques are applied to a series of recent measurements of liquid water distributions inside marine stratocumulus decks; these are found to be multifractal over scales ranging from ≈60 m to ≈60 km. Finally, we define the “mean multifractal plane” and show it to be a simple yet comprehensive tool with many applications including data intercomparison, (dynamical or stochastic) model and retrieval validations.

Independent Pixel and Monte Carlo Estimates of Stratocumulus Albedo

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 sol...

The Landsat Scale Break in Stratocumulus as a Three-Dimensional Radiative Transfer Effect: Implications for Cloud Remote Sensing

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 k25/3, but from there down to the smallest observable scales (50‐100 m) follows another k2b law with b . 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 Dz of the cloud, or with the ‘‘transport’’ mean free path lt 5 [(1 2 g)s]21, which incorporates the effect of forward scattering (s is extinction and g the asymmetry factor of the phase function). The smoothing scale is found to be approximately ltDz at cloud top; this is the prediction of diffusion ˇ theory which applies when (1 2 g)t 5D z / l t * 1( tis optical thickness). Since the scale break is a tangible effect of net horizontal radiative fluxes excited by the fluctuations of t, 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.

Scale Invariance of Liquid Water Distributions in Marine Stratocumulus. Part I: Spectral Properties and Stationarity Issues

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...

Cirrus cloud detection from Airborne Imaging Spectrometer data using the 1.38 µm water vapor band

Thin cirrus clouds are difficult to detect, particularly over land, in images taken from current satellite platforms. Using spectral images acquired by the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) at 20 km altitude, we show that wavelengths close to the center of the strong 1.38 µm water vapor band are useful for detecting thin cirrus clouds. The detection makes use of the fact that cirrus clouds are located above almost all the atmospheric water vapor. Because of the strong water vapor absorption in the lower atmosphere, AVIRIS channels near 1.38 µm receive little scattered solar radiance from the surface or low level clouds. When cirrus clouds are present, however, these channels receive large amounts of scattered solar radiance from the cirrus clouds. Our ability to determine cirrus cloud cover using space-based remote sensing will be improved if channels near the center of the 1.38 µm water vapor band are added to future satellites.

An algorithm using visible and 1.38-/spl mu/m channels to retrieve cirrus cloud reflectances from aircraft and satellite data

The Moderate Resolution Imaging Spectro-Radiometer (MODIS) on the Terra spacecraft has a channel near 1.38 /spl mu/m for remote sensing of high clouds from space. The implementation of this channel on MODIS was primarily based on previous analysis of hyperspectral imaging data collected with the Airborne Visible Infrared Imaging Spectrometer (AVIRIS). We describe an algorithm to retrieve cirrus bidirectional reflectance using channels near 0.66 and 1.38 /spl mu/m. It is shown that the apparent reflectance of the 1.38-/spl mu/m channel is essentially the bidirectional reflectance of cirrus clouds attenuated by the absorption of water vapor above cirrus clouds. A practical algorithm based on the scatterplot of 1.38-/spl mu/m channel apparent reflectance versus 0.66-/spl mu/m channel apparent reflectance has been developed to scale the effect of water vapor absorption so that the true cirrus reflectance in the visible spectral region can be obtained. To illustrate the applicability of the present algorithm, results for cirrus reflectance retrievals from AVIRIS and MODIS data are shown. The derived cirrus reflectance in the spectral region of 0.4-1 /spl mu/m can be used to remove cirrus contamination in a satellite image obtained at a visible channel. An example of such an application is shown. The spatially averaged cirrus reflectances derived from MODIS data can be used to establish global cirrus climatology, as is demonstrated by a sample global cirrus reflectance image.