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## Abstract

The delta–*M* method represents a natural extension of the recently proposed delta–Eddington approximation to all orders *M* of angular approximation. It relies essentially on matching the first 2*M* phase function moments and using a Dirac delta–function representation of forward scattering. Computed fluxes are remarkably accurate at very low orders *M* of approximation, even when the phase function is strongly asymmetric; thus the associated *M* × *M* matrix computations remain small and manageable. Flux is automatically conserved, making phase function “renormalization” unnecessary. Phase function truncation is effected in a much more attractive manner than in the past; furthermore, truncation tends to zero as *M* → ∞. Errors are shown to oscillate with (roughly) exponentially decreasing amplitude as *M* increases; which has the curious consequence that increasing *M* by small amounts does not necessarily reduce error. Mie computations associated with the δ–*M* method can be considerably reduced, based on a simple technique for phase function moment calculations proposed herein.

## Abstract

The delta–*M* method represents a natural extension of the recently proposed delta–Eddington approximation to all orders *M* of angular approximation. It relies essentially on matching the first 2*M* phase function moments and using a Dirac delta–function representation of forward scattering. Computed fluxes are remarkably accurate at very low orders *M* of approximation, even when the phase function is strongly asymmetric; thus the associated *M* × *M* matrix computations remain small and manageable. Flux is automatically conserved, making phase function “renormalization” unnecessary. Phase function truncation is effected in a much more attractive manner than in the past; furthermore, truncation tends to zero as *M* → ∞. Errors are shown to oscillate with (roughly) exponentially decreasing amplitude as *M* increases; which has the curious consequence that increasing *M* by small amounts does not necessarily reduce error. Mie computations associated with the δ–*M* method can be considerably reduced, based on a simple technique for phase function moment calculations proposed herein.

## Abstract

The nonsphericity of many atmospheric particles is often raised as an objection to radiative transfer analyses which assume sphericity. This paper studies the behavior of extinction and absorption cross sections, as well as direct backscattering, for rotationally symmetric nonspherical particles of the form *r*=*r*_{0}[1+ε*T*_{n}(cosθ)]*T*
_{n} a Chebyshev polynomial. For *n*=2 and 4, −0.2≤ε≤0.2 and size parameters up to 10, we compare the various nonspherical scattering parameters in both fixed and random orientation (calculated exactly using the Extended Boundary Condition Method) with those for equal-volume and equal-projected-area spheres.

We find that:

1) The equal-volume-sphere approximation becomes increasingly poor above size parameter 5 unless the oscillations in the spherical curves are smoothed out, either by high absorption or size-averaging.

2) Orientation-averaging of extinction and absorption cross sections reduces spherical-nonspherical differences by an order of magnitude; size-averaging also reduces these differences, but not nearly as much.

3) Equivalent spheres give a better approximation to non-spherical absorption cross section than to extinction cross section or backscattering.

4) Concavity systematically elevates the cross section for larger particles.

5) Backscattering exhibits a magnified sensitivity to particle shape for nearly transparent particles, e.g., a 10% deviation from sphericity may produce a 100% change in backscattering; but this sensitivity is dramatically reduced when the particles have significant absorption.

6) Increasing the absorption *always* improves the agreement with equivalent spheres.

## Abstract

The nonsphericity of many atmospheric particles is often raised as an objection to radiative transfer analyses which assume sphericity. This paper studies the behavior of extinction and absorption cross sections, as well as direct backscattering, for rotationally symmetric nonspherical particles of the form *r*=*r*_{0}[1+ε*T*_{n}(cosθ)]*T*
_{n} a Chebyshev polynomial. For *n*=2 and 4, −0.2≤ε≤0.2 and size parameters up to 10, we compare the various nonspherical scattering parameters in both fixed and random orientation (calculated exactly using the Extended Boundary Condition Method) with those for equal-volume and equal-projected-area spheres.

We find that:

1) The equal-volume-sphere approximation becomes increasingly poor above size parameter 5 unless the oscillations in the spherical curves are smoothed out, either by high absorption or size-averaging.

2) Orientation-averaging of extinction and absorption cross sections reduces spherical-nonspherical differences by an order of magnitude; size-averaging also reduces these differences, but not nearly as much.

3) Equivalent spheres give a better approximation to non-spherical absorption cross section than to extinction cross section or backscattering.

4) Concavity systematically elevates the cross section for larger particles.

5) Backscattering exhibits a magnified sensitivity to particle shape for nearly transparent particles, e.g., a 10% deviation from sphericity may produce a 100% change in backscattering; but this sensitivity is dramatically reduced when the particles have significant absorption.

6) Increasing the absorption *always* improves the agreement with equivalent spheres.

## Abstract

New formulas for the backscattered fraction in two-stream theory are derived. They express this fraction, for either isotropically or monodirectionally incident radiation, as a single integral over the scattering phase function, thereby effecting a substantial simplification over the customary multiple-integral definitions. From these formulas the globally averaged backscatter of the earth due to typical aerosols is shown to depend primarily on the *forward* part (0° to 90°) of the scattering phase function, where the disagreement between spherical-and nonspherical-particle scattering is smallest. The new formulas also lead to connections, in terms of standard elliptic integrals, between the backscatter and the phase function asymmetry factor; while rigorously correct only for the Henyey-Greenstein phase function, these relations are shown to be remarkably accurate for *all* spherical-particle phase functions. The detailed relationship between backscatter and asymmetry factor is shown to be multi-valued; thus two-stream and Eddington approximations cannot be uniquely related.

The common approximation of the globally averaged backscatter, or Bond albedo, by the backscatter for radiation incident at solar zenith angles of O° or 60° is shown to lead, for a wide range of particle sizes and optical properties, to systematic and often large underestimates. The solar-spectrum-integrated enhancement of the Bond albedo due to a uniform, optically thin aerosol layer is examined, holding the total mass of aerosol fixed and varying the particle radii and optical properties over wide ranges. The particle radius at which maximum albedo enhancement occurs decreases from 0.3 µm down to about 0.08 µm as the particle absorptivity increases. Also, increasing the absorption of particles smaller than 0.1 µm actually raises the albedo in contrast to the usual situation where absorption suppresses backscattering.

## Abstract

New formulas for the backscattered fraction in two-stream theory are derived. They express this fraction, for either isotropically or monodirectionally incident radiation, as a single integral over the scattering phase function, thereby effecting a substantial simplification over the customary multiple-integral definitions. From these formulas the globally averaged backscatter of the earth due to typical aerosols is shown to depend primarily on the *forward* part (0° to 90°) of the scattering phase function, where the disagreement between spherical-and nonspherical-particle scattering is smallest. The new formulas also lead to connections, in terms of standard elliptic integrals, between the backscatter and the phase function asymmetry factor; while rigorously correct only for the Henyey-Greenstein phase function, these relations are shown to be remarkably accurate for *all* spherical-particle phase functions. The detailed relationship between backscatter and asymmetry factor is shown to be multi-valued; thus two-stream and Eddington approximations cannot be uniquely related.

The common approximation of the globally averaged backscatter, or Bond albedo, by the backscatter for radiation incident at solar zenith angles of O° or 60° is shown to lead, for a wide range of particle sizes and optical properties, to systematic and often large underestimates. The solar-spectrum-integrated enhancement of the Bond albedo due to a uniform, optically thin aerosol layer is examined, holding the total mass of aerosol fixed and varying the particle radii and optical properties over wide ranges. The particle radius at which maximum albedo enhancement occurs decreases from 0.3 µm down to about 0.08 µm as the particle absorptivity increases. Also, increasing the absorption of particles smaller than 0.1 µm actually raises the albedo in contrast to the usual situation where absorption suppresses backscattering.

## Abstract

An expression for the surface area of a nonspherical particle described by the equation*r* = *r*
_{0}[1 ± ε*T _{n}*(cos θ)]is derived, and radii of various equivalent spheres are calculated.

## Abstract

An expression for the surface area of a nonspherical particle described by the equation*r* = *r*
_{0}[1 ± ε*T _{n}*(cos θ)]is derived, and radii of various equivalent spheres are calculated.

The horizons of atmospheric science are undergoing a considerable expansion as a result of intense interest in problems of climate. This has caused somewhat of a renaissance in hitherto-neglected subfields of atmospheric science. Focusing on atmospheric radiation as the renascent subfield of most direct concern to us, we describe the exciting research and educational challenges that lie ahead in this subfield, and offer possible ways in which these challenges might be met.

The horizons of atmospheric science are undergoing a considerable expansion as a result of intense interest in problems of climate. This has caused somewhat of a renaissance in hitherto-neglected subfields of atmospheric science. Focusing on atmospheric radiation as the renascent subfield of most direct concern to us, we describe the exciting research and educational challenges that lie ahead in this subfield, and offer possible ways in which these challenges might be met.

## Abstract

A method for detecting cirrus clouds in terms of brightness temperature differences between narrowbands at 8, 11, and 12 µm has been proposed by Ackerman et al. In this method, the variation of emissivity with wavelength for different surface targets was not taken into consideration. Based on state-of-the-art laboratory measurements of reflectance spectra of terrestrial materials by Salisbury and D'Aria, it is found that the brightness temperature differences between the 8- and 11-µm bands for soils, rocks and minerals, and dry vegetation can vary between approximately −8 and +8 K due solely to surface emissivity variations. The large brightness temperature differences are sufficient to cause false detection of cirrus clouds from remote sensing data acquired over certain surface targets using the 8-11-12-µm method directly. It is suggested that the 8-11-12-µm method should be improved to include the surface emissivity effects. In addition, it is recommended that in the future the variation of surface emissivity with wavelength should be taken into account in algorithms for retrieving surface temperatures and low-level atmospheric temperature and water vapor profiles.

## Abstract

A method for detecting cirrus clouds in terms of brightness temperature differences between narrowbands at 8, 11, and 12 µm has been proposed by Ackerman et al. In this method, the variation of emissivity with wavelength for different surface targets was not taken into consideration. Based on state-of-the-art laboratory measurements of reflectance spectra of terrestrial materials by Salisbury and D'Aria, it is found that the brightness temperature differences between the 8- and 11-µm bands for soils, rocks and minerals, and dry vegetation can vary between approximately −8 and +8 K due solely to surface emissivity variations. The large brightness temperature differences are sufficient to cause false detection of cirrus clouds from remote sensing data acquired over certain surface targets using the 8-11-12-µm method directly. It is suggested that the 8-11-12-µm method should be improved to include the surface emissivity effects. In addition, it is recommended that in the future the variation of surface emissivity with wavelength should be taken into account in algorithms for retrieving surface temperatures and low-level atmospheric temperature and water vapor profiles.

## Abstract

This paper presents a rapid yet accurate method, the “delta-Eddington” approximation, for calculating monochromatic radiative fluxes in an absorbing-scattering atmosphere. By combining a Dirac delta function and a two-term approximation, it overcomes the poor accuracy of the Eddington approximation for highly asymmetric phase functions. The fraction of scattering into the truncated forward peak is taken proportional to the square of the phase function asymmetry factor, which distinguishes the delta-Eddington approximation from others of similar nature. Comparisons of delta-Eddington albedos, transnmissivities and absorptivities with more exact calculations reveal typical differences of 0–0.022 and maximum differences of 0.15 over wide ranges of optical depth, sun angle, surface albedo, single-scattering albedo and phase function asymmetry. Delta-Eddington fluxes are in error, on the average, by no more than 0.5%0, and at the maximum by no more than 2% of the incident flux. This computationally fast and accurate approximation is potentially of utility in applications such as general circulation and climate modelling.

## Abstract

This paper presents a rapid yet accurate method, the “delta-Eddington” approximation, for calculating monochromatic radiative fluxes in an absorbing-scattering atmosphere. By combining a Dirac delta function and a two-term approximation, it overcomes the poor accuracy of the Eddington approximation for highly asymmetric phase functions. The fraction of scattering into the truncated forward peak is taken proportional to the square of the phase function asymmetry factor, which distinguishes the delta-Eddington approximation from others of similar nature. Comparisons of delta-Eddington albedos, transnmissivities and absorptivities with more exact calculations reveal typical differences of 0–0.022 and maximum differences of 0.15 over wide ranges of optical depth, sun angle, surface albedo, single-scattering albedo and phase function asymmetry. Delta-Eddington fluxes are in error, on the average, by no more than 0.5%0, and at the maximum by no more than 2% of the incident flux. This computationally fast and accurate approximation is potentially of utility in applications such as general circulation and climate modelling.

## Abstract

In an effort to bring more realism cloud-radiation calculations, arising-parcel model of cloud microphysics and a 191 waveband model of atmospheric radiation (ATRAD) have been brought to bear on the problem of cloud absorption of solar radiation, with emphasis on the effect of drops greater than 40–50 μm in radius. The earlier conclusions of Welch and others that such large drops can produce cloud absorptivities in excess of 30% have not been substantiated. Instead we find large-drop enhancements of only 0.02–0.04 in cloud and total atmospheric absorptivities. However, several other, more important influences were uncovered: 1) Large drops make it necessary to know the second and third moments of the drop distribution in order to parameterize the shortwave effect of clouds; parameterizations based only on the third moment (liquid water content) do not consider a wide enough range of variation of drop distribution. 2) Large drops cause a precipitous fall in both cloud and planetary albedo if the supply of liquid water is fixed. 3) Large drops enhance the solar greenhouse effect by distributing solar heating more deeply into the cloud. Plots of spectral heating rate reveal that the spectral regions 1.5–1.8 μm and 1.15–1.3 μm are most important for shortwave heating of clouds.

It is suggested that very large drops may also explain the looming “optical depth paradox,” whereby optical depths deduced from measurements of reflected radiation are much smaller than those calculated from measured liquid water profiles.

## Abstract

In an effort to bring more realism cloud-radiation calculations, arising-parcel model of cloud microphysics and a 191 waveband model of atmospheric radiation (ATRAD) have been brought to bear on the problem of cloud absorption of solar radiation, with emphasis on the effect of drops greater than 40–50 μm in radius. The earlier conclusions of Welch and others that such large drops can produce cloud absorptivities in excess of 30% have not been substantiated. Instead we find large-drop enhancements of only 0.02–0.04 in cloud and total atmospheric absorptivities. However, several other, more important influences were uncovered: 1) Large drops make it necessary to know the second and third moments of the drop distribution in order to parameterize the shortwave effect of clouds; parameterizations based only on the third moment (liquid water content) do not consider a wide enough range of variation of drop distribution. 2) Large drops cause a precipitous fall in both cloud and planetary albedo if the supply of liquid water is fixed. 3) Large drops enhance the solar greenhouse effect by distributing solar heating more deeply into the cloud. Plots of spectral heating rate reveal that the spectral regions 1.5–1.8 μm and 1.15–1.3 μm are most important for shortwave heating of clouds.

It is suggested that very large drops may also explain the looming “optical depth paradox,” whereby optical depths deduced from measurements of reflected radiation are much smaller than those calculated from measured liquid water profiles.

## Abstract

Certain algebraic combinations of single scattering albedo and solar radiation reflected from, or transmitted through, vegetation canopies do not vary with wavelength. These “spectrally invariant relationships” are the consequence of wavelength independence of the extinction coefficient and scattering phase function in vegetation. In general, this wavelength independence does not hold in the atmosphere, but in cloud-dominated atmospheres the total extinction and total scattering phase function vary only weakly with wavelength. This paper identifies the atmospheric conditions under which the spectrally invariant approximation can accurately describe the extinction and scattering properties of cloudy atmospheres. The validity of the assumptions and the accuracy of the approximation are tested with 1D radiative transfer calculations using publicly available radiative transfer models: Discrete Ordinate Radiative Transfer (DISORT) and Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). It is shown for cloudy atmospheres with cloud optical depth above 3, and for spectral intervals that exclude strong water vapor absorption, that the spectrally invariant relationships found in vegetation canopy radiative transfer are valid to better than 5%. The physics behind this phenomenon, its mathematical basis, and possible applications to remote sensing and climate are discussed.

## Abstract

Certain algebraic combinations of single scattering albedo and solar radiation reflected from, or transmitted through, vegetation canopies do not vary with wavelength. These “spectrally invariant relationships” are the consequence of wavelength independence of the extinction coefficient and scattering phase function in vegetation. In general, this wavelength independence does not hold in the atmosphere, but in cloud-dominated atmospheres the total extinction and total scattering phase function vary only weakly with wavelength. This paper identifies the atmospheric conditions under which the spectrally invariant approximation can accurately describe the extinction and scattering properties of cloudy atmospheres. The validity of the assumptions and the accuracy of the approximation are tested with 1D radiative transfer calculations using publicly available radiative transfer models: Discrete Ordinate Radiative Transfer (DISORT) and Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). It is shown for cloudy atmospheres with cloud optical depth above 3, and for spectral intervals that exclude strong water vapor absorption, that the spectrally invariant relationships found in vegetation canopy radiative transfer are valid to better than 5%. The physics behind this phenomenon, its mathematical basis, and possible applications to remote sensing and climate are discussed.

## Abstract

Most cloud radiation models and conventional data processing techniques assume that the mean number of drops of a given radius is proportional to volume. The analysis of microphysical data on liquid water drop sizes shows that, for sufficiently small volumes, this proportionality breaks down; the number of cloud drops of a given radius is instead proportional to the volume raised to a drop size–dependent nonunit power. The coefficient of proportionality, a *generalized drop concentration*, is a function of the drop size. For abundant small drops the power is unity as assumed in the conventional approach. However, for rarer large drops, it falls increasingly below unity. This empirical fact leads to drop clustering, with the larger drops exhibiting a greater degree of clustering. The generalized drop concentration shows the mean number of drops per cluster, while the power characterizes the occurrence frequency of clusters. With a fixed total number of drops in a cloud, a decrease in frequency of clusters is accompanied by a corresponding increase in the generalized concentration. This initiates a competing process missed in the conventional models: an increase in the number of drops per cluster enhances the impact of rarer large drops on cloud radiation while a decrease in the frequency suppresses it. Because of the nonlinear relationship between the number of clustered drops and the volume, these two opposite tendencies do not necessarily compensate each other. The data analysis suggests that clustered drops likely have a stronger radiative impact compared to their unclustered counterpart; ignoring it results in underestimation of the contribution from large drops to cloud horizontal optical path.

## Abstract

Most cloud radiation models and conventional data processing techniques assume that the mean number of drops of a given radius is proportional to volume. The analysis of microphysical data on liquid water drop sizes shows that, for sufficiently small volumes, this proportionality breaks down; the number of cloud drops of a given radius is instead proportional to the volume raised to a drop size–dependent nonunit power. The coefficient of proportionality, a *generalized drop concentration*, is a function of the drop size. For abundant small drops the power is unity as assumed in the conventional approach. However, for rarer large drops, it falls increasingly below unity. This empirical fact leads to drop clustering, with the larger drops exhibiting a greater degree of clustering. The generalized drop concentration shows the mean number of drops per cluster, while the power characterizes the occurrence frequency of clusters. With a fixed total number of drops in a cloud, a decrease in frequency of clusters is accompanied by a corresponding increase in the generalized concentration. This initiates a competing process missed in the conventional models: an increase in the number of drops per cluster enhances the impact of rarer large drops on cloud radiation while a decrease in the frequency suppresses it. Because of the nonlinear relationship between the number of clustered drops and the volume, these two opposite tendencies do not necessarily compensate each other. The data analysis suggests that clustered drops likely have a stronger radiative impact compared to their unclustered counterpart; ignoring it results in underestimation of the contribution from large drops to cloud horizontal optical path.