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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

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

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

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.

## Abstract

Most of the existing cloud radiation models treat liquid water drops of a variety of sizes as an ensemble of particles. The ensemble approach assumes that all drop sizes are well represented in an elementary volume, and its scattering and absorbing properties can be accurately specified through the use of the drop size probability density distribution function. The concentration of large drops, however, can be so low that a chance to capture them in the elementary volume is rare. Thus the drop ensemble assumption is not always true, though classical radiative transfer theory uses this assumption to simplify the radiative transfer process, as if scattering takes place from an “average drop” rather than from a particular drop. The theoretical analysis presented in this paper demonstrates that if a cumulative distribution function is used to describe drop size variability with jumps accounting for the probability of finding large drops in the elementary volume, one obtains an extra term, the Green's function, in the solution of the radiative transfer equation. The analysis of data on cloud drop size distribution acquired during the First International Satellite Cloud Climatology Project (ISCCP) Research Experiment (FIRE) field campaign clearly shows jumps in the cumulative drop size distribution; the magnitudes of the jumps are related to the frequencies of large drop occurrence. This discontinuity is primarily responsible for the additional terms that must be added to the solution to properly account for the photon interaction with the large drops. The enhancement of cloud absorption due to accounting for the “missing solution” exhibits a jump-like response to continuous variation in the concentration of large drops and may exceed the enhancement predicted by the ensemble-based models. The results presented here indicate that the missing term might be plausible to explain the enhanced value of the ratio of the shortwave cloud forcing at the surface to the forcing at top of the atmosphere.

## Abstract

Most of the existing cloud radiation models treat liquid water drops of a variety of sizes as an ensemble of particles. The ensemble approach assumes that all drop sizes are well represented in an elementary volume, and its scattering and absorbing properties can be accurately specified through the use of the drop size probability density distribution function. The concentration of large drops, however, can be so low that a chance to capture them in the elementary volume is rare. Thus the drop ensemble assumption is not always true, though classical radiative transfer theory uses this assumption to simplify the radiative transfer process, as if scattering takes place from an “average drop” rather than from a particular drop. The theoretical analysis presented in this paper demonstrates that if a cumulative distribution function is used to describe drop size variability with jumps accounting for the probability of finding large drops in the elementary volume, one obtains an extra term, the Green's function, in the solution of the radiative transfer equation. The analysis of data on cloud drop size distribution acquired during the First International Satellite Cloud Climatology Project (ISCCP) Research Experiment (FIRE) field campaign clearly shows jumps in the cumulative drop size distribution; the magnitudes of the jumps are related to the frequencies of large drop occurrence. This discontinuity is primarily responsible for the additional terms that must be added to the solution to properly account for the photon interaction with the large drops. The enhancement of cloud absorption due to accounting for the “missing solution” exhibits a jump-like response to continuous variation in the concentration of large drops and may exceed the enhancement predicted by the ensemble-based models. The results presented here indicate that the missing term might be plausible to explain the enhanced value of the ratio of the shortwave cloud forcing at the surface to the forcing at top of the atmosphere.

## Abstract

Nimbus 7 Scanning Multichannel Microwave Radiometer (SMMR) measurements at five frequencies in the region 6.6 to 37 GHz, at a resolution of 155 km, are analyzed to infer precipitation over the global oceans. The microwave data show, on this spatial scale, that the combined liquid water in the clouds and rain increases the brightness temperature almost linearly with frequency in the 6.6 to 18 GHz region, while at 37 GHz such a simple relationship is not noticed. Further, as the atmospheric water vapor absorption and the effects of scattering by precipitation particles are relatively weak at 6.6 and 10.7 GHz, a technique to remotely sense the liquid water content in the atmosphere is developed based on the brightness measurements at these two frequencies. Seasonal mean patterns of liquid water content in the atmosphere derived from SMMR over global oceans relate closely to climatological patterns of precipitation. Based on this, an empirical relationship is derived to estimate precipitation over the global oceans, with an accuracy of about ±30 percent, on a seasonal basis from satellite measurements made during the three years (1979–81) before the recent El Niño event. The deviations from these three-year means in the precipitation, produced by the 1982–83 El Niño event are then deduced from the SMMR measurements. In the Pacific one notices from these deviations that the precipitation over the ITCZ in the north, the South Pacific Convergence Zone, and the oceans around Indonesia is drastically reduced. At the same time a substantial increase in precipitation is observed over the normally dry central and eastern equatorial Pacific Ocean.

## Abstract

Nimbus 7 Scanning Multichannel Microwave Radiometer (SMMR) measurements at five frequencies in the region 6.6 to 37 GHz, at a resolution of 155 km, are analyzed to infer precipitation over the global oceans. The microwave data show, on this spatial scale, that the combined liquid water in the clouds and rain increases the brightness temperature almost linearly with frequency in the 6.6 to 18 GHz region, while at 37 GHz such a simple relationship is not noticed. Further, as the atmospheric water vapor absorption and the effects of scattering by precipitation particles are relatively weak at 6.6 and 10.7 GHz, a technique to remotely sense the liquid water content in the atmosphere is developed based on the brightness measurements at these two frequencies. Seasonal mean patterns of liquid water content in the atmosphere derived from SMMR over global oceans relate closely to climatological patterns of precipitation. Based on this, an empirical relationship is derived to estimate precipitation over the global oceans, with an accuracy of about ±30 percent, on a seasonal basis from satellite measurements made during the three years (1979–81) before the recent El Niño event. The deviations from these three-year means in the precipitation, produced by the 1982–83 El Niño event are then deduced from the SMMR measurements. In the Pacific one notices from these deviations that the precipitation over the ITCZ in the north, the South Pacific Convergence Zone, and the oceans around Indonesia is drastically reduced. At the same time a substantial increase in precipitation is observed over the normally dry central and eastern equatorial Pacific Ocean.

## Abstract

Aircraft measurements of liquid water content (LWC) made at sampling frequencies of 1 and 2 kHz with a particle volume monitor (PVM) probe from horizontal traverses in stratocumulus clouds during the Southern Ocean Cloud Experiment and cumulus clouds during the Small Cumulus Microphysics Study are described. The spectral density of the LWC measurements is calculated and compared to the −5/3 scaling law. The effect of PVM sampling noise is found to be small in most cases. Most measurements follow approximately the −5/3 law until cloud scales decrease below about 5 m in length. Below this length LWC variance can exceed that predicted by the −5/3 law. It is suggested that the enhanced LWC variance at small scales is related to entrainment of environmental air into the clouds, which changes primarily the droplet concentration.

## Abstract

Aircraft measurements of liquid water content (LWC) made at sampling frequencies of 1 and 2 kHz with a particle volume monitor (PVM) probe from horizontal traverses in stratocumulus clouds during the Southern Ocean Cloud Experiment and cumulus clouds during the Small Cumulus Microphysics Study are described. The spectral density of the LWC measurements is calculated and compared to the −5/3 scaling law. The effect of PVM sampling noise is found to be small in most cases. Most measurements follow approximately the −5/3 law until cloud scales decrease below about 5 m in length. Below this length LWC variance can exceed that predicted by the −5/3 law. It is suggested that the enhanced LWC variance at small scales is related to entrainment of environmental air into the clouds, which changes primarily the droplet concentration.