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At least 343 records · Page 19

Multiangle Remote Sensing of Aerosols Over Ocean

Multiangle, multispectral remote sensing observations, such as those anticipated from the Earth Observing System (EOS) Multiangle Imaging SpectroRadiometer (MISR), can significantly improve our ability to constrain aerosol properties from space.

retrieve column optical depth calm ocean particle ↗

A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part 2: Dependence of absorption and extinction on ice crystal morphology

This study builds upon the microphysical modeling described in Part 1 by deriving formulations for the extinction and absorption coefficients in terms of the size distribution parameters predicted from the micro-physical model. The optical depth and single scatter albedo of a cirrus cloud can then be determined, which, along with the asymmetry parameter, are the input parameters needed by cloud radiation models. Through the use of anomalous diffraction theory, analytical expressions were developed describing the absorption and extinction coefficients and the single scatter albedo as functions of size distribution parameters, ice crystal shapes (or habits), wavelength, and refractive index. The extinction coefficient was formulated in terms of the projected area of the size distribution, while the absorption coefficient was formulated in terms of both the projected area and mass of the size distribution. These properties were formulated as explicit functions of ice crystal geometry and were not based on an 'effective radius.' Based on simulations of the second cirrus case study described in Part 1, absorption coefficients predicted in the near infrared for hexagonal columns and rosettes were up to 47% and 71% lower, respectively, than absorption coefficients predicted by using equivalent area spheres. This resulted in single scatter albedos in the near-infrared that were considerably greater than those predicted by the equivalent area sphere method. Reflectances in this region should therefore be underestimated using the equivalent area sphere approach. Cloud optical depth was found to depend on ice crystal habit. When the simulated cirrus cloud contained only bullet rosettes, the optical depth was 142% greater than when the cloud contained only hexagonal columns. This increase produced a doubling in cloud albedo. In the near-infrared (IR), the single scatter albedo also exhibited a significant dependence on ice crystal habit. More research is needed on the geometrical properties of ice crystals before the influence of ice crystal shape on cirrus radiative properties can be adequately understood. This study provides a way of coupling the radiative properties of absorption, extinction, and single scatter albedo to the microphysical properties of cirrus clouds. The dependence of extinction and absorption on ice crystal shape was not just due to geometrical differences between crystal types, but was also due to the effect these differences had on the evolution of ice particle size spectra. The ice particle growth model in Part 1 and the radiative properties treated here are based on analytical formulations, and thus represent a computationally efficient means of modeling the microphysical and radiative properties of cirrus clouds.

Mitchell, David L.↗

Cometary Evolution: Clues on Physical Properties from Chondritic Interplanetary Dust Particles

The degree of diversity or similarity detected in comets depends primarily on the lifetimes of the individual cometary nuclei at the time of analysis. It is inherent in our understanding of cometary orbital dynamics and the seminal model of comet origins by Oort that cometary evolution is the natural order of events in our Solar System. Thus, predictions of cometary behaviour in terms of bulk physical, mineralogical or chemical parameters should contain an appreciation of temporal variation(s). Previously, Rietmeijer and Mackinnon developed mineralogical bases for the chemical evolution of cometary nuclei primarily with regard to the predominantly silicate fraction of comet nuclei. We suggested that alteration of solids in cometary nuclei should be expected and that indications of likely reactants and products can be derived from judicious comparison with terrestrial diagenetic environments which include hydrocryogenic and low-temperature aqueous alterations. In a further development of this concept, Rietmeijer provides indirect evidence for the formation of sulfides and oxides in comet nuclei. Furthermore, Rietmeijer noted that timescales for hydrocryogenic and low-temperature reactions involving liquid water are probably adequate for relatively mature comets, e.g. P/comet Halley. In this paper, we will address the evolution of comet nuclei physical parameters such as solid particle grain size, porosity and density. In natural environments, chemical evolution (e.g. mineral reactions) is often accompanied by changes in physical properties. These concurrent changes are well-documented in the terrestrial geological literature, especially in studies of sediment diagenesis and we suggest that similar basic principles apply within the upper few meters of active comet nuclei. The database for prediction of comet nuclei physical parameters is, in principle, the same as used for the proposition of chemical evolution. We use detailed mineralogical studies of chondritic interplanetary dust particles (IDPS) as a guide to the likely constitution of mature comets traversing the inner Solar System. While there is, as yet, no direct proof that a specific sub-group or type of chondritic IDP is derived from a specific comet, it is clear that these particles are extraterrestrial in origin and that a certain portion of the interplanetary flux received by the Earth is cometary in origin. Two chondritic porous (CP) MPs, sample numbers W7010A2 and W7029Cl, from the Johnson Space Center Cosmic Dust Collection have been selected for this study of putative cometary physical parameters. This particular type of particle is considered a likely candidate for a cometary origin on the basis of mineralogy, bulk composition and morphology. While many IDPs have been subjected to intensive study over the past decade, we can develop a physical parameter model on only these two CP IDPs because few others have been studied in sufficient detail.

Rietmeijer, Frans J. M.↗

Cometary Evolution: Clues on Physical Properties from Chondritic Interplanetary Dust Particles

The degree of diversity or similarity detected in comets depends primarily on the lifetimes of the individual cometary nuclei at the time of analysis. It is inherent in our understanding of cometary orbital dynamics and the seminal model of comet origins that cometary evolution is the natural order of events in our Solar System. Thus, predictions of cometary behaviour in terms of bulk physical, mineralogical or chemical parameters should contain an appreciation of temporal variation(s). Previously, Rietmeijer and Mackinnon [1987] developed mineralogical bases for the chemical evolution of cometary nuclei primarily with regard to the predominantly silicate fraction of comet nuclei. We suggested that alteration of solids in cometary nuclei should be expected and that indications of likely reactants and products can be derived from judicious comparison with terrestrial diagenetic environments which include hydrocryogenic and low-temperature aqueous alterations. In a further development of this concept, Rietmeijer [1988] provides indirect evidence for the formation of sulfides and oxides in comet nuclei. Furthermore, Rietmeijer [1988] noted that timescales for hydrocryogenic and low-temperature reactions involving liquid water are probably adequate for relatively mature comets, e.g. P/comet Halley. In this paper, we will address the evolution of comet nuclei physical parameters such as solid particle grain size, porosity and density. In natural environments, chemical evolution (e.g. mineral reactions) is often accompanied by changes in physical properties. These concurrent changes are well-documented in the terrestrial geological literature, especially in studies of sediment diagenesis and we suggest that similar basic principles apply within the upper few meters of active comet nuclei. The database for prediction of comet nuclei physical parameters is, in principle, the same as used for the proposition of chemical evolution. We use detailed mineralogical studies of chondritic interplanetary dust particles (IDPS) as a guide to the likely constitution of mature comets traversing the inner Solar System. While there is, as yet, no direct proof that a specific sub-group or type of chondritic IDP is derived from a specific comet, it is clear that these particles are extraterrestrial in origin and that a certain portion of the interplanetary flux received by the Earth is cometary in origin. Two chondritic porous (CP) IDPS, sample numbers W701OA2 and W7029CI, from the Johnson Space Center Cosmic Dust Collection have been selected for this study of putative cometary physical parameters. This particular type of particle is considered a likely candidate for a cometary origin on the basis of mineralogy, bulk composition and morphology. While many IDPs have been subjected to intensive study over the past decade, we can develop a physical parameter model on only these two CP IDPs because few others have been studied in sufficient detail.

Reitmeijer, Frans J. M.↗

Cosmic dust particle densities - Evidence for two populations of stony micrometeorites

The existence of two populations of stony micrometeorites of distinctly different densities would result in significantly different orbital evolution properties for particles from each group. The densities inferred from deceleration of meteors in the earth's atmosphere suggest a substantial amount of the meteoric material has densities of 1 g/cu cm or less (Verniani, 1973). However, measurements of microcraters on lunar rock surfaces led Brownlee et al. (1973) to the conclusion that most micrometeoroids impacting the moon had densities in the 2-4 g/cu cm range, and low-density micrometeoroids were rare. The recovery of stony micrometeorites from the earth's stratosphere after atmospheric deceleration provides the opportunity to resolve the discrepancies. Here, the densities of 12 stony micrometeorites are determined, using synchrotron X-ray fluorescence to infer the particle mass and optical microscope measurements of the volumes. The particles fall into two distinct density groups, with mean values of 0.6 and 1.9 g/cu cm. The factor of 3 difference in the mean densities between the two populations implies differences in the orbital evolution time scales.

Flynn, G. J.↗

Lepton flavor asymmetries: from the early Universe to BBN

Large primordial lepton flavor asymmetries with almost vanishing total baryon-minus-lepton number can evade the usual BBN and CMB constraints if neutrino oscillations lead to perfect flavor equilibration. Solving the momentum averaged quantum kinetic equations (QKEs) describing neutrino oscillations and interactions, we perform the first systematic investigation of this scenario, uncovering a rich flavor structure in stark contradiction to the assumption of simple flavor equilibration. We find (i) a particular direction in flavor space, ∆ne ≃ – 2/3 (– 1)∆n μ for normal (inverted) neutrino mass hierarchy, in which the flavor equilibration is efficient and primordial asymmetries are essentially unconstrained, (ii) a minimal washout factor, ∆$n_{e}^{2}$| BBN ≤ 0.03 (0.016) ∑ α ∆$n_{α}^{2}$| ini yielding a conservative estimate for the allowed primordial asymmetries in a generic flavor direction, and (iii) particularly strong or weak washout if one of the initial flavor asymmetries vanishes due to non-adiabatic muon- or electron-driven MSW transitions. These results open up the possibility of a first-order QCD phase transition facilitated by large lepton asymmetries as well as baryogenesis from large and compensated ∆n e = ∆n μ asymmetries. Our systematic approach of deriving momentum averaged QKEs includes collision terms beyond the damping approximation, energy transfer between the neutrino and electron-photon plasma, and provides a fast and reliable way to investigate the impact of primordial lepton asymmetries at the time of BBN. We publicly release the Mathematica code COFLASY-M on https://github.com/mariofnavarro/COFLASY which solves the QKEs numerically.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

The magnificent ACT of flavor-specific neutrino self-interaction

Here we revisit the cosmology of neutrino self-interaction and use the latest cosmic microwave background data from the Atacama Cosmology Telescope (ACT) and the Planck experiment to constrain the interaction strength. In both flavor-universal and nonuniversal coupling scenarios, we find that the ACT data prefers strong neutrino self-interaction that delays neutrino free streaming until just before the matter-radiation equality. When combined with the Planck 2018 data, the preference for strong interaction decreases due to the Planck polarization data. For the combined dataset, the flavor-specific interaction still provides a better fit to the CMB data than ΛCDM. This trend persists even when neutrino mass is taken into account and extra radiation is added. We also study the prospect of constraining such strong interaction by future terrestrial and space telescopes, and find that the upcoming CMB-S4 experiment will improve the upper limit on neutrino self-interaction by about a factor of three.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Recent studies of the optical properties of dust and cloud particles in the Mars atmosphere and the interannual frequency of global dust storms

The results of research with two distinctly separate sets of observations yield new information on the optical properties of particulate scatterers in the Mars atmosphere, and on the interannual variability of the abundance of such scatterers in the Mars atmosphere. The first set of observations were taken by the IRTM (Infrared Thermal Mapper) instrument onboard the Viking Orbiters, during the period 1976 to 1980. Several hundred emission phase function (EPF) sequences were obtained over the Viking mission, in which the IRTM visual brightness channel observed the same area of surface/atmosphere as the spacecraft passed overhead. The 1 to 2 percent accuracy of calibration and the phase-angle coverage that characterizes these data make them ideally suited to determining both the optical depths and optical properties of dust and cloud scatterers in the Mars atmosphere versus latitude, longitude, seasons (L sub s), and surface elevation over the extended period of Viking observations. The EPF data were analyzed with a multiple scattering radiative transfer code to determine dust single scattering albedos which are distinctly higher than indicated by the Viking Lander observations. The second set of observations regard ground-based observations of the 1.3 to 2.6 mm rotational transitions of CO in the Martian atmosphere. The low-to-mid latitude average of the atmospheric temperature profile (0 to 70 km altitude) were derived from a number of such observations over the 1980 to 1990 period.

Clancy, R. T.↗

Advancements in the Representation of Cloud-Aerosol Microphysics in the GEOS-5 AGCM

Despite numerous challenges, the physical parameterization of cloud-aerosol interactions in atmospheric GCMs has become a top priority for advancement because of our need to simulate and understand past, current, and future indirect effects of aerosols on clouds. The challenges stem from the involvement of wide range of cloud-scale dynamics and aerosol activation physical processes. Cloud dynamics modulate cloud areal extent and condensate, while aerosol activation depends on aerosol mass load, size distribution, internal mixing state, and nucleating properties, and ultimately determines cloud optical properties via particle sizes. Both macro- and micro-scale processes are obviously important for cloud-radiation interactions. We will present the main features of cloud microphysical properties in the GEOS- 5 Atmospheric GCM (AGCM) as simulated by the McRAS-AC (Microphysics of Clouds with Relaxed Arakawa-Schubert and Aerosol-Cloud interaction) scheme. McRAS-AC uses Fountoukis and Nenes (2005) aerosol activation for liquid clouds, and has an option for either Liu and Penner (2005) or Barahona and Nenes (2008, 2009) aerosol activation for ice clouds. Aerosol loading (on-line or climatological) comes from GOCART, with an assumed log-normal size distribution. Other features of McRAS-AC are level-by-level cloud-scale thermodynamics, and Seifert-Beheng (2001)-type precipitation microphysics, particularly from moist convection. Results from Single-Column Model simulations will be shown to demonstrate how cloud radiative properties, lifetimes, and precipitation are influenced by different parameterization assumptions. Corresponding fields from year-long simulations of the full AGCM will also be presented with geographical distributions of cloud effective particle sizes compared to satellite retrievals. While the primary emphasis will be on current climate, simulation results with perturbed aerosol loadings will also be shown to expose the radiative sensitivity of the microphysical parameterization.

Lee, D.↗

Global CALIPSO Observations of Aerosol Changes Near Clouds

Several recent studies have found that clouds are surrounded by a transition zone of rapidly changing aerosol optical properties and particle size. Characterizing this transition zone is important for better understanding aerosol-cloud interactions and aerosol radiative effects, and also for improving satellite retrievals of aerosol properties. This letter presents a statistical analysis of a monthlong global data set of Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) lidar observations over oceans. The results show that the transition zone is ubiquitous over all oceans and extends up to 15 km away from clouds. They also show that near-cloud enhancements in backscatter and particle size are strongest at low altitudes, slightly below the top of the nearest clouds. Also, the enhancements are similar near illuminated and shadowy cloud sides, which confirms that the asymmetry of Moderate Resolution Imaging Spectroradiometer reflectances found in an earlier study comes from 3-D radiative processes and not from differences in aerosol properties. Finally, the effects of CALIPSO aerosol detection and cloud identification uncertainties are discussed. The findings underline the importance of accounting for the transition zone to avoid potential biases in studies of satellite aerosol products, aerosol-cloud interactions, and aerosol direct radiative effects.

Varnai, Tamas↗

Graph Neural Networks for Particle Reconstruction in High Energy Physics detectors

Pattern recognition problems in high energy physics are notably different from traditional machine learning applications in computer vision. Reconstruction algorithms identify and measure the kinematic properties of particles produced in high energy collisions and recorded with complex detector systems. Two critical applications are the reconstruction of charged particle trajectories in tracking detectors and the reconstruction of particle showers in calorimeters. These two problems have unique challenges and characteristics, but both have high dimensionality, high degree of sparsity, and complex geometric layouts. Graph Neural Networks (GNNs) are a relatively new class of deep learning architectures which can deal with such data effectively, allowing scientists to incorporate domain knowledge in a graph structure and learn powerful representations leveraging that structure to identify patterns of interest. In this work we demonstrate the applicability of GNNs to these two diverse particle reconstruction problems.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Eyjafjallajokull Volcano Plume Particle-Type Characterization from Space-Based Multi-angle Imaging

The Multi-angle Imaging SpectroRadiometer (MISR) Research Aerosol algorithm makes it possible to study individual aerosol plumes in considerable detail. From the MISR data for two optically thick, near-source plumes from the spring 2010 eruption of the Eyjafjallaj kull volcano, we map aerosol optical depth (AOD) gradients and changing aerosol particle types with this algorithm; several days downwind, we identify the occurrence of volcanic ash particles and retrieve AOD, demonstrating the extent and the limits of ash detection and mapping capability with the multi-angle, multi-spectral imaging data. Retrieved volcanic plume AOD and particle microphysical properties are distinct from background values near-source, as well as for overwater cases several days downwind. The results also provide some indication that as they evolve, plume particles brighten, and average particle size decreases. Such detailed mapping offers context for suborbital plume observations having much more limited sampling. The MISR Standard aerosol product identified similar trends in plume properties as the Research algorithm, though with much smaller differences compared to background, and it does not resolve plume structure. Better optical analogs of non-spherical volcanic ash, and coincident suborbital data to validate the satellite retrieval results, are the factors most important for further advancing the remote sensing of volcanic ash plumes from space.

Kahn, Ralph A.↗

Correlation of thermophoretically-modified small particle diffusional deposition rates in forced convection systems with variable properties, transpiration cooling and/or viscous dissipation

A cooled object (heat exchanger tube or turbine blade) is considered to be immersed in a hot fluid stream containing trace amounts of suspended vapors and/or small particles. Numerical prediction calculations were done for self-similar laminar boundary layers and law-of-the-wall turbulent boundary layers. Correlations are presented for the effect of thermophoresis in the absence of transpiration cooling and viscous dissipation; the effect of real suction and blowing in the absence of thermophoresis; the effect of viscous dissipation on thermophoresis in the absence of transpiration cooling; and the combined effect of viscous dissipation and transpiration cooling on thermophoresis. The final correlation, St/St-sub-zero, is insensitive to particle properties, Euler number, and local mainstream temperature.

Gokoglu, S. A.↗

What we Hope to Learn about Global Mineral Dust Aerosols from EOS Multi-Angle Imaging SpectroRadiometer (MISR)

On global scales, just a few broad atmospheric aerosol compositional groups are commonly observed. Of these, "mineral dust" is the only group which both contains non-spherical particles, and typically has size distributions with enough large particles for particle shape to affect its visible-light-scattering properties. The MISR instrument is scheduled for launch into a 10:30 AM sun-synchronous, polar orbit aboard the EOS Terra satellite in 1999. MISR will measure the upwelling visible radiance from Earth in 4 spectral bands centered at 446, 558, 672, and 866 nm, at each of 9 emission angles spread out in the forward and aft directions along the flight path at +/-70.5 deg, +/-60.0 deg, +/-45.6 deg, +/-26.1deg, and nadir. Over a period of 7 minutes, as the spacecraft flies along, a 360 km wide swath of Earth will successively be viewed by each of the cameras, allowing MISR to sample a very large range of scattering angles; in mid latitudes, the instrument will observe scattering angles between about 60 deg and 160 deg. Global coverage will be acquired about once in 9 days at the equator; the nominal mission lifetime is 6 years. The distinction in single scattering phase function between natural distributions of spherical and randomly oriented, non-spherical particles, with a broad range of aspect ratios, shows up strongly for scattering angles ranging from about 90 deg to near 180 deg. For non-spherical particle distributions, single scattering phase functions tend to be much flatter in this region than for spherical particles. Since MISR samples the relevant range of scattering angles very well, we expect to be able to make critical distinctions between natural distributions of spherical and randomly oriented, non-spherical particles with MISR data. We anticipate that the new multiangle, multispectral data from MISR will also contain other information about particle properties, a major step beyond current spacecraft remote sensing retrievals, which obtain aerosol optical depth based on entirely assumed particle microphysical properties. According to simulations over cloud-free, calm ocean, for pure particles with natural ranges of optical depth, particle size, and indices of refraction, MISR should retrieve column optical depth for all but the darkest particles, to an uncertainty of at most 0.05 or 20%, whichever is larger, even if the particle properties are poorly known. For one common particle type, soot, constraints on the optical depth over dark ocean are very poor. The simulated measurements also should allow us to separate two to four compositional groups based on indices of refraction, and to identify three to four distinct size groups between 0.1 and 2.0 microns characteristic radius at most latitudes. The technique is most sensitive to particle microphysical properties in the "accumulation mode" sizes. where particle scattering undergoes the transition from Rayleigh to large-particle regimes for the MISR wavelengths. Based on these results, we expect to distinguish air masses containing different aerosol types, routinely and globally, with multiangle remote sensing data. Such data complements in situ and field data, which can provide details about aerosol size and composition locally that are needed to assess the radiative effects of aerosols quantitatively. Both field data and correlations in space and time with likely source and sink regions will also be helpful in developing a global picture of mineral dust aerosol budgets. Further work on the expected sensitivity of MISR to natural mixtures of pure particles, including climatologically likely mineral dust components, is currently underway.

Kahn, Ralph↗

Community Radiative Transfer Model: Implementing A New Cloud Scattering Database and Developing the Forward Radar Module

The Mie theory is used by many fast RT models to estimate the optical properties of single particles. The Mie theory assumes spherical shapes for ice or snow particles with mixture of air and ice. However, hydrometeors scattering radiation at microwave frequencies have different shapes, sizes, and orientations. Therefore, using Mie theory to determine their optical properties leads to large uncertainties in all-sky radiative transfer calculations. The discrete dipole approximation (DDA) which approximates the optical properties of large objects in terms of discrete dipoles has shown promises in calculating the scattering properties of particles with different shapes in the microwave frequencies. The goal of the research was to enhance the CRTM scattering calculations for frozen hydrometeors in the microwave frequencies using the DDA technique. Given that such optical properties cannot be practically calculated on the fly, pre-computed look-up tables need to be implemented into the fast RT models to calculate the scattering properties of these particles using the DDA technique and the inputs provided by the users. Therefore, we implemented such pre-computed DDA databases into CRTM. In addition to using stand-alone CRTM calculations using collocated ATMS and reanalysis profiles, the data assimilation experiments conducted using the NOAA FV3GFS forecast system will be used to evaluate the scattering improvements. Additionally, we used the backscattering information from the DDA database to implement a radar simulator into CRTM. The radar operators takes advantage of CRTM different modules to calculate clouds absorption and scattering properties. In addition to the forward model both adjoint and tangent linear of the radar simulator are implemented and evaluated as well. The radar simulator is currently being tested within the JEDI/GEOS data assimilation framework to facilitate the assimilation of radar measurements such as CloudSat CPR and GPM DPR into the NASA GEOS model.

Isaac Moradi↗

Microelectrode Arrays for Electrochemical Cycling of Individual Battery Particles

The electrochemical characterization of battery materials is usually performed on porous electrodes containing conductive additives, binders, and ensembles of redox-active particles at densities as high as 10 15 cm –3 . These complex structures often obscure individual battery particles’ intrinsic properties, hindering insights into the fundamentals of battery electrochemistry. To address this challenge, we developed microelectrode arrays to enable the charge and discharge of individual battery particles in liquid electrolytes. In this Methods and Protocols paper, we describe the fabrication of the microelectrode array chips, the assembly of particles onto microelectrodes, and the different types of electrochemical techniques that can be used, including galvanostatic cycling, cyclic voltammetry, potentiostatic intermittent titration, and electrochemical impedance spectroscopy. We use micron-sized lithium cobalt oxide as a model system to showcase the reliability and versatility of single-particle electrochemical experiments enabled by our microelectrode array. In conclusion, we aim to provide a detailed description to promote future uses of microelectrode arrays in fundamental research of batteries and other electrochemical systems.

25 ENERGY STORAGE↗

Parameterization of Cloud Optical Properties for a Mixture of Ice Particles for use in Atmospheric Models

Based on the single-scattering optical properties that are pre-computed using an improve geometric optics method, the bulk mass absorption coefficient, single-scattering albedo, and asymmetry factor of ice particles have been parameterized as a function of the mean effective particle size of a mixture of ice habits. The parameterization has been applied to compute fluxes for sample clouds with various particle size distributions and assumed mixtures of particle habits. Compared to the parameterization for a single habit of hexagonal column, the solar heating of clouds computed with the parameterization for a mixture of habits is smaller due to a smaller cosingle-scattering albedo. Whereas the net downward fluxes at the TOA and surface are larger due to a larger asymmetry factor. The maximum difference in the cloud heating rate is approx. 0.2 C per day, which occurs in clouds with an optical thickness greater than 3 and the solar zenith angle less than 45 degrees. Flux difference is less than 10 W per square meters for the optical thickness ranging from 0.6 to 10 and the entire range of the solar zenith angle. The maximum flux difference is approximately 3%, which occurs around an optical thickness of 1 and at high solar zenith angles.

Chou, Ming-Dah↗