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NASA NTRS2013

Detection Thresholds of Falling Snow From Satellite-Borne Active and Passive Sensors

There is an increased interest in detecting and estimating the amount of falling snow reaching the Earths surface in order to fully capture the global atmospheric water cycle. An initial step toward global spaceborne falling snow algorithms for current and future missions includes determining the thresholds of detection for various active and passive sensor channel configurations and falling snow events over land surfaces and lakes. In this paper, cloud resolving model simulations of lake effect and synoptic snow events were used to determine the minimum amount of snow (threshold) that could be detected by the following instruments: the W-band radar of CloudSat, Global Precipitation Measurement (GPM) Dual-Frequency Precipitation Radar (DPR)Ku- and Ka-bands, and the GPM Microwave Imager. Eleven different nonspherical snowflake shapes were used in the analysis. Notable results include the following: 1) The W-band radar has detection thresholds more than an order of magnitude lower than the future GPM radars; 2) the cloud structure macrophysics influences the thresholds of detection for passive channels (e.g., snow events with larger ice water paths and thicker clouds are easier to detect); 3) the snowflake microphysics (mainly shape and density)plays a large role in the detection threshold for active and passive instruments; 4) with reasonable assumptions, the passive 166-GHz channel has detection threshold values comparable to those of the GPM DPR Ku- and Ka-band radars with approximately 0.05 g *m(exp -3) detected at the surface, or an approximately 0.5-1.0-mm * h(exp -1) melted snow rate. This paper provides information on the light snowfall events missed by the sensors and not captured in global estimates.

remote sensing↗

DOE OSTI2022

Sizing ice hydrometeor populations using the dual-wavelength radar ratio

Dual-wavelength (3.2 and 0.32 cm, i.e., X- and W-radar bands) radar ratio (DWR) measurements in ice clouds and precipitation using Canada's National Research Council Institute for Aerospace Research airborne radar are compared to closely collocated particle microphysical in situ sampling data in order to develop relations between DWR and characteristic hydrometeor size. This study uses the radar and in situ data sets collected during the In-Cloud ICing and Large-drop Experiment (ICICLE) campaign in midlatitude frontal clouds. Since atmospheric particle scattering at X band is predominantly in the Rayleigh regime and the W-band frequency is the highest frequency usually used for hydrometeor remote sensing, the X–W-band combination provides a relatively strong dual-wavelength reflectivity difference. This study considers radar and in situ measurements conducted in relatively homogeneous cloud and precipitation conditions. Measurements show that under these conditions, the difference between the X-band radar reflectivities observed with vertical and horizontal pointing of the radar beam are generally small and often negligible. However, W-band reflectivities at vertical beam pointing are, on average, larger than those for horizontal beam pointing by about 4 dB, which is a non-Rayleigh scattering effect from preferentially oriented non-spherical particles. A horizontal radar beam DWR–mean volume particle size relation, D v , provides robust estimates of this characteristic size for populations of particles with different habits. Uncertainties of D v retrievals using DWR are around 0.6 mm when D v is greater than approximately 1 mm. Size estimates using vertical radar beam DWRs have larger uncertainties due to smaller dual-wavelength signals and stronger influences of hydrometeor habits and orientations at this geometry of beam pointing. Mean relations among different characteristic sizes, which describe the entire particle size distribution (PSD), such as D v , and other sizes used in various applications (e.g., the mean, effective, and median sizes) are derived, so the results of this study can be used for estimating different PSD characteristic sizes.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRSDate not supplied

The Impact of Ice Sheet Surface Representation on Surface Mass Balance in the Goddard Earth Observing System

Surface conditions and their impacts on surface mass balance (SMB) play important roles in ice sheet dynamics and ice sheet interactions with the overlying atmosphere and surrounding ocean. The NASA Global Modeling and Assimilation Office’s (GMAO) Goddard Earth Observing System (GEOS) – an ecosystem of models and reanalyses – represent ice sheet and glacier SMB components, including prognostic surface albedo evolution, fractional snow cover, and snowpack hydrology and meltwater retention. We find the successful representation of ice sheet surface mass balance in MERRA-2, and similar systems, is partly due to compensating biases in surface energy budget. In the atmospheric system, subsequent changes to cloud microphysics and in the longwave radiative transfer model are found to have reduced biases in surface net longwave radiation fluxes while exacerbating surface net shortwave radiation biases, producing erroneously high near-surface temperatures and surface melt in summer months. These issues can be exacerbated by poor experiment initialization and the use of two-moment cloud microphysics within the GEOS ocean-atmosphere coupled system. Here, we document the spatial and temporal extent and causes of these biases in the ice sheet surface energy budget across GEOS systems and implement a range of model improvements to mitigate these issues. We examine the combined impact on these changes on regional energy budget and SMB using both free running and replay experiments (to simulate the impact in reanalyses). Overall, these improvements increase modeled ice sheet surface realism and provide a strong basis for more accurate ice sheet SMB in GMAO’s future reanalyses and forecasting systems.

Lauren C Andrews↗

NASA NTRS2012

The Kinematic and Microphysical Control of Storm Integrated Lightning Flash Extent

The objective of this preliminary study is to investigate the kinematic and microphysical control of lightning properties, particularly those that may govern the production of nitrogen oxides (NOx) in thunderstorms, such as flash rate, type and extent. The mixed-phase region is where the noninductive charging (NIC) process is thought to generate most storm electrification during rebounding collisions between ice particles in the presence of supercooled water. As a result, prior radar-based studies have demonstrated that lightning flash rate is well correlated to kinematic and microphysical properties in the mixed-phase region of thunderstorms such as updraft volume, graupel mass, or ice mass flux. There is also some evidence that lightning type is associated with the convective state. Intracloud (IC) lightning tends to dominate during the updraft accumulation of precipitation ice mass while cloud-to-ground (CG) lightning is more numerous during the downdraft-driven descent of radar echo associated with graupel and hail. More study is required to generalize these relationships, especially regarding lightning type, in a wide variety of storm modes and meteorological conditions. Less is known about the co-evolving relationship between storm kinematics, microphysics, morphology and three-dimensional flash extent, despite its importance for lightning NOx production. To address this conceptual gap, the NASA MSFC Lightning Nitrogen Oxides Model (LNOM) is applied to North Alabama Lightning Mapping Array (NALMA) and Vaisala National Lightning Detection NetworkTM (NLDN) observations following ordinary convective cells through their lifecycle. LNOM provides estimates of flash type, channel length distributions, lightning segment altitude distributions (SADs) and lightning NOx production profiles. For this study, LNOM is applied in a Lagrangian sense to well isolated convective cells on 3 April 2007 (single cell and multi-cell hailstorm, non-severe multicell) and 6 July 2007 (non-severe multi-cell) over Northern Alabama. The LNOM lightning characteristics are compared to the evolution of updraft and precipitation properties inferred from dual-Doppler and polarimetric radar analyses applied to observations from a nearby Doppler radar network, including the UA Huntsville Advanced Radar for Meteorological and Operational Research (ARMOR, C-band, polarimetric). The LNOM estimated SAD and lightning NOx production profiles are placed in the context of radar derived profiles of vertical motion, precipitation types and amounts. Finally, these analyses are used to determine if storm integrated flash channel extent is as well correlated to volumetric updraft and precipitation ice characteristics in the mixed phase region as flash rate for these individual convective cells.

Carey, Lawrence↗

DOE OSTI2023

Physicochemical characterization of free troposphere and marine boundary layer ice-nucleating particles collected by aircraft in the eastern North Atlantic

Abstract. Atmospheric ice nucleation impacts the hydrological cycle and climate by modifying the radiative properties of clouds. To improve our predictive understanding of ice formation, ambient ice-nucleating particles (INPs) need to be collected and characterized. Measurements of INPs at lower latitudes in a remote marine region are scarce. The Aerosol and Cloud Experiments in the Eastern North Atlantic (ACE-ENA) campaign, in the region of the Azores islands, provided the opportunity to collect particles in the marine boundary layer (MBL) and free troposphere (FT) by aircraft during the campaign's summer and winter intensive operation period. The particle population in samples collected was examined by scanning transmission X-ray microscopy with near-edge X-ray absorption fine structure spectroscopy. The identified INPs were analyzed by scanning electron microscopy with energy-dispersive X-ray analysis. We observed differences in the particle population characteristics in terms of particle diversity, mixing state, and organic volume fraction between seasons, mostly due to dry intrusion events during winter, as well as between the sampling locations of the MBL and FT. These differences are also reflected in the temperature and humidity conditions under which water uptake, immersion freezing (IMF), and deposition ice nucleation (DIN) proceed. Identified INPs reflect typical particle types within the particle population on the samples and include sea salt, sea salt with sulfates, and mineral dust, all associated with organic matter, as well as carbonaceous particles. IMF and DIN kinetics are analyzed with respect to heterogeneous ice nucleation rate coefficients, Jhet, and ice nucleation active site density, ns, as a function of the water criterion Δaw. DIN is also analyzed in terms of contact angles following classical nucleation theory. Derived MBL IMF kinetics agree with previous ACE-ENA ground-site INP measurements. FT particle samples show greater ice nucleation propensity compared to MBL particle samples. This study emphasizes that the types of INPs can vary seasonally and with altitude depending on sampling location, thereby showing different ice nucleation propensities, which is crucial information when representing mixed-phase cloud and cirrus cloud microphysics in models.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Dual-Polarization Radar Fingerprints of Precipitation Physics: A Review

This article reviews how precipitation microphysics processes are observed in dual-polarization radar observations. These so-called “fingerprints” of precipitation processes are observed as vertical gradients in radar observables. Fingerprints of rain processes are first reviewed, followed by processes involving snow and ice. Then, emerging research is introduced, which includes more quantitative analysis of these dual-polarization radar fingerprints to obtain microphysics model parameters and microphysical process rates. New results based on a detailed rain shaft bin microphysical model are presented, and we conclude with an outlook of potentially fruitful future research directions.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2022

Dual-Polarization Radar Fingerprints of Precipitation Physics: A Review

precipitation↗

NASA NTRS2010

Improving the Representation of Snow Crystal Properties Within a Single-Moment Microphysics Scheme

As computational resources continue their expansion, weather forecast models are transitioning to the use of parameterizations that predict the evolution of hydrometeors and their microphysical processes, rather than estimating the bulk effects of clouds and precipitation that occur on a sub-grid scale. These parameterizations are referred to as single-moment, bulk water microphysics schemes, as they predict the total water mass among hydrometeors in a limited number of classes. Although the development of single moment microphysics schemes have often been driven by the need to predict the structure of convective storms, they may also provide value in predicting accumulations of snowfall. Predicting the accumulation of snowfall presents unique challenges to forecasters and microphysics schemes. In cases where surface temperatures are near freezing, accumulated depth often depends upon the snowfall rate and the ability to overcome an initial warm layer. Precipitation efficiency relates to the dominant ice crystal habit, as dendrites and plates have relatively large surface areas for the accretion of cloud water and ice, but are only favored within a narrow range of ice supersaturation and temperature. Forecast models and their parameterizations must accurately represent the characteristics of snow crystal populations, such as their size distribution, bulk density and fall speed. These properties relate to the vertical distribution of ice within simulated clouds, the temperature profile through latent heat release, and the eventual precipitation rate measured at the surface. The NASA Goddard, single-moment microphysics scheme is available to the operational forecast community as an option within the Weather Research and Forecasting (WRF) model. The NASA Goddard scheme predicts the occurrence of up to six classes of water mass: vapor, cloud ice, cloud water, rain, snow and either graupel or hail.

Molthan, Andrew L.↗

DOE OSTI2022

Climatology of the Vertical Profiles of Polarimetric Radar Variables and Retrieved Microphysical Parameters in Continental/Tropical MCSs and Landfalling Hurricanes

Most existing cloud models tend to overestimate the size of cloud ice particles and underestimate their concentration. This emphasizes the need to provide a reliable observational reference to optimize cloud model performance, particularly in areas of high concentration of ice at high altitudes. The dual-polarization radars give the community a unique opportunity to quantify cloud ice with a good accuracy using polarimetric radar retrievals. Here, in this study, we utilize the network of operational WSR-88D radars to build a climatology of the vertical profiles of radar variables, such as radar reflectivity Z, differential reflectivity Z DR , and specific differential phase K DP as well as the radar-retrieved vertical profiles of ice water content (IWC) above the melting layer and liquid water content below it, mean volume diameter D m , and total number concentration N t of ice and liquid particles. Such climatology was created for continental/marine mesoscale convective systems (MCSs) and tropical cyclones including hurricanes. The dataset includes 13 continental MCSs, 10 marine MCSs, and 11 tropical cyclones. Separate statistics of the “background” vertical profiles and the ones associated with high IWC aloft have been obtained in the course of this study. It is shown that continental MCSs exhibit larger size of ice in lower concentration aloft compared to the marine MCSs and especially tropical cyclones/hurricanes. A combination of high KDP and low Z aloft signifies lower D m , higher N t , and often substantial IWC.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Classification of Cloud Particle Imagery from Aircraft Platforms Using Convolutional Neural Networks

Abstract A vast amount of ice crystal imagery exists from a variety of field campaign initiatives that can be utilized for cloud microphysical research. Here, nine convolutional neural networks are used to classify particles into nine regimes on over 10 million images from the Cloud Particle Imager probe, including liquid and frozen states and particles with evidence of riming. A transfer learning approach proves that the Visual Geometry Group (VGG-16) network best classifies imagery with respect to multiple performance metrics. Classification accuracies on a validation dataset reach 97% and surpass traditional automated classification. Furthermore, after initial model training and preprocessing, 10 000 images can be classified in approximately 35 s using 20 central processing unit cores and two graphics processing units, which reaches real-time classification capabilities. Statistical analysis of the classified images indicates that a large portion (57%) of the dataset is unusable, meaning the images are too blurry or represent indistinguishable small fragments. In addition, 19% of the dataset is classified as liquid drops. After removal of fragments, blurry images, and cloud drops, 38% of the remaining ice particles are largely intersecting the image border (≥10% cutoff) and therefore are considered unusable because of the inability to properly classify and dimensionalize. After this filtering, an unprecedented database of 1 560 364 images across all campaigns is available for parameter extraction and bulk statistics on specific particle types in a wide variety of storm systems, which can act to improve the current state of microphysical parameterizations.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2005

Factors Controlling the Properties of Multi-Phase Arctic Stratocumulus Clouds

The 2004 Multi-Phase Arctic Cloud Experiment (M-PACE) IOP at the ARM NSA site focused on measuring the properties of autumn transition-season arctic stratus and the environmental conditions controlling them, including concentrations of heterogeneous ice nuclei. Our work aims to use a large-eddy simulation (LES) code with embedded size-resolved aerosol and cloud microphysics to identify factors controlling multi-phase arctic stratus. Our preliminary simulations of autumn transition-season clouds observed during the 1994 Beaufort and Arctic Seas Experiment (BASE) indicated that low concentrations of ice nuclei, which were not measured, may have significantly lowered liquid water content and thereby stabilized cloud evolution. However, cloud drop concentrations appeared to be virtually immune to changes in liquid water content, indicating an active Bergeron process with little effect of collection on drop number concentration. We will compare these results with preliminary simulations from October 8-13 during MPACE. The sensitivity of cloud properties to uncertainty in other factors, such as large-scale forcings and aerosol profiles, will also be investigated. Based on the LES simulations with M-PACE data, preliminary results from the NASA GlSS single-column model (SCM) will be used to examine the sensitivity of predicted cloud properties to changing cloud drop number concentrations for multi-phase arctic clouds. Present parametrizations assumed fixed cloud droplet number concentrations and these will be modified using M-PACE data.

Fridlind, Ann↗

NASA NTRS1999

Analysis of Cirrus Cloud Microphysical Data

The First International Satellite Cloud Climatology Regional Experiment (FIRE) program has the goal of improving our capabilities to understand, model and detect the properties of climatically-important clouds. This is being undertaken through a three-pronged effort of modeling, long-term observations and short-term intensive field studies. Through examination of satellite and other data it is apparent that stratus and cirrus cloud types have the greatest impact on climate due to their radiative effects and ubiquitous nature. As a result, the FIRE program has developed two paths of investigation, each having its own subset of research objectives and measurement programs. The work conducted under this grant was directed toward furthering our understanding of cirrus cloud systems. While it is known that cirrus are climatically important, the magnitude and even sign of the impact is unclear. Cirrus clouds affect the transfer of radiation according to their physical depth and location in the atmosphere and their microphysical composition. However, significant uncertainties still exist in how cirrus clouds form and how they are maintained, what their physical properties are and how they can be parameterized in numerical models. Better remote sensing techniques for monitoring cirrus cloud systems and improved modeling of radiative transfer through ice particles are also needed. A critical element in resolving these issues is a better understanding of cirrus cloud microphysical properties and how they vary. The focus of the research to be conducted under this grant was th data collected in situ by the University of North Dakota Citation aircraft. The goals of this research were to add to the body of knowledge of cirrus cloud microphysics, particularly at the small end of the size spectrum; and analyze the spatial variation of cirrus clouds.

Poellot, Michael R.↗

NASA NTRSDate not supplied

Observing Ice Phase Clouds and Precipitation with Millimeter- and Submillimeter-wave Radiometry during the IMPACTS field campaign: Algorithm Development for the Emerging Class of Microwave Radiometers

Millimeter- and submillimeter-wave (mmWave, submm) radiometers provide significant information content on atmospheric water in liquid, ice, and vapor phases. Recent sensors, such as the Global Precipitation Measurement (GPM) or the Temporal Experiment for Storms and Tropical Systems Technology – Demonstration (TEMPEST-D), use mmWave bands to improve estimation of ice-phase precipitation, and future missions, such as the Atmosphere Observing System (AOS), will extend to submm to provide additional information on thinner ice clouds, linking clouds and precipitation and therefore elucidating the links between weather, climate, and the water cycle. To support the expanded use of mmWave/submm radiometry and exploit the available information content on condensed water mass, we demonstrate ice and liquid water path retrievals for NASA Goddard Space Flight Center's airborne radiometers the Conical Scanning Millimeter-wave Imaging Radiometer (CoSMIR) and the Configurable Scanning Submillimeter-wave Instrument/Radiometer (CoSSIR, formerly the Compact Scanning Submillimeter Imaging Radiometer). The analysis is based on data collected during the Investigation of Microphysics and Precipitation for Atlantic Coast-Threatening Storms (IMPACTS) field campaign. A simulation study compares and contrasts retrievals for CoSMIR and CoSSIR, with CoSSIR demonstrating higher skill in ice retrievals, particularly for thin clouds, and the CoSMIR frequencies performing better for liquid. Additionally, we look at dual-polarized radiances observed by CoSMIR during the first two IMPACTS deployments to understand the polarization signatures induced by oriented hydrometeors. Results will be presented in the context of upcoming missions and sensors including AOS and the Ice Cloud Imager (ICI). We will describe the objectives for the expected first flight of the newly upgraded CoSSIR during the final IMPACTS deployment, and we will share novel science investigations we plan to undertake with CoSSIR's new capabilities. Popular Summary: Ice in clouds and falling snow is an important component of weather, climate, and water resources. To better use a new class of sensors that provide information on atmospheric ice, we developed an approach to measure ice clouds and falling snow using airborne data taken during the Investigation of Microphysics and Precipitation for Atlantic Coast-Threatening Storms (IMPACTS), a field campaign to study winter storms. We look at the results for two similar instruments to understand how those differences affect these measurements of clouds and snow, and we further investigate the information the sensors can provide on properties of the clouds and snow.

Ian S Adams↗

NASA NTRSDate not supplied

Observing Ice Phase Clouds and Precipitation with Millimeter- and Submillimeter-wave Radiometry during the IMPACTS field campaign: Algorithm Development for the Emerging Class of Microwave Radiometers

Ian S Adams↗

NASA NTRS2007

The Mixed-Phase Arctic Cloud Experiment (M-PACE)

The Mixed-Phase Arctic Cloud Experiment (M-PACE) was conducted September 27 through October 22, 2004 on the North Slope of Alaska. The primary objective was to collect a data set suitable to study interactions between microphysics, dynamics and radiative transfer in mixed-phase Arctic clouds. Observations taken during the 1997/1998 Surface Heat and Energy Budget of the Arctic (SHEBA) experiment revealed that Arctic clouds frequently consist of one (or more) liquid layers precipitating ice. M-PACE sought to investigate the physical processes of these clouds utilizing two aircraft (an in situ aircraft to characterize the microphysical properties of the clouds and a remote sensing aircraft to constraint the upwelling radiation) over the Department of Energy s Atmospheric Radiation Measurement (ARM) Climate Research Facility (ACRF) on the North Slope of Alaska. The measurements successfully documented the microphysical structure of Arctic mixed-phase clouds, with multiple in situ profiles collected in both single-layer and multi-layer clouds over two ground-based remote sensing sites. Liquid was found in clouds with temperatures down to -30 C, the coldest cloud top temperature below -40 C sampled by the aircraft. Remote sensing instruments suggest that ice was present in low concentrations, mostly concentrated in precipitation shafts, although there are indications of light ice precipitation present below the optically thick single-layer clouds. The prevalence of liquid down to these low temperatures could potentially be explained by the relatively low measured ice nuclei concentrations.

Verlinde, J.↗

DOE OSTI2021

COMBLE ARM Mobile Facility (AMF) Measurements of Ice Nucleating Particles Field Campaign Report

During the Cold-Air Outbreaks in the Marine Boundary Layer Experiment (COMBLE), a project with the overarching goal to shed light on the dynamics and microphysical properties of clouds and precipitation in the high-latitude marine boundary layer during cold-air outbreaks (CAOs), we were tasked with providing and assisting the collection of aerosol filter samples for measuring ice nucleating particle (INP) concentrations. Little is known about the properties of CAO cloud systems, how they vary with surface, environmental, and aerosol conditions, the role of cold-air outbreaks in the global atmospheric and ocean circulation, and the accuracy of the treatment of this atmospheric regime in climate models. Uncertainties in climate projections for the arctic and sub-arctic region are large, in part due to the lack of observational guidance to constrain the treatment of aerosol-cloud-precipitation linkages in climate models.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2004

Spatial and Temporal Distribution of Tropospheric Clouds and Aerosols Observed by MODIS Onboard the Terra and Aqua Satellites

Remote sensing of cloud and aerosol optical properties is routinely obtained using the Moderate Resolution Imaging Spectroradiometer (MODIS) onboard the Terra and Aqua satellites. Techniques that are being used to enhance our ability to characterize the global distribution of cloud and aerosol properties include well-calibrated multispectral radiometers that rely on visible, near-infrared, and thermal infrared channels. The availability of thermal channels to aid in cloud screening for aerosol properties is an important additional piece of information that has not always been incorporated into sensor designs. In this paper, we describe the radiative properties of clouds as currently determined from satellites (cloud fraction, optical thickness, cloud top pressure, and cloud effective radius), and highlight the global and regional cloud microphysical properties currently available for assessing climate variability and forcing. These include the latitudinal distribution of cloud optical and radiative properties of both liquid water and ice clouds, as well as joint histograms of cloud optical thickness and effective radius for selected geographical locations around the world. In addition, we will illustrate the radiative and microphysical properties of aerosol particles that are currently available from space-based observations, and show selected cases in which aerosol particles are observed to modify the cloud optical properties.

King, Michael D.↗

NASA NTRS2006

Spatial and Temporal Distribution of Tropospheric Clouds and Aerosols Observed by MODIS Onboard the Terra and Aqua Satellites

Remote sensing of cloud and aerosol optical properties is routinely obtained using the Moderate Resolution Imaging Spectroradiometer (MODIS) onboard the Terra and Aqua satellites. Instruments that are being used to enhance our ability to characterize the global distribution of cloud and aerosol properties include well-calibrated multispectral radiometers that measure in the visible, near-infrared, and thermal infrared. The availability of thermal channels to enhance detection of cloud when estimating aerosol properties is an important improvement. In this paper, we describe the radiative properties of clouds as currently determined from satellites (cloud fraction, optical thickness, cloud top pressure, and cloud particle effective radius) and highlight the global/regional cloud microphysical properties currently available for assessing climate variability and forcing. These include the latitudinal distribution of cloud optical and radiative properties of both liquid water and ice clouds, as well as joint histograms of cloud optical thickness and effective particle radius for selected geographical locations around the world. In addition, we will illustrate the radiative and microphysical properties of aerosol particles (in cloud free regions) that are currently available from space-based observations, and show the latitudinal distribution of aerosol optical properties over both land and ocean surfaces.

King, Michael D.↗