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NASA NTRS2008

Integration of Satellite-Derived Cloud Phase, Cloud Top Height, and Liquid Water Path into an Operational Aircraft Icing Nowcasting System

Operational products used by the U.S. Federal Aviation Administration to alert pilots of hazardous icing provide nowcast and short-term forecast estimates of the potential for the presence of supercooled liquid water and supercooled large droplets. The Current Icing Product (CIP) system employs basic satellite-derived information, including a cloud mask and cloud top temperature estimates, together with multiple other data sources to produce a gridded, three-dimensional, hourly depiction of icing probability and severity. Advanced satellite-derived cloud products developed at the NASA Langley Research Center (LaRC) provide a more detailed description of cloud properties (primarily at cloud top) compared to the basic satellite-derived information used currently in CIP. Cloud hydrometeor phase, liquid water path, cloud effective temperature, and cloud top height as estimated by the LaRC algorithms are into the CIP fuzzy logic scheme and a confidence value is determined. Examples of CIP products before and after the integration of the LaRC satellite-derived products will be presented at the conference.

Haggerty, Julie↗

DOE OSTI2026

Exposing and Reducing Biases of Simulating Mixed-Phase Clouds in the Convection-Permitting E3SM Atmosphere Model: Lessons From an Arctic Cold-Air Outbreak

Mixed-phase clouds modulate the water and energy cycles of high-latitude regions, yet their liquid-ice phase partitioning has long been poorly simulated in climate models. Here, simulations of Arctic mixed-phase clouds by the Simple Cloud-Resolving E3SM Atmosphere Model (SCREAM) are assessed against large-eddy simulations, satellite data, and ground-based observations during the Cold-Air Outbreaks in the Marine Boundary Layer Experiment field campaign. SCREAM simulates nearly completely frozen clouds, which is attributed largely to the unreasonably strong Wegener–Bergeron–Findeisen (WBF) process that converts liquid to ice excessively and partly to the early over-abundant ice production at cold temperatures from a temperature-deterministic deposition ice nucleation scheme. Assuming no subgrid variation for the WBF process in the original formulation particularly conflicts with the instantaneous saturation adjustment assumption in the condensation scheme that assumes subgrid variability, leading to exaggerated WBF process rates. A proposed simple physically-based improvement on the treatment of subgrid cloud overlap substantially increases supercooled liquid water content and notably improves cloud-top phase partitioning, aligning better with observations. Improvement of supercooled liquid water content also converges with increasing horizontal resolution. The deposition ice nucleation scheme is found responsible for a falsely-produced ice cloud aloft that is not observed, biasing the simulated cloud radiative effects and top-of-atmosphere radiative fluxes. This study identifies key deficiencies in cloud parameterizations that continue to challenge convection-permitting models.

Geosciences↗

NASA NTRS1997

A New Parameterization of H2SO4/H2O Aerosol Composition: Atmospheric Implications

Recent results from a thermodynamic model of aqueous sulfuric acid are used to derive a new parameterization for the variation of sulfuric acid aerosol composition with temperature and relative humidity. This formulation is valid for relative humidities above 1 % in the temperature range of 185 to 260 K. An expression for calculating the vapor pressure of supercooled liquid water, consistent with the sulfuric acid model, is also presented. We show that the Steele and Hamill [1981] formulation underestimates the water partial pressure over aqueous H2SOI solutions by up to 12% at low temperatures. This difference results in a corresponding underestimate of the H2SO4 concentration in the aerosol by about 6 % of the weight percent at approximately 190 K. In addition, the relation commonly used for estimating the vapor pressure of H2O over supercooled liquid water differs by up to 10 % from our derived expression. The combined error can result in a 20 % underestimation of water activity over a H2SO4 solution droplet in the stratosphere, which has implications for the parameterization of heterogeneous reaction rates in stratospheric sulfuric acid aerosols. The influence of aerosol composition on the rate of homogeneous ice nucleation from a H2SO4 solution droplet is also discussed. This parameterization can also be used for homogeneous gas phase nucleation calculations of H2SO4 solution droplets under various environmental conditions such as in aircraft exhaust or in volcanic plumes.

Tabazadeh, Azadeh↗

DOE OSTI2020

A modeling examination of cloud seeding conditions under the warmer climate in Utah, USA

Assessment of how the wintertime cloud seeding conditions over Utah's mountains may change provides essential information for the state to work on increasing its snowpack in the warmer climate. Here in this study, data from the National Center for Atmospheric Research's high-resolution, convection-permitting Weather Research and Forecasting model are used to evaluate suitable conditions for wintertime cloud seeding over Utah's mountains, and to estimate future changes to these conditions under a pseudo-global warming scenario. While previous studies have examined historical cloud seeding conditions, this is the first study that explores the impact of future climate change on these conditions. Based on two atmospheric variables commonly used in cloud-seeding operations, i.e. average temperature from the surface to 1 km above ground level (-20 °C≤T≤-6 °C) and vertically integrated supercooled liquid water (SLW > 0.01 mm), Utah's mountains are suitable for seeding more than 20% of the time during winter. A greater rate of suitability exists in the northern and Uinta Mountains. In the warmer climate under a high-emissions scenario, these seeding conditions may wane as the percentage of precipitating clouds suitable for seeding, about 60% under current conditions, would decline to about 40% across the state with significance. This projected decrease is due to rising temperatures and a decreased frequency of precipitation events. These findings imply that climate warming will narrow the window of opportunity for winter cloud seeding operation in Utah.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1990

The 27-28 October 1986 FIRE IFO cirrus case study: Comparison of satellite and aircraft derived particle size

Theoretical calculations predict that cloud reflectance in near infrared windows such as those at 1.6 and 2.2 microns should give lower reflectances than at visible wavelengths. The reason for this difference is that ice and liquid water show significant absorption at those wavelengths, in contrast to the nearly conservative scattering at wavelengths shorter than 1 micron. In addition, because the amount of absorption scales with the path length of radiation through the particle, increasing cloud particle size should lead to decreasing reflectances at 1.6 and 2.2 microns. Measurements at these wavelengths to date, however, have often given unpredicted results. Twomey and Cocks found unexpectedly high absorption (factors of 3 to 5) in optically thick liquid water clouds. Curran and Wu found expectedly low absorption in optically thick high clouds, and postulated the existence of supercooled small water droplets in place of the expected large ice particles. The implications of the FIRE data for optically thin cirrus are examined.

Wielicki, Bruce A.↗

NASA NTRS1981

Homogeneous condensation - Freezing nucleation rate measurements for small water droplets in an expansion cloud chamber

Experimental data on ice nucleation, presented in an earlier paper, are analyzed to yield information about the homogeneous nucleation rate of ice from supercooled liquid and the heights of energy barriers to that nucleation. The experiment consisted of using an expansion cloud chamber to nucleate from the vapor a cloud of supercooled pure water drops and the observation of the fraction of drops which subsequently froze. The analysis employed standard classical homogeneous nucleation theory. The data are used to extract the first experimental measurement (albeit indirect) of the activation energy for the transfer of a water molecule across the liquid-ice interface at temperatures near -40 C. The results provide further evidence that the local liquid structure becomes more icelike as the temperature is lowered.

Hagen, D. E.↗

DOE OSTI2021

Mixed‐Phase Clouds Over the Southern Ocean as Observed From Satellite and Surface Based Lidar and Radar

Abstract This study investigates the occurrence of mixed‐phase clouds (MPC, i.e., cloud layers containing both liquid and ice water at sub‐freezing temperatures) over the Southern Ocean (SO) using space‐ and surface‐based lidar and radar observations. The occurrence of supercooled clouds is dominated by geometrically thin (<1 km) layers that rarely contain ice. We diagnose layers that are geometrically thicker than 1 km to contain ice ~65% and ~4% of the time from below by surface remote sensors and from above by orbiting remote sensors, respectively. We examine the discrepancy in MPC occurrence statistics as diagnosed from below and above the cloud layer. From above, we find that MPC occurrence has a gradient associated with the Antarctic Polar Front near 55°S with a rare occurrence of satellite‐derived MPC south of that latitude. In contrast, surface sensors find ice in 33% of supercooled liquid water layers. We infer using observing system simulation experiments and data analysis that space‐based lidar cannot identify the occurrence of MPC except when secondary ice‐forming processes operate in convection that is, sufficiently strong to loft ice crystals to cloud tops. We conclude that the CALIPSO phase statistics of MPC have a severe low bias in MPC occurrence. Based on surface‐based statistics in the SO, we present a parameterization of the frequency of MPC as a function of cloud top temperature that differs substantially from that used in recent climate model simulations.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2006

Progress in the Development of Practical Remote Detection of Icing Conditions

The NASA Icing Remote Sensing System (NIRSS) has been under definition and development at NASA Glenn Research Center since 1997. The goal of this development activity is to produce and demonstrate the required sensing and data processing technologies required to accurately remotely detect and measure icing conditions aloft. As part of that effort NASA has teamed with NCAR to develop software to fuse data from multiple instruments into a single detected icing condition product. The multiple instrument approach utilizes a X-band vertical staring radar, a multifrequency microwave, and a lidar ceilometer. The radar data determine cloud boundaries, the radiometer determines the sub-freezing temperature heights and total liquid water content, and the ceilometer refines the lower cloud boundary. Data is post-processed with a LabVIEW program with a resultant supercooled liquid water profile and aircraft hazard depiction. Ground-based, remotely-sensed measurements and in-situ measurements from research aircraft were gathered during the international 2003-2004 Alliance Icing Research Study (AIRS II). Comparisons between the remote sensing system s fused icing product and the aircraft measurements are reviewed here. While there are areas where improvement can be made, the cases examined suggest that the fused sensor remote sensing technique appears to be a valid approach.

Reehorst, Andrew↗

NASA NTRS2014

Hydrogen-Bonding Surfaces for Ice Mitigation

Ice formation on aircraft, either on the ground or in-flight, is a major safety issue. While ground icing events occur predominantly during the winter months, in-flight icing can happen anytime during the year. The latter is more problematic since it could result in increased drag and loss of lift. Under a Phase I ARMD NARI Seedling Activity, coated aluminum surfaces possessing hydrogen-bonding groups were under investigation for mitigating ice formation. Hydroxyl and methyl terminated dimethylethoxysilanes were prepared via known chemistries and characterized by spectroscopic methods. These materials were subsequently used to coat aluminum surfaces. Surface compositions were based on pure hydroxyl and methyl terminated species as well as mixtures of the two. Coated surfaces were characterized by contact angle goniometry. Receding water contact angle data suggested several potential surfaces that may exhibit reduced ice adhesion. Qualitative icing experiments performed under representative environmental temperatures using supercooled distilled water delivered via spray coating were inconclusive. Molecular modeling studies suggested that chain mobility affected the interface between ice and the surface more than terminal group chemical composition. Chain mobility resulted from the creation of "pockets" of increased free volume for longer chains to occupy.

Smith, Joseph G., Jr.↗

NASA NTRS1993

MMIC Receiver For Water-Vapor Radiometer

MMIC receiver puts out signal, frequency of which proportioned to brightness temperature of sky at input frequency of 31 GHz. Miniaturization enhances thermal stability and stability of calibration of water-vapor radiometer. Potential for mass production at relatively low cost. Facilitating widespread use of MMIC water vapor radiometers in meteorology and aviation, deployed at several global sites to improve capability of general circulation models and at airports to monitor icing conditions by measuring supercooled liquid water in clouds.

Sukamto, Lin M.↗

DOE OSTI2021

Mixed-phase clouds and precipitation in Southern Ocean cyclones and cloud systems observed poleward of 64°S by ship-based cloud radar and lidar

Mixed-phase clouds (MPCs), composed of both liquid and ice, are prevalent in Southern Ocean cyclones. A characterisation of these clouds on fine vertical scales is required in order to understand the microphysical processes within these clouds, and for model and satellite evaluations over this region. We investigated three examples of cloud systems collected by ship-mounted remote sensing instruments adjacent to East Antarctica at latitudes between 64°S and 69°S. These cases allow us to examine the properties of mid-level MPCs, with cloud tops between 2 and 6 km. Mid-level MPCs contain multiple layers of supercooled liquid water (SLW) embedded within ice during the passage of cyclones. SLW layers are capped by strong temperature inversions and are observed at temperatures as low as - 31°C. Convective generating cells (GCs) are present inside supercooled liquid-topped mid-level MPCs. The horizontal extent, vertical extent and maximum upward Doppler velocity of these GCs were 0.6 – 3.6 km, 0.7 – 1.0 km and 0.5 – 1.0 m s -1 , respectively, and are consistent with observations from previous lower-latitude studies. Ice precipitation is nearly ubiquitous, except in the thinnest clouds at the trailing end of the observed systems. Seeding of lower SLW layers from above leads to periods with either larger ice particles or greater ice precipitation rates. Additionally, periods of supercooled drizzle lasting up to two hours were observed toward the end of two of the three cyclone systems. This supercooled drizzle turns into predominantly ice precipitation as the result of seeding by ice clouds located above the precipitating SLW layer.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Challenging and Improving the Simulation of Mid-Level Mixed-Phase Clouds Over the High-Latitude Southern Ocean

Climate models exhibit major radiative biases over the Southern Ocean owing to a poor representation of mixed-phase clouds. In this study, we used the remote-sensing dataset from the Measurements of Aerosols, Radiation and Clouds over the Southern Ocean (MARCUS) campaign to assess the ability of the Weather Research and Forecasting (WRF) model to reproduce frontal clouds off Antarctica. It focuses on the modeling of thin mid-level supercooled liquid water layers which precipitate ice. The standard version of WRF produces almost fully glaciated clouds and cannot reproduce cloud top turbulence. Our work demonstrates the importance of adapting the ice nucleation parameterization to the pristine austral atmosphere to reproduce the supercooled liquid layers. Once simulated, droplets significantly impact the cloud radiative effect by increasing downwelling longwave fluxes and decreasing downwelling shortwave fluxes at the surface. The net radiative effect is a warming of snow and ice covered surfaces and a cooling of the ocean. Despite improvements in our simulations, the local turbulent circulation related to cloud-top radiative cooling is not properly reproduced, advocating for the need to develop a parameterization for top-down convection to capture the turbulence-microphysics interplay at cloud top.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2019

Use of Polarimetric Radar Measurements to Constrain Simulated Convective Cell Evolution: A Pilot Study with Lagrangian Tracking

To probe the potential value of a radar-driven field campaign to constrain simulation of isolated convection subject to a strong aerosol perturbation, convective cells observed by the operational KHGX weather radar in the vicinity of Houston, Texas, are examined individually and statistically. Cells observed in a single case study of onshore flow conditions during July 2013 are first examined and compared with cells in a regional model simulation. Observed and simulated cells are objectively identified and tracked from observed or calculated positive specific differential phase (K(sub DP)) above the melting level, which is related to the presence of supercooled liquid water. Several observed and simulated cells are subjectively selected for further examination. Below the melting level, we compare sequential cross sections of retrieved and simulated raindrop size distribution parameters. Above the melting level, we examine time series of KDP and radar differential reflectivity (Z(sub DR)) statistics from observations and calculated from simulated supercooled rain properties, alongside simulated vertical wind and supercooled rain mixing ratio statistics. Results indicate that the operational weather radar measurements offer multiple constraints on the properties of simulated convective cells, with substantial value added from derived K(sub DP) and retrieved rain properties. The value of collocated three-dimensional lightning mapping array measurements, which are relatively rare in the continental US, supports the choice of Houston as a suitable location for future field studies to improve the simulation and understanding of convective updraft physics. However, rapid evolution of cells between routine volume scans motivates consideration of adaptive scan strategies or radar imaging technologies to amend operational weather radar capabilities. A 3-year climatology of isolated cell tracks, prepared using a more efficient algorithm, yields additional relevant information. Isolated cells are found within the KHGX domain on roughly 40 % of days year-round, with greatest concentration in the northwest quadrant, but roughly 5-fold more cells occur during June through September. During this enhanced occurrence period, the cells initiate following a strong diurnal cycle that peaks in the early afternoon, typically follow a south-to-north flow, and dissipate within 1 h, consistent with the case study examples. Statistics indicate that ∼ 150 isolated cells initiate and dissipate within 70 km of the KHGX radar during the enhanced occurrence period annually, and roughly 10 times as many within 200 km, suitable for multi-instrument Lagrangian observation strategies. In addition to ancillary meteorological and aerosol measurements, robust vertical wind speed retrievals would add substantial value to a radar-driven field campaign.

Fridlind, Ann M.↗

NASA NTRS2020

Determining Cloud Thermodynamic Phase from the Polarized Micro Pulse Lidar

A method to distinguish cloud thermodynamic phase from polarized Micro Pulse Lidar (MPL) measurements is described. The method employs a simple enumerative approach to classify cloud layers as either liquid water, ice water, or mixed-phase clouds based on the linear volume depolarization ratio and cloud top temperatures derived from Goddard Earth Observing System, version 5 (GEOS-5) assimilated data. Two years of cloud retrievals from the Micro Pulse Lidar Network (MPLNET) site in Greenbelt, MD are used to evaluate the performance of the algorithm. The fraction of supercooled liquid water in the mixed-phase temperature regime (-37 °C – 0 °C) calculated using MPLNET data is compared to similar calculations made using the spaceborne Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) instrument on board the Cloud‐Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) satellite, with reasonable consistency.

Jasper R Lewis↗

NASA NTRS2022

On the Effects of Cloud Water Content on Passive Microwave Snowfall Retrievals

The Bayesian passive microwave retrievals of snowfall often rely on mathematical matching of the observed vectors of brightness temperature with an a priori database of precipitation profiles and their corresponding brightness temperatures. Mathematical proximity does not necessarily lead to consistent retrievals due to limited information content of passive microwave observations. This paper defines imposter (genuine) vectors of brightness temperature as those that are mathematically close but physically inconsistent (consistent) and characterizes them through the Silhouette Coefficient (SC) analysis. The Neyman–Pearson (NP) hypothesis testing is used to separate the imposter and genuine brightness temperatures based on their associated values of cloud ice (IWP) and liquid water path (LWP), given by coincidences of CloudSat Profiling Radar (CPR) and the Global Precipitation Measurement (GPM) Microwave Imager (GMI). The study determines thresholds for IWP and LWP that allow optimal identification of imposter brightness temperatures of non-snowing and snowing clouds, which can mislead the passive microwave retrieval algorithms to falsely detect or miss the snowfall events. It is demonstrated that emission signal of supercooled liquid water in snowing clouds can lead to improved passive microwave retrieval of snowfall and conditioning the retrievals to the cloud IWP and LWP can result in marginal correction of the snowfall detection probability; however, reduce the probability of false alarm by 6%–8% over sea ice and open oceans.

Snowfall↗

DOE OSTI2024

Impacts of Synoptic-Scale Dynamics on Clouds and Radiation in High Southern Latitudes

High-latitudinal mixed-phase clouds significantly affect Earth's radiative balance. Observations of cloud and radiative properties from two field campaigns in the Southern Ocean and Antarctica were compared with two global climate model simulations. A cyclone compositing method was used to quantify “dynamics-cloud-radiation” relationships relative to the extratropical cyclone centers. Observations show larger asymmetry in cloud and radiative properties between western and eastern sectors at McMurdo compared with Macquarie Island. Most observed quantities at McMurdo are higher in the western (i.e., post-frontal) than the eastern (frontal) sector, including cloud fraction, liquid water path (LWP), net surface shortwave and longwave radiation (SW and LW), except for ice water path (IWP) being higher in the eastern sector. Here, the two models were found to overestimate cloud fraction and LWP at Macquarie Island but underestimate them at McMurdo Station. IWP is consistently underestimated at both locations, both sectors, and in all seasons. Biases of cloud fraction, LWP, and IWP are negatively correlated with SW biases and positively correlated with LW biases. The persistent negative IWP biases may have become one of the leading causes of radiative biases over the high southern latitudes, after correcting the underestimation of supercooled liquid water in the older model versions. By examining multi-scale factors from cloud microphysics to synoptic dynamics, this work will help increase the fidelity of climate simulations in this remote region.

58 GEOSCIENCES↗

DOE OSTI2023

Vertical Structure and Ice Production Processes of Shallow Convective Postfrontal Clouds over the Southern Ocean in MARCUS. Part I: Observational Study

A study of the vertical structure of postfrontal shallow clouds in the marine boundary layer over the Southern Ocean is presented here. The central question of this two-part study regards cloud phase (liquid/ice) of precipitation, and the associated growth mechanisms. In this first part, data from the Measurements of Aerosols, Radiation, and Clouds over the Southern Ocean (MARCUS) field campaign are analyzed, starting with a 75-h case with continuous sea surface-based thermal instability, modest surface heat fluxes, an open-cellular mesoscale organization, and very few ice nucleating particles (INPs). The clouds are mostly precipitating and shallow (tops mostly around 2 km above sea level), with weak up- and downdrafts, and with cloud-top temperatures generally around –18° to –10°C. Further, the case study is extended to three other periods of postfrontal shallow clouds in MARCUS. While abundant supercooled liquid water is commonly present, an experimental cloud-phase algorithm classifies nearly two-thirds of clouds in the 0° to –5°C layer as containing ice (cloud ice, snow, or mixed phase), implying that much of the precipitation grows through cold-cloud processes. The best predictors of ice presence are cloud-top temperature, cloud depth, and INP concentration. Measures of convective activity and turbulence are found to be poor indicators of ice presence in the studied environment. The water-phase distribution in this cloud regime is explored through numerical simulations in Part II. Significance Statement: Climate models generally predict a lower albedo than observed over the Southern Ocean, and this is largely attributed to a lack of cloudiness, especially in the postfrontal cold sector of midlatitude cyclones. This in turn may be due to an excess of ice in these simulated clouds, resulting in rapid precipitation fallout and an overly brief cloud lifespan. The objective of this study is to examine whether shallow postfrontal clouds over the Southern Ocean are dominated by supercooled drops, or by snow and ice, using data collected by a U.S. Department of Energy Atmospheric Radiation Measurement Mobile Facility deployed aboard an Australian Antarctic supply vessel. We find that these clouds contain much supercooled liquid, even though cloud-top temperatures generally are around –18° to –8°C, and that about two-thirds of the clouds just above the freezing level contain ice. Much of the precipitation appears to grow through cold-cloud processes above the freezing level, rather than drizzle/rain. Updrafts and/or turbulence in convection or in cloud-top generating cells do not initiate much ice, compared to observations elsewhere in a similar temperature range. This may be attributable to the extremely low concentration of ice nucleating particles in this environment. Ultimately, the deepest clouds with the coldest cloud tops are most likely to be ice dominated.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRSDate not supplied

166 GHz Ice Scattering Signal in Cold-Season Precipitating Cloud Systems

Surface snowfall can be produced by a variety of forcing mechanisms that are accompanied by different in-cloud physical processes and microphysical composition. For instance, shallow cumuliform snowfall forced by marine cold air outbreaks shows distinct geographic and seasonal signatures and plays in important role in annual snowfall production in many regions worldwide. Coincident CloudSat-CALIPSO observations confirm that oceanic regions exhibiting frequent shallow cumuliform snow are typically composed of mixed-phase cloud and precipitation particles and display passive microwave brightness temperature behavior that is dominated by cloud liquid water emission, while combined Global Precipitation Measurement (GPM) radiometer and surface radar studies of intense shallow cumuliform events indicate significant ice particle scattering signals at the highest GPM radiometer frequencies. Passive microwave imager channels in the 37 to 89 GHz range have long been used to detect precipitation-sized ice particle scattering signatures in clouds. Dual polarization plays an important role in distinguishing between scattering signatures and other background variability. The GPM Microwave Imager (GMI) is the first satellite radiometer to possess dual-polarized channels at 166 GHz. These high-frequency channels are more susceptible to emission and attenuation by cloud liquid water and water vapor than lower-frequency channels, but are also more sensitive to ice scattering. This research studies the ice scattering information contained in the 166 GHz channels compared to the historically used 85/89 GHz channels on previous sensors. A 166 GHz scattering index (SI) will be developed to understand how its correlation with non-precipitating ice scattering at higher altitudes also affects the 166 GHz scattering signature in cold-season precipitating cloud systems. The 166 GHz SI will be used to possibly isolate precipitation scattering signature for different snowfall modes (shallow convective vs. deep stratiform). The supercooled liquid water affecting shallow convective systems and drastically reducing the scattering signal could be detected and quantified, providing additional information for convective snow detection and quantification in solid precipitation retrieval algorithms.

Lisa Milani↗