Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Snow model”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Sensitivity of an energy balance climate model with predicted snowfall rates

A snowfall parameterization and a polar-ice-sheet model are developed and applied to the two-level zonally averaged seasonal energy-balance climate model of Held and Suarez (1979), and sensitivity experiments involving changes in insolation are performed both with and without ice sheets. The results are presented in tables and graphs, and the hydrological-cycle response to insolation changes is found to be similar to that predicted by global-circulation models employing prescribed precipitation levels, with a somewhat higher sensitivity in the snow line. The area covered by ice sheets in the ice-sheet models is shown to be greater than that covered by permanent snow in the models without ice sheets, an effect attributed to lower surface temperatures over the ice. It is inferred that an increase in the solar constant can cause increased high-latitude precipitation but not an ice age.

Bowman, K. P.↗
DOE OSTI2026

Flood Estimation under Snowmelt and Rain-on-Snow Processes in Alaska: A Military Installation Perspective

Accurate flood estimation in snow-dominated and high-latitude regions remains challenging due to complex interactions among rainfall, snowmelt, and rain-on-snow (ROS) processes, which are not captured in conventional precipitation-based intensity-duration-frequency (PREC-IDF) curves. This study evaluates the Next-Generation IDF (NG-IDF) framework, an extension of PREC-IDF that incorporates total water available for runoff (precipitation minus changes in snow water equivalent), in two contrasting Alaskan watersheds of Department of Defense (DoD) significance: the Little Chena River Basin (LCRB) in interior Alaska, a tributary of the Chena River that flows through Fort Wainwright and near Eielson Air Force Base, and the Upper Ship Creek Basin (USCB), which drains Joint Base Elmendorf-Richardson near Anchorage. Using long-term SNOTEL observations and event-based rainfall-runoff modeling, NG-IDF and PREC-IDF flood estimates were compared against observation-based flood frequency analyses. Results show that NG-IDF consistently reduces flood-estimation bias by 15–20% relative to PREC-IDF, particularly for snowmelt- and ROS-dominated events. In the interior LCRB, permafrost conditions can substantially amplify flood responses during snowmelt events, an effect not explicitly represented in standard design tools. These findings demonstrate that NG-IDF provides a more physically consistent and transferable framework for flood estimation in cold regions, with potential relevance to mission-critical DoD installation resilience. Projected increases in ROS frequency and permafrost degradation across Alaska further emphasize the need to integrate physics-based hydrologic models that explicitly represent snow and permafrost processes to enhance design resilience and operational readiness for military and other critical infrastructure.

Yan, Hongxiang (ORCID:000000022387403X)↗
NASA NTRS2017

Blowing Snow Sublimation and Transport over Antarctica from 11 Years of CALIPSO Observations

Blowing snow processes commonly occur over the earth's ice sheets when the 10 mile wind speed exceeds a threshold value. These processes play a key role in the sublimation and redistribution of snow thereby influencing the surface mass balance. Prior field studies and modeling results have shown the importance of blowing snow sublimation and transport on the surface mass budget and hydrological cycle of high-latitude regions. For the first time, we present continent-wide estimates of blowing snow sublimation and transport over Antarctica for the period 2006-2016 based on direct observation of blowing snow events. We use an improved version of the blowing snow detection algorithm developed for previous work that uses atmospheric backscatter measurements obtained from the CALIOP (Cloud-Aerosol Lidar with Orthogonal Polarization) lidar aboard the CALIPSO (Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation) satellite. The blowing snow events identified by CALIPSO and meteorological fields from MERRA-2 are used to compute the blowing snow sublimation and transport rates. Our results show that maximum sublimation occurs along and slightly inland of the coastline. This is contrary to the observed maximum blowing snow frequency which occurs over the interior. The associated temperature and moisture reanalysis fields likely contribute to the spatial distribution of the maximum sublimation values. However, the spatial pattern of the sublimation rate over Antarctica is consistent with modeling studies and precipitation estimates. Overall, our results show that the 2006-2016 Antarctica average integrated blowing snow sublimation is about 393 +/- 196 Gt yr(exp -1), which is considerably larger than previous model-derived estimates. We find maximum blowing snow transport amount of 5 Mt km-1 yr(exp -1) over parts of East Antarctica and estimate that the average snow transport from continent to ocean is about 3.7 Gt yr(exp -1). These continent-wide estimates are the first of their kind and can be used to help model and constrain the surface mass budget over Antarctica.

Antarctica Sublimation Ice-sheet↗
DOE OSTI2026

Analyzing historical snow trends in interior Alaska

Study region The Chena River watershed in Interior Alaska, USA Study focus This study examines 40 years (water years 1982–2021) of snowpack characteristics to consider its hydrological implications in the 5350 km² Chena River basin. Using observations and a fine-scale physics model, we analyzed trends of snow water equivalent (SWE), snow onset and disappearance, and snow cover duration (SCD). New hydrological insights for the region Results indicate a decline in SWE across the modeled domain, averaging a decrease of 3 mm per decade, with larger decreases (up to 10 mm per decade) at lower elevations. While domain-averaged SWE trends were not statistically significant, observed SCD showed statistically significant decreases: −5.2, −5.0, and −4.4 days per decade at Teuchet Creek, Fairbanks F.O., and Little Chena Ridge, respectively. Notably, observations at SNOTEL stations and modeling revealed no statistically significant change in domain-averaged Rain-on-Snow (ROS) events over the 40-year period, contrasting some regional future estimates of increased ROS frequency. Peak streamflow did not consistently correlate with peak SWE levels, suggesting that other environmental factors such as ROS events and rapid temperature increases (e.g., a 10°C spike observed in 1992) are key drivers of hydrological outcomes. These findings improve understanding of complex subarctic hydrological processes impacting permafrost and highlight the need for adaptive water resource management to mitigate multi-factor risks like flooding and wildfire, requiring proactive planning.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2021

Data Assimilation of Terrestrial Water Storage Observations to Estimate Precipitation Fluxes: A Synthetic Experiment

The Gravity Recovery and Climate Experiment (GRACE) mission and its Follow-On (GRACE-FO) mission provide unprecedented observations of terrestrial water storage (TWS) dynamics at basin to continental scales. Established GRACE data assimilation techniques directly adjust the simulated water storage components to improve the estimation of groundwater, streamflow, and snow water equivalent. Such techniques artificially add/subtract water to/from prognostic variables, thus upsetting the simulated water balance. To overcome this limitation, we propose and test an alternative assimilation scheme in which precipitation fluxes are adjusted to achieve the desired changes in simulated TWS. Using a synthetic data assimilation experiment, we show that the scheme improves performance skill in precipitation estimates in general, but that it is more robust for snowfall than for rainfall, and it fails in certain regions with strong horizontal gradients in precipitation. The results demonstrate that assimilation of TWS observations can help correct (adjust) the model’s precipitation forcing and, in turn, enhance model estimates of TWS, snow mass, soil moisture, runoff, and evaporation. A key limitation of the approach is the assumption that all errors in TWS originate from errors in precipitation. Nevertheless, the proposed approach produces more consistent improvements in simulated runoff than the established GRACE data assimilation techniques.

Data Assimilation↗
DOE OSTI2023

Contrasting the Biophysical and Radiative Effects of Rising CO 2 Concentrations on Ozone Dry Deposition Fluxes

Abstract The dry deposition of ozone from the atmosphere to ecosystems is an important coupling mechanism between atmospheric chemistry and terrestrial biogeochemical processes. In most Earth system models, dry deposition is simulated using a resistor‐in‐series approach that aims to parameterize the governing biological, chemical, and physical processes through a series of functional approximations. Here, we evaluate the influence of carbon cycle‐climate responses on this parameterization using the results of the Energy Exascale Earth System Model v1.1 Biogeochemistry simulation campaign. This simulation campaign was designed in part to explore the biophysical and radiative effects of rising historical CO 2 concentrations on the Earth system. We find that while the global annual ozone dry deposition is relatively insensitive to these effects, regionally the influence can be up to 10%. The strongest regional sensitivities in ozone dry deposition are predominantly in higher latitudes over land in the northern hemisphere and are dominated by the radiative effect of CO 2 , with little net influence of biophysical responses. Of all the impacts of the radiative effect of CO 2 , we point to the potential importance of accurately representing ozone deposition to snow in Earth System Models and provide recommendations for future simulation campaigns.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1982

Strategies for Using Remotely Sensed Data in Hydrologic Models

Present and planned remote sensing capabilities were evaluated. The usefulness of six remote sensing capabilities (soil moisture), land cover, impervious area, areal extent of snow cover, areal extent of frozen ground, and water equivalent of the snow cover) with seven hydrologic models (API, CREAMS, NWSRFS, STORM, STANFORD, SSARR, and NWSRFS Snowmelt) were reviewed. The results indicate remote sensing information has only limited value for use with the hydrologic models in their present form. Modifications to the models which would enhance their usefulness are discussed.

Peck, E. L.↗
NASA NTRS1979

Large Area Crop Inventory Experiment (LACIE). Evaluation of the LACIE transition year crop calendar model

The author has identified the following significant results. The estimates of developmental stage dates from the LACIE adjustable crop calendar (ACC) winter wheat model was somewhat more accurate than the historical crop calendar after jointing. The ACC winter wheat model was not so accurate for the Texas Panhandle as it was for the other areas of the USPG-7 because dry soil conditions delayed fall planting in the Panhandle. Since the LACIE ACC winter wheat model does not contain a moisture term and it was started with historical planting dates, lengthy delays in planting mean that the ACC model will probably be started early and will estimate the developmental growth stages to occur too early in the season. The LACIE ACC spring wheat model was also started early in most areas because of late planting due to fields wet from melting snow and rain. The starter model used to estimate spring planting dates was not accurate under these wet soil conditions and tended to predict the developmental stages to occur earlier than the dates observed in the fields.

Cheffin, R. E.↗
NASA NTRSDate not supplied

Medium-Range River Flood Forecasts Using a Long Short-Term Memory Network

River flooding and the impacts are a concern for decision makers throughout the United States. Accurate medium-range forecasts (~3-7 days) are critical for providing advanced outlooks to emergency management officials. Unfortunately, accurately forecasting rainfall-runoff and the subsequent rise and fall within rivers remain a challenge in hydrological modeling. While complex physical modeling systems are the standard for representing the hydrological processes, they are computationally demanding and can require extensive calibration. Further, uncertainties remain in the model parameters and input data. The use of machine learning can reduce some of the computational demand while maintaining high accuracy. Therefore, this project makes use of a Long Short-Term Memory (LSTM) network which explicitly accounts for the time-dependent nature of rainfall-runoff modeling. The developed LSTM was trained to predict river gauge height, or stage height, based on time-lagged input features which include: gauge height to initialize the model, the NASA Short-term Prediction Research and Transition Center’s instance of the Land Information System (SPoRT-LIS) relative soil moisture to describe the rainfall infiltration rate, and 6-hr Multi-Radar Multi-Sensor quantitative precipitation estimate (MRMS QPE). The developed LSTM based system is then used to produce 7-day forecasts with a 6-hr temporal resolution using three different quantitative precipitation forecasts (QPF) from the NWS’s Weather Prediction Center (WPC), the NCEP Global Forecast System (GFS) model and the National Blend of Models (NBM). This trained modeling system has been implemented as an experimental product at over 100 different rivers in collaboration with at multiple National Weather Service (NWS) Forecast Offices and River Forecast Centers (RFC) across the eastern half of the United States. The developed LSTM model achieved average Nash-Sutcliffe efficiency (NSE) 0.89 higher than the equivalent medium-range National Water Model ensemble member forecast over a 7-day forecast. In addition to the initial development and evaluation, this project has continued to expand. While the initial model was developed for precipitation dominated basins, expansion of the project has taken it to basins effected by snow melt. This presentation will provide an overview of the project with focus on recent developments on incorporating snow melt processes into the model.

Andrew T. White↗
NASA NTRS2004

Investigation of the Seasonal Freeze/Thaw Cycle of Soils in the GAPP Regions

The objective of this project is to understand the current and future impacts and feedbacks of seasonally frozen ground on the regional and global climate system and the ecosystem as a whole. More specifically, we proposed to: investigate the dynamics of the seasonally freezing and thawing processes and their relations to current climatic conditions through in situ data analysis and numerical modeling; develop a comprehensive algorithm to detect surface soil freeze/thaw status for snow-free land surface using passive microwave remote sensing data and for snow-covered land surface using numerical modeling, and validate the algorithm and numerical model using available ground-based measurements; investigate seasonal and inter-annual variations of frozen soils in the GAPP regions using data generated from the new validated algorithm and their relations to the environmental factors such as air temperature, snow cover, surface morphology, soil type and soil moisture; generate a frozen soil data set which describes the timing, duration, thickness, and area extent of seasonally frozen ground for the period from 1978 to present. The proposed research will be accomplished through data analysis, remote sensing, and numerical modeling.

FREEZE-THAW TESTS↗
NASA NTRS2012

View Angle Effects on MODIS Snow Mapping in Forests

Binary snow maps and fractional snow cover data are provided routinely from MODIS (Moderate Resolution Imaging Spectroradiometer). This paper investigates how the wide observation angles of MODIS influence the current snow mapping algorithm in forested areas. Theoretical modeling results indicate that large view zenith angles (VZA) can lead to underestimation of fractional snow cover (FSC) by reducing the amount of the ground surface that is viewable through forest canopies, and by increasing uncertainties during the gridding of MODIS data. At the end of the MODIS scan line, the total modeled error can be as much as 50% for FSC. Empirical analysis of MODIS/Terra snow products in four forest sites shows high fluctuation in FSC estimates on consecutive days. In addition, the normalized difference snow index (NDSI) values, which are the primary input to the MODIS snow mapping algorithms, decrease as VZA increases at the site level. At the pixel level, NDSI values have higher variances, and are correlated with the normalized difference vegetation index (NDVI) in snow covered forests. These findings are consistent with our modeled results, and imply that consideration of view angle effects could improve MODIS snow monitoring in forested areas.

MODIS↗
NASA NTRS2018

NASA's SnowEx Campaign and Measuring Global Snow from Space

Snow blankets 30% of Earth's land surface (60% of northern hemisphere land) in midwinter, dramatically changing our planet's land surface and affecting our weather for months. Seasonal snow is critically important to society for the management of water resources, natural hazards, water security, and in many economic sectors. The only practical way to estimate the quantity of snow on a global scale is through satellites. Despite 4 decades of satellite observations, the highly variable nature of snow still presents significant challenges toward achieving this goal. For example, current space-based techniques underestimate snow water equivalent (SWE) by as much as 50%, and model-based estimates can differ greatly versus estimates based on remotely-sensed observations. Snow community consensus is that a multi-sensor approach is needed to adequately address global snow, combined with modeling and data assimilation to fill the gaps in space and time. What remains, then, is how best to combine and use the various sensors under different types of snow conditions and confounding factors. NASA's multi-year SnowEx airborne campaign is designed to collect measurements needed to enable algorithm development and to guide those mission trade studies. Year 1 (2017) focused on the distribution of snow-water equivalent (SWE) and the snow energy balance in a forested environment. This paper will discuss the various remote sensing options for snow, the challenging factors, and describe the recently-completed first year of SnowEx in Colorado, USA. Ground-based remote sensing and in situ data collection involved nearly 100 participants over three weeks. The airborne campaign included nine sensors on five aircraft. We will conclude by discussing options for a future snow satellite mission.

campaign↗
NASA NTRS2001

Seasonal and Interannual Variations of Ice Sheet Surface Elevation at the Summit of Greenland: Observed and Modeled

Observed seasonal and interannual variations in the surface elevation over the summit of the Greenland ice sheet are modeled using a new temperature-dependent formulation of firn-densification and observed accumulation variations. The observed elevation variations are derived from ERS (European Remote Sensing)-1 and ERS-2 radar altimeter data for the period between April 1992 and April 1999. A multivariate linear/sine function is fitted to an elevation time series constructed from elevation differences measured by radar altimetry at orbital crossovers. The amplitude of the seasonal elevation cycle is 0.25 m peak-to-peak, with a maximum in winter and a minimum in summer. Inter-annually, the elevation decreases to a minimum in 1995, followed by an increase to 1999, with an overall average increase of 4.2 cm a(exp -1) for 1992 to 1999. Our densification formulation uses an initial field-density profile, the AWS (automatic weather station) surface temperature record, and a temperature-dependent constitutive relation for the densification that is based on laboratory measurements of crystal growth rates. The rate constant and the activation energy commonly used in the Arrhenius-type constitutive relation for firn densification are also temperature dependent, giving a stronger temperature and seasonal amplitudes about 10 times greater than previous densification formulations. Summer temperatures are most important, because of the strong non-linear dependence on temperature. Much of firn densification and consequent surface lowering occurs within about three months of the summer season, followed by a surface build-up from snow accumulation until spring. Modeled interannual changes of the surface elevation, using the AWS measurements of surface temperature and accumulation and results of atmospheric modeling of precipitation variations, are in good agreement with the altimeter observations. In the model, the surface elevation decreases about 20 cm over the seven years due to more compaction driven by increasing summer temperatures. The minimum elevation in 1995 is driven mainly by a temporary accumulation decrease and secondarily by warmer temperatures. However, the overall elevation increase over the seven years is dominated by the accumulation increase in the later years.

Zwally, H. Jay↗
NASA NTRS2023

An Evaluation of A Physics-Based Firn Model and A Semi-Empirical Firn Model Across the Greenland Ice Sheet (1980–2020)

The Greenland Ice Sheet's (GrIS) firn layer buffers the ice sheet's contribution to sea level rise by storing meltwater in its pore space. However, available pore space and meltwater retention capability is lost due to ablation of the firn layer and refreezing of meltwater as near-surface ice slabs in the firn. Understanding how firn properties respond to climate is important for constraining the GrIS's future contribution to sea level rise in a warming climate. Observations of firn density provide detailed information about firn properties, but they are spatially and temporally limited. Here we use two firn models, the physics-based SNOWPACK model and the Community Firn Model configured with a semi-empirical densification equation (CFM-GSFC), to quantify firn properties across the GrIS from 1980 through 2020. We use an identical forcing (Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2) atmospheric reanalysis) for SNOWPACK and the CFM-GSFC in order to isolate firn model differences. To evaluate the models, we compare simulated firn properties, including firn air content (FAC), to measurements from the Surface Mass Balance and Snow on Sea Ice Working Group (SUMup) dataset of snow and firn density. Both models perform well (mean absolute percentage errors of 14 % in SNOWPACK and 16 % in the CFM-GSFC), though their performance is hindered by the spatial resolution of the atmospheric forcing. In the ice-sheet-wide simulations, the 1980–1995 average spatially integrated FAC (i.e., air volume in the firn) for the upper 100 m is 34 645 km 3 from SNOWPACK and 28 581 km 3 from the CFM-GSFC. The discrepancy in the magnitude of the modeled FAC stems from differences in densification with depth and variations in the sensitivity of the models to atmospheric forcing. In more recent years (2005–2020), both models simulate substantial depletion of pore space. During this period, the spatially integrated FAC across the entire GrIS decreases by 3.2 % (−66.6 km 3 yr −1 ) in SNOWPACK and 1.5 % (−17.4 km 3 yr −1 ) in the CFM-GSFC. These differing magnitudes demonstrate how model differences propagate throughout the FAC record. Over the full modeled record (1980–2020), SNOWPACK simulates a loss of pore space equivalent to 3 mm of sea level rise buffering, while the CFM-GSFC simulates a loss of 1 mm. The greatest depletion in FAC is along the margins and especially along the western margin where observations and models show the formation of near-surface, low-permeability ice slabs that may inhibit meltwater storage.

Megan Thompson-Munson↗
DOE OSTI2020

Hydrologic Model Files (PRMS) for Historical Conditions in the East River Watershed, Colorado between 1987-2019

The data package contains model input and output file for the East River watershed (85 km2) located in the headwaters of the Colorado River. The model is the U.S. Geological Survey (USGS) Precipitation-Modeling Runoff System (PRMS) for the historical, or baseline, condition (1987-2019) to assess the efficiency of summer monsoon rains to generate streamflow. The finite difference grid resolution is 100 m with elevations resampled from the USGS National Elevation Dataset. LANDFIRE raster data is used to derive parameters of dominant cover type, summer and winter cover density, canopy interception characteristics for snow and rain and transmission coefficient for shortwave radiation. Climate forcing uses minimum and maximum daily temperature lapse rates defined by the two Snow Telemetry (SNOTEL) stations located in the vicinity of the East River adjusted for aspect. Schofield SNOTEL snowfall is spatially distributed using LiDAR derived snow depth observations from the Airborne Snow Observatory (ASO) flown 4 April 2016. Snow depths are converted to snow water equivalent based on ground surveys and density modeling. Rainfall is spatially distributed using the monthly Parameter-elevation Relationships on Independent Slopes Model 30-year monthly averages. Simulated solar radiation is calibrated to match weather station observations. Model verification of snow water equivalent (SWE) accumulation and ablation relies on the 2018 and 2019 ASO maps and weather station snow depth. The data package contains model input files in text format, model documentation in pfd and a readme.txt file to help users understand the file structure to run the program.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Deep Learning Advances Arctic River Water Temperature Predictions

The accelerated warming in the Arctic poses serious risks to freshwater ecosystems by altering streamflow and river thermal regimes. However, limited research on Arctic River water temperatures exists due to data scarcity and the absence of robust methodologies, which often focus on large, major river basins. To address this, we leveraged the newly released, extensive AKTEMP data set and advanced machine learning techniques to develop a Long Short-Term Memory (LSTM) model. By incorporating ERA5-Land reanalysis data and integrating physical understanding into data-driven processes, our model advanced river water temperature predictions in ungauged, snow- and permafrost-affected basins in Alaska. Our model outperformed existing approaches in high-latitude regions, achieving a median Nash-Sutcliffe Efficiency of 0.95 and root mean squared error of 1.0°C. The LSTM model learned air temperature, soil temperature, solar radiation, and thermal radiation—factors associated with energy balance—were the most important drivers of river temperature dynamics. Soil moisture and snow water equivalent were highlighted as critical factors representing key processes such as thawing, melting, and groundwater contributions. Glaciers and permafrost were also identified as important covariates, particularly in seasonal river water temperature predictions. Our LSTM model successfully captured the complex relationships between hydrometeorological factors and river water temperatures across varying timescales and hydrological conditions. This scalable and transferable approach can be potentially applied across the Arctic, offering valuable insights for future conservation and management efforts.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2000

Coupled Model Simulation of Snowfall Events Over the Black Hills

Although many long-term simulations of snow accumulation and oblation have been made using stand-alone land surface models and surface models coupled with GCMs, less research has focused on short-term event simulations. Actually, accurate event simulations of snow-related processes are the basis for successful long-term simulation. Three advantages of event simulations of snowfall and snow melting are availability of: (1) intensive observation data from field experiments for validation; (2) more physically-realistic precipitation schemes for use in atmospheric models to simulate snowfall; and (3) a more detailed analysis of the snow melting processes. In addition to the complexities of snow related processes themselves, terrain-induced effects on snowfall/snow melting make simulations of snow events more difficult. Climatological observations indicate that terrain features such as the Black Hills of South Dakota and Wyoming can exert important effects on snow accumulation and snow oblation processes. One of the primary effects is that the orography causes forced uplift of airflow and causes atmospheric waves to form both upwind and downwind of it. Airflow often splits around the obstacle, converging on the lee side. This convergence may lead to precipitation enhancement. It also provides an elevated heat and moisture source that enhances atmospheric instability. During the period of April 5-May 5, 1999, the Upper Missouri River Basin Pilot Project (UMRBPP) made intensive observations on precipitation events occurring in the Black Hills. Two moderate snowfall events were captured during the period. The resulting high temporal and spatial resolution data provides opportunities to investigate terrain effects on snowfall amount, distribution, and melting. Successful simulation of snowfall amount, distribution, and evolution using atmospheric models is important to subsequent modeling of snow melting using snow sub-models in land surface schemes. In this paper, a coupled model system, consisting of an atmosphere model (ARPS) and a land-surface model (revised NCAR LSM), is used to simulate one of these UMRBPP cases.

Wang, Jianzhong↗