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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.

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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
NASA NTRS2020

California & Oregon Ecological Forecasting: Detecting and Forecasting Fog Occurrence, Frequency, and Change to Support Coast Redwood (Sequoia sempervirens) Habitat Assessments

Fog and low clouds play an important role in providing moisture to coastal ecosystems. Coast redwood (Sequoia sempervirens) forests are currently distributed along a narrow strip of coastline in California and Oregon and rely on the presence of marine fog for moisture availability during the dry season (June-October). Recent time series analyses presented an uncertain future of fog frequency; however, a decline in fog presence may have adverse effects on the coast redwood habitat. To support Save the Redwoods League, a non-profit organization dedicated to coast redwood forest management, the team analyzed hourly fog data from the Geospatial Operational Environmental Satellite 17 (GOES-17) Advanced Baseline Imager (ABI) and daily cloud cover data from the Moderate Resolution Imaging Spectroradiometer (MODIS) aboard the Terra satellite. To explore present day fog longevity, GOES-17 was utilized to map the number of fog hours per day for the 2019 and 2020 dry seasons. The MODIS cloud flag was used to map the presence or absence of daily fog, which was summarized to create a monthly fog frequency dataset and identify trends in fog presence between 2000-2020. Both datasets were used as inputs into the random forest machine learning algorithm to identify climatic drivers of fog presence and longevity over the landscape. The present-day models suggested that daily temperature difference is a driving force behind fog presence and longevity. Trends in fog presence from 2000-2020 indicated great interannual variability. Finally, fog presence was modeled under a 2080 climate projection to shed light on the future of fog presence under a projected warmer climate. Model results projected an overall decline in fog presence during the dry season in the 2080s. Decreased fog presence as a result of increased temperature difference under a warmer climate remains to be a topic of investigation as to the impact on future redwood habitat suitability.

DEVELOP Project Summary↗
NASA NTRS2020

California & Oregon Ecological Forecasting: Detecting and Forecasting Fog Occurrence, Frequency, and Change to Inform Coast Redwood (Sequoia sempervirens) Habitat Assessments

Fog and low clouds play an important role in providing moisture to coastal ecosystems. Coast redwood (Sequoia sempervirens) forests are currently distributed along a narrow strip of coastline in California and Oregon and rely on the presence of marine fog for moisture availability during the dry season (June-October). Recent time series analyses presented an uncertain future of fog frequency; however, a decline in fog presence may have adverse effects on the coast redwood habitat. To complement ongoing work by Save the Redwoods League, a non-profit organization dedicated to coast redwood forest management, the team analyzed hourly fog data from the Geospatial Operational Environmental Satellite 17 (GOES-17) Advanced Baseline Imager (ABI) and daily cloud cover data from the Moderate Resolution Imaging Spectroradiometer (MODIS) aboard the Terra satellite. To explore present day fog longevity, GOES-17 was utilized to map the number of fog hours per day for the 2019 and 2020 dry seasons. The MODIS cloud flag was used to map the presence or absence of daily fog, which was summarized to create a monthly fog frequency dataset and identify trends in fog presence between 2000-2020. Both datasets were used as inputs into the random forest machine learning algorithm to identify climatic drivers of fog presence and longevity over the landscape. The present-day models suggested that daily temperature difference is a driving force behind fog presence and longevity. Trends in fog presence from 2000-2020 indicated great interannual variability. Finally, fog presence was modeled under a 2080 climate projection to shed light on the future of fog presence under a projected warmer climate. Model results projected an overall decline in fog presence during the dry season in the 2080s. Decreased fog presence as a result of increased temperature difference under a warmer climate remains to be a topic of investigation as to the impact on future redwood habitat suitability.

DEVELOP Technical Paper↗
NASA NTRS2024

Northern Rockies Ecological Conservation: Leveraging Earth Observations to Monitor and Predict Populations of Federally Threatened Whitebark Pine (Pinus albicaulis) across the Intermountain West

Whitebark pine (WBP; Pinus albicaulis) is an ecologically important species in North America. As a federally listed threatened species, an understanding of WBP habitat, distribution, and health is important for the natural resource managers of the National Park Service, United States Forest Service, Bureau of Land Management, Fish and Wildlife Service, and non-profit organizations such as the Whitebark Pine Ecosystem Foundation. Previous attempts to develop models of WBP habitat suitability and distribution lack confidence in their validity and integrity for these organizations. The updated models of habitat suitability and distribution developed by this study would provide managers with a capability to be employed in the conservation and future research direction for WBP. Thus, we developed a habitat suitability model of WBP at a high spatial resolution (Landsat 9 Operational Land Image-2, National Land Cover Database, NASA Shuttle Radar Topography Mission; 30m pixels) using a generalized logistic regression with an area under the curve value of 0.754. We extracted spectral reflectance signatures from overlapped ground sample points and Sentinel-2 Multispectral Instrument. The spectral signature analysis indicates WBP is separable from other tree species. We also utilized a visual validation approach and random forest (RF) modeling to separate WBP from limber pine. Through visual validation the RF classifier successfully identified 8out of 10 WBP trees gathered through ground truth points. Additionally, we achieved an overall accuracy of 91%in our confusion matrix for the distribution model using a dependent validation approach. The derived products from this study allow project partners to assess current suitable habitat and apparent health status in areas of identified WBP occurrence, providing data to aid future research regarding WBP health.

Sentinel-2↗
NASA NTRS1993

Dynamics and control of coherent structures in the turbulent wall layer: An overview

We expand the velocity field in the vicinity of the wall in empirical eigenfunctions obtained from experiment. Truncating our system and using Galerkin projection, we obtain a closed set of non-linear ordinary differential equations with ten degrees of freedom. We find a rich dynamical behavior, including in particular a heteroclinic attracting orbit giving rise to intermittency. The intermittent jump from one attracting point to the other resembles in many respects the bursts observed in experiments. Specifically, the time between jumps and the duration of the jumps, is approximately that observed in a burst; the jump begins with the formation of a narrowed and intensified updraft, like the ejection phase of a burst, and is followed by a gentle, diffuse downdraft like the sweep phase of a burst. The magnitude of the Reynolds stress spike produced during a burst is limited by our truncation. The behavior is quite robust, much of it being due to the symmetries present (Aubry's group has examined dimensions up to 128 with persistence of the global behavior). We have examined eigenvalues and coefficients obtained from experiment, and from exact simulation, which differ in magnitude. Similar behavior is obtained in both cases; in the latter case, the heteroclinic orbits connect limit cycles instead of fixed points, corresponding to cross-stream waving of the streamwise rolls. The bifurcation diagram remains structurally similar, but somewhat distorted. The role of the pressure term is made clear - it triggers the intermittent jumps, which otherwise would occur at longer and longer intervals, as the system trajectory is attracted closer and closer to the heteroclinic cycle. The pressure term results in the jumps occurring at essentially random times, and the magnitude of the signal determines the average timing. Stretching of the wall region shows that the model is consistent with observations of polymer drag reduction. Change of the third order coefficients, corresponding to acceleration or deceleration of the mean flow, changes the heteroclinic cycles from attracting to repelling, increasing or decreasing the stability, in agreement with observations. The existence of fixed points is an artifact introduced by the projection; however, a decoupled model still displays the rich dynamics. Numerous assumptions made in Aubry et al. (1988) can now be proved exactly. Feeding back eigenfuncitons with the proper phase can delay the bursting, (the heteroclinic jump to the other fixed point), decreasing the drag. It is also possible to speed up the bursting, increasing mixing to control separation. Our approach is optimal for short time tracking in control.

Berkooz, Gal↗
DOE OSTI2020

Arctic Tipping Points Triggering Global Change (LDRD Final Report)

The Arctic is warming and feedbacks in the coupled Earth system may be driving the Arctic to tipping events that could have critical downstream impacts for the rest of the globe. In this project we have focused on analyzing sea ice variability and loss in the coupled Earth system Summer sea ice loss is happening rapidly and although the loss may be smooth and reversible, it has significant consequences for other Arctic systems as well as geopolitical and economic implications. Accurate seasonal predictions of sea ice minimum extent and long-term estimates of timing for a seasonally ice-free Arctic depend on a better understanding of the factors influencing sea ice dynamics and variation in this strongly coupled system. Under this project we have investigated the most influential factors in accurate predictions of September Arctic sea ice extent using machine learning models trained separately on observational data and on simulation data from five E3SM historical ensembles. Monthly averaged data from June, July, and August for a selection of ice, ocean, and atmosphere variables were used to train a random forest regression model. Gini importance measures were computed for each input feature with the testing data. We found that sea ice volume is most important earlier in the season (June) and sea ice extent became a more important predictor closer to September. Results from this study provide insight into how feature importance changes with forecast length and illustrates differences between observational data and simulated Earth system data. We have additionally performed a global sensitivity analysis (GSA) using a fully coupled ultra- low resolution configuration E3SM. To our knowledge, this is the first global sensitivity analysis involving the fully-coupled E3SM Earth system model. We have found that parameter variations show significant impact on the Arctic climate state and atmospheric parameters related to cloud parameterizations are the most significant. We also find significant interactions between parameters from different components of E3SM. The results of this study provide invaluable insight into the relative importance of various parameters from the sea ice, atmosphere and ocean components of the E3SM (including cross-component parameter interactions) on various Arctic-focused quantities of interest (QOIs).

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1989

Probabilistic constitutive relationships for material strength degradation models

In the present probabilistic methodology for the strength of aerospace propulsion system structural components subjected to such environmentally-induced primitive variables as loading stresses, high temperature, chemical corrosion, and radiation, time is encompassed as an interacting element, allowing the projection of creep and fatigue effects. A probabilistic constitutive equation is postulated to account for the degradation of strength due to these primitive variables which may be calibrated by an appropriately curve-fitted least-squares multiple regression of experimental data. The resulting probabilistic constitutive equation is embodied in the PROMISS code for aerospace propulsion component random strength determination.

Boyce, L.↗
DOE OSTI2023

Fast estimation of the look-elsewhere effect using Gaussian random fields

Abstract We discuss the use of Gaussian random fields to estimate the look-elsewhere effect correction. We show that Gaussian random fields can be used to model the null-hypothesis significance maps from a large set of statistical problems commonly encountered in physics, such as template matching and likelihood ratio tests. Some specific examples are searches for dark matter using pixel arrays, searches for astronomical transients, and searches for fast-radio bursts. Gaussian random fields can be sampled efficiently in the frequency domain, and the excursion probability can be fitted with these samples to extend any estimation of the look-elsewhere effect to lower p values. In addition, in cases where the Gaussian random field is stationary and the parameter space is Euclidean, the look-elsewhere effect correction can be computed analytically. We demonstrate these methods using two example template matching problems. Finally, we apply these methods to estimate the trial factor of a $$4^3$$ 4 3 accelerometer array for the detection of dark matter tracks in the Windchime project. When a global significance of $$3\sigma $$ 3 σ is required, the estimated trial factor for such an accelerometer array is $$10^{14}$$ 10 14 for a one-second search, and $$10^{22}$$ 10 22 for a 1-year search.

Qin, Juehang (ORCID:0000000182288949)↗
DOE OSTI2026

Data and scripts associated with a manuscript modeling microbial regulation of priming effects

This data package is associated with the publication “Modeling Microbial Regulatory Feedback in Organic Matter Decomposition Identifies Copiotrophic Traits as Key Drivers of Positive Priming” published as a preprint on BioRXiv by Ahamed et al. (2026); https://doi.org/10.1101/2024.08.11.607483. The package contains MATLAB scripts and saved simulation outputs used to implement a cybernetic model of microbial regulation during complex organic matter (OM) decomposition governing priming effects. It includes models of (i) single microbial functional groups (copiotrophic or oligotrophic degraders) and (ii) binary consortia composed of degraders and non-degraders with contrasting or common growth traits. Simulation results were generated using Monte Carlo analyses, with randomized key model parameters across a range of environmental mixing fractions of complex and labile OM. The dataset was created to provide a transparent and reusable computational framework for systematically exploring how microbial growth traits, metabolic regulation, and community composition influence OM decomposition dynamics and priming effects. For details on how to navigate data packages generated by this project, see https://data.ess-dive.lbl.gov/portals/PNNLRiverCorridorSFA/About. In addition to a readme, this data package also includes a file-level metadata (FLMD) file that describes each file and a data dictionary (DD) that describes the variable definitions. This package includes: (1) annotated MATLAB code implementing the system of ordinary differential equations and cybernetic control laws; (2) saved output files containing data (e.g., biomass, substrates, enzyme levels, priming metrics); and (3) scripts for processing saved outputs and regenerating figures. Specifically, the data package contains three main MATLAB scripts: runPrimingModel.m, runPlotData.m, and runPlotSuppFigS1.m, along with this readme and supporting documentation. Users should begin with runPrimingModel.m, which contains the annotated code implementing the system of ordinary differential equations and cybernetic control laws. This script runs the Monte Carlo simulations of microbial OM decomposition and allows users to modify microbial trait definitions, adjust parameter distributions, or define new community configurations. Simulation outputs are automatically saved as .mat files in the folder named SavedData, which stores all pre-generated results included in this package. The second script, runPlotData.m, reads files from the SavedData folder and processes them to regenerate the figures presented in the manuscript. The third script, runPlotSuppFigS1.m, specifically generates Figure S1 in the Supplementary Material of the manuscript. The package also includes the aforementioned files in non-proprietary .txt format. If users intend to use them, they should first save the files in their respective .m or .mat formats prior to execution in MATLAB.

Biomass concentration↗
NASA NTRS1991

Compressed/reconstructed test images for CRAF/Cassini

A set of compressed, then reconstructed, test images submitted to the Comet Rendezvous Asteroid Flyby (CRAF)/Cassini project is presented as part of its evaluation of near lossless high compression algorithms for representing image data. A total of seven test image files were provided by the project. The seven test images were compressed, then reconstructed with high quality (root mean square error of approximately one or two gray levels on an 8 bit gray scale), using discrete cosine transforms or Hadamard transforms and efficient entropy coders. The resulting compression ratios varied from about 2:1 to about 10:1, depending on the activity or randomness in the source image. This was accomplished without any special effort to optimize the quantizer or to introduce special postprocessing to filter the reconstruction errors. A more complete set of measurements, showing the relative performance of the compression algorithms over a wide range of compression ratios and reconstruction errors, shows that additional compression is possible at a small sacrifice in fidelity.

Dolinar, S.↗
NASA NTRS1992

Remote sensing of Earth terrain

Research findings are summarized for projects dealing with the following: application of theoretical models to active and passive remote sensing of saline ice; radiative transfer theory for polarimetric remote sensing of pine forest; scattering of electromagnetic waves from a dense medium consisting of correlated Mie scatterers with size distribution and applications to dry snow; variance of phase fluctuations of waves propagating through a random medium; theoretical modeling for passive microwave remote sensing of earth terrain; polarimetric signatures of a canopy of dielectric cylinders based on first and second order vector radiative transfer theory; branching model for vegetation; polarimetric passive remote sensing of periodic surfaces; composite volume and surface scattering model; and radar image classification.

Kong, J. A.↗
DOE OSTI2023

Statistical Generation of Ocean Forcing With Spatiotemporal Variability for Ice Sheet Models

Melting of ice at the base of floating ice shelves that fringe the Antarctic ice sheet has been identified as a significant source of uncertainty in sea level rise projections. Part of this uncertainty derives from chaotic internal variability of the coupled ocean-atmosphere system. For numerical ice sheet model projections, this uncertainty has not previously been quantified because of the prohibitive computational expense of running large climate model ensembles. Here, we develop and demonstrate a technique that generates independent realizations of internal climate variability from a single climate model simulation. Building on previous developments in model emulation, this technique uses empirical orthogonal function decomposition and Fourier-phase randomization to generate statistically consistent realizations of spatiotemporal variability fields for the target climate variable. The method facilitates efficient sampling of a wide range of climate trajectories, which can also be incorporated within ice sheet or other physical models to represent feedback processes.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1976

The emission-line spectrum above the limb of the quiet sun - 1175-1940 A

Spectra of a quiet solar region obtained at positions within and above the solar white-light limb in the wavelength region from 1175 to 1940 A are discussed. The spectra were obtained by the slit spectrograph (SO82-B) on Skylab. The spectral resolution is 0.06 A, and the projected slit area on the sun was 2 x 60 arcsec (1450 x 43500 km). Relative line intensities are presented for lines formed in the temperature region of the solar atmosphere from about 8000 to 220,000 K. Representative line profiles of both optically thin and optically thick lines are shown as a function of height above the limb. Random mass-motion velocities are deduced from the optically thin lines, and the relative intensities and profiles of the lines are discussed in terms of current theoretical models. A wavelength list with identifications is given for the spectrum obtained at +4 arcsec above the white-light limb.

Doschek, G. A.↗
NASA NTRS2015

Winter-to-Summer Precipitation Phasing in Southwestern North America: A Multi-Century Perspective from Paleoclimatic Model-Data Comparisons

The phasing of winter-to-summer precipitation anomalies in the North American monsoon (NAM) region 2 (113.25 deg W-107.75 deg W, 30 deg N-35.25 deg N-NAM2) of southwestern North America is analyzed in fully coupled simulations of the Last Millennium and compared to tree ring reconstructed winter and summer precipitation variability. The models simulate periods with in-phase seasonal precipitation anomalies, but the strength of this relationship is variable on multidecadal time scales, behavior that is also exhibited by the reconstructions. The models, however, are unable to simulate periods with consistently out-of-phase winter-to-summer precipitation anomalies as observed in the latter part of the instrumental interval. The periods with predominantly in-phase winter-to-summer precipitation anomalies in the models are significant against randomness, and while this result is suggestive of a potential for dual-season drought on interannual and longer time scales, models do not consistently exhibit the persistent dual-season drought seen in the dendroclimatic reconstructions. These collective findings indicate that model-derived drought risk assessments may underestimate the potential for dual-season drought in 21st century projections of hydroclimate in the American Southwest and parts of Mexico.

Variability↗
NASA NTRS2020

Southern Colorado Disasters: Using NASA Earth Observations to Map Aspen Extent and Recovery Due to Wildfire

Quaking aspen (Populus tremuloides) is an important species for wildlife, watershed health, and ecosystem resilience across its range. Heavy ungulate browsing and factors influenced by a changing climate including seasonal temperature changes and moisture deficit have led to reduced post-fire aspen regeneration rates in southern Colorado. This project partnered with Trinchera Ranch and the Colorado State Forest Service to estimate aspen recovery after the Spring Creek Fire, which ignited in June of 2018. The Southern Colorado Disasters team utilized field measurements and satellite imagery from Landsat Operational Land Imager (OLI), Sentinel-2 MultiSpectral Instrument (MSI), and the Shuttle Radar Topography Mission (SRTM) to train and run several random forest models that detect pre- and post-fire aspen extent. Ocular sampling of over 500 points on high-resolution pre-fire and post-fire images identified percentage aspen cover in 30 x 30-meter grid cells. This process provided training data for regression models, which were able to detect aspen across the landscape for both time periods using multiple remote sensing vegetation health indices. In addition, landscape suitability for aspen regeneration was modeled to provide a guide for managers on where to monitor for aspen regeneration post-fire.

DEVELOP Technical Paper↗
DOE OSTI2020

Experimentally Validated Computational Modeling of Creep and Creep-Cracking for Nuclear Concrete Structures

In a Nuclear Power Plant, one of the most important components is the concrete nuclear reactor cavity, which serves both a structural and protective function as the biological radiation shield. Given that creep has been identified as a major knowledge gap in the assessment of nuclear structures (NUREG/CR-7153), this work helps to further the understanding of creep behavior of massive concrete containment structures for decades to enable safe and long-term operation of these facilities. This project has developed a robust, experimentally validated model to predict creep in nuclear concrete structures for up 60 years using short-term creep data thereby enabling a longer service life of critical facilities and early detection of structural failure. The work presented in this report is a pairing of computational and experimental methods. For the first time, the time temperature superposition (TTS) principle was successfully used to generate a uniaxial creep compliance master curve to predict mortar creep response for up to 22,500 days (nearly 60 years) at a reference temperature of 20°C. These data were used as input into finite element analysis (FEA) codes that use highly realistic random, 3D concrete microstructures from reconstructed coarse limestone aggregates. Finite element analysis performed provides the ability to quickly upscale mortar viscoelastic behavior to long-term concrete creep/relaxation data. A master creep compliance curve, constructed from the TTS principle, spanning 27 years, was used to validate two and a half decades of simulated concrete creep. Concurrently, three different simulated wall specimens were designed to mimic the behavior of post-tensioned concrete nuclear containment facility vessel walls over time as a result of concrete creep. The specimens were designed with different thicknesses, transverse and longitudinal reinforcement ratios, and level of post-tensioning stress. Each specimen contained various instrumentation to measure internal concrete temperature, concrete strain, and post-tensioning strain hourly for over 3 years. The concrete creep model developed in this project, based on the FEA concrete simulations, was applied to simulate the structural-scale experiments of prestressed concrete walls conducted in this project using the Grizzly code. These models can represent the effects of reinforcing and prestressing. Although there are some discrepancies with the experimental data, the model can predict the overall trends of the creep response in these experiments. One of these experimental models was also applied to an extended time to demonstrate how the findings from this study can be used to predict the behavior of actual structures of interest that have been in service for extended periods of time.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS2010

Kriging as a Means of Improving WAAS Availability

The Wide Area Augmentation System (WAAS), an augmentation of the Global Positioning System (GPS), provides safe and reliable use of GPS signals for airline navigation over much of North America. Currently the largest source of positioning error in the system is signal delay caused by the ionosphere. To allow the user to take account of such error, WAAS computes and broadcasts ionospheric vertical delays at a set of regularly-spaced grid points. In addition, WAAS computes and broadcasts a safety-critical integrity bound at each ionospheric grid point (IGP) called the Grid Ionospheric Vertical Error (GIVE). GIVEs are constructed to be sufficiently large to protect the user against positioning error due to the presence of ionospheric irregularity. In the initial operating capability (IOC) of WAAS, the vertical delay estimate at each IGP is determined from a planar fit of neighboring slant delay measurements, projected to vertical using an obliquity factor specified by the standard thin-shell model of the ionosphere. In WAAS Follow-On (WFO) Release 3, however, the vertical delay will be estimated by an established, geo-statistical technique known as kriging. Compared to the planar fit model, the kriging model is found, in general, to match better the observed random structure of the vertical delay. This paper presents the kriging methodology that will be used to estimate the vertical delay and its uncertainty at each IGP, and it assesses the subsequent improvement in WAAS availability enabled by kriging.

GPS↗
NASA NTRS1990

Analysis of AVHRR, CZCS and historical in situ data off the Oregon Coast

The original scientific objectives of this grant were to: (1) characterize the seasonal cycles and interannual variability for phytoplankton concentrations and sea surface temperature (SST) in the California Current using satellite data; and (2) to explore the spatial and temporal relationship between these variables and surface wind forcing. An additional methodological objective was to develop statistical methods for forming mean fields, which minimize the effects of random data gaps and errors in the irregularly sampled CZCS (Coastal Zone Color Scanner) and AVHRR (Advanced Very High Resolution Radiometer) satellite data. A final task was to evaluate the level of uncertainty in the wind fields used for the statistical analysis. Funding in the first year included part of the cost of an image processing system to enable this and other projects to process and analyze satellite data. This report consists of summaries of the major projects carried out with all or partial support from this grant. The appendices include a list of papers and professional presentations supported by the grant, as well as reprints of the major papers and reports.

Strub, P. Ted↗
NASA NTRS2002

Visualization of Flows in Packed Beds of Twisted Tapes

A videotape presentation of the flow field in a packed bed of 48 twisted tapes which can be simulated by very thin virtual cylinders has been assembled. The indices of refraction of the oil and the Lucite twisted tapes were closely matched, and the flow was seeded with magnesium oxide particles. Planar laser light projected the flow field in two dimensions both along and transverse to the flow axis. The flow field was three dimensional and complex to describe, yet the most prominent finding was flow threads. It appeared that axial flow spiraled along either within the confines of a virtual cylindrical boundary or within the exterior region, between the tangency points, of the virtual cylinders. Random packing and bed voids created vortices and disrupted the laminar flow but minimized the entrance effects. The flow-pressure drops in the packed bed fell below the Ergun model for porous-media flows. Single-twisted-tape results of Smithberg and Landis (1964) were used to guide the analysis. In appendix A the results of several investigators are scaled to the Ergun model. Further investigations including different geometric configurations, computational fluid dynamic (CFD) gridding, and analysis are required.

Hendricks, R. C.↗