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At least 217 records · Page 12

The New NASA Orbital Debris Engineering Model ORDEM2000

The NASA Orbital Debris Program Office at Johnson Space Center has developed a new computer-based orbital debris engineering model, ORDEM2000, which describes the orbital debris environment in the low Earth orbit region between 200 and 2000 km altitude. The model is appropriate for those engineering solutions requiring knowledge and estimates of the orbital debris environment (debris spatial density, flux, etc.). ORDEM2000 can also be used as a benchmark for ground-based debris measurements and observations. We incorporated a large set of observational data, covering the object size range from 10 mm to 10 m, into the ORDEM2000 debris database, utilizing a maximum likelihood estimator to convert observations into debris population probability distribution functions. These functions then form the basis of debris populations. We developed a finite element model to process the debris populations to form the debris environment. A more capable input and output structure and a user-friendly graphical user interface are also implemented in the model. ORDEM2000 has been subjected to a significant verification and validation effort. This document describes ORDEM2000, which supersedes the previous model, ORDEM96. The availability of new sensor and in situ data, as well as new analytical techniques, has enabled the construction of this new model. Section 1 describes the general requirements and scope of an engineering model. Data analyses and the theoretical formulation of the model are described in Sections 2 and 3. Section 4 describes the verification and validation effort and the sensitivity and uncertainty analyses. Finally, Section 5 describes the graphical user interface, software installation, and test cases for the user.

Liou, Jer-Chyi↗

Development of a Dynamically Configurable, Object-Oriented Framework for Distributed, Multi-modal Computational Aerospace Systems Simulation

The following reports are presented on this project:A first year progress report on: Development of a Dynamically Configurable,Object-Oriented Framework for Distributed, Multi-modal Computational Aerospace Systems Simulation; A second year progress report on: Development of a Dynamically Configurable, Object-Oriented Framework for Distributed, Multi-modal Computational Aerospace Systems Simulation; An Extensible, Interchangeable and Sharable Database Model for Improving Multidisciplinary Aircraft Design; Interactive, Secure Web-enabled Aircraft Engine Simulation Using XML Databinding Integration; and Improving the Aircraft Design Process Using Web-based Modeling and Simulation.

Afjeh, Abdollah A.↗

Computational and Experimental Study of Energetic Materials in a Counterflow Microgravity Environment

Counterflow diffusion flames are studied for various fuels flowing against decomposition products from solid ammonium perchlorate (AP) pellets in order to obtain fundamental understanding of composite propellant flame structure and chemistry. We illustrate this approach through a combined experimental and numerical study of a fuel mixture consisting of C2H4 CO + H2, and C2H2 + C2H4 flowing against solid AP. For these particular AP-fuel systems, the resulting flame zone simulates the various flame structures that are ex+ to exist between reaction products from Ap crystals and a hydrocarbon binder. As in all our experimental studies, quantitative species and temperature profiles have been measured between the fuel exit and AP surface. Species measured included CN, NH, NO, OH, N2, CO2, CO, H2, CO, HCl, and H2O. Temperature was measured using a thermocouple at the exit, spontaneous Raman scattering measurements throughout the flame, OH rotational population distributions, and NO vibrational population distributions. The burning rate of AP was also measured as a function of strain rate, given by the separation distance between the AP surface and the gaseous hydrocarbon fuel tube exit plane. This distance was nominally set at 5 mm, although studies have been performed for variations in separation distance. The measured 12 scalars are compared with predictions from a detailed gas-phase kinetics model consisting of 86 species and 531 reactions. Model predictions are found to be in good agreement with experiment and illustrate the type of kinetic features that may be expected to occur in propellants when AP particle size distributions are varied. Furthermore, the results constitute the continued development of a necessary database and validation of a comprehensive model for studying more complex AP-solid fuel systems in microgravity. Exploratory studies have also been performed with liquid and solid fuels at normal gravity. Because of melting (and hence dripping) and deep thermal wave penetration into the liquid, these experiments were found feasible, but not used for obtaining quantitative data. Microgravity experiments are needed to eliminate the dripping and boiling phenomena of these systems at normal gravity. Microgravity tests in the NASA Glenn 2.2 second drop tower were performed (1) to demonstrate the feasibility of performing propellant experiments using the NASA Glenn microgravity facilities, (2) to develop the operational procedures for safe handing of the energetic materials and disposal of their toxic combustion by-products and (3) to obtain initial measurements of the AP burning rate and flame structure under microgravity conditions. Experiments were conducted on the CH4/AP system previously studied at normal gravity using a modified design of the counterflow burner and a NASA Glenn Pig Rig, i.e., one of the existing drop rigs for general-purpose usage. In these experiments, the AP burning rate was measured directly with a linear variable differential transducer (LVDT) and video imaging of the flame structure was recorded ignition was achieved by hot wires stretched across the AP surfaces. Initial drop tower combustion data show that with the same burner separation distance and flow conditions of the normal gravity experiments, the AP burning rate is approximately a factor of two lower. This difference is likely a result of radiation effects, but further tests with longer test times need to be conducted to verify that steady state conditions were achieved under microgravity conditions.

Takahashi, Fumiaki↗

Aeroheating Characteristics for a Two-Stage-To-Orbit Concept During Separation at Mach 6

An experimental study was conducted to determine the proximity aeroheating characteristics for a two-stage-to-orbit concept in close proximity in the NASA Langley 20-Inch Mach 6 Air Tunnel. A new hybrid discrete thin-film resistance gauge technique was evaluated in this study and used to measure experimental interference heating levels between the booster and the orbiter at a constant freestream Reynolds number of 8.25 x 10(exp 6)/m and a variety of separation and axial offset distances. It was found that, as the orbiter separates from the booster and the booster falls away, the windward centerline heating increased on the orbiter by as much as 13-times over the baseline, single model heating distribution, and on the booster by as much as 6-times. The aeroheating database developed can be used for computational fluid dynamic code validation.

Liechty, Derek S.↗

An Application Programming Interface for Synthetic Snowflake Particle Structure and Scattering Data

The work by Kuo and colleagues on growing synthetic snowflakes and calculating their single-scattering properties has demonstrated great potential to improve the retrievals of snowfall. To grant colleagues flexible and targeted access to their large collection of sizes and shapes at fifteen (15) microwave frequencies, we have developed a web-based Application Programming Interface (API) integrated with NASA Goddard's Precipitation Processing System (PPS) Group. It is our hope that the API will enable convenient programmatic utilization of the database. To help users better understand the API's capabilities, we have developed an interactive web interface called the OpenSSP API Query Builder, which implements an intuitive system of mechanisms for selecting shapes, sizes, and frequencies to generate queries, with which the API can then extract and return data from the database. The Query Builder also allows for the specification of normalized particle size distributions by setting pertinent parameters, with which the API can also return mean geometric and scattering properties for each size bin. Additionally, the Query Builder interface enables downloading of raw scattering and particle structure data packages. This presentation will describe some of the challenges and successes associated with developing such an API. Examples of its usage will be shown both through downloading output and pulling it into a spreadsheet, as well as querying the API programmatically and working with the output in code.

reflectivity properties of snowflakes↗

An Overview of Mesoscale Aerosol Processes, Comparisons, and Validation Studies from DRAGON Networks

Over the past 24 years, the AErosol RObotic NETwork (AERONET) program has provided highly accurate remote-sensing characterization of aerosol optical and physical properties for an increasingly extensive geographic distribution including all continents and many oceanic island and coastal sites. The measurements and retrievals from the AERONET global network have addressed satellite and model validation needs very well, but there have been challenges in making comparisons to similar parameters from in situ surface and airborne measurements. Additionally, with improved spatial and temporal satellite remote sensing of aerosols, there is a need for higher spatial-resolution ground-based remote-sensing networks. An effort to address these needs resulted in a number of field campaign networks called Distributed Regional Aerosol Gridded Observation Networks (DRAGONs) that were designed to provide a database for in situ and remote-sensing comparison and analysis of local to mesoscale variability in aerosol properties. This paper describes the DRAGON deployments that will continue to contribute to the growing body of research related to meso- and microscale aerosol features and processes. The research presented in this special issue illustrates the diversity of topics that has resulted from the application of data from these networks.

Holben, Brent N.↗

Information Power Grid: Distributed High-Performance Computing and Large-Scale Data Management for Science and Engineering

We use the term "Grid" to refer to distributed, high performance computing and data handling infrastructure that incorporates geographically and organizationally dispersed, heterogeneous resources that are persistent and supported. This infrastructure includes: (1) Tools for constructing collaborative, application oriented Problem Solving Environments / Frameworks (the primary user interfaces for Grids); (2) Programming environments, tools, and services providing various approaches for building applications that use aggregated computing and storage resources, and federated data sources; (3) Comprehensive and consistent set of location independent tools and services for accessing and managing dynamic collections of widely distributed resources: heterogeneous computing systems, storage systems, real-time data sources and instruments, human collaborators, and communications systems; (4) Operational infrastructure including management tools for distributed systems and distributed resources, user services, accounting and auditing, strong and location independent user authentication and authorization, and overall system security services The vision for NASA's Information Power Grid - a computing and data Grid - is that it will provide significant new capabilities to scientists and engineers by facilitating routine construction of information based problem solving environments / frameworks. Such Grids will knit together widely distributed computing, data, instrument, and human resources into just-in-time systems that can address complex and large-scale computing and data analysis problems. Examples of these problems include: (1) Coupled, multidisciplinary simulations too large for single systems (e.g., multi-component NPSS turbomachine simulation); (2) Use of widely distributed, federated data archives (e.g., simultaneous access to metrological, topological, aircraft performance, and flight path scheduling databases supporting a National Air Space Simulation systems}; (3) Coupling large-scale computing and data systems to scientific and engineering instruments (e.g., realtime interaction with experiments through real-time data analysis and interpretation presented to the experimentalist in ways that allow direct interaction with the experiment (instead of just with instrument control); (5) Highly interactive, augmented reality and virtual reality remote collaborations (e.g., Ames / Boeing Remote Help Desk providing field maintenance use of coupled video and NDI to a remote, on-line airframe structures expert who uses this data to index into detailed design databases, and returns 3D internal aircraft geometry to the field); (5) Single computational problems too large for any single system (e.g. the rotocraft reference calculation). Grids also have the potential to provide pools of resources that could be called on in extraordinary / rapid response situations (such as disaster response) because they can provide common interfaces and access mechanisms, standardized management, and uniform user authentication and authorization, for large collections of distributed resources (whether or not they normally function in concert). IPG development and deployment is addressing requirements obtained by analyzing a number of different application areas, in particular from the NASA Aero-Space Technology Enterprise. This analysis has focussed primarily on two types of users: the scientist / design engineer whose primary interest is problem solving (e.g. determining wing aerodynamic characteristics in many different operating environments), and whose primary interface to IPG will be through various sorts of problem solving frameworks. The second type of user is the tool designer: the computational scientists who convert physics and mathematics into code that can simulate the physical world. These are the two primary users of IPG, and they have rather different requirements. The results of the analysis of the needs of these two types of users provides a broad set of requirements that gives rise to a general set of required capabilities. The IPG project is intended to address all of these requirements. In some cases the required computing technology exists, and in some cases it must be researched and developed. The project is using available technology to provide a prototype set of capabilities in a persistent distributed computing testbed. Beyond this, there are required capabilities that are not immediately available, and whose development spans the range from near-term engineering development (one to two years) to much longer term R&D (three to six years). Additional information is contained in the original.

Johnston, William E.↗

The Updated IAU MDC Catalogue of Photographic Meteor Orbits

The database of photographic meteor orbits of the IAU Meteor Data Center at the Astronomical Institute SAS has gradually been updated. To the 2003 version of 4581 photographic orbits compiled from 17 different stations and obtained in the period 1936-1996, additional new 211 orbits compiled from 7 sources have been added. Thus, the updated version of the catalogue contains 4792 photographic orbits (equinox J2000.0) available either in two separate orbital and geophysical data files or a file with the merged data. All the updated files with relevant documentation are available at the web of the IAU Meteor Data Center. Keywords astronomical databases photographic meteor orbits 1 Introduction Meteoroid orbits are a basic tool for investigation of distribution and spatial structure of the meteoroid population in the close surroundings of the Earth s orbit. However, information about them is usually widely scattered in literature and often in publications with limited circulation. Therefore, the IAU Comm. 22 during the 1976 IAU General Assembly proposed to establish a meteor data center for collection of meteor orbits recorded by photographic and radio techniques. The decision was confirmed by the next IAU GA in 1982 and the data center was established (Lindblad, 1987). The purpose of the data center was to acquire, format, check and disseminate information on precise meteoroid orbits obtained by multi-station techniques and the database gradually extended as documented in previous reports on the activity of the Meteor Data Center by Lindblad (1987, 1995, 1999 and 2001) or Lindblad and Steel (1993). Up to present, the database consists of 4581 photographic meteor orbits (Lindblad et al., 2005), 63.330 radar determined orbit: Harvard Meteor Project (1961-1965, 1968-1969), Adelaide (1960-1961, 1968-1969), Kharkov (1975), Obninsk (1967-1968), Mogadish (1969-1970) and 1425 video-recordings (Lindblad, 1999) to which additional 817 video meteors orbits published by Koten el al. (2003) were

Porubcan, V.↗

Providing the Persistent Data Storage in a Software Engineering Environment Using Java/COBRA and a DBMS

An investigation was undertaken to build the software foundation for the WHERE (Web-based Hyper-text Environment for Requirements Engineering) project. The TCM (Toolkit for Conceptual Modeling) was chosen as the foundation software for the WHERE project which aims to provide an environment for facilitating collaboration among geographically distributed people involved in the Requirements Engineering process. The TCM is a collection of diagram and table editors and has been implemented in the C++ programming language. The C++ implementation of the TCM was translated into Java in order to allow the editors to be used for building various functionality of the WHERE project; the WHERE project intends to use the Web as its communication back- bone. One of the limitations of the translated software (TcmJava), which militated against its use in the WHERE project, was persistent data management mechanisms which it inherited from the original TCM; it was designed to be used in standalone applications. Before TcmJava editors could be used as a part of the multi-user, geographically distributed applications of the WHERE project, a persistent storage mechanism must be built which would allow data communication over the Internet, using the capabilities of the Web. An approach involving features of Java, CORBA (Common Object Request Broker), the Web, a middle-ware (Java Relational Binding (JRB)), and a database server was used to build the persistent data management infrastructure for the WHERE project. The developed infrastructure allows a TcmJava editor to be downloaded and run from a network host by using a JDK 1.1 (Java Developer's Kit) compatible Web-browser. The aforementioned editor establishes connection with a server by using the ORB (Object Request Broker) software and stores/retrieves data in/from the server. The server consists of a CORBA object or objects depending upon whether the data is to be made persistent on a single server or multiple servers. The CORBA object providing the persistent data server is implemented using the Java progranu-ning language. It uses the JRB to store/retrieve data in/from a relational database server. The persistent data management system provides transaction and user management facilities which allow multi-user, distributed access to the stored data in a secure manner.

Dhaliwal, Swarn S.↗

Recommendations for a service framework to access astronomical archives

There are a large number of astronomical archives and catalogs on-line for network access, with many different user interfaces and features. Some systems are moving towards distributed access, supplying users with client software for their home sites which connects to servers at the archive site. Many of the issues involved in defining a standard framework of services that archive/catalog suppliers can use to achieve a basic level of interoperability are described. Such a framework would simplify the development of client and server programs to access the wide variety of astronomical archive systems. The primary services that are supplied by current systems include: catalog browsing, dataset retrieval, name resolution, and data analysis. The following issues (and probably more) need to be considered in establishing a standard set of client/server interfaces and protocols: Archive Access - dataset retrieval, delivery, file formats, data browsing, analysis, etc.; Catalog Access - database management systems, query languages, data formats, synchronous/asynchronous mode of operation, etc.; Interoperability - transaction/message protocols, distributed processing mechanisms (DCE, ONC/SunRPC, etc), networking protocols, etc.; Security - user registration, authorization/authentication mechanisms, etc.; Service Directory - service registration, lookup, port/task mapping, parameters, etc.; Software - public vs proprietary, client/server software, standard interfaces to client/server functions, software distribution, operating system portability, data portability, etc. Several archive/catalog groups, notably the Astrophysics Data System (ADS), are already working in many of these areas. In the process of developing StarView, which is the user interface to the Space Telescope Data Archive and Distribution Service (ST-DADS), these issues and the work of others were analyzed. A framework of standard interfaces for accessing services on any archive system which would benefit archive user and supplier alike is proposed.

Travisano, J. J.↗

New Satellite Project Aerosol-UA: Remote Sensing of Aerosols in the Terrestrial Atmosphere

We discuss the development of the Ukrainian space project Aerosol-UA which has the following three main objectives: (1) to monitor the spatial distribution of key characteristics of terrestrial tropospheric and stratospheric aerosols; (2) to provide a comprehensive observational database enabling accurate quantitative estimates of the aerosol contribution to the energy budget of the climate system; and (3) quantify the contribution of anthropogenic aerosols to climate and ecological processes. The remote sensing concept of the project is based on precise orbital measurements of the intensity and polarization of sunlight scattered by the atmosphere and the surface with a scanning polarimeter accompanied by a wide-angle multispectral imager-polarimeter. Preparations have already been made for the development of the instrument suite for the Aerosol-UA project, in particular, of the multi-channel scanning polarimeter (ScanPol) designed for remote sensing studies of the global distribution of aerosol and cloud properties (such as particle size, morphology, and composition) in the terrestrial atmosphere by polarimetric and spectrophotometric measurements of the scattered sunlight in a wide range of wavelengths and viewing directions from which a scene location is observed. ScanPol is accompanied by multispectral wide-angle imager-polarimeter (MSIP) that serves to collect information on cloud conditions and Earths surface image. Various components of the polarimeter ScanPol have been prototyped, including the opto-mechanical and electronic assemblies and the scanning mirror controller. Preliminary synthetic data simulations for the retrieval of aerosol parameters over land surfaces have been performed using the Generalized Retrieval of Aerosol and Surface Properties (GRASP) algorithm. Methods for the validation of satellite data using ground-based observations of aerosol properties are also discussed. We assume that designing, building, and launching into orbit a multi-functional high-precision scanning polarimeter and an imager-polarimeter should make a significant contribution to the study of natural and anthropogenic aerosols and their climatic and ecological effects.

remote sensing↗

ControlShell: A real-time software framework

The ControlShell system is a programming environment that enables the development and implementation of complex real-time software. It includes many building tools for complex systems, such as a graphical finite state machine (FSM) tool to provide strategic control. ControlShell has a component-based design, providing interface definitions and mechanisms for building real-time code modules along with providing basic data management. Some of the system-building tools incorporated in ControlShell are a graphical data flow editor, a component data requirement editor, and a state-machine editor. It also includes a distributed data flow package, an execution configuration manager, a matrix package, and an object database and dynamic binding facility. This paper presents an overview of ControlShell's architecture and examines the functions of several of its tools.

Schneider, Stanley A.↗

Windshear certification data base for forward-look detection systems

Described is an introduction to a comprehensive database that is to be used for certification testing of airborne forward-look windshear detection systems. The database was developed by NASA Langley Research Center, at the request of the Federal Aviation Administration (FAA), to support the industry initiative to certify and produce forward-looking windshear detection equipment. The database contains high-resolution three-dimensional fields for meteorological variables that may be sensed by forward-looking systems. The database is made up of seven case studies that are generated by the Terminal Area Simulation System, a state-of-the-art numerical system for the realistic modeling of windshear phenomena. The selected cases contained in the certification documentation represent a wide spectrum of windshear events. The database will be used with vendor-developed sensor simulation software and vendor-collected ground-clutter data to demonstrate detection performance in a variety of meteorological conditions using NASA/FAA pre-defined path scenarios for each of the certification cases. A brief outline of the contents and sample plots from the database documentation are included. These plots show fields of hazard factor, or F-factor (Bowles 1990), radar reflectivity, and velocity vectors on a horizontal plane overlayed with the applicable certification paths. For the plot of the F-factor field the region of 0.105 and above signify an area of hazardous, performance decreasing windshear, while negative values indicate regions of performance increasing windshear. The values of F-factor are based on 1-Km averaged segments along horizontal flight paths, assuming an air speed of 150 knots (approx. 75 m/s). The database has been released to vendors participating in the certification process. The database and associated document have been transferred to the FAA for archival storage and distribution.

Switzer, George F.↗

An Approach to Data Center-Based KDD of Remote Sensing Datasets

The data explosion in remote sensing is straining the ability of data centers to deliver the data to the user community, yet many large-volume users actually seek a relatively small information component within the data, which they extract at their sites using Knowledge Discovery in Databases (KDD) techniques. To improve the efficiency of this process, the Goddard Earth Sciences Distributed Active Archive Center (GES DAAC) has implemented a KDD subsystem that supports execution of the user's KDD algorithm at the data center, dramatically reducing the volume that is sent to the user. The data are extracted from the archive in a planned, organized "campaign"; the algorithms are executed, and the output products sent to the users over the network. The first campaign, now complete, has resulted in overall reductions in shipped volume from 3.3 TB to 0.4 TB.

Lynnes, Christopher↗

Development of a Generic Creep-Fatigue Life Prediction Model

The objective of this research proposal is to further compile creep-fatigue data of steel alloys and superalloys used in military aircraft engines and/or rocket engines and to develop a statistical multivariate equation. The newly derived model will be a probabilistic fit to all the data compiled from various sources. Attempts will be made to procure the creep-fatigue data from NASA Glenn Research Center and other sources to further develop life prediction models for specific alloy groups. In a previous effort [1-3], a bank of creep-fatigue data has been compiled and tabulated under a range of known test parameters. These test parameters are called independent variables, namely; total strain range, strain rate, hold time, and temperature. The present research attempts to use these variables to develop a multivariate equation, which will be a probabilistic equation fitting a large database. The data predicted by the new model will be analyzed using the normal distribution fits, the closer the predicted lives are with the experimental lives (normal line 1 to 1 fit) the better the prediction. This will be evaluated in terms of a coefficient of correlation, R 2 as well. A multivariate equation developed earlier [3] has the following form, where S, R, T, and H have specific meaning discussed later.

Goswami, Tarun↗

Effects of Angle of Attack and Velocity on Trailing Edge Noise

Trailing edge (TE) noise measurements for a NACA 63-215 airfoil model are presented, providing benchmark experimental data for a cambered airfoil. The effects of flow Mach number and angle of attack of the airfoil model with different TE bluntnesses are shown. Far-field noise spectra and directivity are obtained using a directional microphone array. Standard and diagonal removal beamforming techniques are evaluated employing tailored weighting functions for quantitatively accounting for the distributed line character of TE noise. Diagonal removal processing is used for the primary database as it successfully removes noise contaminates. Some TE noise predictions are reported to help interpret the data, with respect to flow speed, angle of attack, and TE bluntness on spectral shape and peak levels. Important findings include the validation of a TE noise directivity function for different airfoil angles of attack and the demonstration of the importance of the directivity function s convective amplification terms.

Hutcheson, Florence V.↗

Effects of Angle of Attack and Velocity on Trailing Edge Noise

Trailing edge (TE) noise measurements for a NACA 63-215 airfoil model are presented, providing benchmark experimental data for a cambered airfoil. The effects of flow Mach number and angle of attack of the airfoil model with different TE bluntnesses are shown. Far-field noise spectra and directivity are obtained using a directional microphone array. Standard and diagonal removal beamforming techniques are evaluated employing tailored weighting functions for quantitatively accounting for the distributed line character of TE noise. Diagonal removal processing is used for the primary database as it successfully removes noise contaminates. Some TE noise predictions are reported to help interpret the data, with respect to flow speed, angle of attack, and TE bluntness on spectral shape and peak levels. Important findings include the validation of a TE noise directivity function for different airfoil angles of attack and the demonstration of the importance of the directivity function s convective amplification terms.

Hutcheson, Florence V.↗

ARES I Aerodynamic Testing at the NASA Langley Unitary Plan Wind Tunnel

Small-scale force and moment and pressure models based on the outer mold lines of the Ares I design analysis cycle crew launch vehicle were tested in the NASA Langley Research Center Unitary Plan Wind Tunnel from May 2006 to September 2009. The test objectives were to establish supersonic ascent aerodynamic databases and to obtain force and moment, surface pressure, and longitudinal line-load distributions for comparison to computational predictions. Test data were obtained at low through high supersonic Mach numbers for ranges of the Reynolds number, angle of attack, and roll angle. This paper focuses on (1) the sensitivity of the supersonic aerodynamic characteristics to selected protuberances, outer mold line changes, and wind tunnel boundary layer transition techniques, (2) comparisons of experimental data to computational predictions, and (3) data reproducibility. The experimental data obtained in the Unitary Plan Wind Tunnel captured the effects of evolutionary changes to the Ares I crew launch vehicle, exhibited good agreement with predictions, and displayed satisfactory within-test and tunnel-to-tunnel data reproducibility.

Erickson, Gary E.↗