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At least 199 records · Page 11

5.5 Hybrid Additively Manufactured Tooling for Large Composite Structures

This program sought to lower tooling costs for tightly toleranced, large composite aerostructures by using large format additive manufacturing (LFAM). LFAM tooling has the potential to lower tooling costs, but the large and unpredictable thermal expansion of polymer-based tooling did not previously allow for the production of large structures made with high performance resins requiring 350 deg F cures. Geometric tolerance issues and demolding concerns continued to arise. The proposed approach for this work was to print a thin shell via additive manufacturing that would be used as a scaffold for a conventional tooling prepreg. The prepreg would be laminated and cured onto both sides of the shell, creating a tool with a sandwich construction. The cured prepreg would then be machined and serve as the tooling surface. In addition to providing a clean, pit-free surface, the prepreg stiffness and low thermal expansion would physically restrain the AM shell and prevent it from expanding during thermal cycling, thus preventing any geometric mismatch between the nominal and actual part dimensions. A stair-step approach was taken to prove out this approach. Coupon level testing was done to ensure a good bond was possible between the AM shell and the tooling prepreg. These samples were also used to determine the amount of prepreg needed to achieve the desired coefficient of thermal expansion (CTE). The project team than progressed to a subscale tool measuring two feet in length. The geometry used was a C-section spar from Airbus’ Wing of Tomorrow program. A hybrid tool, using the AM shell and tooling prepreg, was made, and a composite part was fabricated with the tool. Dimensional scans of the tool and part indicated that the desired CTE was achieved with the tool. A second tooling approach was taken to reduce risk. In this approach, a subscale tool of the same geometry as previously mentioned was created, but the tool was entirely constructed of the tooling prepreg. AM was used to print a “master” that was then used to fabricate the tool by creating a splash off of the master using tooling prepreg. The innovation was that the master was not machined, but rather, was used right off the printer after a small amount of surface preparation. Removing this machining step was thought to make this approach economically viable when compared to conventional master fabrication. As before, the tool was used to fabricate a part and tested for dimensional accuracy. For the conclusion of the program, a ten-foot tool was created using the hybrid tool approach. The geometry also came from the Airbus C-section spar previously mentioned. Some difficulty arose in accurately predicting the geometry of the tool after laminating the tooling prepreg, resulting in multiple material additions and a tool geometry change. A part was fabricated in an autoclave using the tool. A cost savings of almost 50% was achieved on the subscale tool in both the hybrid and master-style tool, but the savings dropped considerably as the tool length increased in the full-scale tool. Given the less than expected cost savings and added complexity of the tool construction, it is suggested that the master-style tool is best for immediate commercialization for larger tools (although it may be appropriate for smaller, table-top size tools). Since its construction is very similar to conventionally made composite tooling, there already exists a broad supply chain for building the tool, and project partner Additive Engineering Solutions (AES) is well acquainted with the design and build of the AM master needed.

36 MATERIALS SCIENCE↗

CTT: Tools for Fine Alignment of Flash X-ray Systems

The CTT code is a compilation of the python code that was prototyped in the interest of making a method for fine alignment of Flash X-ray Systems at the lab. It contains python code for interacting with LTT and running optimization loops using LTT simulations or ray tracing with RaySpace. It Includes classes for creating phantom objects and modular geometry objects in the simulations. It allows one to generate a random phantom that has been optimized to stay with a certain cubic volume and maximize the minimum distance between any two ball phantom centers in all projections. It was primarily made to work on ball phantoms which were simulated as Teflon spheres. It allows the alignment from some nominal position to some displaced position to simulate recovery of a real geometry from the nominal geometry of the system. Fine alignment can be done with the projections in LTT while a rough alignment is faster using rayspace. If rayspace works well for the use case it should be preferred since it is considerably faster and less resource intensive in general. Some Notes about using this code is that one will have to edit the imports for the files to get to the correct LTT path, LTT GUI path and to the CTT path when importing. This was made by using anaconda with python 3 on windows 10. The LTT code was stored in a anaconda environment which seemed to help the python find the LTT although it shouldn’t be necessary if you append the path in your code using sys.path.append(r'add the path you want here’). The sys.path.append method is a quick way to give your python code access to a given folder when running it (such as LTT or CTT). Note that since the code was originally made with paths for a specific machine that a new user will have to edit those paths to make it work in all the files that use the old paths. This is somewhat tedious but will be required to run the code on a new machine. The methodology used to run and make these notebooks starts with opening a command prompt as administrator in windows 10. After this one would activate the anaconda environment in the appropriate directory using “conda activate myenv” in the command prompt. After this open a jupyter-notebook using the keyword “jupyter-notebook". After that a notebook should open in a browser. Note that some libraries used may not come with anaconda so those may have to be installed. There shouldn’t be much though since the main things used are scipy, numpy, matplotlib, and LTT. LTTQuicksetup also requires some paths that will be specific to the computer so that should be changed as well.

42 ENGINEERING↗

Neutron computed tomography of B12W and M8N socket sections of the Arecibo telescope

From neutron user principal investigator: We kindly request the public release of three neutron imaging datasets through ONCat. All datasets were collected from two forensic specimens, B12W and M8N, sectioned from zinc-filled steel-wire sockets recovered from the collapsed Arecibo Telescope. The dataset titled “Neutron radiographs of B12W and M8N socket sections of the Arecibo telescope” contains normalized two-dimensional (2D) neutron radiographs of the specimens, showing the geometry and spatial distribution of the steel wires embedded within the zinc matrix, as well as internal features such as voids and cracks. The dataset titled “Neutron computed tomography of B12W and M8N socket sections of the Arecibo telescope” contains normalized 2D neutron projection images acquired over a range of specimen rotation angles for one selected region of each specimen. These projection images were used to reconstruct three-dimensional (3D) tomographic volumes that reveal the embedded-wire geometry and internal defects. The dataset titled “Bragg edge imaging (BEI) of B12W and M8N socket sections of the Arecibo telescope” contains six time-of-flight (TOF) neutron imaging datasets, three from each specimen, acquired at regions of interest selected based on the radiographs. The spatially resolved 2D TOF images show the zinc matrix and embedded steel wires, and the wavelength-dependent neutron transmission data were used to characterize crystallographic texture within the zinc. All components and their condition are in the public domain as they are the property of the National Science Foundation (NSF). The neutron imaging data, part geometries, and detailed forensic information have been widely published in the Arecibo Telescope Collapse Forensic Report by Thornton Tomasetti Engineers and others (NASA report and NASEM report).

Bilheux, Hassina↗

Feature Based Qualification of 17-4PH Stainless Steel to Evaluate Location-Specific Variability in Wire Arc Additive Manufacturing

Qualifying large-scale metal additive manufacturing (M-AM) technologies such as wire arc additive manufacturing (WAAM) can be challenging. This is especially significant in precipitation hardened martensitic stainless steels like SS 17-4PH, where thermal histories induce location-specific microstructural variability and property anisotropy. The Department of Defense (DOD) and the United States Army Combat Capabilities Development Command Ground Vehicle Systems Center (GVSC) Ground Vehicle Materials Engineering (GVME) aim to build robust and qualified large-scale M-AM workflows that could reduce the time and cost through quick and informed evaluation, testing, and development of feedstock, processes, and parts. The report presents the findings from the collaborative efforts between Oak Ridge National Laboratory (ORNL) and the U.S. Army GVSC GVME. The aim of this project was to develop a geometric feature-based qualification framework for WAAM of SS 17-4PH components. This report outlines selection methodology of representative build geometries, optimization of WAAM process parameters, in-situ monitoring, microstructure-property evaluation, thermal simulations, as well as data visualization techniques incorporated in this project. The results from this project demonstrate a clear understanding of thermal history dependent phase evolution and consequent location-specific property variations in WAAM of SS 17-4PH. These results in conjunction with the data-driven methodologies used in this project are expected to reduce qualification timelines, improve predictability, and accelerate the development of reliable feature-based qualification strategies for part production via large-scale M-AM technologies.

36 MATERIALS SCIENCE↗

Overview of Proposed Summer Project [Slides]

The following are visual aids developed to assist in initial discussion with an intern about their proposed Summer 2021 project. Project Goal: Develop a user friendly code to assess pressure load on a geometry of interest to understand the rigid body effects, and compare to more complex approaches

97 MATHEMATICS AND COMPUTING↗

Simulating Dopant Diffusion in a Detalied Porous Structure

The National Ignition Facility (NIF) at Lawrence Livermore National Laboratory (LLNL) uses highpowered lasers to explore cutting edge fusion technology. Specialized optics, known as fiber lasers, generate and amplify the NIF’s high-powered lasers that facilitate fusion experiments. Fiber lasers are glass fibers that are doped with heavy elements to stimulate light emission. The fiber laser doping processing uses a predeposited layer of porous silica nanoparticles on the surface of the glass to absorb a dopant rich salt solution. Our project simulated the diffusion of dopants into porous silica with prescribed colloidal stacking geometries using COMSOL Multiphysics. With these simulations we identified the porosity and tortuosity of the porous nanoparticle structures as the primary parameters that will dictate how the dopants will diffusive into the porous surface of the glass. Porosity and Tortuosity are geometric parameters that can easily be input into more detailed simulations to estimate an effective diffusivity within the porous layer. Future work should account for surface reactions to accurately simulate the full-scale diffusion of the solution doping process.

36 MATERIALS SCIENCE↗

Capillary-driven Condensation for Heat Transfer Enhancement in Steam Power Plants

A novel condensation method was developed to enhance the efficiency of steam power plants. The method consists of a hierarchical structure on the surface of condenser tubes consisting of a wicking structure overlaid with a porous hydrophobic membrane. The project is divided into two main efforts. First, a structure with highly defined geometry was made utilizing microfabrication on silicon. This structure helped us elucidate the physics of steam condensation at the surface, more easily model the heat transfer coefficient, and demonstrate the first proof-of-concept for our proposed condensation approach. The structures on silicon achieved a heat transfer coefficient ~ 240% higher than the theoretical filmwise value at the same operating condition. Second, the method is explored for scalability by fabricating surfaces with industrial class materials followed by heat transfer measurements on flat surfaces. An HTC enhancement of 50% over the experimental filmwise value was achieved on hierarchical copper surfaces made of commercially available copper foams and meshes, showing the potential of achieving heat transfer enhancement using low cost and scalable materials. Scalable fabrication of the hydrophobic membrane and the porous copper wick were also investigated. Our heat and mass transfer model predicts a > 5x heat transfer enhancement on a scalable version of capillary driven condenser made of membrane covered porous copper. The results demonstrate the promise of capillary-driven condensation surfaces for heat transfer applications.

20 FOSSIL-FUELED POWER PLANTS↗

Validation of Robustness in TCR Design Strategies

Advanced Manufacturing (AM) has been identified as one of key pathways in accelerating deployment of next generation of nuclear power plants. The nuclear industry continues to predominately rely on fabrication techniques and materials that their codes and standards were approved decades ago. This legacy approach in the highly regulated nuclear sector has brought uncertainty to commercial viability of nuclear energy in the U.S. The Transformational Challenge Reactor (TCR), sought to utilize AM methods for fabrication of a High Temperature Gas Reactor (HTGR). The program sought to leverage AM to construct a non-conventional geometry made up of TRISO fuel and hydride moderator in SiC matrix. This project objective was to provide independent analysis of the core configuration with separate set of tools and approaches than utilized by the TCR team in a 2-year time frame. This NEUP project also tackled a question: Can AM be a cost-effective tool in changing the commercial paradigm of nuclear. We investigated this question in-context of a micro HTGR. While respecting the inherit uncertainty in our input parameters, overall we saw that AM has potential in improving economic competitiveness of nuclear energy when employed on multiple fronts (i.e. few percentage cost improvement per component). Given realizing AM requires mass orders and large R&D investment for commercial utilization, Department of Energy should focus its support on few key enabling technologies and support their codes and standards development and qualification cost (including irradiation as applicable) rather than taking a diluted approach.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Ducted Fuel Injection and Cooled Spray Technologies for Particulate Control in Heavy-Duty Diesel Engines

Heavy-duty diesel engine manufacturers are continually seeking simple, low-cost technologies to reduce emissions. Ducted Fuel Injection (DFI) and Cooled Spray (CS) technologies have emerged as promising solutions, offering the potential for nearly sootless operation. These innovations could significantly decrease harmful Particulate Matter (PM) emissions while enabling further optimization to reduce Nitrogen Oxides (NOx) and increase efficiency. While initial experiments across various engine types have shown promise, uncertainties remain regarding the ideal geometry, scaling properties, and effectiveness of these technologies across different operating conditions. This project aims to address these knowledge gaps through experiments in both optical and metal single-cylinder engines.

33 ADVANCED PROPULSION SYSTEMS↗

Analytical evaluation of the impact of broad specification fuels on high bypass turbofan engine combustors

The impact of the use of broad specification fuels on the design, performance durability, emissions and operational characteristics of combustors for commercial aircraft gas turbine engines was assessed. Single stage, vorbix and lean premixed prevaporized combustors, in the JT9D and an advanced energy efficient engine cycle were evaluated when operating on Jet A and ERBS (Experimental Referee Broad Specification) fuels. Design modifications, based on criteria evolved from a literature survey, were introduced and their effectiveness at offsetting projected deficiencies resulting from the use of ERBS was estimated. The results indicate that the use of a broad specification fuel such as ERBS, will necessitate significant technology improvements and redesign if deteriorated performance, durability and emissions are to be avoided. Higher radiant heat loads are projected to seriously compromise liner life while the reduced thermal stability of ERBS will require revisions to the engine-airframe fuel system to reduce the thermal stress on the fuel. Smoke and emissions output are projected to increase with the use of broad specification fuels. While the basic geometry of the single stage and vorbix combustors are compatible with the use of ERBS, extensive redesign of the front end of the lean premixed prevaporized burner will be required to achieve satisfactory operation and optimum emissions.

Lohmann, R. P.↗

Relativistic precessing jets in quasars and radio galaxies - Models to fit high resolution data

The formulation of generalized models tracing the geometry and intensity of the synchrotron emission from precessing, twin, relativistic jets as projected on the plane of the sky is presented. It is shown that neither the shape of the image nor its relative intensities are altered by including the effects of a cosmological redshift and a relative velocity between the source and observer. The models are fitted to the available data for several quasars and radio galaxies and demonstrate the plausibility of the phenomenon. Probable selection effects are considered and diagnostics given for recognizing objects showing this behavior. In the radio galaxies considered, velocities up to about 0.2c and precession periods of 1,000,000 yr are deduced. In the QSOs investigated, velocities of 0.7c and greater are found and periods of order 10,000 yr. In some cases precession cone angles increase with time. Consequences in terms of lifetimes of QSO behavior and binary supermassive objects are discussed.

Gower, A. C.↗

The Development of a Nonequilibrium Radiative Heat Transfer Computational Model for High Altitude Entry Vehicle Flowfield Methods

This final report will attempt to concisely summarize the activities and accomplishments associated with NASA Grant and to include pertinent documents in an appendix. The project initially had one primary and several secondary objectives. The original primary objective was to couple into the NASA Johnson Space Center (JSC) nonequilibrium chemistry Euler equation entry vehicle flowfield code, INEQ3D, the Texas A&M University (TAMU) local thermodynamic nonequilibrium (LTNE) radiation model. This model had previously been developed and verified under NASA Langley and NASA Johnson sponsorship as part of a viscous shock layer entry vehicle flowfield code. The secondary objectives were: (1) to investigate the necessity of including the radiative flux term in the vibrational-electron-electronic (VEE) energy equation as well as in the global energy equation, (2) to determine the importance of including the small net change in electronic energy between products and reactants which occurs during a chemical reaction, and (3) to study the effect of atom-atom impact ionization reactions on entry vehicle nonequilibrium flowfield chemistry and radiation. For each, of these objectives, it was assumed that the code would be applicable to lunar return entry conditions, i.e. altitude above 75 km, velocity greater, than 11 km/sec, where nonequilibrium chemistry and radiative heating phenomena would be significant. In addition, it was tacitly assumed that as part of the project the code would be applied to a variety of flight conditions and geometries.

Carlson, Leland A.↗

NASA Tech Briefs, August 2011

Topics covered include: Miniature, Variable-Speed Control Moment Gyroscope; NBL Pistol Grip Tool for Underwater Training of Astronauts; HEXPANDO Expanding Head for Fastener-Retention Hexagonal Wrench; Diagonal-Axes Stage for Pointing an Optical Communications Transceiver; Improvements in Speed and Functionality of a 670-GHz Imaging Radar; IONAC-Lite; Large Ka-Band Slot Array for Digital Beam-Forming Applications; Development of a 150-GHz MMIC Module Prototype for Large-Scale CMB Radiation; Coupling Between Waveguide-Fed Slot Arrays; PCB-Based Break-Out Box; Multiple-Beam Detection of Fast Transient Radio Sources; Router Agent Technology for Policy-Based Network Management; Remote Asynchronous Message Service Gateway; Automatic Tie Pointer for In-Situ Pointing Correction; Jitter Correction; MSLICE Sequencing; EOS MLS Level 2 Data Processing Software Version 3; DspaceOgre 3D Graphics Visualization Tool; Metallization for Yb14MnSb11-Based Thermoelectric Materials; Solvent/Non-Solvent Sintering To Make Microsphere Scaffolds; Enhanced Fuel-Optimal Trajectory-Generation Algorithm for Planetary Pinpoint Landing; Self-Cleaning Coatings and Materials for Decontaminating Field-Deployable Land and Water-Based Optical Systems; Separation of Single-Walled Carbon Nanotubes with DEP-FFF; Li Anode Technology for Improved Performance; Post-Fragmentation Whole Genome Amplification-Based Method; Microwave Tissue Soldering for Immediate Wound Closure; Principles, Techniques, and Applications of Tissue Microfluidics; Robotic Scaffolds for Tissue Engineering and Organ Growth; Stress-Driven Selection of Novel Phenotypes; Method for Accurately Calibrating a Spectrometer Using Broadband Light; Catalytic Microtube Rocket Igniter; Stage Cylindrical Immersive Display; Vacuum Camera Cooler; Atomic Oxygen Fluence Monitor; Thermal Management Tools for Propulsion System Trade Studies and Analysis; Introduction to Physical Intelligence; Technique for Solving Electrically Small to Large Structures for Broadband Applications; Accelerated Adaptive MGS Phase Retrieval; Large Eddy Simulation Study for Fluid Disintegration and Mixing; Tropospheric Correction for InSAR Using Interpolated ECMWF Data and GPS Zenith Total Delay; Technique for Calculating Solution Derivatives With Respect to Geometry Parameters in a CFD Code; Acute Radiation Risk and BRYNTRN Organ Dose Projection Graphical User Interface; Probabilistic Path Planning of Montgolfier Balloons in Strong, Uncertain Wind Fields; Flight Simulation of ARES in the Mars Environment; Low-Outgassing Photogrammetry Targets for Use in Outer Space; Planning the FUSE Mission Using the SOVA Algorithm; Monitoring Spacecraft Telemetry Via Optical or RF Link; and Robust Thermal Control of Propulsion Lines for Space Missions.

Source record↗

Background Oriented Schlieren Implementation in a Jet-Surface Interaction Test

Many current and future aircraft designs rely on the wing or other aircraft surfaces to shield the engine noise from observers on the ground. However the available data regarding how a planar surface interacts with a jet to shield and/or enhance the jet noise are currently limited. Therefore, the Jet-Surface Interaction Tests supported by NASA's Fundamental Aeronautics Program's Fixed Wing Project were undertaken to supply experimental data covering a wide range of surface geometries and positions interacting with high-speed jet flows in order to support the development of noise prediction methods. Phase 1 of the Test was conducted in the Aero-Acoustic Propulsion Laboratory at NASA Glenn Research Center and consisted of validating noise prediction schemes for a round nozzle interacting with a planar surface. Phased array data and far-field acoustic data were collected for both the shielded and reflected sides of the surface. Phase 1 results showed that the broadband shock noise was greatly reduced by the surface when the jet was operated at the over-expanded condition, however, it was unclear whether this reduction was due a change in the shock cell structure by the surface. In the present study, Background Oriented Schlieren is implemented in Phase 2 of the Jet-Surface Interaction Tests to investigate whether the planar surface interacts with the high-speed jet ow to change the shock cell structure. Background Oriented Schlieren data are acquired for under-expanded, ideally-expanded, and over-expanded ow regimes for multiple axial and radial positions of the surface at three different plate lengths. These data are analyzed with far-field noise measurements to relate the shock cell structure to the broadband shock noise produced by a jet near a surface.

Clem, Michelle M.↗

Additive Manufacturing Infrared Inspection

The Additive Manufacturing Infrared Inspection Task started the development of a real-time dimensional inspection technique and digital quality record for the additive manufacturing process using infrared camera imaging and processing techniques. This project will benefit additive manufacturing by providing real-time inspection of internal geometry that is not currently possible and reduce the time and cost of additive manufactured parts with automated real-time dimensional inspections which deletes post-production inspections.

Gaddy, Darrell↗

Mars Sample Return, Sample Retrieval Lander, Reaction Control System Jet Interaction Supersonic Wind Tunnel Test Overview with CFD Predictions

NASA's Mars Sample Return campaign will be launching several missions over the next decade that will work together to return rock samples from Mars back to Earth. The Sample Retrieval Lander (SRL) will deliver the Mars Ascent Vehicle and fetch rover to the surface of Mars in 2006. Rock samples collected by the Mars 2020 Perseverance rover, landing in early 2021, will be loaded on the the ascent vehicle to be launched into Mars orbit for retrieval by yet another spacecraft. The Sample Retrieval Lander will be a blunt entry capsule similar to past Mars entry vehicles like Mars Science Laboratory and Viking. The vehicle will fly a guided entry, using a small lift vector produced by a non-zero trim angle of attack to eliminate downrange and crossrange position errors at the point of parachute deploy. This energy and heading management is achieved with a reaction control system (RCS) that directs the bank angle of the vehicle and also minimizes unwanted capsule dynamics. The reaction control system and control design is based on the Mars Science Laboratory and Mars 2020 RCS systems. However, due to packaging constraints, the backshell of this new entry vehicle has a different geometry than those earlier designs. To certify the RCS system for flight the project must characterize the jet plume interactions with the capsule backshell that could impair or significantly augment the RCS control authority. This characterization will be done through a combination of computational fluid dynamics (CFD) analysis and wind tunnel test. Two candidate arrangements of the RCS jets have been identified for the SRL vehicle and are currently under evaluation before final selection. The aero/RCS plume interactions of these candidate configurations have been measured in a supersonic wind tunnel test in NASA Langley's Unitary Plan Wind Tunnel. The test was conducted in the fall of 2020 and data is currently being reduced. An overview of the candidate RCS configurations are presented here with an overview of the wind tunnel model design, jet scaling and scaled nozzle design, and the test matrix. Preliminary CFD runs are presented with an assessment of the predicted plumes and their interaction with the wake flow of the vehicle. The predicted effects of the model sting is provided as well. This high fidelity wind tunnel test is being conducted much earlier in the SRL project than would normally be done. The test was funded as part of a CFD evaluation task funded by NASA's Aerosciences Evaluation and Test Capabilities Project. The objective of the evaluation task was to compare the ability of CFD to predict complex flows with data that can be measured in the Langley Unitary Plan Wind Tunnel. RCS Jet interactions were selected as a type of complex flow that is important to NASA missions. In addition to providing useful data to the SRL project, there was added emphasis on quantifying the accuracy of the CFD predictions and wind tunnel test data. An overview of the uncertainty quantification methodologies for computational and experimental portions of this test is presented.

blunt body↗

A computer program for fitting smooth surfaces to an aircraft configuration and other three dimensional geometries

A computer program that uses a three-dimensional geometric technique for fitting a smooth surface to the component parts of an aircraft configuration is presented. The resulting surface equations are useful in performing various kinds of calculations in which a three-dimensional mathematical description is necessary. Programs options may be used to compute information for three-view and orthographic projections of the configuration as well as cross-section plots at any orientation through the configuration. The aircraft geometry input section of the program may be easily replaced with a surface point description in a different form so that the program could be of use for any three-dimensional surface equations.

Craidon, C. B.↗

Geometric Interpretation of the Cluster Location Problem Part I: Theory

We present a new framing of the seismic location problem using principles drawn from differential geometry. Our interpretation relies upon the common assumption that travel times observed across a network are continuous, differentiable functions of source location. In consequence, travel‐time functions constitute a differentiable map between the source region and a Riemannian manifold. The manifold is said to be the image of the source region embedded in a generally high‐dimension travel‐time vector space. A cluster of events in the source region has an image of discrete points on the manifold, that, except in the simplest cases, cannot be viewed directly. However, it is possible to project the image of a cluster into a tangent space of the manifold for direct visualization. The projection operator can be computed directly from the data without a velocity model, but produces a distorted rendering of the cluster geometry. With a model we can predict the distortions and correct them to estimate cluster geometry. We develop these points with the simplest possible example, one for which direct visualization of the manifold is possible, using the example as an introduction to the relevant concepts from differential geometry in a familiar setting. The tangent space, a local linearization of the manifold, plays a key role. We develop a metric to estimate the limits of linearization, that is, to determine when the curvature of the manifold invalidates the linear assumption. We also examine the interplay of model error, inadequate network geometry, and pick error. We then generalize our results from the simple case to the general case of 3D source regions observed by general networks. Although we do suggest a new “project and correct” method for location, we do not develop it into a practical algorithm. In conclusion, our intention rather is to highlight new analytical methods grounded in differential geometry.

East Pacific Ocean Islands↗