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At least 937 records · Page 52
All sourcesNASA NTRSDOE OSTI
NASA NTRS2011

Flood Inundation Mapping in the Logone Floodplain from Multi Temporal Landsat ETM+Imagery

Yearly flooding in the Logone floodplain makes an impact on agricultural, pastoral, and fishery systems in the Lake Chad Basin. Since the flooding extent and depth are highly variable, flood inundation mapping helps us make better use of water resources and prevent flood hazards in the Logone floodplain. The flood maps are generated from 33 multi temporal Landsat Enhanced Thematic Mapper Plus (ETM+) during three years 2006 to 2008. Flooded area is classified using a short-wave infrared band whereas open water is classified by Iterative Self-organizing Data Analysis (ISODATA) clustering. The maximum flooding extent in the study area increases up to approximately 5.8K km2 in late October 2008. The study also provides strong correlation of the flooding extents with water height variations in both the floodplain and the river based on a second polynomial regression model. The water heights are from ENIVSAT altimetry in the floodplain and gauge measurements in the river. Coefficients of determination between flooding extents and water height variations are greater than 0.91 with 4 to 36 days in phase lag. Floodwater drains back to the river and to the northeast during the recession period in December and January. The study supports understanding of the Logone floodplain dynamics in detail of spatial pattern and size of the flooding extent and assists the flood monitoring and prediction systems in the catchment.

Jung, Hahn Chul↗
NASA NTRS2012

Global Assessment of Land Surface Temperature From Geostationary Satellites and Model Estimates

Land surface (or 'skin') temperature (LST) lies at the heart of the surface energy balance and is a key variable in weather and climate models. In this research we compare two global and independent data sets: (i) LST retrievals from five geostationary satellites generated at the NASA Langley Research Center (LaRC) and (ii) LST estimates from the quasi-operational NASA GEOS-5 global modeling and assimilation system. The objective is to thoroughly understand both data sets and their systematic differences in preparation for the assimilation of the LaRC LST retrievals into GEOS-5. As expected, mean differences (MD) and root-mean-square differences (RMSD) between modeled and retrieved LST vary tremendously by region and time of day. Typical (absolute) MD values range from 1-3 K in Northern Hemisphere mid-latitude regions to near 10 K in regions where modeled clouds are unrealistic, for example in north-eastern Argentina, Uruguay, Paraguay, and southern Brazil. Typically, model estimates of LST are higher than satellite retrievals during the night and lower during the day. RMSD values range from 1-3 K during the night to 2-5 K during the day, but are larger over the 50-120 W longitude band where the LST retrievals are derived from the FY2E platform

Reichle, Rolf H.↗
NASA NTRS2012

From EXOSAT to the High Energy Astrophysics Science Archive (HEASARC): X-ray Astronomy Comes of Age

In May 1983 the European Space Agency launched EXOSAT, its first X-ray astronomy observatory. Even though it lasted only 3 short years, this mission brought not only new capabilities that resulted in unexpected discoveries, but also a pioneering approach to operations and archiving that changed X-ray astronomy from observations led by small instrument teams, to an observatory approach open to the entire community through a guest observer program. The community use of the observatory was supported by a small dedicated team of scientists, the precursor to the data center activities created to support e.g. Chandra and XMM-Newton. The new science capabilities of EX OS AT included a 90 hr highly eccentric high earth orbit that allow unprecedented continuous coverage of sources as well as direct communication with the satellite that allowed real time decisions to respond to unexpected events through targets of opportunity. The advantages of this orbit demonstrated by EXOSAT resulted in Chandra and XMM-Newton selecting similar orbits. The three instruments on board the EXOSAT observatory were complementary, designed to give complete coverage over a wide energy band pass of 0.05-50 keY. An onboard processor could be programmed to give multiple data modes that could be optimized in response to science discoveries: These new capabilities resulted in many new discoveries including the first comprehensive study of AGN variability, new orbital periods in X-ray binaries and cataclysmic variables, new black holes, quasi-periodic oscillations from neutron stars and black holes and broad band X-ray spectroscopy. The EXOSAT team generated a well-organized database accessible worldwide over the nascent internet, allowing remote selection of data products, making samples and undertaking surveys from the data. The HEASARC was established by NASA at Goddard Space Flight Center in 1990 as the repository of NASA X-ray and Gamma-ray data. The proven EXOSAT database system became the core of the HEASARC infrastructure. The HEASARC pioneered many concepts now taken for granted including standardized formats using FITS files, restoring data from earlier missions, multi-mission analysis tools and a searchable archive over the world wide web.

White, Nicholas E.↗
NASA NTRS2015

Initial Operation of the Nuclear Thermal Rocket Element Environmental Simulator

The Nuclear Thermal Rocket Element Environmental Simulator (NTREES) facility is designed to perform realistic non-nuclear testing of nuclear thermal rocket (NTR) fuel elements and fuel materials. Although the NTREES facility cannot mimic the neutron and gamma environment of an operating NTR, it can simulate the thermal hydraulic environment within an NTR fuel element to provide critical information on material performance and compatibility. The NTREES facility has recently been upgraded such that the power capabilities of the facility have been increased significantly. At its present 1.2 MW power level, more prototypical fuel element temperatures nay now be reached. The new 1.2 MW induction heater consists of three physical units consisting of a transformer, rectifier, and inverter. This multiunit arrangement facilitated increasing the flexibility of the induction heater by more easily allowing variable frequency operation. Frequency ranges between 20 and 60 kHz can accommodated in the new induction heater allowing more representative power distributions to be generated within the test elements. The water cooling system was also upgraded to so as to be capable of removing 100% of the heat generated during testing In this new higher power configuration, NTREES will be capable of testing fuel elements and fuel materials at near-prototypic power densities. As checkout testing progressed and as higher power levels were achieved, several design deficiencies were discovered and fixed. Most of these design deficiencies were related to stray RF energy causing various components to encounter unexpected heating. Copper shielding around these components largely eliminated these problems. Other problems encountered involved unexpected movement in the coil due to electromagnetic forces and electrical arcing between the coil and a dummy test article. The coil movement and arcing which were encountered during the checkout testing effectively destroyed the induction coil in use at the time and resulted in NTREES being out of commission for a couple of months while a new stronger coil was procured. The new coil includes several additional pieces of support structure to prevent coil movement in the future. In addition, new insulating test article support components have been fabricated to prevent unexpected arcing to the test articles. Additional activities are also now underway to address ways in which the radial temperature profiles across test articles may be controlled such that they are more prototypical of what they would encounter in an operating nuclear engine. The causes of the temperature distribution problem are twofold. First, the fuel element test article is isolated in NTREES as opposed to being in the midst of many other mostly identical fuel elements in a nuclear engine. As a result, the fuel element heat flux boundary conditions in NTREES are far from adiabatic as would normally be the case in a reactor. Second, induction heating skews the power distribution such that power is preferentially deposited near the outside of the fuel element. Nuclear heating, conversely, deposits its power much more uniformly throughout the fuel element. Current studies are now looking at various schemes to adjust the amount of thermal radiation emitted from the fuel element surface so as to essentially vary the thermal boundary conditions on the test article. It is hoped that by properly adjusting the thermal boundary conditions on the fuel element test article, it may be possible to substantially correct for the inappropriate radial power distributions resulting from the induction heating so as to yield a more nearly correct temperature distribution throughout the fuel element.

Emrich, William J., Jr.↗
NASA NTRS2021

Expanding the application of soil moisture monitoring systems through regression-based transformation

Relative to other geophysical variables, soil moisture (SM) estimates derived from land surface models (LSMs) and land data assimilation systems (LDAS) are difficult to transfer between platforms and applications. This difficulty stems from the highly model-dependent nature of LSM SM estimates and differences in the vertical support of discretized SM values. As a result, operational SM estimates generated by one LSM (or LDAS) cannot generally be directly applied to a hydrologic monitoring or forecast system designed around a second LSM. This lack of transferability is particularly problematic for LDAS applications, where the time, expertise, and computational resources required to generate an operational LDAS analysis cannot be practically duplicated for every LSM-specific application. Here, we develop a set of simple regression tools for translating SM estimates between LSMs and multiple LDAS analyses. Results demonstrate that simple multivariate linear regression - utilizing independent variables based on multi-layer and temporally lagged SM estimates - can significantly improve upon baseline transformation approaches using direct percentile matching. The proposed regression approaches are effective for both the LSM-to-LSM and LDAS-to-LDAS transformation of multi-layer SM percentiles. Application of this approach will expand the utility of existing, high-quality (but LSM-specific) operational sources of SM information like the NASA Soil Moisture Active Passive Level-4 Soil Moisture product.

Soil Moisture↗
NASA NTRS2022

Mapping Firn Saturation over Greenland using NASA’s Soil Moisture Active Passive Satellite

Mapping the spatial extent of recently identified englacial hydrological features (i.e., ice slabs and perennial firn aquifers) formed by meters-thick water-saturated firn layers over the percolation facies of the Greenland Ice Sheet using L-band microwave radiometry has recently been demonstrated. However, these initial maps are binary, and do not provide a parameter to estimate the spatial variability in the thickness and volumetric fraction of meltwater stored within the firn pore space. Here, we exploit enhanced-resolution vertical-polarization L-band brightness temperature (T^(B)_(V)) imagery (2015–2019) generated using observations collected over Greenland by NASA’s Soil Moisture Active Passive (SMAP) satellite and a simple two-layer L-band brightness temperature model. We map water-saturated firn layers via a “firn saturation” parameter, and interpret our results together with ice slab and perennial firn aquifer spatial extents, estimates of snow accumulation simulated via the Regional Atmospheric Climate Model (RACMOp2.3), and airborne radar surveys collected via NASA’s Operation IceBridge (OIB) campaigns. We find that variable firn saturation parameter values are mapped in lower snow accumulation ice slab areas in western, northern, and northeastern Greenland, where firn is colder and water-saturated firn layers seasonally refreeze as solid-ice. Higher firn saturation parameter values are mapped in higher snow accumulation perennial firn aquifer areas in southeastern, southern, and northwestern Greenland, where firn is near the melting point, and meters-thick water-saturated firn layers exist. Our results have implications for identifying expansive englacial reservoirs that store significant volumes of meltwater in locations that are vulnerable to meltwater-induced hydrofracturing and accelerated outlet glacier flow year-round.

Greenland ice sheet↗
NASA NTRS2000

IRIS Product Recommendations

This report presents the Applied Meteorology Unit's (AMU) evaluation of SIGMET Inc.'s Integrated Radar Information System (IRIS) Product Generator and recommendations for products emphasizing lightning and microburst tools. The IRIS Product Generator processes radar reflectivity data from the Weather Surveillance Radar, model 74C (WSR-74C), located on Patrick Air Force Base. The IRIS System was upgraded from version 6.12 to version 7.05 in late December 1999. A statistical analysis of atmospheric temperature variability over the Cape Canaveral Air Force Station (CCAFS) Weather Station provided guidance for the configuration of radar products that provide information on the mixed-phase (liquid and ice) region of clouds, between 0 C and -20 C. Mixed-phase processes at these temperatures are physically linked to electrification and the genesis of severe weather within convectively generated clouds. Day-to-day variations in the atmospheric temperature profile are of sufficient magnitude to warrant periodic reconfiguration of radar products intended for the interpretation of lightning and microburst potential of convectively generated clouds. The AMU also examined the radar volume-scan strategy to determine the scales of vertical gaps within the altitude range of the 0 C to -20 C isotherms over the Kennedy Space Center (KSC)/CCAFS area. This report present's two objective strategies for designing volume scans and proposes a modified scan strategy that reduces the average vertical gap by 37% as a means for improving radar observations of cloud characteristics in the critical 0 C to -20 C layer. The AMU recommends a total of 18 products, including 11 products that require use of the IRIS programming language and the IRIS User Product Insert feature. Included is a cell trends product and display, modeled after the WSR-88D cell trends display in use by the National Weather Service.

Short, David A.↗
NASA NTRS2021

Assessing within-Field Corn and Soybean Yield Variability from WorldView-3, Planet, Sentinel-2, and Landsat 8 Satellite Imagery

Crop yield monitoring is an important component in agricultural assessment. Multispectral remote sensing instruments onboard space-borne platforms such as Advanced Very High Resolution Radiometer (AVHRR), Moderate Resolution Imaging Spectroradiometer (MODIS), and Visible Infrared Imaging Radiometer Suite (VIIRS) have shown to be useful for efficiently generating timely and synoptic information on the yield status of crops across regional levels. However, the coarse spatial resolution data inherent to these sensors provides little utility at the management level. Recent satellite imagery collection advances toward finer spatial resolution (down to 1 m) alongside increased observational cadence (near daily) implies information on crops obtainable at field and within-field scales to support farming needs is now possible. To test this premise, we focus on assessing the efficiency of multiple satellite sensors, namely WorldView-3, Planet/Dove-Classic, Sentinel-2, and Landsat 8 (through Harmonized Landsat Sentinel-2 (HLS)), and investigate their spatial, spectral (surface reflectance (SR) and vegetation indices (VIs)), and temporal characteristics to estimate corn and soybean yields at sub-field scales within study sites in the US state of Iowa. Precision yield data as referenced to combine harvesters’ GPS systems were used for validation. We show that imagery spatial resolution of 3 m is critical to explaining 100% of the within-field yield variability for corn and soybean. Our simulation results show that moving to coarser resolution data of 10 m, 20 m, and 30 m reduced the explained variability to 86%, 72%, and 59%, respectively. We show that the most important spectral bands explaining yield variability were green (0.560 µm), red-edge (0.726 µm), and near-infrared (NIR - 0.865 µm). Furthermore, the high temporal frequency of Planet and a combination of Sentinel-2/Landsat 8 (HLS) data allowed for optimal date selection for yield map generation. Overall, we observed mixed performance of satellite-derived models with the coefficient of determination (R^2) varying from 0.21 to 0.88 (averaging 0.56) for the 30 m HLS and from 0.09 to 0.77 (averaging 0.30) for 3 m Planet. R^2 was lower for fields with higher yields, suggesting saturation of the satellite-collected reflectance features in those cases. Therefore, other biophysical variables, such as soil moisture and evapotranspiration, at similar fine spatial resolutions are likely needed alongside the optical imagery to fully explain the yields.

agriculture; yield; within-field; corn; soybean; r↗
NASA NTRS2000

Uncertainty Analysis of Decomposing Polyurethane Foam

Sensitivity/uncertainty analyses are necessary to determine where to allocate resources for improved predictions in support of our nation's nuclear safety mission. Yet, sensitivity/uncertainty analyses are not commonly performed on complex combustion models because the calculations are time consuming, CPU intensive, nontrivial exercises that can lead to deceptive results. To illustrate these ideas, a variety of sensitivity/uncertainty analyses were used to determine the uncertainty associated with thermal decomposition of polyurethane foam exposed to high radiative flux boundary conditions. The polyurethane used in this study is a rigid closed-cell foam used as an encapsulant. Related polyurethane binders such as Estane are used in many energetic materials of interest to the JANNAF community. The complex, finite element foam decomposition model used in this study has 25 input parameters that include chemistry, polymer structure, and thermophysical properties. The response variable was selected as the steady-state decomposition front velocity calculated as the derivative of the decomposition front location versus time. An analytical mean value sensitivity/uncertainty (MV) analysis was used to determine the standard deviation by taking numerical derivatives of the response variable with respect to each of the 25 input parameters. Since the response variable is also a derivative, the standard deviation was essentially determined from a second derivative that was extremely sensitive to numerical noise. To minimize the numerical noise, 50-micrometer element dimensions and approximately 1-msec time steps were required to obtain stable uncertainty results. As an alternative method to determine the uncertainty and sensitivity in the decomposition front velocity, surrogate response surfaces were generated for use with a constrained Latin Hypercube Sampling (LHS) technique. Two surrogate response surfaces were investigated: 1) a linear surrogate response surface (LIN) and 2) a quadratic response surface (QUAD). The LHS techniques do not require derivatives of the response variable and are subsequently relatively insensitive to numerical noise. To compare the LIN and QUAD methods to the MV method, a direct LHS analysis (DLHS) was performed using the full grid and timestep resolved finite element model. The surrogate response models (LIN and QUAD) are shown to give acceptable values of the mean and standard deviation when compared to the fully converged DLHS model.

Hobbs, Michael L.↗
NASA NTRS2021

Large-Eddy Simulation of Variable Speed Power Turbine Cascade with Inflow Turbulence

Numerical results are presented from theNASAGlenn Research Center’sin-house turbomachinery code, Glenn-HT applied to the Variable Speed Power Turbine (VSPT) experiment at the NASA Transonic Turbine Blade Cascade Facility.The main goal of this paper is to implement a digital filtering method to generate turbulence upstream and a sub-grid model (Localized dynamic k-equation model (LDKM)) in the framework of LES in order to investigate the effect of inflow turbulence on the transition seen in the VSPT experimental data at the cruise condition (incidence angle of 40oand Tu = 0.5%, 5%,10%, and 15%). Although the boundary layer on thesuction side and pressure side of the blades is initially laminar due to favorable pressure gradient, the laminar flow can transition to turbulent flow past a separation zoneon the suction sideor by natural or by-pass transition. This process determinesthe total-pressure lossesin the wake. Therefore, it is desirable to develop a reliable prediction tool to accurately capture the transition mechanism in blade rows operated under the conditions of low Reynolds number andat a variety offree stream turbulence conditions. Our numerical studies reveal that the location of separationis rather insensitive to the level of Tu, however the effect ofincreasingTu seems to be in reducing the size and ultimately suppressing the separation bubble. In addition, we performedspectral analysis to identify the peak frequencies in the region where the separation bubble is formed, which provides valuable insights into the transition/separation mechanism.

CFD↗
NASA NTRS2012

NASA Tech Briefs, December 2012

The topics include: Pattern Generator for Bench Test of Digital Boards; 670-GHz Down- and Up-Converting HEMT-Based Mixers; Lidar Electro-Optic Beam Switch with a Liquid Crystal Variable Retarder; Feedback Augmented Sub-Ranging (FASR) Quantizer; Real-Time Distributed Embedded Oscillator Operating Frequency Monitoring; Software Modules for the Proximity-1 Space Link Interleaved Time Synchronization (PITS) Protocol; Description and User Instructions for the Quaternion to Orbit v3 Software; AdapChem; Mars Relay Lander and Orbiter Overflight Profile Estimation; Extended Testability Analysis Tool; Interactive 3D Mars Visualization; Rapid Diagnostics of Onboard Sequences; MER Telemetry Processor; pyam: Python Implementation of YaM; Process for Patterning Indium for Bump Bonding; Archway for Radiation and Micrometeorite Occurrence Resistance; 4D Light Field Imaging System Using Programmable Aperture; Device and Container for Reheating and Sterilization; Radio Frequency Plasma Discharge Lamps for Use as Stable Calibration Light Sources; Membrane Shell Reflector Segment Antenna; High-Speed Transport of Fluid Drops and Solid Particles via Surface Acoustic Waves; Compact Autonomous Hemispheric Vision System; A Distributive, Non-Destructive, Real-Time Approach to Snowpack Monitoring; Wideband Single-Crystal Transducer for Bone Characterization; Numerical Simulation of Rocket Exhaust Interaction With Lunar Soil; Motion Imagery and Robotics Application (MIRA): Standards-Based Robotics; Particle Filtering for Model-Based Anomaly Detection in Sensor Networks; Ka-band Digitally Beamformed Airborne Radar Using SweepSAR Technique; Composite With In Situ Plenums; Multi-Beam Approach for Accelerating Alignment and Calibration of HyspIRI-Like Imaging Spectrometers; JWST Lifting System; Next-Generation Tumbleweed Rover; Pneumatic System for Concentration of Micrometer-Size Lunar Soil.

Source record↗
NASA NTRS1982

Analysis of geologic terrain models for determination of optimum SAR sensor configuration and optimum information extraction for exploration of global non-renewable resources. Pilot study: Arkansas Remote Sensing Laboratory, part 1, part 2, and part 3

Computer-generated radar simulations and mathematical geologic terrain models were used to establish the optimum radar sensor operating parameters for geologic research. An initial set of mathematical geologic terrain models was created for three basic landforms and families of simulated radar images were prepared from these models for numerous interacting sensor, platform, and terrain variables. The tradeoffs between the various sensor parameters and the quantity and quality of the extractable geologic data were investigated as well as the development of automated techniques of digital SAR image analysis. Initial work on a texture analysis of SEASAT SAR imagery is reported. Computer-generated radar simulations are shown for combinations of two geologic models and three SAR angles of incidence.

Kaupp, V. H.↗
NASA NTRS1971

A fixed bias, floating double probe technique with simple Langmuir probe characteristics

A new floating double-probe method is presented which has advantages over other floated-probe systems previously described in literature. The method utilized two electrodes, one of constant area and the other with a variable area. The two-electrode configuration is separated by a fixed bias voltage. The current-voltage characteristics of the new technique, which are generated by varying the area of the one electrode, are identical to those of a simple Langmuir probe, thus coupling all the advantages of a floated-probe system with the simple analysis scheme generally applied to the Langmuir probe for the determination of plasma density and temperature.

Szuszczewicz, E. P.↗
NASA NTRS1979

Sound transmission in ducts containing nearly choked flows

The nonlinear theory previously developed by the authors (1977, 1978) is used to obtain numerical results for sound transmission through a nearly choked throat in a variable-area duct. Parametric studies are performed for different source locations, strengths and frequencies. It is shown that the nonlinear interactions in the throat region generate superharmonics of the fundamental (source) frequency throughout the duct. The amplitudes of these superharmonics increase as the source parameters (frequency and strength) are increased toward values leading to acoustic shocks. For a downstream source, superharmonics carry about 20% of the total acoustic power as shocking conditions are approached. For the source strength levels and frequencies considered, streaming effects are negligible.

Callegari, A. J.↗
NASA NTRS1979

Multivariable control altitude demonstration on the F100 turbofan engine

The control system designed under the Multivariable Control Synthesis (MVCS) program for the F100 turbofan engine is described. The MVCS program, applied the linear quadratic regulator (LQR) synthesis methods in the design of a multivariable engine control system to obtain enhanced performance from cross-coupled controls, maximum use of engine variable geometry, and a systematic design procedure that can be applied efficiently to new engine systems. Basic components of the control system, a reference value generator for deriving a desired equilibrium state and an approximate control vector, a transition model to produce compatible reference point trajectories during gross transients, gain schedules for producing feedback terms appropriate to the flight condition, and integral switching logic to produce acceptable steady-state performance without engine operating limit exceedance are described and the details of the F100 implementation presented. The engine altitude test phase of the MVCS program, and engine responses in a variety of test operating points and power transitions are presented.

Lehtinen, B.↗
NASA NTRS1982

Incompressible Navier-Stokes and parabolized Navier-Stokes solution procedures and computational techniques

Recent developments with finite-difference techniques are emphasized. The quotation marks reflect the fact that any finite discretization procedure can be included in this category. Many so-called finite element collocation and galerkin methods can be reproduced by appropriate forms of the differential equations and discretization formulas. Many of the difficulties encountered in early Navier-Stokes calculations were inherent not only in the choice of the different equations (accuracy), but also in the method of solution or choice of algorithm (convergence and stability, in the manner in which the dependent variables or discretized equations are related (coupling), in the manner that boundary conditions are applied, in the manner that the coordinate mesh is specified (grid generation), and finally, in recognizing that for many high Reynolds number flows not all contributions to the Navier-Stokes equations are necessarily of equal importance (parabolization, preferred direction, pressure interaction, asymptotic and mathematical character). It is these elements that are reviewed. Several Navier-Stokes and parabolized Navier-Stokes formulations are also presented.

Rubin, S. G.↗
NASA NTRS1983

Some lessons from NACA/NASA aerodynamic studies following World War II

An historical account is presented of the new departures in aerodynamic research conducted by NACA, and subsequently NASA, as a result of novel aircraft technologies and operational regimes encountered in the course of the Second World War. The invention and initial development of the turbojet engine furnished the basis for a new speed/altitude regime in which numerous aerodynamic design problems arose. These included compressibility effects near the speed of sound, with attendant lift/drag efficiency reductions and longitudinal stability enhancements that were accompanied by a directional stability reduction. Major research initiatives were mounted in the investigation of swept, delta, trapezoidal and variable sweep wing configurations, sometimes conducted through flight testing of the 'X-series' aircraft. Attention is also given to the development of the first generation of supersonic fighter aircraft.

Spearman, M. L.↗
NASA NTRS1987

An algebraic criterion for the onset of chaos in nonlinear dynamic systems

The correspondence between iterated integrals and a noncommutative algebra is used to recast the given dynamical system from the time domain to the Laplace-Borel transform domain. It is then shown that the following algebraic criterion has to be satisfied for the outset of chaos: the limit (as tau approaches infinity and x sub 0 approaches infinity) of ((sigma(k=0) (tau sup k) / (k* x sub 0 sup k)) G II G = 0, where G is the generating power series of the trajectories, the symbol II is the shuffle product (le melange) of the noncommutative algebra, x sub 0 is a noncommutative variable, and tau is the correlation parameter. In the given equation, symbolic forms for both G and II can be obtained by use of one of the currently available symbolic languages such as PLI, REDUCE, and MACSYMA. Hence, the criterion is a computer-algebraic one.

Unal, A.↗