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NASA NTRS1993

Siderophile element fractionation in meteor crater impact glasses and metallic spherules

Meteor Crater, Arizona provides an opportunity to study, in detail, elemental fractionation processes occurring during impacts through the study of target rocks, meteorite projectile and several types of impact products. We have performed EMPA and INAA on target rocks, two types of impact glass and metallic spherules from Meteor Crater. Using literature data for the well studied Canyon Diablo iron we can show that different siderophite element fractionations affected the impact glasses than affected the metallic spherules. The impact glasses primarily lost Au, while the metallic spherules lost Fe relative to other siderophile elements.

Mittlefehldt, David W.↗

NASA NTRS1992

A field measure of the shade fraction

'Shade' has a technical definition peculiar to linear spectral mixture analysis of imaging spectrometer data: it is the reduction in radiance from a surface due to lighting conditions and geometry, and includes topographic shading described by photometric functions as well as shadowing at all scales. 'Shade' is an important constituent of nearly all remotely sensed images, and is one endmember resolved in spectral mixture analysis, where it is represented as a fraction of the measured radiance and a characteristic spectrum. This spectrum is typically the null vector, provided the data have been corrected for atmospheric and instrument effects: i.e., 'shade' is the radiance from an ideal black surface. In topographic shading, irradiance is reduced - typically in proportion to cos(i), where i (incidence angle) is the angle between the sun and the local surface normal vectors. Therefore, the radiance is lowered by a multiplicative factor. Shadowing occurs when i is greater than 90 deg, or when sunlight is blocked by adjacent high terrain; the only irradiance is down-welling skylight and bounce light from adjacent terrain. In spectral mixture analysis, 'shade' is regarded as an additive term. In this regard, it is an accurate description of the proportion of a scene that consists of ideal shadows ('checkerboard mixing'); however, 'shade' represents the multiplicative cos(i) factor as well, as here it should be interpreted as the proportion of shadow that would darken the scene an equivalent amount. In either case, the 'shade' fraction is lessened by adjacency effects, because the scene has a non-zero reflectivity instead of the ideal black surface generally assumed.

Gillespie, Alan R.↗

NASA NTRS1994

Mars atmospheric escape and isotopic fractionation: Synthesis of data and models

The present Mars atmosphere is relatively thin and cold. It is not at all like that which is presumed to have been responsible for the formation of valley networks and the heavy erosion of craters during the earliest epochs of martian history. An important goal of Mars exploration is to try to understand the properties of the early atmosphere, the initial inventory of volatiles at the planet's surface, the processes by which the atmosphere and climate have evolved over time, and the current location of volatiles presumed to have been in the atmosphere in the earlier times. The current status of understanding of the escape of volatiles to space over geologic time and the resulting fractionation of isotopes of stable atoms remaining in the atmosphere are described, and a scenario for volatile abundance and evolution that is consistent with the available information on the escape and fractionation of each species is constructed. In particular, the evolution of hydrogen, carbon, oxygen, and nitrogen, as contained in atmospheric (and non-atmospheric) water, carbon dioxide, and molecular nitrogen, is examined.

Jakosky, Bruce M.↗

NASA NTRS1994

Fractional crystallization of iron meteorites: Constant versus changing partition coefficients

Analyses of magmatic iron meteorites, plotted on LogC(sub i) vs LogC(sub Ni) diagrams, often form linear arrays. Traditionally, this linearity has been ascribed to fractional crystallization under the assumption of constant partition coefficients (i.e., Rayleigh fractionation). Paradoxically, however, partition coefficients in the Fe-Ni-S-P system are decidedly not constant. This contribution provides a rationale for understanding how trends on LogC(sub i) vs LogC(sub Ni) diagrams can be linear, even when partition coefficients are changing rapidly.

Jones, J. H.↗

NASA NTRS1993

Relationships between evaprorative fraction and remotely sensed vegetation index and microwave brightness temperature for semiarid rangelands

Measurements of the microwave brightness temperature (TB) with the Pushbroom Microwave Radiometer (PBMR) over the Walnut Gulch Experiment Watershed were made on selected days during the MONSOON 90 field campaign. The PBMR is an L-band instrument (21-cm wavelength) that can provide estimates of near-surface soil moisture over a variety of surfaces. Aircraft observations in the visible and near-infrared wavelengths collected on selected days also were used to compute a vegetation index. Continuous micrometeorological measurements and daily soil moisture samples were obtained at eight locations during experimental period. Two sites were instrumented with time domain reflectometry probes to monitor the soil moisture profile. The fraction of available energy used for evapotranspiration was computed by taking the ratio of latent heat flux (LE) to the sum of net radiation (Rn) and soil heat flux (G). This ratio is commonly called the evaporative fraction (EF) and normally varies between 0 and 1 under daytime convective conditions with minimal advection. A wide range of environmental conditions existed during the field campaign, resulting in average EF values for the study area varying from 0.4 to 0.8 and values of TB ranging from 220 to 280 K. Comparison between measured TB and EF for the eight locations showed an inverse relationship. Other days were included in the analysis by estimating TB with the soil moisture data. Because transpiration from the vegetation is more strongly coupled to root zone soil moisture, significant scatter in this relationship existed at high values of TB or dry near-surface soil moisture conditions. The variation in EF under dry near-surface soil moisture conditions was correlated to the amount of vegetation cover estimated with a remotely sensed vegetation index. These findings indicate that information obtained from optical and microwave data can be used for quantifying the energy balance of semiarid areas. The microwave data can indicate when soil evaporation is significantly contributing to EF, while the optical data is helpful for quantifying the spatial variation in EF due to the distribution of vegetation cover.

Kustas, W. P.↗

NASA NTRS1993

Radio observations of D I and fractionation

We report observations to detect the 327 MHz line of atomic deuterium in the primarily molecular clouds towards Cas A and Sgr A, making appropriate choices of spectral and spatial resolution. Our best results for Cas A, based on 1253 hr of observing with the 85 foot (26 m) Hat Creek telescope and 45 hr with the very large array (VLA), yield an upper limit for the fraction of deuterium in atomic form, D I, of 0.14. We present an approximate analytical analysis of deuterium fractionation in translucent clouds, which shows that gas-phase ion-molecule reactions, assisted by dust and HD line self-shielding, are efficient in converting deuterium to HD. We conclude that little atomic deuterium is present in the molecular clumps in the Cas A clouds and that much higher sensitivity observations would be required to detect 327 MHz line in molecular clouds. We have also attempted to detect the 72 GHz line of DCO(+) toward Cas A. The observed upper limit to the DCO(+)/HCO(+) ratio of approximately 0.03 is not much larger than our theoretical estimate, and the DCO(+) line should be detectable with existing instrumentation, unless the temperature of these clouds is much larger than usually assumed.

Heiles, Carl↗

NASA NTRS1996

Chaos in a Fractional Order Chua System

This report studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. Varying the total system order incrementally from 2.6 to 3.7 demonstrates that systems of 'order' less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable.

Lorenzo, Carl F.↗

NASA NTRS1994

Model reduction of nonsquare linear MIMO systems using multipoint matrix continued-fraction expansions

This paper deals with the multipoint Cauer matrix continued-fraction expansion (MCFE) for model reduction of linear multi-input multi-output (MIMO) systems with various numbers of inputs and outputs. A salient feature of the proposed MCFE approach to model reduction of MIMO systems with square transfer matrices is its equivalence to the matrix Pade approximation approach. The Cauer second form of the ordinary MCFE for a square transfer function matrix is generalized in this paper to a multipoint and nonsquare-matrix version. An interesting connection of the multipoint Cauer MCFE method to the multipoint matrix Pade approximation method is established. Also, algorithms for obtaining the reduced-degree matrix-fraction descriptions and reduced-dimensional state-space models from a transfer function matrix via the multipoint Cauer MCFE algorithm are presented. Practical advantages of using the multipoint Cauer MCFE are discussed and a numerical example is provided to illustrate the algorithms.

Guo, Tong-Yi↗

NASA NTRS1996

Multigrid, Fractional-Step Computation Of Flow

Speed of computer code solving three-dimensional Navier-Stokes equations of flow of incompressible fluid by fractional-step method increased significantly by use of multigrid procedures. In method, equations solved on general nonorthogonal curvilinear coordinate grid, using volume fluxes. At each time step, computations performed in two fractional steps. In first step, equations of conservation of momentum solved by use of gradient of pressure from previous time step via explicit approximate-factorization method, yielding approximate flow field that does not satisfy equation of conservation of mass. In second step, discrete Poisson-like equation with Neumann-type boundary conditions, formed by combining equations of conservation of momentum and mass, solved iteratively.

Kwak, Dochan↗

NASA NTRS1998

Insights Into the Fractional Order Initial Value Problem via Semi-Infinite Systems

This paper considers various aspects of the initial value problem for fractional order differential equations. The main contribution of this paper is to use the solutions to known spatially distributed systems to demonstrate that fractional differintegral operators require an initial condition term that is time-varying due to past distributed storage of information.

Hartley, Tom T.↗

NASA NTRS1998

A Solution to the Fundamental Linear Fractional Order Differential Equation

This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

Hartley, Tom T.↗

NASA NTRS1999

The Vertical Linear Fractional Initialization Problem

This paper presents a solution to the initialization problem for a system of linear fractional-order differential equations. The scalar problem is considered first, and solutions are obtained both generally and for a specific initialization. Next the vector fractional order differential equation is considered. In this case, the solution is obtained in the form of matrix F-functions. Some control implications of the vector case are discussed. The suggested method of problem solution is shown via an example.

Lorenzo, Carl F.↗

NASA NTRS2001

Galaxy Cluster Gas Mass Fractions From Sunyaev-Zel'dovich Effect Measurement: Constraints on Omega_M

Using sensitive centimeter-wave receivers mounted on the Owens Valley Radio Observatory and Berkeley-Illinois-Maryland-Association millimeter arrays, we have obtained interferometric measurements of the Sunyaev-Zei'dovich (SZ) effect toward massive galaxy clusters. We use the SZ data to determine the pressure distribution of the cluster gas and, in combination with published X-ray temperatures, to infer the gas mass and total gravitational mass of 18 clusters. The gas mass fraction, f~g, is calculated for each cluster, and is extrapolated to the fiducial radius r_{500} using the results of numerical simulations. The mean f_g within r_{500} is 0.081 + 0.009 - 0.011/(h_{100} (statistical uncertainty at 68% confidence level, assuming OmegaM=0.3, OmegaL=0.7). We discuss possible sources of systematic errors in the mean f 9 measurement. We derive an upper limit for OmegaM from this sample under the assumption that the mass composition of clusters within r_{500} reflects the universal mass composition: Omega_M h <Omega_B/f_g.The gas mass f on cosmology through the angular diameter distance and the r_{500} correction factors. For a flat universe (Omegal, = 1 - OmegaM) and h=0.7, we find the measured gas mass fractions are consistent with OmegaM less than 0.40, at 68% confidence. Including estimates of the baryons contained in galaxies and the baryons which failed to become bound during the cluster formation process, we find OmegaM\approximately 0.25.

Grego, Laura↗

NASA NTRS2001

Fractionation Trends Among IVA Iron Meteorites: Contrasts with IIIAB Trends

A neutron-activation study of 49 group-IVA irons shows much lower negative slopes on Ir-Au and Ir-As diagrams than observed in the larger magmatic group IIIAB. This difference seems to reflect the tendency of D(sub Ir), D(sub Au), and D(sub As) to increase with increasing S content. Contents of S and other volatiles are much lower in IVA irons than IIIAB irons. We show that both groups can be fit with Dx values that depend quadratically on S, with initial IVA S contents about 6X lower than those in IIIAB. The IVA scatter fields show a spread in Au or As at constant Ir that appears to reflect variations in the fraction of trapped melt between 0% and 30%. Copper shows an S-shaped trend that may reflect moderate positive and negative changes in D(sub Cu) as the magma evolved or, less likely, sampling variations in a broad band reflecting fractionation and trapping of melt. Gibeon, the largest IVA iron with a mass greater than 30 tons, shows an appreciable range in compositions consistent either with differences in the degree of magma crystallization or with differences in the content of trapped melt. A striking difference between IVA and IIIAB is observed in the Ir/Au ratios in the most Ir-rich irons in the groups; that in IVA is 40% lower than the IIIAB ratio, and lower than those in other iron-meteorite groups, The IVA Ir/Au ratio is about half the ratios in the chondrite groups. We examined three possible explanations of this anomaly: (1) the high-Ir irons contain large amounts of trapped melt: or (2) half of the IVA core (i.e., the first 50% to crystallize) is missing from the terrestrial set of IVA irons; or (3) the IVA magma formed by incomplete melting of the metal in the chondritic precursor material, with the metal that remained in the mantle having high Ir and low Au contents. Plausibility arguments favor the third possibility. The third scenario is the most plausible, but the second cannot be ruled out. We review recent evidence regarding the cooling rates in group IVA. In contrast to recent interpretations, we note several lines of evidence that indicate constant cooling rates independent of composition, as expected if all IVA meteorites were in the same core when cooling between 900 and 650 K occurred.

Wasson, J. T.↗

NASA NTRS2002

Quantifying Fractional Ground Cover on the Climate Sensitive High Plains Using AVIRIS and Landsat TM Data

The High Plains is an economically important and climatologically sensitive region of the United States and Canada. The High Plains contain 100,000 sq km of Holocene sand dunes and sand sheets that are currently stabilized by natural vegetation. Droughts and the larger threat of global warming are climate phenomena that could cause depletion of natural vegetation and make this region susceptible to sand dune reactivation. This thesis is part of a larger study that is assessing the effect of climate variability on the natural vegetation that covers the High Plains using Landsat 5 and Landsat 7 data. The question this thesis addresses is how can fractional vegetation cover be mapped with the Landsat instruments using linear spectral mixture analysis and to what accuracy. The method discussed in this thesis made use of a high spatial and spectral resolution sensor called AVIRIS (Airborne Visible and Infrared Imaging Spectrometer) and field measurements to test vegetation mapping in three Landsat 7 sub-scenes. Near-simultaneous AVIRIS images near Ft. Morgan, Colorado and near Logan, New Mexico were acquired on July 10, 1999 and September 30, 1999, respectively. The AVIRIS flights preceded Landsat 7 overpasses by approximately one hour. These data provided the opportunity to test spectral mixture algorithms with AVIRIS and to use these data to constrain the multispectral mixed pixels of Landsat 7. The comparisons of mixture analysis between the two instruments showed that AVIRIS endmembers can be used to unmix Landsat 7 data with good estimates of soil cover, and reasonable estimates of non-photosynthetic vegetation and green vegetation. Landsat 7 derived image endmembers correlate with AVIRIS fractions, but the error is relatively large and does not give a precise estimate of cover.

Warner, Amanda Susan↗

NASA NTRS2003

Kinetic Fractionation of Stable Isotopes in Carbonates on Mars: Terrestrial Analogs

An ancient Martian hydrosphere consisting of an alkali-rich ocean would likely produce solid carbonate minerals through the processes of evaporation and/or freezing. We postulate that both (or either) of these kinetically-driven processes would produce carbonate minerals whose stable isotopic compositions are highly fractionated (enriched) with respect to the source carbon. Various scenarios have been proposed for carbonate formation on Mars, including high temperature formation, hydrothermal alteration, precipitation from evaporating brines, and cryogenic formation. 13C and 18O -fractionated carbonates have previously been shown to form kinetically under some of these conditions, ie.: 1) alteration by hydrothermal processes, 2) low temperature precipitation (sedimentary) from evaporating bicarbonate (brine) solutions, and 3) precipitation during the process of cryogenic freezing of bicarbonate-rich fluids. Here we examine several terrestrial field settings within the context of kinetically controlled carbonate precipitation where stable isotope enrichments have been observed.

Socki, Richard A.↗

NASA NTRS2003

Algorithms for the Fractional Calculus: A Selection of Numerical Methods

Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.

Diethelm, K.↗

NASA NTRS2005

Comparison of Monthly Mean Cloud Fraction and Cloud Optical depth Determined from Surface Cloud Radar, TOVS, AVHRR, and MODIS over Barrow, Alaska

A one year comparison is made of mean monthly values of cloud fraction and cloud optical depth over Barrow, Alaska (71 deg 19.378 min North, 156 deg 36.934 min West) between 35 GHz radar-based retrievals, the TOVS Pathfinder Path-P product, the AVHRR APP-X product, and a MODIS based cloud retrieval product from the CERES-Team. The data sets represent largely disparate spatial and temporal scales, however, in this paper, the focus is to provide a preliminary analysis of how the mean monthly values derived from these different data sets compare, and determine how they can best be used separately, and in combination to provide reliable estimates of long-term trends of changing cloud properties. The radar and satellite data sets described here incorporate Arctic specific modifications that account for cloud detection challenges specific to the Arctic environment. The year 2000 was chosen for this initial comparison because the cloud radar data was particularly continuous and reliable that year, and all of the satellite retrievals of interest were also available for the year 2000. Cloud fraction was chosen as a comparison variable as accurate detection of cloud is the primary product that is necessary for any other cloud property retrievals. Cloud optical depth was additionally selected as it is likely the single cloud property that is most closely correlated to cloud influences on surface radiation budgets.

Uttal, Taneil↗