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NASA NTRS1988

Cluster-cluster correlations and constraints on the correlation hierarchy

The hypothesis that galaxies cluster around clusters at least as strongly as they cluster around galaxies imposes constraints on the hierarchy of correlation amplitudes in hierachical clustering models. The distributions which saturate these constraints are the Rayleigh-Levy random walk fractals proposed by Mandelbrot; for these fractal distributions cluster-cluster correlations are all identically equal to galaxy-galaxy correlations. If correlation amplitudes exceed the constraints, as is observed, then cluster-cluster correlations must exceed galaxy-galaxy correlations, as is observed.

Hamilton, A. J. S.↗

NASA NTRS1990

Analysis of stratocumulus cloud fields using LANDSAT imagery: Size distributions and spatial separations

Stratocumulus cloud fields in the FIRE IFO region are analyzed using LANDSAT Thematic Mapper imagery. Structural properties such as cloud cell size distribution, cell horizontal aspect ratio, fractional coverage and fractal dimension are determined. It is found that stratocumulus cloud number densities are represented by a power law. Cell horizontal aspect ratio has a tendency to increase at large cell sizes, and cells are bi-fractal in nature. Using LANDSAT Multispectral Scanner imagery for twelve selected stratocumulus scenes acquired during previous years, similar structural characteristics are obtained. Cloud field spatial organization also is analyzed. Nearest-neighbor spacings are fit with a number of functions, with Weibull and Gamma distributions providing the best fits. Poisson tests show that the spatial separations are not random. Second order statistics are used to examine clustering.

Welch, R. M.↗

NASA NTRS1989

Size distribution of dust grains: A problem of self-similarity

Distribution functions describing the results of natural processes frequently show the shape of power laws, e.g., mass functions of stars and molecular clouds, velocity spectrum of turbulence, size distributions of asteroids, micrometeorites and also interstellar dust grains. It is an open question whether this behavior is a result simply coming about by the chosen mathematical representation of the observational data or reflects a deep-seated principle of nature. The authors suppose the latter being the case. Using a dust model consisting of silicate and graphite grains Mathis et al. (1977) showed that the interstellar extinction curve can be represented by taking a grain radii distribution of power law type n(a) varies as a(exp -p) with 3.3 less than or equal to p less than or equal to 3.6 (example 1) as a basis. A different approach to understanding power laws like that in example 1 becomes possible by the theory of self-similar processes (scale invariance). The beta model of turbulence (Frisch et al., 1978) leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot's (1977) fractal dimension. In the frame of this beta model, it is supposed that on each stage of a cascade the system decays to N clumps and that only the portion beta N remains active further on. An important feature of this model is that the active eddies become less and less space-filling. In the following, the authors assume that grain-grain collisions are such a scale-invarient process and that the remaining grains are the inactive (frozen) clumps of the cascade. In this way, a size distribution n(a) da varies as a(exp -(D+1))da (example 2) results. It seems to be highly probable that the power law character of the size distribution of interstellar dust grains is the result of a self-similarity process. We can, however, not exclude that the process leading to the interstellar grain size distribution is not fragmentation at all. It could be, e.g., diffusion-limited growth discussed by Sander (1986), who applied the theory of fractal geometry to the classification of non-equilibrium growth processes. He received D=2.4 for diffusion-limited aggregation in 3d-space.

Henning, TH.↗

NASA NTRS1991

Formation of planetesimals

Formation of planetesimals is discussed. The following subject areas are covered: (1) nebular structure; (2) aerodynamics of the solid bodies in the nebula; (3) problems with gravitational instability; (4) particle growth by coagulation; properties of fractal aggregates; and (5) coagulation and settling of fractal aggregates.

Weidenschilling, Stuart J.↗

NASA NTRS1991

On representation of fracture profiles by fractional integrals of a Wiener process

An enhancement of the crack diffusion model of brittle fracture is proposed. In the model, certain probabilities are evaluated as averages over the set of all virtual crack paths, and the latter are modeled as realization of a Wiener process. However, commonly observed fracture profiles are smoother. In fact, one can show that their fractal dimension is smaller than the value 3/2 for a Wiener process. Accordingly, it is proposed to represent fracture profiles by graphs of fractional integrals of a Wiener process. This allows one to employ additional fractographic information obtained from fracture profiles, e.g. their fractal dimension, in a predictive formalism, evaluating probabilities of failure.

Kunin, B.↗

NASA NTRS1991

Mathematical morphology for automated analysis of remotely sensed objects in radar images

A symbiosis of pyramidal segmentation and morphological transmission is described. The pyramidal segmentation portion of the symbiosis has resulted in low (2.6 percent) misclassification error rate for a one-look simulation. Other simulations indicate lower error rates (1.8 percent for a four-look image). The morphological transformation portion has resulted in meaningful partitions with a minimal loss of fractal boundary information. An unpublished version of Thicken, suitable for watersheds transformations of fractal objects, is also presented. It is demonstrated that the proposed symbiosis works with SAR (synthetic aperture radar) images: in this case, a four-look Seasat image of sea ice. It is concluded that the symbiotic forms of both segmentation and morphological transformation seem well suited for unsupervised geophysical analysis.

Daida, Jason M.↗

NASA NTRS1992

The conductive propagation of nuclear flames. I - Degenerate C + O and O + Ne + Mg white dwarfs

The paper determines the physical properties - speed, width, and density structure - of conductive burning fronts in degenerate carbon-oxygen (C + O) and oxygen-neon-magnesium (O + Ne + Mg) compositions for a grid of initial densities and compositions. The dependence of the physical properties of the flame on the assumed values of nuclear reaction rates, the nuclear reaction network employed, the thermal conductivity, and the choice of coordinate system are investigated. The occurrence of accretion-induced collapse of a white dwarf is found to be critically dependent on the velocity of the nuclear conductive burning front and the growth rate of hydrodynamic instabilities. Treating the expanding area of the turbulent burning region as a fractal whose tile size is identical to the minimum unstable Rayleigh-Taylor wavelength, it is found, for all reasonable values of the fractal dimension, that for initial C + O or O + Ne + Mg densities above about 9 x 10 exp 9 g/cu cm the white dwarf should collapse to a neutron star.

Timmes, F. X.↗

NASA NTRS1991

The accumulation and structure of comets

The paper reviews evidence for the accumulation of the terrestrial planets and comets from solid grains, with emphasis on the various proposals for the formation of cometary nuclei. With three exceptions, all hypotheses conclude or imply that a single compact object forms. Several hypotheses start with Goldreich-Ward-type gravitational instabilities. The collapse for this case also occurs at low velocities in the cm/s to m/s range. Experiment and theory show that under these conditions, low-density, filamentary clusters form that are fractal aggregates with a fractal dimension approximately equal to 2. In order to form cometary nuclei, the initial temperature must be about 50 K and not undergo a significant temperature rise during the accumulation process. The calculations show that accumulation will occur at low temperatures. Models of cometary nuclei are reviewed, and a simple model of the structure that results fom the accumulation of fluffy aggregates is described.

Donn, Bertram↗

NASA NTRS1993

Size distributions in two porous chondritic micrometeorites

Quantitative size measurements of granular units (GUs), and nm-sized minerals in these units, in two porous chondritic micrometeorites are investigated. The matrix of these micrometeorites consist of loosely packed, 0.1 micron-sized, GUs. These objects were a major component of the solar nebula dust that accreted into protoplanets. The matrix in micrometeorite W7010*A2 has a fractal dimension with a small coefficient that supports efficient sticking of carbon-rich GUs during accretion. The fractal nature of the matrix provides a way to calculate the density using the aggregate size. The resulting very low density for porous chondritic micrometeorites is 0.08-0.14 g/cu cm, which supports the view that they are the solid debris from unconsolidated solar system bodies. Chondritic GUs contain ultrafine olivines, pyroxenes, and sulfides, embedded in hydrocarbons and amorphous carbons. Nanocrystals in the micrometeorites W7010*A2 and U2015*B show log normal size distributions. The high incidence of disk-shaped grains, a changeover from disk-shaped to euhedral grains, the unevolved nature of the size distributions, and multiple populations for grains less than 127 nm in size, are consistent with continuous postaccretion nucleation and growth in amorphous GUs, including coarsening via Ostwald ripening.

Rietmeijer, Frans J. M.↗

NASA NTRS1993

The spectrum of the interplanetary magnetic field near 1.3 AU

A time series of the interplanetary magnetic field measured near 1.3 AU by Phobos 2 is analyzed as a fractal. The fractal dimension of the curves corresponding to the components and to the strength of the magnetic field are found to be close to 5/3. The corresponding spatial spectra are interpreted in the framework of MHD turbulence.

Ruzmaikin, Alexander↗

NASA NTRS1993

Quantifying the effect of rheology on plan-view shapes of lava flows

This study aims at quantifying the effect of rheology on the plan-view shapes of lava flows. Plan-view shapes of lava flows are important because they reflect the processes governing flow emplacement and may provide insight into lava flow rheology and dynamics. In our earlier investigation, it was reported that plan-view shapes of tholeite basalts are fractal, having a characteristic shape regardless of scale. It was also found one could use the fractal dimension (a parameter which quantifies flow margin convolution) to distinguish between the two major types of basalts: a'a and pahoehoe. Encouraged by these earlier results, a similar method for use on silicic flows are being developed and our preliminary work is presented.

Bruno, B. C.↗

NASA NTRS1994

Independent pixel and Monte Carlo estimates of stratocumulus albedo

Monte Carlo radiative transfer methods are employed here to estimate the plane-parallel albedo bias for marine stratocumulus clouds. This is the bias in estimates of the mesoscale-average albedo, which arises from the assumption that cloud liquid water is uniformly distributed. The authors compare such estimates with those based on a more realistic distribution generated from a fractal model of marine stratocumulus clouds belonging to the class of 'bounded cascade' models. In this model the cloud top and base are fixed, so that all variations in cloud shape are ignored. The model generates random variations in liquid water along a single horizontal direction, forming fractal cloud streets while conserving the total liquid water in the cloud field. The model reproduces the mean, variance, and skewness of the vertically integrated cloud liquid water, as well as its observed wavenumber spectrum, which is approximately a power law. The Monte Carlo method keeps track of the three-dimensional paths solar photons take through the cloud field, using a vectorized implementation of a direct technique. The simplifications in the cloud field studied here allow the computations to be accelerated. The Monte Carlo results are compared to those of the independent pixel approximation, which neglects net horizontal photon transport. Differences between the Monte Carlo and independent pixel estimates of the mesoscale-average albedo are on the order of 1% for conservative scattering, while the plane-parallel bias itself is an order of magnitude larger. As cloud absorption increases, the independent pixel approximation agrees even more closely with the Monte Carlo estimates. This result holds for a wide range of sun angles and aspect ratios. Thus, horizontal photon transport can be safely neglected in estimates of the area-average flux for such cloud models. This result relies on the rapid falloff of the wavenumber spectrum of stratocumulus, which ensures that the smaller-scale variability, where the radiative transfer is more three-dimensional, contributes less to the plane-parallel albedo bias than the larger scales, which are more variable. The lack of significant three-dimensional effects also relies on the assumption of a relatively simple geometry. Even with these assumptions, the independent pixel approximation is accurate only for fluxes averaged over large horizontal areas, many photon mean free paths in diameter, and not for local radiance values, which depend strongly on the interaction between neighboring cloud elements.

Cahalan, Robert F.↗

NASA NTRS1994

Cluster growth modeling of plateau erosion

The pattern of erosion of a plateau along an escarpment may be modeled usng cluster growth techniques, recently popularized in models of drainage network evolution. If erosion on the scarp takes place in discrete events at rates subject to local substrate strength, the whole range of behavior is described by a combination of three cluster growth mechanisms: invasion percolation, Eden growth and diffusion-limited aggregation (DLA). These model the relative importance of preexisting substrate strength, background weathering, and seepage weathering and erosion respectively. The rate of seepage processes is determined by the efflux of groundwater at the plateau margin, which in turn is determined by the pressure field in the plateau aquifer. If this process acted alone, it would produce erosion patterns in the form of Laplacian fractals, with groundwater recharge from a distant source, or Poissionian fractals, with groundwater recharge uniform over the plateau. DLA is used to mimic the Laplacian or Poissonian potential field and the corresponding seepage growth process. The scaling structure of clusters grown by pure DLA, invasion percolation, or Eden growth is well known; this study presents a model which combines all three growth mechanisms for the first time. Mixed growth processes create clusters with different scaling properties and morphologies over distinct length scale ranges, and this is demonstrable in natural examples of plateau erosion.

Stark, Colin P.↗

NASA NTRS1995

On the spatial distribution of magnetic fields on the solar surface

Recent measurements of solar surface magnetic fields suggest that the spatial distribution of these fields is fractal. In order to understand the physical basis for such geometric complexity, we study here the advection of magnetic flux tubes relatively simple random motions on the surface of a fluid and investigate the spatial statistics of the resulting surface field. While this study does not directly address the question of why solar surface fields have the observed spatial structure, it is designed to build our intuition about how surface flows lead to complex spatial structuring of magnetic fields. As part of our study, we discuss the various methods by which one can describe the spatial distribution of the surface magnetic flux and relate them mathematically; this turns out to be a crucial point of our work since, as we show, a number of previous analyses have misinterpreted the analysis procedures for determining fractal dimensions. Our principal result is the explicit demonstration that simple random flows lead to magnetic flux spatial distributions with a multifractal dimension spectrum. Furthermore, we demonstrate that this magnetic spatial structure is generic, i.e., is characteristic of a very large class of random flows.

Tao, L.↗

NASA NTRS1995

Windowed and Wavelet Analysis of Marine Stratocumulus Cloud Inhomogeneity

To improve radiative transfer calculations for inhomogeneous clouds, a consistent means of modeling inhomogeneity is needed. One current method of modeling cloud inhomogeneity is through the use of fractal parameters. This method is based on the supposition that cloud inhomogeneity over a large range of scales is related. An analysis technique named wavelet analysis provides a means of studying the multiscale nature of cloud inhomogeneity. In this paper, the authors discuss the analysis and modeling of cloud inhomogeneity through the use of wavelet analysis. Wavelet analysis as well as other windowed analysis techniques are used to study liquid water path (LWP) measurements obtained during the marine stratocumulus phase of the First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment. Statistics obtained using analysis windows, which are translated to span the LWP dataset, are used to study the local (small scale) properties of the cloud field as well as their time dependence. The LWP data are transformed onto an orthogonal wavelet basis that represents the data as a number of times series. Each of these time series lies within a frequency band and has a mean frequency that is half the frequency of the previous band. Wavelet analysis combined with translated analysis windows reveals that the local standard deviation of each frequency band is correlated with the local standard deviation of the other frequency bands. The ratio between the standard deviation of adjacent frequency bands is 0.9 and remains constant with respect to time. This ratio defined as the variance coupling parameter is applicable to all of the frequency bands studied and appears to be related to the slope of the data's power spectrum. Similar analyses are performed on two cloud inhomogeneity models, which use fractal-based concepts to introduce inhomogeneity into a uniform cloud field. The bounded cascade model does this by iteratively redistributing LWP at each scale using the value of the local mean. This model is reformulated into a wavelet multiresolution framework, thereby presenting a number of variants of the bounded cascade model. One variant introduced in this paper is the 'variance coupled model,' which redistributes LWP using the local standard deviation and the variance coupling parameter. While the bounded cascade model provides an elegant two- parameter model for generating cloud inhomogeneity, the multiresolution framework provides more flexibility at the expense of model complexity. Comparisons are made with the results from the LWP data analysis to demonstrate both the strengths and weaknesses of these models.

Gollmer, Steven M.↗

NASA NTRS1995

Brief Communication: Buoyancy-Induced Differences in Soot Morphology

Reduction or elimination of buoyancy in flames affects the dominant mechanisms driving heat transfer, burning rates and flame shape. The absence of buoyancy produces longer residence times for soot formation, clustering and oxidation. In addition, soot pathlines are strongly affected in microgravity. We recently conducted the first experiments comparing soot morphology in normal and reduced-gravity laminar gas jet diffusion flames. Thermophoretic sampling is a relatively new but well-established technique for studying the morphology of soot primaries and aggregates. Although there have been some questions about biasing that may be induced due to sampling, recent analysis by Rosner et al. showed that the sample is not biased when the system under study is operating in the continuum limit. Furthermore, even if the sampling is preferentially biased to larger aggregates, the size-invariant premise of fractal analysis should produce a correct fractal dimension.

Ku, Jerry C.↗

NASA NTRS1997

Environmental Assessment and Monitoring with ICAMS (Image Characterization and Modeling System) Using Multiscale Remote-Sensing Data

With the rapid increase in spatial data, especially in the NASA-EOS (Earth Observing System) era, it is necessary to develop efficient and innovative tools to handle and analyze these data so that environmental conditions can be assessed and monitored. A main difficulty facing geographers and environmental scientists in environmental assessment and measurement is that spatial analytical tools are not easily accessible. We have recently developed a remote sensing/GIS software module called Image Characterization and Modeling System (ICAMS) to provide specialized spatial analytical tools for the measurement and characterization of satellite and other forms of spatial data. ICAMS runs on both the Intergraph-MGE and Arc/info UNIX and Windows-NT platforms. The main techniques in ICAMS include fractal measurement methods, variogram analysis, spatial autocorrelation statistics, textural measures, aggregation techniques, normalized difference vegetation index (NDVI), and delineation of land/water and vegetated/non-vegetated boundaries. In this paper, we demonstrate the main applications of ICAMS on the Intergraph-MGE platform using Landsat Thematic Mapper images from the city of Lake Charles, Louisiana. While the utilities of ICAMS' spatial measurement methods (e.g., fractal indices) in assessing environmental conditions remain to be researched, making the software available to a wider scientific community can permit the techniques in ICAMS to be evaluated and used for a diversity of applications. The findings from these various studies should lead to improved algorithms and more reliable models for environmental assessment and monitoring.

Lam, N.↗

NASA NTRS2003

Applicability of Complexity Theory to Martian Fluvial Systems: A Preliminary Analysis

In the last 15 years, terrestrial geomorphology has been revolutionized by the theories of chaotic systems, fractals, self-organization, and selforganized criticality. Except for the application of fractal theory to the analysis of lava flows and rampart craters on Mars, these theories have not yet been applied to problems of Martian landscape evolution. These complexity theories are elucidated below, along with the methods used to relate these theories to the realities of Martian fluvial systems.

Rosenshein, E. B.↗