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Layout considerations for a large deep space communications antenna array are discussed. A fractal geometry for the antenna layout is described that provides optimal packing of antenna elements, efficient cable routing, and logical division of the array into identical sub-arrays.
We have continued work on realistic numerical models of cratered surfaces, as first reported at last year's LPSC. We confirm the saturation equilibrium level with a new, independent test. One of us has developed a realistic computer simulation of a cratered surface. The model starts with a smooth surface or fractal topography, and adds primary craters according to the cumulative power law with exponent -1.83, as observed on lunar maria and Martian plains. Each crater has an ejecta blanket with the volume of the crater, feathering out to a distance of 4 crater radii. We use the model to test the levels of saturation equilibrium reached in naturally occurring systems, by increasing crater density and observing its dependence on various parameters. In particular, we have tested to see if these artificial systems reach the level found by Hartmann on heavily cratered planetary surfaces, hypothesized to be the natural saturation equilibrium level. This year's work gives the first results of a crater population that includes secondaries. Our model 'Gaskell-4' (September, 1992) includes primaries as described above, but also includes a secondary population, defined by exponent -4. We allowed the largest secondary from each primary to be 0.10 times the size of the primary. These parameters will be changed to test their effects in future models. The model gives realistic images of a cratered surface although it appears richer in secondaries than real surfaces are. The effect of running the model toward saturation gives interesting results for the diameter distribution. Our most heavily cratered surface had the input number of primary craters reach about 0.65 times the hypothesized saturation equilibrium, but the input number rises to more than 100 times that level for secondaries below 1.4 km in size.
Recently, system theorists have recognized that low order systems of nonlinear differential equations can give rise to solutions which are neither periodic, constant, nor predictable in steady state, but which are nonetheless bounded and deterministic. This behavior, which was first described in the study of weather systems, has been termed 'chaotic.' Much study of chaotic systems has concentrated on analysis of the systems' phase space attractors. It has been recognized that invariant measures of the attractor possess inherent information about the system. One such measure is the dimension of the attractors. The dimension of a chaotic attractor has been shown to be noninteger, leading to the term 'strange attractor;' the attractor is said to have a fractal structure. The correlation dimension has become one of the most popular measures of dimension. However, many problems have been identified in correlation dimension estimation from time sequences. The most common methods for obtaining the correlation dimension have been least squares curves fitting to find the slope of the correlation integral and the Takens Estimator. However, these estimates show unacceptable sensitivity to the upper limit on the distance chosen. Here, a new method is proposed which is shown to be rather insensitive to the upper limit and to perform in a very stable manner, at least in the absence of noise. The correlation dimension is also shown to be an effective discriminant in distinguishing between radar returns resulting from weather and those from the ground. The weather returns are shown to have a correlation dimension generally between 2.0 and 3.0, while ground returns have a correlation dimension exceeding 3.0.
Traditionally, long term measurements of atmospherically propagated sound signals have consisted of time series of multiminute averages. Only recently have continuous measurements with temporal resolution corresponding to turbulent time scales been available. With modern digital data acquisition systems we now have the capability to simultaneously record both acoustical and meteorological parameters with sufficient temporal resolution to allow us to examine in detail relationships between fluctuating sound and the meteorological variables, particularly wind and temperature, which locally determine the acoustic refractive index. The atmospheric acoustic propagation medium can be treated as a nonlinear dynamical system, a kind of signal processor whose innards depend on thermodynamic and turbulent processes in the atmosphere. The atmosphere is an inherently nonlinear dynamical system. In fact one simple model of atmospheric convection, the Lorenz system, may well be the most widely studied of all dynamical systems. In this paper we report some results of our having applied methods used to characterize nonlinear dynamical systems to study the characteristics of acoustical signals propagated through the atmosphere. For example, we investigate whether or not it is possible to parameterize signal fluctuations in terms of fractal dimensions. For time series one such parameter is the limit capacity dimension. Nicolis and Nicolis were among the first to use the kind of methods we have to study the properties of low dimension global attractors.
Gamma ray burst time histories, ranging in durations from milliseconds to thousands of seconds, are as varied as the number of bursts. They show a wide array of structures from those that are very smooth to those that contain a seemingly uncountable number of spikes riding on top of other spikes. These profiles have tantalized researchers for years - they obviously hold important information on the nature of GRB's, but to date no one has been successful in analyzing them. For the past year the author has been working on algorithms to analyze these data. Two approaches have been followed in this investigation. The first is an attempt to quantify the amount of structure, or spikiness, in a profile. The second involves applying the latest theorems on chaos and fractals with the aim of extracting useful information from what seems to be a random collection of shot noise.
Data compression has become an increasingly popular option as advances in information technology have placed further demands on data storage capabilities. With compression ratios as high as 100:1 the benefits are clear; however, the inherent intolerance of many compression formats to error events should be given careful consideration. If we consider that efficiently compressed data will ideally contain no redundancy, then the introduction of a channel error must result in a change of understanding from that of the original source. While the prefix property of codes such as Huffman enables resynchronisation, this is not sufficient to arrest propagating errors in an adaptive environment. Arithmetic, Lempel-Ziv, discrete cosine transform (DCT) and fractal methods are similarly prone to error propagating behaviors. It is, therefore, essential that compression implementations provide sufficient combatant error control in order to maintain data integrity. Ideally, this control should be derived from a full understanding of the prevailing error mechanisms and their interaction with both the system configuration and the compression schemes in use.
The Ballistic Missile Defense Organization has been developing the capability to evaluate one or more high-rate sensor/hardware combinations by incorporating them as payloads on a series of Miniature Seeker Technology Insertion (MSTI) flights. This publication represents the final report of a 1993 study to analyze the potential impact f data compression and of related communication system technologies on post-MSTI 3 flights. Lossless compression is considered alone and in conjunction with various spatial editing modes. Additionally, JPEG and Fractal algorithms are examined in order to bound the potential gains from the use of lossy compression. but lossless compression is clearly shown to better fit the goals of the MSTI investigations. Lossless compression factors of between 2:1 and 6:1 would provide significant benefits to both on-board mass memory and the downlink. for on-board mass memory, the savings could range from $5 million to $9 million. Such benefits should be possible by direct application of recently developed NASA VLSI microcircuits. It is shown that further downlink enhancements of 2:1 to 3:1 should be feasible thorough use of practical modifications to the existing modulation system and incorporation of Reed-Solomon channel coding. The latter enhancement could also be achieved by applying recently developed VLSI microcircuits.
We simulate collisional evolution of grains in dense turbulent molecular cloud cores (or Bok globules) in static equilibrium and free-fall collapse, assuming spherical symmetry. Relative velocities are due to thermal motions, differential settling, and turbulence, with the latter dominant for sonic turbulence with an assumed Kolmogorov spectrum. Realistic criteria are used to determine outcomes of collisions (coagulation vs. destruction) as functions of particle size and velocity. Results are presented for a variety of cloud parameters (radial density profile, turbulent velocity) and particle properties (density, impact strength). Results are sensitive to the assumed mechanical properties (density and impact strength) of grain aggregates. Particle growth is enhanced if aggregates have low density or fractal structures. On a timescale of a few Myr, an initial population of 0.1 micrometers grains may produce dense compact particles approximately 1 micrometer in size, or fluffy aggregates approximately 100 micrometers. For impact strengths less than or equal to 10(exp 6) ergs/g, a steady state is reached between coagulation of small grains and collisional disruption of larger aggregates. Formation of macroscopic aggregates requires high mechanical strengths and low aggregate densities. We assume sonic turbulence during collapse, with varied eddy size scales determining the dissipation rate or turbulence strength. The degree of collisional evolution during collapse is sensitive to the assumed small-scale structure (inner sc ale) of the turbulence. Weak turbulence results in few collisions and preserves the precollapse particle size distribution with little change. Strong turbulence tends to produce net destruction, rather than particle growth, during infall, unless inpact strengths are greater than 10(exp 6)ergs/g.
Geophysical data rarely show any smoothness at any scale, and this often makes comparison with theoretical model output difficult. However, highly fluctuating signals and fractal structures are typical of open dissipative systems with nonlinear dynamics, the focus of most geophysical research. High levels of variability are excited over a large range of scales by the combined actions of external forcing and internal instability. At very small scales we expect geophysical fields to be smooth, but these are rarely resolved with available instrumentation or simulation tools; nondifferentiable and even discontinuous models are therefore in order. We need methods of statistically analyzing geophysical data, whether measured in situ, remotely sensed or even generated by a computer model, that are adapted to these characteristics. An important preliminary task is to define statistically stationary features in generally nonstationary signals. We first discuss a simple criterion for stationarity in finite data streams that exhibit power law energy spectra and then, guided by developments in turbulence studies, we advocate the use of two ways of analyzing the scale dependence of statistical information: singular measures and qth order structure functions. In nonstationary situations, the approach based on singular measures seeks power law behavior in integrals over all possible scales of a nonnegative stationary field derived from the data, leading to a characterization of the intermittency in this (gradient-related) field. In contrast, the approach based on structure functions uses the signal itself, seeking power laws for the statistical moments of absolute increments over arbitrarily large scales, leading to a characterization of the prevailing nonstationarity in both quantitative and qualitative terms. We explain graphically, step by step, both multifractal statistics which are largely complementary to each other. The geometrical manifestations of nonstationarity and intermittency, 'roughness' and 'sparseness', respectively, are illustrated and the associated analytical (differentiability and continuity) properties are discussed. As an example, the two techniques are applied to a series of recent measurements of liquid water distributions inside marine stratocumulus decks; these are found to be multifractal over scales ranging from approximately 60 m to approximately 60 km. Finally, we define the 'mean multifractal plane' and show it to be a simple yet comprehensive tool with many applications including data intercomparison, (dynamical or stochastic) model and retrieval validations.
The Gas-Grain Simulation Facility (GGSF) is a multidisciplinary experiment laboratory being developed by NASA at Ames Research Center for delivery to Space Station Freedom in 1998. This facility will employ the low-gravity environment of the Space Station to enable aerosol experiments of much longer duration than is possible in any ground-based laboratory. Studies of fractal aggregates that are impossible to sustain on Earth will also be enabled. Three research areas within exobiology that will benefit from the GGSF are described here. An analysis of the needs of this research and of other suggested experiments has produced a list of science requirements which the facility design must accommodate. A GGSF design concept developed in the first stage of flight hardware development to meet these requirements is also described.
We propose a modified diffusion-limited aggregation (DLA) model for the evolution of fluvial drainage networks. Random walkers are introduced randomly on a grid, and each two-dimensional random walk proceeds until the walker finds a drainage network on which to accrete. This model for headward growth of drainage networks generates drainage patterns remarkably similar to actual drainages. The model also predicts statistical features which agree with actual networks, including the frequency-order (bifurcation) ratio (R(sub b) = 3.98) and the stream length-order (R(sub r) = 2.09). Using the definition of network fractal dimension D = log R(sub b)/log R(sub r), we find that our DLA model gives D = 1.87, near the observed range of D approximately equal to 1.80 - 1.85.
We study the errors brought by finite volume effects and dilution effects on the practical determination of the count probability distribution function P(sub N)(n,l), which is the probability of having N objects in a cell of volume l cubed for a set of average number density n. Dilution effects are particularly revelant to the so-called sparse sampling strategy. This work is mainly done in the framework of the Bailan & Schaeffer scaling model, which assumes that the Q-body correlation functions obey the scaling relation Xi(sub Q)(lambda r(sub l),....lambda r(sub Q) = lambda(exp -(Q-1)gamma) Xi(sub Q)(r(sub 1),....r(sub Q)). We use three synthetic samples as references to perform our analysis: a fractal generated by a Rayleigh-Levy random walk with approximately 3 x 10(exp 4) objects, a sample dominated by a spherical power-law cluster with approximately 3 x 10(exp 4) objects and a cold dark matter (CDM) universe involving approximately 3 x 10(exp 5) matter particles.
The Ulysses spacecraft has gathered data from within flows from the Sun's southern polar coronal hole, the first in situ measurement of this region. We present a brief analysis of the heliospheric magnetic field data from this region, using a fractal method. As is the case near the ecliptic, estimated spectral exponents are near 5/3 on spacecraft scales of seconds to minutes. On longer time scales, however, there appears to be a significantly different population in polar flows, which is similar to that found by the Helios spacecraft in fast solar wind flows at 0.3 AU.
Various concepts from time series analysis are used as the basis for the development of algorithms to assist in the analysis and interpretation of remote sensed imagery. An approach to trend detection that is based upon the fractal analysis of power spectrum estimates is presented. Additionally, research was conducted toward the development of a software architecture to support processing tasks associated with databases housing a variety of data. An algorithmic approach which provides for the automation of the state monitoring process is presented.
Granular materials exhibit a rich variety of dynamical behavior, much of which is poorly understood. Fractal-like stress chains, convection, a variety of wave dynamics, including waves which resemble capillary waves, l/f noise, and fractional Brownian motion provide examples. Work beginning at Duke will focus on gravity driven convection, mixing and gravitational collapse. Although granular materials consist of collections of interacting particles, there are important differences between the dynamics of a collections of grains and the dynamics of a collections of molecules. In particular, the ergodic hypothesis is generally invalid for granular materials, so that ordinary statistical physics does not apply. In the absence of a steady energy input, granular materials undergo a rapid collapse which is strongly influenced by the presence of gravity. Fluctuations on laboratory scales in such quantities as the stress can be very large-as much as an order of magnitude greater than the mean.
We present a brief, preliminary account of the interpretation of dynamic light scattering from fractal colloidal gels. For small scattering angles, and for high initial colloid particle volume fractions, the correlation functions exhibit arrested decay, reflecting the non-ergodic nature of these systems and allowing us to directly determine the elastic modulus of the gels. For smaller initial volume fractions, the correlation functions decay completely. In all cases, the initial decay is not exponential, but is instead described by a stretched exponential. We summarize the principles of a model that accounts for these data and discuss the scaling behavior of the measured parameters.
Master equation gives a more fundamental description of stochastic coagulation processes rather than popular Smoluchowski's equation. In order to examine the effect of the dynamics on the geometry of resulting aggregates, we study Master equation with a rigorous Monte Carlo algorithm. It is found that Cluster-Cluster aggregation model is a good approximation of orderly growth and the aggregates have fluffy structures with a fractal dimension approx. 2. A scaling analysis of Smoluchowski's equation also supports this conclusion.
Dust grains, although a minor constituent, play a very important role in the thermodynamics and evolution of many astronomical objects, e.g., young and evolved stars, nebulae, interstellar clouds, and nuclei of some galaxies. Since the birth of infrared astronomy over two decades ago, significant progress has been made not only in the observations of galactic dust, but also in the theoretical studies of phenomena involving dust grains. Models with increasing degree of sophistication and physical realism (in terms of grain properties, dust formation, emission processes, and grain alignment mechanisms) have become available. Here I review recent progress made in the following areas: (1) Extinction and emission of fractal grains. (2) Dust formation in radiation-driven outflows of evolved stars. (3) Transient heating and emission of very small dust grains. Where appropriate, relevant modeling results are presented and observational implications emphasized.