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NASA NTRS2003

Classifying Urban Land Covers Using Local Indices of Spatial Complexity

The skewed statistical distributions of land cover types in complex, heterogeneous urban areas limits the effectiveness of traditional spectrally based maximum-likelihood classifiers. This work examines the utility of fractal dimension and Moran's I index of spatial autocorrelation in segmenting high-resolution panchromatic and lower-resolution multispectral imagery. Tools available in the Image Characterization and Modeling System (ICAMS) were used to analyze multi-temporal and multi-platform imagery of Atlanta, Georgia. In this example, land cover change trajectories from forest or grassland to built up land covers lead to decreased spatial autocorrelation. In lower resolution imagery such as Landsat MSS, the complex details of forested land covers and urbanized areas are smoothed, and texture-based change detection is less effective. Although segmentation of panchromatic images is possible using fractal dimension or Moran's I, widely differing land covers often yield similar values of these indices. Better results are obtained when a surface of local fractal dimension or spatial autocorrelation is combined as an additional layer in a supervised maximum-likelihood multispectral classification.

Arumugam, Mahesh↗

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 NTRS1992

Three-dimensional kinematic reconnection of plasmoids with nulls

The global nonlinear dynamics of magnetic field lines in plasmoids with a pair of nulls, where B = 0, is studied. The aim of this analysis is to describe the separatrix surfaces on which singularities can occur in ideal magnetohydrodynamics because of topological changes in the field. These separatrix surfaces should locate the boundary layers associated with 3D reconnection in the presence of resistivity or inertia. It is found that the field lines exhibit chaotic scattering with several properties in common with plasmoid models without nulls (in which one component of the magnetic field never changes sign). In particular, the singular surfaces can be fractal, implying complex current density structures down to the dissipation scale. These generic features are expected to exist in typical coronal magnetic geometries exhibiting three-dimensional reconnection and the formation of current sheets.

Lau, Yun-Tung↗

NASA NTRS2000

Fractal Characterization of Multitemporal Remote Sensing Data

Scale is an "innate" concept in geographic information systems. It is recognized as something that is intrinsic to the capture, storage, manipulation, analysis, modeling, and output of space and time data within a GIS purview, yet the relative meaning and ramifications of scaling spatial and temporal data from this perspective remain enigmatic. As GISs become more sophisticated as a product of more robust software and more powerful computer systems, there is an urgent need to examine the issue of scale, and its relationship to the whole body of spatiotemporal data, as imparted in GISs. Scale is fundamental to the characterization of geo-spatial data as represented in GISs, but we have relatively little insight on how to measure the effects of scale in representing data that are acquired in different formats and exist in varying spatial, temporal and radiometric configurations. Moreover, the complexities associated with the integration of multiscaled data sets in a multitude of formats are exacerbated by the confusion of what the term "scale" is from a multidisciplinary perspective. "Scale" takes on significantly different meanings depending upon one's disciplinary background and spatial perspective which lead to substantial confusion in the input, manipulation, analysis, and output operations. Hence, we must begin to look at the universality of scale and begin to develop the theory, methods, and techniques necessary to advance knowledge on the "Science of Scale' across all disciplines that use GISs.

Quattrochi, Dale A.↗

NASA NTRS2008

Mapping Snow Grain Size over Greenland from MODIS

This paper presents a new automatic algorithm to derive optical snow grain size (SGS) at 1 km resolution using Moderate Resolution Imaging Spectroradiometer (MODIS) measurements. Differently from previous approaches, snow grains are not assumed to be spherical but a fractal approach is used to account for their irregular shape. The retrieval is conceptually based on an analytical asymptotic radiative transfer model which predicts spectral bidirectional snow reflectance as a function of the grain size and ice absorption. The analytical form of solution leads to an explicit and fast retrieval algorithm. The time series analysis of derived SGS shows a good sensitivity to snow metamorphism, including melting and snow precipitation events. Preprocessing is performed by a Multi-Angle Implementation of Atmospheric Correction (MAIAC) algorithm, which includes gridding MODIS data to 1 km resolution, water vapor retrieval, cloud masking and an atmospheric correction. MAIAC cloud mask (CM) is a new algorithm based on a time series of gridded MODIS measurements and an image-based rather than pixel-based processing. Extensive processing of MODIS TERRA data over Greenland shows a robust performance of CM algorithm in discrimination of clouds over bright snow and ice. As part of the validation analysis, SGS derived from MODIS over selected sites in 2004 was compared to the microwave brightness temperature measurements of SSM\I radiometer, which is sensitive to the amount of liquid water in the snowpack. The comparison showed a good qualitative agreement, with both datasets detecting two main periods of snowmelt. Additionally, MODIS SGS was compared with predictions of the snow model CROCUS driven by measurements of the automatic whether stations of the Greenland Climate Network. We found that CROCUS grain size is on average a factor of two larger than MODIS-derived SGS. Overall, the agreement between CROCUS and MODIS results was satisfactory, in particular before and during the first melting period in mid-June. Following detailed time series analysis of SGS for four permanent sites, the paper presents SGS maps over the Greenland ice sheet for the March-September period of 2004.

Lyapustin, Alexei↗

NASA NTRS1998

Fractal Characterization of Multitemporal Scaled Remote Sensing Data

Scale is an "innate" concept in geographic information systems. It is recognized as something that is intrinsic to the ingestion, storage, manipulation, analysis, modeling, and output of space and time data within a GIS purview, yet the relative meaning and ramifications of scaling spatial and temporal data from this perspective remain enigmatic. As GISs become more sophisticated as a product of more robust software and more powerful computer systems, there is an urgent need to examine the issue of scale, and its relationship to the whole body of spatiotemporal data, as imparted in GISS. Scale is fundamental to the characterization of geo-spatial data as represented in GISS, but we have relatively little insight on the effects of, or how to measure the effects of, scale in representing multiscaled data; i.e., data that are acquired in different formats (e.g., map, digital) and exist in varying spatial, temporal, and in the case of remote sensing data, radiometric, configurations. This is particularly true in the emerging era of Integrated GISs (IGIS), wherein spatial data in a variety of formats (e.g., raster, vector) are combined with multiscaled remote sensing data, capable of performing highly sophisticated space-time data analyses and modeling. Moreover, the complexities associated with the integration of multiscaled data sets in a multitude of formats are exacerbated by the confusion of what the term "scale" is from a multidisciplinary perspective; i.e., "scale" takes on significantly different meanings depending upon one's disciplinary background and spatial perspective which can lead to substantive confusion in the input, manipulation, analyses, and output of IGISs (Quattrochi, 1993). Hence, we must begin to look at the universality of scale and begin to develop the theory, methods, and techniques necessary to advance knowledge on the "Science of Scale" across a wide number of spatial disciplines that use GISs.

Quattrochi, Dale A.↗

NASA NTRS2001

Fractal Characterization of Multitemporal Remote Sensing Data

Scale is an 'innate' concept in geographical information systems (GIS). It is recognized as something that is intrinsic to the capture, storage, manipulation, analysis, modelling, and output of space and time data within a GIS purview, yet the relative meaning and ramifications of scaling spatial and temporal data from this perspective remain enigmatic. As GIS becomes more sophisticated as a product of more robust software and more powerful computer systems, there is an urgent need to examine the issue of scale, and its relationship to the whole body of spatiotemporal data, as imparted in GIS. Scale is fundamental to the characterization of geo-spatial data as represented in GIS, but we have relatively little insight on how to measure the effects of scale in representing data that are acquired in different formats and exist in varying spatial, temporal and radiometric configurations. Moreover, the complexities associated with the integration of multi-scaled data sets in a multitude of formats are exacerbated by the confusion of what the term 'scale' means from a multidisciplinary perspective. 'Scale' takes on significantly different meanings depending upon one's disciplinary background and spatial perspective, which can lead to substantial confusion in the input, manipulation, analysis, and output operations. Hence, we must begin to look at the universality of scale and begin to develop the theory, methods and techniques necessary to advance knowledge on the 'Science of scale' across all disciplines that use GIS.

Quattrochi, Dale A.↗

NASA NTRS1992

Fractal dimension of alumina aggregates grown in two dimensions

The concepts of fractal geometry are applied to the analysis of 0.4-micron alumina constrained to agglomerate in two dimensions. Particles were trapped at the bottom surface of a drop of a dilute suspension, and the agglomeration process was directly observed, using an inverted optical microscope. Photographs were digitized and analyzed, using three distinct approaches. The results indicate that the agglomerates are fractal, having a dimension of approximately 1.5, which agrees well with the predictions of the diffusion-limited cluster-cluster aggregation model.

Larosa, Judith L.↗

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 NTRS1996

Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics

We postulated that aging is associated with disruption in the fractallike long-range correlations that characterize healthy sinus rhythm cardiac interval dynamics. Ten young (21-34 yr) and 10 elderly (68-81 yr) rigorously screened healthy subjects underwent 120 min of continuous supine resting electrocardiographic recording. We analyzed the interbeat interval time series using standard time and frequency domain statistics and using a fractal measure, detrended fluctuation analysis, to quantify long-range correlation properties. In healthy young subjects, interbeat intervals demonstrated fractal scaling, with scaling exponents (alpha) from the fluctuation analysis close to a value of 1.0. In the group of healthy elderly subjects, the interbeat interval time series had two scaling regions. Over the short range, interbeat interval fluctuations resembled a random walk process (Brownian noise, alpha = 1.5), whereas over the longer range they resembled white noise (alpha = 0.5). Short (alpha s)- and long-range (alpha 1) scaling exponents were significantly different in the elderly subjects compared with young (alpha s = 1.12 +/- 0.19 vs. 0.90 +/- 0.14, respectively, P = 0.009; alpha 1 = 0.75 +/- 0.17 vs. 0.99 +/- 0.10, respectively, P = 0.002). The crossover behavior from one scaling region to another could be modeled as a first-order autoregressive process, which closely fit the data from four elderly subjects. This implies that a single characteristic time scale may be dominating heartbeat control in these subjects. The age-related loss of fractal organization in heartbeat dynamics may reflect the degradation of integrated physiological regulatory systems and may impair an individual's ability to adapt to stress.

NASA Discipline Cardiopulmonary↗

NASA NTRS1990

Models for the hotspot distribution

Published hotspot catalogs all show a hemispheric concentration beyond what can be expected by chance. Cumulative distributions about the center of concentration are described by a power law with a fractal dimension closer to 1 than 2. Random sets of the corresponding sizes do not show this effect. A simple shift of the random sets away from a point would produce distributions similar to those of hotspot sets. The possible relation of the hotspots to the locations of ridges and subduction zones is tested using large sets of randomly-generated points to estimate areas within given distances of the plate boundaries. The probability of finding the observed number of hotspots within 10 deg of the ridges is about what is expected.

Jurdy, Donna M.↗

NASA NTRS2008

Multiscale Computer Simulation of Failure in Aerogels

Aerogels have been of interest to the aerospace community primarily for their thermal properties, notably their low thermal conductivities. While such gels are typically fragile, recent advances in the application of conformal polymer layers to these gels has made them potentially useful as lightweight structural materials as well. We have previously performed computer simulations of aerogel thermal conductivity and tensile and compressive failure, with results that are in qualitative, and sometimes quantitative, agreement with experiment. However, recent experiments in our laboratory suggest that gels having similar densities may exhibit substantially different properties. In this work, we extend our original diffusion limited cluster aggregation (DLCA) model for gel structure to incorporate additional variation in DLCA simulation parameters, with the aim of producing DLCA clusters of similar densities that nevertheless have different fractal dimension and secondary particle coordination. We perform particle statics simulations of gel strain on these clusters, and consider the effects of differing DLCA simulation conditions, and the resultant differences in fractal dimension and coordination, on gel strain properties.

Good, Brian S.↗

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 NTRS1984

Nonlinear problems in flight dynamics

A comprehensive framework is proposed for the description and analysis of nonlinear problems in flight dynamics. Emphasis is placed on the aerodynamic component as the major source of nonlinearities in the flight dynamic system. Four aerodynamic flows are examined to illustrate the richness and regularity of the flow structures and the nature of the flow structures and the nature of the resulting nonlinear aerodynamic forces and moments. A framework to facilitate the study of the aerodynamic system is proposed having parallel observational and mathematical components. The observational component, structure is described in the language of topology. Changes in flow structure are described via bifurcation theory. Chaos or turbulence is related to the analogous chaotic behavior of nonlinear dynamical systems characterized by the existence of strange attractors having fractal dimensionality. Scales of the flow are considered in the light of ideas from group theory. Several one and two degree of freedom dynamical systems with various mathematical models of the nonlinear aerodynamic forces and moments are examined to illustrate the resulting types of dynamical behavior. The mathematical ideas that proved useful in the description of fluid flows are shown to be similarly useful in the description of flight dynamic behavior.

Chapman, G. T.↗

NASA NTRS2004

A Hierarchical Approach to Fracture Mechanics

Recent research conducted under NASA LaRC's Creativity and Innovation Program has led to the development of an initial approach for a hierarchical fracture mechanics. This methodology unites failure mechanisms occurring at different length scales and provides a framework for a physics-based theory of fracture. At the nanoscale, parametric molecular dynamic simulations are used to compute the energy associated with atomic level failure mechanisms. This information is used in a mesoscale percolation model of defect coalescence to obtain statistics of fracture paths and energies through Monte Carlo simulations. The mathematical structure of predicted crack paths is described using concepts of fractal geometry. The non-integer fractal dimension relates geometric and energy measures between meso- and macroscales. For illustration, a fractal-based continuum strain energy release rate is derived for inter- and transgranular fracture in polycrystalline metals.

Saether, Erik↗

NASA NTRS1994

Beyond maximum entropy: Fractal Pixon-based image reconstruction

We have developed a new Bayesian image reconstruction method that has been shown to be superior to the best implementations of other competing methods, including Goodness-of-Fit methods such as Least-Squares fitting and Lucy-Richardson reconstruction, as well as Maximum Entropy (ME) methods such as those embodied in the MEMSYS algorithms. Our new method is based on the concept of the pixon, the fundamental, indivisible unit of picture information. Use of the pixon concept provides an improved image model, resulting in an image prior which is superior to that of standard ME. Our past work has shown how uniform information content pixons can be used to develop a 'Super-ME' method in which entropy is maximized exactly. Recently, however, we have developed a superior pixon basis for the image, the Fractal Pixon Basis (FPB). Unlike the Uniform Pixon Basis (UPB) of our 'Super-ME' method, the FPB basis is selected by employing fractal dimensional concepts to assess the inherent structure in the image. The Fractal Pixon Basis results in the best image reconstructions to date, superior to both UPB and the best ME reconstructions. In this paper, we review the theory of the UPB and FPB pixon and apply our methodology to the reconstruction of far-infrared imaging of the galaxy M51. The results of our reconstruction are compared to published reconstructions of the same data using the Lucy-Richardson algorithm, the Maximum Correlation Method developed at IPAC, and the MEMSYS ME algorithms. The results show that our reconstructed image has a spatial resolution a factor of two better than best previous methods (and a factor of 20 finer than the width of the point response function), and detects sources two orders of magnitude fainter than other methods.

Puetter, Richard C.↗

NASA NTRS2001

Cloud Spatial Structure and 3D Radiative Transfer

Cloud radiative properties are sensitive to drop size and other parameters of cloud micro-structure, but also to cloud shape, spacing, and other parameters of cloud macro-structure, including internal fractal structure. New information on cloud structure is being derived from a variety of cloud radars and lidars. Ongoing field programs such as DoE/ARM are improving the measurement and modelling of physical and radiative properties of clouds. A parallel effort is underway to improve cloud remote sensing, especially from the new suite of EOS (Earth Observing System) instruments which are beginning to provide higher spectral, spatial resolution, and/or angular resolution. Key parameters for improving pixel-scale retrievals are cloud thickness and photon mean-free-path, which together determine the scale of "radiative smoothing" of cloud fluxes and radiances. This scale has been observed as a change in the spatial spectrum of Landsat cloud radiances, and was also recently found with the Goddard micropulse lidar, by searching for returns from directions nonparallel to the incident beam. "Offbeam" Lidar returns are now being used to estimate the cloud "radiative Green's function", (G). G depends on cloud thickness and may be used to retrieve that important quantity. G is also being applied to improving simple estimates of cloud radiative properties that are based on the "Independent Pixel Approximation" or IPA. This and other measurements of 3D transfer in clouds, coupled with Monte Carlo and other 3D transfer methods, are beginning to provide a better understanding of the dependence of radiation on cloud inhomogeneity, and to suggest new retrieval and parameterization algorithms which take account of cloud inhomogeneity. An international "Intercomparison of 3D Radiation Codes" or I3RC, program is underway to coordinate and evaluate the variety of 3D radiative transfer methods now available, and to make them more widely available. Information is on the Web at: http://climate.gsfc.nasa.gov/I3RC. Input consists of selected cloud fields derived from data sources such as radar, microwave and satellite, and from models involved in the GEWEX Cloud Systems Studies. Output is selected radiative quantities that characterize the large-scale properties of the fields of radiative fluxes and heating. Several example cloud fields will be used to illustrate the effects of cloud inhomogeneity and 3D radiation.

Cahalan, Robert F.↗

NASA NTRS1999

Cloud Spatial Structure and 3D Radiative Transfer

Cahalan, Robert F.↗