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NASA NTRS2000

Applications of Fractal Analytical Techniques in the Estimation of Operational Scale

The observational scale and the resolution of remotely sensed imagery are essential considerations in the interpretation process. Many atmospheric, hydrologic, and other natural and human-influenced spatial phenomena are inherently scale dependent and are governed by different physical processes at different spatial domains. This spatial and operational heterogeneity constrains the ability to compare interpretations of phenomena and processes observed in higher spatial resolution imagery to similar interpretations obtained from lower resolution imagery. This is a particularly acute problem, since longterm global change investigations will require high spatial resolution Earth Observing System (EOS), Landsat 7, or commercial satellite data to be combined with lower resolution imagery from older sensors such as Landsat TM and MSS. Fractal analysis is a useful technique for identifying the effects of scale changes on remotely sensed imagery. The fractal dimension of an image is a non-integer value between two and three which indicates the degree of complexity in the texture and shapes depicted in the image. A true fractal surface exhibits self-similarity, a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution, and the slope of the fractal dimension-resolution relationship would be zero. Most geographical phenomena, however, are not self-similar at all scales, but they can be modeled by a stochastic fractal in which the scaling properties of the image exhibit patterns that can be described by statistics such as area-perimeter ratios and autocovariances. Stochastic fractal sets relax the self-similarity assumption and measure many scales and resolutions to represent the varying form of a phenomenon as the pixel size is increased in a convolution process. We have observed that for images of homogeneous land covers, the fractal dimension varies linearly with changes in resolution or pixel size over the range of past, current, and planned space-borne sensors. This relationship differs significantly in images of agricultural, urban, and forest land covers, with urban areas retaining the same level of complexity, forested areas growing smoother, and agricultural areas growing more complex as small pixels are aggregated into larger, mixed pixels. Images of scenes having a mixture of land covers have fractal dimensions that exhibit a non-linear, complex relationship to pixel size. Measuring the fractal dimension of a difference image derived from two images of the same area obtained on different dates showed that the fractal dimension increased steadily, then exhibited a sharp decrease at increasing levels of pixel aggregation. This breakpoint of the fractal dimension/resolution plot is related to the spatial domain or operational scale of the phenomenon exhibiting the predominant visible difference between the two images (in this case, mountain snow cover). The degree to which an image departs from a theoretical ideal fractal surface provides clues as to how much information is altered or lost in the processes of rescaling and rectification. The measured fractal dimension of complex, composite land covers such as urban areas also provides a useful textural index that can assist image classification of complex scenes.

Emerson, Charles W.↗

NASA NTRS1992

The solar cycle; Proceedings of the National Solar Observatory/Sacramento Peak 12th Summer Workshop, Sunspot, NM, Oct. 15-18, 1991

Attention is given to a flux-transport model, the effect of fractal distribution on the evolution of solar surface magnetic fields, active nests on the sun, magnetic flux transport in solar active regions, recent advances in stellar cycle research, magnetic intermittency on the sun, a search for existence of large-scale motions on the sun, and new solar cycle data from the NASA/NSO spectromagnetograph. Attention is also given to the solar cycle variation of coronal temperature during cycle 22, the distribution of the north-south asymmetry for the various activity cycles, solar luminosity variation, a two-parameter model of total solar irradiance variation over the solar cycle, the origin of the solar cycle, nonlinear feedbacks in the solar dynamo, and long-term dynamics of the solar cycle.

Harvey, Karen L.↗

NASA NTRS2011

Numerical Simulations of Single and Multiple Scattering by Fractal Ice Clusters

We consider the scattering model in the form of a vertically and horizontally homogeneous particulate slab of an arbitrary optical thickness composed of widely separated fractal aggregates built of small spherical ice monomers. The aggregates are generated by applying three different approaches, including simulated cluster-cluster aggregation (CCA) and diffusion-limited aggregation (DLA) procedures. Having in mind radar remote-sensing applications, we report and analyze the results of computations of the backscattering circular polarization ratio obtained using efficient superposition T-matrix and vector radiative-transfer codes. The computations have been performed at a wavelength of 12.6 cm for fractal aggregates with the following characteristics: monomer refractive index m=1.78+i0.003, monomer radius r=1 cm, monomer packing density p=0.2, overall aggregate radii R in the range 4<=R<=10 cm and fractal dimensions D(sub f) 2.5 and 3. We show that for aggregates generated with simulated CCA and DLA procedures, the respective values of the backscattering circular polarization ratio differ weakly for D(sub f) 2.5, but the differences can increase somewhat for D(sub f)3, especially in case of an optically semi-infinite medium. For aggregates with a spheroidal overall shape, the dependence of the circular polarization ratio on the cluster morphology can be quite significant and increases with increasing the aspect ratio of the circumscribing spheroid.

Dlugach, Janna M.↗

NASA NTRS1987

Algorithmic commonalities in the parallel environment

The ultimate aim of this project was to analyze procedures from substantially different application areas to discover what is either common or peculiar in the process of conversion to the Massively Parallel Processor (MPP). Three areas were identified: molecular dynamic simulation, production systems (rule systems), and various graphics and vision algorithms. To date, only selected graphics procedures have been investigated. They are the most readily available, and produce the most visible results. These include simple polygon patch rendering, raycasting against a constructive solid geometric model, and stochastic or fractal based textured surface algorithms. Only the simplest of conversion strategies, mapping a major loop to the array, has been investigated so far. It is not entirely satisfactory.

Mcanulty, Michael A.↗

NASA NTRS1991

Environmental analysis using integrated GIS and remotely sensed data - Some research needs and priorities

This paper discusses some basic scientific issues and research needs in the joint processing of remotely sensed and GIS data for environmental analysis. Two general topics are treated in detail: (1) scale dependence of geographic data and the analysis of multiscale remotely sensed and GIS data, and (2) data transformations and information flow during data processing. The discussion of scale dependence focuses on the theory and applications of spatial autocorrelation, geostatistics, and fractals for characterizing and modeling spatial variation. Data transformations during processing are described within the larger framework of geographical analysis, encompassing sampling, cartography, remote sensing, and GIS. Development of better user interfaces between image processing, GIS, database management, and statistical software is needed to expedite research on these and other impediments to integrated analysis of remotely sensed and GIS data.

Davis, Frank W.↗

NASA NTRS1997

Multitemporal and Multiscaled Fractal Analysis of Landsat Satellite Data Using the Image Characterization and Modeling System (ICAMS)

The Image Characterization And Modeling System (ICAMS) is a public domain software package that is designed to provide scientists with innovative spatial analytical tools to visualize, measure, and characterize landscape patterns so that environmental conditions or processes can be assessed and monitored more effectively. In this study ICAMS has been used to evaluate how changes in fractal dimension, as a landscape characterization index, and resolution, are related to differences in Landsat images collected at different dates for the same area. Landsat Thematic Mapper (TM) data obtained in May and August 1993 over a portion of the Great Basin Desert in eastern Nevada were used for analysis. These data represent contrasting periods of peak "green-up" and "dry-down" for the study area. The TM data sets were converted into Normalized Difference Vegetation Index (NDVI) images to expedite analysis of differences in fractal dimension between the two dates. These NDVI images were also resampled to resolutions of 60, 120, 240, 480, and 960 meters from the original 30 meter pixel size, to permit an assessment of how fractal dimension varies with spatial resolution. Tests of fractal dimension for two dates at various pixel resolutions show that the D values in the August image become increasingly more complex as pixel size increases to 480 meters. The D values in the May image show an even more complex relationship to pixel size than that expressed in the August image. Fractal dimension for a difference image computed for the May and August dates increase with pixel size up to a resolution of 120 meters, and then decline with increasing pixel size. This means that the greatest complexity in the difference images occur around a resolution of 120 meters, which is analogous to the operational domain of changes in vegetation and snow cover that constitute differences between the two dates.

Quattrochi, Dale A.↗

NASA NTRS2004

Fractal Variation with Changing Line Length: A Potential Problem for Planetary Lava Flow Identification

Fractals are objects that are generally self similar at all scales. Coastlines, mountains, river systems, planetary orbits and some mathematical objects are all examples of fractals. Bruno et al. used the structured walk model of Richardson to establish that lava flows are fractals and that lava flow morphology could be determined by looking at the fractal dimension of flow margins. They determined that Hawaiian a.a flows have fractal dimensions that range from 1.05 to 1.09 and that the pahoehoe lava flows have a fractal dimension from 1.13 to 1.23. We have analyzed a number of natural and simulated lava flow margins and find that the fractal dimension varies according to the number and length of rod lengths used in the structured walk method. The potential variation we find in our analyses is sufficiently large so that unambiguous determination of lava flow morphology is problematic for some flows. We suggest that the structured walk method can provide meaningful fractal dimensions if rod lengths employed in the analysis provide a best-fit residual of greater than 0.98, as opposed to the 0.95 cutoff used in previous studies. We also find that the use of more than 4 rod lengths per analysis also reduces ambiguity in the results.

Hudson, Richard K.↗

DOE OSTI2025

Incorporating long-range dependence and fractal features in turbulence spectra

We introduce an advanced turbulence spectrum model developed from mathematical foundations from a covariance function class and empirically validated using extensive field data. This model captures the complex dynamics of long-range dependence, and fractal characteristics prevalent in riverine and atmospheric boundary layer (ABL) flows that are ignored by classical spectrum models, such as IEC (International Electrotechnical Commission) von Kármán and Kaimal model. The model delineates scaling behaviors across distinct frequency bands and offers substantial flexibility through five well-defined parameters each characterizing a distinct physical aspect of the velocity time series. A detailed procedure for obtaining each parameter from time series data is outlined. The comprehensive validations with field data from tidal currents and ABL flows substantiate the model’s fidelity in accurately replicating observed phenomena. This validation establishes the reliability of the proposed model and, when incorporated into stochastic full-field simulators such as TurbSim, demonstrates its potential to advance the predictive modeling and analysis of turbulent flows in environmental science and engineering contexts.

Cheng, Shyuan [Univ. of Illinois at Urbana-Champai↗

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 NTRS1993

Implementation and operation of three fractal measurement algorithms for analysis of remote-sensing data

Fractal geometry is increasingly becoming a useful tool for modeling natural phenomena. As an alternative to Euclidean concepts, fractals allow for a more accurate representation of the nature of complexity in natural boundaries and surfaces. The purpose of this paper is to introduce and implement three algorithms in C code for deriving fractal measurement from remotely sensed data. These three methods are: the line-divider method, the variogram method, and the triangular prism method. Remote-sensing data acquired by NASA's Calibrated Airborne Multispectral Scanner (CAMS) are used to compute the fractal dimension using each of the three methods. These data were obtained as a 30 m pixel spatial resolution over a portion of western Puerto Rico in January 1990. A description of the three methods, their implementation in PC-compatible environment, and some results of applying these algorithms to remotely sensed image data are presented.

Jaggi, S.↗

NASA NTRS1992

Multi-resolution processing for fractal analysis of airborne remotely sensed data

Fractal geometry is increasingly becoming a useful tool for modeling natural phenomenon. As an alternative to Euclidean concepts, fractals allow for a more accurate representation of the nature of complexity in natural boundaries and surfaces. Since they are characterized by self-similarity, an ideal fractal surface is scale-independent; i.e. at different scales a fractal surface looks the same. This is not exactly true for natural surfaces. When viewed at different spatial resolutions parts of natural surfaces look alike in a statistical manner and only for a limited range of scales. Images acquired by NASA's Thermal Infrared Multispectral Scanner are used to compute the fractal dimension as a function of spatial resolution. Three methods are used to determine the fractal dimension - Schelberg's line-divider method, the variogram method, and the triangular prism method. A description of these methods and the results of applying these methods to a remotely-sensed image is also presented. Five flights were flown in succession at altitudes of 2 km (low), 6 km (mid), 12 km (high), and then back again at 6 km and 2 km. The area selected was the Ross Barnett reservoir near Jackson, Mississippi. The mission was flown during the predawn hours of 1 Feb. 1992. Radiosonde data was collected for that duration to profile the characteristics of the atmosphere. This corresponds to 3 different pixel sizes - 5m, 15m, and 30m. After, simulating different spatial sampling intervals within the same image for each of the 3 image sets, the results are cross-correlated to compare the extent of detail and complexity that is obtained when data is taken at lower spatial intervals.

Jaggi, S.↗

DOE OSTI2020

Cellular Response to Proton Irradiation: A Simulation Study with TOPAS-nBio

The cellular response to ionizing radiation continues to be of significant research interest in cancer radiotherapy, and DNA is recognized as the critical target for most of the biologic effects of radiation. Incident particles can cause initial DNA damages through physical and chemical interactions within a short time scale. Initial DNA damages can undergo repair via different pathways available at different stages of the cell cycle. The misrepair of DNA damage results in genomic rearrangement and causes mutations and chromosome aberrations, which are drivers of cell death. This work presents an integrated study of simulating cell response after proton irradiation with energies of 0.5–500 MeV (LET of 60–0.2 keV/µm). A model of a whole nucleus with fractal DNA geometry was implemented in TOPAS-nBio for initial DNA damage simulations. The default physics and chemistry models in TOPAS-nBio were used to describe interactions of primary particles, secondary particles, and radiolysis products within the nucleus. The initial DNA double-strand break (DSB) yield was found to increase from 6.5 DSB/Gy/Gbp at low-linear energy transfer (LET) of 0.2 keV/µm to 21.2 DSB/Gy/Gbp at high LET of 60 keV/µm. A mechanistic repair model was applied to predict the characteristics of DNA damage repair and dose response of chromosome aberrations. It was found that more than 95% of the DSBs are repaired within the first 24 h and the misrepaired DSB fraction increases rapidly with LET and reaches 15.8% at 60 keV/µm with an estimated chromosome aberration detection threshold of 3 Mbp. The dicentric and acentric fragment yields and the dose response of micronuclei formation after proton irradiation were calculated and compared with experimental results.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

NASA NTRS1991

Fractal dynamics of bioconvective patterns

Biologically generated cellular patterns, sometimes called bioconvective patterns, are found to cluster into aggregates which follow fractal growth dynamics akin to diffusion-limited aggregation (DLA) models. The pattern formed is self-similar with fractal dimension of 1.66 +/-0.038. Bioconvective DLA branching results from thermal roughening which shifts the balance between ordering viscous forces and disordering cell motility and random diffusion. The phase diagram for pattern morphology includes DLA, boundary spokes, random clusters, and reverse clusters.

Noever, David A.↗

NASA NTRS2022

Optical properties of morphologically complex black carbon aerosols: Effects of coatings

Optical properties are computed for fractal-like aggregate black carbon (BC) aerosols coated with different substances. Two models are used for these aerosols: (i) the coated aggregate model (model I), where coating is added to the voids and surroundings of BC fractal-like aggregates; and (ii) the closed-cell aggregate model (model II), where coating is added concentrically to each monomer of BC fractal-like aggregates. Our results favor choosing the coated aggregate model (model I) to simulate scattering and absorption by coated BC aerosols because this model is morphologically more realistic, and because this model yields mass absorption cross section (MAC) and backscattering linear depolarization ratio (LDR) values that are consistent with field measurements. Moreover the corresponding computed degree of linear polarization (DoLP) and LDR values are very sensitive to changes in the coating volume fraction (fvol) and the coating refractive index (m). With the same absorbing BC core, the MAC value increases steadily with increasing fvol and m values. For example, using a density of 1.8 g/cm 3 and BC refractive index 1.95 + i0.79, the calculated MAC values for uncoated BC aggregates range from 6.2 to 6.8 m 2 /g at 0.55 µm. When coating material is applied to the BC aggregates, the calculated MAC values for model I particles increase to between 9.8 and 13.2 m 2 /g (depending on m) when fvol = 87.5%. The backscattering LDR values also tend to increase with the increasing m values for the shapes and sizes considered in our study. For model I particles, the backscattering LDR values span a wide range of 4.2–27.8% at a wavelength of 0.35 µm at fvol = 87.5% when m increases from 1.33 to 1.55. Our results are relevant to analyses of polarimetric and lidar observations of smoke particles, especially when these particles undergo hygroscopic growth.

Black carbon aerosols↗

DOE OSTI2021

Analysis of convection heat transfer on multiscale rough superhydrophobic and liquid infused surfaces

Multiscale rough superhydrophobic or slippery liquid infused porous surfaces have gained much interest in recent years for their improved transport phenomena properties. While there have been several studies on drag reduction and condensation on non-wetting surfaces, convection heat transfer that is important in many thermal and thermochemical applications has not been addressed systematically. Here, this article utilizes a fractal description of rough surface topographies to develop analytical models for the Nusselt number and the thermal hydraulic factor for fluid flow and heat transfer inside a cylinder with non-wetting surfaces. For air-infused superhydrophobic surfaces, the model considers the dynamic stability of the air/fluid interface in the asperities. Using the analytical formulations and the stability criteria, systematic studies are presented on the effects of the fractal surface parameters, cylinder radius and Reynolds number on the convective heat transfer characteristics, from which surface texture design maps are developed for maximizing the convection heat transfer. It is shown that multiscale non-wetting surfaces are most effective in the range of lower Reynolds number and small cylinder radius for achieving the best convective heat transfer and thermal hydraulic performance. Applying the models to actual non-wetting surface topographies fabricated using electrodeposition and chemical etching, it is shown that contrary to prevailing notion, superhydrophobicity, characterized by the highest contact angles, does not always lead to the maximum convective heat transfer performance, and that under certain fluid flow conditions, hydrophobic surfaces may offer a greater thermal performance.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Machine learning-assisted upscaling analysis of reservoir rock core properties based on micro-computed tomography imagery

Optimum solutions for geologic modeling and reservoir simulation in industries such as oil and gas recovery and carbon capture and storage require accurate characterization of reservoir properties, which are often heterogeneous. In this study, high-quality micro-computed tomography (CT) images (1.475-μm/pixel resolution) of a sandstone core acquired from the Bell Creek oil field, USA, were used to provide nondestructive analysis of pore- and core-scale heterogeneity across measurement scales of 94–566 μm. In addition to characterizing the as-received sample, the core sample was flooded with brine to evaluate the capacity of the core sample to receive injected fluids. The micro-CT images were systematically segmented into pore spaces and grains via machine learning (ML) steps including image preprocessing, label creation using a traditional ML method based on limited manual image annotation, and finally U-Net segmentation. The segmented image stacks were reconstructed into digital cubes of various scales of voxel lengths. The 3D porosity values were calculated for all the digital cubes, and the fractal dimensions of the cubes were estimated using a box-counting method. The results showed that smaller cubes had greater heterogeneity and that the porosity values could be accurately estimated by fractal dimension and voxel lengths using ML models. For the core sample with brine flooding, the ratio of pores filled by brine to the total pore space was related to the porosity and could also be accurately estimated by porosity, fractal dimension, and voxel lengths using ML models. In conclusion, the results of this study demonstrate that the concept of fractal dimension can be a useful vector to perform upscaling analysis of sandstone rock heterogeneity from the pore to core scale and that fractal dimensions can be used to estimate porosity values and pore space-filling capacity across those scales.

58 GEOSCIENCES↗

NASA NTRS1993

Desktop chaotic systems: Intuition and visualization

This paper presents a dynamic study of the Wildwood Pendulum, a commercially available desktop system which exhibits a strange attractor. The purpose of studying this chaotic pendulum is twofold: to gain insight in the paradigmatic approach of modeling, simulating, and determining chaos in nonlinear systems; and to provide a desktop model of chaos as a visual tool. For this study, the nonlinear behavior of this chaotic pendulum is modeled, a computer simulation is performed, and an experimental performance is measured. An assessment of the pendulum in the phase plane shows the strange attractor. Through the use of a box-assisted correlation dimension methodology, the attractor dimension is determined for both the model and the experimental pendulum systems. Correlation dimension results indicate that the pendulum and the model are chaotic and their fractal dimensions are similar.

Bright, Michelle M.↗

NASA NTRS1999

Nanoporous Silica Thermal Insulation for Space Shuttle Cryogenic Tanks: A Case Study

Nanoporous silica (with typical 10-50 nm porous radii) has been benchmarked for thermal insulators capable of maintaining a 150 K/cm temperature gradient. For cryogenic use in aerospace applications, the combined features for low-density, high thermal insulation factors, and low temperature compatibility are demonstrated in a prototype sandwich structure between two propulsion tanks. Theoretical modelling based on a nanoscale fractal structure suggest that the thermal conductivity scales proportionally (exponent, 1.7) with the material density-lower density increases the thermal insulation rating. Computer simulations, however, support the optimization tradeoff between material strength (Young moduli, proportional to density with exponent, 3.7), the characteristic (colloidal silica, less than 5 nm) particle size, and the thermal rating. The results of these simulations indicate that as nanosized particles are incorporated into the silica backbone, the resulting physical properties will be tailored by the smallest characteristic length and their fractal interconnections (dimension and fractal size). The application specifies a prototype panel which takes advantage of the processing flexibility inherent in sol-gel chemistry.

Noever, David A.↗