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All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Fractal dimension of bioconvection patterns

Shallow cultures of the motile algal strain, Euglena gracilis, were concentrated to 2 x 10 to the 6th organisms per ml and placed in constant temperature water baths at 24 and 38 C. Bioconvective patterns formed an open two-dimensional structure with random branches, similar to clusters encountered in the diffusion-limited aggregation (DLA) model. When averaged over several example cultures, the pattern was found to have no natural length scale, self-similar branching, and a fractal dimension (d about 1.7). These agree well with the two-dimensional DLA.

Noever, David A.↗
NASA NTRS1991

Fractal dimension in nonhyperbolic chaotic scattering

In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.

Lau, Yun-Tung↗
NASA NTRS1993

A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes

Sea level changes are related to both climatic variations and tectonic movements. The fractal dimensions of several sea level curves were compared to a modern climatic fractal dimension of 1.26 established for annual precipitation records. A similar fractal dimension (1.22) based on delta(O-18/O-16) in deep-sea sediments has been suggested to characterize climatic change during the past 2 m.y. Our analysis indicates that sea level changes over the past 150,000 to 250,000 years also exhibit comparable fractal dimensions. Sea level changes for periods longer than about 30 m.y. are found to produce fractal dimensions closer to unity and Missourian (Late Pennsylvanian) sea level changes yield a fractal dimension of 1.41. The fact that these sea level curves all possess fractal dimensions less than 1.5 indicates that sea level changes exhibit nonperiodic, long-run persistence. The different fractal dimensions calculated for the various time periods could be the result of a characteristic overprinting of the sediment recored by prevailing processes during deposition. For example, during the Quaternary, glacio-eustatic sea level changes correlate well with the present climatic signature. During the Missourian, however, mechanisms such as plate reorganization may have dominated, resulting in a significantly different fractal dimension.

Hsui, Albert T.↗
DOE OSTI2021

Predicting Fluid Flow Regime, Permeability, and Diffusivity in Mudrocks from Multiscale Pore Characterisation

In geoenergy applications, mudrocks prevent fluids to leak from temporary (H 2 , CH 4 ) or permanent (CO 2 , radioactive waste) storage/disposal sites and serve as a source and reservoir for unconventional oil and gas. Understanding transport properties integrated with dominant fluid flow mechanisms in mudrocks is essential to better predict the performance of mudrocks within these applications. In this study, small-angle neutron scattering (SANS) experiments were conducted on 71 samples from 13 different sets of mudrocks across the globe to capture the pore structure of nearly the full pore size spectrum (2 nm–5 μm). We develop fractal models to predict transport properties (permeability and diffusivity) based on the SANS-derived pore size distributions. The results indicate that transport phenomena in mudrocks are intrinsically pore size-dependent. Depending on hydrostatic pore pressures, transition flow develops in micropores, slip flow in meso- and macropores, and continuum flow in larger macropores. Fluid flow regimes progress towards larger pore sizes during reservoir depletion or smaller pore sizes during fluid storage, so when pressure is decreased or increased, respectively. Capturing the heterogeneity of mudrocks by considering fractal dimension and tortuosity fractal dimension for defined pore size ranges, fractal models integrate apparent permeability with slip flow, Darcy permeability with continuum flow, and gas diffusivity with diffusion flow in the matrix. This new model of pore size-dependent transport and integrated transport properties using fractal models yields a systematic approach that can also inform multiscale multi-physics models to better understand fluid flow and transport phenomena in mudrocks on the reservoir and basin scale.

36 MATERIALS SCIENCE↗
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.↗
NASA NTRS1997

Characterizing Hyperspectral Imagery (AVIRIS) Using Fractal Technique

With the rapid increase in hyperspectral data acquired by various experimental hyperspectral imaging sensors, it is necessary to develop efficient and innovative tools to handle and analyze these data. The objective of this study is to seek effective spatial analytical tools for summarizing the spatial patterns of hyperspectral imaging data. In this paper, we (1) examine how fractal dimension D changes across spectral bands of hyperspectral imaging data and (2) determine the relationships between fractal dimension and image content. It has been documented that fractal dimension changes across spectral bands for the Landsat-TM data and its value [(D)] is largely a function of the complexity of the landscape under study. The newly available hyperspectral imaging data such as that from the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) which has 224 bands, covers a wider spectral range with a much finer spectral resolution. Our preliminary result shows that fractal dimension values of AVIRIS scenes from the Santa Monica Mountains in California vary between 2.25 and 2.99. However, high fractal dimension values (D > 2.8) are found only from spectral bands with high noise level and bands with good image quality have a fairly stable dimension value (D = 2.5 - 2.6). This suggests that D can also be used as a summary statistics to represent the image quality or content of spectral bands.

Qiu, Hong-Lie↗
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.↗
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 NTRS1990

Fractal image analysis - Application to the topography of Oregon and synthetic images.

Digitized topography for the state of Oregon has been used to obtain maps of fractal dimension and roughness amplitude. The roughness amplitude correlates well with variations in relief and is a promising parameter for the quantitative classification of landforms. The spatial variations in fractal dimension are low and show no clear correlation with different tectonic settings. For Oregon the mean fractal dimension from a two-dimensional spectral analysis is D = 2.586, and for a one-dimensional spectral analysis the mean fractal dimension is D = 1.487, which is close to the Brown noise value D = 1.5. Synthetic two-dimensional images have also been generated for a range of D values. For D = 2.6, the synthetic image has a mean one-dimensional spectral fractal dimension D = 1.58, which is consistent with the results for Oregon. This approach can be easily applied to any digitzed image that obeys fractal statistics.

Huang, Jie↗
NASA NTRS1991

Target Detection Using Fractal Geometry

The concepts and theory of fractal geometry were applied to the problem of segmenting a 256 x 256 pixel image so that manmade objects could be extracted from natural backgrounds. The two most important measurements necessary to extract these manmade objects were fractal dimension and lacunarity. Provision was made to pass the manmade portion to a lookup table for subsequent identification. A computer program was written to construct cloud backgrounds of fractal dimensions which were allowed to vary between 2.2 and 2.8. Images of three model space targets were combined with these backgrounds to provide a data set for testing the validity of the approach. Once the data set was constructed, computer programs were written to extract estimates of the fractal dimension and lacunarity on 4 x 4 pixel subsets of the image. It was shown that for clouds of fractal dimension 2.7 or less, appropriate thresholding on fractal dimension and lacunarity yielded a 64 x 64 edge-detected image with all or most of the cloud background removed. These images were enhanced by an erosion and dilation to provide the final image passed to the lookup table. While the ultimate goal was to pass the final image to a neural network for identification, this work shows the applicability of fractal geometry to the problems of image segmentation, edge detection and separating a target of interest from a natural background.

Fuller, J. Joseph↗
NASA NTRS1999

Multi-Scale Fractal Analysis of Image Texture and Pattern

Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.

Emerson, Charles W.↗
NASA NTRS1989

Fractal statistics of cloud fields

Landsat Multispectral Scanner (MSS) and Thematic Mapper (TM) data, with 80 and 30 m spatial resolution, respectively, have been employed to study the spatial structure of boundary-layer and intertropical convergence zone (ITCZ) clouds. The probability distributions of cloud areas and cloud perimeters are found to approximately follow a power-law, with a different power (i.e., fractal dimension) for each cloud type. They are better approximated by a double power-law behavior, indicating a change in the fractal dimension at a characteristic size which depends upon cloud type. The fractal dimension also changes with threshold. The more intense cloud areas are found to have a higher perimeter fractal dimension, perhaps indicative of the increased turbulence at cloud top. A detailed picture of the inhomogeneous spatial structure of various cloud types will contribute to a better understanding of basic cloud processes, and also has implications for the remote sensing of clouds, for their effects on remote sensing of other parameters, and for the parameterization of clouds in general circulation models, all of which rely upon plane-parallel radiative transfer algorithms.

Cahalan, Robert F.↗
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 NTRS1999

Multi-Scale Fractal Analysis of Image Texture and Pattern

Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images of the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimensional-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.

Emerson, Charles W.↗
NASA NTRS1986

A fractal model for crustal deformation

It is hypothesized that crustal deformation occurs on a scale-invariant matrix of faults. For simplicity, a two-dimensional pattern of hexagons on which strike-slip faulting occurs is considered. The behavior of the system is controlled by a single parameter, the fractal dimension. Deformation occurs on all scales of faults. The fractal dimension determines the fraction of the total displacement that occurs on the first-order or primary faults. The value of the fractal dimension can be obtained from the frequency-magnitude relation for earthquakes. The results are applied to the San Andreas fault system in central California. Earthquake studies give D = 1.90. The main strand of the San Andreas fault is associated with the primary faults of the fractal system. It is predicted that the relative velocity across the main strand is 2.93 cm/yr. The remainder of the relative velocity of 5.5 cm/yr between the Pacific and North American plates occurs on higher-order faults. The predicted value is in reasonably good agreement with the value 3.39 + or - 0.29 cm/yr obtained from geological studies.

Turcotte, D. L.↗
NASA NTRS2000

Roughness Perception of Haptically Displayed Fractal Surfaces

Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.

Costa, Michael A.↗