Scattering from a fractal surface: acoustical experiments and comparison with near-nadir models

A homogeneous resin print of the surface of a naturally-fractured granite block with an experimental Hurst exponent H?0.7 is placed in a water tank. An acoustical source insonifies the surface in a near-nadir direction and the backscattered waves are recorded near the source. Such measurements are made at different positions along a profile above the surface in order to obtain the energy spectrum of the mean backscattered wavefield. The wavelength range considered is 0.58?λ?8.72 mm in water. A synthetic energy spectrum is obtained from a model [Shepard and Campbell, Icarus 141 (1999) 156] which represents the rough surface as an ensemble of point scatterers and assumes a single-scattering regime. This model reasonably fits with the experimental spectrum with the largest discrepancies observed for λ<2.5 mm. The fit is improved in the whole wavelength interval by accounting for the long-range topography variations occurring along the averaging profile.     

Lakes on fractal surfaces: A null hypothesis for lake-rich landscapes

A class of stochastic processes known as fractional Brownian motion (fBm) provides strikingly realistic simulations of certain types of terrain, particularly those which appear to be unmodified by geomorphological and geological processes. In addition to their less serious applications in video games and science fiction movies, fractal terrain simulations have proven useful in a number of areas of spatial analysis. For example, they can provide sample data sets for testing the efficiency of data structures and algorithms designed for topographic applications. Previous work has shown that stream networks simulated on fBm surfaces show the same deviations from accepted theories of channel network topology as do real stream networks, implying that such deviations originate in the geometrical constraints of packing channels onto surfaces, rather than from geological or other environmental controls. In effect, this work demonstrates the usefulness of fBm as a null hypothesis for terrain. One difficulty, however, stems from the abundant pits which occur in the simulations, because peaks and pits are equally likely. Flooding of pits on fBm surfaces was simulated to obtain lakes. Lake-rich stream networks were extracted and represented with a suitable integer code. The relative frequencies of various network topologies and groups of topologies were compared to known characteristics of channel networks on real lake-rich landscapes. Lake-string topologies are significantly less abundant than in glaciated landscapes. Lake areas show good fits to hyperbolic distributions, but lake in-degrees do not fit the negative binomial model. fBm surfaces are appropriate null hypotheses of scale-free, lake-rich landscapes.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

Ocean manganese nodules as stromatolite with a fractal like-signature

Deep-sea manganese (Mn) nodules are problematic in terms of factors such as their characteristic form and genesis. There are many reports of bacterial species from manganese nodules. However, the genesis of these nodules has not been fully confirmed. Samples, mainly from the Clarion Clipperton Fracture zone in the Pacific Ocean, were examined by mineralogical methods and X-ray CT. Thin sections of these samples showed columnar stromatolite structures with rhythmic bands. Mineralized bacteria were observed by SEM and TEM. Surface morphology could be described as having a fractal-like nature. The fractal characteristics of spherical to dome-like forms were fundamentally composed of at least four ranks. The 4th order form corresponds to the stromatolite dome top shapes. Similar granular domain units and porous characteristics in manganese nodules were clearly observed by X-ray CT sections. Mathematical simulation based on fractal models reproduced similar morphological characteristics to the natural samples. So, we arrived at the concluding hypothesis that manganese nodules are aggregated stromatolite with fractal-like characteristics. Furthermore, we discussed the possibility that the nature of the layer manganese oxide minerals as the major component of the nodule and associated Fe-oxyhydroxide minerals may become an absorber/scavenger of strategic heavy metals and also toxic metals in the environments.     

Geomorphometry and fractal dimension of a riverine badland in Maharashtra

One of the most serious limitations in studying the surface morphometry of a badland landscape is the nonavailability of a very fine resolution data which is essential for such types of studies. Local relief of most of the badlands in India and also from other parts of the world exhibit limited relief amplitude, often within a few meters. The paper reports a case study carried out in a riverine badland formed along the Western Deccan Trap Region. An attempt has been made in the present paper to extract the morphometric variables of the landscape from the DEMs derived from a high resolution field generated data, because the accuracy of the DEM derived values are dependent on the pixel resolution of the DEM from which they are generated. The size of the pixel resolution should be fixed differently for different landscapes depending on the landscape process in the area. The local relief of the area is around 10 m and for such types of landscapes the topographical maps and also the web-available DEMs are of very coarse resolutions which are not suitable for the analysis. Therefore two well defined tributary catchments were chosen from the area under investigation and theodolite surveys were carried out, contours were generated with 10 cm interval, DEMs were derived by using Arc GIS software. SRTM (Shuttle Radar Topography Mission) 90 m resolution data were utilized to generate DEM for the whole basin. Hypsometric and the drainage basin parameters were extracted from these data by using the same software. Fractal dimension of the whole basin and the sample basins were also obtained for the same data. The morphometric data generated were used to understand the geomorphic processes operating in the area.     

黔北地区牛蹄塘组海相页岩孔隙结构及其非均质性特征

马查拉煤系地层是昌都赋煤带主要煤系地层之一。为明确风化对西藏马查拉煤系地层孔隙结构特征的影响,本文以西藏早石炭世马查拉组剖面不同类型风化岩石为研究对象,利用氮气吸附试验及分形理论对微观孔隙结构及其复杂程度进行研究。试验结果表明：岩石的吸脱附等温线都呈现反“S”型,孔隙结构呈现同质化发展趋势。相对压力(P/P₀)可以根据岩样吸脱附曲线的上升趋势分为小于0.1、0.1~0.45及大于0.45 3个阶段。岩样孔径结构集中分布在1~5 nm范围内,其中泥质碎屑灰岩、粉矿质泥岩、粉砂岩、泥岩和灰岩在1~2 nm范围内孔隙体积较大,煤层的比表面积和孔体积与相邻岩层相比较低。风化煤岩大孔分形维数明显高于小孔,孔隙结构复杂程度更高,煤层及碳质泥岩的孔隙分形维数相较其他类型岩石更低。研究结果可服务于昌都地区开发和灾害防治。     

庞西垌银金矿床分形性研究

庞西垌银金矿床是以银为主,伴有金、铅锌等有益组份的银金多金属矿床。矿体中银元素的品位—频度分维值,反映了银的空间分布特征。通过分维值的大小,可以判断矿体内银品位的分布变化程度。利用储量分形预测法对矿区内银矿的储量进行预测,得到了满意的效果。     

分形方法在洪涝灾害预测中的应用：以广西梧州为例

洪涝灾害是由暴雨洪水形成的一种突发性、最常见的自然灾害,它发发生在时间序列上具有分形特征。以广西梧州解放以来发生的特大洪涝灾害年份建立灾变日期序列,运用分式布朗运动模型中的Ｒ／Ｓ分析对洪涝灾害发生的时间序列进行模拟,计算了Ｈ指数值,建立了（τ）／Ｓ（τ）的幂函数关系式等,并以此为依据,预测下次灾年将在１９９９年出现。     

Elastic-wave velocities for porous media with power-law distribution of pore sizes

 A system consisting of an elastic matrix with randomly distributed pores of uniform shapes but with sizes governed by a power-law distribution is investigated in a theoretical way. The velocities of elastic waves propagating through such a system are calculated. The obtained results are that the velocities decrease with increasing total porosity and for a given porosity velocities moderately decrease with increasing absolute value of the exponent D in a power-law distribution of pore sizes. Received: 3 July 1995 / Accepted: 2 August 1996     

Fractal analysis of tidal channels in the Bahía Blanca Estuary (Argentina)

The fractal dimension (D) was estimated for nine tidal channels depicted in thematic mapper (TM) Landsat-5 imagery to derive information about the degree of geomorphological control on a tidal channel network characteristic of the Bahía Blanca Estuary (Argentina). Two methods, box counting and contiguity, were used to estimate fractal dimensions for each tidal channel. All channels produced D values close to 1, meaning that they are self-affine fractal features. However, these fractal dimensions do not represent the meandering pattern complexity characteristic of the tidal channels analysed. Although both methods allowed for estimation of D, the contiguity method showed that three of the channels actually are not fractal but have sinusoidal characteristics, a condition that was not detected by the former method.     

Are Fractal Dimensions of the Spatial Distribution of Mineral Deposits Meaningful?

It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5–1000 km. In particular, one typically observes a local dimension at spatial scales less than 30–60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30–60 km.