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1.
Fractal trees as a model for drainage systems are described in its generalized non-homogeneous form from the viewpoint of fractal geometry. Box covering techniques are used to show the numerical equivalence between the Hausdorff-Besicovitch dimension and the similarity dimension of the fractally-dominant dust formed by the sources. In this way, the similarity relationD=log (N)/log (1/r) is reinterpreted in terms of bifurcation and length ratio (r B andr L ) asD=log (r B )/log (r L ). We test this relation for non-homogeneous exact fractal trees and two natural drainage systems. The fact thatr B andr L are common parameters in quantitative geomorphology allows a trivial stimation of the fractal dimension of well-known drainage basins.  相似文献   

2.
The perimeter-area fractal model and its application to geology   总被引：14，自引：0，他引：14  
Perimeters and areas of similarly shaped fractal geometries in two-dimensional space are related to one another by power-law relationships. The exponents obtained from these power laws are associated with, but do not necessarily provide, unbiased estimates of the fractal dimensions of the perimeters and areas. The exponent (DAL) obtained from perimeter-area analysis can be used only as a reliable estimate of the dimension of the perimeter (DL) if the dimension of the measured area is DA=2. If DA<2, then the exponent DAL=2DL/DA>DL. Similar relations hold true for area and volumes of three-dimensional fractal geometries. The newly derived results are used for characterizing Au associated alteration zones in porphyry systems in the Mitchell-Sulphurets mineral district, northwestern British Columbia.  相似文献   

3.
Summary Many rock mass classification systems exist to assist the engineer in assessing the rock support requirements for underground design. On-going research in this area is directed at attempting to utilize the fractal dimension and the acoustic emission response of the tuffs at the Nevada Yucca Mountain to further aid in rock mass classification. Acoustic emission response is shown to be correlated with the porosity of the sample. Engineering behaviour of the rock varies dramatically with porosity; events and peak amplitude offer a means to distinguish between fracture porosity and pore porosity and consequently the engineering behaviour of the rock. Fractal dimension is used to characterize the roughness of fracture surfaces. Two fractal dimension calculation methods, one based on the semi-variogram for the surface and the other based on the use of dividers, are applied for this purpose. The divider method is shown to resolve deviation from a straight line; the semi-variogram method is shown to identify statistical similarity to various types of noise.Nomenclature D fractal dimension - AE acoustic emission - b b-value determined from log(frequency) against log(amplitude) plots - (h) semi-variogram function - h lag distance for semi-variogram function - H an exponent term related to fractal dimension asD=2 –H  相似文献   

4.
Summary Thirteen natural rock profiles (Barton and Choubey, 1977) are analyzed for their fractal properties. Most of the profiles were found to approximate fractal curves but some also showed features of specific wavelengths and amplitudes superimposed on fractal characteristics. The profiles showed fractal dimensions from 1.1 to 1.5 covering a range of selfsimilar and self-affine curves. The analysis results suggest a negative correlation between fractal dimension,D, and amplitude,A. Joint roughness coefficients (JRC) show a positive correlation with amplitude,A, and a negative correlation with fractal dimension,D. A numerical model of fracture closure is used to investigate the effects of different profile characteristics (D, A and sample size) on the nature of dilation and contact area, using the natural profiles and synthetic fractional Brownian motion profiles. Smooth profiles (low JRC, highD, lowA) display many small contact regions whereas rough fractures (high JRC, lowD, highA) display few large contact areas. The agreement with published experimental data supports the suggested correlations between JRC and the fractal parameters,A andD. It is suggested that observed scale effects in JRC and joint dilation can be explained by small differential strain discontinuities across fractures, which originate at the time of fracture formation.  相似文献   

5.
The aim of this short note is to test whether the morphological skeletal network (MSN) of water bodies that resembles a river network follows Horton's laws. A fractal relationship of MSN of a water body is also shown. This investigation shows that the MSN of the Nizamsagar reservoir follows Horton's laws. Furthermore, this reservoir has a fractal dimension (D m) of 1.92 which was computed by using two morphometric quantities and the fractal dimension of the main skeletal length (d). This value tallies exactly with the fractal dimension (D f) of the whole MSN computed through box-counting method.  相似文献   

6.
The aim of the presented study is to assess the fractal dimension (D) and the geometrical characteristics (length and width) of the landslides identified in North of Tehran, Iran. At first, the landslide locations (528 landslides) were identified by interpretation of aerial photographs, satellite images and field surveys, and then to calculate the fractal dimension (D), we used the computer programming named as FRACEK. In the next step, geometrical characteristics of each landslide such as length (L) and width (W) were calculated by ArcGIS software. The landslide polygons were digitized from the mentioned landslide inventory map and rotated based on movement direction. The fractal dimension for all landslides varied between 1.665 and 1.968. Subsequently, the relationship between the length/width ratios and theirs fractal D values for 528 landslides was calculated. The results showed that correlation coefficients (R), which are different regression models such as exponential, linear, logarithmic, polynomial, and power, between D and L/W ratio are relatively high, respectively (0.75, 0.75, 0.76, 0.78, and 0.75). It can be concluded that the fractal dimension values and geometry characteristics of landslides would be useful indices for the management of hazardous areas, susceptible slopes, land use planning, and landslide hazard mitigation.  相似文献   

7.
Attenuation of photosynthetically available radiation (PAR) measured using a light meter, was related to Secchi disk, horizontal black disk and horizontal sighting ranges observed in a coastal lagoon of the Southern California Current System. Vertical attenuation coefficient (KPAR) was calculated from radiometric PAR profiles. Vertical (ZD) and horizontal (HS) sighting ranges were measured with white (Secchi depth or ZSD, HS W ) and black (Z BD, HS B ) targets. Empirical power models for the KPAR-ZSD (KPAR=1.47 ZSD −1.13), KPAR-Z BD (KPAR=0.98 Z BD −1.26), KPAR-HS W (KPAR=1.22 HS W −1.14) and KPAR-HS B (KPAR=0.73 HS B −1.07) relationships were developed. The parameters of these models may not apply to other water bodies because their values depend on the range of water reflectance in each case, as reported in the literature. This is the first contribution reporting KPAR-HS empirical relations in an estuarine environment but their application may be limited to this coastal lagoon. While this approach may be universal, more data are needed to explore the variability of the parameters between different water bodies.  相似文献   

8.
Accurate quantification of rock fracture aperture is important in investigating hydro-mechanical properties of rock fractures. Liquefied wood’s metal was used successfully to determine the spatial distribution of aperture with normal stress for natural single rock fractures. A modified 3D box counting method is developed and applied to quantify the spatial variation of rock fracture aperture with normal stress. New functional relations are developed for the following list: (a) Aperture fractal dimension versus effective normal stress; (b) Aperture fractal dimension versus mean aperture; (c) Fluid flow rate per unit hydraulic gradient per unit width versus mean aperture; (d) Fluid flow rate per unit hydraulic gradient per unit width versus aperture fractal dimension. The aperture fractal dimension was found to be a better parameter than mean aperture to correlate to fluid flow rate of natural single rock fractures. A highly refined variogram technique is used to investigate possible existence of aperture anisotropy. It was observed that the scale dependent fractal parameter, K v, plays a more prominent role than the fractal dimension, D a1d, on determining the anisotropy pattern of aperture data. A combined factor that represents both D a1d and K v, D a1d × K v, is suggested to capture the aperture anisotropy.  相似文献   

9.
Ali. O. Oncel  Tom Wilson   《Tectonophysics》2006,418(3-4):205-218
Seismotectonic parameters including the Gutenberg-Richter b-value and multifractal dimensions D2 and D15 of seismicity patterns (both spatial and temporal) were compared to GPS-derived maximum shear and dilatation strains measured in the Marmara Sea region of western Turkey along the Northern Anatolian Fault Zone (NAFZ). Comparisons of seismotectonic parameters and GPS-derived maximum shear and dilatation strain along the NAFZ in the vicinity of the 1999 M7.4 Izmit earthquake reveal a positive correlation (r = 0.5, p = 0.05) between average dilatation and the Gutenberg-Richter b-value. Significant negative correlation (r = − 0.56, p = 0.03 and r = − 0.56, p = 0.02) was also observed between the spatial fractal dimension D2 and GPS-derived maximum geodetic and shear strain. This relationship suggests that, as maximum geodetic and shear strains increase, seismicity becomes increasingly clustered.Anomalous interrelationships are observed in the Marmara Sea region prior to the Izmit event along a bend in the NAFZ near the eastern end of the Marmara Sea known as the Northern Boundary Fault (NBF). An asperity is located near the northwest end of the NBF. Along the 50-km length of the NBF, GPS strains become slightly compressive. The correlation between b-value and GPS-derived dilatation suggests that regions in compression have increased probability of larger magnitude rupture. The NBF appears to serve as an impediment to the transfer of strain from east to west along the NAFZ. Recurrence times for large earthquakes along the NBF are larger than in surrounding areas. Temporal clustering of seismicity in the vicinity of the NBF may represent foreshocks of an impending rupture.  相似文献   

10.
《Engineering Geology》2002,63(1-2):141-155
Fractal theory is used in the present study to develop a more reliable method for rock mass characterization. Field studies have been carried out in opencast mines of dolomite, limestone, fluorite; sandstone and shale in coalmines. Fractal dimension of blasted fragments (Dfrag) and in situ rock blocks (Din situ) is calculated using size distribution curves according to Schumann's model. Based on the co-relation between Uniaxial Compressive Strength (UCS) and Dfrag, it is observed that change in fractal dimension is nominal beyond the UCS value of 20. From the co-relation between Bieniawaski's Rock Mass Rating (RMR) and Din situ, it is found that there is a sharp increase in fractal dimension for RMR greater than 40. Co-relation between RMR and Dfrag/Din situ shows that as RMR increases, Dfrag/Din situ ratio decreases. Rock mass classification based on fractal geometry is suggested.  相似文献   

11.
Summary A wide range of types of contact morphology among mafic and felsic magmas are observed in outcrops on Vegetation Island (Terra Nova Intrusive Complex, Antarctica). Image analysis and fractal geometry techniques were applied for in-depth study of the mafic/felsic interface, with the aim of studying the origin of the varied morphologies. In particular, the length (IPN) and fractal dimension (Dbox) of interfaces were measured. Results indicate that there is a close exponential dependence of IPN on Dbox.The observed morphologies are identical to those observed during viscous fingering processes induced by the displacement of a more viscous fluid by a less viscous one. To test if viscous fingering was responsible in this case too, IPN and Dbox values were measured on viscous fingering structures obtained experimentally using various viscosity ratios (VR) from the literature. Results indicate that, as in the natural case, there is an exponential dependence of IPN on Dbox, leading to the conclusion that the varied interface morphologies between mafic and felsic magmas are the result of viscous fingering dynamics. In addition, experimental studies clearly show that there is an exponential relationship between the viscosity ratio of fluids and the interface fractal dimension (Dbox), and the ratio between the two types of magma was estimated using this relationship. It is shown that viscosity contrasts between mafic and felsic magmas varied considerably, ratios ranging from ca. 6 to 49. These results, together with outcrop evidence, provide indications regarding the evolution of the magmatic system, which generated the actual mafic/felsic associations on Vegetation Island.  相似文献   

12.
Modeling fractured rocks with numerical methods requires some derived parameters, among which the fracture network connectivity and the size of the representative elementary volume (REV) are both of crucial importance. Percolation and REV analyses were made by the RepSim code. The program uses input parameters such as fractal dimension of the fracture midpoints (D c), length exponent (E) and relative dip (α r) data. For percolation analysis, the relative sizes of the largest percolation clusters have been calculated by stochastic realizations of the simulated fracture networks with different parameter triplets. Furthermore, fracture networks can be classified into three major types on the basis of their (E,D c,α r) parameters. For the REV calculations, the porosity of the generated fracture network was calculated. The derived REV size of a fracture network depends essentially on input parameters and shows a decreasing tendency with increasing D and E and vice versa. The method mentioned above was tested on both metamorphic samples of the Pannonian Basin and Variscan granitoid rocks of the Mórágy Complex. Percolation values predicted for the Mórágy granite are highly sensitive to alterations in the input parameters. The amphibolite bodies displayed a modeled fracture network with 80 to 90% of all fractures being interconnected, while the largest achievable percolation cluster size of gneiss is less than 10%. The REV size of the amphibolite is about 20 m as a result of connected fractures filling the whole body, while gneiss has lower porosity and higher REV (approximately 70 m).  相似文献   

13.
The aim of this short note is to test whether the morphological skeletal network (MSN) of water bodies that resembles a river network follows Horton's laws. A fractal relationship of MSN of a water body is also shown. This investigation shows that the MSN of the Nizamsagar reservoir follows Horton's laws. Furthermore, this reservoir has a fractal dimension (D m) of 1.92 which was computed by using two morphometric quantities and the fractal dimension of the main skeletal length (d). This value tallies exactly with the fractal dimension (D f) of the whole MSN computed through box-counting method.  相似文献   

14.
The ongoing continent?Ccontinent collision between Indian and Eurasian plates houses a seismic gap in the geologically complex and tectonically active central Himalaya. The seismic gap is characterized by unevenly distributed seismicity. The highly complex geology with equally intricate structural elements of Himalaya offers an almost insurmountable challenge to estimating seismogenic hazard using conventional methods of Physics. Here, we apply integrated unconventional hazard mapping approach of the fractal analysis for the past earthquakes and the box counting fractal dimension of structural elements in order to understand the seismogenesis of the region properly. The study area extends from latitude 28°N?C33°N and longitude 76°E?C81°E has been divided into twenty-five blocks, and the capacity fractal dimension (D 0) of each block has been calculated using the fractal box counting technique. The study of entire blocks reveal that four blocks are having very low value of D 0 (0.536, 0.550, 0.619 and 0.678). Among these four blocks two are characterized by intense clustering of earthquakes indicated by low value of correlation fractal dimension (D c ) (0.245, 0.836 and 0.946). Further, these two blocks are categorized as highly stressed zones and the remaining two are characterized by intense clustering of structural elements in the study area. Based on the above observations, integrated analysis of the D c of earthquakes and D 0 of structural elements has led to the identification of diagnostic seismic hazard pattern for the four blocks.  相似文献   

15.
Based on the analysis of newly collected data of plate tectonics, distribution of active faults and crustal deformation, the Taiwan area is divided into two seismic regions and six seismic belts. Then, correlation fractal dimensions of all the regions and belts are calculated, and the fractal characteristics of hypocenteral distribution can be quantitatively analyzed. Finally, multifractal dimensions Dq and f(α) are calculated by using the earthquake catalog of the past 11 years in the Taiwan area. This study indicates that (1) there exists a favorable corresponding relationship between spatial images of seismic activity described with correlation fractal dimension analysis and tectonic settings; (2) the temporal structure of earthquakes is not single but multifractal fractal, and the pattern of Dq variation with time is a good indicator for predicting strong earthquake events.  相似文献   

16.
The low-grade Palaeoproterozoic stratabound banded iron ores of the Krivoy Rog basin (Ukraine) underwent strong tectonometamorphic deformation into superimposed folds of several orders, with amplitudes from centimetres to hundreds of metres. The across-strike sections of bed surfaces defining the low-grade ore bodies resemble self-similar fractal curves; hence, a fractal geometrical model was developed in order to quantify the complexity and sinuosity of bed contours. Two different methods of measurement (polygonal approximation and two-dimensional grid cell counting) were used for 5–8 different scales. Factual similarity dimension D and other model parameters have been estimated by means of linear regression and compared for both measurement methods. From the fractal model a sinuosity coefficient of contours of the folded bed surfaces K s and a coefficient of degree of exploration of iron ore bodies K e were constructed. It is pointed out that parameters of the model can be used for determination of the optimal exploration length scales.  相似文献   

17.
Summary A new concept of feature size range of a roughness profile is introduced in the paper. It is shown that this feature size range plays an important role in estimating the fractal dimension,D, accurately using the divider method. Discussions are given to indicate the difficulty of using both the divider and the box methods in estimatingD accurately for self-affine profiles. The line scaling method's capability in quantifying roughness of natural rock joint profiles, which may be self-affine, is explored. Fractional Brownian profiles (self-affine profiles) with and without global trends were generated using known values ofD, input standard deviation, , and global trend angles. For different values of the input parameter of the line scaling method (step sizea 0),D and another associated fractal parameterC were calculated for the aforementioned profiles. Suitable ranges fora 0 were estimated to obtain computedD within ±10% of theD used for the generation. Minimum and maximum feature sizes of the profiles were defined and calculated. The feature size range was found to increase with increasingD and , in addition to being dependent on the total horizontal length of the profile and the total number of data points in the profile. The suitable range fora 0 was found to depend on bothD and , and then, in turn, on the feature size range, indicating the importance of calculating feature size range for roughness profiles to obtain accurate estimates for the fractal parameters. Procedures are given to estimate the suitablea 0 range for a given natural rock joint profile to use with the line scaling method in estimating fractal parameters within ±10% error. Results indicate the importance of removal of global trends of roughness profiles to obtain accurate estimates for the fractal parameters. The parametersC andD are recommended to use with the line scaling method in quantifying stationary roughness. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists. The estimatedC was found to depend on bothD and and seems to have potential to capture the scale effect of roughness profiles.  相似文献   

18.
In the present study, the grain size (d) and shape of 225 magnetite grains, that crystallized at T>600°C in a syntectonic granite (Godhra Granite, India) are evaluated and implications of data to decipher deformation mechanism of magnetite are discussed. Fractal (ruler) dimension (D) analysis of magnetite grains is performed and it is demonstrated that they show fractal behaviour. Smaller magnetite grains tend to be more serrated than the larger ones, which is manifested in the higher fractal (ruler) dimension (D) of the former. Assuming a natural strain rate ranging between 10−10 s−1 and 10−14 s−1, the grain size data fall dominantly in the dislocation creep field of the existing deformation mechanism map of magnetite for 630°C. However, SEM-EBSD studies reveal that subgrains are absent in the magnetite grains and they did not undergo dislocation creep. Thus it is inferred that the shape of magnetite grains was not controlled by dislocation creep. It is concluded that the higher serration and increased fractal dimension of finer magnetite grains implies the importance of diffusion creep as an important deformation mechanism at high-T for magnetite in polymineralic rocks.  相似文献   

19.
At Deobhog, migmatitic gneisses and granulites of the Eastern Ghats Belt are juxtaposed against a cratonic ensemble of banded augen gneiss, amphibolite and calcsilicate gneiss, intruded by late hornblende granite and dolerite. In the migmatitic gneiss unit, early isoclinal folds (syn‐D1M and D2M) are reoriented along N–S‐trending and E‐dipping shear planes (S3M), with (S1M–S3M) intersection lineations having steep to moderate plunges. The near‐peak P–T condition was syn‐D3M (≥900 °C, 9.5 kbar), as inferred from syn‐D3M Grt+Opx‐bearing leucosomes in mafic granulites, and from thermobarometry on Grt (corona)–Opx/Cpx–Pl–Qtz assemblages. The P–T values are consistent with the occurrence of Opx–Spr–Crd assemblages in spatially associated high‐Mg–Al pelites. A subsequent period of cooling followed by isothermal decompression (800–850 °C, c. 7 kbar) is documented by the formation of coronal garnet and its decomposition to Opx+Pl symplectites in mafic granulites. Hydrous fluid infiltration accompanying the retrograde changes is manifested in biotite replacing Opx in some lithologies. The cratonic banded gneiss–granite unit also documents two phases of isoclinal folding (D1B & D2B), with the L2B lineation girdle different from the lineation spread in the migmatitic gneiss unit. Calcsilicate gneiss (Hbl–Pl–Cpx–Scap–Cal) and amphibolite (Hbl–Pl±Grt±Cpx) within banded gneisses record syn‐D2B peak metamorphic conditions (c. 700 °C, 6.5 kbar), followed by cooling (to c. 500 °C) manifested in the stabilization of coronal clinozoisite–epidote. The D3B shear deformation post‐dates granite and dolerite intrusions and is characterized by top‐to‐the‐west movement along N–S‐trending, E‐dipping shear planes. Deformation mechanisms of quartz and feldspar in granites and banded gneisses and amphibole–plagioclase thermometry within shear bands in dolerites document an inverted syn‐D3B thermal gradient with temperature increasing from 350 to 550 °C in the west to ≥700 °C near the contact with the migmatitic gneiss unit. The thermal gradient is reflected in the stabilization of chlorite after hornblende in S3B shears to the west, and post‐D2B neosome segregation along D3B folds and shears to the east. The contrasting lithologies, early structures and peak metamorphic conditions in the two units indicate unconnected pre‐D3P–T –deformation histories. The shared D3 deformation in the two units, the syn‐D3 inverted thermal gradient preserved in the footwall cratonic rocks and the complementary cooling and hydration of the hanging wall granulites across the contact are attributed to westward thrusting of ‘hot’ Eastern Ghats granulites on ‘cool’ cratonic crust. It is suggested that the Eastern Ghats migmatitic gneiss unit is not a reworked part of the craton, but a para‐autochthonous/allochthonous unit emplaced on and amalgamated to the craton.  相似文献   

20.
An iterative least-squares optimization technique is utilized in conjunction with a one-dimensional representation of the mass transport equation to generate theoretical210Pb concentration/depth profiles beneath the water-sediment interface that are best-fit approximations to directly measured210Pb concentration/depth profiles at various locations within the Great Lakes system. The outputs of such an optimization analysis are the diffusion coefficientsD M (molecular) andD B (bioturbation) associated with the transport of210Pb radionuclides in lake bed sediments. For all stations studied, the estimated values ofD B are consistently larger than the estimated values ofD M , emphasizing the importance of accounting for the effects of bioturbation in the modelling of contaminant transport through lake bed sediments.  相似文献   

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