Smaller source earthquakes and improved measuring techniques allow the largest earthquakes in Iceland to be stress forecast (with hindsight)

A group of three earthquakes in 2000 June in SW Iceland included the two largest earthquakes in Iceland in the past 30 yr. Previously, temporal changes in shear-wave splitting had not been recognized before these earthquakes as there were too few small earthquakes to provide adequate shear-wave data, and they were not stress forecast, even with hindsight. These large earthquakes were subject to a special investigation by the European Community funded PREPARED Project during which the seismic catalogue was extended to include smaller magnitude earthquakes. This more detailed data set, together with a semi-automatic programme for measuring the parameters of shear-wave splitting greatly increased the number of time-delay measurements.
The new measurements displayed the typical temporal variations before larger earthquakes as seen elsewhere: a long-term increase in time delays, interpreted as stress accumulation before the earthquake; and a decrease, interpreted as crack coalescence, immediately prior to the earthquake. The logarithms of the durations of both the implied accumulation of stress and the crack coalescence have the same self-similar relationships to earthquake magnitude as found elsewhere in Iceland. This means that, in principle, the time and magnitude of the larger earthquakes could have been stress forecast in real time had the smaller source earthquakes of the extended catalogue and the improved measuring procedures been available at the time.     

Fractal dimensions and geometries of caves

Lengths of all caves in a region have been observed previously to be distributed hyperbolically, like self-similar geomorphic phenomena identified by Mandelbrot as exhibiting fractal geometry. Proper cave lengths exhibit a fractal dimension of about 1.4. These concepts are extended to other self-similar geometric properties of caves with the following consequences.Lengths of a cave is defined as the sum of sizes of passage-filling, linked modular elements larger than the cave-defining modulus. If total length of all caves in a region is a self-similar fractal, it has a fractal dimension between 2 and 3; and the total number of linked modular elements in a region is a self-similar fractal of the same dimension. Cave volume in any modular element size range may be calculated from the distribution.The expected conditional distribution of modular element sizes in a cave, given length and modulus, also is distributed hyperbolically. Data from Little Brush Creek Cave (Utah) agree and yield a fractal dimension of about 2.8 (like the Menger Sponge). The expected number of modular elements in a cave equals approximately the 0.9 power of length of the cave divided by modulus. This result yields an intriguing parlor trick. An algorithm for estimating modular element sizes from survey data provides a means for further analysis of cave surveys.     

A Nonstratified Model for Natural Fractal Curves

The stratified structure of Koch curves limits their value as models of nonstratified curves typical in nature, which may nevertheless have statistically self-similar character. Richardson plots for basic and randomized Koch curves are marked by strong, scale-periodic disruptions, and compare poorly with the smoothly sloping, compact Richardson plots derived for some natural, irregular curves. These disruptions relate to the heirarchies of dominant element sizes created in the discrete steps of Koch curve construction, with element vertices acting as attractors for divider steps. We have developed a curve-generation algorithm that adds individual bend elements to a line segment in a nonstratified manner, explicitly following the self-similar limit relation of curve length (L) to added line segment length (r). Values that may be specified include curve fractal dimension (D) and bend element shape (angle of inclination for the added segments). The procedure allows combinations of fractal dimension and element shape, which encompass the range of D values encountered for natural fractal curves. The nonstratified construction method provides useful standard curves for testing and norming of geometric analysis methods commonly applied to complex natural features. In a preliminary test of a multisampling divider method algorithm, the resulting Richardson plots consistently diverge from linear form, and are likely to lead to an overapproximation of D value.     

A new method of modelling the rock micro-fracturing process in double-torsion experiments using neural networks

Microfracturing of rock is a complicated damage evolution process. Inaccurate prediction of micro-fracturing behaviours suggests a need for the development of a better modelling method. Analysis of acoustic emission (AE) measurements in double-torsion tests indicates that micro-fracturing behaviours during the loading stage have fractal time structures. This fractal behaviour can be described by C(t) ∝ t^D, where D is the correlation exponent, t is the time and C(t) is the correlation integral. Furthermore, by utilizing measured AE data, a new method has been developed to model the AE behaviours of micro-fracturing in rock, in air, and following soaking in water and in a chemical solution of DTAB. The neutral models NN (10,21,2) and NN (10,20,2) were found to describe reasonably well the AE behaviours of micro-fracturing in rock under air and DTAB conditions, and water conditions, respectively. The cumulative AE events and the cumulative AE counts predicted by the neural models agreed well with those measured in experiments. Copyright © 1999 John Wiley & Sons, Ltd.     

非线性成矿预测理论:多重分形奇异性-广义自相似性-分形谱系模型与方法

介绍了“奇异性-广义自相似性-分形谱系”(“3S”: Singularity-generalized self-Similarity-fractal Spectrum)为核心的多重分形现代成矿预测理论与模型(Multifractal Mineralization Prediction Theory and Models)的基本内容和前沿研究方向.讨论了作为非线性、复杂性理论的重要领域之一, 多重分形理论所提供的“奇异性-广义自相似性-分形谱系”等概念和相关的模型.这些新概念和模型不仅能够合理地描述成矿系统、成矿过程、成矿富集规律、矿产资源时空分布, 还提供了定量模拟和识别成矿异常(地质、地球物理、地球化学、遥感异常)的有效模型和实用方法.将多重分形原理与成矿过程、矿产资源分布规律、矿产资源信息获取研究相结合, 可形成具有良好应用前景的现代成矿预测理论与模型.采用该多重分形矿产资源预测理论和在此基础上所开发的专用地学非线性空间信息GeoDASGIS技术, 对国内外多个金属成矿区带进行了矿产资源勘查与评价, 均取得了较理想的预测效果, 表明对开展矿产资源勘查和评价是有效和可行的.      

THE WEST COAST OF BRITAIN: STATISTICAL SELF-SIMILARITY VS. CHARACTERISTIC SCALES IN THE LANDSCAPE

Mandelbrot (Science, 1967, 156, 636–638) used the west coast of Britain as an example of a naturally occurring statistically self-similar fractal. Evidence from this study indicates that the west coast of Britain is not statistically self-similar over the range of scale of measurement, and that complexity reaches maxima at characteristic scales related to identifiable features on the coastline. A fractal analysis is conducted using the divider method, and although the resulting log–log plot of measured length against steplength appears linear, statistical tests for linearity strongly suggest that the coastline is not statistically self-similar. An angle measure technique (AMT) developed by the author to examine changes in line complexity with scale, shows that within the range of scale of measurement there are two peaks in complexity for the west coast of Britain, suggesting that two processes acting at different scales have influenced coastal development. The AMT is also used to identify differences in complexity between northern and southern sections of the coastline. Additionally, high r² values associated with regressions of log L(G) against log G are shown to be insufficient evidence of statistical self-similarity, and apparently linear segments (fractal elements) often found in Richardson plots may contain systematic curvature revealed only by more rigorous tests for non-linearity.     

中国白垩纪矿床时空分布的分形分析

分形概念应用在地球科学中来刻画地质变量与地质体自相似特征。研究表明分形模型常常提供有力工具来刻画地质变量和地质体的基本空间分布结构。许多地质现象的频度和大小之间的分布具有尺度不变性, 如岩石碎片、断层、地震、火山喷发、矿藏和油井等。中国白垩纪的矿床大量地分布在东北、阴山-燕山、华北地台、秦岭-大别山、扬子地台和东南沿海等6个矿集区, 它们是中国白垩纪矿床分布的重要地区, 而西北和西南地区已知的白垩纪矿床尚少; 中国白垩纪矿床分布具有在时间上集中于早白垩世, 在空间上集中于东北和华南的分形特性。以中国白垩纪金矿床为例, 对金矿床空间分布进行了分形建模与预测, 该方法不仅适用于金矿床, 而且还适用于其它矿床, 具有普遍的意义。      

Spatial Distribution,Scaling and Self-similar Behavior of Fracture Arrays in the Los Planes Fault,Baja California Sur,Mexico

We present a case of detailed analysis of fracture arrays spanning four orders of magnitude in length; all of them measured at a single natural site by acquiring images at progressively larger scales. There is a high dispersion of cumulative-length exponents, box dimensions and fracture densities. However, the fractal analysis supports the fractal nature of fracture arrays. Our data indicate the existence of an upper limit for the density parameters, as similarly reported by other authors. We prove that box dimension is in inverse relation with fracture concentration and in direct relation with fracture density. These relations are also observed in our data and additionally there is an upper limit for the box dimensions. We interpret the dispersion in our results as more fundamental than methodological problems. It represents a truncation in the complete evolution of the fracture systems because in natural cases strain initiates overprinting of previous fracture arrays. Considering that larger fractures accommodate strain more efficiently than small fractures, the generation of small fractures is inhibited in the presence of pre-existing larger fractures. Maximum values of fracture density prevent accommodating an excess of strain in a single or restricted range of scales; we claim this condition produces migration of fracturing to larger scales originating fracture scaling.     

非线性矿床模型与非常规矿产资源评价

探讨了建立非线性矿床模型对难识别的非常规矿产资源评价的可能性.首先评述了非线性理论在成矿动力学和矿产勘查中的应用, 以及非常规矿床与非常规矿产资源评价的研究现状.然后引进了以岩浆结晶分异成矿作用为例的多维分形非线性成矿系统模型.在此基础上揭示了非线性成矿系统必然导致元素富集和聚集的奇异分布及矿床分布的广义自相似性规律.广义自相似性刻画了成矿的外在多样性和内在相似性.介绍了2种最新研究的“奇异分析”和“广义自相似性”异常分解方法.采用文中所建议的非线性矿床模型方法对加拿大北部Gowganda地区的热液型Co, Ni, Ag, As, Pb 5种元素矿产进行了预测和评价.结果表明, 以“广义自相似性”和“奇异分析”为基础的非线性矿床模型及GIS信息综合技术对非常规难识别矿产资源评价是有效的.