分形与混沌理论在地震学中的应用与探讨

分形与混沌理论的研究与发展，揭示了自然界中一大类无规几何形体物理过程的内在规律性及其动力演化过程的内在随机性。这给我们探索自然界中分形客体的几何形态及其与内部物理本质的关系提供了一条暂新的途径。特别是这一理论给我们探讨地震问题，例如：探讨地震活动时空分布的确定性图像、地震演化过程的动力学行为、临界特征以及未来大地震的发生等问题，带来了某些希望。本文较详细地综述了近年来分形与混沌理论在地震学中的应用研究。全文共分５部分。第１部分是孕震过程的一般概述。第２部分描述单分形与多分形在地震学中的某些应用．文中第３部分介绍了混沌动力学中的某些研究在测震学与模型中的应用。第４部分综述了自组织临界现象和闪变噪声的研究．最后，第５部分是分形理论与混沌动力学在地球科学中的应用前景与展望．     

Analysis And Simulation Of Surface-Layer Winds Using Multiplicative Cascade Models With Self-Similar Probability Densities

Statistical analysis techniques based on multiplicative cascades are investigated for use with surface-layer wind data sets collected in the atmospheric boundary layer over flat farm land. The data were found to exhibit multiscaling statistics, allowing the surface-layer winds to be simulated with the use of multiplicative random cascades. The study found evidence that, for the surface-layer at least, these cascade models (andhence the methods of multifractal analysis) should be applied in separate ways to the microscale inertial range, and the mesoscale. This is at odds with the view found in the existing literature, which proposes a `universal multifractal' model to replace the widely held view that there exists separate microscale, mesoscale and synoptic scales for which the processes governing each are different. At least two separate ranges of scaling are suggested for surface-layer wind data, corresponding to the microscale inertial range and the mesoscale. For the case of the mesoscale range, a self-similar distribution of weighting factors was found for the wind speed data themselves, rather than for an intermediate (dissipation) field, as is required for themicroscale data.     

Anisotropic Scaling Models of Rock Density and the Earth’s Surface Gravity Field

In this paper we consider an anisotropic scaling approach to understanding rock density and surface gravity which naturally accounts for wide range variability and anomalies at all scales. This approach is empirically justified by the growing body of evidence that geophysical fields including topography and density are scaling over wide range ranges. Theoretically it is justified, since scale invariance is a (geo)dynamical symmetry principle which is expected to hold in the absence of symmetry breaking mechanisms. Unfortunately, to date most scaling approaches have been self-similar, i.e., they have assumed not only scale invariant but also isotropic dynamics. In contrast, most nonscaling approaches recognize the anisotropy (e.g., the strata), but implicitly assume that the latter is independent of scale. In this paper, we argue that the dynamics are scaling but highly anisotropic, i.e., with scale dependent differential anisotropy. By using empirical density statistics in the crust and a statistical theory of high Prandtl number convection in the mantle, we argue that is a reasonable model for the 3-D spectrum (K is the horizontal wavevector and K is its modulus, k _z is a vertical wavenumber), (s,H _z ) are fundamental exponents which we estimate as (5.3,3), (3,3) in the crust and mantle, respectively. We theoretically derive expressions for the corresponding surface gravity spectrum. For scales smaller than ≈100 km, the anisotropic crust model of the density (with flat top and bottom) using empirically determined vertical and horizontal density spectra is sufficient to explain the (Bouguer) g _z spectra. However, the crust thickness is highly variable and the crust-mantle density contrast is very large. By considering isostatic equilibrium, and using global gravity and topography data, we show that this thickness variability is the dominant contribution to the surface g _z spectrum over the range ≈100–1000 km. Using estimates of mantle properties (viscosity, thermal conductivity, thermal expansion coefficient, etc.), we show that the mantle contribution to the mean spectrum is strongest at ≈1000 km and is comparable to the variable crust thickness contribution. Overall, we produce a model which is compatible with both the observed (horizontal and vertical) density heterogeneity and surface gravity anomaly statistics over a range of meters to several thousand kilometers.     

地震空间分布多分维数的计算及初步应用

本文简要介绍了计算多分维数的推广G-P法;讨论了无标度区的选取,认为无标度的上限受边界效应的影响,下限受定位精度的限制;最后给出中国东部几个地区的地震多分维谱及其随时间的变化,结果表明,这些地区的中强震之前多分维数D_q(q=-2,0,2)均有下降过程,特别是q<0的D_q值在震前有明显变化(变化幅度在0.4以上)。这显示D_q谱较好地揭示了震前地震空间分布的演化。     

基于栅格数据的局部奇异性分析迭代方法及其软件实现

近些年,弱缓化探异常识别已成为成矿预测和勘查评价中十分关键的问题。多重分形理论的局部奇异性分析因其崭新的思路、简便的方法、优良的效果而在弱缓异常识别中受到广泛关注。在深入剖析局部奇异性分析基本思想及计算方法的基础上,引入正规化尺度参数L,提出了迭代方法来改进局部奇异性指数的估值。给出了奇异性迭代算法并用C＋＋编程实现,软件功能强劲,操作灵活,不仅适用于各向同性情形,还适用于各向异性尺度的奇异性计算和方向性奇异性计算。软件的动态链接库版本已在Geo DAS矿产资源定量预测专业软件中应用。     

Multifractal structure of small and large scales fluctuations of interplanetary magnetic fields

We consider the multifractal spectrum of fluctuations of the interplanetary magnetic field strengths observed by Advanced Composition Explorer at the Earth's orbit. We have found that the multifractal scaling of magnetic fields is observed both on small and large scales from minutes to days. The obtained multifractal spectrum is asymmetric for small scales, in contrast to a rather symmetric spectrum observed at scales larger than a day. Moreover, we show that the degree of multifractality of the magnetic fields on large scales is correlated with the solar activity and greater than that at the small scales, where the magnetic turbulence may become roughly monofractal.     

The remarkable wide range spatial scaling of TRMM precipitation

The advent of space borne precipitation radar has opened up the possibility of studying the variability of global precipitation over huge ranges of scale while avoiding many of the calibration and sparse network problems which plague ground based rain gage and radar networks. We studied 1176 consecutive orbits of attenuation-corrected near surface reflectivity measurements from the TRMM satellite PR instrument. We find that for well-measured statistical moments (orders 0 < q < 2) corresponding to radar reflectivities with dBZ < 57 and probabilities > 10^− 6, that the residuals with respect to a pure scaling (power law) variability are remarkably low: ± 6.4% over the range 20,000 km down to 4.3 km. We argue that higher order moments are biased due to inadequately corrected attenuation effects. When a stochastic three — parameter universal multifractal cascade model is used to model both the reflectivity and the minimum detectable signal of the radar (which was about twice the mean), we find that we can explain the same statistics to within ± 4.6% over the same range. The effective outer scale of the variability was found to be 32,000 ± 2000 km. The fact that this is somewhat larger than the planetary scale (20,000 km) is a consequence of the residual variability of precipitation at the planetary scales. With the help of numerical simulations we were able to estimate the three fundamental parameters as α ≈ 1.5, C₁ = 0.63 ± 0.02 and H = 0.00 ± 0.01 (the multifractal index, the codimension of the mean and the nonconservation parameter respectively). There was no error estimate on α since although α = 1.5 was roughly the optimum value, this conclusion depended on assumptions about the instrument at both low and high reflectivities. The value H = 0 means that the reflectivity can be modeled as a pure multiplicative process, i.e. that the reflectivity is conserved from scale to scale. We show that by extending the model down to the inner “relaxation scale” where the turbulence and rain decouple (in light rain, typically about 40 cm), that even without an explicit threshold, the model gives quite reasonable predictions about the frequency of occurrence of perceptible precipitation rates.While our basic findings (the scaling, outer scale) are almost exactly as predicted twenty years ago on the basis on ground based radar and the theory of anisotropic (stratified) cascades, they are incompatible with classical turbulence approaches which require at least two isotropic turbulence regimes separated by a meso-scale “gap”. They are also incompatible with classical meteorological phenomenology which identifies morphology with mechanism and breaks up the observed range 4 km–20 000 km into several subranges each dominated by different mechanisms. Finally, since the model specifies the variability over huge ranges, it shows promise for resolving long standing problems in rain measurement from both (typically sparse) rain gage networks and radars.     

Scaling in river corridor widths depicts organization in valley morphology

Landscapes have been shown to exhibit numerous scaling laws from Horton's laws to more sophisticated scaling in topography heights, river network topology and power laws in several geomorphic attributes. In this paper, we propose a different way of examining landscape organization by introducing the “river corridor width” (lateral distance from the centerline of the river to the left and right valley walls at a fixed height above the water surface) as one moves downstream. We establish that the river corridor width series, extracted from 1 m LIDAR topography of a mountainous river, exhibit a rich multiscale statistical structure (anomalous scaling) which varies distinctly across physical boundaries, e.g., bedrock versus alluvial valleys. We postulate that such an analysis, in conjunction with field observations and physical modeling, has the potential to quantitatively relate mechanistic laws of valley formation to the statistical signature that underlying processes leave on the landscape. Such relations can be useful in guiding field work (by identifying physically distinct regimes from statistically distinct regimes) and advancing process understanding and hypothesis testing.     

Characterisation and Simulation of the Multiscaling Properties of the Energy-Containing Scales of Horizontal Surface-Layer Winds

The multiscaling statistics of atmospheric surface-layer winds at low wavenumbers above farmland and in the lee of a mountain range were examined using a hot-wire and lightweight cup anemometer. It was found that the horizontal velocity spectra could be broken into high and low-wavenumber regimes according to the parameters given by this analysis. The low-wavenumber end of the spectrum possessed a spectral slope parameter that varied between values of 0.8 and 1.35 at the farmland site during the period of the experiment, and the high-wavenumber end – corresponding to the inertial range – possessed a spectral slope slightly greater than -5/3. The larger values for this parameter for the low-wavenumber end appeared to coincide with unstable conditions. In the lee of the mountain range, the low-wavenumber spectral slope parameter was larger still, at 1.45. The low-wavenumber signals over farmland were much less intermittent than inertial-range signals, but in the lee of the mountain range the intermittency increased. From this analysis, it was shown that the statistical properties of the recorded wind signal could be reproduced using a bounded random multiplicative cascade. The model was successfully used to simulate the wind velocity field directly, rather than simulating the energy dissipation field. Since the spectral slope parameter for low wavenumbers appeared to be a function of atmospheric stability, the method presented is a simple way of generating wind signals characteristic of a variety of atmospheric conditions.     

Evidence for inherent nonlinearity in temporal rainfall

We examine the underlying structure of high resolution temporal rainfall by comparing the observed series with surrogate series generated by a invertible nonlinear transformation of a linear process. We document that the scaling properties and long range magnitude correlations of high resolution temporal rainfall series are inconsistent with an inherently linear model, but are consistent with the nonlinear structure of a multiplicative cascade model. This is in contrast to current studies that have reported for spatial rainfall a lack of evidence for a nonlinear underlying structure. The proposed analysis methodologies, which consider two-point correlation statistics and also do not rely on higher order statistical moments, are shown to provide increased discriminatory power as compared to standard moment-based analysis.