Walsh列率域中多维分形模型与GIS环境下地球物理信号处理

地球物理信号通常在多个尺度段表现尺度不变性,这些不变性起因于不同的地质、地球物理或成矿过程的自相似性. 利用这种在多个尺度段的尺度不变性可以设计多维分形滤波器,滤波所得信号表征了具尺度不变性的地质地球物理或成矿过程,可以用于成矿预测或环境评价. 本文研究了Walsh变换列率空间地球物理信号的列率功率谱密度与列率之间的分形与多维分形关系, 试验证实了大洋钻探、石油以及煤系地层地球物理测井资料在Walsh域的多维分形性质,提出了用于分解地球物理场,提取有用信号并用于矿产资源勘探或环境评价的多维分形W-A模型. 利用波列率域中的多维分形关系构造了W A图解(W-A Plot). 借助W-A图解可以确定最小平方误差（LS）意义下Walsh功率谱变化的不同自相似性的频率分界点,从而用于设计W-A分形滤波器. 这种滤波器可将地球物理场分解成具有不同自相似性的局部场,并且保留原场的各向异性结构. 与通常使用的基于Fourier变换的滤波技术相比,W-A模型具有许多优点：W-A适用于检测地球物理信号中的突变、线性、环状、局部与纹理结构等弱信号. 同时,由于Walsh变换中只有简单的变号(加法与减法),其计算速度远快于建立在复数乘法之上的Fourier变换,所以W-A计算速度远快于Fourier域的滤波方法,可以用于地球物理信号的现场实时处理. 用加拿大Nova-cotia省西南地区的布格重力异常进行了W-A方法的试算,处理结果反映了地质、矿产分布规律,能够很好地进行矿产预测.     

多维分形奇异功率谱分析与应用

地球物理信号中普遍含有噪声，消除噪声是地球物理信号处理中的关键技术之一.奇异功率谱分析（SSA）是在状态空间（又称相空间）中研究（系统）动力学、非线性科学与混沌现象的方法.本文在状态空间中通过SSA分解，研究、应用地球物理序列的尺度不变性进行多维分形滤波：通过在状态空间的SSA分解，构造了经验正交函数系（EOF）；在EOF子空间中定义了两种尺度与测度后，发现了两种测度与尺度皆在多个尺度范围内存在尺度不变性；利用这种尺度~测度的尺度不变性，设计、实现了多维分形奇异功率谱(MSSA)滤波模型；处理解释了大洋钻探（ODP）1143A孔岩芯自然反射性（NGR）资料；Fourier功率谱分析结果证明，MSSA能有效地压制噪声，提取有用信号.研究得出，嵌入维数对MSSA基本无影响（小于1/1000），多维分形滤波器（MSSA）能有效压制噪声或提取有用信号.     

多维分形奇异功率谱分析与应用

地球物理信号中普遍含有噪声，消除噪声是地球物理信号处理中的关键技术之一.奇异功率谱分析（SSA）是在状态空间（又称相空间）中研究（系统）动力学、非线性科学与混沌现象的方法.本文在状态空间中通过SSA分解，研究、应用地球物理序列的尺度不变性进行多维分形滤波：通过在状态空间的SSA分解，构造了经验正交函数系（EOF）；在EOF子空间中定义了两种尺度与测度后，发现了两种测度与尺度皆在多个尺度范围内存在尺度不变性；利用这种尺度~测度的尺度不变性，设计、实现了多维分形奇异功率谱(MSSA)滤波模型；处理解释了大洋钻探（ODP）1143A孔岩芯自然反射性（NGR）资料；Fourier功率谱分析结果证明，MSSA能有效地压制噪声，提取有用信号.研究得出，嵌入维数对MSSA基本无影响（小于1/1000），多维分形滤波器（MSSA）能有效压制噪声或提取有用信号.     

分形噪声、多维分形滤波及地球物理测井曲线处理应用

白噪声、布郎运动以及其它满足在双对数坐标中呈一直线和信号具分形结构、自相似性或尺度无关性。地球物理信号通常表现为多个这种信号的叠加,表面为多维分形特征,研究、模拟在双对数坐标中成一直线的信号以及如何将多维分形中多个这种信号剖分出来,对于理解相应的地质、地球物理成因机制有重要意义。通过对两口井的地球物理测井曲线处理,可以初步看出,在选定的物理测井曲线中,其具多维分形特征,不同的沉积过程相互叠架在原始测井曲线中,用滤波方法分析分解信号,可以了解不同沉积作用特征并进一步分析可能的环境变化控制机制。开发的图形界面程序使得这一过程极为容易。     

GIS环境下地球物理信号的奇异值分解、多维分形特征与应用

将布格异常作为二维实矩阵对其进行了奇异值分解。用其左特征向量矩阵与右特征向量矩阵的立积构造了一个二维完备(特征空间)正交基。布格异常投影到该正交基上的系数是布格异常矩阵的特征值(奇异值的平方)。奇异值代表了布格异常在其特征空间的一种功率密度。对比了密度分布面数、密度分布面数的变化率、密度分布面数的积分能量后，定义了奇异值谱半径量度下的能量测度。能量测度与能量谱半径符合(简单分形)指数或(多维分形)分段指数变化。利用教优分段方法得到这些分段点，利用这些分段点在特征空间中对地球物理场进行了重建、滤波。编制了与GIS结合的程序。用该方法分析和处理加拿大Nova Scotin的地球物理资料，并将结果与巳知的地质、金矿点进行了对比。结果表明，可以很好地提取地球物理场中的背景、异常场，该结果与岩性、构造、巳知矿点关联，可进行矿产资源评价和靶区预测。该方法还可用于各种地球物理信号的分离、图像处理、图像压缩等。作者开发的结合GIS的应用程序，使得这些分析能快速完成。     

从美国地球物理学会2001秋季年会谈非线性地球物理进展

简要介绍美国地球物理协会2001秋季年会“非线性地球物理”分会的活动情况。它可以大致反映出目前非线性理论和非线性地球物理的发展现状和趋势,尤其是在分形,多维分形,尺度独立性,混沌和自组织临界理论方面的研究。     

地球物理学进展2001年第16卷总目次

第 1期鄂霍茨克海的地球物理场与地质构造郝天珧　ＹｕｒｉＮｅｐｒｏｃｈｎｏｖ　江为为　宋海斌　刘建华　刘伊克　 (1)……………四川盆地综合地质、地球物理研究江为为　刘伊克　郝天珧　宋海斌　 (11)………………………………………………地球物理信号能量 (密度 )多维分形及应用李庆谋　刘少华　 (2 4)…………………………………………………………波形反演法测量Ｑ值的实验室研究叶　林　李伟东　施行觉　 (31)………………………………………………………初论华北东部中生代金成矿的地球动力学背景———以胶东金矿为例　　　………     

空间自相似性与地球物理和地球化学场的分解方法

场的分解是应用勘查地球物理和地球化学数据所必不可少的数理处理工作，这项工作往往是为了分解不同的场来研究和了解不同的地质以及相应的地质过程，比如划分不同岩石类型以及识别和圈定矿化蚀变睡.闰定地质过程所产生的同类型的地质体并不简单地对应于相同的场值和频率，场的分布特征往往还与地质体的几何形态，。埋藏深度以及与其周围地质体的相对差异有关。然而，由于地质过程往往具有多期次和空间相关性等特点，所产生的地质体或者相应的场往往会呈现空间自相似性或统计自相似性。这种自相似性可以帮助分解不同地质过程所产生的同类地质体，文中所介绍的多重分形（multifractal)滤波方法（S－A）根据场的能谱分布的不同空间自相似性形成不规则甚至分形滤波器，并将场分解为不同同的组份，如局部异常和背景场，应用该方法对多处地球化学和地球物理场的处理结果表明，该方法对区分与矿有关的局部异常与背景场是很有效的，对加拿大NOVA SCOTIA省西南部航空放射性铀钍钾（U-Tn-K)数据处理所分解的异常对反映岩体中与铀－钨－锡（U－W－Sn)矿化有关的围岩饰变具有明显的效果。     

大港探区综合地质地球物理研究

遵循沉积盆地的综合地质地球物理研究思路，运用重力、磁力、地震多种数据，通过重磁正反演和综合地质地球物理解释，提供大港探区的深部地球物理场和深部界面的信息，阐明沉积层和基底的磁性特征和密度特征，通过这些地球物理方法的相互印证、相互补充，加深对沉积盆地结构与演化的认识，从而指导油气勘探．     

地震能量时空分布的自仿射分形特征

探讨了小尺度空间范围内地震能量关于时间和宽间分布的统计自仿射分形特征，从中得到了如下认识，地震能量关于时间的分布是自仿射分形的，代表分布结构特征的Ｈ值小于１／２因而时间关联函数Ｃ（ｔ）〈０表明地震能量关于时间分布的变化过程具有负相关效应；地震能量在空间上分布也具有自仿射分形特征，有限震例表明，Ｈ值小于但接近于１／２，表明地震能量的空间分布具有更复杂的变化过程。     

Application of singularity index mapping technique to gravity/magnetic data analysis in southeastern Yunnan mineral district,China

The fractal/multifractal based singularity index mapping technique efficient in characterizing singular physical or chemical properties is applied for the analysis of gravity and aeromagnetic data in southeastern Yunnan mineral district, China. As follow-up after the introduction of singularity theory to geochemical and geological mapping scenarios, this study extends its application to delineate geophysical potential fields. Based on low gravity and low magnetic properties of granitic intrusions in the study area, singularity mapping technique is used as a high-pass filter to emphasize the geophysical anomalies caused by granitic intrusions in support of future mineral exploration. Comparing with the traditionally used band-pass filtering method, it is shown that the new technique provides an improved and simplified approach in geophysical data analysis with the advantage of scale independence.     

地震前兆观测资料的分形处理方法

本文从自仿射分形的角度探讨地震兆汾观测资料的分形处理方法。在对自仿射分形的分维量测方法进行对比分析的基础上，给出了处理地震前兆观测资料较为适用的分形方法和满足计算需要的样本点数，从对部分水氡和地电阻率资料进行了处理分析中，得到了有益的启示：地震前兆时序观测资料可用分形方法处理分析，分维值可反映观测资料的结构变化特征，其作为一个无量纲量为综合对比分析各类前兆现象，探讨地震前兆场的复杂性特征提供了有效     

A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere

Many measurements have shown that the random temperature fields in the stably stratified atmosphere are not locally isotropic (LI). The local axial symmetry (LAS) hypothesis looks more appropriate under these conditions. The object of this paper consists in the development of a flexible tool for spectral studies of LAS scalar fields independently of their origin in stably stratified geophysical flows. A heuristic model of a 3D spatial spectrum is proposed in order to describe and study statistical properties of LAS temperature inhomogeneities from LI structures up to quasi-layered ones. To simplify the solution of this problem, a main assumption was accepted: the consideration is restricted to 3D spectra which may be given on a one-parametric family of surfaces of rotation. Such 3D spectra may be represented by a single function of one variable which is the parameter of the family. This approach allows one to introduce the generalized energy spectrum which describes an energy distribution according to inhomogeneity sizes. The family of surfaces determines the shape of inhomogeneities. The family of ellipsoids of rotation and power-law generalized energy spectrum is used as the simplest example of the model application in order to study the general properties of LAS-structure spectra. The behavior of vertical, horizontal, and oblique 1D spectra and coherency spectra is studied. The relationship between the suggested model and some existing models of temperature spectra is considered. The application of the model for the analysis of experimental data is shown for two sets of measurements. It is shown that the suggested model may quantitatively describe experimental 1D spectra and coherency spectra from a unique point of view. It is noted that the model may be used for both the planning of measurements and data processing. Possible generalizations of the model are considered for random fields with more degenerate symmetry and for space-temporal spectra.     

Visibility graph analysis of geophysical time series: Potentials and possible pitfalls

Recently, complex network approaches to time series analysis have been developed and successfully applied to geophysical records. In this paper, the visibility graph approach is re-considered, which has been found useful as an alternative tool for describing the fractal properties of a time series. The interpretation of various graph-theoretical measures in the context of visibility graphs, their mutual interdependence, and their sensitivity in the presence of missing values and uncertainties (posing typical challenges in geophysical time series analysis) are thoroughly discussed. The obtained results are illustrated for some exemplary records from different fields of geosciences.     

Applying Fractal Dimensions and Energy-Budget Analysis to Characterize Fracturing Processes During Magma Migration and Eruption: 2011–2012 El Hierro (Canary Islands) Submarine Eruption

The impossibility of observing magma migration inside the crust obliges us to rely on geophysical data and mathematical modelling to interpret precursors and to forecast volcanic eruptions. Of the geophysical signals that may be recorded before and during an eruption, deformation and seismicity are two of the most relevant as they are directly related to its dynamic. The final phase of the unrest episode that preceded the 2011–2012 eruption on El Hierro (Canary Islands) was characterized by local and accelerated deformation and seismic energy release indicating an increasing fracturing and a migration of the magma. Application of time varying fractal analysis to the seismic data and the characterization of the seismicity pattern and the strain and the stress rates allow us to identify different stages in the source mechanism and to infer the geometry of the path used by the magma and associated fluids to reach the Earth’s surface. The results obtained illustrate the relevance of such studies to understanding volcanic unrest and the causes that govern the initiation of volcanic eruptions.     

黄庄-高丽营断层的分段性

应用分形几何理论讨论了黄庄－高丽营断层的分段性。用分数维作为定量标志把断层分成了三段（ＳＷ－ＮＥ）;第Ⅰ段和第Ⅱ段以良乡凸起为界,第Ⅱ段和第Ⅲ段以ＮＷ向的南口－孙河断层为界。用地球物理资料、大地测量和断层位移测量资料、地震活动资料等对分段的合理性进行了验证。认为分数维较大、断层活动和地震活动性均较强的第Ⅱ段应为地震危险重点监视区。     

2.5D forward modeling and inversion of frequency-domain airborne electromagnetic data

Frequency-domain airborne electromagnetics is a proven geophysical exploration method. Presently, the interpretation is mainly based on resistivity—depth imaging and one-dimensional layered inversion; nevertheless, it is difficult to obtain satisfactory results for two- or three-dimensional complex earth structures using 1D methods. 3D forward modeling and inversion can be used but are hampered by computational limitations because of the large number of data. Thus, we developed a 2.5D frequency-domain airborne electromagnetic forward modeling and inversion algorithm. To eliminate the source singularities in the numerical simulations, we split the fields into primary and secondary fields. The primary fields are calculated using homogeneous or layered models with analytical solutions, and the secondary (scattered) fields are solved by the finite-element method. The linear system of equations is solved by using the large-scale sparse matrix parallel direct solver, which greatly improves the computational efficiency. The inversion algorithm was based on damping least-squares and singular value decomposition and combined the pseudo forward modeling and reciprocity principle to compute the Jacobian matrix. Synthetic and field data were used to test the effectiveness of the proposed method.     

Generalised scale invariance and multiplicative processes in the atmosphere

Many geophysical fields show highly intermittent fractal structures spanning wide ranges of scale. However, few are isotropic: texture, stratification, as well as variable (scale dependent) orientation of structures is far more common. To deal with such fractals, we must generalise the idea of scale invariance beyond the familiar self-similar (or even self-affine) notions. Taking the atmosphere as our primary example (however, we also model galaxies), we outline the necessary formalism (generalised scale invariance), and show how it can be used to deal with the strongly intermittent structures which result from multiplicative (cascade type) processes concentrating matter or energy into smaller and smaller scales.We illustrate these ideas with rain data from blotting paper and radar, showing first how to directly estimate the elliptical dimension characterising the stratification, and second, how to determine universal scale-independent (invariant) codimension functions that characterise the distribution of the intense rain regions.