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.     

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.     

地球物理信号能量(密度)多维分形及应用

地球物理信号代表的地质地球物理过程在多种尺度上和尺度之间表现为自相似性（self-affinity)或尺度无关性（Scale Invariant)，称为地球物理信号的分形性质，多个分形地球物理信号叠加在一起表现为多维分形特征，研究多维分形地球物理信号的能量或能量密度特征，可以进行时间或空间地球物理信号的校正、奇异性研究分析，或进行不同地球物理动力学过程的分解，本文描述了地球物理时间（空间）信号的多维分形过程和功率谱密度（能量密度）与波数以及重磁场能谱密度及面积（能量）与能谱密度的多维分形关系，并用地球物理测井与重磁资料作了试算。     

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。     

亚洲和北美干湿变化及其与海表温度异常的关系

利用多通道奇异谱方法（MSSA）分析了1953~2003年亚洲和北美Palmer干旱指数（PDSI）与热带和北半球温带海洋海表面温度异常（SSTA）的主要周期振荡特征及其相互联系.结果表明：亚洲和北美PDSI以及SSTA均存在明显的3~6年的年际以及10年左右的年代尺度振荡;此外,亚洲PDSI还存在显著的6~8年的年际振荡.SSTA的年际振荡主要体现了ENSO的变化特征,而其年代尺度振荡的空间分布具有热带太平洋和北太平洋共同作用的类ENSO型.同时,MSSA的分析结果给出了亚洲和北美主要振荡信号的时间和空间演变特征.相关性分析表明,亚洲和北美PDSI的年际及年代尺度振荡均显示明显的对SSTA强迫信号的响应.对于年际振荡,亚洲PDSI对SSTA响应强于北美,但年代尺度振荡则反之.此外,亚洲和北美PDSI对于SSTA信号响应的关键区域也随时间尺度的不同而发生变化.亚洲的西西伯利亚、青藏高原东西两侧以及中西伯利亚东部在年际和年代尺度上均为受SSTA影响最显著的区域;在年际尺度上,北美中部地区的干湿变化与SSTA存在显著相关,而在年代尺度上,美国西部更易受SSTA年代尺度振荡的影响.     

基于经验模态分解的分数维地震随机噪声衰减方法

经验模态分解算法(EMD)是一种基于有效波和噪声尺度差异进行波场分离的随机噪声压制方法,但由于实际地震数据波场复杂,导致模态混叠较严重,仅凭该方法进行去噪很难达到理想效果.本文基于EMD算法对信号多尺度的分解特性,结合Hausdorff维数约束条件,提出一种用于地震随机噪声衰减的新方法.首先对地震数据进行EMD自适应分解,得到一系列具有不同尺度的、分形自相似性的固有模态分量(IMF);在此基础上,基于有效信号和随机噪声的Hausdorff维数差异,识别混有随机噪声的IMF分量,对该分量进行相关的阈值滤波处理,从而实现有效信号和随机噪声的有效分离.文中从仿真信号试验出发,到模型地震数据和实际地震数据的测试处理,同时与传统的EMD处理结果相对比.结果表明,本文方法对地震随机噪声的衰减有更佳的压制效果.     

Hilbert-Huang 变换与大地电磁噪声压制

大地电磁信号具有非线性、非平稳、非最小相位特征,不符合以Fourier变换为基础的传统功率谱估计的基本要求. Hilbert-Huang变换是近年发展起来的处理非线性、非平稳信号的完全局部时频分析方法. 本文在简要介绍Hilbert-Huang变换基本原理与算法基础上,以实际数据分析为例,探讨了它在大地电磁信号处理及噪声压制中的应用. 提出利用Hilbert时-频能量谱对大地电磁信号进行时段筛选,以提高信号品质,增强数据处理的质量和资料的可解释性. 利用经验模态分解方法及其多尺度滤波特征,可以有效地分析MT信号中的噪声分布特征,并进行干扰压制.     

HHT的滤波特性及在声波测井波列信号处理中的应用(英文)

阵列声波信号是典型的非线性、非平稳信号,Hilbert～Huang变换（HHT）是处理非平稳信号的一种比较新的时频分析方法。通过对信号进行经验模态分解（EMD）和对瞬时频率的求解,可以获得声波信号的时一频谱。其关键技术就是进行经验模态分解,任何非平稳的信号都可以分解为有限数目并且具有一定物理意义的固有模态函数。EMD方法可以理解为以声波信号极值特征尺度为度量的时频滤波过程。滤波器充分保留了声波信号本身的非线性和非平稳特征,在声波信号的滤波和去噪中具有很大的优势。文中介绍了HHT时频滤波的实现过程,并列举了一些声波测井波列实例,说明了该方法的有效性。     

GPS坐标时间序列中非构造噪声的剔除方法研究进展c

GPS台站坐标解算中包含多种地球物理参量造成的不确定性、系统误差和随机噪声．回顾了GPS台站坐标噪声分析的研究进展,包括功率谱分析、最大似然估计、区域叠加滤波、主成分变换以及质量负荷下的地壳弹性形变模拟等,并讨论了各种方法在分析噪声的类型和强度、共模误差（common-mode errors）的物理起源等方面的作用及局限性,展望了下一阶段的研究思路．      

Common oscillatory modes in geomagnetic activity, NAO index and surface air temperature records

Detection and extraction of quasi-oscillatory dynamical modes from instrumental records of geophysical data became a useful tool in analyzing variability of observed phenomena reflected in complex, multivariate geophysical signals. Using the extension of the Monte Carlo singular system analysis (MC SSA), based on evaluating and testing regularity of dynamics of the SSA modes against the colored noise null hypothesis, we demonstrate detection of oscillatory modes with period of about 96 months in the long-term records of aa index as well as in the records of surface air temperature from several mid-latitude European locations and in the North Atlantic Oscillation index.     

Preconditioned prestack plane-wave least squares reverse time migration with singular spectrum constraint

Least squares migration can eliminate the artifacts introduced by the direct imaging of irregular seismic data but is computationally costly and of slow convergence. In order to suppress the migration noise, we propose the preconditioned prestack plane-wave least squares reverse time migration (PLSRTM) method with singular spectrum constraint. Singular spectrum analysis (SSA) is used in the preconditioning of the take-offangle-domain common-image gathers (TADCIGs). In addition, we adopt randomized singular value decomposition (RSVD) to calculate the singular values. RSVD reduces the computational cost of SSA by replacing the singular value decomposition (SVD) of one large matrix with the SVD of two small matrices. We incorporate a regularization term into the preconditioned PLSRTM method that penalizes misfits between the migration images from the plane waves with adjacent angles to reduce the migration noise because the stacking of the migration results cannot effectively suppress the migration noise when the migration velocity contains errors. The regularization imposes smoothness constraints on the TADCIGs that favor differential semblance optimization constraints. Numerical analysis of synthetic data using the Marmousi model suggests that the proposed method can efficiently suppress the artifacts introduced by plane-wave gathers or irregular seismic data and improve the imaging quality of PLSRTM. Furthermore, it produces better images with less noise and more continuous structures even for inaccurate migration velocities.     

Reconstruction of groundwater levels to impute missing values using singular and multichannel spectrum analysis: application to the Ardabil Plain,Iran

ABSTRACT

Groundwater-level time series often have a substantial number of missing values which should be taken into consideration before using them for further analysis, particularly for numerical groundwater flow modelling applications. This study aims to comprehensively compare two data-driven models, singular spectrum analysis (SSA) and multichannel spectrum analysis (MSSA), to reconstruct groundwater-level time series and impute the missing values for 25 piezometric stations in Ardabil Plain, northwest Iran. The reconstructed groundwater-level time series are assessed against the complete observed groundwater time series, while the imputed values are appraised against the artificially created gap values. The results show that both SSA and MSSA demonstrate a solid competency in imputation and reconstruction of groundwater-level data. However, depending on the spatial correlation between the piezometers, and the most suitable probability distribution function (pdf) fitted to the time series of each piezometer, the performance may vary from piezometer to piezometer.     

基于奇异谱分析的重磁位场分离方法

奇异谱分析是一种近年兴起的时间序列分析方法,它利用降秩原理实现信号分离.该方法将数据空间投影到不同特征的子空间中,并用奇异值来表征这些子空间的性质,最后通过截取奇异值实现数据的重构.重磁位场分离可以看成一种多信号叠加的分离问题.不同特征的重磁异常具有不同特征的奇异谱,这是奇异谱分析用于解决位场分离问题的应用基础.本文通过建立理论模型,分析重磁异常的奇异谱特征,得出适用于重磁位场分离的最优参数选择方法,并与传统方法进行比较.对比发现,无论是横向叠加模型、垂向叠加模型还是斜向叠加模型,奇异谱分析都具有很好的分离效果.最后,将奇异谱分析用于鄂东南某矿区的重力资料处理中,实现弱异常的识别和分离.     

基于奇异谱分析的静态相对重力观测重力固体潮提取

针对相对重力测量数据中存在的重力固体潮信号,本文提出将奇异谱分析方法（Singular Spectrum Analysis,SSA）应用到相对重力测量数据处理中,在不需要测站坐标等先验信息的条件下从相对重力数据中提取重力固体潮,提供了一种获取重力固体潮的新思路.采用模拟的相对重力数据进行实验,利用SSA方法和小波变换方法分别从模拟信号中提取重力固体潮并进行结果对比,SSA获取的重力固体潮与理论值残差RMS为0.3 μGal,小波方法获取的残差RMS为1.6 μGal.利用CG-5相对重力仪实测数据进行实验,提出一种利用SSA外推时间序列来削弱边界效应的新思路,实验结果显示采用这种方法后重力固体潮值与理论值残差序列的RMS和STD均有所减小.通过实验发现削弱边界效应后SSA提取的重力固体潮与采用Tamura潮波表计算的重力固体潮理论值残差RMS值为2.2 μGal.利用SSA提取的零点漂移值与最小二乘拟合得到的结果基本一致,十天内的差值小于0.4 μGal/d.     

Mathematical model of the spectral decomposition of periodic and non-periodic geophysical stationary random signals

Summary A model is discussed for the spectral decomposition of stationary periodic and non-periodic functions, suitable for studying various geophysical signals. The properties of the individual components of the random spectrum of the periodic stationary random function was shown in the geometry of Hilbert space of random quantities, and these were used to construct the periodic continuation of the random stationary function, given in a finite interval.     

Singular spectrum analysis and forecasting of hydrological time series

The singular spectrum analysis (SSA) technique is applied to some hydrological univariate time series to assess its ability to uncover important information from those series, and also its forecast skill. The SSA is carried out on annual precipitation, monthly runoff, and hourly water temperature time series. Information is obtained by extracting important components or, when possible, the whole signal from the time series. The extracted components are then subject to forecast by the SSA algorithm. It is illustrated the SSA ability to extract a slowly varying component (i.e. the trend) from the precipitation time series, the trend and oscillatory components from the runoff time series, and the whole signal from the water temperature time series. The SSA was also able to accurately forecast the extracted components of these time series.     

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Fractals are famous for their self‐similar nature at different spatial scales. Similar to fractals, solutions of scale invariant processes are self‐similar at different space–time scales. This unique property of scale‐invariant processes can be utilized to translate the solution of the processes at a much larger or smaller space–time scale (domain) based on the solution calculated on the original space–time scale. This study investigates scale invariance conditions of kinematic wave overland flow process in one‐parameter Lie group of point transformations framework. Scaling (stretching) transformation is one of the one‐parameter Lie group of point transformations and it has a unique importance among the other transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process yields a self‐similar solution at different space–time scales. However, the conditions for the process to be scale invariant usually dictate various relationships between the scaling coefficients of the dependent and independent variables of the process. Therefore, the scale invariance of a process does not assure a self‐similar solution at any arbitrary space and time scale. The kinematic wave overland flow process is modelled mathematically as initial‐boundary value problem. The conditions to be satisfied by the system of governing equations as well as the initial and boundary conditions of the kinematic wave overland flow process are established in order for the process to be scale invariant. Also, self‐similarity of the solution of the kinematic wave overland flow under the established invariance conditions is demonstrated by various numerical example problems. Copyright © 2011 John Wiley & Sons, Ltd.     

基于多通道奇异谱分析的GNSS坐标时间序列共模误差的提取

共模误差是区域连续GNSS网中存在的一种与时空相关的主要误差源.为了有效的剔除共模误差,提高坐标时间序列的精度,本文提出了利用多通道奇异谱分析（Multi-channel Singular Spectrum Analysis,MSSA）提取共模误差的新思路,并利用实验区域18个测站9年（2002年到2010年）的GPS坐标时间序列进行实验,分析了共模误差对时间序列的影响和测站噪声特性的影响,并对共模误差序列进行周期探测,结果显示：通过MSSA能够有效的剔除共模误差,提高坐标时间序列的精度.     

Coupled numerical–analytical approach to landscape evolution modeling

The Earth's topography is shaped by surface processes that operate on various scales. In particular, river processes control landscape dynamics over large length scales, whereas hillslope processes control the dynamics over smaller length scales. This scale separation challenges numerical treatments of landscape evolution that use space discretization. Large grid spacing cannot account for the dynamics of water divides that control drainage area competition, and erosion rate and slope distribution. Small grid spacing that properly accounts for divide dynamics is computationally inefficient when studying large domains. Here we propose a new approach for landscape evolution modeling that couples irregular grid‐based numerical solutions for the large‐scale fluvial dynamics and continuum‐based analytical solutions for the small‐scale fluvial and hillslope dynamics. The new approach is implemented in the landscape evolution model DAC (divide and capture). The geometrical and topological characteristics of DAC's landscapes show compatibility with those of natural landscapes. A comparative study shows that, even with large grid spacing, DAC predictions fit well an analytical solution for divide migration in the presence of horizontal advection of topography. In addition, DAC is used to study some outstanding problems in landscape evolution. (i) The time to steady‐state is investigated and simulations show that steady‐state requires much more time to achieve than predicted by fixed area calculations, due to divides migration and persistent reorganization of low‐order streams. (ii) Large‐scale stream captures in a strike‐slip environment are studied and show a distinct pattern of erosion rates that can be used to identify recent capture events. (iii) Three tectono‐climatic mechanisms that can lead to asymmetric mountains are studied. Each of the mechanisms produces a distinct morphology and erosion rate distribution. Application to the Southern Alps of New Zealand suggests that tectonic advection, precipitation gradients and non‐uniform tectonic uplift act together to shape the first‐order topography of this mountain range. Copyright © 2014 John Wiley & Sons, Ltd.