On the estimation of a multi-resolution representation of the gravity field based on spherical harmonics and wavelets

The gravitational potential of the Earth is usually modeled by means of a series expansion in terms of spherical harmonics. However, the computation of the series coefficients requires preferably homogeneous distributed global data sets. Since one of the most important features of wavelet functions is the ability to localize both in the spatial and in the frequency domain, regional and local structures may be modeled by means of a spherical wavelet expansion. In general, applying wavelet theory a given input data set is decomposed into a certain number of frequency-dependent detail signals, which can be interpreted as the building blocks of a multi-resolution representation. On the other hand, there is no doubt that the low-frequency part of the geopotential can be modeled appropriately by means of spherical harmonics. Hence, the main idea of this paper is to derive a combined model consisting of an expansion in spherical harmonics for the low-frequency part and an expansion in spherical wavelets for the remaining medium and high-frequency parts of the gravity field. Furthermore, an appropriate parameter estimation procedure is outlined to solve for the unknown model coefficients.     

Role of Wavelets in the Physical and Statistical Modelling of Complex Geological Processes

— Today, wavelets are recognized to have a wide range of useful properties that allow them to treat effectively multifacet problems, such as data compression, scale-localization analysis, feature extraction, statistics, numerical simulation, visualization, and communication. Second-generation wavelets represent a recent improvement of the wavelet algorithm, allowing for greater flexibility in the spatial domain and other computational advantages. We will show how these wavelets can be employed to extract large-scale coherent structures from (1) three-dimensional turbulent flows and (2) high Rayleigh number thermal convection. We will discuss the concept of modelling via decomposition into coherent and incoherent fields, taking into account the effect of the incoherent field via statistical modelling. We will discuss wavelet properties and how they can be utilized and integrated in handling large-scale problems in earthquake physics and other nonlinear phenomena in the geosciences.     

声子波及其在地震波资料分解中的应用

声子波是由声波波动方程的解构成的一种物理子波,如果不考虑吸收和散射,声子波的传播是相当简单的;相反地,数学子波的传播即使在均匀介质中也是极其复杂的.作为波动方程的解,声子波比一般的数学子波更能有效地应用于复杂声波和地震波的分解和分析.本文从Kaiser的声子波理论出发,给出了通过分别引入点源波形的复时间函数和点源虚时间坐标来构成声子波的两种解释,并对点源模型的合成地震图和实际复杂模型的地震波资料进行了时-空域的声子波变换,说明了声子波应用于地震波资料分解的有效性.     

信号的瞬时参数与正交小波基

对于可表成ｕ（ｔ）＝ａ（ｔ）ｃｏｓθ（ｔ）的信号,称ａ（ｔ）为其瞬时振幅,θ（ｔ）为瞬时位相,为瞬时频率．在许多实际问题中,瞬时振幅、位相、频率等瞬时参数的提取有重要意义．但已知的一些瞬时参数计算方法均各有其局限性．本文构造了一种频谱支集紧且有解析表达式的小波,利用它得到一种计算信号瞬时参数的能抗噪声干扰的快速、局部算法．理论分析和数值模拟的结果表明,该方法是实用而有效的,可望能满足许多实际问题的需要．     

Clustering spatial–temporal precipitation data using wavelet transform and self-organizing map neural network

A data analysis method is proposed to cluster and explore spatio-temporal characteristics of the 22 years of precipitation data (1982–2003) for Taiwan. The wavelet transform self-organizing map (WTSOM) framework combines the wavelet transform (WT) and a self-organizing map (SOM) neural network. WT is used to extract dynamic and multiscale features of the non-stationary precipitation time-series, and SOM is applied to objectively identify spatially homogeneous clusters on the high-dimensional wavelet-transformed feature space. Haar and Morlet wavelets are applied in the data preprocessing stage to preserve the desired characteristics of the precipitation data. A two-level SOM neural network is applied to identify clusters in the wavelet space in the clustering stage. The performance of clustering is evaluated using silhouette coefficients. The results indicate that singularities or sharp transitions are more significant than changes in the periodicity or data structure in the spatial–temporal precipitation data. The WTSOM results show that six clusters are optimal for both Haar and Morlet wavelet functions, but their corresponding geographic locations are different. The geographic locations of clusters based on the Haar wavelet, which captures the occurrence of extreme hydrological events, appear in blocks while those classified by the Morlet wavelet, which indicates periodicity changes and describes fine structures, appear in strips that cross the island of Taiwan. Principal component analysis is applied to the precipitation data of each cluster. The first principal components explain 62–90% of the total variation of data. Characteristics of precipitation data for each cluster are explored using scalogram analysis. The results show that both extreme hydrological events and periodicity changes appear in the spatial and temporal precipitation data but with different characteristics for each cluster. Recognizing homogeneous hydrologic regions and identifying the associated precipitation characteristics improves the efficiency of water resources management in adapting to climate change, preventing the degradation of the water environment, and reducing the impact of climate-induced disasters. Measures for countering the stress of precipitation variation for water resources management are provided.     

分析研究基于小波分析与谱分析提高重力异常的分辨能力

重力数据是所有地下场源产生的重力场的叠加,探测对象经常被淹没在区域背景场之中,因此剩余异常的分离对于重力资料研究至关重要,而近来被引入位场领域的小波算子作为了滤波器和场源分析工具,在这里我们分析研究基于小波分析与谱分析的二维离散小波变换用于提高重力异常的分辨能力,再现出由简单形状场源描述密度不均匀的几何特征.本文先介绍二维多分辨率分析小波的基本理论及其提升算法,利用对数功率谱估计平均深度方法理论,接着对理论模型数据进行多尺度异常分解,估计地质体的形状、大小和深度,最后又对实测重力数据进行分析,并与传统常规方法进行比较分析,结果表明对于实际数据分析其方法也是具有可行性的.     

The simulation of far-field wavelets using frequency-domain air-gun array near-field wavelets

Air-gun arrays are used in marine-seismic exploration. Far-field wavelets in subsurface media represent the stacking of single air-gun ideal wavelets. We derived single air-gun ideal wavelets using near-field wavelets recorded from near-field geophones and then synthesized them into far-field wavelets. This is critical for processing wavelets in marine- seismic exploration. For this purpose, several algorithms are currently used to decompose and synthesize wavelets in the time domain. If the traveltime of single air-gun wavelets is not an integral multiple of the sampling interval, the complex and error-prone resampling of the seismic signals using the time-domain method is necessary. Based on the relation between the frequency-domain phase and the time-domain time delay, we propose a method that first transforms the real near-field wavelet to the frequency domain via Fourier transforms; then, it decomposes it and composes the wavelet spectrum in the frequency domain, and then back transforms it to the time domain. Thus, the resampling problem is avoided and single air-gun wavelets and far-field wavelets can be reliably derived. The effect of ghost reflections is also considered, while decomposing the wavelet and removing the ghost reflections. Modeling and real data processing were used to demonstrate the feasibility of the proposed method.     

基于改进的最佳匹配地震子波的地震资料衰减特性分析

简述了改进的最佳匹配地震子波的小波函数构造及参数的物理含义;根据地震波的高、低频分量在黏弹介质中传播时被地层吸收的差异,给出了一种在时-频域定性估计地震波衰减特性的方法;分别以改进的最佳匹配地震子波的小波及Morlet小波作为母小波分析地层吸收特性,并比较了两种小波函数刻画地层吸收特性的能力;测试了这种方法对噪声的敏感程度. 将文中提出的方法用于某油田的一段实测地震资料衰减分析,得到的吸收特性剖面能较好地反映油气的空间展布.     

逆子波域消除多次波方法研究

SRMA（与表面相关多次波的衰减）算法包含预测和相减两步.相减算法中,当多次波与反射波同相轴相交时,如何有效减去多次波、保留反射波,是面临的主要问题.通过分析非正交性对滤波器（逆子波）的影响,可以证明：逆子波因非正交性产生的误差呈近似的高斯分布.在此基础上,本文提出了在逆子波域（单道自适应相减滤波的滤波算子的集合）,利用其误差的概率分布特征,对逆子波进行估计,用逆子波的估计对逆子波进行校正来消除多次波的方法.其步骤为：首先用SRMA方法预测出表面多次波,并对每一单炮进行单道自适应相减,得到逆子波,形成逆子波域;其次,在逆子波域采用中值滤波,提取接近真实逆子波的逆子波估计;第三,在逆子波域用逆子波估计对畸变的逆子波进行校正;最后采用校正后的逆子波来衰减多次波.通过简单模型和SMARRT模型的测试,该方法不仅能够有效减去多次波,而且在相交的区域,能够保持反射波同相轴的连续性并恢复其正确的振幅.     

MIGRATION IN TERMS OF SPATIAL DECONVOLUTION*

The relationship between two finite-difference schemes (15° and 40°) and the Kirchhoff summation approach is discussed by using closed form solutions of Claerbout's approximate versions of the wave equation. Forward extrapolation is presented as a spatial convolution procedure for each frequency component. It is shown that downward extrapolation can be considered as a wavelet deconvolution procedure, the spatial wavelet being given by the wave theory. Using this concept, a three-dimensional model for seismic data is proposed. The advantages of downward extrapolation in the space-frequency domain are discussed. Finally, it is derived that spatial sampling imposes an upper limit on the aperture and a lower limit on the extrapolation step.     

Geophysical Applications of Multidimensional Filtering with Wavelets

--We present imaging results in geophysics based on using multidimensional Gaussian wavelets as a filter in a 2-D Cartesian domain. Besides decomposing the field into various distinct lengthscales, we have also constructed the 2-D maps describing the spatial distributions of the maximum of the wavelet-transformed L₂-norm E_max (x,y) and its corresponding local wavenumber k_max (x,y), where x and y are the Cartesian coordinates. For geoid anomalies, using a wavelet filter extending to 90 degrees, we have discerned the distinct outlines of convergent and divergent tectonic zones and have conducted a quantitative comparison of the short-wavelength gravitational anomalies at those wavelengths between two different geographical locations. We have also compared the wavelet results with a nonlinear bandpass filter in the spectral domain where a Gaussian filter with the logarithm of the degree l acting as the argument has been employed. A wavelet solution, with a length-scale corresponding to 256 degrees, would need a filter with over 400 spherical harmonics centering around l=157 for an optimal spatial fit. The computational effort with the bandpass filter technique greatly exceeds those associated with wavelets. We have also shown the ability of the wavelets to analyze the vastly different scales present in high Rayleigh number convection and the mixing of passive heterogeneities driven by thermal convection. Wavelets will be a useful tool for rapid analyzing of the large multidimensional fields to be captured in many other geophysical endeavors, such as the upcoming gravity satellite missions and satellite radar interferometry images.     

小波变换及其应用

小波变换是近年来发展起来的一门新的数学分支.它适宜于分析研究信号的局部性质,对于图象数据压缩、奇性检测、非卷积型算子的化简及取样插值定理等方面都有着重要的应用,在地震勘探、大气湍流、语声合成、图象处理等许多领域中都有着广泛应用的前景.本文介绍了小波变换及其正交基的基本概念,并对它的一些重要应用作了概括的介绍.     

Detection and extraction of orientation-and-scale-dependent information from two-dimensional GPR data with tuneable directional wavelet filters

The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engineering, environmental, archaeological and other work. GPR images of the subsurface frequently contain geometric information (constant or variable-dip reflections) from various structures such as bedding, cracks, fractures, etc. Such features are frequently the target of the survey; however, they are usually not good reflectors and they are highly localized in time and in space. Their scale is therefore a factor significantly affecting their detectability. At the same time, the GPR method is very sensitive to broadband noise from buried small objects, electromagnetic anthropogenic activity and systemic factors, which frequently blurs the reflections from such targets.This paper introduces a method to de-noise GPR data and extract geometric information from scale-and-dip dependent structural features, based on one-dimensional B-Spline Wavelets, two-dimensional directional B-Spline Wavelet (BSW) Filters and two-dimensional Gabor Filters. A directional BSW Filter is built by sidewise arranging s identical one-dimensional wavelets of length L, tapering the s-parallel direction (span) with a suitable window function and rotating the resulting matrix to the desired orientation. The length L of the wavelet defines the temporal and spatial scale to be isolated and the span determines the length over which to smooth (spatial resolution). The Gabor Filter is generated by multiplying an elliptical Gaussian by a complex plane wave; at any orientation the temporal or spatial scale(s) to be isolated are determined by the wavelength. λ of the plane wave and the spatial resolution by the spatial aspect ratio γ, which specifies the ellipticity of the support of the Gabor function. At any orientation, both types of filter may be tuned at any frequency or spatial wavenumber by varying the length or the wavelength respectively. The filters can be applied directly to two-dimensional radargrams, in which case they abstract information about given scales at given orientations. Alternatively, they can be rotated to different orientations under adaptive control, so that they remain tuned at a given frequency or wavenumber and the resulting images can be stacked in the LS sense, so as to obtain a complete representation of the input data at a given temporal or spatial scale.In addition to isolating geometrical information for further scrutiny, the proposed filtering methods can be used to enhance the S/N ratio in a manner particularly suitable for GPR data, because the frequency response of the filters mimics the frequency characteristics of the source wavelet. Finally, signal attenuation and temporal localization are closely associated: low attenuation interfaces tend to produce reflections rich in high frequencies and fine-scale localization as a function of time. Conversely, high attenuation interfaces will produce reflections rich in low frequencies and broad localization. Accordingly, the temporal localization characteristics of the filters may be exploited to investigate the characteristics of signal propagation (hence material properties). The method is shown to be very effective in extracting fine to coarse scale information from noisy data and is demonstrated with applications to noisy GPR data from archaeometric and geotechnical surveys.     

基于曲率模态和小波变换的结构损伤位置识别

小波变换具有在时域和频域内表征信号局部特性的能力,能够在不同尺度下对结构响应中的突变信号进行放大和识别.在结构曲率模态基础上,本文提出了一种基于小波变换的结构损伤检测和定位方法.利用双正交小波函数对损伤前后结构的曲率模态进行小波变换,用损伤前后小波变换系数残差建立了结构损伤指标,通过小波变换系数残差的分布统计情况判定损伤的存在并确定其位置.应用简支梁数值模拟结果对该方法进行了验证.     

Seismic buffer recognition using wavelet-based features and neural classification

One of the first operations in a seismic signal processing system applied to earthquake data is to distinguish between valid and invalid records. Since valid signals are characterized by a combination of their time and frequency properties, wavelets are natural candidates for describing seismic features in a compact way. This paper develops a seismic buffer pattern recognition technique, comprising wavelet-based feature extraction, feature selection based on the mutual information criterion, and neural classification based on feedforward networks. The ability of the wavelet transform to capture discriminating information from seismic data in a small number of features is compared with alternative feature reduction techniques, including statistical moments. Three different variations of the wavelet transform are used to extract features: the discrete wavelet transform, the single wavelet transform and the continuous wavelet transform. The mutual information criterion is employed to select a relatively small set of wavelets from the time–frequency grid. Firstly, it is determined whether wavelets can capture more informative data in an equal number of features compared with other features derived from raw data. Secondly, wavelet-based features are compared with features selected based on prior knowledge of class differences. Thirdly, a technique is developed to optimize wavelet features as part of the neural network training process, by using the wavelet neural network architecture. The automated classification techniques developed in this paper are shown to perform similarly to human operators trained for this function. Wavelet-based techniques are found to be useful, both for preprocessing of the raw data and for extracting features from the data. It is demonstrated that the definition of wavelet features can be optimized using the classification wavelet network architecture.     

The attenuated Ricker wavelet basis for seismic trace decomposition and attenuation analysis

The widely used wavelets in the context of the matching pursuit are mostly focused on the time–frequency attributes of seismic traces. We propose a new type of wavelet basis based on the classic Ricker wavelet, where the quality factor Q is introduced. We develop a new scheme for seismic trace decomposition by applying the multi-channel orthogonal matching pursuit based on the proposed wavelet basis. Compared with the decomposition by the Ricker wavelets, the proposed method could use fewer wavelets to represent the seismic signal with fewer iterations. Besides, the quality factor of the subsurface media could be extracted from the decomposition results, and the seismic attenuation could be compensated expediently. We test the availability of the proposed methods on both synthetic seismic record and field post-stack data.     

地震资料处理中小波函数的选取研究

本文给出了常见地震子波的一个模拟公式，可以很好地模拟零相位及混合相位子波，在一定意义上也可以近似模拟最大相位及最小相位子波．模拟出的子波加上适当的修正项后满足允许条件，可用作小波函数．与Ｍｏｒｌｅｔ小波类似，在实际应用中这些修正项在一定条件下可以略去，文中对Ｍｏｒｌｅｔ小波作了改造，使其能更好地适应于地震资料处理．研究了反射波能量及噪声等干扰波在时间－尺度域的分布特征与所选基本小波的关系．提出用地震子波（或与地震子波相近的函数）作为基本小波，对地震资料进行去噪及分频解释的方法．最后用实例证明方法的有效性．     

小波多分辨分析方法在波动方程正演模拟中的应用

初步探讨小波多分辨分析理论在地震波动方程正演模拟中的应用，给出一种基于地震波场局部变化性质而自适应调空间网格点的波动方程数值算法，目前求解波动方程所用的有限差分法、有限元法以及伪谱法都不能根据波场的局部变化性质而动态选择空间网格点的大小、小波基函数在空间域和频域中都具有局部性特征，它的性质优于有限差分法，有限元法和伪谱法中所用的基函数，通过阀值运算，地震波场失发辨表示变得非常稀疏，同时地下介质中的主要信息又不会受到损害，本文将声波场的多分辨表示变得非常稀疏，同时地下介质中的主要信息又不会受到损害，本文将声波方程的矩阵表示形式小波多分辨分析的框架下进行了展开，通过对算子长阵的地震波场矩阵进行多分辨分解和压缩，得到了小波域中地震波场正演模拟算法。     

利用EMD和子波振幅谱与相位谱关系的时变子波提取方法

本文首先分析了地震波在黏弹介质的传播规律,基于黏弹介质地震波动方程总结了时变子波振幅谱和相位谱的关系,从而得出结论,准确估计子波相位谱初值和不同时刻的子波振幅谱是实现时变子波准确提取的必要条件.在此基础上,针对传统方法限制子波振幅谱形态且受限于分段平稳假设的问题,提出了一种利用EMD(Empirical Mode Decomposition)和子波振幅谱与相位谱关系的时变子波提取方法,根据子波对数振幅谱光滑连续而反射系数对数振幅谱振荡剧烈的特点,采用EMD方法将不同时刻地震记录的对数振幅谱分解为一组具有不同振荡尺度的模态分量,通过滤除振荡剧烈分量、重构光滑连续分量提取时变子波振幅谱;再应用子波振幅谱和相位谱的关系提取时变子波相位谱,将分别提取的振幅谱和相位谱逐点进行合成,最终实现时变子波的准确提取.本文方法不需要求取Q值,适用于变Q值的情况,具有良好的抗噪性能.数值仿真和叠后实际资料处理结果表明,相比传统的分段提取方法,利用本文方法提取的时变子波准确度更高,研究成果对提高地震资料分辨率具有重要意义.     

用对称映射ARMA模型的零极点研究子波相位对反射系数序列反演的影响

为研究地震子波相位对反射系数序列反演的影响,在自回归滑动平均(ARMA)模型描述子波的基础上,提出采用z域对称映射ARMA模型零极点的方法构造了一系列相同振幅谱、不同相位谱的地震子波,并结合谱除法对人工合成地震记录进行反射系数序列反演.理论分析表明,子波相位估计不准时反射系数序列反演结果中残留一个纯相位滤波器,该纯相位滤波器的相位谱为真实子波和构造子波的相位谱之差.采用丰度和变分作为评价方法,在反演结果中确定出真实的或准确的反射系数序列.仿真实验和实际数据处理结果也验证了子波相位对反射系数序列反演的影响规律和评价方法的有效性,为进一步提高反射系数序列反演结果精度指明了研究方向.