Directional Wavelets and a Wavelet Variogram for Two-Dimensional Data

This paper describes two new approaches that can be used to compute the two-dimensional experimental wavelet variogram. They are based on an extension from earlier work in one dimension. The methods are powerful 2D generalizations of the 1D variogram that use one- and two-dimensional filters to remove different types of trend present in the data and to provide information on the underlying variation simultaneously. In particular, the two-dimensional filtering method is effective in removing polynomial trend with filters having a simple structure. These methods are tested with simulated fields and microrelief data, and generate results similar to those of the ordinary method of moments variogram. Furthermore, from a filtering point of view, the variogram can be viewed in terms of a convolution of the data with a filter, which is computed fast in O(NLogN) number of operations in the frequency domain. We can also generate images of the filtered data corresponding to the nugget effect, sill and range of the variogram. This in turn provides additional tools to analyze the data further.     

用双正交小波实现有限脉冲信号的滤波

小波分析是一种时间-频率的分析方法，它具有多分辨分析的特点，而且在时频两域都具有表征信号局部特征的功能，因此被称为数学中的“显微镜”。紧支撑正交小波除了Haar小波外不具有线性相位，因此用双正交样条小波和Mallat塔式算法对有限脉冲信号进行了分解与重构。基于小波变换相当于频率域的带通滤波原理，尝试了在时间域对有限脉冲信号利用小波变换进行了滤波，并用一个数学模型做验证，基本上压制了需要消除的波形，并在其中讨论了双正交样条小波的阶次问题。     

Automatic identification of fake patterns caused by short-width wavelets in seismic data

Despite the popularity of using the Haar wavelet filter in many applications, it sometimes introduces fake patterns into the multi resolution analysis (MRA) of seismic data. In this work, we compared different wavelet filters to demonstrate that these patterns are fake and not part of the original waveforms and to show that they are a result of using the Haar wavelet filter as a short-width wavelet. To achieve this, many seismic waveforms from two different sources: the Egyptian National Seismic Network (ENSN) and the High Sensitivity Seismograph Network Japan (Hi-net) are used with different wavelet filters. We propose an algorithm based on an autoregressive (AR) model to detect these patterns automatically and fully.     

基于小波包分析的地震资料联合去噪方法研究

在地震记录中,随机噪声严重影响了有效信号的提取,为此必须进行消噪处理。这里首先使用小波包变换对不同频段的信号进行精细分离,有效信号和噪声经小波包分解后,其小波包系数将表现出不同特性,然后根据这种不同特性进行去噪处理,对小波包分析法处理后的剩余地震信号再进行KL(Karhunen-Loeve)变换,提取相关有效信号,最后对提取的有效信号进行中值滤波处理,进一步去除剩余噪声。经合成地震剖面和实际地震剖面处理实验证明,小波包分析、KL变换和中值滤波联合去噪方法,能有效地消除较强的随机噪声,提高地震剖面信噪比和分辨率。     

Image Filtering by Factorial Kriging—Sensitivity analysis and application to Gloria side-scan sonar images

Factorial Kriging (FK) is a data- dependent spatial filtering method that can be used to remove both independent and correlated noise on geological images as well as to enhance lineaments for subsequent geological interpretation. The spatial variability of signal, noise, and lineaments, characterized by a variogram model, have been used explicitly in calculating FK filter coefficients that are equivalent to the kriging weighting coefficients. This is in contrast to the conventional spatial filtering method by predefined, data-independent filters, such as Gaussian and Sobel filters. The geostatistically optimal FK filter coefficients, however, do not guarantee an optimal filtering effect, if filter geometry (size and shape) are not properly selected. The selection of filter geometry has been investigated by examining the sensitivity of the FK filter coefficients to changes in filter size as well as variogram characteristics, such as nugget effect, type, range of influence, and anisotropy. The efficiency of data-dependent FK filtering relative to data-independent spatial filters has been evaluated through simulated stochastic images by two examples. In the first example, both FK and data-independent filters are used to remove white noise in simulated images. FK filtering results in a less blurring effect than the data-independent fillers, even for a filter size as large as 9 × 9. In the second example, FK and data-independent filters are compared relative to the extraction of lineaments and components showing anisotropic variability. It was determined that square windows of the filter mask are effective only for removing Isotropie components or white noise. A nonsquare windows must be used if anisotropic components are to be filtered out. FK filtering for lineament enhancement is shown to be resistant to image noise, whereas data-independent filters are sensitive to the presence of noise. We also have applied the FK filtering to the GLORIA side-scan sonar image from the Gulf of Mexico, illustrating that FK is superior to the data-independent filters in removing noise and enhancing lineaments. The case study also demonstrate that variogram analysis and FK filtering can be used for large images if a spectral analysis and optimal filter design in the frequency domain is prohibitive because of a large memory requirement.     

基于不同小波变换与同态滤波结合的CBERS-02B卫星CCD图像的薄云去除

卫星CCD图像的去云处理对遥感信息的增强与提取有重要的意义,尤其是在云覆盖严重的低纬度地区。为去除CBERS-02B卫星CCD图像中薄云的影响,分别使用Mallat和à trous 2种小波变换对图像进行分解;利用同态滤波对2种小波分解图像的低频系数进行处理,衰减其低频信息;将处理后的小波低频系数与分解的高频系数进行小波重构,从而达到去云的目的。定量分析基于Mallat和à trous小波变换结合同态滤波法的去云结果表明,经à trous小波变换结合同态滤波法的去云影像所包含信息量大,细节信息丰富,去云效果较好。     

二代小波变换在重力场源分离中的应用

野外采集的重力数据是地下各类地质体重力场的叠加反映,如何有效地分离深源场和浅源场是重力资料处理中的一项重要内容。基于二代小波变换基本理论,在一维的基础上,给出了二维Haar预测算子的构造方法,并利用密度模型正演模拟,证明了二代小波变换实现多尺度分解的有效性,并以苏北某地区重力数据为例,应用二代小波变换开展浅部和深部重力异常场的分离。结果表明,该方法简单实用,在重力数据的场源分离中可以发挥重要的作用,对研究区域性断裂构造特征、划分构造单元、圈定隆坳格局等方面的地质问题具有参考价值。     

The improvement of the Morlet wavelet for multi-period analysis of climate data

The multi-level dynamics of an atmosphere system exhibits temporal structures in different types of climate data. This article addresses two issues in multi-period analysis of climate data. Firstly, the advantages of the modified Morlet wavelet transform (MMWT) for analyzing multi-period structure of time series over Morlet wavelet transform (MWT) are emphasized. Secondly, the multi-period issues of temperature data are studied with MMWT through four steps: the four dominant periods of 60 year temperature data are determined with the wavelet variance; by analyzing the real part of MMWT, the warm and cold stages of the temperature data at different scales are determined, and the time intervals of the warm and cold interchange are singled out; the amplitude of each periodic component is quantitatively characterized by the amplitude of wavelet coefficients; the most intensive oscillation time intervals are computed by the squared modulus of the MMWT (MMPS).     

利用位场连续复小波变换识辨磁场源(上)

对国外最新研究位场连续复小波变换理论的一系列文章进行了系统总结与高度概括，并利用水平面磁荷模型的垂直分量对齐次位场小波变换系数与位场导数向上延拓的等价关系进行了详细推导与归纳。基于Poisson半群核构成一类柯西小波，它将对位场的求导以及位场向上延拓这两种运算相结合，形成一简单算子，在作用于齐次场源的位场时，其小波变换系数与位场导数向上延拓是等同的，并服从小波变换双尺度规律。利用位场小波变换系数模沿极值轴线的变化特征分别反演磁场源的形态、埋深、倾角，同时利用其相位反演磁倾角。此法实用于重磁资料自动化处理与解释。     

复小波频谱分析方法及在储层预测中的应用

针对短时窗傅立叶变换频谱分析方法存在窗函数选取不确定、分辨率固定、频谱分析精度较低等问题,采用了复小波分析方法对地震信号进行时频分析,选择复Morlet小波作为分析小波,对地震信号进行复小波变换,并将尺度域转换到频率域,获得了不同频率的时频特征水平切片;用理论模型说明了复小波频谱分析精度明显高于傅立叶变换的频谱分析.对新疆某油田实际资料的处理结果表明,复小波频谱分析能细致地反映储层的分布、位置及形态,其效果优于短时窗傅立叶变换方法.     

矿井涌水量时间序列的长程相关性分析及分维数估算

根据矿井涌水量实测数据,探讨了矿井涌水量时间序列的两个重要特性,即:长程相关性和统计自相似特性。利用小波分析和分形理论在多尺度分析和自相似本质上的一致性,把小波分析和分形理论引入到矿井涌水量时间序列的分析中,得到矿井涌水量时间序列小波变换系数,在此基础上,提出了矿井涌水量时间序列分形维数的小波计算方法,并对巷道涌水量和回采工作面涌水量时间序列的分维值进行了计算,验证了小波分形维数估计法在提取矿井涌水量中所具有的分形特征信息是稳定的和可靠的。     

小波分析在试验信号消噪方面的应用

利用小波分析对试验数据进行后处理,较好的消除了系统的噪音。与Fourier变换法、10点平均滤波法比较后可知,小波分析方法能较好的保留原始数据的特征信息。对目前仍因被噪音困扰而不能正常进行的试验分析提供了一条解决途径。     

Attenuation of multiple waves using the continuous wavelet transform (CWT)

This paper describes how the continuous wavelet transform is used to filter multiple waveforms in both time and frequency domains. It is well suited to process the stationary signals, and it shows the signal in both time and frequency scales. This new approach was tested first on synthetic data and then on real data. The results obtained on both cases were good. The method consists of identifying the multiples on which we apply a normal move out using the multiple velocity law. The multiples will be aligned and the primary reflections will not be aligned. This operation allows locating the multiples in the time-scale domain. We compute the continuous wavelet transform (CWT for short) in order to focus on the patterns relative to seismic events. To filter the multiples, we define a zone with frequency and time bounds. These bounds are deduced from the projection of the seismic trace. Then an automatic mask is applied to the pattern to be isolated. Filtering in time–frequency domain is done by keeping only the wavelet coefficients that are outside the mask and assigning zero to the coefficients larger than a threshold amplitude inside the defined zone. The mask shape does not matter, which is not the case in classical filtering, where both the window size and shape play a key role. The mask is defined from three parameters: time, frequency, and the wavelet coefficients. To go back to the time domain, one has to compute the wavelet transform inverse of the trace. This procedure is repeated for all traces. To reset the traces to their initial positions, we apply the dynamic correction inverse with the same velocity law as the multiples. It turns out that the attenuation of multiples by the CWT works fine, in particular, the two identified multiples were quasi eliminated (Fig. 10).     

小波变换在确定桩裂缝、断桩及浅部缺陷中的应用

利用小波变换多分辨率分析的Mallat二进小波快速算法,引入Db5小波对桩身具有裂缝、断桩及浅部缺陷的基桩桩顶瞬态响应曲线进行多尺度小波变换。从小波变换结果的高尺度部分可以明显地发现缺陷反射波和桩底反射波,进而精确地确定缺陷位置,为补救措施的制定提供依据。     

Wavelet Transforms Applied to Irregularly Sampled Soil Data

Wavelet transforms have been used widely to analyse environmental data. These data typically comprise a series of measurements taken at regular intervals in time or space. The analysis offers a decomposition of the data that distinguishes components at different spatial scales but also, unlike Fourier analysis, can resolve local intermittent features. Most wavelet methods require the data to be sampled at regular intervals and little attention has been paid to developing methods for data that are not. In this paper, we derive a discrete Haar wavelet transform for irregularly sampled data and show how the resulting wavelet coefficients can be used to estimate contributions of variance. We discuss the interpretation of these statistics using data on apparent soil electrical conductivity of soil measured across a landscape as an example.     

基于反射波特征小波分析的工程基础无损检测

针对现有工程基础无损检测中反射波分析的不足,对Morlet和Marr小波及其小波变换进行了深入分析,指出它们在反射波分析上具有优势互补性。结合两个小波的优势,提出了Morlet小波域能量密度谱辨识动力学特征和Marr小波域模谱提取奇异特征的反射波分析方法,并构建了基于小波域香农熵的Morlet最优基小波确定技术。由反射波的动力学特征可以识别损伤模式,由反射波的奇异特征可以实施损伤定位,两者相结合构成了一条有机的基于反射波特征小波分析的工程基础精细无损检测技术路线。模型实验分析中概括归纳了工程基础的基本损伤模式,并验证了技术路线的可行、有效性。     

小波分析和RBF神经网络在地基沉降预测中的应用研究

地基沉降是一种危害很大的环境灾害。地基沉降的监测数据经常受降雨及工程施工等诸多外界因素的干扰,故而在沉降曲线中存在许多数据突变点。为此,提出基于小波分析与RBF神经网络相结合的新的地基沉降预测方法,首先采用小波分析对对原始监测数据进行数据去噪处理,进而得到反映实际变化的地基沉降曲线,然后采用径向基函数（RBF）神经网络方法对其进行预测,为工程设计提供依据。最后结合工程实例分析,通过多种小波去噪与预测结果的对比研究,表明3次B样条小波的去噪及预测效果最好,与实测值能较好地吻合,具有较好的工程应用前景。     

基于测井数据小波变换的层序划分

探讨了小波变换用于测井层序划分的一种方法。在论述了小波变换对信号时频分析的优势后, 介绍了连续小波变换的定义和一般步骤。以鲁西某钻井为例, 对迄今为止所能获得的分辨率最高、连续性最好的测井地质数据, 进行了连续小波变换, 提取小波变换系数的时频色谱信息。在此基础之上, 将不同时间 (深度) 尺度的旋回清晰地展现出来, 并探测到地层序列中不同级别的突变点, 从而实现对鲁西地区石炭二叠系层序、准层序的划分。这些探索为地层层序的划分提供了一种新的思路和有效途径。      

边坡位移演化的多尺度分析

通过相邻两个时间区间位移观测结果均值差分的积分描述和Harr小波基函数的小波变换方程的对比分析,得到了一个将不同时间尺度下边坡位移均值变化表示为相应尺度下小波变换系数的基本关系。根据这一关系,提出了可以采用小波变换方法确定两个相邻时间区间位移均值在不同时间尺度下变化规律的边坡位移演化的多尺度分析方法。针对卧龙寺新滑坡、三峡永久船闸边坡开挖和隔河岩水电站进水口边坡变形的观测结果,讨论了它们的位移演化多尺度特征。当边坡位移呈现较规则的变化趋势时,在一个尺度上就可以提取它们的时间演化特征。开挖剧烈扰动的影响可以通过给定时间尺度支撑区端点与位移突变点之间递增连的线进行近似。     

小波变换与地震奇异性属性

地震属性是由地震数据提取的有关地震波的几何形态、运动学特征、动力学特征和统计学特征等组成。这里主要讨论了一种新的地震属性(奇异性属性)计算方法。由于地震数据携带着大量的奇异性信息,因此,这一新的地震属性是在偏移地震数据的小波变换和奇异性分析的基础上形成的。小波变换使检测数据中的局部奇异程度成为可能。为能够估计出偏移地震数据每一采样点的局部奇异规律,从小波变换系数的角度来提取地震信号的奇异指数。这种新属性实行单道处理,不需要子波和速度信息,它给出了地下分层情况和断层位置。在地震数据奇异性属性的基础上,用小波变换来划分地层旋回,从而提高了小波变换在地震解释中的应用。