The theory of the continuous wavelet transform in the interpretation of potential fields: a review

We consider the use of the continuous wavelet transform in the interpretation of potential field data. We report its development since the publication of the first paper by Moreau et al . in 1997. Basically, it consists in the interpretation in the upward continued domain since dilation of the wavelet transform is the upward continuation altitude. Thus within a range of altitudes, the wavelet transform of the noise is decreased faster than the wavelet transform of the potential field caused by underground sources; this means that the signal-to-noise ratio is much better than those involved in other enhancing methods (e.g., Euler deconvolution, gradient analysis, or the analytic signals). Similarly to the Euler deconvolution, its first target parameters were the source positions and shape. The method has then been developed to estimate size and directions of extended sources (e.g., faults and dikes of finite dimensions) and also the magnetization direction in the case of magnetic data. Latest developments show that when combined with a Radon transform, the continuous wavelet transform can help in the automatic detection of elongated structures in 3D, simultaneously to the estimation of their strike direction, shape and depth. Several applications to real case studies have been shown before; however for clarity's sake in the present paper, only synthetic cases have been reproduced to clearly sum up the development of the methodology.     

基于小波变换与小波包变换的降噪方法比较

在模拟地震记录信号中加入信噪比为17的高斯白噪声,然后分别采用小波降噪和小波包降噪方法,对含噪信号进行降噪处理。在不同降噪阈值下,比较降噪后信号的信噪比。结果表明:在同一降噪阈值下,小波包降噪后信号的信噪比高于小波降噪后信号的信噪比,而且采用wbmpen方法给定的阈值明显可以提高降噪后信号的信噪比。     

Seismic response in archaeological areas: the case-histories of Rome

Rome is affected by earthquakes associated to three different seismogenic districts: the Central Apennines area, the Colli Albani volcanic area and the Roman area. The major effects were exclusively due to Apennine seismicity and reached in some cases felt intensities up to VII–VIII degree (MCS scale). The predominant role in the damage distribution seems to be played by the local geological conditions. The historical centre of the city is characterized by the presence of two geomorphologic domains: the alluvial plain of Tiber river and the topographic relieves of Roman Hills, where tradition indicates the first site of the city foundation. In particular, the right river side is characterized by the outcropping of the regional bedrock along the Monte Mario–Gianicolo ridge, while the eastern relieves are the remnants of the Sabatini and Albani volcanic plateau, deeply eroded by the Tiber river and its tributaries during the last glacial low-stand (Würm). These domains are characterized by a large difference in seismic response, due to the high impedance contrast between Holocene coarse deposits filling the Tiber Valley and sedimentary and volcanic Plio–Pleistocene units. Seismic damage observed in 150 monuments of downtown Rome was indicating a significant concentration on alluvial recent deposits. This result was confirmed by the geographical distribution of conservation and retrofitting activities subsequent to main earthquakes, mostly related to local geological conditions. The cases of Marcus Aurelius' Column and Colosseum confirmed the influence of the Holocene alluvial network in local seismic response. During 2500 years of history, the monuments of Rome have `memorized' the seismic effects of historical earthquakes. In some cases, the integration of historical and geological research and macroseismic observations may provide original and useful indications to seismologists to define the seismic response of the city. Local site effects represent a serious threat for historical buildings in Rome and in other historical towns with similar cultural heritage and geological characteristics, as in the Mediterranean region, even in areas that are not affected by a local seismic activity.     

第二代小波变换及其在地震信号去噪中的应用

本文讨论了第二代小波变换的基本原理和变换过程，并将第二代小波变换引入到地震资料去噪处理中，基于提升法的小波变换是一种柔性的小波构造方法，它使用线性、非线性或空间变化的预测和更新算子，并能确保变换的可逆性。通过对模拟数据和实际资料的处理，证明了的它对地震信号去噪具有很好的效果。离散信号的小波去噪可分为三步：小波分解，系数缩减（切除噪声部分），信号重建。目前常用的小波去噪的方法有硬阈值法和软阈值法，这里采用软阈值法去噪。本文的提升变换采用的是Deslauriers-Dubuc（4，2）小波，基于以上变换方法，分别对含噪的模拟数据及实际地震数据进行3级可逆提升变换，对每一级上的细节信号按上述的软域值法进行处理，削减小波系数中的噪声部分，从而实现了信号去噪，结果证明去除随机噪声的效果是令人满意的。     

同步提取三参数小波变换及其在储层含气性检测中的应用

随着我国石油勘探程度的不断加深,非常规油气藏逐步成为了油气勘探的重点.由于致密砂岩具有低孔、低渗、非均质性强等特性,因此迫切需要高精度时频分析后处理方法对该类气藏进行有效表征.本文提出了一种同步提取三参数小波变换（Synchroextracting Three-Parameter Wavelet Transform,SETPWT）新方法,该方法具有小波变换多尺度多分辨率的特性,通过改变三个参数来调节小波基的主频、带宽及相位,灵活匹配多种类型的合成信号以及地震子波;与传统时频分析方法相比,同步提取三参数小波变换在三参数小波变换结果的基础上引入同步提取算子（Synchronous Extraction Operator,SEO）,通过SEO提取与信号理想时频谱高度相关的时频信息,去除多余能量,从而实现非平稳信号的高精度时频表征.合成信号算例表明,同步提取三参数小波变换的分辨率不但明显高于常用时频变换,同时具有优于同步提取变换（Synchroextracting Transform,SET）的抗噪性能;将同步提取三参数小波变换应用到地震资料含气性检测中,进一步验证了该时频算法在致密砂岩含气性检测中的潜力.

     

Application of the continuous wavelet transform of gravity and magnetic data to estimate sub‐basalt sediment thickness

The Continuous Wavelet Transform was recently proposed for the interpretation of gravity and magnetic potential data. We utilize the Continuous Wavelet Transform of gravity and magnetic data to address one of the most common issues in exploration geophysics: mapping of sub‐basaltic sedimentary strata. We observe that the magnetic response of the basaltic layer is dominant in a three‐layer case of a basalt‐sediment‐basement, whereas the gravity signal is dominated by the base of the sediment. Thus the Continuous Wavelet Transform of the magnetic data is related to the thickness of the basalt and the Continuous Wavelet Transform of the gravity data is related mostly to the bottom of the sediment. These observations are demonstrated with a synthetic model and a few field examples. Derived depths using Continuous Wavelet Transform are in good agreement with known vertical cross‐sections. Therefore, Continuous Wavelet Transform analysis of both gravity and magnetic data offers a possibility for primary information of sub‐basaltic sediment thickness, which can provide a basis for further detailed modelling.     

Application of the wavelet transform to automatic seismic signal detection

The paper is devoted to such a priority direction of digital seismology as the detection of seismic signals in a noisy medium. Various formulations of the problem of seismic event detection are analyzed. The reasons for the ineffectiveness of the classical formulation of the detection problem are revealed, and an adequate general formulation of the problem in the frequency-time space is proposed. A criterion for the automatic determination of the length of a seismic signal is proposed. The time-scale STA/LTA (Short Time Average to Long Time Average) detector, utilizing the expansion of a seismogram in a packet of wavelets, is proposed. The method eliminates the shortcomings of the standard scheme and enables the detection of seismic events with a low signal-to-noise ratio. The problems of the choice of the mother wavelet and the decomposition level of seismograms are discussed from the standpoint of physical properties of seismic wave fields.     

基于物理小波的频谱分解方法及应用研究

在地震资料频谱分解中,采用匹配地震子波的物理小波,依据地震信号的特征,用振幅、能量衰减率、能量延迟时间及地震子波的中心频率等四类参数构造基本小波,把地震信号分解在小波域,高频分量能够得到精细的刻画.本文以物理小波变换为工具, 给出了该变换中的核函数的选择方法,进而提出了基于物理小波变换的频谱成像方法.我们将此方法用于海上某油田河流相储层的描述,并与常规软件中的小波变换频谱成像结果进行了对比, 结果表明,本文提出的方法更能精细地刻画地质事件.     

小波变换在地电场数据分析中的应用

随着城镇化发展的加快, 上海的地电场观测受到来自周围环境噪声的影响日益严重, 这些噪声干扰对于地震前兆异常的分析判断带来很多不确定性. 本文主要探讨通过小波变换方法来分解地电场观测原始信号, 分析环境噪声对原始观测数据的影响强度. 利用异常信号的小波模值对比分析了滤波结果, 并通过绘制极化方位图检验了该方法在实际运用中的效果. 结果表明： 数字滤波方法能很好地去除日常干扰所带来的背景噪声, 对于重构的地电场信号, 也能较好地反映其原始信号变化特征; 滤波后的自然电场异常信号保留了原始信号中主要的变化特征, 并能反映其原始变化规律; 重构的地电场信号能够突出信号中异常信号极化方位角, 使极化方位收敛有利于实际运用.     

Non-stationary seismic response of MDOF systems by wavelet transform

A wavelet-based formulation has been presented in this paper for the stochastic analysis of a linear multi-degree-of-freedom (MDOF) classically damped system subjected to earthquake ground motion. The ground motion has been modelled as a non-stationary process (both in amplitude and frequency) using wavelets. Closed-form expressions of the moments of the instantaneous Power Spectral Density Function (PSDF) of the response have been derived and used to predict the statistics of the response peak of any desired order. For illustration of the formulation, an example torsionally coupled multistoried building has been considered along with the twenty synthetically generated time-histories corresponding to an example ground motion process. © 1997 John Wiley & Sons, Ltd.     

连续小波变换用于结构地震反应分析

从连续小波变换的基本原理出发，提出了地震地面运动的连续小波模型，以时频特性较好的小波基Meyer小波来表示，并用于多自由度的地震反应分析，给出了相主尖的计算方法及公式计算，实例表明，所提方法是可行的。     

小波变换在TM热红外遥感信息解释中的应用

如何有效利用TM热红外遥感图像是遥感应用和研究人员共同关心的问题．由于空间分辨率不高,其应用与研究程度远不及其它波段获取的图像．小波变换具有信号多尺度分解能力,可以把遥感图像(二维信号)按照近似与细节进行多尺度分解．对TM(或ETM^ )图像进行K-L变换,将可见光及近红外波段信息集中到一个主分量图像上,而后对热红外图像和该主分量图像同时进行多尺度小波分解,使用前者的近似系数与后者的细节系数重建图像,由此得到的合成图像既保留了原热红外图像的基本灰度分布信息,同时分辨率也得到了提高．     

Localization and de-noising seismic signals on SASW measurement by wavelet transform

SASW method is a nondestructive in situ testing method that is used to determine the dynamic properties of soil sites and pavement systems. Phase information and dispersion characteristics of a wave propagating through these systems have a significant role in the processing of recorded data. Inversion of the dispersive phase data provides information on the variation of shear-wave velocity with depth. However, in the case of sanded residual soil, it is not easy to produce the reliable phase spectrum curve. Due to natural noises and other human intervention in surface wave date generation deal with to reliable phase spectrum curve for sanded residual soil turn into the complex issue for geological scientist. In this paper, a time–frequency analysis based on complex Gaussian Derivative wavelet was applied to detect and localize all the events that are not identifiable by conventional signal processing methods. Then, the performance of discrete wavelet transform (DWT) in noise reduction of these recorded seismic signals was evaluated. Furthermore, in particular the influence of the decomposition level choice was investigated on efficiency of this process. This method is developed by various wavelet thresholding techniques which provide many options for controllable de-noising at each level of signal decomposition. Also, it obviates the need for high computation time compare with continuous wavelet transform. According to the results, the proposed method is powerful to visualize the interested spectrum range of seismic signals and to de-noise at low level decomposition.     

Continuous wavelet transform,theoretical aspects and application to aeromagnetic data at the Huanghua Depression,Dagang Oilfield,China

We use the continuous wavelet transform based on complex Morlet wavelets, which has been developed to estimate the source distribution of potential fields. For magnetic anomalies of adjacent sources, they always superimpose upon each other in space and wavenumber, making the identification of magnetic sources problematic. Therefore, a scale normalization factor, a^?n, is introduced on the wavelet coefficients to improve resolution in the scalogram. By theoretical modelling, we set up an approximate linear relationship between the pseudo‐wavenumber and source depth. The influences of background field, random noise and magnetization inclination on the continuous wavelet transform of magnetic anomalies are also discussed and compared with the short‐time Fourier transform results. Synthetic examples indicate that the regional trend has little effect on our method, while the influence of random noise is mainly imposed on shallower sources with higher wavenumbers. The source horizontal position will be affected by the change of magnetization direction, whereas the source depth remains unchanged. After discussing the performance of our method by showing the results of various synthetic tests, we use this method on the aeromagnetic data of the Huanghua depression in central China to define the distribution of volcanic rocks. The spectrum slices in different scales are used to determine horizontal positions of volcanic rocks and their source depths are estimated from the modulus maxima of complex coefficients, which is in good accordance with drilling results.     

Detecting small-scale targets by the 2D inversion of two-sided three-electrode data: application to an archaeological survey

The detecting capabilities of some electrical arrays for the estimation of position, size and depth of small-scale targets were examined in view of the results obtained from 2D inversions of apparent-resistivity data. The two-sided three-electrode apparent-resistivity data are obtained by the application of left- and right-hand pole–dipole arrays that also permit the computation of four-electrode and dipole–dipole apparent-resistivity values without actually measuring them. Synthetic apparent-resistivity data sets of the dipole–dipole, four-electrode and two-sided three-electrode arrays are calculated for models that simulate buried tombs. The results of two-dimensional inversions are compared with regard to the resolution in detecting the exact location, size and depth of the target, showing some advantage for the two-sided three-electrode array. A field application was carried out in the archaeological site known as Alaca Hoyuk, a religious temple area of the Hittite period. The two-dimensional inversion of the two-sided three-electrode apparent-resistivity data has led to locating a part of the city wall and a buried small room. The validity of the interpretation has been checked against the results of subsequent archaeological excavations.     

Theory and seismic applications of the eigenimage discrete wavelet transform

Discrete wavelet transforms are useful in a number of signal processing applications. To improve the scale resolution, a joint function of time, scale and eigenvalue that describes the energy density or intensity of a signal simultaneously in the wavelet and eigenimage domains is constructed. A hybrid method, which decomposes eigenimages in the wavelet domain, is developed and tested on field data with a variety of noise types. Several illustrative examples examine the ability of wavelet transforms to resolve features at several scales. Successful applications to time‐lapse seismic reservoir monitoring are presented. In reservoir monitoring, the scale‐dependent properties of the eigenstructure of the 4D data covariance matrix enable us to extract the low‐frequency time‐lapse signal that is the result of internal diffusive losses caused by fluid flow.     

Non-linear filter, using the wavelet transform, applied to seismological records

As any process in Nature, seismic recordsare affected by noise that the analystwould want to eliminate. One of the mostcommon techniques used to minimise thisnoise effect is the application of linearfilters, which reduce the bandwidth of thesignal. This method is based on the FourierTransform, and therefore any perturbationon the coefficients affects the entirerecord.We have developed a non-linear filter basedon the multiresolution analysis of theDiscrete Time Wavelet Transform (DTWT). Themain idea is to use the time-frequencylocalisation properties of the waveletdecomposition. Each coefficient isassociated to a window on thetime-frequency plane, so any perturbationwould only affect the time and frequencyrange of the correspondent window.The procedure we propose has three stages:periodic noise elimination, spikesreduction and, finally, the non-linearfiltering. The non-linear filter acts bythresholding the wavelet coefficients. Thethresholding estimator will depend on thesignal-noise ratio (SNR) in each of thefrequency bands associated to the waveletdecomposition.We have compared the proposed method to thecoherent structures method (Mallat, 1998)and to two 4^th order linear filterbanks (Butterworth and Elliptic filters),applying all of them to a syntheticdatabase, and a real earthquake databaserecorded by the Short Period ROA Network.The proposed method improves the SNR in the87% of the tested events, being therelative rms error less than three, and themaximum amplitude relative error less than10% in the 90% of the synthetic database.     

A multifractal formalism for vector-valued random fields based on wavelet analysis: application to turbulent velocity and vorticity 3D numerical data

Extreme atmospheric events are intimately related to the statistics of atmospheric turbulent velocities. These, in turn, exhibit multifractal scaling, which is determining the nature of the asymptotic behavior of velocities, and whose parameter evaluation is therefore of great interest currently. We combine singular value decomposition techniques and wavelet transform analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method is calibrated on synthetic self-similar 2D vector-valued multifractal measures and monofractal 3D vector-valued fractional Brownian fields. We report the results of some application of the TWTMM method to turbulent velocity and vorticity fields generated by direct numerical simulations of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.

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Hiibert-Huang transform and wavelet analysis of time history signal

The brief theories of wavelet analysis and Hilbert-Huang transform (HHT) are introduced firstly in the present paper. Then several signal data were analyzed by using wavelet and HHT methods, respectively. The comparison shows that HHT is not only an effective method for analyzing non-stationary data, but also is a useful tool for examining detailed characters of time history signal.     

基于小波分析的结构动力可靠度估计

利用小波分析估计的地震动局部谱,采用虚拟激励法求得了线笥结构非平衡随机响应,结合基于首超破坏准则的几种动力可靠度表达式,对结构的动力可靠度进行了讨论,并与平稳响应的情况进行了比较。