Bayesian inference of the Cole–Cole parameters from time- and frequency-domain induced polarization

The inversion of induced‐polarization parameters is important in the characterization of the frequency electrical response of porous rocks. A Bayesian approach is developed to invert these parameters assuming the electrical response is described by a Cole–Cole model in the time or frequency domain. We show that the Bayesian approach provides a better analysis of the uncertainty associated with the parameters of the Cole–Cole model compared with more conventional methods based on the minimization of a cost function using the least‐squares criterion. This is due to the strong non‐linearity of the inverse problem and non‐uniqueness of the solution in the time domain. The Bayesian approach consists of propagating the information provided by the measurements through the model and combining this information with a priori knowledge of the data. Our analysis demonstrates that the uncertainty in estimating the Cole–Cole model parameters from induced‐polarization data is much higher for measurements performed in the time domain than in the frequency domain. Our conclusion is that it is very difficult, if not impossible, to retrieve the correct value of the Cole–Cole parameters from time‐domain induced‐polarization data using standard least‐squares methods. In contrast, the Cole–Cole parameters can be more correctly inverted in the frequency domain. These results are also valid for other models describing the induced‐polarization spectral response, such as the Cole–Davidson or power law models.     

台站地震资料的时频域自适应极化分析和滤波

提出一种自适应协方差的时频域极化滤波方法。该方法在广义S变换时频方法的基础上,构造时频域自适应协方差矩阵,通过特征分析计算时频域瞬时极化参数,设计极化滤波器,实现多分量地震极化分析和滤波。其优势在于协方差矩阵的分析时窗的长度由多分量地震数据的瞬时频率确定,可以自适应于有效信号的周期,在每个时频点计算极化参数不需要进行插值处理;结合时间频率信息,解决在时间域或频率域波形或频率重叠的信号具有明显的直观性。模型数据及实际三分量台站地震数据处理结果表明,该极化滤波方法在台站地震资料分析和处理方面具有很好的直观性和较高的分辨率。     

Rock fracture compliance derived from time delays of elastic waves

The purpose of this study is to compare the reliability of various methods of estimating normal rock fracture compliance from elastic wave measurements. We compare ultrasonic through‐transmission laboratory measurements for a smooth fracture in a Westerly granite specimen with numerical simulations and analytical solutions. The focus is on deriving compliance from time delays. The influence of specimen and source transducer width was constrained using numerical wave simulations. We find that measured ultrasonic phase delays are better suited to estimate the fracture compliance than group delays. Using the frequency domain instead of the time domain increases the accuracy of the fracture compliance estimates. We further show that for cases where precise phase delay measurements are unavailable, employing first break times in conjunction with numerical simulations can be considered as an alternative.     

An application of waveform denoising for microseismic data using polarization–linearity and time–frequency thresholding

Noise suppression or signal‐to‐noise ratio enhancement is often desired for better processing results from a microseismic dataset. In this paper, a polarization–linearity and time–frequency‐thresholding‐based approach is used for denoising waveforms. A polarization–linearity filter is initially applied to preserve the signal intervals and suppress the noise amplitudes. This is followed by time–frequency thresholding for further signal‐to‐noise ratio enhancement in the S transform domain. The parameterisation for both polarization filter and time–frequency thresholding is also discussed. Finally, real microseismic data examples are shown to demonstrate the improvements in processing results when denoised waveforms are considered in the workflow. The results indicate that current denoising approach effectively suppresses the background noise and preserves the vector fidelity of signal waveform. Consequently, the quality of event detection, arrival‐time picking, and hypocenter location improves.     

临夏台钻孔系统性质的论证

对中国临夏台2013年至2014年两年的井水位和四分量钻孔应变资料进行了预处理以消除趋势与突跳.根据文献(Means,1982;Young and Budynas,2005),相互正交的两条测线的应变观测值之和等于面应变.文献(刘序俨等,1988)证明近地表的面应变的2/3等于体应变,因此,可由4分量钻孔应变观测值得到钻孔体应变,然后根据体应变与井水位观测资料,从时域和频域对该钻孔系统的性质进行了论证.结果表明,在时域,体应变与井水位高度负相关.钻孔系统的灵敏度为—0.1620mm/10-9.把两年中的某两个月份的两者时间坐标轴和纵轴比例尺放大,发现井水位曲线的峰/谷与体应变观测曲线的谷/峰一一对应,两者的相位滞后非常小.在频域内,本文采用Venedikov调和分析方法分别取得了井水位与体应变9个月的半日波与全日波数个波群的逐月潮汐因子与相位滞后,然后作简单计算,得到了钻孔系统对上述波群的灵敏度与相位滞后.结果表明9个月中大多数波群的灵敏度不但十分相近,且非常接近由时域得到的周年频率分量的灵敏度,但相位滞后误差较大,本文对此进行了分析,认为由反正切得到的相位滞后受计算误差影响较大,应以时域经审视所得的相位滞后接近于零为准.通过时域与频域的分析,表明井水位对体应变的响应基本是线性时不变的,论证了临夏台钻孔系统基本上满足了叠加性、齐次性与时不变性,基本上为一线性时不变系统.     

INDUCED POLARIZATION RESPONSE OF A HORIZONTALLY MULTILAYERED EARTH WITH NO RESISTIVITY CONTRAST*

The induced polarization response of a horizontally multilayered earth with no resistivity contrast can rapidly be calculated on a desk calculator or minicomputer for any electrode array. The formulation is a simple series summation of the products of weighting coefficients and the true induced polarization responses for each of the layers. The coefficients are directly derivable from the corresponding resistivity model. This series approach to IP formulation was originally described by Seigel but has not been treated extensively in the present-day geophysical literature. This method can be applied to either time or frequency domain induced polarization measurements. Once the coefficients are known, apparent induced polarization response can readily be obtained by judicious substitution of known, suspected, or assumed values of the true induced polarization of each layer. Basic formulation is presented for the IP potential coefficients (pole-pole or two array) with no resistivity contrast between the layers. From these coefficients, response of any number of layers for any electrode array can be obtained by suitable differentiation. Some examples of Wenner array for a three-layered earth and dipole-dipole array for a four-layered earth are used to illustrate the application. The results of this technique are valid for many natural situations of modest resistivity contrast. However, they definitely cannot be used if there are highly contrasting resistivity layers present. Such an approach is conceptually simple and is useful for survey planning, checking or setting the “depth-of-penetration”of a given array. For field induced polarization data that fits reasonably well to the no-resistivity-contrast model, this simple approach facilitates quantitative interpretation.     

On the determination of marine magnetotelluric receiver orientation using land magnetic observatory data

Marine magnetotelluric measurements using “free‐fall’’ instruments without effective compasses suffer from the problem of unknown orientation of the receivers at the seafloor. While past works indicate that marine magnetotelluric orientation of the instruments can be estimated by reference to land deployments of known orientation using the transfer tensor method, there is limited published information on how this is implemented in practice. We document this method and propose a set of new time‐ and frequency‐domain approaches to solve this orientation problem of the seafloor receivers. We test these methodologies in onshore and offshore magnetotelluric data whose orientations are well known and apply these techniques to marine magnetotelluric data with unknown orientation. For the controlled tests, both time‐ and frequency‐domain approaches produce overall comparable results. To investigate the effects of the subsurface structure distribution on the orientation process, a dimensionality analysis of a controlled dataset is carried out. In subsequent analysis using the available disoriented marine magnetotelluric data from offshore Brazil and from the Vassouras magnetic observatory on the mainland for remote referencing, frequency‐domain methods yield approximate orientation angles among themselves with low standard deviation each. Time‐domain results are consistent for most cases but differ from frequency‐domain results for some situations.     

THE TIME-DOMAIN ELECTROMAGNETIC RESPONSE OF POLARIZABLE BODIES: AN APPROXIMATE CONVOLUTION ALGORITHM1

It is now believed that the negative transients observed in coincident-loop transient electromagnetic (TEM) measurements are caused by polarizable bodies (bodies whose conductivity increases as a function of frequency). Ordinarily the TEM response of polarizable bodies is obtained by calculating the frequency-domain response at many frequencies and transforming it to the time domain via Fourier, Laplace or Hankel transforms. This is normally a computationally laborious task. However, for some simple non-polarizable bodies the time-domain response is analytical and can be computed easily. When these simple bodies are weakly polarizable an approximate response can be obtained by convolving the easily-calculated, non-polarizable response with the impulse response of the polarization. The approximate response is found to be very similar to the exact response for the polarizabilities normally seen in geological materials.     

Equivalent linear method considering frequency dependent characteristics of stiffness and damping

Equivalent linear dynamic response analysis of ground is based on complex moduli and Fourier series expansion; therefore, it is not an equivalent method but an approximate method. Two deficiencies in the conventional equivalent linear method represented by SHAKE are described first. The maximum shear strength is overestimated, resulting in overestimation of the peak acceleration under a strong ground motion, and the amplification is underestimated at high frequency. The latter sometimes results in underestimation of the peak acceleration under weak ground shaking, and gives an incident wave with unrealistic large accelerations or a divergence of analysis in deconvolution analysis under strong ground motion. Both deficiencies are shown to come from the same cause, i.e. computing the effective strain as a constant fraction of the maximum strain. Since this is a key concept of the equivalent linear analysis, one cannot overcome both deficiencies at the same time in the conventional method. An apparent frequency dependence in stiffness and damping is shown to appear in the dynamic response, although soil itself does not show frequency dependent characteristics. Following this observation, the effective strain is expressed in terms of frequency from the similarity concept of the strain–frequency relationship between time domain and frequency domain. This enables the reduction of both deficiencies at the same time, resulting in a marked improvement in the equivalent linear analysis. The accuracy of the proposed method is examined by the simulations of three vertical array records during large earthquakes. The proposed method always gives much better prediction than conventional equivalent linear methods for both convolution and deconvolution analyses, and it is confirmed to be applicable at more than 1% shear strain.     

Low-frequency dilatational wave propagation through fully-saturated poroelastic media

The strong coupling of applied stress and pore fluid pressure, known as poroelasticity, is relevant to a number of applied problems arising in hydrogeology and reservoir engineering. The standard theory of poroelastic behavior in a homogeneous, isotropic, elastic porous medium saturated by a viscous, compressible fluid is due to Biot, who derived a pair of coupled partial differential equations that accurately predict the existence of two independent dilatational (compressional) wave motions, corresponding to in-phase and out-of-phase displacements of the solid and fluid phases, respectively. The Biot equations can be decoupled exactly after Fourier transformation to the frequency domain, but the resulting pair of Helmholtz equations cannot be converted to partial differential equations in the time domain and, therefore, closed-form analytical solutions of these equations in space and time variables cannot be obtained. In this paper we show that the decoupled Helmholtz equations can in fact be transformed to two independent partial differential equations in the time domain if the wave excitation frequency is very small as compared to a critical frequency equal to the kinematic viscosity of the pore fluid divided by the permeability of the porous medium. The partial differential equations found are a propagating wave equation and a dissipative wave equation, for which closed-form solutions are known under a variety of initial and boundary conditions. Numerical calculations indicate that the magnitude of the critical frequency for representative sedimentary materials containing either water or a nonaqueous phase liquid is in the kHz–MHz range, which is generally above the seismic band of frequencies. Therefore, the two partial differential equations obtained should be accurate for modeling elastic wave phenomena in fluid-saturated porous media under typical low-frequency conditions applicable to hydrogeological problems.     

BURGER'S EQUATION AS A MODEL FOR THE IP PHENOMENON1

The dynamic response characterizing the induced-polarization (IP) phenomenon is modelled by a non-linear diffusion equation (Burger's equation) supplemented by relevant initial and boundary values. The analysis of the model yields a voltage step response and an impedance curve in the frequency domain which agree qualitatively with experimental measurements. Curve fits based on the model have been made in the case of electrochemical cell measurements. The diffusion coefficients estimated by means of these curves are of the same order of magnitude as those calculated using experimental measurements. The normalized transient with these diffusion coefficients agrees with observations, but probably has a shorter discharge time. We have also carried out a comparison with predictions obtained from a linear, finite diffusion layer model, thus showing that for most practical situations the nonlinear term modelling the migration effect can be neglected.     

Wavefield Polarization in Fault Zones of the Western Flank of Mt. Etna: Observations and Fracture Orientation Modelling

Ambient noise measurements performed on the western flank of Mt. Etna are analyzed to infer the occurrence of directional amplification effects in fault zones. The data were recorded along short (<500 m) profiles crossing the Ragalna Fault System. Ambient noise records were processed to compute the horizontal-to-vertical noise spectral ratio as a function of frequency and direction of motion. Wavefield polarization was investigated in the time–frequency domain as well. Peaks of the spectral ratios generally fall in the frequency band 1.0–6.0 Hz pointing out directional amplifications that are also confirmed by the results of the time–frequency analysis, the largest amplification occurring with high angle to the fault strike. A variation of the frequency of the spectral peak is observed between the two sides of the fault, possibly related to a damage fault asymmetry. Measurements performed several kilometers away from the fault zone do not show behavior that is as systematic as in the fault zone, and this suggests that the observed directional effects can be ascribed to the fault fabric. We relate the polarization effect to compliance anisotropy in the fault zone, where the presence of predominantly oriented fractures makes the normal component of ground motion larger than the transversal one. In order to test the direction and the type of fractures that are expected in the fault zone, we modeled the brittle deformation pattern of the investigated fault. Theoretical results are in good agreement with field observations of the fracture strike.     

回线源瞬变电磁成像的理论分析及数值计算

进一步提高瞬变电磁法对地探测的解释精度,提出了回线源瞬变电磁成像原理及数值计算方法. 讨论了频率域中水平层状介质中瞬变电磁响应,得到一个以波阻抗为积分核的双重积分式;然后对水平层介质下电磁场的解进行上、下行波分离,得到含有以反射系数序列为未知的线性方程组,并给出了求取波阻抗和反射系数的数值解法：对实测磁场值进行域的变换,以均匀半空间下的等效波阻抗代替积分核函数,经过线性数字滤波后,在频率域求出等效波阻抗;把频率域中的波阻抗转换到时间域,以此为参数,构建方程组,在时间域用线性规划法求出反射系数序列. 最终以反射系数为参数进行成像. 对理论模型的数值计算结果表明,用本文提出的成像方法可以增强瞬变电磁法识别地下电性分界面的能力.     

On planar seismic wavefront modeling for estimating rotational ground motions: Case of 2-D SH line-source

At large hypocentral distances, it is convenient to approximate the curved transient seismic wavefronts as planar to estimate rotational ground motions from the single-station recordings of translational ground motions. In this paper, we investigate whether and when this approximation, referred to as the ‘plane-wave’ approximation, can be considered adequate close to the source. For this, we consider a simplistic source model comprising a two-dimensional, kinematic shear dislocation SH line-source buried in a homogenous, elastic half-space and assume this to be an equivalent representation of a finite-sized fault. The ‘plane-wave’ rotational motion is then synthesized from the exact translational motion solution to the assumed model and compared with the exact rotational motion solution for this model. The comparison between the two sets of rotational amplitudes in frequency domain suggests that the plane-wave approximation may be adequate, when the wavelengths of the seismic waves are much smaller than the source depth. When this is not true, the plane-wave approximation is seen to underestimate the Fourier amplitudes close to the source by several orders, particularly when the fault planes are vertically oriented. A similar comparison in the time domain indicates that a severe underestimation may also occur when the source rise time is longer than the shear-wave arrival time at the epicenter. Significant discrepancies are also observed between the waveforms of the exact and plane-wave rotational motions.     

福建省井水位与体应变观测一致性分析

本文从时域和频域分别对福州台和永安台的体应变和井水位资料观测的一致性进行了分析,并对其力学机制进行了深入探究,在此基础上阐明了二者观测一致性的物理含义.时域内的分析结果显示:福州台体应变与井水位的相关系数为0.7939,钻孔系统的灵敏度为0.0558 mm/10-9,井水位与体应变观测曲线十分相似;永安台体应变与井水位的相关系数为-0.7543,钻孔系统的灵敏度为-0.0005 mm/10-9,井水位与体应变曲线呈镜像对称,这表明两台站的井水位与体应变观测高度相关,且二者的相位滞后近似为零.在频域内,分别对福州台2012年3—5月及永安台2014年3—5月的井水位和体应变资料进行维尼迪柯夫调和分析,分别获取半日波、全日波数个波群的逐月潮汐因子和相位滞后,并计算得到了钻孔系统的灵敏度和井水位相对于体应变的相位滞后.结果表明,3个月中大多数波群的灵敏度十分相近,而且与时域内所得年周期低频分量的灵敏度较接近,但相位滞后误差较大,这可能与采用反正切公式计算相位滞后受计算误差影响有关,即非线性的反正切公式对计算误差具有放大累积效应,因此相位滞后应以时域分析所得接近于零的结果为准.综上,时域和频域的分析结果表明,福州台和永安台的井水位对体应变的响应基本上是线性时不变的,二者观测一致性的根本原因在于两台站的承压含水层井水位观测系统基本上是满足叠加性、齐次性和时不变的物理可实现因果系统.      

THE INTERPRETATION OF GRADIENT ARRAY CHARGEABILITY ANOMALIES *

Empirical equalities derived from time domain induced polarization scale modelling with the gradient array over simple geometries, and from the potential field functions for equivalent simple charge configurations are similar. The function for the dyke like body is analogous to the magnetic case allowing both total and vertical magnetic field interpretation techniques to be applied to gradient array chargeability anomalies.     

Evaluation of transport and storage properties in the soil and groundwater zone from induced polarization measurements1

Spectral induced polarization as well as complex electrical measurements are used to estimate, on a non-invasive basis, hydraulic permeability in aquifers. Basic laboratory measurements on a variety of shaly sands, silts and clays showed that the main feature of their conductivity spectra in the frequency range from 10^-3 to 10³ Hertz is a nearly constant phase angle. Thus, a constant-phase-angle model of electrical conductivity is applied to interpret quantitatively surface and borehole spectral induced polarization measurements. The model allows for the calculation of two independent electrical parameters from only one frequency scan and a simple separation of electrical volume and interface effects. The proposed interpretation algorithm yields the true formation factor, the cation exchange capacity and the surface-area-to-porosity ratio, which corresponds to the inverse hydraulic radius. Using a Kozeny–Carman-like equation, the estimation of hydraulic permeability is possible.     

Time-Domain Spectral Induced Polarization Based on Pseudo-random Sequence

To reduce noise during electrical prospecting, we hereby propose a new method using correlation identification technology and conventional electrical exploration devices. A correlation operation can be carried out with the transmitted pseudo-random sequence and received time signal to suppress the random noise, and the time-domain impulse response and frequency response of the frequency domain of the underground media can be obtained. At the same time, using a dual Cole–Cole model to fit a complex resistivity spectrum, which is close to the frequency response, we can get a variety of induced polarization parameters and electromagnetic parameters of subsurface, which can provide more useful information for the exploration of mineral resources. This time domain prospecting method can effectively improve the efficiency of the spectral induced polarization method. In this article, we have carried out theoretical calculations and a simulation to prove the feasibility of such a method.     

海底油气藏及天然气水合物的时频电磁辨识

提出了伪随机多频海洋电磁法观测方案.采用伪随机多频信号作为激励场源,多偏移距同线偶极-偶极同时观测,相关辨识海底地电系统的频率特性和冲激响应,可以在时间域和频率域同时辨识海底高阻薄层.在时间域,瞬变冲激时刻可以直接指示海底地层电导率的变化;在频率域,利用多个频率的电场响应计算的频散率及其道闻变化量,相对相位道间变化量对高阻薄层有很好的反映.从而实现对海底油气及天然气水合物的多参数辨识.