Analysis of different parameterisations of waveform inversion of compressional body waves in an elastic transverse isotropic Earth with a vertical axis of symmetry

In a multi‐parameter waveform inversion, the choice of the parameterisation influences the results and their interpretations because leakages and the tradeoff between parameters can cause artefacts. We review the parameterisation selection when the inversion focuses on the recovery of the intermediate‐to‐long wavenumbers of the compressional velocities from the compressional body (P) waves. Assuming a transverse isotropic medium with a vertical axis of symmetry and weak anisotropy, analytical formulas for the radiation patterns are developed to quantify the tradeoff between the shear velocity and the anisotropic parameters and the effects of setting to zero the shear velocity in the acoustic approach. Because, in an anisotropic medium, the radiation patterns depend on the angle of the incident wave with respect to the vertical axis, two particular patterns are discussed: a transmission pattern when the ingoing and outgoing slowness vectors are parallel and a reflection pattern when the ingoing and outgoing slowness vectors satisfy Snell's law. When the inversion aims at recovering the long‐to‐intermediate wavenumbers of the compressional velocities from the P‐waves, we propose to base the parameterisation choice on the transmission patterns. Since the P‐wave events in surface seismic data do not constrain the background (smooth) vertical velocity due to the velocity/depth ambiguity, the preferred parameterisation contains a parameter that has a transmission pattern concentrated along the vertical axis. This parameter can be fixed during the inversion which reduces the size of the model space. The review of several parameterisations shows that the vertical velocity, the Thomsen parameter δ, or the Thomsen parameter ε have a transmission pattern along the vertical axis depending on the parameterisation choice. The review of the reflection patterns of those selected parameterisations should be done in the elastic context. Indeed, when reflection data are also inverted, there are potential leakages of the shear parameter at intermediate angles when we carry out acoustic inversion.     

声波在两种多孔介质界面上的反射和透射

本文导出了声波在两种多孔介质界面上反射、透射的一般计算公式.作为例子,数值计算了P₁波入射于界面时,P₁、P₂和S波的反射、透射系数与声波频率、入射角等量之间的关系.结果表明,各种模式波的反射、透射系数与入射角、多孔介质性质有关,在Biot特征频率附近与频率有关,并用界面两侧的法向能流相等验证了结果的正确性.若把多孔介质当作均匀固体处理,将会得到显著不同的结果.     

Seismic reflection and transmission coefficients at an air-water interface of saturated porous soil

Based on the modified Biot＇s theory of two-phase porous media, a study was presented on seismic reflection and transmission coefficients at an air-water interface of saturated porous soil media. The major differences between air-saturated soils and water-saturated soils were theoretically discussed, and the theoretical formulas of reflection and transmission coefficients at an air-water interface were derived. The characteristics of propagation and attenuation of elastic waves in air-saturated soils were given and the relations among the frequency, the angle of incidence and the reflection, transmission coefficients were analyzed by using numerical methods. Numerical results show that the propagation characteristic of the wave in air-saturated soils is great different from that in water-saturated soils. The frequency and the angle of incidence can have great influences on the reflection and transmission coefficients at interface. Some new cognition about the wave propagation is obtained and the study suggests that we may carefully pay attention to the influence of air on the dynamic analysis of seismic wave.     

声波方程频率域有限元参数反演

推导出频率域有限元声波正演方程,为了消除边界反射,将Clayton-Engquist旁轴波动方程吸收边界条件引入频率域,并对有限元刚度矩阵和质量矩阵进行压缩存储,利用广义共轭梯度法求解有限元方程获得正演解.在此基础上,推导出在某一频率下波场数据残差δU与单元物性参数修改量δλ之间关系的Jacobi矩阵,反演方法允许利用地面二维炮集全波场资料与给出初始模型参数的正演值的差值δU,迭代求得δλ.由于计算机内存的限制,方法计算不允许有过多数目的未知数个数,因此还提出了对同一介质物性单元的Jacobi矩阵元素进行压缩组装的措施,从而使反演的未知量个数减少,结合采用共轭梯度迭代法,使得只需利用有效波频段的少数一些频率即可进行迭代反演.正演和反演理论模型的数值模拟结果表明方法是有效的.     

Generalized Hooke'slow and New Imaging Method in Seismic Exploration

The classical Hook's law is a experimental law, and in small region it can be mathemahcallyexpressed by the followhg linear formulawhere y is the stress, z the strain, and k the elashc coefficient Note that all the experimentalresults are based artificial measurements. Any facial measurement is in static state beforehundreds years. These results can not be expressed directly in drutalc state asThe logic of "from (l) tO (2)" is wrong. All the classical laws need to revise in dynamic state. [1…     

超声反射法套管-地层环空间介质波阻抗反演方法与成像实例

针对套管井反射式超声波水泥固井质量检测问题，本文利用超声反射波频谱中套管共振透射窗内的复合反射系数，提出了定量反演套管-地层环空间内介质的波阻抗的方法. 利用复合反射系数中多个频率点的相移反正切函数主值，给出了波阻抗的计算公式，并把套管-地层环空间介质中波的相移的整、实特性作为波阻抗反演收敛的判据. 针对井下实际测井环境中超声换能器性能受温度和压力等的影响而发生变化，导致的声源波形（子波）和复合反射系数不确定的困难，提出了用井下原地（in situ）实测波形定量估计声源波形和复合反射系数的方法. 通过对合成数据的处理，分别考察了已知介质的声学和几何参数存在的偏差对套管-地层环空间内介质的波阻抗反演结果的影响. 本文对多组模型井检验数据进行了目的层介质的波阻抗反演与成像，成像结果与真实情况吻合，验证了反演方法的有效性. 最后对现场实测资料进行了套管-地层环空间内介质的波阻抗定量反演与成像.     

THE NON-LINEAR INVERSION OF SEISMIC WAVEFORMS CAN BE PERFORMED EITHER BY TIME EXTRAPOLATION OR BY DEPTH EXTRAPOLATION1

The classical aim of non-linear inversion of seismograms is to obtain the earth model which, for null initial conditions and given sources, best predicts the observed seismograms. This problem is currently solved by an iterative method: each iteration involves the resolution of the wave equation with the actual sources in the current medium, the resolution of the wave equation, backwards in time, with the current residuals as sources; and the correlation, at each point of space, of the two wavefields thus obtained. Our view of inversion is more general: we want to obtain a whole set of earth model, initial conditions, source functions, and predicted seismograms, which are the closest to some a priori values, and which are related through the wave equation. It allows us to justify the previous method, but it also allows us to set the same inverse problem in a different way: what is now searched for is the best fit between calculated and a priori initial conditions, for given sources and observed surface displacements. This leads to a completely different iterative method, in which each iteration involves the downward extrapolation of given surface displacements and tractions, down to a given depth (the‘bottom’), the upward extrapolation of null displacements and tractions at the bottom, using as sources the initial time conditions of the previous field, and a correlation, at each point of the space, of the two wavefields thus obtained. Besides the theoretical interest of the result, it opens the way to alternative numerical methods of resolution of the inverse problem. If the non-linear inversion using forward-backward time propagations now works, this non-linear inversion using downward-upward extrapolations will give the same results but more economically, because of some tricks which may be used in depth extrapolation (calculation frequency by frequency, inversion of the top layers before the bottom layers, etc.).     

Research and application of spectral inversion technique in frequency domain to improve resolution of converted PS-wave

Multi-wave exploration is an effective means for improving precision in the exploration and development of complex oil and gas reservoirs that are dense and have low permeability. However, converted wave data is characterized by a low signal-to-noise ratio and low resolution, because the conventional deconvolution technology is easily affected by the frequency range limits, and there is limited scope for improving its resolution. The spectral inversion techniques is used to identify λ/8 thin layers and its breakthrough regarding band range limits has greatly improved the seismic resolution. The difficulty associated with this technology is how to use the stable inversion algorithm to obtain a high-precision reflection coefficient, and then to use this reflection coefficient to reconstruct broadband data for processing. In this paper, we focus on how to improve the vertical resolution of the converted PS-wave for multi-wave data processing. Based on previous research, we propose a least squares inversion algorithm with a total variation constraint, in which we uses the total variance as a priori information to solve under-determined problems, thereby improving the accuracy and stability of the inversion. Here, we simulate the Gaussian fitting amplitude spectrum to obtain broadband wavelet data, which we then process to obtain a higher resolution converted wave. We successfully apply the proposed inversion technology in the processing of high-resolution data from the Penglai region to obtain higher resolution converted wave data, which we then verify in a theoretical test. Improving the resolution of converted PS-wave data will provide more accurate data for subsequent velocity inversion and the extraction of reservoir reflection information.     

基于VSP直达波资料的粘弹介质参数遗传算法反演

粘弹性参数变得越来越重要,其反演算法也逐渐成为众多研究者的研究热点。而遗传算法是一种随机、自适应、启发式的算法, 具有很好的鲁棒性和全局收敛性, 本文基于VSP直达波方程,引入了遗传算法来进行粘弹参数反演, 首先将频率域直达波方程表示为复速度的函数,然后通过遗传算法反演出复速度。而复速度和品质因子又是复速度的函数,从而便可很容易的得出。但若直接反演复速度, 反演参数太多, 不容易实现, 所以又将复速度表示成参数C0和C∞的函数,以减少反演参数数量。最后给出了理论模型实验,以证明该算法的有效性。     

介质密度反演偏导矩阵的精确计算方法

实现反演偏导矩阵的计算是基于导数最优化反演方法的关键,然而目前的地震反演几乎都是基于Zoeppritz方程近似实现的,使计算精度和适应范围受到限制.本文利用Zoeppritz方程建立了反射系数对地层介质密度比偏导方程,导出了Zoeppritz方程矩阵元对介质密度比的导数.通过求解偏导方程获得了反射系数对介质密度比偏导数的精确计算(考虑了速度中含介质密度的问题).利用数值算例分析了反射系数对介质密度比偏导数的变化特点.本文采用直接解法求解偏导矩阵方程组,获得了快的计算速度和高的计算精度,为实现地层介质密度反演(包括大角度反演)提供了偏导矩阵的计算方法.     

Marine seismic wavefield measurement to remove sea-surface multiples

We propose a new method for removing sea-surface multiples from marine seismic reflection data in which, in essence, the reflection response of the earth, referred to a plane just above the sea-floor, is computed as the ratio of the plane-wave components of the upgoing wave and the downgoing wave. Using source measurements of the wavefield made during data acquisition, three problems associated with earlier work are solved: (i) the method accommodates source arrays, rather than point sources; (ii) the incident field is removed without simultaneously removing part of the scattered field; and (iii) the minimum-energy criterion to find a wavelet is eliminated. Pressure measurements are made in a horizontal plane in the water. The source can be a conventional array of airguns, but must have both in-line and cross-line symmetry, and its wavefield must be measured and be repeatable from shot to shot. The problem is formulated for multiple shots in a two-dimensional configuration for each receiver, and for multiple receivers in a two-dimensional configuration for each shot. The scattered field is obtained from the measurements by subtracting the incident field, known from measurements at the source. The scattered field response to a single incident plane wave at a single receiver is obtained by transforming the common-receiver gather to the frequency–wavenumber domain, and a single component of this response is obtained by Fourier transforming over all receiver coordinates. Each scattered field component is separated into an upgoing wave and a downgoing wave using the zero-pressure condition at the water-surface. The upgoing wave may then be expressed as a reflection coefficient multiplied by the incident downgoing wave plus a sum of scattered downgoing plane waves, each multiplied by the corresponding reflection coefficient. Keeping the upgoing scattered wave fixed, and using all possible incident plane waves for a given frequency, yields a set of linear simultaneous equations for the reflection coefficients which are solved for each plane wave and for each frequency. To create the shot records that would have been measured if the sea-surface had been absent, each reflection coefficient is multiplied by complex amplitude and phase factors, for source and receiver terms, before the five-dimensional Fourier transformation back to the space–time domain.     

P-SV波AVO方法研究进展

为有效地利用AVO(amplitude versus offset)信息来反演岩性参数和预测油气储层，详细介绍了P—SV转换波AVO方法的研究现状和主要进展，并针对几种具有代表性的方法的基本思路、方法特点及参数反演等方面进行了对比和评述。同时，给出了两个不同形式的近似公式，并讨论了利用这两种近似公式进行AVO分析的思路和参数反演的方法，通过对比和讨论可以得出：不同近似的主要目的是为了提高近似精度并体现不同的岩性参数对反射系数的影响、敏感程度及其在参数反演中的意义；根据弹性模量和波速之间的关系，目前所有的PP波和P—SV波反射系数近似公式都可以统一表示成射线参数幂级数的形式，这些结论对于AVO理论研究和参数反演都具有重要意义。     

Frequency characteristics of coda and determination of medium parameters in Lingwu region, China

In this paper, considering the influences of source spectrum, the scattering property of medium and instrument response on the dominant frequency of coda, a method of using the coda of local earthquake to determine the correlation length of medium andQ-value is given. We find the following formula as: {fx719-1} wheret* =t/Q, f is the dominant frequency of coda,u ₁ andu ₂ are the parameters depend on the correlation length and the corner frequency of the source spectrum respectively,I(f) is a function of instrument response. If the source parameter is given, we can obtain the correlation length andQ-value by means of the inversion of observed curves off-t of coda. We processed the data of coda wave of more than 40 earthquakes from 1982 to 1989 in Lingwu region, China, determined the correlation length andQ-value, and preliminarily studied the temporal change of correlation length before and after moderately strong earthquakes. The result suggests that there are indications that the correlation length of medium decreases before the moderate earthquake. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,14, 62–70, 1992.     

频率域波形反演中与频率相关的影响因素分析

波动方程深度偏移是解决复杂地质体成像的关键技术,基于波动方程的速度建模为其提供更为精确的速度模型.频率域波形反演是目前研究最为广泛的波动方程速度建模方法之一,它推动了波形反演在勘探尺度下的应用.本文通过对频率域波形反演的实现,分析对比了其有效执行过程中与频率相关的影响因素.介绍了时间域的多尺度反演方法在频率域的一种实现方式,对比分析了输入数据的频点带宽和应用的子波频带范围不同时对反演结果的影响.本文通过设计的山地地质模型对频率域波形反演进行了测试和对比,得到的结论为频率域波形反演的有效计算提供了依据和参考.     

Recursive impedance inversion of ground-penetrating radar data in stochastic media

The travel time and amplitude of ground-penetrating radar (GPR) waves are closely related to medium parameters such as water content, porosity, and dielectric permittivity. However, conventional estimation methods, which are mostly based on wave velocity, are not suitable for real complex media because of limited resolution. Impedance inversion uses the reflection coefficient of radar waves to directly calculate GPR impedance and other parameters of subsurface media. We construct a 3D multiscale stochastic medium model and use the mixed Gaussian and exponential autocorrelation function to describe the distribution of parameters in real subsurface media. We introduce an elliptical Gaussian function to describe local random anomalies. The tapering function is also introduced to reduce calculation errors caused by the numerical simulation of discrete grids. We derive the impedance inversion workflow and test the calculation precision in complex media. Finally, we use impedance inversion to process GPR field data in a polluted site in Mongolia. The inversion results were constrained using borehole data and validated by resistivity data.     

地震逆散射波场和算子的谱分解

本文对地震逆散射的研究,旨在于为抑制层间多次波和地震波场多重散射对一次反射干扰效应提供理论依据.这对薄互层地层滤波的高频恢复、保幅弹性反演、衍射地震勘探及海洋地震勘探中的干扰消除皆具重要意义.本文基于上下行波分解及弹性波互易定理,导出横向变速介质条件下线性预测算子的表达式和反射数据的广义谱分解方程. 文中先由上覆地层广义反射透射矩阵的元素定义线性预测算子,并将其表示成一系列单程波算子的线性组合,之后将横向变速介质条件下线性预测方程表达为反射数据与线性预测算子及其逆的乘积. 对该方程的求解可获得上覆地层的线性预测算子,从而可借以求出相应的反射透射算子. 本文先将水平层状介质条件下垂直入射的一维线性预测方程推广到斜入射的情况,以此为参照,导出横向非均匀介质条件下反射数据的地震逆散射广义谱分解方程.文中也揭示了单程波地震逆散射算子、反射透射算子的性态.本文还针对水平层状介质条件,给出斜入射的数值结果.     

The form of the reflection operator and extended image gathers for a dipping plane reflector in two space dimensions

The reflection operator for a simple flat‐lying interface can be thought of as the set of all its plane‐wave reflection coefficients or as the set of virtual surveys with sources and receivers along the interface. When there is dip, however, it is necessary to include the varying effects of propagation between the virtual‐survey level and the interface. Hence, step one in this paper is to derive the reflection operator for a dipping plane interface as observed at a datum level some distance away. The key assumption is that the aperture at the datum level is sufficient to characterize the reflector properties around a particular point. This translates into an assumption that the dip is moderate, though no explicit small‐angle approximation is required. The second step is to find the apparent reflection operator that would relate data that have been extrapolated from the datum towards and possibly beyond the reflector using an assumed migration velocity. This apparent reflection operator is closely related to extended common‐image gathers. The apparent reflection operator may be analysed asymptotically in terms of rays and other signals, shedding light on the structure of extended image gathers. In keeping with the virtual‐survey idea, the results are considered in a subsurface space‐time or slowness‐time domain at various extrapolation levels around the interface. An important distinction is drawn between using subsurface midpoint‐offset coordinates and the wavefield coordinates of the incident and reflected waves. The latter reveal more clearly the effects of dip, because they lead to a more asymmetric apparent reflection operator. Properties such as an up‐dip shift of a traveltime minimum and its associated curvature theoretically provide information about the reflector location and dip and the migration‐velocity error. The space‐time form of the reflection operator can be highly intricate around the offset‐time origin and it was described for a simple flat interface in a background paper. To avoid a layer of mathematics, the reflection‐operator formulas presented here are in the intermediate space‐frequency domain. They are analysed by considering their stationary‐phase and branch‐point high‐frequency contributions. There is no Born‐like assumption of weak reflector contrast and so wide‐angle, total reflection and head‐wave effects are included. Snell’s law is an explicit part of the theory. It is hoped that the work will therefore be a step towards the goal of unifying amplitude‐versus‐offset, imaging and waveform inversion.     

频率-波数域递推计算海底多分量地震记录中的反射系数

在频率-波数域中采用解析法,解出多层条件下海底实测的多分量地震数据分解成上行和下行P波和S波的算法,导出海底各层地震反射系数随入射角变化(简称RVA)的递推计算公式,为海底多波多分量AVO弹性参数的反演及流体因子预测提供基础数据.合成数据的计算结果表明,本文给出的算法能较可靠地从海底多波多分量记录中提取RVA信息.     

断层衍射波及其应用ht

本文从衍射波的物理定义出发,简化了Trorey提出的断层上Kirchhoff-Helmholtz衍射方程的解法;计算了不同深度、不同测线方向下的断层衍射波理论地震图;计算了衍射波振幅谱与相位谱,给出了利用衍射波求断层位置的公式;得出了一些新的结论,通过人工地震测深与地震勘探的实例,对断层衍射波的特性有了更明确的认识。 本文的结果表明:断层衍射波发生在地球介质剧烈变化处;衍射点两侧的衍射波走时曲线呈双曲线状;波初动清晰且半周期小;其优势频率振幅谱与反射波的相同,在衍射波与反射波走时曲线相切处附近,记录图中出现衍射波最大振幅,且波反相;视断点与真实断点一般不重合。上述特点可能为判定衍射波并确定断层位置提供判据。