Seismic amplitude inversion for the transversely isotropic media with vertical axis of symmetry

Transverse isotropy with a vertical axis of symmetry is a common form of anisotropy in sedimentary basins, and it has a significant influence on the seismic amplitude variation with offset. Although exact solutions and approximations of the PP-wave reflection coefficient for the transversely isotropic media with vertical axis of symmetry have been explicitly studied, it is difficult to apply these equations to amplitude inversion, because more than three parameters need to be estimated, and such an inverse problem is highly ill-posed. In this paper, we propose a seismic amplitude inversion method for the transversely isotropic media with a vertical axis of symmetry based on a modified approximation of the reflection coefficient. This new approximation consists of only three model parameters: attribute A, the impedance (vertical phase velocity multiplied by bulk density); attribute B, shear modulus proportional to an anellipticity parameter (Thomsen's parameter ε−δ); and attribute C, the approximate horizontal P-wave phase velocity, which can be well estimated by using a Bayesian-framework-based inversion method. Using numerical tests we show that the derived approximation has similar accuracy to the existing linear approximation and much higher accuracy than isotropic approximations, especially at large angles of incidence and for strong anisotropy. The new inversion method is validated by using both synthetic data and field seismic data. We show that the inverted attributes are robust for shale-gas reservoir characterization: the shale formation can be discriminated from surrounding formations by using the crossplot of the attributes A and C, and then the gas-bearing shale can be identified through the combination of the attributes A and B. We then propose a rock-physics-based method and a stepwise-inversion-based method to estimate the P-wave anisotropy parameter (Thomsen's parameter ε). The latter is more suitable when subsurface media are strongly heterogeneous. The stepwise inversion produces a stable and accurate Thomsen's parameter ε, which is proved by using both synthetic and field data.     

Distributions of density and elastic parameters in the Earth

The velocity curve previously obtained for P waves in the Earth is used to determine the distributions of the density and elastic parameters. The density distribution in the new model differs from that in the standard PREM model only in the inner core. The distributions of the bulk and shear moduli can differ, depending on physical processes in the Earth. In particular, the bulk modulus can have a negative jump at the outer-inner core boundary, whereas the shear modulus can differ from zero in the lower part of the outer core.     

Two-dimensional models of elastic tides in a horizontally heterogeneous medium

The possibility of determining local horizontal inhomogeneities of the shear and bulk moduli from data on amplitudes and phases of tidal tilts and strains is considered. Simple analytical formulas determining the sought effects are derived by the perturbation method, and their numerical estimates are determined for the simplest 2-D models. It is shown that relative variations in the shear and bulk moduli affect not only the amplitudes and phases of variations in tidal parameters but also the shape of curves of tidal amplitude anomalies versus the horizontal coordinate. Given adequate spatial resolution, this can significantly facilitate the elastic moduli inversion from tidal data.     

Array analysis of lateral inhomogeneities in the deep mantle

P-wave slowness and azimuth anomalies at LASA are critically dependent on the array configuration. This agrees with the interpretation that the anomalies arise by scattering at small-scale randomly distributed inhomogeneities in the crust and uppermost mantle beneath the array. Of particular importance is the result that numerous configurations can be chosen which yield ( dT/dΔ, ?) anomalies which are inconsistent with recent interpretations including a lateral inhomogeneity at the base of the mantle beneath Hawaii. Also configurations giving ( dT/dΔ, ?) anomalies inconsistent with the existence of mantle plumes under Iceland or the Galapagos Islands are found.     

基于叠前地震纵横波模量直接反演的流体检测方法

流体因子是储层流体识别的重要方法,而叠前地震反演是获得流体因子的有效途径之一.本文从流体因子的构建出发,基于多孔弹性介质岩石物理模型,建立了流体因子与纵横波模量之间的直接关系,避免了流体因子计算所需的密度参数无法准确求取的问题,通过推导基于纵横波模量的Zeoppritz近似方程及弹性阻抗方程,探讨了基于弹性阻抗的纵横波模量直接反演方法,模型与实际应用表明,基于弹性阻抗的纵横波模量直接反演方法合理、可靠,减少了常规方法间接计算纵横波模量带来的累积误差,基于纵横波模量的流体因子计算方法有较好的实际应用效果.     

Seismic wave scattering inversion for fluid factor of heterogeneous media

Elastic wave inverse scattering theory plays an important role in parameters estimation of heterogeneous media. Combining inverse scattering theory, perturbation theory and stationary phase approximation, we derive the P-wave seismic scattering coefficient equation in terms of fluid factor, shear modulus and density of background homogeneous media and perturbation media. With this equation as forward solver, a pre-stack seismic Bayesian inversion method is proposed to estimate the fluid factor of heterogeneous media. In this method, Cauchy distribution is utilized to the ratios of fluid factors, shear moduli and densities of perturbation media and background homogeneous media, respectively. Gaussian distribution is utilized to the likelihood function. The introduction of constraints from initial smooth models enhances the stability of the estimation of model parameters. Model test and real data example demonstrate that the proposed method is able to estimate the fluid factor of heterogeneous media from pre-stack seismic data directly and reasonably.     

Direct inversion for a fluid factor and its application in heterogeneous reservoirs

Prestack seismic inversion plays an important role in estimating elastic parameters that are sensitive to reservoirs and fluid underground. In this paper, a simultaneous inversion method named FMR‐AVA (Fluid Factor, Mu (Shear modulus), Rho (Density)‐Amplitude Variation with Angle) is proposed based on partial angle stack seismic gathers. This method can be used for direct inversion for the fluid factor, shear modulus and density of heterogeneous reservoirs. Firstly, an FMR approximation equation of a reflection coefficient is derived based on poroelasticity with P‐ and S‐wave moduli. Secondly, a stable simultaneous AVA inversion approach is presented in a Bayesian scheme. This approach has little dependence on initial models. Furthermore, it can be applied in heterogeneous reservoirs whose initial models for inversion are not easy to establish. Finally, a model test shows the superiority of this FMR‐AVA inversion method in stability and independence of initial models. We obtain a reasonable fluid factor, shear modulus and density even with smooth initial models and moderate Gaussian noise. A real data case example shows that the inverted fluid factor, shear modulus and density fit nicely with well log interpretation results, which verifies the effectiveness of the proposed method.     

Stepwise joint inversion of surface wave dispersion,Rayleigh wave ZH ratio,and receiver function data for 1D crustal shear wave velocity structure

Accurate determination of seismic velocity of the crust is important for understanding regional tectonics and crustal evolution of the Earth. We propose a stepwise joint linearized inversion method using surface wave dispersion, Rayleigh wave ZH ratio (i.e., ellipticity), and receiver function data to better resolve 1D crustal shear wave velocity (v _S) structure. Surface wave dispersion and Rayleigh wave ZH ratio data are more sensitive to absolute variations of shear wave speed at depths, but their sensitivity kernels to shear wave speeds are different and complimentary. However, receiver function data are more sensitive to sharp velocity contrast (e.g., due to the existence of crustal interfaces) and v _P/v _S ratios. The stepwise inversion method takes advantages of the complementary sensitivities of each dataset to better constrain the v _S model in the crust. We firstly invert surface wave dispersion and ZH ratio data to obtain a 1D smooth absolute v _S model and then incorporate receiver function data in the joint inversion to obtain a finer v _S model with better constraints on interface structures. Through synthetic tests, Monte Carlo error analyses, and application to real data, we demonstrate that the proposed joint inversion method can resolve robust crustal v _S structures and with little initial model dependency.     

Estimation of Dry Fracture Weakness,Porosity, and Fluid Modulus Using Observable Seismic Reflection Data in a Gas-Bearing Reservoir

Fracture detection and fluid identification are important tasks for a fractured reservoir characterization. Our goal is to demonstrate a direct approach to utilize azimuthal seismic data to estimate fluid bulk modulus, porosity, and dry fracture weaknesses, which decreases the uncertainty of fluid identification. Combining Gassmann’s (Vier. der Natur. Gesellschaft Zürich 96:1–23, 1951) equations and linear-slip model, we first establish new simplified expressions of stiffness parameters for a gas-bearing saturated fractured rock with low porosity and small fracture density, and then we derive a novel PP-wave reflection coefficient in terms of dry background rock properties (P-wave and S-wave moduli, and density), fracture (dry fracture weaknesses), porosity, and fluid (fluid bulk modulus). A Bayesian Markov chain Monte Carlo nonlinear inversion method is proposed to estimate fluid bulk modulus, porosity, and fracture weaknesses directly from azimuthal seismic data. The inversion method yields reasonable estimates in the case of synthetic data containing a moderate noise and stable results on real data.     

Linearized elastic parameter sections1

Contrasts in elastic parameters can be estimated directly from seismic data using offset-dependent information in the PP reflection coefficient. A linear approximation to the PP reflection coefficient including three coefficients is fitted to the data, and relative contrasts in various elastic parameters are obtained from linear combinations of the estimated coefficients. Linearized elastic parameter sections for the contrasts in P-wave impedance, P-wave velocity, density, plane-wave modulus, and the change in bulk modulus and shear modulus normalized with the plane-wave modulus are estimated. If the average P- to S-wave velocity ratio is known, linearized parameter sections including the contrast in the average P- to S-wave velocity ratio and a fluid factor section can be computed. Applied to synthetic data, visual comparison of the estimated and true elastic parameter sections agree qualitatively, and the results are confirmed by an analysis of the standard deviation of the estimated parameters. The parameter sections obtained by inversion of a shallow seismic anomaly in the Barents Sea are promising, but the reliability is uncertain because neither well data nor regional trends are available.     

Review of correlations between SPT N and shear modulus: A new correlation applicable to any region

A low strain shear modulus plays a fundamental role in earthquake geotechnical engineering to estimate the ground response parameters for seismic microzonation. A large number of site response studies are being carried out using the standard penetration test (SPT) data, considering the existing correlation between SPT N values and shear modulus. The purpose of this paper is to review the available empirical correlations between shear modulus and SPT N values and to generate a new correlation by combining the new data obtained by the author and the old available data. The review shows that only few authors have used measured density and shear wave velocity to estimate shear modulus, which were related to the SPT N values. Others have assumed a constant density for all the shear wave velocities to estimate the shear modulus. Many authors used the SPT N values of less than 1 and more than 100 to generate the correlation by extrapolation or assumption, but practically these N values have limited applications, as measuring of the SPT N values of less than 1 is not possible and more than 100 is not carried out. Most of the existing correlations were developed based on the studies carried out in Japan, where N values are measured with a hammer energy of 78%, which may not be directly applicable for other regions because of the variation in SPT hammer energy. A new correlation has been generated using the measured values in Japan and in India by eliminating the assumed and extrapolated data. This correlation has higher regression coefficient and lower standard error. Finally modification factors are suggested for other regions, where the hammer energy is different from 78%.     

Numerical simulations of thermal convection in a rapidly rotating spherical shell cooled inhomogeneously from above

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code. The spherical shell of Boussinesq fluid has inner and outer radii the same as those of the Earth's liquid outer core. The Taylor number is 10⁷, the Prandtl number is 1, and the Rayleigh number R is 5R_c (R_c is the critical value of R for the onset of convection when the top boundary is isothermal and R is based on the spherically averaged temperature difference across the shell). The shell is heated from below and cooled from above; there is no internal heating. The lower boundary of the shell is isothermal and both boundaries are rigid and impermeable. Three cases are considered. In one, the upper boundary is isothermal while in the others, temperature anomalies with (l,m) = (3,2) and (6,4) are imposed on the top boundary. The spherically averaged temperature difference across the shell is the same in all three cases. The amplitudes of the imposed temperature anomalies are equal to one-half of the spherically averaged temperature difference across the shell. Convective structures are strongly controlled by both rotation and the imposed temperature anomalies suggesting that thermal inhomogeneities imposed by the mantle on the core have a significant influence on the motions inside the core. The imposed temperature anomaly locks the thermal perturbation structure in the outer part of the spherical shell onto the upper boundary and significantly modifies the velocity structure in the same region. However, the radial velocity structure in the outer part of the shell is different from the temperature perturbation structure. The influence of the imposed temperature anomaly decreases with depth in the shell. Thermal structure and velocity structure are similar and convective rolls are more columnar in the inner part of the shell where the effects of rotation are most dominant.     

基于基追踪弹性阻抗反演的深部储层流体识别方法

深部储层地震资料通常照明度低、信噪比低、分辨率不足,尤其是缺乏大角度入射信息,对深部储层流体识别存在较大影响.Gassmann流体项是储层流体识别的重要参数,针对深层地震资料的特点,本文首先在孔隙介质理论的指导下,推导了基于Gassmann流体项与剪切模量的两项AVO近似方程.通过模型分析,验证了该方程在小角度时与精确Zoeppritz方程误差很小,满足小角度入射条件下的近似精度要求.然后借助Connolly推导弹性阻抗的思想,推导了基于Gassmann流体项与剪切模量的两项弹性阻抗方程.针对深部储层地震资料信噪比差的特点,利用奇偶反射系数分解实现了深部储层基追踪弹性阻抗反演方法,最后提出了基于基追踪弹性阻抗反演的Gassmann流体项与剪切模量的求取方法,并将提取的Gassmann流体项应用于深部储层流体识别.模型测试和实际应用表明该方法稳定有效,具有较好的实用性.     

Seismic inversion with generalized Radon transform based on local second-order approximation of scattered field in acoustic media

Sound velocity inversion problem based on scattering theory is formulated in terms of a nonlinear integral equation associated with scattered field. Because of its nonlinearity, in practice, linearization algorisms (Born/single scattering approximation) are widely used to obtain an approximate inversion solution. However, the linearized strategy is not congruent with seismic wave propagation mechanics in strong perturbation (heterogeneous) medium. In order to partially dispense with the weak perturbation assumption of the Born approximation, we present a new approach from the following two steps: firstly, to handle the forward scattering by taking into account the second-order Born approximation, which is related to generalized Radon transform (GRT) about quadratic scattering potential; then to derive a nonlinear quadratic inversion formula by resorting to inverse GRT. In our formulation, there is a significant quadratic term regarding scattering potential, and it can provide an amplitude correction for inversion results beyond standard linear inversion. The numerical experiments demonstrate that the linear single scattering inversion is only good in amplitude for relative velocity perturbation ( \( \delta_{c}/c_{0} \) ) of background media up to 10 %, and its inversion errors are unacceptable for the perturbation beyond 10 %. In contrast, the quadratic inversion can give more accurate amplitude-preserved recovery for the perturbation up to 40 %. Our inversion scheme is able to manage double scattering effects by estimating a transmission factor from an integral over a small area, and therefore, only a small portion of computational time is added to the original linear migration/inversion process.     

Refraction traveltime and amplitude corrections for very near- surface inhomogeneities

The performance of refraction inversion methods that employ the principle of refraction migration, whereby traveltimes are laterally migrated by the offset distance (which is the horizontal separation between the point of refraction and the point of detection on the surface), can be adversely affected by very near‐surface inhomogeneities. Even inhomogeneities at single receivers can limit the lateral resolution of detailed seismic velocities in the refractor. The generalized reciprocal method ‘statics’ smoothing method (GRM SSM) is a smoothing rather than a deterministic method for correcting very near‐surface inhomogeneities of limited lateral extent. It is based on the observation that there are only relatively minor differences in the time‐depths to the target refractor computed for a range of XY distances, which is the separation between the reverse and forward traveltimes used to compute the time‐depth. However, any traveltime anomalies, which originate in the near‐surface, migrate laterally with increasing XY distance. Therefore, an average of the time‐depths over a range of XY values preserves the architecture of the refractor, but significantly minimizes the traveltime anomalies originating in the near‐surface. The GRM statics smoothing corrections are obtained by subtracting the average time‐depth values from those computed with a zero XY value. In turn, the corrections are subtracted from the traveltimes, and the GRM algorithms are then re‐applied to the corrected data. Although a single application is generally adequate for most sets of field data, model studies have indicated that several applications of the GRM SSM can be required with severe topographic features, such as escarpments. In addition, very near‐surface inhomogeneities produce anomalous head‐wave amplitudes. An analogous process, using geometric means, can largely correct amplitude anomalies. Furthermore, the coincidence of traveltime and amplitude anomalies indicates that variations in the near‐surface geology, rather than variations in the coupling of the receivers, are a more likely source of the anomalies. The application of the GRM SSM, together with the averaging of the refractor velocity analysis function over a range of XY values, significantly minimizes the generation of artefacts, and facilitates the computation of detailed seismic velocities in the refractor at each receiver. These detailed seismic velocities, together with the GRM SSM‐corrected amplitude products, can facilitate the computation of the ratio of the density in the bedrock to that in the weathered layer. The accuracy of the computed density ratio improves where lateral variations in the seismic velocities in the weathered layer are known.     

Surface waves in an elastic semi-space with a stable nonhomogeneity

Summary Surface waves in an isotropic nonhomogeneous elastic semi-space (plane strain) are studied. It is assumed that the Poisson's ratio and the density of the medium are constant but shear modulus is a monotonic function of the depth. Use of the stress equation of motion is made to reduce the problem to an eigenvalue problem for a differential equation of fourth order with polynomial coefficients. A series solution of the problem is obtained and dependence of the Rayleigh velocityC _R on the wavelength and the nonhomogeneity of the medium, is studied. Group velocity period curves are also obtained.

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Zusammenfassung Oberflächenwellen in einem isotropen Nicht-homogenen elastischen Halbraum (Für den ebenen Verzerungszustand) werden untersucht. Die Poissonsche Querzahl und die Dichte werden als konstant vorausgesetzt während der Schubmodul ein monotone Funktion der Tiefe ist. Unter Benutzung der Spannungstensorsbewegungsgleichungen reduziert sich das problem zu einem Eigenwertproblem einer Differentialgleichung der vierten Ordnung mit Polynomkoeffizienten. Eine Reihenentwicklung wird als Lösung des problems erhalten und die Abhängigkeit der RayleighgeschwindigkeitC _R von der Wellenlänge und die Nicht-homogeneität des mediums werden untersucht. Ausserdem werden Gruppengeschwindigkeit und Perioden kurven erhalten.

     

基于Russell近似的纵横波联合反演方法研究

PP波和PS波联合反演方法作为有效的地震技术,比单纯纵波反演精度要高,能够提高地震储层识别的精度.以Russell近似理论为基础,推导了新的转换波AVO近似公式,双层模型界面的反射特征数值模拟显示,新公式具有较高的近似精度,且具备直接反演流体因子f、剪切模量μ和密度ρ等参数的优势,有效避免间接反演带来的误差.结合纵横波联合反演理论,提出了基于贝叶斯理论的新型联合反演算法.在实际应用中,对纵波和转换波角道集进行同相轴匹配处理,综合利用纵波和转换波资料携带的信息,实现基于Russell近似的多波联合反演.模型数据和实际资料测试结果表明,反演结果与真实值或测井结果匹配度较高,证实该方法真实有效.     

Far field elastodynamic Born scattering revisited

This paper summarizes part of an ongoing feasibility study that investigates the possible use of the full elastic Born approximation in multipole borehole acoustics. As a first step we exclude the fluid-filled borehole with the motivation that one or two wavelengths away from the fluid-filled borehole, radiating borehole mode amplitudes (e.g., Stoneley wave, formation dipole wave, etc.) are small compared to body wave amplitudes (P-, SV- and SH-waves). Consequently, for scatterers one or two wavelengths away from the fluid-filled borehole, it suffices to only consider their interaction with body waves.In this paper we apply the contrast-source stress-velocity forward scattering (integral equation) formulation for solid configurations in its first order (Born-) approximation (De Hoop, 1995) assuming a multipole force source excitation in a zero-offset configuration. To scrutinize the validity of the Born approximation, we consider the simplest type of scatterer, i.e., one characterized by a (Heaviside) step function change in one or more of the contrast (perturbation) parameters and we derive analytic zero-offset formulas for the scattered wave particle velocity and displacement in both the space–frequency and space–time domain, respectively. We assume the scatterer to be located in the far-field. More complicated layered configurations can easily be derived by superposition of the given solution types. Explicit results are given for the dipole and quadrupole excitation, where the former is allowed to have an arbitrary orientation relative to the scatterer and where the latter one is located in a plane perpendicular to the scatterer. In the time domain it is shown, how the scattered wave field decomposes in a specular and diffuse wave field (two terms borrowed from ‘Optics’), where the former contribution vanishes in the absence of an imaging condition and where the latter is always present. For the dipole case, we subject our results to a sensitivity analysis with respect to the three independent perturbation parameters (i.e., density and two compliance parameters) and we compare these results to a full waveform benchmark code that has implemented the reflectivity method (Kennett, 1983), for a ‘horizontally’ stratified elastic medium. This allowed us to pinpoint the root cause of the observed (small) differences. As it turns out these deviations could be traced back to the inaccurateness of the first order Born scattering coefficients. An additional confirmation of this fact is provided through a comparison between the zero-offset scattering coefficients and the corresponding Zoeppritz reflection coefficients. Most notably, it was found that the PP first order scattering coefficient needs a higher than quadratic correction in two of the three independent perturbation parameters, i.e., the two compliance parameters, δΛ and δM. With respect to the density perturbation parameter (δρ) the PP scattering coefficient correction is quadratic with respect to this perturbation parameter, as is to be expected for a first order approximation. Moreover, also the SS first order scattering coefficient only needs a quadratic correction with respect to its associated perturbation parameters, i.e., δρ and δM.Finally, we give a brief outline on how to numerically implement the Born approximation (employing arbitrary offsets) in a configuration where a source–receiver pair is moving continuously relative to the ‘contrast’ (Geology), as is the case in borehole acoustic applications.     

T-matrix approach to seismic forward modelling in the acoustic approximation

Forward seismic modelling in the acoustic approximation, for variable velocity but constant density, is dealt with. The wave equation and the boundary conditions are represented by a volume integral equation of the Lippmann-Schwinger (LS) or Fredholm type. A T-matrix (or transition operator) approach from quantum mechanical potential scattering theory is used to derive a family of linear and nonlinear approximations (cluster expansions), as well as an exact numerical solution of the LS equation. For models of 4D anomalies involving small or moderate contrasts, the Born approximation gives identical numerical results as the first-order t-matrix approximation, but the predictions of an exact T-matrix solution can be quite different (depending on spatial extention of the perturbations). For models of fluid-saturated cavities involving large or huge contrasts, the first-order t-matrix approximation is much more accurate than the Born approximation, although it does not lead to significantly more time-consuming computations. If the spatial extention of the perturbations is not too large, it is practical to use the exact T-matrix solution which allows for arbitrary contrasts and includes all the effects of multiple scattering.     

Computation of the key parameters of radio signals propagating through a perturbed ionosphere in the land-satellite channel

An analysis of the key parameters of HF/UHF radio signals was carried out for land-satellite radio channels, which determine the effects of fading in a perturbed ionosphere. Using the parameters of the perturbed plasma, the effects of the absorption and phase fluctuations of radio signals are analyzed for a channel with fading. For the evaluation of the effect of scattering of a radio signal by ionospheric inhomogeneities in an approximation of small-scale scintillations, expressions for the root-mean-square (RMS) magnitude of signal intensity and phase scintillations are presented. Scintillation index σ _I ² that corresponds to variations in a signal under the conditions of multipath propagation with fading is investigated by using experimental data. It is shown that roughly ～10% of inhomogeneities of the electron concentration in the F region of the ionosphere, perturbed during a magnetic storm, yield strong quickly fading radio signals in the VHF/UHF range with significant fluctuations (up to 1%) in the intensity of the signal and phase fluctuations (up to hundreds of radians). The calculated magnitudes of the scintillation index are in good agreement with experimentally observed data.