Thomsen Parameters Research Articles

Numerical Modeling of Scholte Wave in Acoustic-Elastic Coupled TTI Anisotropic Media

Numerical modeling of acoustic-elastic media is helpful for seismic exploration in the deepwater environment. We propose an algorithm based on the staggered grid finite difference to simulate wave propagation in the interface between fluid and transversely isotropic media, where the interface does not need to consider the boundary condition. We also derive the stability conditions of the proposed method. Scholte waves, which are generated at the seafloor, exhibit distinctly different propagation behaviors than body waves in ocean-bottom seismograms. Numerical examples are used to characterize the wavefield of Scholte waves and discuss the relationship between travel time and the Thomsen parameters. Thomsen parameters are assigned clear physical meanings, and the magnitude of their values directly indicates the strength of the anisotropy in the media. Numerical results show that the velocity of the Scholte wave is positively correlated with ε and negatively correlated with δ. And the curve of the arrival time of the Scholte wave as a whole is sinusoidal and has no symmetry in inclination. The velocity of the Scholte wave in azimuth is positively related to the polar angle. The energy of the Scholte wave is negatively correlated with the distance from the source to the fluid-solid interface. The above results provide a basis for studying oceanic Scholte waves and are beneficial for utilizing the information provided by Scholte waves.

Anisotropic parameters and bulk waves propagation in arbitrarily-anisotropic materials

To reflect the anisotropic degree of materials and to further reveal its relevance with the elastic wave propagation in materials, the Thomsen's parameters were proposed for transversely-isotropic materials with weakly anisotropy and were generalized to materials with lower symmetries. This paper derives the complete 21 anisotropic parameters of arbitrarily-anisotropic (triclinic) materials to reflect their anisotropic degree by virtue of the Christoffel equation, as the propagation of generally coupled bulk waves is considered. First, the derivation is conducted analytically on basis of the coordinate transformations to the propagation direction of general bulk waves and to the stiffness matrix of arbitrarily-anisotropic materials involved in the Christoffel equation, but does not require the concrete expanded expression of the Christoffel equation. Then, the influences of any anisotropic parameters on the anisotropy characteristics on arbitrary planes and on the dispersion characteristics along any directions of bulk waves are investigated in detail, which are reflected by the variations of the slowness/phase velocity curves and the comprehensive dispersion curves, respectively. Finally, the influential rules are illustrated and summarized with combination of numerical examples about two typical exemplified anisotropic materials. It is found that the 21 anisotropic parameters for triclinic materials, which can actually be utilized to describe the anisotropic degree for all materials, are divided into five groups according to their influential patterns on the anisotropy characteristics of bulk waves. Furthermore, the dispersion characteristics of each bulk wave mode along specific direction are affected by three anisotropic parameters.

Estimation of anisotropy parameters in vertical transverse isotropy media using the ray‐based tomography

AbstractIn seismic exploration, it is very important to consider the presence of anisotropy in order to image the subsurface correctly. The knowledge of anisotropic parameters leads to more precise characterization of reservoir, fracture density and flow paths. However, conventional geophysical methods do not directly measure these parameters, and it is useful to have a method to estimate them from seismic data. In the weak transverse isotropy approximation, the fields to be considered are the vertical and horizontal velocity components and the Thomsen parameters and . We present a method for estimating the anisotropic Thomsen parameters in the presence of weak vertical transverse isotropy using P‐wave traveltime tomography based on anisotropic ray tracing. Depending on the available information, we propose different approaches to retrieve the unknowns. A conventional three‐dimensional traveltime tomography algorithm has been extended to include anisotropic ray tracing and using the algebraic reconstruction technique or modified simultaneous iterative reconstruction technique to retrieve the unknowns. We test the method on synthetic examples for the inversion of transmitted and reflected traveltimes, and we evaluate the sensitivity of the tomographic results to the available information. Furthermore, we also consider the case of tilted transverse isotropy in a seismic reflection example.

A microstructure-based parameterization of the effective anisotropic elasticity tensor of snow, firn, and bubbly ice

Abstract. Quantifying the link between microstructure and effective elastic properties of snow, firn, and bubbly ice is essential for many applications in cryospheric sciences. The microstructure of snow and ice can be characterized by different types of fabrics (crystallographic and geometrical), which give rise to macroscopically anisotropic elastic behavior. While the impact of the crystallographic fabric has been extensively studied in deep firn, the present work investigates the influence of the geometrical fabric over the entire range of possible volume fractions. To this end, we have computed the effective elasticity tensor of snow, firn, and ice by finite-element simulations based on 391 X-ray tomography images comprising samples from the laboratory, the Alps, Greenland, and Antarctica. We employed a variant of Eshelby's tensor that has been previously utilized for the parameterization of thermal and dielectric properties of snow and utilized Hashin–Shtrikman bounds to capture the nonlinear interplay between density and geometrical anisotropy. From that we derive a closed-form parameterization for all components of the (transverse isotropic) elasticity tensor for all volume fractions using two fit parameters per tensor component. Finally, we used the Thomsen parameter to compare the geometrical anisotropy to the maximal theoretical crystallographic anisotropy in bubbly ice. While the geometrical anisotropy clearly dominates up to ice volume fractions of ϕ≈0.7, a thorough understanding of elasticity in bubbly ice may require a coupled elastic theory that includes geometrical and crystallographic anisotropy.

Wavefield simulation of acoustic VTI wave equation based on adaptive-coefficient finite-difference frequency-domain method

Abstract Many simulation methods have been developed for P-waves in vertically transversely isotropic (VTI) media. These methods are based on the acoustic approximation. The finite-difference frequency-domain (FDFD) method stands out for its ability to simulate multi-shot or narrowband seismic data. It has no temporal dispersion, facilitates attenuation modelling, and enables parallelization. The optimal FDFD method is commonly used to simulate the acoustic VTI wave equation, but it applies the same FDFD coefficients for different frequencies and model velocities, which cannot fully minimize the numerical dispersion error. To enhance its accuracy and effectiveness, we develop an adaptive-coefficient FDFD method specifically for the acoustic VTI wave equation. The FDFD coefficients depend on two factors: the number of wavelengths in each grid and the Thomsen parameters. The dispersion analysis reveals that the proposed FDFD method can achieve a reduction in the necessary number of grid points from 4 to 2.5 compared to the optimal 9-point average derivative method (ADM), while maintaining a maximum dispersion error of 1%. From three numerical examples, the developed FDFD method can obtain more accurate wavefield results than the ADM optimal FDFD method, while taking comparable computational time and memory.

Physical properties variations in a shaly formation across a fault core

Summary Faults in general, and in clay materials in particular, have complex structures that can be linked to both a polyphased tectonic history and to the anisotropic nature of the intact rock. Drilling through faults in shaly materials allows measuring properties such as the structure, mineralogical composition, stress orientation and physical properties. We combine different petrophysical measurements on core samples retrieved from a borehole drilled perpendicularly to a fault zone affecting Toarcian shales from the Tournemire underground research laboratory (France). The borehole is cross-cutting the entire fault thickness which is of the order of ten meters. We perform several types of measurements: density, porosity, saturation directly in the field, and P-wave velocities together with P-waves anisotropy on core samples taken at regular intervals. Special protocols were developed to preserve as much as possible the saturation state of the samples. From our measurements, we were able to track the increase of damage, characterized by a smooth decrease in elastic moduli from the intact zone to the fault core. We then calculated Thomsen's parameters to quantify the elastic anisotropy evolution across the fault. Our results show strong variations of the elastic anisotropy with the distance to the fault core as well as the occurrence of anisotropy reversal.

Estimating the anisotropy of the vertical transverse isotropy coal seam by rock physics model–based inversion

AbstractCoal seams exhibiting nearly horizontal bedding, and fractures can be characterized as transversely isotropic media with a vertical axis of symmetry, known as vertical transverse isotropy coal seams. The resulting anisotropy cannot be overlooked in high‐precision seismic velocity analysis, migration imaging and pre‐stack inversion. Therefore, we estimate the anisotropy of the vertical transverse isotropy coal seam (using the anisotropic Thomsen's parameters ε, γ and δ) by inverting the horizontal P‐ and SH‐wave velocities and fracture density based on a rock physics model. The Mori–Tanaka model and Brown–Korringa formula were first used to quantify the anisotropy of the vertical transverse isotropy coal seams impacted by dry and fluid‐saturated fractures. Subsequently, we formulated an equation for the inversion of horizontal P‐ and SH‐wave velocities, considering the measured vertical P‐ and SH‐wave velocities as constrained parameters. This approach is generally applied in vertical drilling scenarios. We tested the inversion method using the ultrasonic test results of coal samples collected from the southern margin of the Qinshui Basin, China and then used it on the full waveform logging data from a vertical coalbed methane well. Both the inversion results of horizontal P‐ and SH‐wave velocities and the estimated anisotropic parameters (ε and γ) were in good agreement with the ultrasonic test results of coal samples, although the accuracy of δ was slightly lower. Therefore, we believe that the proposed method can be extended to estimate the anisotropy of vertical transverse isotropy medium (assuming suitable rock physics models) for the ultrasonic testing of rock samples and full waveform logging along the vertical direction in near‐horizontal formations.

Linearized waveform inversion for vertical transversely isotropic elastic media: Methodology and multi-parameter crosstalk analysis

Seismic migration and inversion are closely related techniques to portray subsurface images and identify hydrocarbon reservoirs. Seismic migration aims at obtaining structural images of subsurface geologic discontinuities. More specifically, seismic migration estimates the reflectivity function (stacked average reflectivity or pre-stack angle-dependent reflectivity) from seismic reflection data. On the other hand, seismic inversion quantitatively estimates the intrinsic rock properties of subsurface formulations. Such seismic inversion methods are applicable to detect hydrocarbon reservoirs that may exhibit lateral variations in the inverted parameters. Although there exist many differences, pre-stack seismic migration is similar with the first iteration of the general linearized seismic inversion.Usually, seismic migration and inversion techniques assume an acoustic or isotropic elastic medium. Unconventional reservoirs such as shale and tight sand formation have notable anisotropic property. We present a linearized waveform inversion (LWI) scheme for weakly anisotropic elastic media with vertical transversely isotropic (VTI) symmetry. It is based on two-way anisotropic elastic wave equation and simultaneously inverts for the localized perturbations (ΔVp0/Vp0,ΔVs0/Vs0,Δϵ,Δδ) from the long-wavelength reference model. Our proposed VTI-elastic LWI is an iterative method that requires a forward and an adjoint operator acting on vectors in each iteration. We derive the forward Born approximation operator by perturbation theory and adjoint operator via adjoint-state method. The inversion has improved the quality of the images and reduces the multi-parameter crosstalk comparing with the adjoint-based images. We have observed that the multi-parameter crosstalk problem is more prominent in the inversion images for Thomsen anisotropy parameters. Especially, the Thomsen parameter δ is the most difficult to resolve. We also analyze the multi-parameter crosstalk using scattering radiation patterns.The linearized waveform inversion for VTI-elastic media presented in this article provides quantitative information of the rock properties that has the potential to help identify hydrocarbon reservoirs.

Dispersion computation of guided waves in a layered transversely isotropic elastic medium sandwiched between two half‐spaces

AbstractA transversely isotropic elastic medium with a vertical axis of symmetry is considered. We obtain dispersion equations in real terms for guided Love and Rayleigh waves in a such a medium consisting of horizontal layers sandwiched between two half‐spaces by brief modifications of the available literatures on dispersion equations in elastic layered media through transfer matrix. To illustrate the applicability, dispersion curves of guided waves are computed for three‐ and five‐layered symmetric models with transversely isotropic coal seam in the middle. The Airy phase is marked by a minimum or maximum of group velocity in a dispersion curve, and this phase is important to get seam structure for mining safety. For the first three modes, the effect of Thomsen anisotropy parameters γ, ε and δ of a coal seam on frequency (fA) and group velocity (UA) of the Airy phase is similar in three‐ and five‐layered models. For guided Love waves, fA and UA have nearly a uniform increase with the increase of γ. For guided Rayleigh waves, the increase of ε causes fA and UA to increase; however, the increase of δ causes fA and UA to decrease.

A Machine Learning Approach to Estimate Geo-mechanical Parameters from Core Samples: A Comparative Approach

Geo-mechanical parameters and Thomsen parameters play very important roles to design a stable wellbore in a challenging environment. The main objective of this paper is to estimate Thomsen's parameters (ε, γ, δ) and geo-mechanical properties from the core samples by Machine Learning and a comparative analysis with the conventional mathematical approach; to place emphasis on the use of Machine Learning and Artificial Intelligence in the Oil & Gas industry and to highlight its future potential to help in the digital transformation of the industry. Two different Machine Learning models, the Ordinary Least Square method and the Random Forest method, were used to predict the aforementioned geo-mechanical properties from the wave velocity and confining pressure data. In this study, it has been observed that the approaches employed in the estimate of geo-mechanical properties are rapid and reliable (about 93.5 percent accuracy) and may be applied in geo-mechanical modeling for wellbore stability analysis for safe and cost-effective well plan and design on a large scale. The analysis in this work indicates that Young’s modulus and Poisson’s ratio are heavily influenced by the anisotropy parameters. Finally, a comparison is made with mathematical approaches. The machine learning and artificial intelligence approaches shown here are excellently matched with mathematical approaches. The geo-mechanical parameters and Thomsen parameters and be computed with reasonable accuracy with the help of our proposed ML algorithms. Our proposed ML model can predict the geo-mechanical parameters and Thomsen parameters from the velocity profile directly without complex mathematical computation. The mathematical model would have required us to first determine the stiffness constants for the prediction of that parameters. Additionally, we may conclude that a machine learning model needs to be trained with more modeling data to predict the right values with a smaller error margin. The number of data points required to train a model has a significant impact on the model's overall accuracy. Therefore, additional modeling data is needed to learn about and comprehend the intricacies, patterns, and interactions between provided input and output variables.

Anisotropy and petrophysics of Floyd Shale, Alabama

Floyd Shale has a measured anisotropy that is consistent with transverse isotropy but with a magnitude of nearly 40% in Thomsen parameters γ and ε. The measured anisotropy is very strong, with a relation between γ and ε but not δ. This requires incorporating strong anisotropy while locating any microseismic events induced during fracture stimulation. It makes velocities deduced from sonic logs in deviated wellbores considerably different than vertical velocities. Floyd Shale looks very similar to Barnett Shale in many of its petrophysical properties except mineralogy. Production to date has been disappointing. We review key petrophysical and rock-physics properties of 103 ft of Floyd Shale recovered from the McShan Timberlands well. Based on these analyses, Floyd looks very promising (i.e., total organic carbon is in excess of 4 wt%, and porosities average 6%). However, mineralogy is dominated by clays (typically greater than 50 wt%), and Barnett has generally less than 29% clay. High-pressure mercury injection capillary pressure and nuclear magnetic resonance indicate that pore throats and pore bodies are small (i.e., less than 15 nm). Physical inspection of the recovered core reveals fresh slickensides (i.e., evidence of hydraulic fracturing) and compromised calcite-filled fractures. If this core is representative of Floyd Shale, we suspect that any hydrocarbons generated during maturation likely migrated during later tectonic fracturing.

Thomsen-type parameters and attenuation coefficients for constant-Q transverse isotropy

Transversely isotropic (TI) media with the frequency-independent quality-factor elements (also called “constant- Q” transverse isotropy) are often used to describe attenuation anisotropy in sedimentary rocks. The attenuation coefficients in constant- Q TI models can be conveniently defined in terms of the Thomsen-type attenuation-anisotropy parameters. Recent research indicates that not all those parameters for such constant- Q media are frequency-independent. Here, we present concise analytic formulas for the Thomsen-type attenuation parameters for Kjartansson’s constant- Q TI model and show that one of them ([Formula: see text]) varies with frequency. The analytic expression for [Formula: see text] helps evaluate the frequency dependence of the normalized attenuation coefficients of P- and SV-waves by introducing the newly defined “dispersion factors.” These factors are frequency-independent and expressed in terms of the Thomsen and Thomsen-type parameters defined at a specified reference frequency. Viscoacoustic constant- Q transverse isotropy is also discussed as a special case, for which the elliptical condition and simplified expressions for the parameters [Formula: see text] and [Formula: see text] are derived. Our results show that, in the presence of significant absorption, the attenuation coefficients of the constant- Q model vary with frequency for oblique propagation with respect to the symmetry axis. This variation needs to be taken into account when applying the spectral-ratio method and other attenuation-analysis techniques.

2-D PP and PS imaging for VTI with weak equation-converted S-waves

For elastic reverse time migration (ERTM) in inhomogeneous isotropy (ISO), the key is to deal with conversions caused by media inhomogeneity. However, for ERTM in vertical transverse isotropy (VTI), researchers also need to handle equation-converted waves (such as S-wave generated with pure explosive source exciting) caused by undecoupling of elastic wave equation generated coincident with incident wave (such as P-wave generated with pure explosive source exciting) in homogeneous VTI, leading to crosstalk artifacts caused by unseparation of waves of the same mode. To analyze equation-converted waves, we define five pseudospatial differential operators and rewrite elastic wave equation of VTI into a form represented by Thomsen parameters. The rewrited equation shows the relationship between equation-converted wave and Thomsen parameters including three anisotropic parameters and two velocity parameters (α and β). Further studies indicate that when explosive source excites in homogeneous weak-anisotropy VTI with α/β = 1.73, the ratio between the energy of equation-converted S-waves and incident P-waves is generally <0.5 and we define equation-converted S-wave whose energy is less than the half of incident P-wave as weak equation-converted S-wave. On this basis, we propose an ERTM method for VTI with weak equation-converted S-waves and its effectiveness is demonstrated by the imaging results for two-layer models and Marmousi2 model.

Bayesian seismic travel-time cross-hole tomography in vertically transversely isotropic media

We apply a transdimensional, hierarchical Markov chain Monte Carlo sampling algorithm (McMC) for 2-D cross-hole travel-time tomography in transversely isotropic media with vertical symmetry axis. The McMC approach has several advantages compared to classical inversion approaches: It is a global search, the high number of tested models allows the statistical analysis including the calculation of a reference model as well as uncertainty estimation, no initial models or regularization parameters are needed, the amount of data noise is automatically determined, and the model parametrization is data dependent and self-adjusting. For the forward solution a FD Fast Marching method utilizing second-order Godunov schemes is used. The performance of the approach is first tested on synthetic datasets to evaluate the potential and possible limitation to recover anisotropic models. We have shown that the recovery of models described by 2 anisotropic parameters (Thomsen parameters) and the vertical velocity is possible for observation scenarios with good distribution of sources and receivers. For more realistic observational geometries (i.e. cross-hole experiments), the recovery of the 3 parameters is limited, but still possible for example for the elliptical anisotropic case (ε = δ) or regarding the horizontal velocity. Finally we applied the McMC approach to a well-studied real cross-hole data set from the MALLIK 2002 research program and compared the results with previous conventional inversions.

Research on Rock Physics Modeling Methods for Fractured Shale Reservoirs

The Sichuan Basin is a significant region for exploration and development of shale gas in China, and it is essential to clarify the impact of deep shale gas reservoir parameters on cost-effective development at scale to ensure national energy security. Rock physics modeling is a significant means of communicating the physical and elastic parameters of rocks. A rock physics modeling method applicable to fractured shale gas reservoirs is proposed for the current situation of complex fluid relationships in shale gas reservoirs and unclear characteristics of gas identification seismic response. In this paper, based on the Self Consistent Approximation (SCA) model and the differential effective medium (DEM) model, the anisotropic source of shale is used as a starting point to add bound water, kerogen, clay, and brittle minerals, the Schoenberg linear slip theory is used to add fracture disturbance effects, and then the Brown–Korringa model is used to perform fluid replacement under anisotropic conditions. Finally a rock physics model applicable to fractured shale gas reservoirs is obtained, and the established rock physics model is used for analysis of elastic parameters, Thomsen parameters, and fracture weakness parameters. Rock physics tests were performed on shale in southern Sichuan as an example. The experimental results show that the model established by the process can accurately invert the longitudinal and transverse wave velocities of the shale, which can provide a conceptual basis for the study of fractured shale gas reservoirs.

Petroelastic simulation as a method for analysing the anisotropy of elastic properties on the scale of seismic surveys

Introduction. Reservoirs composed of thin layers with different elastic properties exhibit the anisotropy of elastic properties. Anisotropic environments are studied using special seismic positioning systems that require large financial and time resources. However, neglecting pronounced anisotropy leads to misinterpretation of the data. Evaluation of the degree of anisotropy of a thin-layer stratum allows researchers to determine whether anisotropy could be neglected by conditionally considering the stratum to be isotropic.Aim. To estimate the anisotropy of carbonate rocks on the scale of seismic surveys based on the data of geophysical well logging and petroelastic simulation.Materials and methods. Models of carbonate rocks with different characteristics of pore space were created based on the Berryman and Backus methods. For these models, an analysis of the degree of anisotropy by Thomsen parameters was carried out. Acoustic and density logging data of a well located in Western Siberia were used for comparison with actual data.Results. A numerical characteristic was found, the analysis of which helped to determine the significance of anisotropy before the application of the Backus method thus significantly reducing the number of necessary computational processes.Conclusion. The obtained parameter of the degree of rock heterogeneity correlates well with Thomsen parameters responsible for anisotropy, which indicates the possibility of its use for assessing the anisotropy of strata.

Weak-anisotropy approximation of P-wave reflection coefficient in anisotropic media of arbitrary symmetry and tilt

Applicability of the approximate expression for the P-wave reflection coefficient at a weak-contrast reflector separating two weakly anisotropic half-spaces of arbitrary symmetry is extended to media of arbitrary symmetry and tilt. The reflection coefficient consists of the approximate P-wave reflection coefficient at a weak-contrast interface separating two reference isotropic half-spaces and a correction term due to anisotropy. Along an arbitrarily chosen profile, the “isotropic” term depends on the density and P- and S-wave contrasts, and the correction term depends linearly on the contrasts of four profile weak-anisotropy (WA) parameters specifying anisotropy along the profile. In addition, both terms depend on the polar angle of incidence. In each half-space, the four profile WA parameters can be expressed as a linear combination of 12 of 21 global WA parameters specifying anisotropy of the half-space. Coefficients of this linear combination are functions of the azimuth of the profile. WA parameters are a generalization of Thomsen parameters to arbitrary anisotropy and represent an alternative to 21 independent elements of the stiffness tensor. WA parameters can be used for the approximation of other related concepts such as the reflection moveout or the geometric spreading. Presented tests illustrate high accuracy and flexibility of the proposed formula for the P-wave reflection coefficient. They also show that very accurate results can be obtained for contrasts and anisotropy, which cannot be considered weak.

Acoustic Forward Model for Guided Wave Propagation and Scattering in a Pipe Bend.

The sections of pipe bends are hot spots for wall thinning due to accelerated corrosion by fluid flow. Conventionally, the thickness of a bend wall is evaluated by local point-by-point ultrasonic measurement, which is slow and costly. Guided wave tomography is an attractive method that enables the monitoring of a whole bend area by processing the waves excited and received by transducer arrays. The main challenge associated with the tomography of the bend is the development of an appropriate forward model, which should simply and efficiently handle the wave propagation in a complex bend model. In this study, we developed a two-dimensional (2D) acoustic forward model to replace the complex three-dimensional (3D) bend domain with a rectangular domain that is made artificially anisotropic by using Thomsen parameters. Thomsen parameters allow the consideration of the directional dependence of the velocity of the wave in the model. Good agreement was found between predictions and experiments performed on a 220 mm diameter (d) pipe with 1.5d bend radius, including the wave-field focusing effect and the steering effect of scattered wave-fields from defects.

Fracture Detection With Azimuthal Seismic Amplitude Difference Inversion in Weakly Monoclinic Medium

The rock with a vertical group of aligned fractures in the tilted transverse isotropy (TTI) background medium is regarded as an effective monoclinic medium. Thomsen parameters and fracture weaknesses can reflect the development degree of TTI background anisotropy and vertical fractures, respectively. Thus, this paper first establishes the approximate stiffness matrix of the monocline medium through the Bond transformation from Thomsen weak anisotropy theory and effective medium theory. Moreover, considering the Born scattering theory, a linearized approximate formula of PP wave reflection coefficient is further deduced. This approximate formula contains the parameters of incident angle, azimuth, tilted angle, compressional and shear moduli, density, fracture weaknesses, and Thomsen parameters. Then, the contribution of each parameter on the reflection coefficient by numerical simulation is analyzed respectively. Finally, from the perspective of Bayesian inference, an azimuthal seismic amplitude difference inversion approach incorporating Cauchy prior information and low-frequency regularization of model parameters is proposed to realize the simultaneous estimation of fracture weaknesses and Thomsen parameters. The synthetic test demonstrates that the inversion approach proposed in this article can obtain information on vertical fractures and TTI background medium and can be applied to azimuthal seismic data with moderate noise well. Furthermore, the inversion results of the synthetic test and field data illustrate the stability and feasibility of the inversion approach and also confirm the fracture weaknesses and Thomsen parameters of the monoclinic medium can be reasonably estimated by the inversion equation derived in this paper.

The dynamic stiffness matrix method for seismograms synthesis for layered transversely isotropic half-space

Natural rocks and soils show significant anisotropic behavior owing to sedimentation and weathering, particularly apparent in the crust. However, seismograms synthesis for the anisotropic medium is scarce, especially bottlenecks for large-scale and broadband seismic wave propagation. In this paper, seismogram synthesis for a multi-layered transversely isotropic half-space due to seismic point sources is addressed by using a dynamic stiffness matrix method. The critical step of the proposed method lies in the construction of the global dynamic stiffness matrix in frequency-wavenumber domain, which avoids the accumulation of errors that appeared in the reflection and transmission matrix method. Exponential divergent terms related to wavenumber and layer thickness in dynamic stiffness matrix are extracted for fast simulating large-scale and high-frequency wavefield. The seismic point sources are equated to the external loads imposed at the introduced auxiliary surfaces, through which the dynamic stiffness matrix can be utilized to obtain the solutions. The accuracy is verified by comparing the proposed method results with three examples in the publications, involving different source types and depth scales. Two numerical examples regarding Thomsen parameters are given to illustrate the mechanism of wave propagation and ground motion in an anisotropic media. Numerical results show that the anisotropic characteristics affect the wave propagation time, waveform and wave-front energy distribution, further leading to shear wave splitting phenomena, which remarkably differ from the isotropic case. A case study using a simplified three-layered crustal structure model in Tibet elucidates that peak ground displacement and its distribution difference within the epicenter of 10 km by comparing with the equivalent isotropic model without considering anisotropic properties. The results show a 10.91% deviation in the location of the peaks, with amplitude differences of up to 120% due to a 45° dip-slip source at different frequencies (from 1 Hz to 15 Hz).