A note on the effects of material discontinuities on spatial variations of surface displacements from static dislocations

Post-earthquake observations show that permanent static and transient dynamic displacements in the near field of shallow faults can be complex and quite different from theoretical predictions for homogeneous half-space. These complexities are associated with many departures from the linear, elastic, homogeneous, and isotropic representations, and include the consequences of complex three-dimensional (3D) geology surrounding faults. In this paper we illustrate these complexities for a two-dimensional (2D) model of shallow strike-slip fault when it is inside, on the border, and outside a sedimentary basin.     

Dynamic response of an alluvial valley consists of three types of soil

Dynamic response of an alluvial valley consisting of three different types of soil was studied. In the two-dimensional model, the alluvial valley was assumed to have half-cylindrical shape. The alluvial valley contained three different types of soil with different shapes. The valley was surrounded by a half-space. All of the soil types and the half-space were assumed to be isotropic, homogeneous and linear elastic. The half-space was excited by simple harmonic SH-waves radiated from a strike-slip fault. The fault was assumed to have circular-arc shape in the mathematical model. The governing equations were solved by using wave function expansion method where boundary conditions were relatively simple. In the regions where boundary conditions were more complex, finite difference method was employed. Consequently, the wave propagation problem was solved semi-analytically. The obtained numerical results showed that surface displacement amplitudes are significantly affected by the material properties of the different soil types of the alluvial valley. It was also observed that the shapes of the soil types in the alluvial valley played a considerable role in surface displacements.     

Boundary integral equations for dynamic rupture propagation on vertical complex fault system in half-space:Theory

The boundary integral equation method(BIEM)is now widely used in numerical studies on earthquake rupture dynamics,and is proved to be a powerful tool to deal with problems on complex fault system.However,since this method heavily lies on the specific forms of Green's function and only the Green's function in full-space has a closed analytic expression,it is usually limited to a full-space medium.In this study,as a first step to extend this method to an arbitrary complex fault system in half-space,the boundary integral equations(BIEs)for dynamic strike-slip on vertical complex fault system in half-space are derived based on exact Green's function for isotropic and homogeneous half-space.Effect of the geometry of the complex fault system are dealt with carefully.Final BIEs is composed of two parts:contribution from full-space,which has been thoroughly investigated by Aochi and his co-workers by using the Green's function for full-space,and that from free surface,which is studied in detail in this study.     

Coseismic and postseismic subsurface displacements and strains for a vertical strike-slip fault in a three-layer elastic medium

A three-layer elastic-gravitational fault displacement model using dislocation theory has been developed and used to examine the effect of layering of earth elastic moduli on surface and subsurface displacement fields for a vertical strike-slip fault. The model has been used to examine the effect of depth variation of elastic properties at coseismic and postseismic time scales. For pure strike-slip motion the effect of gravity on coseismic and postseismic horizontal deformation is negligible. For coseismic deformation the model predicts that (for constant Poisson's ratio) an increase in elastic moduli with depth attenuates the displacements within the upper layers with respect to displacement distribution for a uniform half-space, while an inclusion of a soft layer between the top layer and lower half-space amplifies upper layer displacements. The effect of variation in Poisson's ratio on surface and subsurface displacements has also been examined.The effect of postseismic stress relaxation on surface and subsurface displacements for a three-layer model has been calculated and compared with that of a uniformly relaxed half-space model. Layer 1 is assumed to correspond to the upper crust, layer 2 the lower crust and layer 3 the upper mantle. The effect of postseismic stress relaxation within a uniform half-space and within just the lower crust and upper mantle has been examined. Stress relaxation within the whole half-space decreases the amplitude and shortens the wavelength of displacements, while stress relaxation within the lower two layers increases the amplitude and broadens the wavelength of displacements. The difference between uniform and layered postseismic relaxation is particularly pronounced at the base of the crust.Coseismic and postseismic normal and volumetric strains for a vertical strike-slip fault have also been examined. For a uniformly relaxed half-space model, an increase in normal strains is shown with respect to the coseismic elastic solution, whereas the postseismic volumetric strain is effectively zero. For a three-layer model with stress relaxation in the lower layers only, the normal and volumetric strains within the top elastic layer resemble coseismic strains, while in the lower layers which suffer a rigidity decrease, the postseismic volumetric strain is effectively zero.     

Coseismic deformation and slip distribution of the 1997 7.5 Manyi, Tibet, earthquake from InSAR measurements

We use interferometric synthetic aperture radar (InSAR) observations to investigate the coseismic deformation and slip distribution of the 1997 M_w7.5 Manyi earthquake, a left-lateral strike-slip earthquake occurred on the west portion of the Kunlun fault in the northern Tibet, China. The fault trace is constrained by the combination of interferometric coherence image and azimuth offset image. The total length of the identified fault is about 170 km. We estimate the source parameters using a seven-segment fault model in a homogeneous elastic half-space. We first use a uniform slip model to estimate the slip, width, dip and rake for each segment, resulting in a maximum slip of 5.5 m with a depth of 11 km on the fourth segment. The average dip of the uniform slip model is about 93° northward and the average rake is about −2°. We then use a distributed slip model to estimate the pure strike-slip and oblique slip distribution, respectively. In the distributed slip model, the fault plane is discretized into 225 patches, each of them 4 km × 4 km. We fix the optimal geometric parameters and solve for the slip distribution using a bounded variable least-squares (BVLS) method. We find a geodetic moment of 1.91 × 10²⁰ Nm (M_w7.5), of which almost 68% released in the uppermost 8 km and 82% in the uppermost 12 km. For all the models used in this study, the synthetic profiles along strike show asymmetric displacements on the opposite sides of the fault, which are in agreement with the observations. This suggests that a linear elastic model with variable and non-vertical dips is also reasonable for the mechanism of the Manyi earthquake.     

Surface wave propagation in a transversely isotropic elastic layer overlying a liquid saturated porous solid half-space and lying under the uniform layer of liquid

Dispersion of Rayleigh-type surface wave is studied in a homogeneous transversely isotropic elastic layer overlying a nondissipative liquid-saturated porous solid half-space and lying under a uniform layer of homogeneous liquid. The frequency equation in the form of ninth-order determinant is obtained.Special cases have been deduced by reducing the depth of the layers to zero and by changing the transverse isotropic layer to an isotropic layer. Dispersion curves for the phase velocity have been plotted for a particular model.     

REFLECTION OF ELASTIC WAVES FROM PERIODICALLY STRATIFIED MEDIA WITH INTERFACIAL SLIP*

A periodically stratified elastic medium can be replaced by an equivalent homogeneous transverse isotropic medium in the long wavelength limit. The case of a homogeneous medium with equally spaced parallel interfaces along which there is imperfect bonding is a special instance of such a medium. Slowness surfaces are derived for all plane wave modes through the equivalent medium and reflection coefficients for a half-space of such a medium are found. The slowness surface for the SH mode is an ellipsoid. The exact solution for the reflection of SH-waves from a half-space with parallel slip interfaces is found following the matrix method of K. Gilbert applied to elastic waves. Explicit results are derived and in the long wavelength limit, shown to approach the results for waves in the equivalent homogeneous medium. Under certain conditions, a half-space of a medium with parallel slip interfaces has a reflection coefficient independent of the angle of incidence and thus acts like an acoustic reducing mirror. The method for the reflection of P- and SV-waves is fully outlined, and reflection coefficients are shown for a particular example. The solution requires finding the eigenvalues of a 4 × 4 transfer matrix, each eigenvalue being associated with a particular wave. At higher frequencies, unexpected eigenvalues are found corresponding to refracted waves for which shear and compressional parameters are completely coupled. The two eigenvalues corresponding to the transmitted wavefield give amplitude decay perpendicular to the stratification along with up- and downgoing phase propagation in some other direction. Much of this work was performed while the author was at the Department of Geophysics and Planetary Sciences, Tel-Aviv University, Ramat-Aviv, Israel. The author is grateful for illuminating discussions with K. Helbig and K. Gilbert.     

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.     

Ground motion amplification due to elastic inclusions in a half-space

Scattering of plane harmonic SH, P, SV and Rayleigh waves by several inclusions of arbitrary shape, completely embedded into an elastic half-space, is considered. Perfect bonding between the half-space and the inclusions is assumed. The problem is investigated for linear, isotropic and homogeneous elastic materials. The displacement field is evaluated throughout the elastic medium so that the continuity conditions between the half-space and the inclusions are satisfied in mean-square sense. Numerical results of the surface displacement field are evaluated for single and two elliptic inclusions. The results show the following: (a) presence of a subsurface inhomogeneity may lead to large amplifications of the surface ground motion; (2) different surface displacement patterns emerge for different incident waves; (3) the presence of an additional inclusion may change significantly the surface displacement response of a single inclusion; (4) the surface motion extremes strongly depend upon (i) angle of incidence; (ii) frequency of incident field; (iii) embedment depth of the inclusions; (iv) separation distance between the inclusions; (v) material properties of the half-space and the inclusions; and (vi) location of observation point on the surface of the half-space.     

Two dimensional transient fundamental solution due to suddenly applied load in a half-space

A transient Green function due to suddenly applied line loads in an isotropic and homogeneous half-space is reported in this paper. The derivation of the half-space Green function in the Laplace and the Fourier transform spaces is first reviewed. Following an explicit inversion of the Fourier transform, the inverse Laplace transform is implemented along the contour integral on the p-complex plane in an integral form. The half-space Green function consists of full-space Green functions and a singularity-free complementary term. It can be easily incorporated into current transient boundary elements using the transient full-space Green function. Combined with finite elements, the half-space Green function can be used in a hybrid procedure to solve transient half-space problems without discretization of the free surface. Numerical results are presented to illustrate transient wave propagation in a half-space.     

Quasi-static deformation of a viscoelastic half-space by a displacement dislocation

The correspondence principle is used to get the Laplace transformed solution of the problem of the quasi-static deformation of a viscoelastic half-space by a shear displacement dislocation from the corresponding static elastic results. The transformed solution is inverted for the Voigt and the Maxwell viscoelastic models. It is shown that, for a vertical dip-slip fault, the surface displacements for the viscoelastic case are identical with the elastic displacements. In the case of a vertical strike-slip fault, detailed numerical results are obtained for both a point source and a finite rectangular source. It is found that the results for the viscoelastic models differ significantly from the corresponding elastic results.     

Two-dimensional scattering of SH waves in a soil layer underlain with a sloping bedrock

A two-dimensional analysis is applied to examine the effect that a sloping bedrock half-space has on the amplification of an anti-plane shear wave. The direct boundary integral equation method is used for the two-dimensional analysis. The particular soil–rock configuration investigated includes a homogeneous soil layer underlain by a sloping rock half-space. The rock half-space dips for a horizontal distance L and then becomes horizontal so that the overlying soil layer has a thickness H that remains constant from this point to infinity. The materials in the soil–rock configuration are considered viscoelastic except in the rock half-space below soil layer thickness H, which is considered elastic. This limitation in damping is due to the correction used for the truncation of the half-space boundary. Four cases are used to study the relationship between rock slope and surface displacement, vertical, 1:2, 1:4, 1:8. Surface displacements are determined for each of these cases for half-space incidence angles of 90, 75, and 60°. To allow for applicability to a wide range of problems, results are determined as a function of dimensionless parameters. In addition, solutions from a one-dimensional analysis are compared with the results of the two-dimensional analysis to establish limits outside of which a one-dimensional analysis suffices.     

Transient disturbance produced in an elastic half-space by impulsive shearing traction distributed over a circular region

The displacement produced in a semi-infinite, homogeneous, isotropic elastic medium by the application of shearing traction over a circular portion of the half-space has been evaluated in exact form by Cagniard's Technique (Cagniard, 1962;Gakenheimer andMiklowitz, 1969).     

Optimum strong-motion array geometry for source inversion—II

Optimum strong-motion array geometry for source inversions is again determined for each of three types of earthquake faults: strike-slip, dip-slip and offshore subduction thrust. The method is the same as employed in a previous study;¹ however, use of a complete Green's function in an elastic half-space provides better results for engineering practice. It is found that the complete Green's function is capable of stabilizing the accuracy of an inversion solution obtained using theoretical seismograms, regardless of the differences in array configuration. The optimum strong-motion array for a strike-slip fault is characterized by stations well distributed in azimuth, while the optimum array for a dip-slip event has stations arranged in a grid-shaped form. The array geometries obtained here are grossly similar to those in the previous study,¹ which were derived using only the far-field S waves, and are more consistent with those proposed at the 1978 International Workshop on Strong-Motion Earthquake Instrument Arrays.².     

利用InSAR资料反演缅甸Mw6.8地震断层滑动分布

2011年3月24日缅甸东北部发生M_w6.8级地震.本文利用覆盖该地区的升降轨ALOS PALSAR数据,获取了该次地震的同震形变场,并采用灰度配准技术获取了其地表破裂位移.针对影像中因轨道不精确造成的非线性长波长误差,本文采用二次多项式曲面法予以去除,获取了更为精确的同震形变场.最后,基于弹性半空间形变模型反演了该地震的断层滑动分布.结果表明,该地震断层滑动以左旋走滑为主,兼具少许的倾滑运动分量,断层滑动主要集中分布在断层面0~10 km深度范围,最大滑动量达5 m,位于地表以下5 km深处.反演获得的地震标量矩为2.49×10¹⁹N·m,震级约为M_w6.8级.     

A fundamental solution due to a periodic point force in the interior of an elastic half-space

The fundamental solution for a periodic point force in the interior of a three-dimensional, homogeneous, isotropic, elastic half-space is derived. The method of synthesis and superposition is employed to obtain the solution in the Laplace transform as well as the frequency domain. These correspond to the dynamic equivalent of Mindlin's static half-space point force solutions. It is reduced, for certain limiting conditions, to the dynamic equivalent of Boussinesq's and Cerruti's problems of a normal and tangential periodic point force respectively, on the boundary of a half-space. Also, static solutions of Mindlin, Boussinesq and Cerruti are recovered for small frequency parameters. Finally, results are presented and compared with other available solutions.     

Out-of-plane (SH) soil-structure interaction: Semi-circular rigid foundation revisited

The model studied in this paper presents an extension of previous work for a shear wall on a semi-circular rigid foundation in an isotropic homogeneous and elastic half-space. The objective is to develop a soil-structure interaction model that can later be applied to the case of a flexible foundation. As shown in the Introduction below, Luco considered the case of a rigid foundation subjected to vertical incident plane SH waves, and Trifunac extended the solution for the same rigid foundation subjected to SH waves but for arbitrary angles of the incidence. In this paper, a new approach and model are presented for the same semi-circular rigid foundation with a tapered-shape (instead of rectangular) superstructure. The analytical expression for the deformation of the semi-circular rigid foundation below this tapered shear wall with soil-structure interaction in an isotropic homogeneous and elastic half-space is thus derived. Results are then compared with those of Trifunac discussed in the section below. This problem formulation can and will later be extended in the case of a flexible foundation that is semi-circular or arbitrarily shaped.     

Dynamic interaction analysis of a circular cylindrical shell of finite length in a half-space

A study on the transient response of a circular cylindrical shell of finite length embedded in a homogeneous, isotropic and linear elastic half-space is presented. The soil-structure system is subjected to suddenly applied explosion waves. The numerical method employed is a combination of the time domain semi-analytical boundary element method used for the semi-infinite soil medium and the finite strip method used for the circular cylindrical shell. The two methods are combined through equilibrium and compatibility conditions at the soil-structure interface. The dynamic responses at the interface between the soil medium and the structure for every time step are obtained. Numerical examples are presented in detail to demonstrate the use and versatility of the proposed method. The following parameters are found to affect the response: (1) the slenderness ratio of the length over the diameter of the shell, L/D; (2) the relative wall thickness, h/a; (3) the relative stiffness ratio between the shell and the medium, E_s/E_m; and (4) the incidence angle of the explosion wave, α.     

基于Sentinel-1A数据反演漾濞MS6.4地震的同震形变场及断层几何参数

2021年5月21日晚21时48分,云南省大理州漾濞县(震中:25.67°N,99.87°E)发生M_S6.4地震,震源深度8 km。为快速获得此次地震同震形变场及断层几何参数,研究该次地震的发震构造等,文章基于震前、震后的sentinel-1A卫星升降轨SAR数据进行二轨法差分雷达干涉测量(DInSAR),并基于Okada弹性半空间位错模型反演断层几何参数。研究结果如下:(1)此次地震造成的同震形变场长约19 km,宽约20 km;(2)升轨雷达视线向最大形变约为8.2 cm,降轨雷达视线向最大形变约为8.7 cm;(3)地震断层走向为313.7°,倾角为87°,滑动角为175°,为右旋走滑型断层,最大滑动量为0.79 m,反演得出的地震矩为1.48×10~(18) N·m,矩震级为M_W6.1。在川滇块体向南挤出的构造背景下,块体西边界的维西—乔后断裂、红河断裂发生右旋走滑,本次地震便是维西—乔后断裂南段分支断裂右旋走滑活动的体现。     

THE INSENSITIVITY OF REFLECTED SH WAVES TO ANISOTROPY IN AN UNDERLYING LAYERED MEDIUM1

Propagation in the plane of mirror symmetry of a monoclinic medium, with displacement normal to the plane, is the most general circumstance in anisotropic media for which pure shear-wave propagation can occur at all angles. Because the pure shear mode is uncoupled from the other two modes, its slowness surface in the plane is an ellipse. When the mirror symmetry plane is vertical the pure shear waves in this plane are SH waves and the elliptical SH sheet of the slowness surface is, in general, tilted with respect to the vertical axis. Consider a half-space of such a monoclinic medium, called medium M, overlain by a half-space of isotropic medium I with plane SH waves incident on medium M propagating in the vertical symmetry plane of M. Contrary to the appearance of a lack of symmetry about the vertical axis due to the tilt of the SH-wave slowness ellipse, the reflection and transmission coefficients are symmetrical functions of the angle of incidence, and further, there exists an isotropic medium E with uniquely determined density and shear speed which gives exactly the same reflection and transmission coefficients underlying medium J as does monoclinic medium M. This means that the underlying monoclinic medium M can be replaced by isotropic medium E without changing the reflection and transmission coefficients for all values of the angle of incidence. Thus no set of SH seismic experiments performed in the isotropic medium in the symmetry plane of the underlying half-space can reveal anything about the monoclinic anisotropy of that underlying half-space. Moreover, even when the underlying monoclinic half-space is stratified, there exists a stratified isotropic half-space that gives the identical reflection coefficient as the stratified monoclinic half-space for all angles of incidence and all frequencies.