Václav Vavryčuk

3D Heterogeneous Staggered-grid Finite-difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities

We analyze the problem of a heterogeneous formulation of the equation of motion and propose a new 3D fourth-order staggered-grid finite-difference (FD) scheme for modeling seismic motion and seismic-wave propagation. We first consider a 1D problem for a welded planar interface of two half-spaces. A simple physical model of the contact of two media and mathematical considerations are shown to give an averaged medium representing the contact of two media. An exact heterogeneous formulation of the equation of motion is a basis for constructing the corresponding heterogeneous FD scheme. In a much more complicated 3D problem we analyze a planar-interface contact of two isotropic media (both with interface parallel to a coordinate plane and interface in general position in the Cartesian coordinate system) and a nonplanar-interface contact of two isotropic media. Because in the latter case 21 elastic coefficients at each point are necessary to describe the averaged medium, we consider simplified boundary conditions for which the averaged medium can be described by only two elastic coefficients. Based on the simplified approach we construct the explicit heterogeneous 3D fourth-order displacement-stress FD scheme on a staggered grid with the volume harmonic averaging of the shear modulus in grid positions of the stress-tensor components, volume harmonic averaging of the bulk modulus in grid positions of the normal stress-tensor components, and volume arithmetic averaging of density in grid positions of the displacement components. Our displacement-stress FD scheme can be easily modified into the velocity-stress or displacement-velocity-stress FD schemes. The scheme allows for an arbitrary position of the material discontinuity in the spatial grid. Numerical tests for 12 configurations in four types of models show that our scheme is more accurate than the staggered-grid schemes used so far. Numerical examples also show that differences in thickness of a soft surface or interior layer smaller than one grid spacing can cause considerable changes in seismic motion. The results thus underline the importance of having a FD scheme with sufficient sensitivity to heterogeneity of the medium. Manuscript received 21 May 2001.

Inversion for parameters of tensile earthquakes

The tensile source model generalizes the shear source model by assuming that the slip vector can be arbitrarily oriented with respect to the fault and is not constrained to lie within the fault plane. The proposed inversion for the parameters of tensile sources is based on the evaluation of the isotropic (ISO), compensated linear vector dipole (CLVD), and double-couple (DC) components in seismic moment tensors. The most significant parameters inverted are the λ/μ ratio at the fault (denoted as the κ parameter) and the inclination α of the slip vector from the fault. The κ parameter is significant for discriminating noisy moment tensors of shear earthquakes from those of tensile earthquakes. The inclination α can be accurately determined from the DC component in the moment tensor because the DC component rapidly decreases with increasing α. For example, the inclination of 20° causes DC being ∼50–60% only. The inversion is applied to earthquakes which occurred in January 1997 in West Bohemia, Czech Republic. It is shown that some of these earthquakes display tensile faulting. The κ parameter is ∼0.1. The inclination of the slip from the fault attains values of up to 20°. This inclination is a result of tensile traction and reduced shear traction along the fault and high-fluid pressure in the region.

PP-wave reflection coefficients in weakly anisotropic elastic media

Approximate PP-wave reflection coefficients for weak contrast interfaces separating elastic, weakly transversely isotropic media have been derived recently by several authors. Application of these coefficients is limited because the axis of symmetry of transversely isotropic media must be either perpendicular or parallel to the reflector. In this paper, we remove this limitation by deriving a formula for the PP-wave reflection coefficient for weak contrast interfaces separating two weakly but arbitrarily anisotropic media. The formula is obtained by applying the first-order perturbation theory. The approximate coefficient consists of a sum of the PP-wave reflection coefficient for a weak contrast interface separating two background isotropic half-spaces and a perturbation attributable to the deviation of anisotropic half-spaces from their isotropic backgrounds. The coefficient depends linearly on differences of weak anisotropy parameters across the interface. This simplifies studies of sensitivity of such coefficients to the parameters of the surrounding structure, which represent a basic part of the amplitude-versus-offset (AVO) or amplitude-versus-azimuth (AVA) analysis. The reflection coefficient is reciprocal. In the same way, the formula for the PP-wave transmission coefficient can be derived. The generalization of the procedure presented for the derivation of coefficients of converted waves is also possible although slightly more complicated. Dependence of the reflection coefficient on the angle of incidence is expressed in terms of three factors, as in isotropic media. The first factor alone describes normal incidence reflection. The second yields the low-order angular variations. All three factors describe the coefficient in the whole region, in which the approximate formula is valid. In symmetry planes of weakly anisotropic media of higher symmetry, the approximate formula reduces to the formulas presented by other authors. The accuracy of the approximate formula for the PP reflection coefficient is illustrated on the model with an interface separating an isotropic half-space from a half-space filled by a transversely isotropic material with a horizontal axis of symmetry. The results show a very good fit with results of the exact formula, even in cases of strong anisotropy and strong velocity contrast.

Moment tensor decompositions revisited

The decomposition of moment tensors into isotropic (ISO), double-couple (DC) and compensated linear vector dipole (CLVD) components is a tool for classifying and physically interpreting seismic sources. Since an increasing quantity and quality of seismic data allow inverting for accurate moment tensors and interpreting details of the source process, an efficient and physically reasonable decomposition of moment and source tensors is necessary. In this paper, the most common moment tensor decompositions are revisited, new equivalent formulas of the decompositions are derived, suitable norms of the moment tensors are discussed and the properties of commonly used source-type plots are analysed. The Hudson skewed diamond plot is introduced in a much simpler way than originally proposed. It is shown that not only the Hudson plot but also the diamond CLVD–ISO plot and the Riedesel–Jordan plot conserve the uniform distribution probability of moment eigenvalues if the appropriate norm of moment tensors is applied. When analysing moment tensor uncertainties, no source-type plot is clearly preferable. Since the errors in the eigenvectors and eigenvalues of the moment tensors cannot be easily separated, the moment tensor uncertainties project into the source-type plots in a complicated way. As a consequence, the moment tensors with the same uncertainties project into clusters of a different size. In case of an anisotropic focal area, the complexity of moment tensors of earthquakes prevents their direct interpretation, and the decomposition of moment tensors must be substituted by that of the source tensors.

Ray velocity and ray attenuation in homogeneous anisotropic viscoelastic media

Asymptotic wave quantities such as ray velocity and ray attenuation are calculated in anisotropic viscoelastic media by using a stationary slowness vector. This vector generally is complex valued and inhomogeneous, and it predicts the complex energy velocity parallel to a ray. To compute the stationary slowness vector, one must find two independent, real-valued unit vectors that specify the directions of its real and imaginary parts. The slowness-vector inhomogeneity affects asymptotic wave quantities and complicates their computation. The critical quantities are attenuation and quality factor ( Q -factor); these can vary significantly with the slowness-vector inhomogeneity. If the inhomogeneity is neglected, the attenuation and the Q -factor can be distorted distinctly by errors commensurate to the strength of the velocity anisotropy — as much as tens of percent for sedimentary rocks. The distortion applies to strongly as well as to weakly attenuative media. On the contrary, the ray velocity, which defin...

Can unbiased source be retrieved from anisotropic waveforms by using an isotropic model of the medium

Abstract Anisotropy is frequently present in geological structures, but usually neglected when source parameters are determined through waveform inversion. Due to the coupling of propagation and source effects in the seismic waveforms, such neglect of anisotropy will lead to an error in the retrieved source. The distortion of the mechanism of a double-couple point source located in an anisotropic medium is investigated when inverting waveforms using isotropic Greens functions. The anisotropic medium is considered to be transversely isotropic with six levels of anisotropy ranging from a fairly weak to rather strong anisotropy, up to about 24% in P waves and 11% in S waves. Inversions are based on either only direct P waves or both direct P and S waves. Two different algorithms are employed: the direct parametrization (DIRPAR, a nonlinear algorithm) and the indirect parametrization (INPAR, a hybrid scheme including linear and nonlinear steps) of the source. The orientation of the double-couple mechanism appears to be robustly retrieved. The inclination of the resulting nodal planes is very small, within 10° and 20° from the original solution, even for the highest degree of anisotropy. However, the neglect of anisotropy results in the presence of spurious isotropic and compensated linear-vector dipole (CLVD) components in the moment tensor (MT). This questions the reliability of non-double-couple components reported for numerous earthquakes.

Asymptotic Green's function in homogeneous anisotropic viscoelastic media

An asymptotic Greens function in homogeneous anisotropic viscoelastic media is derived. The Greens function in viscoelastic media is formally similar to that in elastic media, but its computation is more involved. The stationary slowness vector is, in general, complex valued and inhomogeneous. Its computation involves finding two independent real-valued unit vectors which specify the directions of its real and imaginary parts and can be done either by iterations or by solving a system of coupled polynomial equations. When the stationary slowness direction is found, all quantities standing in the Greens function such as the slowness vector, polarization vector, phase and energy velocities and principal curvatures of the slowness surface can readily be calculated. The formulae for the exact and asymptotic Greens functions are numerically checked against closed-form solutions for isotropic and simple anisotropic, elastic and viscoelastic models. The calculations confirm that the formulae and developed numerical codes are correct. The computation of the P-wave Greens function in two realistic materials with a rather strong anisotropy and absorption indicates that the asymptotic Greens function is accurate at distances greater than several wavelengths from the source. The error in the modulus reaches at most 4% at distances greater than 15 wavelengths from the source.

Calculation of the slowness vector from the ray vector in anisotropic media

The wave quantities needed in constructing wave fields propagating in anisotropic elastic media are usually calculated as a function of the slowness vector, or of its direction called the wave normal. In some applications, however, it is desirable to calculate the wave quantities as a function of the ray direction. In this paper, a method of calculating the slowness vector for a specified ray direction is proposed. The method is applicable to general anisotropy of arbitrary strength with arbitrary complex wave surface. The slowness vector is determined by numerically solving a system of multivariate polynomial equations of the sixth order. By solving the equations, we obtain a complete set of slowness vectors corresponding to all wave types and to all branches of the wave surface including the slowness vectors along the acoustic axes. The wave surface can be folded to any degree. The system of equations is further specified for rays shot in the symmetry plane of an orthorhombic medium and for a transversely isotropic medium. The system is decoupled into two polynomial equations of the fourth order for the P–SV waves, and into equations for the SH wave, which yield an explicit closed-form solution. The presented approach is particularly advantageous in constructing ray fields, ray-theoretical Green functions, wavefronts and wave fields in strong anisotropy.

Approximate Relation Between the Ray Vector and the Wave Normal in Weakly Anisotropic Media

Determination of the ray vector (the unit vector specifying the direction of the group velocity vector) corresponding to a given wave normal (the unit vector parallel to the phase velocity vector or slowness vector) in an arbitrary anisotropic medium can be performed using the exact formula following from the ray tracing equations. The determination of the wave normal from the ray vector is, generally, a more complicated task, which is usually solved iteratively. We present a first-order perturbation formula for the approximate determination of the ray vector from a given wave normal and vice versa. The formula is applicable to qP as well as qS waves in directions, in which the waves can be dealt with separately (i.e. outside singular directions of qS waves). Performance of the approximate formulae is illustrated on models of transversely isotropic and orthorhombic symmetry. We show that the formula for the determination of the ray vector from the wave normal yields rather accurate results even for strong anisotropy. The formula for the determination of the wave normal from the ray vector works reasonably well in directions, in which the considered waves have convex slowness surfaces. Otherwise, it can yield, especially for stronger anisotropy, rather distorted results.

SH-wave Green tensor for homogeneous transversely isotropic media by higher-order approximations in asymptotic ray theory

Tensor equations of the ray theory for homogeneous anisotropic elastic media are presented. For a point source, an explicit solution of the transport equation is obtained, thus the additional as well as principal components of the ray amplitudes for higher-order ray approximations are expressed only by differential operators of lower-order terms. Possibility of analytical calculation of higher-order approximations is exemplified for SH waves in a transversely isotropic medium. The ray series of the SH-wave Green tensor for the transversely isotropic medium involves only two non-zero terms and the complete ray solution coincides with an exact solution obtained by other complicated procedures.