Thomson-Haskell Method Research Articles

Calculation of wave fields in a layered half-space with absorption based on the Thomson-Haskell method

Calculation of wave fields in a layered half-space with absorption based on the Thomson-Haskell method

Rayleigh wave dispersion equation with real terms in a layered transversely isotropic half space

We consider transversely isotropic media with vertical axis of symmetry (VTI). Solutions of P-SV equation of motion in a homogeneous VTI medium contain depth decay factors r 1 and r 2 , which sometimes become complex depending on VTI parameters. In this case, r 1 and r 2 are complex conjugates. Using this property, we obtain dispersion equation of Rayleigh waves with real terms for a layered VTI half space through Thomson-Haskell method with reduced delta matrix. Phase and group velocities as well as surface ellipticity of Rayleigh waves are computed in real domain for a few oceanic and continental VTI structures of the earth. Present computation in real domain is similar to that in a layered isotropic half space using the same method. Thus it is presumed that such computation in a layered VTI half space will allow efficient evaluation of a VTI structure of the earth.

Shear building stiffness estimation by wave traveling time analysis

A novel damage identification technique to estimate stiffness in a multistory building supported on solid ground is presented. Based on a shear building model, a 1-dimensional wave equation for a vertically propagating shear wave is derived. A Ricker pulse is used as excitation signal and propagated through the building. Wave propagation in the building is based on the Thomson–Haskell method, where each story is represented as a single layer in a multiple stratum model. The wave arrival times of the pulse at each story are used to calculate the stiffness of the columns. The involved calculations in this method grow only linearly with the number of stories, as opposed to other identification methods, as modal analysis, that grow geometrically; this makes this approach an interesting alternative method to asses building integrity. Simulation result for a building with heterogeneous characteristics across the stories confirms the feasibility of the proposal.

An analysis of multi-channel surface wave’s Thomson-Haskell forward modelling

Multi-channel Analysis of Surface Wave is a geophysical method based on the dispersion curve obtained from Rayleigh wave to predict the stiffness of subsurface structure. To do this, one have to solve the forward modeling first. Thomson-Haskell method which was introduced by Thomson and Haskell is among the first methods used for this purpose. Unfortunately, Thomson-Haskell method has a problem on the stability when it is implemented numerically. In this papers, authors want to code Thomson-Haskell method to examine the stability property of Thomson-Haskell method which hasn’t been discussed clearly in other Thomson-Haskell’s relating papers. This will put the conerete stone for the next step of solving the inversion problem of examining near surface soil.

The computations of reflection coefficients of multilayer structure based on the reformulation of Thomson-Haskell method

A reformulation of the Thomson-Haskell method is presented for calculating the reflection coefficients of multilayer structure immersing in the coupling fluid. Instead of directly multiplying the layer propagator matrix, the new method splits the layer propagator matrix and excursively determines the interface stiffness matrix starting from the bottom half-space with known stiffness. A formulation for the reflection coefficients is derived based on the obtained interface stiffness matrix of the top layer. This scheme can be applied to a single solid layers or layered structures containing both fluid and solid layers. It keeps the simplicity but naturally excludes the exponential growth term and thus can be applied at any frequency range. Its validity and feasibility were experimentally proved by the measurement of the reflection coefficients of a three layered structure of aluminum–glass–aluminum and a sandwiched layer structure of two 250μm stainless plates filled with 100μm deionized water based on the inversion of V(z, t) technique. The result of experiments is consistent with the theoretical calculation. The reformulation of the Thomson-Haskell method offers an efficient and effective solution for calculating the acoustic reflection coefficients of multilayer structures of any configurations.

Generalized transfer matrix method for propagation of surface waves in layered azimuthally anisotropic half-space

SUMMARY This paper presents a systematic and efficient method, namely the generalized transfer matrix method, for evaluating the dispersion curves and eigenfunctions of surface waves in multilayered azimuthally anisotropic half-space. Apart from avoiding the well-known numerical difficulties associated with the existing Thomson–Haskell method, the generalized transfer matrix method possesses the robust determination of independent polarization vectors by using the singular value decomposition (SVD) approach, the explicit inversion of the 6 × 6 eigencolumn matrix without any resort to numerical inversion and the efficient computation of eigenfunctions for layered azimuthally anisotropic media. By means of straightforward transformations, the generalized transfer matrix method leads to a twofold recursive algorithm: (1) for the recursive computation of phase velocities it starts from the bottom half-space to the top layer and (2) for the recursive solution of eigenfunctions it starts from the top layer to the bottom half-space. While keeping the simplicity of the Thomson–Haskell transfer matrix method, the generalized transfer matrix method is of unconditional stability and computational efficiency. The related numerical examples demonstrate that the generalized transfer matrix method is a powerful and robust tool for simulating the propagation of elastic surface waves in the layered azimuthally anisotropic half-space.

Efficient Reformulation of the Thomson-Haskell Method for Computation of Surface Waves in Layered Half-Space

This paper presents an efficient and effective scheme for reforming the original Thomson–Haskell transfer matrix method for the stable modeling of surface waves in a layered half-space. The proposed scheme splits the layer propagator matrix and then determines the interface stiffness instead of directly building the propagator matrix as the original Thomson–Haskell method does. For the Love wave, an explicit solution for the interface stiffness is obtained. The reformulation of the Thomson–Haskell method keeps the simplicity of the original method but naturally excludes its exponential growth terms. Hence, the modified Thomson–Haskell method is computationally stable for high frequency and increasing layer thickness. Moreover, both analytical and numerical results indicate the modified Thomson–Haskell method is the most efficient of all the complex arithmetic algorithms available, including the original Thomson–Haskell method and generated reflection and transmission matrix method. The computational speed of the modified Thomson–Haskell method rises by more than 15% for the Love and Rayleigh waves. Furthermore, a discussion of the effect of frequency and layer thickness on the phase velocity of Gutenberg Earth model indicates the modified Thomson–Haskell method is a powerful candidate for efficiently and effectively simulating surface waves in a layered half-space.

Elastic wave scattering by an inclusion in a multilayered medium submerged in fluid. I. Plane strain model

Response of plane multilayered media enclosing an elastic inclusion of arbitrary shape and submerged in fluid is investigated. The plane strain model is assumed. The steady-state free-field response is modeled by the Thomson-Haskell method using delta-matrix modification and the scattered wave field by a boundary method. The transient response is evaluated using the Fourier synthesis. Numerical results are presented for a single layer plate with an elliptical inclusion and a single layer with an inclusion bounded by an elastic half-space. For the steady-state fluid-solid interface response the results show that (i) presence of the half-space greatly influences the response; (ii) the response is very sensitive upon the geometry and the embedment depth of the inclusion; (iii) the response is sensitive upon the angle and frequency of incident wave; and (iv) impedance contrast between the plate and inclusion is important for overall response pattern. Transient results for vertical incidence demonstrate that the presence of the inclusion is detected very clearly by both predominant and nonpredominant components of motion.

Elastic wave scattering by an inclusion in a multilayered medium submerged in fluid. II. Three-dimensional model

Response of plane three-dimensional multilayered medium enclosing an elastic inclusion of arbitrary shape and submerged in fluid is investigated. The system is subjected to an oblique plane incident wave from within the fluid. The steady-state free-field response is modeled by the Thomson-Haskell method using delta-matrix modification and the scattered wave field by a boundary method. The transient response is evaluated using the Fourier synthesis. Numerical results are presented for a single layer plate with an ellipsoidal inclusion. The steady-state results show that the response along the fluid-solid interface is sensitive to the aspect ratio and location of the inclusion, impedance contrast, and the angle of incidence. In addition, the results of the three-dimensional analysis for the nonaxisymmetric problem indicate significant variation of the radial and azimuthal displacement components with the change in the azimuthal angle. Comparison of the two-dimensional and three-dimensional models indicates that for the harder inclusion, both the models show similar qualitative trends in the transient and steady-state responses. However, for a softer inclusion, the steady-state responses of the two models are found to differ significantly from each other. The transient studies clearly indicate the horizontal wave traveling at the solid-fluid interface with the acoustic wave velocity of the fluid. The effect of the scatterer on the vertical displacement is found to be significant only in the vicinity of the inclusion.

Christchurch seismic hazard pilot study

A pilot zoning study has been carried out for ground shaking hazard in the city of Christchurch, using the grade-3 procedures of the draft manual on seismic zonation for geotechnical hazards, being developed by Technical Committee TC-4 of the ISSMFE. Because of limited site data, we were not able to produce a complete zoning map for the city, but ground motions were estimated at six distinct sites for which borehole data was available, and this provided a good test for the procedures of the manual. The city of Christchurch is situated on nearly 1 km of volcanic rock and alluvial sediments overlying greywacke basement, on the edge of the main seismic region of New Zealand. In addition to being an interesting site from the point of view of ground shaking, there are also liquefaction and slope stability hazards in the alluvial and beach sands found throughout the city, and in the recent loess deposits mantling the adjacent Port Hills. In the pilot study of ground shaking hazard, a seismicity model for the central South Island region published recently by Elder et al. (1991) was combined with the attenuation model of Kawashima et al. (1984) recommended in the draft manual, to estimate rock motion at Christchurch, described by its acceleration response spectrum. Transfer functions were computed for the site response to estimate motions at the six selected sites using the Thomson-Haskell method. The results were highly sensitive to details of. the upper 30 m or so of the soil profile. In the local context, the large disparity between our estimates of ground shaking at Christchurch and those implicit in the draft revision of NZS 4203 are disturbing. According to our study the draft code underestimates shaking by about a factor of two or more.

Ultrasonic transmission of adhesively bonded structures for large FD: Ultrasonics International 91, Le Touquet (France), 1–4 Jul. 1991. pp. 443–446. Butterworth Heinemann (1991)

This paper presents the application of the Thomson-Haskell method with bulk heterogeneous waves to the prediction of waveforms transmitted by adhesively bonded structures. This numerical method is still stable for large values of the product of the frequency by the thickness of the medium (FD) and permits a simple simulation of transmission through interface with varying boundary conditions. These conditions are not always weak conditions like in the presence of defects, but also some surface preparations like sandblasting, enhance the transmission when the adherent surfaces are roughen. Experimental verifications are given for steel/epoxy/steel stratified media.

Bulk heterogeneous plane waves propagation through viscoelastic plates and stratified media with large values of frequency domain

This paper introduces the bulk heterogeneous waves concept into the well known Thomson-Haskell method for computing transmission/reflection coefficients through stratified media. If one of the layers is absorbing, bulk heterogeneous waves are generated at interfaces with other layers and the generalized Snell's laws for heterogeneous modes have to be used. This method was reported to be unstable for large values of frequency domain (FD), the product of the frequency and the medium thickness. The new expression for the matrix transfer between interfaces given in this paper is tested with large values of FD, without showing instability. Large values of FD imply significant effects of attenuation, which is frequency dependent. One particular effect is the interface transmission of modes beyond the limit angle if it is defined with the homogeneous waves concept. This is shown with acquired and simulated waveforms, after transmission through an epoxy layer. Waveforms transmitted by adhesive joints are also presented.

Acoustic multipole logging in transversely isotropic poroelastic formations

The properties of the modes generated by multipole sources in a fluid-filled borehole embedded in (radially layered) formations that include transversely isotropic poroelastic layers are investigated. These layers are modeled following Biot’s and homogenization theories, under the assumption that the principal axes of symmetry of both the transversely isotropic skeleton and the complex permeability tensor are parallel to the vertical axis of the borehole. A general formulation based on the Thomson–Haskell method accounts for any combination of fluid, elastic, and poroelastic layers, either isotropic or transversely isotropic. The investigation is achieved through the computation of dispersion and attenuation curves, and frequency-dependent sensitivity coefficients with respect to the parameters. In the presence of a fast or slow radially semi-infinite transversely isotropic poroelastic formation with an impermeable borehole wall, the shear-wave transverse isotropy of the skeleton can be determined using the low-frequency parts of the fundamental modes generated by a monopole source and one of higher order. With a permeable interface, the fluid flow effects refer to the horizontal mobility (horizontal permeability/saturant fluid viscosity) and disables the determination of the shear-wave transverse isotropy. Without this information, any horizontal mobility determination is incorrect. Whatever the nature of any fluid–poroelastic interface, the presence of radial layering decreases the reliability of any estimation of parameter related to the virgin formation except for the velocity and attenuation of the vertically propagating SV wave. Whatever the configuration, the transverse isotropy of the compressional wave velocity cannot be determined nor can that of the complex permeability tensor or the anelastic attenuation due to the polarization of the (quasi) body waves involved. The results of this work also show that, in contrast to the pure elastic situation, a transverse isotropy of the complex permeability tensor leads to slightly different quasi-SV-wave velocities along the principal directions of propagation at ultrasonic frequencies.

Shear wave logging in elastic formations

The analysis of the dispersion and attenuation of the modes generated both by a dipole and a quadrupole source in a fluid-filled borehole surrounded by a (multilayered) elastic formation is performed. The displacement-stress vectors are propagated through the layers using the Thomson–Haskell method. Only the well-bonded configuration is investigated. In the time domain, the effects of the source center frequency on synthetic full waveform microseismograms computed using the discrete wavenumber method are investigated. Whatever the formation (fast or slow) and the configuration, the low-frequency part of both the flexural and screw modes follows the virgin formation shear wave characteristics. Their high-frequency parts behave like that of the Stoneley wave excited by a monopole source. As a result, the fundamental modes are very sensitive to the properties of the inner layers at intermediate and high frequencies. In the presence of an invaded zone, the internal dynamics of the waveforms can be significantly modified, while the useful starting energy of the fundamental modes is shifted toward lower frequencies. The reverse phenomenon occurs when the borehole is cased. For this last configuration, it is only with low source center frequencies that multipole sources can log a slow formation shear wave due to the leaky character of the fundamental modes. Such behaviors emphasize the need for useful low-frequency energy for the actual logging tools.

Effects of radial layering when logging in saturated porous formations

In a fluid-filled borehole embedded in a radially semi-infinite saturated porous formation with a permeable borehole wall, numerical simulations have shown that the Stoneley wave dispersion and attenuation are sensitive to the in situ permeability. In this article, the configuration of a damaged (invaded or flushed) zone resulting from a general formulation, based on the Thomson–Haskell method, is used to propagate the wave field through several saturated porous layers. The two phase media are modeled following a modified Biot’s theory. Whatever the boundary conditions at the borehole wall, the Stoneley wave integrates the properties of the inner layer in the entire frequency range. With a permeable borehole wall, the Stoneley wave dispersion and attenuation are mainly representative of the rheological properties of the first layer. Such a coupling increases with increasing thickness and/or decreasing velocities, porosity, and saturant fluid mobility of the inner layer. Of course, it also increases with increasing frequency. As a result, the estimation of the in situ permeability of the virgin formation, based on the Stoneley wave characteristics, is ill-posed. Whatever the configuration, the phase velocity and attenuation of the first pseudo-Rayleigh mode start at those of the virgin formation. The low-frequency (high-velocity) part of the pseudo-Rayleigh wavetrain can then be used for the indirect estimation of the virgin formation shear wave characteristics. Higher in frequency, the inner shell becomes the controlling layer. Full wave synthetic microseismograms illustrate these behaviors in the time domain.

Frequency-Domain Inversion of Receiver Transfer Functions For Crustal Structure

A linearized frequency domain inversion of receiver transfer functions is investigated as a means for determining crustal structure. The forward problem is solved using a Thomson-Haskell method which includes the effects of attenuating layers. The function to be fit consists of the deconvolved radial component of teleseismic P waves (to remove source effects) which is sensitive mainly to variations in shear velocity and density. Compressional velocity and attenuation are adjusted by assuming a Poisson solid. Complex partial derivatives are computed with respect to the complex shear velocity in each layer using a finite difference approach. Thus, it is possible to simultaneously invert for shear velocity and shear attenuation. Test cases show that for short path lengths, the effects of attenuation are relatively small compared to the effects of velocity contrasts at layer interfaces. Test cases utilizing synthetic and real data demonstrate the impressive resolving power of the technique for the determination of crustal thickness and the importance of density on the calculated receiver functions. Examination of partial derivatives for the receiver transfer functions with respect to various layer parameters further demonstrates that the effect of density is nearly as important as that due to shear velocity. The test cases also bring out the importance of an accurate trial solution, especially in the presence of noise. Thus, in actual practice it is necessary to combine the inversion with trial and error calculations.

Comparison of synthetic seismograms for anelastic media by asymptotic ray theory and the Thomson-Haskell method

abstractSynthetic seismograms for P-SV teleseismic waves in a layered anelastic medium are computed by asymptotic ray theory and compared with those computed by a modification of the Thomson-Haskell matrix method for anelastic media, due to Silva (1976). The synthetic seismograms for the two methods agree well with each other, which verifies the validity of the modification of the Thomson-Haskell method. A useful feature of the ray approach is the ability to identify the rays which make up the various events on the seismograms.

Extension of the Thomson--Haskell method to non-homogeneous spherical layers

Summary. Amplitude spectra of Rayleigh and Love waves in a layered nongravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson-Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical- and flat-earth models have been investigated in the range 20-300-s period and expressed in polynomials in the period.

Matrix methods for generally stratified media

Summary The extension of Thomson–Haskell method to generally stratified media is studied. The separation of SH and P-SV waves is shown to be possible only for spherical, circular cylindrical and parallel plane stratifications. For stratifications differing but slightly from these types a perturbation method is proposed. It implies the simultaneous evaluation of the effects of a dislocation field and of a body force field.

On the equivalence of thomson-haskell matrices and knopoff's method for love waves

The relation between Thomson-Haskell matrices and Knopoff's method for Love waves is studied. In comparing boths methods one draws on the physical formulation of the problem, used in Thomson-Haskell method.