Windowed f-k spectra of a three-dimensional wavefield excited by a point source in a two-dimensional multilayered elastic medium

When full 3-D modelling is too costly or cumbersome, computations of 3-D elastic wave propagation in laterally heterogeneous, multilayered 2-D geological structures may enhance considerably our ability to predict strong ground motion for seismological and engineering purposes. Towards this goal, we extend the method based on the combination of the thin-layer finite-element and boundary-element methods (TLFE-BEM) and calculate windowend f - k spectra of the 3-D wavefield. The windowed f - k spectra are spatially localized spectra from which the local properties of the wavefield can be extracted. The TLFE-BEM is particularly suited for calculating the complete wavefield where surface waves are dominant in multilayered media. The computations are performed in the frequency domain, providing the f - k spectra directly. From the results for the 3-D wavefield excited by a point source in a 2-D multilayered, sloped structure, it can be said that the phase velocity of the fundamental-mode Rayleigh wave in a laterally heterogeneous multilayered medium, estimated from the windowed f - k spectra, varies with the location of the point source. For the model calculated in this article, the phase velocity varies between the value for the flat layered structure of the thick-layer side and that for the structure just under the centre of the window. The exact subsurface structure just under the centre of an array in a laterally heterogeneous medium cannot be obtained if we use the f - k spectral analysis assuming a flat layered structure.     

Backscattering from spherical elastic inclusions and accuracy of the Kirchhoff approximation for curved interfaces

The Kirchhoff (or tangent plane) approximation, derived from the theoretically complete Kirchhoff–Helmholtz integral representation for the seismic wavefield, has been used extensively for the analysis of seismic-wave scattering from irregular interfaces; however, the accuracy of this method for curved interfaces has not been rigorously established. This paper describes an efficient Kirchhoff algorithm to simulate scattered waves from an arbitrarily curved interface in an elastic medium. Synthetic seismograms computed using this algorithm are compared with exact synthetics computed using analytical formulae for scattering of plane P waves by a spherical elastic inclusion. A windowing technique is used to remove strong internal reverberations from the analytical solution. Although the Kirchhoff method tends to underestimate the total scattering intensity, the accuracy of the approximation improves with increasing value of the wavenumber-radius product, kR . The arrival times and pulse shapes of primary reflections from the sphere are well approximated using the Kirchhoff approach regardless of curvature of the scattering surface, but the amplitudes are significantly underestimated for kR ≤ 5. The results of this work provide some new guidelines to assess the accuracy of Kirchhoff-synthetic seismograms for curved interfaces.     

Synthetic SH seismograms in a laterally varying medium by the discrete wavenumber method

Summary. We present a new method to calculate the SH wavefield produced by a seismic source in a half-space with an irregular buried interface. The diffracting interface is represented by a distribution of body forces. The Green's functions needed to solve the boundary conditions are evaluated using the discrete wavenumber method. Our approach relies on the introduction of a periodicity in the source-medium configuration and on the discretization of the interface at regular spacing. The technique developed is applicable to boundaries of arbitrary shapes and is valid at all frequencies. Some examples of calculation in simple configurations are presented showing the capabilities of the method.     

Effect of a simple mountain range on underground seismic motion

Summary. The influence of a simple mountain range on seismic ground motion is studied. A two-dimensional model of the medium and vertically incident plane SH -waves are considered. Attention is devoted not only to the wavefield along the Earth's surface, but also within the medium. The wavefield is computed in two steps: (1) the computation of the impulse response by the finite-difference method, (2) the computation of the response to a time variation of the incident wave. approximately corresponding to a 'hundred-year' local earthquake at the site. Numerical results (the impulse response, the transfer function, the accelerograms, and their spectra) indicate strong spatial variability of the wavefield due to the topographical anomaly. The differences as large as 100 per cent in the peak amplitudes of the accelerograms, and of the order of 5 Hz in their predominant frequencies, at closely (∼ 200 m) spaced internal points of the medium have been found. Attention is focused also on the effect of causal absorption. Even unrealistically strong absorption seems to be unable to reduce significantly the spatial variability of the ground motion, caused by the topography. A variability like this. implying the occurrence of underground differential strains, might be of engineering importance in the antiseismic design of underground structures (tunnels, for example) in mountainous seismic regions. The ground-motion variability along the surface of the mountain is considerably smaller than within the medium.     

Inverse scattering of surface waves: imaging of near-surface heterogeneities

An efficient inverse scattering method is developed for imaging near-surface heterogeneities using scattered surface waves. Three dimensional elastodynamic wave propagation and scattering in a laterally invariant embedding medium is considered. The Born Approximation is used and the scattered wavefield is expressed as a domain type integral representation. The computation time of Green's tensor elements is reduced by considering the radial symmetry of the medium. The method is validated by numerical tests. Ultrasonic laboratory data obtained from a scale model experiment are used for imaging the near-surface inhomogeneities caused by an epoxy-filled hole in the surface of an aluminum block. Both synthetic and the scale model tests show that the location, the actual density contrast and the depth of the inhomogeneities are reasonably well estimated.     

Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method

Summary. Several approaches to computing body wave seismograms in 2–D and 3–D laterally inhomogeneous layered structures are suggested. They are based on the Gaussian beam method, which has been recently applied to the evaluation of time-harmonic high-frequency wavefields in inhomogeneous media. Three variants are discussed in some detail: the spectral method, the convolutory method and the wave-packet method. The most promising seems to be the wave-packet approach. In this approach, the wavefield, generated by a source, is expanded into a system of wave packets, which propagate along rays from the source in all directions. The wave packets change their properties due to diffusion, spreading, reflections/transmissions, etc. The resulting seismogram at any point of the medium is then obtained as a superposition of those packets which propagate close to the point. The final expressions in all the three methods are regular even in regions, in which the ray method fails, e.g. in the vicinity of caustics, in the critical region, at boundaries between shadow and illuminated regions, etc. Moreover, they are not as sensitive to the minor details of the medium as the ray method and, what is more, they remove the time-consuming two-point ray tracing from computations. Numerical examples of synthetic seismograms computed by the wave-packet approach are presented.     

Multicomponent near-surface correction for land VSP data

Multicomponent seismic data collected using directional sources are degraded by the wave excitation process due to inaccurate control of the ground motion. unequal activation strengths or ground couplings between differently oriented sources, and misalignment of the pad. These acquisition uncertainties are exacerbated by the complicated near-surface scattering present in most seismic areas. Neither group of effects should be neglected in multicomponent analyses that make use of relative wavefield attributes derived from compressional and shear waves. These effects prevent analysis of the direct and reflected waves using procedures based on standard scalar techniques or a prima facia interpretation of the vector wavefield properties, even for the seemingly straightforward case of a near-offset vertical seismic profile (VSP). Near-surface correction, using a simple matrix operator designed from the shallowest recordings, alleviates many of these interpretational difficulties in near-offset VSP data. Results from application of this technique to direct waves from a nine-component VSP shot at the Conoco test-site facility, Oklahoma, are encouraging. The technique corrects for unexpected compressional-wave energy from shear-wave vibrators and collapses near-surface multiples, thus facilitating further processing for the upgoing wavefield. The method provides a simple and effective processing step for routine application to near-offset VSP analyses.     

Wavepath traveltime tomography

The elastic-wave equation is used to construct sensitivity kernels relating perturbations in elastic parameters to traveltime deviations. Computation of the functions requires a correlation of the forward-propagating seismic wavefield with a backward propagation of the residual wavefield. The computation of the wavefields is accomplished using a finite difference algorithm and is efficiently executed on a CM-2 parallel processor. The source and receiver locations have maximum sensitivity to velocity structure. The sensitivity kernels or wavepaths are well suited for transmission traveltime inversion such as cross-borehole tomography and vertical seismic profiling. Conventional ray tomography and wavepath tomography are applied to a set of P -wave arrival times, from a cross-borehole experiment at Kesterson, California. Because the wavepaths have increased sensitivity near the source and receiver there are differences in resolution of the velocity structure. Both techniques recover the same relative variations in velocity where the coverage is adequate. The wavepath solution is more laterally continuous and the dominant variation is vertical, as is expected for the layered sediments in this region.     

Two efficient algorithms for iterative linearized inversion of seismic waveform data

We present two equivalent algorithms for iterative linearized waveform inversion for 3-D Earth structure with respect to an arbitrary 3-D starting model; one is a matrix formulation, and the second is a wavefield formulation. Both algorithms require the computation of accurate synthetic seismograms, but neither requires that any particular method be used to compute the synthetics. The matrix formulation is equivalent to our previously published algorithm (Hara, Tsuboi & Geller 1991), but requires less than 10 per cent of the CPU time of the previous algorithm. The wavefield algorithm is equivalent to that of Tarantola (1986) and Mora (1987), but appears to be substantially more efficient.     

Regional tomographic inversion of the amplitude and phase of Rayleigh waves with 2-D sensitivity kernels

In this study, we test the adequacy of 2-D sensitivity kernels for fundamental-mode Rayleigh waves based on the single-scattering (Born) approximation to account for the effects of heterogeneous structure on the wavefield in a regional surface wave study. The calculated phase and amplitude data using the 2-D sensitivity kernels are compared to phase and amplitude data obtained from seismic waveforms synthesized by the pseudo-spectral method for plane Rayleigh waves propagating through heterogeneous structure. We find that the kernels can accurately predict the perturbation of the wavefield even when the size of anomaly is larger than one wavelength. The only exception is a systematic bias in the amplitude within the anomaly itself due to a site response.
An inversion method of surface wave tomography based on the sensitivity kernels is developed and applied to synthesized data obtained from a numerical simulation modelling Rayleigh wave propagation over checkerboard structure. By comparing recovered images to input structure, we illustrate that the method can almost completely recover anomalies within an array of stations when the size of the anomalies is larger than or close to one wavelength of the surface waves. Surface wave amplitude contains important information about Earth structure and should be inverted together with phase data in surface wave tomography.     

Inferring the relative measure of principal stress components

The phase velocity and the attenuation coefficient of compressional seismic waves, propagating in poroelastic, fluid-saturated, laminated sediments, are computed analytically from first principles. The wavefield is found to be strongly affected by the medium heterogeneity. Impedance fluctuations lead to poroelastic scattering; variations of the layer compressibilities cause inter-layer flow (a 1-D macroscopic local flow). These effects result in significant attenuation and dispersion of the seismic wavefield, even in the surface seismic frequency range, 10–100 Hz. The various attenuation mechanisms are found to be approximately additive, dominated by inter-layer flow at very low frequencies. Elastic scattering is important over a broad frequency range from seismic to sonic frequencies. Biot's global flow (the relative displacement of solid frame and fluid) contributes mainly in the range of ultrasonic frequencies. From the seismic frequency range up to ultrasonic frequencies, attenuation due to heterogeneity is strongly enhanced compared to homogeneous Biot models. Simple analytical expressions for the P -wave phase velocity and attenuation coefficient are presented as functions of frequency and of statistical medium parameters (correlation lengths, variances). These results automatically include different asymptotic approximations, such as poroelastic Backus averaging in the quasi-static and the no-flow limits, geometrical optics, and intermediate frequency ranges.     

Inverse kinematical problems of reflection seismology—II. Problems of downward field continuation

Summary. This paper discusses the inverse (downward) continuation of a wavefield into a medium, as a means of estimating seismic velocities and the imaging of the geometry of reflecting objects. The notion of a kinematically equivalent (K-equivalent) operator of the field continuation is introduced, which allows both the possibility of replacing the initial Lamé equation by a simpler one (for example, a wave equation) and also the existence of an infinite set of continuation operators which permit us to construct a wavefield with proper characteristics. Any of the K-equivalent operators can be used in the task of imaging of reflectors given the known velocities. If velocities are unknown and reflectors are irregular, it is possible to improve the regularity of reflections by continuing the field downward to a fictitious surface in the media with a velocity V_o, that differs from the true velocity V . The conditions needed to untie loops in the true patterns in the continued field are also given.
Another approach is connected with calculations of dynamic sections using different values of velocity (say, an extremal velocity) under which the amplitudes of reflectors on sections are brought to a maximum due to the phenomena of focusing. Extremal velocities can be also used for solving inverse kinematical problems.     

A data-mapping approach to 3-D wavefield redatuming

In case of a complex overburden, the seismic data can be greatly improved by applying a full wavefield redatuming procedure. In practice, the application of the redatuming process to 3-D data acquired by conventional acquisition designs is non-trivial. Because of the large amount of data involved in the 3-D redatuming process and because of the sparseness of these data, it is impossible to apply conventional wave equation datuming directly.
We present a data mapping approach to redatuming (DMR), which follows the concept of Kirchhoff data mapping. A simplified background medium where no ray bending occurs is assumed for the medium below the datum in order to map an input data set referenced to the acquisition surface to an output data set referenced to the new datum level. The DMR method can be interpreted as a simplified version of the Kirchhoff summation redatuming (KSR) method, where one of the 2-D integrals over the acquisition coordinates can be solved analytically. Consequently, in this approach fewer traces are involved in the computation of one time sample (a 2-D integral is computed instead of a 4-D integral), which makes it particularly attractive for the application to 3-D data sets.
In this paper the theory underlying data mapping redatuming is discussed and the proposed approach is tested on fully sampled 2-D and 3-D synthetic data from models with both simple and complex velocity distributions in the subsurface.
The tests clearly show that the objective of producing results that are comparable to the conventional KSR has been achieved. The redatumed traces are dynamically and kinematically correct. Furthermore, these results confirm that the dependency of the new approach on the assumed medium below the datum level is, indeed, weak because the assumption of a velocity medium where no ray bending occurs is already sufficient to produce correct results.     

A hybrid calculation technique of the Indirect Boundary Element Method and the analytical solutions for three-dimensional problems of topography

We present a new approach of the Indirect Boundary Element Method (IBEM) for 3-D topographic problems which can be used to deal with an infinitely spread free surface owing to the introduction of a reference solution, that is the analytical solution for the half-space with a flat free surface. This approach is an efficient countermeasure for the non-physical waves owing to the domain truncation which contaminates the computed results in the ordinary approach. Theoretical consideration shows that this newly proposed approach is a higher-grade approximation than some existing ones and achieves a higher efficacy and accuracy than those of existing ones. The discretization of the resulting boundary integral equation for this formulation is carried out with triangular elements. Their contributions to the solution are calculated by Gaussian numerical integration except in the case where the wavefield is evaluated on the source element itself. For this case, we present an analytical formula based on the reasonable assumption that the elements are much smaller than the wavelengths appearing in the calculation. Several numerical examples used for validation show acceptably precise results.     

One-way representations of seismic data

A general one-way representation of seismic data can be obtained by substituting a Green's one-way wavefield matrix into a reciprocity theorem of the convolution type for one-way wavefields. From this general one-way representation, several special cases can be derived.
By introducing a Green's one-way wavefield matrix for primaries , a generalized Bremmer series representation is obtained. Terminating this series after the first-order term yields a primary representation of seismic reflection data. According to this representation, primary seismic reflection data are proportional to a reflection operator, 'modified' by primary propagators for downgoing and upgoing waves. For seismic imaging, these propagators need to be inverted. Stable inverse primary propagators can easily be obtained from a one-way reciprocity theorem of the correlation type.
By introducing a Green's one-way wavefield matrix for generalized primaries , an alternative representation is obtained in which multiple scattering is organized quite differently (in comparison with the generalized Bremmer series representation). According to the generalized primary representation, full seismic reflection data are proportional to a reflection operator, 'modified' by generalized primary propagators for downgoing and upgoing waves. Internal multiple scattering is fully included in the generalized primary propagators {either via a series expansion or in a parametrized way). Stable inverse generalized primary propagators can be obtained from the one-way reciprocity theorem of the correlation type. These inverse propagators are the nucleus for seismic imaging techniques that take the angle-dependent dispersion effects due to fine-layering into account.     

Gaussian beams in elastic 2-D laterally varying layered structures

Summary. In a paper by Červený & Pšenčik, high-frequency Gaussian beams in elastic 2-D, laterally inhomogeneous, smooth media were investigated as asymptotic high-frequency solutions of elastodynamic equations, concentrated close to rays of P - and S -waves. This paper generalizes the above results for 2-D, laterally inhomogeneous, layered structures. Gaussian beams concentrated close to any multiply-reflected, possibly converted, ray are investigated. Gaussian beams are regular everywhere, including caustic regions. The paraxial ray approximation, which allows the wavefield in the zero-order ray approximation to be evaluated not only directly on the ray, but also in its vicinity, is derived as a limiting case of the Gaussian beams.     

Time-lapse traveltime change of singly scattered acoustic waves

We present a technique based on the single-scattering approximation that relates time-lapse localized changes in the propagation velocity to changes in the traveltime of singly scattered waves. We describe wave propagation in a random medium with homogeneous statistical properties as a single-scattering process where the fluctuations of the velocity with respect to the background velocity are assumed to be weak. This corresponds to one of two end-member regimes of wave propagation in a random medium, the first being single scattering, and the second multiple scattering. We present a formulation that relates the change in the traveltime of the scattered waves to a localized change in the propagation velocity by means of the Born approximation for the scattered wavefield. We validate the methodology with synthetic seismograms calculated with finite differences for 2-D acoustic waves. Potential applications of this technique include non-destructive evaluation of heterogeneous materials and time-lapse monitoring of heterogeneous reservoirs.