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.     

Small-scale heterogeneities below the Gräfenberg array, Germany from seismic wavefield fluctuations of Hindu Kush events

Small-scale elastic heterogeneities (<5  km) are found in the upper lithosphere underneath the Gräfenberg array, southeast Germany. The results are based on the analysis of broadband recordings of 17 intermediate-depth (201–272  km) events from the Hindu Kush region. The wavefront of the first P arrival and the following 40  s coda are separated into coherent and incoherent (scattered) parts in the frequency range from 0.05 to 5  Hz. The frequency-dependent intensities of the mean and fluctuation wavefields are used to describe the scattering characteristics of the lithosphere underneath the receivers. It is possible to discriminate a weak-fluctuation regime of the wavefield in the frequency range below approximately 1.5–2.5  Hz and a strong-fluctuation regime starting at 2.0–2.5  Hz and continuing to higher frequencies. In order to explain the observed wavefield fluctuations, an approach with seismic scattering at random media-type structures is proposed. The preferred model contains heterogeneities with 3–7 per cent perturbations in seismic velocity and correlation lengths of 0.6–4.8  km in the crust. This is compatible with models from active seismic experiments. Scattering in the lithospheric mantle is not required, but cannot be excluded at weak velocity contrasts (<3 per cent).     

Statistical analysis of spatial point patterns on deep seismic reflection data: a preliminary test

Spatial point patterns generated from bitmaps of images of processed reflection seismic profiles are analysed to quantify the reflectivity patterns. The point process characteristics for two different regions of a deep seismic reflection profile in northwestern Canada demonstrate that in both cases the points are not randomly distributed and that the point pattern distribution is different between the regions. The cluster effects for small point pair distances are stronger for the region of data where there is strong sedimentary layering than for the region where the layering is less distinct. As a result, it appears that future developments in point pattern analysis may provide a new tool for analysing spatial variations in reflection data.     

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.     

Inversion of seismic refraction data in planar dipping structure

Summary. A new method is presented for the direct inversion of seismic refraction data in dipping planar structure. Three recording geometries, each consisting of two common-shot profiles, are considered: reversed, split, and roll-along profiles. Inversion is achieved via slant stacking the common-shot wavefield to obtain a delay time–slowness (tau– p ) wavefield. The tau– p curves from two shotpoints describing the critical raypath of refracted and post-critically reflected arrivals are automatically picked using coherency measurements and the two curves are jointly used to calculate velocity and dip of isovelocity lines iteratively, thereby obtaining the final two-dimensional velocity model.
This procedure has been successfully applied to synthetic seismograms calculated for a dipping structure and to field data from central California. The results indicate that direct inversion of closely-spaced refraction/wide-aperture reflection data can practically be achieved in laterally inhomogeneous structures.     

Effects of volcano topography on seismic broad‐band waveforms

Volcano seismology often deals with rather shallow seismic sources and seismic stations deployed in their near field. The complex stratigraphy on volcanoes and near‐field source effects have a strong impact on the seismic wavefield, complicating the interpretation techniques that are usually employed in earthquake seismology. In addition, as most volcanoes have a pronounced topography, the interference of the seismic wavefield with the stress‐free surface results in severe waveform perturbations that affect seismic interpretation methods. In this study we deal predominantly with the surface effects, but take into account the impact of a typical volcano stratigraphy as well as near‐field source effects. We derive a correction term for plane seismic waves and a plane‐free surface such that for smooth topographies the effect of the free surface can be totally removed. Seismo‐volcanic sources radiate energy in a broad frequency range with a correspondingly wide range of different Fresnel zones. A 2‐D boundary element method is employed to study how the size of the Fresnel zone is dependent on source depth, dominant wavelength and topography in order to estimate the limits of the plane wave approximation. This approximation remains valid if the dominant wavelength does not exceed twice the source depth. Further aspects of this study concern particle motion analysis to locate point sources and the influence of the stratigraphy on particle motions. Furthermore, the deployment strategy of seismic instruments on volcanoes, as well as the direct interpretation of the broad‐band waveforms in terms of pressure fluctuations in the volcanic plumbing system, are discussed.     

Seismic reflection traveltimes in two-dimensional statistically anisotropic random media

Velocity estimation remains one of the main problems when imaging the subsurface with seismic reflection data. Traveltime inversion enables us to obtain large-scale structures of the velocity field and the position of seismic reflectors. However, as the media currently under study are becoming more and more complex, we need to know the finer-scale structures. The problem is that below a certain range of velocity heterogeneities, deterministic methods become difficult to use, so we turn to a probabilistic approach. With this in view, we characterize the velocity field as a random field defined by its first and second statistical moments. Usually, a seismic random medium is defined as a homogeneous velocity background perturbed by a small random field that is assumed to be stationary. Thus, we make a link between such a random velocity medium (together with a simple reflector) and seismic reflection traveltimes. Assuming that the traveltimes are ergodic, we use 2-D seismic reflection geometry to study the decrease in the statistical traveltime fluctuations as a function of the offset (the source–receiver distance). Our formulae are based on the Rytov approximation and the parabolic approximation for acoustic waves. The validity and the limits are established for both of these approximations in statistically anisotropic random media. Finally, theoretical inversion procedures are developed for the horizontal correlation structure of the velocity heterogeneities for the simplest case of a horizontal reflector. Synthetic seismograms are then computed (on particular realizations of random media) by simulating scalar wave propagation via finite difference algorithms. There is good agreement between the theoretical and experimental results.     

Inversion of a normally incident reflection seismogram by the Gopinath–Sondhi integral equation

Summary. The one-dimensional acoustic wave equation has been transformed to two coupled first-order equations whose inverse solution is obtained through application of the Gopinath and Sondhi integral equation. A scattering solution of the Schrödinger wave equation for an explosive source leads us to express the kernel of the Gopinath–Sondhi integral equation in terms of a seismic reflection response. A numerical solution of the integral equation obtained by a trapezoidal rule yields a continuous impedance profile whose derivative has step-like discontinuities. The method is illustrated with computer model studies.     

Non-linear Bayesian joint inversion of seismic reflection coefficients

Inversion of seismic reflection coefficients is formulated in a Bayesian framework. Measured reflection coefficients and model parameters are assigned statistical distributions based on information known prior to the inversion, and together with the forward model uncertainties are propagated into the final result. This enables a quantification of the reliability of the inversion. Quadratic approximations to the Zoeppritz equations are used as the forward model. Compared with the linear approximations the bias is reduced and the uncertainty estimate is more reliable. The differences when using the quadratic approximations and the exact expressions are minor. The solution algorithm is sampling based, and because of the non-linear forward model, the Metropolis–Hastings algorithm is used. To achieve convergence it is important to keep strict control of the acceptance probability in the algorithm. Joint inversion using information from both reflected PP waves and converted PS waves yields smaller bias and reduced uncertainty compared to using only reflected PP waves.     

Combined seismic reflection and refraction profiling in southwest Germany - detailed velocity mapping by the refraction survey

Summary. Within the framework of site survey studies of the Deep Drilling Program of the Fedral Republic of Germany (KTB), coincident deep-seismic reflection and refraction experiments in the Black Forest, southwest Germany, were carried out. The simultaneous interpretation of the reflection and the refraction data reveals in particular both a strong velocity reduction in the upper crust and a laterally varying laminated structure of the lower curst. Additional refraction lines result in a three-dimensional crustal model which shows two distinct crustal types of different seismic properties. These crustal types seem to correlate with the major geologic units of Southwest Germany. Variations of Poisson's ratio derived from clearly recorded shear wave data show a similar trend.     

Generalized Born inversion of seismic reflection data

Summary . Born inverse methods give accurate and stable results when the source wavelet is impulsive. However, in many practical applications (reflection seismology) an impulsive source cannot be realized and the inversion needs to be generalized to include an arbitrary source function. In this paper, we present a Born solution to the seismic inverse problem which can accommodate an arbitrary source function and give accurate and stable results. It is shown that the form of the generalized inversion algorithm reduces to a Wiener shaping ***filter, which is solved efficiently using a Levinson recursion algorithm. Numerical examples of synthetic and real field data illustrate the validity of our method.     

Synoptic interpretation of seismic reflection and refraction data

Summary. The structure of the upper lithosphere beneath southern Germany, northern Switzerland and west-central Utah (U.S.A.) has been investigated in detail by various geophysical methods. A synoptic interpretation of travel time and amplitude data obtained in seismic refraction and wide-angle reflection surveys, combined with near-normal incidence reflection observations, now permits the elucidation of the fine structure in a more quantitative and unified manner. With this scheme it is possible to unambiguously identify low-velocity zones and to deduce velocity gradients if reliable amplitude information is included in the inversion process.     

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.     

A general correspondence principle for time-domain electromagnetic wave and diffusion fields

A general correspondence principle is presented that relates any time-domain electromagnetic diffusion field to an electromagnetic wavefield in a 'corresponding' configuration. The principle applies to arbitrarily inhomogeneous and anisotropic media and arbitrary transmitters and receivers. For the correspondence between the two types of electromagnetic fields to hold, the electric conductivity in the diffusive case and the permittivity in the wavefield case should have the same spatial variation, while the permeability distributions in space in the two cases are to be identical. Essential steps in the derivation of the correspondence principle are the use of the time Laplace transformation of causal signals, taken at real, positive values of the transform parameter, the Schouten-Van der Pol theorem in the theory of the Laplace transformation, and the reliance upon Lerch's theorem of the uniqueness of the interrelation between causal field quantities and their time-Laplace-transform representations at real, positive values of the transform parameter. Correspondence is then established between the tensorial Green's functions in the two cases, where the Green's functions are the point-receiver responses (either electric or magnetic field) to point-transmitter excitations (either electric- or magnetic-current source).
Through the correspondence principle, all transient electromagnetic wavefields (where losses are neglected) have as a counterpart a transient diffusive electromagnetic field (where the electric displacement current is neglected). The interrelation yields the tool to compare quantitatively the potentialities of the two types of fields in transient electromagnetic geophysical prospecting.
Finally, a general medium-parameter scaling law for time-domain electromagnetic wavefields is presented.     

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.     

Observations on seismic wave equation and reflection coefficient symmetries in stratified media

Summary. The equations describing seismic waves in a stratified earth have a number of symmetry properties, one of which has recently been used by Garmany to derive a simple expression for the inverse of the matrix of eigenvectors appearing in the solution of the equations. We review these symmetries of the wave equation in several notations to demonstrate that the property used by Garmany is distinct from the usual symmetries found in the seismological literature. Like the others, the new symmetry has implications for the reflection and transmission properties of a medium. These implications are briefly reviewed in order to show how the new symmetry is placed relative to the others. A limited discussion of the physical origins of the symmetries is given and, though the overall picture is incomplete, it is noted that the new symmetry yields conservation of energy for reflection/transmission at a single interface in all types of media (anisotropic, lossy, etc.).     

Scattering characteristics in heterogeneously fractured reservoirs from waveform estimation

Offset-dependent characteristics of seismic scattering are useful for characterizing fractured reservoirs. We use two models that have different background medium properties and different azimuthal AVO responses to study elastic wave propagation and scattering in gas-saturated, heterogeneously fractured reservoirs. Heterogeneous fracture density distributions are built through stochastic modelling. Synthetic seismograms are generated by 3-D finite difference modelling, and waveforms along crack-normal and strike directions are considered in this paper. The multiple signal classification (MUSIC) frequency estimator is used in waveform estimation to provide frequency-domain attributes related to seismic wave scattering by fracture heterogeneity. Our results indicate that the strength of the scattering field is a function of the background medium. The strength also increases with increasing fracture scatterer density and with decreasing correlation length of spatial variations of fracture density. The scattering field is weak at the top of the fractured reservoir. The first-order results are dominated by velocity anisotropy of the mean fracture density field. However, the base of the fractured reservoir corresponds to a strong scattering field on which fracture heterogeneity has a larger effect and is characterized by the loss of coherence.     

Lithoprobe seismic reflection profiling across Vancouver Island: results from reprocessing

Summary. Multichannel seismic reflection sections recorded across Vancouver Island have revealed two extensive zones of deep seismic reflections that dip gently to the northeast, and a number of moderate northeasterly dipping reflections that can be traced to the surface where major faults are exposed. Based on an integrated interpretation of these data with information from gravity, heat flow, seismicity, seismic refraction, magnetotelluric and geological studies it is concluded that the lower zone of gently dipping reflections is due to underplated oceanic sediments and igneous rocks associated with the current subduction of the Juan de Fuca plate, and that the upper zone represents a similar sequence of accreted rocks associated with an earlier episode of subduction. The high density/high velocity material between the two reflection zones is either an underplated slab of oceanic lithosphere or an imbricated package of mafic rocks. Reprocessing of data from two of the seismic lines has produced a remarkable image of the terrane bounding Leech River fault, with its dip undulating from >60° near the surface to 20° at 3 km depth and ∼38° at 6 km depth.     

Deep seismic reflection profiles across the western Barents Sea

Summary. The continental crust beneath the western Barents Sea has been acoustically imaged down to Moho depths in a large scale deep seismic reflection experiment. A first-order pattern of crustal reflectivity has been established and the thickness of the crust determined. A number of features with important implications for the tectonics of the area have been discovered. The results are presented in the form of two transects.     

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.