Theoretical Seismograms Research Articles

Reflection seismograms in a 3-D elastic model: an isochronal approach

The reflection and diffraction of elastic waves by surfaces in three dimensions and scattering by thin scatterers can be combined in a common formulation. This approach is derived by using an integral formulation of the elastic wavefield together with ray approximations for wave propagation between source or receiver and reflecting surface or scatterer. For a scatterer, first order scattering is assumed and at a reflecting surface, reflection and transmission effects are estimated using the assumption of a locally plane interface. With these approximations, the reflected seismic field can be represented as the convolution of an approximate source function with a weight function for a particular source and receiver configuration. The weight function at a particular time may be evaluated by a contour integral along the isochronal curve for which the total time from source to receiver via points along the curve on the reflecting surface or on the median surface of the scatterer is equal to the specified time. The kernel of this integral contains information on the scattering or reflection coefficients for the incident wave, the angular effects of the incoming and outgoing waves with respect to the surface normal and the speed of advance of the isochron on the surface. By transfering temporal derivatives to the source function in the convolution, a very stable numerical scheme can be formulated for the generation of synthetic seismograms. This method is illustrated by the calculation of P-P reflections for a seismic line oblique to an anticline and for a variety of parameter contrasts for a simple scatterer model. Reflections from multilayered models can be generated for one surface at a time and the effect of a number of scatterers will be additive within the first order approximation employed. The facility to calculate theoretical seismograms for both surface reflections and scattering is exploited to look at the nature of reflection from the crust-mantle boundary in three dimensions. Very similar seismic responses are obtained for the P-waves returned from an irregularly corrugated Moho surface and a model of the crust-mantle transition as a set of scatterers in the lowermost crust with mantle properties. Thus with current resolution there will be inherent ambiguities in interpretations of the character of the crust mantle transition.

Fréchet Derivatives For Curved Interfaces In the Ray Approximation

SUMMARY The sensitivity of ray and beam theoretical seismograms to changes in velocity models and curved interfaces is discussed in this paper. Previous results from Nowack & Lutter (1988) give the derivatives of travel-time and ray amplitude with respect to changes of smoothly varying velocities. These derivatives are required to perform linearized maximum likelihood inversions for structure. In the ray approximation, smooth interfaces are incorporated by applying Snell’s law locally, correcting wavefront curvature, and using local plane-wave reflection and transmission coefficients. The partial derivatives for travel-time are directly calculated along the original ray trajectory using Fermat’s principle. For perturbations of the ray amplitude of a reflected/transmitted ray, the ray shift of the perturbed two-point ray trajectory must be accounted for. The approach followed here is to calculate the approximately perturbed two-point ray using perturbation theory without additional ray tracing. The perturbed ray amplitudes are then computed directly, including modified reflection/transmission coefficients and geometric spreading, along this approximate twopoint ray. Several numerical experiments are conducted which invert for velocity and interface shape using both travel-time and amplitude in order to test the derived partial derivative operators. Travel-time and amplitude inversion results are also contrasted with amplitude being less sensitive to larger scale features and more sensitive to heterogeneity curvature.

Lg-wave propagation in heterogeneous media

Abstract The Lg -wave phase, which is of considerable interest for nuclear discrimination problems, is normally observed after propagation through a few hundred kilometres. This phase is dominantly guided in the crustal waveguide, which is known to be a region with a very significant horizontal variability in properties. The effect of heterogeneous crustal structures on Lg waves has been determined by using a “coupled-mode” technique in which the local seismic wavefield in the real medium is expressed as a horizontally varying combination of the modal eigenfunctions of a stratified reference structure. Departures of the seismic properties in the medium from those of the reference medium lead to coupling between the various amplitude coefficients in the modal expansion. The evolution of these modal weighting factors with horizontal position are described by a coupled set of ordinary differential equations. This approach provides a calculation scheme for studying guided wave propagation over extended distances, at frequencies of 1 Hz and above. The heterogeneity models that have been used are two-dimensional and calculations are carried out for one frequency at a time. A sequence of models with varying levels of heterogeneity have been considered in order to determine the merits and limitations of the computation scheme. The coupled mode technique works well with heterogeneous models in which the local seismic velocities differ from the stratified reference model by up to two per cent and there are no significant distortions of the main discontinuities (e.g., the crust-mantle boundary). The approach can be used for higher levels of heterogeneity and with distorted interfaces but a large number of modes needs to be considered with consequent high computation costs. If the level of heterogeneity is not too large, the interaction between modes can be restricted, rather than extending over the whole mode set, with consequent reduction in computation cost. One of the major effects of crustal heterogeneity is to introduce the possibility of smearing out the main amplitude peak in the Lg wave train over a band of group velocities. As a result, an effective measure of the energy content of the Lg waves will be to consider the integrated amplitude along the trace between group velocities of 3.6 and 3.3 km/sec. The effects of heterogeneity vary between different parts of the Lg wave train and the representation of the wavefield in terms of modal contributions allow a detailed analysis in terms of the group velocity components, which can be illustrated by constructing theoretical seismograms (with a narrow bandwidth in frequency) for the heterogeneous models.

On two-dimensional dynamical dislocations: Theoretical seismograms.

The problem of a two-dimensional dynamical dislocation in an isotropic, homogeneous, unbounded, elastic medium has been discussed by the present authors in a recent paper (SINGH and SIKKA, 1988), assuming harmonic time-dependence. The aim of the present paper is to generalize the results to arbitrary time-dependence. Explicit theoretical expressions for the displacement and the stress components for a two-dimensional dislocation source have been obtained. These expressions can be used to calculate the theoretical seismograms due to a two-dimensional dislocation in an unbounded medium.

TIME‐DOMAIN COMPUTATION OF NON‐NORMAL INCIDENCE WAVEFIELDS IN PLANE LAYERED MEDIA1

ABSTRACTA new time‐domain method is introduced for the calculation of theoretical seismograms which include frequency dependent effects like absorption. To incorporate these effects the reflection and transmission coefficients become convolutionary operators. The method is based on the communication theory approach and is applicable to non‐normal incidence plane waves in flat layered elastic media. Wave propagation is simulated by tracking the wave amplitudes through a storage vector inside the computer memory representing a Goupillaud earth model discretized by equal vertical transit times. Arbitrary numbers of sources and receivers can be placed at arbitrary depth positions, while the computational effort is independent of that number. Therefore, the computation of a whole plane‐wave vertical seismic profile is possible with no extra effort compared to the computation of the surface seismogram. The new method can be used as an aid to the interpretation of plane‐wave decomposed reflection data where the whole synthetic vertical seismic profile readily gives the interpreter the correct depth position of reflection events.

Synthetic seismograms for marine sediments and determination of porosity and permeability

We present numerical simulations of vertical seismic profiles (VSPs) of marine sediments. The theoretical seismograms, which are computed for vertically incident waves in flat layered poroelastic media, include the effects of dispersion and attenuation predicted by Biot theory. According to Biot theory, fast and slow compressional waves are excited and there is mode conversion at the interfaces. We include this effect in the calculation of reflection and transmission coefficients as an energy‐loss mechanism through the slow compressional waves. Finally, we examine the spectral ratio method for determining porosity and permeability from synthetic seismogram data. Analytical expressions for the velocity and specific attenuation are found based on the weak‐frame approximation. Once the frequency‐dependent velocity and specific attenuation are calculated, the porosity and permeability of marine sediments can be determined by using the proposed weak‐frame approximations. Spectral ratio calculations on synthetic examples show that Biot’s theory incorporated in the VSP method can be used to determine the porosity and permeability of marine sediments.

Linearized inversion for fault rupture behavior: Application to the 1984 Morgan Hill, California, earthquake

We present a technique to infer the rupture history of an earthquake from near‐source records of ground motion. Unlike most previous studies, each point on the fault is assumed to slip only once, when the rupture front passes, with a spatially variable slip intensity. In this parameterization the data are linearly related to slip intensity but nonlinearly related to rupture time. We perform a linearized inversion for slip intensity and rupture time by iteratively perturbing an assumed starting model. The inverse problem for the model perturbation is solved using a tomographic back projection technique. Smoothing and inequality constraints are applied to ensure that the resulting solution is stable and feasible. Asymptotic ray theory is used to calculate theoretical seismograms and partial derivatives with respect to model parameters. In test cases with noisy synthetic data sets we found that it is possible to distinguish between rupture models having variable slip amplitude and models having variable rupture velocity, provided the station coverage is adequate. We apply the technique to recordings of the April 24, 1984, Morgan Hill earthquake. The results indicate that slip amplitude on the fault plane was extremely variable and that the rupture front did not propagate uniformly away from the hypocenter. In particular, rupture was delayed on a 12 km2 section of the fault approximately 14 km to the southeast of the hypocenter. The rupture front surrounded the region, which subsequently failed with a component of rupture propagation back toward the hypocenter. Similar behavior has been observed in dynamic rupture models with stress or strength inhomogeneities. This segment of the fault ruptured with a large slip amplitude, releasing 12% of the total seismic moment from 4% of the total area of the aftershock zone. The surface trace of this section of the fault is characterized by a complex left step that could act to increase the normal stress acting across the fault. However, the distribution of aftershocks suggests that the fault at depth is simpler and that it may bend to the right. In either case, our rupture model suggests that this segment of the fault represents an asperity, which initially resisted rupture but eventually ruptured massively. We estimate the shear fracture energy for this earthquake to be 2×106 J/m2.

Gaussian beam methods and theoretical seismograms

SUMMARY Two Gaussian beam methods for the computations of theoretical seismograms are suggested and tested on model problems. The first is based on using of space-time Gaussian beams instead of stationary ones. The second consists in replacing the Fourier-frequency integral by a sum of residues at the poles of spectral density of a time pulse (some additional analytic restrictions on the spectral density are required). The main advantages of the methods are as follows (1) they enable us to avoid the computations of Fourier-frequency transform; (2) they overcome caustic problems as does the Gaussian beam method in stationary problems. Numerical experiments confirm the efficiency of the methods for the computations of non-stationary elastic wavefields.

Developments toward computations of synthetic seismograms in laterally inhomogeneous anelastic media

Observations of short-period surface waves, Lg waves (Panza and Calcagnile, 1974), indicate that there are large differences between propagation properties in various geographic regions. One example of this is the North Sea, shown in Fig. 1. We have evaluated the possibility of explaining these observations by making computations of theoretical seismograms in various laterally homogeneous models. By summation of as many as 205 Rayleigh-wave modes we have generated synthetic seismograms for those S and P waves that arrive at times later than a specified phase velocity. The procedure, which has been thoroughly described by Panza (1985), works up to frequencies of 10 Hz Marson et al.,

Modeling of marine seismic profiles in the t-x and τ-p domains

In many cases, comparison of real data with synthetic seismograms provides additional constraints on the velocity‐depth profiles obtained with simple inversion techniques. Obtaining a satisfactory match between the real and computed data usually requires several trials with different models but can be performed rapidly if the theoretical seismograms are themselves easily interpretable, i.e., if the major contributions which make up the synthetic traces can be identified and separated. In horizontally stratified media, modeling is further simplified and is faster if the simulation techniques are implemented in the domain of the intercept time τ and the ray parameter p. The generalized reflection and transmission matrix method is well suited for these purposes and may be used to generate synthetic seismograms in both the t-x and τ-p planes. Computation of plane‐wave seismograms is straightforward and merely corresponds to an inverse Fourier transform of the overall reflectivity matrix for each ray parameter. The construction of point‐source seismograms can be carried out in several ways. In this paper, I combine the generalized reflection and transmission matrix method with a discrete wavenumber integration of the reflectivity function, extending previous work to include fluid‐solid interfaces. The introduction of scaling parameters also simplifies the reflectivity matrices. Simple numerical experiments demonstrate that the relations between the earth model and the corresponding seismic response are simpler in the τ-p domain than in the t-x domain. In particular, calculation of the PP, SP, and SS contributions to the complete seismic response shows that shear and converted waves may have a clearer expression in the τ-p plane than in the t-x plane and can in some cases provide discrimination between several earth models. Finally, the main features of a real marine seismic profile are well reproduced by synthetic sections in both the t-x and τ-p domains.

Spontaneous growth and autonomous contraction of a two-dimensional earthquake fault

We numerically study a model of an earthquake as a spontaneous rupture on a fault. We assume that a two-dimensional antiplane shear crack initiates at a point in an infinite, homogeneous and isotropic elastic medium and subsequently propagates with variable velocity under the influence of nonuniform stress drop on the crack and nonuniform cohesive resistance at the crack edges. To begin with, we analyze the dynamical rupture process immediately after nucleation. We determine the subsequent extension of each edge by solving an ordinary differential equation of the first order, as well as the dynamical stress in the regions between each edge and the nearest wave front. Application of this procedure to both edges of the crack in an alternating manner yields the complete history of extension of the crack. We use the critical stress-intensity fracture criterion to determine the conditions on propagation and arrest of the crack. To verify the accuracy and the stability of our numerical technique we compare it with some special cases in which analytical solutions are available. The comparison indicates that the numerical results are in good agreement with the analytical ones obtained previously by Knopoff and Chatterjee (1982) and Chatterjee and Knopoff (1983). We apply this technique to the computation of the dynamical extension of the crack for several representative cases. Among them are: 1. (1) uniform stress drop but nonuniform cohesion (a barrier model), 2. (2) nonuniform stress drop but uniform cohesion (an asperity model) and 3. (3) nonuniform stress drop and nonuniform cohesion (a combination of both the barrier and the asperity models). The numerical results indicate that the heterogeneities of both the stress drop and the cohesion are the main factors which control the growth, cessation and healing of the crack, and that the complexities in the seismic radiation are caused by the complex healing process as well as by the complex rupture propagation. In contrast to the asperity model, the barrier model is characterized by abrupt changes in the slope of the source-time function and the theoretical seismogram. The collision of the stress pulses from each pair of barriers is the potential source of the crack fission. In the general model, the fission process occurs in a complex sequence, and the effects of this process have to be taken into account in the interpretation of observations.

Observational and theoretical constraints on crustal and upper mantle heterogeneity

The effects of crustal and mantle heterogeneity are reflected in the relative complexity of short-period seismograms at regional to teleseismic distances. Observations of events at regional ranges from Indonesia and New Guinea at the Warramunga array in Northern Australia allow a separation of the propagation through the mantle by concentrating on the coherent signal across the medium aperture array. Most of the incoherent features on the seismograms arise from signal generated noise in the general vicinity of the array with scale lengths of the order of 1–2 km or smaller. Partially coherent phases with relatively rapid changes in waveform across the array are to be associated with organised heterogeneity. This is most likely to arise from features in the crust or at the crust-mantle boundary near the receiver, but could in some cases be due to structural effects near the source. The coherent signals show considerable complexity for phases travelling large portions of their path above 80 km. As the depth of sources increases this coherent portion of the seismic field tends to become simpler, which suggests that the scale-length of heterogeneity is larger at depth. The use of horizontally stratified models has provided a convenient framework for the understanding of many seismic wave phenomena via the calculation of synthetic seismograms. However, further development is needed for calculating the response of the laterally heterogeneous Earth. The convenience of physical interpretation provided by working with the reflection and transmission properties of the medium can be retained if the structure can be split into a stratified reference model with superimposed heterogeneity. With a plane-wave decomposition of the wavefield, the effect of lateral variations can be introduced by forcing mixing of wavenumber components. A wide variety of phenomena can be simulated by varying the statistical description of the heterogeneity field in the wavenumber domain. This varies the cross-coupling between wavenumber components and so changes the nature of the resulting theoretical seismograms. Such an approach allows a quantitative study of the degradation of the major seismic phases compared to the predictions for stratified models, and also the characterisation of the codas of these phases.

スラブの下部マントル沈み込みについての最近の地震学的研究

The fate of lithospheric slabs descending along island arcs is an important issue on mantle dynamics. The slab penetration into the lower mantle implies the flow of upper mantle material into the lower mantle. Travel time anomalies observed from deep-focus earthquakes suggest the extension of the high velocity slabs into the lower mantle in the western Pacific subduction zones. The depth extent of the slabs is not definitely determined by using only travel time data, because the path coverage of body waves is bad in the lower mantle below subduction zones. Observations of anomalously broad and complex S waveforms from deep-focus earthquakes are consistent with the lower mantle slab penetration hypothesis. Lower mantle slab models which quantitatively explain the observed waveform broadening have not been derived. Appropriate methods for calculation of theoretical seismograms in strongly heterogeneous media such as subduction zones are required to analyze the observed waveforms and confirm the hypothesis. Deployments of global network of broadband seismographs with wide dynamic range are also essential to reveal the lower mantle slab structure.

Reduction of deep layers in surface wave computation

Summary The Rayleigh wave dispersion function in a plane layered medium has been described by a reduced-delta matrix which is expressed as a sum of five terms. For given values of phase velocity and frequency, it may happen in a layer that one of the five terms become significant and others can be neglected. The topmost layer in which this happens can be considered as a terminating half-space thereby reducing all layers below it. A homogeneous layer becomes a terminating half-space when its shear wave velocity is more than its phase velocity and layer thickness is large compared to wavelength. In other words, such a thick layer is equivalent to a terminating half-space in surface wave computation. This reduction procedure generates dispersion curves of surface waves consisting of crustal waves and coupled channel waves. This layer reduction procedure can be used in computing theoretical seismograms by normal mode summation or by a reflectivity method.

Exact evaluation of the response of a layered elastic medium to an explosive point source using leaking modes

AbstractA simple and rigorous theory of leaking modes for liquid layer or SH elastic wave propagation problems is presented. By taking the frequency ω and wave-number k simultaneously as complex variables and choosing appropriate paths of integration in the ω-plane, the integration with respect to k is performed exactly using Cauchy residue theory. The remaining integration with respect to ω is then carried out by use of the Fast Fourier transform. The method is simply to apply, accurate, and computationally efficient. There are no spurious arrivals, and provided the number of points in the Fast Fourier transform can be taken sufficiently large, there are no restrictions on distance.The theoretical results established in the paper show conclusively that complete seismograms, including all possible body waves, can be expressed simply as a superposition of modes. No branch line integrals are required, contrary to the widespread supposition in the literature.The application of the theory is illustrated by producing complete theoretical seismograms for a model consisting of six liquid layers.

Seismic Wave Propagation In Stratified Media B. L. N. Kennett, Cambridge University Press, 1985 12.50, $24.95

Publisher Summary This chapter focuses on elastic wave propagation in stratified media. The development of the theory of elastic wave propagation in stratified media has been strongly influenced by the problems of seismic wave propagation and the nature of the seismograms recorded from earthquakes. For purely analytic developments of elastic wave propagation, the level of manageable algebraic complexity is reached in a model with one or two uniform layers overlying a uniform half space. This chapter shows how the excitation of elastic waves, within a horizontally stratified structure, can be conveniently developed in terms of reflection and transmission matrices. This procedure has allowed the construction of the full response of the medium or approximations with desired properties so that theoretical seismograms may be calculated for realistic distributions of elastic parameters. Although this development has been for isotropic media, nearly all the results apply directly to the case of full anisotropy if 3 × 3 reflection and transmission matrices allowing coupling between all wave types are employed. This development of the wavefield for both source and receiver within the stratification may be used for other classes of wave propagation.

A Hybrid Collocation Method For Calculating Complete Theoretical Seismograms In Vertically Varying Media

A numerical method is presented for calculating complete theoretical seismograms, under the assumption that the earth models have velocity, density and attenuation profiles which are arbitrary piece-wise continuous functions of depth only. Solutions for the stress-displacement vectors in the medium are expanded in terms of orthogonal cylindrical functions. Our method for solving the resulting two-point boundary value problems differs from that of other investigators in three ways. First, collocation is used in traditionally troublesome situations, e.g. for highly evanescent waves, at turning points, and in regions having large gradient in material properties. Second, in some situations (high frequencies and small gradients) P and S-waves decouple and we use a different solution method for each wave type, instead of trying to force a single method to find all solutions. For example, above the P- and S-waves turning points an approximate fundamental matrix may be used for each wave type. At the P-wave turning point, the fundamental matrix may be used for the S-wave components but collocation is used for the P-wave. Between the P- and S-wave turning points collocation is used for the evanescent P-wave and the fundamental matrix is used for the S-wave. At the S-wave turning point and below, collocation is used for both. Third, the computational algorithm chooses the appropriate solution method and depth domain upon which it is employed based upon a specified error tolerance and the known inaccuracies of the various approximations employed. Once solutions of the boundary value problems are obtained, a Fourier—Bessel transform is then applied to get back into the space-time domain.

Theoretical vertical seismic profiling seismograms

Offset vertical seismic profiling (VSP) theoretical seismograms which include multiples and mode conversions can be computed using a modified “reflectivity” method. In this method, the transformed displacement potentials are first calculated by multiplying the source spectrum by the composite reflectivity function. Integration over wavenumber, followed by inverse Fourier transformation over the frequency range of the signal, yields the synthetic trace. The composite reflectivity function for a buried receiver is derived from Kennett’s matrices (Kennett, 1974, 1979) which are synthesized to form phase‐related reflection and transmission coefficients from a layer stack. Both conventional fixed source‐moving receiver and fixed receiver‐walkaway source (multioffset) VSP geometries can be handled easily. The method can also readily accommodate deviated‐hole VSP. The method is general in that no ray needs to be specified. Because the order of the multiples can be controlled, wraparound problems with the discrete Fourier transform can be avoided. The normal‐incidence VSP seismograms can be rapidly generated as a special case. Several examples illustrate the method. Some classes of laterally varying structures can be approximately handled by restricting the range of ray‐angle integration and by using the principle of superposition.

Theoretical seismograms at the ocean bottom due to underwater explosions.

A new method of calculating theoretical seismograms at the ocean bottom due to underwater explosions has been developed. The formulation is based on a method using numerical integration in the complex wave-number domain proposed by SATO and HIRATA (1980). The ray expansion method for a liquid layer (TANIMOTO and SATO, 1980) is also adopted. A delta matrix, which consists of 2 x 2 subdeterminants of the propagator matrix divided into frequency-dependent and frequency-independent parts in each layer, is introduced to avoid numerical errors. The method is effective for the practical use in studying both high frequency initial phases and low frequency boundary waves at the ocean bottom.By using the theoretical seismograms, characteristic phases expected for an oceanic crustal structure are investigated. P to S and/or S to P converted waves generated at the bottom of the sedimentary layer and two kinds of boundary waves at the liquid-solid boundary, pseudo-Rayleigh waves and Stoneley waves, can be well identified in the theoretical seismograms. The Stoneley waves with a period of 2-15 s show clear dispersion due to the soft sediment overlying the oceanic crust.Though converted waves are often found in recorded ocean bottom seismograms, long period dispersive waves have not been reported as yet. This study suggests a good possibility of obtaining important information on the oceanic crust if long period waves are well observed.

Propagation of complex discontinuities with piecewise constant and variable velocities along curvilinear and branching trajectories

The method of functionally invariant solutions of wave equations utilizing the principle of superposition is used to construct exact solutions for a system of complex discontinuities propagating with piecewise constant velocities along curvilinear and branching trajectories. A passage to the limit is shown in the course of which the solution constructed yields a solution for the case of the propagation of discontinuities with variable velocities along smooth curvilinear trajectories. It is shown that if the propagation of a major discontinuity (crack) begins with the formation of a pure shear element, then as the velocity increases, dislocation components of the displacement vector will appear at the discontinuity; if on the other hand the propagation of a major crack begins with the formation of a pure dislocation element, then as the velocity of propagation of the crack increases, it will branch and considerable shear components of the displacement vector will form on the branched segments of the crack. The minimum values of the branching angles are determined. Theoretical seismograms are given computed for the curvilinear and branching cracks consisting of alternating elements with dislocation and shear components of the displacement vector at the discontinuity.