A method for the computation of finite frequency body wave synthetic seismograms in laterally varying media

Summary. Body wave synthetic siesmograms for laterally varying media are computed by means of a slowness implementation of the extended WKBJ (EWKBJ) theory of Frazer & Phinney. An EWKBJ seismogram is computed by first tracing rays through a particular model to obtain conventional ray information (travel time, ray end point, ray slowness) and then using these data in the finite frequency integral expression for the EWKBJ seismogram. The EWKBJ seismograms compare favourably to geometrical ray theory (GRT) seismograms but are significantly better because of the finite frequency nature of the EWKBJ calculation. More realistic behaviour is obtained with EWKBJ seismograms at normal seismic frequencies near caustics, where the GRT amplitude is infinite, and within geometrical shadow zones where GRT predicts zero amplitudes. In addition the EWKBJ calculation is more sensitive than GRT to focuses and defocuses in the ray field. The major disadvantage of the EWKBJ calculation is the additional computer time over that of GRT, necessary to calculate one seismogram although an EWKBJ seismogram costs much less to compute than a reflectivity seismogram. Another disadvantage of EWKBJ theory is the generation of spurious, non-geometrical phases that are associated with rapidly varying lateral inhomogeneities. Fortunately the amplitudes of these spurious phases are usually much lower than that of neighbouring geometrical phases so that the spurious phases can usually be ignored. When this observation is combined with the moderately increased computational time of the EWKBJ calculation then the gain in finite frequency character significantly outweighs any disadvantages.     

Computation of high-frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media

Summary. An algorithm for the computation of travel times, ray amplitudes and ray synthetic seismograms in 3-D laterally inhomogeneous media composed of isotropic and anisotropic layers is described. All 21 independent elastic parameters may vary within the anisotropic layers. Rays and travel times are evaluated by numerical solution of the ray tracing equations. Ray amplitudes are determined by evaluating reflection/ transmission coefficients and the geometrical spreading along individual rays. The geometrical spreading is computed approximately by numerical measurement of the cross-sectional area of the ray tube formed by three neighbouring rays. A similar approximate procedure is used for the determination of the coefficients of the paraxial ray approximation. The ray paraxial approximation makes computation of synthetic seismograms on the surface of the model very efficient. Examples of ray synthetic seismograms computed with a program package based on the described algorithm are presented.     

A method for calculating synthetic seismograms in laterally varying media

Summary An effective algorithm for computing synthetic seismograms in laterally inhomogeneous media has been developed. The method, based on zero-order asymptotic ray theory, is primarily intended for use in refraction and reflection studies and provides an economical means of seismic modelling.
A given smoothed velocity-depth-distance model is divided into small squares with constant seismic parameters and first-order interfaces are represented by an arbitrary number of dipping linear segments. The computation of ray propagation and amplitudes through such a model does not involve complicated analytic expressions and therefore minimizes computer time.
Amplitudes are determined by geometrical spreading of spherical wave-fronts and energy partitioning at interfaces. Synthetic seismograms calculated for laterally homogeneous models are in good agreement with those obtained by the Reflectivity Method.     

The accuracy of models derived by WKBJ waveform inversion

Summary . In this paper the accuracy of velocity-depth profiles derived by matching WKBJ seismograms to observations is quantitatively evaluated. Seismograms computed with the WKBJ method are generally quite reliable but possess predictable, systematic inaccuracies in the presence of strong velocity gradients. The effects of these inaccuracies on models derived through WKBJ waveform inversion are studied, using reflectivity seismograms as 'data'. The velocity structure used is an oceanic lithosphere model that contains several transition regions separated by relatively homogeneous layers, producing partially-reflected reverberations in the reflectivity synthetics that are absent from the WKBJ seismograms. The inversion incorporates the 'jumping' strategy to solve for the smoothest models consistent with the data. We find these solutions to be independent of the starting model and to have a stable basic structure that agrees well with the correct model. The differences, everywhere less than a seismic wavelength, depend on the frequency content of the seismograms. Reverberations in the reflectivity seismograms that are well separated from WKBJ arrivals are treated as 'noise' in the inversion.     

Limitations of Waveform Modelling of Long-Period Seismograms

In recent years the use of synthetic seismograms calculated for radially stratified models has gained increasing popularity as a means of placing further constraints on the velocity structure of the Earth's mantle. Such synthetics do. however, have a number of limitations. At short periods (∼1 s) the amplitudes as well as the wave shapes of travel-time branches are affected by seismograph siting, the structure immediately beneath the seismograph and any laterally heterogeneous structure in the mantle. Later arrivals can also be masked by signal-generated noise and by extended source times. At longer periods (∼-20 s) the larger wavelengths reduce the sensitivity of amplitudes and waveforms to contaminating effects. As a result the use of long-period synthetics can only lead to the resolution of the gross features of the Earth's interior.     

Ray-reflectivity method for SH-waves in stacks of thin and thick layers

Summary Reflectivity and ray theories are united to produce a hybrid technique of computing synthetic seismograms for a plane layered medium in subcritical regions. Numerical experiments have indicated that this technique is useful when the depth structure is one composed of thick layers separated by finely layered zones. As the theory for wave propagation in a plane layered medium is well known, the simple SH case is investigated so that the basic idea of the method may be conveyed without an excess of mathematics that would be necessitated if the P-SV problem were considered.
In computing the ray-reflectivity seismogram, the thick layers are treated using asymptotic ray theory while the thin-layered zones are treated as quasiinterfaces where analogues of reflection and transmission coefficients called reflectivities and transmittivities are calculated utilizing a Thomson-Haskell formulation. A stationary phase approximation is employed when evaluating the integral which gives the displacement due to an arbitrary ray propagating in the thick layers of the above-mentioned medium, and the validity of this approximation is discussed.
A comparison of ray, numerical integration (reflectivity) and ray-reflectivity synthetic sections indicates that this method yields quite acceptable results for subcritical reflection work and is suitable for application in seismic interpretation as individual arrivals associated with ray-paths in the thick layers may be identified. Furthermore, the method is quite cost efficient and may be extended to a medium where the thick layers are non-planar using asymptotic ray theory in these layers.     

A global study of S-to-P and P-to-S conversions from the upper mantle transition zone

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Long-period data of the Global Digital Seismograph Network (GDSN) recorded over the three-year period from 1984 to 1986 were studied for the occurrence of S-P and P-S conversions from the upper mantle transition zone that appear as precursors to teleseismic S arrivals. Conversions of this type were identified on a large number of single-station records. Simple stacking of many records enhanced the appearance of converted phases and demonstrated that no major lateral variations in the nature of the transition zone exist between various tectonic regions. S-P and P-S conversions from the 400 km discontinuity were best observed at distances between 70 and 85 while conversions from the 670 km discontinuity showed up best at distances beyond 87. The analysis of published source mechanisms and comparison with synthetic seismograms suggests that the appearance of converted phases is primarily governed by the earthquake radiation pattern. Phases that have undergone S-P conversions beneath the receiver are best observed from dip-slip events that radiate strong SV - and weak P -waves towards the station. P-S conversions beneath the source area, on the other hand, are frequently observed from events that radiate strong P and little SV energy towards the station, and also from some strike-slip events. Comparison of observed with synthetic seismograms suggests that the PREM model of Dziewonski & Anderson (1981) explains most of the observations. Observed S-P and P-S conversions from the 670 km discontinuity, however, often have larger amplitudes than in the synthetics. Constructive interference of converted waves with the P -wave coda, source radiation effects and a velocity contrast across the 670 km discontinuity which is higher than in PREM may all contribute to the discrepancy.     

The theory of finite frequency body wave synthetic seismograms in inhomogeneous elastic media

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A new method is presented by means of which one can compute finite frequency synthetic seismograms for media whose velocity and density are continuous functions of two or three spatial variables. Basically, the method is a generalization of the familiar phase integral method, to which it reduces in a stratified medium. For a given source location the travel-time and distance functions needed to compute synthetics are obtained by numerically tracing rays through the model. This information is then used to evaluate a double integral over frequency and take-off angle at the source. The solution obtained reduces to the geometrical optics solution wherever that is valid but it also works in shadows and at caustics without knowing explicitly where these may be located. The method can be used as a spectral method, in which the integral over take-off angle is evaluated first, or as a slowness method, in which the frequency integral is evaluated first.     

Wave propagation in transversely isotropic and periodically layered isotropic media

Summary. A set of stable algorithms for computing synthetic seismograms in attenuating transversely isotropic media is presented. The structures of these algorithms for anisotropic media are formally equivalent to their counterparts for isotropic media. The seismic responses of a periodically layered isotropic medium are compared with those of its long-wave equivalent transversely isotropic medium. The synthetics for the two media show observable differences in the range of frequencies considered. The differences are small in the P -waves, but partly large in later arrivals.     

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.     

Chaotic ray behaviour in regional seismology

Rays propagating through strongly laterally varying media exhibit chaotic behaviour. This means that initially close rays diverge exponentially, rather than according to a power law. This chaotic behaviour is especially pronounced if the medium contains laterally varying interfaces. By studying simple 2-D and 3-D versions of models with laterally varying interfaces, the importance of chaotic ray behaviour is determined. A model of the Moho below Germany produces sharp variations with epicentral distance of the number of arrivals. In addition, the number of caustics grows dramatically: up to 1200 caustics are present between a distance of 0 and 800 km. Using the theory of Hamiltonian systems, a more in-depth study of the chaotic character of the ray equations is obtained. It is found that for realistic heterogeneous models most of the relevant rays will exhibit chaotic behaviour. The degree of chaos is quantified in terms of predictability horizons. Beyond the predictability horizons ray tracing cannot be carried out accurately. For the models under consideration, the length from the source to the predictability horizon has an order of magnitude of 1000 km. The chaotic behaviour of the rays makes it necessary to use extensions of asymptotic ray theory, such as Maslov theory, to compute seismic waveforms. It is shown that pseudo-caustics, an important obstacle in computing Maslov synthetics, are a generic feature of the 2-D laterally varying models that are studied. Eventually, the use of asymptotic methods is restricted because of the inaccuracy in the computation of the ray paths.     

Earthquake seismograms that show Doppler effects due to crack propagation

Summary. Examples are presented of earthquake P -wave pulses seen on broadband seismograms, to show that on such recordings the pulse shapes are more clearly seen than on conventional short-period and long-period seismograms. Most of the broadband seismograms have been chosen because they show marked differences between the pulse lengths of P and those of the surface reflections. In addition some of the pulses appear to have smooth onsets and abrupt trailing edges so that the onset of the pulse is difficult to observe and the largest amplitude arrivals seen on the seismogram coincide not with the onset of motion but with the termination of motion: that is the large arrivals mark stopping phases of motion.
We assume that the differences in pulse length are due to the effects of a moving source – that is a Doppler effect – and that the pulses with smooth onsets and abrupt trailing edges can be modelled simply by a source propagating on a line with low radiation amplitude at the start of motion. A trial and error method guided by a published fault plane solution is then used to obtain a fit between observed and computed seismograms for one of the earthquakes. This process leads to an estimate of the crack speed of about 1.4 times the 5-wave speed.
The errors that may arise in estimating source depths and orientation, if stopping phases are not recognized as such, is discussed.     

Three-dimensional geometrical ray theory and modelling of transmitted seismic energy of data from the Nevada Test Site

This paper presents a geometrically based algorithm for computing synthetic seismograms for energy transmitted through a 3-D velocity distribution. 3-D ray tracing is performed to compute the traveltimes and geometrical spreading (amplitude). The formulations of both kinematic and dynamic ray-tracing systems are presented. The two-point ray-tracing problem is solved by systematically updating the initial conditions and adjusting the ray direction until the ray intersects the specified endpoint. The amount of adjustment required depends on the derivatives of the position with respect to the given starting angles between consecutive rays. The algorithm uses derivatives to define the steepest-descent direction and to update the initial directions. The convergence rate depends on the complexity of the model.
Test seismograms compare favourably with those from a 2-D asymptotic ray theory algorithm and a 3-D Gaussian-beam algorithm. The algorithm is flexible in modelling arbitrary source and recorder geometries for various smoothly varying 3-D velocity distributions. The algorithm is further tested by simulating surface-to-tunnel vibroseis field data. Shear waves as well as compressional waves may be approximately included. Application of the algorithm to a data set from the Rainier Mesa of the Nevada Test Site produced a good fit to the transmitted (first arrival) traveltimes and amplitudes, with approximately 15 per cent variation in the local 3-D velocity.     

Synthetic seismograms for the case of the receiver within the reflectivity zone

Synthetic seismograms are shown and discussed for the case of the receiver within the medium. Most of the discussion is on the reflectivity method with the receiver within the reflectivity zone, but results using the ray method are shown for comparison. Such synthetic seismograms can be used to interpret data from Oblique Seismic Experiments where shots generated on the surface up to large ranges are recorded in crustal boreholes.     

Dynamic ray tracing on Lagrangian manifolds

Summary. We show that Maslov's extension of the WKBJ method allows an extension of the dynamic ray tracing to wavefields involving caustics of arbitrary form. If the receiver lies off the caustics, then the synthetic seismogram can be obtained by integrating the DRT system along a single ray joining the receiver to the source which may touch caustics. If the receiver-lies in the vicinity of a caustic then DRT has to be carried out along a bunch of rays covering a neighbourhood of the receiver. Our approach encompasses pre-stressed and/or anisotropic media. Initial boundary conditions for a point source embedded in an anisotropic elastic medium are also presented.     

Validity of the great circular average approximation for inversion of normal mode measurements

Summary. Synthetic seismograms based upon first-order perturbation theory are analysed to test the validity of assumptions which form the basis of current velocity inversion procedures. It is found that the lowest order geometrical optics approximation, namely that measured normal mode eigen-frequencies reflect the average structure underlying the source–receiver great circle path, becomes less valid near nodes in the source radiation pattern and near the surface wave foci at the source and its antipode. These failures are a consequence of singlet interference within an isolated normal mode multiplet. The technique of determing frequency by fitting a single resonance peak to a multiplet yields results which agree well with the first-order theory for slow and fast paths where excitation is dominated by one pair of singlets but on intermediate paths where singlet interference is more of a problem, agreement is not as good. Inversion of small data sets is particularly sensitive to frequency fluctuations near radiation nodes, while larger sets are influenced more by antipodal deviations from geometrical optics. The latter leads to inversions which fail to recover the short wavelength structure of the starting model. Basing inversions directly upon first-order theory shows promise of improving recovery of short wavelengths.     

Efficient calculation of Gaussian-beam seismograms for two-dimensional inhomogeneous media

Summary. This paper discusses several aspects of the calculation of theoretical seismograms for two-dimensional inhomogeneous media with the method of Gaussian beams. The most important steps of this method, kinematic and dynamic ray tracing, can be performed very efficiently, if the model cross-section is subdivided into triangles with linear velocity laws. Each Gaussian beam is characterized by a complex beam constant ε which determines its width and phase-front curvature. Various possibilities to choose ε are discussed, including cases where beam properties at the beam endpoint (and not at the beginning) are prescribed; for instance, the beam width at the endpoint can be specified. In such cases the beam constant is a function of the radiation angle at the source, and the decomposition of a cylindrical wave into beams has to take this into account by weighting the beams differently, at least in principle. The exact weight function is derived and shown to be reasonably well approximated by the weight function, corresponding to angle-independent ε Theoretical seismograms are presented for a laterally heterogeneous model of the crust–mantle transition which is characterized by complications in the reflection from the transition and in the refraction from below. These complications are modelled by and large with success. The seismograms, however, depend to a certain extent on the choice of the beam constant. Moreover, according to the reciprocity principle calculations with source and receiver interchanged should have the same results as calculations for the original configuration. In practice this is not so, and the difference increases with the strength of lateral heterogeneities. Hence, for a successful application of Gaussian beams the model should not vary too strongly in lateral direction.     

Time-domain computation of scattering from a heterogeneous layer

Adopting Born and ray approximations, time-domain synthetic seismograms for P-P and P-S scattering from a plane wave incident on a thin, laterally heterogeneous layer are presented in this paper. The time-domain P coda is a convolution between a structure function and the second-order derivative of the time function of the incident P wave. Examples of synthetic seismograms are given using a time function from a computed short-period seismogram for a point explosive source in a half-space. These show that it is impossible, with realistic values of the parameters involved, to generate significant codas when only single scattering is involved.     

Fault plane solutions using relative amplitudes of P and pP

Summary. One way of finding the fault plane orientations of small shallow earthquakes is by the generation of theoretical P -wave seismograms to match those observed at several distant stations. Here, a technique for determining the uniqueness of fault plane solutions computed using the modelling method of Douglas et al . is described. Relative amplitudes of P and pP , and their polarities if unambiguous, are measured on the observed seismograms to be modelled, and appropriate confidence limits are assigned to each measurement. A systematic search is then made for all fault plane orientations which satisfy these observations.
Examples show that if P and pP are not severely contaminated by other arrivals, a well-defined and unique fault plane orientation can often be computed using as few as three stations well distributed in azimuth. Further, even if pP is not identifiable on a particular seismogram, then an upper bound on its amplitude – deduced from the observed coda – still places a significantly greater constraint on the fault plane orientation than would be provided by a P onset polarity alone. Modelling takes account of all such information, and is able to further eliminate incompatible solutions (e.g. by the correct simulation of sP ). It follows that if solutions can be found which satisfy many observed seismograms, this places high significance on the validity of the assumed double-couple source mechanism.
This relative amplitude technique is contrasted with the familiar first motion method of fault plane determination which requires many polarity readings, whose reliabilities are difficult to quantify. It is also shown that fault plane orientations can be determined for earthquakes below the magnitude at which first motion solutions become unreliable or impossible.     

Investigation of 3D wavefields of reflected shear-waves and converted waves: mathematical modelling and reflection data processing

Summary. Seismic investigations using shear-wave and converted wave techniques show that very often reflected PS - and SS -waves have anomalous polarizations ( accessory components ). This phenomenon cannot be explained in terms of isotropic models with dipping boundaries. Computations of synthetic seismograms of reflected PS - and SS -waves were made for different models of transversely isotropic media with dipping anisotropic symmetry axes not normal to the boundaries. Synthetic seismograms were computed by ray techniques using an optimization algorithm to construct all rays arriving at a given receiver. These computations indicate that accessory components arise when the medium above the boundary is anisotropic, where they are caused by the constructive interference of qSV - and qSH -waves. If a low-velocity layer is present, displacement vectors of both waves have horizontal projections which are approximately orthogonal. The algorithm for wave separation is presented and some results of its use are given.