Solution of the H-polarization induction problem by a wavenumber integral equation approach

Summary. The H -polarization induction problem is solved in terms of an integral equation, which in the horizontal direction is transformed into the wavenumber domain. By this transformation the usual complicated integral expressions for the Green's tensor elements are removed. By extracting asymptotic features from the system of linear equations, we reduce the number of equations considerably independent of whether the horizontal variation in the conductivity is continuous or discontinuous. Likewise we reformulate the problem so that arbitrary conductivity contrasts may be studied. The method is finally tested by comparing with analytic solutions, and good agreement is achieved. Furthermore the numerical results indicate that a small amount of wavenumbers is required.     

Scattering of surface waves modelled by the integral equation method

The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.     

Inversion of reflection data for layered media: a review of exact methods

Summary. This paper is a summary of known exact methods for inferring the acoustical impedance of a horizontally layered medium from its reflection response to a vertical sound signal. It includes reviews of linear and nonlinear methods in the time and frequency domains.     

The fine structure of the normal incidence synthetic seismogram

Summary . The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient c _o, where | c _o|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, c_o =±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well.
If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.     

Waveform inversion of mantle Love waves:the Born seismogram approach

Summary. Normal mode theory, extended to the slightly laterally heterogeneous earth by the first-order Born approximation, is applied to the waveform inversion of mantle Love wave (200–500 s) for the Earth's lateral heterogeneity at l = 2 and a spherically symmétric anelasticity ( Q _μ) structure. The data are from the Global Digital Seismograph Network (GDSN). The l =2 pattern is very similar to the results of other studies that used either different méthods, such as phase velocity measurements and multiplet location measurements, or a different data set, such as mantle Rayleigh waves from different instruments. The results are carefully analysed for variance reduction and are most naturally explained by heterogeneity in the upper 420 km. Because of the poor resolution of the data set for the deep interior, however, a fairly large heterogeneity in the transition zones, of the order of up to 3.5 per cent in shear wave velocity, is allowed. It is noteworthy that Love waves of this period range cannot constrain the structure below 420 km and thus any model presented by similar studies below this depth are likely to be constrained by Rayleigh waves (spheroidal modes) only.
The calculated modal Q values for the obtained Q _μ model fall within the error bars of the observations. The result demonstrates the discrepancy of Rayleigh wave Q and Love wave Q and indicates that care must be taken when both Rayleigh and Love wave data, including amplitude information, are inverted simultaneously.
Anomalous amplitude inversions of G2 and G3, for example, are observed for some source-receiver pairs. This is due to multipathing effects. One example near the epicentral region, which is modelled by the obtained l = 2 heterogeneity, is shown.     

Inversion of a refraction wavefield by imaging in the p-x and v-z planes

Summary. The slowness-distance ( p, x ) plane is an alternative to the slowness-time intercept ( p, τ ) plane as the intermediate image space in inversion of seismic refraction data. The production of a ( p, x ) image from travel time-distance ( T, x ) data has been presented elsewhere so emphasis here is on ( p, x ) to velocity-depth ( v, z ) transformation. Iterative downward continuation of a ( p, x ) image converges to the correct ( v, z ) image in a manner similar to that in the widely used ( p, τ ) to ( v, z ) process. Application to a real refraction data set from the Imperial Valley of southern California gives a similar ( v, z ) solution via both ( p, x ) and ( p, τ ) images.     

Inversion of the travel-time curve of a reflected wave

Summary. A general theory of inversion of the reflected travel-time curve is developed. The properties of the travel-time curve and its analytical continuation are examined. The inversion formulae are presented and the structure of the set of velocity—depth functions is studied in detail. The inverse problem or a finite number of reflectors and the problem of separation of the reflected and multiple travel-time curves are both considered.     

Recognizing explosion sites without seismogram readings: neural network analysis of envelope-transformed multistation SP recordings 3–6 Hz

Seismic-waveform similarities for closely spaced earthquakes and explosions in particular are well established observationally. In many industrialized countries of low seismicity more than 90 per cent of seismic event recordings stem from chemical explosions and thus contribute significantly to the daily analyst workload. In this study we explore the possibility of using envelope waveforms from a priori known explosion sites (learning) for recognizing subsequent explosions from the same site excluding any analyst interference. To ensure high signal correlation while retaining good SNRs we used envelope-transformed waveforms, including both the P and Lg arrivals. To ensure good spatial resolution we used multistation (network) recordings. The interpolation and approximation neural network (IANN) of Winston (1993) was used for teaching the computer to recognize new explosion recordings from a specific site using detector output event files of waveforms only. The IANN output is a single number between 0 and 1, and on this scale an acceptance threshold of 0.4 proved appropriate. We obtained 100 per cent correct decisions between two sets of 'site explosions' and hundreds of 'non-site' explosions/earthquakes using data files from the Norwegian Seismograph Network.     

The generalization of the Mie—Grüneisen equation of state

Summary. A generalized Mie—Grüneisen equation of state is found without thermodynamical limits of validity. The range of applicability of the classical Mie—Grüneisen equation is discussed and results are presented of a particular case of the present equation. The various definitions of the Grüneisen's parameter are analysed in comparison with their generalized theoretical expression, given here.     

Kirchhoff–Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium – II. Computations

Summary. High-frequency reflection and refraction seismograms for laterally variable multi-layered elastic media are computed by using the frequency domain elastic Kirchhoff–Helmholtz (KH) theory of Frazer and Sen. Both source and receiver wavefields are expanded in series of generalized rays and then elastic (KH) theory is applied to determine the coupling between each source ray and each receiver ray at each interface. The motion at the receiver is given as a series of integrals, one for each generalized ray. We use geometrical optics and plane wave reflection and transmission coefficients for rapid evaluation of the integrand. When the source or the receiver ray field has caustics on the surface of integration geometrical ray theory breaks down and this gives rise to singularities in the KH integrand. We repair this using methods suggested by Frazer and Sen.
Examples of reflection seismograms for 2-D structures computed by elastic KH theory are shown. Those for a vertical fault scarp structure are compared with the seismograms obtained by physical modelling. Then OBS data obtained from the mid-America trench offshore Guatemala area are analysed by computing KH synthetics for a velocity model that has been proposed for that area. Our analysis indicates the existence of a small low-velocity zone off the trench axis.
No head wave arrivals are obtained in our KH synthetics since we do not consider multiple interactions of a ray with an interface. The nearly discontinuous behaviour of elastic R/T coefficients near the critical angle causes small spurious phases which arrive later than the correct arrivals.