Total Wave Field Research Articles

Seismic modeling and inversion

The seismic method of petroleum exploration has always been an inverse technique. The explorationist has used data from seismic and other sources to derive an indirect description of a potential reservoir. Key to his success has been the use of forward-modeling methods to aid him in relating what he sees in the seismic data to what he expects to see, based on his hypothesized geologic model These forward-modeling methods have developed from simple ray-tracing procedures that determine travel time, to more sophisticated methods that model the total wavefield. These methods include extensions of ray techniques, along with propagator and finite-difference methods. The formation of subsurface images from the seismic data has traditionally involved the use of inversion principles. The operations of deconvolution and migration are inverse methods that aid in the production of enhanced images of the subsurface. In recent years, the development of more direct methods of inversion has been initiated, based on scattering theory. These methods seek to estimate the acoustic or elastic parameters of the earth for ever more general model assumptions. Good progress has already been made using the assumption that the processed seismic data emulate the response of a horizontally stratified earth to a plane-wave excitation. Research is continuing to generalize the assumptions to include point-source excitation and a general scattering model of the subsurface.

WAVEFRONT DIVERGENCE, MULTIPLES, AND CONVERTED WAVES IN SYNTHETIC SEISMOGRAMS*

AbstractSynthetic seismograms can be very useful in aiding understanding of wave propagation through models of real media, verification of geologic models derived from interpretation of field seismic data, and understanding the nature and complexity of wave phenomena. If meaningful results are to be obtained from synthetic seismograms, the method of their computation must, in general, include three‐dimensional geometrical spreading of wavefronts associated with highly concentrated (i.e., point) sources. The method should also adequately represent the seismic response of solid‐layered media by including enough primaries, multiples, and converted phases to accurately approximate the total wavefield. In addition to these features, it is also very helpful, although not always essential, if the method of seismogram computation provides for explicit identification of wave type and ray path for each arrival. Various seismograms, computed via asymptotic ray theory and an automatic ray generation scheme, are presented for a highly simplified North Sea velocity structure. This is done to illustrate the importance of the above features and to demonstrate the inadequacy of the plane‐wave synthesis method of seismogram computation for point sources and the limitations of acoustic models of solid‐layered media.

Back reflection of ultrasonic waves from a liquid–solid interface

A new acoustic phenomenon has recently been observed in experiments where a bounded beam of ultrasound is incident upon a smooth liquid–solid interface. A significant amount of coherent radiation is found to be backscattered in the general direction of incidence. The angle of back reflection is observed to be equal to the critical Rayleigh angle or leaky wave angle. Most of these observations were made during experiments on the Schoch displacement effect, and therefore it has been tacitly assumed that the back reflection is strongly dependent upon the angle of incidence, as is the case for the beam shifting in the Schoch effect. We present a theoretical basis for this new phenomenon. A two-dimensional incident beam of Gaussian profile is considered. By a careful analysis we isolate that part of the field on the interface which has Fourier components corresponding to backward propagating waves in the liquid. This subset of the total wave field is then considered separately and it is shown to display a maximum in a certain direction, close to the critical Rayleigh angle. This peak in the angular pattern of the scattered field corresponds to an evanescent reflection boundary. We discuss the dependence of the effect upon certain parameters. The amplitude is shown to decrease as the beam width is increased, and it increases with increasing Schoch displacement. This backscattering is present for all angles of incidence; there is nothing inherently special about the Rayleigh angle.

A mild‐slope wave equation and its application to tsunami calculations

A wave equation correct to first order in wave amplitude and bottom slope is used to calculate the wave field around an island. This is of circular cylindrical shape, and is situated on a paraboloidal shoal in an ocean of constant depth (Figure 1). The sides of the island are assumed fully reflecting. The incident waves are plane, periodic, and of small amplitude. Periods up to 30 min are investigated, and the Coriolis force is neglected. The wave equation is solved analytically, and a great number of numerical computations are carried through. The total wave field over the shoal is presented for two discrete periods in the upper end of the tsunami frequency range. The amplitudes at the middle of the front face of the island, and at the middle of the lee face, are given as functions of the wave period, and the existence of “resonance”; periods is thus demonstrated. Comparison with solutions to the linearized long‐wave equation is made, and the validity range of the shallow water theory is estimated. The g...

Propagation of waves along an impedance boundary

A theoretical analysis of the scalar wave field due to a point source above a plane impedance boundary is presented. A surface wave is found to be an essential component of the total wave field. It is shown that, as a result of ducting of energy by the surface wave, the amplitude of the total wave near the boundary can be greater than it would be if the boundary were perfectly reflecting. Asymptotic results, valid near the boundary, are obtained both for the case of finite impedance (the soft-boundary case) and for the limiting case in which the impedance becomes infinite (the hard-boundary case). In the latter, the wave amplitude in the farfield decreases essentially as r−1, where r is the horizontal propagation distance; whereas in the former (if the surface-wave term is neglected) it decreases as r−2. The results obtained here are compared with results of experimental studies of sound propagation along boundaries, and it is suggested that some apparently anomalous experimental findings might be explained in terms of the present theory.

Partial Ray Expansion Required to Suitably Approximate the Exact Wave Solution

Summary Whenever synthetic seismograms are computed for layered media with the aid of ray theories only a partial ray expansion is possible. We present criteria that may be incorporated in a computer algorithm to obtain an automatic selection of rays which will minimize the error in the partial ray expansion below a preselected level. The program, results of which are presented in this paper, can also be used for the study of multiple reflected and converted phases or for media with dipping interfaces. 1. Introduction Various ray theories which decompose the total wave field in a layered medium into contributions attributed to individual rays have been used to advantage in the generation and analysis of synthetic seismograms. Among these are generalized ray theory (Spencer 1960) in which the displacement field is decomposed into contributions from an infinite set of rays, each ray being evaluated by a numerical solution of the impulse response (Bortfield 1967; Muller 1968, 1970) or by the Cagniard-de Hoop method (Helmberger 1968; Gilbert & Helmberger 1972). The solution for each ray is exact but the computations are very complex so that only a limited number of rays are used in the solution. An alternative method is asymptotic ray theory (Hron & Kanasewich 1971; Hron 1972) in which the field quantities are expressed as an infinite power series of reciprocal frequency combined with a spacedependent vector which is independent of frequency. The displacement field also becomes decomposed into an infinite set of rays but a greater number may be taken into the approximation by limiting the power series of each to the first non-zero term in the expansion. Although different techniques are used to evaluate the ray amplitudes, the final result is always a synthetic seismogram in which the kinematic and dynamic characteristics of particular rays may be studied. Ray theories are limited invariably by computational difficulties associated with the selection of the phases in the partial ray expansion. Very often the selection is based on the personal intuition or experience of the seismologist. I€ the selection is poor then the seismograms are misleading for rays with significant amplitudes will be omitted. An initial attempt at producing a computer algorithm that would generate systematically a set of rays, determine their kinematic and dynamic analogues and test for significant amplitude levels was reported by Hron & Kanasewich (1971). The algorithm for ray generation proved to be very efficient so that seismograms based on the evaluation of more than 150 000 individual rays were not exceptional (Hron 1972). However, no estimate of the accuracy of this partial ray expansion was possible because the convergence of the complete expansion had not been established. This

Multiple Scattering of Thermal Neutrons by a Perfect Crystal. I. General Foundations

AbstractThe wave function of a neutron in a regular crystal is built up as a superposition of a primary plane wave and outgoing spherical waves. The elastic scattering properties of the nuclei are given exactly by their complex scattering amplitude f. It is carefully distinguish‐ed between the total wave field and the arriving field at any scattering centre. This results in a correct and consistent description of absorption effects. In the case of an infinite crystal slab the construction of the wave field is simplified by a lattice plane method: all the spherical waves coming out from such an infinite lattice plane add to plane waves, most of them are strongly damped with increasing distance from the plane. The undamped waves give the far‐field, which contains all the interesting informations about the collective interference effects. Neglect of the damped waves gives a good starting point for the approach given in the following two parts of this paper.

Back Reflected Intensity for the Laue Case

The exact formulation of the dispersion equation in the dynamical electron diffraction theory of Bethe leads to the existence of four wave fields in the crystal in the two-beam Laue case. At low energies it is necessary to consider all four wave fields within the crystal and the resulting interferences between them. Computer calculations for the vacuum electron currents in the exact two-beam case of low energy electron diffraction for a thin parallel slab are shown. The current variation due to interferences between all the crystal wave fields for the elastic case and for the case where absorption is introduced by means of a complex Fourier coefficient of the potential are demonstrated. The importance of the structure due to the additional solutions for primary electron energies below 50 eV shows that any elastic theory of electron diffraction must contain the total wave field to be valid at these low energies.

Absorption of Sound by Patches of Absorbent Materials

Circular patches, or long strips, of sound absorbent material are on the surface of an infinite perfectly reflecting plane. Exact solutions are found for the absorption of perpendicularly incident plane waves, and of randomly incident waves, on a circular piston-like absorber. An exact solution is found for the absorption of perpendicularly incident plane waves on an infinitely long strip-piston absorber. The basis of the technique is representation of the total wave field by superposition of an incident plane wave, a specularly reflected plane wave, and a scattered wave. The last-named is in turn represented by a superposition of plane waves for the strip piston, and by a superposition of cylindrical waves for the circular piston absorber. The superposed wave fields are adjusted to meet the boundary conditions.