Abstract We derive a rigorous analytic description of multiple reflection/transmission phenomena of 3D scalar harmonic wave fields in two inhomogeneous half spaces separated by a smooth curvilinear interface of arbitrary shape. The resulting wave field is expressed in terms of a branching sequence of events connected by recurrence relationships. An individual event is a result of the perturbation of boundary conditions by the previous event of the sequence that, in turn, again perturbs the boundary conditions. Such an event is described by the surface Helmholtz integral of any fundamental solution satisfying conditions at infinity. Functions of the current point of the interface and their normal derivatives (Cauchy data) are described by the surface integrals with the fundamental solutions for an absolutely absorbing interface. The Cauchy data of the latter integrals are obtained from the subsequent transformations. The perturbation of the interface and its normal derivatives are mapped into the spatial frequency domain by means of the Fourier transform where they are multiplied by the corresponding reflection/transmission coefficients and then is mapped back to an individual point on the interface by means of the inverse Fourier transform. Parameters of these transformations depend on the point where the result is computed. Therefore the computation at a new point of the interface requires the repetition of these transformations.

Abstract We present an improved method for modeling 3D acoustic wavefields scattered at smooth curved interfaces. The approach is based on a high-frequency approximation of surface integral propagators and a correct description of their boundary values in terms of transmission operators. The main improvement is a uniform local approximation of these operators in the form of effective reflection and transmission coefficients. We show that the effective coefficients represent a generalization of the plane-wave coefficients widely used in conventional seismic modeling, even for the case of curved reflectors, nonplanar wave fronts, and finite frequencies. The proposed method is capable of producing complex wave phenomenas, such as caustics, edge diffractions, and head waves. Seismograms modeled for even simple models reveal significant errors implicit in the plane-wave approximation. Comparison of modeling based on effective coefficients with the analytic solution reveals errors less than 4% in peak amplitude at seismic frequencies.

Abstract Propagating waves represent an aspect of more general forms of motion that can be described by the exact equation of continuum mechanics. Although a study of general forms of motion is practically possible only by means of numerical methods, their wave aspects can be expressed in terms of an elementary theory. The basic concepts of such a theory were established as a generalization of experimental facts long before mechanics itself appeared as a branch of mathematical physics. Only much later were they derived as consequences of the exact equations of mechanics. This system of concepts has an important property — it allows us to build a simple wave-propagation theory common to waves of any kind. That is why the basic concepts of wave-propagation theory can be introduced by considering the simplest equation of motion and then generalizing to more complex situations. Here we consider these concepts in the case of the scalar wave equation where f * and F * are functions of space coordinates ( x 1 , x 2 , x 3 ) and time t and c is a function of space coordinates only.

Abstract A line formed by points of discontinuity of the interface or its first tangential derivatives is called an edge. A point of the edge is considered to be regular if the corresponding line is continuous along with its first tangential derivative. The edge is considered smooth if its points are all regular. In the near vicinity of any point of a smooth edge, we can neglect its curvature and approximate the interface by two half planes touching at the edge (Figure 1 ). Then reflected/transmitted wavefields must satisfy the boundary conditions at both half planes. By using cylindrical coordinates ( r , θ , z ) where axis z coincides with the edge, we can write the boundary conditions as where θ ℓ is the coordinate of the ℓ th half plane. Then the reflection/transmission problem reduces to the integration of equations 139 of Chapter 1 under conditions 1 above. Also rewrite equation 136 of Chapter 1 in cylindrical coordinates. Let points x 1 = x 2 = x 3 = 0 and r = z = 0 coincide, axes x 1 and z coincide, and planes x 2 = 0 and θ = 0 coincide. Then and the wave-vector components 137 of Chapter 1 can be written as where k rm is the projection of k m on the plane z = constant and α m is the angular cylindrical coordinate of this projection (Figure 2). By substituting these equations into 136 of Chapter 1 , we obtain

Abstract The descriptions of edge diffraction that one can use to solve practical problems depend on the character of the problem considered. In some cases, consideration of an individual edge wave can be of practical importance. It is often sufficient in such situations to describe the edge wave in the framework of the geometric theory of diffraction. Sometimes it is necessary to use a more complicated description, which involves combining the formulas of the geometric theory of diffraction and the boundary-layer approximation. However, of greatest practical importance is the case in which edge waves can be regarded as factors interfering with regular reflections/transmissions representing basic geophysical information. In such situations, it is possible to use the simplest description of edge-diffraction phenomena. All the following sections deal only with that kind of situation. Let us begin with general considerations on the representation of wavefields in media consisting of regions and interfaces. Description of a stationary wavefield in such media is based on separation of the wavefield into individual waves caused by the consecutive reflection/transmission phenomena at the interfaces. It can be written in the form of the superposition of individual waves: where m is the index of the individual wave f m , Φ m is its amplitude, τ m is its eikonal, c m is the propagation velocity, and ω is the frequency of oscillations. If there are diffracting edges at the interfaces, this description is not sufficient because of shadow zones in the individual

Abstract Interfaces in 3D block media are represented by the surfaces of curvilinear polyhedrons. Such surfaces include vertices where several diffracting edges can converge. This means that edges are not smooth in 3D block media. Each smooth part of the edge has two terminal points called tips. A joint of two tips is called a breakpoint of the edge. A joint of two or more tips is a vertex. The existence of such points puts some limitations on the previous discussion of edge diffraction. The essence of these limitations becomes clear from the following example. Figure 1a shows the geometry of the shadow boundary of a reflected/transmitted wave in the case of a broken edge. The edge consists of two semi-infinite parts, RA and RB . Point R is a breakpoint because the tangent to the edge is not single valued. The shadow boundary consists of two smooth parts, RAT and RBT . Figure 1b and 1c shows the edge wavefronts arising at semiedges RB and RA . The diffracted rays, generated by each individual semiedge, form a congruence. However, the unification of two sets of diffracted rays emanating from both semiedges is not a congruence. Each of these congruences exists only on one side of the cone of diffracted rays spreading from point R . Such a cone acts as a shadow boundary of the corresponding edge wave generated by the semiedge. The edge wavefield has a discontinuity because there is no edge wave in the region where diffracted rays do not exist. The

Abstract Edge- and tip-wave theories were developed during a time when computational power was not readily available for verification by comparing with full wave solutions. However, physical modeling of wave propagation was common in several Soviet laboratories, including the Institute of Geophysics in Novosibirsk, where the initial theory and algorithms were developed (Klem-Musatovet al., 1972, 1975, 1976, 1982; Aizenberg and Klem-Musatov, 1980; Aizenberg, 1982). The first section of this chapter reviews experiments made by Russian scientists to compare their theoretical calculations against experimental data in simple 2D and 3D models (Klem-Musatov, 1980; Landa and Maksimov, 1980; Luneva and Kharlamov, 1990). Because theory and applications of edge and tip waves were published in Western journals (Klem-Musatov and Aizenberg, 1984, 1985, 1989), several groups pursued their own implementation, e.g., Pajchel et al. (1987, 1988, 1989) in Norway, Hoffmann et al. (1993) and Klaeschen et al. (1994) in Germany, Hron and Chan (1995) in Canada, and Wang and Waltham (1995) in the United Kingdom. As ray-method applications developed as tools in geophysical prospecting, edge-wave theory was discovered to be a convenient remedy for limitations of the ray approach in handling model discontinuities. We devote the second section of this chapter to one of the first practical implementations of edge-wave theory: the 2D software package of Pajchel et al. (1987). This implementation was used widely for practical exploration problems in the North Sea, where discontinuities in geologic structures and diffractions are common features of seismic sections. Edge-wave theory fails where the ray-theory field changes rapidly

Abstract In Edge and Tip Diffractions: Theory and Applications in Seismic Prospecting (SEG Geophysical Monograph Series No. 14), the theoretical framework of the edge and tip wave theory of diffractions has been elaborated from fundamental wave mechanics. Seismic diffractions are inevitable parts of the recorded wavefield scattered from complex structural settings and thus carry back to the surface information that can be exploited to enhance the resolution of details in the underground. The edge and tip wave theory of diffractions provides a physically sound and mathematically consistent method of computing diffraction phenomena in realistic geologic models. In this book, theoretical derivations are followed by their numerical implementation and application to real exploration problems. The book was written initially as lecture notes for an internal course in diffraction modeling at Norsk Hydro Research Center, Bergen, Norway, and later was used for a graduate course at Novosibirsk State University in Russia. The material is drawn from several previous publications and from unpublished technical reports. Edge and Tip Diffractions will be of interest to geoscientists, engineers, and students at graduate and Ph.D. levels.