Computation of Ray Integrals and Ray Amplitudes in Radially Symmetric Media

A new approximation of the velocity-depth distribution in radially symmetric media is suggested. This approximation guarantees the continuity of velocity and its first and second derivatives, and does not generate false low-velocity layers. It removes false anomalies from the amplitude-distance curve and considerably increases its stability. The evaluation of ray integrals and ray amplitudes using this velocity-depth approximation does not require the computation of any transcendental function and is, therefore, very fast. Numerical examples are presented.     

Numerical modelling of time-harmonic seismic wave fields in simple structures by the Gaussian beam method. Part II

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Numerical properties of low-frequency expansions for the reflection and transmission coefficients from transition layers

Summary The numerical properties of the low-frequency expansions for the reflection and transmission coefficients of SH-waves from transition layers, derived in [1], are studied. It is shown that the expansions are suitable for computations only when the thickness d of the transition layer is small in comparison with the wavelength of the incident wave (d0.5). For thicker transition layers, certain modifications of the method are suggested.     

Approximate expression for the hilbert transform of a certain class of functions and their application to the ray theory of seismic waves

Summary Approximate expressions for the Hilbert transform of the functionf(t)=exp(- ₀ ² t ²/²) cos( ₀ t+v) are determined. This function, given a suitable choice of the three parameters ₀, and v, approximates a wide class of seismic signals very well. The approximate expressions for the Hilbert transform enable very simple formulae to be given for the elementary seismograms of the individual seismic body waves (in the zero approximation of the ray theory). This accelerates the computation of ray theoretical seismograms considerably.     

A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

Tunneling of seismic body waves through thin high-velocity layers in complex structures

Summary The hybrid ray-reflectivity method is applied to the problem of the transmission of the reflected wave field through a thin high-velocity layer (or through a thin stack of high velocity layers), situated in the overburden of the reflector. In the hybrid ray-reflectivity method, the standard ray method is applied in the smooth parts of the model, and the reflectivity method is used locally at the thin high-velocity layer. With the exception of small epicentral distances, the standard ray method itself fails in such computations. The reason is that a considerable part of the energy for overcritical angles of incidence may be tunneled through the thin high-velocity layer along complex ray-paths, corresponding to inhomogeneous waves. The reflectivity method, applied locally at the thin high-velocity layer, automatically includes all inhomogeneous wave contributions. Thus, the hybrid ray-reflectivity method removes fully the limitations of the standard ray method, but still retains its main advantages, such as its applicability to 2-D and 3-D complex layered structures, flexibility, and low-cost computations. In the numerical examples, the hybrid ray-reflectivity synthetic seismograms are compared with standard ray synthetic seismograms and with full reflectivity computations. The numerical examples show that the hybrid ray-reflectivity method describes the tunneling of seismic energy through a thin high-velocity layer with sufficient accuracy.     

Radiation patterns of point sources situated close to structural interfaces and to the earth's surface

The seismic wave field is considerably influenced by local structures close to the source and to the receiver. This applies to sources and receivers situated close to localized inhomogeneities, to structural interfaces, to the earth's surface, etc. In this paper we concentrate our attention mainly to the ray-theoretical radiation patterns of point sources situated close to the structural interfaces and to the earth's surface. In numerical modeling of high-frequency seismic wave fields by the ray method, the interaction of the source with the earth's surface has not usually been taken into account.The proposed procedure of the computation of the radiation patterns of point sources situated directly on structural interfaces and on the earth's surface is based on the zero-order approximation of the ray method, assuming that the length of the ray between the source and the receiver is long. The derived equations are extended to point sources located close to structural interface, to the earth's surface and to thin transition layers using the hybrid ray-reflectivity method, seeervený (1989). The thin layer need not be homogeneous; it may include an arbitrary inner layering (transition layers, laminas, etc.) The only requirement is for the layer to be thin. Roughly speaking, we require its thickness to be less than one quarter of the prevailing wavelength. The hybrid ray-reflectivity method describes well even certain non-ray effects (tunneling.S ^* waves, etc.). Explicit analytical expressions for radiation patterns for all above listed point sources are found. These expression have a local character and may be easily implemented into computer codes designed for the routine computation of ray amplitudes and synthetic ray seismograms in 2-D and 3-D, laterally varying isotropic layered and block structures by the ray method.Numerical examples of radiation patterns ofP andS waves of point sources situated close to the earth's surface and to a thin low-velocity surface layer are presented and discussed. The explosive point source (center of dilatation) and the vertical and horizontal single force point sources are considered. It has been ascertained that the radiation patterns of point sources depend drastically on the depth of the source below the surface even if the depths vary within one quarter of the prevailing wavelength.