An asymptotic procedure for the computation of wave fields in two-dimensional laterally inhomogeneous media is proposed. It is based on the simulation of the wave field by a system of Gaussian beams. Each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all Gaussian beams arriving in some neighbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails (the vicinity of caustic, critical point, etc.). Numerical examples are given.
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The first-order perturbation method is used to evaluate approximate phase velocities and polarization vectors in elastic anisotropic media. Formulae are given which make possible computations of perturbations of these parameters for quasi-compressional as well as quasi-shear waves, no matter whether the unperturbed medium is isotropic or anisotropic. Approximate results for an extremely anisotropic material and relatively large deviations of parameters of unperturbed and perturbed media closely resemble the results computed exactly. It is, therefore, expected that the application of the perturbation method to realistic media with generally weaker anisotropy and for smaller deviations between unperturbed and perturbed medium parameters should give satisfactory results. The method will find the most important applications in the investigation of high-frequency wave propagation in inhomogeneous anisotropic media and in solving inverse problems for anisotropic structures. Several possible applications are listed and briefly discussed.
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.
S U M M A R YProperties of homogeneous and inhomogeneous plane waves propagating in an unbounded viscoelastic anisotropic medium in an arbitrarily specified direction N are studied analytically. The method used for their calculation is based on the so-called mixed specification of the slowness vector. It is quite universal and can be applied to homogeneous and inhomogeneous plane waves propagating in perfectly elastic or viscoelastic, isotropic or anisotropic media. The method leads to the solution of a complex-valued algebraic equation of the sixth degree. Standard methods can be used to solve the algebraic equation. Once the solution has been found, the phase velocities, exponential decays of amplitudes, attenuation angles, polarization vectors, etc., of P, S1 and S2 plane waves, propagating along and against N, can be easily determined.Although the method can be used for an unrestricted anisotropy, a special case of P, SV and SH plane waves, propagating in a plane of symmetry of a monoclinic (orthorhombic, hexagonal) viscoelastic medium is discussed in greater detail. In this plane the waves can be studied as functions of propagation direction N and of the real-valued inhomogeneity parameter D. For inhomogeneous plane waves, D = 0, and for homogeneous plane waves, D = 0. The use of the inhomogeneity parameter D offers many advantages in comparison with the conventionally used attenuation angle γ . In the N, D domain, any combination of N and D is physically acceptable. This is, however, not the case in the N, γ domain, where certain combinations of N and γ yield non-physical solutions. Another advantage of the use of inhomogeneity parameter D is the simplicity and universality of the algorithms in the N, D domain.Combined effects of attenuation and anisotropy, not known in viscoelastic isotropic media or purely elastic anisotropic media, are studied. It is shown that, in anisotropic viscoelastic media, the slowness vector and the related quantities are not symmetrical with respect to D = 0 as in isotropic viscoelastic media. The phase velocity of an inhomogeneous plane wave may be higher than the phase velocity of the relevant homogeneous plane wave, propagating in the same direction N. Similarly, the modulus of the attenuation vector of an inhomogeneous plane wave may be lower than that for the relevant homogeneous plane wave. The amplitudes of inhomogeneous plane waves in anisotropic viscoelastic media may increase exponentially in the direction of propagation N for certain D. The attenuation angle γ cannot exceed its boundary value, γ * . The boundary attenuation angle γ * is, in general, different from 90 • , and depends both on the direction of propagation N and on the sign of the inhomogeneity parameter D. The polarization of P and SV plane waves is, in general, elliptical, both for homogeneous and inhomogeneous waves. Simple quantitative expressions or estimates for all these effects (and for many others) are presented. The results of the numerical treatment are presented in a companion paper (Paper II,...
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