Luděk Klimeš

Seismic ray method: Recent developments

The seismic ray method has found broad applications in the numerical calculation of seismic wavefields in complex 3-D, isotropic and anisotropic, laterally varying layered structures and in the solution of forward and inverse problems of seismology and seismic exploration for oil. This chapter outlines the basic features of the seismic ray method, and reviews its possibilities and recent extensions. Considerable attention is devoted to ray tracing and dynamic ray tracing of S waves in heterogeneous anisotropic media, to the coupling ray theory for S waves in such media, to the summation of Gaussian beams and packets, and to the selection of models suitable for ray tracing.

Second-Order and Higher-Order Perturbations of Travel Time in Isotropic and Anisotropic Media

The partial derivatives of travel time with respect to model parameters are referred to as perturbations. Explicit equations for the second-order and higher-order perturbations of travel time in both isotropic and anisotropic media are derived. The perturbations of travel time and its spatial derivatives can be calculated by simple numerical quadratures along rays.

The relation between Gaussian beams and Maslov asymptotic theory

РезюмеПрuложенuе aсuмnmоmuческоŭ mеорuu Мaсловa к общему mрехмерному смещaнному nо¶rt;nросmрaнсmву щесmuмерного фaзового nросmрaнсmвa nре¶rt;ложено ¶rt;ля nолученuя uнmегрaльных суnерnозuцuŭ Гaуссовых naкеmов u nучков. Суnерnозuцuя nлоскuх волн u лучевоŭ меmо¶rt; являюmся сnецuaльнымu nре¶rt;ельнымu случaямu nре¶rt;ложенного nо¶rt;хо¶rt;a. Те же сaмые высокочaсmоmные aсuмnmоmuческuе формулы былu рaньще nолучены меmо¶rt;ом Гaуссовых nучков [8].SummaryThe application of Maslov asymptotic theory in a general 3-D mixed subspace of 6-D complex phase space is proposed to obtain the integral superpositions of Gaussian packets and beams. The ray method and the superposition of plane waves (Maslov method of Chapman and Drummond [7]) are special limiting cases of the above mentioned approach. The same high-frequency asymptotic expansion formulae for seismic body waves were derived previously in [8] using the Gaussian beam method.

Transformations for dynamic ray tracing in anisotropic media

Abstract Six-dimensional dynamic ray tracing in (phase-space) Cartesian coordinates was introduced by Cervený [Geophys. J.R. Astr. Soc. 29, 1–13 (1972)]. Hanyga [Tectonophysics 90, 243–251 (1982)] showed that it reduces to 4-dimensional dynamic ray tracing in (phase-space) ray-centered coordinates. This paper concentrates on the explicit transformation equations of dynamic ray tracing between Cartesian and ray-centred coordinates. Many of the transformation equations have not been published before even for isotropic medium. Also proposed is an efficient way of reducing the number of equations being solved when numerically evaluating the paraxial-ray propagator matrices, both in Cartesian and ray-centred coordinates.

Optimization of the Shape of Gaussian Beams of a Fixed Length

The procedure of choosing the shape of Gaussian beams in order to minimize a given object function of a certain kind is proposed. The general form of the object function enables both the average square of the quadratic variation of the phase and the average square of the beamwidth to be minimized along the central ray. The error of the transformation of the Gaussian beams at the structural interfaces may also be taken into account. Most of the hitherto published suggestions of how to chose the shape of Gaussian beams are special cases of the described procedure. The aim of this paper is not to propose the object function to be minimized, but only to describe the minimization of a given object function. The minimization assumes the a priori known lengths of the central rays of the Gaussian beams (i.e. the lengths of the beams are not free parameters in the minimization procedure).

Grid travel-time tracing: Second-order method for the first arrivals in smooth media

A new computational scheme for calculating the first-arrival travel times on a rectangular grid of points is proposed. The new proposed method is of second-order accuracy. This means that the error of the calculated travel time is proportional to the second power of the grid spacing. The method should be sufficiently accurate for all applications in smooth seismic models. On the other hand, the method is not, in its present form, proposed for models with structural interfaces which make the method unstable and generate travel-time errors of the first order. Equations are also presented for the appropriate evaluation of the errors of calculated travel times to check their accuracy, and the proposed method is compared with other numerical methods. The method is developed, described and demonstrated in 2-D, but may also be extended to 3-D models and to general models with structural interfaces.

Comparison of Quasi-Isotropic Approximations of the Coupling Ray Theory with the Exact Solution in the 1-D Anisotropic “Oblique Twisted Crystal” Model

The coupling ray theory bridges the gap between the isotropic and anisotropic ray theories, and is considerably more accurate than the anisotropic ray theory. The coupling ray theory is often approximated by various quasi-isotropic approximations.Commonly used quasi-isotropic approximations of the coupling ray theory are discussed. The exact analytical solution for the plane S wave, propagating along the axis of spirality in the 1-D anisotropic “oblique twisted crystal” model, is then numerically compared with the coupling ray theory and its three quasi-isotropic approximations. The three quasi-isotropic approximations of the coupling ray theory are (a) the quasi-isotropic projection of the Green tensor, (b) the quasi-isotropic approximation of the Christoffel matrix, (c) the quasi-isotropic perturbation of travel times. The comparison is carried out numerically in the frequency domain, comparing the exact analytical solution with the results of the 3-D ray tracing and coupling ray theory software. In the oblique twisted crystal model, the three studied quasi-isotropic approximations considerably increase the error of the coupling ray theory. Since these three quasi-isotropic approximations do not noticeably simplify the numerical implementation of the coupling ray theory, they should deffinitely be avoided. The common ray approximations of the coupling ray theory do not affect the plane wave, propagating along the axis of spirality in the 1-D oblique twisted crystal model, and should be studied in more complex models.

Weak‐contrast reflection–transmission coefficients in a generally anisotropic background

Explicit equations for approximate linearized reflection–transmission coefficients at a generally oriented weak‐contrast interface separating two generally and independently anisotropic media are presented. The equations are derived also for all singular directions and are thus valid in degenerate cases (e.g., in an isotropic background). The equations are expressed in general Cartesian coordinates, with arbitrary orientation of the interface. The explicit equations for linearized reflection–transmission coefficients have a very simple form—much simpler than the equations published previously. The equations for all reflection–transmission coefficients, with the exception of the unconverted transmitted wave, have a common form. The form of the equations is very suitable for inversion and for analyzing the sensitivity of seismic data to discontinuities in individual elastic moduli. The factors of proportionality of the contrasts of elastic moduli and density are expressed in terms of the slowness and polari...

Lyapunov Exponents for 2-D Ray Tracing Without Interfaces

The Lyapunov exponents quantify the exponential divergence of rays asymptotically, along infinitely long rays. The Lyapunov exponent for a finite 2-D ray and the average Lyapunov exponents for a set of finite 2-D rays and for a 2-D velocity model are introduced. The equations for the estimation of the average Lyapunov exponents in a given smooth 2-D velocity model without interfaces are proposed and illustrated by a numerical example. The equations allow the average exponential divergence of rays and exponential growth of the number of travel-time branches in the velocity model to be estimated prior to ray tracing.

Applications of dynamic ray tracing

Abstract Dynamic ray tracing has recently found broad application in the numerical modelling of high-frequency seismic wave fields in complex 2-D and 3-D layered structures. It involves solving a system of ordinary differential equations (dynamic ray tracing system) along a ray. This paper briefly explains the main principles of dynamic ray tracing in ray centred coordinates, introduces the ray propagator matrix and describes its properties, summarizes the most important applications of dynamic ray tracing and of the ray propagator matrix. Together with known applications, several new applications are discussed in greater detail. Among these, a method to determine the geometrical spreading from the travel-time measurements is proposed.