First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

International audienceWe propose approximate equations for P-wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used. The first-order ray theory P-wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent

Seismic ray theory

In this paper, the main principles of seismic ray theory for isotropic and anisotropic inhomogeneous media with curved structural interfaces are reviewed. Some extensions and modifications of seismic ray theory are also briefly mentioned

Moveout approximation for a P-SV wave in a moderately anisotropic homogeneous DTI layer

International audienceWe derive and test the moveout formula for converted P-SV (or SV-P) waves in a homogeneous, weakly or moderately anisotropic, transversely isotropic layer with axis of symmetry perpendicular to a dipping reflector underlying it. Instead of commonly used Taylor series expansion of the squared traveltime with respect to the offset, we use weak-anisotropy approximation of the exact traveltime formula. We replace the exact traveltime along the actual ray of a converted wave by its first-order weak-anisotropy approximation along a ray in a nearby reference isotropic medium. The traveltime formula is derived in the plane defined by the source-receiver line and the normal to the reflector. The source and the receiver may be situated arbitrarily in the model, the flat reflector may have an arbitrary 3D orientation. The accuracy of the proposed moveout formula is tested on models with varying strength of anisotropy for a varying dip of the reflector. In isotropic media, the formula yields highly accurate results with relative traveltime errors not exceeding 0.5%. In anisotropic media of P- and SV-wave anisotropy of about 25% and 12%, respectively, relative traveltime errors do not exceed 2%

First-order ray computations of coupled S waves in inhomogeneous weakly anisotropic media

International audienceWe propose an approximate procedure for computing coupled S waves in inhomogeneous weakly anisotropic media. The procedure is based on the first-order ray tracing (FORT) and dynamic ray tracing (FODRT), which was originally developed for P waves. We use the so-called common ray tracing concept to derive approximate ray tracing and dynamic ray tracing equations, and an approximate solution of the transport equation for coupled S waves propagating in laterally varying, weakly anisotropic media. In our common ray tracing, ray equations are governed by the first-order Hamiltonian formed by the average of first-order eigenvalues of the Christoffel matrix, corresponding to the two S-wave modes propagating in anisotropic media. The solution of the transport equation for the coupled S waves leads to a system of two coupled frequency-dependent, linear ordinary differential equations for amplitude coefficients, which is evaluated along the S-wave common ray. For derivation of the FORT and FODRT equations, we use the perturbation theory in which deviations of anisotropy from isotropy are considered to be first-order perturbations. To derive the coupled differential equations for S-wave amplitudes, we assume that the first-order perturbations are of order O(ω-1), where ω is the circular frequency. This makes it possible to express the amplitude coefficients in the coupled differential equations in terms of geometrical spreading and other quantities related to the common ray. The proposed procedure removes problems of most currently available ray tracers, which yield distorted results or even collapse when shear waves propagating in weakly anisotropic media are computed. The first-order approximation leads to simpler ray tracing and dynamic ray tracing equations than in the exact case. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. If the anisotropy vanishes, the equations reduce to standard, exact ray tracing and dynamic ray tracing equations for S waves propagating in isotropic media. The proposed ray tracing and dynamic ray tracing equations, corresponding traveltimes and geometrical spreading are all given to the first order. The accuracy of the traveltimes along S-wave first-order common rays can be increased by calculating a second-order traveltime correction

PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt

International audienceUse of unconverted reflected PP waves leads to moveout formulae, which relate the moveout to only a limited number of parameters specifying anisotropy. Use of converted reflected PS waves extends the number of parameters to, generally, a complete set of twenty-one parameters. It leads, on the other hand, to more complicated formulae and phenomena unknown from studies of unconverted PP waves. In anisotropic media one has to deal with the existence of two S waves and their singularities, acoustic approximation cannot be used. We present and test approximate formulae for the reflection moveout of PS waves converted at a horizontal reflector underlying a homogeneous layer of arbitrary anisotropy symmetry and orientation. For their derivation, we use the combination of weak-anisotropy approximation and A-parameters specifying anisotropy. The complete set of twenty-one A-parameters represents an alternative to twenty-one independent elements of the stiffness tensor. In addition to the moveout formulae for separate PS1 and PS2 waves, we also present moveout formula for a P to common S wave whose traveltimes represent an average of traveltimes of PS1 and PS2 waves. Use of the P to common S wave reflection leads to several simplifications. Presented tests indicate that for most offsets and models with P-wave anisotropy up-to to 26% and S-wave anisotropy over 25%, the relative traveltime errors do not exceed 2.5%

First-order reflection/transmission coefficients for unconverted plane P waves in weakly anisotropic media

International audienceWe present approximate formulae for the plane-wave displacement reflection/transmission (R/T) coefficients for interfaces of arbitrary contrast, separating two homogeneous, weakly anisotropic media. They result from boundary conditions requiring continuity of displacement vector and traction, in which coupled S waves are considered as a single S wave and exact quantities are replaced by first-order quantities used in first-order ray tracing. Specifically, the phase velocities, slowness and polarization vectors of P and coupled S waves appearing in the boundary conditions are of the first-order with respect to the deviations of anisotropy from isotropy. Application of the derived R/T coefficients transforms the amplitude of an incident P wave into amplitudes of reflected/transmitted P or coupled S waves. Coefficients can be computed for any incidence angle between 0° and 90°, and for any azimuth. In this paper, we test the accuracy of the derived R/T coefficients of unconverted plane P waves. We show that, except for critical regions, first-order coefficients approximate the exact coefficients with accuracy comparable or better than accuracy of linearized weak-contrast coefficients, which are, however, applicable only in subcritical regions

Coupled S waves in inhomogeneous weakly anisotropic media using first-order ray tracing

International audienceWe present a modification of our recently proposed approximate procedure for computing coupled S waves in inhomogeneous weakly anisotropic media. The new procedure can be used to compute S waves propagating in smooth inhomogeneous isotropic or anisotropic media. In isotropic media, it reduces to standard ray theory procedure for S waves. In anisotropic media, it can be used to study coupled as well as decoupled S waves. As the previous procedure, the new one is also based on the approximately computed common S-wave ray. First-order ray tracing and dynamic ray tracing, originally developed for computations of P-wave fields, is used to compute common S-wave rays and the dynamic ray tracing along them. The principal difference between the previous and new procedure consists in implicit incorporation of the second-order common S-wave traveltime correction and more accurate estimate of traveltime difference in the modified coupling equations. This leads to a substantial increase of accuracy of the coupling equations, which are solved along the common ray to evaluate S-wave amplitudes. The new coupling equations provide, first of all, more accurate traveltimes, but they also allow modelling of decoupled S waves, which could hardly be done with the original coupling equations. There is no need for a choice of a reference medium. The reference medium is determined uniquely from the actual medium. The new procedure has all the advantages of the previous procedure. Among the basic advantages is that it can describe the coupling of S waves. The procedure eliminates problems with ray tracing in the vicinity of singularities; the common S-wave ray tracing is as stable as P-wave ray tracing. Due to the use of perturbation formulae, the ray tracing, dynamic ray tracing and coupling equations are much simpler and more transparent than in the exact case. As a byproduct of both coupling procedures, we get formulae for approximate evaluation of traveltimes of separate S waves. These formulae can find applications in migration and traveltime tomography. The accuracy of the previous and modified coupling procedures is studied on several models of varying strength of anisotropy. First, we investigate the accuracy of perturbation formulae in homogeneous models, in which coupling does not exist. Then we study both coupling and accuracy of perturbation formulae in inhomogeneous models. We compare the results obtained by the coupling procedures with the results of the quasi-isotropic approach and standard ray theory

P-wave reflection-moveout approximation for horizontally layered media of tilted weak-to-moderate orthorhombic symmetry

International audienceAn approximate nonhyperbolic P-wave moveout formula applicable to horizontally layered media of weak-to-moderate orthorhombic or higher-symmetry anisotropy of arbitrary orientation is presented and tested. Weak-anisotropy approximation is used for the derivation of the formula, in which the square of the reflection traveltime is expanded in terms of weak-anisotropy (WA) parameters specifying the anisotropy of the layers. Due to the use of the weak-anisotropy approximation, the proposed formula is applicable to any offset. Its accuracy decreases with increasing strength of anisotropy. The relation between traveltimes and WA parameters in the formula is transparent and relatively simple. For any offset along an arbitrarily chosen single surface profile, the formula depends in each layer on its thickness, on the auxiliary reference P-wave velocity, on, at most, six WA parameters describing P-wave orthorhombic or higher-symmetry anisotropy, and on three Euler angles specifying the orientation of symmetry elements in the layer. Performed tests indicate that the maximum relative traveltime error does not exceed 2.5% for P-wave anisotropy about 25%. Often, the errors are considerably lower