First-order perturbation method for anisotropic media

Jech, Jiří; Pšenčı́k, Ivan

doi:10.1111/j.1365-246x.1989.tb01694.x

Cited by 158 publications

(122 citation statements)

References 15 publications

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“…When averaging the volume of rock typically sampled by a seismic wave, linearized solutions are a good approximation and illustrate well the first-order influence of each elastic coefficient on the wavefield. A complete treatment by the application of perturbation theory to the Christoffel equation at first order in anisotropy is presented in Jech and Psencik (1989), including perturbation of the polarization vectors.…”

Section: Weak Anisotropy Approximationmentioning

confidence: 99%

Theory and Observations - Seismic Anisotropy

Maupin

Park

2015

Treatise on Geophysics

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Section: Weak Anisotropy Approximationmentioning

confidence: 99%

Theory and Observations - Seismic Anisotropy

Maupin

Park

2015

Treatise on Geophysics

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“…Only some simple symmetry, such as lateral transversely isotropic medium, has analytical expression (DALEY and HRON, 1977). Based on the fact that most anisotropic media are weakly anisotropic, the perturbation theory developed by Č ERVENÝ (1982) and JECH and PSˇENCˇI´K (1989) can be used to calculate the travel time in anisotropic media. When isotropic unperturbed media are taken as the reference media, phase velocity of qP wave can be expressed as a linearized function of weakly anisotropic (WA) parameters (PSˇENCˇI´K and GAJEWSKI, 1998).…”

Section: Introductionmentioning

confidence: 99%

Inversion of Travel Times in Weakly Anisotropic Rock Samples

Zheng¹,

Tomaskova²

2004

Pure and Applied Geophysics

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Based on the perturbation theory, inversion formulae for travel time of qP and qS waves in arbitrary weak anisotropy media are presented. The inversion formulae are linear expressions of elastic parameters expressed in terms of weak anisotropy (WA) parameters. The formulae of qS 1 and qS 2 waves have the same form and they can be used without identifying which wave is considered. A synthetic experiment similar to the measurement of rock sample in the laboratory is carried out to illustrate the efficiency of the presented inversion formulae. Two data sets for qP wave travel time from rock samples in the laboratory are inverted and 15 WA parameters are obtained.

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“…Jech and Pšenčík (1989) give the traveltime error by 1.4) where the ∆a ijkl are the deviations of the density-normalised elastic parameters from those of the unperturbed medium, p i are the slownesses and g j the polarisation vectors. The following procedure is the same as for isotropic media.…”

Section: Sensitivity To Velocity Errorsmentioning

confidence: 99%

Traveltime based true‐amplitude migration

Vanelle

Gajewski

2000

SEG Technical Program Expanded Abstracts 2000

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Amplitude-preserving migration based on a weighted diffraction stack is a task of high computational costs. Three factors determine the major contributions to the computational effort: The diffraction time surface must be computed for each image point as well as proper weight functions. Finally, the summation is carried out over the whole aperture of the experiment. I propose a new strategy for amplitude-preserving migration that is entirely based on traveltimes, thus called traveltime-based true-amplitude migration. Its foundation is a hyperbolic traveltime expansion. High accuracy is achieved because second-order spatial derivatives are included in order to acknowledge the curvature of the wavefront. The algorithm permits the determination of the interpolation coefficients from traveltime tables sampled on a coarse grid, thus reducing the requirements in data storage. The method also provides a tool for the interpolation between sources. Application to various velocity models confirms that savings up to a factor of 10 5 are possible in data storage with no significant loss in accuracy. Also, the interpolation is 5-6 times faster than the calculation of traveltime tables using a fast finite differences eikonal solver. Since it is possible to express true-amplitude weight functions in terms of second-order traveltime derivatives a traveltime-based relationship for the weights can be established. As a consequence, the weight functions can be directly computed from the interpolation coefficients. This reduces the need in computational time, and particularly storage, because the weights are computed on-the-fly. Application of the weight functions shows good agreement between numerical and analytical results for the simple type of models considered in this work. For complex models the good accuracy of the geometrical spreading, which can be computed using the same interpolation coefficients as for the weight functions, was also demonstrated. This indicates that the migration technique will also perform well for complex models. A further significant reduction of the computational effort involved in the migration can be achieved if the summation is carried out only over those traces that really contribute to the stack, i.e. by limiting the migration aperture. Moreover, the optimum migration aperture can also be determined from the traveltime coefficients. Examples show that limited aperture migration reduces the requirements in computational time by 80% in 2-D, and by more than 90% in 3-D media. At the same time, the image quality is enhanced by suppressing migration noise. Since the foundation of the method, the hyperbolic traveltime expansion, is not limited to isotropic models the technique has a high potential to be extended to anisotropic media.iii iv

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First-order perturbation method for anisotropic media

Cited by 158 publications

References 15 publications

Theory and Observations - Seismic Anisotropy

Theory and Observations - Seismic Anisotropy

Inversion of Travel Times in Weakly Anisotropic Rock Samples

Traveltime based true‐amplitude migration

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