This repository contains Mathematica notebooks and supporting scripts for my earlier work on kinematic and dynamic approximations in anisotropic media.
The codes implement traveltime, moveout, CRS, diving-wave, and geometrical-spreading formulas in VTI and orthorhombic media developed in the papers listed below.
The repository is organized by topic rather than by single paper. Each folder contains notebooks that reproduce key figures, test the accuracy of the approximations, and provide symbolic and numerical evaluation routines.
Curvature and anisotropy estimation using the Common-Reflection-Surface (CRS) approximation.
- Xu & Stovas, 2015, Curvature and anisotropy estimation through the CRS approximation, Journal of Geophysics and Engineering, 12, 934–945.
Contents:
- CRS traveltime expansions in anisotropic media.
- Estimation of curvature and effective anisotropy parameters.
- Synthetic examples demonstrating parameter recovery.
Imaging moveout and parameter estimation from diving waves in factorized VTI media.
- Xu, Stovas & Alkhalifah, 2016, Estimation of the anisotropy parameters from imaging moveout of diving wave in a factorized VTI medium, Geophysics, 81, C139–C150.
Contents:
- Factorized VTI background model and parameterization.
- Analytical diving-wave moveout series.
- Inversion routines for Thomsen parameters from imaging moveout.
Traveltime smoothing and CRS-type kinematics in orthorhombic media.
- Xu & Stovas, 2017, Preserved traveltime smoothing in orthorhombic media, Geophysical Prospecting, 65, 1205–1217.
Contents:
- Orthorhombic CRS/CRP traveltime formulations.
- Traveltime smoothing operators and tests on layered ORT models.
- Comparison between smoothed and exact traveltimes.
High-order moveout and converted-wave traveltime approximations in VTI and orthorhombic media.
Relevant papers:
- Xu, Stovas & Hao, 2017, Perturbation-based moveout approximations in anisotropic media, Geophysical Prospecting, 65, 1218–1230.
- Xu & Stovas, 2019, Traveltime approximation for converted waves in elastic orthorhombic media, Geophysics, 84, C229–C237.
- Xu & Stovas, 2019, Estimation of the conversion point position in elastic orthorhombic media, Geophysics, 84, C15–C25.
- Xu & Stovas, 2018, Triplications on traveltime surface for pure and converted wave modes in elastic orthorhombic media, GJI, 215, 677–694.
Contents:
- Perturbation-based moveout series for VTI and ORT.
- qP–qSV converted-wave traveltime and conversion-point approximations.
- Analysis of triplications and multivalued traveltime surfaces.
Relative geometrical-spreading approximations in VTI media.
Relevant papers:
- Xu & Stovas, 2018, Generalized non-hyperbolic approximation for qP-wave relative geometrical spreading in layered VTI media, Geophysical Prospecting, 66, 1290–1302.
- Xu & Stovas, 2017, Three-dimensional generalized nonhyperbolic approximation for relative geometrical spreading, GJI, 211, 1162–1175.
- Xu & Stovas, 2018, Fresnel zone in VTI and orthorhombic media, GJI, 213, 181–193.
Contents:
- 2D and 3D generalized non-hyperbolic formulas for relative geometrical spreading.
- Evaluation of Fresnel zones and illumination patterns in VTI.
- Comparison with ray-traced amplitudes.
Geometrical-spreading and anelliptic corrections in orthorhombic and TI media.
Relevant papers:
- Xu, Stovas & Sripanich, 2018, An anelliptic approximation for geometrical spreading in transversely isotropic and orthorhombic media, Geophysics, 83, C37–C47.
- Xu & Stovas, 2019, The kinematics of S waves in acoustic orthorhombic media, Geophysical Prospecting, 67, 1983–2002.
- Xu & Stovas, 2019, Singularity point in effective orthorhombic medium computed from zero- and infinite-frequency limit, GJI, 217, 319–330.
- Xu & Stovas, 2017, A new parameterization for acoustic orthorhombic media, Geophysics, 82, C229–C240.
Contents:
- Anelliptic geometrical-spreading formulas for TI and ORT.
- Analysis of S-wave kinematics in acoustic orthorhombic media.
- Computation of singularity points from zero- and infinite-frequency limits.
- Orthorhombic parameterization utilities used across the notebooks.
The codes are written in Mathematica and were developed as research prototypes to derive formulas, generate synthetic examples, and reproduce figures from the publications.
Open the corresponding notebook in each folder, evaluate the cells from top to bottom, and adapt model parameters as needed for your own tests.
If you use these notebooks or derived formulas in your work, cite the relevant paper(s) listed above. A minimal generic reference is:
Xu, S., and co-authors, 2015–2020: Traveltime, moveout, and geometrical-spreading approximations in VTI and orthorhombic media. See the individual journal articles listed in the repository README for full bibliographic details.