News and Announcements
ARTICLE
A numerical modeling of wave propagation that is independent of coordinate transformation
LUC T. IKELLE
Show Less
CASP Project, Department of Geology and Geophysics, Texas A&M University, College Station, TX 77843-3115, U.S.A.,
JSE 2012, 21(2), 153–176;
Submitted: 14 August 2011 | Accepted: 10 February 2012 | Published: 1 May 2012
© 2012 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract
It is a remarkable fact that Maxwell’s equations under any coordinate transformation can be written in an identical mathematical form as the ones in Cartesian coordinates. However, in some particular coordinate transformations, like the cylindrical coordinate transformations, the physical properties becomes anisotropic, even if they are isotropic in the Cartesian coordinates. Even the permittivity can be anisotropic. We here review these fundamental results. The remarkable invariance of Maxwell’s equations under coordinate transformation can also extend to elastodynamic wave equations by rewriting them in a new form. We have used this new form of the elastodynamic wave equations to describe a numerical solution of elastic wave propagation which is independent of coordinate transformation.
Keywords
elastodynamic equations
Maxwell’s equations
curvilinear coordinates
natural coordinates
physical coordinates
finite-difference solution
References
- Aki, K. and Richards, P.G., 1980. Quantitative Seismology: theory and methods (2 vols.). W.H. ©Freeman and Co., San Francisco.
- Berenger, J.P., 1994. A perfectly matched layer for the absorption of electromagnetic waves. J.Comput. Phys., 114: 185-200.
- Carcione, J.M., 1994. The wave equation in generalized coordinates. Geophysics, 59: 1911-1919.
- Cerjian, C., Kosloff, D., Kosloff, R. and Reshef, M., 1985. A nonreflecting boundary conditionfor discrete acoustic-wave and elastic-wave equations. Geophysics, 50: 705-708.
- Chew, W.C. and Liu, Q.H., 1996. Perfectly matched layers for elastodynamics: A new absorbingboundary condition: J. Comput. Acoust., 4: 341-359.de Hoop, A.T., 1995. Handbook of Radiation and Scattering of Waves. Academic Press, San Diego.
- Fornberg, B., 1988. The pseudospectral method: Accurate representation of interfaces in elasticwave calculations. Geophysics, 53: 625-637.
- Friis, H.A., Johansen, T.A., Haveraaen, M., Munthe-Kaas, H. and Drottning, A., 2001. Use ofcoordinate-free numerics in elastic wave simulation. Appl. Numer. Mathemat., 39: 151-171.
- Gangi, A.F., 2000. Constitutive equations and reciprocity. Geophys. J. Internat., 143: 311-318.
- Hestholm, S.0. and Ruud, B.O., 1994. 2-D finite difference elastic wave modelling including 23surface topography. Geophys. Prosp., 42: 371-390.
- Ikelle, L.T. and Amundsen, L., 2005. An Introduction to Petroleum Seismology. Investigations inGeophysics, SEG, Tulsa.
- Ikelle, L.T., 2010. Coding and Decoding: Seismic Data. Elsevier Science Publishers, Amsterdam.
- Ikelle, L.T., 2012. On elastic-electromagnetic equivalences. Geophys. J. Internat., in press.
- Kausel, E., 2006. Fundamental Solutions in Elastodynamics: A Compendium. Cambridge UniversityPress, Cambridge.
- Komatitsch, D., Coutel, F. and Mora, P., 1996. Tensorial formulation of the wave-equation formodeling curved interfaces. Geophys. J. Internat., 127: 156-168.
- Liu, Q.H., 1999. Perfectly matched layers for elastic waves in cylindrical and spherical coordinates.J. Acoust. Soc. Am., 105: 2075-2084.
- Milton, G.W., Briane, M. and Willis, J.R., 2006. On cloaking for elasticity and physical equationswith a transformation invariant form. New J. Phys., 8: 248-267.
- Nielsen, P., Flemming, I., Berg, P. and Skovgaard, 0., 1994. Using the pseudospectral techniqueon curved grids for 2D acoustic forward modelling. Geophys. Prosp., 42: 321-341.
- Pendry, J.B., Schurig, D. and Smith, D.R., 2006. Controlling electromagnetic fields. Science, 312:1780-1782.
- Pissarenko, D., Reshetova, G. and Tcheverda, V., 2009. 3D finite-difference synthetic acoustic login cylindrical coordinates. Geophys. Prosp., 57: 367-377.
- Post, E.J., 1962. Formal Structure of Electromagnetics. Wiley & Sons, New York.
- Stratton, J.A., 1941. Electromagnetic Theory. McGraw-Hill, New York.
- Tessmer, E., Kosloff, D. and Behle, A., 1992. Elastic wave propagation simulation in the presenceof surface topography. Geophys. J. Internat., 108: 621-632.176 IKELLE
- Tessmer, E. and Kosloff, D., 1994. 3-D elastic modeling with surface topography by a Chebychevspectral method. Geophysics, 59: 464-473.
- Yan, W., Yan, M., Ruan, Z. and Qiu, M., 2008. Coordinate transformations make perfectinvisibility cloaks with arbitrary shape. New J. Phys., 10: 1-13.
