Theory and modeling of constant-<i>Q</i> P- and S-waves using fractional time derivatives

Geophysics - Tập 74 Số 1 - Trang T1-T11 - 2009
José M. Carcione1
1Istituto Nazionale di Oceanografia e di Geofisica Sperimentale, Trieste, Italy

Tóm tắt

I have developed and solved the constant-[Formula: see text] model for the attenuation of P- and S-waves in the time domain using a new modeling algorithm based on fractional derivatives. The model requires time derivatives of order [Formula: see text] applied to the strain components, where [Formula: see text] and [Formula: see text], with [Formula: see text] the P-wave or S-wave quality factor. The derivatives are computed with the Grünwald-Letnikov and central-difference fractional approximations, which are extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general material-property variability. I verified the results by comparison with the 2D analytical solution obtained for wave propagation in homogeneous Pierre Shale. Moreover, the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experiment.

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Tài liệu tham khảo

Auld, B. A. , 1990, Acoustic fields and waves in solids, Vol. 1: Robert E. Krieger, Publishing Co.

Bland, D. R. , 1960, The theory of linear viscoelasticity: Pergamon.

10.1007/BF02820620

10.1190/1.1444692

Carcione, J. M. , 2007, Wave fields in real media. Theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media: Elsevier.

10.1007/s00024-002-8705-z

10.1111/j.1365-246X.1988.tb06706.x

Christensen, R. M. , 1982, Theory of viscoelasticity: Academic Press.

10.1098/rsta.1956.0010

10.1121/1.428250

Fornberg, B., 1996, A practical guide to pseudospectral methods: Cambridge University Press.

Gorenflo R., 1997, 277

10.1007/978-3-7091-2664-6_5

Gorenflo R., 1998, Fractional Calculus & Applied Analysis, 1, 167

Grünwald A. K., 1867, Zeitschrift für Angewandte Mathematik und Physik, 12, 441

10.1111/j.1365-246X.1995.tb06449.x

10.1098/rspa.2001.0904

10.1111/j.1365-246X.2003.02086.x

10.1029/JB084iB09p04737

10.1190/1.1441288

Letnikov A. V., 1868, Matematiceskij Sbornik, 3, 1

10.1016/j.jcp.2005.03.008

Mainardi F., 1997, Annali di Geofisica, 40, 1311

10.1190/1.1438489

Oldham, K. B. , and J. Spanier, 1974, The fractional calculus: Academic Press.

Pilant, W. L. , 1979, Elastic waves in the earth: Elsevier Publishing Company.

Podlubny, I., 1999, Fractional differential equations: Academic Press.

10.1111/j.1365-246X.2004.02337.x

Scott-Blair, G. W. , 1949, Survey of general and applied rheology: Pitman.

10.1016/j.jcp.2006.05.030

Toksöz, M. N. , and D. H. Johnston, 1981, Seismic wave attenuation, SEG.

10.1002/zamm.19950750826

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Geophysics

Tập 74 Số 1

T1-T11

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