×

One-way wave propagation with amplitude based on pseudo-differential operators. (English) Zbl 1231.35328

Summary: The one-way wave equation is widely used in seismic migration. Equipped with wave amplitudes, the migration can be provided with the reconstruction of the strength of reflectivity. We derive the one-way wave equation with geometrical amplitude by using a symmetric square root operator and a wave field normalization. The symbol of the square root operator, \(\omega\sqrt{\frac{1}{c(x,z)^2}-\frac{\xi^2}{\omega^2}}\), is a function of space-time variables and frequency \(\omega\) and horizontal wavenumber \(\xi\). Only by matter of quantization does it become an operator, and because quantization is subjected to choices it should be made explicit. If one uses a naive asymmetric quantization an extra operator term will appear in the one-way wave equation, proportional to \(\partial_xc\). We propose a symmetric quantization, which maps the symbol to a symmetric square root operator. This provides geometrical amplitude without calculating the lower order term. The advantage of the symmetry argument is its general applicability to numerical methods. We apply the argument to two numerical methods. We propose a new pseudo-spectral method, and we adapt the 60 degree Padé type finite-difference method such that it becomes symmetrical at the expense of almost no extra cost. The simulations show in both cases a significant correction to the amplitude. With the symmetric square root operator the amplitudes are correct. The \(z\)-dependency of the velocity leads to another numerically unattractive operator term in the one-way wave equation. We show that a suitably chosen normalization of the wave field prevents the appearance of this term. We apply the pseudo-spectral method to the normalization and confirm by a numerical simulation that it yields the correct amplitude.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35L05 Wave equation
53D50 Geometric quantization
47N20 Applications of operator theory to differential and integral equations

Software:

RODAS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jin, S.; Wu, R.-S.; Peng, C., Seismic depth migration with pseudo-screen propagator, Comput. Geosci., 3, 321-335 (1999) · Zbl 0977.74590
[2] Kiyashchenko, D.; Plessix, R.-E.; Kashtan, B.; Troyan, V., Improved amplitude multi-one-way modeling method, Wave Motion, 43, 99-115 (2005) · Zbl 1231.35308
[3] Stolk, C. C., A pseudodifferential equation with damping for one-way wave propagation in inhomogeneous acoustic media, Wave Motion, 40, 111-121 (2004) · Zbl 1163.74444
[4] Zhang, Y.; Zhang, G.; Bleistein, N., True amplitude wave equation migration arising from true amplitude one-way wave equations, Inverse Problems, 19, 1113-1138 (2003) · Zbl 1048.35118
[5] Zhang, Y.; Zhang, G.; Bleistein, N., Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration, Geophysics, 70, 4, E1-E10 (2005)
[6] Margrave, G. F.; Ferguson, R. J., Wavefield extrapolation by nonstationary phase shift, Geophysics, 64, 4, 1067-1078 (1999)
[7] Gazdag, J.; Sguazzero, P., Migration of seismic data by phase shift plus interpolation, Geophysics, 49, 2, 124-131 (1984)
[8] Biondi, B., Stable wide-angle Fourier finite-difference downward extrapolation of 3-D wavefields, Geophysics, 67, 03, 872-882 (2002)
[9] Ristow, D.; Ruhl, T., Fourier finite-difference migration, Geophysics, 59, 12, 1882-1893 (1994)
[10] Xie, X.-B.; Wu, R.-S., A depth migration method based on the full-wave reverse-time calculation and local one-way propagation, SEG Tech. Program Expanded Abstr., 25, 1, 2333-2337 (2006)
[11] Graves, R. W.; Clayton, R. W., Modeling acoustic waves with paraxial extrapolators, Geophysics, 55, 3, 306-319 (1990)
[12] Claerbout, J. F., Imaging the Earth’s Interior (1985), Blackwell Scientific Publications, Inc.
[13] Cohen, J. K.; Bleistein, N., Velocity inversion procedure for acoustic waves, Geophysics, 44, 6, 1077-1087 (1979)
[14] Beylkin, G., Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys., 26, 99-108 (1985)
[15] Halpern, L.; Trefethen, L. N., Wide-angle one-way wave equations, J. Acoust. Soc. Am., 84, 4, 1397-1404 (1988)
[16] Biondi, B. L., 3D seismic imaging, Soc. Explor. Geophys. (2006)
[17] Wild, A. J.; Hobbs, R. W.; Frenje, L., Modelling complex media: an introduction to the phase-screen method, Phys. Earth Planet. Inter., 120, 219-225 (2000)
[18] de Hoop, M. V.; Rousseau, J. H.L.; Wu, R. S., Generalization of the phase-screen approximation for the scattering of acoustic waves, Wave Motion, 31, 43-70 (2000) · Zbl 1074.76614
[19] Alinhac, S.; Gérard, P., Pseudo-differential Operators and the Nash-Moser Theorem (2007), American Mathemathical Society · Zbl 1121.47033
[20] Hörmander, L., The Analysis of Linear Partial Differential Operators III (1994), Springer-Verlag
[21] Gazdag, J., Wave-equation migration with the phase-shift method, Geophysics, 43, 7, 1342-1351 (1978)
[22] Duistermaat, J. J., Fourier Integral Operators, Birkhäuser (1996) · Zbl 0841.35137
[23] Wong, M. W., An Introduction to Pseudo-differential Operators (1999), World Scientific · Zbl 0940.35216
[24] Taylor, M. E., Pseudodifferential Operators (1981), Princeton University Press · Zbl 0453.47026
[25] Lamoureux, M. P.; Margrave, G. F., An Introduction to Numerical Methods of Pseudodifferential Operators, Pseudo-Differential Operators, 1949, 79-133 (2008) · Zbl 1171.35132
[26] Kincaid, D.; Cheney, W., Numerical Analysis: Mathematics of Scientific Computing (2002), Brooks/Cole, Thomson Learning, Inc.
[27] Bao, G.; Symes, W. W., Computation of pseudo-differential operators, SIAM J. Sci. Comput., 17, 2, 416-429 (1996), ISSN 1064-8275 · Zbl 0848.35146
[28] Hairer, E.; Norsett, S. P.; Wanner, G., Solving ordinary differential equations I. Solving ordinary differential equations I, Nonstiff Problems (1993), Springer · Zbl 0789.65048
[29] Hairer, E.; Wanner, G., Solving ordinary differential equations II. Solving ordinary differential equations II, Stiff and Differential-Algebraic Problems (1996), Springer · Zbl 0859.65067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.