- One-way wave propagation with amplitude based on pseudo-differential operators
- Wave Motion
- Volume | Issue number
- 47 | 2
- Pages (from-to)
- Document type
- Faculty of Science (FNWI)
- Korteweg-de Vries Institute for Mathematics (KdVI)
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 root 1/c(x.z)(2) - 2/omega(2). is a function of space-time variables and frequency (j) and horizontal wave number. Only by matter of quantization it becomes 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 derivative(x)c. 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 Pade 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 lead 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.
- go to publisher's site
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library, or send a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.