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Optimized one-way wave equation operator for VTI

For isotropic media, the dispersion relation for the one-way wave equation can be represented as
 (1)
where is the circular frequency, v=v(x,y,z) is the velocity, kz is the wavenumber, is the radial wavenumber, and kx, ky are wavenumbers for x and y respectively. Let , and . The square-root function can be approximated by a series of rational functions:
 (2)
The coefficients and can be obtained by Taylor-series analysis or rational factorization. If we consider the second-order approximation (n=1) and , ,we obtain the traditional equation. The coefficients and can also be obtained by least-squares optimization, and a more accurate finite-difference scheme like the equation can be obtained Lee and Suh (1985).

For VTI media, the true dispersion relation requires solving a quartic equation numerically Shan and Biondi (2005). With the assumption that the S-wave velocity is much smaller than the P-wave velocity, the dispersion relation for VTI media can be obtained analytically and represented as follows:
 (3)
where vp=vp(x,y,z) is the vertical velocity, and and are the anisotropy parameters defined by Thomsen (1986):

where Cij are elastic stiffness moduli. Let and . This dispersion relation can be further simplified under the weak anisotropy assumption, and it can be approximated as
 (4)
where and Ristow and Ruhl (1997). The coefficients and are obtained analytically by Taylor-series analysis.

As in the isotropic case, the coefficients and can also be obtained by least-squares optimization. The advantage of least-squares approximation is that I do not have to derive an explicit approximated expression for the dispersion relation analytically. This is especially useful for anisotropic media. For VTI media, I can use the true dispersion relation, and no assumption of small S-wave velocity and weak anisotropy is necessary.

Generally, the Padé approximation suggests that if the function , then Sz(Sr) can be approximated by a rational function Rn,m(Sr):
 (5)
where

and

are polynomials of degree n and m, respectively. The coefficients ai and bi can be obtained either analytically by Taylor-series analysis or numerically by least-squares fitting.

 kz1 Figure 1 Dispersion relation: curve A is the true dispersion relation; B is the aprroximate dispersion relation by Tayor-series analysis; C is the approximate dispersion relation by optimization.

 err1 Figure 2 Relative dispersion error: curve D is the relative dispersion error of the approximation by Taylor-series analysis; E is the relative dispersion error of the approximation by optimization.

We can obtain the coefficients ai and bi by solving the following optimization problem:
 (6)
where is the maximum optimization angle. This problem can be changed to
 (7)
The optimization problem (7) can be solved by a least-squares method. Given and , we can solve ai and bi from equation (7), and we can approximate kz as follows:
 (8)
As Ma (1981) suggested, if m=n, equation (8) can be further split into a rational-function series as follows:
 (9)

The dispersion error of approximation (9) is given by
 (10)
The relative dispersion error is defined by .

For the second-order approximation (m=1,n=1), Figure shows the true and approximated dispersion relation, given and . In Figure , curve A is the true dispersion relation curve. B is the approximated dispersion suggested by Ristow and Ruhl (1997), in which and . C is the approximated dispersion relation by the least-squares optimization, in which and . The dispersion relation by optimization (C) approximates the true dispersion relation better than the approximation using Taylor-series analysis and the weak anisotropy assumption.

Figure shows the relative dispersion error. D is the relative dispersion error of the approximation using the Taylor-series analysis. E is the relative dispersion error of the optimized one-way wave operator. Figure shows that opimization greatly improves the dispersion relation. If we accept a one-percent dispersion error, the optimized one-way wave-equation is accurate to while the approximation using Taylor-series analysis is accurate to only .

impulse
Figure 3
Impulse responses: (a) Optimized finite-difference method; (b) Finite-difference method by Tayor-series analysis; (c) Phase-shift method.

Next: Table-driven implicit finite-difference migration Up: Shan: Implicit migration for Previous: Introduction
Stanford Exploration Project
4/5/2006