Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference |

where , and . Anisotropic parameter relates the vertical P-wave velocity with the NMO velocity , while the anellipticity parameter relates the horizontal velocity with the NMO velocity . Shan (2009) suggests that the exact dispersion relationship 1 can be approximated by a rational function :

where

and

Moreover, when the polynomials in equations 3 and 4 are of the same degree, namely , dispersion relationship 2 can be further split as follows:

The coefficients and can be obtained by solving the least-square problem below:

For second-order coefficients the exact and approximated dispersion curves are shown in Figure 1(a), given and . Curve A is the exact dispersion relation from Equation 1. Curve B is obtained from a previous estimation by Ristow and Ruhl (1997), and curve C is obtained using the optimized coefficients. Apparently, the dispersion relation using the optimized coefficients is a better approximation compared with the previous method which uses Taylor expansion and assumes weak anisotropy. The relative errors between these two approximated curves and the exact dispersion curve are plotted in Figure 1(b). Within a tolerance of 1% relative error in the dispersion relation, the optimized dispersion is accurate up to , while the Taylor approximation is only accurate up to .

kz1,err1
(a) Dispersion relation curves: A, exact dispersion relation curve from equation 1;
B, approximated dispersion curve from weak anisotropy and Taylor expansion; C, approximated dispersion curve from optimization. (b) Relative dispersion
error: D, relative error between B and A; E, relative error between C and A.
Figure 1. |
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The tables for coefficients and for ranging from 0 to and ranging from to are shown in Figure 2. In general, parameter is more sensitive to the change in than to the change in . Parameter has similar sensitivities to both and .

coef
(a) Table for
and (b) table for
at discrete
and
locations.
Figure 2. |
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Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference |

2012-05-10