Theory of interval traveltime parameter estimation in layered anisotropic media

Comparison with the method of Xu et al. (2005)

Under the weak anisotropy assumption, Xu et al. (2005) show that the azimuthally dependent quartic coefficients in each layer can be approximated by

$$A(\alpha) = -\frac{1}{2T^2_0 V^4(\alpha)}(\eta_2\cos^2 \alpha - \eta_3\cos^2 \alpha \sin^2 \alpha +\eta_1 \sin^2 \alpha)~,$$ (65)

where $V$ denotes the interval azimuthally varying NMO velocity (equation 63) and $\eta_i$ denotes the interval $\eta$ value. Vasconcelos and Tsvankin (2006) suggest that in the case of a mild azimuthal anisotropy, one can also use equation 62 to approximate the effective value of $A$ . Therefore, the independent parameters in each layer are substituted by their azimuthally variant counterparts (equations 63 and 65). Analogous to the case of equation 64, the effective $A$ is then multiplied by the source-receiver distance along the CMP line in order to evaluate traveltime in polar coordinates.

Figure 1 shows the resultant azimuthal error plots of the quartic term $a_{ijkl}x_ix_jx_kx_l/r^4$ where $r=\sqrt{x^2_1+x^2_2}$ denotes the offset along the CMP line. Two models of three-layered aligned orthorhombic models with different degree of anisotropy are considered and the model parameters are given in Table 2. qP reflection traveltimes and errors from including traveltime approximation up to hyperbolic and quartic terms are shown in Figure 2 and 3. The VTI based approximation produces small errors when applied in weak anisotropic media. However, the errors increase noticeably with the strength of anisotropy of the media. Note that the observed errors result from the cumulative effect of the use of the pseudoacoustic approximation (equations 58-60) and of the VTI averaging formula (equation 62). The separate effects from pseudoacoustic approximation alone are shown in the same figures and are denoted with large-dashed blue lines. It can be seen from the results that the error from pseudoacoustic approximation dominates the error from the VTI averaging formula in media with a higher degree of anisotropy. The total error on the quartic term $a_{ijkl}x_ix_jx_kx_l$ can be computed from the azimuthal error amplified by $r^4$ and grows with larger distance $r$ along the CMP line.

errorweak,error
Figure 1. Azimuthal relative error of the quartic term $a_{ijkl}x_ix_jx_kx_l/r^4$ where $r^2=x^2_1+x^2_2$ denoting the source-receiver distance along the CMP line in a) layered model 1 and b) layered model 2 based on methods by Al-Dajani et al. (1998) (small-dashed green) and Xu et al. (2005) (Solid red). The large-dashed blue line denotes the errors solely from pseudoacoustic approximation with the correct averaging formulas (equations 45-47). The azimuthal error shown remains constant for all offsets. The total error is equal to the multiplication of azimuthal error with $r^4$ .

exacttwtimeweak,nonhyperweak,nonhyperqweak
Figure 2. a) The exact reflection traveltime ($4 t^2$ ) for Model 1 given in Table 2, b) the difference between the exact reflection traveltime and the hyperbolic moveout, and c) the difference between the exact reflection traveltime and the quartic moveout. Since the reflector depth at the bottom of the third layer is equal to 1 in this case, the nonhyperbolicity effects should have noticeable when $x_1$ and $x_2$ are greater than 1. Taking into account the quartic term improves the accuracy of traveltime approximation at large offsets.

exacttwtime,nonhyper,nonhyperq
Figure 3. a) The exact reflection traveltime ($4 t^2$ ) for Model 2 given in Table 2, b) the difference between the exact reflection traveltime and the hyperbolic moveout, and c) the difference between the exact reflection traveltime and the quartic moveout. Since the reflector depth at the bottom of the third layer is equal to 1 in this case, the nonhyperbolicity effects should have noticeable when $x_1$ and $x_2$ are greater than 1. Taking into account the quartic term improves the accuracy of traveltime approximation at large offsets.

Theory of interval traveltime parameter estimation in layered anisotropic media