HORIZONTAL REFLECTOR IN A HOMOGENEOUS VTI MEDIUM

To exemplify the use of weak anisotropy, let us consider the simplest case of a homogeneous anisotropic medium with a horizontal reflector. In the isotropic case, the reflection traveltime curve is an exact hyperbola, which follows directly from Pythagoras's theorem (see Figure 2):  

$$t^2(h) = {{z^2 + h^2} \over V_z^2} = t_0^2 + {h^2 \over V_z^2}\;,$$ (13)

where z denotes the depth of the reflector, h is the half-offset, t₀=t(0) is the zero-offset traveltime, and V_z corresponds to half of the actual isotropic velocity. In the case of a homogeneous VTI medium, the velocity V_z in formula (13) is replaced by the angle-dependent group velocity V_g. This replacement leads to the exact traveltimes, if no approximation for the group velocity is used, since the ray trajectories in homogeneous VTI media remain straight, and the reflection point does not shift because of the vertical axis symmetry. We can also obtain an approximate traveltime using the approximate velocity V_g defined in equation (1) or (9). From the simple trigonometric considerations, the ray angle $\psi$ in this case is defined by the equation  

$$\sin^2{\psi} = {{h^2} \over {z^2 + h^2}}\;.$$ (14)

Substituting equation (14) into (9) and linearizing the expression  

$$t^2(h) = {{z^2 + h^2} \over V_g^2(\psi)}$$ (15)

with respect to anisotropic parameters $\delta$ and $\eta$, we arrive at the three-parameter nonhyperbolic approximation Tsvankin and Thomsen (1994)  

$$t^2(h) = t_0^2 + {h^2 \over V_n^2} - {{2\,\eta\,h^4} \over {V_n^2\,\left(V_n^2 t_0^2 + h^2\right)}}\;,$$ (16)

where the normal moveout velocity V_n is defined by equation (6). At small offsets $(h \ll z)$, the influence of the parameter $\eta$ is negligible, and the traveltime curve is nearly hyperbolic. At large offsets $(h \gg z)$, the third term of equation (16) has a clear influence on the behavior of the traveltime. The Taylor series expansion of formula (16) in the vicinity of the vertical zero-offset ray has the form  

$$t^2(h) = t_0^2 + {h^2 \over V_n^2} - {{2\,\eta\,h^4} \over ... ...\,t_0^2}} + {{2\,\eta\,h^6} \over {V_n^6\,t_0^4}} - \ldots \;.$$ (17)

When the offset h approaches infinity, the traveltime approximately satisfies the intuitively reasonable relationship  

$$\lim_{h \rightarrow \infty} t^2(h) = {h^2 \over V_x^2}\;,$$ (18)

where the horizontal velocity V_x is defined by (5). Approximation (16) is analogous, within the weak anisotropy assumption, to the ``skewed hyperbola'' formulas Byun et al. (1989); Harlan (1995), which use the three velocities V_z, V_n, and V_x as the parameters of the approximation, as follows:  

$$t^2(h) = t_0^2 + {h^2 \over V_n^2} - {{h^4} \over {V_n^2 t_0^2 + h^2}}\, \left({1 \over V_n^2} - {1 \over V_x^2}\right)\;.$$ (19)

The accuracy of formula (16) for many realistic situation lies within 1% error and can be further improved at a finite offset by modifying the denominator of the third term Alkhalifah and Tsvankin (1995); Grechka and Tsvankin (1996).

 

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Figure 2 Reflected rays in a homogeneous layer with a horizontal reflector (a scheme).

The anelliptic moveout approximation suggested by Muir and Dellinger 1985 has the form  

$$t^2(h) = {{t_0^4 + (1 + f)\,{h^2 \over V_n^2} + f^2 {{h^4} \... ...-f)\,h^4} \over {V_n^2\,\left(V_n^2 t_0^2 + f\,h^2\right)}}\;,$$ (20)

where f is a dimensionless parameter of anellipticity. At large offsets, formula (20) approaches  

$$\lim_{h \rightarrow \infty} t^2(h) = f\,{h^2 \over V_n^2}\;.$$ (21)

Comparing equations (18) and (21), we can establish the correspondence  

$$f = {{V_n^2} \over {V_x^2}} = {{1 + 2\,\delta} \over {1 + 2\,\epsilon}} \approx 1 - 2\,\eta\;.$$ (22)

Taking this correspondence into account, we can see that formula (20) is approximately equivalent to formula (16) in the sense that their difference has the order of $\eta$ squared.