The accuracy of the VTI equation for horizontal reflectors

Setting x-x₀=0 in the offset-midpoint traveltime equation yields a formula that describes the moveout of reflections from horizontal reflectors. Since many of these moveout equations exist Hake et al. (1984); Tsvankin and Thomsen (1994), I will compare the offset axis of our VTI offset-midpoint traveltime equation with these equations, as well as with the exact solution computed numerically. Specifically, I will measure the difference in traveltimes between the various moveout approximations and the numerically computed solution. This difference is given in terms of the percentage error in traveltime as a function of offset. Clearly all approximations yield the exact solution for zero-offset, since they are derived based on this limit.

For a horizontal reflector x-x₀=0, as well as p_x=0, in equations (12) and (13), and thus they reduce to  

$$T(\tau,h,v,\eta)=2\,h\,{p_h} + \tau \,{\sqrt{1 - {\frac{{v^2}\,{{{p_h}}^2}} {1 - 2\,{v^2}\,\eta \,{{{p_h}}^2}}}}},$$ (17)

where

$$p_h={\frac{{X^2}\,\left( {X^6} - 6\,{X^4}\,{v^2}\,\left( -1 ... ...eta \right) \,{{\tau }^4} + 4\,{v^6}\,{{\tau }^6} \right) }},$$

and X is the offset.

Hake et al. (1984) derived a three-term Taylor series expansion for the moveout of reflections from horizontal interfaces in homogeneous, VTI media. Their traveltime equation can be simplified when expressed in terms of $\eta$ and v Alkhalifah and Tsvankin (1995), as follows:  

$$t^2(X)= \tau^2 + \frac{X^2}{v^2} - \frac{2 \eta X^4}{t^2_{0} v^4}.$$ (18)

The first two terms on the right correspond to the hyperbolic portion of the moveout, whereas the third term approximates the nonhyperbolic contribution. Note that the third term (fourth-order in X) is proportional to the anisotropy parameter $\eta$, which therefore controls nonhyperbolic moveout directly.

Tsvankin and Thomsen (1994) derived a correction factor to the nonhyperbolic term of Hake et al's 1984 equation that increases the accuracy and stabilizes traveltime moveout at large offsets in VTI media. The more accurate moveout equation, when expressed in terms of $\eta$ and v Alkhalifah and Tsvankin (1995), and slightly manipulated, is given by  

$$t^2(X)= \tau^2 + \frac{X^2}{v^2} - \frac{2 \eta X^4}{v^2 [t^2_{0} v^2+ (1+2 \eta) X^2]}.$$ (19)

 

Error2eta
Figure 3 Percentage errors in traveltime moveout from a horizontal reflector as a function of offset. The solid gray curve corresponds to using Hake et al's equation (18), the solid black curve corresponds to using equation (19), and the dashed gray curve corresponds to using our new equation (17). The medium is VTI with $\eta=0.2$ (left), and $\eta=0.4$(right). The velocity is 2 km/s and the vertical traveltime is 2 s.

Figure 3 shows the percentage error in traveltime moveout as a function of offset for the three moveout equations given above. Clearly, the offset-axis component of the new VTI pyramid equation (dashed gray curve) has less errors and is by far more accurate than the traveltime moveout given by either the three-term Taylors series equation (18) (solid gray curve) or the modified traveltime moveout equation (19) (solid black curve). In fact, the errors in equation (17) for moderate anisotropy, given by $\eta=0.2$ (left), and relatively strong anisotropy, given by $\eta=0.4$ (right), are practically zero for offsets-to-depth ratio up to 3, shown here.

 

Erroreta1 Figure 4 Same as Figure 3, but with $\eta=1.0$, which is an extremely high value for $\eta$, not common in the subsurface.

We have to use a model with $\eta$ equals the huge value of 1, as shown in Figure 4, before observing any sizable errors in the new equation. Even for such a huge anisotropy, the errors are again practically zero for offsets-to-depth ratio below 1.5. Therefore, for all intent and purposes in seismic applications, equation (8) for horizontal reflectors is practically exact. Next, we test the accuracy of the midpoint-offset pyramid equation for dipping reflectors. We do that by prestack migrating synthetic data that include dipping reflections using this new equation.