Curved reflector in a homogeneous VTI medium

In the case of a dipping curved reflector in a homogeneous VTI medium, the ray trajectories of the incident and reflected waves are straight, but the location of the reflection point is no longer controlled by the isotropic laws. In order to obtain analytic expressions for this case, we use the theorem, which connects the derivatives of the common-midpoint traveltime curves with the derivatives of the forward traveltimes for the imaginary wave originating at the reflection point of the central ray. This theorem has been introduced for the second-order derivatives by Chernjak and Gritsenko 1979 and is usually referred to as the normal incidence point (NIP) theorem Hubral and Krey (1980); Hubral (1983). Though the original proof didn't address the case of anisotropy, it is applicable to this case as well, being based on the fundamental Fermat's principle. The normal incidence point in the anisotropic case should be replaced by the point of incidence for the central ray (which is in general not normal to the reflector surface). In the appendix, we review the NIP theorem as well as its extension for high-order traveltime derivatives Fomel (1994).

Two basis formulas derived in the appendix take the following form:

$$\begin{eqnarray} \left.{{\partial^2 t} \over {\partial h^2}}\right\vert _{h=0} &... ...left({{\partial^3 T} \over {\partial y^2\,\partial x}}\right)^2\;,\end{eqnarray}$$ (73) (74)

where T(x,y) is the traveltime of the direct wave propagating from the reflector point x to the point y at the surface z=0. All the derivatives in formulas (73) and (74) are evaluated in the vicinity of the central (zero-offset) ray. Both formulas are based solely on Fermat's principle and therefore remain valid in any type of medium for reflectors of arbitrary shape, assuming that the traveltimes possess the required order of smoothness. It is especially convenient to use formulas (73) and (74) in the case of homogeneous media, because the forward traveltime in this case has an explicit expression.

In order to apply formulas (73) and (74) to the VTI case, we need to start by tracing the central ray. According to Fermat's principle, the ray trajectory must correspond to the extremum value of the traveltime. For the central ray, this simply means that in the vicinity of the central ray, the traveltime of the direct ray satisfies the equation  

$${{\partial T} \over {\partial x}} = 0\;,$$ (75)

where  

$$T(x,y) = {\sqrt{z^2(x) + (x-y)^2} \over {V_g(\psi(x,y))}}\;,$$ (76)

z(x) describes the reflector surface, and $\psi$ is the ray angle, which satisfies the evident trigonometric relationship (see Figure 5)  

$$\cos{\psi(x,y)} = {z(x) \over \sqrt{z^2(x) + (x-y)^2}}\;.$$ (77)

Substituting approximate equation (9) for the group velocity V_g into formula (76) and linearizing with respect to the anisotropic parameters $\delta$ and $\eta$, we can solve equation (75) for y, obtaining  

$$y = x + z\,\tan{\alpha}\,(1 + 2\,\delta + 4\,\eta\,\sin^2{\alpha})$$ (78)

or, in other terms,  

$$\tan{\psi} = \tan{\alpha}\,(1 + 2\,\delta + 4\,\eta\,\sin^2{\alpha})\;,$$ (79)

where $\alpha$ is the local dip angle of the reflector at the reflection point x. Equation (79) clearly shows that in VTI media the central ray angle $\psi$ differs from the dip angle $\alpha$. As one can expect, the difference is approximately proportional to Thomsen's anisotropic parameters.

 

nmoray Figure 5 Zero-offset reflection from a curved reflector in a VTI medium (a scheme). Note that the ray angle is not equal to the local dip angle.

Now we can apply formula (73) to evaluate the second term of the Taylor series expansion (26) for the case of a curved reflector. The linearization in anisotropic parameters in this case leads to the expression  

$$a_1 = {1 \over V_n^2} = {\cos^2{\alpha} \over {V_z^2\, \lef... ...ha}) + 6\,\eta\,\sin^2{\alpha}\,(1+\cos^2{\alpha})\right)}}\;,$$ (80)

which is equivalent to Tsvankin's result Tsvankin (1995). As in the isotropic case, the normal moveout velocity does not depend on the curvature. Its dip dependence is an important indicator of anisotropy, especially in areas of conflicting dips Alkhalifah and Tsvankin (1995).

Finally, we can apply formula (74) to determine the third coefficient of the Taylor series. After linearization in anisotropic parameters and lengthy algebra, the resulting expression takes the form  

$$a_2 = {A \over {V_n^4\,t_0^2}}\;,$$ (81)

where

$$\begin{eqnarray} A = G\,\tan^2{\alpha} + 2\,\delta\,G\,\sin^2{\alpha}\,(2 + \ta... ...\,\cos^2{\alpha} + \sin^2{\alpha}\,(\tan^2{\alpha}-3\,G)\right)\;,\end{eqnarray}$$ (82)

and the coefficient G is defined by equation (65). In the case of a zero curvature (a plane reflector), G is also equal to zero, and the only term remaining in formula (82) is  

$$A = - 2\,\eta\,(1 - 4\,\sin^2{\alpha})\;.$$ (83)

In the general case of a curved reflector, we can rewrite the isotropic formula (70) in the form  

$$t^2(h) = t_0^2 + {h^2 \over V_n^2} + {{A\,h^4} \over {V_n^2\,\left(V_n^2 t_0^2 + G\,h^2\right)}}\;,$$ (84)

where the normal moveout velocity V_n is given by (80). Equation (84) approximates nonhyperbolic moveouts in a VTI medium with a curved reflector. In the isotropic case, it reduces to formula (70). In the case of a small curvature, the accuracy of formula (84) at finite offsets can be increased by modifying the denominator term.