Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium

Velocity Independent $\tau$ -p moveout in VTI

Under the quasi-acoustic approximation (Alkhalifah, 1998,2000) in a VTI homogeneous layer, the vertical slowness $q$ can be expressed as a function of the horizontal slowness $p$ :

$$q(p)=\frac{1}{\hat{V}_{P0}}\sqrt{\dfrac{1-\hat{V}_{H}^{2}p^{2}}{1-[\hat{V}_{H}^{2}-\hat{V}_{N}^{2}]p^{2}}},$$ (5)

where $\hat{V}_{H}$ and $\hat{V}_{N}$ are the horizontal and normal moveout velocity in the layer. In the following equations, the hat superscript $(\symbol{94})$ indicates layer or interval parameters. Considering a stack of $N$ horizontal homogeneous layers with horizontal symmetry planes, we can insert equation 5 into equation 4 to obtain an expression for the $\tau$ -$p$  reflection time from the bottom of the $N$ -th layer, as follows:

$$\tau (p)=\sum\nolimits_{n=1}^{N}\sqrt{\dfrac{1-\hat{V}_{H,n}^{2}p^{2}}{1-[\hat{V}_{H,n}^{2}-\hat{V}_{N,n}^{2}]p^{2}}}\Delta \tau _{0,n}$$ (6)

where $\Delta \tau _{0,n}$ is the two way vertical time for the $n$ -th layer. To simplify the following theoretical derivations, we assume that, instead of having a layered velocity model, interval parameters are vertically-varying continuous profiles. Therefore, we replace the summation in formula 6 with an integral along the vertical time $\tau _{0}$ and arrive at the following $\tau$ -$p$  moveout formula for a vertically-heterogeneous VTI medium:

$$\tau (p)=\displaystyle \int \limits_{0}^{\tau _{0}}\sqrt{ \dfrac{... ...^{2}(\xi )p^{2} } {1-[\hat{V}_{H}^{2}(\xi )-\hat{V}_{N}^{2}(\xi )]p^{2}}}d\xi ,$$ (7)

where $\hat{V}_{N}=\hat{V}_{N}(\xi)$ and $\hat{V}_{H}=\hat{V}_{H}(\xi)$ are (smooth) functions for interval NMO and horizontal velocities, and $\tau _{0}$ is the vertical time. The vertical heterogeneity is measured as a function of $\tau _{0}$ . The anellipticity parameter $\hat{\eta}=\frac{1}{2}\left( \frac{\hat{V}^2_{H}}{\hat{V}^2_{N}}-1\right)$ is also a function of the vertical time $\tau _{0}$ . Using effective parameters, equation 7 can be approximated by

$$\tau (p)\approx \tau _{0}\sqrt{\dfrac{1-V_{H}^{2}(\tau _{0})p^{2}}{1-[V_{H}^{2}(\tau _{0})-V_{N}^{2}(\tau _{0})]p^{2}}},$$ (8)

where the effective NMO $V_{N}$ and horizontal $V_{H}$ velocity are related to the interval parameters through the second- and fourth-order average velocities (Taner and Koehler, 1969; Ursin and Stovas, 2006) by the following direct Dix-type formulas:

$$V_{N}^{2}(\tau _{0})=\dfrac{1}{\tau _{0}}\displaystyle \int \limits_{0}^{\tau _{0}}\hat{V}_{N}^{2}(\xi )d\xi ,$$ (9)

$$S(\tau _{0})V_{N}^{4}(\tau _{0})=\dfrac{1}{\tau _{0}}\displaystyle \int \limits_{0}^{\tau _{0}}{}\hat{S}(\xi )\hat{V}_{N}^{4}(\xi )d\xi ,$$ (10)

where $S$ is the ratio between the fourth- and second-order moments or the heterogeneity factor (Alkhalifah, 1997; Siliqi and Bousquié, 2000; de Bazelaire, 1988).

Equation 8 is basically the four-parameters rational approximation defined in $\tau$ -$p$ domain Stovas and Fomel (2010)

$$\tau (p)\approx \tau _{0}\sqrt{1-V_{N}^{2} p^{2} + \frac{A V_{N}^{4} p^4}{1-B V_{N}^{2} p^2 } } ,$$ (11)

with parameter $A=(1-S)/4$ defined from the Taylor series expansion of the exact $\tau$ -$p$  function (Ursin and Stovas, 2006). Under the acoustic VTI approximation, $B=-A$ and the equation 11 now depends on three parameters only. Finally, to be consistent with equation 8, the heterogeneity coefficient becomes $S=4\dfrac{V_H^2}{V_N^2}-3$ . In principle, it is possible to use any other three-parameters approximation in $\tau$ -$p$  domain apart from the rational approximation 11 like, for example, the shifted ellipse approximation given by Stovas and Fomel (2010). The reason for choosing the approximation 8 is that it accurately describes the $\tau$ -$p$   moveout for a single VTI layer (blue line in figure 2b). Nevertheless, approximation 8 remains valid for vertically heterogeneous VTI media with a decrease in accuracy for larger angles (large values of $p$ ) because of the Dix averaging

Letting $R$ represent the slope $\tau ^{\prime }(p)$ and $Q$ the curvature $\tau ^{\prime \prime }(p)$ , we differentiate equation 8 once

$$R(\tau ,p)=-\frac{{\tau (V_{H}^{2}-Y)p}}{{\left( {1-p^{2}Y}\right) \left( {1-p^{2}V_{H}^{2}}\right) }},$$ (12)

and twice

$$Q(\tau ,p)=-\frac{{\tau F(p)(V_{H}^{2}-Y)}}{{\left( {1-p^{2}Y}\right) ^{2}\left( {1-p^{2}V_{H}^{2}}\right) ^{2}}},$$ (13)

where $F(p)=1+2p^{2}Y-3p^{4}V_{H}^{2}Y$ with $Y=V_{H}^{2}-V_{N}^{2}.$ Equations 12 and 13 provide an analytical description of the slope and curvature fields for given effective values $V_{N}$ and $V_{H}$ . Here we have omitted the ${\tau _{0}}$ dependency for clarity in the notation. Since $\tau$ -$p$  and $t$ -$X$   domains are mapped by the linear transformation in equation 2, we observe that

$$\tau ^{\prime }(p)=R=-x.$$ (14)

Thus, the negative of the slope $R$ has the physical meaning of emerging offset, as pointed out by van der Baan (2004). Moreover, when the curvature $Q$ changes sign, there is an inflection point in the $\tau$ -$p$ wavefront that is as a condition for caustics in $t$ -$X$ domain. (Roganov and Stovas, 2011).

Given slope $R$ and curvature $Q$ fields in a $\tau$ -$p$  CMP gather, we can eliminate the velocity $V_{N}$ and the parameter $Y$ in equations 12 and 13, thus obtaining a `` velocity-independent'' (Fomel, 2007b) moveout equation in the $\tau$ -$p$  domain:

$$\tau _{0}(\tau ,p)=\tau \sqrt{\frac{{\tau pQ+3\tau R-3pR^{2}}}{{\tau pQ+3\tau R+pR^{2}}}.}$$ (15)

Equation 15 describes a direct mapping from events in the prestack $\tau$ -$p$  data domain to zero-slope time $\tau _{0}$ . This equation represents the oriented or slope-based moveout correction. As follows from equations 12 and 13, the effective parameters, if needed for other tasks, are given by the following relations as a function of the slope and curvature estimates (see Table 1):

$$V_{N}^{2}(\tau, p) = -\frac{1}{p}\frac{{16\tau R^{3}}}{{ND}},$$ (16)

$$V_{H}^{2}(\tau, p) = \frac{1}{{p^{2}}}\dfrac{{N-4\tau R}}{N},$$ (17)

and

$$\eta(\tau, p)=\frac{1}{p}\frac{{N(4\tau R-D)}}{{32\tau R^{3}}}.$$ (18)

In the above equations, $N={\tau pQ+3\tau R-3pR^{2}}$ and $D={\tau pQ+3\tau R+pR^{2}}$ represent the terms in the numerator $N$ and denominator $D$ of the square root in equation 15. In the isotropic or elliptically anisotropic case ($V_N=V_H$ or $\eta=0$ ), equations 15 to 18 simplify to equations

$$\tau _{0}(\tau ,p) = \sqrt{\tau ^{2}-\tau pR}$$ (19)

and

$$V_{N}^{2}(\tau ,p) = \frac{R}{{p(pR-\tau )}}\;,$$ (20)

previously published by Fomel (2007b).

The anisotropic parameters $V_{N}$ and $V_{H}$ (or $\eta$ ) are no longer a requirement for the moveout correction, as in the case of conventional NMO processing, but rather they are data attributes derived from local slopes and curvatures. Moreover, these parameters are mappable directly to the appropriate zero-slope time $\tau _{0},$ according to equation 15.

Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium