Angle gathers in wave-equation imaging for transversely isotropic media

The depth slice approach for VTI media

Relations between image coordinates and reflection (scattering) angles at reflecting interfaces can be extracted by analyzing the geometry of reflections in the simple case of a dipping reflector in a locally homogeneous medium (Fomel, 2004). The geometry of the reflection ray paths in 2-D is depicted in Figure 1(a).

raysr,rayparameters
Figure 1. (a) A schematic plot showing angle $\theta$ . Although the model depicts a homogeneous setting, the development will rely on the ray parameters defined in the immediate vicinity of the the reflection point, as shown in b. (b) A schematic plot depicting the relation between the source and receiver ray-parameter vectors ( $\mathbf{p}_s$ and $\mathbf{p}_g$ ) and the offset and midpoint vectors ( $\mathbf{p}_h$ and $\mathbf{p}_m$ )

According to elementary rules of geometry for the ray configuration in Figures 1(a) and 1(b), with the wavenumber vector given by $\mathbf{k} =\omega\,\mathbf{p}$ as it relates to the ray-parameter vector for a given angular frequency $\omega$ , reflection 2opening (scattering) phase angle $\theta$ is represented by the following relation (Fomel, 2004; Sava and Fomel, 2005)

$$k_{\text{hx}}^2+k_{\text{hz}}^2 = k_s^2+k_r^2 - 2 k_r k_s \cos (\theta )\;,$$ (1)

where $k_{\text{hx}}$ and $k_{\text{hz}}$ are horizontal and vertical components of the offset wave number, and $k_s$ and $k_r$ are source and receiver wavenumber amplitudes related to their components as follows: $k_s^2 \equiv k_{\text{sx}}^2+k_{\text{sz}}^2$ , $k_r^2 \equiv k_{\text{rx}}^2+k_{\text{rz}}^2$ , with $k_{\text{hx}} \equiv k_{\text{rx}} - k_{\text{sx}}$ , $k_{\text{mx}} \equiv k_{\text{rx}} + k_{\text{sx}}$ , as suggested by Figure 1(b), where $k_{\text{mx}}$ is the horizontal component of the midpoint wavenumber.

To complete the system of equations necessary to relate angle $\theta$ to midpoint and offset 2horizontal wavenumbers, we use the dispersion relation developed by Alkhalifah (1998) to define each of $k_{\text{sz}}$ and $k_{\text{rz}}$ as follows:

$$k_{\text{sz}}^2 \equiv (\omega \frac{\partial t_{\text{s}}}{\partial z})^2=\frac{\omega ... ...(2 \omega ^2-v^2 \eta \left(k_{\text{hx}}-k_{\text{mx}}\right){}^2\right)}\;,$$ (2)

$$k_{\text{rz}}^2 \equiv (\omega \frac{\partial t_{\text{r}}}{\partial z})^2=\frac{\omega ... ...(2 \omega ^2-v^2 \eta \left(k_{\text{hx}}+k_{\text{mx}}\right){}^2\right)}\;,$$ (3)

2where $v$ is the NMO velocity. Using equation (1) in its expanded form and after some manipulation and collecting terms with the same power of $\cos \theta$ , we end up with the following quadratic equation:

$$a \cos ^4(\theta )+b \cos ^2(\theta )+c=0,$$ (4)

with solutions given by

$$\theta = \cos^{-1}\left(\pm \sqrt{\frac{-b\pm \sqrt{b^2-4 a c}}{2 a}}\right)\;.$$ (5)

Analytical representation of the coefficients is shown in Table 1. The four solutions of equation (5) are controlled by the sign of the offset wavenumber and its magnitude compared with the midpoint wavenumber. In the frequency-wavenumber domain, equation (5) can be used to map offset 2(horizontal) wavenumbers to angle gathers for a specific frequency, midpoint 2(horizontal) wavenumber, and depth slice. A description of an algorithm to use with the mapping equation, in the case of an isotropic medium, is given by Fomel (2004).

Setting $\eta =0$ yields mapping for elliptical anisotropy with coefficients of equation (5) given by Table 2. The coefficients are represented by much simpler formulas. In the isotropic case, $\eta =0$ and $v_z=v$ , Table 1 reduces to Table 3 and, if substituted into the mapping formula of equation (5), is equivalent to the corresponding mapping equation of Fomel (2004).

Angle gathers in wave-equation imaging for transversely isotropic media