A transversely isotropic medium with a tilted symmetry axis normal to the reflector

Moveout analysis

If we restrict the observation to the immediate vicinity of the reflection point, which means that we consider the moveout surface in a small range of lags, we can approximate the typical irregular wavefront in complex media by a plane, although the shapes of wavefronts are arbitrary in heterogeneous media. Following the derivation of Yang and Sava (2009) and using the geometry shown in Figure 2, the source and receiver plane waves are described by:

$$\textbf n_{\bf s}\cdot {\bf x} = v(\theta) t \;,$$ (8)

$$\textbf n_{\bf r}\cdot \left ({\bf x}- 2d{\bf n}\right) = v(\theta) t \;,$$ (9)

where $\textbf n_{\bf s}$ and $\textbf n_{\bf r}$ are the unit direction vectors of the source and receiver plane waves, respectively, ${\bf n}$ is the unit vector orthogonal to the reflector at the image point, and vector ${\bf x}$ indicates the image point position. $v$ is defined as the phase velocity in the locally homogeneous medium around the reflection point, and thus it is identical for both wavefields. $\theta$ is half the scattering angle (reflection angle).

We can also obtain the shifted source and receiver plane waves by introducing the space- and time-lags

$$\textbf n_{\bf s}\cdot \left ({\bf x}+ {\boldsymbol{\lambda}} \right) = v(\theta) \left (t + \tau \right)\;,$$ (10)

$$\textbf n_{\bf r}\cdot \left ({\bf x}- 2d{\bf n}- {\boldsymbol{\lambda}} \right) = v(\theta) \left (t - \tau \right)\;.$$ (11)

Solving the system of equations 10-11 leads to the expression

$$\left (\textbf n_{\bf s}- \textbf n_{\bf r}\right)\cdot {\bf x}= ... ...bf r}\right)\cdot {\boldsymbol{\lambda}} - 2d \textbf n_{\bf r}\cdot {\bf n}\;,$$ (12)

which characterizes the moveout function (surface) of space- and time-lags at a common-image point.

Furthermore, we have the following relations for the reflection geometry:

$$\textbf n_{\bf s}- \textbf n_{\bf r} = 2 {\bf n}\cos \theta \;,$$ (13)

$$\textbf n_{\bf s}+ \textbf n_{\bf r} = 2 {\textbf{q}}\sin \theta \;,$$ (14)

where ${\bf n}$ and ${\textbf{q}}$ are unit vectors normal and parallel to the reflection plane, respectively, and $\theta$ is the reflection angle. Vector ${\textbf{q}}$ characterizes the line representing the intersection of the reflection and the reflector planes. Combining equations 12-14, we obtain the moveout function for plane waves:

$$z \left ( {\boldsymbol{\lambda}} , \tau \right)= d_0 - \frac{\tan... ...ldsymbol{\lambda}} \right)}{n_z} + \frac{{v(\theta) \tau}}{n_z\cos \theta } \;.$$ (15)

The quantity $d_0$ is defined as

$$d_0 = \frac{d - \left ( {\bf c}\cdot {\textbf{n}}\right)}{n_z} \;,$$ (16)

and represents the depth of the reflection corresponding to the chosen 2common-image gather (CIG) location. This quantity is invariant for different plane waves, thus assumed constant here. The vector ${\bf {c}}$ is along the Earth's surface given by $(x,y,0)$ .

When incorrect velocity is used for imaging, and thus, an inaccurate reflection angle is assumed, based on the analysis in the preceding section, we can obtain the moveout function

$$z \left ( {\boldsymbol{\lambda}} , \tau \right)= d_{0f} - \frac{\... ...z}} + \frac{v_{m}(\theta_m) \left (\tau - t_d\right)}{n_{mz}\cos \theta _m} \;,$$ (17)

where $d_{0f}$ is the focusing depth of the corresponding reflection point, $v_{m}$ is the migration velocity, $t_d$ is the focusing error, ${\textbf{n}}_{m}$ and ${\textbf{q}}_m$ are vectors normal and parallel to the migrated reflector, respectively. Equation 17 describes the extended images moveout for a single seismic experiment and it is essentially identical to the similar formula obtained by Yang and Sava (2009) for isotropic media, but for using the phase velocity instead of the isotropic velocity.

A transversely isotropic medium with a tilted symmetry axis normal to the reflector