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

Angle decomposition

In downward continuation methods, theoretical analysis of angle gathers can be reduced to analyzing the geometry of reflection in the simple case of a dipping reflector in a locally homogeneous medium (Sava and Fomel, 2005). The behavior of plane waves in the vicinity of the reflection point is sufficient for deriving relationships for local reflection traveltime derivatives (Goldin, 1986). The geometry of the reflection ray paths is depicted in Figure 2.

rayparameters
Figure 3. A schematic plot depicting the relation between source and receiver ray-parameter vectors ( ${{\bf p}_{\bf s}}$ and ${{\bf p}_{\bf r}}$ ) and that of the space-lag and position ( ${{\bf p}_ {\boldsymbol {\lambda }} }$ and ${{\bf p}_{\bf x}}$ ). The angle $\theta$ corresponds to the phase angle direction of the plane wave.

Using the standard notations for the source and receiver coordinates: ${\bf s}={\bf x}+ {\boldsymbol{\lambda}}$ and ${\bf r}={\bf x}- {\boldsymbol{\lambda}}$ , the traveltime from a source to a receiver is a function of all spatial coordinates of the seismic experiment $t=t({\bf x}, {\boldsymbol{\lambda}} )$ . Differentiating $t$ with respect to all components of the vectors ${\bf x}$ and ${\boldsymbol{\lambda}}$ , and using the standard notations to represent slownesses ${{\bf p}_{\alpha}}=\nabla_{\alpha} t$ , where $\alpha=({\bf x}, {\boldsymbol{\lambda}} ,{\bf s},{\bf r})$ , we can write:

$${{\bf p}_{\bf x}} = {{\bf p}_{\bf r}}+ {{\bf p}_{\bf s}}\;,$$ (18)

$${{\bf p}_ {\boldsymbol{\lambda}} } = {{\bf p}_{\bf r}}- {{\bf p}_{\bf s}}\;.$$ (19)

By analyzing the geometric relations of various vectors at an image point (Figures 3), we can write the following trigonometric expressions:

$$\vert{{\bf p}_ {\boldsymbol{\lambda}} }\vert^2 = \vert{{\bf p}_{\bf s}}\vert^2 +\vert{{\bf p}_{\bf r}}\vert^2 - 2 \vert{{\bf p}_{\bf s}}\vert \vert{{\bf p}_{\bf r}}\vert \cos(2 \theta)\;,$$ (20)

$$\vert{{\bf p}_{\bf x}}\vert^2 = \vert{{\bf p}_{\bf s}}\vert^2 +\vert{{\bf p}_{\bf r}}\vert^2 + 2 \vert{{\bf p}_{\bf s}}\vert \vert{{\bf p}_{\bf r}}\vert \cos(2 \theta)\;.$$ (21)

Defining ${{\bf k}_{\bf x}}$ and ${{\bf k}_ {\boldsymbol{\lambda}} }$ as the position and lag (or offset) wavenumber vectors, we can replace ${\bf p}={\bf k}/\omega$ . Using the trigonometric identities

$$1-\cos(2\theta) = 2\sin^2(\theta)\;,$$ (22)

$$1+\cos(2\theta) = 2\cos^2(\theta)\;,$$ (23)

and assuming $\vert{{\bf p}_{\bf s}}\vert=\vert{{\bf p}_{\bf r}}\vert=s(\theta)$ , where $s(\theta)=1/v_p(\theta)$ is the phase slowness as a function of phase angle at an image location, we obtain the following relations:

$$\vert{{\bf k}_ {\boldsymbol{\lambda}} }\vert^2 = (2 \omega s(\theta) \sin(\theta))^2\;,$$ (24)

$$\vert{{\bf k}_{\bf x}}\vert^2 = (2 \omega s(\theta) \cos(\theta))^2\;,$$ (25)

$$\,\,\,\,\, \; \; \;{{\bf k}_ {\boldsymbol{\lambda}} }. {{\bf k}_{\bf x}}= 0\;.$$ (26)

We can eliminate from equations 24- 25 the dependence on the depth axis and obtain an angle decomposition formulation prior to imaging. Thus, if we eliminate ${k_z}$ and $k_{\lambda_z}$ , we obtain the expression:

$$({k_x}^2 + {k_y}^2) (2 \omega s(\theta) \sin{\theta})^2 + (k_{\lambda_x}^2+k_{\lambda_y}^2) (2 \omega s(\theta) \cos{\theta})^2 =$$

$$({k_x}k_{\lambda_y}-{k_y}k_{\lambda_x})^2 + (2 \omega s(\theta) \sin{\theta})^2 (2 \omega s(\theta) \cos{\theta})^2 \;.$$ (27)

The quadratic equation 27 can be used to map data from space-lag gathers ( $k_{\lambda_x}$ , $k_{\lambda_y}$ ) into angle coordinates $\theta$ , prior to imaging. For 2-D 22D data, equation 27 takes the simpler form

$${k_x}^2 (2 \omega s(\theta) \sin{\theta})^2 + k_{\lambda_x}^2 (2 \omega s(\theta) \cos{\theta})^2 =$$

$$(2 \omega s(\theta) \sin{\theta})^2 (2 \omega s(\theta) \cos{\theta})^2,$$ (28)

which can be solved for an explicit mapping of $k_{\lambda_x}$ to $\theta$ .

Note that the angle decomposition formula 28 reduces to a form simpler than that shown by Alkhalifah and Fomel (2009) for VTI media. This angle decomposition is particularly useful in imaging via downward continuation, as discussed next.

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