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Vector wavefields

A similar approach can be used for decomposition of the reflectivity as a function of incidence and reflection angles for elastic wavefields imaged with extended imaging conditions equations [*] or [*]. The angle $\theta_e$ characterizing the average angle between incidence and reflected rays can be computed using the expression (Sava and Fomel, 2005)

\begin{displaymath}
\tan^2\theta_e = \frac
{ \left (1+\gamma \right)^{2} \vert{\...
...ight)^{2} \vert{\bf k}_ {\boldsymbol{\lambda}} \vert^2 } \; ,
\end{displaymath} (12)

where $\gamma$ is the velocity ratio of the incident and reflected waves, e.g. $V_P/V_S$ ratio for incident P mode and reflected S mode. Figure 1 shows the schematic and the notations used in equation [*], where $\mathbf{\vert p_x\vert=\vert k_x\vert}/\omega$, $\mathbf{\vert p_\lambda\vert=\vert k_\lambda\vert}/\omega$, and $\omega$ is the angular frequency at the imaging location ${\bf x}$. The angle decomposition equation [*] is designed for PS reflections and reduces to equation [*] for PP reflections when $\gamma=1$.

Angle decomposition using equation [*] requires computation of an extended imaging condition with 3D space lags ( $\lambda_x,\lambda_y,\lambda_z$), which is computationally costly. Faster computation can be done if we avoid computing the vertical lag $\lambda_z$, in which case the angle decomposition can be done using the expression (Sava and Fomel, 2005):

\begin{displaymath}
\tan\theta_e = \frac
{ \left (1+\gamma\right)\left (a_{\lamb...
...ambda_x}+b_{x}\right)\left (a_{x}+b_{\lambda_x}\right)} } \;,
\end{displaymath} (13)

where $a_{\lambda_x}=\left (1+\gamma\right)k_{h_x}$, $a_{x}=\left (1+\gamma\right)k_{m_x}$, $b_{\lambda_x}=\left (1-\gamma\right)k_{h_x}$, and $b_{x}=\left (1-\gamma\right)k_{m_x}$. Figure 2 shows a model of five reflectors and the extracted angle gathers for these reflectors at the location of the source. For PP reflections, they would occur in the angle gather at angles equal with the reflector slopes. However, for PS reflections, as illustrated in Figure 2, the reflection angles are smaller than the reflector slopes, as expected.
next up previous [pdf]

Next: Examples Up: Angle decomposition Previous: Scalar wavefields

2013-08-29