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Computation of the local snell parameter

Inversion methods which use the directional dependence of the reflection coefficient to estimate the elastic parameters of the medium (AVO inversion) use in general an angular functionality to express such a dependence. This choice is not the more convenient because the angle estimation is strongly dependent on the macro model that was used in the estimation process. Moreover, the propagation angles depend on the elastic perturbations that it is been used to estimate. A more appropriate choice for expressing the directional dependence of the reflection coefficient is the local Snell parameter, which is defined as the component of the slowness parallel to the ``reflector plane" at each position of the underground. Evidently, not all points of the subsurface can be considered as a reflector, but at all points of interest; that is, where the the upcoming wavefronts intercepts the downgoing wavefront, a ``reflector plane can be defined". As defined, the local Snell parameter is conserved for any perturbation in the local elastic parameters. Although its estimated value will be still dependent on the macro model, it will be much less sensitive to errors in the model than the angle.

Figure [*] shows a descending (incident) wavefield crossing an ascending (reflected) wavefield at a given time step of the backward propagation part of the scheme. The crossing point defines the point of the interface that was imaged at that time, and the angles $\alpha$ and $\beta$i are measured, respectively, from the tangent and from the normal to the interface at that point.

Figure 1
The points where the ascending and descending wavefronts overlap define the location of the reflector. The reflection angle can be determined by the gradients of the two wavefields at the reflection point, at the time when the reflection occurred.

From the figure we get the following relation,

\cos (2\alpha) = {\bf \hat{i} \cdot \hat{r}},\end{displaymath}

where the unit vectors ${\bf \hat{i}}$ and ${\bf \hat{r}}$ represent, respectively, the directions of incidence and reflection. I define the local Snell parameter $\tilde{p}$ as the slowness component parallel to the interface at the reflection point  
\tilde{p} \; \; = \; \; {\sin (\theta) \over V_p} \; \; = \; \;
\sqrt{{\bf \hat{i} \cdot \hat{r}} + 1 \over 2 V_p^2},\end{displaymath} (3)
where Vp is the P wave group velocity at that particular location. This definition is restricted to an isotropic assumption, but a more general definition can be formulated that includes the anisotropic extension of Snell's law.

The unit vectors ${\bf \hat{i}}$ and ${\bf \hat{r}}$ are estimated from the particle velocity field, by using the following equations:
{\bf \hat{i}} & = & \mbox{sign}({\partial v^d \over \partial t}...
 ...l v^u \over \partial t}) 
{\nabla v^u \over \mid \nabla v^u \mid},\end{eqnarray}
where vd and vu are, respectively, the downgoing and the upcoming scalar particle velocity wavefields, defined by the equation
v^d & = & \mbox{sign}(\dot{u}_z) \sqrt{\dot{u}_x^2 + \dot{u...
 ...& = & \mbox{sign}(\dot{w}_z) \sqrt{\dot{w}_x^2 + \dot{w}_z^2}.\end{eqnarraystar}
The sign of the time derivative in equation (4) is calculated by

\mbox{sign}({\partial v^d \over \partial t}) =
{\partial \mb...
 ...z) {\partial \sqrt{\dot{u}_x^2 +
\dot{u}_z^2} \over \partial t}\end{displaymath}

which results in

\mbox{sign}({\partial v^d \over \partial t}) = \mbox{sign}(\...
 ...{sign}({\bf \dot{u}} {\partial {\bf\dot{u}} \over \partial t}).\end{displaymath}

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