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Physical interpretation

This section presents a brief physical interpretation of the various members of relation (17).

Consider that we have recorded two wavefields at the top and bottom of a depth slab: W0, the wavefield at the top of the slab which has not propagated through the anomalous region; W1, the wavefield at the bottom of the slab which incorporates scattering effects caused by the slowness anomaly inside the slab. The goal of WEMVA is to extract the slowness perturbation $\Delta s$ from W0 and W1.

We can imagine that the linearized process can be thought of as a succession of four steps.

Continuation of W0 and W1 to a level inside the slab where we can compare the two wavefields. This level can be either at the top, bottom or anywhere in between:
W_1e^{ -\xi k_z\left(s \right)} = 
W_0e^{\left(1-\xi\right)k_z\left(s \right)} \end{displaymath} (37)
kz represents the depth wavenumber and is a function of the arbitrary slowness inside any given depth slab, and $\xi=0 \dots 1$ is a scalar which defines where inside the slab we continue the two wavefields.
Linearization of W0 and W1 with respect to the slowness perturbation:
W_1e^{ -\xi k_z\left(s_0\right)}{\left[1- \xi \b\Delta s\rig...
 ...\left(s_0\right)}{\left[1+\left(1-\xi\right)\b\Delta s\right]},\end{displaymath} (38)
where $\b$ is the function defined in Equation (12).
Datuming of the linearized wavefields to the bottom of the slab:
W_1{\left[1- \xi \b\Delta s\right]} &=& 
\\  &=& 
W_b{\left[1+\left(1-\xi\right)\b\Delta s\right]}\end{eqnarray} (39)

Subtraction of the wavefield propagated through the perturbed medium from the wavefield propagated through the background medium:  
\Delta W= W_1- W_b= W_b\frac{ \b \Delta s}{1-\xi \b\Delta s} \end{displaymath} (41)

All three cases in Equation (14) can be derived from Equation (41) as follows:
\xi=0 &\rightarrow& \Delta W= W_b\b \Delta s
\\ \xi=\frac{1}{2}...
 ...xi=1 &\rightarrow& \Delta W= W_b\frac{ \b \Delta s}{1-\b\Delta s}.\end{eqnarray} (42)

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