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.

1.

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:

$$W_1e^{ -\xi k_z\left(s \right)} = W_0e^{\left(1-\xi\right)k_z\left(s \right)}$$ (37)

k_z 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.

2.

Linearization of W₀ and W₁ with respect to the slowness perturbation:

$$W_1e^{ -\xi k_z\left(s_0\right)}{\left[1- \xi \b\Delta s\rig... ...\left(s_0\right)}{\left[1+\left(1-\xi\right)\b\Delta s\right]},$$ (38)

where $\b$ is the function defined in Equation (12).

3.

Datuming of the linearized wavefields to the bottom of the slab:

$$\begin{eqnarray} W_1{\left[1- \xi \b\Delta s\right]} &=& W_0{\left[1+\left(1-\x... ...\right)} \\ &=& W_b{\left[1+\left(1-\xi\right)\b\Delta s\right]}\end{eqnarray}$$ (39) (40)

4.

Subtraction of the wavefield propagated through the perturbed medium from the wavefield propagated through the background medium:  

$$\Delta W= W_1- W_b= W_b\frac{ \b \Delta s}{1-\xi \b\Delta s}$$ (41)

All three cases in Equation (14) can be derived from Equation (41) as follows:

$$\begin{eqnarray} \xi=0 &\rightarrow& \Delta W= W_b\b \Delta s \\ \xi=\frac{1}{2}... ...xi=1 &\rightarrow& \Delta W= W_b\frac{ \b \Delta s}{1-\b\Delta s}.\end{eqnarray}$$ (42) (43) (44)