Introducing velocity perturbations

If a velocity perturbation is applied at some depth level, a perturbed wavefield $\Delta{\mathcal U}$ can be derived from the background wavefield by application of the chain rule to equation 1,  

$$\Delta {\mathcal U}_{z+\Delta z}= {\mathbf E}_z[\Delta {\mathcal U}_z]+\Delta {\mathcal V}_{z+\Delta z},$$ (8)

where $\Delta {\mathcal V}_{z+\Delta z}$ represents the scattered wavefield generated at $z+ \Delta z$by the interaction of the velocity model at depth z. Field $\Delta {\mathcal U}_{z+\Delta z}$is the accumulated wavefield perturbation corresponding to the slowness perturbations at all levels above. It is computed by extrapolating the wavefield perturbations from the level above $\Delta {\mathcal U}_z$,plus the scattered wavefield at this level, $\Delta {\mathcal V}_{z+\Delta z}$.

Equation 8 is also a recursive equation that can be written in matrix form

$$\begin{eqnarray} \left[ \begin{array} {cccccc} \mathbf{1} & 0 & 0 & ... & 0 & 0 ... ...{\mathcal U}_2 \\ \vdots \\ {\mathcal U}_n\end{array}\right], & \,\end{eqnarray}$$ (9) (10)

or in more compact notation as,

$$\left( \mathbf{1} - {\mathbf E}\right) \Delta {\mathcal U}= \Delta {\mathbf E}{\mathcal U}.$$ (11)

Operator $\Delta {\mathbf E}$ denotes a perturbation of the extrapolation operator ${\mathbf E}$, while quantity $\Delta {\mathbf E}{\mathcal U}$ represents a scattered wavefield and is a function of the medium perturbation given by the scattering relationship derived in Appendix A. For single scattering we write,

$$\Delta {\mathcal V}_{z+\Delta z}\equiv \Delta {\mathbf E}_z[... ...athbf E}_z[ {\mathbf S}_z( \tilde{\mathcal U}_z) [ \Delta s] ],$$ (12)

where ${\mathbf S}_z$ is the scattering operator, and $\Delta {\mathbf s}$ is slowness perturbation. This expression yields a recursive relationship that can be written in matrix form:

$$\left[ \begin{array} {cccccc} \mathbf{1} & 0 & 0 & ... & 0 &... ...\ \vdots \\ \Delta {\mathcal U}_n\end{array}\right] = \nonumber$$

 

$$\left[ \begin{array} {cccccc} 0 & 0 & 0 & ... & 0 & 0\\ {\ma... ...a s_1 \\ \Delta s_2 \\ \vdots \\ \Delta s_n\end{array}\right] ,$$ (13)

or in more compact notation

$$\left( \mathbf{1}-{\mathbf E}\right) \Delta{\mathcal U}= {\mathbf E}\S \Delta s,$$ (14)

where vector $\Delta s$ denotes slowness perturbations at all depths.

Finally, introducing  

$$\mathbf{L}= \left(\mathbf{1}- {\mathbf E}\right)^{-1} {\mathbf E}\S,$$ (15)

we can write a simple relationship between slowness $\Delta s$ and wavefield $\Delta{\mathcal U}$ perturbations:

$$\Delta{\mathcal U}= \mathbf{L}\Delta s.$$ (16)

This expression represents the wavefield scattering caused by the interaction of the background wavefield with the a medium perturbation. The total modeled field $\Psi_m$ is defined as,

$$\Psi_m({\mathbf r},{\mathbf s}) = \tilde{{\mathcal U}}({\mathbf r},{\mathbf s}) + \Delta{\mathcal U}({\mathbf r},{\mathbf s}),$$ (17)

where $\tilde{{\mathcal U}}$ is the background wavefield modeled by equation 3.