Wavefield perturbations

A perturbation of the wavefield at some depth level can be derived from the background wavefield by a simple application of the chain rule to Equation ():

$$\delta \u_{z+\Delta z}= {\bf E}_z\left[\delta \u_{z } \right] + \delta {\bf E}_z \left[\u_{z } \right].$$ (16)

This is also a recursive equation which can be written in matrix form as

$$\begin{eqnarray} \left(\matrix { {{\bf 1}} & {\bf 0}& {\bf 0}&\cdots& {\bf 0}& ... ...x{ \u_0\cr \u_1\cr \u_2\cr\vdots\cr \u_n \cr } \right), \nonumber \end{eqnarray}$$

or in a more compact notation as:

$$\left({\bf 1}- {\bf E}\right)\delta \u= \delta {\bf E}\u,$$ (17)

where the operator $\delta {\bf E}$ stands for a perturbation of the extrapolation operator ${\bf E}$.

() show that, at every depth level, we can write the operator $\delta {\bf E}$ as a chain of the extrapolation operator ${\bf E}$ and a scattering operator ${\bf S}$applied to the slowness perturbation $\delta s_z$:

$$\delta {\bf E}_z \left[\u_{z } \right] = {\bf E}_z\left[{\bf S}_z\left[\delta s_z \right] \right].$$ (18)

The expression for the wavefield perturbation $\delta \u$ becomes

$$\delta \u_{z+\Delta z}= {\bf E}_z\left[\delta \u_{z } \right] + {\bf E}_z\left[{\bf S}_z\left[\delta s_z \right] \right],$$ (19)

which is also a recursive relation that can be written in matrix form as

$$\begin{eqnarray} \left(\matrix { {{\bf 1}} & {\bf 0}& {\bf 0}&\cdots& {\bf 0}& ... ...bf \Delta s_2\cr\vdots\cr \bf \Delta s_n \cr } \right), \nonumber \end{eqnarray}$$

or in a more compact notation as:

$$\left({\bf 1}- {\bf E}\right)\delta \u= {\bf E}{\bf S}\bf \Delta s.$$ (20)

The vector $\bf \Delta s$ stands for the slowness perturbation.

If we introduce the notation

$${\bf G}= \left({\bf 1}- {\bf E}\right)^{-1} {\bf E}{\bf S},$$ (21)

we obtain a relation between a slowness perturbation and the corresponding wavefield perturbation:  

$$\delta \u= {\bf G}\bf \Delta s.$$ (22)