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 (1):

$$\delta \u_{z+\Delta z}= {\bf E}_z\left[\delta \u_{z } \right] + \delta {\bf E}_z \left[\u_{z } \right].$$ (5)

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:

$$\left({\bf 1}- {\bf E}\right)\delta \u= \delta {\bf E}\u,$$ (6)

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

Biondi and Sava (1999) 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 $\delta s_z$:

$$\delta {\bf E}_z \left[\u_{z } \right] = {\bf E}_z\left[{\bf S}_z\left[\delta s_z \right] \right].$$ (7)

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

$$\delta \u_{z+\Delta z}= {\bf E}_z\left[\delta \u_{z } \right] + {\bf E}_z\left[{\bf S}_z\left[\delta s_z \right] \right],$$ (8)

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

$$\begin{eqnarray} \left(\matrix { {{\bf 1}} & {\bf 0}& {\bf 0}&\cdots& {\bf 0}& ... ...s_1\cr \delta s_2\cr\vdots\cr \delta s_n \cr } \right), \nonumber \end{eqnarray}$$

or in a more compact notation as:

$$\left({\bf 1}- {\bf E}\right)\delta \u= {\bf E}{\bf S}\delta s.$$ (9)

The vector $\delta s$ stands for the slowness perturbation.

If we introduce the notation

$${\bf G}= \left({\bf 1}- {\bf E}\right)^{-1} {\bf E}{\bf S},$$ (10)

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

$$\delta \u= {\bf G}\delta s.$$ (11)