Recursive wavefield extrapolation

Imaging by wavefield extrapolation (WE) is based on recursive continuation of the wavefields ($\UU$) from a given depth level to the next by means of an extrapolation operator ($\Eop$):  

$$\UU\atzz= \Eop_z\lb \UU\atzo \rb \;.$$ (45)

Here and hereafter, I use the following notation conventions: ${\bf A}\lb x \rb$ means operator ${\bf A}$ applied to x, and $f\lp x \rp$ means function f of argument x. The subscripts $\zz$ or $\zz+\DEL\zz$ indicate quantities corresponding to the depth levels $\zz$ and $\zz+\DEL\zz$, respectively.

This recursive down can also be explicitly written in matrix form as -_0-_1-_n-1 _0_1_2_n = _0 0 0 0 ,

or in a more compact notation as:

$$\lp \Iop - \Eop \rp \UU = \DD,$$ (46)

where the vector $\DD$ stands for data, $\UU$ for the extrapolated wavefield at all depth levels, $\Eop$ for the extrapolation operator and $\Iop$ for the identity operator. Here and hereafter, I make the distinction between quantities measured at a particular depth level (e.g. $\UU\atzo$), and the corresponding vectors denoting such quantities at all depth levels (e.g. $\UU$).

After wavefield extrapolation, we can obtain an image by applying, at every depth level, an imaging operator ($\Mop_z$) to the extrapolated wavefield $\UU\atzo$:

$$\RR\atzo= \Mop_z\lb \UU\atzo \rb \;,$$ (47)

where $\RR\atzo$ stands for the image at some depth level. A commonly used imaging operator ($\Mop_z$) involves summation over the temporal frequencies. We can write the same relation in compact matrix form as:

$$\RR = \Mop{\UU} \;.$$ (48)

$\RR$ stands for the image, and $\Mop$ stands for the imaging operator which is applied to the extrapolated wavefield $\UU$ at all depth levels.

A perturbation $\Delta \UU$ of the wavefield at some depth level can be derived from the background wavefield by a simple application of the chain rule to down:  

$$\Delta \UU\atzz= \Eop_z\lb \Delta \UU\atzo \rb + {\DEL \VV}\atzz\;,$$ (49)

where ${\DEL \VV}\atzz= \DEL \Eop_z \lb \UU\atzo \rb$ represents the scattered wavefield generated at $\zz+\DEL\zz$ by the interaction of the wavefield $\UU\atzo$ with a perturbation of the velocity model at depth $\zz$.$\Delta \UU\atzz$ is the accumulated wavefield perturbation corresponding to slowness perturbations at all levels above. It is computed by extrapolating the wavefield perturbation $\Delta \UU\atzo$from the level above, plus the scattered wavefield ${\DEL \VV}\atzz$at this level.

wavpert is also a recursive equation which can be written in matrix form as -_0-_1-_n-1 _0_1_2_n = _0 _1_n-1 _0_1_2_n ,

or in a more compact notation as:

$$\lp \Iop - \Eop \rp \Delta \UU= \DEL \Eop\UU.$$ (50)

The operator $\DEL \Eop$ stands for a perturbation of the extrapolation operator $\Eop$.The quantity $\DEL \Eop\UU$ represents a scattered wavefield, and is a function of the perturbation in the medium by the scattering relations derived in Appendix C. For the case of single scattering, we can write that

$${\DEL \VV}\atzz\equiv \DEL \Eop_z \lb \UU\atzo \rb = \Eop_z\lb \Sop_z\lp{\widetilde{\UU}}\atzo\rp \lb{\Delta s}\atzo\rb \rb.$$ (51)

The expression for the total wavefield perturbation $\Delta \UU$ from wavpert becomes

$$\Delta \UU\atzz= \Eop_z\lb \Delta \UU\atzo \rb + \Eop_z\lb \Sop_z\lp{\widetilde{\UU}}\atzo\rp \lb{\Delta s}\atzo\rb \rb,$$ (52)

which is also a recursive relation that can be written in matrix form as -_0-_1-_n-1 _0_1_2_n = _0 _1 _n-1 _0 _1 _2 _n s_0s_1s_2s_n ,

or in a more compact notation as:

$$\lp \Iop - \Eop \rp \Delta \UU= \Eop \Sop \Delta s.$$ (53)

The vector $\Delta s$ stands for the slowness perturbation at all depths.

Finally, if we introduce the notation

$$\Gop = \lp \Iop - \Eop \rp^{-1} \Eop \Sop,$$ (54)

we can write a simple relation between a slowness perturbation $\Delta s$and the corresponding wavefield perturbation $\Delta \UU$: 

$$\Delta \UU= \Gop \Delta s.$$ (55)

This expression describes wavefield scattering caused by the interaction of the background wavefield with a perturbation of the medium.