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# Recursive wavefield extrapolation

Imaging by wavefield extrapolation (WE) is based on recursive continuation of the wavefields () from a given depth level to the next by means of an extrapolation operator ():
 (45)
Here and hereafter, I use the following notation conventions: means operator applied to x, and means function f of argument x. The subscripts or indicate quantities corresponding to the depth levels and , 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:
 (46)
where the vector stands for data, for the extrapolated wavefield at all depth levels, for the extrapolation operator and for the identity operator. Here and hereafter, I make the distinction between quantities measured at a particular depth level (e.g. ), and the corresponding vectors denoting such quantities at all depth levels (e.g. ).

After wavefield extrapolation, we can obtain an image by applying, at every depth level, an imaging operator () to the extrapolated wavefield :
 (47)
where stands for the image at some depth level. A commonly used imaging operator () involves summation over the temporal frequencies. We can write the same relation in compact matrix form as:
 (48)
stands for the image, and stands for the imaging operator which is applied to the extrapolated wavefield at all depth levels.

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 down:
 (49)
where represents the scattered wavefield generated at by the interaction of the wavefield with a perturbation of the velocity model at depth . is the accumulated wavefield perturbation corresponding to slowness perturbations at all levels above. It is computed by extrapolating the wavefield perturbation from the level above, plus the scattered wavefield 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:
 (50)
The operator stands for a perturbation of the extrapolation operator .The quantity 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
 (51)

The expression for the total wavefield perturbation from wavpert becomes
 (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:
 (53)
The vector stands for the slowness perturbation at all depths.

Finally, if we introduce the notation
 (54)
we can write a simple relation between a slowness perturbation and the corresponding wavefield perturbation :
 (55)
This expression describes wavefield scattering caused by the interaction of the background wavefield with a perturbation of the medium.

Next: WEMVA objective function Up: Wave-equation migration velocity analysis Previous: Introduction
Stanford Exploration Project
11/4/2004