In the presence of the background wavefield (*U*), a perturbation in
slowness () will generate a scattered wavefield (), which can, by the same method as the background field, be downward
continued () and imaged (), as shown in
Figure 1.

Figure 1

We can take the perturbation in image () and apply to it the adjoint operation. Doing so creates an adjoint perturbation in wavefield (), an adjoint scattered field (), and eventually an adjoint perturbation in slowness (), as the bottom panel of Figure 1 shows. Considering a first-order Born relation between the perturbation in slowness and the scattered wavefield, we can establish a direct linear relation between the perturbation in image () and the perturbation in slowness (). It follows that if we can obtain a better focused image, we can iteratively invert for the perturbation in slowness that generated the improvement in focusing. This is the foundation of our wave-equation MVA method.

In the next two sections, we briefly present the mathematical relations that form the basis of our method. A more detailed mathematical description appears in Appendices A and B.

- Background field: Forward operator
- Perturbation field: Forward operator
- Perturbation field: Adjoint operator

6/1/1999