Linear theory

In migration by downward continuation, illustrated in Figure 1, data measured at the surface (D) are recursively propagated down in depth to generate the complete wavefield (U). Downward continuation requires us to make an assumption about the magnitude of the slowness field (S). Once the wavefield is known, we can apply the imaging condition, which gives us the wavefield at time zero, or in the other words, the image or reflectivity map at the moment the reflectors explode (R).

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

 

chart
Figure 1 A summary-chart of our MVA method. The upper panel describes the computations done with respect to the background field, while the bottom panel refers to the computations done with respect to the perturbation field.

We can take the perturbation in image ($\Delta R$) and apply to it the adjoint operation. Doing so creates an adjoint perturbation in wavefield ($\Delta U'$), an adjoint scattered field ($\Delta W'$), and eventually an adjoint perturbation in slowness ($\Delta S'$), 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 ($\Delta R$) and the perturbation in slowness ($\Delta S$). 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.