Linearized downward continuation - purpose and description

Let us define a slowness perturbation $\Delta s=s-s_0$ as a difference between two slowness models, one of which (s₀) is named the ``background slowness''. By undertaking several approximations, the most notable of which is Born, the mixed-domain downward continuation operator can be written as an explicit function of the slowness perturbation. This allows an explicit relation between the slowness perturbation and the wavefield. In conjunction with the imaging condition, this allows writing an explicit relation between the slowness perturbation and the image. This relation is the basis of Wave-Equation Migration Velocity Analysis (WEMVA), a complete flowchart of which is presented in Figure 3. This procedure finds the velocities in the following way: using the recorded data and the background slowness, it creates a background image. This image is then improved so that it is closer to the optimally focused one, then the two images are subtracted to create an image perturbation. This is transformed into a wavefield perturbation through an inverse imaging condition, then is upward continued, to create an adjoint scattered wavefield. This in turn is transformed into a slowness update by inverting the linearized downward continuation operator. This operator is linearized so that its inversion will be computationally cheap. A complete derivation is provided by Biondi and Sava (1999), with more explanations in Sava (2000).

 

wemva Figure 3 WEMVA flowchart, provided for illustrating the use of linearized downward continuation.

If we denote the wavefield as U, the linearized downward continuation (complexified local Born-Fourier method), according to Appendix B in Vlad (2002), is given by:  

$$U_{z = n\Delta z} = \left( {\prod\limits_1^n {\mathcal T} } ... ...( {\prod\limits_1^j {\mathcal T} } \right)U_{z = 0} } \right]},$$ (2)

where ${\mathcal T}$ is the background wavefield downward continuation operator:  

$${\mathcal T} = e^{i \Delta z \sqrt{{\omega}^2{s_o}^2 - {(1 - i \eta)}^2 {\bf \left\vert k_m \right\vert}^2}} ,$$ (3)

and ${\mathcal S}$ is the scattering operator:  

$${\mathcal S} = \frac{i \Delta z {\omega}^2 s_o}{\sqrt{{\omeg... ...s_o}^2 - {(1 - i \eta)}^2 {\bf \left\vert k_m \right\vert}^2}}.$$ (4)