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# Methodology

In its original formulation Biondi and Sava (1999), wave-equation migration velocity analysis relates a perturbation of the slowness model () to its corresponding perturbation of the seismic image (). Mathematically, this relation can be expressed as the linear fitting goal
 (1)
is the WEMVA operator that is constructed as a linearization of downward continuation operators involving the Born approximation Sava and Fomel (2002). We obtain the slowness perturbation from Equation (1) by applying either the adjoint or the least-squares inverse of to the image perturbation .

The critical quantity in Equation (1) is the perturbation of the seismic image . For the purpose of this equation, this is the known quantity and various techniques can be used to derive it.

 4Dscheme Figure 1 Different 4-D datasets imaged using the same slowness model produce different seismic images, from which we can extract image differences for WEMVA.

In 4-D seismic monitoring, the image perturbation is defined as the difference between the images at various acquisition times with respect to the reference image. For example, suppose that at time t=0 we record a reference dataset which is imaged with the migration slowness to produce the reference image . At later times, repeat surveys produce new datasets which are different from and, therefore, reflect the changes in the reservoirs.

After imaging using the same slowness model , we obtain the images which are different from the reference image (Figure 1). The image differences or perturbations are obtained by simply subtracting the reference image from each of the repeat images. Once we have created the image perturbations , we can invert for slowness perturbation using Equation (1).

Next: Example Up: Sava et al.: 4-D Previous: Introduction
Stanford Exploration Project
11/11/2002