Methodology

In its original formulation (), wave-equation migration velocity analysis relates a perturbation of the slowness model ($\Delta {\bf S}$) to its corresponding perturbation of the seismic image ($\Delta \r$). Mathematically, this relation can be expressed as the linear fitting goal  

$${\bf L}\Delta {\bf S}\approx \Delta \r.$$ (10)

${\bf L}$ is the WEMVA operator that is constructed as a linearization of downward continuation operators involving the Born approximation (). We obtain the slowness perturbation $\Delta {\bf S}$ from Equation () by applying either the adjoint or the least-squares inverse of ${\bf L}$ to the image perturbation $\Delta \r$.

The critical quantity in Equation () is the perturbation of the seismic image $\Delta \r$. 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 $\d_0$which is imaged with the migration slowness ${\bf S}$ to produce the reference image $\r_0$. At later times, repeat surveys produce new datasets $\d_1,\d_2\dots$ which are different from $\d_0$ and, therefore, reflect the changes in the reservoirs.

After imaging using the same slowness model ${\bf S}$, we obtain the images $\r_1,\r_2\dots$ which are different from the reference image $\r_0$ (Figure ). 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 $\Delta \r_1,\Delta \r_2\dots$, we can invert for slowness perturbation $\Delta {\bf S}$ using Equation ().