Wave-equation migration velocity analysis

Wave-equation migration velocity analysis (WEMVA) is based on a linear relation established between perturbations of the slowness model $\delta {\bf s}$ and perturbations of migrated images $\delta \r$.$\delta \d$ and $\delta \r$ correspond, respectively, to $\delta {\bf m}$ and $\delta \d$ in equation (3).

Formally, we can write

$$\delta \r= {\bf L}_0\delta {\bf s}\;,$$ (5)

where ${\bf L}_0$ is the linear first-order Born wave-equation MVA operator. The operator ${\bf L}_0$ incorporates all first-order scattering and extrapolation effects for media of arbitrary complexity. The major difference between WEMVA and wave-equation tomography is that $\delta \d$ is formulated in the image space for the former as opposed to the data space for the later. Thus, with WEMVA we are able to exploit the power of residual migration in perturbing migrated images - a goal which is much harder to achieve in the space of the recorded data.

By construction, the linear operator ${\bf L}_0$ depends on the wavefield computed by extrapolation of the surface data using the background slowness, which corresponds to ${\bf m}_0$ in equation (2). Thus, the operator ${\bf L}_0$ depends directly on the type of recorded data and its frequency content, and it also depends on the background slowness model. Thus, the main elements that control the shape of the sensitivity kernels are

• the frequency content of the background wavefield,
• the type of source from which we generate the background wavefield (e.g. point source, plane wave), and
• the type of perturbation introduced in the image space, which for this problem corresponds to the data space.

In our examples, we define two types of image perturbations: a purely kinematic type $\delta \r_k$, implemented simply as a derivative of the image with respect to depth, which can be implemented as a multiplication in the depth wavenumber domain Sava and Biondi (2004a,b) as follows:  

$$\delta \r_k = -i k_z \r \;,$$ (6)

and a purely dynamic type $\delta \r_a$, implemented by scaling the reference image $\r$ with an arbitrary number:  

$$\delta \r_a = \epsilon \r \;.$$ (7)

In both cases, the perturbations are limited to a small portion of the image. The main difference between $\delta \r_k$ and $\delta \r_a$is given by the $90^\circ$ phase-shift between the two image perturbations.