Migration velocity analysis based on linearization of the two-way wave equation

Interpretation

There are several ways to interpret equations (14) and (15). For simplicity, let us first break each perturbation into two components, one from the source side and one from the receiver side. So, for equation (14), the first component will be as following:

$$\Delta I_S(\mathbf x, \mathbf h) = \sum_{\omega ,\mathbf x_s} \left\lbrace \sum_{\mathbf y} -2 \omega^2 s_0(\mathbf y) G_0^*(\m... ...\Delta s(\mathbf y) G_0^*(\mathbf x - \mathbf h,\mathbf y,\omega) \right\rbrace$$

$$\left\lbrace \sum_{\mathbf x_r} G_0^*(\mathbf x + \mathbf h,\mathbf x_r,\omega) d(\mathbf x_r,\mathbf x_s,\omega) \right\rbrace.$$ (16)

Now, we can further break equation (16) into two components that we define as follows:

$$\Delta S(\mathbf x - \mathbf h,\mathbf x_s,\omega) = -2 \omega^2 ... ...bf x_s,\omega) \Delta s(\mathbf y) G_0(\mathbf x - \mathbf h,\mathbf y,\omega),$$ (17)

and

$$R_0(\mathbf x + \mathbf h,\mathbf x_s,\omega) = \sum_{\mathbf x_r... ...^*(\mathbf x + \mathbf h,\mathbf x_r,\omega) d(\mathbf x_r,\mathbf x_s,\omega).$$ (18)

We can see that equation (17) represents the Born-modeled-wavefield due to the slowness perturbation and equation (18) represents the background receiver wavefield. So, we can now present the source side of the image perturbation as the following:

$$\Delta I_S(\mathbf x, \mathbf h) = \sum_{\omega ,\mathbf x_s} \De... ... - \mathbf h,\mathbf x_s,\omega) R_0(\mathbf x + \mathbf h,\mathbf x_s,\omega).$$ (19)

Now, we can perform a similar analysis on the other component of equation (14), which is:

$$\Delta I_R(\mathbf x, \mathbf h) = \sum_{\omega ,\mathbf x_s} G^*_0(\mathbf x - \mathbf h,\mathbf ... ...bf y) G_0^*(\mathbf x + \mathbf h,\mathbf y,\omega) \Delta s(\mathbf y) \rbrace$$

$$\lbrace \sum_{\mathbf x_r} G_0^*(\mathbf y,\mathbf x_r,\omega) d(\mathbf x_r,\mathbf x_s,\omega) \rbrace.$$ (20)

Again, let us define a perturbed receiver wavefield and a background source wavefield as the following:

$$R_0(\mathbf y,\mathbf x_s,\omega) = \sum_{\mathbf x_r} G_0^*(\mathbf y,\mathbf x_r,\omega) d(\mathbf x_r,\mathbf x_s,\omega) ,$$ (21)

$$\Delta R(\mathbf x + \mathbf h,\mathbf x_s,\omega) = -2 \omega^2 ... ...thbf h,\mathbf y,\omega) \Delta s(\mathbf y) R_0(\mathbf y,\mathbf x_s,\omega),$$ (22)

$$S_0(\mathbf x - \mathbf h,\mathbf x_s,\omega) = G_0(\mathbf x - \mathbf h,\mathbf x_s,\omega).$$ (23)

This enables us to represent the receiver side of the image perturbation as the following:

$$\Delta I_R(\mathbf x, \mathbf h) = \sum_{\omega ,\mathbf x_s} S_0... ...athbf h,\mathbf x_s,\omega) \Delta R(\mathbf x + \mathbf h,\mathbf x_s,\omega).$$ (24)

As for equation (15), we do the same analysis to arrive at the following gradient formulae:

$$\Delta s_R(\mathbf y) = \sum_{\omega ,\mathbf x_s} S_0(\mathbf y,\mathbf x_s,\omega) \Delta R^*(\mathbf y,\mathbf x_s,\omega) ,$$ (25)

$$\Delta s_S(\mathbf y) = \sum_{\omega ,\mathbf x_s} \Delta S(\mathbf y,\mathbf x_s,\omega) R_0^*(\mathbf y,\mathbf x_s,\omega),$$ (26)

where the residual wavefields are defined as the following:

$$\Delta R(\mathbf y,\mathbf x_s,\omega) = -2 \omega^2 \sum_{\mathb... ...) \Delta I(\mathbf x, \mathbf h) R_0(\mathbf x + \mathbf h,\mathbf x_s,\omega),$$ (27)

$$\Delta S(\mathbf y,\mathbf x_s,\omega) = -2 \omega^2 \sum_{\mathb... ...) G_0(\mathbf x - \mathbf h,\mathbf x_s,\omega) \Delta I(\mathbf x, \mathbf h).$$ (28)

In summary, the tomographic operator computes the image perturbation or slowness perturbation by correlating background and residual wavefields of both source and receiver sides.

Migration velocity analysis based on linearization of the two-way wave equation