Scattering and migration velocity analysis

The generalization analyzed in the preceding sections can also be used in the area of wave-equation migration velocity analysis Biondi and Sava (1999); Sava and Biondi (2000).

The downward continued wavefield in Equation (6) can be rewritten as  

$$U_{z+\Delta z}= \mathcal TU_z\left\{ 1+ \mathcal S\left(s - s_o\right)\right\}$$ (13)

where, as before, $\mathcal T$ is the downward continuation operator applied in the $\omega-{\bf k}$ domain, and $\mathcal S$ is the scattering operator, applied in the $\omega-{\bf x}$ domain. The scattered wavefield created at depth $z+\Delta z$ under the influence of the background wavefield (U_z) by a slowness perturbation at depth z ($\Delta s_z=s-s_o$) is  

$$\Delta W_{z+\Delta z}= \mathcal TU_z\mathcal S\Delta s_z,$$ (14)

where the general form of the scattering operator derived from FFD is  

$$\mathcal S= i\Delta z\omega\left[1- \frac{s}{s_o} \left(\sum... ...aystyle{\frac{1}{2} \choose n} \S^{2n} \delta_n \right)\right].$$ (15)

When computed using the operator in Equation (15), the scattered wavefield, Equation (14), exhibits a nonlinear relation to the slowness perturbation. A straightforward linearization is to approximate $\mathcal S$ for the constant background slowness. Biondi and Sava (1999) implement the prestack version of the scattering operator using a 4^th order Taylor series expansion, under the constant velocity assumption ($\delta_n=2n-1$ and s=s_o) for all the terms in the sum:  

$$\mathcal S\approx i\Delta z\omega\left(1+ \SQR4exp{\left[\frac{\left\vert \bf k_m\right\vert}{\omega s_o}\right]} \right).$$ (16)

 

sc Figure 3 Comparison of various approximation for the linear scattering operator. The solid lines correspond to scattering operators computed using the spatially variable slowness function (s), and the dashed lines correspond to the scattering operator computed using only the background slowness (so). The accuracy of the operator improves when using the variable slowness function. The horizontal axis is the ratio $\S$, and the vertical axis is the relative error of the linear scattering operator with respect to the true one, in logarithmic scale.

Figure  shows a comparison of various approximation for the linear scattering operator, Equation (15). As expected, the relative error of the backprojection operator increases with increasing ratio $\S$. Also, Muir's continuous fraction expansion gives a significantly better approximation to the scattering operator for the same order of the expansion. The solid horizontal line corresponds to $10\%$ relative error of the approximate to the true scattering operator.

After we apply the imaging condition to the downward continued scattered wavefield Equation (13), we can formulate the relationship between the image ($\Delta {\bf R}$) and slowness ($\Delta {\bf S}$) perturbations as  

$$\Delta {\bf R}\approx \mathcal L\Delta {\bf S}$$ (17)

where $\mathcal L$ is the wave-equation migration velocity analysis operator Biondi and Sava (1999).

Next section presents a simple example, in which I simulate $\Delta {\bf R}$ and compute $\Delta {\bf S}$using a prestack backprojection operator derived from Equation (16).