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Least-squares reverse time migration (LSRTM)

In OBS acquisition, pre-stack images are created from data in the common-receiver domain. This intrinsically requires that each trace in the CRG is de-signatured. In RTM, the migration image is a linear operator applied to the recorded data:

$\displaystyle \mathbf m_{mig} (\mathbf x) =\sum_{\mathbf x_r,\mathbf x_s, \omeg...
...x,\omega) G^*(\mathbf x, \mathbf x_s,\omega) d(\mathbf x_r, \mathbf x_s,\omega)$ (1)

where $ \omega$ is frequency, $ m(\mathbf x)$ represents the reflectivity at image point $ \mathbf x$ , $ f_s(\omega)$ is the source waveform, and $ G(\mathbf x_s, \mathbf x,\omega)$ is the Green's function that is the solution to the two-way acoustic constant-density equation. In practice, the Green's function is calculated using a finite-difference time-domain technique, and the multiplication in the frequency domain is replaced by a zero-lag cross-correlation in the time domain.

The difference between mirror imaging and higher-order mirror imaging lies within the incident wavefield, $ U_s(\mathbf x_r,\mathbf x,\omega)=\omega^2 f_s(\omega) G(\mathbf x_r, \mathbf x,\omega) $ , which is calculated differently to the method described in the previous two sub-sections.

To obtain a better reflectivity image, we go beyond migration by formulating the imaging problem as a least-squares inversion problem. The solution $ m_{inv}(\mathbf x)$ is obtained by minimizing the objective function $ S(\mathbf m)$ , which is defined as the least-squares difference between the forward modeled data $ \mathbf d^{mod}$ and the recorded data $ \mathbf d^{r}$ .

$\displaystyle S(\mathbf m) = \parallel \mathbf d^{mod} - \mathbf d \parallel_{2} = \parallel \mathbf L \mathbf m - \mathbf d \parallel_{2}$ (2)

In least-squares reverse time migration (LSRTM), the forward modeled data is defined to be the Born approximation of the linearized acoustic wave equation:

$\displaystyle d^{mod}(\mathbf x_r, \mathbf x_s,\omega) = \sum_{\mathbf x} U_s(\mathbf x_r,\mathbf x,\omega) G(\mathbf x,\mathbf x_r,\omega) \mathbf m(\mathbf x)$ (3)

It is important to point out that the forward modeling operator $ \mathbf L$ is the adjoint of the reverse-time migration operator $ \mathbf L^T$ . Even for the case of modeling certain classes of surface-related multiples, the operator is still linear with respect to $ m(\mathbf x)$ . This is because $ \mathbf L$ simulates only events that would interact with the model space once.

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Next: Field example Up: Theory Previous: Higher order mirror imaging