Least-squares reverse time migration for the Cascadia ocean-bottom dataset

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:

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

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

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

Least-squares reverse time migration for the Cascadia ocean-bottom dataset