Joint imaging with streamer and ocean bottom data

Linearized Inversion

We pose the imaging problem as an inversion problem by linearizing the wave-equation with respect to our model ( $m ({\mathbf x})$ ). Assuming that the earth behaves as a constant-density acoustic isotropic medium, we linearize the wave equation and apply the first-order Born approximation to get the following forward modeling equation:

$$d^{mod}({\mathbf x_r},{\mathbf x_s},\omega) = \sum_{{\mathbf x}} ... ...({\mathbf x_s},{\mathbf x},\omega) m({\mathbf x}) G({\mathbf x},{\mathbf x_r}),$$ (1)

where $d^{mod}$ represents the forward modeled data, $\omega$ is the temporal frequency, $m ({\mathbf x})$ represents the reflectivity at image point ${\mathbf x}$ , $f_s(\omega)$ is the source waveform, and $G({\mathbf x_s},{\mathbf x})$ is the Green's function of the two-way acoustic constant-density wave equation. Note that $G$ is actually $\omega$ dependent. It is important to point out that the adjoint of the forward modeling operator is the migration operator:

$${\mathbf m_{mig}} ({\mathbf x}) = \sum_{{\mathbf x_r},{\mathbf x_... ...},\omega) G^*({\mathbf x},{\mathbf x_r}) d({\mathbf x_r},{\mathbf x_s},\omega).$$ (2)

The inversion problem is defined by minimizing the least-squares difference between the synthetic and the recorded data: One can use various types of propagators to formulate the Green's function. In our study, we use the two-way propagator. In this case, the migration operator is equivalent to reverse time migration (RTM). The inverted image ( $m ({\mathbf x})$ ) is better than the migration image in the sense that its forward-modeled data fits the recorded data. Next, we discuss how to apply linearized inversion to jointly image with streamer and OBN data.

Joint imaging with streamer and ocean bottom data