Imaging with multiples using linearized full-wave inversion

Theory

LFWI poses the imaging problem as an inversion problem by linearizing the wave-equation with respect to our model ( $m({\mathbf x})$ ). We define our model to be a weighted difference between the migration slowness ( $s_o({\mathbf x})$ ) and the true slowness ( $s({\mathbf x})$ ):

$$m ({\mathbf x}) = (s({\mathbf x}) - s_o({\mathbf x})) s_o({\mathbf x})$$ (1)

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:

$$d^{mod}({\mathbf x_r},{\mathbf x_s},\omega) = \sum_{{\mathbf x}} ... ...G({\mathbf x_s},{\mathbf x},\omega) m({\mathbf x}) G({\mathbf x},{\mathbf x_r})$$ (2)

where $d^{mod}$ represents the forward modeled data, $\omega$ is the temporal frequency, $m({\mathbf x})$ is a function of the 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 over the migration slowness. Note that $G$ is actually $\omega$ -dependent and is a function of $s_o({\mathbf x})$ only. It is important to point out that the adjoint of the forward-modeling operator is the migration operator:

$${\mathbf m_{mig}} ({\mathbf x}) = \sum_{{\mathbf x_r},{\mathbf x_... ...x},\omega) G^*({\mathbf x},{\mathbf x_r}) d({\mathbf x_r},{\mathbf x_s},\omega)$$ (3)

The inversion problem is defined as minimizing the least-squares difference between the synthetic and the recorded data:

$$S({\mathbf m}) = \Vert {\mathbf L}{\mathbf m} - {\mathbf d} \Vert^2 = \Vert {\mathbf d}^{mod} - {\mathbf d}\Vert^2$$ (4)

where ${\mathbf L}$ is the forward-modeling operator that corresponds to equation 2.

At first glance, equation 2 seems to only generate singly scattered events (e.g. Figure 1 a). To clarify, the term scattering includes both diffraction and reflection. However, if we construct our propagator ( $G({\mathbf x},{\mathbf y})$ ) using the two-way wave equation, equation 2 can actually generate multiply scattered events. In figure 1 b, the ray path reflects off a salt flank and then the horizontal reflector. If the sharp salt-flank boundary already exists in the migration velocity, then the scattering off the salt flank is automatically generated by the propagator (Green's function). Figure 1 (c) and (d) shows two triply scattered events. Single circles (in purple) show scattering off the migration velocity, while double circles (in green) show scattering off the model $m({\mathbf x})$ .

Imaging with multiples using linearized full-wave inversion