Imaging with multiples using linearized full-wave inversion

Next: Multiple imaging with towed Up: Wong et al.: Linearized Previous: Introduction

# Theory

LFWI poses the imaging problem as an inversion problem by linearizing the wave-equation with respect to our model ( ). We define our model to be a weighted difference between the migration slowness ( ) and the true slowness ( ):

 (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:

 (2)

where represents the forward modeled data, is the temporal frequency, is a function of the image point , is the source waveform, and is the Green's function of the two-way acoustic constant-density wave equation over the migration slowness. Note that is actually -dependent and is a function of only. It is important to point out that the adjoint of the forward-modeling operator is the migration operator:

 (3)

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

 (4)

where 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 ( ) 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 .

Subsections
 Imaging with multiples using linearized full-wave inversion

Next: Multiple imaging with towed Up: Wong et al.: Linearized Previous: Introduction

2012-05-10