Elastic Born modeling in an ocean-bottom node acquisition scenario

Elastic Born modeling

The elastic isotropic wave equation in index notation reads:

$$\partial_i \sigma_{ii} + \partial_j \sigma_{ij} + f_i (\mathbf x , t) = \rho \partial_t v_i,$$ (2)

where $\sigma_{ii}$ are the normal stresses, $\sigma_{ij}$ are the transverse stresses, $f_i$ is the source function in direction $i$ , $\mathbf x$ is the spatial source location operating at time $t$ , $\rho$ is density and $v_i$ is the particle velocity in direction $i$ . The stresses are propagated using the stress-displacement relation:

$$\sigma_{ii} = (\lambda + 2 \mu) \partial_i v_i + \lambda \partial_j v_j,$$

$$\sigma_{ij} = \mu \left( \partial_j v_i + \partial_i v_j \right),$$ (3)

where $\lambda$ and $\mu$ are the Lame elastic constants.

In the staggered time grid methodology for elastic propagation (Virieux, 1986) the stresses and particle velocities are always half a time step apart. Therefore equation 2 and equation 3 are solved in alternation during the propagation.

For elastic Born modeling, these equations must be linearized. Beylkin and Burridge (1990) show a full derivation of linearized scattering for an elastic solid. Using $r = \frac{1}{\rho}$ to denote the specific volume, we have three models for the elastic isotropic case:

$$r = r^0 + \Delta r, \quad \lambda = \lambda^0 + \Delta \lambda, \quad \mu = \mu^0 + \Delta \mu,$$ (4)

where $\lambda^0$ , $\mu^0$ and $r^0$ are the smooth models, and $\Delta \lambda$ , $\Delta \mu$ and $\Delta r$ are the perturbed models.

We also have the incident and scattered stress and particle velocity fields:

$$v^0_i, \quad \Delta v_i, \quad \sigma^0_{ij}, \quad \Delta \sigma_{ij}$$ (5)

The incident stress and particle velocity fields need to propagated with the smooth models so that they do not generate reflections. At each time step, the perturbed models must be multiplied by these incident fields, and then injected as an additive source function into the scattered fields. As a result of the staggered time grid, the injection must be done alternately into the stress and particle velocity fields. The scattered fields themselves must also be propagated with the smooth models, so that they do not generate any additional scattering.

Elastic Born modeling in an ocean-bottom node acquisition scenario