Elastic Born modeling in an ocean-bottom node acquisition scenario

Wave mode separation and imaging

The Helmholtz amplitude separation is based on the assumption that any isotropic vector field can be described as a combination of a scalar and vector potential fields:

$$\mathbf u = \nabla \Phi + \nabla \times \mathbf \Psi,$$ (6)

Where $\Phi$ is the scalar potential field and $\mathbf \Psi$ is the vector potential. $\mathbf u$ is the elastic displacement vector wavefield. The scalar potential generates pressure waves, and the vector potential generates shear waves. Therefore, the Helmholtz method of separating the P-wave amplitude from the S-wave amplitude is to apply a divergence operator and a curl operator to the displacement field:

$$P = \nabla \cdot \mathbf u = \nabla^2 \Phi;$$ (7)

$$\mathbf S = \nabla \times \mathbf u = -\nabla^2 \Psi.$$ (8)

Equations 7 and 8 work only for an isotropic medium. Dellinger and Etgen (1990) extend these operators for an anisotropic medium. Yan and Sava (2008) use these separated P and S-wave modes to formulate an imaging condition for vector potentials in an isotropic medium:

$$I_{ij}( \mathbf x ) = \int_t \alpha_{si} \left ( \mathbf x , t \right) \alpha_{ri} \left ( \mathbf x , t \right) dt,$$ (9)

where the indices $i,j$ denote the wave mode (P or S) of the wavefield $\alpha$ , and $s,r$ denote the source and receiver fields. Therefore $I_{PP}(\mathbf x)$ represents the cross-correlation of the source P field with the receiver P field, while $I_{PS}(\mathbf x)$ represents the cross-correlation of the source P field with the receiver S field. In addition to the conventional PP image, the PS image can supply more information regarding medium parameters. This does, however, depend on the quality of the shear wave data, and on the accuracy of the acoustic velocity, shear velocity and density models.

Elastic Born modeling in an ocean-bottom node acquisition scenario