Solving 3D Anisotropic Elastic Wave Equations on Parallel GPU Devices

Theory

The equations governing elastic wave propagation in 3D transversely isotropic media, when assuming linearized elasticity theory and a stress-stiffness tensor formulation, are fairly straightforward to implement in a numerical scheme. Our goal is to develop finite-difference (FD) operators of 2$^{nd}$ - and 8$^{th}$ -order temporal and spatial accuracy, respectively, that are well-suited for GPU hardware by virtue of being a compact stencil with a regular memory access pattern.

The linear theory of elasticity [e.g., Landau and Lifshitz (1986)] establishes a relationship between a vector seismic wavefield displaced infinitesimally from rest and the dimensionless linear strain tensor. In indicial notation we write

$$\epsilon_{kl} = \frac{1}{2}\left[ \partial_k u_l + \partial_l u_k \right], \quad \quad k,l=1,2,3,$$ (1)

where $\epsilon_{kl}$ is an element of the linear strain tensor, $\partial_k$ is the spatial derivative in the $k^{th}$ direction, and $u_l$ is the $l^{th}$ component of wavefield displacement. Herein, we assume Cartesian geometry where the $x$ -, $y$ - and $z$ -axes are represented by indices $i=1,2,3$ , respectively, and use summation notation for repeated indices.

The linear strain tensor, $\epsilon_{kl}$ , is related to the Cauchy stress tensor, $\sigma_{ij}$ , through a constitutive relationship that describes the elastic material properties through a fourth-order stiffness tensor $c_{ijkl}$ :

$$\sigma_{ij}=c_{ijkl}\epsilon_{kl}.$$ (2)

The above equations can be combined into the equations of motion, derivable from Newton's second law, that describe wave propagation through an anisotropic elastic medium:

$$\rho \partial^2_{tt}u_i = \partial_j \sigma_{ij} + F_i,$$ (3)

where $F_i$ is the body force per unit volume (that can be implemented as an equivalent stress source), $\rho$ is material density, and $\partial^2_{tt}$ is the second-order temporal derivative.

Solving 3D Anisotropic Elastic Wave Equations on Parallel GPU Devices