Acoustic Staggered Grid Modeling in IWAVE

Acoustodynamics

The pressure-velocity form of acoustodynamics consists of two coupled first-order partial differential equations:

$$\rho \frac{\partial {\bf v}}{\partial t} = - \nabla p$$ (1)

$$\frac{1}{\kappa}\frac{\partial p}{\partial t} = -\nabla \cdot {\bf v} + g$$ (2)

In these equations, $p({\bf x},t)$ is the pressure (excess, relative to an ambient equilibrium pressure), ${\bf v}({\bf x},t)$ is the particle velocity, $\rho({\bf x})$ and $\kappa({\bf x})$ are the density and bulk modulus respectively. Bold-faced symbols denote vectors; the above formulation applies in 1, 2, or 3D.

The inhomogeneous term $g$ represents externally supplied energy (a ``source''), via a defect in the acoustic constitutive relation. A typical example is the isotropic point source

$$g({\bf x},t) = w(t) \delta({\bf x}-{\bf x}_s)$$

at source location ${\bf x}_s$.

Virieux (1984) introduced finite difference methods based on this formulation of acoustodynamics to the active source seismic community. Virieux (1986) extended the technique to elastodynamics, and Levander (1988) demonstrated the use of higher (than second) order difference formulas and the consequent improvement in dispersion error. Many further developments are described in the review paper Moczo et al. (2006). IWAVE's acoustic application uses the principles introduced by these authors to offer a suite of finite difference schemes, all second order in time and of various orders of accuracy in space.

The bulk modulus and buoyancy (reciprocal density) are the natural parameters whose grid samplings appear in the difference formulae. These are the parameters displayed in the figures below, rather than, say, velocity and density, which might seem more natural.

Acoustic Staggered Grid Modeling in IWAVE