Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation

Acoustic Wave Extrapolation

The acoustic wave equation is widely used in forward seismic modeling and reverse-time migration (Bednar, 2005; Etgen et al., 2009):

$$\frac{\partial^2p}{\partial t^2} = v(\mathbf{x})^2 \nabla^2p\;,$$ (1)

where $p(\mathbf{x},t)$ is the seismic pressure wavefield and $v(\mathbf{x})$ is the wave propagation velocity.

Assuming the model is homogeneous $v(\mathbf{x}) \equiv v_0$, after a Fourier transform in space, we get the following explicit expression in the wavenumber domain:

$$\frac{d^2\hat{p}}{dt^2} = -v_0^2\vert\mathbf{k}\vert^2\hat{p}\;,$$ (2)

where

$$\hat{p}(\mathbf{k},t)=\int^{+\infty}_{-\infty}{p(\mathbf{x},t)e^{i\mathbf{k}\cdot\mathbf{x}}d\mathbf{x}}\;.$$ (3)

Equation 2 has the following analytical solution:

$$\hat{p}(\mathbf{k},t+\Delta t) = e^{\pm i\vert\mathbf{k}\vert v_0\Delta t}\hat{p}(\mathbf{k},t)\;,$$ (4)

which leads to the well-known second-order time-marching scheme (Etgen, 1989; Soubaras and Zhang, 2008) :

$${p(\mathbf{x},t+\Delta t)+p(\mathbf{x},t-\Delta t) = }$$

$$2\int^{+\infty}_{-\infty}{\hat{p}(\mathbf{k},t)\cos(\vert\mathbf{k}\vert v_0\Delta t)e^{-i\mathbf{k}\cdot\mathbf{x}}d\mathbf{k}}\;.$$ (5)

Equation 5 provides a very accurate and efficient solution in the case of a constant-velocity medium with the aid of FFTs. When the seismic wave velocity varies in the medium, equation 5 turns into a reasonable approximation by replacing $v_0$ with $v(\mathbf{x})$, and taking small time steps, $\Delta t$. However, FFTs can no longer be applied directly to evaluate the inverse Fourier transform, because a space-wavenumber mixed-domain term appears in the integral operation:

$$W(\mathbf{x},\mathbf{k})=\cos(\vert\mathbf{k}\vert v(\mathbf{x})\Delta t).$$ (6)

As a result, a straightforward numerical implementation of wave extrapolation in a variable velocity medium with mixed-domain matrix 6 will increase the cost from $O(N_xlogN_x)$ to $O(N_x^2)$, the original cost for the homogeneous case, in which $N_x$ is the total size of the three-dimensional space grid. A number of numerical methods (Fomel et al., 2010; Du et al., 2010; Song et al., 2013,2011; Song and Fomel, 2011; Etgen and Brandsberg-Dahl, 2009; Liu et al., 2009; Zhang and Zhang, 2009; Fomel et al., 2012) have been proposed to overcome this mixed-domain problem.

In the case of orthorhombic acoustic modeling, we derive a new phase operator $\phi(\mathbf{x},\mathbf{k})$ to replace $\vert\mathbf{k}\vert v(\mathbf{x})$ of the isotropic model. We describe the details in the next section.

Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation