Wave extrapolation in viscoacoustic media

We first briefly review the basic theory of lowrank one-step wave extrapolation in viscoacoustic media, and derive its adjoint operator for applications to RTM and LSRTM.

A constant-$Q$ model (Kjartansson, 1979) describes an attenuating medium whose quality factor $Q$ is constant in frequency (but may vary in space), indicating that the attenuation coefficient is linear in frequency. Zhu and Harris (2014) derived the following approximate constant-$Q$ wave equation with decoupled fractional Laplacians for modeling and imaging in viscoacoustic media:

$$\frac{1}{c^2}\frac{\partial^2 P}{\partial t^2} = \nabla^2... ...tau \frac{\partial}{\partial t}(-\nabla^2)^{\gamma+1/2} P \;,$$ (1)

where

$$\eta(\mathbf{x}) = -c_0^{2\gamma(\mathbf{x})}\omega_0^{-2\gamma(\mathbf{x})}\cos(\pi \gamma(\mathbf{x})) \;,$$ (2)

$$\tau(\mathbf{x}) = -c_0^{2\gamma(\mathbf{x})-1}\omega_0^{-2\gamma(\mathbf{x})}\sin(\pi \gamma(\mathbf{x})) \;,$$ (3)

$$c^2(\mathbf{x}) = c_0^2(\mathbf{x})\cos^2(\pi\gamma(\mathbf{x})/2) \;,$$ (4)

$$\gamma(\mathbf{x}) = \arctan(1/Q(\mathbf{x}))/\pi \;.$$ (5)

Here $\gamma$ is a dimensionless parameter that ranges between $0$ to $1/2$. $P(\mathbf{x},t)$ is the pressure wavefield, $c_0(\mathbf{x})$ is the acoustic velocity model defined at a reference frequency $\omega_0$. The $\beta_1$ and $\beta_2$ parameters act like on/off switches that control velocity dispersion and amplitude loss effects, respectively (Zhu and Harris, 2014). For simplicity of notation, in the rest of the paper the fractional Laplacian operators are denoted as $\mathbf{L} = (-\nabla^2)^{\gamma+1}$ and $\mathbf{H} = (-\nabla^2)^{\gamma+1/2}$.

Setting both $\beta_1$ and $\beta_2$ to one, equation 1 leads to the viscoacoustic dispersion relation with fractional powers of the wave number:

$${\frac{\omega^2}{c^2}} = {-\eta \vert\mathbf{k}\vert^{2\gamma +2} - i \omega\tau \vert\mathbf{k}\vert^{2\gamma +1}} \;,$$ (6)

Solving for $\omega$ in equation 6 yields:

$$\omega = \frac{-ip_1 + p_2}{2} \; ,$$ (7)

where:

$$p_1 = \tau c^2\vert\mathbf{k}\vert^{2\gamma+1} \; ,$$ (8)

$$p_2 = \sqrt{-\tau^2c^4\vert\mathbf{k}\vert^{4\gamma+2}-4\eta c^2\vert\mathbf{k}\vert^{2\gamma+2}} \;.$$ (9)

The phase function $\phi (\mathbf{x},\mathbf{k},\Delta t)$ that determines the phase shift of the wavefield for propagation in time is then defined as

$$\phi (\mathbf{x},\mathbf{k},\Delta t) \approx \frac{-ip_1 + p_2}{2} \Delta t \; .$$ (10)

The one-step wave extrapolation provides an approximate solution to equation 1 by incorporating the phase function defined in equation 10 into the Fourier integral operator (FIO):

$$P(\mathbf{x},t+\Delta t) = \int \hat{P}(\mathbf{k},t) e^... ...{x} + i \phi(\mathbf{x},\mathbf{k},\Delta t)} d\mathbf{k}\;,$$ (11)

where $\hat{P}$ is the spatial Fourier transform of $P$. The accuracy of the approximation increases with smaller $\Delta t$ (Fomel et al., 2013). The adjoint of operator in equation 11 can be expressed as

$$\hat{P}(\mathbf{k},t) = \int P(\mathbf{x},t+\Delta t) e^... ... \bar{\phi}(\mathbf{x},\mathbf{k},\Delta t)} d\mathbf{x}\; ,$$ (12)

where $\bar{\phi}$ denotes the complex conjugate of $\phi$.

The FIOs introduced in equations 11 and 12 can be efficiently applied using the lowrank one-step wave extrapolation Sun et al. (2015), which we also refer to as the lowrank PSPI operator because of its resemblance to the well-known PSPI method for solving the one-way wave equation (Gazdag and Squazzero, 1984; Margrave and Ferguson, 1999; Kesinger, 1992). The detailed formulation of lowrank PSPI operator, as well as the derivation of its adjoint operator, the lowrank NSPS operator, is shown in the appendix. RTM and LSRTM in viscoacoustic media can therefore be constructed using the forward and adjoint operators.