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A Generic Wave Equation

Following the notation of Du et al. (2014), a generic linear second-order in time wave equation can be expressed in the following form

$\displaystyle \left( \frac{\partial^2}{\partial t^2} + \mathbf{A}\right) \mathbf{u}(\mathbf{x},t) = 0 \;,$ (1)

where $ \mathbf{u}$ is the wavefield, $ \mathbf{x}$ is the spatial location, $ t$ is time, and $ \mathbf{A}$ is the a matrix operator containing material parameters and spatial derivative operators. Equation 1 can also be expressed using the first-order system

$\displaystyle \frac{\partial}{\partial t} \begin{bmatrix}\mathbf{u}\\ \mathbf{u...
...ix} \equiv \mathbf{B}\begin{bmatrix}\mathbf{u}\\ \mathbf{u}_t \end{bmatrix} \;,$ (2)

where $ \mathbf{u}_t \equiv \partial \mathbf{u}/\partial t$ . The solution of equation 2 can be formulated using the definition of the matrix exponential:

$\displaystyle \begin{bmatrix}\mathbf{u}(t) \\ \mathbf{u}_t(t) \end{bmatrix} = e^{\mathbf{B}t} \begin{bmatrix}\mathbf{u}(0) \\ \mathbf{u}_t(0) \end{bmatrix}$ (3)

Defining $ \Phi \equiv \sqrt{\mathbf{A}}$ , the eigenvalue decomposition of $ \mathbf{B}$ can be written as (Du et al., 2014)

$\displaystyle \mathbf{B}= \S \Lambda \S^{-1} \;,$ (4)

    $\displaystyle \S = \frac{1}{\sqrt{2}} \begin{bmatrix}\mathbf{I}& \mathbf{I}\\ i\Phi & -i\Phi \end{bmatrix} \;,$ (5)
    $\displaystyle \S^{-1} = \frac{1}{\sqrt{2}} \begin{bmatrix}\mathbf{I}& -i\Phi ^{-1} \\ \mathbf{I}& i\Phi ^{-1} \end{bmatrix} \;,$  
    $\displaystyle \Lambda = \begin{bmatrix}i\Phi & 0 \\ 0 & -i\Phi \end{bmatrix} \;.$  

The solution to the first-order system 3 can now be written as

$\displaystyle \begin{bmatrix}\mathbf{u}(t) \\ \mathbf{u}_t(t) \end{bmatrix} = \...
...da t} \S^{-1} \begin{bmatrix}\mathbf{u}(0) \\ \mathbf{u}_t(0) \end{bmatrix} \;.$ (6)

To simplify the system, we can define the analytical wavefield

$\displaystyle \begin{bmatrix}\hat{\mathbf{u}}_1(t) \\ \hat{\mathbf{u}}_2(t) \en...
...1}\mathbf{u}_t(t) \\ \mathbf{u}(t)+i\Phi ^{-1}\mathbf{u}_t(t) \end{bmatrix} \;.$ (7)

The solution to equation 1 finally takes the form

$\displaystyle \begin{bmatrix}\hat{\mathbf{u}}_1(t) \\ \hat{\mathbf{u}}_2(t) \en...
...\begin{bmatrix}\hat{\mathbf{u}}_1(0) \\ \hat{\mathbf{u}}_2(0) \end{bmatrix} \;.$ (8)

Selecting the first of the two decoupled solutions of equation 8 leads to a time extrapolation operator

$\displaystyle \hat{\mathbf{u}}(\mathbf{x},t+\Delta t) = e^{i\Phi \Delta t} \hat{\mathbf{u}}(\mathbf{x},t) \;,$ (9)

where $ \hat{\mathbf{u}} =(\mathbf{u}-i\Phi ^{-1}\mathbf{u}_t)/\sqrt{2}$ .

For acoustic isotropic constant-density wave equations for pressure waves, $ \mathbf{A}= v^2\vert\mathbf{k}\vert^2$ where $ v$ is velocity and $ \vert\mathbf{k}\vert^2$ is the Laplacian operator. This corresponds to the one-step extrapolation method proposed by (Zhang and Zhang, 2009).

The wavefield $ \hat{\mathbf{u}}$ is an analytical signal, with its imaginary part being the Hilbert transform of its real part (Zhang and Zhang, 2009). To see this, we can perform Hilbert transform to the real-valued wavefield $ \mathbf{u}$ in the frequency domain, and use the dispersion relation $ \omega^2 = \Phi^2$ and derivative property of Fourier Transform

$\displaystyle \mathbf{u}(t) \xrightarrow{\mathscr{F}} \mathbf{u}(\omega) \xrightarrow{\mathscr{H}} i\,$sign$\displaystyle (\omega) \mathbf{u}(\omega) = \frac{i\,\omega}{\vert\omega\vert} ...
...athbf{u}(\omega) \xrightarrow{\mathscr{F}^{-1}} = \Phi ^{-1}\mathbf{u}_t(t) \;,$ (10)

where $ \mathscr{F}$ and $ \mathscr{F}^{-1}$ denotes forward and inverse Fourier transform in time. The output corresponds to the imaginary part of analytical wavefield $ \hat{\mathbf{u}}$ .

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