next up previous [pdf]

Next: Variable velocity and anisotropy Up: Theory Previous: Theory

Analytical solutions in constant velocity

When V is constant, after Fourier transform in space, the wave equation takes the form

$\displaystyle \left( \frac{\partial^2}{\partial t^2} + V^2 \vert\mathbf{k}\vert^2 \right) P(\mathbf{k},t)=0\; ,$ (2)

where $ \mathbf{k}$ is the spatial wavenumber and $ P(\mathbf{k},t)$ is the spatial Fourier transform of $ p(\mathbf{x},t)$ :

$\displaystyle P(\mathbf{k},t) = \frac{1}{(2\pi)^3}\int p(\mathbf{x},t) e^{-i\mathbf{k}\cdot\mathbf{x}}d\mathbf{x}\; .$ (3)

The analytical solution to equation 2 can be expressed as

$\displaystyle P = A_1 e^{i\vert\mathbf{k}\vert Vt} + A_2 e^{-i\vert\mathbf{k}\vert Vt} = P_1 + P_2 \; ,$ (4)

where $ P_1$ represents the forward-propagating wavefield, i.e., positive frequencies, and $ P_2$ represents the backward-propagating wavefield, i.e., negative frequencies. The time derivative of $ P$ has the following form:

$\displaystyle \frac{\partial P}{\partial t}=i\vert\mathbf{k}\vert V(P_1-P_2)\;.$ (5)

Zhang and Zhang (2009) used the Hilbert transform to define an additional function:

$\displaystyle Q(\mathbf{k},t)=\frac{1}{\psi}\frac{\partial P(\mathbf{k},t)}{\partial t}\;,$ (6)

where $ Q(\mathbf{k},t)$ is the Hilbert transform of $ P(\mathbf{k},t)$ , and $ \psi=V\vert\mathbf{k}\vert$ . Combining equations 4, 5 and 6, $ P_1$ and $ P_2$ can be expressed as
$\displaystyle P_1=\frac{1}{2}\left(P - iQ \right) \; ,$     (7)
$\displaystyle P_2=\frac{1}{2}\left(P + iQ \right) \; .$     (8)

Equation 2 can be split into a pair of first-order equations and expressed in the following matrix form:

$\displaystyle \frac{\partial}{\partial t}\left[ \begin{array}{c} P  P_t \end{...
...end{array} \right] \; \left[ \begin{array}{c} P  P_t \end{array} \right] \; .$ (9)

With the help of the Hilbert transform and equations 7 and 8, a more symmetric expression can be achieved:

$\displaystyle \frac{\partial}{\partial t}\left[ \begin{array}{c} P  iQ \end{a...
...\end{array} \right] \; \left[ \begin{array}{c} P  iQ \end{array} \right] \; .$ (10)

We can further decompose the first matrix on the right-hand side as follows:
\begin{displaymath}\left[
\begin{array}{cc}
0 & -i\psi \\
-i\psi & 0 \end{ar...
...{array}{cc}
1/2 & -1/2 \\
1/2 & 1/2 \end{array} \right] \; .\end{displaymath}     (11)

Substituting equation 11 into equation 10, and using equations 7 and 8, we arrive at:

$\displaystyle \frac{\partial}{\partial t}\left[ \begin{array}{c} P  iQ \end{a...
...d{array} \right] \; \left[ \begin{array}{c} P_1  P_2 \end{array} \right] \; .$ (12)

In RTM, only one branch of the total wavefield is needed at one time. The two parts of wave propagation decouple according to

$\displaystyle \frac{\partial}{\partial t}\left[ \begin{array}{c} P_1  P_2 \en...
...d{array} \right] \; \left[ \begin{array}{c} P_1  P_2 \end{array} \right] \; .$ (13)

Modeling seismic wave propagation requires the source function. Letting the source function be $ f(\mathbf{x},t)$ , wave equation 2 can be rewritten in the following form:

$\displaystyle \left( \frac{\partial^2}{\partial t^2} + \psi^2 \right) P(\mathbf{k},t)=\hat{f}(\mathbf{k},t) \; .$ (14)

Correspondingly, equation 13 becomes:

\begin{displaymath}\frac{\partial}{\partial t}\left[
\begin{array}{c}
P_1 \\
P_2 \end{array} \right]\end{displaymath} $\displaystyle =$ \begin{displaymath}\left[
\begin{array}{cc}
1/2 & -1/2 \\
1/2 & 1/2 \end{arra...
...}{c}
0 \\
\frac{i}{\psi} \hat{f} \end{array} \right] \right\}\end{displaymath} (15)
  $\displaystyle =$ \begin{displaymath}\left[
\begin{array}{cc}
i\psi & 0 \\
0 & -i\psi \end{arr...
...,\hat{f} \\
\frac{i}{2\psi} \hat{f} \end{array} \right] \; .\end{displaymath}  

The application of operator $ -i/2\psi$ can be implemented in either time domain or Fourier domain; it can also be directly incorporated into the definition of source functions. For example, operator $ 1/\psi$ can be regarded as $ (i\omega/\vert\omega\vert) \cdot (1/i\omega)$ , which in the time domain corresponds to cascading the Hilbert-transform with the first-order integration.

In constant velocity, the forward-propagating wavefield away from the source at the next time step $ t+\Delta t$ can be expressed as:

$\displaystyle p_1(\mathbf{x},t+\Delta t) = \int P_1(\mathbf{k},t) e^{i [\mathbf{k} \cdot \mathbf{x} + V \vert\mathbf{k}\vert \Delta t]} d\mathbf{k}\;.$ (16)


next up previous [pdf]

Next: Variable velocity and anisotropy Up: Theory Previous: Theory

2016-11-16