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Waveray implementation

In the waveray method, the wavelength-dependent averaging of the velocity is done dynamically as a function of the position and orientation of the plane wavefronts. The velocity averaging is done using a Gaussian weight curve, centered at the wavepath location (see Figure 1). Equation 1 expresses the wavelength averaged velocity $\overline{v}$ at a point ( $ \vec{\bf x}_{\nu} $) for a wave period $\bf T$ (Lomax, 1994),

\begin{displaymath}
\overline{v}  (\vec{\bf x}_{\nu},{\bf T\/}) = \frac{
\dis...
...displaystyle \int_{-\infty}^{\infty}
\omega(\gamma) d\gamma}
\end{displaymath} (1)

where $\gamma$ is the arc length along the wavefront away from $ \vec{\bf x}_{\nu} $ expressed in wavelengths. $c  (\vec{\bf x})$ is the velocity at point $\vec{\bf x}$ and $\omega(\gamma)$ is the Gaussian weight curve:
\begin{displaymath}
\omega(\gamma) = e^{-4 \ln 2  \cdot  (\gamma/\alpha)^{2}}
\end{displaymath} (2)

where $\alpha$ specifies the half width of the Gaussian bell in wavelengths.

( $\vec{\bf x}(\gamma,{\bf T\/})$) is the position along the instantaneous straight wavefront given by the recursive relation (Lomax, 1994):

\begin{displaymath}
\vec{\bf x}(\gamma, {\bf T\/}) = \vec{\bf x}_{\nu} + \frac{...
...gamma  ', {\bf T\/})]\
{\bf\hat{n}}({\bf T\/})\
d\gamma  '
\end{displaymath} (3)

where ${\bf\hat{n}}$ is the unit normal to the wavepath at point $ \vec{\bf x}_{\nu} $.

The discrete representation of equation 1 is given by equation 4:

\begin{displaymath}
\overline{v}  (\vec{\bf x}_{\nu},{\bf T\/}) = \frac{
\dis...
... n}({\bf T\/})] }
{\displaystyle \sum_{n=-N}^{N}
\omega_{n}}
\end{displaymath} (4)

where the integral has been replaced by a finite sum over $2 N+1$ control points. The position of the control points along the wavefronts are given by equations 5 and 6.
\begin{displaymath}
\vec{\bf x}_{\nu} = \vec{\bf x}_{\nu}^{  0}
\end{displaymath} (5)


\begin{displaymath}
\vec{\bf x}_{\nu}^{  n} = \left\{ \begin{array}{lll}
\disp...
...{\bf\hat{n}} &
\; &
n=-1,-2, \ldots, -N. \end{array} \right.
\end{displaymath} (6)

These two equations are the discrete version of equation 3, but, the dependence on the wavelength has been made explicit. Notice that the subscript $\nu$ of $\vec{\bf x}_{\nu}^{  n}$ runs along the wavepath and the superscript $n$ runs along the wavefront. $\gamma_{max}$ specifies the largest distance in wavelengths along the wavefront at which smoothing is applied.

The discrete equivalent of the Gaussian weight function is:

\begin{displaymath}
\omega_{n} = e^{-4 \ln 2  \cdot  (\gamma_{n}/\alpha)^{2}}
\end{displaymath} (7)

where the distance $\gamma_{n}$ along the wavefront in wavelengths is expressed as:
\begin{displaymath}
\gamma_{n} = \frac{n  \gamma_{max}}{N}
\end{displaymath} (8)

The motion of the waverays along the direction of propagation is expressed by the following equation:

\begin{displaymath}
\vec{\bf x}_{\nu+1} = \vec{\bf x}_{\nu} +
\overline{v}_{\nu}  \Delta t\; {\bf\hat{s}}
\end{displaymath} (9)

where $\overline{v}_{\nu} = \overline{v}  (\vec{\bf x}_{\nu},{\bf T\/})$, $\Delta t$ is the time step and ${\bf\hat{s}}$ is a unit vector that moves along the direction of propagation.

lomax2
Figure 2.
Waveray wavepath calculation. Huygen's principle is used to obtain the bending $\Delta {\bf\hat{s}}$ of the wavepath from points $\vec{\bf x}_{\nu}^{  1}$ and $\vec{\bf x}_{\nu}^{  -1}$. Adapted from Lomax (1994).
lomax2
[pdf] [png] [xfig]

The change in direction of the waverays is approximated by the difference in movement between the first control point on either side of the wave location $ \vec{\bf x}_{\nu} $ as shown in Figure 2:

\begin{displaymath}
\Delta {\bf\hat{s}} = - \left( \frac{
\overline{v}_{\nu}^{...
...f x}_{\nu}^{  -1} \vert} \right)  
\Delta t \; {\bf\hat{n}}
\end{displaymath} (10)

where $\overline{v}_{\nu}^{  1}$ and $\overline{v}_{\nu}^{  -1}$ are the wavelength averaged velocities at the wavefront points $\vec{\bf x}_{\nu}^{  1}$ and $\vec{\bf x}_{\nu}^{  -1}$ respectively.

Finally, the half width parameter $\alpha$ and the truncation parameter $\gamma_{max}$ are set at $\alpha = 2.0$ and $\gamma_{max} = 1.5$, based on Lomax (1994) calibration. The number of control points $N$ is set proportional to the ratio $T/\Delta t$ of the wave period over the time step.

Figure 3 shows the significant differences between the waveray and ray methods. Notice how a high frequency ray is scattered by the small velocity anomaly, while the waveray's wavepath is little deflected. Note also, how the third ray (from right to left) is not perturbed by the low velocity anomaly, while the waveray wavepath is deflected. The wavelength-dependent velocity averaging smoothes out small velocity variations and causes the wavepath to be affected from velocity variations away from it.

lomaxsbs
lomaxsbs
Figure 3.
Left frame shows a fan of high frequency ray paths. Right frame shows a fan of 12 Hz waveray wavepaths. The straight segments perpendicular to the waverays represent the instantaneous wavefronts. The velocity model is defined by two circular anomalies drawn in a homogeneous background. The black dot (located at 1000 m. by 1250 m. in depth) depicts a low velocity anomaly. The white circle a high one.
[pdf] [png]


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Next: NORSAR WAVEFRONT CONSTRUCTION Up: LOMAX'S WAVERAYS Previous: LOMAX'S WAVERAYS

2013-03-03