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NORSAR WAVEFRONT CONSTRUCTION

The main idea behind the ``NORSAR'' method (Vinje et al., 1993) is to compute ray parameters along wavefronts instead of computing them from independently traced rays, as conventional ray methods do. Wavefronts are defined as isochron traveltime curves (lines in 2-D) from the source. New wavefronts are constructed from previous ones, by ray tracing over a time step. As wavefronts expand out, new rays are interpolated between rays that go further apart than a predefined distance DSmax. Figure 4 illustrates how wavefronts expand out, by ray tracing from a time $\tau$ to a time $\tau + \Delta \tau$. The dashed line on Figure 4 represents the new wavefront. The solid dots represent the end points of the rays. The distance between contiguous end points is checked against the predefined maximum distance DSmax. If they are located further than DSmax, a new point (empty dots) is interpolated.

The interpolation of these new points over the wavefront is done using a vectorial third order polynomial ${\bf\vec{x}}(s) = {\bf\vec{a}} s^{3} + {\bf\vec{b}} s^{2} +
{\bf\vec{c}} s + {\bf\vec{d}}$. The polynomial is evaluated as a function of the normalized distance $s$ between points ${\bf\vec{x}}^{  i}$ and ${\bf\vec{x}}^{  i+1}$. Also a scalar third order polynomial is used to interpolate amplitude values and the ray's angle of direction [see Vinje et al. (1993)].

The key property of this procedure is that it produces a fairly constant density of rays over C1 models (Vinje et al., 1993) (see Figure 8), illuminating zones with high geometrical spreading where conventional ray tracing have shadow zones.

norsar1
Figure 4.
New wavefronts (dashed lines) are constructed from the previous wavefront (solid line), by ray tracing a fix number of time steps. New rays are interpolated between points on the wavefront that lay further than a predefined distance. Adapted from Vinje et al. (1993).
norsar1
[pdf] [png] [xfig]

Rays are eliminated if they go out of the model boundaries. They may also be eliminated if a wavefront crosses over itself, as shown on Figure 5. The ``self-crossing'' of the wavefronts may correspond to a caustic or to the intersection of rays from different parts of the model. Once again, as in the Lomax algorithm, for the sake of computational time and also of memory, a ``first arrival'' mode should be used. This first arrival mode removes all later arrivals. When the number of points in a wavefront becomes less than a certain value (e.g. 4 points), the algorithm stops.

Traveltimes and amplitudes are interpolated into a rectangular grid. Ray cells, defined as the area enclosed by a pair of contiguous rays and wavefronts, are checked for the presence of grid points (see Figure 6). Traveltimes at the receivers are estimated by computing the following quantities:

  1. the distances $d_{1}$ and $d_{2}$ from the receiver perpendicular to the two rays.
  2. the normalized distance $s$ along the wavefront $s = d_{1}/(d_{1}+d_{2})$.
  3. the interpolated point ${\bf\vec{x}}(s)$ over the old wavefront.
  4. the distance $l_{r}$ from ${\bf\vec{x}}(s)$ to the receiver
  5. the velocity $v_{mid}$ in the midpoint of the segment $l_{r}$.

The traveltime at the receiver is then estimated to be:

\begin{displaymath}
t_{rec} = t + \frac{l_{r}}{v_{mid}}
\end{displaymath} (11)

where $t$ is the traveltime to the old wavefront.

norsar2
Figure 5.
The new wavefront crosses itself. If only first arrivals are wanted, the points behind the crossing (points no. 7, 8, 9) are removed from the wavefront. Adapted from Vinje et al. (1993).
norsar2
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In computing amplitudes, the geometrical spreading factor $\sqrt{(r_{1}+r_{2})/(R_{1}+R_{2})}$, gives the ratio between the amplitude of one wavefront to the next one. $R_{1}$, $R_{2}$, $r_{1}$ and $r_{2}$ are shown on Figure 7. The amplitude estimation at the receivers is also obtained in this way, where the distances $d_{1}$ and $d_{2}$ are used for $R_{1}$ and $R_{2}$.

Figure 8 shows an example of the Norsar method run over a highly contrasted velocity model. The velocity model is a pair of Gaussian bell curves. The distance between the peaks is 48 meters with a drop of 4 km/s in velocity.

norsar3
Figure 6.
Traveltimes and amplitudes are found at receivers by interpolating within each ray cell. The ray cell is defined by $Ray_{1}$ and $Ray_{2}$, and by the new wavefront and the previous wavefront. Adapted from Vinje et al. (1993).
norsar3
[pdf] [png] [xfig]

A final point on Norsar's method is, as said by Vinje et al. (1993): ``the way the ray tracing between each wavefront is performed is irrelevant to the idea of the wavefront construction''. We notice that all along the discussion on the NORSAR method, ray tracing was kept as an abstract idea. With this in mind we proceed to merge the Lomax algorithm, as the ray tracing algorithm for the NORSAR method. Another advantage of the NORSAR method is that the estimation of ray parameters (as traveltimes, amplitudes, etc.) does not come from a posteriori interpolation between single, separate rays, but instead directly from previously constructed wavefronts.

norsar4
Figure 7.
Amplitudes are computed from the previous wavefront. The geometrical spreading factor gives the ratio between the amplitude $A_{i}$ at the previous wavefront and the new amplitude value $A_{i+1}$.
norsar4
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gauss-wv
gauss-wv
Figure 8.
The ``NORSAR'' or wavefront construction method covers the whole model with wavefronts and rays. The model presents a strong variation in velocity, given by a Gaussian bell with a maximum amplitude of 5000 m/s and a second Gaussian bell with a minimum amplitude of 1000 m/s. The background is 3000 m/s.
[pdf] [png] [scons]


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Next: WAVERAYS AND WAVEFRONTS Up: Urdaneta: Waverays and wavefronts Previous: Waveray implementation

2013-03-03