Wavefront direction and energy velocity

In ordinary wave propagation, energy propagates perpendicular to the wavefront. When there is anisotropy dispersion, the angle won't be perpendicular.

The apparent horizontal velocity seen along the earth's surface is dx/dt. The apparent velocity along a vertical, e.g., as seen in a borehole, is dz/dt. By geometry, both of these apparent speeds exceed the wave speed. The vector perpendicular to the wavefront with a magnitude inverse to the velocity is called the slowness vector:

$$\hbox{slowness vector}\eq \left( {dt \over dx }\ , \ {dt \over dz }\, \right)$$ (14)

The phase velocity vector is defined to go in the direction of the slowness vector, but have the speed of the wavefront normal. More precisely, the phase velocity vector is the slowness vector divided by its squared magnitude:

$$\hbox{phase velocity} \ \eq \ { \left( {dt \over dx }\ ,\ {... ... }} \, \right)^2 \ +\ \left( { {dt \over dz }} \, \right)^2 }$$ (15)

For a disturbance of sinusoidal form, namely, $\exp (i \phi )\ =$$\exp (-i \omega t \,+\,ik_x x \,+\, i k_z z )$,the phase $\phi$ may be set equal to a constant:

$$0 \eq d \phi \eq - \ \omega \, dt \ +\ k_x \, dx \ +\ k_z \, dz$$ (16)

Thus, in Fourier space the slowness vector is  

$$\hbox{slowness vector} \eq \left( {k_x \over \omega }\ , \ {k_z \over \omega }\, \right)$$ (17)

The direction of energy propagation is somewhat more difficult to derive, but it comes from the so-called group velocity vector:  

$$\hbox{group velocity} \eq \left( {\partial \ \ \over \partia... ...rtial \ \ \over \partial k_z } \, \right) \ \omega (k_x , k_z )$$ (18)

For the scalar wave equation $\omega^2 / v^2 = k_x^2 \ +\ k_z^2$,the group velocity vector and the phase velocity vector turn out to be the same, as can be verified by differentiation and substitution. The most familiar type of dispersion is frequency dispersion, i.e. different frequencies travel at different speeds. Later in this section it will be shown that the familiar (15$^\circ$, 45$^\circ$, etc.) extrapolation equations do not exhibit frequency dispersion. That is, as functions of $\omega$ and angle $k_x / \omega$, the velocities in these equations do not depend on $\omega$.In other words, the elliptical and heart shapes in Figure 8 are not frequency-dependent.

An interesting aspect of anisotropy dispersion is that energy appears to be going in one direction when it is really going in another. An exaggerated instance of this occurs when the group velocity has a downward component and the phase velocity has an upward component. Figure 10, depicting the dispersion relation of the 45$^\circ$extrapolation equation, shows an example. A slowness vector, which is in the direction of the wavefront normal, has been selected by drawing an arrow from the origin to the dispersion curve. The corresponding direction of group velocity may now be determined graphically by noting that group velocity is defined by the gradient operator in equation (18). Think of $\omega$ as the height of a hill from which k_z points south and k_x points east. Then the dispersion relation is a contour of constant altitude. Different numerical values of frequency result from drawing Figure 10 to different scales. The group velocity, in the direction of the gradient, is perpendicular to the contours of constant $\omega$.

 

group15
Figure 10 Dispersion relation for downgoing extrapolation equation showing group velocity vector and slowness vector. (Rothman)

The anisotropy-dispersion phenomenon can be most clearly recognized in a movie, although it can be understood on a single frame, as in Figure 11. The line drawing interprets energy flow from the top, through the prism, reflecting at the 45$^\circ$ angle, reflecting from the side of the frame, and finally entering an area of the figure that is sufficiently large and uncluttered for the phase fronts to be recognizable as energy apparently propagating upward but really propagating downward.

 

prism
Figure 11 Plane waves of four different frequencies propagating through a right, 45$^\circ$ prism. Left is the wavefield. Right is a ray interpretation that illustrates different directions of energy and wavefront normal. (Estevez)

That neither energy nor information can propagate upward in Figure 11 should be obvious when you consider the program that calculates the wavefield. The program does not have the entire frame in memory; it produces one horizontal strip at a time from the strip just above. Thus the movie's phase fronts, which appear to be moving upward, seem curious. Theoretically, wave extrapolation using the 45$^\circ$ equation is not expected to handle angles to 90$^\circ$.Yet the example in Figure 11 shows that these extreme cases are indeed handled, although in a somewhat perverted way.

 

overthrust
Figure 12 Ray reflected from the underside of an overthrust.

I once saw a similar circumstance on reflection seismic data from a geologically overthrusted area. The data could not be made available to me at the time, and by now is probably long lost in the owner's files, so I can only offer the line drawing in Figure 12, which is from memory. The increasing velocity with depth causes the ray to bend upward and reflect from the underside of the overthrust. To see what is happening in the wave equation, it is helpful to draw the dispersion curve at two different velocities, as in Figure 13.

 

disper2z
Figure 13 Dispersion curve at two different velocities, $v_{\rm fast}$ and $v_{\rm slow}$.

Downward continuation of a bit of energy with some particular stepout $dt/dx = k_x / \omega$ begins at an ordinary angle on the near-surface, slow-velocity dispersion curve. But as deeper velocity material is encountered, that same stepout implies a negative phase velocity. Although the thrust angle is unlikely to be quantitatively correct, the general picture is appropriate. It is like Figure 11.