Evanescence and ground roll

Completing the physical derivation of the dispersion relation,  

$$k_x^2 \ +\ k_z^2 \quad =\quad{ \omega^2 \over v^2 }$$ (29)

we can now have a new respect for it. It carries more meaning than could have been anticipated by the earlier geometrical derivation. The dispersion relation was originally regarded merely as $\sin^2 \theta + \cos^2 \theta = 1$where $\sin \theta = v k_x / \omega$.There was no meaning in $\sin \, \theta$ exceeding unity, in other words, in $v \, k_x$ exceeding $\omega$.Now there is. There was a hidden ambiguity in two of the previous migration methods. Since data could be an arbitrary function in the (t,x)-plane, its Fourier transform could be an arbitrary function in the $( \omega , k_x )$-plane. In practice then, there is always energy with an angle sine greater than one. This is depicted in Figure 15. What should be done with this energy?

 

evtheory Figure 15 The triangle(s) of reflection energy $\vert \omega \vert \gt v(z) \, \vert k_x \vert$ become narrower with velocity, hence with depth. Ground roll is energy that is propagating at the surface, but evanescent at depth.

When $v \, k_x$ exceeds $\omega$, the familiar downward-extrapolation expression is better rewritten as  

$$e^{ \pm \,i\, \sqrt { \omega^2 / v^2 \,-\, k_x^2 } \, z } \quad =\quad e^{ \pm \sqrt { k_x^2 \,-\, \omega^2 / v^2 } \, z }$$ (30)

This says that the depth-dependence of the physical solution is a growing or a damped exponential. These solutions are termed evanescent waves. In the most extreme case, $\omega =0$, k_x is real, and $k_z = \pm i k_x$.For elastic waves, that would be the deformation of the ground under a parked airplane. Only if the airplane can move faster than the speed of sound in the earth will a wave be radiated into the earth. If the airplane moves at a subsonic speed the deformation is said to be quasi-static.

Perhaps a better physical description is a thought experiment with a sinusoidally corrugated sheet. Such metallic sheet is sometimes used for roofs or garage doorways. The wavelength of the corrugation fixes k_x. Moving such a sheet past your ear at velocity V you would hear a frequency of oscillation equal to $V \, k_x$, regardless of whether V is larger or smaller than the speed of sound in air. But the sound you hear would get weaker exponentially with distance from the sheet unless it moved very fast, V > v, in which case the moving sheet would be radiating sound to great distances. This is why supersonic airplanes use so much fuel.

What should a migration program do with energy that moves slower than the sound speed? Theoretically, such energy should be exponentially damped in the direction going away from the source. The damping in the offending region of $( \omega , k_x )$-space is, quantitatively, extremely rapid. Thus, simple exploding-reflector theory predicts that there should be almost no energy in the data at these low velocities.

The reality is that, instead of tiny amounts of energy in the evanescent region of $( \omega , k_x )$-space, there is often a great deal. This is another breakdown of the exploding-reflector concept. The problem is worst with land data. Waves that are evanescent in deep, fast rocks of interest can be propagating in the low-velocity soil layer. This energy is called ground roll. Figure 16 shows an example.

 

evdata Figure 16 Florida shallow marine profile, exhibiting ground roll with frequency dispersion. (Conoco, Yedlin)

This data is not a zero-offset section. The shot is on the left, and the traces to the right are from geophones at increasing distances from the shot. First draw a line on the figure going through origin and at a slope of the water velocity, 1.5 km/sec. Steeper slopes are slower surface waves. Geological events have lesser slopes. Like the surface of the earth, which varies greatly from place to place, the immediate subsurface which controls the ground roll varies substantially. So although Figure 16 is a nice example, no example can really be typical. In this figure there are two types of ground roll, one at about half of water velocity, and a stronger one at about a quarter of water velocity. The later and stronger one shows an interesting feature known as frequency dispersion. Viewing the data from the side, you should be able to notice that the high frequencies arrive before the low frequencies.

Ground roll is unwanted noise since its exponential decay effectively prevents it from being influenced by deep objects of interest. In practice, energy in the offending region of $( \omega , k_x )$-space should be attenuated. A mathematical description is to say that the composite mapping from model space to data space and back to model space again is not an identity transformation but an idempotent transformation.

EXERCISES:

1. Why does phasemig() use a negative sign on k_z whereas phasemod() uses a positive sign even though both are dealing with upcoming waves?
2. The wave modeling program sketch assumes that the exploding reflectors are impulse functions of time. Modify the program sketch for wave modeling to include a source waveform s(t).
3. The migration program sketch allows the velocity to vary with depth. However the program could be speeded considerably when the velocity is a constant function of depth. Show how this could be done.
4. Define the program sketch for the inverse to the Stolt algorithm--that is, create synthetic data from a given model.
5. The Stolt algorithm can be reorganized to reduce the memory requirement of zero padding the time axis. First Fourier transform x to k_x. Then select, from the (t,k_x )-plane of data, vectors of constant k_x. Each vector can be moved into the space of a long vector, then zero padded and interpolated. Sketch the implied program.
6. Given seismic data that is cut off at four seconds, what is the deepest travel time depth from which 80$^\circ$ dips can be observed?