Spatial aliasing various space axes

Aliasing can occur on the axes of time, depth, geophone, shot, midpoint, offset, or crossline. In practice, aliasing is the worst on the horizontal space axes. Figure 19 provides an illustration. Looking at that figure, you get confused about whether the dip is to the left or right. Mathematical analysis has the same difficulty. The dispersion relation of the wave equation enables us to compute the vertical spatial frequency k_z from the temporal frequency $\omega$,the velocity v, and the horizontal spatial frequency k_x using the semicircle relation $k_z ( \omega , k_x ) =$$\sqrt { \omega^2 / v^2 \ -\ k_x^2 }$.Sampling on the x-axis sets an upper limit on k_x equal to the Nyquist frequency $\pi / \Delta x$.Both frequency-domain methods and finite-difference methods treat higher frequencies as though they were folded back at the Nyquist frequency. Thus the semicircle dispersion relation is replicated above the Nyquist frequency, as shown in Figure 19.

 

kxalias Figure 19 The effective dispersion relation of the wave equation when the horizontal axis is sampled. Frequencies are given for typical zero-offset migration.

The problem of spatial aliasing begins when two circles touch each other, as shown at 20 Hz in Figure 19. This occurs when a half-wavelength v/2f equals the spatial sample interval $\Delta x$.The exploding-reflector model implies that the velocity to use is half the rock velocity. Thus the aliasing problem is avoided if $2 f \Delta x< {1\over 2}\ v_{\rm rock}$.For a rock velocity equal to 2 km/sec, the safe frequencies are listed in the table below:

$$\triangle x$$ safe frequency

standard 25m <20Hz

reconnaissance 50m <10Hz

3-D cross line 100m <5Hz

Another view of the spatial aliasing problem is that steeply dipping waves are suppressed by the geophone group. (This disregards shot-space aliasing). From this standpoint the limit past which spatial aliasing begins should be thought of in terms of angles at which energy is missing from the data. Taking the ray angle to be 30$^\circ$ instead of 90$^\circ$ doubles horizontal wavelengths. Thus, for 30$^\circ$ and a rock velocity of 2 km/sec, to ensure safety from aliasing, frequencies should be in the ranges listed below:

$$\triangle x$$ safe frequency

standard 25m <40Hz

reconnaissance 50m <20Hz

3-D cross line 100m <10Hz

Because data usually has good signal above 40 Hz, wide-angle processing is often frustrated by spatial aliasing.

The problem of spatial aliasing usually overshadows the difference between the 15$^\circ$ and the 90$^\circ$ equations. Aliased energy does not move between hyperbola flanks and the apex. Aliased energy tends to stay in place. This is illustrated on Figure 20 which shows a 90$^\circ$ hyperbola and a 15$^\circ$ hyperboloid from a finite difference equation. Overall, there is little difference. Look at the amplitude of the hyperbolic arrival. It is dropping off faster than predicted by spherical spreading and the obliquity function. This is because the dispersion curve semicircles overlap one another. There can be no angles of propagation beyond that which aliases x. Since waves can't go so steeply, they don't. The pulse doesn't spread properly.

 

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Figure 20 One second of synthetic hyperbolas with $\Delta t = 4\$ms, $\Delta x = 25\$meters, and velocity 2 km/sec. Fourier domain 90$^\circ$ hyperbola (top) and 15$^\circ$ finite difference hyperboloid (bottom).