Vibroseis: sweeps and distortion

A vibroseis truck presses a baseplate against the ground and drives it with a hydraulic servo. The driving signal, the pilot, is a sweep: a sinusoid whose frequency climbs from $f\_1$ to $f\_2$ over $T$ seconds, typically 8 to 20 s on land. Every reflection returns its own copy of that long sweep, so the raw record is a pile of overlapping chirps in which no reflection can be seen. The processor correlates the record with the pilot, and each chirp collapses to a short, zero-phase wavelet at its reflection time.

In Figure 1.3, lengthen the sweep and watch the reflections climb out of the noise in (b); then change the band, the tapers and the distortion, and watch what each does to the wavelet in (c) and to the ghosts in (b).

The opening sweep, 4 s from 8 to 60 Hz, lifts reflection 1 from −20 dB in the raw record to 3.5 dB after correlation, a gain of 23.5 dB. That is the time-bandwidth product at work. Correlation with the pilot is a matched filter, and for noise inside the sweep’s band its gain is about $G = 10\\log\_{10}(TB)$ dB, with $B = f\_2 - f\_1$: here $10\\log\_{10}(4 \\times 52) = 23.2$ dB.

The Klauder wavelet

Correlating one reflection’s copy of the sweep with the pilot gives the pilot’s autocorrelation, the Klauder wavelet of plate (c). It is zero phase, and its amplitude spectrum is the square of the pilot’s. Its central peak is about $1/(f\_1 + f\_2)$ wide between zero crossings, so it is set by the centre frequency $(f\_1 + f\_2)/2$, not by the bandwidth $B$: 15 ms for 8 to 60 Hz and 8 ms for 2 to 120 Hz, while 5 to 40 Hz gives 22 ms, wider than the 17 ms of 20 to 40 Hz although its band is wider. The number of octaves, $\\log\_2(f\_2/f\_1)$, sets its side lobes: a one-octave sweep, 20 to 40 Hz, leaves side lobes of 84 % of the peak, so every reflection rings, while three octaves bring them to about 40 %. That is why a sweep starts as low as the vibrator can drive.

Sweep design trade-offs

• Low frequency $f\_1$: more octaves, a cleaner wavelet and deeper penetration. The reaction mass must travel further for the same force as the frequency falls (its stroke grows as $1/f^2$), so a vibrator’s force is limited at the low end, and practical sweeps start at 2 to 8 Hz, often at reduced force.
• High frequency $f\_2$: a narrower central peak and finer shallow resolution, but high frequencies attenuate fastest (Section 0.10), and the baseplate flexes and loses contact with soft ground as the frequency rises.
• Sweep length $T$: energy. The gain grows as $10\\log\_{10}T$, 3 dB per doubling, just as stacking $n\_{\\mathrm{s}}$ sweeps adds $10\\log\_{10}n\_{\\mathrm{s}}$ (Section 0.8). The figure’s first exercise quadruples $T$ from 4 to 16 s: the law adds 6 dB, and the recording time per VP grows from 7 to 19 s.
• End tapers: a raised-cosine ramp of a few hundred milliseconds at each end lets the baseplate start and stop smoothly and removes the ripple an abrupt start leaves across the pilot’s spectrum, about 2.5 dB peak to peak in plate (d). A long taper is not free: it narrows the band that carries full force, so the side lobes grow a little and the gain falls by up to a dB.

Peak force and energy

Peak force and total energy are separate budgets. A vibrator’s peak force is capped by its hold-down weight: push harder than the truck bears down and the baseplate lifts off, so a large vibrator, rated at 60,000 to 80,000 lbf (about 270 to 360 kN), is driven just below the weight on its baseplate. Force adds in phase across a group of $N\_{\\mathrm{v}}$ vibrators, so two trucks double the amplitude and add 6 dB. Sweep length and the number of sweeps add energy against the same noise at 3 dB per doubling. The radiated energy grows as the square of the group’s force times the total sweep time: force is bought with trucks, energy with time, as the fifth exercise shows.

Harmonic distortion

Neither the hydraulics nor the ground is perfectly linear, so the ground force carries harmonics of the pilot frequency, commonly 10 to 30 % of the fundamental at low frequencies. The second harmonic of a frequency $f$ is $2f$, which an up-sweep reaches later, so correlation puts its energy at negative lag: a smeared harmonic ghost ahead of every reflection. The ghost of the first strong arrival falls before time zero and is discarded with it. A down-sweep puts every ghost behind its parent, on top of the weaker reflections that follow (plate (b) and the fourth exercise), which is the main reason vibrators sweep up. Against its parent the ghost is weak, about 32 dB down for the opening sweep, because it does not compress into a wavelet; it matters beneath strong early energy such as ground roll, which can stand 40 dB or more above the deep reflections.

The vibrator’s controller estimates the ground force from the weighted sum of the baseplate and reaction-mass accelerations and phase-locks its fundamental to the pilot, which keeps the Klauder wavelet centred on the reflection time. The harmonics themselves are reduced by sweep design and in processing: sweeps whose starting phase is rotated from one to the next, each correlated with its own pilot, cancel chosen harmonics when they are summed, and harmonic-removal filters subtract what remains.

References

• Crawford, J. M., Doty, W. E. N., Lee, M. R. (1960). Continuous signal seismograph. Geophysics, 25(1), 95-105.
• Sallas, J. J., Weber, R. M. (1982). Comments on “The amplitude and phase response of a seismic vibrator” by W. E. Lerwill. Geophysical Prospecting, 30, 935-938.
• Sallas, J. J. (1984). Seismic vibrator control and the downgoing P-wave. Geophysics, 49(6), 732-740.
• Pritchett, W. C. (1990). Acquiring Better Seismic Data. Chapman & Hall.
• Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data (2 vols.). SEG Investigations in Geophysics 10.