chapter 2 vibroseis, nonlinearity, and harmonics

Chapter 2 Vibroseis, Nonlinearity, and Harmonics

The ear is, however, endowed with a property which the eye does not possess, namely that of creating waves of entirely new frequencies out of the disturbances which fall on it.

— James Jeans (1877–1946)

If we were to survey the universe of seismic sources in use today for production-seismic data in the petroleum industry, it would reveal only a few serious contenders (Liner, 2008). Since the beginning of commercial seismology in about 1930, many seismic sources have been developed, tested, and tossed into the Darwinian struggle for market survival as a reliable commercial source. At present, only three sources account for the vast majority of data acquisition.

In marine seismic applications, the air gun is ubiquitous. Several dispersive phe-nomena arise in the use of air guns and air-gun arrays, including ghosting and radia-tion patterns. Recall that we are using the term dispersion in a generalized sense mean-ing frequency-dependent phenomena, not just seismic-velocity variation with fre-quency.

The ghost is an interesting example of dispersion wherein the physical source interacts with the ocean surface to form a plus-minus dipole whose time signal is a strong function of frequency and propagation angle (Krail and Shin, 1990). For a given source depth, the radiated field can have one or several interference notches at frequen-cies that vary with angular direction away from the source. These show up in the mea-sured seismic data as spectral nulls called ghost notches. To further complicate the picture, ghosting occurs on both the source and receiver sides of acquisition. The radia-tion pattern associated with an air-gun array is an exercise in the theory of antenna design and analysis, again complicated by dipole characteristics caused by ghosting.

For land seismic data, two major sources are in worldwide use — explosives and vibroseis. In principle, the explosive source has the weakest dependence on frequency. Certainly the recorded signal has a bandwidth determined by shot characteristics, local geology, and recording filters, but it is an approximately impulsive point source. A bur-ied explosive shot will generate ghosting as the marine air gun does but with a descrip-tion generally more complicated than that of the simple dipole. Moreover, ghosting is often not as well developed in the land case, likely because of lateral variations in near-surface elastic properties and topography.

The other significant land source is vibroseis (Crawford et al., 1960). It is hard to imagine a vibroseis source that does not exhibit some form of dispersive behavior. For a single vibrator, we mention two fascinating phenomena — radiation pattern and harmonics. The theory of radiation from a vertically vibrating circular disk on an

Distinguished Instructor Short Course • 23

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isotropic elastic earth was developed by several investigators in the 1950s, most notably Miller and Pursey (1954). They show that the total power emitted in various wave types (P-, S-, Rayleigh) ultimately depends only on the Poisson ratio of the subsurface near the vibrator baseplate. It is theoretically not a function of frequency and is therefore non-dispersive. However, in real-world applications, it is common to use a vibroseis source array, which invariably radiates seismic waves in a way that strongly depends on fre-quency and direction. Pursey (1956) shows that the theory of radiation from a vibroseis source array can be used to advantage by arranging multiple vibrators in such a way as to, say, minimize surface wave generation.

A vibroseis unit injects a source signal (sweep) into the earth over the course of several seconds (Liner, 2008). The sweep is defined by time-frequency (TF) characteris-tics. For simplicity, here we will consider a linear upsweep, which is common in prac-tice. The emitted signal bounces around in the earth and is recorded by an array of surface sensors, with the resulting time series being an uncorrelated seismic trace.

Conceptually, when this uncorrelated time trace is transformed into the TF plane by a suitable spectral-decomposition method, we should see a representation of the sweep with a decaying tail of reflection energy. That is observed, but we also commonly see a series of other linear TF features at frequencies higher than those contained in the sweep at any given time. Those are vibroseis harmonics that can be attributed to various factors, including nonlinear effects in the vibrator itself, nonlinear coupling of the baseplate to the ground, an inadequate feedback system, and so forth (Li, 1997). Because the observed uncorrelated seismic trace is the summation of all frequencies in the TF plane, those harmonics can interfere with and distort weak reflection events. Further-more, the harmonics result in correlated noise trains known as harmonic ghosts (Seriff and Kim, 1970) that can interfere with detailed seismic interpretation.

A vibroseis unit is a complicated and delicately tuned machine that contains com-plex coupled mechanical, electrical, and hydraulic systems (Sallas, 2010). Harmonics arise from various processes in the vibrator as it executes the sweep and from baseplate interaction with the earth.

When the mechanical apparatus of the vibrator pushes down against the earth, it is resisted by the elastic nature of the near surface, whereas on the upstroke, the motion is unimpeded. This means that up-down asymmetry develops in the motion or stroke of the baseplate. In this chapter, we follow Liner (2008) to show that data asymmetry of that kind results in the generation of harmonic frequencies that generally are considered to be undesirable. The harmonics occur despite some rather amazing control and feed-back engineering in the vibroseis system (Sallas, 2010). With modern time- frequency methods, various algorithms can be derived to reduce the harmonics by processing of uncorrelated data traces (Okaya et al., 1990; Liner, 2008). An ongoing discussion also exists about how to benefit from the harmonics rather than filter them out.

A field test

In this section, we consider uncorrelated vibroseis traces from a field test (Liner, 2008) with data courtesy of Geokinetics. Table 1 shows pertinent acquisition parameters.

Elements of Seismic Dispersion

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The first 2 s of uncorrelated data in a 3D shot record is shown in Figure 1a. Because the sweep begins at 8 Hz and proceeds to 96 Hz during 12 s, the sweep rate is 7.3 Hz/s. By 2 s, the sweep frequency is passing through 8 + 2 × 7.3 = 22.6 Hz. The real part of the Fourier transform of those data (Figure 1b) shows energy in the band of frequen-cies emitted by the source (8 to 23.6 Hz, bracketed by solid horizontal lines) as well as harmonics outside the source frequency band. The harmonics are observed mainly at near offsets.

Figure 2 displays 12-s uncorrelated traces at far offset (Figure 2a) and near offset (Figure 2b). The feature of inter-est here is symmetry: Far-offset trace 320 is highly symmetrical, whereas near-offset trace 388 exhibits extreme asymmetry, termed harmonic distor-tion (Wei et al., 2010).

A close view of the first 2 s of trace 388 reveals a complicated superposition of low and high frequencies (Figure 3a). With a sweep of 8 to 96 Hz progressing across 12 s, the frequency is a slowly varying function of time. At 2 s, the sweep is passing through 20 Hz, but a Fourier amplitude spectrum at about that time shows the spectrum depicted in Figure 3b. In it, we see a clear peak at the 20-Hz sweep frequency, along with a series of diminishing peaks at multiples of 20 Hz. Those are vibro-seis harmonics, a frequency-dependent phenomenon with an interesting history.

Harmonics and hearing

Since 1992, a footnote in my “Seismos” column in The Leading Edge has said some-thing about the value of reading really old books. Old is a relative term, but one “Seis-mos” column (Liner, 2008) discusses a wonderful 1937 James Jeans book, Science and



Table 1. Vibroseis field-test information.

Parameter Value

Field site Ponca City, Oklahoma

Sweep type Linear upsweep

Sweep range 8 to 96 Hz

Sweep length 12 s

Listen time 3 s

Sensor OYO 30CT

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Figure 1. Uncorrelated vibroseis data and har-monics. (a) First 2 s of data from a near-offset receiver line in a 3D shot record (parameters are shown in Table 1). (b) Real part of Fourier trans-form over time axis. Solid horizontal lines show frequencies in the design sweep from start until 2 s. Duplicate features at higher frequencies are vibroseis harmonics.

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Music (Jeans, 1937). Jeans was one of those erudite British physicists who wrote so well and seemed to know everything.

Jeans’ book is full of details that get lost in the compression of knowl-edge into modern textbooks, including a table showing the frequency of musi-cal pitches from the fourth suboctave of C (16 Hz) to the fifth octave of B (7732 Hz). The book includes a good discussion of interference phenomena, such as beats and sum/difference tones.

Pertaining to wind instruments, Jeans (1937) tells us about eddies and whirlpools that form behind a fixed obstacle (say, a wire) when air moves by it. That is the physical basis of reed vibration that initiates sound in reed wind instruments. Turbulent flow seems to be complex and unpredict-able, but Jeans explains that certain aspects of the eddies are in fact predict-able and indeed desirable. They form at a regular distance downwind of the wire (about 5.4 times the diameter). By forming alternately on either side of the wire, they impart a shock into the air with accompanying sound. When the wind speed and wire size are just so, we can hear this as a whistling

sound, familiar to anyone who has spent time on a sailboat. The alternating eddies also can be seen in the flapping of a flag during a strong wind. The flag furls and unfurls along eddies that form downwind of the flagpole.

Another detail in Jeans’ book concerns the speed of sound. A charming analogy is made to an army of foot soldiers (remember, it was 1937) walking around randomly. The soldiers are like molecules in the air, and the sound wave is like a note of informa-tion passed by hand from soldier to soldier and making its way across the ranks. Jeans tells us the net effect is that the speed of sound is “about 74%” of the average speed of molecules whizzing randomly in the air.

Beyond that interesting fact, Jeans (1937) reports a fascinating theory formulated by W. C. Sabine of Harvard University about absorption of sound and reverberation time in rooms and concert halls. Did you know that a seated audience absorbs a middle



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Figure 2. Far- and near-offset traces from the re-ceiver line shown in Figure 1. The far-offset trace (320) shows minor visual asymmetry, whereas the near-offset trace (388) shows extreme asymmetry, termed harmonic distortion, particularly before 4 s.

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Figure 3. Uncorrelated vibroseis data from a nearest-to-shot receiver line of a 3D shot record. (a) At 2 s, the sweep is passing through 20 Hz, but (b) the Fourier spectrum reveals 20 Hz and multiples of 20 Hz (harmonics). When correlated against the sweep, those harmonics result in undesirable noise trains called harmonic ghosts that complicate seismic interpretation.

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C pitch (256 Hz) 10 times more than a pile carpet does? Sabine did, and today the Sabine is a unit of sound absorp-tion.

The section of Jeans’ (1937) book titled “Tones Created by the Ear” can really make a modern seismologist sit up and pay attention (Liner, 2008). As you might know, the eardrum is a thin membrane with air on the outside and a small bony structure and viscous fluid on the inside. Jeans explains that when a pure tone (single-frequency sine wave) interacts with the eardrum, something unexpected happens. Away from the ear, the sine wave strokes “fore and aft” equally because it finds equal resistance from the air in all directions. However, when the sine wave strikes the eardrum, it is impeded in one direction (toward the ear) and unimpeded in the other. Thus the concept of acoustic impedance is born.

Figure 4 shows the situation. The solid line represents a full-stroke, unimpeded sine wave, and the dashed line shows a wave that has unequal maximum deflection on the upstroke and downstroke, simulating resistance of the eardrum (or resistence of the earth to a vibroseis baseplate). We can quantify the nonlinearity as a dimensionless number that is zero when the maximum upstroke and downstroke displacements are equal and is the compression ratio on the downstroke when they differ. For the dashed-line example in Figure 4, the nonlinearity is 0.3, or 30%.

It is worth noting that not all kinds of linearity come from asymmetry with re-spect to up-and-down motion (Liner, 2008). The cases of vibroseis and human hearing are examples of nonlinear interaction at a boundary, leading to peak-trough asymmetry, as we have discussed. It is also possible to have nonlinearity in the wave prop agation itself, a situation termed nonlinear media. The characteristic behavior here is that wave speed depends on the amplitude of the wave.

In common materials such as air and water, that is a situation that occurs only for extremely high-energy waves. The net effect is not peak-trough asymmetry but one in which the higher-amplitude features outrun those with low amplitude, transforming an initial sine wave into a kind of seesaw shape. If the initial waveform is, say, a 100-Hz sine wave, propagation through nonlinear media will cause it to deviate from the sine shape yet still repeat 100 times per second (Liner, 2008).

As we have seen earlier, this means that a Fourier transform of the deformed sine wave will contain the original 100-Hz frequency as well as some pattern of harmonics and possibly subharmonics. In addition, water does suffer from imbalance of upstrokes and downstrokes because it is easier to push water than to pull it. A hard pull creates a vacuum in a phenomenon known as cavitation (G. Cambois, personal communica-tion, 2009).



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Figure 4. Example of linear (solid lines) and nonlinear (dashed lines) waveforms. After Liner (2008).

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To reiterate the lesson of Figure 4, if the sound wave incident on the ear is a pure 100-Hz sine wave (solid curve), it contains only one frequency, 100 Hz, and therefore repeats 100 times per second. The response of the ear (dashed curve) also repeats 100 times per second, but it is not a pure sine wave because of asymmetry in upstroke and downstroke. How can something repeat 100 times per second and not be a pure sine wave?

The sound-reception problem was studied by Helmholtz (1863). He pointed out that interaction of a pure tone with the eardrum in this way is a nonlinear process, in the sense that particle displacement is different on the fore and aft strokes of the response curve. The compressed sine wave is characteristic of nonlinear wave phenom-ena of this type. We need to be clear on the nature of this nonlinearity. It does not represent propagation of nonlinear waves but nonlinear interaction. In other words, there are linear waves in the air outside the ear and linear waves in the fluid within the ear, but nonlinear interaction occurs at the boundary between the two.

If sound is measured as motion of the eardrum, this asymmetrical process must create new frequencies to generate the distorted (compressed) sine wave — frequencies that are not present in the original pitch. However, not just any frequencies will do. They must repeat 100 times per second (to stay with our example). That can be true only for 200 Hz, 400 Hz, and so on. In other words, this nonlinear response of the eardrum results in the generation of harmonic frequencies.

That seems to go against everything we have learned about waves. Physics 101 tells us that if a source contains a certain range of frequencies, only those frequencies can be present in the output. Some of the original frequencies can be missing as a result of attenuation or interference, but we certainly would be skeptical if frequencies show up in the output that are not in the input. That is true of linear wave propagation, but the situation is different for nonlinear processes, which can produce new frequencies not generated by the source (Liner, 2008).

Figure 5 illustrates this in the time and Fourier domains (Liner, 2008). The upper plots show a pure 100-Hz wave (Figure 5a) and its Fourier amplitude spectrum (Figure 5b) displayed in decibels. The spectrum of this sine wave shows a single peak at 100 Hz. In Figure 5c and 5d, the same displays are given for the waveform shown in Figure 4, which is a 30% nonlinear distortion of the sinusoid. The spectrum shows the funda-mental frequency (100 Hz) and the harmonics that Helmholtz understood in the 1860s. It is only this distribution of frequencies with these relative strengths that can recon-struct the original nonlinear waveform in the time domain.

Jeans (1937) discusses the many consequences of this nonlinearity, but we cannot follow that path without a much longer chapter. Read Jeans’ book, if you can find it. There is a modern connection to be made, however, to the vibroseis harmonics dis-cussed earlier. If we look at uncorrelated vibroseis data in the time-frequency domain, the fundamental sweep is seen, but there are also faint (or not so faint) repetitions of the sweep with frequencies that are multiples of the fundamental. Those are vibroseis harmonics, and Figure 6 shows an excellent example.

Wei and Hall (2011) have shown differences in action of hard and soft ground on generation of vibroseis harmonics, as shown here in Figures 7 and 8. Clearly, soft ground is more conducive to generation of harmonics, so measurement of harmonic signal strength might hold promise as a method of estimating near-surface stiffness.



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It is certainly true that vibroseis harmonics are common and important, but our treatment of the subject here is aimed at understanding the nature of such phenomena. The key characteris-tics are harmonic input (a single fre-quency) to a physical system and ob-servation of new frequencies in the output. Those arise from developing asymmetry in the harmonic input as it propagates through or interacts with the medium. Linear acoustic or elastic media transmit harmonic input per-fectly to deliver harmonic output, meaning that no new frequencies are present in the output. However, nonlin-ear interaction or propagation through



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Figure 5. Linear sinusoidal waveforms contain only one frequency, whereas nonlinear distortions of the sinusoid contain new frequencies in the form of harmonics. (a) 100-Hz sine wave. (b) Fourier amplitude spectrum of 100-Hz sinusoid. (c) Nonlinear distorted 100-Hz wave with downstroke sup-pressed by 30%. (d) Amplitude spectrum of the distorted waveform showing development of har-monics. After Liner (2008).

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Figure 6. Time-frequency decomposition of field vibroseis data (GF = ground force) showing fun-damental sweep and several harmonics. The first harmonic is about 40 dB weaker than the funda-mental, implying nonlinearity of about 5%. After Bagaini, 2008, Figure 5b.

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nonlinear media returns a waveform that still repeats at the pace of the harmon-ic input but possesses asymmetries that require harmonics in their Fourier description (Liner, 2008).

Uncorrelated vibroseis records are useful for discus-sion of harmonics simply because the uncorrelated data are one of the few situations in which we commonly input a nearly pure frequency into the earth. Of course, we actu-ally input the sweep, but because the sweep is several seconds long, the input is very nearly a pure fre-quency for short periods of time, leading to straightfor-ward observation of har-monics. Harmonics are much more difficult to detect and analyze after vibroseis correlat ion or in the earth response to a broadband source such as dynamite or an air gun.



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Figure 8. Time-frequency decomposition of field vibroseis data performed on soft ground. (a) Weighted-sum ground force. (b) Response of surface geophone 1 m from vibrator baseplate After Wei and Hall, 2011, Figure 2.

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chapter 2 vibroseis, nonlinearity, and harmonics

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