Review of vibroseis data acquisition and processing for...

Geophysical Prospecting, 2010, 58, 41–53 doi: 10.1111/j.1365-2478.2009.00841.x

Review of vibroseis data acquisition and processing for betteramplitudes: adjusting the sweep and deconvolving for thetime-derivative of the true groundforce

Anton Ziolkowski∗PGS, Geoscience and Engineering, Birch House, 10 Bankhead Crossway South, Edinburgh EH11 4EP, UK

Received February 2009, revision accepted September 2009

ABSTRACTThe goal of vibroseis data acquisition and processing is to produce seismic reflectiondata with a known spatially-invariant wavelet, preferably zero phase, such that anyvariations in the data can be attributed to variations in geology. In current practicethe vibrator control system is designed to make the estimated groundforce equal tothe sweep and the resulting particle velocity data are cross-correlated with the sweep.Since the downgoing far-field particle velocity signal is proportional to the time-derivative of the groundforce, it makes more sense to cross-correlate with the time-derivative of the sweep. It also follows that the ideal amplitude spectrum of thegroundforce should be inversely proportional to frequency. Because of non-linearitiesin the vibrator, bending of the baseplate and variable coupling of the baseplate tothe ground, the true groundforce is not equal to the pre-determined sweep and variesnot only from vibrator point to vibrator point but also from sweep to sweep at eachvibrator point. To achieve the goal of a spatially-invariant wavelet, these variationsshould be removed by signature deconvolution, converting the wavelet to a muchshorter zero-phase wavelet but with the same bandwidth and signal-to-noise ratioas the original data. This can be done only if the true groundforce is known. Theprinciple may be applied to an array of vibrators by employing pulse coding techniquesand separating responses to individual vibrators in the frequency domain. Variousapproaches to improve the estimate of the true groundforce have been proposed orare under development; current methods are at best approximate.

INTRODUCTION

The theory of the elastic radiation from a vibrator has beenstudied extensively for over half a century, following the sem-inal papers of Bycroft (1956) and Miller and Pursey (1954).Geyer (1989) provided a comprehensive collection of paperson vibroseis. In a landmark paper that provoked much dis-cussion, including the famous comments by Sallas and Weber(1982), Lerwill (1981) compared the amplitude and phase re-sponse of a model vibrator with the performance of a realvibrator in the field and came to the conclusion that the main

∗E-mail: [email protected]

characteristics of the Earth’s response could be obtained byappropriate measurements ‘at the output of the vibrator’. Thisoutput, we now understand to be the ‘groundforce’, which isthe force of the vibrator baseplate on the ground. It is, in fact,the boundary condition at the Earth’s surface. Its measure-ment, it turns out, is not simple.

This paper considers the measurements and processing stepsthat are required to produce seismic reflection data witha known spatially-invariant wavelet, preferably zero phase,such that any variations in the data can be attributed to vari-ations in geology. A major step towards this goal was madeby Sallas and Allen (1998a,b) who solved the problem of de-termining the impulse responses between individual vibratorsin a vibrator array and any geophone. The solution requires

C© 2009 European Association of Geoscientists & Engineers 41

42 A. Ziolkowski

the force of the vibrator plate on the ground, the ‘ground-force’, to be known for each vibrator in the array. Estimationof the groundforce has been studied extensively since the fa-mous ‘weighted sum’ invention of Castenet and Lavergne waspublished in 1965.

Conventionally the recorded vibroseis data are cross-correlated with the sweep. The control system of the vibra-tor attempts to make the estimated groundforce equal to thesweep but it is well-known that the control system is imper-fect and the groundforce and the sweep are not the same:not only are there non-linearities in the vibrator itself, caus-ing harmonics of the instantaneous frequency to be generated(e.g., Bagaini 2008), the groundforce measurement itself isnot perfect, as discussed below. The groundforce varies fromplace to place, so the result of cross-correlation is to gener-ate a wavelet that depends on the vibrator position. Giventhe true groundforce at each vibrator, it is possible with de-convolution to produce data in which the wavelet is laterallyinvariant.

An important issue that appears to have been largely over-looked in the acquisition of vibroseis data is that the down-going particle velocity wavefield is basically proportional tothe time-derivative of the groundforce (Baeten, Fokkema andZiolkowski 1988). To recover the full bandwidth response ofthe Earth from the measured data, it is most efficient to illumi-nate the Earth with a broad bandwidth, preferably white, spec-trum. It also follows that the ideal amplitude spectrum of thegroundforce should be inversely proportional to frequency.This is counter-intuitive, since it emphasizes the low frequen-cies rather than the high frequencies and is not in agreementwith current practice.

The convolutional model for a single vibrator is consid-ered first to establish the framework for discussion. Thecross-correlation process is then compared with deconvolu-tion and a method for designing deconvolution operators isproposed. I discuss sweep design for an amplitude spectruminversely proportional to frequency. It turns out that physi-cal limitations of the vibrator at low frequencies increase thetime that has to be spent beyond what would be expected(Bagaini 2008). The invention of Sallas and Allen (1998a,b)is then briefly described, highlighting the need for estimatesof the true groundforce for each vibrator. It turns out thatdetermination of the true groundforce is not solved exactly;but there are several approaches that, combined, may allowus to converge on a satisfactory approximation. Finally, Idiscuss the consequences for data quality of the proposalfor groundforce amplitude to be inversely proportional tofrequency.

CONVOLUTIONAL M ODEL FOR ONEVIBRATOR

Baeten et al. (1988) modelled the wavefield of a vibrator at thesurface of a homogeneous isotropic elastic half-space. Becausethe half-space is elastic, no energy is absorbed. In the farfield below the vibrator the surface waves disappear and thedowngoing particle displacement components are essentiallyproportional to the integrated surface traction components, orgroundforce. It follows that the far-field downgoing particlevelocity components are proportional to the time-derivativeof the groundforce. This needs to be put into the formulationof the problem.

Appendix B of Baeten et al. (1988) defines the far field.There are two components. First, there is a constraint thatcomes straight from geometry: the far-field is at a distancethat is large compared with the source dimensions (equa-tions (B3) and (B4)); for a single vibrator the far-field wouldbe much less than 100 m; for an array of vibrators it wouldbe about 1 km. Second, it is also required that the far fieldbe at a distance greater than a wavelength of both P-wavesand S-waves (equation (B9)). Using a maximum velocity of3000 m/s and a lowest frequency of 5 Hz, gives a far-fieldof 600 m. Taking the two constraints together, we are in thefar field at ranges greater than about 600–1000 m below thesource. This is shallower than most targets of interest. Thatis, most targets of interest are in the far-field.

Figure 1 shows a vertical vibrator source and geophonereceivers at the surface of the Earth. Although the geophonesmeasure particle velocity, it is convenient to start with thedisplacement. The particle displacement at the kth geophone

Figure 1 Vibrator source and geophone receivers at the surface of theearth.

C© 2009 European Association of Geoscientists & Engineers, Geophysical Prospecting, 58, 41–53

Review of vibroseis data acquisition and processing 43

is

uk(t) = f (t) ∗ gk(t) + mk(t), (1)

in which, uk(t) is the particle displacement at the geophone,f (t) is the force of the vibrator on the ground, the asterisk∗ denotes convolution, gk(t) is the particle displacement re-sponse at the receiver to an impulsive force acting verticallydownwards at the vibrator position and mk(t) is displacementnoise at the receiver in the absence of the source. Note thatgk(t) is not some ‘reflectivity series’; it is the complete impulseresponse and contains P-waves, S-waves, surface waves in-cluding ground roll, reflections, refractions, diffractions, etc.

The particle velocity at the kth geophone is obtained bydifferentiating equation (1) to give

vk(t) = u′k(t) = f ′(t) ∗ gk(t) + nk(t), (2)

in which the prime indicates time-derivative and nk(t) is thetime-derivative of mk(t). Note that only one of f (t) or gk(t)are differentiated. We choose to differentiate the groundforcef (t) because of the result of Baeten et al. (1988), resultingin a better description of the convolutional model in the farfield. Krohn (2006) noted in paragraph 45: “Use of the timederivative of the ground force signal results in less noise andartifacts and less phase distortion than using the ground forcesignal itself.”

Equation (2) is the starting point for the subsequent discus-sion. We will be concentrating on trying to obtain the bestpossible estimate of gk(t) from the measurement of the par-ticle velocity vk(t). We are particularly interested in the farfield of the source where our targets are. The noise nk(t) isthe noise we would measure in the absence of our signal andhas the normal characteristics of seismic noise measured withgeophones. We will make no assumptions about this noise;we note that it is a problem for the deconvolution and that itis probably not white.

CROSS -CORRELA T I ON , DE C ON V OL UTIONAND T HE SPECTRUM OF T HE SWEEP

Cross-correlation

In conventional vibroseis processing, the data are cross-correlated with the sweep s(t). Cross-correlation is convo-lution with the time-reverse, so this can be expressed as

s(−t) ∗ vk(t) = s(−t) ∗ f ′(t) ∗ gk(t) + s(−t) ∗ nk(t). (3)

The wavelet in the data is now

p(t) = s(−t) ∗ f ′(t), (4)

which is the cross-correlation of the sweep with the time-derivative of the groundforce. The vibrator control system isusually designed to make the estimate of the groundforce equalto the sweep. Normally, however, the true groundforce is notexactly equal to the sweep. Even if it were, the wavelet wouldnot be zero phase because of the time-derivative operator.

To obtain a better approximation to a zero-phase waveletin the far-field it would be better to cross-correlate with thetime-derivative of the sweep, which can be expressed as

s ′(−t) ∗ vk(t) = s ′(−t) ∗ f ′(t) ∗ gk(t) + s ′(−t) ∗ nk(t). (5)

The wavelet in the data is now

p′(t) = s ′(−t) ∗ f ′(t). (6)

If the groundforce is in-phase with the sweep, this waveletwill be more nearly zero phase. We recognize that althoughthis makes sense for the far-field data, it is less good for theshallow near-field data.

Ideally, we should like to know the groundforce. If f (t) isknown, the deconvolution is straightforward.

Deconvolution

The objective of deconvolution is to replace the source sig-nature with an impulse or delta function δ(t). First transformequation (2) to the frequency domain

Vk(ω) = −iωF (ω)Gk(ω) + Nk(ω), (7)

in which ω is angular frequency, Vk(ω), F (ω), Gk(ω) and Nk(ω)are the Fourier transforms of vk(t), f (t), gk(t) and nk(t), respec-tively, the time-derivative becomes −iω and the convolutionbecomes a multiplication. The Fourier transform of δ(t) is 1,so signature deconvolution in the frequency domain is simplydivision by the Fourier transform of the signature, which inthis case is −iωF (ω), yielding

− Vk(ω)iωF (ω)

= Gk(ω) − Nk(ω)iωF (ω)

, (8)

in which the Fourier transform of the Earth’s impulse responseGk(ω) is recovered without contamination by any wavelet butthe noise blows up at frequencies where |iωF (ω)| becomesclose to zero. Since the groundforce function F (ω) is band-limited, |iωF (ω)| is zero, or close to zero, at frequencies out-side this band.

A stabilized deconvolution of equation (7) for Gk(ω) canbe performed as described by Baeten and Ziolkowski (1990,Ch. 7). Consider the filter z(t) with Fourier transform Z(ω)


44 A. Ziolkowski

defined as

− D(ω)iωF (ω)

≈ i F̄ (ω)D(ω)

ω |F (ω)|2 + ε= Z(ω), (9)

in which F (ω) is the complex-conjugate of F (ω), ε is a smallpositive constant and D(ω) is the spectrum of some short de-sired wavelet d(t). Appendix A discusses the design of thiswavelet: it is preferably zero phase and as close to a delta-function as is possible within the available bandwidth. Its am-plitude spectrum |D(ω)| is smooth and, in this case, approx-imately equal to |iωF (ω)|. Therefore |Z(ω)| is approximatelya constant over the entire bandwidth. Multiplication of thenumerator and denominator by the complex conjugate of thespectrum of the wavelet iω F̄ (ω) makes the denominator realand positive and the small positive constant ε prevents over-flow when |F (ω)| is very small (Stoffa and Ziolkowski 1983).Multiplication of (7) by this filter gives

Z(ω)Vk(ω) = −Z(ω)iωF (ω)Gk(ω) + Z(ω)Nk(ω)

≈ D(ω)Gk(ω) + Z(ω)Nk(ω).(10)

The result is to replace the time-derivative of the ground-force f ′(t) with the short desired zero-phase wavelet d(t), es-sentially without affecting the noise, since |Z(ω)| is essentiallya constant. This is the best we can do. Clearly, it is critical toknow the groundforce.

Since the objective of the deconvolution is to replace f ′(t)with an impulse, which has a flat amplitude spectrum, weconclude that it is desirable that f ′(t) has a flat spectrum.Therefore f (t) should have an amplitude spectrum that decaysinversely with frequency; that is,

|F (ω)| ≈ Aω

, (11)

in which A is a constant. Equation (11) has implications forsweep design and vibroseis operations that are considered be-low.

Adjusting the sweep

A sweep can be described as

s(t) = e(t) sin (θ (t)) , (12)

where e(t) is an envelope function and θ (t) is the instantaneousphase. The derivative of the phase θ ′(t) is the instantaneousangular frequency ω(t) and the second derivative θ ′′(t) is thesweep rate. Rietsch (1977) showed that the amplitude spec-trum of the sweep is related to the sweep rate as

|S(ω)| = 12

e(t)[θ ′′(t)

]− 12 . (13)

As shown in Appendix B, defining |S(ω)| = A/ω in equa-tion (13), where A is a function of the sweep parameters andwith the envelope function equal to one, yields the followingsweep

s(t) = sin(

θ (0) + ω1ω2T(ω1 − ω2)

ln[1 + (ω1 − ω2)t

ω2T

]), (14)

where θ (0) is the phase at t = 0 and ω1 and ω2 are the instan-taneous angular frequencies at t = 0 and t = T, respectively.Figure 2 shows this sweep, its autocorrelation function andits amplitude spectrum, for an instantaneous frequency thatvaries from ω1/(2π ) = 5 Hz to ω2/(2π ) = 80 Hz in a timeT = 2 s. The ripples in the amplitude spectrum are the well-known Gibbs’ phenomenon caused by the sharp corners of theenvelope function. A taper is usually applied at each end ofthe envelope to reduce these ripples (e.g., Bagaini 2008). An-dersen (1994) developed a method for designing sweeps withsimple autocorrelation functions having minimal side lobe en-ergy. This was achieved partly by extending the bandwidth tofrequencies as low as 1 Hz. Andersen’s (1994) method did notaddress the issue of harmonics introduced by non-linearitiesin the vibrator, bending of the baseplate and variable groundcoupling.

The autocorrelation of the sweep of equation (14) is notvery compact but this is to be expected since the amplitudespectrum is not flat. The time-derivative of this sweep,

s ′(t) = cos(

θ (0) + ω1ω2T(ω1 − ω2)

ln[1 + (ω1 − ω2)t

ω2T

])

× ω1ω2Tω2T − (ω2 − ω1)t

, (15)

however, does have a flat amplitude spectrum, apart fromGibbs’ phenomenon ripples, as shown in Fig. 3. It also has amore compact autocorrelation function, as expected.

As mentioned above, if the data are cross-correlated, ratherthan deconvolved, it makes more sense for identification offar-field targets to cross-correlate with the time-derivative ofthe sweep than with the sweep itself. Even so, the wavelet inthe data will not be zero phase because the groundforce is notequal to the sweep and varies from place to place. To achievecorrelated data that have a spatially-invariant wavelet, prefer-ably zero phase, it is necessary to perform the deconvolutionafter cross-correlation.

Deconvolution after cross-correlation

After cross-correlation with the sweep, the wavelet is given byequation (4); after cross-correlation with the time-derivative



Figure 2 a) Sweep 5–80 Hz, computed with 2 ms sampling interval, b) autocorrelation of (a), c) instantaneous frequency for sweep (a) and d)amplitude spectrum of sweep (a).

of the sweep, as proposed here, the wavelet is given by equa-tion (6). In either case the wavelet is different for every vi-brator position but is known if the groundforce is known.Letting this wavelet be h(t), with Fourier transform H(ω), thedeconvolution filter would be y(t), with Fourier transform

Y(ω) = H̄(ω)D(ω)

|H(ω)|2 + ε, (16)

in which |D(ω)| is fixed for the survey and is essentially asmooth version of |H(ω)|, as described in Appendix A. A newfilter y(t) should be calculated for every sweep at every vibra-tor position. Deconvolution of the correlated data with thisfilter will replace the wavelet with the fixed desired waveletd(t).

Practical issues at low frequencies

Creating a vibroseis sweep with an amplitude spectrum in-versely proportional to frequency is more difficult in practice

than in theory. Limits to the vibrator reaction mass maxi-mum displacement and pump flow rate both impede gener-ation of low- frequency energy (Bagaini 2007; Wei 2008).For frequencies below about 8 Hz, the envelope functione(t) cannot be a constant: it must decrease with decreasingfrequency.

This problem has been recognized for a number of years. In-dependent of the argument presented here, the need for morelow-frequency vibroseis energy has been noted for exampleby Andersen (1994) and Jeffryes and Martin (2003). Bagaini(2007, 2008) and Bagaini et al. (2008) developed an approachto vibrator sweep design that accommodates the physical lim-itations at low frequencies. It requires lower sweep rates be-cause the amplitude is limited by the reaction mass strokeat low frequencies. If this is combined with the approachproposed here it will lead to even lower sweep rates at lowfrequencies than are shown in Fig. 2(c). Some days the sundoes not shine.


46 A. Ziolkowski

Figure 3 a) Time-derivative of sweep in Fig. 2(a), b) amplitude spectrum of (a) and c) autocorrelation of (a).

The low-frequency limitations of vibrators have not beenignored by manufacturers and there is at least one vibratoron the market that is designed specifically for low frequencies(but can also operate up to 100 Hz), with a reaction massstroke of plus or minus 20 cm and an output of 4,500 kg force(10 000 lbf) at 1 Hz (E. Christensen, pers. comm. 2009).

An important consideration when working in built-up ar-eas is property damage. If more time and effort is spent atlow frequencies the risk of damage may increase. The riskmay be mitigated by using pseudo-random sweeps instead ofa swept-frequency sine wave. The damage is caused by excit-ing resonance of the building. A pseudo-random sweep has arandomly-varying instantaneous frequency, so in a given timewindow only a small percentage of the energy is near the reso-nance frequency. In a swept-frequency sine wave the energy ina corresponding time window is always concentrated arounda narrow band of frequencies and, if the resonance frequencyis within the bandwidth, there is greater potential to excite theresonance. An early use of pseudo-random vibroseis sweeps(but for a different purpose) was described by Payton, Watersand Goupillaud (1979).

Normally pseudo-random sweeps and pseudo-random bi-nary sequences have flat amplitude spectra over a de-fined bandwidth. In our application we need an amplitudespectrum that is inversely proportional to frequency. Theproblem of producing pseudo-random sequences with desiredspectra has been solved. Tough and Ward (1999), for ex-ample, developed a procedure that produces a non-Gaussianrandom sequence with a desired autocorrelation function bya non-linear transformation of a Gaussian process.

A R R A Y O F V I B R A T O R S

Model of the seismogram

Normally an array of vibrators is used, rather than a singlevibrator. Figure 4 shows an array of M vibrators and an arrayof geophones. Sallas and Allen (1998a,b) invented a methodknown as high fidelity vibratory source or HFVS that solvesthe intra-array vibrator interaction problem and permits theresponses to individual vibrators in an array to be extractedfrom the data. In the following, the equations of Sallas and



Figure 4 Array of vibrators and array of geophones.

Allen are modified to include the time-derivative operator de-scribed above.

Let the displacement impulse response of the Earth for theith vibrator and the kth geophone be gik(t). The response atthe kth geophone to all M vibrators is a superposition ofconvolutions, similar to the convolution in equation (2), plusnoise,

vk(t) =M∑

i=1

f ′i (t) ∗ gik(t) + nk(t), (17)

in which fi (t) is the groundforce of the ith vibrator and theprime indicates the time-derivative. Clearly we need to findthe gik(t).

Separation of responses to individual vibrators in an array

Silverman (1979) began the investigation of the problem ofseparating the responses to individual vibrators in an array byconsidering the case of two vibrators A and B and two sweeps.In the first sweep the geophone measures the response to A+B.Then, in the second sweep, the polarity of the second vibratoris reversed so that the geophone measures the response to A-B.Adding the two measurements gives twice the response to A;subtracting the second measurement from the first gives twicethe response to B. Following Silverman, there were a numberof patent applications on phase encoding, including Landrum(1987).

Sallas and Allen (1998a,b) recognized that phase encodingis very important but it is not sufficient for complete separa-tion of individual responses because it relies on the responseof a vibrator being repeatable from sweep to sweep. This is

rarely the case. Nevertheless, Sallas and Allen (1998a,b) rec-ognized that the impulse responses gik(t) could be found byencoding the M vibrator signals, using M or more sweeps, andtransforming the data to the frequency domain. In the timedomain the measurements at the kth geophone are

v jk(t) =M∑

i=1

f ′j i (t) ∗ gik(t) + njk(t), j= 1, 2,...,M or more,

(18)

in which j is the number of the sweep and f ji (t) is the ground-force of the ith vibrator in the jth sweep. Transforming (18)to the frequency domain converts the convolutions to multi-plications and gives

Vjk(ω) = −iωM∑

i=1

F ji (ω)Gik(ω) + Njk(ω),

j = 1, 2,...,M or more,(19)

which is M or more simultaneous equations for each fre-quency. In the absence of noise only M sweeps are requiredand, for a given frequency, the equations may be written inthe matrix form⎡⎢⎢⎢⎢⎢⎣

V1k

V2k

...

VMk

⎤⎥⎥⎥⎥⎥⎦

= −iω

⎡⎢⎢⎢⎢⎢⎣

F11 F12 · · · F1M

F21 F22 · · · F2M

...... · · ·

...

FM1 FM2 · · · FMM

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

G1k

G2k

...

GMk

⎤⎥⎥⎥⎥⎥⎦

. (20)

Given the measurements Vjk(ω) and the groundforce coef-ficients F jk(ω), the equations can be solved for the Gik(ω) foreach frequency. Adding more sweeps gives an over-determinedsystem of equations and allows the transfer functions Gik(ω)to be recovered in the presence of noise with errors that canbe estimated.

Transforming the Gik(ω) to the time domain yields the de-sired impulse responses gik(t). The solution of equations (20)is a frequency domain deconvolution and Gik(ω) should bemultiplied by D(ω) before transformation back to time. Thiswill ensure that the wavelet is known, is the same on all tracesand has a bandwidth consistent with the source.

Note that each vibrator has a separate source static, whichcan be estimated using a common vibrator gather: which is,keeping i fixed and varying k. Unfortunately, the situation isoften more complicated than described here. The ground oftencompacts in response to the vibrations of the vibrator, so theimpulse responses gik(t) depend not only on the source andreceiver positions but also on the sweep number j. So solutionof equations (20) gives a sort of average impulse responsefrom the M or more sweeps.


48 A. Ziolkowski

In the formulation of Sallas and Allen (1998a,b) there is aperiod of listening time after each sweep, so that the measure-ments are complete convolutions, as described by equation(18). Krohn and Johnson (2006) noticed that it is possible tosave time by cascading the different sweeps together into acontinuous sequence for each vibrator and having only oneperiod of listening time at the end. Provided the impulse re-sponses are shorter than the individual sweeps, it is possibleto sort out the data in processing and still recover the gik(t).

GROUNDFORCE

In all the foregoing we see that it is essential to knowthe groundforce. Castenet and Lavergne (1965) invented amethod to estimate the groundforce using the simplified modelof the vibrator shown in Fig. 5 in which the baseplate has massMp, the reaction mass has mass Mr , the vertical upwards dis-placements of the two masses are up and ur , respectively andw(t) is the force applied by the vibrator drive system to thetwo masses. The groundforce is f (t). The force of the plateacting downward is equal to the force of the ground actingupward, by Newton’s third law. So the groundforce in Fig. 5is the same force as that shown in Fig. 1.

Applying Newton’s second law of motion to the reactionmass and the baseplate yields

Mr u′′r (t) = w(t), (21)

Mpu′′p(t) = f (t) − w(t), (22)

Figure 5 Simplified diagram of a vertical vibrator (after Castenet andLavergne 1965).

Adding these two equations gives the famous ‘weighted sum’estimate of the groundforce:

Mr u′′r (t) + Mpu′′

p(t) = f (t). (23)

That is, the sum of the reaction mass and the baseplate,each weighted by its vertical upwards acceleration, is equalto the groundforce. The two masses Mr and Mp are fixedand known; the two accelerations u′′

r (t) and u′′p(t) need to be

measured. This method is still widely used today to estimatethe groundforce. Bagaini (2007, 2008) showed clearly thatthe groundforce signal estimated in this way has clear har-monics that are not present of course in the sweep; that is,the groundforce is not equal to the sweep. The harmonics areattributed to non-linearities in the hydraulics of the vibrator,bending of the baseplate and to the different response of thesoil to loading and unloading.

The weighted-sum method assumes the baseplate is rigid.In fact it is not and Sallas, Amiot and Alvi (1985) were thefirst to show that the bending of the plate causes a loss offorce, especially at high frequencies. There are three knownapproaches to this problem: 1) make the baseplate stiffer, 2)model the baseplate deformation to obtain the stress distribu-tion at the interface between the baseplate and the earth and3) measure the stress at the interface between the baseplateand the earth.

The first of these is rather obvious. Work is currentlybeing done on this, for example Wei (2008). The bound-ary condition that the displacement directly under the base-plate is uniform has been studied extensively (Bycroft 1956;Awojobi and Grootenhuis 1965; Robertson 1966; vanOnselen 1982; Tan 1985): it requires a stress singularity atthe edge of the plate, which is impossible in practice. In the-ory, then, no amount of engineering is ever going to makeequation (23) exact. In practice, the stiffer the plate, the betterapproximation equation (23) becomes at higher frequencies.Sallas et al. (1985) did a series of experiments that showedhow the modulus and phase of both the acceleration and thetraction varied under the baseplate. Figures 3.3.1 and 3.3.2 inBaeten and Ziolkowski (1990) showed some of these results.

Baeten (1989) was the first to attempt to model the base-plate deformation analytically: his model predicted the stressunderneath the plate from the measurement of the verticalacceleration at the centre of the plate, assuming perfect con-tact with the ground and no horizontal stresses. This method,known as the ‘flexural rigidity method’ requires a detailedmodel of the baseplate. It successfully predicts the attenuationof high frequencies as determined by stress measurements un-der the plate.



John Sallas (1984) proposed measuring the stress at thebaseplate-ground interface using ‘load cells’. I like this ap-proach the best because it gets to the heart of the problem:it measures the boundary condition. Although these stressmeasurements have been made in controlled test-bed condi-tions and have produced excellent results, they have not beenadopted by the industry for vibroseis surveys. I understandfrom an anonymous reviewer that there appear to be two rea-sons for this: the cost of the baseplates (which is not great)and the added mass to the baseplate. Perhaps industry shouldreconsider the option to use load cells to measure the stressand obtain the true groundforce.

There is more and more focus on the use of accelerometermeasurements on the baseplate to determine the stress underthe plate. This is essentially an inverse problem and is there-fore likely to be difficult to solve. The forward problem is tocompute the accelerations on the surface of the baseplate fromapplied stresses and a model of the plate. The inverse problemis to compute the applied stresses (and thus the groundforce)given the model of the baseplate and measured accelerationson the surface of the plate.

DISCUSS ION: WI LL T H E D A T A BE BETTER?

What I am proposing is expensive: it will take more time to ac-quire the data, especially with most existing vibrators. Will theseismic data be better if the sweep has an amplitude spectruminversely proportional to frequency instead of a conventionalflat spectrum? If the data are better, is the increased cost justi-fied? What assumptions are we making about the noise? Andwhat about intrinsic attenuation?

The argument for the amplitude spectrum being inverselyproportional to frequency is straightforward. It is well-knownthat for a homogeneous isotropic unbounded elastic mediumthe far-field particle displacement is in-phase with the appliedforce: Aki and Richards (1980) said on p. 73 that Stokesfirst derived this in 1849. There is, therefore, really nothingremarkable about the result of Baeten et al. (1988): it is consis-tent with and rigorously based on, classical elasticity theory.The time-derivative comes in because we measure particle ve-locity, not displacement. Since our target is, in general, in thefar field of the source, the downgoing particle velocity wavefrom the vibrator source is deficient in low-frequency energyif the groundforce has a flat spectrum. The response gk(t)we are trying to measure at the kth geophone is unknownbut contains all the frequencies we are able to generate. It iswell-known from linear system theory that the recovery of thesystem response is best achieved with an input signal that has

a flat spectrum. To make the particle velocity spectrum flatin the far field, the spectrum of the force must be of the formA/ω.

Suppose we measure particle acceleration instead of parti-cle velocity, which we do, sometimes. Should we make thespectrum of the force of the form A/ω2?

The answer is no. Take the time-derivative of equation (2):

v′k(t) = u′′

k(t) = f ′(t) ∗ g′k(t) + n′

k(t), (24)

where v′k(t) is the particle acceleration, n′

k(t) is the time-derivative of the noise and g′

k(t) is the time-derivative of gk(t).Only one of the two functions in the convolution f ′(t) ∗ gk(t)is differentiated. We differentiate the impulse response, notthe source, because we are still focusing on the far field. Ingoing from equation (1) to equation (2), we were accountingfor propagation effects from the source to the far field. Inequation (24) we simply have the time-derivative of equation(2): particle acceleration instead of particle velocity, particleacceleration noise instead of particle velocity noise and thetime-derivative of the Earth’s impulse response instead of theEarth’s impulse response. After deconvolution we would re-cover the time-derivative of the Earth’s impulse response, con-volved with the desired wavelet d(t). To recover the Earth’simpulse response, we would need to integrate. This is allstraightforward.

Theoretically, as argued above, the data should be better ifwe use a force with amplitude spectrum inversely proportionalto frequency. Will the data actually be better? Or will therebe ‘a nice lot of ground roll and very little reflection data’ asone reviewer has suggested?

We can do two experiments to find out. The first, less costly,experiment would be to compute synthetic seismograms (us-ing, for example, a 3D finite-difference elastic modelling code)for a force source at the surface of an elastic Earth and lookat the particle velocity seismograms generated at the Earth’ssurface. We could convolve the calculated Earth’s impulse re-sponses with a normal linear sweep, add real noise and processthe resulting data as described above. We could also convolvethe same Earth’s impulse responses with the sweep of equa-tion (14), add the same noise, process the data and comparethe result with the linear sweep. If the results look promisingwe could do the second experiment, which would be field testswith real vibrators.

If the real data are better, the issue of cost then needs to beconsidered and there will be immediate pressure for vibratorsto deliver more power at low frequencies.

I am making no assumptions about the noise. The signaturedeconvolution described above is essentially a multiplication


50 A. Ziolkowski

in the frequency domain by an operator with an amplitudespectrum that is essentially a constant. It increases the resolu-tion of the data by compressing the wavelet in time but leavesthe spectrum of the uncorrelated noise unchanged. This is eas-ily seen in equation (10). The ground roll is part of the impulseresponse gk(t). As the low-frequency content of the reflectiondata is increased, the amplitude of the low-frequency groundroll is increased. All the methods we use in acquisition andprocessing to enhance reflections and reduce ground roll willwork just as well as they do with existing data but the ampli-tudes of both reflections and ground roll at low frequencieswill be greater.

If the far-field input signal has a flat spectrum, as I propose,the effect of constant-Q intrinsic attenuation is to apply alinear negative slope to the logarithmic amplitude as a functionof frequency. I expect this effect to be more easily observablewith a A/ω groundforce spectrum.

CONCLUSIONS

The goal of vibroseis data acquisition and processing is to pro-duce seismic reflection data with a known spatially-invariantwavelet, preferably zero phase, such that any lateral variationsin the data can be attributed to variations in geology.

The control system of the vibrator is usually designed tomake the estimated groundforce signal equal to the prede-termined sweep and the resulting particle velocity data areusually cross-correlated with the sweep. Since the downgoingfar field particle velocity signal is proportional to the time-derivative of the groundforce, it makes more sense to cross-correlate with the time-derivative of the sweep. It also followsthat the ideal amplitude spectrum of the groundforce shouldbe inversely proportional to frequency.

Because of non-linearities in the vibrator, bending of thebaseplate and variations in ground coupling, the true ground-force is not equal to the pre-determined sweep and variesnot only from vibrator point to vibrator point but also fromsweep to sweep at each vibrator point. The wavelet in thedata therefore varies from sweep to sweep, both before andafter the cross-correlation process. So we do not want to stackyet. To achieve the goal of a spatially-invariant wavelet, thesewavelet variations should be removed by signature deconvo-lution, converting the wavelet to a much shorter zero-phasewavelet but with the same bandwidth and signal-to-noise ratioas the original data. This can be done if the true groundforceis known.

A deconvolution study presented in Appendix A shows thatthere is plenty of scope for improving the resolution of vibro-seis data without altering the bandwidth of the source.

If the sweep is designed to give the groundforce an ampli-tude spectrum that is inversely proportional to frequency, thedata ought to be better but the acquisition costs will go upbecause the sweep time will be longer. The ability to gener-ate low-frequency energy is also limited in most vibrators bythe pump flow rate and the reaction-mass stroke, so this fur-ther increases the sweep time. There is scope to reduce thelow-frequency limitations on the vibrator.

Sallas and Allen (1998a,b) solved the problem of determin-ing the impulse responses between individual vibrators in avibrator array and a given geophone for the case in which thegroundforce under each baseplate is known and there is noground compaction between sweeps. In practice the ground-force is not known exactly and there is often compactionthat is variable from place to place. Load cells can measurethe groundforce directly but are not used. The reasons fornot using load cells are weak. A combination of stiffer base-plates and solving the inverse problem for the determinationof the stresses on the plate from measured surface accelera-tions might lead to significantly better groundforce estimates.

We do not know how good our current estimates of thegroundforce are. To find out we must compare current esti-mates with true groundforce measurements. That is, we needto make the load cell measurements.

ACKNOWLEDGEMENTS

I am very grateful to the organizers of the EAGE Vibro-seis Workshop 2008 (Claudio Bagaini, Julien Meunier, PeterPecholcs, John Sallas and Peter Scholz) for inviting me topresent this paper. It is some time since I worked on vibroseiswith Bill Lerwill (in the late 1970s) and Guido Baeten andFloris Strijbos (1980s and early 1990s) and I had become outof date. John Sallas rescued me and pointed me in the right di-rection. I am very grateful to Claudio Bagaini and two otherreviewers–who prefer to remain anonymous–for their com-ments and suggestions for ‘minor revision’. These resulted ina complete overhaul of the paper and doubling of its length.I thank my colleagues Bruce Hobbs and Andrew Long forreviewing the various manuscripts and advising on improve-ments. I thank Guido Baeten for clarifying the definition of farfield and I thank Steve McLauchlin and Bernie Mulgrew forinteresting discussions on pseudo-random binary sequences. Ithank PGS for permission to publish the paper.



REFERENCES

Aki K. and Richards P.G. 1980. Quantitative Seismology. W.H. Free-man.

Andersen K.D. 1994. Shaped-sweep technology. US Patent 5,347,494.Awojobi A.O. and Grootenhuis P. 1965. Vibration of rigid bodies

on semi-infinite elastic media. Proceedings of the Royal Society ofLondon, Series A 287, 27–63.

Baeten G.J.M. 1989. Theoretical and practical aspects of the vibroseismethod. PhD thesis, Delft University of Technology, The Nether-lands.

Baeten G.J.M., Fokkema J.T. and Ziolkowski A.M. 1988. Seismicvibrator modelling. Geophysical Prospecting 36, 22–65.

Baeten G.J.M. and Ziolkowski A.M. 1990. The Vibroseis Source.Elsevier.

Bagaini C. 2007. Enhancing the low-frequency content of vibroseisdata with maximum displacement sweeps. 69th EAGE meeting,London, UK, Expanded Abstracts, B004.

Bagaini C. 2008. Low-frequency vibroseis data with maximum dis-placement sweeps. The Leading Edge 27, 582–591.

Bagaini C., Dean T., Quigley J. and Tite G.A. 2008. Systems andmethods for enhancing the low-frequency content in vibroseis ac-quisition. US Patent 7,327,633.

Berkhout A.J. 1974. Related properties of minimum-phase and zero-phase time functions. Geophysical Prospecting 22, 683–709.

Bycroft G.N. 1956. Forced vibrations of a rigid circular plate on asemi-infinite elastic space and on an elastic stratum. PhilosophicalTransactions of the Royal Society, Series A 248, 327–368.

Castenet A. and Lavergne M. 1965. Vibrator controlling system. USPatent 3,208,550.

Geyer R.L. 1989. Vibroseis. Geophysics Reprint Series No. 11. SEG.Jeffryes B.P. and Martin J.E. 2003. Method and apparatus for in-

creasing low-frequency content during vibroseismic acquisition. UKPatent No. 0415518.0.

Krohn C.E. 2006. Shaped high frequency vibratory source. US PatentApplication US 2006/0250891 A1.

Krohn C.E. and Johnson M.L. 2006. Method for continuous sweepingand separation of multiple seismic vibrators. US Patent ApplicationUS 2006/0164916 A1.

Landrum R.A. 1987. Simultaneous performance of multiple seismicvibratory surveys. US Patent 4,715,020.

Lerwill W.E. 1981. The amplitude and phase response of a seismicvibrator. Geophysical Prospecting 29, 503–528.

Miller G.F. and Pursey H. 1954. The field and radiation impedance ofmechanical radiators on the free surface of a semi-infinite isotropicsolid. Proceedings of the Royal Society A 223, 521–541.

Payton C.E., Waters K.H. and Goupillaud P.L. 1979. Simultaneoususe of pseudo-random control signals in vibrational explorationmethods. US Patent 4164185.

Rietsch E. 1977. Vibroseis signals with prescribed power spectrum.Geophysical Prospecting 25, 613–620.

Robertson I.A. 1966. Forced vertical vibration of a rigid circulardisk on a semi infinite elastic solid. Proceedings of the CambridgePhilosophical Society 62, 547–553.

Sallas J.J. 1984. Seismic vibrator control and the downgoing P-wave.Geophysics 49, 732–740.

Sallas J.J. and Allen K.P. 1998a. Method and apparatus for sourceseparation of seismic vibratory signals. US Patent 5,719,821.

Sallas J.J. and Allen K.P. 1998b. High fidelity vibratory source seismicmethod with source separation. US Patent 5,721,710.

Sallas J.J., Amiot E.A. and Alvi H.I. 1985. Ground force control of aP-wave vibrator. SEG Seismic Field Techniques Workshop, 13–16August 1985, Monterey, California.

Sallas J.J. and Weber R.M. 1982. Comments on ‘The amplitude andphase response of a seismic vibrator’ by W.E. Lerwill. GeophysicalProspecting 30, 935–938.

Silverman D. 1979. Method of three-dimensional seismic prospecting.US Patent 4,159,463.

Stoffa P.L. and Ziolkowski A.M. 1983. Seismic source decomposition.Geophysics 48, 1–11.

Tan T.H. 1985. The elastodynamic field of N interacting vibrators(two-dimensional theory). Geophysics 50, 1229–1252.

Tough R.J.A. and Ward K.D. 1999. The correlation properties ofgamma and other non-Gaussian processes generated by memorylessnonlinear transformation. Journal of Physics D: Applied Physics32, 3075–3084.

Trantham E.C. 1993. Minimum uncertainty filters for pulses. Geo-physics 58, 853–862.

Van Onselen C. 1982. Vibroseismic excitation of elastic waves in asemi-infinite solid. Internal Report No. 1982–10, Laboratory ofElectromagnetic Research, Department of Electrical Engineering,Delft University of Technology.

Wei Z. 2008. Reducing harmonic distortion on vibrators–stiffeningthe vibrator baseplate. 70th EAGE meeting, Rome, Italy, ExpandedAbstracts, B006.

Ziolkowski A., Sneddon Z. and Walter L. 1999. Determination oftube-wave to body-wave ratio for Conoco borehole orbital source.69th SEG meeting, Houston, Texas, USA, Expanded Abstracts.

APPENDIX A: DES IGN OF THE D ES IREDWAVELET

Trantham (1993) gave a general method for constructing min-imum uncertainty filters from analytical functions. The un-certainty of a filter, or wavelet, is dimensionless: it is theproduct of its length in time and bandwidth in frequency.Trantham (1993) said “Note that data processing objectivesmay require a specific deconvolution impulse response that isnot minimum uncertainty”. That is the case here. I describehere an approach I have used for many years (for example,Baeten and Ziolkowski 1990, Ch. 7; Ziolkowski, Sneddon andWalter 1999) to design a desired wavelet. The approach ofKrohn (2006) is similar.

If the amplitude spectrum of a wavelet has a lot of detailedvariation, many samples are required to define it in the fre-quency domain and there must be at least as many correspond-ing samples in the time domain. The number of time-domainsamples also depends on the phase. If the wavelet is zero


52 A. Ziolkowski

Figure 6 Elements of the design of a desired wavelet: a) same as Fig. 3(a) but plotted –0.1 to +0.1 s, b) amplitude spectrum of (a), c) desiredzero-phase wavelet, d) amplitude spectrum of (c), e) comparison of input wavelet and desired wavelet plotted –0.1 to +0.1 s and f) comparisonof amplitude spectra of input wavelet and desired wavelet.

phase the number of time samples is minimum and the zero-phase wavelet is the shortest wavelet with a given amplitudespectrum (Berkhout 1974). If the phase is zero and the ampli-tude spectrum is smooth, few samples are required to specifythe spectrum and there are few corresponding samples in thetime domain. In other words, the wavelet is short.

I illustrate my trial-and-error approach to the design of adesired wavelet for signature deconvolution using the auto-correlation function shown in Fig. 3(c). It is reproduced againin Fig. 6(a) but plotted from −1 to +1 seconds. Its ampli-tude spectrum is shown in Fig. 6(b). This is the power spec-trum of the sweep shown in Fig. 3(a) or the square of the



amplitude spectrum shown in Fig. 3(b). This spectrum has alot of wiggles, so it needs many samples to specify it in thefrequency domain. We see the corresponding wiggles inthe time domain in Fig. 6(a). We need a desired amplitudespectrum that is smooth and which is close to the amplitudespectrum of Fig. 6(b), such that the ratio of the amplitudes isof the order of 1. A possible design is shown in Fig. 6(d), withthe corresponding zero-phase time-domain wavelet shown inFig. 6(c). To achieve this, I set the flanks of the desired spec-trum to be the same as those of the input spectrum, from 0–5.5Hz and from 80.0–250.0 Hz (the sampling interval was 2 ms)and I set two control points with amplitudes of 15 000 at 15Hz and 55 Hz. In-between the flanks and these control pointsI interpolated with a cubic-spline interpolator. Comparing theoriginal spectrum with the desired spectrum (Fig. 6f), we seethat, of course the ratio is 1 at frequencies below 5.5 Hz andabove 80 Hz. In-between the ratio is of the order of 1–perhapsvarying between 0.3– 3.0. We could make the ratio closer to 1by making the desired spectrum flatter in the middle but thiswould introduce shoulders in the spectrum and force extraoscillations in the time domain.

Figure 6(e) shows the original wavelet and the desiredwavelet superposed: the desired wavelet is much shorter andsmoother than the original.

The wavelet I used for this illustration is much shorter thanwavelets we will measure if we try to measure the true ground-force. There is therefore a great deal of scope for improvementof the resolution of vibroseis data before we attempt to extendthe bandwidth of the source.

APPENDIX B: SWEEPS

Following Rietsch (1977), a sweep s(t) can be defined as

s(t) = e(t) sin (θ (t)) , (B1)

where e(t) is an envelope function and the phase function θ (t)is given by

θ (t) = θ (0) +∫ t

0θ ′(τ )dτ , (B2)

where θ (0) is the phase at t = 0 and θ ′(τ ) = ω(τ ) is the in-stantaneous angular frequency at time τ . The instantaneousfrequency changes with time at a rate known as the sweep ratewhich, in general, is frequency-dependent:

ω′(t) = θ ′′(t) = a(ω). (B3)

Rietsch (1977) showed that the amplitude spectrum of thesweep |S(ω)| is related to the sweep rate as

|S(ω)| = 12

e(t)[θ ′′(t)

]− 12 . (B4)

We require |S(ω)| to be of the form B/ω, where B is aconstant. We let the envelope function e(t) = 1. Then, fromequation (B4), we have

θ ′′(t) = ω′(t) = Aω2, (B5)

where A = 1/(4B2) is a constant that depends on the sweepparameters. We rewrite equation (B5) as

dω(t)ω2

= Adt. (B6)

Integrating equation (B6) leads to

θ ′(t) = ω(t) = ω1ω2Tω2T − (ω2 − ω1)t

, (B7)

where ω(0) = ω1, ω(T) = ω2 and, therefore, A = (ω2 −ω1)/(Tω2ω1).

Integrating equation (B7) gives

θ (t) =∫ t

0ω(τ )dτ = ω1ω2T

(ω1 − ω2)ln

[1 + (ω1 − ω2)t

ω2T

](B8)

and the sweep is

s(t) = sin(

θ (0) + ω1ω2T(ω1 − ω2)

ln[1 + (ω1 − ω2)t

ω2T

]). (B9)


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