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2 Performance ofPiezoelectricTransducers inTerms ofAmplitude andWaveform
Abstract
For seismic physical modeling, mostly piezo-electric transducers (PETs) are used as sourcesand receivers. Their properties have sig-nificant effects on the data, especially ifthey are to be processed as seismic data:1. Strong resonance at one frequency causes aringy signal and a narrow frequency band. 2.The pronounced directionality effectively limitsthe offsets at which energy arrives. 3. Be-cause the dimension of the transducer with 12mm is bigger than the wavelength (1.5-10 mm),the recorded waveform changes with offset. Toreduce the pronounced directionality of the trans-ducers at ultrasonic frequencies, we have de-signed PETs that have a smaller effective diame-ter than traditional ones. To test their applicabil-ity for laboratory seismic profiling, we test theirfrequency sensitivity, their directionality, and thechange of waveform as a function of offset dueto their size compared to the wavelengths. Theexperiments show that the PETs produce theirbest quality data at frequencies around 350-550kHz and source-receiver offsets≤ 14 cm. Forthese frequencies, the amplitudes decay to ring-ing noise level at incidence angles of<35◦; fora 10 cm deep reflector that results in a 14 cmsource-receiver offset. For these offsets and fre-quencies, the spacious dimension of the PETs
does not cause the waveform to change such thatfurther processing is compromised. Also, wepresent an analytical solution to the changingwaveform problem which predicts the temporaldivergence of the signal as an additional resolu-tion limit to the Fresnel effect; the loss of highfrequencies is not only caused by attenuation, butis also due to the spacious dimension of the sen-sors.
2.1 Introduction
Since the 1920s, seismic physical modeling hasbeen a successful tool for research in wave phe-nomena (i.e. the kinematics of wave propaga-tion and the validation of wave theoretical pre-dictions). In the first experiments, optical meth-ods were used to record surface motion (Tsuboi,1994) or wavefronts through transparent media(Rieber, 1936, 1937; Schmidt, 1939).
These experiments were performed on modelssuch as rods (1-D), or elastic plates (2-D and 3-D) (e.g., Berryman et al., 1958; Redwood, 1960;Purnell, 1986; Zhang et al., 1996; Wandler et al.,2007, among many others). However, before2001 virtually all models were made of solid ma-terials, and thus were static. Dynamic modelsin which the material is deformed while mon-itoring require viscous or granular media suchas sand, but severe attenuation and scatteringof seismic waves in sand prevented the appli-cation of seismic imaging methods on sandboxmodels (Purnell, 1986). Sherlock (1999) andSherlock and Evans (2001) were the first to tryto overcome these problems and performed zero-offset seismic surveys at the mm-scale usingpiezoelectric transducers (PET) on sandbox mod-els.
One reason to repeatedly try to perform seis-mic imaging on sandbox models is that thesedynamic sandbox experiments have providedqualitative and quantitative insights into spe-cific geological problems (e.g., Koyi, 1995,1997; Storti et al., 2000; Lohrmann et al., 2003;
5
6 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
Gartrell et al., 2005; Hoth et al., 2007). So far,the internal structure of sandbox models can onlybe directly observed in 2-D profiles along glasswalls confining the 2-D models or indirectly bysurface observations (e.g. PIV - particle im-age velocimetry; Adam et al., 2005) or by X-ray tomography (Colletta et al., 1991). An im-proved seismic imaging system could providenon-invasive albeit less resolved 3-D informa-tion.
To achieve this objective, a new small-scaleseismic apparatus for laboratory use was de-signed and developed, composed of a water tank,a PC control unit including PETs, and a position-ing system (Krawczyk et al., 2007) with the ulti-mate goal to apply 3-D seismic and seismologi-cal imaging methods to sandbox models subjectto deformation. These models are made of satu-rated granular materials so that deformation cantake place. Resulting structures like shear bandsare 2-3 mm wide and the layers have a thicknessof a few millimeters to centimeters. Hence, thesource frequencies need to be between 250 kHzand 1 MHz to generate waves with wavelengthsbetween 6 to 1.5 mm, so that they are able to re-solve these structures. Higher frequencies gener-ate wavelengths that are approximately as big asthe grain size of the material, so that each grainscatters arriving energy causing high attenuation.
When doing seismic physical modeling ofsolid or granular models, three aspects requireparticular attention: Scaling, transducer and ma-terial properties. In contrast to the continuingdiscussion about scaling factors within the ge-ological physical modeling community, scalingfor seismic physical models is trivial: Lengthand time scale factors are arbitrary, as long asthe ratio of geological feature size to wavelengthis the same in both the field and the model(Ebrom and McDonald, 1994). In nature as wellas in the model, only the Nyquist criterion mustbe obeyed for temporal and spatial sampling.Regarding the equipment, most experimentersused electromechanical transducers, beginning
with the work of Kaufman and Roever (1994).Sometimes sparks (Kaufman and Roever, 1994;Hilterman, 1970) were used as seismic sources,but mostly PETs have served both as sourcesand receivers, such that the results are in thesame form as field records (Riznichenko, 1994;O’Brien and Symes, 1994). However, the properscaling of source and receiver dimension towavelength is inevitably violated (in the field, in-dividual sources and receivers are generally smallcompared to a wavelength), which imposes lim-itations on the use of PETs in seismic physicalmodeling:
• strong resonance at one frequency (i.e. re-stricted bandwidth),
• pronounced directionality, and
• source and receiver dimensions in the sameorder of magnitude as the wavelengths.
Furthermore, the material available for seismicphysical modeling has several limiting effects:The parameter ranges for velocity and density arelimited to those materials that are available or canbe fabricated. Additionally, the attenuative prop-erties and scattering effects of modeling materialscause substantial weakening of the received sig-nals. Good source and receiver coupling to themodel can be achieved by performing the experi-ments in a water tank.
The effects of the source and receiver dimen-sion have been frequently neglected or dismissedin published physical modeling studies, despitethe fact that they can have a first order effecton the data. Dellinger and Vernik (1994) nu-merically modeled whether experiments to mea-sure velocities of layered rocks are more likelyto measure the group velocity or the phase ve-locity of p- and s-waves. In their models theyaddressed the effect of a spatial source on wavepropagation, and their Figure 5 shows nicely thewaveform divergence with increasing offset orincreasing source and receiver size at zero off-set. However, they explained it solely as a result
2.2. Effects of piezoelectric transducers in ultrasonic experiments 7
of the anisotropic wave propagation in a layeredmedium with a slope and did not stress, that partof this divergence occurs even in a homogeneousmedium.
Within the scope of this publication, we testthe properties of our PETs with respect to theiruse in seismic reflection surveys of models andgeometries as described above. We analyze theabove mentioned properties analytically and ex-perimentally and discuss the limitations and op-portunities imposed by them.
2.2 Effects of piezoelectrictransducers in ultrasonicexperiments
The three limitations of piezoelectric transducers(PET), namely the strong resonance at one fre-quency, large dimensions and directionality, havethe following effects on the seismic signal:
1. ringy signal at resonance/eigen frequency,
2. changing waveform with offset (temporaldivergence),
3. high attenuation with offset (spatial diver-gence) in the plane parallel to the emittersurface.
2.2.1 Resonance frequency
Due to the piezoelectric effect, a piezocrys-tal can be excited to oscillate by applying AC.The amplitude of the oscillation is dependenton the source frequency and has its maxi-mum at the resonance frequency of the PET(Krautkrämer and Krautkrämer, 1986). In orderto reduce the oscillation subsequent to the excita-tion period, particularly at resonance frequency,the PETs are damped by a tungsten/resin filling.However, some ringing remains. This can be ei-ther reduced by a deconvolution filter, or the sig-nal can be recorded at all angles and then used asa cross-correlation wavelet.
2.2.2 Changing waveform
The shape of a wavelet propagating from a sourceto a receiver changes with offset when their di-mensions are bigger or of the same size as theemitted wavelength, because the length differ-ence of the rays may be well over a wavelengthλ , as illustrated by the two sample raysr1 andr2 in Figure 2.1. Hence, we have to considereach point of the source as an individual sourcepoint and assume that it emits energy at the sametime (or at least within one or two sampling in-tervals) as all other source points. Due to differ-ent raypath lengths to any of the receiver points,the energy emitted at one instant arrives over acontinuous time period which is increasing withoffset. We call this effect "temporal divergence"following the concept of "spherical divergence"where energy propagating from a source pointis distributed over a greater area. This effect isvery similar to the Fresnel zone effect, only thatthe Fresnel zone is defined as the area to withinhalf a wavelength around the reflection point thatis contributing to the signal. In this case, it isthe area of the source and the receiver that con-tribute; depending on the wavelength these areascan also have a radius of more than half a wave-length. This has to be taken into account addi-tionally to the Fresnel zone when considering thespatial resolution.
The problem with the changing waveform isthat stacking as an important step in the seis-mic imaging process assumes that the waveformstays constant, otherwise the superposition of thesignals may not add constructively. Hence, weneed to determine the critical offset below whichstacking improves the signal quality for any ap-plied source frequency.
The waveform as a function of offset canbe predicted either numerically, e.g. byfinite-element or finite-difference solutions(Savic and Ziolkowski, 1994), or analytically.The solutions depend highly on the geometry ofthe emitting surface. Therefore, we derive in thefollowing a 3D-semi-analytical solution for the
8 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
Offset
r1
r2 z
r1
- r2
~ λ
DD
λ
Source
transducer
Receiver
transducer
Figure 2.1: Sketch to illustrate that the waveform changeswith offset, when high frequency energy propagates be-tween two spatial transducers of diameterD. The two sam-ple raysr1 and r2 show the maximum and the minimumraypath lengths possible for the given source-receiver ge-ometry. The length difference can be greater than a wave-lengthλ , so that the shape of the recorded wavelet can dif-fer significantly from the emitted one.
energy-time-distribution as a function of offsetand then compare it to experimental data in theresults section.
3D-semi-analytical solution
The general idea for this solution is, that a certainamount of source areaAs =
∫
xs
∫
ysdxsdys con-
tributes linearly to the energyE arriving at a cer-tain timet at the receiver areaAr =
∫
xr
∫
yrdxrdyr
(Figure 2.2). Assuming an isotropic medium andperfect coupling, the arrival timet is equivalentto the ray path lengthl . Hence, we have to solvethe following integral equation:
E(l) =
∫
xs
∫
ys
∫
xr
∫
yr
δ (xs,ys,xr ,yr)dxsdysdxrdyr ,
(2.1)
ϕ
lmin
lmax
z
y
x
Receiver
area
zd
(xs,ys)
y
x
y‘
x‘
α
r d
a
(xs,ys)
ϕd
Source
area
Receiver
area
Source
area
s‘
(xr’,yr’)
s
s
a
Figure 2.2: Geometrical sketch of all variables needed tocalculate the arc lengtha. For detailed explanation, seetext and equations 2.1 to 2.11.
with
δ (xs,ys,xr ,yr) =
∞ if (xr −xs)2+
(yr −ys)2+
z2d = l2
0 else,
(2.2)
wherezd is the vertical distance between thesource and the receiver. Each point(xs,ys) of thesource areaAs acts as a point source and the raypath lengthl to any point of the receiver(xr ,yr)can be calculated geometrically. Furthermore,the set of(xr ,yr) that arel apart from(xs,ys) de-scribe an arca on the area of the receiver. Hence,for each source point, we calculate the arc lengtha that contributes to the energy recorded at a cer-tain timet, i.e. l . In this manner, instead of calcu-
2.2. Effects of piezoelectric transducers in ultrasonic experiments 9
lating an area, we calculate the arc lengthsa foreachl and integrate overAs:
A(l) =
∫
As
a(xs,ys, l)dAs. (2.3)
Then, for a given source point(xs,ys), an offsets to the receiver center and a depthzd betweensource and receiver, the arc lengtha can thus beexpressed as a function ofl :
axs,ys(l) = 2r(l)α, (2.4)
r(l)2 = l2−z2
d, (2.5)
α(r) = tan−1(
x′ry′r
)
, (2.6)
y′r =1
2s′(r2
−d2 +s′2), (2.7)
x′r = r2−y′2r , (2.8)
s′ = (s−ys)cos(ϕ), (2.9)
ϕ = tan−1(
xs
s−ys
)
, (2.10)
whered is the radius of both source and re-ceiver plane, andr the projection ofl onto the x-y-plane. The arc angleα is determined by the in-tersection(x′r ,y
′r) of the arc and the receiver out-
line in a new coordinate system(x′,y′). This newcoordinate system is offset by(xs,ys) and rotatedby ϕ to the old one(x,y). s′ is the distance tothe center of the receiver plane from the origin of(x′,y′) and thus the offset of the receiver to thesource point(xs,ys).
Since the integration ofa over As is not triv-ial, we evaluatea(l) at regularly spaced, discrete(xs,ys) positions:
A(l) = ∑ys
∑xs(ys)
a(xs,ys, l). (2.11)
As long as the spacingdx is smaller than theNyquist theorem requests to prevent spatial alias-ing, the discretized solution toA(l) is propor-tional to the integral solution. Hence, the eval-uated waveform and its amplitude are also pro-portional to a purely analytical solution.
Source and receiver radiusd: 12 mmVertical displacementzd: 100 mmSpatial discretizationdx: 0.075 mmSampling ratedt: 50 nsOffsetss: every 6 mm
from 0 to 120 mmSource frequencies: 175, 250, 350, 500,
750, 1000 kHZ
Table 2.1: List of parameters used to calculate the wave-forms for different offsets and frequencies. The geometrycorresponds to that used in the experiments, and the sam-pling rate is equal to that of the recording equipment in thelaboratory.
To obtain the waveform theoretically recordedat offsets and depthzd, A(l) is convolved withthe signal emitted by the point source onAs, as-suming that each point ofAs has the same sig-nal. This assumption obviously does not holdtrue in reality, because the waveform is, during itscourse from a digital signal to its emission, sub-ject to several interactions that change its shapeinherently. To adjust the analytical waveform toresemble the real one, one needs an appropri-ate weighting functionω(xs,ys) to apply to thesource areaa(xs,ys). However, we refrain fromapplying a weighting function or other correc-tions to the emission function and analyze thechanging waveform rather qualitatively. Thus,the energy function is computed for offsets rang-ing from 0 to 120 mm and different frequenciesusing the parameters listed in Table 2.1.
The resulting waveforms are shown in Fig-ure 2.3. As it can be clearly seen, the waveformsfor smaller offsets (blue) are much more compactand resemble the original waveform, whereas thefar offset waveforms (yellow to red) have smalleramplitudes (first column), and later peaks andcontinue over a longer period of time (centercolumn). The frequency spectra (right column)show that the center frequency of the receivedsignal decreases as the offset and the frequencyincreases.
10 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
0 0.005 0.010
0.5
1.0Energy divergence
0 0.005 0.010
1
2
Normalized divergence
0 500 10000
0.005
0.010Frequency spectrum
0 0.01 0.02
−5
0
5
Relative waveform (175 kHz)
0 0.01 0.02
0.5
0
0.5
1.0Normalized waveform
0 100 200 3000
10
20
0 0.01 0.02
−5
0
5Relative waveform (250 kHz)
0 0.01 0.02
−
0 200 4000
5
10
0 0.01 0.02
−5
0
5Relative waveform (350 kHz)
0 0.01 0.02 0 200 400 6000
2
4
0 0.005 0.01 0.015−5
0
5Relative waveform (500 kHz)
0 0.005 0.01 0.015−1
0
1
0 500 10000
1
2
0 0.005 0.01 0.015−4−2
024
Relative waveform (700 kHz)
0 0.005 0.01 0.015−1
0
1
0 500 10000
0.4
0.8
0 0.005 0.01 0.015
− 2
0
2
Relative waveform (1000 kHz)
Time (ms)0 0.005 0.01 0.015
−1
0
1
Time (ms)0 1000 2000
0
0.1
0.2
Frequency (kHz)
Offset (mm)
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Normalized waveform
Normalized waveform
Normalized waveform
Normalized waveform
Normalized waveform
Frequency spectrum
Frequency spectrum
Frequency spectrum
Frequency spectrum
Frequency spectrum
Frequency spectrum
6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120
−
0.5
0
0.5
1.0
0.5
0
0.5
1.0
−
Figure 2.3: Resulting waveforms of the analytical solutionfor two spatial circular transducers according to the geometrydescribed in Table 2.1 for offsets between 6 (blue) and 120 mm(red). First row: Temporal energy divergence (relative andnormalized) and its frequency spectrum. Second row to last row: Relative and normalized waveforms (source signal con-volved with energy divergence function) and the (relative)frequency spectrum for six different source signal frequencies.The vertical black line in the third column delineates the center frequency of the source, the curved line shows the actualfrequency maxima being smaller than the source frequency. The waveforms of smaller offsets are much more compactwhereas the far offset waveforms have smaller amplitudes and later peaks and continue over a longer period of time.
2.2. Effects of piezoelectric transducers in ultrasonic experiments 11
De
pth
(m
m)
−100 −50 0 50 100
0
50
100
1500
0.5
1
1.5
Distance to center axis (mm)
De
pth
(m
m)
−100 −50 0 50 100
0
50
100
1500
0.5
1
1.5
Pre
ssu
re (
dB
)P
ressu
re (
dB
)
a)
b)
Figure 2.4: The calculated pressure field (see eq. 2.12) fora diameter of 12 mm as a function of depth and angle for200 kHz (a) and 800 kHz (b). The higher the frequency themore focussed is the beam.
2.2.3 Directionality
For circular sources with a flat surface, thespatial divergence of the pressure field canbe analytically described as a function of dis-tance to the emitting plane and angle fromthe axis through the center of the plane byKrautkrämer and Krautkrämer (1986):
p(p0,D,λ ,z,γ) = 4p0J1(X)
Xsin
(
πD8λz
)
,
(2.12)with
X =πDλ
sinγ, (2.13)
where p0 is the initial pressure, i.e. amplitude,D the diameter of the emitter,λ the wavelength,z the distance to the emitting plane,γ the angleto the cylinder axis, andJ1 the Bessel function.Hence, the higher the applied frequencies and theshorter the wavelengths are, the more directed isthe pressure field of a circular transducer as illus-trated in Figure 2.4. This implies that less energypropagates at high angles, i.e. far offsets.
However, the emitting plane of the PETs usedfor this study is made of a piezocrystal of 5-mm
Stainless-steel
cylinder
Synthetic resin
with tungsten
Piezocrystal
Stainless-steel lid
Earth cable
Agglutinated cable
Brass bottom
Conductive glue
Shrink foil
5 mm12 mm
2 m
m
20
mm
1 m
m
0.8
mm
Figure 2.5: Sketch of the cross section of the piezoelectrictransducers. The piezocrystal has a diameter of 5 mm anda height of 2 mm and is glued onto a brass plate of 12 mmdiameter. For damping, the cylinder is filled with a mixof resin and tungsten. The transducer is in an unexpandedstate and the emission area flat.
diameter glued onto a brass plate of 12-mm di-ameter (Figure 2.5). Hence, both the piezocrys-tal and the brass plate contribute to the emittedwavefield, such that it has a broader beam (Fig-ure 2.6) at the expense of a smaller amplitudecompared to usual PETs of this diameter. Sincethis is the first time these PETs are used, we de-termine the effective diameter experimentally bymeasuring the pressurefield at a fixed distance.We define the effective diameter (Figure 2.6) asthe diameter which, when inserted into equa-tion 2.12, gives the best match to the observedamplitudes. This effective diameter is frequencydependent.
12 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
2.3 Transducer design and ex-perimental setup
For this study, we used 15 piezoelectric trans-ducers (PET) which were manually build in Ger-many. The transducers have a 5-mm piezocrystalglued onto a brass plate of 12-mm diameter and0.8-mm thickness underneath the piezocrystal ac-cording to Figure 2.5. The brass plate is verythin to keep the effect of internal reflections be-tween the crystal and the plate reasonably small.The resin/tungsten filling is supposed to dimin-ish the resonance ringing of the PET. In orderto decrease the effective diameter of our PET,such that its directionality is less pronounced,the emitting plane differs from traditional ones:Commonly, the piezocrystal’s diameter is that ofthe emitting plane (Figure 2.6, left panel) andequation 2.12 applies. In our case, the piezocrys-tal’s diameter of 5 mm is smaller than the 12-mmdiameter of the emitting plane. Hence, both thepiezocrystal and the brass plate contribute to theemission such that the effective diameter may beanywhere between 5-12 mm, as illustrated in Fig-ure 2.6. In fact, the bulging emitting surface cre-ates a directivity pattern that is similar to that ofa normal transducer of an even smaller diameter,which we define as the effective diameter of ourtransducers. In the results section, we determinethe effective diameter experimentally.
A schematic illustration of the major compo-nents of the laboratory seismic system is shownin Figure 2.7 and the specifications are listedin Table 2.2. Briefly summarized, a well de-fined digital wavelet is converted to an analogelectric current and sent to the piezocrystal ofthe source PET, which in turn expands and con-tracts according to (the time derivative of) thecurrent. However, despite of the damping mate-rial, some ringing remains. Also, interaction be-tween the piezocrystal and the brass plate affectsthe emitted source wavelet. The effectively emit-ted source wavelet is therefore different from thedigitally inserted signal. After passing through
D > Deff
α < α
Common
transducers
Our
transducers
eff
Figure 2.6: Sketch of a traditional piezoelectric transducerversus one of ours with a bulging emitter surface (not toscale). The geometry of an unexpanded piezocrystal is out-lined in gray, whereas the expanded geometry is outlinedwith the thin black line. The traditional transducer has a di-ameterD and a beam angleα, whereas the bulging emittersurface causes the effective beam angleαe f f to be widerand hence the effective diameterDe f f to be smaller (out-lined by the dotted black line).
the brass plate, the pulses propagate as pressurewaves through room-temperature tap water andare received by a PET of the same build. Theprocess of a digital signal to an analog pressurepulse is reversed for the receiver and the digitalsignal is recorded and stored in SEG-Y format.
As a source signal we used 2-4 periods of asine function of the center frequency tapered witha cosine4 for six different center frequencies. Theshape of the source signal was the same for allapplied frequencies. The frequencies ranged be-tween 175 kHz and 1 MHz, the sampling ratewas 0.05µs and, because the signal showed re-markable consistency, we improved it by verticalstacking 256 times to overcome the weak poweroutput of the transducers.
To determine the directionality as a functionof incidence angle and the waveform as a func-tion of offset, two different experiment setupswere used as illustrated in Figure 2.8. The ge-ometry for these setups is comparable to that inthe sandbox experiments. For these experiments
2.3. Transducer design and experimental setup 13
P C control uni t Watertank
Transientrecorder
Receiver
Signal generator
SenderSignal amplifier
Powersupply Preamplifier
Figure 2.7: Schematic illustration of the functions performed for these experiments by the PC control unit and the posi-tioning system modified after Krawczyk et al. (2007).
PC system with control unit: Industry PC (type IPC-9401).Signal generator: PCI-board (type MI6030);
Max. output 125 MHz (14 bit);Max. 8 Msamples;Max. output amplitude± 3 V.
Signal amplifier: AC voltage signal amplifier;Input -2 to +2 V;Input resistor 200 Ohm;Output -141 to +141 V;Output resistance 2 kOhm;Band width 20 Hz-500 kHz (-3 dB), 20 Hz-1000 kHz (-6 dB).
Preamplifier: (Type VV30) 30 dB voltage amplification and impedance tuning;Frequency range 1 kHz-2 MHz;Max. output amplitude± 3 V.
Transient recorder: Three 4-channel PCI-boards (type MI4022);For each channel signal amplifier and AD-converter;Max. sampling 20 MHz (14 bit);Max. memory 2 Msamples/channel.
Table 2.2: Technical specifications of the components in theultrasonic recording system (Krawczyk et al., 2007).
the transducers have been tested one by one andat water depths and distances to container wallssuch that reflections arrive much later in timethan the direct waves that are to be analyzed.
The directionality was determined experimen-tally by opposing two PETs (one source, one re-ceiver) with a constant distance of 10 cm, and ro-tating the source with a precision of at least 0.25◦;the rotation axis was positioned at the emittingend of the source perpendicular to the cylinderaxis (Figure 2.8a). The measurements were re-peated approximately every 2.15◦ from direct in-cidence up to 43.8◦ for six different source fre-quencies. The maximum amplitudes for each in-cidence angle and each PET were then automati-cally picked by a computer.
We define the effective diameter (Figure 2.6)as the diameter that gives the best match betweenthe observed and the theoretical amplitudes from
equation 2.12. For this, we implemented a least-squares inversion routine to findD, p0, and λwhich best match these amplitudes:
erf(p0,D,λ ) = (2.14)
∑γ
[pobs(λ ,γ)− p(p0,D,λ ,z,γ)]2,
withλ = cw/ fsrc, (2.15)
where the initial pressurep0, the diameterD, andthe wavelengthλ are the PET and signal parame-ters, which the error function minimizes for. Ad-ditionally, p0 is left to be a variable, because thetransducers do not respond with equal amplitudesto the electric signal due to their variable sensi-tivity to frequencies.λ is calculated via the wavevelocity in watercw (1500 m/s) and the centerfrequency of the source signalfsrc. However, be-cause the recorded frequency maxima are smaller
14 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
Directivity: Experimental setup
α
Source Receiver
Waveform: Experimental setup
Rotating-
source
transducer
10 cmOffset 0 - 120 mm
Mobile-
receiver
transducer
Receiver
transducer
a)
b)
10 cm
Figure 2.8: Experiment setup for measuring a) the direc-tionality as a function of incidence angleα, and b) thewaveform as a function of offset.
than the digital center frequencies, we also allowto minimize forλ within according boundaries.
The distance between source and receiverzand the angle to the cylinder axisγ are deter-mined by the experiment setup. We consis-tently replace the pressure given in decibel byKrautkrämer and Krautkrämer (1986) by ampli-tudes in mV as they are recorded by our system.Hence, in equation 2.15 we minimize for the ef-fective diameterD, the initial pressurep0, and,within reasonable boundaries, forλ .
The waveform experimental setup (Fig-ure 2.8b) provides a method for recording thesignal as a function of offset by moving the re-ceiving transducer parallel to the emitter surfaceof the source transducer. The distancez was 10cm and the offset varied between 0 and 12 cm.To compare the actual waveforms at differentoffsets, the arrival time delay due to increasing
200 400 600 800 10000
0.5
1.0
1.5
2.0
2.5
3.0
Frequency (kHz)
Am
plit
ud
e (
V)
250 kHz 675 kHz
425 kHz
Figure 2.9: Frequency spectrum of piezoelectric transduc-ers used here. The transducers are most sensitive at 425±
25 kHz with a half-power bandwidth of 250 to 675± 25kHz.
offsets is subtracted accordingly. The receivingtransducer was the same for all experiments. Thezero-offset traces were used to determine thesensitivity of the PETs to different frequenciesand the resonance frequency.
2.4 Experimental results
2.4.1 Resonance
The measured amplitudes for different source fre-quencies are displayed in Figure 2.9. The piezo-electric transducers (PET) are most sensitive at400-450 kHz with a half-power bandwidth of 250to 675 kHz.
2.4.2 Waveform
Before analyzing the waveforms, we looked atthe length of the source signal, i.e. the number ofperiods in the source wavelet. We used the wave-form experimental setup (Figure 2.8b) and proce-dure with a source signal of two, three, and fourperiods of a 350 kHz sine-function tapered witha cosine4. Theoretically, for a signal created asdescribed above, the bandwidth is broader, as the
2.4. Experimental results 15
2 PeriodsT
ime
(m
s)
0 50 100
0.10
0.11
0.12
0.13
0 200 400 600
5
10
15Frequency Spectrum
Am
plit
ud
e
3 Periods
Tim
e (
ms)
0 50 100
0.10
0.11
0.12
0.13
0 200 400 600
4
8
12
Am
plit
ud
e
4 Periods
Tim
e (
ms)
Offset (mm)0 50 100
0.10
0.11
0.12
0.13
0 200 400 600
4
8
12
Frequency (kHz)
Am
plit
ud
e
Off
se
t (m
m)
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
Frequency Spectrum
Frequency Spectrum
Figure 2.10: Recorded seismograms (350 kHz source fre-quency) and their frequency spectra for source signals of 2,3, and 4 periods length as a function of offset (color coded).The maximum frequencies are delineated in gray.
signal shortens. However, Figure 2.10 shows thatthe bandwidth of the recorded signal is equallybroad for three different source signal lengths,but the three-period source signal (2nd row pan-els) produces a better quality spectrum and thesecond positive phase is clearly defined. Thefour-period signal requires deconvolution for fur-ther use. Hence, for the following experiments,we show the results of the three-period signal ex-periments.
To analyze the recorded waveforms, we ap-plied different source frequencies using the sameexperimental setup (Figure 2.8b). In Figure 2.11,we compare the waveform and frequency contentfor different source frequencies. Aside from theamplitude decay (i.e. directionality), which wedeal with in the next section, it is noteable thatthe first peak arrives later in time for higher off-sets, and later phases interfere destructively withvarying offsets as is predicted by the analyticalsolution.
The recorded waveforms look very different tothe predicted ones, because the analytical solu-
tion does not account for the interactions betweenthe piezocrystal, the damping material and thecylinder walls. The influence of the brass bottomand the glue on the wave propagation are thoughtto be negligible because their thickness is muchsmaller than the wavelength. In any case, theseinteractions can hardly be predicted and mayeven vary from transducer to transducer. How-ever, the waveforms and amplitudes are compa-rable in their frequency dependence and thereforethe conclusion remains the same:
For all frequencies< 700 kHz, the first peakand trough along all offset-traces are in-phase,whereas the end of the signal and the followingringing vary over the offsets. Since we stackedthe signal 256 times, we can assume that the ring-ing is coherent for each offset and use that toour advantage: Since only the first two phasesinterfere constructively, NMO-stacking reducesthe ringing. However, because of the narrow fre-quency band of the signal, a traditional velocityanalysis produces non-unique results. One reflec-tion causes several semblance maxima, one foreach peak in the signal at every period of 2-3µs,at different velocities. Hence, further processingcould be greatly improved by knowing the ve-locities of the media within a model in advance.Since we know the media that we use to buildthe model, we can as well measure their veloc-ity beforehand by transmission through a knownthickness of the medium.
The strong dependency on the eigenfrequencyis revealed by the frequency spectra (Figure 2.11,right column): Each spectrum contains severallocal frequency maxima; the highest one delin-eated by the dotted gray line differs from the cen-ter frequency of the source (gray line) by± 100kHz for source frequencies< 1000 kHz. Therecorded spectra fit the source frequencies whenthey are between 350-500 kHz. The local max-ima listed in Table 2.3 show that many maximaare multiples (± 10 kHz) of 110 kHz. Hence,110 kHz is the main eigenfrequency. Only themaxima at 165 and 190 kHz cannot be associated
16 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
0.1 0.12
-1
0
1
Record (175 kHZ)
Am
plit
ude (
V)
0.1 0.12 0.14-1
0
1Normalized
0 200 400 600
2
6
Freqency spectrum
Am
plit
ude
0.1 0.11
-1012
Record (250 kHZ)
0.1 0.11 0.12
-0.500.5
Normalized
0 200 400 600
26
10
Freqency spectrum
0.1 0.11-2
0
2Record (350 kHZ)
0.1 0.11-1
0
1Normalized
0 200 400 600
26
10
Freqency spectrum
0.1 0.105
-2
0
2Record (500 kHZ)
0.1 0.105 0.11-1
0
1Normalized
0 200 400 600
26
10
Freqency spectrum
0.1 0.105
-101
Record (700 kHZ)
0.1 0.105 0.11-1
0
1Normalized
0 500 1000
2
4Freqency spectrum
0.1 0.104
-0.500.5
Record (1000 kHZ)
0.1 0.104-1
0
1Normalized
0 500 1000
0.2
0.4Freqency spectrum
Time (ms) Time (ms) Frequency (kHz)
Offset (mm)6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102108114120
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude
Am
plit
ude (
V)
Am
plit
ude (
V)
Am
plit
ude (
V)
Am
plit
ude (
V)
Am
plit
ude (
V)
Figure 2.11: Change of recorded waveforms with offset (color code) and frequency (row 1-6) seen in the recorded traces(left), the normalized traces (center), and frequency spectra (right). The stack of all traces is shown in black. The verticalline delineates the source signal frequency, the dotted line the recorded maximum frequencies for all offsets. The firstpeak and trough are in-phase whereas the end of the signal andthe following ringing vary over the offsets.
2.4. Experimental results 17
Source frequency Recorded frequency maxima (kHz)(kHz)175 105, 165, 190, 225, 340, 440, 545250 105, 165, 190, 230,325, 450, 545350 340, 440, 540500 340,445, 550700 445,5501000 550
Table 2.3: Recorded peak frequencies for different sourcefrequencies. The major maxima are bold.
with the eigenfrequency and can be due to oscil-lations in a different plane.
Hence, for the preferred source frequencies of250-675 kHz with respect to the sensitivity, thefirst two phases are not affected by the chang-ing waveform. Destructive interference for laterphases works to our advantage in further dataprocessing. The spectra fit the source frequencybest for source frequencies of 350 and 500 kHz,and both contain a major peak at 550 kHz. Thisreduces the suitable frequency range to 350-550kHz, even though all frequencies between 100 to1000 kHz can be applied to the transducers.
2.4.3 Directionality
The seismograms of a 5-mm PET for six differ-ent source frequencies, i.e. wavelengths, illus-trate how the amplitudes decay with increasingincidence angles (Figure 2.12). This effect of di-rectionality is more pronounced with higher fre-quencies. At 500 kHz source frequency, the sig-nal cannot be distinguished from ringing noise atan incidence angle< 35◦, whereas for 1000 kHz,the limiting angle is around 23◦. Note, that laterphases of the waveform change at around 20◦ forfrequencies> 350 kHz. This is due to the chang-ing geometry with the rotation.
Continuing with this sample PET, we dis-play the maximum amplitudes (Figure 2.13, solidlines) as a function of incidence angle for alltested frequencies (color coded) together with thebest-fit curves (dashed lines). These best-fit am-plitudes are the result of inserting the parametersof the best-fit solution (eq. 2.15) into the pressure
Tim
e (
µs)
0 10 20 30 40
200
250
0 10 20 30 40
200
220
0 10 20 30 40
190
200
210
0 10 20 30 40
195
200
205
210
0 10 20 30 40
195
200
205
210
F = 175 kHz, λ = 8.57 mm
F = 250 kHz, λ = 6 mm
F = 350 kHz, λ = 4.29 mm
F = 500 kHz, λ = 3 mm
F = 750 kHz, λ = 2 mm
F = 1000 kHz, λ = 1.5 mm
Incidence angle (°)/offset (mm)
0 10/36 20/72 30/116 40/168
200
204
208
Tim
e (
µs)
Tim
e (
µs)
Tim
e (
µs)
Tim
e (
µs)
Tim
e (
µs)
Figure 2.12: Recorded seismograms of a sample transduceras a function of incidence angle for different source fre-quencies (F), that, in water, are equivalent to the givenwavelengths (λ ). The gain has been maintained at a con-stant value. The incidence angle increases from left (direct)to right (44◦). The respective offset indicated is calculatedassuming a reflector depth of 10 cm. The dashed line in-dicates the picked maximum amplitude used for the mini-mization.
18 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
10/36 20/72 30/116 40/168
0
500
1000
1500
2000
Incidence angle (°)/offset (mm)
Am
plit
ud
e (
mV
)Amplitude decay and best-fit curves
175 kHz
250 kHz
350 kHz
500 kHz
750 kHz
1000 kHz
Figure 2.13: Amplitude decay of the sample transducer(solid) and the best-fit theoretical curves (dashed) for dif-ferent source center frequencies (color coded). The inci-dence angle increases from left (direct) to right (44◦). Therespective offset indicated is calculated assuming a reflec-tor depth of 10 cm.
field equation 2.12. The variability of the direct-incidence values illustrates the sensitivity of thetransducers to frequency. The amplitude decay ismost prominent for frequencies where the direct-incidence values are high.
The maximum amplitudes of all PETs areshown in Figure 2.14. Each panel represents onesource frequency. The amplitudes of the 5-mmtransducers (gray) vary± 500 mV for 350-500kHz; however, the amplitude decay follows thesame course for all frequencies. The mean de-cay course is delineated in red and the blue linesrepresent the best-fit curves of the mean effectiveparameters. Hence we conclude, that the 5-mmtransducers are comparable in their directionalityand, for higher frequencies, in their sensitivity.For comparison, the amplitude decay of a tested7-mm PET (black) has much higher amplitudesat direct incidence and is much more directed.Although the energy output of the 7-mm trans-ducer is better than that of the 5-mm transducers,its strong directivity disinclines us to use it formultiple-offset surveys.
0 10 20 30 400
1
Am
plit
ud
e (
V)
Recorded maximum amplitudes (175 kHz)
0 10 20 30 400
1
Am
plit
ud
e (
V)
Recorded maximum amplitudes (250 kHz)
0 10 20 30 400
1
2
Am
plit
ud
e (
V)
Recorded maximum amplitudes (350 kHz)
0 10 20 30 400
1
2A
mp
litu
de
(V
)
Recorded maximum amplitudes (500 kHz)
0 10 20 30 400
0.5
1
Am
plit
ud
e (
V)
Recorded maximum amplitudes (750 kHz)
0 10/36 20/72 30/116 40/1680
0.5
Angle (˚)/offset (mm)
Am
plit
ud
e (
V)
Recorded maximum amplitudes (1000 kHz)
Figure 2.14: Comparison of the amplitude decay of the5-mm piezoelectric transducers (PET) delineated in gray,their mean (red), their best-fit curve (blue) to the mean re-sults from Figure 2.15, for six different source frequencies.Also shown is the amplitude of a 7-mm PET (black).
The best-fit parameters (De f f, p0, andλ ) of allPETs (gray) and their mean (blue) are shown as afunction of frequency in Figure 2.15. When thesemean values are inserted into equation 2.12, theresulting amplitudes are nearly coincident with
2.5. Discussion 19
200 400 600 800 10002
4
6
8
10D
eff
(m
m)
200 400 600 800 1000
14
16
18
P0
(V
)
200 400 600 800 1000
2468
101214
λe
ff (
mm
)
200 400 600 800 1000
0.1
0.2
0.3
Re
sid
ua
l (V
)
Frequency (kHz)
Effective diameter
Effective initial pressure
Effective wavelength
Residual
Figure 2.15: Effective parameters (diameterDe f f , wave-length λ , and initial pressurep0) determined from mea-surements with different source frequencies. The resultsfor each individual piezoelectric transducer (PET) are dis-played in gray and their mean value in blue.
the mean amplitude decay in Figure 2.14. The in-version for the 175 kHz source signal was poorlyconditioned and did not converge, therefore the175 kHz results are not very meaningful and arenot discussed in the following. The relativelyhigh standard deviation of the best-fit parametersfor the lower frequencies is also a result of thevariation in the data. In the high frequency range,the transducers are more similar in their behaviorand the inverted parameters are more consistent.
For all transducers, the determined effectivediameter is decreasing with increasing frequency(see Table 2.4) and at high frequencies it in factdoes go below the piezocrystal’s diameter of 5mm (Deff = 4.85 mm at fsrc = 700 kHz). Hence,
Source frequency (kHz) De f f (mm)175 3.36± 1.30250 9.54± 3.12350 9.37± 1.81500 7.96± 0.82700 4.85± 0.301000 2.51± 0.17
Table 2.4: Mean and standard deviation of effective diam-etersDe f f for different source frequencies.
our design in fact reduces the effective diameterof the PETs.
2.5 Discussion
The experiments tested the piezoelectric trans-ducers (PET) for their frequency sensitivity, andfor the change of waveform and directivity asfunctions of offset and frequency. The results aresummarized in Table 2.5.
All frequency requirements considered, thebest quality results were achieved with a 350-550kHz source frequency, because the PETs are mostsensitive in that range and the recorded spectrabest match the source frequency spectrum. Thewaveform changes more rapidly the higher thesource frequencies are, however, the first twophases of the recorded wavelet are in phase forup to 12 cm in offset. Later phases interfere de-structively to some degree, so that stacking ac-tually focuses the signal. With the experimentalsetup used here, where the receiver is opposingthe source, the effect of the offset on the wave-form is not as big as it will be when both sourceand receivers are on the surface while survey-ing sandbox models. On the other hand, the ve-locities in sand and other granular material arearound 1700 m/s (Sherlock and Evans, 2001), sothat the wavelengths increase at the water-modelinterface. That, and the smaller path differenceat deeper offsets counter-balance the waveformchange at reflections within the model. Hence,the fit between the first two phases is sufficientlyclose, so that the changing waveform remains tobe a minor problem when both source and re-ceivers are at the surface. During further pro-
20 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
PET property Effect target Solution / opportunity
Sensitivity Narrow bandwidth 250-675 kHz Apply 250-675 kHzResonance Ringy signal n×110 kHz Damping, deconvolutionSpectrum Does the signal match Conditionally, Apply 350-550 kHz, andcharacteristics the source spectrum ? yes. apply 3-period signalLarge dimension Changing waveform ≤ 700 kHz, - Positive interference
(temporal for first peak and troughdivergence) - Negative interference
for later phases- A-priori knowledge ofmaterial velocity recommended
Directivity Less energy < 35◦ - Increase water depthat high offsets < 160 mm - ReduceDe f f
Table 2.5: List of the piezoelectric transducer (PET) properties, the resulting target frequencies, incidence angles, andoffsets, and other solutions or alternatives.
cessing steps, the image quality could be greatlyimproved by knowing the velocities of the mediawithin a model in advance. Since we know themedia that we use to build the model, we can aswell measure their velocity beforehand by sound-ing through a known thickness of the medium.
An alternative to this approach is to use the a-priori knowledge of the waveform to our advan-tage: Because the waveform is a function of off-set and reflector depth, the energy distribution isunique for each offset, and we can use it to deter-mine the reflector depth. Additionally, if the spe-cific waveform for each offset is known, a shap-ing filter or cross-correlation wavelet can be de-signed for each offset, so that after filtering, thesignals are sufficiently similar for a stack. Onlywe need to know the waveform for each offsetand depth. The disadvantage of this solution isthat the amplitude decays more than the wave-form changes with increasing offset. Hence, itis more applicable to stay within the offset lim-its imposed by the directionality. The analyticalsolution to the changing waveform problem pre-dicts the temporal divergence of the signal andshows that the loss of high frequencies is dueto different path lengths and not to attenuationalone. The shape of the signal though is verydifferent to the recorded shapes because the an-alytical function does not respect the interactionsbetween the piezocrystal, the damping material,and the cylinder walls. Hence, the analytical so-lution in no way can replace experiments.
For all our experiments, we chose a waterdepth of 10 cm comparable to the water depth inexperiments where sandbox models are seismi-cally surveyed. At this water depth level, a 20-cmoffset is equivalent to an incidence angle of 45◦.The directivity results though suggest, that theincidence angle should not fall below 35◦ for asource frequency of 500 kHz, which correspondsto an offset of 14 cm. To increase applicable off-sets, the water depth has to be increased. Fur-thermore, an increased water depth reduces thetemporal divergence due to smaller length differ-ences of raypaths.
In conclusion, we are restricted to 350-550kHz and water depths≥ 10 cm. Hence, the emit-ted wavelength is between 4.2-2.7 mm, allowingto resolve features of∼2-1.5 mm vertical dimen-sion. This frequency range is very suitable toimage the structures in sandbox models. Higherfrequencies generate wavelengths that approxi-mately as big as the grain size of the material, sothat each grain causes a reflection and obscuresthe image of larger structures, whereas smallerfrequencies do not resolve the structures in sand-box models. The lateral resolution is dependenton the p-wave velocity of the medium, the depthof the reflection, and in this case of the effectivediameter of the PETs, which has to be added asa constant to the normal fresnel zone calculation.In the worst case, i.e. the lowest source frequencyof 350 kHz source signal and the deepest reflec-tion at 15 cm through wet sand, it is going to be
2.6. Conclusion 21
about 46 mm. The models, therefore, should con-tain lateral variation longer than that, and shearbands should cut through at least a 46-mm width.
To do amplitude-versus-offset (AVO) analyseson field data, very often a plane-wave solution(Zoeppritz) is used, although the wave propaga-tion is spherical. Alhussain et al. (2008) haveexperimentally confirmed the spherical wave ef-fect on the AVO response by laboratory ultra-sonic measurements using omni-directional p-wave transducers with a 220 kHz dominant fre-quency. The plane-wave solution (Zoeppritz) thatis widely used for AVO analysis agrees well formoderate incidence angles up to 25◦ (their Fig-ure 2), but is not valid at large angles. However,within the beam our PETs produce something be-tween a plane wave and a spherical wave, andhence the AVO implementations are applicable todata collected with our PETs approximately up toincidence angles of 32◦.
Further improvement of the sources and re-ceivers can only be achieved by using smallertransducers that have the same or higher energyoutput and the same frequency bandwidth. Un-fortunately, those transducers cannot be fabri-cated yet. However, our design of a smallerpiezocrystal being glued onto the brass plate didreduce the effective diameter, at high frequencieseven to diameters less than that of the piezocrys-tal. Further improvement of this technique pos-sibly will lead to transducers with zero effectivediameter and high energy output.
The next step on the way towards seismicimaging of sandbox models, is to find suitablegranular material. In the experiments reportedhere, the transmitting medium was water. Theenergy losses at the grain-to-grain contacts arevery high, so that the penetration depth is not ex-pected to be more than a few centimeters. Carefulsieving and saturation with boiling or hot watercan reduce attenuation caused by unwanted scat-tering or attenuation due to remaining air bub-bles. In a two layer model, energy arriving at thefirst material interface should be reflected and re-
fracted to approximately equal amounts, so thata sufficient amount of energy passes into the sec-ond medium to resolve structures within. To cre-ate such models, we can measure the velocityand density of available materials to find suitableimpedance contrasts, and at the same time we canbypass a velocity analysis in the processing.
2.6 Conclusion
In this study, we tested the properties of spe-cially designed piezoelectric transducers (PET)for their usefulness in seismic reflection sur-veys of sandbox models, covering three aspects,namely the frequency sensitivity, the directional-ity, and the changing waveform. We presented ananalytical solution to the changing waveform andcompared it to the recorded ones.
The special design of our PETs, where asmaller diameter (5 mm) piezocrystal is gluedonto a bigger diameter (12 mm) brass plate, infact reduced the directionality, so that the effec-tive diameter is smaller than that of traditionalPETs. The directivity experiments show, that theincidence angle should not fall below 35◦ for asource frequency of 350-550 kHz, which corre-sponds to an offset of 14 cm for a water depth of10 cm. The changing waveform is a minor prob-lem for those offsets and frequencies. However,an a-priori knowledge of the p-wave velocity ofthe materials is recommended to bypass the ve-locity analysis during further processing. The an-alytical solution to the changing waveform prob-lem predicted the temporal divergence of the sig-nal and showed that the loss of high frequenciesis due to different path lengths and not to atten-uation alone. However, it did not reproduce therecorded waveforms because it does not respectthe internal dynamics of the PETs.
With respect to the PETs, we have shown thatreflection processing on such a small scale is fea-sible for source frequencies of 350-550 kHz andfor incidence angles<35◦. This frequency range
22 2. Performance of Piezoelectric Transducers in Terms of Amplitude and Waveform
allows to resolve structures of∼2-1.5 mm dimen-sion, which is sufficient to resolve the structureswithin sandbox models. However, the bandwidthof the generated signals is relatively small forseismic surveying. In order to broaden the spec-trum, we recommend to pulse in several of thesenarrow frequency bands and stack those as a pre-processing step.
