INSTANTANEOUS VARIATIONS IN VELOCITY AND ATTENUATION OF SEISMIC
WAVES IN A FRIABLE MEDIUM IN SITU UNDER PULSATORY DYNAMIC LOADING:
AN EXPERIMENTAL STUDY
N.I. Geza, G.V. Egorov, Yu.V. Mkrtumyan, and V.I. Yushin
Institute of Geophysics, Siberian Branch of the Russian Academy of Sciences, prosp. Akad.
Koptyuga 3, Novosibirsk, 630090, Russia
Instantaneous variations in velocity and attenuation of seismic waves in a friable medium subject
to dynamic loading have been studied by new experimental techniques using a powerful seismic
vibrator. The half-space below the vibrator baseplate was scanned by short-period elastic waves,
and the records were exposed to the stroboscopic analysis. The velocity and attenuation pulses are
synchronous with the pulsatory loading but are out of phase with the latter and with each other.
Instantaneous velocity and attenuation variations depend on the magnitude and absolute rate of
strain, which always reduces elastic wave velocities. Some weak reflectors show especially high
sensitivity to dynamic loading in experiments (effect of induced reflectors) but the physics of this
effect remains unclear.
Nonlinear medium, ground, dynamic nonlinearity, instantaneous variations, seismic velocity,
attenuation.
INTRODUCTION
A number of nonlinear effects (violation of proportionality principle, dependence of resonance
frequency of 'vibrator-ground' system on amplitude of vibrations and static load, relaxation, etc.)
observed in studies of conversion of mechanic vibrations into seismic waves [1] were investigated
by near-field pulse transmission scanning of the halfspace under the vibrator's baseplate [2, 3].
Strong vibrations were found out to influence the velocity of acoustic waves, both averaged over a
period of low-frequency vibrations and the instantaneous values corresponding to specific phases
of dynamic loading. Experiments show a decrease in mean velocity of acoustic waves (up to 5 %
in our case [2, 3]) and a considerable (3-5 times) increase in period-averaged attenuation
(accordingly absorption) accompanied by instantaneous velocity and absorption variations
coherent to the long-period vibrations. The extremes of the instantaneous variations are out of
phase, i.e., at least one value does not coincide with the phase of maximum ground compression
under the vibrator. The recovery of both velocity and absorption in the off mode takes a long time
but velocity recovery is faster.
A decrease in mean seismic velocity in sandy loam under a vibrator can be found in earlier
experiments [4]. This decrease eluded special attention in [4] but a later paper by the same authors
[5] contains P-wave velocity vs. dynamic loading plots (может лучше - plot of the P-wave
velocity vs. dynamic loading?).
These, however, contradict the plot in [4] and our data, showing an about 30 % jump in
instantaneous velocity (against ~ 3 % in our results) and maximum instantaneous velocity at
dynamic load much above the mean in the absence of vibrations (which never happened in our
experiments). In [2] we unexpectedly observed that the upper and lower combination frequencies
of amplitude and phase modulated waves were not equal. An analytical model of this unusually
modulated signal, satisfactory in terms of the theory of signals, was obtained [2] on an assumption
that velocity and attenuation absorption modulate the frequency of sounding waves in different
phases of dynamic loading. The method of combination frequencies revealed the out-of-phase
behavior of parameters that influence velocity and attenuation absorption but unfortunately could
not allow for their unique correspondence to specific vibration phases. This prompted us to
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develop a subtler experimental method to highlight instantaneous changes in elastic parameters at
different phases of pulsatory dynamic loading, which is the subject of this paper.
Пропущен абзац:
In the short wording, in work were put two problems. The First - purely methodical, -to
value in practice efficiency of new method of experimental dynamic tense condition study.
The Second - to get concrete quantitative evaluations of integral and instant change a
physicist-mechanical features of loam in sufficiently greater volume of able natural lying(?)
under the action of variable mechanical extension.
EXPERIMENT LAYOUT AND METHOD
The experiment layout (Fig. 1) includes a low-frequency unbalanced vibrator (V) providing
pulsatory loading up to 35 tons on ground below it, a high-frequency acoustic source (AS) for
pulse transmission scanning of the ground, and several boreholes. The vibrator is connected to
frequency- and phase-controlled electric drivers and is positioned (mounted?) on a baseplate
with area about of 9 m2 [2]. The piezoceramic acoustic source has a natural resonance frequency
of 900 Hz, a baseplate area of 0.2 m sup 2 and a full power up to 1 kVA. The boreholes are cased
by plastic pipes filled with diesel fuel. Acoustic waves are recorded by borehole-emplaced
pressure-sensitive PDS-21 (sea seismic prospecting geophones). A special geophone controls
acoustic signals at the wellhead of borehole 2 and an accelerometer is placed on the surface near
the hole. The geophones easily move up and down and need not be secured tightly to the hole
walls. Additional boreholes (Fig. 1 ) are used for detailed investigation of the seismic profile.
Seismic rays are from 5 to 20 m and a mean wavelength is 2 m. Therefore, the offset encompasses
several vibration wavelengths, and the effects on contacts of rock with the source and the receiver
are negligible.
Thus a powerful long-period vibrator is used as a controlled source of periodic pulsatory dynamic
load on a relatively large (representative) volume of ground below it. Unlike static load, dynamic
loading is easier to monitor with an array of geophones and pressure and strain sensors. Strain in
ground beneath a vibrator was investigated in [2]. The main problem was to measure the
instantaneous bulk modulus and related elastic parameters of the medium at each loading phase.
To measure P- and S-wave velocities and attenuation, the ground should be scanned by controlledsource seismic waves of frequencies above the vibration bandwidth (we call them acoustic waves
proceeding from their spectral composition). The broader the scanning spectrum the higher the
traveltime resolution and the higher the sensitivity to variations of other parameters. At the same
time, the long-period signals are inevitably accompanied by additional noise which is produced by
the vibrator and by acoustic emission in the ground at the acoustic bandwidth and can be several
times the pulse transmission signal. This makes direct recording of acoustic signals practically
impossible. A broader band at the account of higher-frequency signals does not help being
associated with stronger attenuation. Therefore, the main technical difficulty in measuring
instantaneous variations of elastic parameters was caused by incompatibility between very short
measurement time at each loading phase and long time needed to achieve a satisfactory signal-tonoise ratio. Indeed, cyclic loading cannot be stopped to measure the instantaneous elastic
parameters at a certain phase. Therefore, the energy of vibrations should be accumulated in small
portions corresponding to a specific phase in each loading cycle. This is possible using the socalled correlation stroboscopy method.
The method is applied to records obtained for a medium subject to periodic vibrations at some
constant frequency (in our case within 3 to 10 Hz) and scanned simultaneously by acoustic waves
(slow smooth sweep signals from 50 to 500 Hz). To be converted into pulses, received signals
should be convoluted (correlated) to source signals. Received acoustic signals bear information on
all states of the medium during the experiment, and the task is to pick the portion corresponding to
a certain loading phase. For this amplified and digitized signals are first strobed at the low
frequency of the vibrations in their respective phase, and then correlated. A strobe is a pulse
sequence with a relative pulse duration equal to the user-specified number of instantaneous phases
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per one loading/unloading cycle. Strobes have the same period as the vibrations and any phase
(relative delay) can be selected. A strobed acoustic signal is a sort of a “comb” consisting of
vibration fragments that bear information only on the selected loading phases. Strobing causes no
distortion to correlated records at certain parameters of acoustic and vibration signals. Repeated
correlation of an acoustic record strobed at different phases to the continuous vibration signal
yields a series of pulse acoustic records for all loading phases, as is required. Then it only remains
to pick amplitude and traveltime variations. Taking into account that traveltime variations can be
below the sampling rate, lags are picked using an interpolation method (method of crosscorrelation phase spectra).
Consider a near-field wave pattern of variations in acoustic signals to correlate them with the
stress-strain behavior of the ground (loam) subject to the dynamic loading. The baseplate velocity
is measured simultaneously with pulse transmission (70-350 Hz); stress and strain of the ground
under the baseplate associated with the long-period pulsatory loading are measured in a separate
experiment, simultaneously with baseplate velocity. The baseplate motion is recorded by a
velocity seismometer (2 Hz natural frequency, almost zero attenuation) which has a zero velocity
phase delay at 6-8 Hz, and the integral of its signal is thus proportional to baseplate displacement
also without delay. Stress in the ground about 20 cm under the baseplate is measured by a
membrane pressure sensor placed in a non-vertical borehole. After the sensor has been installed,
the borehole is filled with ground excavated during drilling and is tightly tamped layer by layer.
The sensor and the amplifier operate in a dc mode and cause no phase distortions to long-period
signals. Strain is measured by an induction-conversion strain meter placed on the vertical wall of
the pit at the edge of the baseplate, at an offset of 70 cm.
Simultaneous recording shows that baseplate displacement, pressure, and ground motion are in
phase (to counter-phase +- 10) at 6-7 Hz where vibrations are purely sine-shaped. The
measurements become impossible above 7.5 Hz when vibrations increase sharply in amplitude and
become saw-shaped because of a worse contact between the membrane and the ground. Therefore,
the instantaneous parameters of the ground are hereafter described only by baseplate displacement,
as it is assumed to be in phase with stress and strain.
RESULTS
Wavefield. The seismic profile of borehole 2 (Fig. 2) obtained with a vertical source and a
pressure sensor (Fig. 1) shows one direct and two reflected P waves in the acoustic wavefield. The
direct wave arriving at the day surface at the wellhead of borehole 2 at 20 ms is much weaker than
the following reflected waves as it is located near the minimum of the P-wave source radiation
pattern. Depth-dependent amplitude increase of the first arrival records a classical radiation pattern
in elastic half-space corresponding to the given acoustic source. The first reflection arriving at the
day surface at 45 ms is from an aquifer at a depth of 8 m, another stronger reflection at 60 ms is
from the roof of the Paleozoic basement at 10-15 m. The mean P-wave velocity in the 8 m upper
section is 340 m/s, attaining 370 m/s immediately beneath the baseplate, possibly, because the
ground is compacted by vibrations.
Experiment 1. Wave patterns in unloaded and loaded ground. The vertical profile of borehole
2 (Fig. 2) was repeated in the presence of long-period vibrations. The vibrator in this experiment
operated at 7.83 Hz producing pulsatory 20-25 kPa pressure under the baseplate and transient
strain in the upper 1m ground layer of an order 2 x 10 sup –4, at a vibration amplitude 3 mm. Pulse
transmission was performed as 5 min long linear sweep signals at a bandwidth of 78 to 156 Hz.
The result (Fig. 3) shows overlapped correlated records for the dynamic (vibrator on, bold traces)
and static (vibrator off, thin traces) modes, obtained without instantaneous phase splitting so that
all instantaneous variations of the wavefield are averaged by the correlation integral. When the
vibrator is on, the amplitudes of all waves are smaller and the traveltimes are longer, i.e., wave
velocity decreases and attenuation increases. Thus the experiment confirms the earlier inference of
strong variations of elastic parameters in dynamically loaded friable medium [2]. The variations
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can be estimated quantitatively from records in Fig. 3, b, same as in Fig. 3, a but with 2.2 times
greater amplitudes in the dynamic mode (this must be the magnitude of attenuation increase in
presence of pulsatory loading, judging by the ratio of mean square values). This scaling visually
cancels the additional attenuation in the loaded medium and makes evident that the traveltime
delay relative to the static mode increases with traveltime. Therefore, the observed effect is not
localized, for instance, at the source/ground contact but originates in the ground. Converted from
traveltimes, seismic velocities in dynamically loaded medium in this experiment show a decrease
about 5 %.
Experiment 2. Biharmonic sounding. This experiment illustrating the parametric effect of
vibration was carried out following the traditional method of amplitude and phase modulation to
keep consistency with the previous experiment. The vibrator and the acoustic source were operated
simultaneously, each at its main frequency (7.83 and 235 Hz, respectively). Analyzed were the
vibration spectra recorded at the face of borehole 2 at 6 m below the surface. Note that vibration at
7.83 Hz, which is a speed-up regime, produces saw-shaped signals (Fig. 4). Vibrations are
repeated every two periods, i.e., the vibrator generates a subharmonic (half-frequency harmonic).
The plots in Fig. 4 (a for baseplate displacement, b for vibration amplitude spectrum and its two
derivatives (velocity and acceleration), c for acoustic spectra, near-field and at 6 m in borehole 2)
show that the vibration signal containing the first, second and third harmonics modulates the highfrequency acoustic signal, and the higher the harmonic the stronger the modulation effect. This
persistent regularity is illustrated in Table 1 showing that the level of low-frequency harmonics in
the combination signal corresponds to their ratio in velocity rather than displacement, and even
approaches acceleration. Therefore, the role of stress-strain nonlinearity which is at base of many
existing models of nonlinear phenomena [6] (on the background of some more complicated
nonlinearities) is not so important in the effects observed in dynamically loaded friable medium.
The lower plot in Fig. 4 c also shows that the right and left combination frequencies of the same
vibration harmonics are different. Earlier we explained this effect theoretically (in terms of the
theory of signals) by the out-of-phase behavior of amplitude and phase modulations [2], but could
not check this hypothesis experimentally. In this study it is for the first time illustrated by
experiments.
Experiment 3. Splitting of P wave records into phases of baseplate motion: short rays,
normal (linear) vibration. The experiment aimed at checking whether the higher harmonics of
baseplate motion, i.e., the components of double, triple, etc, frequency, are responsible for higher
harmonics in velocity and attenuation. Experiment 2 implied the negative answer proceeding from
comparison of harmonic levels, but a direct proof will be more convincing. In this experiment
(Fig. 5) ground motion under the baseplate should be perfectly harmonic, i.e., the loading effect
should be free from higher harmonics. This was achieved at frequencies below 6.8 Hz, namely at
6.2 Hz at which higher harmonics are less than 1 %, but the scanning ray was directed through the
region of strongest deformation to compensate for possible weakening of nonlinear effects at
weaker vibration. The acoustic source was positioned horizontally and tightly secured by a jack at
the pit wall near the vibrator. Thus the radiation pattern of the acoustic source was oriented
horizontally, closer to the baseplate. The signals were received by a borehole-emplaced pressure
sensor at a depth of 1 m on the opposite side of the baseplate.
Figure 5 shows amplitude and traveltime variations of a direct P wave (Fig. 5, a) and a record split
into phases of baseplate motion (Fig. 5, b). The uppermost trace was obtained when vibrator was
off; traces from 1 to 8 correspond to eight successive equally spaced phases of baseplate motion.
The split traces are duplicated to highlight the periodicity, but the repeated traces were obtained
independently by repeated soundings and are not just copies of the original eight traces, which
demonstrates high reproducibility. The amplitude, traveltime, and baseplate motion plots (Fig. 5,
a) correspond to the split traces (X is directed downward and shows the time of two pulse periods
and the number corresponding to the phase of baseplate motion).
Under pulsatory dynamic loading the mean P wave velocity decreases and the instantaneous
velocity varies simultaneously with loading-unloading cycles, and even its highest value at
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maximum load does not reach the off-mode level. The mean amplitude also decreases, and the
instantaneous amplitude contains a specific component of double frequency and two local negative
and positive peaks: minimums at baseplate up and down, and maximums at phases of its maximum
velocity.
Thus, although the dynamic effect of baseplate force and motion is roughly sine-shaped, the
variations in the parameters of the medium are not sine. It means that either the very nonlinearity
of the medium (for instance, stress dependence of velocity) is nonlinear or other factors interfere
into the process. The Fourier series contain second and third harmonics in both amplitude and
traveltime variations, with phases different even for the same (especially second) harmonics. They
may be rather related with the dependence of the parameters on absolute strain rate.
As mentioned above, the asymmetry of lateral spectral lines in Experiment 2 (Fig. 4, c) is
explained in terms of the theory of signals by the out-of-phase behavior of long-period amplitude
and phase modulation of the acoustic signal. In terms of elasticity mechanics this may be caused,
for instance, by different relaxation times of velocity and attenuation.
At longer rays, especially when the observed signal results from interference of several waves, the
pattern of variations may differ considerably from that in Figure 5. However, the velocity and
attenuation variations are almost always out of phase, which is exemplified by the experiment
below.
Experiment 4. Splitting of acoustic records into loading phases: face of borehole 2, speed-up
vibration regime. The layout of this experiment was the same as in Experiments 1 and 2. The Pwave trace obtained at a depth of 6 m in borehole 2 (lowermost trace in Fig. 2) was split into 16
phases (Fig. 6, a); Fig. 6, b shows the corresponding amplitude (A, in absolute arbitrary units) and
traveltime (... t, deviation from the absolute time, in ms) of the strongest arrival at 40 ms and the
function of baseplate motion (B). Unlike the previous experiment, the shape of the latter differs
notably from the sine. Amplitude and velocity reach their maximum near the maximum loading
phase but not simultaneously: Amplitude lags behind velocity both in the loading phase and at the
local peak in the unloading phase. This confirms the difference in their relaxation times [2]
observed when the vibrator was off (the static velocity recovered much faster).
The shape of traveltime variations, namely the presence of a strong second harmonic, attests that
the velocity of elastic waves in a friable medium of the type of loam decreases considerably with
absolute strain rate. A sharp velocity decrease at the onset of failure was recorded in model
experiments with loam [7], which agrees with our hypothesis.
The obtained experimental data make a basis for a simplified model of dynamic dependence of Pwave velocity v on instantaneous dynamic strain ... :
where v sub 0 is velocity in unloaded medium, ... is instantaneous strain rate, ... is time-averaged
absolute strain rate, and a sub 0, a sub 1, a sub 2 are positive constants.
Here a sub 0 is defined by nonlinearity of the static stress-strain function and has a physical sense
of a standard nonlinearity parameter [4-6]. The coefficient a sub 1 corresponds to relative
instantaneous velocity variations, and a sub 2 represents mean velocity decrease. Of course, it
should be kept in mind that negative strain ... is associated with loading. Therefore, the
instantaneous velocity of elastic waves is the highest at maximum compression. In reality the
maximum is somewhat shifted because of relaxation. It is also hypothesized that the largest sum of
all strain-dependent variable terms is much below the unity. The greatest value (0.05) was
observed in a sub 2 which depends on the mean absolute strain rate and represents mean wave
velocity variations.
Experiment 5. Detection of unstable acoustic (seismic) reflectors. A striking feature in the
gather of records in Fig. 5 is periodic splitting of one signal into two waves in the unloading phase
and their merging in the loading phase. A wave arriving at 60 ms in Fig. 6, a strongly varies from
phase to phase on the background of relatively stable signals. We neglected it in our analysis
believing that it is not this wave that determines the most important nonlinear properties of the
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medium. At the same time, this wave always manifested itself by its high sensitivity to loading
phase associated with changes of pressure, velocity, or acceleration sensors, as well as with
changes of radiation pattern, in different arrays. Figure 7 shows a record split into loading phases
obtained from a horizontally oriented acoustic source and a receiver (accelerometer) positioned,
also horizontally, on the opposite side of the vibrator. A weak but well pronounced wave at 65-75
ms shows much stronger traveltime variations than the first arrival, only once per period, and
disappears in the upper trace corresponding to relaxation. This behavior cannot be explained by
interference, as there are no other intense waves in its vicinity but the first arrival. Unfortunately,
we could not reliably reconstruct the corresponding ray path, as the wavefield in the vertical
profile (Fig. 2) was investigated at a vertical orientation of the source and pressure sensors
insensitive to shear waves. Perhaps, it is a converted wave from a relatively deep weak reflector
which becomes better pronounced under dynamic loading. The physics of this phenomenon, which
we provisionally called “induced reflector effect”, remains unclear and requires purposeful studies.
CONCLUSIONS
1. A new method for investigation of dynamically disturbed medium in situ implies
controlled-source pulsatory loading using a powerful long-period vibrator and simultaneous
short-period pulse transmission scanning of the loaded ground. The method allows analyzing
mean variations in wave velocity and attenuation as well as their changes at different
instantaneous phases of dynamic loading.
2. Experiments within the limits of the method showed that the state of the medium under
pulsatory dynamic loading (at least, friable medium of the type of loam) cannot be presented
by averaged instantaneous static states that compose transient strain. Ground under dynamic
loading is rather in a specific state, both qualitatively and quantitatively, in which any
instantaneous P-wave velocity is lower and any instantaneous attenuation is higher relatively
to that in unloaded medium.
3. Velocity and attenuation of elastic waves in a friable medium subject to pulsatory loading
vary in magnitude simultaneously with the loading cycles but are out of phase relative to each
other, which indicates that the two parameters have different relaxation times. As a rule, wave
velocity is more responsive to strain changes than attenuation, whereas attenuation is much (an
order of magnitude) more sensitive to dynamic loading.
4. Instantaneous variations in velocity and attenuation decrease with absolute strain rate,
which causes intense higher harmonics.
5. Some weak seismic reflectors are especially sensitive to dynamic loading, which we
suggest to provisionally call “induced reflector effect”. The physics of induced reflectors
remains unclear and requires purposeful studies.
We wish to thank V. Velinskii, V. Man’kovskii, V. Nosov, and V. Savvinykh for their aid in
preparation and effectuation of field experiments. The paper profited much from discussions and
constructive criticism by Academician S. Gol’din.
The study was supported by grants 98-05-64948, 99-05-79083, and 00-05-65276 from the Russian
Foundation for Basic Research.
REFERENCES
[1]
[2]
V.I. Yushin, ..., Russian Geology and Geophysics, vol. 35, no. 5, p. 161(), 1994.
V.I.Yushin, G.V.Egorov, N.F.Speranskii, and V.N.Astafiev, Acoustic study of nonlinear
and rheological phenomena in the near zone of seismic vibrator, Geologiya i Geofizika
(Russian Geology and Geophysics), vol. 37, no. 9, p. 156(152), 1996.
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[3]
[4]
[5]
[6]
[7]
G.V. Egorov, V.M. Nosov, and V.V. Man’kovskii, Experimental estimation of nonlinear
elastic parameters of dry and saturated porous media, Russian Geology and Geophysics,
vol. 40, no. 3, p. 457(), 1999.
A.S. Aleshin and V.V. Kuznetsov, in: Studies of the Earth with non-explosion seismic
sources [in Russian], Moscow, p. 267, 1981.
A.S. Aleshin and I.Ya. Koval’skaya, Voprosy Inzhenernoi Seismologii, issue 30, p. 90,
1984.
V.V. Gushchin and G.M. Shalashov, in: Studies of the Earth with non-explosion seismic
sources [in Russian], Moscow, p. 144, 1981.
S.V. Gol’din, Yu.I. Kolesnikov, and S.V. Polozov, Fizicheskaya Mekhanika, vol. 2, no. 6,
p. 105, 1999.
Recommended 17 January 2001
By S.V. Gol’din
Received 22 August 2000
FIGURE CAPTIONS
Fig. 1. Layout of experiment.
Fig. 2. Vertical profile of borehole 2 in two ( a, b) amplitude scales.
Fig. 3. Near-field pulse transmission records.
a – vibrator off (thin line) and on (bold line), b –same with mean square normalized amplitudes
(about 2.2 times greater for operating vibrator).
Fig. 4. Baseplate motion.
a – baseplate motion, mm; b – amplitude spectra of displacement (bold line), velocity (thin
line), and acceleration (dotted line); c – near-field pulse transmission spectra in borehole 2 at 6
m deep.
Fig. 5. Splitting of P-wave records into eight phases of pulsatory dynamic loading.
a – variations in amplitude ... A and traveltime ... t of a direct P wave plotted together with
baseplate motion function (B); b – split record for two periods of vibrations. Traces: 0 - vibrator
off, 1 – baseplate up, minimum load, 5 – baseplate down, maximum load.
Fig. 6. Variations in wave pattern (a) and parameters of P wave (b) under pulsatory loading.
a – pulse transmission record for ground under vibrator (depth 6 m) split into 16 phases of
dynamic loading; b – parameters of P wave at 40 ms: A is amplitude in absolute arbitrary units,
... t is traveltime variations, in ms, B is vertical displacement of baseplate, in mm.
Fig. 7. Effect of induced reflector: unstable wave at 65-75 ms.
Trace: 1 - baseplate up, minimum load, 8 - baseplate down, maximum load. Sampling rate is
1/15 vibration cycle.
Table 1.
Higher harmonics in Vibrations and in Combination Frequency Spectra
Point of measurement
Baseplate Borehole 2, depth 6 m
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Measured parameter
Displacement
Velocity
bandwidth
Acceleration
Acoustic pressure at combination
Relative levels of higher harmonics
Second/first
Third/first
Third/second
Left
Right
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