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Geophysical Prospecting, 2010, 58, 19–31 doi: 10.1111/j.1365-2478.2009.00811.x

Modelling and modal analysis of seismic vibrator baseplate

Zhouhong Wei∗

ION Geophysical Corp., 12300 Parc Crest Drive, Stafford TX 77477, USA

Received November 2008, revision accepted March 2009

ABSTRACTThe vibroseis method must be extended to its limits as the search for oil and gascontinues on land. To successfully improve vibroseis data quality, it is crucial toevaluate each element in the vibroseis data acquisition system and ensure that thecontribution from each element is successful. Vibroseis systems depend greatly uponthe ability of vibrators to generate synchronous, repeatable ground-force sweeps overa broad frequency range. This requires that the reaction mass and the baseplate of thevibrator move as rigid bodies. However, rigid-body motion is not completely true forhigh- frequency vibrations, especially for the vibrator baseplate. In order to accuratelyunderstand the motion of the vibrator baseplate, a finite element analysis model of thevibrator baseplate and the coupled ground has been developed. This model is usefulfor simulating the vibrator baseplate dynamics, evaluating the impact of the baseplateon the coupled ground and vibrator baseplate design. Model data demonstrate thatthe vibrator baseplate and its stilt structure are subject to six significant resonantfrequencies in the range of 10–80 Hz. Due to the low rigidity of the baseplate, thebaseplate stilt structure experiences severe rocking motions at lower frequencies andthe baseplate pad experiences severe flexing motions at higher frequencies. Flexingmotions cause partial decoupling, which gives rise to increased levels of harmonicdistortion and less useable signal energy. In general, the baseplate pad suffers morebending and flexing motions at high frequencies than low frequencies, leading lessefficiency in transmitting the useable energy into the ground.

INTRODUCTION

The enhancement of seismic vertical resolution dependsgreatly on the ability to extend the emitted signal bandwidthtoward low and high frequencies. Extending to low frequen-cies aids in the inversion of seismic trace data and acquiringthem can greatly improve the accuracy of surface-derived ve-locity sections (Bell 1986). High frequencies are most usefulfor enhancing spatial and temporal resolution as found inshallow or vertical seismic profiling (VSP) surveys. Successfulreservoir delineation depends on both of these attributes. If thevibrator shakes the earth with much higher-fidelity, reachingon the high-side into the 100–150 Hz bandwidth, this band-

∗E-mail: [email protected]

width could be transmitted deep into the Earth and it wouldimprove subsurface resolution to about 20 m or less, well intothe realm of reservoir characterization and stratigraphic-trapdefinition. Certainly, vibrators must be capable of generat-ing the required frequencies with sufficient fundamental forceto retain reflection signal above noise at the objective targetdepths. Unfortunately, for many decades the frequency band-width of land seismic images has remained between approxi-mately 10 and 60 Hz. This bandwidth translates to a verticalsubsurface resolution of about 50 m or so. This is mainly be-cause of the limitations existing in vibrator mechanical andhydraulic components. The vibrator can only maintain suffi-cient force amplitude in a narrow frequency range. At very lowfrequencies the vibrator performance is constrained by the vi-brators’ physical limits such as the reaction mass peak-to-peakstroke and at high frequencies the vibrator performance is

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restrained by the rigidity and weight of the vibrator baseplate(Wei 2008a). Other limiting factors such as a limited servovalve frequency bandwidth and the baseplate coupling withthe ground cause degradation of the vibrator performance athigh frequencies as well.

The weighted-sum method is used to approximate theground force (Sallas 1984). This method is based upon a rigidbody assumption for both the reaction mass and the base-plate. Unfortunately, as the sweep frequency goes up, thisestimate for the vibrator force output is in error. In gen-eral the weighted-sum approximation overestimates the ac-tual force output of the vibrator. Many researchers have ver-ified this by making comparisons between the weighted-sumground force and the direct-force measurement using load cells(Sallas, Amiot and Alvi 1985; Baeten and Strijbos 1988;Baeten and Ziolkoski 1990). The disparity is due to boththe flexure and the heavy weight of the vibrator baseplate.The vibrator baseplate weight plays a large role in achiev-ing good vibrator performance at high frequencies. Unfortu-nately, this factor has long been forgotten in modern vibratordesign. Actually, a lightweight and stiff baseplate assures thatthe force generated by the reaction mass is used to drive theearth and not expended driving the baseplate mass. Further,the lightweight and stiff baseplate will also increase the res-onant frequency of the earth-vibrator, which provides addi-tional benefit at high frequencies.

To understand the baseplate dynamic motion more accu-rately, a vibrator baseplate finite element analysis model wasdeveloped. This finite element analysis model was employedto identify key vibrator vibration modals that influence thevibrator output energy at low and high frequencies. The mainobjective is to provide a useful set of guidelines for the vibra-tor baseplate design in order to generate high quality vibratoroutput energy.

F INITE ELEMENT A NALYSIS M ODEL

The vibrator baseplate finite element analysis model is basedupon an ION (previously I/O) AHV-IV vibrator and con-structed using the ANSYS Structure software including bothstatic and dynamic analysis packages. This model is shownin Fig. 1. Each part of the vibrator baseplate included inFig. 1 was modelled based on real geometries of componentson the vibrator baseplate. For example, the baseplate pad wasmodelled in a rectangular type of structure 2.338 m in length,1.068 m in width and 0.175 m in height. Due to compli-cations in meshing and computation, the reaction mass wasremoved from the model. The reason for this is that the in-

Figure 1 The finite element analysis model of the AHV-IV vibratorbaseplate and the ground.

terface between the surface area of the tie-rod tubes and theinternal surface area of the reaction mass through which thesetubes slide creates a need in the software to mesh this areaas points of contact. This complication applies to the contactareas around the piston surface as well. Furthermore, the com-putation itself is complicated by the dynamic nature of thesecontact areas. The tubes and piston slide within the reactionmass body when the structure undergoes motion, forcing thesoftware to dynamically adjust the contact area between thetwo surfaces. These impediments prevent the simulation fromarriving at a convergent solution. Since the model is primar-ily concerned with the motion and structure of the baseplate,the removal of the reaction mass has a relatively small im-pact on the accuracy of the simulation’s results and does notpresent a major concern as it is necessary for the simulation tosucceed.

The ground was modelled as a solid cylinder 6.096 m in di-ameter and 3.048 m in height. The model size of this groundis chosen to be big enough to contain the radiation mass orthe captured ground mass. The radiation mass or the capturedground mass is defined as the ground mass that participates inthe motion of the vibrator baseplate as it vibrates. The shape ofthe cylinder was chosen to ensure that the ground would havean even response to the force applied. In addition, absorbingboundaries were included in the form of mode damping andnode-to-node damping such that the energy from the vibra-tion would be dissipated properly. The stress and deformation

C© 2009 European Association of Geoscientists & Engineers, Geophysical Prospecting, 58, 19–31

Modelling and modal analysis of vibrator baseplate 21

distribution in ground layers including the top and bottomsurfaces are simulated as the vibrator baseplate moves up anddown. Knowledge of the stress and deformation distributionin ground layers, allows the radiation energy to be visualized.In addition to the stress and deformation distribution, it is im-portant to accurately represent the material properties of theground. Moduli of elasticity and density are three very impor-tant parameters used to characterize these properties. Withdifferent values for these three parameters the ground suchas mud, sand, concrete and so on can be modelled. In orderto make ‘apple-to-apple’ data comparisons between the finiteelement analysis model and the experimental measurement,the material of concrete was chosen for the ground in thisfinite element analysis simulation. This is because the densityand elastic moduli of the concrete are constant while valuesof these three parameters for other materials such as mud andsand vary due to weather conditions. Fortunately, a vibratortesting pad made in concrete has been built at the ION Sealytesting facility. Due to space limitations it is difficult to showthis concrete testing pad graphically.

To determine the utility of the model in predicting thedynamics of the vibrator baseplate, a data comparison ofthe baseplate acceleration between the finite element analysismodel and the field measurement was made at 5, 10, 20, 40,80 and 120 Hz. The AHV-IV vibrator sitting on the concretepad performed a set of monochromatic sweeps. Each sweep is5 seconds long with a 0.5 s start and 0.5 s end cosine taper.Considering that the actual vibrator has the vibrator truckframe connecting to the baseplate through isolation airbagswhile the finite element analysis model of the vibrator base-plate does not include the vibrator truck frame, a 0.5 s longtaper is used at each end of the sweep to avoid ruining thesteady-state vibration by exceeding the operational limits ofthe vibrator at the start and end of the sweep. Meanwhile, a5 s sweep length is used to ensure that steady-state vibratoroutput is achieved. The baseplate acceleration and the differ-ential pressure across the piston in the reaction mass chamberwere recorded. For the AHV-IV vibrator the baseplate ac-celerometer is mounted on the top cross of the baseplate stiltstructure. Applying the differential pressure measured fromthe vibrator as an input signal to the piston in the finite ele-ment analysis model, the baseplate acceleration of the finiteelement analysis model is then calculated. The transient re-sponse of the baseplate acceleration during the tapered sec-tions of the sweep is not taken into account when the dataare compared for different frequencies. Figure 2 comparesthe baseplate acceleration trace to the finite element analy-sis model data at six different monochromatic frequencies.

Each plot is as follows: a) the upper-left window shows thebaseplate acceleration traces at 5 Hz, b) the upper-right win-dow shows the baseplate acceleration traces at 10 Hz, c) themiddle-left window shows the baseplate acceleration traces at20 Hz, d) the middle-right window shows the baseplate accel-eration traces at 40 Hz, e) the bottom-left window shows thebaseplate acceleration traces at 80 Hz and f) the bottom-rightwindow shows the baseplate acceleration traces at 120 Hz.The measured baseplate acceleration is represented using themagenta line while the finite element analysis model baseplateacceleration is the blue line. The baseplate acceleration tracesshown in Fig. 2 are steady-state outputs and the transient por-tions of the sweep are discarded. The data comparison showsthat the finite element analysis model captures the main dy-namic motion-peaks of the AHV-IV vibrator baseplate up to80 Hz. However, a clear discrepancy was observed at 40 Hz.The baseplate acceleration from the FEA model has a strongsecond harmonic while the experimental data show a strongthird harmonic. It is unclear as to the definite source of thisdiscrepancy. However, it may be possible that the source ofthis discrepancy lies in the non-idealities of a real world con-tact area between the baseplate and the ground. In the simu-lation, the ground and baseplate are taken to be perfectly flat,leading to an ideal contact area. In this situation the primarymode of motion is the bending motion and will cause strongsecond harmonics. However, in real world applications, thecontact area between these two structures is not ideal. Thebaseplate pad yields to be an invisible bow-shape due to hy-draulic preloads. Uneven surface of the baseplate and groundslope lead to non-uniformity in the contact area and cause ad-ditional modes of motion, which may contribute to the strongthird harmonic present in the experimental data. At 120 Hz,the baseplate acceleration of the finite element analysis modelleads the field measurement about 90 degrees in phase. Thereason for this discrepancy is not clear although the com-plex modes of the stilt structure exhibited at high frequenciesmay be a major contributor to the differences between thesimulation and experimental data. Overall, there is a goodagreement between the baseplate acceleration calculated bythe finite element analysis model and the measured baseplateacceleration in the 80 Hz sweep bandwidth. It is important tonote that although there are some discrepancies between thesimulation results and the experimental data, these discrepan-cies represent minor variations and the simulation successfullydepicts the primary characteristics of the baseplate accelera-tion. For the dynamic motion study of the baseplate below100 Hz, the utility of this finite element analysis model isvalid.


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Figure 2 Base plate accelerations are compared for the measured data and the finite element analysis model data. a) 5 Hz, b) 10 Hz, c) 20 Hz,d) 40 Hz, e) 80 Hz and f) 120 Hz.

VIBRATION MODES OF T H E V I BR A T ORBASEPLATE

Vibration modes of the vibrator baseplate are characteristicpatterns or shapes in which the coupling system of the vibratorbaseplate and the ground will vibrate. Certainly the vibratorbaseplate has many modes of vibration and it is the task ofthe finite element analysis modal analysis to determine thesemodal shapes. The actual vibration of the baseplate is always acombination or mixture of all the vibration modes. However,they need not all be excited to the same degree. For instance,if the vibrator shakes at a very low force (drive) level, we seeprimarily the vertical vibration mode of the baseplate but ifthe vibrator shakes at a higher force (drive) level, other modesare excited and we see some other motions such as rockingand flexing of the baseplate in addition to the vertical motion.

Figure 3 shows six vibration modes of the baseplate domi-nantly below 100 Hz using the finite element modal analysis.These figures are generated to illustrate the motion and rela-tive displacement of the baseplate and captured ground masswhen the baseplate undergoes motion. Maximum displace-ment is represented in these figures as red while minimum dis-placement is shown in blue. Because the images are generatedstrictly through modal analysis with the intent of illustratingthe modes of motion only, the scale of the displacement isnot provided as the displacement magnitude will vary in realworld applications. Each plot is as follows: a) the upper-leftplot shows the baseplate has a front-rear rocking mode at14.9 Hz, b) the upper-middle plot shows the baseplate hasa side-to-side rocking mode at 15.1 Hz, c) the upper-rightplot shows the baseplate has a twisting mode at 19 Hz, d)the bottom-left plot shows the baseplate and the ground have



Figure 3 a) Baseplate front-rear rocking, b) baseplate side-side rocking, c) baseplate twisting, d) baseplate vertical vibration, e) lateral motionof the baseplate stilt structure and f) baseplate vertical vibration and ground flapping.

a vertical vibration mode at 61.92 Hz, e) the bottom-middleplot shows the baseplate has a lateral motion mode of thestilt structure at 80 Hz and f) the bottom-right plot shows thebaseplate has a vertical motion mode while the ground is ina flapping mode at 100 Hz. Catching natural frequencies ofthe vibrator baseplate is necessary for determination of thevibrator performance and operational frequency range. Basedon the results of the vibrator baseplate finite element analysissimulation performed, it is necessary to do some experimen-tal tests on the vibrator and identify these natural frequencymodes. Figure 4 shows an AHV-IV vibrator baseplate on con-crete. The surface of the concrete is not perfectly flat. Threepieces of hard rubber-pad are inserted under the baseplate toimprove the contact between the baseplate and the concrete.The AHV-IV vibrator is equipped with a 4905 Kg reactionmass and a 1771 Kg baseplate. The baseplate area is 2.5 m2.The concrete is approximately 5.5 m long, 3.4 m wide and2 m deep. Its stiffness and viscosity are 0.89 × 1010 N/m and3.0 × 106 N-s/m, respectively. The captured ground mass isabout 900 Kg and concrete only (Wei 2008b).

Our experiment was divided into three phases. The firstphase of the experiment was designed to measure the front-rear rocking and the side-side rocking using a 1–21 Hz, 10-s sweep and six magnetic accelerometers. Among them, twomagnetic accelerometers were positioned vertically in a diago-

nal on the top cross of the baseplate. Another two magnetic ac-celerometers were attached on the front and rear middle of thetop cross horizontally pointing forward and rear, respectively.The last two were attached on the right and left side middle ofthe top cross horizontally pointing right and left, respectively.Figure 5 shows four acceleration traces recorded by mag-netic accelerometers on the top cross of the baseplate. Trace 1shown in red was recorded by the front accelerometer measur-ing the front-rear rocking motion. Since the data measured bythe front accelerometer and the rear accelerometer were iden-tical, only the front accelerometer data were plotted. Trace 2depicted in blue was measured by the right-side accelerometershowing the side-to-side rocking. For the same reason only theright-side accelerometer data were shown. Trace 3 and Trace4 are shown in brown and magenta, respectively. They weremeasured by two vertical accelerometers. It is obvious thatthey are identical. The reason for showing these vertical accel-erations together is to demonstrate that there is no unexpectedbending motion on the top cross. As clearly illustrated inFig. 5, the amplitudes of trace 1 and trace 2 get larger be-tween 4 and 5 seconds when the frequency is in the range of 9–11 Hz. These indicate that the top cross of the baseplate expe-riences a front-rear rocking motion and a side-to-side rockingmotion approximately at 10 Hz. Compared with the finiteelement analysis simulation results for front-rear rocking and


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Figure 4 AHV-IV vibrator locates on concrete.

Figure 5 Measured baseplate accelerations on the top cross for the1–21 Hz, 10-s sweep.

side-to-side rocking, the field measurements are about 5 Hzlower. This is because the modal analysis of finite elementanalysis simulation treats the coupling of the baseplate andthe ground as a bonded condition while in reality the base-plate is not completely coupled with the ground. This bondedcondition in the finite element analysis model enhanced thestiffness of the baseplate so that the frequencies of the rockingmotions performed from the finite element analysis simulationwere higher than the actual measurements.

In the second phase of the experiment, a 20-s long sweepfrom 5 to 105 Hz was used to measure the vertical vibrationmode of the baseplate and the ground. This vertical vibra-tion mode maximally represents the interaction between thevibrator baseplate and the ground. Generally, this interac-tion is simplistically described by a second-order system andrecorded by the reaction mass accelerometer and the baseplateaccelerometer. To clarify the data measured for this experi-ment, it should be noted that the reaction mass accelerationand the baseplate acceleration were taken using the Pelton re-action mass and baseplate accelerometers. They were mountedon the top of the reaction mass and the top cross of the base-plate, respectively. Figure 6 shows a frequency response plot-ted using the reaction mass acceleration (input) and the base-plate acceleration (output). In Fig. 6 the magnitude ratio plotshows that the amplitude curve starts with a slope of approxi-mately 40 dB/dec. Then, at 65 Hz the slope of the curve startschanging and a peak is formed. The phase plot illustrates thatthe phase curve starts at approximately zero degrees. At 65Hz the phase curve starts shifting to –180 degrees. The fre-quency at the peak in the magnitude plot and the frequencyat the turning point in the phase plot are very close to eachother. This means that frequency is the resonant frequencyor natural frequency of the vertical vibration of the baseplateand the ground. This measured frequency nearly matches thefrequency of 61.92 Hz shown by plot ‘d’ in Fig. 3.



Figure 6 Frequency response of the reaction mass-baseplate/groundmodel. a) Magnitude ratio plot and b) phase plot.

Figure 7 Assessment of the motion of the upper stilt structure using four accelerometers placed on the top cross of the baseplate. a) Magnituderatio plot and b) phase plot.

In the last phase of the experiment, we assessed the motionof the upper stilt structure itself. Figure 7 shows a comparisonbetween four magnetic accelerometers placed on the top crossof the baseplate. A frequency response was then taken betweeneach of these magnetic accelerometers and the Pelton top-crossaccelerometer as a reference. The black and blue traces rep-resent the measure lateral motion in-line with the front-backdirection of the vibrator. The red and green traces correspondto lateral motion in the side-to-side direction of the vibrator.As can be seen from Fig. 7, the stilt structure obviously experi-ences a significant amount of lateral motion as the frequencyresponse of these with respect to the Pelton top-cross verti-cal accelerometer is relatively high. These four lateral-motionaccelerometers experience equivalent magnitude of motion ata frequency of 80 Hz. At this high-frequency range, it is notpossible to distinguish the motion with the eye. However, itmay be correlated to a higher-order mode of motion like plot‘e’ depicted in Fig. 3.

It is very difficult to measure the baseplate twisting and thebaseplate vertical vibration-ground flapping shown in Figs.3(c) and 3(f), respectively. When the vibrator was relocatedon the ground of soft soil, rotating prints were found beneaththe baseplate pad. This implied that the baseplate experiencedtwisting motions. The rotation angle of prints was very small.


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For the movement represented by plot ‘f’ in Fig. 3 it requiresour imagination to picture this mode. As the vibrator baseplatemoves up-down, the surface of ground flaps in and out.

VIBRATOR BASE PL A T E FLEXUR EAND W EIGHTED- SUM GR OUN D FOR CE

The vibrator baseplate functionally acts like a transmitter,transmitting a hydraulic force into the earth. Due to low rigid-ity, the baseplate is subject to flexural vibrations at higherfrequencies. The modes of the flexural vibration are highlydiversified depending on ground conditions. When the base-plate is on very uneven surfaces, the flexural vibration can bedescribed as the baseplate jumping up-down. Often times fullor partial decoupling occurs leading to poor vibrator perfor-mance and a reduction of useable transmitted signal energy.When the baseplate is on flat surfaces, the flexural vibra-tion can be described as a bird flapping its wings. Under thiscondition, partial decoupling occurs but the decoupling areais small. Figure 8 demonstrates the flexural vibration of thebaseplate in the bird flapping mode at 80 Hz using the finiteelement analysis simulation. The plots in Fig. 8 are exagger-ated to make a clear demonstration on the motion and rela-tive displacement of the baseplate when the baseplate under-goes motion. Maximum displacement is represented in thesegraphs as red while minimum displacement is shown in blue.The scale of the displacement is not provided as the mesh sizeapplied to the modelled ground leads to a situation in whichthe displacement of particular areas of the baseplate mightnot be perfectly accurate. This larger mesh size was necessarybecause using a small mesh size to obtain a detailed look atthe ground response prevented the software from successfully

Figure 8 Vibrator baseplate bending and flexing. a) Vibrator in compressing and b) vibrator in releasing.

completing a dynamic simulation. In addition, the figure isintended to depict the general type of motion that is occurringrather than to quantify specific displacement magnitudes. Theplot of Fig. 8(a) shows that the baseplate is pushed down-wards. It can be seen that the hydraulic force is applied oncentral areas and causes high stress and big deformation. Thestress and deformation quickly fades out from the central areato the side area. The plot also clearly demonstrates that theportions of the baseplate at the four corners move upwards,opposite to the downward motion of the central area of thebaseplate. The plot of Fig. 8(b) illustrates that the baseplatetakes the shape of an arch as it is pulled upwards. When thishappens a decoupling occurs in the central area of the plate.The portions of the baseplate at the four corners are mov-ing in a downward direction opposite to the motion of thecentral area. From Fig. 8, the conclusion can be drawn thatdifferent locations within the baseplate generally produce dif-ferent motions. Placing accelerometers in different places onthe baseplate will measure different accelerations. It is verydifficult to use only one baseplate accelerometer to determinethe exact vertical acceleration that the baseplate is in duringmotion. It is difficult as well to determine how much baseplatemass participates in transmitting useable signal energy.

The vibrator ground force can be approximated by using a‘weighted sum’ of accelerations of the reaction mass and thebaseplate (Sallas 1984; Safar 1984; Baeten and Ziolkowski1990). The weighted-sum method is built on the rigid bodyassumption of the baseplate. As demonstrated above, thebaseplate is subject to flexural vibrations when the vibra-tor shakes at higher frequencies. Different locations withinthe baseplate have different motions causing the accelerationrecorded by the single baseplate accelerometer to be an invalid



Figure 9 a) The vibrator is on load cells and b) magnetic accelerometer positions on the baseplate pad.

Figure 10 Weighted-sum ground force versus load cell force.

representation of the baseplate motion. Therefore, the valid-ity of the weighted-sum ground force becomes questionable.Figure 9(a) shows an example with the AHV-IV vibrator lo-cated on load cells. The load cell is a direct force measurementsensor and is very rigid enabling it to measure the true groundforce. There are a total of 8 load cells under the baseplate. Theload cell is mounted on the concrete. On the top of each loadcell there are two pieces of rubber pad to protect the load cellsfrom metal-to-metal contact. Under this loading condition,the baseplate is well coupled with the load cells. However,the contact area is small. Figure 9(b) shows a top view of thebaseplate pad. A, D, J and K are all six inches from the cornersin the X and Y directions. B, C, H and I are near the leg tubepositions. E and G are on the centreline midway between theedge and stilt tubes. F is at the centre of the pad. A total of

6 magnetic accelerometers are used to measure the motion ofthe baseplate pad. The locations are grouped in the driver side(A, B, E, F, H and J) and passenger side (D, C, G, F, I and K).In general, the data from the driver’s and passenger’s side arevery similar. Thus, this document uses the driver side for itsanalysis.

An experiment was designed using a 1–201 Hz 20-s sweepto compare the weighted-sum ground forces and the trueground force. The weighted-sum ground forces were calcu-lated by using different baseplate accelerations recorded withmagnetic accelerometers on different locations and the trueground force was measured by load-cell sensors. Figure 10shows the magnitude ratio and phase spectra (frequency-response or bode plots) of the weighted-sum ground forces,compared to the ground force measured by the load-cell


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sensors. The spectra were calculated by using the weighted-sum ground forces as the output and the actual ground forcemeasured by load cells as the input. In theory, if the weighted-sum and measured ground forces are equal, the magnituderatio spectrum is 0 dB and the phase spectrum remains atzero degrees. It should be noted that none of weighted-sumground forces are in agreement with the true ground force inthe 200 Hz frequency range. The broken line named FBK is theweighted-sum ground force calculated from Pelton accelerom-eters against the true ground force. The Pelton baseplate ac-celerometer is mounted on the top cross of the baseplate stiltstructure.

The positions of F and H are totally abnormal. Theweighted-sum ground forces calculated from these positionsare completely different than the true ground force. This phe-nomenon indicates that the central area on the baseplate isdecoupling a substantial amount of the time. Conversely, theweighted-sum ground forces calculated from positions A, B, Eand J agree with the true ground force in the frequency rangebelow 50 Hz. For frequencies above 50 Hz, the measurementshave shown that the weighted-sum approximation is not rep-resentative of the vibrator output. This is also true for FBKthat the baseplate accelerometer is located on the top cross ofthe stilt structure. This has been confirmed as well by Sallaset al. (1985) and Allen, Johnson and May (1998). Tradition-ally, most of these discrepancies are attributed to the base-plate flexural vibration. However, the complexity is furtherstrengthened because the relationship between the weighted-sum ground force and the true ground force is variable due todifferent ground coupling conditions. In practice the groundforce is not known exactly and varies from place to place.In general the weighted-sum approximation overestimates theactual force output of the vibrator. The direct force measure-ment such as the load cell is needed for real ground force.Unfortunately, the load cell is not yet robust enough for rou-tine use.

BASEPLATE RI GI DI T Y A N D G R OUN DFORCE HARMON I C S

The true ground force is defined as the total compressiveforce exerted by the earth upon the vibrator baseplate. Theground force is the contact force at the interface of the earthand the vibrator baseplate and can be calculated by the com-pressive stress integrated over the surface area that contactsthe baseplate. In reality, the ground force suffers severe har-monic distortion. Besides harmonic distortion caused by non-linearities exiting in the hydraulic servo-valve system and the

non-linearity of the ground itself, the low rigidity of the vibra-tor baseplate also produces a significant amount of harmonicdistortion in the ground force. In addition, the low rigidity ofthe baseplate ensures that in practice the ground stress anddeformation are never uniform. Rather than a uniform dis-tribution, the low rigidity causes an uneven distribution ofconcentrated areas of ground stress and deformation. In thecase of an ION AHV-IV vibrator, high ground stress and de-formation are concentrated at the area of the ground directlybeneath the centre baseplate area defined by the four tie-rodlegs. To the left and right sides of this centre area, the groundstress and deformation are drastically reduced. To study theimpact of the baseplate rigidity on the ground force harmonicdistortion it was necessary to develop a more realistic, com-plicated model of the interaction between the baseplate andthe captured ground mass.

Figure 11 depicts a complicated vibrator-ground model inwhich an emphasis is put on expressing the rigidity of thebaseplate as part of the non-ideal contact stiffness presentat the boundary interaction of the two surfaces. This pro-posed model can serve as a more realistic representation of thevibrator-ground interaction. This model has been confirmedthat it is capable of describing a wide range of non-linearcontact behaviour such as a partial contact and a full contact(Lebedev and Beresnev 2004). In Fig. 11, the ground modelis described as a linear and second-order system that consistsof a ground mass, ground stiffness and ground viscosity. The

Figure 11 Example of a complicated vibrator-ground model.



vibrator system is also treated as a linear and rigid body. Inthis model, the baseplate is considered to have a mass only andits stiffness is distributed to become a part of the contact stiff-ness. The highlight in this model is the contact stiffness locatedin-between the vibrator baseplate and the ground. The contactstiffness here is defined as a group of springs connecting thevibrator baseplate and the ground and its value depends on thenumber of springs that physically connect the baseplate andthe ground during vibrator vibration. Therefore, the contactstiffness is a variable stiffness. As we know, partial decou-pling often occurs as the vibrator shakes at high frequenciesdue to the low rigidity of the vibrator baseplate and it be-comes even worse on uneven ground. When the vibrator isin compressing mode, there are more contact areas betweenthe vibrator baseplate and the ground. More contact areasmean more springs and more stiffness. As the vibrator goes toreleasing mode, very often partial decoupling happens. Thismeans that the baseplate looses some contact with the groundso that the contact stiffness is reduced. Generally speaking, thecontact stiffness is reduced halfway through the compressioncycle until halfway through the release cycle and its value alsodecreases as the sweep frequency increases. When the vibratoris located on uneven ground, the vibrator baseplate is subjectto many motions such as the baseplate bending, flexing andtwisting so that the contact stiffness becomes unpredictableand the harmonic distortion is severe.

Two simulation tests were designed using monochromaticfrequency sweeps and the finite element analysis model tocalculate the true ground force and the ground deformationon the top surface of the ground. The finite element analysismodel was shown earlier in Fig. 1. This model is purposelydesigned as a linear system except for the contact area be-tween the baseplate and the ground. The contact area can bedescribed as a flat, even contact and considered to be wellcoupled. However, the model allows decoupling or partialdecoupling to happen if any local force generated by the hy-draulic force exceeds any local hold-down force produced bythe total hold-down force. The model also allows the base-plate to bend and flex if the local vibration force exceedsthe local material strength. A sine-wave force was chosen asthe hydraulic force to avoid the harmonic distortion createdby the non-linearity in the hydraulic servo-valve system. Thisforce is applied on the vibrator piston at a level of 91 628.8N (20 600 lbs). For the AHV-IV vibrator, this is equivalent toa differential pressure of 1000 psi across the piston.

Figure 12 demonstrates a data comparison of the sine-wavehydraulic force and the ground force calculated by the finiteelement analysis model in two cycles at 10 Hz. In Fig. 12

Figure 12 Comparison of the sine-wave hydraulic force and the finiteelement analysis model ground force at 10 Hz.

the sine-wave force is shown by a red line while the groundforce computed from the finite element analysis simulation isshown by a blue line. The sine-wave force pushes and pullsthe vibrator piston in positive and negative cycles, respec-tively. Then, through the stilt structure to which the pistonis connected, this sine-wave force is passed to the baseplatepad and drives the baseplate and coupled ground to move upand down. Eventually, the sine-wave force is transmitted intothe ground. When the force acts on the ground, the groundis stressed and deformed. By integrating the average groundstress and the stress distributed area the finite element anal-ysis model ground force can be obtained. Ideally, the finiteelement analysis model ground force should be identical tothe sine-wave force. However, in Fig. 12 the finite elementanalysis model ground force is obviously not a pure sine-wave. It has been distorted and contains a small portion ofeven harmonics although the fundamental content of the finiteelement analysis model ground force is very close to the sine-wave force. In addition, approximately 3% of the sine-waveforce is exhausted in driving the baseplate. Figure 13 showsthe average deformation of the ground. It can be seen that theaverage deformation of the ground is not a pure sine-waveeither and is severely distorted as well. It contains a certainamount of the second and third harmonics. As mentioned ear-lier, the only non-linearity present in the finite element analy-sis model is the contact area in-between the vibrator baseplateand the ground. The contact stiffness in the compressing half-cycle is different from the contact stiffness in the releasinghalf-cycle. This difference causes even harmonics in the finiteelement analysis model ground force and the average defor-mation of the ground. Even harmonics in the finite element


30 Z. Wei

Figure 13 Average ground deformation calculated from finite elementanalysis model at 10 Hz.

analysis model ground force further cause additional harmon-ics in the average deformation of the ground. However, at10 Hz the contact stiffness does not vary much in compress-ing and releasing half-cycles of vibration because the vibratorbaseplate is relatively stiff at low frequencies. This is the mainreason why we do not see significant harmonic distortion inthe finite element analysis model ground force and the grounddeformation.

Another simulation example uses an 80 Hz sine-wave forceas input. Figure 14 shows a wiggle-trace comparison of thesine-wave force and the FEA model ground force in two cyclesmade at 80 Hz. The red curve represents the sine-wave forceand the blue curve represents the finite element analysis modelground force. The sine-wave force keeps the same drive levelas the 10-Hz sine-wave force does. The finite element analysis

Figure 14 Comparison of the sine-wave hydraulic force and the finiteelement analysis model ground force at 80 Hz.

Figure 15 Average ground deformation calculated from finite elementanalysis model at 80 Hz.

model ground force is severely corrupted and contains a largeamount of harmonic distortion. Comparing the peaks of thesine-wave force and the finite element analysis model groundforce, the finite element analysis model ground force equals toapproximately 74% of the sine-wave force. This means thatroughly 26% of the sine-wave force is used to accelerate andbend the baseplate. Figure 14 also shows that in the finite ele-ment analysis model ground force there is another 25% of theground force spent in generating harmonics. The remainderof the finite element analysis model ground force producesthe fundamental force. There is a large amount of the sine-wave force exhausted in unrelated baseplate activities due tolow rigidity of the baseplate and the weight of the baseplate.Figure 15 demonstrates that the ground deformation is alsoseverely contaminated by harmonic distortion. This is dueto harmonic distortion present in the ground force. The av-erage deformation of the ground shown in Fig. 15 shows aslight phase lag compared to the finite element analysis modelground force in Fig. 14. Meanwhile, the behaviour in the dot-ted circle in Fig. 15 is very distorted. The reason for this isunclear. One possible explanation is that the contact betweenthe baseplate and the ground is set as a rough contact. Thismeans that the contact is treated as a non-linear contact bythe ANSYS program.

Comparing the finite element analysis model ground forcespresented in Figs. 12 and 14, more fundamental force is pro-duced at 10 Hz than at 80 Hz. Meantime, the harmonic dis-tortion is less at 10 Hz than at 80 Hz. There is also moreground deformation and less harmonic distortion yields at10 Hz than at 80 Hz. These are due to the contact stiffnessvarying much more significantly at high frequencies than lowfrequencies.



CONCLUSIONS

A vibrator finite element analysis model was developed andit has been proven to be very useful for studying the vibra-tor baseplate and the coupled ground. Through finite elementanalysis simulation, some dynamic motions of the baseplatestilt structure have been realized and quantified. The base-plate pad suffers more bending and flexing motions at highfrequencies than low frequencies. Model data also demon-strate that the true ground force and the ground deformationare distorted by harmonics due to the variable contact stiffnessbetween the baseplate and the ground. Although the simula-tion of the baseplate acceleration does exhibit some discrep-ancies at 40 Hz and 120 Hz when compared to the experi-mental data, the model successfully characterizes the primaryaspects of the baseplate acceleration. Further investigationswill be pursued in an attempt to discover the source of thediscrepancies. The variable contact stiffness is attributed tolow rigidity of the vibrator baseplate and the surface uneven-ness of the ground. Improving the vibrator baseplate stiffnessand coupling should be considered as a primary variable dur-ing vibrator design. The limiting factor in this endeavour isthen the increased costs associated with achieving very highstiffness while maintaining a very low baseplate mass.

REFERENCES

Allen K.P., Johnson M.L. and May J.S. 1998. High fidelity vibratoryseismic (HFVS) method for acquiring seismic data. 68th SEG meet-ing, New Orleans, Louisiana, USA, Expanded Abstracts, 140–143.

Baeten G. and Strijbos F. 1988. Wave field of a vibrator on a lay-ered half-space: Theory and practice. 58th SEG meeting, Anaheim,California, USA, Expanded Abstracts, 92–96.

Baeten G. and Ziolkowski A. 1990. The Vibroseis Source. ElsevierPress. ISBN 0444888799.

Bell D.W. 1986. Acquisition and utilization broadband signals con-taining 2–8 Hz reflection energy. 55th SEG meeting, Houston,Texas, USA, Expanded Abstracts, 443–446.

Lebedev A.V. and Beresnev I.A. 2004. Nonlinear distortion of signalsradiated by vibroseis sources. Geophysics 69, 968–977.

Safar M.H. 1984. On the determination of the downgoing P-wavesradiated by the vertical seismic vibrator. Geophysical Prospecting32, 392–405.

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

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

Wei Z. 2008a. Pushing the vibrator envelope: extending low and highfrequency limits. First Break 26, 37–43.

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