Modelling and modal analysis of seismic vibrator baseplate

Transcription

1 Geophysical Prospecting, 2010, 58, doi: /j x Modelling and modal analysis of seismic vibrator baseplate Zhouhong Wei ION Geophysical Corp., Parc Crest Drive, Stafford TX 77477, USA Received November 2008, revision accepted March 2009 ABSTRACT The vibroseis method must be extended to its limits as the search for oil and gas continues on land. To successfully improve vibroseis data quality, it is crucial to evaluate each element in the vibroseis data acquisition system and ensure that the contribution from each element is successful. Vibroseis systems depend greatly upon the ability of vibrators to generate synchronous, repeatable ground-force sweeps over a broad frequency range. This requires that the reaction mass and the baseplate of the vibrator move as rigid bodies. However, rigid-body motion is not completely true for high- frequency vibrations, especially for the vibrator baseplate. In order to accurately understand the motion of the vibrator baseplate, a finite element analysis model of the vibrator baseplate and the coupled ground has been developed. This model is useful for simulating the vibrator baseplate dynamics, evaluating the impact of the baseplate on the coupled ground and vibrator baseplate design. Model data demonstrate that the vibrator baseplate and its stilt structure are subject to six significant resonant frequencies in the range of Hz. Due to the low rigidity of the baseplate, the baseplate stilt structure experiences severe rocking motions at lower frequencies and the baseplate pad experiences severe flexing motions at higher frequencies. Flexing motions cause partial decoupling, which gives rise to increased levels of harmonic distortion and less useable signal energy. In general, the baseplate pad suffers more bending and flexing motions at high frequencies than low frequencies, leading less efficiency in transmitting the useable energy into the ground. INTRODUCTION The enhancement of seismic vertical resolution depends greatly on the ability to extend the emitted signal bandwidth toward low and high frequencies. Extending to low frequencies aids in the inversion of seismic trace data and acquiring them can greatly improve the accuracy of surface-derived velocity sections (Bell 1986). High frequencies are most useful for enhancing spatial and temporal resolution as found in shallow or vertical seismic profiling (VSP) surveys. Successful reservoir delineation depends on both of these attributes. If the vibrator shakes the earth with much higher-fidelity, reaching on the high-side into the Hz bandwidth, this band- john.wei@iongeo.com width could be transmitted deep into the Earth and it would improve subsurface resolution to about 20 m or less, well into the realm of reservoir characterization and stratigraphic-trap definition. Certainly, vibrators must be capable of generating the required frequencies with sufficient fundamental force to retain reflection signal above noise at the objective target depths. Unfortunately, for many decades the frequency bandwidth of land seismic images has remained between approximately 10 and 60 Hz. This bandwidth translates to a vertical subsurface resolution of about 50 m or so. This is mainly because of the limitations existing in vibrator mechanical and hydraulic components. The vibrator can only maintain sufficient force amplitude in a narrow frequency range. At very low frequencies the vibrator performance is constrained by the vibrators physical limits such as the reaction mass peak-to-peak stroke and at high frequencies the vibrator performance is C 2009 European Association of Geoscientists & Engineers 19

2 20 Z. Wei restrained by the rigidity and weight of the vibrator baseplate (Wei 2008a). Other limiting factors such as a limited servo valve frequency bandwidth and the baseplate coupling with the ground cause degradation of the vibrator performance at high frequencies as well. The weighted-sum method is used to approximate the ground force (Sallas 1984). This method is based upon a rigid body assumption for both the reaction mass and the baseplate. Unfortunately, as the sweep frequency goes up, this estimate for the vibrator force output is in error. In general the weighted-sum approximation overestimates the actual force output of the vibrator. Many researchers have verified this by making comparisons between the weighted-sum ground 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 both the flexure and the heavy weight of the vibrator baseplate. The vibrator baseplate weight plays a large role in achieving good vibrator performance at high frequencies. Unfortunately, this factor has long been forgotten in modern vibrator design. Actually, a lightweight and stiff baseplate assures that the force generated by the reaction mass is used to drive the earth and not expended driving the baseplate mass. Further, the lightweight and stiff baseplate will also increase the resonant frequency of the earth-vibrator, which provides additional benefit at high frequencies. To understand the baseplate dynamic motion more accurately, a vibrator baseplate finite element analysis model was developed. This finite element analysis model was employed to identify key vibrator vibration modals that influence the vibrator output energy at low and high frequencies. The main objective is to provide a useful set of guidelines for the vibrator baseplate design in order to generate high quality vibrator output energy. FINITE ELEMENT ANALYSIS MODEL The vibrator baseplate finite element analysis model is based upon an ION (previously I/O) AHV-IV vibrator and constructed using the ANSYS Structure software including both static and dynamic analysis packages. This model is shown in Fig. 1. Each part of the vibrator baseplate included in Fig. 1 was modelled based on real geometries of components on the vibrator baseplate. For example, the baseplate pad was modelled in a rectangular type of structure m in length, m in width and m in height. Due to complications in meshing and computation, the reaction mass was removed from the model. The reason for this is that the in- Figure 1 The finite element analysis model of the AHV-IV vibrator baseplate and the ground. terface between the surface area of the tie-rod tubes and the internal surface area of the reaction mass through which these tubes slide creates a need in the software to mesh this area as points of contact. This complication applies to the contact areas around the piston surface as well. Furthermore, the computation itself is complicated by the dynamic nature of these contact areas. The tubes and piston slide within the reaction mass body when the structure undergoes motion, forcing the software to dynamically adjust the contact area between the two surfaces. These impediments prevent the simulation from arriving at a convergent solution. Since the model is primarily concerned with the motion and structure of the baseplate, the removal of the reaction mass has a relatively small impact on the accuracy of the simulation s results and does not present a major concern as it is necessary for the simulation to succeed. The ground was modelled as a solid cylinder m in diameter and m in height. The model size of this ground is chosen to be big enough to contain the radiation mass or the captured ground mass. The radiation mass or the captured ground mass is defined as the ground mass that participates in the motion of the vibrator baseplate as it vibrates. The shape of the cylinder was chosen to ensure that the ground would have an even response to the force applied. In addition, absorbing boundaries were included in the form of mode damping and node-to-node damping such that the energy from the vibration would be dissipated properly. The stress and deformation

3 Modelling and modal analysis of vibrator baseplate 21 distribution in ground layers including the top and bottom surfaces are simulated as the vibrator baseplate moves up and down. Knowledge of the stress and deformation distribution in ground layers, allows the radiation energy to be visualized. In addition to the stress and deformation distribution, it is important to accurately represent the material properties of the ground. Moduli of elasticity and density are three very important parameters used to characterize these properties. With different values for these three parameters the ground such as mud, sand, concrete and so on can be modelled. In order to make apple-to-apple data comparisons between the finite element analysis model and the experimental measurement, the material of concrete was chosen for the ground in this finite element analysis simulation. This is because the density and elastic moduli of the concrete are constant while values of these three parameters for other materials such as mud and sand vary due to weather conditions. Fortunately, a vibrator testing pad made in concrete has been built at the ION Sealy testing facility. Due to space limitations it is difficult to show this concrete testing pad graphically. To determine the utility of the model in predicting the dynamics of the vibrator baseplate, a data comparison of the baseplate acceleration between the finite element analysis model and the field measurement was made at 5, 10, 20, 40, 80 and 120 Hz. The AHV-IV vibrator sitting on the concrete pad performed a set of monochromatic sweeps. Each sweep is 5 seconds long with a 0.5 s start and 0.5 s end cosine taper. Considering that the actual vibrator has the vibrator truck frame connecting to the baseplate through isolation airbags while the finite element analysis model of the vibrator baseplate does not include the vibrator truck frame, a 0.5 s long taper is used at each end of the sweep to avoid ruining the steady-state vibration by exceeding the operational limits of the vibrator at the start and end of the sweep. Meanwhile, a 5ssweeplengthisusedtoensurethatsteady-statevibrator output is achieved. The baseplate acceleration and the differential pressure across the piston in the reaction mass chamber were recorded. For the AHV-IV vibrator the baseplate accelerometer is mounted on the top cross of the baseplate stilt structure. Applying the differential pressure measured from the vibrator as an input signal to the piston in the finite element analysis model, the baseplate acceleration of the finite element analysis model is then calculated. The transient response of the baseplate acceleration during the tapered sections of the sweep is not taken into account when the data are compared for different frequencies. Figure 2 compares the baseplate acceleration trace to the finite element analysis model data at six different monochromatic frequencies. Each plot is as follows: a) the upper-left window shows the baseplate acceleration traces at 5 Hz, b) the upper-right window shows the baseplate acceleration traces at 10 Hz, c) the middle-left window shows the baseplate acceleration traces at 20 Hz, d) the middle-right window shows the baseplate acceleration traces at 40 Hz, e) the bottom-left window shows the baseplate acceleration traces at 80 Hz and f) the bottom-right window shows the baseplate acceleration traces at 120 Hz. The measured baseplate acceleration is represented using the magenta line while the finite element analysis model baseplate acceleration is the blue line. The baseplate acceleration traces shown in Fig. 2 are steady-state outputs and the transient portions of the sweep are discarded. The data comparison shows that the finite element analysis model captures the main dynamic motion-peaks of the AHV-IV vibrator baseplate up to 80 Hz. However, a clear discrepancy was observed at 40 Hz. The baseplate acceleration from the FEA model has a strong second harmonic while the experimental data show a strong third harmonic. It is unclear as to the definite source of this discrepancy. However, it may be possible that the source of this discrepancy lies in the non-idealities of a real world contact area between the baseplate and the ground. In the simulation, the ground and baseplate are taken to be perfectly flat, leading to an ideal contact area. In this situation the primary mode of motion is the bending motion and will cause strong second harmonics. However, in real world applications, the contact area between these two structures is not ideal. The baseplate pad yields to be an invisible bow-shape due to hydraulic preloads. Uneven surface of the baseplate and ground slope lead to non-uniformity in the contact area and cause additional modes of motion, which may contribute to the strong third harmonic present in the experimental data. At 120 Hz, the baseplate acceleration of the finite element analysis model leads the field measurement about 90 degrees in phase. The reason for this discrepancy is not clear although the complex modes of the stilt structure exhibited at high frequencies may be a major contributor to the differences between the simulation and experimental data. Overall, there is a good agreement between the baseplate acceleration calculated by the finite element analysis model and the measured baseplate acceleration in the 80 Hz sweep bandwidth. It is important to note that although there are some discrepancies between the simulation results and the experimental data, these discrepancies represent minor variations and the simulation successfully depicts the primary characteristics of the baseplate acceleration. For the dynamic motion study of the baseplate below 100 Hz, the utility of this finite element analysis model is valid.

4 22 Z. Wei 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 THE VIBRATOR BASEPLATE Vibration modes of the vibrator baseplate are characteristic patterns or shapes in which the coupling system of the vibrator baseplate and the ground will vibrate. Certainly the vibrator baseplate has many modes of vibration and it is the task of the finite element analysis modal analysis to determine these modal shapes. The actual vibration of the baseplate is always a combination 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 see primarily the vertical vibration mode of the baseplate but if the vibrator shakes at a higher force (drive) level, other modes are excited and we see some other motions such as rocking and flexing of the baseplate in addition to the vertical motion. Figure 3 shows six vibration modes of the baseplate dominantly below 100 Hz using the finite element modal analysis. These figures are generated to illustrate the motion and relative displacement of the baseplate and captured ground mass when the baseplate undergoes motion. Maximum displacement is represented in these figures as red while minimum displacement is shown in blue. Because the images are generated strictly through modal analysis with the intent of illustrating the modes of motion only, the scale of the displacement is not provided as the displacement magnitude will vary in real world applications. Each plot is as follows: a) the upper-left plot shows the baseplate has a front-rear rocking mode at 14.9 Hz, b) the upper-middle plot shows the baseplate has a side-to-side rocking mode at 15.1 Hz, c) the upper-right plot shows the baseplate has a twisting mode at 19 Hz, d) the bottom-left plot shows the baseplate and the ground have

5 Modelling and modal analysis of vibrator baseplate 23 Figure 3 a) Baseplate front-rear rocking, b) baseplate side-side rocking, c) baseplate twisting, d) baseplate vertical vibration, e) lateral motion of the baseplate stilt structure and f) baseplate vertical vibration and ground flapping. a vertical vibration mode at Hz, e) the bottom-middle plot shows the baseplate has a lateral motion mode of the stilt structure at 80 Hz and f) the bottom-right plot shows the baseplate has a vertical motion mode while the ground is in a flapping mode at 100 Hz. Catching natural frequencies of the vibrator baseplate is necessary for determination of the vibrator performance and operational frequency range. Based on the results of the vibrator baseplate finite element analysis simulation performed, it is necessary to do some experimental tests on the vibrator and identify these natural frequency modes. Figure 4 shows an AHV-IV vibrator baseplate on concrete. The surface of the concrete is not perfectly flat. Three pieces of hard rubber-pad are inserted under the baseplate to improve the contact between the baseplate and the concrete. The AHV-IV vibrator is equipped with a 4905 Kg reaction mass and a 1771 Kg baseplate. The baseplate area is 2.5 m 2. The concrete is approximately 5.5 m long, 3.4 m wide and 2 m deep. Its stiffness and viscosity are N/m and N-s/m, respectively. The captured ground mass is about 900 Kg and concrete only (Wei 2008b). Our experiment was divided into three phases. The first phase of the experiment was designed to measure the frontrear rocking and the side-side rocking using a 1 21 Hz, 10- s sweep and six magnetic accelerometers. Among them, two magnetic accelerometers were positioned vertically in a diagonal on the top cross of the baseplate. Another two magnetic accelerometers were attached on the front and rear middle of the top cross horizontally pointing forward and rear, respectively. The last two were attached on the right and left side middle of the top cross horizontally pointing right and left, respectively. Figure 5 shows four acceleration traces recorded by magnetic accelerometers on the top cross of the baseplate. Trace 1 shown in red was recorded by the front accelerometer measuring the front-rear rocking motion. Since the data measured by the front accelerometer and the rear accelerometer were identical, only the front accelerometer data were plotted. Trace 2 depicted in blue was measured by the right-side accelerometer showing the side-to-side rocking. For the same reason only the right-side accelerometer data were shown. Trace 3 and Trace 4 are shown in brown and magenta, respectively. They were measured by two vertical accelerometers. It is obvious that they are identical. The reason for showing these vertical accelerations together is to demonstrate that there is no unexpected bending motion on the top cross. As clearly illustrated in Fig. 5, the amplitudes of trace 1 and trace 2 get larger between 4 and 5 seconds when the frequency is in the range of 9 11 Hz. These indicate that the top cross of the baseplate experiences a front-rear rocking motion and a side-to-side rocking motion approximately at 10 Hz. Compared with the finite element analysis simulation results for front-rear rocking and

6 24 Z. Wei Figure 4 AHV-IV vibrator locates on concrete. Figure 5 Measured baseplate accelerations on the top cross for the 1 21 Hz, 10-s sweep. side-to-side rocking, the field measurements are about 5 Hz lower. This is because the modal analysis of finite element analysis simulation treats the coupling of the baseplate and the ground as a bonded condition while in reality the baseplate is not completely coupled with the ground. This bonded condition in the finite element analysis model enhanced the stiffness of the baseplate so that the frequencies of the rocking motions performed from the finite element analysis simulation were higher than the actual measurements. In the second phase of the experiment, a 20-s long sweep from 5 to 105 Hz was used to measure the vertical vibration mode of the baseplate and the ground. This vertical vibration mode maximally represents the interaction between the vibrator baseplate and the ground. Generally, this interaction is simplistically described by a second-order system and recorded by the reaction mass accelerometer and the baseplate accelerometer. To clarify the data measured for this experiment, it should be noted that the reaction mass acceleration and the baseplate acceleration were taken using the Pelton reaction mass and baseplate accelerometers. They were mounted on the top of the reaction mass and the top cross of the baseplate, respectively. Figure 6 shows a frequency response plotted using the reaction mass acceleration (input) and the baseplate acceleration (output). In Fig. 6 the magnitude ratio plot shows that the amplitude curve starts with a slope of approximately 40 db/dec. Then, at 65 Hz the slope of the curve starts changing and a peak is formed. The phase plot illustrates that the phase curve starts at approximately zero degrees. At 65 Hz the phase curve starts shifting to 180 degrees. The frequency at the peak in the magnitude plot and the frequency at the turning point in the phase plot are very close to each other. This means that frequency is the resonant frequency or natural frequency of the vertical vibration of the baseplate and the ground. This measured frequency nearly matches the frequency of Hz shown by plot d in Fig. 3.

7 Modelling and modal analysis of vibrator baseplate 25 Figure 6 Frequency response of the reaction mass-baseplate/ground model. a) Magnitude ratio plot and b) phase plot. In the last phase of the experiment, we assessed the motion of the upper stilt structure itself. Figure 7 shows a comparison between four magnetic accelerometers placed on the top cross of the baseplate. A frequency response was then taken between each of these magnetic accelerometers and the Pelton top-cross accelerometer as a reference. The black and blue traces represent the measure lateral motion in-line with the front-back direction of the vibrator. The red and green traces correspond to lateral motion in the side-to-side direction of the vibrator. As can be seen from Fig. 7, the stilt structure obviously experiences a significant amount of lateral motion as the frequency response of these with respect to the Pelton top-cross vertical accelerometer is relatively high. These four lateral-motion accelerometers experience equivalent magnitude of motion at a frequency of 80 Hz. At this high-frequency range, it is not possible to distinguish the motion with the eye. However, it may be correlated to a higher-order mode of motion like plot e depictedinfig.3. It is very difficult to measure the baseplate twisting and the baseplate vertical vibration-ground flapping shown in Figs. 3(c) and 3(f), respectively. When the vibrator was relocated on the ground of soft soil, rotating prints were found beneath the baseplate pad. This implied that the baseplate experienced twisting motions. The rotation angle of prints was very small. Figure 7 Assessment of the motion of the upper stilt structure using four accelerometers placed on the top cross of the baseplate. a) Magnitude ratio plot and b) phase plot.

8 26 Z. Wei For the movement represented by plot f in Fig. 3 it requires our imagination to picture this mode. As the vibrator baseplate moves up-down, the surface of ground flaps in and out. VIBRATOR BASEPLATE FLEXURE AND WEIGHTED-SUM GROUND FORCE The vibrator baseplate functionally acts like a transmitter, transmitting a hydraulic force into the earth. Due to low rigidity, the baseplate is subject to flexural vibrations at higher frequencies. The modes of the flexural vibration are highly diversified depending on ground conditions. When the baseplate is on very uneven surfaces, the flexural vibration can be described as the baseplate jumping up-down. Often times full or partial decoupling occurs leading to poor vibrator performance and a reduction of useable transmitted signal energy. When the baseplate is on flat surfaces, the flexural vibration can be described as a bird flapping its wings. Under this condition, partial decoupling occurs but the decoupling area is small. Figure 8 demonstrates the flexural vibration of the baseplate in the bird flapping mode at 80 Hz using the finite element analysis simulation. The plots in Fig. 8 are exaggerated to make a clear demonstration on the motion and relative displacement of the baseplate when the baseplate undergoes motion. Maximum displacement is represented in these graphs as red while minimum displacement is shown in blue. The scale of the displacement is not provided as the mesh size applied to the modelled ground leads to a situation in which the displacement of particular areas of the baseplate might not be perfectly accurate. This larger mesh size was necessary because using a small mesh size to obtain a detailed look at the ground response prevented the software from successfully completing a dynamic simulation. In addition, the figure is intended to depict the general type of motion that is occurring rather than to quantify specific displacement magnitudes. The plot of Fig. 8(a) shows that the baseplate is pushed downwards. It can be seen that the hydraulic force is applied on central areas and causes high stress and big deformation. The stress and deformation quickly fades out from the central area to the side area. The plot also clearly demonstrates that the portions of the baseplate at the four corners move upwards, opposite to the downward motion of the central area of the baseplate. The plot of Fig. 8(b) illustrates that the baseplate takes the shape of an arch as it is pulled upwards. When this happens a decoupling occurs in the central area of the plate. The portions of the baseplate at the four corners are moving in a downward direction opposite to the motion of the central area. From Fig. 8, the conclusion can be drawn that different locations within the baseplate generally produce different motions. Placing accelerometers in different places on the baseplate will measure different accelerations. It is very difficult to use only one baseplate accelerometer to determine the exact vertical acceleration that the baseplate is in during motion. It is difficult as well to determine how much baseplate mass 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 the baseplate (Sallas 1984; Safar 1984; Baeten and Ziolkowski 1990). The weighted-sum method is built on the rigid body assumption of the baseplate. As demonstrated above, the baseplate is subject to flexural vibrations when the vibrator shakes at higher frequencies. Different locations within the baseplate have different motions causing the acceleration recorded by the single baseplate accelerometer to be an invalid Figure 8 Vibrator baseplate bending and flexing. a) Vibrator in compressing and b) vibrator in releasing.

9 Modelling and modal analysis of vibrator baseplate 27 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 validity of the weighted-sum ground force becomes questionable. Figure 9(a) shows an example with the AHV-IV vibrator located on load cells. The load cell is a direct force measurement sensor and is very rigid enabling it to measure the true ground force. There are a total of 8 load cells under the baseplate. The load cell is mounted on the concrete. On the top of each load cell there are two pieces of rubber pad to protect the load cells from 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 the baseplate pad. A, D, J and K are all six inches from the corners in the X and Y directions. B, C, H and I are near the leg tube positions. E and G are on the centreline midway between the edge and stilt tubes. F is at the centre of the pad. A total of 6 magnetic accelerometers are used to measure the motion of the 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 are very similar. Thus, this document uses the driver side for its analysis. An experiment was designed using a Hz 20-s sweep to compare the weighted-sum ground forces and the true ground force. The weighted-sum ground forces were calculated by using different baseplate accelerations recorded with magnetic accelerometers on different locations and the true ground force was measured by load-cell sensors. Figure 10 shows the magnitude ratio and phase spectra (frequencyresponse or bode plots) of the weighted-sum ground forces, compared to the ground force measured by the load-cell

10 28 Z. Wei sensors. The spectra were calculated by using the weightedsum ground forces as the output and the actual ground force measured by load cells as the input. In theory, if the weightedsum and measured ground forces are equal, the magnitude ratio spectrum is 0 db and the phase spectrum remains at zero degrees. It should be noted that none of weighted-sum ground forces are in agreement with the true ground force in the 200 Hz frequency range. The broken line named FBK is the weighted-sum ground force calculated from Pelton accelerometers against the true ground force. The Pelton baseplate accelerometer is mounted on the top cross of the baseplate stilt structure. The positions of F and H are totally abnormal. The weighted-sum ground forces calculated from these positions are completely different than the true ground force. This phenomenon indicates that the central area on the baseplate is decoupling a substantial amount of the time. Conversely, the weighted-sum ground forces calculated from positions A, B, E and J agree with the true ground force in the frequency range below 50 Hz. For frequencies above 50 Hz, the measurements have shown that the weighted-sum approximation is not representative of the vibrator output. This is also true for FBK that the baseplate accelerometer is located on the top cross of the stilt structure. This has been confirmed as well by Sallas et al. (1985) and Allen, Johnson and May (1998). Traditionally, most of these discrepancies are attributed to the baseplate flexural vibration. However, the complexity is further strengthened because the relationship between the weightedsum ground force and the true ground force is variable due to different ground coupling conditions. In practice the ground force is not known exactly and varies from place to place. In general the weighted-sum approximation overestimates the actual force output of the vibrator. The direct force measurement such as the load cell is needed for real ground force. Unfortunately, the load cell is not yet robust enough for routine use. non-linearity of the ground itself, the low rigidity of the vibrator baseplate also produces a significant amount of harmonic distortion in the ground force. In addition, the low rigidity of the baseplate ensures that in practice the ground stress and deformation are never uniform. Rather than a uniform distribution, the low rigidity causes an uneven distribution of concentrated areas of ground stress and deformation. In the case of an ION AHV-IV vibrator, high ground stress and deformation are concentrated at the area of the ground directly beneath the centre baseplate area defined by the four tie-rod legs. To the left and right sides of this centre area, the ground stress and deformation are drastically reduced. To study the impact of the baseplate rigidity on the ground force harmonic distortion it was necessary to develop a more realistic, complicated model of the interaction between the baseplate and the captured ground mass. Figure 11 depicts a complicated vibrator-ground model in which an emphasis is put on expressing the rigidity of the baseplate as part of the non-ideal contact stiffness present at the boundary interaction of the two surfaces. This proposed model can serve as a more realistic representation of the vibrator-ground interaction. This model has been confirmed that it is capable of describing a wide range of non-linear contact behaviour such as a partial contact and a full contact (Lebedev and Beresnev 2004). In Fig. 11, the ground model is described as a linear and second-order system that consists of a ground mass, ground stiffness and ground viscosity. The BASEPLATE RIGIDITY AND GROUND FORCE HARMONICS The true ground force is defined as the total compressive force exerted by the earth upon the vibrator baseplate. The ground force is the contact force at the interface of the earth and the vibrator baseplate and can be calculated by the compressive stress integrated over the surface area that contacts the baseplate. In reality, the ground force suffers severe harmonic distortion. Besides harmonic distortion caused by nonlinearities exiting in the hydraulic servo-valve system and the Figure 11 Example of a complicated vibrator-ground model.

11 Modelling and modal analysis of vibrator baseplate 29 vibrator system is also treated as a linear and rigid body. In this model, the baseplate is considered to have a mass only and its stiffness is distributed to become a part of the contact stiffness. The highlight in this model is the contact stiffness located in-between the vibrator baseplate and the ground. The contact stiffness here is defined as a group of springs connecting the vibrator baseplate and the ground and its value depends on the number of springs that physically connect the baseplate and the ground during vibrator vibration. Therefore, the contact stiffness is a variable stiffness. As we know, partial decoupling often occurs as the vibrator shakes at high frequencies due to the low rigidity of the vibrator baseplate and it becomes even worse on uneven ground. When the vibrator is in compressing mode, there are more contact areas between the vibrator baseplate and the ground. More contact areas mean more springs and more stiffness. As the vibrator goes to releasing mode, very often partial decoupling happens. This means that the baseplate looses some contact with the ground so that the contact stiffness is reduced. Generally speaking, the contact stiffness is reduced halfway through the compression cycle until halfway through the release cycle and its value also decreases as the sweep frequency increases. When the vibrator is located on uneven ground, the vibrator baseplate is subject to many motions such as the baseplate bending, flexing and twisting so that the contact stiffness becomes unpredictable and the harmonic distortion is severe. Two simulation tests were designed using monochromatic frequency sweeps and the finite element analysis model to calculate the true ground force and the ground deformation on the top surface of the ground. The finite element analysis model was shown earlier in Fig. 1. This model is purposely designed as a linear system except for the contact area between the baseplate and the ground. The contact area can be described as a flat, even contact and considered to be well coupled. However, the model allows decoupling or partial decoupling to happen if any local force generated by the hydraulic force exceeds any local hold-down force produced by the total hold-down force. The model also allows the baseplate to bend and flex if the local vibration force exceeds the local material strength. A sine-wave force was chosen as the hydraulic force to avoid the harmonic distortion created by the non-linearity in the hydraulic servo-valve system. This force is applied on the vibrator piston at a level of N ( lbs). For the AHV-IV vibrator, this is equivalent to a differential pressure of 1000 psi across the piston. Figure 12 demonstrates a data comparison of the sine-wave hydraulic force and the ground force calculated by the finite element analysis model in two cycles at 10 Hz. In Fig. 12 Figure 12 Comparison of the sine-wave hydraulic force and the finite element analysis model ground force at 10 Hz. the sine-wave force is shown by a red line while the ground force computed from the finite element analysis simulation is shown by a blue line. The sine-wave force pushes and pulls the vibrator piston in positive and negative cycles, respectively. Then, through the stilt structure to which the piston is connected, this sine-wave force is passed to the baseplate pad and drives the baseplate and coupled ground to move up and down. Eventually, the sine-wave force is transmitted into the ground. When the force acts on the ground, the ground is stressed and deformed. By integrating the average ground stress and the stress distributed area the finite element analysis model ground force can be obtained. Ideally, the finite element analysis model ground force should be identical to the sine-wave force. However, in Fig. 12 the finite element analysis model ground force is obviously not a pure sinewave. It has been distorted and contains a small portion of even harmonics although the fundamental content of the finite element analysis model ground force is very close to the sinewave force. In addition, approximately 3% of the sine-wave force is exhausted in driving the baseplate. Figure 13 shows the average deformation of the ground. It can be seen that the average deformation of the ground is not a pure sine-wave either and is severely distorted as well. It contains a certain amount of the second and third harmonics. As mentioned earlier, the only non-linearity present in the finite element analysis model is the contact area in-between the vibrator baseplate and the ground. The contact stiffness in the compressing halfcycle is different from the contact stiffness in the releasing half-cycle. This difference causes even harmonics in the finite element analysis model ground force and the average deformation of the ground. Even harmonics in the finite element

12 30 Z. Wei Figure 15 Average ground deformation calculated from finite element analysis model at 80 Hz. Figure 13 Average ground deformation calculated from finite element analysis model at 10 Hz. analysis model ground force further cause additional harmonics in the average deformation of the ground. However, at 10 Hz the contact stiffness does not vary much in compressing and releasing half-cycles of vibration because the vibrator baseplate is relatively stiff at low frequencies. This is the main reason why we do not see significant harmonic distortion in the finite element analysis model ground force and the ground deformation. Another simulation example uses an 80 Hz sine-wave force as input. Figure 14 shows a wiggle-trace comparison of the sine-wave force and the FEA model ground force in two cycles made at 80 Hz. The red curve represents the sine-wave force and the blue curve represents the finite element analysis model ground force. The sine-wave force keeps the same drive level as the 10-Hz sine-wave force does. The finite element analysis Figure 14 Comparison of the sine-wave hydraulic force and the finite element analysis model ground force at 80 Hz. model ground force is severely corrupted and contains a large amount of harmonic distortion. Comparing the peaks of the sine-wave force and the finite element analysis model ground force, the finite element analysis model ground force equals to approximately 74% of the sine-wave force. This means that roughly 26% of the sine-wave force is used to accelerate and bend the baseplate. Figure 14 also shows that in the finite element analysis model ground force there is another 25% of the ground force spent in generating harmonics. The remainder of the finite element analysis model ground force produces the fundamental force. There is a large amount of the sinewave force exhausted in unrelated baseplate activities due to low rigidity of the baseplate and the weight of the baseplate. Figure 15 demonstrates that the ground deformation is also severely contaminated by harmonic distortion. This is due to harmonic distortion present in the ground force. The average deformation of the ground shown in Fig. 15 shows a slight phase lag compared to the finite element analysis model ground force in Fig. 14. Meanwhile, the behaviour in the dotted circle in Fig. 15 is very distorted. The reason for this is unclear. One possible explanation is that the contact between the baseplate and the ground is set as a rough contact. This means that the contact is treated as a non-linear contact by the ANSYS program. Comparing the finite element analysis model ground forces presented in Figs. 12 and 14, more fundamental force is produced at 10 Hz than at 80 Hz. Meantime, the harmonic distortion is less at 10 Hz than at 80 Hz. There is also more ground deformation and less harmonic distortion yields at 10 Hz than at 80 Hz. These are due to the contact stiffness varying much more significantly at high frequencies than low frequencies.

13 Modelling and modal analysis of vibrator baseplate 31 CONCLUSIONS A vibrator finite element analysis model was developed and it has been proven to be very useful for studying the vibrator baseplate and the coupled ground. Through finite element analysis simulation, some dynamic motions of the baseplate stilt structure have been realized and quantified. The baseplate pad suffers more bending and flexing motions at high frequencies than low frequencies. Model data also demonstrate that the true ground force and the ground deformation are distorted by harmonics due to the variable contact stiffness between the baseplate and the ground. Although the simulation of the baseplate acceleration does exhibit some discrepancies at 40 Hz and 120 Hz when compared to the experimental data, the model successfully characterizes the primary aspects of the baseplate acceleration. Further investigations will be pursued in an attempt to discover the source of the discrepancies. The variable contact stiffness is attributed to low rigidity of the vibrator baseplate and the surface unevenness of the ground. Improving the vibrator baseplate stiffness and coupling should be considered as a primary variable during vibrator design. The limiting factor in this endeavour is then the increased costs associated with achieving very high stiffness while maintaining a very low baseplate mass. REFERENCES Allen K.P., Johnson M.L. and May J.S High fidelity vibratory seismic (HFVS) method for acquiring seismic data. 68 th SEG meeting, New Orleans, Louisiana, USA, Expanded Abstracts, Baeten G. and Strijbos F Wave field of a vibrator on a layered half-space: Theory and practice. 58 th SEG meeting, Anaheim, California, USA, Expanded Abstracts, Baeten G. and Ziolkowski A The Vibroseis Source. Elsevier Press. ISBN Bell D.W Acquisition and utilization broadband signals containing 2 8 Hz reflection energy. 55 th SEG meeting, Houston, Texas, USA, Expanded Abstracts, Lebedev A.V. and Beresnev I.A Nonlinear distortion of signals radiated by vibroseis sources. Geophysics 69, Safar M.H On the determination of the downgoing P-waves radiated by the vertical seismic vibrator. Geophysical Prospecting 32, Sallas J.J Seismic vibrator control and the downgoing P-wave. Geophysics 49, Sallas J., Amiot E. and Alvi H Ground force control of a P-wave vibrator. SEG Seismic Field Techniques Workshop, August 1985, Monterrey, California. Wei Z. 2008a. Pushing the vibrator envelope: extending low and high frequency limits. First Break 26, Wei Z. 2008b. Estimation of ground stiffness, ground viscosity and captured ground mass using vibrator field measurements. 70 th EAGE meeting, Rome, Italy, Expanded Abstracts.