round motion through geophones and MEMS accelerometers: sensor comparison in theory modeling and field data Michael Hons* Robert Stewart Don Lawton and Malcolm Bertram CREWES ProjectUniversity of Calgary Summary Digital sensors based on micro electro mechanical systems (MEMS) accelerometers are one of the newest technologies being used in seismic acquisition. s such some confusion remains surrounding similarities and differences relative to the coil-over-magnet geophone. n understanding of the functioning of these sensors and how to compare them can be facilitated by deriving transfer functions which relate the data acquired through each sensor to actual ground motion. n equation is then derived to calculate acceleration comparable to unprocessed MEMS data from unprocessed geophone data. The inverse of this equation may be used to calculate geophone data from MEMS data. The effects of sensors on zero and minimum phase wavelets are modeled demonstrating that the raw output from the sensors should be similar. The minimum phase wavelets are convolved with a random reflectivity series to test deconvolution of impulsive source data. Deconvolution produces geophone and MEMS processed traces that appear similar and constant phase rotation of MEMS data after deconvolution cannot correct all remaining differences. The geophone-to-mems transfer equation will exactly transfer between sensors only in the absence of instrument noise. Comparisons between MEMS and geophones recording the same shots and ground motion domains calculated from those records show that the data is very similar in frequency content when the same domain is considered and MEMS records will not necessarily have a larger magnitude contribution from low frequencies than geophones. Introduction Seismic data from a geophone is voltage induced from the velocity of the magnet relative to the coil. In the time domain it does not identically represent displacement velocity or acceleration of the ground. However through transfer functions derived from the simple harmonic oscillator equation it can be corrected in phase and amplitude to represent any domain of ground motion desired. Seismic data from MEMS accelerometers is a forcefeedback voltage directly related to the displacement of the proof mass detected by a capacitor (Maxwell et al. 1). It is also directly related to the acceleration of the ground through its sensitivity constant (expressed in /g). This is because its very high resonant frequency allows a simplification of the full transfer function and the resulting amplitude and phase spectra are essentially flat. This flat response has been considered an advantage for MEMS sensors but in the absence of instrument noise geophone data can be amplitude and phase corrected with a transfer function to achieve a similar end. lthough when the signal has been acquired below the geophone s noise floor this recovery is challenged. To compare MEMS and geophones in a consistent way the same domain should be considered. Transfer functions derived from the simple harmonic oscillator equation allow any domain to be calculated from any data (Havskov and lguacil 6). Theory The motion of the ground and the motion of the proof mass within a seismic sensor are related by the simple harmonic oscillator equation: x x u + λω + ω x= where u is the displacement of the ground x is the displacement of the proof mass ω is the resonant frequency and λ is the damping factor. To represent the transfer characteristics this is rearranged to a transfer function of the form: B ( ω) = H ( ω) ( ω) where B is the output is the input and H is the transfer function. In a geophone voltage B is given by dx/dt (proof mass velocity) multiplied by sensitivity in s/m. In an accelerometer voltage B is given by x (proof mass displacement) multiplied by sensitivity in /m. This voltage is the recorded data and does not change when a different input domain is considered. Taking the Fourier transform of the simple harmonic oscillator equation allows us to replace time derivatives with iω. Rearranging as required for a geophone gives: X iω U ω + iλωω + ω where is the data from the geophone and S is the sensitivity. The amplitude and phase response are shown in Figure 1. One can find results relating to ground velocity and displacement simply by multiplying by iω replacing one d/dt on the right hand side. The velocity result is referred to as the geophone equation and has historically
round motion through geophones and MEMS accelerometers been of use because it shows that correcting the phase of the geophone-acquired data to zero gives a high-pass version of ground velocity (Figure ). There is no physical reason however that a geophone must acquire only velocity and not acceleration or displacement. Rearranging for a MEMS accelerometer gives: 1 U X ω + iλω ω+ ω where is the data from the accelerometer and S is the sensitivity in units of /m. gain equations for ground velocity and displacement can be found by multiplying by (iω) n where n is the difference in domains. In seismic exploration the resonant frequencies of MEMS devices are very high compared to the seismic signal band (>1 khz) so the above equation can be reduced to: g 1 U S U X = ω 9. 81 where S g is sensitivity in /g which is how the sensitivity of these devices is most commonly reported. If the goal is then to convert data from a geophone into data from a MEMS sensor with amplitudes intact the transfer function can be written as: g S = 9.81S λωω i( ω ω ) noting that the simplified was used so ω and λ are parameters of the geophone. This is essentially the geophone acceleration equation inverted multiplied by a scaling factor. The inverse of the final result can of course be used to transform MEMS data into geophone data. ω Modeling We now have a method to calculate acceleration from geophone output or match amplitudes with MEMS output (providing other gains in the recording system are accounted for). Why is this valuable? This can be seen in the shape of the geophone acceleration transfer function (Figure 1): both high and low frequencies must be boosted to correct geophone data to acceleration. If an embedded wavelet exists its acceleration shape will be narrower than the shape in velocity or displacement domain. The effects of a geophone and MEMS accelerometer on an input acceleration both zero and minimum phase are shown in Figures 3 and 4. ll traces have been normalized. The output from a MEMS closely overlies the acceleration. Output from a geophone resembles MEMS output. Recall that acceleration can be recovered from geophone data so differences between geophones and MEMS should also exist between raw geophone data and acceleration calculated from the geophone. However the embedded wavelet looks the purpose of deconvolution is to remove it and return a reflectivity estimate. Figure 5 shows results of several minimum phase wavelets convolved with a ms random reflectivity series and processed with a simple 4 ms spiking deconvolution. The integrated acceleration trace was integrated after convolution (i.e. after acquisition) and before decon. Integration is only physically meaningful before deconvolution as afterwards the ground motionrelated wavelet has been removed. ll synthetics perform Figure 1. Example amplitude and phase spectra for a 1 Hz.7 λ geophone relative to ground acceleration. Figure 3. Zero phase wavelets: through a geophone (red) and MEMS (green). Based on 5 Hz Ricker displacement wavelet (gray). Figure. Example amplitude and phase spectra for a 1 Hz.7 λ geophone relative to ground velocity. Figure 4. Minimum phase wavelets: through a geophone (red) and MEMS (green). Based on 5 Hz displacement wavelet (gray).
round motion through geophones and MEMS accelerometers well at recovering the larger impedance contrasts but the smaller detail is best matched by the acceleration synthetic. This analysis requires that the acceleration be recovered without overbearing noise in the additional bandwidth. Figure 6 shows the deconvolved acceleration trace phase rotated by -9-45 and degrees and compared to the deconvolved geophone trace. Each matches with the geophone data in some areas but none matches overall. If a similar output from deconvolution is the goal (for example if a MEMS line must be tied in to a geophone project) it is sensible to calculate directly comparable data prior to deconvolution. Field Data The following data was acquired in December 5 near iolet rove lberta Canada in the Pembina oil field. Four sensors (three 3C geophones and one Sercel DSU3 MEMS) were simultaneously laid out at 8 stations with a separation of ~1 m from each other and a m receiver spacing. The ground was solidly frozen when the sensors were laid out and warm water was used to soften the earth so the sensors could be planted. The sensors then froze into the earth after planting so in all cases coupling was excellent. total of 5 dynamite shots were recorded. The vertical components of the sensors showed exceptional similarity between geophones and MEMS. This provides an excellent test case to observe differences between geophone and MEMS data and whether acceleration data can be accurately recovered from geophones. cceleration was calculated from one geophone record and is compared to the MEMS data and the original geophone data in Figure 7. part from a small time delay likely related to different anti-alias filters between the geophone and MEMS calculated acceleration closely resembles MEMS data. High-frequency signals acquired by MEMS and geophones are compared in Figure 8; more coherent events in the MEMS record are not apparent. Low frequencies were compared by examination of amplitude spectra (Figure 9). verage spectra were extracted from 5 records: geophone (red) geophone corrected to velocity (black) geophone corrected to acceleration (yellow) MEMS (green) and MEMS corrected to velocity (blue). The results show the transfer functions perform well over the band 1-1 Hz where the two velocity traces (black and blue) and the two acceleration traces (green and yellow) overlap. The similarity in magnitude and character suggest the calculated data is valid and acceleration data of similar quality to MEMS data can be obtained from geophone records. closeup of the low frequency range is shown in Figure 1. The low frequencies are significantly larger relative to the dominant frequency in velocity spectra than acceleration spectra. Even the raw geophone has a larger contribution than acceleration down to ~5 Hz. This makes sense because the correction to acceleration boosted frequencies away from 1 Hz. s a result the dominant frequency being a similar number of octaves from 1 Hz was boosted a similar amount as 5 Hz. Indeed velocity amplitudes are reduced by a factor of ω relative to acceleration so in velocity the dominant frequency is reduced relative to acceleration emphasizing low frequencies. In cases where low frequencies are paramount such as impedance inversion (Bell 1986; Martin and Stewart 1994) using calculated velocity or displacement as an attribute may be useful. Overall bandwidth was estimated by creating zero phase wavelets from the amplitude spectra. Broader bandwidth should result in smaller sidelobes relative to the peak and a narrow appearance in general (Martin and Stewart 1994). Figure 11 shows acceleration calculated from the geophone is the narrowest with smallest sidelobes followed by MEMS velocity calculated from the geophone velocity calculated from the MEMS and the raw geophone is the broadest wavelet. Conclusions This paper has outlined how geophones and MEMS accelerometers relate and what can be expected from MEMS data relative to geophone data: In time domain geophone data is not a direct representation of ground motion; in amplitudes only it represents a high-pass version of velocity MEMS data is directly representative of ground acceleration MEMS data can be calculated from geophone data and vice versa and quality of low-frequencies appears to be similar though more study is required Deconvolved geophone and MEMS data appear similar but not exactly the same cceleration data can be expected to give the narrowest wavelets and best resolution elocity and displacement will greatly emphasize low frequencies relative to the dominant frequency and may be of use in some applications. cknowledgements The authors wish to thank the sponsors of the CREWES Project for their continuing support of research in advanced seismic exploration methods.
round motion through geophones and MEMS accelerometers Figure 5. Results of 4 ms spking deconvolution. Reflectivity series (red) (for comparison) geophone wavelet trace (yellow) velocity wavelet trace (green) acceleration wavelet trace (purple) acceleration integrated prior to deonvolution (pink). Figure 9. verage amplitude spectra: geophone record (red) velocity calculated from geophone (black) acceleration calculated from geophone (yellow) MEMS record (green) velocity calculated from MEMS (blue). Figure 6. eophone trace deconvolved (red) red phase rotated -45 degrees (yellow) red phase rotated -9 degrees (green) compared to acceleration trace (purple). Figure 1. verage amplitude spectra from Figure 1: close-up of -5 Hz. Figure 7. Field data: Red are geophone traces orange are acceleration traces calculated from geophone and blue are MEMS traces. Figure 11. Zero phase wavelets calculated from average amplitude spectra. Figure 8. Highpass (>35 Hz) geophone (left) and MEMS (right).
1 3 4 Bell D.W. 1986 Low seismic frequencies: acquisition and utilization of broadband signals containing -8 Hz reflection energy 56 th nnual International Meeting SE Expanded bstracts 443-446. Havskov J. and. lguacil 4 Instrumentation in Earthquake Seismology Springer. Martin N. and Stewart R. R. 1994 The effect of low frequencies on seismic analysis CREWES Research Report 6 :1-:8 Maxwell P. J. Tessman and B. Reichert 1 Design through to production of a MEMS digital accelerometer for seismic acquisition First Break 19 141-144.