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High precision geophone calibration.

Abstract

Geophones are commonly used in seismic signal detection and are widely sold with response parameters with specified tolerances. High precision seismic analysis can extract subtle information from seismic records, but requires knowledge of the in-situ response parameters of the sensors. The geophone response is specified at room temperature; however, geophone response changes with the orientation with respect to gravity, temperature and aging. The changes due to temperature and tilt may exceed the tolerance limits, so there is a need to perform in-situ calibration to know the exact response. In the impedance calibration method, the impedance of a geophone can be measured by injecting current. From the impedance, it is possible to describe all of the response parameters except for the moving mass. The dynamic method is also known for calibration of a vibration sensor relative to a reference sensor. The reference sensor can be fabricated with a pre-measured moving-coil mass so that all the geophone parameters can be determined. Alternatively the reference sensor can be calibrated by using a reciprocity method making use of three vibration sensors. We have developed unique geophones that output signals proportional to acceleration. Complete calibration schemes have been established, and all the geophones are fully calibrated during the manufacturing process. A method has been described to re-evaluate the geophone's moving mass to make in-situ calibration in the working environment with the impedance method.

Keywords: Geophone, Seismic sensors, Moving-coil accelerometer, Calibration, Impedance method

1. Introduction

A geophone consists of a pair of moving coils suspended in a magnetic field by means of a pair of springs as shown in Fig. 1. The spring is usually pre-stressed to compensate the natural displacement of the coil due to gravitational force so that the coil is centered in the magnetic field.

The output of a geophone is usually terminated by a shunt resistance R as shown in Fig. 2 to provide external damping. The current flowing in the moving coil reduces coil motion. The amount of shunt resistance is chosen so that the total damping factor is 70%.

The equation of motion for the moving coil relative to the magnetic flux for given external displacement u may be written as;

m [d.sup.2][xi]/[dt.sup.2] + c d[xi]/dt + k [xi] = m [[d.sup.2] u/[dt.sup.2] - Bli (1)

where [xi]: coil displacement; k: spring constant; m: moving mass of coil; c: friction coefficient; g: gravitational acceleration; B: magnetic flux density; l: length of coil wire; i: current; u: external displacement.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[xi] is the relative position of the moving coil inside the geophone. The first term in the left hand side of equation (1) is the inertial force, the second term is the friction force proportional to the velocity, and the last term is the spring force. The summation of the three forces balances with the force due to external displacement and the damping force caused by the electric current.

The electric signal generated by the moving coil is expressed in terms of magnetic flux density, length of coil wire and velocity of the coil as;

[e.sub.g[ = bl d [xi]/dt (2)

For a given external displacement, a sin([omega]t], the response may be expressed as;

[e.sub.g] A sin ([omega]t-[theta] (3)

The amplitude response and phase may be found from equations (1), (2) and (3) as;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4, 5)

where

[[omega].sub.0] = [square root] of k/m], [zeta] = [[zeta].sub.0] + [S.sup.2.sub.0]/2(r + R), [[omega].sub.0], [[zeta].sub.0] = c/2m[[omega].sub.0], and [S.sub.0] = Bl.

The geophone parameters are the natural frequency [f.sub.0], the open circuit sensitivity S0, the open circuit damping, the coil resistance r, and the moving mass m. The response of a geophone is determined with four geophone parameters and the shunt resistance R.

For [omega]> [[omega].sub.0], equation (3) may be approximated as [e.sub.g] = [S.sub.0] (a [omega]) cos([omega]t). The sensitivity is proportional to the velocity (a [omega]) of the vibration at above the natural frequency. Figure 3 shows the amplitude response [e.sub.g[/[S.sub.0](a [omega]) and phase response of a geophone with 10 Hz natural frequency and 0.7 damping.

[FIGURE 3 OMITTED]

2. Geophone Response in Measurement Environment

Coil resistance is almost double at 200[degrees]C and open circuit damping is reduced by 20% as shown in Figs. 4 and 5. Natural frequency [f.sub.0], and the open circuit sensitivity S0 change by a few percent each. (Figs. 6 and 7)

The geophone parameters are measured when the geophone is positioned vertically. In practice, geophones are planted by stamping into the ground by foot, and so are not necessarily vertical. In borehole seismic acquisition, a downhole tool with geophones is deployed in a borehole that may not be vertical. Figures 8, 9 and 10 show measured results of a geophone under tilt. There is a few percentage points change observed at [+ or -]30[degrees] tilt.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

3. Geophone Accelerometer

We have developed a unique geophone that responds to acceleration between 1/10 x [f.sub.0] and 10 x [f.sub.0]. A light moving coil is suspended in a strong magnetic flux density. The imaginary short circuit of an operational amplifier applies a large damping current so that the geophone responds to acceleration at near its natural frequency. See the schematic diagram in Fig. 11. For ??= ??0, equation (3) may be approximated as

[e.sub.g] (a[[omega].sup.2]) [S.sub.0]/2[zeta][[omega].sub.2] sin (a[omega). (6)

Equation (6) shows that a response is in proportion to acceleration. In general, equation (3) is rewritten as acceleration form (7);

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

When the damping factor is large, the damping term in equation (7) becomes dominant and the frequency range becomes wider. The natural frequency is chosen to be in the middle of the seismic band at about 20 Hz. For this natural frequency, the natural displacement of the coil due to gravity is not large and the GAC design is omni-tiltable. (See Fig. 12.)

The GAC was originally introduced for borehole seismic acquisition to obsolete the need for a gimbal-mount mechanism. A second generation GAC sensor is now bring used in seabed and land seismic recording.

4. Geophone Calibration Method

4.1 Impedance Method

A current runs into the moving coil that is suspended in a magnetic flux B. The force acting to the moving coil is Bli, where B is the magnetic flux density, l is the effective wire length of the moving coil and i is the current. The equation of motion for the moving coil may be written as;

m [d.sup.2][xi]/[dt.sup.2]+cd[xi]/dt+k[xi] = Bli (8)

Since the output signal is (see Fig. 13);

[e.sub.0] = [e.sub.g] + ri (9)

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

Equation (8) may be rewritten in terms of output signal as;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The impedance may be found from the Laplace transformation of equation (10) as

Z(j[omega])=r+[S.sub.0] j([omega]/[[omega].sub.0}/m[[omega].sub.0] {1- ([omega]/[[omega].sub.0].sup.2]}+2j[[xi].sub.0]([omega]/[[omega].sub.0] (11)

The real and imaginary parts of the impedance equation show a circle on a complex plane as shown in Fig. 14. At [omega]= [[omega].sub.0] and [omega]= 0, the imaginary part is zero. At [omega]= 0, the real part is the DC resistance of the coil.

The absolute value of equation (11) yields an expression for the impedance circle as;

[a.sup.2] = [{Re(Z)-c}.sup.2] + [{Im(Z)}.sup.2] (12)

The impedance is calculated by the output signal [e.sub.0] and the current injected in a geophone as;

Z = -[e.sub.0]/i (13)

Measurement of impedance at two frequencies, ??1 and ??2 gives four equations by considering the real and imaginary parts of the impedance.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

From equations (14-17), it is possible to find the four geophone parameters, r, [[xi].sub.0], [[omega].sub.0], and [S.sub.0].

Inserting [Z.sub.1] and [Z.sub.2] into Equation (12) gives the center and radius of the impedance circle.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

The DC resistance r is found to be;

r = c - a (20)

From impedance equations (14-17), [[xi].sub.0], [[omega].sub.0], and [S.sub.0] may be found as,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

4.2 Dynamic Method

In the dynamic calibration method, a geophone is mounted on a shaker with a reference sensor with a known response. The sensitivity may be evaluated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sub.g] = [e.sub.0]/[e.sub.r] [S.sub.r]

[S.sub.g] : sensitivity of a geophone to be calibrated

[S.sub.r] : sensitivity of the reference geophone

[e.sub.0] : output signal from the geophone to be calibrated

[e.sub.r]: output signal from the reference geophone.

Calibrated accelerometers are commonly sold in the market; however, their calibration is usually to 1% accuracy. It is known that the temperature coefficient of a piezoelectric accelerometer is about 0.1%/[degrees]C; however, the coefficient is not calibrated nor warranted by manufacturers. We do not know if this temperature coefficient can be applied to any calibrated accelerometers we can purchase. A five-degree temperature change can cause 0.5% error, and it is difficult to measure the temperature of the accelerometer, since the temperature of the housing may not be the same as the temperature of the piezoelectric element.

[FIGURE 15 OMITTED]

4.3 Reciprocity Calibration

An alternative to the dynamic method is the reciprocity method of calibration to determine geophone sensitivities. In the reciprocity method, three geophones are mounted together as shown in Fig. 16. A current may be injected into a geophone to shake the entire assembly and the output from the other geophones may be measured. Let [e.sub.ij] denote a signal from geophone i responding to signal input to shaker geophone j. Geophones G2 and G3 output signals are then e21 and e31 when shaking G1. Then by injecting signal into Geophone G2 we get [e.sub.32] as the output from G3. A geophone shakes the entire mass M along with three geophones. For the driving signal [e.sub.0] sin [omega]t, the force may be approximated as;

F = [S.sub.j]e/[Z.sub.j]([omega]) sin [omega]t (25)

Outputs, [e.sub.21], [e.sub.31], and [e.sub.32] can be expressed in terms of the driving signal. The sensitivity [S.sub.i] of each geophone may be solved as

[S.sub.1] = [square root of ([e.sub.12][e.sub.13]M [{[Z.sub.1] ([[omega].sub.1]}.sup.2]]/[e.sub.21] [e.sub.0] [Z.sub.2] ([[omega].sub.i][X.sub.i]([[omega].sub.i]) (26)

[S.sub.2] = [square root of ([e.sub.12][e.sub.23]M [{[Z.sub.2] ([[omega].sub.1]}/[e.sub.13] [e.sub.0] [X.sub.i] ([[omega].sub.i]) (27)

[S.sub.3] = [square root of ([e.sub.13][e.sub.23]M [{[Z.sub.2] ([[omega].sub.1]}/[e.sub.12] [e.sub.0] [X.sub.1] ([[omega].sub.i] (28)

where

[X.sub.i] ([omega]) = [[omega].sup.3] /[([[omega].sup.2.sub.i] - [[omega].sup.2]).sup.2] + [(2[[zeta].sub.i][[omega].sub.i][omega]).sup.2]

It is then possible to calibrate the sensitivities of three geophones by knowing their impedances.

4.4 Moving Mass Determination and in-situ Calibration

The dynamic method requires a reference geophone for high precision calibration. The moving coil mass of the reference geophone is measured during assembly. By knowing the amount of moving mass, the absolute sensitivity can be calibrated by the impedance method. The reciprocity method can also be used to provide absolute sensitivity calibration. With the reference geophone, the absolute sensitivity of a geophone may be obtained by the dynamic method.

In the impedance method, the amount of the moving mass, m was assumed, and the sensitivity was derived based on the assumed moving mass. If the absolute sensitivity is obtained by the dynamic or reciprocity method, it is then possible to re-evaluate the moving mass. The absolute moving mass, m0 may be found by using equations (23) and (24) as

[m.sub.0 = m[([S.sub.g]/[S.sub.0].sup.2] (30)

The moving mass is a constant that does not change with temperature or tilt of the geophone. Once the moving mass is known, a geophone can be tested in-situ any time by the impedance method by injecting current.

5. Conclusion

A high precision geophone calibration method has been established. The method integrates the impedance and dynamic methods and determines the amount of moving mass. All the Geophone Accelerometers, GACs are fully calibrated during the manufacturing process to determine DC resistance, natural frequency, open circuit damping, open circuit sensitivity, and moving mass by impedance and dynamic methods. Since the moving mass does not change with temperature or tilt, it is also possible to make in-situ calibration in the working environment by just injecting calibration signals. The moving mass does not change in temperature or in tilt. It is also possible to make in-situ calibration at working environment by just injecting calibration signals.

References

1) A. Obuchi, M. Kamata: "Moving Coil Accelerometer", Japanese patent P3098045, H12, Oct.10

2) A. Obuchi and T. Fujinawa: "Moving Coil Dynamic Accelerometer", 53rd EAEG Meeting, May, 1991 (B021)

3) M. Kamata: "An Accelerometer for Seismic Signal Detection", Proc. 6th Conference of Underground and Civil Engineering Acoustic Emission, May 1999, Tohoku University, Sendai, Japan, pp. 25-26.

4) Peter W. Rodgers, Aaron J. Martin, Michelle C. Robertson, Mark M. Hsu, and David B. Harris: "Signal Coil Calibration of E-M seismometers", Bulletin of the Seismological Society of America, 85, (3), 845-850, June 1995.

5) Mark Harrison, A.O. Sykers, and Paul G. Marcotte: "The Reciprocity Calibration of Piezoelectric Accelerometers", Journal of Acoustical Society of America, 24, (4), July 1952.

6) Thomas B. Gabrielson: "Apparatus and Method for Calibration of Sensing Transducers", U.S. patent No. 5,644,067, July 16, 1996.

MASAHIRO KAMATA

Schlumberger K.K., 2-2-1 Fuchinobe, Sagamihara, Kanagawa 229-0006 Japan
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Author:Kamata, Masahiro
Publication:Journal of Acoustic Emission
Date:Jan 1, 2005
Words:2487
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