ISSN 20751087, Gyroscopy and Navigation, 2015, Vol. 6, No. 4, pp. 288–293. © Pleiades Publishing, Ltd., 2015. Published in Giroskopiya i Navigatsiya, 2015, No. 3, pp. 41–51.
Calibration of the ChekanAM Gravimeter by a Tilting Method A. V. Sokolova, A. A. Krasnova, L. S. Elinsona, V. A. Vasil’eva, and L. K. Zheleznyakb a
Concern CSRI Elektropribor, JSC, ITMO University, St. Petersburg, Russia Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences, Moscow email: anton
[email protected]
b
Received July 10, 2015
Abstract—A method for calibration of the ChekanAM gravimeter, the one that does not require any bench testing equipment, is proposed. Special software providing for automatic calibration and processing of the calibration results has been developed. The experimental verification of the proposed techniques has con firmed the feasibility of periodic calibration of the ChekanAM gravimeter by the new method. DOI: 10.1134/S2075108715040112
INTRODUCTION Mobile gravimeters of the ChekanAM series designed by the CSRI Elektropribor are now exten sively used both by Russian and foreign organizations in marine and airborne gravimetric surveys aimed at oil and gas prospecting and studying the Earth’s grav itational field [1–9]. The ChekanAM gravimeter is a certified measur ing instrument (Certificate no. 43837) with a 2year interval between calibrations. The main procedure of periodic calibration consists in determining coeffi cients of the gravity sensor calibration curve, or, in other words, calibration of the gravimetric sensing ele ment. It is common practice that gravimeters are cali brated by tilting. In this case, the known gravity incre ments are given by changes in the position of the grav ity sensor measuring axis relative to the local vertical [10, 11]. Tilt angles are set and determined on high precision tilt rotary benches. Therefore, the gravime ter has to be demounted and delivered to the manufac turer for periodic calibration. The aim of this work was to develop and implement a new technology for gravimeter calibration using only its hardware and software.
dulum angle of rotation Δϕ, rad, characterize gravity changes Δg, mGal. For the purpose of measuring angle Δϕ, the pendu lums have two mirrors welded to them. The mirror planes are parallel to the pendulum axes and are at a small angle to each other [12]. An autocollimationtype optoelectronic converter (OEC) with two lineartype charge coupled devices (CCD), which serve as photodetectors, is used to mea sure angle Δϕ [13]. Autocollimation images P1 and P2 are formed on the CCD lightsensitive surfaces (Fig. 2). The linear displacements of images L1, L2 TV–camera Beam splitter
LED
Lens
TILTING METHOD The sensing element of the ChekanAM gravime ter is based on a torsiontype double quartz elastic sys tem (DQES), which consists of two torsion systems made of veryhighpurity quartz glass, contained in a common housing (Fig. 1). The torsion systems are deployed in a horizontal plane through 180° relative to each other. Each system has a quartz frame with a torsion and a pendulum with a proof mass welded to this torsion. The housing of the elastic system is filled with organic silicon fluid to pro vide damping of the pendulums. Changes in the pen 288
Proof mass Quartz torsion Quartz frame
Window DQES housing Mirror Damping fluid
Fig. 1. A schematic of the gravity sensor.
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L1 P1 ΔL
d
are proportional to the changes in the pendulum angles of rotation Δϕ1, Δϕ2: L1 = 2nFΔϕ1, (1) L2 = 2nFΔϕ2, (2) where n is the refractive index of the fluid (n = 1, 4); F is the focal distance of the objective lens, mm. The OEC output are m1, m2 in pixels, related to the linear values L1, L2 as m1 = L1/δ, m2 = L2/δ, (3) where δ is the pixel size in millimeters. The resultant coefficient K of the pendulum deflec tion angle ϕ conversion to m is defined by the equa tion: K = m/ϕ = 2nF/δ. (4) The torsion is previously twisted by angle Φ so that the initial position of the pendulum is close to the hor izontal one, wherein λΦ = g, (5) where λ is the rigidity of the quartz system. The gravity change by Δg makes the pendulum turn by angle ϕ. Then the following condition holds: λ(Φ + ϕ) = (g + Δg)cosϕ. (6) For small angles ϕ, the relation between Δg and ϕ has the form: Δg = λϕ + gϕ2/2. (7) This is the main equation for the gravimeter. In practice, it is represented as Δg = bm + am2, (8) where b = λ/K, (9) 2 a = g/2K (10) are linear and quadratic coefficients of the gravimeter calibration curve. Coefficients b and a have different physical natures: whereas the first one is determined by the rigidity of the quartz system and the OEC parameters, the second one only depends on the OEC parameters, which are identical for all ChekanAM gravimeters. The OEC parameters of the ChekanAM gravime ter are the following: F = 150 mm, δ = 7 μm; the cal culated value of the conversion factor K = 60428. Hence, the calculated values of the coefficient are the following: a0 = 136.9 nGal/pixel2 for a single elastic system, a = 34.23 nGal/pixel2 for a double elastic system. The torsion twisting angle Φ is 150°. The calculated value of the elastic system constant λ is 1.85 mGal/arcsec. Consequently, the sensitivity of the elastic system is written as μ = 1/λ = 0.54 arcs/mGal. The calculated values of the linear coefficient of the calibration curve are the following: b0 = 6.14 mGal/pixel for a single elastic system, b = 3.07 mGal/pixel for a double elastic system.
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P2 L2
Fig. 2. Autocollimation images on the CCDs.
The actual values of coefficients b and a are deter mined experimentally for each individual gravimeter by calibration using the tilting method. In practice, calibration is performed on a stationary base with a highprecision tilt rotary bench. In [10, 11] it was shown that the coefficients of the calibration curve for the torsiontype elastic system can be determined from the fundamental equation of the gravimeter when it is tilted in an arbitrary azimuth γ: b(m – mh) + a(m – mh)2 + dθ(m – mh) + eθ = Δg + δg, (11) where d = gcosγ/K, (12) e = g(mh cosγ/K + εsinγ), (13) mh is the gravimeter readings in the horizon plane, pixel; m is the gravimeter readings at tilting, pixel; θ is the tilt angle of the gravimeter measuring axis, rad; γ is the azimuth of the quartz system rotation axes rel ative to tilt plane of the bench table, rad; ε is the deviation of the quartz system rotation axes from horizon, rad; d, e are the coefficients caused by the gravimeter ori entation on the bench table; Δg is the specified gravity difference, mGal; δg is the correction for the specified difference deter mined by the initial settings on the table and tilt angle, mGal. The value of Δg is calculated by the equation: Δg = g(1 – cosθ), (14) where g is the gravity value at the place of the calibra tion site. After a set of tilts, the sought coefficients b and a are determined by solving a system of equations of type (11) by the least squares method [14]. Since Equation (11) does not take into account all influencing factors, e.g., wire sagging, calibration is performed in several stages at different azimuths. The gravimeter is turned on the table so that the inclination plane can be changed. After the coprocessing of all data at different
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where m+, m– are the measurements obtained for the gravimeter inclined by angle θ in positive or negative direction, respectively. Coefficient Kε characterizes angle ν of the cosine curve symmetry axis deviation from the ordinate axis. At small angles Δγ, it can be assumed that the relation Kε = f(γ) is linear. Therefore, to determine the azimuth γcal of the tilt plane for each of the quartz systems, it is sufficient to determine value Kε at two azimuth angles, γ1 and γ2, spaced on the opposite sides of the plane γ = 90° at angle Δγ equal to 10°. Then 20 (18) γ =γ + K .
θ
ν
cal1
m1 m2
m, pix
Fig. 3. A plot of f(θ) for two quartz elastic systems of a gravimeter.
azimuths, the coefficients are determined for each of the quartz systems and the gravimeter as well. Since the calibration procedure is rather complicated, it can be carried out only with special equipment and highly skilled personnel. To improve the calibration accuracy, the gravimeter should be aligned in the horizon and azimuth, which coincides with the quartz system axis of rotation. The table is tilted several times pairwise above and below the horizon. As a result, mdependence on tilt angle θ is obtained: m = f (θ). (15) A plot for this ratio, which is usually called a cosine curve, is given in Fig. 3. The top of the cosine curve determines the zero position of the level, which is fixed. Angle ν between the cosine axis of symmetry and the ordinate axis is sinusoidal, depending on the table inclination plane. When the table is tilted in the plane of the pendulums, angle ν reaches its maximum, but when it is tilted in the plane of the pendulum axis of rotation, ν = 0. The tilt plane is determined by finding the zero value of the coefficient: ε − ε iθ1 (16) K ε = iθ 2 , θ 2 − θ1 where ε θ2 , ε θ1 are the deflections of the sensitive axes of each of the quartz systems from the vertical calcu lated for the gravimeter tilted by angles θ2 and θ1, respectively. Values εθ are calculated by the equation: ε iθ1 = −
θ mi + − mi − × 1, mi + + mi − − 2mi0 2
(17)
1
ε1
K ε1 − K ε'1 γ cal1 + γ cal2 (19) γ= 2 is taken as the azimuth of the tilt plane of the double elastic system. This procedure describes the determination of the calibration plane with an error of no more than 1 minute of arc. Error δg is due to angle 2 ε between the axes of rota tion of two elastic systems and error Δγ of the gravime ter azimuth alignment. In the first approximation, it is given as δg = 2gεϕΔγ. (20) Angle ε is not regulated in the gravimeter and can reach 10 minutes of arc. When the gravity changes by 5 Gal, angle 2ϕ changes approximately by 1.5°. Deviation Δγ of the bisector between the pendulum rotation axes from the gravimeter tilt plane should be less than 20 arcmin, which is provided by the manu facturer, lest error δg exceed 0.2 mGal. To determine the coefficients of the calibration curve, the gravimeter is consistently tilted by angles ±1°50′, ±2°30′, ±3°10′, ±3°40′, ±4°10′; ±4°30′, ±4°50′, and ±5°10′. The val ues of tilt angles were selected so that when the tilt angle θ changed, Δg varied approximately by 500 mGal. For proper values of angles ε, ϕ, γ, Δγ, remaining within the tolerances, and identity of two quartz systems in sensitivity, calibration can be per formed with the reduced equation (11): Δg = b(m – mh) + a(m – mh)2 (21) both for each of the quartz systems and a double quartz elastic system. If the latter is the case, m is a summary reading. From (21), coefficient b is determined as b = Δg/ ( m – m h ) – a ( m – m h ) (22) 2 = gθ ( m – m h )/ 2 – a ( m – m h ). Since the second component a(m – mh) of (22) is only 2% from the range measurement, the relative error of coefficient b can be calculated by differentia tion of the first term (22) db/b = dg/g + dθ/θ + dm/(m – mh). (23)
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Assuming that the value g at the point of calibration is known with an error of dg < 1 mGal, the first com ponent (23) is less than 10–4%. The measurement error is given as dm <0.02 pixel. The maximum change of m at an inclination of 5 Gal is ~ 800 pixels. Hence, the second error component does not exceed 0.0025%. The error in setting the tilt angle θ has the main impact on the error in determining coefficient b. For the test tools used in the calibration of gravimeters, the error is given as dθ = 3 arcsec. The maximum tilt angle is 5°. Hence, the relative error is written as dθ/θ ~ 0.02%. Coefficient a is determined only by the geometrical parameters of the OEC and the thermostating temper ature. It can be calculated by the manufacturing toler ances of the optical elements, taking into account the fact that the quadratic term of Equation (21) does not exceed 90 mGal at the edge of the range. In accordance with (21), the gravimeter measure ments are a quadratic function of the readings. There fore, coefficient b of the calibration curve refers to the initial reading mh, which is determined by the time and place of gravimeter calibration. The most correct way of determining the coeffi cient b is at m0, which corresponds to the horizontal position of the pendulums in the elastic system. For the ChekanAM gravimeter, the CCD line center is taken as the reference point of m0.. It is easy to show that the value b0, corresponding to m0, is calculated by the formula: b0 = b + 2a(m0 – mh). (24) The sensing elements of ChekanAM gravimeters have exacting tolerances on the angular position of each of the quartz systems with respect to the outer housing. The gravity sensor position also provides alignment of the gimbal ring roll axis and the quartz system axis of rotation. If the gravity sensor tilts together with the gimbal ring in the plane of the quartz system axes of rotation, it can be calibrated without removal from the gyro platform. SOFTWARE AND HARDWARE The tilting method described above is used for all types of relative gravimeters for calibration of the sens ing element. The fact that there is no need to use a highprecision tilt rotary bench is the main advantage of the new calibration technology. The gravity sensor can be tilted owing to the biaxial gyro platform included in the gravimeter (Fig. 4). The gyro platform design has made it possible to omit determination of a tilt azimuth since error Δγ of the gravity sensor setting relative to the tilt plane of the gimbal rings does not exceed 20 arcmin, and error δθ of the gyro platform readout devices does not exceed 3 arcsec within the required measurement range. GYROSCOPY AND NAVIGATION
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Fig. 4. Gravimeter calibration using an inhouse twoaxis gyrostabilizer.
Calibration of the gravity sensor installed in the gyro platform is supported by a specially developed Calibr.exe program, which provides both automatic tilting of the gravity sensor by predetermined angles at regular intervals and processing of measurement results. During the entire measurement cycle, the file with the current gravimeter readings and the gyro platform tilt angles is recorded (Fig. 5). The duration of the measurements at each angle of the platform inclination is 30 minutes. The whole cal ibration lasts 9 hours. Data processing is performed by the least squares method by a simplified formula (21), which is followed by bringing the scaling factor b to the value b0 in accor dance with (24). To eliminate the effect of deviations of the gravimeter axis of sensitivity from the vertical, coeffi cient b is calculated using measurements m = (m+ + m–)/2, (25) where m+, m– are the measurements obtained from the gravimeter inclined by angle θ in positive or negative direction, respectively. Upon completion of the program, a file of calibr_XX.pdf is formed, where XX is a serial number of the ChekanAM gravimeter. It contains the follow ing parameters: a) specified gravity increments dacal i, mGal, and measurement results dgi, mGal, for each angle θi of the gyro platform tilt; b) deviations δgi, mGal, of measurement results dgi from specified values dacal i, mGal;
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Fig. 5. Display of Calibr.exe in the measurement mode.
c) reduced error θg,%, of the calibration curve, represented by ratio of the maximum (in absolute value) of the obtained values δgi to the upper limit of the gravimeter measurement range Δ0 = 5 Gal; d) quadratic coefficient a and linear coefficients b1, b2 of the calibration curve for each of the quartz sys tems;
e) deviations δb1,%, δb2,% of the calculated values b1, b2 from the initial values b01, b02; f) gravimeter measurement range margin. The protocol formed by the Calibr.exe. program is a necessary and sufficient document for the registra tion of a verification certificate of a mobile Chekan AM gravimeter. EXPERIMENTAL DATA
Table Tilts: set 1 Tilts: set 2 Gravime ter serial number max error RMS error max error RMS error δg, mGal of δg, mGal δg, mGal of δg, mGal 02
–1.34
0.74
–1.03
0.66
06
⎯1.10
0.63
–1.80
0.82
12
–1.40
0.74
1.20
0.68
26
–1.18
0.67
1.23
0.70
27
–1.57
0.92
–1.62
0.95
31
–1.44
0.95
1.34
0.90
34
–1.33
0.82
–1.35
0.85
The proposed tilting method using a gyro platform was tested on seven commercially produced Chekan AM gravimeters. The calibration results are given in the table. Fig. 6 shows a comparison of errors in the calibra tion curves of the same gravimeter during calibration in the gyro platform with the use of a highprecision Acutronic bench. The analysis of the results shows that the residual error of calibration in the case when a gyro platform was used is, on the average, 1.5–2 times higher than that with the use of the Acutronic bench. Nonetheless, the proposed method allows for calibration of Che kanAM gravimeters with the required accuracy.
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(a)
1 0 –1 –2 0
2
4
6
8
mGal 2
10
12
14
16
18
12
14
16
18
(b)
1 0 –1 –2 0
2
4
6
8
10
Fig. 6. Calibration results of the ChekanAM gravimeter serial no. 12 on an Acutronic bench (a) and inside the gyro platform (b).
CONCLUSIONS A new technology for calibration of ChekanAM gravimeters by the tilting method using the gravimeter gyrostabilization system and special software has been proposed and tested. The new method allows for peri odic calibration of ChekanAM gravimeters in the field without any highprecision bench equipment. ACKNOWLEDGMENTS This work was supported by the Russian Science Foundation, project no. 142900160. REFERENCES 1. Krasnov, A.A., Sokolov, A.V., and Elinson, L.S., Oper ational experience with the ChekanAM gravimeters, Gyroscopy and Navigation, 2014, vol. 5, no. 3, pp. 181– 185. 2. Krasnov, A.A., Sokolov, A.V., and Elinson, L.S., A new airsea gravimeter of the Chekan series, Gyroscopy and Navigation, 2014, no. 3, pp. 131–137. 3. Sokolov, A.V., Usov, S.V., and Elinson, L.S., The expe rience of conducting gravity surveys in the conditions of marine seismic operations, Giroskopiya i Navigatsiya, 2000, no. 1, pp. 39–50.
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4. Krasnov, A.A., Sokolov, A.V., and Usov, S.V., Modern equipment and methods for gravity investigation in hardtoreach regions, Gyroscopy and Navigation, 2011, vol. 2, no. 3, pp. 178–183. 5. Drobyshev, N.V., Koneshov, V.N., Koneshov, I.V., and Solov’ev, V.N., Development of an aircraft laboratory and methods of gravimetric surveying in the arctic con ditions, Vestnik Permskogo universiteta. Seriya Geologiya, 2011, no. 3, pp. 37–50. 6. Atakov, A.I., Lokshin, B.S., Prudnikov, A.N., and Shkatov M.Yu., The results of integrated geophysical survey at the ushakovskonovozemel’skaya prospective area in the Kara Sea, Proc. IAG International Sympo sium on Terrestrial Gravimetry: Static and Mobile Mea surements (TGSMM2010), 2010, pp. 33–35. 7. Lygin, V.A., Gravity surveys in transition zones with the use of hovercraft, Proc. IAG International Symposium on Terrestrial Gravimetry: Static and Mobile Measurements (TGSMM2010), 2010, pp. 47–49. 8. Forsberg, R., Olesen, A., and Einarsson, I., Airborne gravimetry for geoid determination with Lacoste Rom berg and ChekanAM gravimeters. Proc. IAG Sympo sium on Terrestrial Gravimetry: Static and Mobile Mea surements, St. Petersburg: Elektropribor, 2013, p. 10. 9. Barthelmes, F., Petrovic, S., and Pflug, H., First expe riences with the GFZ new mobile gravimeter Chekan AM. Proc. IAG Symposium on Terrestrial Gravimetry: Static and Mobile Measurements, St. Petersburg: Elek tropribor, 2013, pp. 18–19. 10. Zheleznyak, L.K. and Elinson, L.S., Calibration of gravimeters with two elastic torsiontype systems by a tilting method, Fizikotekhnicheskaya gravimetriya, Moscow: IFZ AN SSSR, Nauka, 1982, pp. 110–124. 11. Zheleznyak, L.K., Elinson, L.S., and Boyarskii, E.A., Calibration by tilting gravimeters with an elastic tor siontype system, Graviinertsial’nye issledovaniya, Moscow: IFZ AN SSSR, Nauka, 1983, pp. 144–180. 12. Evstifeev, M.I., Krasnov, A.A., Sokolov, A.V., Sta rosel’tseva, I.M., Elinson, L.S., Zheleznyak, L.K., and Koneshov, V.N., A new generation of gravimetric sen sors. Measurement Techniques, 2014, vol. 57, no. 9, pp. 967–972. 13. Berezin, V.B., Berezin, V.V., Sokolov, A.V., and Tsytsu lin, A.K., Adaptive image sensing in an astronomical system on a CCD, Izvestiya vysshikh uchebnykh zavede nii. Radioelektronika, 2004, no. 4, p. 36. 14. Stepanov, O.A., Osnovy teorii otsenivaniya s priloz heniyami k zadacham obrabotki navigatsionnoi infor matsii (Fundamentals of the Estimation Theory with Applications to the Problems of Navigation Informa tion Processing), Part 1, Vvedenie v teoriyu otsenivaniya (Introduction to the Estimation Theory), St. Peters burg: TsNII Elektropribor, 2010.
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