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The Earth surface deformations caused by air pressure variations
Journal of Geodynamics 35 (2003) 541–551 www.elsevier.com/locate/jog
The Earth surface deformations caused by air pressure variations L.A. Latynina*, I.M. Vasil’ev Institute of Physics of the Earth, Russian Academy of Sciences, B.Gruzinskaya Str. 10, Moscow, 123995 Russia
Abstract The relationship between the tilts and strains of rocks and air pressure variations is investigated using deformation measurements from the Protvino observatory. Deformations due to air pressure are large in the observation area. The deformation response to the observed air pressure variations is compared with calculated values. In addition, the effective elasticity of the rock is determined from measured deformations in the underground galleries of the observatory after creating a load on the Earth surface. The observed tilts are considerably larger than the calculated values, for local as well as atmospheric loading, and the rock shear modulus determined from the tilt data is anomalously small compared to the mean shear modulus value for the crust. The observed horizontal strain data indicate no anomaly in the rock shear modulus value. The cavity effects at the observatory cannot explain this phenomenon. It may be caused by the rocks high fracturing. It is supposed, that the influence of cracks on the rock horizontal deformations is small if the cracks are closed and have the vertical direction. In case of large wavelength the large tilts correspond to the great displacements. These displacements may worsen the accuracy of geodetic measurements, especially when measuring vertical crustal movements. # 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Traditionally, experimental geophysics has been interested in crustal deformations under influence of atmospheric pressure. The importance of this problem has increased due to the rapid development of space geodesy and the new prospects for investigating geodynamical processes. The response of the crust to the atmospheric pressure variations also brings information on the crustal structure and geological heterogeneities, when the parameters of these variations are known (Kroner and Jentzsch, 1998; Onoue and Takemoto, 1998).
* Corresponding author. Fax: +7-95-255-60-40. E-mail address: [email protected] (L.A. Latynina). 0264-3707/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0264-3707(03)00013-9
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L.A. Latynina, I.M. Vasil’ev / Journal of Geodynamics 35 (2003) 541–551
In this investigation, an attempt is made to detect and evaluate local heterogeneities and to determine the relationship between air pressure and deformation variations based on data from the Protvino observatory.
2. Protvino observatory Deformation measurements are carried out in an area located on the Russian plain in the region of the East European Platform. Protvino observatory (37.21 E, 56.88 N) is located 100 km south of Moscow within the territory of the Serpukhov accelerator. Fig. 1 shows the location of the sensors in the underground chamber of the observatory. The chamber is located at a depth of 15 m and consists of three galleries oriented in North–South (NS), East–West (EW) and South–West (SW) directions with each a length of 20 m. The geological section of rocks in the observatory area is as follows: first, there is a sand layer down to 8 m. Then, there is fractured limestone rock down to 50 m.
Fig. 1. Scheme of the underground galleries in the Protvino observatory. Location of the sensors: tiltmeters T1, T2, T3, extensometers Ens-1, Ens-2, Eew-1, Eew-3, barometer B, thermometers Th, and water level in bore-hole W. Successive sites of the load in the experiment (tractor).
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The observatory has existed since the 1970s. The instruments that are nowadays in operation in the underground galleries will be discussed here. There are four quartz extensometers, each 16–17 m long, equipped with capacitive transducers. To increase data reliability and to detect instrumental errors the extensometers are installed in pairs in two galleries (NS and EW). A comparative analysis based on a two-year measurement series carried out in 2000–2001 confirms their high agreement within the frequency range from 1 to 0.002 cph. The distinction of the amplitudes of the main tidal waves O1 and M2 is 2% (Boyarsky et al., 2001) and is about 20% for periods of 10–30 days (Latynina and Vasil’ev, 2001). Two-component tiltmeters (T1, T2, T3) are located at the ends of the galleries (Latynina et al., 1997). The sensing device of the tiltmeter is a vertical pendulum in combination with a capacitive transducer. Four thermometers are installed to be able to take the influence of temperature into account: one thermometer measures the ambient air temperature, another measures the soil temperature at a depth of 0.6 m and the two remaining instruments measure the temperature at the ends of the galleries (Th). A microbarograph measures the variations of atmospheric pressure in the underground chamber. The water level of the main water-bearing horizon in the well (W) is measured at a depth of 25 m. All information from the instruments is automatically collected and stored by a system running on two PCs with 16bits ADC. The data is collected at intervals of 30 s and 10 min. The sensitivity and other technical data of the equipment has been reported (Latynina et al., 1997; Boyarsky et al., 2001).
3. Determination of the elastic modulus of the rock 3.1. Loading experiment In order to determine the rock properties a load was created on the Earth surface and the resulting deep-laid strains and tilts recorded. The load was generated by an 8-t tractor at several points in succession (see Fig. 1). At a distance of 16–17 m from the loading point the tiltmeters monitored tilts of tens of milliseconds (mas) and the extensometers recorded linear strains of about 1 nstr (109). The measured values of tilts and strains were compared with theoretical values calculated using the Boussinesq formulae (Love, 1935). These formulae provide the solution for the deformation of a homogeneous elastic half-space with z<0 induced by a vertical point wise force. The question arises whether it is correct to apply the Boussinesq formulae to the heterogeneous fractured medium. In this case the geophysical object is simulated by a homogeneous elastic halfspace. The elasticity estimate received from the Boussinesq formula gives the in-situ (effective) elasticity of the geophysical object. The effective elasticity depends on the object geological structure, its heterogeneities, and crack density and orientation. Provided that the Lame´ constants l and are equal to each other, and a force P acts on the Earth surface in a point with coordinates (0, 0, 0), the displacement in the x, y, z frame can be given by: W ¼ P=4 z2 =r3 þ 3=2r ; ð1Þ U ¼ P=4 xz=r3 x=2rðz þ rÞ ; V ¼ P=4 yz=r3 y=2rðz þ rÞ ;
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where W is the vertical component, and U and V the horizontal components of the displacement. r is the distance from the point of loading to the measurement point. Expressions for tilts in x and y direction are given by the derivatives dW/dx and dW/dy. Substituting experimental data into these expressions will give the equations for the determination of the shear modulus . The components of linear strains exx and eyy are calculated from the differences in rocks displacement on the base of each extensometer: ð2Þ exx ¼ Uðx1; y1Þ Uðx2; y1Þ =L1; Vðx3; y2Þ Vðx3; y3Þ =L2; where xi and yi are the horizontal coordinates of the extensometers end points. The vertical coordinates, zi, of all points are equal to the galleries depth, and L1 and L2 are the lengths of the extensometers. By substituting the measured deformations exx and eyy into (2), can be determined. 3.2. Experimental evaluation of m The values for obtained from the tilt data in two points of the observatory (Fig. 1; Latynina and Vasil’ev, 2001) are: ¼ 5:9103 bar in point 1; ¼ 1:5103 bar in point 2;
ð3Þ
The values for in the galleries NS and EW calculated from extensometer data are: ¼ 2:5105 bar in NS direction; ¼ 3:0105 bar in EW direction:
ð4Þ
The mean shear modulus of the crust is equal to 3.105 bar, and equal to 3.104 bar for the upper crustal layers. The values for calculated from the tilt data (3) are considerably smaller than the mean shear modulus value of the crust. The calculations are carried out for a single point force. Using a loading distribution one can increase the calculated value of by a factor of 1.5–2.0, but it still remains an order of magnitude less than the crustal shear modulus of the upper layers. The question arises whether the values for calculated from the tilt data are smaller because of cavity and topography effects, instead of medium properties. The topography in the plain region of the Protvino area induces deformation changes of about 3% (Kramer et al., 1985). The horizontal strains along the tunnels caused by air pressure are determined without any distortion. However, the tunnel can induce perturbations in the strain field when a point load acts. We need a rough estimate of these effects. For that, we use calculated models of the different underground chambers (Harrison, 1976; Dal Moro et al., 2001). The cavity does not increase the anomalous effects to a great extent when a violent stress concentrator (for example a deep narrow fault) is absent. In a homogeneous strain field the cavity effects vanish for tilts in the center of the chamber. The observed tilts (vector modulus) at points 1 and 2 are 11.5 mas and 44.5 mas. The ratio of the observed tilt to the calculated value is 41 at point 1 and 158 at point 2. Such great anomalies cannot be explained by cavity effects. They can be caused by the rock structure. The extensometers measure linear strains along the tunnel. In the case of air pressure, the cavity has no effect on the extensometer measurements. Looking at the obtained data the following conclusions can be made: 1. The observatory is located in a weakened zone where the shear modulus is, according to the tilt measurements, much smaller than its mean value for the upper crustal layer.
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2. This zone is heterogeneous, as according to the tilt measurements the elastic modulus of the rock is larger at point 1 than at point 2. The mobility of point 2 is probably caused by rock fractures, which were caused by natural or technical influences (Molodensky, 2000; Latynina and Boyarsky, 2000). 3. The measured tilt vector deviates from the radial direction (with the center at the load point) towards the NS direction. This means that the rigidity of the medium varies depending on the direction. The large tilts in NS direction could be explained by crack orientation and local crack strikes. 4. In the case of extensometer measurements, unlike the tilt measurements, the resulting value for is close to the value of undisturbed limestone and equals (2–3).105 bar. 5. The value for , obtained from extensometer data is noticeably different from the value calculated from the tiltmeter data. We explain this difference by the mechanism of a vertical load acting upon the fractured rocks. The measured tilt is caused by continuous deformation of the rock and by relative displacements of separate blocks. The experimental value of is the effective shear modulus of rocks, which reflects both processes. The influence of vertical cracks on the horizontal linear strain is small in the case of the closed cracks. Moreover, the tilts are measured at short bases (0.5 m), while the strains are measured at longer bases (20 m).
4. Spectral density of vibration processes The spectral analysis of the tilt and air pressure data sets at the Protvino Observatory was discussed in previous work (Latynina and Vasil’ev, 2001). The power spectral density S of the tilts varies from 0.1 (mas)2/cph up to 30 000 (mas)2/cph at point 1, and up to 200 000 (mas)2/cph at point 2 in a range from 2 to 500 h. The value S depends on the period T with power three. Fig. 2 illustrates the power spectral density of 6 month data sets (10.09.2000–10.03.2001) in the frequency range from 0.002 cph to 0.5 cph. The linear strains in NS and EW directions, the tilts in NS direction at point 1, and pressure variations are analyzed. The spectra in Fig. 2 are presented in different scales for clearness. Even at the stable point 1, the tilt spectrum curve lies higher than the strain curves. For a frequency of 0.003 cph the power spectral densities are equal to 5.1015/ cph for strains and to 150.1015/cph for tilts. The spectral densities S of the strains increase with T to power 2.5–3.0. There are peaks in the monotonous build-up of the spectral density of tilts and strains corresponding to the diurnal and semidiurnal tidal waves. For pressure variations, S peaks at T close to the duration of cyclones (10–15 days). Maybe, the analysis of longer sets of pressure data will allow us to explain the other spectral peaks. The relationship between S and T at Protvino observatory (as T3) is typical for low frequency noises and corresponds to Peterson’s ‘‘New Low Noise Model’’. It can be applied when the spectra of deformations are dominantly determined by atmospheric pressure variations. Similar results were obtained also with the superconducting gravimeter at the Geodynamic Observatory Moxa in Germany (Kroner et al., 2001). These spectra can be distinguished from the deformation spectra in active tectonic zones, such as the Friuli area in Italy (Braitenberg, 1999). Maybe, the contribution of air pressure in deformation processes is not very large in the last case.
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Fig. 2. The power spectrum of variations in N–S horizontal strains (curve 1, unit 0.1.(nstr)2/cph), in E–W horizontal strains (curve 2, unit 10.(nstr)2/cph), in N–S tilts (curve 3, unit 103.(nrad)2/cph), and in air pressure (curve 4, unit 103.(mbar)2/cph).
5. The relationship between the crustal deformations and the variations in air pressure 5.1. Estimation of air pressure effects A quantitative evaluation of displacements, tilts and deformations of the Earth crust has been performed for a cyclone model, where the cyclone is presented as axial-symmetric about the z-axis of the load applied to the Earth surface. The pressure on the axis corresponds to the pressure at the cyclone center. The distributed load has been numerically calculated using the Boussinesq formulae. The displacements and deformations of the Earth surface are proportional to the excess pressure P0 in the cyclone epicenter, the radius R of the loaded area, and a combination of the Lame´ constants l and . For P0=50 mbar, 2R=1000 km, and l==3.105 bar maximum values are found of 3 cm for the vertical displacement W, 5.5.108 mas for tilts ’, 0.4 cm for horizontal displacements U, and 3.0.108 for strains . The ratio of the maximum surface tilts and strains to the maximum pressure P0 is:
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’=P0 ¼ 1:1109 =mbar;
"=P0 ¼ 0:6109 =mbar
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ð5Þ
5.2. Results The vertical displacements caused by air pressure are one order of magnitude larger than the horizontal displacements. The vertical displacements peak in the cyclone epicenter and the area with the maximum horizontal displacements is shifted by R/2 relative to the epicenter. The horizontal strains peak in the epicenter.
Fig. 3. The long-period variations of air pressure (curve 1), N–S horizontal strains (curve 2), and E–W horizontal strains (curve 3).
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5.3. Experimental data analysis The similarity between the curves of the linear strains, tilts and atmospheric pressure is significant as is shown in Fig. 3, where the measurements for the period December 2000–February 2001 are presented. Variations with periods from 2 to 40 h (Earth tides) and from 500 to 4000 h (slow drift) are excluded. During pressure build-up periods dilatation is observed in EW direction and compression in NS direction. 5.4. Coherency function for air pressure and strain variations The relationship between the tilts or strains and pressure variations is given by the frequency depended transfer function (Bendat and Piersol, 1989). The contribution of barometric tilts or strains to general geophysical processes is given by the square modulus of a coherency function, here called gamma(f). The function gamma(f) is shown in Fig. 4. It reflects how the air pressure variations reflect on the strain components for a frequency range from 0.003 to 0.2 cph based on the data set 09.2000–03. 2001. In the range from 0.003 to 0.02 cph the average gamma(f) is 0.5 for NS strain and 0.2–0.3 for EW strain and NS tilt. This means that approximately 50% of the signal power of the NS strain and about 30% of the EW strain is related to pressure variations.
Fig. 4. Squared modulus of the coherence spectrum gamma(f) between pressure and strains N–S (curve 1), between pressure and strains E–W (curve 2), and between pressure and tilts N–S (curve 3) in point 1.
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This analysis confirms the earlier results reported in (Latynina and Vasil’ev, 2001): the reduction of gamma(f) to a value of 0.10–0.15 for frequencies higher than 0.3 cph. 5.5. Amplitude–frequency transfer function Fig. 5 shows the amplitude–frequency transfer function. This function, relating barometric and tilt variations, is equal to 5.109/mbar within the range of 0.002–0.02 cph (for periods of more than 2 days). This value is 5 times larger than the ratio of the maximum surface tilt and maximum
Fig. 5. Amplitude transfer function A(f) between pressure and strains N–S (curve 1), pressure and strains E–W (curve 2), and pressure and tilts N–S (curve 3).
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pressure for the cyclone model. For horizontal strains within the same frequency range the transfer function is equal to (0.6–1.0).109/mbar. This experimental value is close to the calculated one. Thus, the experimental data show that air pressure variations cause Earth surface tilts that are larger than horizontal strains and model values.
6. Conclusion The Protvino observatory location is traditionally considered favorable for investigating deformation processes. Plain topography, heavy sedimentary series, and the absence of regional faults create conditions for a homogenous stress–strain state. However, when a load was created to the Earth surface, the rock effective shear modulus calculated from the tilt data was significantly smaller than the mean crustal modulus. Its value depends on the measurement point and the tilt direction. The topography and cavity effects did not explain these anomalies. Moreover, the values of the shear modulus calculated from the horizontal strain data are one order of magnitude larger and are close to the mean value for the Earth crust. This means that the rock effective shear modulus not only reflects continuous deformations but also the relative displacements of separate blocks. At the same time the influence of vertical cracks on horizontal strains is small. High correlation between atmospheric pressure variations and tilt and strain variations is detected. Within a period of 2–15 days about 50% of the power of the strain variations is related to atmospheric pressure variations. This relation is probably less in zones where the tectonic movements are more intensive. Using the transfer function it has been shown that the tilts with barometric origin reach 5.109/mbar in a stable point in the Protvino observatory area, and the model value equals 1.1.109/mbar. The experimental horizontal strains of 0.6.109/mbar are approximately equal to the calculated strains. As the barometric tilt is given by the change of vertical displacement along a horizontal element, the large tilts at given wavelength imply large displacements. Therefore, the barometric effects may worsen the accuracy of geodetic measurements, especially concerning vertical crustal movements. References Bendat, J.S., Piersol, A.G., 1989. Random Data. Analysis and Measurement Procedures. Mir, Moscow. Boyarskyi, E.A., Vasil’ev, I.M., Suvorova, I.I., 2001. The study of tilts and strains at the Protvino Geophysical station. Izvestiya Physics of the Solid Earth 37 (N9), 764–770. Braitenberg, C., 1999. The Friuli (NE-Italy) tilt/strain gauges and short term observations. Annali di Geofisica 42 (N4), 637–664. Dal Moro, G., Ebblin, C., Zadro, M., 2001. The FEM in the interpretation of tilt/strainmeter observations in a cave: air pressure loading effects. Journal of the Geodetic Society of Japan 47 (1), 88–94. Harrison, J.C., 1976. Cavity and topographic effects in tilt and strain measurement. Journal of Geophysical Research 81 (N2), 319–328. Kramer, M.V., Karmaleeva, R.M., Latynina, L.A., Molodensky, S.M., 1985. Evaluation of the tide strain disturbance in conditions of the plain topography. Phizika Zemli N7, 80–86. (in Russian). Kroner, C., Jentzsch, G., 1998. Comparison of air pressure methods and discussion of other influences on gravity. In: Proc. of the Thirteenth International Symposium of Earth Tides, Brussels. pp. 423–430. Kroner, C., Jahr, T., Jentzsch, G., 2001. Comparison of data sets recorded with the dual sphere superconducting
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gravimeter CD 034 at the Geodynamic Observatory Moxa. Jornal of the Geodetic Society of Japan 47 (N1), 398– 403. Latynina, L.A., Boyarskii, E.A., Vasil’ev, I.M., Sorokin, V.L., 1997. Tiltmeter measurements at the Protvino station, Moscow region. Izvestiya Physics of the Solid Earth 33 (N11), 949–956. Latynina, L.A., Boyarsky, E.A., 2000. Seasonal Earth tide variations as a model of Earthquake. Precursor. Volc. Seis 21, 587–595. Latynina, L.A., Vasil’ev, I.M., 2001. Crustal deformations induced by atmospheric pressure. Izvestiya Physics of the Solid Earth 37 (N5), 392–401. Love, A.E.H., 1927. A Treatise on the Mathematical Theory of Elasticity, fourth ed. Cambridge Univeristy Press, Cambridge. Translated under the title Matematicheskaya teoriya uprugosti, Leningrad, ONTI,1935.. Molodensky, S.M., 2000. On the local effects of the thermo-elastic Earth strains in the tidal range of the frequency. Izvestiya Physics of the Solid Earth 36 (N5), 364–368. Onoue, K., Takemoto, S.,1998. Atmospheric pressure effect on ground strain observation at Donzurubo Observatory, Nara, Japan. In: Proc. of the Thirteenth International Symposium of Earth tides, Brussels. pp. 157–164. Earth. 33(N11), 949–956.
The Earth surface deformations caused by air pressure variations L.A. Latynina*, I.M. Vasil’ev Institute of Physics of the Earth, Russian Academy of Sciences, B.Gruzinskaya Str. 10, Moscow, 123995 Russia
Abstract The relationship between the tilts and strains of rocks and air pressure variations is investigated using deformation measurements from the Protvino observatory. Deformations due to air pressure are large in the observation area. The deformation response to the observed air pressure variations is compared with calculated values. In addition, the effective elasticity of the rock is determined from measured deformations in the underground galleries of the observatory after creating a load on the Earth surface. The observed tilts are considerably larger than the calculated values, for local as well as atmospheric loading, and the rock shear modulus determined from the tilt data is anomalously small compared to the mean shear modulus value for the crust. The observed horizontal strain data indicate no anomaly in the rock shear modulus value. The cavity effects at the observatory cannot explain this phenomenon. It may be caused by the rocks high fracturing. It is supposed, that the influence of cracks on the rock horizontal deformations is small if the cracks are closed and have the vertical direction. In case of large wavelength the large tilts correspond to the great displacements. These displacements may worsen the accuracy of geodetic measurements, especially when measuring vertical crustal movements. # 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Traditionally, experimental geophysics has been interested in crustal deformations under influence of atmospheric pressure. The importance of this problem has increased due to the rapid development of space geodesy and the new prospects for investigating geodynamical processes. The response of the crust to the atmospheric pressure variations also brings information on the crustal structure and geological heterogeneities, when the parameters of these variations are known (Kroner and Jentzsch, 1998; Onoue and Takemoto, 1998).
* Corresponding author. Fax: +7-95-255-60-40. E-mail address: [email protected] (L.A. Latynina). 0264-3707/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0264-3707(03)00013-9
542
L.A. Latynina, I.M. Vasil’ev / Journal of Geodynamics 35 (2003) 541–551
In this investigation, an attempt is made to detect and evaluate local heterogeneities and to determine the relationship between air pressure and deformation variations based on data from the Protvino observatory.
2. Protvino observatory Deformation measurements are carried out in an area located on the Russian plain in the region of the East European Platform. Protvino observatory (37.21 E, 56.88 N) is located 100 km south of Moscow within the territory of the Serpukhov accelerator. Fig. 1 shows the location of the sensors in the underground chamber of the observatory. The chamber is located at a depth of 15 m and consists of three galleries oriented in North–South (NS), East–West (EW) and South–West (SW) directions with each a length of 20 m. The geological section of rocks in the observatory area is as follows: first, there is a sand layer down to 8 m. Then, there is fractured limestone rock down to 50 m.
Fig. 1. Scheme of the underground galleries in the Protvino observatory. Location of the sensors: tiltmeters T1, T2, T3, extensometers Ens-1, Ens-2, Eew-1, Eew-3, barometer B, thermometers Th, and water level in bore-hole W. Successive sites of the load in the experiment (tractor).
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The observatory has existed since the 1970s. The instruments that are nowadays in operation in the underground galleries will be discussed here. There are four quartz extensometers, each 16–17 m long, equipped with capacitive transducers. To increase data reliability and to detect instrumental errors the extensometers are installed in pairs in two galleries (NS and EW). A comparative analysis based on a two-year measurement series carried out in 2000–2001 confirms their high agreement within the frequency range from 1 to 0.002 cph. The distinction of the amplitudes of the main tidal waves O1 and M2 is 2% (Boyarsky et al., 2001) and is about 20% for periods of 10–30 days (Latynina and Vasil’ev, 2001). Two-component tiltmeters (T1, T2, T3) are located at the ends of the galleries (Latynina et al., 1997). The sensing device of the tiltmeter is a vertical pendulum in combination with a capacitive transducer. Four thermometers are installed to be able to take the influence of temperature into account: one thermometer measures the ambient air temperature, another measures the soil temperature at a depth of 0.6 m and the two remaining instruments measure the temperature at the ends of the galleries (Th). A microbarograph measures the variations of atmospheric pressure in the underground chamber. The water level of the main water-bearing horizon in the well (W) is measured at a depth of 25 m. All information from the instruments is automatically collected and stored by a system running on two PCs with 16bits ADC. The data is collected at intervals of 30 s and 10 min. The sensitivity and other technical data of the equipment has been reported (Latynina et al., 1997; Boyarsky et al., 2001).
3. Determination of the elastic modulus of the rock 3.1. Loading experiment In order to determine the rock properties a load was created on the Earth surface and the resulting deep-laid strains and tilts recorded. The load was generated by an 8-t tractor at several points in succession (see Fig. 1). At a distance of 16–17 m from the loading point the tiltmeters monitored tilts of tens of milliseconds (mas) and the extensometers recorded linear strains of about 1 nstr (109). The measured values of tilts and strains were compared with theoretical values calculated using the Boussinesq formulae (Love, 1935). These formulae provide the solution for the deformation of a homogeneous elastic half-space with z<0 induced by a vertical point wise force. The question arises whether it is correct to apply the Boussinesq formulae to the heterogeneous fractured medium. In this case the geophysical object is simulated by a homogeneous elastic halfspace. The elasticity estimate received from the Boussinesq formula gives the in-situ (effective) elasticity of the geophysical object. The effective elasticity depends on the object geological structure, its heterogeneities, and crack density and orientation. Provided that the Lame´ constants l and are equal to each other, and a force P acts on the Earth surface in a point with coordinates (0, 0, 0), the displacement in the x, y, z frame can be given by: W ¼ P=4 z2 =r3 þ 3=2r ; ð1Þ U ¼ P=4 xz=r3 x=2rðz þ rÞ ; V ¼ P=4 yz=r3 y=2rðz þ rÞ ;
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where W is the vertical component, and U and V the horizontal components of the displacement. r is the distance from the point of loading to the measurement point. Expressions for tilts in x and y direction are given by the derivatives dW/dx and dW/dy. Substituting experimental data into these expressions will give the equations for the determination of the shear modulus . The components of linear strains exx and eyy are calculated from the differences in rocks displacement on the base of each extensometer: ð2Þ exx ¼ Uðx1; y1Þ Uðx2; y1Þ =L1; Vðx3; y2Þ Vðx3; y3Þ =L2; where xi and yi are the horizontal coordinates of the extensometers end points. The vertical coordinates, zi, of all points are equal to the galleries depth, and L1 and L2 are the lengths of the extensometers. By substituting the measured deformations exx and eyy into (2), can be determined. 3.2. Experimental evaluation of m The values for obtained from the tilt data in two points of the observatory (Fig. 1; Latynina and Vasil’ev, 2001) are: ¼ 5:9103 bar in point 1; ¼ 1:5103 bar in point 2;
ð3Þ
The values for in the galleries NS and EW calculated from extensometer data are: ¼ 2:5105 bar in NS direction; ¼ 3:0105 bar in EW direction:
ð4Þ
The mean shear modulus of the crust is equal to 3.105 bar, and equal to 3.104 bar for the upper crustal layers. The values for calculated from the tilt data (3) are considerably smaller than the mean shear modulus value of the crust. The calculations are carried out for a single point force. Using a loading distribution one can increase the calculated value of by a factor of 1.5–2.0, but it still remains an order of magnitude less than the crustal shear modulus of the upper layers. The question arises whether the values for calculated from the tilt data are smaller because of cavity and topography effects, instead of medium properties. The topography in the plain region of the Protvino area induces deformation changes of about 3% (Kramer et al., 1985). The horizontal strains along the tunnels caused by air pressure are determined without any distortion. However, the tunnel can induce perturbations in the strain field when a point load acts. We need a rough estimate of these effects. For that, we use calculated models of the different underground chambers (Harrison, 1976; Dal Moro et al., 2001). The cavity does not increase the anomalous effects to a great extent when a violent stress concentrator (for example a deep narrow fault) is absent. In a homogeneous strain field the cavity effects vanish for tilts in the center of the chamber. The observed tilts (vector modulus) at points 1 and 2 are 11.5 mas and 44.5 mas. The ratio of the observed tilt to the calculated value is 41 at point 1 and 158 at point 2. Such great anomalies cannot be explained by cavity effects. They can be caused by the rock structure. The extensometers measure linear strains along the tunnel. In the case of air pressure, the cavity has no effect on the extensometer measurements. Looking at the obtained data the following conclusions can be made: 1. The observatory is located in a weakened zone where the shear modulus is, according to the tilt measurements, much smaller than its mean value for the upper crustal layer.
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2. This zone is heterogeneous, as according to the tilt measurements the elastic modulus of the rock is larger at point 1 than at point 2. The mobility of point 2 is probably caused by rock fractures, which were caused by natural or technical influences (Molodensky, 2000; Latynina and Boyarsky, 2000). 3. The measured tilt vector deviates from the radial direction (with the center at the load point) towards the NS direction. This means that the rigidity of the medium varies depending on the direction. The large tilts in NS direction could be explained by crack orientation and local crack strikes. 4. In the case of extensometer measurements, unlike the tilt measurements, the resulting value for is close to the value of undisturbed limestone and equals (2–3).105 bar. 5. The value for , obtained from extensometer data is noticeably different from the value calculated from the tiltmeter data. We explain this difference by the mechanism of a vertical load acting upon the fractured rocks. The measured tilt is caused by continuous deformation of the rock and by relative displacements of separate blocks. The experimental value of is the effective shear modulus of rocks, which reflects both processes. The influence of vertical cracks on the horizontal linear strain is small in the case of the closed cracks. Moreover, the tilts are measured at short bases (0.5 m), while the strains are measured at longer bases (20 m).
4. Spectral density of vibration processes The spectral analysis of the tilt and air pressure data sets at the Protvino Observatory was discussed in previous work (Latynina and Vasil’ev, 2001). The power spectral density S of the tilts varies from 0.1 (mas)2/cph up to 30 000 (mas)2/cph at point 1, and up to 200 000 (mas)2/cph at point 2 in a range from 2 to 500 h. The value S depends on the period T with power three. Fig. 2 illustrates the power spectral density of 6 month data sets (10.09.2000–10.03.2001) in the frequency range from 0.002 cph to 0.5 cph. The linear strains in NS and EW directions, the tilts in NS direction at point 1, and pressure variations are analyzed. The spectra in Fig. 2 are presented in different scales for clearness. Even at the stable point 1, the tilt spectrum curve lies higher than the strain curves. For a frequency of 0.003 cph the power spectral densities are equal to 5.1015/ cph for strains and to 150.1015/cph for tilts. The spectral densities S of the strains increase with T to power 2.5–3.0. There are peaks in the monotonous build-up of the spectral density of tilts and strains corresponding to the diurnal and semidiurnal tidal waves. For pressure variations, S peaks at T close to the duration of cyclones (10–15 days). Maybe, the analysis of longer sets of pressure data will allow us to explain the other spectral peaks. The relationship between S and T at Protvino observatory (as T3) is typical for low frequency noises and corresponds to Peterson’s ‘‘New Low Noise Model’’. It can be applied when the spectra of deformations are dominantly determined by atmospheric pressure variations. Similar results were obtained also with the superconducting gravimeter at the Geodynamic Observatory Moxa in Germany (Kroner et al., 2001). These spectra can be distinguished from the deformation spectra in active tectonic zones, such as the Friuli area in Italy (Braitenberg, 1999). Maybe, the contribution of air pressure in deformation processes is not very large in the last case.
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Fig. 2. The power spectrum of variations in N–S horizontal strains (curve 1, unit 0.1.(nstr)2/cph), in E–W horizontal strains (curve 2, unit 10.(nstr)2/cph), in N–S tilts (curve 3, unit 103.(nrad)2/cph), and in air pressure (curve 4, unit 103.(mbar)2/cph).
5. The relationship between the crustal deformations and the variations in air pressure 5.1. Estimation of air pressure effects A quantitative evaluation of displacements, tilts and deformations of the Earth crust has been performed for a cyclone model, where the cyclone is presented as axial-symmetric about the z-axis of the load applied to the Earth surface. The pressure on the axis corresponds to the pressure at the cyclone center. The distributed load has been numerically calculated using the Boussinesq formulae. The displacements and deformations of the Earth surface are proportional to the excess pressure P0 in the cyclone epicenter, the radius R of the loaded area, and a combination of the Lame´ constants l and . For P0=50 mbar, 2R=1000 km, and l==3.105 bar maximum values are found of 3 cm for the vertical displacement W, 5.5.108 mas for tilts ’, 0.4 cm for horizontal displacements U, and 3.0.108 for strains . The ratio of the maximum surface tilts and strains to the maximum pressure P0 is:
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’=P0 ¼ 1:1109 =mbar;
"=P0 ¼ 0:6109 =mbar
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ð5Þ
5.2. Results The vertical displacements caused by air pressure are one order of magnitude larger than the horizontal displacements. The vertical displacements peak in the cyclone epicenter and the area with the maximum horizontal displacements is shifted by R/2 relative to the epicenter. The horizontal strains peak in the epicenter.
Fig. 3. The long-period variations of air pressure (curve 1), N–S horizontal strains (curve 2), and E–W horizontal strains (curve 3).
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5.3. Experimental data analysis The similarity between the curves of the linear strains, tilts and atmospheric pressure is significant as is shown in Fig. 3, where the measurements for the period December 2000–February 2001 are presented. Variations with periods from 2 to 40 h (Earth tides) and from 500 to 4000 h (slow drift) are excluded. During pressure build-up periods dilatation is observed in EW direction and compression in NS direction. 5.4. Coherency function for air pressure and strain variations The relationship between the tilts or strains and pressure variations is given by the frequency depended transfer function (Bendat and Piersol, 1989). The contribution of barometric tilts or strains to general geophysical processes is given by the square modulus of a coherency function, here called gamma(f). The function gamma(f) is shown in Fig. 4. It reflects how the air pressure variations reflect on the strain components for a frequency range from 0.003 to 0.2 cph based on the data set 09.2000–03. 2001. In the range from 0.003 to 0.02 cph the average gamma(f) is 0.5 for NS strain and 0.2–0.3 for EW strain and NS tilt. This means that approximately 50% of the signal power of the NS strain and about 30% of the EW strain is related to pressure variations.
Fig. 4. Squared modulus of the coherence spectrum gamma(f) between pressure and strains N–S (curve 1), between pressure and strains E–W (curve 2), and between pressure and tilts N–S (curve 3) in point 1.
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This analysis confirms the earlier results reported in (Latynina and Vasil’ev, 2001): the reduction of gamma(f) to a value of 0.10–0.15 for frequencies higher than 0.3 cph. 5.5. Amplitude–frequency transfer function Fig. 5 shows the amplitude–frequency transfer function. This function, relating barometric and tilt variations, is equal to 5.109/mbar within the range of 0.002–0.02 cph (for periods of more than 2 days). This value is 5 times larger than the ratio of the maximum surface tilt and maximum
Fig. 5. Amplitude transfer function A(f) between pressure and strains N–S (curve 1), pressure and strains E–W (curve 2), and pressure and tilts N–S (curve 3).
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pressure for the cyclone model. For horizontal strains within the same frequency range the transfer function is equal to (0.6–1.0).109/mbar. This experimental value is close to the calculated one. Thus, the experimental data show that air pressure variations cause Earth surface tilts that are larger than horizontal strains and model values.
6. Conclusion The Protvino observatory location is traditionally considered favorable for investigating deformation processes. Plain topography, heavy sedimentary series, and the absence of regional faults create conditions for a homogenous stress–strain state. However, when a load was created to the Earth surface, the rock effective shear modulus calculated from the tilt data was significantly smaller than the mean crustal modulus. Its value depends on the measurement point and the tilt direction. The topography and cavity effects did not explain these anomalies. Moreover, the values of the shear modulus calculated from the horizontal strain data are one order of magnitude larger and are close to the mean value for the Earth crust. This means that the rock effective shear modulus not only reflects continuous deformations but also the relative displacements of separate blocks. At the same time the influence of vertical cracks on horizontal strains is small. High correlation between atmospheric pressure variations and tilt and strain variations is detected. Within a period of 2–15 days about 50% of the power of the strain variations is related to atmospheric pressure variations. This relation is probably less in zones where the tectonic movements are more intensive. Using the transfer function it has been shown that the tilts with barometric origin reach 5.109/mbar in a stable point in the Protvino observatory area, and the model value equals 1.1.109/mbar. The experimental horizontal strains of 0.6.109/mbar are approximately equal to the calculated strains. As the barometric tilt is given by the change of vertical displacement along a horizontal element, the large tilts at given wavelength imply large displacements. Therefore, the barometric effects may worsen the accuracy of geodetic measurements, especially concerning vertical crustal movements. References Bendat, J.S., Piersol, A.G., 1989. Random Data. Analysis and Measurement Procedures. Mir, Moscow. Boyarskyi, E.A., Vasil’ev, I.M., Suvorova, I.I., 2001. The study of tilts and strains at the Protvino Geophysical station. Izvestiya Physics of the Solid Earth 37 (N9), 764–770. Braitenberg, C., 1999. The Friuli (NE-Italy) tilt/strain gauges and short term observations. Annali di Geofisica 42 (N4), 637–664. Dal Moro, G., Ebblin, C., Zadro, M., 2001. The FEM in the interpretation of tilt/strainmeter observations in a cave: air pressure loading effects. Journal of the Geodetic Society of Japan 47 (1), 88–94. Harrison, J.C., 1976. Cavity and topographic effects in tilt and strain measurement. Journal of Geophysical Research 81 (N2), 319–328. Kramer, M.V., Karmaleeva, R.M., Latynina, L.A., Molodensky, S.M., 1985. Evaluation of the tide strain disturbance in conditions of the plain topography. Phizika Zemli N7, 80–86. (in Russian). Kroner, C., Jentzsch, G., 1998. Comparison of air pressure methods and discussion of other influences on gravity. In: Proc. of the Thirteenth International Symposium of Earth Tides, Brussels. pp. 423–430. Kroner, C., Jahr, T., Jentzsch, G., 2001. Comparison of data sets recorded with the dual sphere superconducting
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