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Slab residual gravity anomaly: gravity reduction due to subducting plates beneath the Japanese Islands
Journal of Geodynamics 36 (2003) 497–514 www.elsevier.com/locate/jog
Slab residual gravity anomaly: gravity reduction due to subducting plates beneath the Japanese Islands N. Furuse1, Y. Kono* Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan Received 29 April 2002; accepted 31 March 2003
Abstract Gravity contributions due to subducting plates (slabs) beneath the Japanese Islands were calculated using a three-dimensional slab model. The slab model is constructed from the depth of deep seismic planes. A density contrast between the slabs and the surrounding asthenosphere is assumed to be 0.065 g/cm3 and the thicknesses of the Pacific and the Philippine Sea plates are postulated to be 90 km and 40 km, respectively. Some other cases, such as the different plate thicknesses, the different density contrasts, layering structure within the slab, etc., are also examined in the course of the present study. The belts of the highest gravity contributions appear along the eastern and southern coasts of Japan. The contributions gradually decrease toward the Japan Sea, with the deepening of the subducting plates. The gravity highs reach over 220 mGal in the northeastern and over 80 mGal in the southwestern Honshu (main island of Japan), respectively. The Slab Residual Gravity Anomaly (SRGA) is obtained by subtracting the slab contribution from the observed gravity anomaly. The SRGA represents the gravity contributions mainly due to the crustal structures. The SRGA over the Japanese Islands reveals a stronger negative correlation between topography and the gravity anomaly. # 2003 Elsevier Ltd. All rights reserved.
1. Introduction Horizontal variations of gravity anomalies over the Japanese Islands were interpreted only in terms of the variation of crustal structures four decades ago (e.g., Kanamori, 1963; Mikumo, 1966). Later studies, however, indicated that the gravity anomalies over the Islands are strongly affected by not only the crustal structures but also heterogeneous density distribution within the upper mantle (e.g., Yoshii, 1972). * Corresponding author. Tel. +81-76-264-5731; fax.: +81-76-264-5746. E-mail address: [email protected] (Y. Kono). 1 Present address: Mitsubishi Space Software Co. Ltd., Kamimachiya, 524-3 Kamakura, Kanagawa 247-0065, Japan. 0264-3707/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0264-3707(03)00062-0
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Explosion seismological experiments revealed detailed crustal structures along several sections crossing the Japanese Islands (e.g., Yoshii and Asano, 1972; Aoki et al., 1972; Iwasaki et al., 1994, 2001; Moriya et al., 1998). Yoshii (1972) attempted to calculate a theoretical gravity anomaly due to the crustal structure with an empirical relationship between seismic wave velocities and densities of rocks. A significant discrepancy was found between the observed and calculated gravity anomalies. He showed that a substantial part of the anomaly could be explained by gravity contributions due to descending plates. He then proposed to call this difference the Residual Gravity Anomaly (RGA) (Yoshii, 1973), which should directly provide information on the mass anomalies within the upper mantle. However, in Yoshii’s work, features of the gravity anomaly caused by the upper mantle structure were discussed for a limited case where only a few reliable crustal sections had been observed from seismological experiments. Fujimoto and Ito (1996) calculated the RGA along a profile across northeastern Japan considering the effect of the Pacific plate. They interpreted the RGA arises from the anomalous mantle having low density beneath the Japanese Islands. This implicitly means that the asthenosphere contact directly with the Moho of the Islands. Precise locations of earthquakes around Japan shows the depth distribution of the upper surfaces of subducting plates (i.e., slabs) beneath the Japanese Islands (e.g., Mizoue et al., 1983; Ishida and Hasemi, 1988; Ishida, 1992; Noguchi, 1996; Nakamura et al., 1997). These studies provide, as a first approximation, a detailed three-dimensional configurations of slabs beneath the Japanese Islands. It is anticipated that such configurations of subducting plates produce long wavelength positive gravity anomalies over the island arcs, since the slabs are slightly denser than the surrounding mantle (i.e., asthenosphere). Denser slabs were postulated from the analyses of seismic wave velocities and attenuation by Utsu (1967, 1971) for the first time and later by many authors (e.g., Yoshii, 1973; Hirahara, 1977; Hasemi et al., 1984; Umino and Hasegawa, 1984). It is now widely accepted that the gravity field of the Japanese Islands should be strongly affected by gravity contribution due to subducting plates beneath them. Hereafter, we use a term, slab contribution, instead of gravity contribution due to subducting plates. Therefore, in order to interpret crustal or geological structures in terms of gravity anomalies, we need a quantitative evaluation of slab contributions. Hagiwara (1986) first estimated the slab contributions beneath central part of the Japanese Islands using a two-dimensional approximation. Later, Furuse (1990) computed gravity contribution of slabs based on a more realistic three-dimensional slab configuration for most regions of the Japanese Islands. The present paper is extended version of his studies. Recently, Ryoki (1999) reported gravity anomalies caused by slabs beneath southwestern and central Japan based on the different modeling of subducting slabs. In this study, we report (1) the results of the calculations of the gravity contributions of the slabs based on the detailed three-dimensional configurations of slabs and (2) the new residual gravity anomalies (Slab Residual Gravity Anomaly; SRGA) which presumably indicate gravity contributions due only to crustal structures of the Japanese Islands.
2. Construction of a three dimensional model of subducting plates beneath the Japanese Islands Dense seismic networks in Japan greatly increased the knowledge of the configuration of the Wadati-Benioff zones beneath the Japanese Islands. The upper boundaries of the slabs, as the
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first approximation, are usually defined by deep earthquake-planes. On the other hand, the depths of the lower boundary of the slabs were reported by several researchers but the values are still inconclusive. Thicknesses of the Pacific and the Philippine Sea plates have been estimated from the analyses of surface wave dispersions (Kanamori and Press, 1970; Yoshii, 1975a; Seekings and Teng, 1977; Yamanaka et al., 1992), travel time analyses of distant earthquakes (Fukao, 1977), long-range explosion experiments (Asada and Shimamura, 1976; Shimamura et al., 1983), thickening plate theories (e.g., Yoshii, 1973; Parker and Oldenburg, 1973; Kono and Yoshii, 1975; Yoshii et al., 1976) and converted phases of deep earthquakes (e.g., Nakanishi, 1980; Nakamura et al., 1998). We used iso-depth contour maps of the deep seismic planes derived from micro-earthquake studies as cited in the previous section, and maps of deep seismic planes compiled by Utsu (1971) and Yoshii (1979). In order to confirm and revise these data, we drew 249 cross sections of distribution of hypocenters by using the hypocenter data files from the Japan Meteorological Agency for the period of 1960–1998, the International Seismological Centre for the period of 1960–1989, and Japan University Network Earthquake Catalog for 1998. Referring to the hypocenter maps and these profiles, detailed contour maps of deep seismic planes are reconstructed. Fig. 1 shows the contours of the upper planes of the Pacific (A) and the Philippine Sea (B) plates thus determined. A wedge shaped region in the uppermost mantle between the aseismic front (Yoshii, 1975b) and the trench axis (Fig. 2) can be interpreted to be a part of a high Q–high V zone (Research Group for Explosion Seismology, 1977; Hirahara et al., 1989). A term of the aseismic front is defined by Yoshii (1975b) as a surface trace of landward limit of seismic active zone within a mantle wedge. This runs roughly parallel to both the volcanic front and the Japan Trench in northeastern Japan. The density in this region is assumed to be the same as that of the subducting plates obtained from seismological studies. The aseismic front is not traceable along the Nankai Trough in southwestern Japan.
3. Formulation Bodies of subducted plates are approximated by an assembly of parallel-piped prisms. The size of the larger prisms is taken to be 1 1 along the latitudinal and longitudinal directions. Smaller prisms of 100 150 (about 2020 km) are also employed in regions where the upper surface of slabs is shallower than 100 km. The total number of prisms is more than 1600. Since exact solutions of the gravity attraction of a parallel-piped prism in a spherical coordinate system are difficult to formulate, the computation of the contribution of each prism is done in a Cartesian coordinate system. The system is defined by a pair between each prism and a station at the Earth’s surface. Details on formulation are explained in the Appendix. The accuracy of the calculation is governed mainly by the following terms; grid sizes of prisms, truncation errors of the calculation, differences arising from not using the spherical coordinate system, and application of the line-mass method (an approximate solution of attraction of a prism at far distance) (Bott, 1958). We evaluated the total accuracy by comparing with theoretical solutions for a simplified subducting plate model. The results show that the total accuracy is better than 3%, which corresponds to a maximum of about 6 mGal.
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Fig. 1. Depth contour map of the upper plane of the subducting plates: (A) Pacific plate; (B) Philippine Sea plate. Contour unit is in kilometers.
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Fig. 2. Schematic illustration to show the present model to calculate gravity contribution due to a subducting plate.
Since upper several tens of kilometers of the Earth are occupied by continental or oceanic crusts with variable thicknesses, we compute the gravity contributions only from the part deeper than 30 km (Fig. 2). 30 km is chosen here because the largest crustal thickness of the Japanese Islands is reported to be about this value. Therefore, strictly speaking, our computation is done to obtain ‘‘gravity contributions due to density anomalies within the upper mantle deeper than 30 km’’. In other words, this is a requirement to extract density distributions within the upper part (shallower than 30 km) of the Earth including the Earth’s crust.
4. Results Fig. 3A–C show examples of the calculated gravity attraction due to the slabs. A constant thickness is postulated to be 80, 90 or 100 km for the Pacific plate, and 30 or 40 km for the Philippine Sea plate, respectively. In each case a density contrast between the slabs and the surrounding mantle (asthenosphere) is assumed to be 0.065 g/cm3 (Yoshii, 1973) based on P-wave velocity-density empirical relationship (Nafe and Drake, 1957; Barton, 1986). Fig. 3A shows the results for the smallest plate thickness combination, Fig. 3B for the thickest combination, and Fig. 3C an intermediate combination. All of these figures indicate that the largest gravity contribution exists over the Pacific Ocean sides of the Japanese Islands, i.e., an asymmetrical distribution of gravity contribution with respect to the elongation direction of the Islands. In northeastern Japan, a gravity contribution higher than 200 mGal appears along the Pacific coastline. A belt of the high gravity contributions is located about 200 km west of the Japan Trench. The highest region is situated in the southern extension, where the Philippine Sea plate subducts over the Pacific plate at a shallow depth (see Fig. 1). These gravity contributions decrease gradually toward the Japan Sea, and an average gradient is about 25 mGal per 100 km over the northeastern Honshu.
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Fig. 3. Contour maps of calculated gravity contributions due to the slabs. Contour interval is 20 mGal and an assumed density contrast is 0.065 g/cm3. Thickness of Pacific and Philippine Sea plates are assumed to be 80 km and 30 km for case A; 100 km and 40 km for case B; and 90 km and 30 km for case C, respectively. The case C is the preferred model in this study.
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.
Fig. 3. (continued.)
In southwestern Japan, a high gravity belt over 80 mGal appears parallel to Nankai Trough. The distance between the axes of the belt and the Trough is Gravity contribution due only to the Philippine Sea plate is 40 mGal at most since the thickness of the slab is nearly a half of that of the Pacific plate and angle is smaller.
the axis of the about 100 km. in our models, the subducting
5. Effect of thickness and density structures of slabs Gravity contributions due to a slab depends on several parameters such as the thickness of the slab, the density contrast between the slab and the asthenosphere, thickness and density variations of the slab during descent. These values are not quantitatively well known. Therefore to examine their effects, we calculated gravity contributions for some typical cases by using the twodimensional method (Talwani et al., 1959). Even the concept is not well defined, thicknesses of the Pacific plate is estimated between 80 and 100 km. In this study, the thickness of plate is defined as seismic velocity gaps between the surrounding asthenosphere. Effects of varying the plate thickness on the gravity contribution are shown in Fig. 4. The density difference is assumed to be 0.065 g/cm3. Varying the plate thickness by 10 km results in a differences of 10 mGal in gravity. Fig. 5 shows the results for three different density contrast values for the same slab thickness. Calculated gravity contribution curves are drawn for the density contrasts of 0.05, 0.065 and 0.08 g/cm3.
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Fig. 4. Effect of plate thickness on the gravity contribution. Density difference is assumed to be 0.065 g/cm3.
Fig. 5. Effect of density contrast on the gravity contribution. Three curves in the upper part of the figure are calculated by assuming the density contrast between the subducting plate and the surrounding asthenosphere as 0.05, 0.065 and 0.08 g/cm3, respectively. AF: aseismic front (see the text); !: trench axis.
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The results show that patterns of gravity contribution are similar, though the difference becomes about 40 mGal for the variation of density contrast of 0.015 g/cm3. Changes of gravity anomalies due to changes of plate thicknesses and/or density differences are roughly proportional to amount of their changes, but not completely proportional because their gravity contributions from deeper parts are smaller. Thinning of a slab as it goes deeper is thought to be mainly governed by the thermal effect (Toksoz et al., 1971). Curve b in Fig. 6 shows gravity contribution in this case. The assumed density contrast is 0.065 g/cm3. Curve a in Fig. 6 shows gravity contributions with a constant thickness and the same density contrast. An amplitude of the curve a is 40 mGal larger than that of the curve b. Effect of an oceanic crust on the gravity contributions is also examined. Its density is assumed to be smaller by 0.3 g/cm3 than that of the asthenosphere. The oceanic crust with 5 km thickness is postulated to subduct to a depth of 150 km. The result show that the maximum effect due to the oceanic crust becomes about 15 mGal. This effect reduces the total slab contributions. As mentioned previously, since the slab contribution from the Philippine Sea plate alone is 40 mGal at most, the effect due to oceanic layer would be critical for the evaluation of the slab contributions due to the subducting Philippine Sea plate. Grow and Bowin (1975) and Ahrens and Schubert (1975) pointed out the importance of phase transitions within the slab, such as the transition of oceanic basalt (3.0 g/cm3) to eclogite (3.55 g/cm3) (Ringwood and Green, 1966; Ito and Kennedy, 1971). Starting depth of transition from basalt to eclogite is estimated to be about 100 km. Peacock and Wang (1999) discussed the basalt–eclogite transition within subducted oceanic layer beneath the Japanese Islands and estimated it to occur at a shallower depth in the Philippine Sea plate than in the
Fig. 6. Thinning slab thickness model. The thickness of the slab at its leading edge is a half of that shown in Fig. 5. Assumed density contrast is 0.065 g/cm3. Curve a is the gravity contribution of the constant thickness model as was shown in Fig. 4. Curve b is calculated by using the structure shown in the lower part of the figure.
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Pacific plate. Following these propositions, we evaluate the effect of the transition on the gravity field. According to our calculations, even when the transformation occurs within the oceanic crust from a depth of 100 km and deeper, only a negligibly small gravity effect is observed, primarily a total mass is conserved even the phase transition occurs. From these numerical evaluations, it is concluded that the amplitude of gravity contributions are essentially sensitive to slab thickness and density contrast. In order to construct more reliable model, we need to carefully examine each effect on the slab contributions. Even though great uncertainty still remains, we choose Fig. 3C in the previous section as one of the plausible approximation of the slab contributions over the Japanese Islands, because this model gives a better approximation of the general trend of gravity anomaly over the Japanese Islands.
6. Slab residual gravity anomaly map over the Japanese Islands Fig. 7A illustrates a Bouguer anomaly map over the Japanese Islands and the adjacent sea. This is a simplified version of the detailed gravity anomaly map that has been published by us (Kono and Furuse, 1989a,b) with over half a million gravity measurements. Higher Bouguer anomalies appear over oceanic areas, particularly along the Pacific Ocean side, and major low anomalies appear over the central part of the Honshu Island, which is a high mountainous region. By subtracting the gravity contributions due to subducting plates from the observed gravity anomalies, we obtain a new gravity anomaly. Because the gravity contributions calculated in this study are those due to density structures beneath earth’s crust, this should be principally attributed to the gravity contributions due only to crustal structures. We propose to call this Slab Residual Gravity Anomaly (SRGA). We note again that the SRGA in this paper represents the gravity anomaly due to subsurface structures shallower than 30 km. Fig. 7B shows distribution of the SRGA (Slab Residual Bouguer Anomaly, in this case) over the Japanese Islands in a simplified form. As mentioned above, these patterns presumably indicate gravity contributions mainly from crustal structures of the Japanese Islands. The new gravity anomaly map shows more symmetrical distributions with topographic relief in and around the Japanese Islands. Levels of anomalies are nearly equal in Pacific Ocean and Japan Sea sides. These features indicate that our calculations of gravity contributions due to subducting plates were performed properly. In southwestern Japan, the SRGA is larger by about 50 mGal compared with that over central and northeastern Japan. This might infer that our model underestimates the slab contributions due to the Philippine Sea plate. In the present models, however, gravity contribution of oceanic layer within the slab is not taken into consideration. As mentioned previously, when oceanic layer exists at the top of the subducting plate, it reduces the slab contribution to 15 mGal. Furthermore, we have no data to employ thicker plate of the Philippine Sea plate than used here. Therefore apparent higher SRGA over southwestern Japan remains a future problem. For the purpose of demonstrating the effect of slab contribution more clear, we present in Fig. 8 cross-sections of the subducting Pacific plate, surface topographies, the calculated gravity contributions of the subducting plate, the observed gravity anomalies (Bouguer anomalies over both land and sea), and the SRGA across northeastern Japan. Notice that each gravity anomaly is not
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Fig. 7. Simplified gravity anomaly maps. The Slab Residual Gravity Anomaly (SRGA) is a gravity anomaly calculated by subtracting the gravity contribution due to subducting plates from observed gravity anomalies (Bouguer anomalies, in this case): (A) Bouguer anomaly; (B) Slab Residual Gravity Anomaly (SRGA). Contour interval is 10 mGal.
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Fig. 7. Continued
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plotted by their absolute values but by relative ones. We find that the general trend of the observed gravity anomalies on land primarily originates from the subducting plate. The pattern and magnitude of the slab contributions are concordant with the RGA by Yoshii (1973), which was calculated along the close location as shown in Fig. 8.
Fig. 8. Cross sections over northeastern Japan (roughly parallel to 39 N). From top to bottom: the Slab Residual Gravity Anomaly (SRGA); the observed gravity anomalies (Bouguer anomalies over both land and sea); the gravity contribution due to the subducting plate; topographic relief; and cross sections of the subducting Pacific plate (lower part of 30 km). Note that each gravity anomaly is not plotted by their absolute values but by relative ones.
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Recent precise multi-channel reflecting studies around the Nankai Trough (for example, Kodaira et al., 2002) are suggesting that upper boundaries of subducting plates are somewhat, say 10 km at the most, shallower than upper limit of deep seismic planes. The differences between the upper surfaces of the slabs and the deep seismic planes presumably be changing place by place along the trench axis. Therefore, we might need minor modification on even the configuration of slabs in the future.
7. Conclusions We evaluated the gravity field caused by the slabs beneath the Japanese Islands based on the plausible three-dimensional plate configuration models as shown in Fig. 1A and B. One of the preferred model is as follows: the thickness of the Pacific and the Philippine Sea plates are 90 and 30 km, respectively, and the density contrast between the subducting plates and the asthenosphere is 0.065 g/cm3 (Fig. 3C). These parameters are consistent with those in Yoshii (1972). The maximum ridges of the gravity contributions due to the slabs run parallel landward to the trench axes and the contributions decrease gradually toward the Japan Sea. The obtained gravity contribution is concordant with the Residual Gravity Anomalies in northeastern Japan calculated by Yoshii (1972) based on controlled seismological studies. By subtracting the gravity contributions from the observed gravity anomalies, we obtain the SRGA (Fig. 8B). The SRGA gives more direct information on the crustal structures. Threedimensional configuration of the upper boundaries of subducting plates are better constrained by employing deep seismic planes than three dimensional crustal structures of the Japanese Islands. Therefore, the SRGA is one of the important constrains of the three dimensional crustal structure, if they could be calculated properly. The following problems, however, still remain to be solved for more detailed discussions: we needed more precise evaluation of shape and thickness of the subducting Pacific and Philippine Sea plates, density contrast between slabs and asthenosphere, and the density layering within slabs. Combining data sets from explosion seismological experiments with the SRGA, we can proceed one step further to construct more realistic and reliable three-dimensional crustal structure of the Japanese Islands.
Acknowledgements We would like to thank Profs. S. Uyeda, T. Yoshii, and I. Kawasaki. They critically read the manuscript and gave us valuable suggestions. Mr. N. Maruyama greatly contributed for constructions of both 3D-subducting plate models and a prototype program to calculate gravity contribution due to subducting plates. Fig. 7(A) and (B) are drawn by using the Generic Mapping Tool (Wessel and Smith, 1998). The authors acknowledge valuable comments by reviewers (Dr. Keiling Wang and an anonymous reviewer).
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Appendix. Formulation for calculating a gravity contribution from parallelepiped prism For a parallelepiped prism whose upper and lower planes are parallel and both sides are rectangular, i.e., a rectangular parallelepiped prism, an analytical solution to give the gravity attraction at the origin of the coordinate system was derived by Banarjee and Gupta (1970) as 2 1=2 2 1=2 x þ y2 þ z2 þy x þ y2 þ z2 þx G þ y log gP ¼ ½x log 1=2 1=2 2 2 2 2 2 2 2 ðx þ y þ z Þ y ðx þ y þ z Þ x ðA1Þ 2z tan1
xy
x2 y2 z2
z ðx2 þ y2 þ z2 Þ1=2 x1 y1 z1
where each side is bounded by x=x1, x=x2, y=y1, y=y2, z=z1 and z=z2. G is a gravitational constant and is a density. In more general cases, the authors treat a parallelepiped prism whose top and bottom planes are not parallel. The solution is: " #z¼zb ð x2 ð y2 1 1 cos tan1 dxdy ðA2Þ gP ¼ G 1=2 2 2 ðx2 þ y2 Þ1=2 z¼z x1 y1 ðx þ y Þ t
where zt and zb are equations of top and bottom planes. A general form of these planes is zðx; yÞ ¼ a0 þ a1 x þ a2 y þ a3 xy
ðA3Þ
where a0 to a3 are determined with four local coordinates of the apexes. gP of Eq. (A2) is obtained by using the Gaussian numerical integration method (Abranowich and Stegun, 1970). To shorten the calculation time, we employ the line-mass approximation method (Bott, 1958) for prisms, which are in a distant region from the station P: 1 1 ðA4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gP ¼ GS1 S2 R R2 þ d 2 where, S1 ¼ jx2 x1 j ; S2 ¼ y2 y1 ; R ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x þ x 2 y þ y 2 1 2 1 2 þ 2 2
ðA5Þ
We used the expressions from (A1)–(A5) to calculate the expected gravity contributions. Effects of curvature of the Earth’s surface and fan-shaped deformation of prisms in the Cartesian coordinate system are taken into consideration as corrections of Eqs. (A1)–(A5). References Abranowitz, M., Stegun, I.A., 1970. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Pub, New York.
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Kono, Y., Yoshii, T., 1975. Numerical experiments on the thickening plate model. J. Phys. Earth 23, 63–75. Mikumo, T., 1966. A study on crustal structure in Japan by the use of seismic and gravity data. Bull. Earthq. Res. Inst., Univ. Tokyo 44, 965–1007. Mizoue, M., Nakamura, M., Sato, N., Ishiketa, Y., 1983. Three layered distribution of microearthquake in relation to focal mechanism variation in the Kii Peninsula, Southwestern Honshu. Jpn. Bull. Earthq. Res. Inst., Univ. Tokyo 58, 287–310. Moriya, T., Okada, H., Matsushima, T., Asano, S., Yoshii, T., Ikami, A., 1998. Collision structure in the upper crust beneath the southwestern foot of the Hidaka Mountains, Hokkaido, Japan, as derived from explosion seismic observations. Tectonophysics 290, 181–196. Nafe, J.E., Drake, C.L., 1957. Variation with depth in shallow and deep water marine sediments of porosity, density and the velocity of the compressional and shear waves. Geophysics 22, 523–552. Nakamura, M., Watanabe, H., Konomi, T., Kimura, S., Miura, K., 1997. Characteristic activities of subcrustal earthquakes along the outer zone of southwestern Japan. Bull. Dis. Prev. Res. Inst., Kyoto Univ. 40B-1, 1–20 (in Japanese with English abstract).. Nakamura, M., Ando, N., Ohkura, T., 1998. Fine structure of deep Wadati-Benioff Zone in the Izu-Bonine region estimated from S-to-P converted phase. Phys. Earth Planet. Int. 106, 63–74. Nakanishi, I., 1980. Precursors to ScS phases and dipping interface in the upper mantle beneath southwestern Japan. Tectonophysics 69, 1–35. Noguchi, S., 1996. Geometry of the Philippine Sea slab and convergent tectonics in the Tokai district, Japan. J. Seismol. Soc. Japan 49, 295–325 (in Japanese with English abstract). Parker, R.L., Oldenburg, D.W., 1973. Thermal model of ocean ridges. Nature 242, 137–139. Peacock, S., Wang, K., 1999. Seismic consequences of warm versus cool subduction metamorphism: examples from southwest and northeast Japan. Science 286, 937–939. Research Group for Explosion Seismology, 1977. Regionality off the upper mantle around Northeastern Japan derived from explosion seismic observations and its seismological implication. Tectonophysics 37, 117–130. Ringwood, A.E., Green, D.H., 1966. An experimental investigation of the gabbro-eclogite transition and some geophysical implications. Tectonophysics 3, 383–427. Ryoki, K., 1999. Three-dimensional depth structure of the crust and uppermost mantle beneath southwestern Japan and its regional gravity anomalies. J. Seismol. Soc. Jpn. 52, 51–63 (in Japanese with English abstract). Seekings, L.C., Teng, T., 1977. Lateral variations in the structure of the Philippine Sea plate. J. Geophys. Res. 82, 317–324. Shimamura, H., Asada, T., Suyehiro, K., Yamada, T., Inatani, H., 1983. Long-shot experiments to study velocity anisotropy in the oceanic lithosphere of the northwestern Pacific. Phys. Earth Planet. Inter. 31, 348–362. Talwani, M., Worzel, J.L., Landisman, M., 1959. Rapid gravity computation for two dimensional bodies with application to the Mendocino Submarine Fracture Zone. J. Geophys. Res. 64, 49–59. Toksoz, M.N., Minear, J.W., Julian, B.R., 1971. Temperature filed and geophysical effects of a down-going slab. J. Geophys. Res. 76, 1113–1138. Umino, N., Hasegawa, A., 1984. Three-dimensional Qs structure in the northeastern Japan arc. J. Seismol. Soc. Jpn. 37, 217–228 (in Japanese with English abstract). Utsu, T., 1967. Anomalies on seismic wave velocity and attenuation associated with deep earthquake zone (1). J. Fac. Sci. Hokkaido Univ. 1 (Ser.), 1–25. Utsu, T., 1971. Seismological evidence for anomalous structure of island arcs with special reference to the Japanese region. Rev. Geophys. Space Phys. 9, 839–890. Wessel, P., Smith, W.H.F., 1998. New, improved version of generic mapping tools released. EOS Trans. Amer. Geophys. U. 79 (47), 579. Yamanaka, T., Miyake, T., Hirahara, K., 1992. Three-dimensional S-wave velocity structure beneath the Japan island—configuration of the subducting slab. Bull. Earth. Res. Int., Univ. Tokyo 76, 265–302 (in Japanese with English abstract). Yoshii, T., 1972. Feature of upper mantle around Japan as inferred from gravity anomalies. J. Phys. Earth 20, 23–34. Yoshii, T., 1973. Upper mantle structure beneath the north Pacific and the marginal seas. J. Phys. Earth 21, 313–328. Yoshii, T., 1975a. Regionality of group velocities of Rayleigh waves in the Pacific and thickening of the plate. Earth Planet. Sci. Lett. 25, 305–312.
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Yoshii, T., 1975b. Proposal of the ‘‘Aseismic Front’’. J. Seismol. Soc. Jpn. 28, 365–367 (in Japanese with English abstract). Yoshii, T., 1979. Compilation of geophysical data around the Japanese Islands (I). Bull. Earthq. Res. Inst., Univ. Tokyo 54, 75–117 (in Japanese with English abstract). Yoshii, T., Asano, S., 1972. Time-term analyses of explosion seismic data. J. Phys. Earth 20, 47–57. Yoshii, T., Kono, Y., Ito, K., 1976. Thickening of the oceanic lithosphere. Am. Geophys. Union, Geophys. Monogr. 19, 423–430.
Slab residual gravity anomaly: gravity reduction due to subducting plates beneath the Japanese Islands N. Furuse1, Y. Kono* Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan Received 29 April 2002; accepted 31 March 2003
Abstract Gravity contributions due to subducting plates (slabs) beneath the Japanese Islands were calculated using a three-dimensional slab model. The slab model is constructed from the depth of deep seismic planes. A density contrast between the slabs and the surrounding asthenosphere is assumed to be 0.065 g/cm3 and the thicknesses of the Pacific and the Philippine Sea plates are postulated to be 90 km and 40 km, respectively. Some other cases, such as the different plate thicknesses, the different density contrasts, layering structure within the slab, etc., are also examined in the course of the present study. The belts of the highest gravity contributions appear along the eastern and southern coasts of Japan. The contributions gradually decrease toward the Japan Sea, with the deepening of the subducting plates. The gravity highs reach over 220 mGal in the northeastern and over 80 mGal in the southwestern Honshu (main island of Japan), respectively. The Slab Residual Gravity Anomaly (SRGA) is obtained by subtracting the slab contribution from the observed gravity anomaly. The SRGA represents the gravity contributions mainly due to the crustal structures. The SRGA over the Japanese Islands reveals a stronger negative correlation between topography and the gravity anomaly. # 2003 Elsevier Ltd. All rights reserved.
1. Introduction Horizontal variations of gravity anomalies over the Japanese Islands were interpreted only in terms of the variation of crustal structures four decades ago (e.g., Kanamori, 1963; Mikumo, 1966). Later studies, however, indicated that the gravity anomalies over the Islands are strongly affected by not only the crustal structures but also heterogeneous density distribution within the upper mantle (e.g., Yoshii, 1972). * Corresponding author. Tel. +81-76-264-5731; fax.: +81-76-264-5746. E-mail address: [email protected] (Y. Kono). 1 Present address: Mitsubishi Space Software Co. Ltd., Kamimachiya, 524-3 Kamakura, Kanagawa 247-0065, Japan. 0264-3707/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0264-3707(03)00062-0
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Explosion seismological experiments revealed detailed crustal structures along several sections crossing the Japanese Islands (e.g., Yoshii and Asano, 1972; Aoki et al., 1972; Iwasaki et al., 1994, 2001; Moriya et al., 1998). Yoshii (1972) attempted to calculate a theoretical gravity anomaly due to the crustal structure with an empirical relationship between seismic wave velocities and densities of rocks. A significant discrepancy was found between the observed and calculated gravity anomalies. He showed that a substantial part of the anomaly could be explained by gravity contributions due to descending plates. He then proposed to call this difference the Residual Gravity Anomaly (RGA) (Yoshii, 1973), which should directly provide information on the mass anomalies within the upper mantle. However, in Yoshii’s work, features of the gravity anomaly caused by the upper mantle structure were discussed for a limited case where only a few reliable crustal sections had been observed from seismological experiments. Fujimoto and Ito (1996) calculated the RGA along a profile across northeastern Japan considering the effect of the Pacific plate. They interpreted the RGA arises from the anomalous mantle having low density beneath the Japanese Islands. This implicitly means that the asthenosphere contact directly with the Moho of the Islands. Precise locations of earthquakes around Japan shows the depth distribution of the upper surfaces of subducting plates (i.e., slabs) beneath the Japanese Islands (e.g., Mizoue et al., 1983; Ishida and Hasemi, 1988; Ishida, 1992; Noguchi, 1996; Nakamura et al., 1997). These studies provide, as a first approximation, a detailed three-dimensional configurations of slabs beneath the Japanese Islands. It is anticipated that such configurations of subducting plates produce long wavelength positive gravity anomalies over the island arcs, since the slabs are slightly denser than the surrounding mantle (i.e., asthenosphere). Denser slabs were postulated from the analyses of seismic wave velocities and attenuation by Utsu (1967, 1971) for the first time and later by many authors (e.g., Yoshii, 1973; Hirahara, 1977; Hasemi et al., 1984; Umino and Hasegawa, 1984). It is now widely accepted that the gravity field of the Japanese Islands should be strongly affected by gravity contribution due to subducting plates beneath them. Hereafter, we use a term, slab contribution, instead of gravity contribution due to subducting plates. Therefore, in order to interpret crustal or geological structures in terms of gravity anomalies, we need a quantitative evaluation of slab contributions. Hagiwara (1986) first estimated the slab contributions beneath central part of the Japanese Islands using a two-dimensional approximation. Later, Furuse (1990) computed gravity contribution of slabs based on a more realistic three-dimensional slab configuration for most regions of the Japanese Islands. The present paper is extended version of his studies. Recently, Ryoki (1999) reported gravity anomalies caused by slabs beneath southwestern and central Japan based on the different modeling of subducting slabs. In this study, we report (1) the results of the calculations of the gravity contributions of the slabs based on the detailed three-dimensional configurations of slabs and (2) the new residual gravity anomalies (Slab Residual Gravity Anomaly; SRGA) which presumably indicate gravity contributions due only to crustal structures of the Japanese Islands.
2. Construction of a three dimensional model of subducting plates beneath the Japanese Islands Dense seismic networks in Japan greatly increased the knowledge of the configuration of the Wadati-Benioff zones beneath the Japanese Islands. The upper boundaries of the slabs, as the
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first approximation, are usually defined by deep earthquake-planes. On the other hand, the depths of the lower boundary of the slabs were reported by several researchers but the values are still inconclusive. Thicknesses of the Pacific and the Philippine Sea plates have been estimated from the analyses of surface wave dispersions (Kanamori and Press, 1970; Yoshii, 1975a; Seekings and Teng, 1977; Yamanaka et al., 1992), travel time analyses of distant earthquakes (Fukao, 1977), long-range explosion experiments (Asada and Shimamura, 1976; Shimamura et al., 1983), thickening plate theories (e.g., Yoshii, 1973; Parker and Oldenburg, 1973; Kono and Yoshii, 1975; Yoshii et al., 1976) and converted phases of deep earthquakes (e.g., Nakanishi, 1980; Nakamura et al., 1998). We used iso-depth contour maps of the deep seismic planes derived from micro-earthquake studies as cited in the previous section, and maps of deep seismic planes compiled by Utsu (1971) and Yoshii (1979). In order to confirm and revise these data, we drew 249 cross sections of distribution of hypocenters by using the hypocenter data files from the Japan Meteorological Agency for the period of 1960–1998, the International Seismological Centre for the period of 1960–1989, and Japan University Network Earthquake Catalog for 1998. Referring to the hypocenter maps and these profiles, detailed contour maps of deep seismic planes are reconstructed. Fig. 1 shows the contours of the upper planes of the Pacific (A) and the Philippine Sea (B) plates thus determined. A wedge shaped region in the uppermost mantle between the aseismic front (Yoshii, 1975b) and the trench axis (Fig. 2) can be interpreted to be a part of a high Q–high V zone (Research Group for Explosion Seismology, 1977; Hirahara et al., 1989). A term of the aseismic front is defined by Yoshii (1975b) as a surface trace of landward limit of seismic active zone within a mantle wedge. This runs roughly parallel to both the volcanic front and the Japan Trench in northeastern Japan. The density in this region is assumed to be the same as that of the subducting plates obtained from seismological studies. The aseismic front is not traceable along the Nankai Trough in southwestern Japan.
3. Formulation Bodies of subducted plates are approximated by an assembly of parallel-piped prisms. The size of the larger prisms is taken to be 1 1 along the latitudinal and longitudinal directions. Smaller prisms of 100 150 (about 2020 km) are also employed in regions where the upper surface of slabs is shallower than 100 km. The total number of prisms is more than 1600. Since exact solutions of the gravity attraction of a parallel-piped prism in a spherical coordinate system are difficult to formulate, the computation of the contribution of each prism is done in a Cartesian coordinate system. The system is defined by a pair between each prism and a station at the Earth’s surface. Details on formulation are explained in the Appendix. The accuracy of the calculation is governed mainly by the following terms; grid sizes of prisms, truncation errors of the calculation, differences arising from not using the spherical coordinate system, and application of the line-mass method (an approximate solution of attraction of a prism at far distance) (Bott, 1958). We evaluated the total accuracy by comparing with theoretical solutions for a simplified subducting plate model. The results show that the total accuracy is better than 3%, which corresponds to a maximum of about 6 mGal.
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Fig. 1. Depth contour map of the upper plane of the subducting plates: (A) Pacific plate; (B) Philippine Sea plate. Contour unit is in kilometers.
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Fig. 2. Schematic illustration to show the present model to calculate gravity contribution due to a subducting plate.
Since upper several tens of kilometers of the Earth are occupied by continental or oceanic crusts with variable thicknesses, we compute the gravity contributions only from the part deeper than 30 km (Fig. 2). 30 km is chosen here because the largest crustal thickness of the Japanese Islands is reported to be about this value. Therefore, strictly speaking, our computation is done to obtain ‘‘gravity contributions due to density anomalies within the upper mantle deeper than 30 km’’. In other words, this is a requirement to extract density distributions within the upper part (shallower than 30 km) of the Earth including the Earth’s crust.
4. Results Fig. 3A–C show examples of the calculated gravity attraction due to the slabs. A constant thickness is postulated to be 80, 90 or 100 km for the Pacific plate, and 30 or 40 km for the Philippine Sea plate, respectively. In each case a density contrast between the slabs and the surrounding mantle (asthenosphere) is assumed to be 0.065 g/cm3 (Yoshii, 1973) based on P-wave velocity-density empirical relationship (Nafe and Drake, 1957; Barton, 1986). Fig. 3A shows the results for the smallest plate thickness combination, Fig. 3B for the thickest combination, and Fig. 3C an intermediate combination. All of these figures indicate that the largest gravity contribution exists over the Pacific Ocean sides of the Japanese Islands, i.e., an asymmetrical distribution of gravity contribution with respect to the elongation direction of the Islands. In northeastern Japan, a gravity contribution higher than 200 mGal appears along the Pacific coastline. A belt of the high gravity contributions is located about 200 km west of the Japan Trench. The highest region is situated in the southern extension, where the Philippine Sea plate subducts over the Pacific plate at a shallow depth (see Fig. 1). These gravity contributions decrease gradually toward the Japan Sea, and an average gradient is about 25 mGal per 100 km over the northeastern Honshu.
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Fig. 3. Contour maps of calculated gravity contributions due to the slabs. Contour interval is 20 mGal and an assumed density contrast is 0.065 g/cm3. Thickness of Pacific and Philippine Sea plates are assumed to be 80 km and 30 km for case A; 100 km and 40 km for case B; and 90 km and 30 km for case C, respectively. The case C is the preferred model in this study.
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.
Fig. 3. (continued.)
In southwestern Japan, a high gravity belt over 80 mGal appears parallel to Nankai Trough. The distance between the axes of the belt and the Trough is Gravity contribution due only to the Philippine Sea plate is 40 mGal at most since the thickness of the slab is nearly a half of that of the Pacific plate and angle is smaller.
the axis of the about 100 km. in our models, the subducting
5. Effect of thickness and density structures of slabs Gravity contributions due to a slab depends on several parameters such as the thickness of the slab, the density contrast between the slab and the asthenosphere, thickness and density variations of the slab during descent. These values are not quantitatively well known. Therefore to examine their effects, we calculated gravity contributions for some typical cases by using the twodimensional method (Talwani et al., 1959). Even the concept is not well defined, thicknesses of the Pacific plate is estimated between 80 and 100 km. In this study, the thickness of plate is defined as seismic velocity gaps between the surrounding asthenosphere. Effects of varying the plate thickness on the gravity contribution are shown in Fig. 4. The density difference is assumed to be 0.065 g/cm3. Varying the plate thickness by 10 km results in a differences of 10 mGal in gravity. Fig. 5 shows the results for three different density contrast values for the same slab thickness. Calculated gravity contribution curves are drawn for the density contrasts of 0.05, 0.065 and 0.08 g/cm3.
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Fig. 4. Effect of plate thickness on the gravity contribution. Density difference is assumed to be 0.065 g/cm3.
Fig. 5. Effect of density contrast on the gravity contribution. Three curves in the upper part of the figure are calculated by assuming the density contrast between the subducting plate and the surrounding asthenosphere as 0.05, 0.065 and 0.08 g/cm3, respectively. AF: aseismic front (see the text); !: trench axis.
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The results show that patterns of gravity contribution are similar, though the difference becomes about 40 mGal for the variation of density contrast of 0.015 g/cm3. Changes of gravity anomalies due to changes of plate thicknesses and/or density differences are roughly proportional to amount of their changes, but not completely proportional because their gravity contributions from deeper parts are smaller. Thinning of a slab as it goes deeper is thought to be mainly governed by the thermal effect (Toksoz et al., 1971). Curve b in Fig. 6 shows gravity contribution in this case. The assumed density contrast is 0.065 g/cm3. Curve a in Fig. 6 shows gravity contributions with a constant thickness and the same density contrast. An amplitude of the curve a is 40 mGal larger than that of the curve b. Effect of an oceanic crust on the gravity contributions is also examined. Its density is assumed to be smaller by 0.3 g/cm3 than that of the asthenosphere. The oceanic crust with 5 km thickness is postulated to subduct to a depth of 150 km. The result show that the maximum effect due to the oceanic crust becomes about 15 mGal. This effect reduces the total slab contributions. As mentioned previously, since the slab contribution from the Philippine Sea plate alone is 40 mGal at most, the effect due to oceanic layer would be critical for the evaluation of the slab contributions due to the subducting Philippine Sea plate. Grow and Bowin (1975) and Ahrens and Schubert (1975) pointed out the importance of phase transitions within the slab, such as the transition of oceanic basalt (3.0 g/cm3) to eclogite (3.55 g/cm3) (Ringwood and Green, 1966; Ito and Kennedy, 1971). Starting depth of transition from basalt to eclogite is estimated to be about 100 km. Peacock and Wang (1999) discussed the basalt–eclogite transition within subducted oceanic layer beneath the Japanese Islands and estimated it to occur at a shallower depth in the Philippine Sea plate than in the
Fig. 6. Thinning slab thickness model. The thickness of the slab at its leading edge is a half of that shown in Fig. 5. Assumed density contrast is 0.065 g/cm3. Curve a is the gravity contribution of the constant thickness model as was shown in Fig. 4. Curve b is calculated by using the structure shown in the lower part of the figure.
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Pacific plate. Following these propositions, we evaluate the effect of the transition on the gravity field. According to our calculations, even when the transformation occurs within the oceanic crust from a depth of 100 km and deeper, only a negligibly small gravity effect is observed, primarily a total mass is conserved even the phase transition occurs. From these numerical evaluations, it is concluded that the amplitude of gravity contributions are essentially sensitive to slab thickness and density contrast. In order to construct more reliable model, we need to carefully examine each effect on the slab contributions. Even though great uncertainty still remains, we choose Fig. 3C in the previous section as one of the plausible approximation of the slab contributions over the Japanese Islands, because this model gives a better approximation of the general trend of gravity anomaly over the Japanese Islands.
6. Slab residual gravity anomaly map over the Japanese Islands Fig. 7A illustrates a Bouguer anomaly map over the Japanese Islands and the adjacent sea. This is a simplified version of the detailed gravity anomaly map that has been published by us (Kono and Furuse, 1989a,b) with over half a million gravity measurements. Higher Bouguer anomalies appear over oceanic areas, particularly along the Pacific Ocean side, and major low anomalies appear over the central part of the Honshu Island, which is a high mountainous region. By subtracting the gravity contributions due to subducting plates from the observed gravity anomalies, we obtain a new gravity anomaly. Because the gravity contributions calculated in this study are those due to density structures beneath earth’s crust, this should be principally attributed to the gravity contributions due only to crustal structures. We propose to call this Slab Residual Gravity Anomaly (SRGA). We note again that the SRGA in this paper represents the gravity anomaly due to subsurface structures shallower than 30 km. Fig. 7B shows distribution of the SRGA (Slab Residual Bouguer Anomaly, in this case) over the Japanese Islands in a simplified form. As mentioned above, these patterns presumably indicate gravity contributions mainly from crustal structures of the Japanese Islands. The new gravity anomaly map shows more symmetrical distributions with topographic relief in and around the Japanese Islands. Levels of anomalies are nearly equal in Pacific Ocean and Japan Sea sides. These features indicate that our calculations of gravity contributions due to subducting plates were performed properly. In southwestern Japan, the SRGA is larger by about 50 mGal compared with that over central and northeastern Japan. This might infer that our model underestimates the slab contributions due to the Philippine Sea plate. In the present models, however, gravity contribution of oceanic layer within the slab is not taken into consideration. As mentioned previously, when oceanic layer exists at the top of the subducting plate, it reduces the slab contribution to 15 mGal. Furthermore, we have no data to employ thicker plate of the Philippine Sea plate than used here. Therefore apparent higher SRGA over southwestern Japan remains a future problem. For the purpose of demonstrating the effect of slab contribution more clear, we present in Fig. 8 cross-sections of the subducting Pacific plate, surface topographies, the calculated gravity contributions of the subducting plate, the observed gravity anomalies (Bouguer anomalies over both land and sea), and the SRGA across northeastern Japan. Notice that each gravity anomaly is not
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Fig. 7. Simplified gravity anomaly maps. The Slab Residual Gravity Anomaly (SRGA) is a gravity anomaly calculated by subtracting the gravity contribution due to subducting plates from observed gravity anomalies (Bouguer anomalies, in this case): (A) Bouguer anomaly; (B) Slab Residual Gravity Anomaly (SRGA). Contour interval is 10 mGal.
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Fig. 7. Continued
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plotted by their absolute values but by relative ones. We find that the general trend of the observed gravity anomalies on land primarily originates from the subducting plate. The pattern and magnitude of the slab contributions are concordant with the RGA by Yoshii (1973), which was calculated along the close location as shown in Fig. 8.
Fig. 8. Cross sections over northeastern Japan (roughly parallel to 39 N). From top to bottom: the Slab Residual Gravity Anomaly (SRGA); the observed gravity anomalies (Bouguer anomalies over both land and sea); the gravity contribution due to the subducting plate; topographic relief; and cross sections of the subducting Pacific plate (lower part of 30 km). Note that each gravity anomaly is not plotted by their absolute values but by relative ones.
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Recent precise multi-channel reflecting studies around the Nankai Trough (for example, Kodaira et al., 2002) are suggesting that upper boundaries of subducting plates are somewhat, say 10 km at the most, shallower than upper limit of deep seismic planes. The differences between the upper surfaces of the slabs and the deep seismic planes presumably be changing place by place along the trench axis. Therefore, we might need minor modification on even the configuration of slabs in the future.
7. Conclusions We evaluated the gravity field caused by the slabs beneath the Japanese Islands based on the plausible three-dimensional plate configuration models as shown in Fig. 1A and B. One of the preferred model is as follows: the thickness of the Pacific and the Philippine Sea plates are 90 and 30 km, respectively, and the density contrast between the subducting plates and the asthenosphere is 0.065 g/cm3 (Fig. 3C). These parameters are consistent with those in Yoshii (1972). The maximum ridges of the gravity contributions due to the slabs run parallel landward to the trench axes and the contributions decrease gradually toward the Japan Sea. The obtained gravity contribution is concordant with the Residual Gravity Anomalies in northeastern Japan calculated by Yoshii (1972) based on controlled seismological studies. By subtracting the gravity contributions from the observed gravity anomalies, we obtain the SRGA (Fig. 8B). The SRGA gives more direct information on the crustal structures. Threedimensional configuration of the upper boundaries of subducting plates are better constrained by employing deep seismic planes than three dimensional crustal structures of the Japanese Islands. Therefore, the SRGA is one of the important constrains of the three dimensional crustal structure, if they could be calculated properly. The following problems, however, still remain to be solved for more detailed discussions: we needed more precise evaluation of shape and thickness of the subducting Pacific and Philippine Sea plates, density contrast between slabs and asthenosphere, and the density layering within slabs. Combining data sets from explosion seismological experiments with the SRGA, we can proceed one step further to construct more realistic and reliable three-dimensional crustal structure of the Japanese Islands.
Acknowledgements We would like to thank Profs. S. Uyeda, T. Yoshii, and I. Kawasaki. They critically read the manuscript and gave us valuable suggestions. Mr. N. Maruyama greatly contributed for constructions of both 3D-subducting plate models and a prototype program to calculate gravity contribution due to subducting plates. Fig. 7(A) and (B) are drawn by using the Generic Mapping Tool (Wessel and Smith, 1998). The authors acknowledge valuable comments by reviewers (Dr. Keiling Wang and an anonymous reviewer).
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Appendix. Formulation for calculating a gravity contribution from parallelepiped prism For a parallelepiped prism whose upper and lower planes are parallel and both sides are rectangular, i.e., a rectangular parallelepiped prism, an analytical solution to give the gravity attraction at the origin of the coordinate system was derived by Banarjee and Gupta (1970) as 2 1=2 2 1=2 x þ y2 þ z2 þy x þ y2 þ z2 þx G þ y log gP ¼ ½x log 1=2 1=2 2 2 2 2 2 2 2 ðx þ y þ z Þ y ðx þ y þ z Þ x ðA1Þ 2z tan1
xy
x2 y2 z2
z ðx2 þ y2 þ z2 Þ1=2 x1 y1 z1
where each side is bounded by x=x1, x=x2, y=y1, y=y2, z=z1 and z=z2. G is a gravitational constant and is a density. In more general cases, the authors treat a parallelepiped prism whose top and bottom planes are not parallel. The solution is: " #z¼zb ð x2 ð y2 1 1 cos tan1 dxdy ðA2Þ gP ¼ G 1=2 2 2 ðx2 þ y2 Þ1=2 z¼z x1 y1 ðx þ y Þ t
where zt and zb are equations of top and bottom planes. A general form of these planes is zðx; yÞ ¼ a0 þ a1 x þ a2 y þ a3 xy
ðA3Þ
where a0 to a3 are determined with four local coordinates of the apexes. gP of Eq. (A2) is obtained by using the Gaussian numerical integration method (Abranowich and Stegun, 1970). To shorten the calculation time, we employ the line-mass approximation method (Bott, 1958) for prisms, which are in a distant region from the station P: 1 1 ðA4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gP ¼ GS1 S2 R R2 þ d 2 where, S1 ¼ jx2 x1 j ; S2 ¼ y2 y1 ; R ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x þ x 2 y þ y 2 1 2 1 2 þ 2 2
ðA5Þ
We used the expressions from (A1)–(A5) to calculate the expected gravity contributions. Effects of curvature of the Earth’s surface and fan-shaped deformation of prisms in the Cartesian coordinate system are taken into consideration as corrections of Eqs. (A1)–(A5). References Abranowitz, M., Stegun, I.A., 1970. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Pub, New York.
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