Surface potential and gravity changes due to internal dislocations in a spherical earthöii. Application to a nite fault

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1 Geophys. J. Int. (998) 32, 79^88 Surface potential and gravity changes due to internal dislocations in a spherical earthöii. Application to a nite fault Wenke Sun and Shuhei Okubo 2 Department of Geodesy and Photogrammetry, Royal Institute of Technology, S-0044 Stockholm, Sweden. sunw@geomatics.kth.se 2 Earthquake Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo 3, Japan Accepted 997 July. Received 997 May 20; in original form 996 October 3 SUMMARY We present a numerical formulation for computing elastic deformations caused by a dislocation on a nite plane in a spherically symmetric earth. It is based on our previous work for a point dislocation (Sun & Okubo 993). The formulation enables us to compute the displacement, potential and gravity changes due to an earthquake modelled as spatially distributed dislocations. As an application of the nite-fault dislocation theory, we make a case study of the theoretical and observed gravity changes. The computed results are in excellent agreement with the observed gravity changes during the earthquake. The gravity changes in the near eld can reach some 00 ]gal, which can be easily detected by any modern gravimeter. In the far eld they are still signi cantly large: jdgj > 0 ]gal within the epicentral distance h < 6 0 ; jdgj > ]gal within h < 6 0 ; jdgj > 0: ]gal within h < 40 0 ;andjdgj > 0:0 ]gal globally. We also calculate the geoid height changes caused by the 964 Alaska earthquake and by the same earthquake with revised parameters and an assumed barrier. We nd that the earthquake should have caused geoid height changes as large as.5 cm. Key words: coseismic change, dislocation, geoid, gravitational potential, gravity. INTRODUCTION The dislocation theory has been successfully applied to model the coseismic displacement eld (Steketee 958; Maruyama 964; Saito 967; Okada 985). Studies on the potential and gravity changes caused by various dislocations started in the late 970s. Hagiwara (977) rst investigated the elevation and gravity changes due to an explosive source. Rundle (978) gave the relations of gravity changes and the Palmdale uplift. Hagiwara et al. (985) detected signi cant gravity changes due to large earthquakes: the 978 Izu-Oshima-kinkai earthquake (M7.0) and the 980 Izu-hanto-toho-oki earthquake (M6.7). Sasai (986, 988) studied the surface displacement, gravity and magnetic changes associated with multiple tensile cracks of a Gaussian distribution. Okubo (99, 992) studied the problem of potential and gravity changes caused by point dislocations and by faulting on a nite plane in a semi-in nite medium. He derived all sets of expressions in closed forms. They enable us to evaluate coseismic changes in surface gravity and geoid height. His numerical simulation shows that a great earthquake could cause a geoid height change of the order of 0. m. Okubo et al. (99) detected a signi cant gravity change due to the 989 earthquake swarm and the submarine eruption o Ito. They successfully explained the gravity change using Okubo's (992) theory. The gravity change caused by the 986 eruption of the Izu-Oshima volcano was also studied by Okubo et al. (988). Okubo (993) presented a reciprocity theorem for computing the static deformation due to a point dislocation buried in a spherically symmetric earth. Sun and Okubo (Sun 992a,b; Sun & Okubo 993, 994, 995; Sun, Okubo & Van c ek 996) studied the deformations of a spherical earth by a point dislocation. They de ned a set of dislocation Love numbers and Green's functions. They derived spherical harmonic expressions for the shear and tensile dislocations, which can be expressed by four independent solutions: vertical strike-slip, vertical dip-slip, a tensile opening on a horizontal plane and a tensile opening on a vertical plane. In the following, Sun & Okubo (993) is abbreviated to SO93. The theoretical studies on coseismic global potential, gravity and displacement indicated that coseismic deformations could be detectable by modern geodetic techniques at very large epicentral distances (SO93; Sun et al. 996). Ma & Kusznir (994) derived subsurface displacements for faults in a three-layered elastic-gravitational medium and examined coseismic and post-seismic surface and subsurface displacement during continental extensional faulting. Piersanti et al. (995) and Sabadini, Piersanti & Spada (995) studied the displacement and rates induced by a dislocation in viscoelastic, strati ed earth models, taking sphericity and ß998RAS 79 GJI000 22//98 :48:02 3B2 version 5.20

2 80 W. Sun and S. Okubo self-gravitation into account. Their theoretical studies on postseismic global deformations suggested a similar conclusion: horizontal motions associated with great earthquakes are detectable in the far eld. Recently, Piersanti, Spada & Sabadini (997) studied the post-seismic rebound caused by nite dislocations in a viscoelastic earth. They found that the post-seismic rebound due to the 964 Alaska earthquake plays a relevant role in VLBI baseline changes. In this paper, we will present a numerical recipe to compute the surface deformation due to faulting on a nite plane. This is achieved by slightly extending the point-source theory of SO93. The new formulation makes it possible to estimate coseismic deformation in the near eld where the source dimension is no longer negligible. 2 THEORY FOR A POINT DISLOCATION In the following we brie y outline the point-source theory of SO93. Let us rst take the conventional polar coordinate system with h and denoting the colatitude and the longitude, respectively. Consider now a point dislocation at D(h, ) with h ~ ~0 0 (see Fig. in SO93). This is equivalent to considering an earthquake located on the polar axis, keeping the fault line along the Greenwich meridian and UdS~; U is the dislocation on an in nitesimal fault of area ds. Choose then an observation point at P(h 2, 2 ) (see Fig. 4 in SO93). We also de ne the fault plane by its dip angle, d,withrespect to the Earth's surface, and its slip angle, j, measured in the fault plane counterclockwise from the horizontal. Then the solutions of the two excitation problemsöshear and tensileö are expressed as follows. Shear dislocation: Z (s) ~ cos j[{z 2 sin d{z 23 cos d] z sin j 2 (Z33 {Z 22 )sin2d{z 23 cos 2d : () Tensile dislocation: Z (t) ~Z 33 {(Z 33 {Z 22 )sin 2 d{z 23 sin 2d, (2) where Z ij is a set of four independent solutions. The superscripts ij (i, j~, 2, 3) stand for a dislocation component l i on a plane of in nitesimal area ds with a normal n j. Z 2, Z 23, Z 22 and Z 33 thus stand for deformation due to vertical strike-slip, vertical dip-slip, a tensile opening on a horizontal plane, and a tensile opening on a vertical plane, respectively. The asterisk denotes the complex conjugate. Note that Z is a generic name for the displacement u r, potential t, gravity changes *g (without free-air correction) and dg (with free-air correction) on the Earth's surface. With the four independent solutions, we can derive the expressions for a dislocation at an arbitrary point on the Earth's surface. We denote (see Fig. 4 in SO93) the angular distance between the dislocation D(h, ) and the observation point P(h 2, 2 )byr, and de ne the fault plane by its strike azimuth z on the Earth's surface, measured clockwise from the north. The azimuth of the calculated point, with respect to the point D(h, ), is denoted z 2, again measured clockwise from the north. The azimuth z of P with respect to the fault line is z~z {z 2. (3) The angular distance r and the azimuth z 2 are given by the following spherical trigonometric formulae: cos r~ cos h cos h 2 z sin h sin h 2 cos ( 2 { ), (4) sin z 2 ~ sin r sin h 2 sin ( 2 { ), (5) cos z 2 ~ sin h sin r (cosh 2{ cos h cos r). (6) Then we get the expressions of, for example, the potential caused by a point dislocation at an arbitrary point: t (s) (a, r, z; d s ) ~ cos j[{^t 2 (a, r)sindsin 2zz^t 32 (a, r)cosdcos z] z sin j 2 sin 2d[^t 33 (a, r){^t 22,0 (a, r)z^t 2 (a, r)cos2z] z^t 32 (a, r)cos2dsin z, (7) t (t) (a, r, z; d s ) ~^t 33 (a, r)cos 2 dz(^t 22,0 (a, r){^t 2 (a, r)cos2z)sin 2 d {^t 32 (a, r)sin2d sin z, (8) where d s is the depth of the dislocation. Note that ^t ij (a, r) are implicit functions of depth d s. Similar forms apply to the expressions for displacement u r and gravity changes *g and dg; we have only to replace t in (7) and in (8) with u r, *g or dg. In numerical calculations, we obtain the surface deformation by using the following procedure. () Compute epicentral distance r and azimuth z of the observation point P(h 2, 2 ) from the dislocation at D(h, ) after (3)^(6). (2) Compute ^u ij r (a,r), ^tij (a,r), *^g ij (a,r) and d^g ij (a,r) as describedinso93. (3) Compute u ij r (a, r, z; d s), t ij (a, r, z; d s ), *g ij (a, r, z; d s ) and dg ij (a, r, z; d s ) using (7)^(8). These deformation elds are due to a point dislocation of unit magnitude UdS~. (4) Multiply the results of the above item by the magnitude of the dislocation UdS. 3 COMPUTATIONS FOR A FINITE FAULT When the epicentral distance is much larger than the fault dimension, the above point-dislocation theory can be used immediately. In the case of the near eld, however, we may no longer regard the source as being a point dislocation. We must take the distribution of dislocations on a nite fault into account. In fact, this can be simply done by integrating the point dislocations over the actual fault plane. In the following we discuss how to derive the deformation in the near eld from the results concerning point dislocations. Fig. shows a seismic fault located at a point (h, ; d s ), with length L, widthw, dip angle d and strike angle z,where d s ~a{r is the source depth of the upper-left corner of the nite fault ds(r, h, ). Notice that we take here the upperleft corner of the fault as being the origin for computational convenience. GJI000 22//98 :49:0 3B2 version 5.20

3 Surface potential and gravity changesöii 8 of the integral kernels (u ij r, tij, *g ij and dg ij ) are expressed in terms of linear combinations of the four types of independent solutions. Note that there are implied sums over i and j in (0)^(3). Since there are no analytical forms of u ij r, tij, *g ij and dg ij,we have to perform numerical integration to evaluate (0)^(3) by dividing the fault plane into nite elements. The elements should be small enough so that the dislocation on each small element can be regarded as a point source. When the spherical coordinate of the centre of each kernel is obtained, the integral kernel in (0)^(3) can be derived from (7)^(8). For a detailed derivation, see Appendix A. The nal formulae obtained from (0)^(3) are as follows: u r (a, h, )~ t(a, h, )~ XP p~ XP p~ X Q q~ X Q q~ u (st) r (a, ' pq, $ pq ; D pq s!us ), (4) t (st) (a, ' pq, $ pq ; D pq s )!US, (5) *g(a, h, )~ dg(a, h, )~ XP p~ XP p~ X Q q~ X Q q~ *g (st) (a, ' pq, $ pq ; D pq s )!US, (6) dg (st) (a, ' pq, $ pq ; D pq s )!US, (7) where the superscript (st) takes either (s) (shear dislocation) or (t) (tensile dislocation). Figure. Geometry of a nite fault model. The upper gure shows the fault geometry in spherical coordinates with a division of the fault plane. d s is the source depth, d is the dip angle. The lower gure indicates the projection of the fault plane onto the E^N plane. Since a general solution for a point dislocation can be expressed as Z~Z ij l i n j UdS, (9) we have the following formulae for a nite fault: u r (a, h, )~ u ij r (a, ', $; D s)l i (r, h, )n j UdS(r, h, ), S (0) t(a, h, )~ t ij (a, ', $; D s )l i (r, h, )n j UdS(r, h, ), S () *g(a, h, )~ *g ij (a, ', $; D s )l i (r, h, )n j UdS(r, h, ), S (2) dg(a, h, )~ dg ij (a, ', $; D s )l i (r, h, )n j UdS(r, h, ), S (3) where S denotes the fault area, ' the angular distance between the surface element ds at (r, h, ) and the observation point, and $ the azimuth of the observation point with respect to the fault line. D s is the source depth of the centre of each element (see Fig. ) and is determined later. The 3 3~9 components 4 GRAVITY CHANGES CAUSED BY THE 964 ALASKA EARTHQUAKE As an illustration, we apply our theory to estimate the gravity changes caused by the 964 Alaska earthquake (M w ~9:2). Comparison of the computed gravity change with the observed one should provide an opportunity to test the validity of our formulae. Before and after the earthquake, gravity surveys were made at some stations in the area with a LaCoste and Romberg gravity meter of reading sensitivity 0.0 mgal (Barnes 966). The 0 reoccupied stations are distributed over a wide part of both the uplifted and the depressed area (Fig. 2). The gravity changes at these stations are also given in Fig. 2 with contour lines. The gravity changes versus the elevation changes obtained from levelling or shoreline studies near these stations are shown in Fig. 3 (Barnes 966). We try to explain the gravity changes caused by the Alaska earthquake. The fault parameters of the earthquake were given by Savage & Hastie (966) (Table ). Using the fault parameters in Table, the 066A earth model and the dislocation theory in Section 3, we calculate the gravity changes at the 0 stations. The results are shown in Figs 4 and 5. To compare easily the observed and predicted gravity changes we plot the observed and calculated results in Fig. 6, showing the observed value on the x-axis and the predicted value on the y-axis. We see that the computed gravity changes are in excellent agreement with the observed ones. The agreement indicates that our theory explains well the deformation caused by an earthquake. We also consider the far- eld gravity changes. Since the epicentral distance of a remote area is much larger than GJI000 22//98 :49:32 3B2 version 5.20

4 82 W. Sun and S. Okubo Figure 2. Locations of gravity stations that were reoccupied in 964 and observed gravity changes caused by the Alaska earthquake (after Barnes 966). the geometrical size of the earthquake, the point-dislocation theory can be used directly for the far- eld calculation. We calculate and plot the far- eld gravity changes in Fig. 7, which shows that jdgj > 0 ]gal within the epicentral distance h < 6 0, jdgj > ]gal within h < 6 0, jdgj > 0: ]gal within h < 40 0, and jdgj > 0:0 ]gal globally. This observation implies that a coseismic gravity step caused by a large earthquake, as large as the 964 Alaska earthquake, can be detected by an absolute gravimeter and a superconducting gravimeter, especially when h < AN APPLICATION TO POTENTIAL/ GEOID HEIGHT CHANGES As an application to the potential change, we estimate geoid height changes due to a large earthquake in this section. The principle of the geoid height change is very simple. Since we have already obtained the potential changes due to dislocations, we may easily calculate the geoid height changes from the potential changes by dividing by the gravity value on Earth's surface, i.e. f~ t g 0. (8) We investigate two models. 5. Geoid height change caused by the 964 Alaska earthquake We begin with a calculation of the geoid height change caused by the Alaska earthquake (964, M w ~9.2). The calculations Figure 3. Observed gravity changes versus elevation changes (after Barnes 966). Table. Fault parameters of the 964 Alaska earthquake M w ~9:2. L Length 600 km W Width 200 km d s Depth 20 km d Dip angle 9 0 z Strike N35 0 E U Dislocation 0 m M 0 Moment 0: dyn.cm GJI000 22//98 :49:52 3B2 version 5.20

5 Surface potential and gravity changesöii 83 Figure 4. Calculated gravity changes caused by the Alaska earthquake. The calculation is made by using the fault parameters in Table, the 066A earth model and the dislocation theory in Section 3. are performed along two pro les: one is parallel to the fault line (pro le ); the other (pro le 2) is perpendicular to pro le. Both pro les pass through the centre of the fault plane. Using the seismic parameters in Table and the theory of the present study, we calculate the geoid height changes for the two pro les (Figs 8 and 9). The gures show that the dominant changes occur over the fault plane. The absolute change reaches 0.8 cm and decreases rapidly as the epicentral distance increases. 5.2 Geoid height changes caused by a nite fault with a barrier We assume a rectangular fault with a barrier buried in the 066A earth model at a depth of 2 km (see Fig. 0). The fault has the same size as that of the 964 Alaska earthquake, L W~ km with a dislocation of 0 m. The barrier of km is placed in the centre of the fault. The dip angle d and slip angle j are 60 0 and 90 0 respectively. Note that the assumed model with a barrier is not relevant for the Alaska earthquake, even though we use similar earthquake Figure 5. Calculated gravity changes versus elevation changes. Figure 6. Observed gravity changes versus predicted gravity changes caused by the Alaska earthquake at 0 stations. Units are mgal. GJI000 22//98 :50:3 3B2 version 5.20

6 84 W. Sun and S. Okubo Figure 7. Far- eld gravity changes due to the 964 Alaska earthquake. The x-axis is the angular distance; the y-axis is the gravity change in ]gal. dg 0 is the result for an azimuth of 0 0 ; dg 90 is the result for an azimuth of Figure 8. Geoid changes caused by the Alaska earthquake for pro le, which is parallel to the fault line and passes through the centre of the fault plane. The x-axis indicates the epicentral distance, the y-axis shows the geoid changes in cm. Figure 9. Geoid changes caused by the Alaska earthquake for pro le 2, which is perpendicular to pro le and passes through the centre of the fault plane. The x-axis indicates the epicentral distance, the y-axis shows the geoid changes in cm. GJI000 22//98 :50:6 3B2 version 5.20

7 Surface potential and gravity changesöii 85 Figure 0. A nite fault model with a barrier. This model is assumed to be the same size as the fault plane of the 964 Alaska earthquake with some parameters changed: the source depth is 2 km, the dip angle and slip angle are 60 0 and 90 0, respectively. A barrier is assumed just in the centre of the plane. parameters. The purpose of considering a barrier is to investigate how sensitive and how large the geoid height changes are to a barrier in a fault. We rst estimate the geoid height changes by considering two cases: a change f b caused by a fault with a barrier, and a change f a due to a fault without a barrier. We can then observe the di erence between the two. We calculate f a and f b along two pro les: A^B and C^D (see Fig. 0). We show the results of f a and f b along pro le A^B in Fig., and those along C^D in Fig. 2. The results show that the geoid height change due to a large earthquake can reach.5 cm. The barrier in the fault has an evident e ect on the geoid height change. 6 CONCLUSIONS The point-dislocation theory presented in SO93 can be used when the epicentral distance is much larger than the geometrical size of an earthquake, that is it is valid only for far- eld calculations. Figure. Geoid changes of line A^B of Fig. 0. f a shows the geoid change due to the fault L W; f b is the geoid change caused by the fault L W with a barrier. GJI000 22//98 :50:9 3B2 version 5.20

8 86 W. Sun and S. Okubo Figure 2. Geoid changes of line C^D of Fig. 0. f a shows the geoid change due to the fault L W; f b is the geoid change caused by the fault L W with a barrier. For the near eld, we have to take the geometrical size of the earthquake into account. In Section 3, we extended the pointdislocation theory to that of a nite fault. We presented expressions for calculating the potential and gravity changes caused by dislocations on a nite fault. As an application of the nite-fault dislocation theory, we calculated (in Section 4) the gravity changes caused by the 964 Alaska earthquake (M w ~9:2), and compared the theoretical gravity changes for the 964 Alaska earthquake. The excellent agreement between the computed and observed gravity changes con rms the validity of our formulation. We also nd that the gravity changes reach some 00 ]gal, which can be easily detected by any modern gravimeter. The gravity changes even in the far eld are still signi cantly large: jdgj > 0 ]gal within the epicentral distance h < 6 0, jdgj > ]gal within h < 6 0, jdgj > 0: ]gal within h < 40 0 and jdgj > 0:0 ]gal globally. We also calculated and discussed geoid height changes caused by dislocations. We considered two models: the 964 Alaska earthquake and a nite fault with a barrier. The results show that a great earthquake causes a large change in geoid height (this can reach.5 cm). ACKNOWLEDGMENTS This research was mostly carried out when the rst author (WS) was working at the Earthquake Research Institute of the University of Tokyo, Japan, and was partially supported by the Japanese Government Scholarship. Two anonymous reviewers are acknowledged for their helpful comments and suggestions. REFERENCES Barnes, D.F., 966. Gravity changes during the Alaska earthquake, J. geophys. Res., 7, 45^456. Hagiwara, Y., 977. The Mogi model as a possible cause of the crustal uplift in the eastern part of Izu Peninsula and the related gravity change, Bull. Earthq. Res. Inst., 52, 30^309. Hagiwara, Y., Tajima, H., Izutsuya, S., Nagasawa, K., Murata, I., Okubo, S. & Endo, T., 985. Gravity change in the Izu Peninsula in the last decade, J. geod. Soc. Japan, 3, 220^235. Ma, X.Q. & Kusznir, N.J., 994. E ects of rigidity layering, gravity and stress relaxation on 3-D subsurface fault displacement elds, Geophys. J. Int., 8, 20^220. Maruyama, T., 964. Statical elastic dislocations in an in nite and semi-in nite medium, Bull. Earthq. Res. Inst., 42, 289^368. Okada, Y., 985. Surface deformation due to shear and tensile faults in a half-space, Bull. seism. Soc. Am., 75, 35^54. Okubo, S., 99. Potential and gravity changes raised by point dislocations, Geophys. J. Int., 05, 573^586. Okubo, S., 992. Potential and gravity changes due to shear and tensile faults in a half-space, J. geophys. Res., 97, 737^744. Okubo, S., 993. Reciprocity theorem to compute the static deformation due to a point dislocation buried in a spherically symmetric Earth, Geophys. J. Int., 5, 92^928. Okubo, S., Watanabe, H., Tajima, H., Sawada, M., Sakashita, S., Yokoyama, I. & Maekawa, T., 988. Gravity change caused by the 986 eruption of Izu-Oshima volcano, Bull. Earthq. Res. Inst., 63, 3^44. Okubo, S., Hirata, Y., Sawada, M. & Nagasawa, K., 99. Gravity change caused by the 989 earthquake swarm and submarine eruption o Ito, Japanötest on the magma intrusion hypothesis, J. Phys. Earth, 39, 29^230. Piersanti, A., Spada, G., Sabadini, R. & Bonafede, M., 995. Global post-seismic deformation, Geophys. J. Int., 20, 544^566. Piersanti, A., Spada, G. & Sabadini, R., 997. Global post-seismic rebound of a viscoelastic earth: theory for nite faults and application to the 964 Alaska earthquake, J. geophys. Res., 02, 477^492. Rundle, J.B., 978. Gravity changes and the Palmdale uplift, Geophys. Res. Lett., 5, 4^44. Sabadini, R., Piersanti, A. & Spada, G., 995. Toroidal/poloidal partitioning of global post-seismic deformation, Geophys. Res. Lett., 2, 985^988. Saito, M., 967. Excitation of free oscillations and surface waves by a point source in a vertically heterogeneous earth, J. geophys. Res., 72, 3689^3699. Sasai, Y., 986. Multiple tension crack model for dilatancy, surface displacement, gravity and magnetic change, Bull. Earthq. Res. Inst., 6, 429^473. Sasai, Y., 988. Correction to the paper `Multiple tension crack model for dilatancy, surface displacement, gravity and magnetic change', Bull. Earthq. Res. Inst., 63, 323^326. Savage, J.C. & Hastie, L.M., 966. Surface deformation associated with dip-slip faulting, J. geophys. Res., 7, 4897^4904. GJI000 22//98 :50:23 3B2 version 5.20

9 Surface potential and gravity changesöii 87 Steketee, J.A., 958. On Volterra's dislocations in a semi-in nite elastic medium, Can. J. Phys., 36, 92^205. Sun, W., 992a. Potential and gravity changes raised by dislocations in radially heterogeneous earth models, PhD thesis, University of Tokyo, Japan. Sun, W., 992b. Potential and gravity changes caused by dislocations in spherically symmetric earth models, Bull. Earthq. Res. Inst., 67, 89^238. Sun, W. & Okubo, S., 993. Surface potential and gravity changes due to internal dislocations in a spherical earthöi. Theory for a point dislocation, Geophys. J. Int., 4, 569^592. Sun, W. & Okubo, S., 994. Spheroidal displacement due to point dislocations in a spherical earth,, Theory, Acta geophys. Sin., 37, 298^30. Sun, W. & Okubo, S., 995. Spheroidal displacement due to point dislocations in a spherical earth, 2, Dislocation Love numbers, Acta geophys. Sin., 38, 89^0. Sun, W., Okubo, S.& Van c ek, P., 996. Global displacement caused by dislocations in a realistic earth model, J. geophys. Res., 0, 856^8577. APPENDIX A: FORMULAE FOR NUMERICAL INTEGRATION OVER A FINITE FAULT To perform numerical integration to evaluate (0)^(3), we divide the faultplaneintop Q nite elements with (see Fig. ): P~ L d/k, (A) Q~ W d/k, (A2) where d is the nearest distance from the calculating point to the fault plane. (A) and (A2) imply that the small elements have a dimension of (d/k); k should be taken larger than 0 in order that the dislocation on a small element can be regarded as a point source. The side lengths of each element are C L ~ L P, C W ~ W Q. (A3) (A4) We can obtain the spherical coordinate (h pq, pq ; Dpq s )ofthe centre of each element: h pq ~h 0{ c a cos (z za), (A5) pq ~ 0z c a sin h pq sin (z za), (A6) D pq s ~d sz Q{ C W sin d, (A7) 2 with ( c~ P{ 2 C W cos d z Q{ ) 2 =2 C L 2 2, (A8) P{ C W cos d a~ tan { 2 Q{, (A9) C L 2 where p~, 2,..., P and q~, 2,..., Q. The angular distance ' pq between the centre of ds and the observation point (h 2, 2 ), and the azimuth z pq 2 of the line connecting the two points with respect to the fault line are given as cos ' pq ~ cos h pq cos h 2z sin h pq sin h 2 cos ( 2 { pq ), (A0) sin z pq 2 ~ sin r pq sin h 2 sin ( 2 { pq ), (A) cos z pq 2 ~ sin h pq sin (cosh 2{ cos h pq rpq cos rpq ), (A2) and $ pq ~z {z pq 2. (A3) Now that we have the arguments (' pq, $ pq ; D pq s ) of each element ds at (h pq, pq ; D pq s ), the integral kernels (u ij r, tij, *g ij, dg ij ) are derived from (7)^(8). They are as follows. For shear dislocation: u (s) r (a, 'pq, $ pq ; D pq s ) ~u ij r (a, 'pq, $ pq ; D pq ~ cos j[{^u 2 r (a, 'pq )sindsin 2$ pq z^u 32 r (a, 'pq )cosdcos $ pq ] z sin j sin 2d[^u33 r 2 (a,'pq ){^u 22,0 r (a,' pq ) z^u 2 r (a, 'pq )cos2$ pq ]z^u 32 r (a, 'pq )cos2dsin $ pq, (A4) t (s) (a, ' pq, $ pq ; D pq s ) ~t ij (a, ' pq, $ pq ; D pq ~ cos j[{^t 2 (a, ' pq )sindsin 2$ pq z^t 32 (a, ' pq )cosdcos $ pq ] z sin j 2 sin 2d[^t 33 (a, ' pq ){^t 22,0 (a, ' pq ) z^t 2 (a, ' pq )cos2$ pq ]z^t 32 (a, ' pq )cos2d sin $ pq, (A5) *g (s) (a, ' pq, $ pq ; D pq s ) ~*g ij (a, ' pq, $ pq ; D pq ~ cos j[{*^g 2 (a, ' pq )sind sin 2$ pq z*^g 32 (a, ' pq )cosdcos $ pq ] z sin j 2 sin 2d[*^g33 (a, ' pq ){*^g 22,0 (a, ' pq ) z*^g 2 (a, ' pq )cos2$ pq ]z*^g 32 (a, ' pq )cos2d sin $ pq, dg (s) (a, ' pq, $ pq ; D pq s ) ~dg ij (a, ' pq, $ pq ; D pq ~ cos j[{d^g 2 (a, ' pq )sind sin 2$ pq zd^g 32 (a, ' pq )cosdcos $ pq ] z sin j 2 sin 2d[d^g33 (a, ' pq ){d^g 22,0 (a, ' pq ) zd^g 2 (a, ' pq )cos2$ pq ]zd^g 32 (a, ' pq )cos2d sin $ pq. (A6) (A7) GJI000 22//98 :50:46 3B2 version 5.20

10 88 W. Sun and S. Okubo For tensile dislocation: u (t) r (a, 'pq, $ pq ; D pq s )~uij r (a, 'pq, $ pq ; D pq ~^u 33 r (a, 'pq )cos 2 dz[^u 22,0 r (a, ' pq ) {^u 2 r (a, 'pq )cos2$ pq ]sin 2 d {^u 32 r (a, 'pq )sin2d sin $ pq, t (t) (a, ' pq, $ pq ; D pq s )~tij (a, ' pq, $ pq ; D pq ~^t 33 (a, ' pq )cos 2 dz[^t 22,0 (a, ' pq ) {^t 2 (a, ' pq )cos2$ pq ]sin 2 d {^t 32 (a, ' pq )sin2d sin $ pq, A8) (A9) *g (t) (a, ' pq, $ pq ; D pq s )~*gij (a, ' pq, $ pq ; D pq ~*^g 33 (a, ' pq )cos 2 dz[*^g 22,0 (a, ' pq ) {*^g 2 (a, ' pq )cos2$ pq ]sin 2 d {*^g 32 (a, ' pq )sin2d sin $ pq, dg (t) (a, ' pq, $ pq ; D pq s )~dgij (a, ' pq, $ pq ; D pq ~d^g 33 (a, ' pq )cos 2 dz[d^g 22,0 (a, ' pq ) {d^g 2 (a, ' pq )cos2$ pq ]sin 2 d {d^g 32 (a, ' pq )sin2d sin $ pq, (A20) (A2) where the independent solutions, or Green's functions [^u ij r (a, 'pq ), ^tij (a, ' pq ), *^g ij (a, ' pq ) and d^g ij (a, ' pq )], can be obtained from the numerical tables (SO93) by interpolation. GJI000 22//98 :5:3 3B2 version 5.20