Rotation of the Principal Stress Directions Due to Earthquake Faulting and Its Seismological Implications

Bulletin of the Seismological Society of America, Vol. 85, No. 5, pp. 1513-1517, October 1995 Rotation of the Principal Stress Directions Due to Earthquake Faulting and Its Seismological Implications by Z.-M. Yin and G. C. Rogers Abstract Earthquake faulting results in stress drop over the rupture area. Because the stress drop is only in the shear stress and there is no or little stress drop in the normal stress on the fault, the principal stress directions must rotate to adapt such a change of the state of stress. Using two constraints, i.e., the normal stress on the fault and the vertical stress (the overburden pressure), which do not change before and after the earthquake, we derive simple expressions for the rotation angle in the at axis. For a dip-slip earthquake, the rotation angle is only a function of the stressdrop ratio (defined as the ratio of the stress drop to the initial shear stress) and the angle between the at axis and the fault plane, but for a strike-slip earthquake the rotation angle is also a function of the stress ratio. Depending on the faulting regimes, the al axis can either rotate toward the direction of fault normal or rotate away from the direction of fault normal. The rotation of the stress field has several important seismological implications. It may play a significant role in the generation of heterogeneous stresses and in the occurrence and distribution of aftershocks. The rotation angle can be used to estimate the stress-drop ratio, which has been a long-lasting topic of debate in seismology. Introduction Earthquake rupture, which is characterized by shear faulting, causes the tectonic stress and strain in the crust to release. The change in the stress field during the earthquake is usually termed the co-seismic stress field to distinguish from the pre-earthquake background tectonic stress field. The state of co-seismic stress at the focus is well represented by the double-couple force model (Kasahara, 1981, pp. 28-52). Thus, the state of co-seismic stress over the whole rupture area can be viewed as it comprises many double-couple forces, which is the foundation of the elasticity theory of dislocation (Steketee, 1958a, b). According to the doublecouple force model, the stress drop (or release) is only in the shear stress and there is no stress drop in the normal stress on the fault. Therefore, after seismic rupture, the principal stress directions must rotate to adapt such a change of the state of stress on and surrounding the fault. The rotation of the stress field due to seismic rupture has been observed through stress inversion in some California earthquake sequences, such as the 1983 Coalinga earthquake (Michael, 1987), the 1986 Oceanside earthquake (Hauksson and Jones, 1988), the 1989 Loma Prieta earthquake (Michael et al., 199), and the 1992 Joshua Tree and Landers earthquake (Hauksson, 1994). The purpose of this article is to derive simple expressions to quantify the change in the principal stress directions and to discuss its seismological significance. Here, we consider the case where the a 2 axis is par- allel to the fault plane and oriented either vertically or horizontally (the equations derived in this article are therefore applicable to this case only). However, the analysis can be readily extended to the case where the o- 2 axis is parallel to the fault plane and arbitrarily oriented, but it cannot be extended to the general case where the o- 2 axis is oriented arbitrarily both in space and with respect to the fault plane, because all the three principal stress axes may rotate in threedimensional space and cannot be constrained. Besides, the analysis presented in this article is valid only for the major part of a rupture where there is a stress drop and cannot be applied to the areas close to and beyond the edge of the rupture where there is a stress concentration. Mechanical Constraints The normal stress on the fault does not change during seismic faulting, as mentioned above. The vertical stress (i.e., the overburden or lithostatic pressure) is also invariant before and after the earthquake. Let o-v, 3, and a denote the vertical stress, the shear, and the normal stress on the fault before the earthquake, and o-v*, r*, and o-* denote the vertical stress, the shear, and the normal stress after the earthquake (hereafter we use the symbol * to denote the corresponding parameters after the earthquake). Thus, we have 1513

1514 Short Notes O'v* "~- "v z* = z- Az = z(1 - Go), "* : ", (1) m x3, n where Az is the stress drop and e = Az/z is the ratio of the stress drop to the initial shear stress. We term e the stressdrop ratio, to is related with the seismic efficiency ~/by ~/ = eo/(2 - t) (Kasahara, 1981, p. 14). Equation (1) can be used to derive expressions for rotation of the principal stress directions. Throughout this article, the parameters defined above and in the sequel are taken as a function of position. In the case where confusion might arises, we shall use "average" in front of these parameters to denote their average value over the fault or a segment of the fault. Dip-Slip Earthquakes Figure 1 shows the parameters defined for dip-slip faulting. When fl >, the fault is thrust. When fl <, the fault is normal. The pre-earthquake shear and normal stress acting on the fault are, respectively (Jaeger and Cook, 1969, pp. 87-91), p, Figure 1. A dip-slip fault (F) with normal n and dip under the tectonic stress field with the al axis and the a 3 axis denoted by x 1 and x a, respectively (the a2 axis is parallel to the fault plane), m is a unit vector denoting the vertical direction. The maximum stress axis makes an angle a with the fault plane and an angle fl with the vertical vector. sin 2a* XI F (1 - e) sin 2a 1 - cos 2a* - 2 cos z r* 1 - cos 2a - 2 cos z fl" (6) 1 = 2 ("' -- "3) sin 2a 1 1 " = = (", + "~) - : ("1 - "9 cos 2a. Z Z Because the vertical stress has the following relation with the principal stresses (Yin and Ranalli, 1992), (2) "v = pgz(1-2) = (", - "3)cos2fl + "3, (3) where pgz(1-2) is the effective overburden pressure (p is the density, g the acceleration of gravity, z depth, and 2 the pore fluid factor), the normal stress in equation (2) can be rewritten as 1 " = ~ ("1 -- "3)( 1 -- COS 2a - 2 cos: r) + pgz(1-2). The postearthquake shear and normal stress on the fault have the same form of expressions shown in equations (2) and (4). Thus, substituting equations (2) and (4) in equation (1) and cancelling the effective overburden pressure yields (a,* - a3*) sin 2a* = (1 - G)(-1 - "3) sin 2a (",* -- "3")(1 -- COS 2a* -- 2 COS 2 fl*) (4) = ("i -- "3)( 1 -- cos 2a -- 2 cos 2fl). (5) Cancelling the stress differences in equation (5), it becomes We define the rotation angle Aa = a* - a. Thus, Aa is positive if the a, axis rotates toward the fault normal n, and it is negative if the -, axis rotates away from n. With reference to Figure 1, one can find that for thrust faulting r* = fl - Aa. For normal faulting, however, there are two different cases. When the a, axis lies on the left side of the vertical vector m, r* = fl - Aa. When the al axis lies on the right side of m, r* = fl + Aa. Consequently, solving equation (6) for Aa, we derive the expressions for the rotation angle for the above two different cases, _eo sin 2a cos (a --_fl).~ for r* = Aa = ~tan -1 ~eosin2asin(a - r) -] fl - Aa \ cos (a + ~ / 1,{ eosin2acos(a + r) ~ fort* = Aa = ~tan- /eos~n~+ ~ -] fl + Aa. \ cos (a - ~ / Equation (7) shows that the rotation angle is a function of the stress-drop ratio (t), the angle between the -1 axis and the fault plane (a), and the dip of the fault () (note that fl is a function of a and ). The rotation angle is independent of the magnitude of both the initial stress and the stress drop. Given the orientation of a fault, an initial angle a and a stress-drop ratio e, the rotation angle can be calculated using equation (7). Figure 2 shows two examples for thrust and normal faulting under the tectonic stress fields where the initial -1 axis is oriented horizontally and vertically, respectively. For the thrust faulting, the -, axis rotates toward the fault normal n. The negative values of Aa in Figure 2b in- (7)

ShonNo~s 1515 8 7 6 5 We define the sign of the rotation angle (Aa = a* - a) in the same way as in the case of dip-slip faulting, that is, Aa is positive if the a~ axis rotates toward n, and it is negative if the a~ axis rotates away from n. Solving equation (9) for a*, we obtain 3 2 1-1 -3-4 -5-6 -7-8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 (X Figure 2. Rotation angle (Aa) of the a~ axis versus the angle (a) between the initial a~ axis and the fault plane for (a) thrust faulting in the case where the initial c h axis is horizontal and (b) normal faulting in the case where the initial a~ axis is vertical. (A) The stress-drop ratio is eo =.1; (B) eo =.3; (C) eo =.5; (D) eo =.7; (E) eo =.9. dicate that the al axis rotates away from n for the normal faulting. For both cases, the absolute value of rotation angle, which could be as large as 8, increases with the stress-drop ratio eo. 2(1 - ~)" - a, (1) where b = [1 - cos (2a) - 26]/[(1 - eo) sin (2a)], i.e., the reciprocal of the term on the right side of equation (9). Comparing equations (7) and (1), one may find that, for strike-slip faulting, the rotation angle is not only a function of the stress-drop ratio and the initial orientation of the "1 axis with respect to the fault plane, but also a function of the preseismic and postseismic stress ratio. Since the stress ratio ~ may change during seismic rupture, the rotation of the stress field due to strike-slip faulting is more complicated than that caused by dip-slip faulting. A few examples shown in Figure 3 indicate that, in the case of strike-slip faulting, the rotation angle Aa can be both positive (the a~ axis rotates toward n) and negative (the a~ axis rotates away from n). The rotation of the stress field caused by earthquake faulting is a universal phenomenon. Therefore, the above analysis is applicable to all earthquakes as long as they belong to the type of Coulomb-Navier shear fractures. The analysis is valid no matter whether the distribution of slip (or stress drop) is uniform or heterogeneous as long as there is a stress drop on the fault. However, when applying the formulations to real cases, two limitations should be taken into consideration. One is that, as mentioned before, the formulations do not apply to the areas close to and beyond the edge of the rupture area, because in those areas there is a stress concentration rather than a stress drop. The second is that the formulations apply to large earthquakes better than small earthquakes. Because the rupture area of small earthquakes is very small, the rotation (and disturbance) of the stress field is limited to a very small area around the focus. Strike-Slip Earthquakes For strike-slip faulting, the vertical stress is identical to the intermediate principal stress, i.e., av = a2. Using the relation -2 = a3 + 6(a~ - a3) (~ is usually termed the stress ratio), the least principal stress can be expressed as (Yin and Ranalli, 1992) a3 = pgz(1-2) - ~(o- 1 - -3). (8) Because equations (1) and (2) still hold for strike-slip faulting, combining equations (1), (2), and (8) yields sin 2a* (1 - eo) sin 2a 1 - cos 2a* - 26* 1 - cos2a- 26" (9) Seismological Implications Heterogeneity of Postearthquake Stress Field Some earthquake case studies show that the focal mechanism solutions of foreshocks are usually well accounted for by a uniform stress tensor, but those of aftershocks are poorly or sometimes cannot be represented by a uniform stress tensor (Michael, 1987; Michael et al., 199; Beroza and Zoback, 1993). In other words, the earthquake rupture causes the stress field to be more heterogeneous. From the above analysis of stress rotation, it is not difficult to understand this process. It can be seen from equations (7) and (1) that a (the initial angle between the al axis and the fault plane) and eo (the ratio of the stress drop to the initial shear

1516 Short Notes 4 3 2O N 1o <~ <1-1 3 2 1-1 -3 (a) I J I I I I I ~ 1 I I I I I 1 2 3 4 5 6 7 8 (x A I i A i 1 2 3 4 5 6 7 8 (X 2 [ ' J i i i m 1 ~_ (c) A -1 B -3-4 I 2 3 4 5 6 (x r I 7 8 Figure 3. Rotation angle (Aa) of the cq axis versus the angle (a) between the initial a 1 axis and the fault plane for strike-slip faulting in the case of (a) the stress ratio 8 = 8" =.2, (b) 8 = ~* =.5, and (c) 8 = 8" =.8. A, B, C, D, and E as in Figure 2. stress) play an important role in the rotation of the stress field (note that both a and eo are a function of position). We know for sure that a varies along the strike and the dip direction of the rupture because of complicated fault geometry. This results in the variation of the rotation angle with position, and consequently gives rise to the heterogeneous distribution of the principal stress directions. From the slip distribution, we also know that the stress drop on the fault is heterogeneous, but we know little about the distribution of the initial shear stress. Because ~ is the ratio of the stress drop (At) to the initial shear stress (z), both a uniform distribution of z and a heterogeneous distribution of z that is uncorrelated with Az are guaranteed to give rise to heterogeneity in the principal stress directions. Aftershocks The mechanism of aftershocks is still not very clear, although several theories have been proposed (Scholz, 199). On the basis of calculation of the co-seismic stress field using a dislocation model, previous workers have noticed that the change in the magnitude of the stress field during the earthquake (such as the stress concentration due to dynamic slip) can affect the occurrence and distribution of aftershocks (Scholz, 199, pp. 259; Segall and Du, 1993; Du and Aydin, 1993). Our above analysis shows that the change in the directions of the stress field can also affect the occurrence and distribution of aftershocks. First, the seismic rupture causes the stress field to be more heterogeneous, which enhances the occurrence of small earthquakes. Second, because the orientation of fault planes with respect to the principal stress directions is changed during the mainshock, some faults become "weakening" and some are "strengthening". Estimation of the Stress-Drop Ratio Using seismological methods, we can only determine the stress drop, but not the initial shear stress. This is because the elastic waves generated and the energy released by the earthquake are irrelevant to the initial stress. Consequently, a debate arises about whether the observed average stress drops account for, on the average, a small percentage or a large percentage of the initial shear stress. This debate can be translated as whether the level of tectonic stresses in the deep upper crust is of the order of tens or hundreds of megapascals. In this article, we propose a simple method to estimate the stress-drop ratio. In equations (7) and (1), the rotation angle Aa is expressed as a function of the stressdrop ratio e. Given eo, Aa can be calculated, such as those shown in Figures (2) and (3). On the contrary, eo can be obtained provided that Aa is known. Since the average rotation angle can be determined from stress inversion using focal mechanism solutions of foreshocks and aftershocks, equations (7) and (1) can be used to estimate the average stress-drop ratio (and/or the seismic efficiency) for a particular fault or fault segment. Here, we show two examples: the 1992 Joshua Tree and the 1992 Landers earthquakes. Hauksson (1994) did a detailed study of the change of the stress field before and after the Joshua Tree and the Landers earthquakes from inversion of thousands of foreshock and aftershock focal mechanism solutions. In Hauksson's (1994) work, both seismic faults were divided into several segments and the principal stress directions and the stress ratio were determined for each of the segments. We use the inversion results obtained for the middle part of the faults (i.e., the

Short Nows 1517 Joshua Tree segment for the Joshua Tree earthquake and the Homestead Valley segment for the Landers earthquake) to estimate the average stress-drop ratio; specifically, for Joshua Tree segment, a = 4, a* = 48, t~ =.54, t~* =.7, and for Homestead Valley segment, a -= 43, a* = 56, t~ =.5, 3" =.89 (all refer to their average value). The average stress-drop ratio estimated is eo =.41 for the Joshua Tree segment and eo =.84 for the Homestead Valley segment. Although both these two examples are strikeslip earthquakes, the analysis can also be applied to dip-slip earthquakes. For dip-slip earthquakes, the estimation of the stress-drop ratio is easier and more accurate, because the rotation angle in this case is independent of the stress ratios (3 and 3*). Conclusions The rotation of the stress field due to seismic faulting has been observed in several large earthquake sequences. However, a general theory about the rotation of the stress directions has not been formulated so far. In this article, on the basis of the double-couple force model, we have shown that the stress field must rotate during seismic faulting as a result of shear faulting. Using two mechanical constraints [that is, the normal stress on the fault and the vertical stress (overburden pressure) do not change before and after the earthquake], we have derived simple expressions for the rotation angle of the al axis. For dip-slip earthquakes, the rotation angle is a function of the initial angle between the al axis and the fault plane and the stress-drop ratio (the ratio of the stress drop to the initial shear stress) only. It is also a function of the stress ratio for strike-slip earthquakes. We have discussed the seismological implications of the rotation of the stress field. The rotation in the principal stress directions causes the stress field to be more heterogeneous. It may play an important role in the occurrence and distribution of aftershocks. In addition, the expressions derived for the rotation angle can be used to estimate the average stress-drop ratio. Taking the Joshua Tree and the Landers earthquakes as examples, we have calculated the average stress-drop ratio over the middle segment of these two seismic ruptures. Acknowledgments This work has been supported by a NSERC postdoctoral fellowship to Yin and by the Geological Survey of Canada at the Pacific Geoscience Centre. The authors are grateful to an anonymous reviewer for suggesting improvements to the manuscript. References Beroza, G. C. and M. D. Zoback (1993). Mechanism diversity of the Loma Prieta aftershocks and the mechanics of mainshock-aftershock interaction, Science 259, 21-213. Du, Y. and A. Aydin (1993). Stress transfer during three sequential moderate earthquakes along the central Calaveras fault, California, J. Geophys. Res. 98, 9947-9962. Hauksson, E. (1994). State of stress from focal mechanisms before and after the 1992 Landers earthquake sequence, Bull. Seism. Soc. Am. 84, 917-934. Hauksson, E. and L. M. Jones (1988). The July 1986 Oceanside (M L = 5.3) earthquake sequence in the continental borderland, southern California, Bull. Seism. Soc. Am. 78, 1885-196. Jaeger, J. C. and N. G. Cook (1969). Fundamentals of Rock Mechanics (Third Ed.), Chapman and Hall, London. Kasahara, K. (1981). Earthquake Mechanics, Cambridge University Press, Cambridge, United Kingdom. Michael, A. J. (1987). Stress rotation during the Coalinga aftershock sequence, J. Geophys. Res. 92, 7963-7979. Michael, A. J., W. L. Ellsworth, and D. H. Oppenheimer (199). Coseismic stress changes induced by the 1989 Loma Prieta, California earthquake, Geophys. Res. Lett. 17, 1441-1444. Scholz, C. H. (199). The mechanics of earthquakes and faulting, Cambridge'University Press, Cambridge, United Kingdom. Segall, P. and Y. Du (1993). How similar were the 1934 and 1966 Parkfield earthquakes? J. Geophys. Res. 98, 4527-4538. Steketee, J. A. (1958a). On Volterra's dislocations in a semi-infinite medium, Can. J. Phys. 36, 1925. Steketee, J. A. (1958b). Some geophysical applications of the elasticity theory of dislocations, Can. J. Phys. 36, 1168-1198, Yin, Z.-M. and G. Ranalli (1992). Critical stress difference, fault orientation and slip direction in anisotropic rocks under non-andersonian stress systems, J. Struct. GeoL 14, 237-244. Pacific Geoscience Centre Geological Survey of Canada Sidney, British Columbia Canada V8L 4B2 (Z.-M.Y.) (G.C.R.) School of Earth and Ocean Sciences University of Victoria Victoria, British Columbia Canada (Z.-M.Y.) (G.C.R.) Manuscript received 11 November 1994.