Overestimates of Earthquake Prediction Efficiency in a "Post-Prediction"

Transcription

1 Letter J. Phys. Earth, 41, 41-43, 1993 Overestimates of Earthquake Prediction Efficiency in a "Post-Prediction" State Shozo Matsumura National Research Institute for Earth Science and Disaster Prevention, Tsukuba 305, Japan A basic procedure in earthquake prediction research is to find out possible relationship between precursory phenomena and earthquake occurrences. After the occurrence of strong earthquakes, we usually arrange observed data on a common temporal axis, and carefully examine if there has been some anomalous phenomena prior to the events. The result obtained from such procedure is so-called "postprediction" state. It should be noted that "post-prediction" often states more than the truth, because, in the above procedure of finding possible precursors, it is necessary to introduce several free parameters, such as threshold levels, precursor time (duration or preceding time of precursor) and others, and they are liable to be intentionally selected to make the final result optimum. After all, such phenomenological handling will result in exaggeration of prediction efficiency compared with "true-prediction." The aim of this study is to clarify this situation, and to give a numerical estimation of it by taking a simplified model, where only precursor time Tp is treated as a free parameter. A model pattern of precursor and earthquake sequence is given in Fig. 1, where five precursors, or candidates of precursors, p1, c,p5, precede occurrences of five earthquakes, el, c,e5. It seems that whoever sees this pattern will have an impression of good correlation between appearance of pi and occurrence of ei. One way to judge whether or not the impression is true, is to execute a statistical test for a hypothesis that the pattern happens by accident. For this purpose, a probability P that at least n earthquakes among e1, c, em accidentally fall within specified periods is used to be calculated, and compared with a level of statistical significance. For the example of Fig. 1, those periods are specified as shown in Fig. 2, where four among five earthquakes fall in the heavy-lined periods. In this case, the total duration of specified periods is 5Tp, so P = 5C4p4q + 5C5p5, where p = 5Tp /To (= 0.25), and q=1-p. Now, the calculated value of P, 0.016, looks small enough to conclude that the hypothesis should be rejected. However, in this case, the value of Tp has been intentionally selected to make P minimum. In other words, P is biased from a reasonable value. Then, how much value of P should be recognized to be reasonable, or what value should be adopted for the significance test? In order to solve this problem, we have carried out a computer simulation as follows. Two extreme cases are investigated, and compared. One is case-a, where the free parameter is intentionally selected in order to get an optimum solution. The other, case-b, corresponds to non-intentional selection. First, five ei (earthquakes) and five pi Received July 22, 1992; Accepted December 24,

2 42 S. Matsumura Fig. I. An example pattern presenting five earthquakes (e1, c,e5), and five candidates of precursors (pi, c,p5). Fig. 2. A free parameter Tp (duration or preceding time of precursor) is brought into the pattern of Fig. 1. The value of Tp is intentionally selected to get an optimum solution. Fig. 3. Accumulated appearance probability functions of P for possible patterns composed of five earthquakes and five precursors obtained for both cases. The solid line corresponds to the case of intentional selection for Tp (case-a). The dotted line corresponds to non-intentional selection (case-b). The broken line is proportional to P itself given as a reference. (candidates of precursor) are independently generated on a time axis of unit length. At each trial, Tr is selected to make P minimum in case-a, while Tp is selected at random on a unit time length in case-b, and the values of P are calculated for both cases. Finally after 100,000 trials, appearance probabilities of P are obtained, and compared. The results are shown in Fig. 3, where accumulated probability functions F(P) are drawn in a solid line for case-a, and in a dotted line for case-b. The latter approximately equals to a function of P2. This can be intuitively understood because the appearance probability of P must be proportional to itself, so that the integrated and normalized function becomes P2. Here, let us re-evaluate the significance of the above-introduced J. Phys. Earth

3 Letter J. Phys. Earth, 41, 45-56, 1993 Critical Size of the Nucleation Zone of Earthquake Rupture Inferred from Immediate Foreshock Activity Mitiyasu Ohnaka Earthquake Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113, Japan A recent study (Ohnaka, 1992) shows that an intrinsic part of earthquake dynamic rupture is a nucleation process that is itself an immediate earthquake precursor. The nucleation process consists of two parts, (i) an initial quasistatic nucleation which leads to (ii) quasidynamic nucleation at a later stage. As nucleation proceeds in a finite localized zone along the fault, it is crucial for the attainment of short-term (or immediate) earthquake prediction to understand where the rupture nucleus for a potential major earthquake is formed in the fault zone, and to estimate how large is the critical size of the nucleation zone. If the critical size of the nucleation zone for a real major earthquake is small enough, it may be meaningless to regard the nucleation process as an effective tool for the immediate earthquake prediction, and in this sense it is particularly important to know how large is the critical size of the nucleation zone for a real major earthquake. As a first attempt, I here estimate the critical size of the nucleation zone for a major earthquake (MJMA 7.0) that occurred in 1978 near the Izu Oshima Island, central Japan. It is inferred that the critical size of the mainshock nucleation zone was as large as 10km and that the nucleation zone expanded with an apparent speed of the order of 1cm/s to several tens of cm/s. A series of experimental (Ohnaka et al., 1986; Ohnaka and Kuwahara, 1990) and theoretical (Yamashita and Ohnaka, 1991; Matsu'ura et al., 1992) studies have revealed that even in the purely brittle regime, slip-failure instability giving rise to dynamically propagating rupture over the entire fault is locally preceded by a stable and quasistatic nucleation process, leading to quasidynamic rupture nucleation at a later stage, when the resistance to rupture growth is nonuniformly distributed on the fault. The rupture growth resistance is a physically distinct concept in the context of fracture mechanics, and it may be regarded as the shear fracture energy required for the rupture front to further advance. The critical slip displacement Dc, which is defined as the slip displacement required for the breakdown near a rupture front, is indicative of the rupture growth resistance. The shear strength Ėp on the fault is also indicative of the rupture growth resistance, because the breakdown begins to occur after the stress at the rupture front attains the strength. Recent studies (Zoback et al., 1987; Byerlee, 1990, 1992; Lachenbruch and Sass, 1992; Rice, 1992) support the idea that the fluid pressure at the center of the fault zone is much greater than the hydrostatic fluid pressure in the country rock, and the residual frictional stress after the breakdown will be Received October 15, 1992; Accepted February 15,

4 46 M. Ohnaka negligible under the supra-hydrostatic fluid pressure. If this is the case, the shear fracture energy is given roughly by the product of the shear strength and the critical slip displacement. Experimental studies (Ohnaka et al., 1986; Ohnaka and Kuwahara, 1990) also revealed that such a quasistatic to quasidynamic nucleation begins to occur at a point where the resistance to rupture growth has a minimum value. These facts indicate that there exists a stable and quasistatic preparatory process prior to earthquake dynamic shear rupture along a fault in the lithosphere, because such inhomogeneities commonly exist in parts of the fault zone where a potential for earthquakes prevails. For example, seismicity itself in a fault zone at a shallow crustal depth, such as the San Andreas (Eaton et al., 1970), is a direct reflection of local to small scale inhomogeneities in the brittle seismogenic layer. Another notable example of the inhomogeneities is the overall depth variation of the rupture growth resistance in the lithosphere (Ohnaka, 1992), which is prescribed primarily by depth variations of pressure and temperature profiles. The preparatory process prior to earthquake dynamic rupture has been referred to as earthquake source nucleation, and a physical model for the earthquake rupture nucleation has recently been put forward based on facts found in laboratory experiments on slip failure nucleation and physical principles (Ohnaka, 1992). The model is summarized below, but is more fully described in an earlier paper (Ohnaka, 1992). The basic process during which earthquake slip failure nucleates stably and quasistatically, and eventually develops to unstable dynamic rupture, is modelled conceptually as shown in Fig. 1. Figure 1 shows how rupture grows, from the point where a slip failure nucleus is formed, stably and quasistatically at an initial stage (I) and pseudostably and quasidynamically at a later stage (II), to the critical point A beyond which rupture propagates unstably and dynamically at a speed close to sonic velocities. The critical size of the nucleation zone is prescribed by an inhomogeneous distribution of the rupture growth resistance on the fault (Yamashita and Ohnaka, 1991). The hatched portion in Fig. 1 indicates the breakdown zone where slip-weakening proceeds (right-hand side of Fig. 2). The breakdown zone is defined as the zone over which local shear stresses near the propagating rupture front on the fault decrease from the peak value Ėp to a residual friction stress level Ėr with ongoing slip (see Fig. 2). During the earthquake nucleation ((1) in Fig. 1), local shear stresses decrease gradually in the breakdown zone, and at the same time the corresponding premonitory slip also proceeds in the zone, since slip-weakening occurs during the nucleation process ((1) in Fig. 2). Premonitory stress (or strain) changes can also be expected outside (but adjacent to) the nucleation zone; that is, local shear stresses on the remaining unslipped parts adjacent to the nucleation zone ((2) in Fig. 1) increase with time because the unslipped segments must bear extra stress loads that have been sustained by the slipped parts ((2) in Fig. 2). These gradual and subsequent, accelerating changes in both local stress and slip are inevitable precursors that occur locally in or adjacent to the zone of earthquake nucleation. By contrast, no such precursory slip and stress degradation may be expected to occur in a region distant from the nucleation zone ((3) in Figs. 1 and 2). The zone of earthquake nucleation is characterized as the zone where slip failure deformation is concentrated and accelerated, so that dynamic instabilities of local to small scale are expected to occur during the nucleation process, and they are confined J. Phys. Earth

5 Letter 47 Fig. 1. A model of the rupture nucleation based on laboratory experiments (Ohnaka, 1992). The hatched portion indicates the breakdown zone where slip-weakening proceeds (see the right-hand side of Fig. 2). Xc is the breakdown zone size, and Tc is the breakdown time (defined as the time required for the shear stress to decrease from its peak value to a residual frictional stress level). Changes in local shear stresses and slip displacements with time at the representative locations (1), (2) and (3) along the fault are shown in Fig. 2. For details see text. in the nucleation zone, when the rupture growth resistance on the fault varies on a local to small scale in the brittle seismogenic layer. In general, such local to small scale dynamic instabilities do not necessarily result in a dynamic instability giving rise to rupture over the entire fault, if the rupture growth resistance on the fault varies conspicuously on a local to small scale. Thus, it is logically concluded that the entire quasistatic to quasidynamic nucleation process prior to overall dynamic rupture can include dynamic instabilities of local to small scale. In fact, Sammonds et al. (manuscript in preparation) recently observed a great number of microseismic (or acoustic emission) activities during the overall quasistatic slip-weakening process for granite sliding surfaces of fractal roughnesses in the laboratory experiments. Dynamic instabilities of local to Vol. 41, No. 1, 1993

6 48 M. Ohnaka Fig. 2. Changes in local shear stresses and the corresponding slip displacements with time at different locations along the fault. (1) Location in the nucleation zone (see (1) in Fig. 1), (2) location adjacent to the nucleation zone ((2) in Fig. ), and (3) location distant from the nucleation zone ((3) in Fig. 1). Shown 1 on the right-hand side are the slip-weakening relations, which are assumed to be the constitutive relation governing the earthquake source processes throughout the quasistatic nucleation to dynamic rupture. During the breakdown process, the shear strength degrades from its peak value Ėp to a residual stress level Ėr with ongoing slip displacement. small scale that occur in the process of the quasistatic to quasidynamic nucleation of the ensuing overall dynamic earthquake instability recognized as a mainshock, are immediate foreshocks in a strict sense, and such local to small scale instabilities (or immediate foreshocks) may be regarded as a part of premonitory slip during the mainshock nucleation. In other words, the mainshock nucleation during which slip-weakening proceeds occurs necessarily in a zone that includes the hypocentral region of immediate foreshocks. This gives a physical explanation for seismicity observations made for decades that immediate foreshocks are concentrated in the vicinity of the epicenter of the pending mainshock. Immediate foreshock activities do not necessarily occur during the nucleation process of the pending mainshock. Whether or not immediate foreshocks occur during the mainshock nucleation depends upon how the rupture growth resistance varies on a local to small scale in the nucleation zone. As demonstrated experimentally (Ohnaka J. Phys. Earth

7 Letter 49 et al., 1986; Ohnaka and Kuwahara, 1990) and theoretically (Yamashita and Ohnaka, 1991; Matsu'ura et al., 1992), the nucleation begins to occur at a location where the rupture growth resistance is at a minimum on the fault, and a dynamic instability giving rise to seismic rupture occurs at a position where the energy released with further rupture growth exceeds its growth resistance. If this dynamic instability results in seismic rupture over the entire fault, it will be recognized as mainshock earthquake. However, the dynamic rupture may be arrested at a barrier where the rupture growth resistance is conspicuously high, and where the dynamic rupture may again return to stable, quasistatic growth. Such a local to small scale dynamic instability will be recognized as a foreshock during the nucleation for the pending mainshock. Once the tips of the nucleation zone penetrate through the barrier patch of greater resistance, other local to small scale dynamic instabilities may occur. Thus, earthquakes can carry foreshock activity during their nucleation, when inhomogeneous short-wavelength variations of the rupture growth resistance are superimposed on a non-uniform long-wavelength trend of the resistance in the fault zone (Ohnaka, 1992). Stable, quasistatic rupture penetration into a barrier patch of greater resistance may be aided by stress corrosion in a wet environment (Das and Scholz, 1981). Whether or not a sizable zone of stable, quasistatic to quasidynamic nucleation appears prior to earthquake dynamic shear instability depends on how nonuniformly the rupture growth resistance (in other words, the strength and the critical slip displacement) prevails on the fault in the lithosphere. Such spatial nonuniformities of rupture growth resistance may be caused by geometrical or topographical settings of the fault (zone), rock type in the fault zone, the presence or absence of water, ambient temperature and effective pressure at crustal depths. More specifically, overall depth variation of the rupture growth resistance in the lithosphere results primarily from non-uniform distributions of temperature and effective pressure at crustal depths, whereas local to small scale variations of the rupture growth resistance in the brittle seismogenic layer may be caused by fractal roughnesses of fault surfaces, heterogeneous distribution of rock type in the fault zone, and local to small scale fluctuation of pore water pressure (Ohnaka, 1992). If spatial distribution of the rupture growth resistance is given for a fault of particular geological and tectonic settings in the lithosphere, the earthquake rupture nucleation can be computer-simulated and the critical size of the nucleation zone for the fault can be estimated specifically (Yamashita and Ohnaka, 1992). In general, however, spatial distribution of the rupture growth resistance is not known for faults that have the potential to cause major earthquakes and this makes it difficult to evaluate the critical size of the nucleation zone for a real earthquake. For an earthquake that has carried a great number of immediate foreshocks, there is another possible way to infer the critical size of the nucleation zone: that is to utilize immediate foreshock activity itself. Immediate foreshocks are interpreted as dynamic instabilities of local to small scale that occur in the process of the quasistatic to quasidynamic nucleation leading to the mainshock rupture and hence they are a part of the mainshock nucleation process. This enables one to infer the mainshock nucleation process and its zone size by monitoring the spatio-temporal distribution of the hypocenter locations of its immediate foreshocks. Practically, however, there is a problem of how to identify immediate foreshocks; Vol. 41, No. 1, 1993

8 50 M. Ohnaka specifically, what period and size should be employed as the time and space windows, respectively, to identify immediate foreshocks. This is a difficult question to answer. Jones and Molnar (1979), and Scholz (1988) pointed out that typical immediate foreshocks, that are concentrated close to the hypocentral region, begin to appear less than about 10 days before the mainshock. This empirical observation may give an indication of a suitable time window to identify immediate foreshocks. A specific size for the space window to be used to identify immediate foreshocks may be difficult to generalize because it should differ according to the critical size of the nucleation zone for individual earthquake mainshocks, which in turn depends on spatial distribution of the rupture growth resistance. A computer simulation (Yamashita and Ohnaka, 1992) of earthquake nucleation along the San Andreas transform fault boundary indicates that the critical size of the nucleation zone could be of the order of 10 km, though it depends on specific spatial distribution of the critical slip displacement. This suggests that the size for the space window to identify immediate foreshocks for a major earthquake may be as large as 10 km. A major earthquake (MJMA 7.0) occurred on 14 January 1978 (JST) in an area between the Izu Peninsula and the Oshima Island (for the location of which see Fig. 3), and this earthquake was preceded by swarm-like immediate foreshock activity off the west coast of Oshima Island. This earthquake, often called the 1978 Izu Oshima Kinkai earthquake, was a typical intraplate earthquake that nucleated within the brittle seismogenic layer and is known for its conspicuous immediate foreshock activity Fig. 3. Location map of the area of Izu Peninsula and Oshima Island. The star mark indicates the epicentral location of the 1978 Izu Oshima Kinkai earthquake. J. Phys. Earth

9 Letter 51 Table 1. Comparison of the origin time and hypocentral coordinates of the 1978 Izu Oshima Kinkai mainshock earthquake determined by earlier authors. (Tsumura et al., 1978). We here infer the critical size of the mainshock nucleation zone for this earthquake from the immediate foreshock activity. The origin time and the hypocentral location of the mainshock earthquake are listed in Table 1. One may notice from Table 1 that the values for the origin time and the hypocentral location given by Shimazaki and Somerville (1979) are particularly different from those obtained by the other authors. This is because Shimazaki and Somerville (1979) interpreted the origin time and hypocentral coordinates determined by Tsumura et al. (1978) using data at 27 stations which lie within an epicentral distance of 200 km (see Table 1) as those of a precursory earthquake that took place on the mainshock fault plane about 6s before the mainshock occurrence. This sequence, however, is regarded as a single main event by other authors (see Table 1). This latter position will be taken in the present analysis. Another point to be suggested from Table 1 is that reliability may be low for the hypocentral depth. I here simply assume the hypocentral depth of the mainshock to be 3.8 km, following the earthquake catalog by Matsu'ura et al. (1988). However, the position and assumption employed here does not affect the conclusion which will be given below. The earthquake catalog "List of Earthquakes in the Kanto Area and Its Vicinity" (Matsu'ura et al., 1988) provides a set of the most reliable data for the present purpose and hence is used for the present study. From this catalog, earthquakes of M>2.0 of which hypocenters had been determined from the data at 6 or more than 6 stations were selected, and since these selected data are considered most reliable, they are used for the present analysis. According to these data, immediate foreshocks of M>2.0 for the Izu Oshima Kinkai earthquake began to occur at about 17h40m on 13 January 1978 (JST) in an area near the hypocenter of the ensuing mainshock. The entire period from this time to the time of the mainshock occurrence may be divided into three phases (see Figs. 4 and 5): phase I (time interval of 15 h and 20 min from 17h140m on 13 January) in which foreshocks were concentrated within a very small limited area; phase II (time interval of 1 h and 10 min) in which foreshocks began to migrate toward its environs; and phase III (time interval of 2 h and 14 min) in which foreshocks continued to migrate Vol. 41, No. 1, 1993

10 52 M. Ohnaka Fig. 4. Epicentral locations of foreshocks (dots) in successive phases I, II, and III, and mainshock (star) for the 1978 Izu Oshima Kinkai earthquake. The arrow indicates the direction of the mainshock rupture (Shimazaki and Somerville, 1979). J. Phys. Earth

11 Letter 53 Fig. 5. Depth distribution of hypocenters of foreshocks (dots) in successive phases I, II, and III, and mainshock (star) for the 1978 Izu Oshima Kinkai earthquake. The arrow indicates the direction of the mainshock rupture (Shimazaki and Somerville, 1979). along the strike of the mainshock fault until the mainshock took place. Phase I may be regarded as the initial stage of immediate foreshock activity, and phase III as the final stage. Hypocentral locations of these foreshocks were restricted to lie within 10 km depth, and the mainshock hypocenter was located at a depth of 4 km, about 10 km west of the Izu Oshima Island (Fig. 5). Its aftershock activities were also restricted to shallow depths with a maximum depth of 10 km (Ohnaka, 1992). These suggest that the base of the seismogenic layer in this particular region is at a depth of 10 km, which is compatible with depth profiles of the shear resistance, the critical slip displacement and the breakdown stress drop in a wet environment (Ohnaka, 1992). The foreshock activity migrated primarily along the mainshock fault strike in the east-west direction (Figs. 4 and 5), and this migration of foreshock activity means that the nucleation zone developed with time in that direction. Development of the nucleation zone eventually led to the mainshock dynamic rupture, which propagated westward unilaterally in the east-west direction from its hypocenter (star mark in Figs. 4 and 5) (Shimazaki and Somerville, 1979; Kikuchi and Sudo, 1984). The lateral migration of immediate foreshock activity along the mainshock fault strike indicates that the Vol. 41, No. 1, 1993

12 54 M. Ohnaka Fig 6. Spatio-temporal view of the foreshock migration (dots) for the 1978 Izu Oshima Kinkai earthquake. Star mark represents the epicenter of the mainshock and the thick arrows indicate the direction of the mainshock rupture (Shimazaki and Somerville, 1979). The hatched portion denotes the inferred lower bound of the nucleation zone. nucleation zone for the 1978 Izu Oshima Kinkai earthquake developed in the east-west direction with time. Figure 6 shows the spatio-temporal view of the foreshock migration for the 1978 Izu Oshima Kinkai earthquake. The distance on the lateral axis in Fig. 6 is taken along the mainshock fault strike. The dots in the figure denote the immediate foreshocks, and the star indicates the onset of the mainshock dynamic rupture. I assume that the region where immediate foreshocks were concentrated is a part of the zone of the mainshock nucleation. Under this assumption, the hatched portion in the figure, inferred from the immediate foreshock activity, may indicate the lower bound of the nucleation zone for the 1978 Izu Oshima Kinkai earthquake. It is inferred from Fig. 6 that the critical size of the mainshock nucleation zone was as large as 10 km, and that the tips of the nucleation zone grew with an apparent speed of 1 to 40 cm/s. The zone J. Phys. Earth

13 Letter 55 size of the order of 10 km is large enough for the nucleation to be utilized as an immediate precursor for the short-term earthquake prediction, and hopefully this will pave the way for the attainment of the short-term earthquake prediction. Peter Sammonds reviewed this paper and gave helpful comments, for which I am grateful. I thank Ms. Y. Kotake for providing original computer-drawn map of Fig. 3. REFERENCES Byerlee, J., Friction, overpressure and fault normal compression, Geophys. Res. Lett., 7, , Byerlee, J., The change in orientation of subsidiary shears near faults containing pore fluid under high pressure, in Earthquake Source Physics and Earthquake Precursors, ed. T. Mikumo, K. Aki, M. Ohnaka, L.J. Ruff, and P.K.P. Spudich, Special Issue of Tectonophysics, 211, , Das, S. and C.H. Scholz, Theory of time-dependent rupture in the earth, J. Geophys. Res., 86, , Eaton, J.P., W.H.K. Lee, and L.C. Pakiser, Use of microearthquakes in the study of the mechanics of earthquake generation along the San Andreas fault in central California, Tectonophysics, 9, , Japan Meteorological Agency, Seismological Bulletin for January 1978, pp. 1-67, Jones, L.M. and P. Molnar, Some characteristics of foreshocks and their possible relationship to earthquake prediction and premonitory slip on faults, J. Geophys. Res., 84, , Kikuchi, M. and K. Sudo, Inversion of teleseismic P waves of Izu-Oshima, Japan earthquake of January 14, 1978, J. Phys. Earth, 32, , Lachenbruch, A.H. and J.H. Sass, Heat flow from Cajon Pass, fault strength and tectonic implications, J. Geophys. Res., 97, , Matsu'ura, M., H. Kataoka, and B. Shibazaki, Slip-dependent friction law and nucleation processes in earthquake rupture, in Earthquake Source Physics and Earthquake Precursors, ed. T. Mikumo, K. Aki, M. Ohnaka, L.J. Ruff, and P.K.P. Spudich, Special Issue of Tectonophysics, 211, , Matsu'ura, S.R., I. Karakama, and K. Tsumura, List of Earthquakes in the Kanto Area and Its Vicinity, Earthquake Research Institute, Univ. Tokyo, Tokyo, Ohnaka, M., Earthquake source nucleation: a physical model for short-term precursors, in Earthquake Source Physics and Earthquake Precursors, ed. T. Mikumo, K. Aki, M. Ohnaka, L.J. Ruff, and P.K.P. Spudich, Special Issue of Tectonophysics, 211, , Ohnaka, M. and Y. Kuwahara, Characteristic features of local breakdown near a crack-tip in the transition zone from nucleation to unstable rupture during stick-slip shear failure, in Earthquake Source Processes, ed. S. Das and M. Ohnaka, Special Issue of Tectonophysics, 175, , Ohnaka, M., Y. Kuwahara, K. Yamamoto, and T. Hirasawa, Dynamic breakdown processes and the generating mechanism for high-frequency elastic radiation during stick-slip instabilities, in Earthquake Source Mechanics, ed. S. Das, J. Boatwright, and C.H. Scholz, M. Ewing Vol. 6, Geophys. Monograph 37, pp , American Geophysical Union, Vol. 41, No. 1, 1993

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