Bulletin of the Seismological Society of America, Vol. 92, No. 8, pp. 3318–3320, December 2002
Short Notes Fractal Dimensions of Small, Intermediate, and Large Earthquakes D. Legrand*
Abstract The spatial fractal dimension D of earthquakes (or faults) is often correlated with the slope b of the Gutenberg–Richter law, independently of earthquake size. An already classical formula is Aki’s D ⳱ 3b/c ⳱ 2b. This formula implies the three following hypothesis: (1) the Gutenberg–Richter law log10N ⳱ a ⳮ bM is satisfied; (2) the seismic moment M0 is related to the surface magnitude Ms as log10 M0 ⳱ cMs Ⳮ d with a typical value of c ⳱ 1.5; and (3) the static self-similarity scaling law is satisfied, that is, M0 ⬀ L3, where L is the characteristic dimension of the fault. Hypothesis (3) implies that events are small or intermediate and break on a square plane (i.e., M0 ⬀ L3). Nevertheless, for large events, this hypothesis is not satisfied because the shape of large events is a rectangle and not a square (i.e., M0 ⬀ L2). Therefore, for large events the formula D ⳱ 3b/c should not be used; the formula D ⳱ 2b/c should be used instead. In hypothesis (2), c depends upon event sizes: c ⳱ 1, 1.5, and 2 for small, intermediate, and large events, respectively, therefore resulting in D ⳱ 3b, D ⳱ 2b, and D ⳱ b, respectively. As a consequence, small earthquakes (or small faults) are distributed within volumes, whereas large earthquakes (or large faults) are distributed along lines. Introduction The size distribution of earthquakes in space and time is not a random phenomena. Its complexity can be described by a fractal behavior in both space and time. For aftershocks, the quantity of small and large events are related by the Ishimoto–Iida/Gutenberg–Richter law (Ishimoto and Iida, 1939; Gutenberg and Richter, 1944). The temporal distribution of aftershocks often follows Omori’s law (Omori, K 1895): N ⳱ (i.e., the number N of aftershocks de(1 Ⳮ t)p cays with time t after the mainshock). K is a constant. Many studies have been done relating D and b-values, showing that D and b relations are not so clear (e.g., Hirata, 1989; Turcotte, 1992; Henderson et al., 1994; Barton et al., 1999; Scholz, 2002). I show in this article that the relation between the fractal dimension D and the b-value of the Gutenberg–Richter law depends on earthquake magnitudes.
Case of Intermediate Earthquakes The b-value has already been interpreted in terms of the geometrical distribution of events, as a fractal dimension D (Aki, 1981) and in terms of tectonism (King, 1983). Aki (1981) related the b-value to the fractal dimension D of an earthquake spatial distribution with the three following assumptions: 1. The number of events follows the Ishimoto–Iida/Gutenberg–Richter law (Ishimoto and Iida, 1939; Gutenberg and Richter, 1944): log10N(t) ⳱ a(t) ⳮ bM, when N(t) is the number of earthquakes of magnitude ⱖM recorded during the time interval t, a(t) is the log10 of the number of earthquakes of magnitude ⱖ0 recorded during the time interval t, and b is a constant over the time interval t. 2. The scalar seismic moment M0 is related to surface magnitude Ms as follows:
*Present address: La Pens’Yves, 7 ter Rue de la Vigne, Miniac-Morvan, 35 540, France.
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log10M0 ⳱ cMs Ⳮ d,
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where c and d depend on earthquake sizes. A typical value for intermediate earthquakes is c ⳱ 1.5 (Kanamori and Anderson, 1975). 3. The static self-similarity scaling law x2 (Aki, 1967) is respected (in practice for small and intermediate earthquakes). This supposes a constant stress drop, hence the scalar seismic moment is given by M0 ⳱ lSDu ⬀ L3, where l is the shear modulus, S ⳱ LW is the surface of the fault with L the length and W the width of the fault and Du is the dislocation over the fault. This property characterizes a self-similarity in both L and W directions (squarish area fault with L ⬇ W). Combining 1, 2, and 3 leads to: N ⬀ Lⳮ3b/c. The fractal dimension D is defined by Mandelbrot (1975) as: N ⬀ LⳮD. Therefore Aki (1981) deduced that D ⳱ 3b/c. For a typical value of c ⳱ 1.5, the relation becomes D ⳱ 2b and is limited to intermediate events. Intermediate events are those for which the relation M0 ⬀ L3 is satisfied. Aki’s relation in combination with Kanamori and Anderson’s study is limited to intermediate earthquakes for which c ⳱ 1.5. It cannot be employed for small events for which c ⳱ 1, nor for large events for which the similitude law is not respected.
Case of Small Earthquakes For small events, Kanamori and Anderson (1975) showed that Ms ⬀ log10L3. In that case, using the similitude law (3) described before, the corresponding c-value is c ⳱ 1; hence the fractal dimension becomes D ⳱ 3b/c ⳱ 3b.
Case of Large Earthquakes The hypothesis (3) is not true for large earthquakes because their width W of the rupture area is restricted by the local depth of the seismogenic zone. The shape of the surface of faulting affected by large earthquakes is not squarish as for small or intermediate events but is a rectangle; therefore M0 ⬀ L2 (Scholz, 1982, 1994; Shimazaki, 1986; Pacheco et al., 1992). Large events are those for which the relation M0 ⬀ L2 is satisfied. Using hypotheses (1) and (2) and M0 ⬀ L2, the formula D ⳱ 2b/c must be used for large events. For large events, Kanamori and Anderson (1975) showed that Ms ⬀ log10L and used the similitude law (3) described above. Hence, they deduced that c ⳱ 3. Later Scholz (1982) showed with his L-model that the relation (3) should be replaced by M0 ⬀ L2, so that for large events one can demonstrate that c ⳱ 2. In such a case, the fractal dimension becomes: D ⳱ 2b/c ⳱ b. Table 1 shows a summary of all of these cases.
Discussion One important hypothesis in D-value calculation of earthquakes is to use a source point model for earthquake locations and not a finite-dimension source model (L. Dorbath, personal comm.). In theory, it should be better to use a finite-dimension source for the fault with its corresponding orientation in space. This information can be obtained via the focal mechanism. In practice, focal mechanisms of moderate to large events can be determined, but this information is almost impossible to obtain for all small events. Hence, a correct estimation of D in a three-dimensional space is almost impossible to achieve. Another difficulty is the correlation of spatial complex¨ ncel et al., 2001). ity of active faults and earthquakes (e.g., O For such studies, the fractal dimension D is not calculated from earthquake locations but from active faults detected in the field from geological and/or from spatial studies. As the
Table 1 Ms, M0, L, D and b Relations for Small, Intermediate, and Large Events Small events
Intermediate events
Large events
Ms ⬀ log10L3*
Ms ⬀ log10L2*
Ms ⬀ log10L*
logM0 ⬀ MS with assumption M0 ⬀ L3*,†
3 MS 2 with assumption M0 ⬀ L3*,†
logM ⬀ 2MS with assumption M0 ⬀ L2†
Ms and L relation M0 and Ms relation deduction
c-value of relation: logM0 ⬀ cMs Ⳮ d Fractal dimension D
logM0 ⬀
1
1.5
2
3b D⳱ ⳱ 3b c
3b D⳱ ⳱ 2b c
2b D⳱ ⳱ b c
*Kanamori and Anderson (1975) † Scholz (1982).
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extension of faults in depth is unknown for many of them, these studies are done in a two-dimensional horizontal drawing corresponding to the map of the faults. D is then compared with the b-value calculated from earthquakes distributed in a three-dimensional space. In order to compare D and b, different approaches are used. One classical way of doing this is to add 1 to the fractal dimension calculated from a two-dimensional map of faults to simulate a fractal dimension in a three-dimensional space. But by doing so, we assume that the fractal behavior in depth of the fault is 1; therefore, the corresponding D value is overestimated. Another way of doing this is to repartition the earthquakes inside cylinders to simulate a two-dimensional space and compare the corresponding b-value to the D value calculated ¨ ncel et al., 2001). But by from a two-dimensional space (O doing so, we are not sure that the Gutenberg–Richter law is followed inside the cylinder, because large and small events may not be distributed approprietely in the corresponding cylinder (usually many events are distributed along a fault plane, which means on a square or a rectangle, and not a cylinder). As a limit, if a typical value of b ⳱ 1 is considered, then the fractal dimension varies from 3 for small events to 1 for large events, and is 2 for intermediate events. This means that small events are distributed within volumes, whereas large events are distributed along lines and intermediate events over surfaces. A similar conclusion can be deduced for faults: small faults will be distributed within volumes, whereas big faults (typically old faults) are more linearly distributed.
Conclusions The spatial fractal dimension D of earthquakes varies with event size. For small, intermediate and large events, D ⳱ 3b, D ⳱ 2b, and D ⳱ b, respectively. The difference between intermediate and large events is related to the seismogenic zone thickness. Hence, small earthquakes (respectively small faults) are more distributed within volumes, whereas large earthquakes (respectively large faults) are more generally distributed along lines. Intermediate earthquakes are distributed over surfaces. Some caution should be taken in comparing D values calculated from fault traces on a two-dimensional map with respect to b-values calculated from the size distribution of earthquakes in a three-dimensional space. This may cause discrepancies which are due to the way in which D and bvalues are estimated and are not due to a real geophysical behavior.
Acknowledgments I thank the Institut Franc¸ais d’Etudes Andines (IFEA), the French Ministe`re des Affaires Etrange`res (MAE), and Ambassade de France in Quito
to support a part of this study. I would like to thank A. Cisternas, L. Dorbath, M. Hall, J-Ph. Eissen, and an anonymous reviewer for helpful discussions and correcting this article.
References Aki K. (1967). Scaling law of seismic spectrum, J. Geophys. Res. 72, 1217– 1231. Aki K. (1981). A probabilistic synthesis of precursory phenomena, in Earthquake Prediction: An International Review, Maurice Ewing Series 4, D. W. Simpson and P. G. Richards (Editors), American Geophysical Union, Washington, D.C., 566–574. Barton, D., G. Foulger, J. Henderson, and B. Julian (1999). Frequencymagnitude statistics and spatial correlation dimensions of earthquakes at Long Valley caldera, California, Geophys. J. Int. 138, 563–570. Gutenberg, B., Ch. Richter (1944). Frequency of earthquakes in California, Bull. Seism. Soc. Am. 34, 185–188. Henderson, J., I. Main, R. Pearce, and M. Takeya (1994). Seismicity in north-eastern Brazil: fractal clustering and the evolution of the b value, Geophys. J. Int. 116, 217–226. Hirata, T. (1989). A correlation between the b value and the fractal dimension of earthquakes, J. Geophys. Res. 94, 7507–7514. Ishimoto, M., and K. Iida (1939). Observations sur les se´ismes enregistre´s par le micro-se´ismographe construit dernie`rement (1), Bull. Earthquake Res. Inst. Univ. Tokyo 17, 443–478. Kanamori, H., L. Anderson (1975). Theoretical basis of some empirical relations in seismology, Bull. Seism. Soc. Am. 65, 1073–1095. King, G., 1983: The accommodation of large strain in the upper lithosphere of the earth and other solids by self-similar faults systems: The geometrical origin of b-value, Pure Appl. Geophys., 121, 761–815. Mandelbrot, B. (1975). Les objets fractals: forme, hasard et dimension, Flammarion, Paris. Omori, F. (1895). On the aftershocks of earthquakes, Coll. Sci. Imper. Univ. Tokyo 7, 111–200. ¨ ncel, A., Th. Wilson, and O. Nishizawa (2001). Size scaling relationships O in the active fault networks of Japan and their correlation with Gutenberg–Richter b values, J. Geophys. Res. 106, 21,827–21,841. Pacheco, J., C. Scholz, and L. Sykes (1992). Changes in frequency-size relationship from small to large earthquakes, Nature 355, 71–73. Scholz C. (1982). Scaling laws for large earthquakes: consequences for physical models, Bull. Seism. Soc. Am. 72, 1–14. Scholz, C. (1994). A reappraisal of large earthquake scaling, Bull. Seism. Soc. Am. 84, 215–218. Scholz, C. (2002). The mechanics of earthquakes and faulting, Second Ed., Cambridge University Press, Cambridge, U.K., 471 pp. Shimazaki, K. (1986). Small and large earthquakes: the effect of the thickness of the seismogenic layer and the free surface, in Earthquake Source Mechanics, S. Das, J. Boatwright, and C. Scholz (Editors), American Geophysical Monograph 37, 209–216. Turcotte, D. (1992). Fractals and chaos in geology and geophysics, Cambridge University Press, New York.
Institut Franc¸ais d’Etudes Andines (IFEA) Whymper 442 y Corun˜a Quito, Ecuador
[email protected] Instituto Geofı´sico de la Escuela Polite´cnica Nacional (IG-EPN) Apartado 2759 Quito, Ecuador
Manuscript received 18 January 2002.
