641Bulletin of the Seismological Society of America, Vol. 92, No. 2, pp. 641–655, March 2002Aftershock Zone Scalingby Yan Y. KaganAbstract We investigate the distribution of aftershock zones for large earthquakes(scalar seismic moment M ⱖ1019.5N m, moment magnitude, m ⱖ7). Mainshocksare selected from the Harvard centroid moment tensor catalog, and aftershocks areselected from the Preliminary Determination of Epicenters (NEIC) catalog. The af-tershock epicenter maps are approximated by a two-dimensional Gaussian distribu-tion; the major ellipse axis is taken as a quantitative measure of the mainshock focalzone size. The dependence of zone length, l, on earthquake size is studied for threerepresentative focal mechanisms: thrust, normal, and strike slip. Although the num-bers of mainshocks available for analysis are limited (maximum a few tens of eventsin each case), all earthquakes show the same scaling (M ⬀ l3). No observable scalingbreak or saturation occurs for the largest earthquakes (M ⱖ1021Nm,m ⱖ8). There-fore, it seems that earthquake geometrical focal zone parameters are self-similar.IntroductionEarthquake rupture is characterized by three geometricquantities: length of the rupture, l, width, W, and averageslip, u. Seismic moment, M, is defined through these quan-tities asMuWl=µ,(1)where l is the elastic shear modulus.The scaling relations between M and the geometricalparameters in (1) have attracted seismologists’ attention formany years. Kanamori and Anderson (1975), Geller (1976),and Sato (1979) proposed that for small and moderate earth-quakes the scalar moment M is proportional to the cube ofits focal length ldM ⬀ l , (2)with d ⳱ 3. However, they suggested that for larger earth-quakes, this scaling relation breaks down, with d ⳱ 2 forlarge earthquakes or even d ⳱ 1 for the largest ones.The reason for postulating such a breakdown is usuallyformulated as follows. For small earthquakes the rupturepropagates entirely within the brittle crust, whereas for largeearthquakes rupture is confined to the upper crust layer ofthickness W. Thus, ifWW==0const,(3)the rupture is forced to propagate in a relatively thin brittlelayer.We use notation m for the moment magnitude:mM=−23610log ,(4)where M is measured in N m. The magnitude calculated by(4) is used here only for illustration, all pertinent computa-tions being carried out with the moment M values.Although other geometrical parameters of earthquakerupture have been correlated with seismic moment, thelength of the rupture is determined with significantly betteraccuracy than width or slip. This is reflected, for example,in a higher correlation coefficient of M versus l compared tothe correlation of the moment with W or u (Wells and Cop-persmith, 1994): q ⳱ 0.95, 0.84, and 0.75, respectively.Thus, we only use the M versus l correlation in this study.There is a significant amount of literature on the scalingrelation of the aforementioned variables for various faultsand earthquakes (Kanamori and Anderson, 1975; Geller,1976; Sato, 1979; Romanowicz, 1994; Wells and Copper-smith, 1994; Abercrombie, 1995; Pegler and Das, 1996;Scholz, 1997, 1998; Bodin and Brune, 1996; Wang and Ou,1998; Mai and Beroza, 2000; Stock and Smith, 2000; Fujiiand Matsu’ura, 2000; Shaw and Scholz, 2001, and refer-ences therein). Abercrombie (1995) showed no slope breakin composite data sets for small and moderate earthquakesand a slope of 3, albeit with a large scatter. Although thescaling exponent d ⳱ 3 for m ⱕ 6 events is now generallyaccepted, various empirical and theoretical values for d havebeen proposed for larger earthquakes: d ⳱ 1.0, 1.5, 2.0,and 3.0.Two models have been proposed for scaling of slip inlarge earthquakes: the W-model (Romanowicz, 1994, and642 Y. Y. Kaganreferences therein), which assumes that slip is proportionalto the rupture width W and consequently is constant as longas condition (3) holds. A second model, called the L-model(Scholz, 1997; 1998, and references therein), assumes thatu ⬀ l. Assuming that fault width is essentially constant, asdiscussed previously, the W-model predicts d ⳱ 1, whereasthe L-model requires d ⳱ 2. Various empirical data havebeen used to confirm or refute the aforementioned hypoth-eses. Because of the poor quality of the data and inappro-priate control over data selection, the results have been in-conclusive.Two problems are obvious in previous solutions of theearthquake scaling relation: (1) Statistical techniques: Manymagnitude estimates are derived from intensity data or fromother less-than-reliable sources. Since their accuracy is low,one must take into account uncertainty in both magnitudeand rupture length. (2) Data: As Pegler and Das (1996) sug-gest, data on rupture length come from different, sometimesunreliable sources that often yield multiple solutions. Theaccuracy, systematic errors, and selection biases of eachlength and magnitude determination are not well controlled.Wang and Ou (1998) reanalyzed the data of Wells andCoppersmith (1994). In addition to the ordinary least-squareregression (moment versus length) they used the major-axisleast-squares method, which assumes that both regressionvariables have an associated experimental error. For the or-dinary least-squares the value of the d exponent is 2.2–2.5,but it increases to 2.4–3.0 when the more appropriate (major-axis) method is used.Stock and Smith (2000) analyzed a set of 550 earth-quake scaling data that included the Wells and Coppersmith(1994) list. They also took into account uncertainties in bothvariables. They obtained d ⳱ 3 scaling for all earthquakesother than large strike-slip events (M ⬎ 1019N m), for whichthey suggest d ⳱ 2 as a more appropriate scaling. Their dataset contains 14 strike-slip earthquakes with M ⱖ 1020Nm(m ⱖ 7.33), most of which form a point cloud, justifying theapproximation d ⳱ 2 for large earthquakes. However, oncloser inspection the list reveals that all but one (the 16 July1990 Philippines) earthquake occurred before 1977 (theNobi, Japan, earthquake time is 28 October 1891, not 28October 1981). Thus, almost all the large strike-slip earth-quakes on their list are from a predigital or even preinstru-mental era, so the scalar moment values were determinedwith a large error. For preinstrumental earthquakes, the mo-ment value is usually calculated by assuming a W value in(1) equal to 15 or 20 km. Thus, the scaling break is, in effect,postulated. Surface-wave