Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact?

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1 Bulletin of the Seismological Society of America, 90, 1, pp , February 2000 Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact? by Ian Main Abstract It has been suggested that the finite width of the seismogenic lithosphere can have a strong effect on the frequency-moment relation for large earthquakes. Theories have been proposed in which large earthquakes have either a shallower or a steeper power-law slope in the incremental frequency-moment distribution, at characteristic seismic moments corresponding to earthquakes that rupture the entire seismogenic depth. However, many authors have applied the predicted double-slope distribution, which requires five independent parameters, to cumulative frequency data, and used the location of the break of slope to make inferences on characteristic size effects in the earthquake source in different seismotectonic regions. Here we examine the problem in a forward modeling mode by adding a realistic degree of statistical scatter to ideal incremental frequency-moment distributions of various commonly used forms. Adopting a priori the assumption of a piecewise linear distribution, we find in each case apparently statistically distinct breaks of slope that are not present in the parent distribution. These breaks of slope are artifacts produced by a combination of (a) high-frequency noise introduced by the random statistical scatter, (b) the more gradual natural roll-over in the cumulative frequency data near the maximum seismic moment, and (c) a systematic increase in the apparent regression coefficient due to the natural smoothing effect of the use of cumulative-frequency data. Therefore, if there is no apparent break of slope in the incremental distribution, it is unwise to interpret the cumulative-frequency data uniquely in terms of a break in slope. Until such breaks of slope can be distinguished in incrementalfrequency data, we conclude that alternative methods (inversion of fault length and width from individual source mechanisms/aftershock sequences etc.) should be preferred for the examination of finite-depth effects, and that simpler solutions to the frequency-moment problem should be adopted for seismic-hazard applications. Introduction The nature of the earthquake frequency-magnitude distribution is of fundamental importance in probabilistic seismic-hazard calculations (Reiter, 1991), and accordingly has attracted a great deal of attention since it first became apparent that, to first order, earthquakes followed a power-law distribution of quantitative source parameters such as radiated seismic energy, seismic moment, or fault length (e.g. Richter, 1958; Turcotte, 1992). In fact, the Gutenberg-Richter frequency-magnitude law (corresponding to a power-law in seismic moment or energy) is still in common use in much standard earthquake-hazard analysis, testament to the ubiquity of power-law scaling (at least in a finite range of seismic moment) in a variety of tectonic settings and depth ranges, and whether the seismicity is natural or induced. This experience has been based on catalogs spanning a few decades for digital seismic-moment determinations, or a hundred years or so of magnitude determinations from analog instrumental data. As a consequence, we can be relatively sure that the power-law scaling of the Gutenberg-Richter law holds for small, and hence well-sampled, earthquakes, apparently universally. In this article, we concentrate instead on examining the behavior of the frequency-moment distribution for the less well-sampled large earthquakes. Such large events occur rarely when compared to the timespan of modern instrumental earthquake catalogs and hence have a greater statistical scatter, even in cases where historical or paleoseismic data are available. Nevertheless, although fewer in number, these events usually dominate the total release of seismic moment or energy (e.g., Main, 1995). The problem of how to deal with large earthquakes has attracted a great deal of attention since it first became obvious that the Gutenberg- Richter law could not continue indefinitely to higher magnitudes, due to the physical requirement of finite-energy flux, 86

2 Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact? 87 or equivalently a finite-seismic moment-release rate (e.g. Anderson, 1979; Molnar, 1979). Upper bounds to momentrelease rates may be determined from plate tectonic slip rates for interplate earthquakes, or from observed intraplate deformation rates, the latter predominantly for the mechanically weaker continental lithosphere. Both types of deformation, now amenable to accurate determination from continuous monitoring of global satellite data, often provide a surprisingly good match both when compared to seismic moment tensor determinations of relative motion from shortterm earthquake data (e.g., Jackson et al., 1994; Bilham et al., 1997; Clarke et al., 1997) or even longer-term plate models (e.g., DeMets, 1995). Such observed tectonic moment rates can therefore be used to constrain the behavior of the seismic frequency-moment distribution for the largest earthquakes (e.g. Main and Burton, 1984, 1986; Kagan, 1997), including the determination of the maximum credible magnitude (Kagan, 1993, Main, 1995) as commonly used in probabilistic seismic-hazard calculations (Reiter, 1991). The form of the frequency-moment distribution for large earthquakes thus has an important and direct application in seismic-hazard analysis, particularly when it can be combined with longer-term tectonic, historical, or paleoseismic data. In recent years, several authors have examined the cumulative frequency-moment distribution for large earthquakes, and found evidence for a systematic and sudden change in the scaling properties. This break of slope has been plausibly explained on theoretical grounds as being due to the finite thickness of the seismogenic lithosphere (e.g., Pacheco et al., 1992, Scholz, 1997; Triep and Sykes, 1997). However, the statistical significance of this inference has been challenged by Sornette et al. (1996), who developed an objective test for systematic changes in slope using rankordering statistics. In this article we examine this problem in a forward-modeling mode in order to investigate the statistical origin for the apparent breaks in scaling, and hence the validity of the inversion approach, given the current relative incompleteness of earthquake catalogs for large earthquakes. First we review the evidence for a break in scaling between small and large earthquakes due to the finite depth (width) of the seismogenic lithosphere. Finite-Size Effects in Earthquake Scaling The concept of a seismogenic zone of finite width is consistent either with the onset of thermally activated plastic deformation, or the transition to a velocity strengthening rheology with increasing hypocentral depth (e.g., Scholz, 1990). If the maximum depth of an earthquake is fixed, then above a critical size, a larger event can only grow along strike (length L), rather than down the dip (width W) ofthe fault (Pacheco et al., 1992, Fig. 1). This implies that small earthquakes may grow in size according to self-similar principles (W L), and hence with a constant aspect ratio L/W, but that large earthquakes (W W* constant, where W* is the finite-seismogenic depth) can occur only by increasing the aspect ratio of the fault. If the mean-fault slip u scales as fault width (known as a W-model), then the slip is fixed for large events, u is constant, and seismic moment M equal to llwu scales linearly with length L, where l is the shear modulus of the crust, typically 30 GPa. Alternatively, if the slip scales as fault length (known as the L-model, then slip can continue to increase, and M L 2 for large earthquakes (Scholz, 1982). There has been some debate on the applicability of the model proposed by Pacheco et al. (1992) to regional scales, due partly to a lack of data on earthquakes at the scale of the transition W*, but also to a large degree of inherent statistical scatter in the observed scaling relationships themselves. For example early studies of the relationship between fault length and width, predominantly from aftershock studies (Abe, 1975; Geller, 1976), found that even large earthquakes had a fairly constant aspect ratio of around 2, but with a large scatter of values predominantly between 1 and 4. In a larger study, Purcaru and Berckhemer (1982) came to similar conclusions, that is, that L 2W, with the notable exception of large strike-slip earthquakes, where W W* held, implying a significant change of scaling between small and large earthquakes. This observation implies that sensitivity to the finite-seismogenic width may be related to the direction of the slip vector, but also casts doubt on the W W* hypothesis for large earthquakes with a significant dipslip component, in particular large subduction zone events. An alternate method of examining this problem is to consider the scaling of fault length and seismic moment. Romanowicz (1992) examined this for quasi-vertical strikeslip faults, finding an apparent change of slope from M L 3 to M L, as predicted by the W-model. For dip-slip events (and a significant proportion of nominally unusual strikeslip events), there was no significant break of slope, consistent with Purcaru & Berckhemer s (1982) earlier observations, partly because of a large degree of overlap in the data sets used. Romanowicz interpreted the difference as due to the spatial variability of W* for dip-slip faults in different parts of the world (predominantly large thrust faults in subduction zones). Scholz (1994) challenged this interpretation, arguing instead that the location of the break of slope determined by Romanowicz (1992) at W* 60 km was a factor three or four larger than the known seismogenic thickness of the lithosphere. By removing subduction zone events (with variable W*), and adding a set of normal faulting earthquakes, he also showed that the best fit to the data for L 20 km was M L 2, as predicted for large earthquakes by the L-model. Unfortunately, this data set contained no information for small earthquakes (L 20 km) so no break of slope could be resolved. The difference between these two interpretations could be put down to a more or less subjective choice of data set in each case, combined with a large degree of inherent statistical scatter in the primary data. However, Pegler and Das (1996) qualitatively confirmed Scholz s (1994) observations of no significant break of slope by using an unbiased, uniform data set for shallow earthquakes with

3 88 I. Main Figure 1. (a) Synthetic incremental (P i ) and cumulative (P c ) probability distributions for an ideal gamma distribution with a positive exponential coefficient M*. The data are collected in incremental logarithmic bins of width dlog(m) 0.15, corresponding to a linear incremental magnitude bin of width dm 0.1, as commonly used in earthquake-hazard analysis. The parameters used to form this distribution are noted in Table 1. (b) Ideal probability distributions with random Zn statistical noise added. Note increase in scatter for rarer large events. In this case there is one empty bin (note break in incremental curve). (c) Interpretation of this ideal distribution assuming a priori the existence of a break in slope. The parameters for the two straight lines shown are listed in Table 2. L 15 km, constructed from modern centroid moment tensor (CMT) determinations of the seismic moment, and using 1-day earthquake aftershock data (including relocated events) to determine L. Perhaps surprisingly, the aftershock data presented by Pegler and Das (1996) was not used independently to determine fault width W in order to provide as direct a test as possible of the hypothesis illustrated in Figure 1. The third method of examining the small/large earthquake scaling problem is to examine the earthquake frequency-moment distribution for sudden breaks of slope in the power-law scaling (e.g., Main and Burton, 1986; Turcotte, 1992; Pacheco et al., 1992, Triep and Sykes, 1997; Scholz, 1997), the main subject of this article. There are two aspects to this question, namely (1) the theoretical basis for expecting a change of scaling, based on calculating a probability density function or incremental frequency, and (2) the observational evidence, universally based on cumulativefrequency data. In this article we mainly challenge the second of these, based on a simple forward modeling approach that takes into account a realistic degree of statistical scatter in a finite-sized data set. We now consider the theoretical basis for a break in scaling. Main and Burton (1986) noticed an apparent change in the slope of the cumulative frequency-magnitude relation from b 1tob 0.5 for southern California at a local

4 Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact? 89 magnitude m L 6, corresponding to a source dimension approximately equal to the width of the seismogenic crust in this area of 10 to 15 km. Using a simple tiling argument first proposed by Kanamori and Anderson (1975), they speculated that this might be due to a change in scaling in the length distribution from N(L) L 2 (a two-dimensional fault to N(L) L 1 (a one-dimensional fault), consistent with Figure 1 of Pachao et al. (1992). At this point it is worth noting that all variants of the tiling model are framed in terms of the incremental-frequency distribution. However Turcotte (1992) suggested that the break of slope inferred by Main and Burton (1996) may instead be entirely spurious, due to the statistical scatter in the data caused by the paucity of events of magnitude greater than 6 this century, a conclusion that in retrospect can quickly be confirmed by examining the absence of a clear break of slope in the incremental frequency data (Main and Burton, 1984). In this article, we examine Turcotte s suggestion quantitatively by producing forward models of apparent breaks of slope, which are known to be unrelated to the underlying incremental-frequency distribution. The model of Main and Burton (1986) does not take into account the additional effect of finite-fault width of the scaling of moment and length. When this is done correctly, and adding the constraint of scale-invariance (Rundle, 1989; Romanowicz & Rundle, 1993), the tiling argument still predicts a shallowing of the slope from b 1tob 0.75 (B 2/3 to 1/2) for the L-model, but a steepening from b 1tob 1.5 (B 2/3 to B 1) for the W-model. [The relationship between the Gutenberg-Richter b-value and the exponent B of the frequency-moment distribution (defined in equation 1) is covered in standard textbooks, for example, Turcotte, 1992]. However, by dropping the constraint of strict scale-invariance, for example by adopting a more general multifractal geometry, Sornette and Sornette (1994) showed, using a strain diffusion model, that this steepening of the slope, from B 2/3 to B 1, due to a finite seismogenenic depth, could occur even with L-model scaling. Recently Scholz (1997) provided a third model for the steepening of the frequency-moment slope from B 2/3 to B 1, by adopting a further variant on the tiling model that uses the principle of temporal as well as geometric scaleinvariance, and adopting the characteristic earthquake model for a single fault as the basis for the distribution of a population of faults. In this model earthquakes on a single fault are distributed according to a power-law distribution with B 2/3 up to a width W* or length L*, and postulated a single characteristic earthquake of size L L*, so that there is a gap between the biggest and the next biggest earthquake on a single fault (Schwartz and Coppersmith, 1984). The statistical properties of the large earthquakes (L L*) are then determined by using the observed length distribution of large faults [N(L) L 1 )], and assuming all earthquakes of size L L* are equally likely except for the constraints of spatial and temporal scale-invariance. Two potential problems with this model are (1) the concept of a characteristic earthquake for a single fault has yet to be proven unambiguously in a statistical sense (e.g., Kagan, 1993), and (2) that the magnitude gap between the biggest and the next biggest event, at least in aftershock sequences, tends to be fixed at around one or two magnitude units (Wesnousky et al., 1983). This implies that the size of the next largest earthquake to the characteristic earthquake need not be coincident with W*, as assumed by Scholz (1997). In fact, when the characteristic earthquake model is used to predict the frequency-magnitude distribution in this way for Japan (Wesnousky et al., 1983), no obvious break of slope is introduced. In summary, there are a variety of physical models that give more or less plausible explanations for a break in slope in earthquake scaling laws. There is an ongoing debate on this question, which will not be resolved until there is sufficient and unambiguous independent evidence for the break in slope observed in the scaling of fault width and length, moment and length, or frequency and moment, in decreasing order of direct proof. I argue here, that in the absence of direct proof for a double-slope frequency-magnitude distribution (which requires five independent parameters), a simpler solution to the problem of seismic-hazard estimation should be adopted in the meantime. The gamma distribution subsequently described predicts a more gradual curve in the frequency-moment distribution, rather than the sudden break of slope described previously, and requires the minimum possible number (three) of independent parameters. A second advantage of this approach is that longer-term localdeformation rates can be used to constrain the hazard calculations in a straightforward way (e.g.,kagan, 1993; Main, 1995), with a surprising degree of accuracy when applied to large enough data sets (Kagan, 1997). The Earthquake Frequency-Moment Distribution The incremental frequency-moment distribution [F(M)] is essentially a histogram of the number of times an event in a bin of finite size centered on a moment M occurs within a time interval T. The bin width may be either linear (M dm/2) or logarithmic [log M dlog(m/2)], so it is important to define a basic probability density p(m) from which a frequency or a cumulative frequency can be calculated, irrespective of this arbitrary choice. Most physical models for any distribution are determined by deriving a probability density p, and it is then straightforward to calculate the incremental and cumulative frequencies. We now consider different postulated forms of the density distribution for the frequency-moment relation. Band-Limited Power Law (Gutenberg-Richter Relation) The most commonly used form of frequency-moment distribution is power-law scaling of moment implied by the Gutenberg-Richter law, of the form B1 p(m) AM (1)

5 90 I. Main Assuming a small bin width dm, and applying the binomial expansion, the incremental frequency is then MdM/2 B1 F(M) NT p(m)dm ANTM dm (2) MdM/2 where the event rate N T is the total number of events in the earthquake catalogue divided by its duration T. Thus the frequency is related to the density distribution to a good approximation by F(M) Np(M)dM (3) T a result which holds generally for small dm. The cumulative frequency distribution is then finite A, so the power-law scaling is strictly restricted within a limited bandwidth (M min, M max ). For M max k M min, equations (5) and (7) reduce to: 1B B B Ṁ NM T max M min (8) (B 1) for example, Kagan (1993). Given (7), and noting that N T N(M min ), the distribution (4) has three independent parameters, namely N T, B, and M max. Double-Slope Distribution The double-slope distribution has the form B11 p(m) AM 1 : Mmin M M break (9a) M max B B (M M max) N(M) NT p(m)dm AN T (4) B M The exponent B here is proportional to the Gutenberg-Richter b-value, usually according to b (2/3)B, where typically 0.5 b 1.5, or equivalently 1/3 B 1 (e.g., Turcotte, 1992). In this case the power-law exponent (or equivalently the slope on a log-log plot of frequency or cumulative frequency and moment) of F and N are different. This is not the case if logarithmic bins are used, where it can similarly be shown that both slopes are equal to B, implying parallel slopes on a log-log plot. The most common form of this is the frequency-magnitude distribution, where linear magnitude bins correspond to logarithmic moment bins, and the slope of the incremental and cumulative frequencies are found to be parallel (e.g., see Main, 1995, Fig. 1). In (4) we have had to introduce a finite-maximum moment M max,so that the total seismic moment release rate M max 1B 1B max min T T (M M ) Ṁ N Mp(M)dM AN (5) (B 1) M min remains finite for the observed range B 1. The constant A can be determined from the normalization criterion or M max p(m)dm 1, (6) M min B A B B. (7) M M min In this case a finite minimum M min is also now needed for max B21 p(m) AM 2 : Mbreak M M max (9b) where M break llw*u is the size of the largest small earthquake or the smallest large earthquake, and the slopes of the distributions above and below this value are B 1 and B 2, respectively. The incremental frequency for small dm is then given simply by (3). The cumulative frequency is a little more complicated B1 B1 (M M break) N(M) NTA1 B 1 B2 B2 (Mbreak M max ) A : M M M (10a) 2 min break B 2 B2 B (M M 2 max ) T 2 B 2 break N(M) N A : M M M max Applying the normalization criterion (6), we obtain B 1 B1 B1 (Mmin M break) A 1 B2 B (M M 2 break max ) 2 B 2 (10b) A 1. (11) Multiplying throughout by N T, this is equivalent to N N N (12) 1 2 T where N 1 and N 2 are respectively the number of earthquakes per unit time interval in the moment ranges M min M M break and M break M M max. From continuity at M break we have, from (9a) and (9b),

6 Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact? 91 or B11 B21 AM 1 break AM 2 break (13a) Ṁ ANT B1 M/M* M M exp dm. (18) M min A AM B2B1 1 2 break (13b) We can split this integral into two parts: Solving (11) and (13) gives A B 1 B1 B2 B B (Mmin M break) Mbreak B2 B2 1 (Mbreak M max ) B 2. (14) A 1 can then be determined from (13). The distribution again requires a finite-maximum moment so that the moment release rate Mbreak Mmax T Ṁ N Mp(M)dM Mp(M)dM (15) or from (9) Ṁ N Mmin (B 1) Mbreak 1B1 1B (M M 1 break min ) T A1 1 1B2 1B (M M 2 max break ) 2 (B 1) 2 A (16) remains finite for B 2 1. Therefore, the whole distribution is specified by the five parameters N T, M break, M max, B 1 and B 2 [again noting that N T N(M min )]. The Gamma Distribution The gamma distribution for the frequency-moment distribution was postulated on the basis of information theory by Shen and Mansinha (1983) and Main and Burton (1984). Main and Burton (1984) interpreted the distribution physically in terms of a simple statistical mechanical model for earthquakes, based on a Boltzmann distribution of possible energy transitions on a model fault, modulated by a geometric degeneracy term determined by the two-dimensional tiling model already discussed. The general utility of the gamma distribution, and its basis in the statistical mechanics of earthquakes, was recently reviewed by Main (1996), and its application to seismic hazard analysis discussed by Kagan (1993, 1997) and Main (1995). The form of the probabilitydensity distribution is B1 M/M* p(m) AM exp (17) where M* is a characteristic moment where the probability density falls to a level 1/e less than that predicted by the power-law or Gutenberg-Richter trend. M* is determined directly by the seismic moment release rate, as follows: B M/M* Ṁ AN M exp dm T 0 M min B M/M* M exp dm. (19) 0 The contribution of the smallest earthquakes using this distribution for B 1 is negligible (e.g., Main, 1995), so the second integral can be equated to zero to a very good approximation. After a change of variables to x M/M*, dx dm/m*, we find Ṁ AN M* x exp dx T (1B) B x 0 (1B) ANTM* C(1 B), (20) where C is the standard gamma function, which holds for 1 B 0orB 1, that is, within the normal range of observations of this parameter. The normalization criterion (6) gives A B1 M/M* M exp dm 1 (21) M min For M min K M* and B 1 the total probability is dominated by the power-law term in the integration kernel, so to a good approximation we can neglect the exponential term, whence B A B. (22) M min From (20) and (22), the distribution is (1B) B Ṁ NM* T Mmin BC(1 B). (23) [This is equivalent to equation (14) of Kagan (1993), since C(2 B) (1 B)C(1 B).] The distribution is therefore completely specified by the three independent variables N T, B, and M*, with a straightforward relationship between the moment-release rate and the characteristic moment M*. It has the same number of independent parameters as the Gutenberg-Richter law (three), but avoids the physically unrealistic sudden cut-off to the distribution at high magnitudes. This distribution is more general than the Gutenberg-Richter law in the sense

7 92 I. Main that the latter is recovered [the exponential term in (17) vanishes] for M*, and a characteristic distribution results for M* 0. In both of these cases a finite-maximum moment, representing a sharp cut-off in the probability density distribution, is needed at M max. The characteristic distribution requires four independent parameters, N T, B, M* and M max. We now examine the effect of adding a degree of statistical scatter to the ideal frequency-moment distributions previously described. Method Having reviewed the main types of distribution, we now apply these to the forward modeling of cumulativefrequency data. To simplify matters we consider probabilities only, and neglect the effect of the actual event rate. We calculate an ideal density distribution to determine the incremental probability P(M) p(m) dlogm, where dlogm The bin size was taken to be logarithmic in moment, (or equivalently linear in magnitude: corresponding to the common dm 0.1) so that power-laws in the cumulative and incremental distributions should be parallel on a log-log plot of frequency and moment. We then add a random component to the incremental probability dp ar(1,1)zp, where R is a random number between 1 and 1, and smaller probabilities have a greater scatter according to a square root function. A standard random number generator available in the Excel 5 Spreadsheet was used, so that the exercise could be checked by any reader with access to a PC. The scaling constant a 1 was chosen so that at most, only a few empty bins were included. (In fact it is common in earthquake catalogs to have several empty bins at high moment, so this represents a conservative approach to the problem; the real case is likely at the present time to be more scattered than the simulations presented here). Results Model runs were carried out for the parameters shown in Table 1, and are reproduced in Figures 1 5. Each figure contains three diagrams: In each case (a) represents the ideal synthetic distribution with no errors; (b) shows the synthetic distribution with random errors added; and (c) shows the interpretation based on the a priori assumption that the distribution includes one or more breaks of slope in the cumulative probability plot. [Diagram (b) has been included despite the associated repetition, so that the reader can make his/her own interpretation, and emphasizes the implicit, subjective intermediary step that is currently used in inverting such data for apparent breaks of slope.] Table 2 summarizes the results of interpreting the data with two or three straightline segments, using a standard regression analysis. Although we note that a standard regression although is inappropriate for cumulative-frequency data (even a random-frequency distribution will have a nonzero regression coefficient when plotted in cumulative form), it is current standard practice, which we follow here in a forward modeling mode, before suggesting criteria for a more appropriate inversion scheme. First we examine cases where there is no break of slope in the underlying distribution. Figure 1 shows that an underlying gamma distribution can give rise to an apparent double-slope distribution. This arises because of the natural roll-over in band-limited cumulative frequencies, enhanced by the exponential tail to the gamma distribution at high magnitudes (Fig. 2a), combined with a modest amount of statistical scatter in the probabilities (Fig. 2b). A naive comparison of the two slopes would imply a statistically significant break of slope from B to B at log 10 (M) 17.95, with errors being quoted at one standard deviation. However, in this case the two apparently distinct slopes are an artifact. Figure 2 shows similar results for a bandlimited powerlaw, or Gutenberg-Richter, distribution. In this case the natural roll-over in bandlimited cumulative frequency data (Figure 2a) also contributes to an apparent steepening of the slope at high magnitudes, even though there is no exponential tail to the distribution as in Figure 1. Again, an apparent double slope distribution results, with an apparent break of slope from B to B at log 10 (M) Again a naive comparison would lead to an apparently statistically significant break of slope, which we know is not present in the underlying distribution. Figure 3 shows a similar exercise for a bandlimited gamma distribution with a negative exponential term, leading to a characteristic-type distribution. This distribution produces two apparent breaks in slope, from B Model Name Table 1 Summary of Model Parameters Used Text Parameters (excluding event rate) Equation Figure No. log(m min ) B or B 1 log(m max ) log(m*) B 2 log(m break ) No. of free parameters (a) Gamma () E 19 3 (b) Power-law (c) Gamma () E 19 4 (d) Double-slope () (e) Double-slope ()

8 Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact? 93 Figure 2. Probability distributions for the powerlaw (Gutenberg-Richter) frequency-moment distribution. See Table 1 for parameters used, and the caption to Figure 1 for explanation of figure content. Figure 3. Probability distributions for the gamma distribution with a negative exponential coefficient M* to simulate the characteristic-earthquake distribution. See Table 1 for parameters used and the caption to Figure 1 for explanation of figure content.

9 94 I. Main Figure 4. Probability distributions for the double power-law distribution, assuming an increase in slope. See Table 1 for parameters used and the caption to Figure 1 for explanation of Figure content. Figure 5. Probability distributions for the double power-law distribution, assuming an decrease in slope. See Table 1 for parameters used and caption to Figure 1 for explanation of figure content.

10 Apparent Breaks in Scaling in the Earthquake Cumulative Frequency-Magnitude Distribution: Fact or Artifact? 95 Table 2 Summary of Line Fits* Model Name Figure No. log(m min ) log(m 1 ) A 1 (r A1 ) B 1 (r B1 ) log(m 2 ) A 2 (r A2 ) B 2 (r B2 ) log(m 3 ) A 3 (r A3 ) B 3 (r B3 ) (a) Gamma () (0.088) 0.708(0.005) (1.167) 1.266(0.062) (b) Power-law (0.047) 0.659(0.003) (0.471) 0.954(0.025) (c) Gamma () (0.219) 0.577(0.013) (0.250) 0.178(0.014) (7.783) 1.465(0.400) (d) Double-slope () (0.103) 0.707(0.006) (0.204) 0.992(0.011) (6.625) 2.380(0.343) (e) Double-slope () (0.190) 0.635(0.012) (0.164) 0.483(0.009) (2.846) 1.612(0.147) *The data for each magnitude interval were fitted to a regression equation of the form Y A BX, where Y log 10 (P C ) and X log 10 (M). Uncertainties in A and B are quoted to one standard deviation. Two or three lines are fitted to the cumulative-frequency data, and subscripts 1, 2, 3 refer to individual lines to a shallower slope, B at log 10 (M) 17.50, and then to a steeper slope B at log 10 (M) None of these breaks of slope in these three figures are present in the underlying distribution and are hence artifacts of the procedure used. In each case they result from a combination of (1) the natural roll-over in cumulative-frequency data for a band-limited data set; (2) a small degree of random high-frequency noise with amplitude Zn in the incremental frequency; and (3) the inappropriate application of a naive regression analysis to cumulative-frequency data, that is, neglecting the systematic increase in regression coefficient due to the natural smoothing effect of plotting cumulativefrequency data. What then of the cases where a break in slope is really present in the underlying distribution? Figure 4 shows the case for a double power-law with an increasing slope at high magnitudes, with an underlying distribution that changes from a slope B at log 10 (M) to B The corresponding regression solution gives a reasonable estimate of the first change, from B at log 10 (M) to B , but also introduces a third slope of B above log 10 (M) So again, with a modest amount of statistical noise, a break of slope (marked by an arrow) near that of the underlying distribution is preserved where the statistical scatter is relatively low, but an additional increase is seen at high magnitudes, again because of the roll-over effect combined with the increasing scatter in the synthetic data. Sornette et al. (1996) came to a similar conclusion for this case, using synthetic data analysed by rank-ordering statistics. A similar exercise for a decrease in slope from B at log 10 (M) to B (Figure 5) also preserves approximately the position of the break of slope in the underlying distribution, but introduces a spurious second break of slope at higher magnitudes. In this case the regression analysis gives a decrease in slope from B at to B , but at a lower moment corresponding to log 10 (M) A second break of slope that is not present in the underlying distribution is seen log 10 (M) 19.00, to B , again because of the natural roll-over in bandlimited data sets. Discussion The previously described results cast doubt on the utility of the a priori interpretation of apparent breaks of slope in cumulative-frequency data. Completely spurious but apparently statistically significant breaks of slope can be introduced, and even when there is an underlying break of slope it can be misplaced by a naive regression analysis based on forcing a series of straight lines on the cumulative-frequency data. The reasons are identified as a combination of a natural roll-over in bandlimited data sets with high-frequency noise introduced because of the limitations of the data available to us at present, especially at high magnitudes, and overestimation of the statistical significance of the slopes due to the use of cumulative data. In many cases a spurious better fit to the cumulative-frequency data can be found with the assumption of a break of slope, partly because of the aforementioned effect, but also because of the increase in the number of free parameters used in the line fit. The simple conclusion is that, if there is no break of slope apparent in the incremental distribution, it is unwise to interpret the cumulative frequency in terms of a break in slope. Triep & Sykes (1997) also acknowledge problems with the interpretation of a break of slope in cumulative-frequency data (some of their individual line fits are made to as few as three data points), in particular that standard statistical tests would fail to distinguish the break of slope. As a consequence, they devised a bootstrap model for assessing the error in the two slopes, involving a random regeneration of different statistical realizations using the best-fit model as a parent distribution. The problem with this approach, although it acknowledges some of the points raised in this article, is that the model assumed a priori the presence of an underlying double-slope distribution and specifically ignored the effect of additional parameters in the overall-line fit. Thus the method does not put at risk the hypothesis being tested (namely the existence and location of a change point).

11 96 I. Main In a separate article, Main et al. (1999) examine the problem of model complexity for the case of data pairs (x i, y i ) applied to fracture width and length data in which a break of slope involving an increase in model parameters from 2 to 4, can be inferred objectively using a Bayesian (Schwartz s) Information Criterion. A model with a change point is similar to the double-slope distribution discussed here, except that the present work is applied to frequency data. Accordingly, the method of Main et al. (1999) is currently being adapted for use on frequency data in order to allow an objective test of whether adding extra model parameters improves the fit, despite the additional penalty incurred by additional degrees of freedom in the model (some of the fits in Table 2 imply seven independent parameters). This question was addressed by Pacheco et al. (1992), who used Akaiki s information criterion to test the hypothesis that a double-slope distribution was better than a singleslope distribution for global-cumulative frequency-moment data. It is a general principle that, given the choice of a more complicated or a more simple interpretation of the data, it is usually best to choose the simplest model consistent with the data, a principle known as Ockham s razor. That is, a naive rule that only examines a reduction of the residual sum of squares is likely to result in a best fit for the model with the largest number of parameters. In common parlance, you can fit an elephant with sufficient free parameters in the model. The information criterion described by Main et al. (1999), penalizes this quantitatively, allowing an objective test of which model is statistically the most likely. The exercise carried out here highlights the need for an equivalent objective criterion for frequency data, testing all of the previously described possible models. We have seen that spurious double-slope cumulative frequency distributions in bandlimited synthetic data sets can occur for just about any underlying frequency distribution. If anything, the approach taken here is conservative real data sets are likely to have much more statistical scatter (and also empty bins) than modeled here. No special effort was needed to produce the results shown here, so spurious breaks of slope are not only possible but typical. In any case, the calculations presented here provide reason for caution in uniquely assigning any breaks of slope in cumulativefrequency data to a physical cause, at least given the current data sets available to us. In the meantime we conclude that, because of this fundamental ambiguity, we should use other methods to test the break-of-scaling hypothesis uniquely, such as the use of aftershock studies for the determination of fault length and width, or the direct seismological inversion for finite source area from near-field and far-field seismograms. In terms of seismic-hazard analysis, in the absence of definitive conclusions on the validity or otherwise of the W W* constant model for large earthquakes, we should in the meantime prefer the simplest distribution consistent with a finite-deformation rate and that requires the fewest number of independent parameters, that is, the gamma distribution (Kagan, 1993; Main, 1992, 1995). Conclusions The appearance of distinct breaks of slope in cumulative-frequency-moment data can occur even when this is not present in the underlying incremental-frequency distribution, due to (1) the natural roll-over at high magnitude in bandlimited data sets; (2) high-frequency Zn fluctuations in incremental frequency due to the finite timespan of earthquake catalog; and (3) the inappropriate application of regression analysis to cumulative-frequency data. This in turn implies that the inversion of cumulative-frequency data to test hypotheses of distinct breaks of scaling at depths corresponding to the seismogenic depth is open to question, unless the break of slope can be seen in the underlying incremental distribution, with an appropriate penalty for the additional number of free parameters required. Until we have enough data to resolve this ambiguity, other more direct methods should be used to objectively test the hypothesis that large earthquakes are different from small ones, and simpler distributions should be preferred in applications to seismic-hazard analysis. Acknowledgments I am grateful to Richard Holme for suggesting the method of deriving the seismic-moment rate for the gamma distribution; to Yan Kagan, Bob Geller, and Francesco Mulargia for a lively discussion on some of the points raised in this article, to the latter for his kind hospitality at the University of Bologna during the writing of this article; and finally to two anonymous reviewers for their constructive criticism of an earlier draft. References Abe, K. (1975). Reliable observation of the seismic moment for large earthquakes, J. Phys. Earth 23, Anderson, J. G. (1979). Estimating seismicity from geological structure for seismic risk studies, Bull. Seism. Soc. Am. 69, Bilham, R., K. Larson, J. 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