UPPER-TRUNCATED POWER LAW DISTRIBUTIONS

Transcription

1 Fratals, Vol. 9, No. (00) 09 World Sientifi Publishing Company UPPER-TRUNCATED POWER LAW DISTRIBUTIONS STEPHEN M. BURROUGHS and SARAH F. TEBBENS College of Marine Siene, University of South Florida, St. Petersburg, Florida 70, USA Department of Chemistry and Physis, The University of Tampa, Tampa, Florida 0, USA Reeived Otober, 000; Aepted November 0, 000 Abstrat Power law umulative number-size distributions are widely used to desribe the saling properties of data sets and to establish sale invariane. We derive the relationships between the saling exponents of non-umulative and umulative number-size distributions for linearly binned and logarithmially binned data. Cumulative number-size distributions for data sets of many natural phenomena exhibit a fall-off from a power law at the largest objet sizes. Previous work has often either ignored the fall-off region or desribed this region with a different funtion. We demonstrate that when a data set is abruptly trunated at large objet size, fall-off from a power law is expeted for the umulative distribution. Funtions to desribe this fall-off are derived for both linearly and logarithmially binned data. These funtions lead to a generalized funtion, the upper-trunated power law, that is independent of binning method. Fitting the upper-trunated power law to a umulative number-size distribution determines the parameters of the power law, thus providing the saling exponent of the data. Unlike previous approahes that employ alternate funtions to desribe the fall-off region, an upper-trunated power law desribes the data set, inluding the fall-off, with a single funtion.. INTRODUCTION Cumulative number-size distributions have been employed in a wide variety of appliations inluding seismology, forest fire area, and fault lengths and offsets. When a umulative number-size distribution of data follows a power law, the data set is often onsidered fratal sine both power laws and fratals are sale-invariant. A number of fators should be onsidered when fitting a funtion to a umulative number-size distribution, suh as whether the data has been binned and, if binned, where to plot the binned values. Interpretation of the results takes further onsideration, espeially if there is a deviation from a power law (straight line on a log-log plot) at either the upper or lower limits of the data. 7 In Se., we present the ontinuous nonumulative and umulative power law funtions. In Se., we derive the funtions that desribe disrete power law distributions that have been linearly and 09

2 0 S. M. Burroughs & S. F. Tebbens logarithmially binned. We demonstrate that the saling exponent of a umulative number-size distribution of logarithmially binned data is equal to the saling exponent of the non-umulative numbersize distribution of the same data. In ontrast, the saling exponent of a umulative number-size distribution of linearly binned data differs by one from the saling exponent of the non-umulative numbersize distribution for the same data. In Se., we also derive the funtions for umulative power law distributions that are abruptly trunated at large objet size. In Se., we present the funtion that desribes the umulative distribution independent of binning method, the upper-trunated power law. In Se., we disuss how to plot binned data as nonumulative number-size distributions. In Se., we apply the upper-trunated power law to a syntheti data set. In the final setions, we disuss the results and summarize our onlusions.. CONTINUOUS NUMBER-SIZE FUNCTIONS Non-umulative and umulative number-size funtions are the basis of all equations used in this work and are defined below. The terms power funtion and power law are used interhangeably and refer to ontinuous funtions. The term distribution is used to desribe a set of disrete objets. Objets are the items measured to onstrut a data set, suh as earthquakes, faults and forest fires. Eah measured objet has a harateristi size suh as earthquake magnitude, faultoffset length, or forest fire area. Non-umulative distributions inlude only the number of objets within eah data bin. These distributions sometimes are referred to as density distributions and may be plotted as histograms. Cumulative distributions inlude the number of objets within eah data bin plus all larger objets. Cumulative number-size distributions sometimes are plotted as rank-order distributions or Zipf plots. 8. Non-Cumulative Number-Size Funtions For a set of data points where the number-size distribution follows a power law, the number of objets, n(r), with harateristi size r, is n(r) =r d. () On a log-log plot of n(r) vs. r, thisequationisa straight line where d equals the slope and is a onstant equal to the number of objets with size r =.. Cumulative Number-Size Funtions Cumulative number-size funtions are ommonly used in fratal analysis of data sets to find the fragmentation dimension. The number-size distribution for a large number of objets may be fratal if the umulative number of objets, N(r), with harateristi size greater than or equal to r satisfies the power law relation N(r) =Cr D. () Equation () is a straight line on a log-log plot, where D is the slope and C is a onstant equal to the number of objets with size r. This umulative relation applies when r is a ontinuous set of values. The term D is the saling exponent of the power law. When r represents a one-dimensional quantity, suh as height, D is the fratal dimension of the distribution. If r represents a two-dimensional or three-dimensional quantity, suh as area or volume, the fratal dimension is D or D, respetively. Equations () and () are in a form onsistent with the work of many authors. 9,0 In Se., we examine relationships between these equations for linearly and logarithmially binned data and the effets of upper trunation.. ANALYZING DISCRETE DATA Disrete data sets are omprised of individual, nonontinuous objets. To derive the umulative distribution from the non-umulative distribution for a disrete data set, we must onsider the binning interval used to reord the objet size. We will onsider two types of binning: linear and logarithmi. Unless otherwise noted, all bins ontain data.. Linear Binning Interval For linearly binned data, eah bin has the same width, r. This bin width may represent the width of a histogram bar. We will derive the equations that desribe the non-umulative and umulative distributions of linearly binned data.

3 Upper-Trunated Power Law Distributions.. Linear binning interval with disrete data If objet sizes are sorted in rank order with r i the ith largest value, then for any objet size r, r i = r + i r. () The objets in Eq. () are disrete, so the sum of the objets will not be equivalent to the ontinuous funtion N(r) in Eq. (). We all the sum of disrete objets that are binned linearly N (r). When the sum is taken from any objet size, r, to the maximum objet size in the data set, we all this sum (max) (r). N (r) is alulated for any objet size, r, by summing all individual objets of size greater than or equal to r, thus max (r) = ri d. () i=0 The term r max is the largest value of r in the data set (see Se...). If we take the sum from any value of r to infinity, we all this sum (r). Changing the upper limit of the sum in Eq. () to infinity and substituting Eq. () for r i into Eq. () gives (r) = (r + i r) d. () i=0 Equation () may be rewritten (r) =r d + (r + i r) d. () i= If r is small relative to the range of the values of r, then the sum in Eq. () may be approximated by the integral (r + i r) d i= r r x d dx. (7) For the sum in Eq. (7) to onverge to a finite value, we must onsider only the ase where d>. The right-hand-side of Eq. (7) may be evaluated as r r x d dx = x d r( d) r = r d, for d>. (8) r(d ) Substituting the final term of Eq. (8) for the sum ineq.(),wehave (r) r d + r d (d ) r. (9) Fatoring the right-hand-side of Eq. (9), we obtain ) (+ (r) r r d. (0) (d ) r If r is small relative to r, then ( r r ) is muh greater than, and r + (d ) r r (d ) r. () The error introdued by this approximation will be demonstrated graphially in Se.... Substituting the approximation from Eq. () into Eq. (0), and re-arranging, we have ( ) r d (r). () (d ) r With the above approximations, the sum of individual objets approahes the ontinuous funtion of Eq. () so (r) N(r). Comparing Eq. (), writtenintermsof and d, to Eq. (), written in terms of C and D, for linearly binned data, with r small relative to the range of values of r, wehave ( ) r d N(r) = () (d ) r where and C = (d ) r () D = d. () Equations () and () give the relationship between the onstants and oeffiients for the nonumulative and umulative distributions of the same linearly binned data. The umulative distribution saling exponent, D, differs by from the non-umulative distribution saling exponent, d... Linear binning interval with disrete data trunated at large r When the umulative power law is applied to individual, disrete objets, Eq. () predits that an

4 S. M. Burroughs & S. F. Tebbens objet size exists suh that N(r) =. The objet size for whih the umulative funtion equals, we all r N,sothatCrN D =. When the largest objet found in a data set, r max,islessthanr N,we onsider the data set to be upper-trunated. We now onsider a linearly binned uppertrunated data set. The funtion that desribes the result of summing the disrete objets of Eq. () evaluated from r to r max,weallm (r). If in Eq. (8) we integrate from r to r max,wehave M (r) =r d + Equation () may be re-written as M (r) =r d + r d max ( d) r r d ( d) r. () (d ) r (r d r d max ). (7) Equation (7) is an approximation of the umulative number of objets greater than or equal to size r, binned linearly, with a maximum observed value r max. Equation (7) is a good approximation to Eq. (), (r), for values of r lose to r max, as will be shown graphially in Se.... Substituting from Eqs. () and () into Eq. (7) yields M (r) =CD r(r (D+) ) + C(r D r D max ). (8) Equation (8) desribes a linearly binned umulative number-size distribution with a maximum observed value for r. It is useful to onsider one bin larger than r max, whih we all r T, so that r T = r max + r. Replaing r max with r T r, when r is small, Eq. (8) beomes M (r) C(r D r D T ). (9) Equation (9) desribes a linearly binned power law distribution that is upper-trunated. Notie that M (r T ) = 0. Thus, there are no objets of size r T or larger. Equation (9) represents the power law, Cr D, with the objets of size r T and larger removed. Fitting Eq. (9) to a set of umulative data points provides the oeffiient, C, saling exponent, D, and trunation term, r T. Sine there will probably be a satter of the data points about the best fit of Eq. (9), the determined value of r T may not be larger than the largest objet size in the data set... Graphial illustration for linear binning interval We now graphially illustrate the funtions derived in Se.... Starting with a umulative power funtion in the form of Eq. (), we arbitrarily hoose C = 0 and D = 0. son(r) = 0r 0. (top line in Fig. ). To simulate linear binning, we sample this funtion by seleting objet sizes in the range 0. r with r = 0. [see Eq. ()]. The solid boxes along the top line in Fig. represent the number of objets greater than or equal to eah seleted objet size. With our hosen umulative parameters C and D, and binning interval r, Eqs. () and () yield the non-umulative parameters =.8 andd =.. Therefore Eq. () beomes n(r) =.8r. (bottom line in Fig. ). The solid diamonds on this line represent the number of objets of eah seleted size. The umulative distribution, (r), is alulated for eah value of r by adding together all objets of size greater than or equal to r in the hosen range 0. r. This proess is the sum in Eq. () with a maximum objet size, r max, equal to. This umulative distribution, (r), is shown by solid triangles in Fig.. The funtion that approximates this umulative distribution, M (r) [Eq. (8)], is alulated from the hosen values for the umulative parameters and plotted as open boxes in Fig.. M (r) provides a good approximation to the umulative distribution, (r), when r is near r max. As predited by the derivations in Se..., the approximation is less aurate for smaller values of r. Notie the prominent fall-off of (r) fromapowerlawas r approahes r max in the range log r max < log r log r max. This setion demonstrates relationships between non-umulative and umulative linearly binned funtions. Methods for applying these funtions to disrete values in data sets will be presented in Se... Logarithmi Binning Interval Logarithmi binning means that bin width inreases as the objet size inreases so that the ratio between suessive bin widths is onstant. Logarithmi binning is ommonly used in many sientifi disiplines, suh as studies of earthquake magnitude and sediment grain size.

5 Upper-Trunated Power Law Distributions Linear Sampling Interval 00 Number of Objets 0 n (r) N (r) (max) N (r) M (r) Objet size, r Fig. Behavior of umulative funtions with linear binning. Note that this figure illustrates the behavior of funtions, not data sets. In this example N(r) = 0r 0.,son(r) =.8r. and their slopes differ by. Values of r are hosen in the range 0. r atintervalsof r =0.. Eah plotted point represents the value of a funtion at these seleted values of r. Evaluating n(r) ateahseletedr and adding these values from eah r to r max =.0 produes (r). M(r) isthe approximation of (max) (r) as alulated from Eq. (8). M(r) approximates N (r) very well when r is near rmax. (r) exhibits a signifiant fall-off from the straight line trend of a power law in the largest half-magnitude (r >inthis example). (r) approximates a straight line below the largest half-magnitude (r in this example), but the slope of (max) (r) in this region is steeper than the slope of the power law, N(r). Attempts to fit a power law to N (r) inthe nearly straight portion of the graph will not yield the orret exponent for the power law... Logarithmi binning interval with disrete data To ompute the umulative distribution for objet size r with a logarithmi binning interval, let r j = r0 ja (0) where there are j measured sizes, r j, greater than r, andwherea is the log 0 of the ratio of suessive bin widths. We will all the sum of disrete objets that are binned logarithmially N (r). When the sum is taken from any objet size, r, tothemaximum objet size in the data set, we will all this sum (max) (r). N (r) is alulated for any objet size, r, by summing all disrete objets of size greaterthanorequaltor as follows max (r) = rj d. () j=0 As in the ase of linear binning, r max is the largest value of r inthedataset. Ifwetakethesumfrom any value of r to infinity, we all this sum (r). Changing the upper limit of the sum in Eq. () to infinity and substituting Eq. (0) for r j into Eq. () gives (r) = (r0 ja ) d. () j=0

6 S. M. Burroughs & S. F. Tebbens The right-hand-side of Eq. () is a geometri series. Evaluating this series gives (r) = r d. () 0 ad The right sides of Eqs. () and () have the same form, so for logarithmially binned data, (r) =N(r) where and C = 0 ad () D = d. () Note that N(r) isnottheintegralofn(r) for logarithmially binned data. The exponents of both the umulative funtion N(r) and the non-umulative funtion n(r) arethesame,d = D... Logarithmi binning with disrete data trunated at large r Using an approah similar to that taken for linearly binned data (Se...), we now onsider a logarithmially binned data set, trunated at r max, where r max is less than r N. We wish to find the funtion that desribes the result of summing the disrete objets of Eq. (). We will all this funtion M (r). Equation () evaluated at r max is (r max) = r d max 0 ad. () Equation () represents the umulative distribution from r max to infinity, exlusive of r max.the umulative distribution evaluated from r to infinity, minus the umulative distribution evaluated from r max to infinity, is obtained by subtrating Eq. () from Eq. (). This differene exludes the value of n(r max ). Adding to this differene the value of n(r max ), where n(r max )=rmax, d produes the umulative distribution from r to r max inlusive. Following these steps, we obtain ( M (r) = 0 ad ) (r d r d max) + r d max. (7) Re-writing Eq. (7) in terms of C and D from Eqs. () and (), gives M (r) =C(r D r D max) + C( 0 ad )r D max. (8) Equation (8) desribes a logarithmially binned umulative number-size distribution with a maximum value for r. For logarithmi binning, r max = r T 0 a, so Eq. (8) beomes M (r) =C(r D r D T ). (9) Equation (9) desribes a logarithmially binned power law distribution that is upper-trunated with no objets of size r T or larger... Graphial illustration for logarithmi binning interval The equations derived above for logarithmially binned data will now be shown graphially. We start with the same umulative power funtion we used for the linear binning example in Se..., N(r) = 0r 0.. To simulate logarithmi binning, we hoose a logarithmi sampling interval with a = 0.. Equations () and () yield the values = 0. andd = 0.. The non-umulative funtion is therefore n(r) =0.r 0.. Thisisnot the non-umulative funtion we obtained when the same umulative funtion was binned linearly in Se.... The non-umulative funtion depends on the binning method. In Fig., the non-umulative funtion is plotted as the lower line (solid dots) and the umulative funtion is plotted as the upper line (solid squares). The umulative distribution, (r), is alulated for eah value of r, using the approah in Se.... Here we hoose the range 0. r. orresponding to 0.0 r 0 0. with the exponent hanging in steps of a = 0.. This umulative distribution is plotted as solid triangles in Fig.. The mathematial evaluation of the sum, M (r), in the range 0. r. isgivenby Eq. (8) and plotted as open boxes in Fig. (using the hosen values of a, C, D, r, andr max ). Sine no approximations were made in evaluating the sum in Eq. (), (r) equalsm (r) and the solid triangles and open boxes plot at the same points in Fig..

7 Upper-Trunated Power Law Distributions Logarithmi Sampling Interval Number of Objets n (r) N (r) (max) N (r) M (r) Objet Size, r Fig. Behavior of umulative funtions with logarithmi binning. In this example N(r) = 0r 0.,son(r) =0.r 0. and they have the same slope. Values of r are hosen in the range 0.0 r 0 0. (0. r.), with the exponent hanging in steps of a = 0.. Eah plotted point represents the value of a funtion at these seleted values of r. Evaluating n(r) ateahseletedr and adding these values from eah r to r max =. produes (r). M(r) gives the result of (r) plot at the same points sine no approximations are (r) exhibits a signifiant fall-off from the straight line trend of a power law in the largest (r) approximates a straight line below the largest half-magnitude (r in (r) in this region is steeper than the slope of the power law, N(r). Attempts to fit a (r) in the nearly straight portion of the graph will not yield the orret exponent for the power law. this summation as alulated from Eq. (8). M (r) and made in deriving M (r). half-magnitude (r >inthisexample). this example), but the slope of power law to. THE UPPER-TRUNCATED POWER LAW When the bin width is small relative to the range of r, the umulative distributions are the same for both linear binning [M (r), Eq. (9)] and logarithmi binning [M (r), Eq. (9)], sine under these onditions M (r) =M (r). We all this equation the upper-trunated power law, M(r), where M(r) =C(r D r D T ). (0) Equation (0) is appropriate for fitting a umulative distribution governed by an underlying power law, regardless of the binning method. To find the values of C and D for the umulative power funtion, we find the values of C, D,andr T that provide the best fit of the upper-trunated power law [Eq. (0)] to a umulative number-size distribution (see Se..).. ANALYTICAL TECHNIQUES. Analysis of Syntheti Disrete Data The points plotted on the umulative and nonumulative lines in Figs. and were hosen to illustrate the effet of the binning interval. The distribution of these points is not like that of any

8 S. M. Burroughs & S. F. Tebbens Traditional Method Upper-trunated Power Law Comparison Cumulative Number of Objets Syntheti Data Set Trunated at r = A Traditional Method C =., D = B 8 0. Power Law from Best Fit of M (r) C = 9.9, D =.0 Best Fit of M (r) C = 9.9, D =.0 8 Objet Size, r C Traditional Method C =., D = Best Fit of M (r) C = 9.9, D =.0 8 (a) (b) () Fig. Syntheti umulative data with hosen values of C =0andD =. trunated at r max = for linear binning with r =0.. Cumulative data points are plotted at the beginning of eah data bin. (a) Traditional method. A power law fit to all data points provides inorret values for C and D. (b) The upper-trunated power law. The upper-trunated power law fit to all data points produes C =9.9 andd =., in good agreement with the hosen values. () Comparison of traditional and upper-trunated power law methods. The traditional method overestimates the value of D whereas the upper-trunated power law provides a better fit to the data and yields values of C and D in lose agreement with the hosen values. real data points we have observed for naturally ourring power distributions. For example, as the objet size beomes large, natural data sets tend to have fewer points than shown in Figs. and and sometimes there are empty data bins. To demonstrate the method of fitting an uppertrunated power law to a distribution, we reate a syntheti data set that repliates harateristis similar to those observed for real disrete umulative data. Starting with a umulative power law, we alulate a set of values for r suh that for eah suessive r, the umulative funtion dereases by one. We arbitrarily hoose C = 0, D =., and trunate the data set at a maximum objet size of r max = [Fig. (a)]. To reate a syntheti umulative data set, we bin the r values linearly with r = 0.. Tofitapowerlawtoaumu- lative data set, previous authors have fit a power funtion diretly to some portion of the umulative distribution. We all this approah the traditional method and show the results for the syntheti data in Fig. (a). The traditional method over-estimates the value of D in the umulative power funtion. A more aurate approah is to fit the umulative distribution with an upper-trunated power law. The best fit of M(r) to the umulative distribution is shown in Fig. (b). Figure () ompares the results of this fitting method to the results of the traditional method. Fitting the distribution with an upper-trunated power law aurately determines C and D of the power law.. Fitting the Upper-Trunated Power Law to a Data Set To find the best fit of the upper-trunated power law [Eq. (0)] to a data set, we need to simultaneously solve for the parameters C, D, andr T. It is useful to plot a graph that is expeted to yield a straight line so that least-squares fitting an be employed. There are two ways to do this. First, for eah objet size in the data set, the atual value of the umulative number, N, is plotted against the umulative number alulated from the uppertrunated power law, M(r) [Fig. (a)]. If the data set follows an upper-trunated power law and the parameters are hosen orretly, the resulting graph will be a straight line of slope and interept 0. Adjusting the parameters of the upper-trunated power law to give the best fit to this straight line will yield the appropriate values for C, D and r T.This proedure emphasizes the largest values of N, and therefore the smallest values of objet size, when

9 Upper-Trunated Power Law Distributions 7 Cumulative Numner, N M (r) 00 Objet Size, r r (N).0.0 (a) (b) Fig. Demonstration of how to fit the upper-trunated power law to a umulative distribution. (a) For eah objet size in the data set, the atual value of the umulative number, N, is plotted against the umulative number alulated from the upper-trunated power law, M(r) [Eq. (0)]. (b) For eah value of N in the data set, the atual objet size, r, is plotted against the alulated value r(n) [Eq. ()]. The parameters C, D, andr T are adjusted to provide the best fit to a straight line of slope one and interept zero for both (a) and (b). In this ase, using the data shown in Fig., we find C =9.9, D =., and r T =.9 whih is in lose agreement with the hosen values. determining the three parameters. This tehnique may not plae enough emphasis on the largest objets and thus may fail to represent the fall-off. A seond method plaes more emphasis on the largest objets by re-writing the upper-trunated power law as r(n), where r(n) = ( N C + r D T ) D. () For eah value of N in the data set, the atual objet size, r, is plotted against the alulated value r(n) [Fig. (b)]. This proedure yields a straight line of slope and interept 0 if the distribution follows an upper trunated power law and if the parameters are orretly hosen. Alternating between these two proedures will onverge on the parameters of C, D and r T that yield the best fit of the upper-trunated power law to all values in the data set.. Plotting Binned Data The value of r hosen as the plotting loation for binned data may signifiantly affet interpretation of the data. For umulative distributions, eah plotted point represents all values of the plotted size or larger. To orretly represent all the objets in the interval, the data should be plotted at the beginning (lower end) of the bin. In ontrast, for non-umulative linearly binned data, plotting the number of objets in a bin at the bin s enter is a good approximation as long as the width of the bin is small relative to the objet size (see Appendix A.). For non-umulative logarithmially binned data, the number of objets in a bin should be plotted at the beginning of the bin (see Appendix A.). Failure to follow these plotting guidelines may lead to misinterpretation of the data.. COMPARISON TO OTHER METHODS Several methods have been employed to analyze umulative number-size distributions that exhibit a fall-off at large objet size. Common approahes either ignore the fall-off and fit a power law to the entire distribution, or exlude the largest event sizes where the fall-off is observed and fit a power law to the remaining smaller event sizes., If the fall-off is aused by upper trunation, these two methods over-estimate the fratal dimension of the distribution (see Se..). Methods that aount for the fall-off inlude: fitting different funtions to different portions of the same distribution; fitting multiple power laws

10 8 S. M. Burroughs & S. F. Tebbens to different setions of the same distribution; differentiating the umulative distribution to determine a non-umulative distribution that is then analyzed;, and estimating the number of large objets missing from the upper end of the data set.,7 The first two methods suggest that different funtions, or different forms of the same funtion, desribe different portions of the distribution. If an upper-trunated power law is found to fit the entire distribution, then this single funtion provides a simpler interpretation than fitting multiple funtions to the data set. The third method,, differentiating the umulative distribution, has been shown to yield the same results as fitting an uppertrunated power law diretly to the umulative distribution for forest fire areas in the Australian Capital Territory. To obtain eah point in the nonumulative distribution, the umulative distribution is differentiated over a hosen range of points. The resulting non-umulative distribution depends on the hoie of range. The hosen range sometimes must be adjusted to use fewer points where the data are sparse, suh as at large objet size. An upper-trunated power law is simpler to apply, as it is fit diretly to all the data points in the umulative distribution and does not require hoosing and adjusting a range of points for differentiation. The fourth method,7 aounts for the fall-off by estimating the number of objets missing above the upper end of the data set, alled the ensoring orretion, then adding the ensoring orretion to the umulative number assoiated with eah objet size in the data set. This proess is repeated until fall-off is no longer observed in the resulting orreted umulative distribution. Fitting a power law to this orreted distribution yields a value for D. However, the value for D an be obtained diretly by fitting an upper-trunated power law, M(r), to the unorreted umulative distribution. These two methods should produe the same results beause the term CrT D in M(r) [Eq. (0)] is equivalent to the ensoring orretion approximated by iteration. Fitting an upper-trunated power law diretly to the umulative distribution is a simpler approah to finding D. 7. CONCLUSIONS Many umulative number-size distributions of natural data exhibit fall-off from a power law at large objet size. These distributions may be desribed with a single funtion, the upper trunated power law [Eq. (0)]. Fitting the upper-trunated power law to a umulative number-size distribution yields the values C and D of the power law, and thus provides the saling exponent that desribes the data. NOTATIONS a C d D i j M(r) M (r) log of ratio between suessive bin widths for logarithmially binned data oeffiient of non-umulative power law oeffiient of umulative power law exponent of non-umulative power law is ( d) exponent of umulative power law is ( D) index for summing linearly binned objets to obtain the umulative distribution index for summing logarithmially binned objets to obtain the umulative distribution Upper-trunated power law ontinuous funtion desribing a umulative number-size distribution of linearly binned data trunated at large objet size M (r) ontinuous funtion desribing a umulative number-size distribution of logarithmially binned data trunated at large objet size n(r) N(r) N (r) (r) (r) ontinuous funtion desribing a non-umulative number-size distribution of data ontinuous funtion desribing a umulative number-size distribution of data umulative number-size distribution of disrete, linearly binned objet sizes umulative number-size distribution for linearly binned sizes found by ounting to a very large objet size umulative number-size distribution for linearly binned sizes found by ounting to the largest objet size in the data set N (r) umulative number-size distribution of disrete, logarithmially binned objet sizes

11 Upper-Trunated Power Law Distributions 9 (r) umulative number-size distribution for logarithmially binned sizes found by ounting to a very large objet size (r) umulative number-size distribution for logarithmially binned sizes found by ounting to the largest objet size in the data set r objet size r sampling interval or bin width for linear binning r b objet size at the small end of a linear bin r objet size at bin enter r e objet size at the large end of a linear bin r s objet size at the small end of a logarithmi bin r f objet size at the large end of a logarithmi bin r max largest objet size in a data set r N objet size where the umulative power law equals r PLIN plotting point for linearly binned non-umulative data r PLOG plotting point for logarithmially binned non-umulative data r T trunation objet size, in an upper-trunated distribution there are no objets of this size or larger ACKNOWLEDGMENTS Kevin Dove and Emilio Toro of the University of Tampa provided invaluable help reviewing the mathematial analysis. This manusript benefited from disussions with Robert Byrne and Chris Barton. SB was supported in part by a Knight Researh Fellowship awarded by the USF College of Marine Siene. REFERENCES. B. Gutenberg and C. F. Rihter, Seismiity of the Earth and Assoiated Phenomena (Prineton University Press, Prineton, 99).. B. D. Malamud, G. Morein and D. L. Turotte, Siene 8, 80 8 (998).. C. H. Sholz and J. C. Contreras, Geology, (998).. T. Villemin, J. Angelier and C. Sunwoo, in Fratals in the Earth Sienes, eds. C. Barton and P. R. LaPointe (Plenum Press, New York, 99), pp. 0.. B. A. Brooks and R. W. Allmendinger, J. Geophys. Res. 0, (99).. G. Pikering, J. M. Bull, D. J. Sanderson and P. V. Harrison, in Fratals and Dynami Systems in Geosiene, ed. J. H. Kruhl (Springer-Verlag, Berlin, 99), pp G. Pikering, J. M. Bull and D. J. Sanderson, Tetonophysis 8, 0 (99). 8. B. B. Mandelbrot, Fratals and Saling in Finane (Springer, New York, 997), and referenes therein. 9. D. L. Turotte, Fratals and Chaos in Geology and Geophysis, nd ed. (Cambridge Press, Cambridge, 997). 0. J. Feder, Fratals (Plenum Press, New York, 988).. C. K. Wentworth, J. Geology 7, 07 (9).. M. Matsushita, J. Phys. So. Jpn., (98).. P. Jakson and D. J. Sanderson, Tetonophysis 0, (99).. A. Sornette, P. Davy and D. Sornette, J. Geophys. Res. 98, 9 (99).. C. H. Sholz and P. A. Cowie, Nature, (990).. S. M. Burroughs and S. F. Tebbens, Pure Appl. Geophys. (in press). APPENDIX A A. Plotting Non-Cumulative Linearly Binned Data When analyzing and plotting linearly binned nonumulative data, a single objet size must be hosen for eah bin to represent all objets within the bin. The hoie of effetive objet size for eah bin affets the results of any fitting equation. If objets are measured with a muh better resolution than the width of the data bin, the average objet size within eah bin an be alulated and used as the effetive size of all objets in the bin. If the resolution of measurement is omparable to the width of the data bin, the position of objets within the bin will not be known, so alulating an average size in the bin may not be possible. However, if the bin width is small (small r), the frational hange in objet size within the bin will be small and the average objet size within a bin will be lose to the size at bin enter. The proper effetive size to use in plotting the non-umulative number of objets in a bin an be found by setting the non-umulative

12 0 S. M. Burroughs & S. F. Tebbens funtion equal to the number of objets in the bin. The proper plotting loation we are seeking for linear binning we all r PLIN.Weusethesymbolsr b and r e to represent the small end and large end of a linear bin so r = r e r b. The number of objets in a bin is the differene between the umulative number at the beginning of the bin and the umulative number at the end of the bin, or N(r b ) N(r e ). We wish to find r PLIN suh that n(r PLIN )=N(r b ) N(r e ). (A.) Substituting the appropriate power funtions into Eq. (A.) yields rplin d = Cr D b Cre D. (A.) From Eqs. () and (), Eq. (A.) an be rewritten as CD( r)r (D+) PLIN Solving for r PLIN gives r PLIN = = Cr D b ( D( r) r D b r D e Cr D e. (A.) ) (D+). (A.) 00 Linear Binning of Syntheti Data Power Law N (r) = 0 r. with r = 0. Yields =.8, d =. Number of Objets 0 Number in eah bin N (r) (max) N (r) n (r) Objet Size, r Fig. A. Example for linearly binned data. A syntheti data set is generated from a power law trunated at r max =and linearly binned with r = 0.. This yields the same umulative distribution as shown in Fig.. Dots on the line representing the umulative power funtion, N(r), are loated at the enter of eah oupied data bin (upper line). The spaing between these dots inreases for large values of r, indiating that some bins ontain no data. For this example, the umulative number, (r), is determined from the power law, N(r) (see Se..). For eah data bin entered on r, the umulative number, (r), is plotted at the beginning of the bin. The non-umulative number within eah bin is plotted at the average of the r values in the bin. The parameters of n(r) are obtained by solving Eqs. () and () using the hosen parameters for N(r). Following these plotting guidelines, the non-umulative distribution (solid diamonds) is well-represented by the funtion for n(r) (dashed line).

13 If the bin width and exponent of the umulative power funtion are known, Eq. (A.) gives the objet size for whih the non-umulative funtion is equal to the number of objets in the bin. If the umulative power law is not known, the approximate value of r PLIN an still be found for small (but nonzero) bin width. If we all the enter of the bin r, then r b = r r and r e = r + r, and the denominator of Eq. (A.) beomes (r r ) D (r + r ) D. Expanding these terms, and ignoring seond-order terms and higher (powers of r greater than one), we obtain the following approximation: Upper-Trunated Power Law Distributions ( r r ) D ( r + r ) D D( r)r (D+). (A.) Equation (A.) therefore beomes r PLIN = ( D( r) ( r D b r D e D( r) D( r)r (D+) ) (D+) ) (D+) (A.) Logarithmi Binning of Syntheti Data Number of Objets 00 0 Number in eah bin N (r) (max) N (r) n (r) Power Law N (r) = 0 r. with a = 0. Yields =., d = Objet Size, r Fig. A. Example for logarithmially binned data. A syntheti data set is generated from a power law trunated at r max =. and logarithmially binned with a =0.. Dots on the graph of the umulative power funtion, N(r), indiate the enter of eah oupied data bin. In this example there are no unoupied bins within the range of the data. For eah data binenteredonr, the umulative number, (r), is plotted at the beginning of the bin, not at r. The non-umulative number for eah bin is also plotted at the beginning of eah bin sine it an be shown that, for logarithmi binning, the number of points within eah bin is equal to the value of the assoiated non-umulative funtion at the start of the bin (see Appendix A.). The parameters of n(r) are obtained by solving Eqs. () and () using the hosen parameters for N(r). Following these plotting guidelines, the non-umulative distribution (solid diamonds) is well-represented by the funtion for n(r) (dashed line).

14 S. M. Burroughs & S. F. Tebbens or, for nonzero bin width ( r 0), r PLIN r. (A.7) When plotting non-umulative linearly binned data, plotting the number of objets in a bin at the bin s enter is a good approximation as long as the width of the bin is small relative to the size of objets in the bin. This approximation is not valid for two ases. First, at small objet sizes, the width of the bin may not be small relative to the objet size being measured. In this ase, the frational hange in objet size within a bin may be signifiant, produing a signifiant onentration of objets toward the small end of the bin. Seond, when there are only a few objets in a bin, their sizes may be sattered about within the bin range and not neessarily have an average size near the bin s enter. This may our for both the largest and smallest objets in the data set. For small objets, the data set may be inomplete. For large objets, there may be only a few objets in the bin (or one objet, or none at all). The largest and smallest objet sizes should not be onsidered when fitting a power law to a non-umulative distribution of data points (see Fig. A.). A. Plotting Non-Cumulative Logarithmially Binned Data For logarithmially binned non-umulative data, the hoie of effetive objet size for eah bin affets the oeffiient of the funtion obtained when a power law is fit to the data. The proper objet size to use for plotting logarithmially binned nonumulative data we all r PLOG.Weuser to represent the enter of the bin in log spae and the symbols r s and r f to represent the small end and large end of a bin. For logarithmi binning, r s = r 0 ( a ) and r f = r 0 +( a ),sowemaywriter f = r s 0 a. The number of objets in a bin is the differene between the umulative number at the start of the bin and the umulative number at the finish of the bin, or N(r s ) N(r f ). We wish to find the value of r PLOG that makes the non-umulative funtion evaluated at r PLOG equal to the number of objets in the bin, so n(r PLOG )=N(r s ) N(r f ). (A.8) Substituting the appropriate power funtions into Eq. (A.8) yields rplog d = Cr D s Cr D f. (A.9) From Eqs. () and () and the above relationships between r s and r f, Eq. (A.9) an be rewritten as C( 0 ad )rplog D = Cr D s C(r s 0 a ) D (A.0) so, for nonzero bin width (a 0), r PLOG = r s. (A.) For logarithmi binning, the number of objets in a bin equals the value of the non-umulative funtion at the start of the bin (see Fig. A.).

15