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Using Zipf's Law to Predict Future Earthquakes in KansasAuthor(s): Daniel F. Merriam and John C. DavisSource: Transactions of the Kansas Academy of Science, 112(1/2):127-129. 2009.Published By: Kansas Academy of ScienceDOI: http://dx.doi.org/10.1660/062.112.0218URL: http://www.bioone.org/doi/full/10.1660/062.112.0218

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Using Zipf’s Law to predict future earthquakes in KansasDANIEL F. MERRIAM

1AND JOHN C. DAVIS

2

1. University of Kansas, 1930 Constant Avenue, Campus West, Lawrence, Kansas 66047(e-mail: dmerriam@ku.edu)

2. DAVCON, Box 353,Baldwin City, Kansas 66006

Zipf’s Law states that there is a relation between size and rank of discrete phenomena.This relationship has been noted in other areas, but only recently has been used todescribe geological events; in this example, the occurrence of earthquakes in Kansas.Forty earthquakes recorded in Kansas were used to test and evaluate the law as adescription model. It was determined that Zipf’s Law can describe the size and rankdistribution of earthquakes, including those with magnitudes not yet recorded, but itcannot predict when they will occur.

Keywords: earthquake magnitude, area affected, prediction model, power law

TRANSACTIONS OF THE KANSAS

ACADEMY OF SCIENCE

Vol. 112, no. 1/2p.127-129(2009)

INTRODUCTION

In solving any geological problem look for the simplest solution: It is probably most nearly correct.

attributed to J.M. van Tuyl (K.F. Dallmus, 1958)

Zipf’s Law describes the relationship betweensize and rank of discrete phenomena (Zipf,1949, Li, 2003). It has gained favor recentlyamong mineral-resource investigators fordescribing both the locations and time ofemplacement of mineral and petroleum deposits(Merriam, Drew, and Schuenemeyer, 2004).The law’s applicability in these areas hasprovided the inspiration for investigatingwhether it also might be useful for predictingearthquakes, which are discrete phenomena ofknown size (in the form of earthquakemagnitude) and hence can be ranked. One ofthe results of the study by Merriam, Drew,and Schuenemeyer was that Zipf’s Law coulddescribe the relation between the magnitudesof earthquakes and the areas affected by them.Here, we apply that concept to the occurrenceof earthquakes in Kansas.

Merriam, Drew, and Schuenemeyer (2004)applied Zipf’s Law to earthquake size,

relating area affected to earthquake magnitudefor the Central Stable Region of the U.S. Theyasserted that Zipf’s Law adequately describedthe distribution of earthquakes in the region,but made no predictions about future quakes.Studies such as Sornette and others (1996)and Abe and Suzuki (2002, 2004) also foundZipf’s Law relationships in the record ofearthquake occurrences but used differentmetrics.

Earthquakes in Kansas have been of interestand investigated for numerous years (Merriam,1956, 2006; Steeples and Brosius, 1996;Wilson, 1979). There have been a few largeones (5 on the Richter Magnitude Scale) andswarms of small ones (2.5). Some of the smallermore local quakes have been ascribed to oil-field activity or quarry blasts.

ZIPF’S LAW

We see the form of Zipf’s Law (1949) thatstates:

fn = 1/na

where fn is the frequency of the nth event, anda>0 is close to unity.

128 Merriam and Davis

Figure 1. Log plot of area affected (squaremiles) versus Richter magnitude; R=0.72. Thedeviation of the largest earthquake (intensity of8.5) from the fitted Zipf’s Law line can beaccounted for by several alternatives. If theintensity was correctly recorded, then the areaaffected may be under estimated. If theintensity was actually greater or not recordedcorrectly, then the observation is erroneous;the most likely scenario is that the areaaffected was under reported.

The law can be used to describe either temporalor spatial events that occur frequently (or rarely,depending on the quantity used). In spatialresource analysis, an example of frequentevents would be the occurrence of oil fields. Anexample of infrequent or rare events would bethe presence of ore deposits. In either situation,the principle is that the largest feature (a fieldor deposit) is the anchor, the second-ranked ornext smaller feature is half the size of thelargest, the third-ranked feature is a third thesize of the largest, and so on. By plotting allthe known features, it should be possible toestimate how many remain undiscovered andto estimate their sizes.

EARTHQUAKES IN KANSAS

Forty earthquakes measured in Kansas wereused in the study. The magnitudes and areasaffected for each quake were recorded and areshown as a log-log plot in Figure 1, where

earthquake magnitudes are shown on theabscissa and the log of the areas affected bythe quakes are the ordinate. A linear least-squares fit to the observations has a correlationcoefficient of r = 0.72, so the line shown onthe plot accounts for 52% of the variation inlog area. The slope coefficient a = 1.10, whichis close to unity. The fit is statisticallysignificant (Clauset and Newman, 2009).Note that the fit “underestimates” the biggestevent.

A plot of the Zipf number of each earthquake(i.e., the rank order of the earthquakemagnitudes from largest to smallest) versusearthquake size in square miles is shown inFigure 2. Both axes have logarithmic scales.The small dots along the line are the Zipfnumbers and those quakes that actually haveoccurred are shown as large dots. Note theconcentration at the lower end of the scaleand the paucity of larger quakes. If Zipf’s Lawholds for this distribution, then more largeearthquakes may be expected in the future.

Although Zipf’s Law can model the distributionof earthquakes in Kansas, including themagnitudes of hypothetical quakes not yetrecorded, it cannot predict whether the next

Figure 2. Plot of Zipf’s number versus size insquare miles. Note paucity of intenseearthquakes. Caption below figures and abovetables.

Transactions of the Kansas Academy of Science 112(1/2), 2009 129

quake will be large or small. A similarconundrum was noted by Abe and Suzuki(2004), who studied the time intervals betweenearthquakes and determined a Zipf’s Lawrelationship; although it described thedistribution of the delays between events, itshed no light on the time before the nexttremor.

LITERATURE CITED

Abe, S. and Suzuki, N. 2004. Zipf-MandelbrotLaw for time intervals of earthquakes:NASA Astrophysics Data System archive0208344, 13 pp.

Abe, S. and Suzuki, N. 2004. Scale-freestatistics of time interval between successiveearthquakes. Physica A: StatisticalMechanics and its Applications 350(2-4):588–596.

Clauset, A., Shalizi, C.R. and Newman, M.E.J.(in press) Power-law distributions inempirical data. SIAM

Dallmus, K.F. 1958. Mechanics of basinevolution and its relation to the habitat ofoil in the basin. pp. 883-931 in Weeks, L.G.(ed.), Habitat of Oil: American Associationof Petroleum Geologists, Tulsa, Oklahoma.

Li, W. 2003. Zipf’s Law everywhere.Glottometrics 5:14-21.

Merriam, D.F. 1956. History of earthquakes inKansas. Seismological Society of AmericaBulletin 46(2):87-96.

Merriam, D.F. 2006. The Conrad Discontinuityin the Midcontinent (USA). KansasAcademy of Science, Transactions 109(3/4):125-130.

Merriam, D.F. Drew, L.J. and Schuenemeyer,J.H. 2004. Zipf’s law: a viable geologicalparadigm? Natural Resources Research13(4):265-271.

Newman, M.E.J. 2006. Power laws, Paretodistributions and Zipf’s law. NASAAstrophysics Data System archive0412004v3, 28 pp.

Sornette, D., Knopoff, L. Kagan, Y. andVanneste, C. 1996. Rank-orderingstatistics of extreme events: Application tothe distribution of large earthquakes. Journalof Geophysical Research 101(B6):13883-13894.

Steeples, D.W. and Brosius, L. 1996.Earthquakes. Kansas Geological SurveyPublic Information Circular 3, 5 pp.

Wilson, F.W. 1979, Earthquakes in Kansas?The Journal of the Kansas GeologicalSurvey 1(3):5-11.

Zipf, G.K. 1949. Human behavior and thePrinciple of Least Effort: An introductionto human ecology. Addison-Wesley, Reading,Massachusetts, 573 pp.