Similarity of Magmatism and Volcanism

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104,NO. B7, PAGES 15,425-15,438,JULY 10, 1999

Statistical self-similarity of magmatism and volcanism

Jon D. Pelletier Division of Geologicaland Planetary Science,California Institute of Technology,Pasadena

Abstract. Magraatismand volcanismexhibit spatial and temporal clusteringon a wide range of scales.Using the spatial pair-correlationfunction, the number of pairs ofmagmatic or volcanicevents whose separation isbetween r- «Ar andr + «Ar per unit area, we quantify the spatial clusteringof raagraatismand volcanismin several data sets. Statistically self-similarclustering characterized by power law spatial pair-correlation functions is observed. The temporal pair-correlation function is usedto identify self-similartemporal clusteringof raagraatismand volcanismin the Radiometric Age Data Bank of 11,986 dated intrusive and extrusive rocks in the North American Cordillera. The clusteringof raagraatismand volcanismin space and time in this data set is found to be statistically self-similarand identical to thoseof distributed seismicity.The frequency-sizedistributions of eruption volume and areal extent of basaltic flows are also found to be self-similarwith power laws analogousto the Gutenburg-Richterdistribution for earthquakes.In an attempt to understandthe origin of statisticalself-similarity in raagraatismand volcanismwe presentone end-membermodel in which the ascentof magma through a disordered crust of variable macroscopicpermeability is modeled with a cellular-automaton model to create a distribution of magma supply regionswhich erupt with equal probability per unit time. The model exhibits statistical self-similarity similar to that observed in the real data sets.

1. Introduction magmatic activity when the whole Cordillerean region is viewed during later Cenozoic time." Spatiotempo- Magmatism and volcanismare phenomenacharacter- ral clusteringof volcanismis recognizedin models in ized by variable intensityon a wide rangeof spatial and which the volcanicintensity of a region dependson the temporal scales. For example, volcanic eruptions are intensityof neighboringregions [Conner and Hill, 1995; episodic: like great earthquakes,they often occur with- Gonditand Conner,1996]. How magmasupplied from out warning after a long period of quiescence. Such a subductingslab upwellsthrough the continentalcrust episodicityoccurs over a broad range of timescalesfrom to erupt episodically at localized points is not well un- historic timescalesto much longer timescalesrecorded derstood. The purposeof this paper is to quantify the in deep-seasediment cores as discussedby Kenneff et aL variability of magmatic and volcanicintensity in space [1977],Kenneff [1981], and Kenneff [1989]. They docu- and time, including the frequency-sizedistribution and mentedepisodes of intensevolcanism separated by long the spatiotemporalclustering of events,using a variety quiescentperiods on the timescalesof millionsof years of techniquesand data sets. In all cases,the measures in PacificOcean sedimentcores. Similarly, volcanism is of magmatismand volcanismare found to exhibit power localizedin space. Magma eruptsin volcaniczones with law relationshipsindicative of statistical self-similarity. up to hundredsof individualvolcanic vents separated by Self-similarbehavior in spaceand time has prevously largedistances with little or no volcanism[Cuffanti and been documentedin hot spot volcanismby Shaw and Weaver,1988; Conner,1990; $irnkin, 1993]. The rate Ghouet[1991]. A modelof fluid invasionin porousme- and spatial concentrationof magmaticand volcanicac- dia has been introduced in the physicsliterature which tivity dependon the scaleof observation.For example, gives rise to statistically self-similar behavior. We pro- with respectto the Cenozoicmagmatic history of the pose this model as one possiblemodel of large-scale NorthAmerican Cordillera, Armstrong and Ward[1991, magma migration through a crust of variable macro- p. 13203]observed that "althoughmagmatism is always scopicpermeability for the formation of magma supply locally episodic,the episodesblur into a continuum of regions. The organization of the text is as follows. First, ev- i•• i• ,,••+• +•o+ +• ...... •+i•,• frequency-size Copyright1999 by the American GeophysicalUnion. distribution of volcanic eruptions and basalt flows is a power law analogousto the Gutenberg-Richterlaw for Paper number 1999JB900109. earthquakes. Since there is evidence that the size of an 0148-0227/99/1999JB900109509.00 eruption scaleswith the size of the magma supply re- 15,425 15,426 PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM gion that producedit, theseresults suggest that magma canic output is observedto be a decreasingpower law supply regionsmay also exhibit a power law cumulative function of the timescale of observation. This indicates frequency-sizedistribution. that as the timescale is increased,more long periods of The spatial clusteringof volcanicvents and basaltic quiescenceare included, reducingthe rates of magma- flows is then considered. We utilize the spatial pair- tism and volcanism. This relationship is another way to correlation function, the number of pairs of volcanic characterizethe episodicityand intermittency of mag- eventswhose separation isbetween r- «Ar andr + •1A r matism and volcanism. per unit area. We find that power laws are applicableto Last, we introducea modelfor the upwellingof magma the spatial pair-correlationfunction for many data sets, throughthe continentalcrust that producesself-affine indicating that the clusteringis statisticallyself-similar. variations in temperature and a self-similarfrequency- As a model for the size distributionand clustering size distribution and spatiotemporal clustering. Our of volcanic events and magma supply regions,we uti- model considersthe penetration of magma in a crust lize the observationthat temperature variations in the of variable macroscopicpermeability. The variable per- uppermost mantle are self-affine. Self-affine functions meability tends to roughen the isothermscharacteriz- are those which have a power spectrum with a power ing the ascent, allowing magma to be advected more law dependenceon wavenumber, k: $(k) c• k-z, easily in regionsof high permeability. Self-affinevaria- where/• is a characteristic exponent. Self-affinetem- tions in isothermic depths are produced by this model. perature variations are inferred from the observation This is analogousto experiments in which fluids pen- of self-affineregional variations in seismicwave speed etrating porous media developinterfaces that are self- at depth within Earth's mantle [Passierand $neider, affine[Buldyrev et al., 1992].The magmasupply regions 1995]. In our model,magma supply regions are those formed in this model of upwellingmagma are allowedto where the isothermic depth is larger than a threshold erupt with a random probability. As a result of the pen- value required for the melt fraction to be greater than etration of magma into new regions, magmatism and the percolation threshold. Synthetic magma supply re- volcanism migrate over time in a realistic way in this gions and distributions of volcanic rocksare constructed model. Spatiotemporal data from a model of magmatic and their statistics compared to those from the western and volcanic history are produced and the statistics are Great Basin. Close agreementis obtained betweenthe comparedto those of the Radiometric Age Data Bank. statistics of real and model magmatic and volcanic re- Identical clustering statistics are obtained, suggesting gions. that this model is one possible end-member model for Spatiotemporalclustering of magmatismin the North magmatic and volcanic processes. American Cordillera is considered with the Radiomet- ric Age Data Bank of 11,986 radiometrically-datedin- trusive and extrusive, basaltic and nonbasalticigneous 2. Frequency-Size Distribution rocks. The large number of rocks in the database en- of Volcanic Eruptions ables us to examine the clusteringin time as a function of the distance separating rocks and, conversely,the The frequency-sizedistribution of volcanicevents from clustering in space as a function of time. It is found a variety of geologicand marine environmentsis ob- that power law statistics are applicable to the tem- servedto be a power law analogousto the Gutenberg- poral pair-correlation function but that the exponent Richter law for earthquakes. The Gutenberg-Richter of the temporal pair-correlationfunction characterizing law states that the number of earthquakes with a seis- the strength of the clustering depends on the spatial mic moment greater than Mo is a power law function separation such that locations close together in space with an exponentclose to -• 2. N(• Mo) • M• -B, have highly correlatedsequences of eventsin time. Sim- whereB • •2 [Kanamoriand Anderson, 1975]. Sirakin ilarly, the clusteringin spaceis strongerfor rockssepa- [1993]has compileddata on Holoceneeruptions from rated by a small interval of time. The pair-correlation small, monthly events to globally catastrophic events functionswe compute with the Radiometric Age Data such as the 1815 eruption of Tambora and the eruption Bank are identical to those computed for distributed of the basalt flowsof Yellowstone2 Ma to obtain an ap- seismicityby Kagan and Knopoff[1980] and Kagan and proximate frequency-sizedistribution of volcanic erup- Jackson[1991], indicating that distributedvolcanism tions. The volcanicexplosivity index (VEI), definedto and distributed seismicity may have similar spatiotem- be the logarithm (base10) of the eruptedtephra vol- poral dynamics. ume,was used. Sirekin [1993] found an inverserelation- Next, we discussthe dependenceof rates of magma ship between eruption volume and frequency of events emplacement and volcanic output on the timescale over with an exponent of approximately-1. which the rate is computed. As with other episodicphe- As further evidence for a power law frequency-size nomenain Earth sciencesuch as sedimentation[Sadlet, distribution in volcanism, we consider the distribution 1981; Sadlet and Strauss, 1990; Strauss and Sadlet, of basaltic rocks in the western Great Basin, mapped in 1989]and morphologicalchanges in species[Gingerich, Figure I of Yogodzinskiet al. [1996]. We digitizedthe 1983, 1993], the rate of magma emplacementand vol- basaltic regions from the map west of 117ø longitude PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM 15,427 and computed the cumulative frequency-sizedistribu- 1000 tion of area (Figure 1), the numberof areaslarger than an area A. The result is plotted on logarithmicscales as the solidcircles in Figure 2. The data from Figure 2 fit a power law distribution with an exponent of-0.8, lOO N(• A) o• A -ø'søñø'øs,indicated by the solidline. The (>A)øcA-0'8 power law fit was obtained with a linear least squares regressionto the logarithms(base 10) of the data. The same procedurewas used to obtain all of the power law lO trends quoted in the analysesin this paper. The uncertainty in the exponentwas obtained by dividing the population of basalt flows into two halves (each flow was randomlyassigned to one groupor the other) and 1 1 o 1 oo 1 ooo calculating a least squaresfit to the frequency-sizedis- A (kin2) tributions of each half of the data. The difference in the exponents between the two halves of the data is Figure 2. Cumulative frequency-sizedistribution of the estimate of the uncertainty. The uncertainties cal- basalticregions N(• A) (solidcircles), the numberof culated in this way are comparableto standard errors regionswith an area greater than A, for the distribution in Figure 1. The data are plotted on logarithmic obtained through the least squaresprocedure and pro- scalesso that a straight line is indicative a power law vide an independent evaluation of the uncertainties of relationship. A least squaresfit to the logarithms of the the exponents associated with the size of the data set. datayields the relationshipN(• A) o•A -ø'sø shown by the straight line.

This procedure for uncertainty estimation has been re- peated for all of the uncertainties quoted in this paper.

3. Spatial Clustering of Volcanoes and Basaltic Flows

We have also consideredthe spatial clusteringof the basaltic flows from the western Great Basin. In addi- tion to the broad size distribution apparent in Figure I with many small regions and a few large ones, the basaltic regions are also clustered in space with large distancesseparating the zones of concentratedvolcan- i ism. We have utilized the spatial pair-correlation func- ß tion to characterize the clustering. The spatial pair- . ß correlationfunction c(r) is the numberof pairsof events whoseseparation is between r - «At andr + «At per

ß unit area[e.g., Turcotte, 1997]. One point is picked,and

411 the distances to all other points are determined. The samething is done for the secondpoint and for all other points. The number of pairs with separations between r-• zAr andr - •z Ar is dividedby far to obtainc(r) . For a one-dimensionaldistribution suchas the temporal pair-correlation function that we will consider shortly, the number of pairs in each interval At is divided by At to obtain c(t). The spatialpair-correlation function has beenused by Kagan and Jackson[1991] to studyearth- quakeclustering and by Pelletlet and Turcotte[1996] to study the clustering of wells showing hydrocarbons Figure 1. Distribution of basaltic rocks • 8 Myr old in the westernGreat Basin (west of 117ø longitude)of in the Denver and Powder River basins. The point set southwestern Nevada and southeastern California. Dis- in our analysis is the set of grid points in which basalt tribution digitized from Figure I of Yogodzinskiet al. occurs. Alternatively, one could compute the centroid [1996]. of each distinct basaltic region and use that point set 15,428 PELLETIER: STATISTICALSELF-SIMILARITY OF MAGMATISMAND VOLCANISM

to compute the spatial pair-correlation function. We surface performed both analyses and obtained similar results. The spatial pair-correlation function for the basaltic regionsof Figure 1 is presentedin Figure3 on logarithmic scales.The data are well approximatedby a powerlaw v functionof the formc(r) ocr-ø's5+ø'ø5, indicating that the basaltic regionsare clusteredin a statisticallyself- similar way. Figure 4. Illustration of the model relating variations Beforewe considerother data setsfor the spatial dis- in isothermicdepth to the geometryof magma supply tribution of volcanicactivity we will showhow magma regions(shaded). The variationsin isothermicdepth supply regionscan be related to regionsof high melt- are given by the Brownianwalk, generatedby succes- ing intensity. The power spectrum of regional varia- sive coin flips, labeled T -- To. If the isothermicdepth tions in seismic wave speed of the upper mantle has is less than a critical value where the pressure P -- Pc beenconsidered by Passierand Snieder[1995]. They is such that the melt fraction is greater than the per- computedvariations in seismicwave speedat a given colationthreshold, large-scale advection of magma will occur, forming a magma chamberand initiating active depthin Europeand the Mediterraneanwith techniques volcanismin that region. of seismictomography and performedpower spectral analyseson the resultant models. Variations in seismic wave speedare associatedprincipally with variationsin sociatedwith regionsof active volcanism.To make the temperature[Hearn et al., 1991]. Passletand $nieder three dimensionaltryof our model clear, we have also [1995]found seismicwave speeds to havea powerlaw illustrated the constructionof synthetic magma supply powerspectrum with an exponentof approximately-2, regionsin three dimensionsin Figure 5. A shadedrelief $(k) ock -•, from harmonicdegree I • 30 to I • 300. image of a syntheticBrowntan walk surfaceis shownin Sincethis power spectrumis similar to that of a Brow- Figure 5a. Only thoseregions with a value greaterthan nian walk, we have usedthis result to constructa geo- a threshold value are included in the image in Figure -metrical model of magma supplyregions. The model is 5b. These regionsare the synthetic magma supply re- illustrated in Figure 4. In Figure 4 the Browntanwalk gions. It is not necessaryin this model that variationsin representsa transect of variations in isothermicdepth seismicwave speed be associatedsolely with variations labeledas T = To. If the isothermicdepth at any point in crustal temperature as suggestedin Figure 4. We along the transect is lessthan a critical value for which have associatedthe Browntan walk of Figure 4 with an the melt fraction (a functionof temperatureand criti- isotherm for simplicity but it may also representcompo- cal depth Pc) is greaterthan the percolationthreshold, sitional variations that influencemelting intensity. The large-scaleadvection of magmawill occur,forming a essentialfeature of the model is that magma supply re- magma supply region in the crust and resultingin ac- gionswill form when high melting intensity occursat a tive volcanism. Magma supply regionsin the model shallowenough depth. are shaded in Figure 4. In this model, domains of a We have created synthetic magma supply regionsby Browntansurface greater than a thresholdvalue are as- constructinga Browntanwalk surface(• - 2) usingthe Fourier-filteringtechnique described by Turcotte[1997]. Domains greater than a threshold value are shown in lOO Figure 6 in map view. The threshold value was chosen such that the percentage of area magmatically active matched that of the percentageof area with basalts in the westernGreat Basin (6%). The sizedistribution of areas in Figure 6 is presentedin Figure 7. The size dis-

lO tribution of areas is identical to that of basaltic regions in the GreatBasin: N(• A) ocA -ø'søñø'ø3. This result can also be shown theoretically. Kondev and Henley [1995]have related the length distributionof contour lengths of Gaussiansurfaces to the Hausdorffmeasure Ha. The Hausdorff measure is related to the power spectralexponent • by the relation• - 2Ha -•-1 [e.g., 10 100 1000 Turcotte,1997]. Kondevand Henley[1995] have given r (krn) the sizedistribution of contourlengths (the probability that a randomlychosen contour loop has a length s) as Figure 3. Spatial pair-correlation function of basaltic regionsc(r) as a functionof the separationdistance of N(s) ocs -•, where• - 1 -•- (2 - Ha)/D and D is the the pair. The data are plotted on logarithmic scales, fractal dimension of the contours. The cumulative dis- and the straight line representsthe power law fit to the tribution (the numberof contourswith length greater data with c(r) ocr- 085ß . than s) is the integralof the noncumulativedistribution, PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM 15,429

Figure 6. Distribution of model basaltic regionspro- duced with the model of Figure 4. A Brownian walk surfacewith a powerlaw powerspectrum and/• - 2 was generatedwith the Fourier-filteringmethod [e.g. Tur- cotte,1997] to representvariations in isothermicdepth in the upper mantle. Model magma supply regionsare associated with domains of the Brownian walk surface with depth lessthan a thresholdvalue. The threshold Figure 5. Illustration of the model relating variations value was chosensuch that the percentageof area as a in isothermicdepth to the geometryof magma supply supply region was equal to the percentagein Figure 1 regionsin three dimensions.(a) Shadedrelief image (6%). of syntheticBrownian walk surface. (b) Shadedre- lief imageof syntheticmagma supply regions where the isothermic depth is lessthan a threshold value. pair-correlationfunctions. The spatial pair-correlation function of the synthetic volcanic regionsof Figure 6 is presented in Figure 8. The data fit a power law trendof the formc(r) ocr-ø's7, in closeagreement with N(> s) ocs -(2-•t•)lv. Sincethe lengthof a contour the pair-correlation function of the Great Basin basalts is relatedto the areait enclosesby s ocA D/2 (by def- shown in Figure 3 which has a least squaresregression inition), the cumulativedistribution of areasenclosed to ½(r)ocr -0'85+0'04. by contoursis N(> A) ocA -(2-•t•)/2. For a surface ooo with/• - 2, Ha - 1/2, andwe have N(• A) ocA -3/4, nearly consistentwith the size distribution of Figures 2 and 7. Since the Hausdorff measure of the interface is related to the exponent of the frequency-sizedistribu- lOO tion through a one-to-onefunction, the observedexpo- N(>A)o•A-0'8 nents are consistent with the Brownian walk variations of temperature at a given depth and the model for vol- z canic regions as extremal domains of a Brownian walk •oF surface. Pelletier [1997] applied these calculationsto the statistics of cumulus cloud fields and inferred from them the self-affinevariability of the top of the convec- tive boundary layer. The similarity of the cumulative frequency-sizedis- 1 lO lOO lOOO tribution of synthetic magma supply regions and that ^ 2) of the basalt flows from the western United States suggeststhat the size of eruption volume and the size of the Figure 7. Cumulative frequency-sizedistribution of magma supply region that producedit may be strongly modelbasaltic regions N(• A) (solidcircles), the number of regionswith an area greaterthan A, for the distri- correlated.Smith [1979]has arguedsuch a relationship bution in Figure 6. A least squaresfit to the logarithms for ash flow magmatism. Further similarity between of the data yield the relationshipN(> A) ocA -ø'8ø the distribution of model magma supply regions and shown by the straight line. This distribution is identi- basaltic flows is given by the similarity of their spatial cal to that of basalts from the western Great Basin. 15,430 PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM

lOO lOO i_ i ' C(r)•r-0'70

1 lO 1 1 o 1 oo 1 ooo r (kin) r (km) Figure 8. Spatial pair-correlation function of model Figure9. Spatialpair-correlation function o•volcanic basalticregions c(r) (solid circles). The straight line events from four time intervals from 1.75 to 0.75 Ma in representsthe power law fit to the data with c(r) o• 0.25 Myr incrementsin the Springervillevolcanic field, •-0.s7, nearlyidentical to that of westernGreat Basin Arizona. The events were digitized from Figure 9 of basalts. Condit and Conner[1996]. The data from the oldest time interval to the most recent are presentedas solid circles,open circles,crosses, and squares,respectively. Despite the fact that the loci of activity vary signifi- We have also calculated the spatial pair-correlation cantly from one part of the volcanicfield to othersover function for two other published distributions of vol- time, the pair-correlationfunction is remarkablyconsis- tent with c(r) • r -ø'7ø. canic activity. In Figure 9 we presentthe pair-correlation function of volcanic events from four time intervals of the history of the Springervillevolcanic field in Arizona spacingsbetween volcanoes,that volcanoeswere ran- digitizedfrom Figure 9 of Condit and Conner[1996]. domly distributed with no clustering. The differencein The pair-correlation functions from the oldest time in- our conclusionscan most likely be ascribedto the differ- terval to the most recent are presented as solid circles, ent methods we employ. The pair-correlation function open circles, crosses,and squaresin Figure 9, respec- uses data for the spacing between every pair of vol- tively. Activity in the Springervillevolcanic field over canoes, not just those between nearest neighbors, and time from 1.75 to 0.75 Ma in 0.25 Myr increments as therefore givesa completerepresentation of the original illustratedin Figure 9 of Condit and Conner[1996] is distribution while nearest-neighbordistances maintain a particularly striking example of migration of volcanic only a small subset of the information contained in the activity from one region to another. Despite the fact data set (N versusN (N - 1)/2, whereN is the number that the loci of activity are quite different from one of points). time increment to another the pair-correlation function for each time increment is nearly identical, suggesting that statistically self-similarclustering is not a transient lO4 feature but a fundamental characteristic of volcanism. The observationof statistical self-similarclustering of hydrothermal veins at scalesof centimetersto meters [Manning,1994; Magde et al., 1995]suggests that self- • c(r)•r--0'5 similar clusteringholds on more than one range of spatial scales. lO5 . The second data set is cinder cones from the Mauna Kea volcano,Hawaii, mappedby Porter [1972]. The pair-correlation function of this distribution is presented in Figure 10. Statistically self-similarclustering is observed as with the previous data sets. The exponent, however,is -0.50:l:0.05,smaller in magnitudethan the lO2 , i i i i i i i I i i i i i i i i exponents previously obtained. lO -1 lO0 lO1 The conclusionswe have reachedon clusteringof vol- r (km) canism are inconsistent with those of de Buemond d'Ars Figure 10. Spatial pair-correlation function of cinder et al. [1995].They consideredthe spatialdistribution of conesfrom the Mauna Kea volcano, Hawaii, digitized volcanoesin active margins and have concluded,based from Porter [1972]. The data fit a powerlaw function on an analysis of the distribution of nearest-neighbor of the formc(r) o•r -ø'5. PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM 15,431

4. Temporal Clustering of Igneous and Eruption Events

In this sectionwe considerthe RadiometricAge Data Bank, a collectionof locationsand agesof 11,986 dated igneousrocks from the North AmericanCordillera [Zari- man et al., 1976; Armstrong and Ward, 1991; Ward, 1995]. Sincethe databaseconsists of both basalticand nonbasalticrocks, the analysisof clustering in this data set yields additional information than the clusteringof strictly basaltic rocksin Figure 1. The spatial distribu- Figure 12. Age of radiometricallydated igneousrocks tion of rocks in this database, obtained from P. Ward from northern California to Texas projected onto a pro- file parallel to the plate margin from Cape Mendocino, (personalcommunication, 1996), is shownin Figure 11. California, to Guadalupe,Mexico, and shownas a func- A representationof the data which illustrates the vari- tion of distancefrom Cape Mendocino. The location of ability of magmatism through time is presentedin Fig- major cities within the United States are shown pro- ure 12. Figure 12 plots the distributionof igneousrocks jected onto the crosssection for reference.From Ward in time against the position projected along the plate [1991]. margin from Cape Mendocino to Guadalupe and suggeststhat both spatial and temporal clusteringoccur in this data set. There are two potential biasesin the sampling of ig- sity data were obtained from the Global Demography neousrocks: proximity of the samplelocations to roads Project[Tobler et al., 1995].The two distributionfunc- and differentialerosion and/or depositionwhich may tions are presentedin Figure 13a with the distribution for all land areas as the solid line, and the distribu- remove or bury rocks in certain locations more than in others. If, for example, there is a higher frequencyof tion for only land areas where rocks are sampled as the dashed line. The two distributions have nearly the samplingin areas closerto roads the Radiometric Age Data Bank may not be a uniform record of magma- same shape. However, sincethe dashedline lies above tism and volcanism in the North American Cordillera. the solid line, the frequencyof sampled rock in areas We have ruled out the possibilityof significantsampling with a populationdensity of at least1 person/km2 is slightly higher than the frequencyof all suchland ar- bias due to the proximity of roads by computing the cu- eas. There are two possiblereasons for this. One is that mulative probability distribution of population density for all land areasin the 15øx15ø squarefrom 120øW rock samplingis slightlybiased toward more populated to 105øWlongitude and 35øN to 50øN latitude. This areas. Another possibilityis that the igneousrocks hap- pen to be clusteredin regionswith, on average,slightly distribution can be compared to that for only those land areas where rocks are sampled. Population den- higher population density. This is likely since population density is tightly clusteredwith the extremely de- populated areas of the study area concentratedin the 7O ' I ' basin and range. Two clustereddistributions may not overlap even if they are independent of one another. ß• .,...•.-.•',;•:..'. •..' '...... ; :•,".? :-??;'-:.,".•,!.•:.?.•,ß Nevertheless,we concludefrom the close similarity of the distributionsin Figure 13a that population density ,:' '.._•? ß..-'...... ' .,,.-•...,•,'-'•' • representsat most a small bias in the samplingof rocks ,- ...;,...•..,.... •..-. in the Radiometric Age Data Bank. •:•.?".•.•.J•t•"'';• To test for possiblesampling bias due to differential .-..•.."?-,.•'.-,.;!. '. erosionand/or deposition,we have computedthe cu- ß".'. •'"?-, :•'-• .' '-:i' •': •'....;"•.¾!.'..;. •' •. mulative probability distribution function as a function ß 40 of elevation. Relief alone is an accurate predictor of de- ß..!•., ..•..,•,.,•!,;..• .-,...: •,,. .: -•.•',.'t ' ' '..' nudationrates [Ahnert, 1970; Pinet andSouriau, 1988]. -•.'•.•,.,,:'-..,,,•.. ; ..'.!',•?..' ,. We have taken advantageof this correlationto use ele-

ß - •;0 ß vation as a proxy for erosion and deposition. The two .. distributions, analogousto those for population den-

ß : ß sity, are presented in Figure 13b. The elevation data 2O , I , , • I , , , I • , ' I used were extracted from the ETOP05 digital elevation 1 60 I • n I '•0 I "'"' ß•uu• L•O•-•-•, -•ouj. x z•e closecorresponuence longitude(degrees W) of the two distributions indicates that no denudational samplingbias exists in the data set. Figure 11. Spatial distribution of igneousrocks from The large number of rocksin the databaseenables us the RadiometricAge Data Bank (RADB). to quantify the clustering in spacefor different ranges 15,432 PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM

1.0 by a small distance. The clusteringweakens, i.e., the pair-correlationfunction is flatter, as the radial separa- -• o.8 tion increases. The same analysishas been performed on earthquakecatalogs by Ka9an and Jackson[1991] i with strikingly similar results. They observedthe pair- •- 0.6 o correlationfunction to dependon time as c(t) o• t -z for earthquakepairs with smallseparations. A progres- II sively weaker,power law dependencewas observedfor A 0.4 i ! . o larger separations.This is consistentwith the results for the pair-correlationanalyses of Ka#an and Jackson o 0.2 [1991]for smallspatial separations. Identical clustering statistics for distributed volcanism and distributed seis- 0.0 ...... , micity suggesta similar dynamicsfor the two phenom- lO lOO ena. Clustering of distributed seismicityis related to populotiondensity (#/km 2) strain weakeningon faults [Dieterich,1992]..We point out the similarity in spatiotemporalclustering of dis-

1.0 tributed seismicity and magmatism and volcanismin the hope that understandingof one phenomenonmay be broughtto bear on the other eventhough they are c 0.8 o .... very different physicalprocesses. The complementaryanalysis of the spatialpair-correlation -• 0.6 function for rocks separatedby different time rangesis II presentedin Figure 15. The time rangescorresponding to the different curvesare, from top to bottom, t = 0.03- c 0.4 o o• 0.1 Myr, 0.1-0.3Myr, 0.3-1 Myr, 1-3 Myr, 3-10Myr, and

o 10-30Myr, respectively.The samebehavior is obtained •- 0.2 for c(r) as for c(t): the clusteringfor the smallesttime separationsis givenby c(r) o• r -•, with progressively 0.0 weakerclustering characterizing rocks with greaterage 0 500 1000 1500 2000 2,500 ,5000 differences. elevotion (m)

Figure 13. (a) Cumulative probability distribution functionof all land area (solidline) and only thoseland areaswhere rocks were sampled (dashed line) as a function of populationdensity for the 15øx15ø squarefrom lO2 120øWto 105øWlongitude and 35øN to 50øNlatitude. Gridded population density data were obtained from the Global DemographyProject [Tobleret al., 1995]. -u, 1ol (b) Cumulativeprobability distributionfunction of all landarea (solid line) and only those land areas where x x x ø c(t)'•t-1 rockswere sampled (dashed line) as a functionof eleva- • oo tion for the same area as Figure 13a. The ETOP05 dig- 1 ! [] [] x o ß ital elevationmodel [JLou#hrid#e,1986] was used. The - [] similarity of the distributionsbetween all areasand only those areas where rocks were sampled indicates that the • lO-1 sampling was unbiasedwith respect to population density and elevation. -2 lO i lO -1 lO0 lO1 lO2 t (Myr) of age separationbetween the rocks and, conversely,to quantify the clusteringin time for differentspatial sepa- Figure 14. Temporal pair-correlationfunction of ig- rations. The temporal pair-correlation function in time neous rocks from the North American Cordillera com- is presentedin Figure 14 for radial separationsr = 0.3-1 piled in the RadiometricAge Data Bank for radial sepa- km, 1-3 km, 3-10 km, 10-30 km, 30-100 km, and 100-300 rations r = 0.3-1 km, 1-3 km, 3-10 km, 10-30 km, 30-100 km, presentedfrom top to bottom, respectively. The km, and 100-300 km, presentedfrom top to bottom, respectively. The strongestclustering, characterizedby temporal pair-correlationfunction has alsobeen used by c(t) o• t -z, is observedfor pairsof rocksseparated by Pelletier[1999] to characterizethe clusteringof geomag- a small distance. The clusteringweakens, i.e., the pair- netic reversals. The strongest clustering, characterized correlation function is flatter, as the radial separation by c(t) • t -z, is observedfor pairsof rocksseparated increases. PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM 15,433

lO2 by this model is to calculatethe volcanicflux as a function of the timescale of observation.

• ß We will next discussthe relationship between mag- •, lO1 ' 0 matic output and time span based on a model with a deterministic fractal distribution of quiescenceperiods. • o o c(r)•r-1 We consider a vertical sequenceof volcanic beds. The age of volcanic sequencesin this model is given as a ß• lOø function of depth in Figure 17a. As illustrated, the vertical segments(beds) are of equal thickness. The • lO-1 positions of the transitions from beds to hiatuses are given by a second-orderCantor set. Eight kilometers 10-2 ...... , , of volcanic rock have been deposited in this model se- 100 101 102 1'0'3 quence in a period of 9 Myr so that the mean rate of volcanismis R (9 Myr) = 8 km/9 Myr = 0.89 mm/yr over this period. However, there is a major unconfor- Figure 15. Spatial pair-correlationfunction in space miry at a depth of 4 km. The rocks immediately above of igneousrocks from the North American Cordillera this unconformity have an age of 3 Ma and the rocks compiled in the Radiometric Age Data Bank for time immediately below it have an age of 6 Ma. There are ranges t = 0.03-0.1 Myr, 0.1-0.3 Myr, 0.3-1 Myr, 1-3 no rocks in the volcanic sequencewith ages between 3 Myr, 3-10 Myr, and 10-30 Myr, from top to bottom, and 6 Ma. In terms of the Cantor set this is illustrated respectively.The samebehavior is obtainedfor c(r) as for c(t): the clusteringfor the smallesttime separations in Figure 17b. The line of unit length is divided into is givenby c(r) ocr-z, with progressivelyweaker clus- three parts, and the middle third, representingthe pe- tering characterizingrocks with greater age differences. riod without volcanism, is removed. The two remaining parts are placed on top of each other as shown. During the first 3 Myr of eruptions (the lower half of the volcanicsequence) the mean rates of volcanism 5. Scaling of Volcanic Activity are R (3 Myr)= 4 km/3 Myr = 1.33 mm/yr. Thus Rate Versus Time the rate of volcanism increasesas the period considered decreases.This is shown in Figure !7c. The observationof similar clusteringbehavior in space There is also an unconformity at a depth of 2 km. The and time suggeststhat temporal variations in volcanic rocks immediately above this unconformity have an age flux at a single location may have statistically self- of ! Ma, and rocksbelow have an age of 2 Ma. Similarly, similar behavior just as spatial variations in volcanic there is an unconformity at a depth of 6 km, the rocks flux at a single instant of time are statistically self- above this unconformity have an age of 7 Ma, and rocks similar. For instance, considerFigure 4 to represent below have an age of 8 Ma. There are no rocks in the variationsin the isothermicdepth at a singlelocation as pile with ages between 8 and 7 Ma or between 2 and ! a function of time rather than variations in isothermic depth along a transect at an instant of time. The shaded regionsnow representperiods of volcanic activity which 10 can turn on and switch off through time. A plot of the cumulative volcanicflux Qcum producedwith this model is presentedin Figure 16. To constructFigure 16 we have generated a fractional Brownian walk with the Fourier-filteringmethod of T•rcotte[1997]. A fractional Brownian walk is a time serieswith a power law powerspectrum with 1 < fi < 2 [T•rcott½,1997]. We have constructedthe time serieswith fi = 1.8, slightly smaller that 2, since the model consideredin section 6 producestemporal variations with fi = 1.8. An eruption occurs when the isothermic depth is above a criti- i i i i I ! i i i I ! ! s s i .... i .... i .... i , cal value. Eruptionsoccur instantaneously, releasing a 5 10 15 20 25 30 flux of magmaequal to the area of the shadedregion. T (time step / 1000) Eruptionscan be identifiedby vertical lines in Figure 16. When the. isothermic depth is below Figure 16. History of cumulative volcanic output Q cum as a function of time T in a model in which value, no volcanismoccurs and a period of quiescence, variations in isothermic depth at a single location as a indicated by a horizontal line of constant cumulative functionof time area Brownianwalk with $(f) volcanicoutput in time, is observed.One way to char- and eruptionstake placewhen the isothermicdepth fails acterize the episodicity of magmatic output produced below a threshold value. 15,434 PELLETIER:STATISTICAL SELF-SIMILARITY OF MAGMATISMAND VOLCANISM

-1 Age T, Ma lO I I I I I I I I 0 9 6 3 '-2•_0 .E 10-2 -4Depth y,km .v.. i !i: "::. ß• s ; ,, ß ' ' , ' * O ß 10 (a) - 8 .o_ 2

ß ß • --4 ø ß 10 ..... :.,, , ...... 100 101 102 103 104 R1.5 309 T (time steps) (mm/yr)I • Figure18. Averagevolcanic flux R asas function of 1.0 time spanT for the volcanicsequence of Figure16.

1 T, Myr 10 Hausdorffmeasure: a o• T •/". The rate of changeof the time seriesfor a giventime intervalT is then the vol- (b) (c) canicflux R = a/T ocT Ha-z. For/• = 1.8,Ha = 0.4, and the volcanicflux is then R ocT -ø'•ø, in agreement Figure 17. Illustration of a model for volcanicse- with the numerical result in Figure 18. quencesbased on a Devil's staircaseassociated with a The results of this model can be comparedto pub- second-orderCantor set. (a) Age of volcanicrocks T lishedvolcanic fluxes. Crisp [1984]has compiledesti- as a functionof depth y. (b) Illustrationof how the Cantor set is used to construct the volcanic sequence. mated rates of magmaticoutput alongwith the area (c) Averagevolcanic flux R as a functionof the period and time over which the output occured.We have esti- T considered. mated the volcanic flux as the volcanic rate estimated by Crisp[1984] divided by the areaof the region.The volcanicflux for active marginsestimated in this way is plottedas a functionof the timescaleof observation Ma. This is clearly illustrated in Figure 17a. In terms in Figure19. Sincethe data are collectedfrom many of the Cantor set (Figure 17b) the two remainingline regionsin the world,significant scatter in the dataset segmentsof length1/3 areeach divided into three parts is observed. However,fluxes have a clear inversede- and the middle thirds are removed. The four remaining pendenceon timescale.A leastsquares fit to the data segmentsof length1/9 are placedon top of eachother yieldsthe relationshipR oc T -ø'•ñø'ø•,indicated by as shown. During the periods9 to 8, 7 to 6, 3 to 2, and I to 0 Myr the ratesof volcanismare R (1 Myr) - 2 km/1 Myr - 2 mm/yr. This rate is alsoincluded in 102 I i i i i Figure 17c. The volcanicflux clearlyhas a powerlaw dependence 101 with respectto the lengthof the time intervalconsid- ered. The resultsillustrated in Figure 17 are basedon 100 a second-order Cantor set but the construction can be extendedto any orderdesired and the powerlaw results 10-1 givenin Figure17c would be extendedto shorterand shorter time intervals. 10-2 F' '' %R,•T-0.65 One thousand samplesof volcanicflux calculated betweenpairs of randomly choseninstants in Figure 10-3

16 are plotted in Figure 18. The volcanicflux has ß ß a powerlaw dependenceon time spanwith exponent 10-4 -0.6' R o• T -ø'aø+ø'ø3. The decreasein rate results 10-4 10-3 10-2 10-1 100 lO1 lO2 lO3 m (Myr) from the appearanceof longer hiatusesin longer time intervals is illustrated in Figure 16. These results can Figure 19. Estimatedvolcanic and magmaticfluxes also be obtained from theoretical fractal relations. Frac- as a function of the timescale over which the flux is tional BrownJanwalks have the property that the stan- estimatedfrom the data set of Crisp[1984]. Volcanic dard deviation of the time series has a power law de- flux hasan inversedependence on timescalewith a least pendenceon time with a fractionalexponent Ha, the squaresfit to R o•T -ø'•5. PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM 15,435 the solid line, in closeagreement with the model result (,) (b) R ocT -ø'6ø describedabove. On historicaltimescales, fractal behavior characterizedby a power law distribution of quiescenceperiods in the histories of basaltic volcanoeshas been identified by Dubois and Cheminee [•].

6. Self-affine Model of Magmatic Upwelling Through the Continental Crust (a)

Experimentsand modelsof fluid migration in a disordered porousmedium have been consideredby B•ldyrcv ½t al. [1992]. In their experiments,fluid was injected into one end of a sponge-likematerial called oasisand commonlyused by florists. After sometime had passed, the material was dissected, and the interface between the injected fluid and the displaced air was scanned. The final interface was rough' the distance of the in- Figure 20. Illustrationof the modelof Buldyrevetal. terface from the injection point was highly variable, [1992]for fluidinvasion in porousmedia. Cellswhere despite the fact that the medium was macroscopically fluid has penetratedare shaded,while cellsthat have uniform. The reasonfor the roughinterface is that cap- not beenpenetrated are randomlyblocked with proba- illary forcesroughen the interfaceby advectingthe fluid bility p (indicatedby circles)or unblockedwith proba- into the pore spaces. As the fluid penetrates into the bility (l-p) (indicatedby a verticalline). The interface betweenthe magmaand the countryrock is shownby material, it encountersmore pore spaceswhich further a thickline. (a)t=0, (b)t=l, (c)t=2, and (d)t=3. From roughenthe surfacecreating a macroscopicallyvariable Barabasiand Stanley[1995]. Reprintedwith the per- interface over time. Surface tension tends to smooth the missionof Cambridge University Press. interface. The result is a dynamic steady state between these two competing effects. The statistics of the interface were studied in this experiment and were found to We havecombined the modelof Buldyrevetal. [1992] be characteristicof a Brownianwalk with $(k) ock -2. with eruption dynamicsas a model for the upwelling A cellular-automatonmodel for this phenomenonwas of magma through the disorderedcrust to obtain a consideredby Buldyrevetal. [1992]. The rulesof the modelhistory of volcanismin spaceand time. Isotherms model are illustrated for a two-dimensional cross section evolveaccording to the dynamicsof the modelof Buldyrev in Figure 20. On a cubic lattice a fraction p of cellsare etal. [1992]. The probabilityp was chosento be "blocked" to represent zones of low macroscopicper- 0.4, slightly below the critical probabilitypc • 0.47 meability into which the fluid cannot easily penetrate. where the interface will encounter a connected sequence "Unblocked" cells are those in which the fluid can pen- of blocked sites. At each time step, potentially erup- etrate more easily. A portion of a transect of the model tive magmachambers are identifiedas domains with an at t - 0, where the interface is a fiat surface at its isothermicdepth lessthan a thresholdvalue as in Fig- maximum depth, is shown in Figure 20a. The blocked ure 5. Duringthat time step,each magma chamber has sites are denoted by circles and the unblockedsites are an equalprobability to erupt. Sincethe magmacham- denotedby vertical lines. At t - i a cell (labeledX bers have a power law frequency-sizedistribution, an in Figure 20b) is randomlychosen from amongthe un- equalfailure probabilityper unit time for eachmagma blocked cells adjacent to the interface. Fluid is advected chamberimplies a powerlaw cumulativefrequency-size into cell X, and any cells that are below it in the same distributionof the formN(> V) o• V -ø's consistent column. This process is then iterated. For example, with the observationsof a frequency-sizedistribution of Figure 20c showsthat at t - 2 we choosea secondun- eruptions.As the isothermsascend in the model,more blockedcell, cell Y, to fill, while Figure 20d showsthat cellsare added at the top so that the model maintains a at t - 3 we fill cell Z and also cell Z' below it. With long-termaverage crustal density in the model.Magma this rule, blocked sites are eventually filled, but it takes chambersrefill immediately after eruption. longer to penetrate them in accordancewith their lower In Figure21 we presenta map of eruptedmagma sup- permeability. This model producesself-affine behavior ply regionsfrom two consecutivetime intervalsof equal in both space and time. Varia.tions in interfacial depth duration. The regionsthat erupt duringthe first time alonga transocthave a powerspectrum S(k) ock -2, sliceare shownshaded in grey,while those from the sec- and variations in time at a single point have a power ond time slice are shownin black. Figure 21 illustrates spectrum$(f) ocf-l.8 [Buldyrevetal., 1992;Havlin et the migrationof volcanismin the model. The volcan- al., 1995]. ism from the secondtime slice correlatesstrongly with 15,436 PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM

One popular model for generating spatial patterns of volcanismin active margins is the buoyancyinsta- bility [Marsh, 1982;Kerr and Lister, 1988; Whitehead andHelffich, 1991]. Depending on the initial conditions and assumptionsused, this model can generate periodic [Whiteheadand Helffich, 1991],random [de Buemond et al., 1995], or clustered[Kelly and Bercovici,1997] distributions. In contrast, the effect of variable macro- ,3, scopicpermeability acts only to cluster, consistentwith the empirical observationswe have documented. We must stress, however, that the model describedin this sectionis only one possibleend-member model for magmatic ascensionthrough the crust.

7. Conclusions

The purpose of this paper was to show that complex Figure 21. Distribution of model volcanicrocks from variability in spaceand time in magmatism and volcan- two consecutive time intervals. The most recent vol- ism is characterizedby robust self-similarity. Statistical canic rocks are black, while the older rocks are shaded. self-similarityarises in four ways: (1) the frequency- Migrationof volcanismhas occured as the magmahas penetrated into weak regionsin the crust. size distributionof eruptionsand basalticregions, (2) the clusteringof eventsin space,(3) the clusteringof eventsin time, and (4) the relationshipof volcanicflux with the timescale of observation. We have presenteda model of magma migration through a disorderedcrust of variable macroscopicpermeability which generates that of the first, but the magma chambershave evolved self-affine variations in isothermic depth in space and somewhatduring the two time slices.The migrationhas time. Extreme values of these variations in space and occuredbecause the isotherm has penetratedinto weak time can be associatedwith active regions and erup- zonesat the locationsof the new regionsof magmatism. tions, respectively, and the statistics of those regions The model migration is similar to that of the western related to the self-affinity of the isothermic depth in a United States (e.g., Figure 1-51 of Turcotteand Schu- variety of ways. The model is, however,only one possi- bert [1982]). As in Figure 21, real volcanismexhibits ble model consistent with the data. persistencein that activity in the two consecutivetime intervals are highly correlated but migration has occured to displacethe volcanismin many places. The linear track of basaltsalong the SnakeRiver Plain has been taken as evidencethat this regionis underplated by a hot spot. Such behavior is. not consideredin the model we have proposed,which assumesa steady,uniform supply of magma from below. For this reason, the model we have presentedmay be useful as a null _ [] o c(r)•r-1 model of magmatic ascensionthrough the crust. This model may therefore be used to determine the likeli- hoodthat observedvolcanic histories reflect patterns in magma supply rate in the form of hot spotsor arc seg- mentation as opposedto suggestivepatterns that are, however,produced only as a result of the historyof as- cension. 3 10- We havecalculated spatial and temporalpair-correlation 100 lO1 lO2 lO3 functionson model volcaniceruptions of a long syn- r (lattice spacing) thetic history of volcanism produced with this model. The result is shownin Figure 22. It is quite consistent Figure 22. Pair-correlationfunction in spaceof model with Figure 15, suggestingthat the model is consistent volcanicdomains for time rangest - 103- 3x10s, t - 3x103-10 a, t -- 104 -3x10,47 t -- 3x104 -10, 5 t ---- with the dynamicsof magmaticand volcanicdynamics 105- 3x105,and t - 3x105- 106time stepsfrom top in the continental crust. Similar results were obtained to bottom, respectively.This is nearly identical to that for the temporal pair-correlation function in time for calculatedfor the igneousrocks of the RadiometricAge different radial separations. Data Bank in Figure 14. PELLETIER: STATISTICAL SELF-SIMILARITY OF MAGMATISM AND VOLCANISM 15,437

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