scaling geology: landformsand earthquakes6698 colloquium paper: turcotte a, 2 d=1.25 x;%^1 v iiii...

Proc. Natl. Acad. Sci. USAVol. 92, pp. 6697-6704, July 1995Colloquium Paper

This paper was presented at a colloquium entitled "Physics: The Opening to Complexity," organized by Philip W.Anderson, held June 26 and 27, 1994, at the National Academy of Sciences, in Irvine, CA.

Scaling in geology: Landforms and earthquakesDONALD L. TURCOTrEDepartment of Geological Sciences, Cornell University, Ithaca, NY 14853

ABSTRACT Landforms and earthquakes appear to beextremely complex; yet, there is order in the complexity. Bothsatisfy fractal statistics in a variety of ways. A basic questionis whether the fractal behavior is due to scale invariance or isthe signature of a broadly applicable class of physical pro-cesses. Both landscape evolution and regional seismicity ap-pear to be examples of self-organized critical phenomena. Avariety of statistical models have been proposed to modellandforms, including diffusion-limited aggregation, self-avoiding percolation, and cellular automata. Many authorshave studied the behavior of multiple slider-block models,both in terms of the rupture of a fault to generate anearthquake and in terms of the interactions between faultsassociated with regional seismicity. The slider-block modelsexhibit a remarkably rich spectrum of behavior, two sliderblocks can exhibit low-order chaotic behavior. Large numbersof slider blocks clearly exhibit self-organized critical behavior.

Landscapes are both beautiful and complex; yet, there isconsiderable order in the complexity. Landscapes satisfy frac-tal statistics in a variety of ways (1). The length of a coastlinescales as a power of the length of the measuring rod andspectral studies of topography show that it is Brown noise toa good approximation. A fundamental question is whether thisfractal dependence is simply a consequence of scale invariantbehavior or the signature of one or more underlying physicalprocesses.

Earthquakes are also a complex natural phenomenon; yetearthquakes satisfy the Guttenberg-Richter frequency-mag-nitude relation under a wide variety of conditions: the numberof earthquakes scales as a power of the area of the rupturezone. Fractal scalings have also been proposed for the spatialand temporal distributions of earthquakes.The concept of self-organized criticality was introduced by

Bak et al. (2) and is defined as a natural system in a marginallystable state; when perturbed from that state, it will evolve backto the state of marginal stability. Energy input to the system iscontinuous, but the energy loss is in a discrete set of events thatsatisfy fractal frequency-size statistics. Bak et al. (2) made ananalogy between their cellular-automata model and a sand pileon a table. Sand grains are dropped on the pile continuouslyand are lost from the table in discrete sand slides; some sandslides are large and some are small, but they satisfy a fractaldistribution. This is a simplistic model for the evolution oflandforms, and a variety of more sophisticated models havebeen proposed. Earthquakes are also taken as an example ofself-organized criticality. There is a continuous input of energy(strain) through the relative motion of tectonic plates; thisenergy is dissipated in a fractal distribution of earthquakes.Scholz (3) has argued that Earth's entire crust is in a state ofself-organized criticality. He makes the point that wherever a

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large dam is built, induced seismicity results from the filling ofthe reservoir. Thus the crust is everywhere on the brink offailure. Earthquakes have been modeled by using connectedarrays of slider blocks. The slipping statistics of the sliderblocks are generally fractal and also appear to exhibit self-organized criticality.

Landform Statistics

Topography is created by a variety of tectonic processes. In thebrittle, near-surface rocks, deformation is dominated by dis-placements on preexisting faults. These displacements are

generally associated with earthquakes. At greater depths con-tinuum deformation is dominant, and plastic-viscous rheolo-gies are applicable. Topography is destroyed by erosion. Rockis softened by mechanical and chemical weathering, and theresulting sediments are then transported by rivers that form a

drainage network. The rivers actively participate in the erosionby creating gullies and river valleys.

Mandelbrot (4) introduced the concept of fractals in termsof the length of the west coast of Great Britain (his result isgiven in Fig. 1A). Mandelbrot observed that the length P = Nrof a rocky coastline is proportional to a power of the length r

of the measuring rod. Thus, the number of steps taken, N,satisfies the following equation:

N = Cr-D, [1]

where D is the fractal dimension, which is independent of thelength r chosen, and in this case D = 1.25. Similar results areobtained for the length of elevation contours on topographicmaps, three examples being given in Fig. 1 for diverse geolog-ical settings. The fractal dimensions of topographic contoursare not sensitive to the geological setting or age. The fractaldimension of relatively smooth old mountains (the Adiron-dacks in New York) is not significantly different from that ofjagged young mountains (i.e., the Transverse Ranges in Cal-ifornia). Similarly, the number-size statistics of both lakes andislands are fractal in that the numberN with area greater thanA satisfies relation 1 by taking r = A1/2 and D = 1.30 (5).

Coastlines and topographic contours are examples of self-similar fractals. Topography is also an example of self-affinefractals. If a spectral analysis results in a power-law relationbetween the spectral power density S and the wave number k(1), then

S = Ck-1 [2]

and a self-affine fractal results with constraints on the value of,B and D = (5 - ,B)/2. Spectral expansions of the globaltopography of Earth (6) and Venus have been carried out; theresults are shown in Fig. 2. In both cases, a good correlationwith f3 = 2 (D = 1.5) is found-i.e., Brown noise. Topographyis truly self-similar in the sense that the spectral amplitude is

Abbreviation: DLA, diffusion-limited aggregation.

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6698 Colloquium Paper: Turcotte

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, 2 D= 1.25

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2100 100r, m

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FIG. 1. (A) Length P of the west coast of Great Britain as afunction of the length r of the measuring rod (4); (B-D) lengths P ofspecified elevation contours in several mountain belts as a function ofthe length r of the measuring rod. (B) Three thousand-foot contour ofthe Cobblestone Mountain quadrangle, Transverse Ranges, CA. (C)Ten thousand-foot contour of the Byers Peak quadrangle, RockyMountains, CO. (D) One thousand-foot contour of the Silver Bayquadrangle, Adirondack Mountains, NY. Correlations are with thefractal relation Eq. 1.

proportional to the spectral wavelength for Brown noise. It issomewhat surprising that the results for the Earth and Venusare similar since erosion is dominant in the evolution of manylandforms on the earth, while tectonic processes are dominanton Venus because erosion is essentially absent. This suggeststhat the tectonic processes that build topography and theerosional processes that destroy topography both give Brown-noise statistics.The Brown-noise behavior of topography has been found

over a wide range of scales and has been applied to the verticalrelief of oil-producing layers in sedimentary basins. This is anexplanation for the fractal distribution of oil pools illustratedin Fig. 3 (7). The wells drilled are nearly space filling with D= 1.86, whereas those that struck oil haveD = 1.49, consistentwith a fractal Brown-noise structure for the trapping cap rock.

Essential features of most landforms on Earth are drainagenetworks. Drainage networks are classic examples of fractaltrees. It is standard practice to use the Strahler (8) orderingsystem. When two like-order streams meet, they form a streamwith one higher order than the original. Thus, two first-orderstreams combine to form a second-order stream, two second-order streams combine to form a third-order stream, and soforth. The bifurcation ratio Rb is defined by the following:

R =Nn [3]b =Nn' [3

where Nn is the number of streams of order n. The length-orderratio Rr is defined by:

5 10 30 6090120 180

1O0- 1 110- 1i0-4 lo-, 10-2

k, km-'

B

1015

1014

1013

102

1011

1010

I

k, km-'

FIG. 2. Power spectral density, S, as a function of wave number, k,for spherical harmonic expansions of topography (degree 1) for Earth(A) and Venus (B). Correlations are with Eq. 2 by taking ,B = 2.

[4]R = rn+lrn

where rn is the mean length of streams of order n. Both Rb andRr are found to be nearly constant for a range of stream ordersin a drainage basin (9). From Eq. 1, the fractal dimension ofa drainage network is:

D = In Rb [5]

An example of a drainage network is given in Fig. 4A4; this isthe drainage network in the Volfe and Bell Canyons, SanGabriel Mountains, near Glendora, CA. The number-lengthstatistics for this network are given in Fig. 5A; a good corre-lation with Eq. 5 is obtained taking D = 1.81.

Landform Models

A variety of random-walk models for drainage networks wereproposed in the 1960s. This work was reviewed by Smart (10),but the results were generally not very satisfactory. In the past5 years, there has been a rebirth of interest in the problem.Kondoh and Matsushita (11), Meakin et al. (12), and Masek

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Proc. Natl. Acad. Sci. USA 92 (1995) 6699

All drilled wells Producing and showing wells

1000

100

10 _.

1i0.011 0.1 1

Size of counting box, 8 (1/miles)

FIG. 3. Maps of all wells (Upper Left) and hydrocarbon-producing and showing wells (UpperRight) in a 40 x 40 mi2 section of the Denver basin.Shown below each map is the fractal plot of the spatial clustering of wells in that map.

and Turcotte (13) have introduced DLA models. Stark (14)developed a model based on self-avoiding percolation clusters.Chase (15), Inaoka and Takayasu (16), Leheny and Nagel (17),Rinaldo et al. (18), and Prigozhin (19) introduced modelsbased on self-organized criticality. Willgoose et al. (20) andKramer and Marder (21) developed advection-diffusion mod-els based on equations that yield a network structure. A varietyof other models have also been proposed (22-27).As a specific example, we consider the DLA model of Masek

and Turcotte (13), the mechanics of which are illustrated inFig. 6. A square grid of 15 x 15 cells is used in this illustrationand five seed cells are introduced at random points on thelower boundary. The evolving network must grow from theseseed cells. For the example shown, 16 cells have been accretedto the seed cells. Cells are allowed to accrete if one (and onlyone) of the four nearest neighbor cells is part of the preexistingnetwork. Prohibited sites which already have two neighboringsites occupied are identified by stars. Sites available foraccretion to the network are indicated by open circles. Arandom walker is introduced at a random cell on the grid, andthe hypothetical path is traced by the solid line. After 28

random walks, it accretes to the network at the hatched cell.A random walk proceeds until the walker accretes to thenetwork, exits the grid, or lands on a prohibited cell. In eachcase, the walk is terminated and a new walker is introduced ona new, randomly selected cell. The iteration of this basicprocedure results in a branching network composed of linkeddrainage cells. A simulation carried out on a 256 x 256 grid isillustrated in Fig. 4B: 20,000 random walkers have beenintroduced. The simulated and actual drainage networks arereasonably similar. The number-length statistics for the sim-ulated network are given in Fig. SB; a good correlation withEq. 1 is obtained by taking D = 1.85. Again, good agreementis obtained between the simulated and real networks.The random walkers can be interpreted as floods that flow

over a relatively flat surface until they find a gully. When theflood enters the gully, it further erodes the gully and extendsthe network headwards. This type of headward gully evolutionhas often been proposed for actual drainage networks (28).The DLA model can also be used to generate synthetictopography if an empirical power-law relation is assumedbetween stream order and gradient. It should be emphasized

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FIG. 4. (A) Drainage network in the Volfe and Bell Canyons, SanGabriel Mountains, near Glendora, CA, obtained from field mapping.(B) Illustration of a diffusion-limited aggregation (DLA) model for adrainage network.

that the DLA model is only one of several that generate fractalnetworks that simulate the evolution of topography.Earthquake Statistics

To a first approximation earthquakes can be considered to bepoint events in a five-dimensional space: x, y, z (position), t(time), and m (magnitude). Stresses are applied to Earth'scrust by the relative motion of the surface plates. At highstresses, the crust deforms by displacements on preexistingfaults; these displacements result in earthquakes. Althoughthis deformation is clearly complex, there is order. In partic-ular, the frequency-magnitude statistics of earthquakes aregiven by the Gutenberg-Richter relation:

logN = -bm + a, [6]

where N is the number of earthquakes per unit time in aspecified region with magnitudes greater than m. The b value

012 _4

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FIG. 5. Dependence of the number of streams of various orders ontheir mean length for the example illustrated in Fig. 4A (A) and themodel illustrated in Fig. 4B (B).

is generally in the range of b = 0.9 ± 0.1. Eq. 6 is equivalentto a fractal (power-law) relation between frequency andrupture area A (29):

N= CA-D/2, [7]

withD = 2b. This fractal relation is recognized to be applicableunder a wide variety of tectonic settings from midocean ridgesto continental collision zones, such as Tibet. However, the

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FIG. 6. Illustration of the mechanism for network growth in theDLA model. A random walker is randomly introduced to an unoccu-pied cell. The random walk proceeds until a cell is encountered withone (and only one) of the four nearest neighbors occupied (hatchedcell). The new cell is accreted to the drainage network.

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fractal dimension is not diagnostic since it has such a smallvariation.

Earthquakes occur on preexisting faults. It is accepted as a

working hypothesis that each fault is associated with a char-acteristic earthquake. Thus, a fractal distribution of earth-quakes implies a fractal distribution of faults. It does notfollow, however, that the fractal dimension for the frequency-size distribution of faults is the same as that for earthquakessince this would imply that the interval of time betweenearthquakes is independent of scale. The intervals would alsobe expected to have a power-law dependence on scale. Thereis also evidence that the spatial and temporal distributions ofseismicity are also fractal (30, 31).

In China it has been generally accepted that one earthquakecould remotely trigger another. The evidence put forth was

discounted in the United States until the m = 7.3 Landers(California) earthquake occurred on June 28, 1992. Thisearthquake clearly triggered seismicity as far as 1100 km fromthe epicenter (32). This activity is illustrated in Fig. 7. Anintriguing question is how this seismicity was triggered. Onesuggestion is that seismically generated surface waves triggerthe events. However, the stresses associated with the surfacewaves are less than the stresses associated with the solid earthtides, and there is no apparent correlation between earth-quakes and earth tides. But once again Earth's crust appearsto be near a critical state, and the temporal correlations ofearthquakes are taken as further evidence that distributedseismicity is a "critical" phenomenon.

Earthquake Models

There is observational evidence that earthquakes occur on afractal distribution of faults. Each fault is associated with a

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characteristic earthquake, although there is certainly some

variation. The fractal distribution of earthquakes is associatedwith a hierarchy of faults and is not the distribution applicableto a single fault.Sammis et al. (33) have proposed that comminution is

responsible for the creation of the fractal distribution of faults.The comminution hypothesis implies that the direct contactbetween two blocks of near equal size during tectonic defor-mation will result in the breakup of one of the blocks. It isunlikely that small blocks will break large blocks or that largeblocks will break small blocks. A deterministic model forcomminution is given in Fig. 8. Two diagonally opposed blocksare retained at each scale such that no two blocks of equal sizeare in contact with each other. We have two blocks with r =

1/2 and 12 blocks with r = 1/4, so that D = ln 6/In2 = 2.585.Turcotte (34) has shown that many experimental studies offragmentation yield fractal frequency-size distributions withD

2.5. Assuming each side of a block in Fig. 8 to be a fault, we

have a fractal distribution of fault sizes with D = ln 3/ln 2 =

1.585. Barton and Hsieh (35) considered the distribution ofexposed joints and fractures near Yucca Mountain, Nevada,and found good agreement with a fractal distribution of blockexposures by taking D = 1.6.There are certainly similarities between the cellular autom-

ata model for self-organized criticality given by Bak et al. (2)and earthquakes (36, 37). In this analogy, the addition ofparticles is analogous to an increase in the regional tectonicstrain. The events are analogous to earthquakes, and thenumber of particles present is analogous to the mean regionalstress. The applicability of the Gutenberg-Richter frequency-magnitude relation to earthquakes is taken as evidence thatEarth's crust exhibits self-organized criticality.

0 50 150 200Days since January 1, 1992

FIG. 7. Cumulative seismic moment in selected zones beginning January 1, 1992. Numbers in parentheses are distances (in km) from the Landers

earthquake. Total seismic moment (in dyne-cm) for each zone is shown at right. Vertical lines mark times of the Petrolia (Cape Mendocino) (m= 7.1) and Landers (m = 7.3) earthquakes.

Coso - Indian Wells (165 to 205)

Little Skull Mtn., NV (240)

White Mtns., CA (380 to 420) 6x 1023

MonoBasin (450) 2.6 x 1021

|Cedar City, UT (490) 2.6 x 1022|

|Westem Nevada (450 to 650) 2.6 x 1023 l

Lassen (840) 6.4 x 1020

uYello e(121.4 x1522Cascade, ID (1100) 6.2 x 1019 C

|Yellowstone (1250)_~~~~~~~~~~~~~~~. x 102


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FIG. 8. Illustration of a discrete model for comminution. Diago-nally opposite blocks are retained at each scale. The blocks satisfy a

fractal relationship with D = In 6/ln 2 = 2.585. The block surfacessatisfy a fractal relationship with D = ln 3/ln 2 = 1.585.

Although there are important similarities between the cel-lular-automata model discussed above and distributed seis-micity, there are also important differences. The model doesnot generate "characteristic" earthquakes. A specified box inthe grid may participate in a small event in one time step anda much larger event at a later time step. Large events arerandomly distributed over the grid. The model does notgenerate either foreshocks or aftershocks. A sequence ofaftershocks is a universal feature of crustal earthquakes.To address these differences, Barriere and Turcotte (38)

introduced a cellular-automata model in which the boxes havea scale-invariant distribution of sizes. The objective was tomodel a scale-invariant distribution of fault sizes. When a

redistribution from a box occurs, it is equivalent to a charac-teristic earthquake on the fault. A redistribution from a smallbox (a foreshock) may trigger an instability in a large box (themain shock). A redistribution from a large box always triggersmany instabilities in the smaller boxes (aftershocks).Although the simple cellular-automata models have a num-

ber of quite remarkable similarities to distribute seismicity,they lack much of the basic physics, such as stick-slip behaviorand elastic rebound. A model that includes both was intro-duced by Burridge and Knopoff (39). They considered a seriesof slider blocks which were pulled over a surface by driversprings and connected to each other by connector springs. Thesimplest form of this model consists of two slider blockscoupled to each other and to a constant velocity driver bysprings. If the friction law is velocity weakening, the sliderblocks exhibit stick-slip behavior. Huang and Turcotte (40, 41)and Narkounskaia and Turcotte (42) have shown that any

asymmetry in this model results in classical chaotic behavior;the Feigenbaum period-doubling route of chaos is observed.A model that combines the analog features of slider blocks

and the high-order aspects of cellular-automata models in-volves the use of many slider blocks. Carlson and Langer (43)considered linear arrays of slider blocks, with each blockconnected by springs to the two neighboring blocks and to a

constant velocity driver. They used a velocity-weakening fric-tion law and considered up to 400 blocks. Slip events involvinglarge numbers of blocks were observed, the motion of allblocks involved in a slip event were considered, and the ap-

plicable equations of motion were solved simultaneously.Although the system is completely deterministic, the behaviorwas apparently chaotic. Frequency-size statistics were ob-

tained for slip events, and the events fell into two groups:smaller events obeyed a power-law (fractal) relationship andan anomalously large number of large events included all theslider blocks. This model was considered to be a model for thebehavior of a single fault, not a model for distributed seismic-ity. The large events were associated with characteristic earth-quakes on the fault, and the smaller events were associatedwith background seismicity on the fault between characteristicearthquakes.

Nakanishi (44,45) proposed a model that combined featuresof the cellular-automata model and the slider-block model. Alinear array of slider blocks was considered, but only one blockwas allowed to move in a slip event. The slip of one block couldlead to the instability of either or both of the adjacent blocks,which would then be allowed to slip in a subsequent step or

steps until all blocks were again stable. Brown et al. (46)proposed a modification of this model involving a two-dimensional array of blocks. A variety of other models of thistype have been considered (47-59).We now give some details of a two-dimensional slider-block

model (60, 61). The model is illustrated in Fig. 9, and it isassumed that during the sliding of one block all other blocksremain stationary. This requirement limits the system to near-

est neighbor interactions, and an analytic expression can bewritten for the displacements.To minimize the complexity, a discontinuous static-dynamic

friction law is considered. The governing parameters are a =

kl/kl (kc is the spring constant of the connector springs, k1 is thespring constant of the driver springs), where a is a measure ofthe stiffness of the system; = FS/Fd (the ratio of the static F,to dynamic Fd friction); and N is the number of blocksconsidered. In this model the parameter can be eliminatedby rescaling. Thus, for large systems (N very large) the onlyscaling parameter is the stiffness a. Frequency-size statisticsfor a 50 x 50 (N = 2500) array are given in Fig. 10 for severalvalues of the stiffness parameter a. A good correlation isobtained with the fractal relation (Eq. 7) with D = 2.72. Thefrequency-size relation shows a roll off from the power lawnear the larger end of the scaling region. This deviation isreduced as the stiffness a is increased. Frequency-size statisticsfor several different size arrays are given in Fig. 11. When theparameter a/N'12 is greater than one, we observe an excessnumber of catastrophic events that include the failure of allblocks. For a very stiff system, the array of blocks act as a singleblock; however, a fractal distribution of small events remains.The failure statistics of these multiple-block systems clearly

kc

FIG. 9. Illustration of the two-dimensional slider-block model. Anarray of blocks, each with mass m, is pulled across a surface by a driverplate at a constant velocity v. Each block is coupled to the adjacentblocks with either leaf or coil springs (constant kj), and to the driverplate with a leaf spring (constant k1). The extension of the (i, j) pullingspring is xij.

1'4.::

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FIG. 10. The r;total number of e)blocks that particilevent). Results areThe solid line is thEq. 1.

indicate a self-oxsimilar to distritThe fundame

also directly relassessment strattof large earthququakes only occsmall earthquakihazard. Johnstoiassess the hazarn(63, 64) has discgeneral associatquakes provideshazard assessmefrom the applica

Precursory seprediction (1-5 3at the Internati4Prediction and Idirection of V. I

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tion includes increases in regional seismicity, increases in theclustering of earthquakes, and changes in aftershock statistics.Premonitory seismicity patterns were found for strong earth-quakes in California and Nevada (algorithm CN) and forearthquakes withm 2 8 worldwide (algorithm M8). Assumingself-similarity of the properties of seismicity, both algorithmswere tested in seismically active regions. During the last 5years, at least three strong earthquakes (Spitak, Loma-Prieta,and Costa Rica) were predicted in advance. Although thisapproach to earthquake prediction is certainly controversial, itis consistent with the behavior of a system that is in a state ofself-organized criticality.

Conclusions

I I _ __ I______________ A variety of models have been shown to exhibit a behavior that0.6 1.2 1.8 2.4 3.0 is defined to be self-organized critical. A primary characteristic

log Nf is that a series of discrete events are observed that obey fractalfrequency-magnitude statistics. Some models are completely

atio of the number of events N with size Nf to the deterministic, such as the array of slider blocks; other modelsvents No is plotted against Nf (Nf is the number of are stochastic, such as the sand-pile model.pate in an event and is a measure of the area of an Earthquakes also appear to be an example of self-organized- given for 4) = 1.5 and a = 10, 15, 20, 30, and 40. critical behavior. Such a conclusion has important implicationsie correlation with the power-law (fractal) relation for estimating the earthquake hazard and, possibly, for making

reliable earthquake predictions. There is increasing observa-rganized critical behavior and are remarkably tional evidence that earthquakes interact with each other over)uted seismicity. very large distances; this is a characteristic of critical phenom-ntal aspects of self-organized criticality are ena. An implication is that earthquakes will cluster in time overevant to earthquake prediction and hazard very large distances, possibly over the entire planet.egies. An important question is the association Landforms also obey fractal statistics in a variety of ways.iakes with small earthquakes. Do large earth- Because earthquakes are important in the creation of land-ur where small earthquakes occur? If so, the forms, it would not be surprising if the creation of topographyes can be used to assess the regional seismic is an example of self-organized criticality. However, the mor-

phology of landforms is generally dominated by erosion. An indtheNewa62M adrid, Missohisaprea.Trcoe variety of models have been used to simulate the erosionalussedthis aproac insoregeeal Terms.A evolution of landforms. Again this process may fall in the

general class of self-organized critical phenomena.clon O1 iarge eartnquaKes wiLt small earmn-a rational basis for probabilistic earthquakent. Such an approach would follow naturallyibility of self-organized criticality.ismicity is the basis for intermediate-termyears) pattern recognition methods developedonal Institute for the Theory of EarthquakeFheoretical Geophysics in Moscow under theI. Keilis-Borok (65-67). The pattern recogni-

-2.01

-4.010

-6.01

0.0 0.5 1.0 1.5 2.0log Nf

2.5 3.0 3.5

FIG. 11. The ratio of the number of events N with size Nf to thetotal number of events No is plotted against Nf. Results are given forsystems of size 20 x 20, 30 x 30, 40 x 40, and 50 x 50 with parameters= 1.5 and a = 50. The peaks at log Nf = 2.60, 2.95, and 3.20

correspond to catastrophic events involving the entire system.

1. Turcotte, D. L. (1992) Fractals and Chaos in Geology and Geo-physics (Cambridge Univ. Press, Cambridge, U.K.).

2. Bak, P., Tang, C. & Wiesenfeld, K. (1988) Phys. Rev. A 38,364-374.

3. Scholz, C. H. (1991) in Spontaneous Formation of Space-TimeStructures and Criticality, eds. Riste, T. & Sherrington, D. (Klu-wer, The Netherlands), pp. 41-56.

4. Mandelbrot, B. (1967) Science 156, 636-638.5. Mandelbrot, B. (1975) Proc. Natl. Acad. Sci. USA 72, 3825-3828.6. Rapp, R. H. (1989) Geophys. J. Int. 99, 449-455.7. Barton, C. C. & Scholz, C. H. (1994) in Fractals in Petroleum

Geology and Earth Processes, eds. Barton, C. C. & LaPointe, P. R.(Plenum, New York), in press.

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Slope= -1.36

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scaling geology: landformsand earthquakes6698 colloquium paper: turcotte a, 2 d=1.25 x;%^1 v iiii...

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