Implications of chaos, scale-invariance, and fractal statistics in geology

Palaeogeography, Palaeoclimatology, Palaeoecology(Global and Planetary Change Section), 89 (1990) 301-308

301

Elsevier Science Publishers B.V., Amsterdam

Implications of chaos, scale-invariance, and fractal statistics in geology D.L. T u r c o t t e Department o/Geological Sciences, Cornel] University, Ithaca, NY 14853, USA (Received by publisher July 30, 1990)

ABSTRACT

Turcette, D.L., 1990. Implications of chaos, ecale-invariance, and fractal statistics in geology. Palaeogeogr., Palaeoclimatol., Palaeoecol. (Global Planet. Change Sect.). 89: 301-308. Many natural phenomena can be clas~fied as events with reasonably well defined magnitudes. An obvious example is an earthquake where frequency-magnitude statistics fit the Guttenburg-Richter empirical correlation for both regional and world-wide data. Another example is volcanic eruptions. However, volcanic eruptions are more difficult to quantify in terms of magnitude. Many meteorological events can be reasonably well quantified on a statistical basis. Floods are one example. It has been shown that the empirical correlation for earthquakes is in fact a fractal relation between the number of earthquakes and the linear dimension for the rupture zone. This fractal behavior is evidence for scale-invariance since fractal statistics are the only statistics without a characteristic dimension. Fractal statistics for the frequency-magnitude behavior of earthquakes provides a basis for estimating seismic bRT~rds; the observed statistics for small earthquakes can be extrapolated to estimate the probability of occurrence of large earthquakes. It is also recognized t h a t the earth's topography generally satisfies fractal statistics. Topography is primarily generated by erosion but a complete mathematical theory for erosion is not available, certainly chemical and mechanical weathering play a role but the actual topography is dominated by drainage systems. It is often hypothesized t h a t great storms and floods dominate the formation of these systems. Storms and floods are likely to obey fractal statistics and be dominated by the largest events just as in earthquakes and volcanoes.

I. I n t r o d u c t i o n

The weather, climate, and lithosphere deformation are examples of complex, non-linear problems. In each case, however, the governing equations are deterministic. For the weather and climate, turbulence is clearly important. Fluids t h a t satisfy the deterministic Navier-Stokes equations are often turbulent and the variables must be treated statistically. Steady solutions may satisfy the governing equations and the boundary conditions but are unstable to small disturbances. In 1963 Lorenz obtained numerical solutions to a set of the three first-order, non-linear equations that exhibited deterministic chaos. Solutions with nearly identical initial conditions were 0921-8181/90/$03.50

shown to diverge exponentially so t h a t predictability decreased with time. However, the solutions exhibited a well-defined statistical behavior, in fact this statistical behavior as quantified in Poincar6 sections in fractal. The Lorenz equations were derived to represent thermal convection in a fluid layer heated from below, but the spontaneous reversals obtained are clearly relevant to revers0]s ~ in the earth's magnetic field (Chillingworth and Holmes, 1980). In fact a parametrized set of dynamo equations proposed by Rikitake (1958) has been shown to exhibit deterministic chaos and spontaneous reversals (Allan, 1962; Cook and Roberts, 1970; Robbius, 1977). Although a number of simple mechanical systems have been shown to satisfy a low order set of equations t h a t exhibit deterministic chaos,

© 1990 - Elsevier Science Publishers B.V.

302

problems of interest in geology generally have much higher orders. Fortunately, m a n y complex non-linear systems satisfy fractal statistics. One example is fragmentation. T h e physics of fragmentation is extremely complex involving the initiation and propagation of fractures. Yet under a wide variety of circumstances the frequency-size distribution of fragments exhibit fractal, power-law statistics (Turcotte, 1986). T h e only requirement for fractal statistics is t h a t the applicable physics be scale invariant. Clearly this condition will be satisfied only over a finite range of scales. For example, from the size of the body t h a t is fragm e n t e d to the grain size. Another example of fractal statistics is the size-frequency distribution of earthquakes (Aki, 1981). This implies t h a t the size-frequency distribution of faults is also fractal. Again the physics of deformation of the lithosphere associated with earthquakes is extremely complex. Although the linear equations of elasticity are applicable in part, the behavior of faults is certainly non-linear. Non-linear friction laws are probably applicable. Yet, despite the complexities of the physics and the creation and death of active faults, fractal scale-invariant statistics are satisfied. An i m p o r t a n t question is whether other major geological events satisfy fractal statistics, i.e. volcanic eruptions, floods, etc. Time series with power spectral densities t h a t have a power law dependence on frequency (wave number) can be shown to be fractal under a variety of circumstances. One example is topography. Many authors have shown t h a t both topography and b a t h y m e t r y satisfy Brown noise statistics. This is taken as evidence t h a t the evolution of surface morphology satisfies nonlinear equations t h a t yield self-similar solutions. Lorenz equations

Lorenz (1963) derived a set of three non-linear total differential equations t h a t exhibited deterministic chaos. Although other sets of equations also exhibit deterministic chaos, the Lorenz equations have been thoroughly studied and clearly illustrate the characteristics of

D.L.

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Fig. 1. A solution of the Lorenz Eqs. 1-3 with Pr = 10, r = 28, and b = 4/3 in the phase plane A, B and C projected into the A- B plane. The unstable fixed points A = :+6 V~, B = + 6vf2, C = 27 are shown by crosses.

chaotic behavior (Sparrow, 1982). T h e Lorenz equations are: dA dr - -PrA dB dr dC dr

+ PrB

(1)

AC+rA-B

-

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T h e Lorenz equations were derived to represent thermal convection in a fluid layer heated from below. T h e variables A, B and C are coeffi-

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IMPLICATIONS

OF CHAOS, SCALE-INVARIANCE,

AND FRACTAL

cients in highly truncated Fourier expansions of the nondimensional stream function and temperature and • is the nondimensional time. The parameters are the Prandtl number Pr, the ratio of the Rayleigh number to the critical Rayleigh number r, and an aspect ratio parameter b. A numerical solution of the Lorenz Eqs. 1-3 for P r = 10, r = 28, and b = 4 / 3 projected on the A - B plane in the three-dimensional A - B - C phase space is given in Fig. 1. The unstable fixed 0

a

..r

T

STATISTICS

IN GEOLOGY

points A : +_6v~, B = _+6V~, C = 27 are shown by crosses. The solution is chaotic in the sense that solutions infinitesimally close together diverge exponentially; the fixed points behave as strange attractors. The dependence of the parameter B on time is given in Fig. 2. It is seen that the cellular flow randomly jumps from clockwise to counterclockwise rotation and back. It should be emphasized that this r~mdom behavior is superimposed on the well defined period

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of the cellular flow. These solutions exhibit fractal behavior in terms of Poincard solutions; the distribution of crossing points on the A - B , A - C , and B - C planes is fractal. T h e random reversals of the earth's magnetic field are very similar to the reversals illustrated in Fig. 2. In fact a set of parameterized d y n a m o equations proposed by Rikitaki (1958) has been shown to exhibit chaotic behavior similar to t h a t of the Lorenz equations (Allan, 1962; Cook and Roberts; 1970; Robbins, 1977). An interesting question is whether the evolution of climate associated with ice ages is also a chaotic phenomena. T h e Milankovitch cycle could correspond to the rotational frequency of Lorenz, the two fixed points of Lorenz could correspond to metastable ice ages and metastable interglacial periods. T h e importance of the Lorenz equations is t h a t t h e y demonstrated t h a t deterministic equations with deterministic initial conditions can yield stocastic solutions with fractal statistics. Log/stic map May (1976) showed the remarkable behavior of the logistic map. T h e logistic equation is the simple recursion relation xn+,=axn(1-x,)

0
(4)

For 0 < a < 1 (4) gives x = 0 in the limit n -~ oo; this behavior is illustrated for a = 0.8 in the iterative m a p given in Fig. 3a. For 1 < a < 3 (4) gives x~o = 1 - 1/a; this behavior is illustrated for a = 2.5, x~ = 0.6 in Fig. 3b. At a = 3 a flip bifurcation occurs and for 3 < a < 3.544090 a two-root ( n = 2) limit cycle is found; this behavior for a = 3.1, xoo = 0.558, 0.765 is illustrated in Fig. 3c. F u r t h e r bifurcations occur and for 3.569946 < a < 4 windows of chaotic behavior and multi-root cycles appear. Fully chaotic behavior for a = 3.9 is illustrated in Fig. 3d, 1000 cycles are given. T h e systematic bifurcations of the solution strongly resemble the f~actal Cantor set. T h e logistic m a p is often associated with the diversity of species so t h a t spontaneous chaotic behavior m a y be independent of driving functions.

Fractal distributions T h e definition of a fractal distribution is 1

N-

r--~

(5)

where N is the n u m b e r of objects with a characteristic size greater t h a n r and D is the fractal dimension. For a discrete set N becomes N / a n d r becomes ri. Any scale invariant process will yield a fractal distribution over the range of scale invariance. I t should be pointed out t h a t linear equations do not yield fractal behavior because t h e y always include a characteristic dimension, either internal to the equations or imposed by b o u n d a r y conditions. Fractal behavior follows naturally from sets of nonlinear equations t h a t exhibit chaotic behavior, i.e. the Lorenz equations, and from recursion relations such as the logistic map. It should be pointed out t h a t M a n d e l b r o t (1967) introduced the concept of fractals in terms of the length of a coast line; the length of a coast line has a power law dependence on the length of the measuring stick. Fragmentation Fragmentation is an extremely complex phenomena, b u t under a wide variety of conditions the frequency size distribution of fragments is

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than r for broken coal (Bennett, 1936), broken granite from a 61 kt underground nuclear detonation (Schoutens, 1979), and impact ejecta due to a 2.6 k m / s polycarbonate projectile impacting on basalt (Fujiwara et al., 1977). The best fit fractal distribution from Eq. 5 is shown for each data set.

IMPLICATIONS OF CHAOS, SCALE-INVARIANCE, AND FRACTAL STATISTICS IN GEOLOGY

305

ity of fractal statistics greatly reduced the number of parameters t h a t any theory must specify.

Seismicity One major resu]t of observational seismology

is the applicability of the Gutenberg-Richter frequency-magnitude relation (Gutenberg and Richter, 1954). log N

2 Fig. 5. Illustration of a renormalization group model for fragmentation.

fractal (Turcotte, 1986). Three examples of fractal distributions of fragments are given in Fig. 4. It has been suggested by a number of authors (e.g. Donnison and Sugden, 1984) t h a t meteorites and asteroids have fractal distributions. A simple renormalization group model gives a fractal distribution of fragments; this model is illustrated in Fig. 5. A cube with dimensions h is divided into eight cubes with dimensions h/2, each of these cubes is also divided into eight cubes with dimensions h/4, and so forth. A fractal distribution is obtained if the probability f t h a t a cube will break is the same on all scales relevant to the fractal'distribution. If initially there are NO cubes with a linear dimension r0 the number of fragments N~ at a scale r, is

N. = (8f)n(z

-

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-=- 2--ff

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(10)

where N is the number of earthquakes per unit time with magnitude greater than m in a specified area. Aki (1981) showed that Eq. 10 is equivalent to the fractal relation

(11)

-- r - 2 b

so t h a t D

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=

(12)

Well defined fractal distributions of earthquakes are found both regionally and globally. The observed range of b is small, generally 0.8 < b < 1 so t h a t 1.6 < D < 2.

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b=0.89 D= 1.78

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=

(8)

And comparison with Eq. 5 gives the fractal dimension in terms of the probability of frag-

mentation D= ln8f in-mE

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This is an illustration of how a recursion relation gives a fractal distribution. The applicabil-

id3

4 m Fig. 6. N u m b e r of e a r t h q u a k e s occurring per y e a r N w i t h a surface wave m a g n i t u d e greater t h a n m as a function of m. T h e open s q u a r e s are d a t a for S o u t h e r n California from 1932-1972 ( M a i n a n d B u r t o n , 1986); t h e solid circle is the expected r a t e of occurrence of great e a r t h q u a k e s in S o u t h e r n California (Sieh, 1978).

306

D.L.

T he general applicability of the fractal relation for seismicity can provide the basis for a quantitative seismic hazard assessment (Turcotte, 1989). The number of earthquakes per year in Southern California with magnitude greater than m is given as a function of m is given in Fig. 6 (Main and Burton, 1986), the data period was 1932-1972. Also included is the magnitude and repetition rate of great earthquakes based on the paleoseismic studies of Sieh (1978). This figure provides a basis for extrapolating the statistics gathered from small earthquakes on a short time scale to estimate the recurrence interval for great earthquakes. This approach has obvious difficulties. The maximum magnitude may not a priori be known so one basis for extrapolation is not known. Nevertheless a global map of a would be a quantitative estimate of seismic hazards t ha t is not now available. Another important question is whether other natural phenomena obey fractal statistics. It is generally accepted t hat the greatest volcanic eruptions contain a large fraction of the energy. However, data on volcanic eruptions is not sufficiently quantitative to establish fractal statistics. The same comments are applicable to meteorological phenomena such as floods and severe storms.

Topography

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Fractal concepts can also be applied to the spatial or temporal variations of a single variable. Examples in geology included topography and gravity. These are known as self-affine rather than self-similar fractals. For self-affine fractals the spectral energy density S must have a power law dependence on the wave number k

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IMPLICATIONS OF CHAOS, SCALE-INVARIANCE, AND FRACTAL STATISTICS IN GEOLOGY

A n u m b e r of a u t h o r s have carried out 1 - D Fourier transform studies of t o p o g r a p h y and b a t h y m e t r y and this work has been summarized b y Fox and Hayes (1985). These studies found a good correlation with fl = 2 ( D = 1.5) for wavelengths from 0.1 to 103 km; t o p o g r a p h y appears to be Brown noise. Four examples of linear tracks of t o p o g r a p h y adjacent to the Colorado Plateau in Arizona are given in Fig. 8. I t is seen t h a t the best fit correlations from Eq. 13 give fractal dimensions near 1.5. T h e mechanisms responsible for the generation of t o p o g r a p h y are poorly understood. In most areas erosion is dominant. Culling (1965) proposed t h a t erosional t o p o g r a p h y satisfies the h e a t equation. This would be the case if transp o r t is proportional to slope. However, the heat equation is linear and does not produce fractal topography; small scale features would erode more rapidly t h a n large scale features. T h e fact t h a t erosional t o p o g r a p h y has fractal statistics is evidence t h a t it is scale invariant. This suggests t h a t t h e governing physics is nonlinear and m a y be related to a fractal distribution of storms and floods t h a t generate and

T h e power law is related to the fractal dimension b y (Voss, 1985) D = 5 - fl 2

(14)

Such spectra are known as fractional Brownian noise. Examples of synthetically generated fractional Brownian noise are given in Fig. 7. T h e m e t h o d used to generate these r a n d o m series was as follows: (1) R a n d o m numbers based on a Gaussian probability distribution are assigned to 512 adj a c e n t intervals. This is a Gaussian white noise sequence. (2) T h e Fourier transform of the sequence is taken. (3) T h e resulting Fourier coefficients are filtered b y multiplying b y k-~/2. (4) A inverse Fourier transform is taken using the filtered coefficients. In order to remove edge effects only the central portion of the series should be retained. Brown noise corresponds to fl = 2 ( D = 1.5) and is illustrated in Fig. 7d. O

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log(k, k m -1) Fig. 8. Plots of spectral energy density versus wave number for four linear tracks of topography adjacent to the Colorado Plateau in Arizona. The tracks were 72 km long and each contained 512 digitized data point~. The fractal dimermionfor the best fit straight line is given for each data set.

308

renew erosional features such as gullies and drainage systems.

Conclusions Sets of nonlinear total differential equations and recursion relations have been shown to yield chaotic solutions. Infinitesimal differences in initial conditions lead to order one difference in the evolving solutions. Many now accept that the turbulent behavior of the oceans and atmosphere is chaotic. The conclusion is that the evolution of this system is not predictable in a deterministic sense but must be treated statistically. For a variety of problems in geology the basic applicable equations are not known. One example is the deformation of the earth's crust. We do not know how to formulate fault displacements. Nevertheless, we have ample evidence that the frequency-magnitude statistics of earthquakes is fractal. It is known that this requires scale invarient behavior. Fractal behavior is also generally associated with chaotic solutions. It appears entirely reasonable to hypothesize that the deformation of the crust is a chaotic process. A conclusion is that seismicity is not predictable in a deterministic sense but must be treated in a statistical manner.

Acknowledgements This work has been supported by the National Aeronautics and Space Administration under grant NAG 5-860. This is contribution 845 of the Department of Geological Sciences, Cornell University.

References Aki, K., 1981. A probabilistic synthesis of precursory phenomena. In:D.W. Simpson and P.G. Richards (Editors) Earthquake Prediction. Am. Geophys. Union, Washington, D.C., pp. 566-574. Allan, D.W., 1962. On the behavior of coupled dynamos. Proc. Camb. Philos. Soc., 58: 671-693.

D.L. T U R C O T T E

Bennett, J.G., 1936. Broken coal. J. Inst. Fuel, 10: 22-39. Chillingworth, R.J. and Holmes, P.J., 1980. Dynamical systems and models for reversals of the earth's magnetic field. Math. Geol., 12: 41-59. Cook, A.E. and Roberts, P.H., 1970. The Rikitake two-disc dynamo system. Proc. Camb. Philos. Soc., 68: 547-569. Culling, W.E.H., 1965. Theory of erosion on soil-covered slopes. J. Geol., 73: 230-254. Donnison, J.R. and Sugden, R.A., 1984. The distribution of asteroid diameters. Mon. Not. R. Astron. Soc., 210: 673682. Fox, C.G. and Hayes, D.E., 1985. Quantitative methods for analyzing the roughness of the seafloor. Rev. Geophys., 23: 1-48. Fujiwara, A., Kamimoto, G. and Tsukamoto, A., 1977. Destruction of basaltic bodies by high-velocity impact. Icarus, 31: 277-288. Gutenberg, B. and Richter, C.F., 1954. Seismicity of the E a r t h and Associated Phenomena. Princeton Univ. Press, Princeton, NJ, 310 pp. Lorenz, E.N., 1963. Deterministic nonperiodic flow. J. Atoms. Sci., 20: 130-141. Main, I.G. and Burton, P.W., 1986. Long-term earthquake recurrence constrained by tectonic seismic moment release rates. Seismol. Soc. Am. Bull., 76: 297-304. Mandelbrot, B., 1967. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156: 636-638. May, R.M., 1976. Simple mathematical models with very complicated dynamics. Nature, 261: 459-467. Rikitake, T., 1958. Oscillations of a system of disc dynamos. Proc. Camb. Philos. Soc., 54: 89-105. Robbins, K.A., 1977. A new approach to subcritical instability and turbulent transitions in a simple dynamo. Math. Proc. Camb. Philos. Soc., 82: 309-325. Schoutens, J.E., 1979. Empirical analysis of nuclear and high-explosive cratering and edecta. In: Nuclear Geoplosics Sourcebook, Vol. LV, Part II, Section 4. Defense Nuclear Agency Rep. DNA 65 01H-4-2-. Sieh, K.E., 1978. Prehistoric large earthquakes produced by slip on the San Anreas fault at Pallett Creek, California. J. Geophys. Res., 83: 3907-3939. Sparrow, C., 1982. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer, New York, NY, 269 pp. Turcotte, D.L., 1986. Fractals and fragmentation. J. Geophys. Res., 91: 1921-1926. Turcotte, D.L., 1989. A fractal approach to probabilistic seismic hazard assessment. Tectonophysics, 167: 171-177. Voss, R.F., 1985. Random fractals: characterization and measurement. In: R. Pynn and A. Skjeltorp, (Editors), Scaling Phenomena in Disordered Systems. Plenum Press, New York, NY, pp. 1-11.