Mathematical Geology, Vol. 12, No. 4, 1980

A Stochastic Model for Strike-Slip Faulting I G. R a n a l l i 2

The frequency o f length o f strike-slip faults in continental crust follows a lognormal probability distribution, and a nonlinear positive correlation exists between length and offset. These results appear to be scale-independent. A n explanation o f the observations is presented in terms o f a stochastic model which treats the occurrence o f faulting as a Kolmogorov-type process obeying the law o f proportionate effect. This model accounts for the length distribution o r faults. Tentatively, the correlation between length and offset is ascribed to an allometric law relating the relative growth rates o f these two parameters. The possibility o f the application o f the concepts o f continuum damage mechanics to the problem is also briefly explored as a way to introduce time-space averages o f tectonic stress and strain-rate into the model.

KEY WORDS: stochastic process, structural geology, continuum mechanics. INTRODUCTION A physical system is deterministic "if its properties can be quantitatively predicted, by accepted deterministic principles, to a satisfactory level of accuracy" (Smart, 1979). Such an operational description has the advantage of leaving open the question of whether physical events are intrinsically random, or if their apparent randomness is due to the observer's imperfect knowledge of the system parameters. This question has been debated not only in physics but also in the earth sciences (as an example, one may mention fluvial geomorphology); however, it leads to statements that cannot be tested by present scientific methods, and consequently adds nothing to the understanding of a given physical system. From this operational viewpoint, the deterministic description of a system becomes increasingly difficult as the complexities and uncertainties in the boundary conditions of the system increase. The system under consideration here consists of the e n s e m b l e of strike-slip faults occurring in the continental 1Manuscript received 14 November 1979. This paper was presented at Symposium S. 12.2.4 "Advances in Mathematical Geology," held as part of the 26th International Geological Congress in Paris, France, July 1980. 2Departrnent of Geology, Carleton University, Ottawa, Canada K1S 5B6, and Institut ftir Meteorologic und Geophysik, Goethe-Universit~t, 6000 Frankfurt a. Main, BRD. 399 0020-5958/80/0800-0399503.00/0 © 1980 Plenum Publishing Corporation

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crust on a local, regional, or global scale. While the mechanics of a single fault can be studied by "accepted deterministic principles" (in this case, besides the usual conditions of conservation of matter and momentum, what is needed is a rheological constitutive equation incorporating a failure criterion), a description of the ensemble cannot be given in deterministic terms. Mechanical principles predict that fracture and slippage occur when a critical state is reached; this state depends on material properties and stress conditions, both of which vary in space and time in what must be called a "random" way. Therefore, a global description of the system, in the sense of a mental model accounting for the observable properties of the ensemble, must necessarily be of a probabilistic character. In this paper, a model for the observed distributions of lengths and offsets in strike-slip faults in continental crust is given in terms of the laws of proportionate effect and of allometric growth. Although in the course of the author's work the statistical analysis of data has preceded the construction of the model, the presentation follows an inverse order, to emphasize the existence of a fundamental basis in the theory of probability for the observed relations. LENGTH DISTRIBUTION OF STRIKE-SLIP FAULTS Faulting is a process that occurs at all scales, from the crystalline to the continental. There are similarities in the geometry and evolution of fault systems from laboratory experiments to geological processes, that is, over a range of over 106 in linear scale (Tchalenko, 1970; King, 1978). A useful starting point for a model of strike-slip faulting is to assume that, at a given time, there exist in the crust potential faulting regions, or volumes, where a critical state is reached. The critical state (leading to the formation of a new fault or to slippage along a preexisting fault) is a function of the distribution of strain energy density, rock strength, friction, and so on. It is therefore possible to assign to each fault a positive measure S representing the critical volume surrounding it. If fault length is related to volume, the problem then becomes that of setting up a stochastic model for the distribution of fault volume. The analysis is restricted to strike-slip faults mainly because of the availability of data; furthermore, in this case the crust can readily be envisaged as consisting of blocks separated by vertical shear planes (major faults) along which motion occurs in a horizontal direction, and cross-cut by other shear planes (minor faults). The Lognormal Distribution

Many positive variates have a lognormal probability distribution, and therefore some of its properties are reviewed here. A variate X(O < x < oo) is lognor-

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mally distributed if the transformed variate in X is normally distributed

e{x<<.x) = a ( x [ . , o 2) where A(x) =NOn x) and # and 02 are the mean and variance of in X, respectively. The probability distribution of X is therefore dA(x) = 1/xo(Zzr) ~/2 exp [-(1/2o2)Onx - U)2 ] dx

(1)

and its moments and related parameters can easily be derived (el., e.g., Aitchison and Brown, 1957). There is a formal correspondence between the properties of additive variates, which lead to the normal distribution, and the properties of multiplicative variates, which lead to the lognormal distribution. Two properties that will be used in the sequel are reproducibility and asymptotic lognormality, here stated without proof (el. Aitchison and Brown, 1957, for further details).

Reproducibility If {Xi} is a sequence of independent variates with distribution A(,ul, o/2),

{hi} a sequence of constants and c = exp (a) a positive constant, then, provided that the relevant summations converge, the product c 11] X f i is A(a + E] b]#], E i b~ o~). (Thus, in particular, if X is A ~ , o 2), then cX b is A(a + bg, b 2 o2 .) Asymptotic Lognormality (Central Limit Theorem) If (Xi} is a sequence of positive independent variates having distributions whose parameters satisfy some very general existence and convergence requirements, then the product II]X i is asymptotically A(E] ~], E i a~). The lognormal distribution finds application in many different fields; economics, sociology, anthropometry, industrial engineering, biology, ecology, astronomy, and literary criticism. In the earth sciences (Agterberg, 1974) it has been applied to sedimentology (grain-size distribution in clastic rocks), geochemistry and petrology (element concentrations in rocks and mineral deposits), and economic and resource geology (mineral assay and mine valuation, size of oil and gas fields). In this and in previous papers (Ranalli, 1976, 1977) its range of applicability is extended to a geodynamic process. The Law of Proportionate Effect and Kolmogorov's Theory of Breakage Applied to Faulting The basis of most existing theories on the genesis of the lognormal distribution is the multiplicative central limit theorem. Physically, the equilibrium distribution of a positive variate is lognormal when the values of the variate are the outcome of a discrete random process (usually visualized as taking place in time),

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at each step of which the change in the variate is a random proportion of its previous value

xj- xi_, = ixi-,

(2)

where {S'i} is a sequence of mutually independent variates also independent of

(xj}. A variate subject to a process of change according to eq. (2) is said to follow the law o f proportionate effect. Its resulting lognormality was demonstrated by Kapteyn (1903). The lognormality follows immediately from eq. (2) and the central limit theorem. If X o and X n are the initial and final value of the variate, respectively, and ~'; = 1 + ~'j one can write

X n = X o Hi ~';

(3)

and Xn will be asymptotically lognormal. Restated in terms of distribution functions rather than variate values, the law of proportionate effect has originally been put forward by Kolmogorov (1941) to explain the particle size distribution of crushed ores. In this form, it is known as Kolmogorov's theory o f breakage. Given an initial distribution function F o (x), let Gi (x [u) be the distribution function, at step j, of elements arising from elements of size u. The law of proportionate effect stipulates that Gi(x[u)=Hi(x/u ) (i.e., the function Gi depends only on the ratio x/u) and therefore

ei(x) :

Ju Hi(x/u) dFi-I (u)

(4)

which is the distribution function equivalent of eq. (2) or (3). An analogous line of reasoning is followed in economics, in the analysis of income distribution (cf. Aitchison and Brown, 1957). There, a different formalism is used (using a matrix of transition probabilities) but, if the income range can be satisfactorily assumed to be a continuum, the final equilibrium distribution of income is lognormal. The law of proportionate effect, therefore, seems to have a very wide range of applicability to processes, involving positive variates subject to a sequence of changes. This generality is an indication of its fundamental nature. If the above arguments are adapted to a description of the faulting process, it is clear that the previously defined "critical volume" 8 is the physical quantity that can be postulated to follow the law of proportionate effect (Ranalli, 1976). The model based on this postulate is formally identical to Kolmogorov's theory of breakage; consequently, if the model is correct, the present distribution of S is the outcome of a time-sequence of independent "breakage" operations according to eq. (3), and is therefore lognormal. The critical volume is not directly measurable, but fault length is, e.g., from geological maps, satellite photographs, and so on. The relation between fault length and volume can be assumed to be

A Stochastic Model for Strike-Slip Faulting

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of the form x = c i S C~

(5)

and therefore, by the reproductive properties of the lognormal distribution, the variate X is also lognormal. (Note that, here and in the sequel, X indicates fault length.) The lognormality of fault length can be tested on suitable data sets. Data and Results on Fault Length

Two sets of data are considered here. The first (from Ranalli, 1976) consists of strike-slip faults of regional and continental scale; the second, of local fractures tentatively identified as small-scale strike-slip faults. The first set of data has been obtained from a search of the literature and includes faults with x ~> 50 kin, for a total sample size n = 176, occurring worldwide. The large lower limit is intended to minimize geographical bias due to uneven coverage, although this and many other factors of uncertainty (including fault length itself, as determined geologically) are undoubtedly present in the sample. Figure 1 shows the observed and the expected frequency. To estimate the parameters of the lognormal distribution, Fisher's maximum likelihood method for the truncated normal distribution was applied to the transformed variate In X (Ranalli, 1976). The results are = 5.057,

~2 = 1.313

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mission of the National Research Council of Canada).

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and a X2 test of goodness o f fit leads to the conclusion that the hypothesis that X is A(~, ~2) cannot be rejected at the 5% significance level. Given the uncertainties in the data, this is a satisfactory result. The second set of data has been obtained from a map o f local faults in Catalonia (So16 Sugra~es, 1978). The sample consists of n = 285 faults, only eight o f which have x > 8 km. The statistical analysis has been carried out for three different class intervals, Ax = 1.0, 0.5,0.25 km; the results are stable. Figure 2 shows the observed and expected frequency for Ax = 0.5 km; the estimated parameters are = 0.382,

~2 = 0.870

and a X2 test shows that the null lognormality hypothesis cannot be rejected at the 5% significance level. The two sets of data cover a very wide range of scale. The mean, median, and mode of X in the large-scale fault sample are 303.0,157.1, and 42.3 km, respectively; in the small-scale sample, they are 2.3, 1.5, and 0.6 km, respectively. The coefficient o f variation r/(ratio of standard deviation to mean o f X ) is different in the two cases: ~ = 1.65 for large-scale and 1.18 for small-scale faults. f (x)' 70

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Incidentally, the fact that the coefficient of variation is not the same at different scales excludes the possibility that the observed populations consist of a mixture of subpopulations graded according to size and having identical 7. Furthermore, a geographical subpopulation grouping is unlikely, since large strike-slip faults appear to occur without any geographical bias. It can therefore be excluded that the lognormality of fault length is only apparent and due to a mixture of subpopulations with a constant coefficient of variation (a factor that can cause a spurious positive skewness; cf Vistelius, 1960). A stochastic model based on the law of proportionate effect appears at present to be the most logical explanation of the properties of the ensemble of fault lengths, independently of scale. CORRELATION BETWEEN LENGTH AND OFFSET

The other obvious parameter of strike-slip faults is their offset Y. By "offset" is meant the geologically determined total relative horizontal displacement between the two sides of a fault; Y as well as X are therefore the results of many episodes of slippage and tectonic creep. The determination of offset is subject to even larger uncertainties than fault length. It has been observed that, on very large strike-slip faults, offset is roughly speaking about one-tenth of fault length (King, 1978). The positive correlation between length and offset has been studied by the author (Ranalli, 1977), who has also given an empirical kinematic model to account for the observations. Here this model is integrated into the general stochastic model of the faulting process, and it is proposed that offset is related to length by an allornetric growth law. Again, there is a remarkable convergence in the probabilistic description of vastly different systems. Huxley's Law of Allometric Growth The law of allometric (or heterogonic) growth is a concept that so far has found application primarily in biology. Huxley (1932), while studying the growth of organisms, originally formulated it by stating that the relative growth rate of a part of an organism is a constant fraction of the relative growth rate of the organism. This statement can, of course, be extended to physical systems which evolve with time. For such systems, and using a symbolism of immediate significance to the faulting model, the law ofallometric growth can be written as

Y(t)/Y (t) = b [X(t)/X(t)]

(6)

where X(t) and Y(t) are the sizes of the system and of its part, respectively, and the dots represent differentiation with respect to time. Rather than deterministic quantities, X and Y can be taken to be stochastic variables (and indeed they were in the cases studied by Huxley). In such cases, eq. (6) establishes a statisti-

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cal relation between two variates; in the model at hand these are length and offset of strike-slip faults. Integrating eq. (6) one obtains Y ( t ) = a [X(t)] b

(7)

from which it immediately follows that, if X is A ~ , oZ), Y is A(a' + b#, b2o:), a' = In a. By analogy, if a relation between X and Y such as eq. (7) is found to hold in nature, fault offset can also be thought of as a stochastic variable following the law of proportionate effect. The parameter b in eqs. (6) and (7) is the ratio between the relative growth rate of a part of the system and the relative growth rate of the system, and is termed the "relative growth ratio." According to whether 1 > b < 1, the relative growth rate of Y is larger or smaller than X. The relevance of these relations to the faulting process will be discussed in the sequel to this paper. Data and Results on the Correlation Between Length and Offset

A nonlinear positive correlation between length and offset of large (40 ~< x < 1600 km) strike-slip faults in continental crust (n = 131) has been found by the author (Ranalli, 1977). The data are presented in Figure 3, which shows that there is no apparent variation according to character (dextral or sinistral) and age of faulting. Least-square analysis leads to the relation y = 0.05X 1.17

(8)

and the correlation coefficient between the logarithmically transformed variates is significantly larger than that for the variates themselves. This implies that relation (8) is a significantly better fit than a linear relation (b = 1). Naturally, as the scatter of Y about X can be as large as one order of magnitude, the standard error of estimate is large, but this is a reflection of the random character of the data. Equation (8) is the outcome of an analysis based on the largest sample size available to date, and as such it must be accepted as a working hypothesis. The common assertion that offset in strike-slip faults is about one order of magnitude less than length is therefore approximately valid, but misleading if interpreted as implying a linear correlation between the two variates. Although the deviations from linearity are not very large (b ~-- 1.2), they can be justified physically in terms of actualistic kinematic quantities. By analyzing the evolution of a single fault, and relating the concepts of seismic rupture length and seismic slip (cf., e.g., Kanamori and Anderson, 1975) to fault length and offset, the following relation can be derived (RanaUi, 1977) Y = a ( 1 +p~, 71(X) ~ X

(9)

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where c~ is the (approximately constant; see King, 1978) ratio between seismic slip and rupture length, p the ratio of tectonic creep to seismic slip, and 71 (X) and 72 (X) are functions inversely related to the fractions of seismic slip and rupture length that do not contribute to an increase in total fault offset and length, respectively. This lack of contribution arises from the fact that different earthquake slips do not necessarily occur along the same segment of the fault, and, since many earthquakes occur along preexisting faults, not all seismic ruptures along a fault increase its overall length. Under reasonable assumptions (cf. Ranalli, 1977), some restrictions can be placed on 71 and 72 (0 < 71 ~< 1 , 0 < 72 ~< 1 ; both are decreasing functions of X). However, only the combined dependence of the ratio 71/72 c a n be obtained from the data. Comparing eqs. (8) and (9) 71(X)/72(X) cc X b -i ~--tnX

(10)

This relation shows that, as the length of a fault increases, seismic activity decreases its relative contribution to fault length more rapidly than to total offset. Noting that the dependence of 71/72 on X is the same as that of dY/dX, the semiquantitative model above is seen to be a particular realization of the allometric growth law. The latter does not deal with a single fault, but with an ensemble of faults whose space-time average it describes by means of an allometric relation between two lognormally distributed stochastic variables. A relative growth ratio larger than unity implies that, over the whole ensemble, the longer the fault, the larger the proportion of tectonic and seismic activity that increases offset rather than length: d Y / d X increases with X at a decreasing rate, which is what should be expected on the basis of intuitive considerations on the geometry of fault systems. (The lack of convergence of offset and its rate of change for x -+ o~ is of no consequence in practice, since fault length has an upper limit of 103 km as an order of magnitude.) It should perhaps be pointed out here that a mechanical model for earthquake occurrence in a given region, which assumes a lognormal distribution of seismic fault length, accounts satisfactorily for the empirical magnitude and moment frequency distributions (Caputo, 1977). This constitutes an indirect confirmation of the present model.

POSSIBLE DEVELOPMENTS OF THE MODEL: CONTINUUM DAMAGE MECHANICS

The purpose of this section is to indicate some lines of thought which may prove fruitful for the further development of a stochastic model of faulting; the intention is merely to draw attention to the possibility of introducing continu-

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ous field quantities related to tectonic parameters such as stress and strain rate. These field quantities should satisfy two conditions: they should be defined in feasible and simple operational terms; and their statistical properties should be derivable from the properties o f known variates. Besides permitting alternative descriptions o f the faulting process, their introduction could lead to useful applications. The following argument is intended to show that this is theoretically possible, and therefore worth pursuing.

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Obviously, classical mechanical failure criteria do not have the required properties-indeed, it was because of the impossibility of giving a description of the ensemble of faults on such a basis that the present model was developed. With a change of scale, the same problem exists in material, science, where the "deterioration" of a medium, caused by a multitude of microscopic defects, is of interest. A useful approach there is afforded by continuum damage mechanics (CDM), which is based on the concept of "damage," a continuous field quantity expressing the deterioration of the medium (cf. Hult, 1979, for a discussion of the principles involved and some examples). The concept of damage is based on the distinction between the area across which an external load is applied (for in-

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stance, the cross section of a rod in tension), and the "net load-carrying area" resulting from the microstructural deterioration of the specimen. It is possible to approach the problem of faulting in an analogous way, with modifications accounting for the complex nature of the crust. Suppose that a tectonic region, the dimensions of which will depend on the scale of the problem considered, is subject to a tectonic stress aT. The tectonic stress is caused by large-scale geodynamic processes (plate tectonics) and for the present purposes can be considered constant; if may be thought of as related to the average tectonic strain rate by a relation iT =f(aT). If A is fault area, let AL be the "net load-carrying area," that is, the part of fault area that carries most of the load, either because of its orientation with respect to the tectonic stress field, or because it is "locked" by geological factors. Damage can then be defined as d ~ = -dAL/A L

that is =lnA/AL

(11)

Fault area A is a lognormal variate because so is fault length X. If it is now assumed that AL is lognormal too (a reasonable assumption in light of the lognormality of seismic fault length; Caputo, 1977), it follows that ~ is a normally distributed variate. Analogously, a lognormally distributed "net stress" can be defined as OL = eT(A/A L ) = aT exp (~2)

(12)

For the distributions N(co) and A(eL) to be derivable from A(x), it is necessary to specify a relation between A and A L , and to assign a constant value to OT. Whereas an educated guess about the latter may come from plate tectonics, there is at present no clear way to solve the former problem satisfactorily. However, the possibility comes to mind of subdividing the region into a number of sectors, and then determining a heuristic relation of the type ALl = 6 iAi for each sector on the basis of geological and seismological information (orientation of faults, locked segments, and so on). Under fairly general conditions, the distribution in each sector will be of the same type as the overall distribution (Aitchison and Brown, 1957); and this procedure would open up the possibility of producing regional maps of the spatial variation of ~2 (and eL) from the spatially averaged mean values in each sector. Such maps could, for instance, integrate seismic risk maps obtained from statistical analysis of past seismicity and pattern recognition analysis. A justification for presenting an idea that is still operationally vague lies in its generality and potential usefulness. In this case, space-averaged representations of damage and related quantities, being independent of scale, could be applied to a wide range of problems, from dislocation density and motion in crystals to mineralized fracture patterns in economic geology. Moreover, it is of

A Stochastic Model for Strike-Slip Faulting

411

course possible, though of little use at present, to specify a time-dependence of local average damage under constant tectonic stress, fZ(t); it is this temporal variation in damage (whether defined in terms of pore water pressure, friction, etc.), and not changes in tectonic stress, that causes many earthquakes. It was felt, therefore, that the topic is sufficiently important to be offered for debate. CONCLUSIONS By considering the ensemble of strike-slip faults in continental crust as the realization of a stochastic process, a model has been given to account for the observed distributions of fault length and offset. Fault length is a stochastic variable with distribution A(xlg, a2). This can be explained by assuming that X is the outcome of a Kolmogorov-type process of fracturing, described by the law of proportionate effect. Fault offset is related to length by an allometric growth law Y aN b, b > 1, and consequently is A(Y[a' +bg, b2o2), a' =ln a. Besides giving a satisfactory fit to empirical data, the model can be interpreted in terms of kinematic geological and seismic source parameters. It is also theoretically possible to give alternate descriptions of the faulting process, by expressing the model in terms of continuous field quantities such as damage fZ and net local stress Oz. There are still practical difficulties with this approach, which is, however, of potentially wide applicability. The model draws upon probabilistic concepts and laws of a very general nature, and of course the lognormal distribution has already found several uses in the earth sciences. Its extension to cover a geodynamic process such as faulting should be taken as an encouragement to seek further applications of the stochastic approach to geological systems that have hitherto been treated as deterministic. =

ACKNOWLEDGMENTS Over the years, discussions with Andrea Fabbri of the Geological Survey of Canada have stimulated the author's interest in unorthodox applications of probability theory to geology. The assistance of Olga Kukal, Department of Biology, Carleton University, has been very valuable. This work was completed while the author was an Alexander von Humboldt Fellow at the University of Frankfurt. Financial assistance was provided by the Natural Sciences and Engineering Research Council of Canada and by the yon Humboldt Foundation. REFERENCES Agterberg, F. P., 1974, Geomathematics: Amsterdam: Elsevier. Aitchison, J. and Brown, J. A. C., 1957, The lognormal distribution: Cambridge, University Press.

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Caputo, M., 1977, A mechanical model for the statistics of earthquakes, magnitude, moment, and fault distribution: Bull. Seism. Soc. Amer., v. 67, p. 849-861. Hult, J., 1979, CDM-Capabilities, limitations, and promises, in Easterling, K. E., ed., Mechanisms of deformation and fracture: Oxford, Pergamon Press, p. 233-247. Huxley, J. S., 1932, Problems of relative growth: New York, Dial Press. Kanamori, H. and Anderson, D. L., 1975, Theoretical basis of some empirical relations in seismology: Bull Seism. Soc. Amer., v. 65, p. 1073-1095. Kapteyn, J. C., 1903, Skew frequency curves in biology and statistics: Groningen, Noordhoff. King, G. C. P., 1978, Geological faults: fracture, creep and strain: Proc. Roy. Soc. London, A-288, p. 197-212. Kolmogorov, A. N., 1941, IJber das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstiickelung: C. R. Acad. Sci. U.R.S.S.v. 31, p. 99-101. Ranalli, G., 1976, Length distribution of strike-slip faults and the process of breakage in continental crust: Can. Jour. Earth Sci., v. 13, p. 704-707. Ranalli, G., 1977, Correlation between length and offset in strike-slip faults: Tectonopbysics, v. 37, T1-T7. Smart, J. S., 1979, Determinism and randomness in fluvial geomorphology: EOS, Trans. Amer. Geophys. Union, v. 60, 651-655. Sole Sugrafies, L., 1978, Alineaciones y fracturas en el sistema Catalan segun las imagenes LANDSAT-I: Tecniterrae, v. 22, p. 1-11. Tchalenko, J. S., 1970, Similarities between shear zones of different magnitudes: Bull. Geol. Soc. Amer., v. 81, p. 1625-1640. Vistelius, A. B., 1960, The skew frequency distribution and the fundamental law of the geochemical processes: Jour. Geol., v. 68, p. 1-22.