A large number of new results on fractals, chaos, and self-organized criticality applied to problems in geology and geophysics have appeared since the first edition of this book was published in 1992. Evidence for this comes from the large number of new references included in this edition, increasing from 160 to over 500. Because of the rapid advances in knowledge it became evident shortly after the publication of the first edition that a second edition would be required in a few years.

A large number of additions and some deletions have been made in preparing this second edition. In Chapter 2 a comprehensive treatment of the completeness of the sedimentary record has been used to introduce the application of fractal techniques to geological problems. To make this textbook more complete, a brief introduction to probability and statistics has been included in Chapter 3. Chapter 4 on seismicity and tectonics has been extensively revised to include the work that has been recently carried out on the spatial distributions of earthquakes and fractures. In Chapter 5 the elegant work by Claude Allègre and his associates explaining power-law (fractal) distributions of mineral deposits has been added.

A major addition to the second edition is the comprehensive treatment of multifractals in Chapter 6. Also added to this chapter on fractal clustering are pair-correlation techniques and lacunarity. One of the most extensive revisions concerns the treatment of self-affine fractals in Chapter 7.

Chaos

The concept of deterministic chaos is a major revolution in continuum mechanics (Bergé et al., 1986). Its implications may turn out to be equivalent to the impact of quantum mechanics on atomic and molecular physics. Solutions to problems in solid and fluid mechanics have generally been thought to be deterministic. If initial and boundary conditions on a region are specified, then the time evolution of the solution is completely determined. This is in fact the case for linear equations such as the Laplace equation, the heat conduction equation, and the wave equation.

However, the problem of fluid turbulence has remained one of the major unsolved problems in physics. Turbulent flows govern the behavior of the oceans and atmosphere. The appropriate Navier–Stokes equations can be written down, but solutions yielding fully developed turbulence cannot be obtained. It is necessary to treat turbulent flows statistically and to carry out spectral analyses.

The concept of deterministic chaos bridges the gap between stable deterministic solutions to equations and deterministic solutions that are unstable to infinitesimal disturbances. Chaotic solutions must also be treated statistically; they evolve in time with exponential sensitivity to initial conditions. A deterministic solution is defined to be chaotic if two solutions that initially differ by a small amount diverge exponentially as they evolve in time. The evolving solutions are predictable only in a statistical sense. A necessary condition that a solution be chaotic is that the governing equations be nonlinear.

ATTRACTOR A point in phase space toward which a time history evolves as transients die out.

BASIN OF ATTRACTION Some dynamical systems have more than one fixed point. The region in phase space in which solutions approach a particular fixed point is known as the basin of attraction of that fixed point. The boundaries of a basin of attraction are often fractal.

BIFURCATION A change in the dynamical behavior of a system when a parameter is varied.

BROWNIAN WALK The running sum of a sequence of random values usually obtained from a normal distribution.

CANTOR DUST A fractal set generated by subdividing a line into parts.

CHAOS Solutions to deterministic equations are chaotic if adjacent solutions diverge exponentially in phase space; this requires a positive Lyapunov exponent.

CLUSTER A group of particles with nearest-neighbor links to other particles in the cluster.

DETERMINISTIC A dynamical system whose equations and initial conditions are fully specified and are not stochastic or random.

DIFFERENCE EQUATION An equation that relates a value of a function xn+1 to a previous value xn. A difference equation generates a discrete set of values of the function x.

DIFFUSION-LIMITED AGGREGATION (DLA) Diffusing (random-walking) particles accrete to a seed particle to form a dendritic structure.

DIMENSION The usual definition of dimension is the topological dimension. The dimension of a point is zero, of a line is one, of a square is two, of a cube is three. In this book we have introduced the concept of fractional (noninteger) dimensions, or fractals.

The earth's magnetic field is attributed to the electrically conducting outer core, which acts as a dynamo. The liquid outer core is primarily composed of iron, which is an excellent electrical conductor at core conditions. Electrical currents in the core generate a magnetic field. Buoyancy forces in the core, due to either temperature or composition, drive a fluid flow. The flowing electrical conductor in the magnetic field induces an electric field. This is a self-excited dynamo.

Paleomagnetism is the study of the earth's past magnetic field from the records preserved in magnetized rocks. Rocks containing small amounts of ferromagnetic minerals such as magnetite and hematite can acquire a weak permanent magnetism when they are formed. This fossil magnetism in a rock is referred to as natural remanent magnetism.

Many volcanic rocks at the surface of the earth can be magnetized because of the presence of minerals such as magnetite. When these volcanic rocks were cooled through the Curie temperature, they acquired a permanent magnetism from the earth's field at the time of cooling. Paleomagnetic studies of remanent magnetism have provided a variety of remarkable conclusions. These studies have traced the movement of the rocks due to plate tectonics and continental drift over periods of hundreds of millions of years. They have shown that the magnetic field at the surface of the earth has been primarily a dipole, as it is today, and has remained nearly aligned to the earth's axis of rotation.

Renormalization

In the first eight chapters of this book we considered the fractal behavior of natural systems. This behavior was generally statistical and the physical causes were generally inaccessible. In the six chapters that followed we considered low-order dynamical systems that exhibit chaotic behavior. Because of the low order, the examples are generally quite far removed from natural systems of interest. In this chapter and the next we take a collective view of natural phenomena and consider some applications in geology and geophysics.

Thermodynamics represents the standard approach to collective phenomena. System variables are defined, that is, temperature, pressure, density, entropy; and the evolution of these variables is determined from the first law of thermodynamics (conservation of energy) and the second law of thermodynamics (variation of entropy). Statistical mechanics provides the rational microscopic basis for much of thermodynamics.

In general, neither thermodynamics nor statistical mechanics yields fractal statistics or chaotic behavior. Exceptions include critical points and phase changes. A characteristic feature of a phase change is a discontinuous (catastrophic) change of macroscopic parameters of the system under a continuous change in the system's state variables. For example, when water freezes its viscosity changes from a very small value to a very large value with no change in temperature.

The renormalization group method has been used successfully in treating a variety of phase change and critical-point problems (Wilson and Kagut, 1974). This method often produces fractal statistics and explicitly utilizes scale invariance.

The scale invariance of geological phenomena is one of the first concepts taught to a student of geology. It is pointed out that an object that defines the scale, i.e., a coin, a rock hammer, a person, must be included whenever a photograph of a geological feature is taken. Without the scale it is often impossible to determine whether the photograph covers 10 cm or 10 km. For example, self-similar folds occur over this range of scales. Another example would be an aerial photograph of a rocky coastline. Without an object with a characteristic dimension, such as a tree or house, the elevation of the photograph cannot be determined.

Sand-pile models

In the last chapter we considered the renormalization group method for treating large interactive systems. By assuming scale invariance a relatively small system could be scaled upward to a large interactive system. The approach is often applicable to systems that have critical point phenomena. In this chapter we consider the alternative approach to large interactive systems. This approach is called self-organized criticality. A system is said to be in a state of self-organized critically if it is maintained near a critical point (Bak et al., 1988). According to this concept a natural system is in a marginally stable state; when perturbed from this state it will evolve naturally back to the state of marginal stability. In the critical state there is no longer a natural length scale so that fractal statistics are applicable.

The simplest physical model for self-organized criticality is a sand pile. Consider a pile of sand on a circular table. Grains of sand are randomly dropped on the pile until the slope of the pile reaches the critical angle of repose. This is the maximum slope that a granular material can maintain without additional grains sliding down the slope. One hypothesis for the behavior of the sand pile would be that individual grains could be added until the slope is everywhere at an angle of repose. Additional grains would then simply slide down the slope. This is not what happens. The sand pile never reaches the hypothetical critical state.

Seismicity

A variety of tectonic processes are responsible for the creation of topography. These include discontinuous processes such as displacements on faults and continuous processes such as folding. Tectonic processes are extremely complex but they satisfy fractal statistics. Earthquakes are of particular concern because of the serious hazard they present; earthquakes also satisfy fractal statistics in a variety of ways. Seismicity is a classic example of a complex phenomenon that can be quantified using fractal concepts (Turcotte, 1993, 1994a, 1995).

According to plate tectonic theory, crustal deformation takes place at the boundaries between the major surface plates. In the idealized plate tectonic model plate boundaries are spreading centers (ocean ridges), subduction zones (ocean trenches), and transform faults (such as the San Andreas fault in California). Relative displacements at subduction zones and transform faults would occur on well-defined faults. Displacements across these faults would be associated with great earthquakes such as the 1906 San Francisco earthquake. However, crustal deformation is more complex and is usually associated with relatively broad zones of deformation. Take the western United States as an example: Although the San Andreas fault is the primary boundary between the Pacific and the North American plates, significant deformation takes place as far east as the Wasatch Front in Utah and the Rio Grande Graben in New Mexico. Active tectonics is occurring throughout the western United States. Distributed seismicity is associated with this mountain building. Even the displacements associated with the San Andreas fault system are distributed over many faults.

Drainage networks

In the previous chapters we concluded that landscapes generally obey fractal statistics. Analyses of shore lines and topographic contours using ruler or box-counting methods provide statistical correlations with the self-similar fractal relation (2.1). Spectral studies of topography and bathymetry correlate well with the self-affine fractal relation (7.41). We will show in this chapter that drainage networks are fractal trees. Landforms evolve as a result of the tectonic processes that produce them and the erosional processes that destroy them. Landforms are a classic example of a complex phenomenon that can be quantified using fractal concepts (Turcotte 1993, 1994b, 1995).

We will begin our study of geomorphic processes by considering drainage networks. Drainage networks are a universal feature of landscapes on the earth. Over a large fraction of land areas, water eventually flows into an ocean. The surface area that drains into an ocean through a river defines the drainage basin of that river. The streams and rivers that drain into the Mississippi River and eventually drain into the Gulf of Mexico define the Mississippi River basin. Small streams merge to form larger streams, large streams merge to form rivers, and so forth. A typical example of a drainage network is given in Figure 8.1, from the Volfe and Bell Canyons in the San Gabriel Mountains near Glendora, California. We will show that drainage networks are classic examples of fractal trees (Tarboton et al., 1988; Beer and Borgas, 1993; Garcia-Ruiz and Otalora, 1992).

Ore-enrichment models

Statistical treatments of ore grade and tonnage for economic ore deposits have provided a basis for estimating ore reserves. The objective is to determine the tonnage of ore with grades above a specified value. The grade is defined as the ratio of the mass of the mineral extracted to the mass of the ore. Evaluations can be made on either a global or a regional basis. Much of the original work on this problem was carried out by Lasky (1950). He argued that ore grade and tonnage obey log-normal distributions.

Other authors, however, have suggested that a linear relation is obtained if the logarithm of the tonnage of ore with grades above a specified value is plotted against the logarithm of the grade. The latter is a fractal relation. A fractal relation would be expected if the concentration mechanism is scale invariant. Many different mechanisms are responsible for the concentrations of minerals that lead to economic ore deposits. Probably the most widely applicable mechanisms are associated with hydrothermal circulations.

We first consider two simple models that illustrate the log-normal and power-law distributions for tonnage versus grade. De Wijs (1951, 1953) proposed the model for mineral concentration that is illustrated in Figure 5.1 (a). In this model an original mass of rock M0 is divided into two equal parts each with a mass M1 = M0/2. The original mass of the rock has a mean mineral concentration C0, which is the ratio of the mass of mineral to mass of rock.

The Lorenz equations are a low-order expansion of the full equations applicable to thermal convection in a fluid layer heated from below. For the range of parameters in which chaotic behavior is obtained, the low-order expansion is not valid; higher-order terms should be retained. Nevertheless, the chaotic behavior of the low-order analog is taken as a strong indication that the full equations will also yield chaotic solutions. Numerical solutions of the full equations are strongly time dependent for high Rayleigh number flows; these solutions appear to be turbulent or chaotic.

It is generally accepted that thermal convection is the primary means of heat transport in the earth's mantle. Heat is produced in the mantle due to the decay of the radioactive isotopes of uranium, thorium, and potassium. Heat is also lost due to the cooling of the earth. The surface plates of plate tectonics are the thermal boundary layers of mantle convection cells. The plates are created by ascending mantle flows at ocean ridges. The plates become gravitationally unstable and founder into the mantle at ocean trenches (subduction zones). Intraplate hot spots such as Hawaii are attributed to mantle plumes that ascend from the hot unstable thermal boundary layer at the base of the convection mantle.

An important question with regard to the earth is whether mantle convection is chaotic. The earth's solid mantle behaves as a fluid on geological time scales because of thermally activated creep.

Background

To illustrate how fractal distributions are applicable to real data sets, we consider fragmentation. Fragmentation plays an important role in a variety of geological phenomena. The earth's crust is fragmented by tectonic processes involving faults, fractures, and joint sets. Rocks are further fragmented by weathering processes. Rocks are also fragmented by explosive processes, both natural and man made. Volcanic eruptions are an example of a natural explosive process. Impacts produce fragmented ejecta. Although fragmentation is of considerable economic importance and many experimental, numerical, and theoretical studies have been carried out on fragmentation, relatively little progress has been made in developing comprehensive theories of fragmentation. A primary reason is that fragmentation involves the initiation and propagation of fractures. Fracture propagation is a highly nonlinear process requiring complex models for even the simplest configuration. Fragmentation involves the interaction between fractures over a wide range of scales. Fragmentation phenomena have been discussed by Grady and Kipp (1987) and Clark (1987). If fragments are produced with a wide range of sizes and if natural scales are not associated with either the fragmented material or the fragmentation process, fractal distributions of number versus size would seem to be expected. Some fractal aspects of fragmentation have been considered by Turcotte (1986a).

Probability and statistics

Clearly the distribution of fragment sizes is a statistical problem. This will also be the case in other applications we will consider. Thus at this point it is appropriate to introduce some of the fundamental concepts of probability and statistics.