The incorporation of fractal concepts into the natural sciences, especially physics, has been swift. This growth has also occurred, albeit at a much slower pace, in the earth sciences, including geomorphology, sedimentology, stratigraphy, and petroleum geology (Korvin, 1992; Turcotte, 1992, 1994a; Barton and LaPointe 1995a,b). Nevertheless, many earth scientists remain unfamiliar with fractal concepts and applications and are perhaps even suspicious that it is a fad. In this chapter we will demonstrate that fractal models and methods allow the geologist to quantify many concepts that have long been intuitive, while also providing new and fruitful ways of looking at data. In addition, as we will show in later chapters, fractals are important in the interpretation of chaos and nonlinear dynamics. We believe that fractal methods will eventually become part of the standard toolkit of any quantitatively oriented geologist, to the same extent that calculus or statistics are currently.

In the following discussion we will use the term “model” as a short form of “mathematical model”, that is, a set of equations that are designed to represent some aspect of the real world. We recognize that other types of models are possible, e.g., conceptual models that cannot be exactly quantified, or physical models, that are generally models built of real materials, but at a smaller scale than the real phenomenon. We can recognize two different extreme types of mathematical models.

As discussed in Chapter 1, time series whose power spectra exhibit 1/∫ dependency (1/∫, flicker, or pink noise) have long been known to be ubiquitous in natural and experimental systems (Mandelbrot and Wallis, 1968, 1969a, b; Schroeder, 1991). For example, this behavior has been recognized in turbulent velocity fluctuation as measured in laboratory flumes (Nordin et al., 1972; Mollo-Christensen, 1973; Nowell, 1978). Mandelbrot and Wallis (1968, 1969a, b) demonstrated that fractional Brownian motions and discrete fractional Gaussian noises provide mathematical models for these 1/∫ noises (see Chapter 1). In addition, the last few years have witnessed renewed attempts to produce general physical models, such as self-organized criticality (SOC), to explain this behavior (Bak and Chen, 1989; Chapter 2, this volume). In addition to spectral analysis, other techniques, such as rescaled range, autocorrelation, and geostatistics have been applied to the analysis of fractal series. In this chapter we describe these methods and illustrate their use in the analysis of real and synthetic data sets with fractal structure.

Most methods of analysing time series carry out the numerical analysis on the time series itself. This is because the investigator is basically interested in the variable x = f ( t ) that has actually been measured. But an alternative approach is to suppose that the variable that was measured is merely one of several that might be measured as output from a dynamical system, which might indeed be better characterized using some other variable. For example, fluid converting in a box might be characterized by a probe measuring temperature or velocity at one point in the box, or by some other measurement, as a function of time. If the governing equations for the system are not known, it is also not possible to identify the fundamental variables which should be measured, or even how many of them there are. In this case, one might think that it is impossible to use measurements made on a single, arbitrarily chosen variable to reconstruct any important properties of the full multidimensional system. As we have seen, it is generally not even clear from simply examining the output x , whether or not the system producing x is stochastic or a nonlinear dynamical system of relatively low dimension. But if, in fact, the signal is a product of a low dimension deterministic system then it turns out that it is possible to reconstruct all the major topological properties of the system, by a technique known as embedding. The basic idea seems to have been discovered independently

Science is based on the principle of repeatability: each time a system experiences similar conditions—both internal to the system and forces exerted externally on the system—we expect the system to exhibit a similar response. Forecasting exploits this principle by using the observed behavior of a system to predict behavior when similar conditions recur. Even if the equations describing a system are unknown, we can nevertheless use forecasting to learn about the system. For some purposes— such as weather forecasting, financial forecasting, or noise reduction—predicting the future is the primary goal of the forecasting. For the purpose of characterizing system dynamics, in contrast, predictions are made in an exploratory manner to learn what kinds of models perform best. For a preview of how the forecasting procedure works, we can consider the Lorenz system described in Chapter 2. Three approaches could be used to predict the future of this 85-system. First, we could measure the initial conditions (nonlinearity of vertical temperature gradient, temperature difference between rising and falling fluid, and intensity of convection) and use the three coupled equations (Equations 2.13) to predict the values of the three variables for successive steps in time. A second approach could be employed if the governing equations were unknown, but sequential observations of the system were available. We could use the sequential observations to plot the 3-dimensional attractor (Figure 2.1), locate each predictee (a point whose three coordinates are given by the three variables that define the state of the system), identify nearby points on

Abstract The intention of these notes is to provide sedimentary geologists with an introduction to the new techniques for analyzing experimental and observational data provided by the rapid development of those disciplines generally known as Fractals and Nonlinear Dynamics (chaos theory). A general introduction to a minimum of theory is given, but most of the space is devoted to show how these ideas are useful for interpreting sedimentary data. The main applications are likely to be time series or spatial profiles or two-dimensional maps or images. Sedimentary geologists deal every day with actual time series, such as measurements of current velocity or suspended concentration at a station, or with virtual time series, such as stratigraphic sections, well logs, or topographic profiles yet few geologists know much about the new numerical techniques available to analyze such data.

Abstract Some knowledge of fluid dynamics is essential to an adequate understanding of sediment movement. We’re not assuming any previous knowledge on your part. Chapters 1, 3, and 5 present a very selective treatment of some of the topics in fluid dynamics that are important in the study of sediment movement. This material is not a substitute for a more substantial background in fluid flow, but it allows a level of discussion of many of the important ideas in the mechanics of sediment transport that would not otherwise be possible. Fluids are substances that deform continuously and permanently when subjected to forces that vary in magnitude or direction from point to point. The nature of the relationship between the deforming forces and the geometry of deformation varies from fluid to fluid; as discussed in this chapter, the relationship is a simple linear one for air and water, the two fluids most important in sedimentology Liquids and gases differ greatly in their structure on the molecular scale. How is it that the macroscopic motions of these two kinds of fluid need not be considered separately? The answer is that fluids can be treated as if they were continua--as if their constituent matter, which is actually distributed discontinuously as atoms and molecules, were smeared uniformly throughout space. The justification for this approach is that it works extremely well for fluid flows on scales that are much larger than the intermolecular spacing. This includes most problems in sediment movement down to the range of colloida1 sizes (fractions of a micrometer). In this chapter we’ll develop some ideas in fluid mechanics by looking at three kinds of flow: shearing of fluid between parallel plates, flow past a sphere, and flow down an inclined plane. The last two are of direct importance to sedimentology, and we’ll build upon the results in later chapters.

Abstract This chapter deals with some basic ideas about settling of sediment particles through fluids. This is a good topic in sediment transport to start with in these notes, because settling is an important aspect of sediment transport, and Chapter 1 has provided enough background for substantial progress. But complexities that require greater understanding of fluid flow will soon arise, and Chapter 3 is therefore devoted to several important topics in fluid flow. Chapter 4 is a continuation of material on settling. If placed in suspension in a viscous fluid, a sediment particle will settle toward the lower boundary of the fluid, provided that the weight of the particle is not much smaller than the random forces exerted on the particle by bombardment by the fluid molecules in thermal motion. All mineral particles larger than colloidal sizes of hundredths of a micrometer are in this category. When such a particle is released from rest in a still fluid, it accelerates in response to the force of gravity, but as its velocity increases, the oppositely directed drag force exerted by the fluid grows until it equals the weight of the particle. When the weight and the drag are in balance the particle no longer accelerates but falls at its terminal velocity, called the fall velocity or settling velocity. Particles of sand size and smaller attain terminal velocity over very short times and distances. With respect to natural sedimentary environments, the settling of a sphere in a still fluid is obviously a great oversimplification with respect both to particle shape and to the state of motion of the fluid, but it will lead to development of some important ideas and point the way toward consideration of nonspherical particles and flowing fluids.

Abstract So far we’ve been able to cover a lot of ground with a minimum of material on fluid flow. At this point we need to present some more topics in fluid dynamics before returning in the next chapter to flow past spheres at Reynolds numbers higher than the Stokes range. We’ll look at inviscid fluid flow, the Bernoulli equation, turbulence, boundary layers, and flow separation. This material will also provide some of the necessary background for discussion of dynamics of sediment movement in Chapter 6.

Abstract In Chapter 2 we outlined the basic law for settling of a spherical grain through a still, viscous fluid. This was done by establishing, partly on theoretical grounds but mainly from dimensional analysis and experimentation, a relationship between drag coefficient and Reynolds number for a sphere moving through a viscous fluid. it was seen that the nature of the relationship changes as the Reynolds number increases, corresponding to a change from a viscous regime of flow to a regime in which the motion of the particle results in the formation of a turbulent wake. In this chapter we will examine a little more closely the phenomenon of wake formation, and then consider the modifications of settling behavior that arise from changes in shape and concentration of the settling grains.

Abstract Almost all flows that transport sediment are turbulent. We’ve mentioned turbulence at several points already and devoted an introductory section in Chapter 3 to it; this chapter deals in more detail with some aspects of turbulent flow most relevant to sediment transport. We’ll concentrate on two related problems: the nature of the resistance force exerted on the turbulent flow by the boundary, and the profile of time-average velocity along a line normal to the boundary. With turbulent flow it’s not possible to solve the equations of motion to obtain exact solutions for such things as boundary resistance or velocity profiles. The reason for this is basically similar to, although more general than, the reason why in Chapter 1 we weren’t able to obtain a solution for turbulent flow down a plane: we know what equations we have to solve but we can’t solve them because of the uncertainty that turbulence introduces into the application of these equations. The great number of equations to be found in textbooks and papers on turbulent flow are semi-empirical: the general form of the equation may be suggested by physical reasoning, but the numerical constants in the equation, and therefore its specific form, must be found from experiments. And in many cases not even the general form of the equation is known, and the curve must be obtained entirely by experiment.