ABSTRACT The Mars Exploration Rover Opportunity has investigated bedrock outcrops exposed in several craters at Meridiani Planum, Mars, in an effort to better understand the role of surface processes in its geologic history. Opportunity has recently completed its observations of Victoria crater, which is 750 m in diameter and exposes cliffs up to ∼15 m high. The plains surrounding Victoria crater are ∼10 m higher in elevation than those surrounding the previously explored Endurance crater, indicating that the Victoria crater exposes a stratigraphically higher section than does the Endurance crater; however, Victoria strata overlap in elevation with the rocks exposed at the Erebus crater. Victoria crater has a well-developed geomorphic pattern of promontories and embayments that define the crater wall and that reveal thick bedsets (3–7 m) of large-scale cross-bedding, interpreted as fossil eolian dunes. Opportunity was able to drive into the crater at Duck Bay, located on the western margin of Victoria crater. Data from the Microscopic Imager and Panoramic Camera reveal details about the structures, textures, and depositional and diagenetic events that influenced the Victoria bedrock. A lithostratigraphic subdivision of bedrock units was enabled by the presence of a light-toned band that lines much of the upper rim of the crater. In ascending order, three stratigraphic units are named Lyell, Smith, and Steno; Smith is the light-toned band. In the Reference Section exposed along the ingress path at Duck Bay, Smith is interpreted to represent a zone of diagenetic recrystallization; however, its upper contact also coincides with a primary erosional surface. Elsewhere in the crater the diagenetic band crosscuts the physical stratigraphy. Correlation with strata present at nearby promontory Cape Verde indicates that there is an erosional surface at the base of the cliff face that corresponds to the erosional contact below Steno. The erosional contact at the base of Cape Verde lies at a lower elevation, but within the same plane as the contact below Steno, which indicates that the material above the erosional contact was built on significant depositional paleotopography. The eolian dune forms exposed in Duck Bay and Cape Verde, combined with the geometry of the erosional surface, indicate that these outcrops may be part of a larger-scale draa architecture. This insight is possible only as a result of the larger-scale exposures at Victoria crater, which significantly exceed the more limited exposures at the Erebus, Endurance, and Eagle craters.

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.