Stochastic and fractal methods in coastal morphodynamics
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A stochastic process X is a process consisting of a collection of random time-dependent variables X(t), where t can be discrete or continuous (and may represent a parameter different from time). A realisation of X, also called a sample path, corresponds to a possible outcome of X (Ross, 1983). Continuous stochastic processes may have independent increments or stationary increments. Increments are independent if for an increasing set of times t0, t1, …, tn, the random variables ΔX(ti)=X(ti) - X(ti-1), i = 1, 2, ..., n are independent. Increments are stationary if ΔX(t)=X(t+s) - X(t), has the same distribution for all t. An important class of stochastic processes are counting processes, where X consists of only one random variable, N(t) say, (with t positive), corresponding to the number of events that have occurred up to time t. An example of a counting process is the Poisson process.
When stochastic modeling is used, a prediction becomes a forecast, as the outcome is not deterministic anymore, but instead has a certain probability of occurrence. In coastal models, probabilistic techniques have been used, for instance, to find different outcomes, all equally possible, for different equally possible data sets (that is, independent random processes, as described above). These data sets are constructed by removing one experimental data value form the original dataset and using the reduced dataset to make the predictions. This method is called the jack-knife method, and has been extensively used to make forecasts of bathymetric evolution. Stochastic modeling in coastal morphodynamics may also consist on making forecasts for the statistics of the dynamics rather than forecasts of the dynamics themselves (Reeve and Fleming, 1997). In such approaches, the probability distribution for the wave climate is selected and the evolution of the sea floor or shoreline are calculated for a random selection of wave climates satisfying that distribution. The statistics of the outcome distribution are then determined, and forecasts made on the statistics. Some more detailed examples are presented below.
Applications in coastal modeling
Probabilistic approaches were first applied to coastal research by Vrijling and Meijer (1992), in the context of shoreline evolution. Since this seminal work, several other workers have adopted this framework. For instance, Reeve and Spivack (2004)  analysed as well the evolution of a shoreline under random waves via a one-line model, which was first proposed by Pelnard-Considere (1956) and is described more in detail in Long-term modelling using 1-line models - GENESIS and new extensions. The forcing was assumed to fluctuate, and equations for the mean and the variance of the shoreline position were then deduced. The angle of the breaking wave crests, α, with respect to base profile, and the longshore transport, K, were both assumed to be random functions of time, while the shoreline position was taken as a stochastic variable. In order to be able to find the statistics of the solution, the longshore transport was divided into an ensemble average, ‹K(t)›, and a known fluctuating quantity, δ(t), with zero mean and stationary increments -- note that an ensemble average is an “average taken over all possible realizations” (Reeve and Spivack, 2004). The authors determine several statistical quantities: The mean and the variance of the shoreline positions, the departure of all possible shoreline positions form the mean, and the variance of the morphological forcing, ie the variance of δ(t). The method was applied to hindcasted and transformed wave data, together with the CERC formula to determine the diffusion coefficient.
Another application of stochastic modeling to shoreline dynamics was that by Dong and Chen (1999),  who proposed a model combining long-shore and cross-shore transport effects on the bathymetric evolution of a beach under idealised wave climate conditions. The shoreline variable was assumed to be stochastic. The initial shoreline configuration at any longshore position was assumed to be a sloping plane. Then, a probability distribution of shoreline positions was calculated for a set of idealized wave climates. Following long-term statistical methods, a maximum shoreline recession variable was defined and used as the beach erosion variable. The authors apply the technique to shoreline evolution near an impermeable groyne on a straight beach, but the method can potentially be applied to different scenarios, for instance to long-term scour around structures. However, it is important that longshore and cross-shore transport do not interact strongly, as these interactions are neglected in the model. In fact, a further simplification of the model was later proposed where the longshore and cross-shore transport are assumes to be statistically independent (Dong and Chen 2000).
The effects of shoreface geometry and substrate composition on long-term coastal erosion constitute one of the issues most recently analysed with probability theory (Cowell et al. 2006),  in order to assess the impact of climate change. As these authors indicate, climate change is likely to influence not only the sea level, but also the wave, current and sediment dynamics. Also, a probability approach should consider as well probable fluctuations on the climate itself, and a probability of occurrence of relevant processes on decadal time scales. In contrast with the examples mentioned previously, here the authors concentrate on bathymetric profile evolution. A model is set up which includes time-dependent substrate properties, cross-shore sediment transport with source (and sink) terms, and shoreface geometry. The approach here is different too: to account for uncertainty in the processes, probability distributions of the variables are taken, but the properties of the probability distribution pertaining to a deterministic model are based on behaviour-based estimates rather that taken from process based approaches. The same applies to relevant parameters, which are estimated by extrapolation from laboratory or field studies. However, the procedure to calculate the probability distribution of quantities of interest (ie beach recession, morphological change, etc) is the same as for the examples previously described: a set of possible variate values is chosen randomly from the probability distributions of these variates and an outcome is computed for each of these sets, allowing for the determination of a probability distribution of the outcomes. In statistics, this procedure is usually called a Monte Carlo simulation. It is noteworthy that allowing for time-dependent substrate properties and shoreface geometry increases the numerical complexities, as it removes the equilibrium shoreface assumption adopted, for instance, in Cowell et al (1995). However, the possibility of studying very general shoreface geometries and substrate compositions and distributions allows for important advances in the understanding of beach and bathymetry evolution due to sea level changes.
Fractal Dimension Methods
What do we mean by fractal methods? Here we define them as those methods which characterize the fractal properties of a coastal system. Such methods calculate a metric dimension, of the phenomenon being observed. Well known examples of metric dimensions are the Lyapunov exponents or the Hurst exponents. Lyapunov exponents allow to determine the stable and the unstable directions of the space, that is, the directions in which the system is either squeezed (or folded) or stretched, respectively. Positive Lyapunov exponents indicate the directions in which the system is explanding, whereas negative Lyapunov exponents relate to the directions corresponding to squeezing (Gilmore and Lefranc, 2003). The Hurst exponent, on the other hand, constitutes a measure of self-affinity.
The concept of Hurst exponent and the process of Brownian motion are intrinsically related. Brownian motion is the random walk that very small particles are subject to, and was first observed by Robert Brown in 1872 when experimenting with pollen suspended on a liquid surface. In standard Brownian motion, or Wiener process, the probability for a particle to move from a position x to a position x+ Δx in a time step Δt is (following notation of Kantz and Schreiber, 1997)
with , the variance of the position increments, being proportional to the square root of the time step of the observations, that is .
Now, not all particle motions are as simple as those described above. Even when the particles may be weightless, they still can display a variety of movements different from Brownian. However, some of these motions may still be described by a probability distribution as given above, but the variance of the increments may have a more general expression, . where the constant H is called the Hurst exponent, after Harold Hurst (1990-1978). Such types of motion are termed anomalous diffusion processes (Kantz and Schreiber, 1997; J. C. Sprott, 2003), because when the particles have the tendency to continue in the same direction they were going in the previous time step, diffusion is enhanced. In this case the process is nonstationary, as the variance increases without bounds as increases. When however, the particles have the tendency to change the direction of their trajectories, thus reducing diffusion. The first type of motion is called Lévy flight, or persistent behaviour, and produce anomalous diffusion; the latter is characteristic of antipersistent behaviour. As indicated by Sprott (2003), the Hurst exponent is a measure of fractality, and it is with this property in mind that it has been used in coastal studies (see Southgate and Moller, 2000 ).
Applications to coastal morphodynamics
Although Reeve (2002)  recognised the potential importance of chaos in the evolution of beaches, very few attempts have been made so far to quantify the chaotic properties of coastal systems. In particular, the application of Hurst exponents to coastal morphodynamics has been limited up to date, and has related to the analysis of the bathymetry dynamics at Duck Site, North Carolina, USA. This data has been extensively used for many studies for several reasons, both practical and scientific. These reasons include: a) data is available to the public through a web portal; b) the data extends from the year 1981 up to our days, and c) systematic measurements have been performed at least every three months for at least 10 transects (profiles for which the long-shore component is constant), and for at least 4 transects cross-shore measurements have been performed from the beach to at least 500 meters, that is, beyond the depth of closure (which is about 7 meters). Also, wave climate measurements have been performed continuously and from 1981 too, although only at one position (by the end of the pier present at Duck). Therefore, as indicated by Southgate et al. (2003)), this data set is an ideal test case for fractal analysis, as it allows to analyse the slow time variations of this spatially extended system, for which fractal methods are particularly suitable. Southgate and Moller (2000) determined the Hurst exponent following various equivalent methods and divided the cross-shore profiles according to their self-organisational properties. They also determined periods when these properties were dominant, which appeared to be correlated to time intervals when there were not many storms. Thus, when wave conditions were weak or moderate, self-organisational (internal) processes determined the dynamics of the sea floor, but the waves determined the behaviour of the sea floor when the waves were strong. Fractal methods have also been used to analyse the self-organisational behaviour of shorelines, both by Southgate and Moller (2000) for the shoreline at Duck, and by Southgate and Beltran (1996) for the shoreline evolution of a Lincolnshire beach (in the UK). It is worth noting here that Reeve et al. (1999) used a different nonlinear method to classify beaches; the classification was based on the intrinsic properties of the beach, which could be chaotic, stochastic, fully wave-dependent. Reeve et al. (1999) characterised three shorelines according to the values of two parameters, which were extracted from a singular component analysis. The first parameter measured the overall variability of the data while the second measured the importance of the second element of the singular value decomposition with respect to the first (where the elements in the diagonal of the singular value decomposition matrix have been ordered in descending order, without loss of generality). The shorelines analyses where: Lincolnshire, Rye and Milford-on-Sea. No comparison between the conclusions from Reeve et al. (1999) Southgate and Beltran (1996) and for the behaviour of Lincolnshire has been performed, at least to the knowledge of this author. The main handicap of the Hurst exponent method presented above, shared in general by all metric methods, is that a large dataset is usually necessary. Southgate and Moller (2000) and Southgate et al. (2003) claim Hurst exponent computations only require a few hundred data values to obtain reliable results, but they give no physical evidence to support this statement. Also, as noted by Lefranc et al. (1992), metric methods are not very robust and there are many sources of error and bias.
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