Nonlinear Analysis of Unevenly Sampled Shoreline Position Signals

Time series signals for Mean High Water shoreline positions were generated by sampling cross-shore profiles surveyed over the past twenty years by the Army Corps of Engineers at Duck, NC. Because the surveying frequency was uneven, these signals are not amenable to traditional time series analysis. Instead, we have used two different approaches to study these data: (1) fractal measures of scaling and persistence using spectral analysis techniques for unevenly sampled data, and (2) stochastic modeling with Monte Carlo methods. In stochastic modeling and Monte Carlo Methods, the time series are statistically modeled based upon cumulative distribution functions on the probability of specific occurrences in the field data. First, we attempted to synthesize the signal with a probability of 0.5 (coin flip) for erosion or accretion events. A second set of analyses used a randomly selected probability conditioned by a cummulative density function for the magnitude of erosional and accretional events observed in the field data for all occurrences of the two discrete event possibilities. Exhaustive runs of sequences from the Matsumoto Merseme Twister random number generator were inadequate in with this simple first order constraint on the magnitude of change in a "Brownian" walk. This, along with tightly clustered positive sequential correlation exhibited from up to 8 order lag plots of the data, implies that this signal is neither random noise nor "Brownian", but appears to have memory and is self-organized with persistence. Analysis of the data also reveals trend runs of varying length in either direction (erosion or accretion). These run lengths ranged from 0 (where subsequent shoreline movement changes direction ) up to 7 sequential surveys with movement in the same direction. In addition, functions for the magnitudes of individual increments of erosion and accretion and cumulative magnitudes of change correlate with run lengths. The run length constraints add persistence to the model producing synthetic time signals statistically identical to the field data. Application of Quasi-Monte Carlo methods yield even better fits to the observed data. The synthetic time series with evenly spaced values can be analyzed for nonlinear behavior and fractal dimensions.