Detecting nonlinearity in run-up on a natural beach

Abstract

Abstract. Natural geophysical timeseries bear the signature of a number of complex, possibly inseparable, and generally unknown combination of linear, stable non-linear and chaotic processes. Quantifying the relative contribution of, in particular, the non-linear components will allow improved modelling and prediction of natural systems, or at least define some limitations on predictability. However, difficulties arise; for example, in cases where the series are naturally cyclic (e.g. water waves), it is most unclear how this cyclic behaviour impacts on the techniques commonly used to detect the nonlinear behaviour in other fields. Here a non-linear autoregressive forecasting technique which has had success in demonstrating nonlinearity in non-cyclical geophysical timeseries, is applied to a timeseries generated by videoing the waterline on a natural beach (run-up), which has some irregular oscillatory behaviour that is in part induced by the incoming wave field. In such cases, the deterministic shape of each run-up cycle has a strong influence on forecasting results, causing questionable results at small (within a cycle) prediction distances. However, the technique can clearly differentiate between random surrogate series and natural timeseries at larger prediction distances (greater than one cycle). Therefore it was possible to clearly identify nonlinearity in the relationship between observed run-up cycles in that a local autoregressive model was more adept at predicting run-up cycles than a global one. Results suggest that despite forcing from waves impacting on the beach, each run-up cycle evolves somewhat independently, depending on a non-linear interaction with previous run-up cycles. More generally, a key outcome of the study is that oscillatory data provide a similar challenge to differentiating chaotic signals from correlated noise in that the deterministic shape causes an additional source of autocorrelation which in turn influences the predictability at small forecasting distances.

Observations of nonlinear runup patterns on plane and rhythmic beach morphology

“…(The results reported here are not sensitive to the lag time choice, and Δ t = 1 gives very similar results; however, the larger lag times increase computational speed dramatically). Disregarding the results where d < 12 s (the runup period), which are driven by variability of the shape of the swash (see Bryan and Coco [2007]), a plaquette size m of four is optimal in the bays at high tide, which equates to the plaquette spanning 4Δ t = 40 points or 20 s, which is larger than a swash cycle (approximately two swash cycles). At the horns, the plaquette spans ∼25 s, which is slightly more than two swash cycles.…”

“…Although nonlinear forecasting has been used for a wide range of time series, including many examples of natural time series, the extension to naturally cyclic series is not straightforward, since it is necessary to control for the confounding effect of the autoregressive behavior which results from the shape of the cycle [ Bryan and Coco , 2007]. A forecast in which the prediction distance d is less than T , where T is the wave period, will be better forecast with a smaller k , because there is enough information in the plaquette or sequence x ( t − j Δ t ) on the evolution of that cycle to train the model by finding similar cycles from the training data set (if the plaquette contains a small trough, the training data set will indicate that a small crest is likely to follow).…”

“…A forecast in which the prediction distance d is less than T , where T is the wave period, will be better forecast with a smaller k , because there is enough information in the plaquette or sequence x ( t − j Δ t ) on the evolution of that cycle to train the model by finding similar cycles from the training data set (if the plaquette contains a small trough, the training data set will indicate that a small crest is likely to follow). In this case, local behavior in the series (although still useful information) cannot be attributed to a nonlinear process but simply to the degree of random variability in wave cycles in the time series [ Bryan and Coco , 2007]. Conversely, in the case of a forecast in which d > T , the plaquette contains no information from the cycle to be predicted, and so the local model will only be better than the global model if there are local sequences of wave cycles that are repeated in the training data set (for example, a sequence of two small crests and then a large crest).…”

“…The last case was included to artificially induce localness in the time series. Bryan and Coco [2007] show other types of synthetic tests designed to test whether the results were significant relative to a time series with cycles of similar parabolic shape, but with random ordering of swash cycles (following recommendations for autocorrelated time series, e.g., Small and Tse [2002]).…”

“…A technique that has been developed to assess the relative contribution of the global versus the local properties in a time series is nonlinear forecasting [ Farmer and Sidorowich , 1987; Sugihara and May , 1990]. This has been used to examine the behavior of a wide range of natural time series, for which techniques that require stationarity have provided limited insight: for example, phytoplankton population dynamics [ Sugihara , 1994], the electrical precursor signals to earthquakes [ Cuomo et al , 1998], synthetic swash time series [ Bryan and Coco , 2007], surf zone bar behavior [ Holland et al , 1999; Pape and Ruessink , 2008] and surf zone suspended sediment patterns [ Jaffe and Rubin , 1996]. The technique is based on the premise that, if the statistical properties of the time series vary locally, a simple data‐driven model will perform better if only trained using a fraction of the available training data set.…”