Viewing Sea Level by a One-Dimensional Random Function with Long Memory

Sea level fluctuation gains increasing interests in several fields, such as geoscience and ocean dynamics. Recently, the long-range dependence (LRD) or long memory, which is measured by the Hurst parameter, denoted by H, of sea level was reported by Barbosa et al. (2006). However, reports regarding the local roughness of sea level, which is characterized by fractal dimension, denoted by D, of sea level, are rarely seen. Note that a common model describing a random function with LRD is fractional Gaussian noise (fGn), which is the increment process of fractional Brownian motion (fBm) (Beran (1994)). If using the model of fGn, D of a random function is greater than 1 and less than 2 because D is restricted by H with the restriction D = 2 - H. In this paper, we introduce the concept of one-dimensional random functions with LRD based on a specific class of processes called the Cauchy-class (CC) process, towards separately characterizing the local roughness and the long-range persistence of sea level. In order to achieve this goal, we present the power spectrum density (PSD) function of the CC process in the closed form. The case study for modeling real data of sea level collected by the National Data Buoy Center (NDBC) at six stations in the Florida and Eastern Gulf of Mexico demonstrates that the sea level may be one-dimensional but LRD. The case study also implies that the CC process might be a possible model of sea level. In addition to these, this paper also exhibits the yearly multiscale phenomenon of sea level.