Creating Synthetic Water Level Time Series from the Scaling Exponents of Water Level Records from Atlantic, Gulf of Mexico, and Pacific Coastal Stations and the North American Great Lakes

A linear differential equation Frequency Response Model of water level fluctuations of Atlantic, Gulf of Mexico, and Pacific Coastal Stations and the North American Great Lakes is created from the scaling behavior of water level time series in the frequency domain. The Frequency Response Model is then used to create a synthetic time series that captures the underlying dynamics and physical properties of the original time series. In this study, NOAA verified hourly water level data ranging from 20 to 30 years in duration for nine stations in the Great Lakes and twelve North American (Atlantic, Pacific, and Gulf of Mexico) coastal stations are analyzed. For each station, the time series is converted to the frequency domain using a Fourier transform and then expressed as Power Spectral Density (PSD) in terms of frequency versus power. A power scaling exponent ( β ) is determined by fitting a power function to a log-log plot of frequency ( f ) or period ( 1/f ) versus power in the frequency domain. Bode Analysis is a method of fitting transfer functions in the frequency domain to explain variations in scaling behavior ( β ) by examining the patterns of change in amplitude and phase across frequencies. A transfer function representing the output of the system divided by the input is derived from a Bode magnitude plot of the data using Laplace transforms. Bode analysis results in a series of two transfer function equations, one for magnitude and one for phase, for each distinct β over a specified period range. The type of differential equation controls the slope ( β ) while the constant (k) in the differential equation controls the position (period) of transitions in scaling behavior (i.e., corner frequencies or inflection points). Combining the transfer functions of each distinct scaling regime for all frequencies yields a Frequency Response Model and provides a basis to determine how a system will respond to any given input. This technique captures the scaling behavior of the system under study and the synthetic time series created from the Frequency Response Model contains the same statistical properties and scaling exponents ( β ) of the original time series from which it was created. For water level change in the Oceans and Great Lakes, the complex pattern of scaling versus period is well approximated by a combination of linear differential equations or transfer functions over the entire period range. The Frequency Response Model for these physical systems is used to generate a synthetic time series that is statistically identical to the original Ocean and Great Lakes water level time series. Synthetic time series can enhance probability forecasts, risk assessments, and provide insight into the physical processes responsible for the original time series behavior. This method introduces a quantitative, equation-based model of a self-affine time series with single or multiple scaling and a technique to create synthetic yet accurate renditions of these time series.