Publication Date


Document Type


Committee Members

Christopher Barton (Committee Member), John Flach (Committee Member), Paul Seybold (Committee Member), Sarah Tebbens (Advisor), Brian Tsou (Committee Member)

Degree Name

Doctor of Philosophy (PhD)


Time series that exhibit multiple scaling properties in the frequency domain are common in natural systems (e.g., temperature through geologic time). NOAA verified hourly water level data ranging from 20 to 30 years in duration for nine stations in the North American Great Lakes is converted to the frequency domain using a complex discrete fast Fourier transform (FFT) and then expressed as a power spectrum in terms of frequency versus power. To quantify power law scaling behavior, a scaling exponent (β) is determined by fitting a power function to a log-log plot of frequency (f) or period (T) versus power in the frequency domain. For water level fluctuations in the Great Lakes, the frequency domain exhibits four distinct regions of power law scaling.

The mathematical relationship of the scaling exponent (β) to 1/f time series behavior is examined employing Bode analysis. Variations in scaling behavior of water level data, indicated by the patterns of change in amplitude and phase across frequencies, can be expressed through transfer functions. The transfer functions are created using Laplace transforms. Each Laplace term (s) has a fractional exponent based on the scaling exponent (β) derived from the Bode magnitude plot. Convolution of the transfer function in the time domain is equivalent to multiplication in the frequency domain (Laplace space). Combining the transfer functions for all frequencies yields a Frequency Response Model and provides a basis to determine how the system that created the time series will respond to any given input over all frequencies. For water level fluctuations in the Great Lakes, the scaling behavior pattern is well approximated by a combination of four linear differential equations or transfer functions, one primary equation for each distinct scaling region. The collective interactions of all equations over all frequencies create the Great Lakes Frequency Response Model and represent the underlying physical dynamics of the Great Lakes system. Incorporating the Laplace term (s) and the scaling exponent (β) into 1/s-noise transfer functions yields a quantitative, equation-based Frequency Response Model of a self-affine time series with single or multiple scaling behaviors and an innovative technique to generate synthetic yet accurate time series simulations.

Page Count


Department or Program

Department of Earth and Environmental Sciences

Year Degree Awarded