Bode Analysis and Modeling of Water Level Change in the Great Lakes
Abstract
Power Spectral Density calculated from a fast Fourier transform expresses a time series in terms of power in the corresponding frequency domain. The 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. Anthropogenic and natural fluctuations including precipitation, runoff, snowmelt, water retention time, evaporation, and outflow all contribute to changes in water levels recorded in the Great Lakes. In this study, NOAA verified hourly water level data ranging from 20 to 30 years in duration for five stations in Lake Michigan and four stations in Lake Superior were analyzed. Water level time series in the Great Lakes are found to exhibit power law scaling and are thus self-affine over four distinct period ranges, each with a different beta value. With this information, a model of the original time series may be generated using an approach which draws from concepts in control theory and feedback systems. Bode Analysis can be applied in the frequency domain to explain variations in the scaling behavior ( β ) of water level data by examining the patterns of change in amplitude and phase across frequencies. A Bode magnitude plot of the system is created from the data of power versus frequency converting the amplitude to 20log dB magnitude. A transfer function representing the output of the system divided by the input is then derived based on the data using Laplace transforms and solved for magnitude and phase. Bode analysis results in a series of two transfer function equations, one for magnitude and one for phase, for each distinct beta value over the 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) and are characteristics of the system. Combining the transfer functions for all frequencies yields a Frequency Response Model of the underlying internal dynamics of the system and provides a basis to determine how a system will respond to any given input. For water level change in the Great Lakes, the complex pattern of scaling versus period can be well approximated by a combination of linear differential equations or transfer functions representing magnitude and phase at each frequency within each period range. The linear differential equation Frequency Response Model describes water levels of the Great Lakes system as two spring inertial systems coupled together and provides a quantitative measure of variations observed in the Power Spectral Density plots. The Frequency Response Model is also used to generate a synthetic time series which is statistically identical to the original Great Lakes water level time series. This method introduces a novel way to generate a quantitative, equation-based model of self-affine time series data.
- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2009
- Bibcode:
- 2009AGUFMNG33B1082T
- Keywords:
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- 1872 HYDROLOGY / Time series analysis;
- 4255 OCEANOGRAPHY: GENERAL / Numerical modeling;
- 4400 NONLINEAR GEOPHYSICS;
- 4445 NONLINEAR GEOPHYSICS / Nonlinear differential equations