It is standard practice to remove harmonic "tides" from measured water levels and then model the remaining "residual." The residual can be as low as about 10% percent of the total water level in the open ocean and coastal regions and up to about 60% inside complex estuaries. This decomposition presupposes that water levels are a superposition of independent linear components, despite the fact that the inherently nonlinear Navier-Stokes equations are used to describe these motions. Recent studies have shown that the dynamics of water levels (all surface fluctuations that occur on time scales greater than three minutes) at coastal and estuary tide stations can be treated as low dimensional chaotic systems. The argument put forward here is that if water levels are composed of superimposed linear subsystems, then the dynamical characteristics of the individual subsystems ought to have a relationship that is consistent with simple superposition. However, using data from five "tide" stations an analysis of the measured water levels, the associated harmonic models, the residuals, and a simulation with additive noise, suggests that removing the harmonic components by linear decomposition destroys the dynamical characteristics of the residuals. This result is consistent with the historical difficulty of characterizing and modeling the residuals.

"Chaos and Predictability in Ocean Water Level Measurements," Ted Frison, Henry Abarbanel, Marshall Earle, John Schultz, and Wolfgang Sheerer, Journal of Geophysical Review, Vol 104, C4, pp. 7,935 - 7,951, April 15, 1999.

Ocean water levels are all fluctuations that are greater than a few minutes duration. Our interest in nonlinear methods stems from the historical difficulty of making accurate predictions of water depth in ports, harbors, and estuaries. This study, commissioned by NOAA, examined data from seven "tide" stations located along the South East U.S. coast. The integer embedding space (d_E) for data collected from the "tide" stations exposed to the open ocean is five. Four dynamical degrees of freedom (d_L) are required to describe the observed dynamics in a state space reconstructed solely from the observations themselves. In a complex estuary (Chesapeake Bay) d_E = 6 and d_L = 5. The largest global Lyapunov exponents (₁), a measure of predictability, vary from 0.57 for a station exposed directly to the ocean to 4.6 for a station well inside the estuary. The larger values are generally associated with stations that are less predictable, which is consistent with the errors of the astronomically driven estimator currently used by the U.S. government to generate tide predictions and with the interpretation of the Lyapunov exponents as a measure of dynamical predictability. The paper also examines using the Lyapunov exponents for characterizing water level variability and classifying tide zones.

"Intersection Ocean Water Level Prediction Using Methods of Nonlinear Dynamics," Ted Frison, Marshall Earle, Henry Abarbanel, and Wolfgang Sherer, Journal of Geophysical Review. Vol 104, C6, pp. 13,653-13,666, June 15, 1999.

Using data from seven "tide" stations along the south eastern U.S. coast, we demonstrate how measurements from one sensor can be used to predict the current value of another sensor (a "virtual sensor"). This is an important concept because it can be used to reduce the number of sensors in a system and to self monitor the health of each sensor.

The classic description of the ocean surface as stochastic is often correct. There are times, however, when the seemingly irregular behavior is deterministic. Analysis of ocean gravity waves at the Harvest Platform off the California coast show unambiguous low dimensional dynamics. These data are compared to wave measurements taken in a large (> 100 ' diameter) hydrodynamic test tank where thewaves are guaranteed to be correlated noise. The characteristics of correlated noise are also discussed.

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Turbulence

These studies relate to a desire to mitigate flow noise and control turbulent flow.

"Nonlinear Analysis of High Reynolds Number Flows over a Buoyant Axisymmetric Body", Henry Abarbanel, Richard Katz, Joan Cembrola, Tom Galib and Ted Frison, Phys. Rev. E, Vol 49, No. 5,May 1994, pp. 4003 - 4018.

"High Reynolds Number Boundary Layer Chaos",Henry Abarbanel, Richard Katz, Joan Cembrola, Tom Galib and Ted Frison, Physical Review Letters, Vol 72, April 1994.

These papers show that fully developed hydrodynamic turbulence has an embedding dimension of 9. A subsequent study by Prof. Abarbanel further explored the physics of the phenomena and developed an explaination for the behavior observed in this paper.

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EKG and Machinery Diagnostics

"Chaos in Gearbox Vibrations", Proceedings of the 1st NUWC Conference on Applied Chaos, Mystic, CT April 1993, AIP Press, AIP Conference Proceedings 296.

This paper was a prelude to the development of improved diagnostics and "condition based maintenance" methods.

Shows how significant diagnostic information can be derived in only two or three dimensions, even if the system is of much higher dimension. Although we used R-R intervals for this paper, subsequent investigations show that full waveform EKG data contains the relevant information and ought to be used. For a further example, see the example below.

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Private Communications

A digital CODEC (encoder/decoder) that we built for the Air Force.
Information is encoded using feedback from a chaotic osciallator.

The advantages of using a chaotic oscillator for information encoding is that it is cheap, and, if the proper methods are used, is badwidth efficient. The problem with most schemes is that they fill the channel with chaos, not information.

This project is now part of our work on agile, ad-hoc networked communications.

"Detection of broad-band communications signals", in Chaos in Communications, Louis M. Pecora,Editor, Proc. SPIE 2038, 252-261 (1993).

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Links

Papers

Recommended Overviews

Abarbanel, H.D.I., T. Frison, and L. Sh. Tsimring, "Time Domain Analysis of Signals from Nonlinear Sources,"
IEEE Signal Processing Magazine, Vol 15, No. 3, pp. 49 Ð 65, May 1998.

Abarbanel, Henry D. I., Analysis of Observed Chaotic Data, Springer-Verlag New York, 1996.

Papers by chaotic.com® Authors

Frison, Ted, "Dynamics of the residuals in estuary water levels," Physics and Chemistry of the Earth, Part B Vol 25, No. 4, pp. 359-364, May 2000.

Frison, Ted, "Chaotic Dynamics of Ocean Water Levels," Proceedings of the 3rd Conference on Coastal Atmospheric and Oceanic Prediction and Processes AMS, New Orleans, Nov 3-5, 1999.

Frison, Ted, Henry D. I. Abarbanel, Marshall Earle, and Wolfgang Sheerer, "Interstation ocean water level predictions using methods of nonlinear dynamics," Journal of Geophysical Review - Oceans Vol 104, No. C6, pp. 13,653 - 13,666, June, 1999.

Frison, Ted, Henry D. I. Abarbanel, Marshall Earle, and Wolfgang Sherer, "Chaos and predictability in ocean water level measurements," Journal of Geophysical Review - Oceans, Vol 104, No. C4, pp. 7935 - 7951, April 1999.

Frison, Ted W. and H.D.I. Abarbanel, "Identification and Quantification of Nonstationary Chaotic Behavior," Proceedings, IEEE-ICASSP (April 1997), Munich.

Frison, Ted and H. D. I. A. Abarbanel, "Ocean gravity waves: A nonlinear analysis of observations," Journal of Geophysical Review - Oceans, Vol. 102 C1, pp. 1051-1059, Jan. 15, 1997.

Frison, Ted W., H.D.I Abarbanel, J. Cembrola, and R. Katz, "Nonlinear analysis of environmental distortions of continuous wave signals in the ocean," Journal of the Acoustical Society of America, Vol. 99 No. 1, pp. 139-146, January 1996.

Frison, Ted W., H.D.I Abarbanel, J. Cembrola, and B. Neales, "Chaos in ocean ambient 'noise'," Journal of the Acoustical Society of America, Vol. 99 No. 3, pp. 1527-1539, March 1996.

Abarbanel, H. D. I., R. Katz, J. Cembrola, T. Galib and T. Frison, "Nonlinear analysis of high Reynolds number flows over a buoyant axisymmetric body," Phys. Rev. E, Vol. 49, No. 5, pp. 4003 - 4018, May 1994.

Abarbanel, H. D. I., R. Katz, J. Cembrola, T. Galib and T. Frison, "High Reynolds number boundary layer chaos," Phys. Rev. Lett., Vol. 72, No. 15, 11 April 1994, pp. 2383 - 2386.

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Figure 1. The signal trace of a biological process.	Figure 2. The reconstructed attractor.
Figure 3. The average mutual information for the signal of Figure 1. By prescription[1], the optimum time delay (Topt) for the attractor reconstruction is the first minimum, which is the case is 23 samples. But, the minimum is very shallow and a larger value may be appropriate.	Figure 4. Global embedding dimension calculations for the example signal. The method examines the percentage of close neighbors in d that are also close in d+l [2]. These are "false" neighbors in d. The minimum embedding dimension, dE, is the point where as dimensions are added, the percentage of false neighbors is minimized. For this case, dE = 5 or 6.
Figure 5. The Local dynamical dimension, dL, is the dimension where projections forward in time becomes independent of the dimensions and the number of neighbors in a neighborhood[3]. A good estimate of dL is 5, or for the bold, 4.	Figure 6. The local Lyapunov exponents for the signal of Figure 1 and 2[4]. There are dL exponents and the positive exponent is evidence that the signal comes from a chaotic system. The Lyapunov exponents quanitify the inherent predictability of the system

[2] Kennel, Mathew B., R. Brown, and H. D. I. Abarbanel, "Determining embedding dimension for phase-space reconstruction unsing a geometrical construction," Phy. Rev. A 45 pp. 3403-3411, 15 March 1992.

[3] Abarbanel, H.D.I., and M.B. Kennel, "Local false nearest neighbors and dynamical dimensions from observed chaotic data", Phy. Rev. E., 47, 3057-3068, 1993.

[4] Abarbanel, H.D.I., R. Brown, amd Mathew Kennel, "Local Lyapunov exponents computed from oberved data", J. Nonlinear Science, 2 pp. 343-365, Sept. 1992.

chaos, software, chaotic, prediction, Lyapunov, chaos, software, chaotic, prediction, Lyapunov, nonlinear time series, tides, chaos theory, noise reduction, oceanography
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