spectral representation in oceanography: observation and ... · references chu, p.c., l.m. ivanov,...
TRANSCRIPT
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Calhoun: The NPS Institutional Archive
Faculty and Researcher Publications Faculty and Researcher Publications
2004
Spectral representation in
oceanography: observation and modeling
Chu, P.C.
Chu, P.C., and L.M. Ivanov, Spectral representation in oceanography: observation and
modeling, First Annual Asia-Oceanic Geophysical Society Conference, Singapore, 5-9 July
2004 (invited).
http://hdl.handle.net/10945/42431
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Spectral Representation in OceanographyObservation and Modeling
Peter C ChuNaval Postgraduate School
Monterey, CA [email protected]
http://www.oc.nps.navy.mil/~chu
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Part-1 Optimal Spectral Decomposition (OSD)
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References Chu, P.C., L.M. Ivanov, T.P. Korzhova, T.M. Margolina, and O.M. Melnichenko, 2003a: Analysis of sparse and noisy ocean current data using flow decomposition. Part 1: Theory. Journal of Atmospheric and Oceanic Technology, 20 (4), 478-491.
Chu, P.C., L.M. Ivanov, T.P. Korzhova, T.M. Margolina, and O.M. Melnichenko, 2003b: Analysis of sparse and noisy ocean current data using flow decomposition. Part 2: Application to Eulerian and Lagrangian data. Journal of Atmospheric and Oceanic Technology, 20 (4), 492-512.
Chu, P.C., L.M. Ivanov, and T.M. Margolina, 2004: Rotation method for reconstructing process and field from imperfect data. International Journal of Bifurcation and Chaos, in press.
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Observational Data (Sparse and Noisy)
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Most Popular Method for Ocean Data Analysis: Optimum Interpolation (OI)
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Three Necessary Conditions For the OI Method
(1) First guess field
(2) Autocorrelation functions
(3) Low noise-to-signal ratio
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Ocean velocity data
(1) First guess field (?)
(2) Unknown autocorrelation function
(3) High noise-to-signal ratio
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It is not likely to use the OI method to process ocean velocity data.
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Spectral Representation - a Possible Alternative Method
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Two approaches to obtain basis functions
EOFs
Eigenfunctions of Laplace Operator
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Spectral Representation for Velocity
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Flow Decomposition
2 D Flow (Helmholtz)
3D Flow (Toroidal & Poloidal): Very popular in astrophysics
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3D Incompressible Flow When
We have
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Flow Decomposition
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Boundary Conditions
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Basis Functions (Closed Basin)
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Basis Functions (Open Boundaries)
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Spectral Decomposition
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Optimal Mode Truncation
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Vapnik (1983) Cost Function
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Optimal Truncation
Gulf of Mexico, Monterey Bay, Louisiana-Texas Shelf
Kopt = 40, Mopt = 30
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Determination of Spectral Coefficients (Ill-Posed Algebraic Equation)
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Rotation Method (Chu et al., 2004)
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Part-2 Application in Data Analysis Current Reversal in Louisiana-Texas Continental Shelf (LTCS)
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ReferenceChu, P.C., L.M. Ivanov, and O.V. Melnichenko, 2004: Fall-winter current reversals on the Taxes-Lousianacontinental shelf, Journal of Physical Oceanography, in press.
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Ocean Velocity Data 31 near-surface (10-14 m) current meter moorings during LATEX from April 1992 to November 1994
Drifting buoys deployed at the first segment of the Surface Current and Lagrangian-drift Program (SCULP-I) from October 1993 to July 1994.
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Surface Wind Data
7 buoys of the National Data Buoy Center (NDBC) and industry (C-MAN) around LATEX area
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Moorings and Buoys
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Reconstructed and observed circulations at Station-24.
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LTCS current reversal detected from SCULP-I drift trajectories.
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Probability of TLCS Current Reversal for Given Period (T)
n0 ~0-current reversal n1~ 1-current reversaln2~ 2-current reversalsm ~ all realizations
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Fitting the Poison Distribution
µ is the mean number of reversal for a single time interval
µ ~ 0.08
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Dependence of P0, P1, P2 on T
For observational periods larger than 20 days, the probability for no currentreversal is less than 0.2.
For 15 day observational period, the probability for 1-reversal reaches 0.5
Data – Solid CurvePoison Distribution Fitting –Dashed Curve
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Time Interval between Successive Current Reversals (not a Rare Event)
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LTCS current reversal detected from the reconstructed velocity data
December 30, 1993
January 3, 1994
January 6, 1994
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EOF Analysis of the Reconstructed Velocity Filed
0.81.10.76
2.31.41.15
4.63.31.44
6.95.63.93
9.39.510.12
74.477.180.21
10/05/94-11/29/9412/19/93-04/17/9401/21/93-05/21/93
Variance (%)EOF
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Mean and First EOF Mode
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Mean Circulation
1. First Period (01/21-05/21/93)
2. Second Period 12/19/93-04/17/94)
3. Third Period (10/05-11/29/94)
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EOF1
1. First Period (01/21-05/21/93)
2. Second Period 12/19/93-04/17/94)
3. Third Period (10/05-11/29/94)
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Calculated A1(t) Using Current Meter Mooring (solid)and SCULP-1Drifters (dashed)
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8 total reversals observed
Uals ~ alongshore wind
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Morlet Wavelet
A1(t)
Uals
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Regression betweenA1(t) and Surface Winds
Solid Curve (reconstructed)Dashed Curve (predicted using winds)
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Part-3 Application in Modeling
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How Long Can a Model Predict?
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ReferencesChu, P.C., L.M. Ivanov, T. M. Margolina, and O.V. Melnichenko, On probabilistic stability of an atmospheric model to various amplitude perturbations. Journal of the Atmospheric Sciences, 59, 2860-2873.Chu, P.C., L.M. Ivanov, C.W. Fan, 2002: Backward Fokker-Planck equation for determining model valid prediction period. Journal of Geophysical Research, 107, C6, 10.1029/2001JC000879.Chu, P.C., L. Ivanov, L. Kantha, O. Melnichenko, and Y. Poberezhny, 2002: Power law decay in model predictability skill. GeophysicalResearch Letters, 29 (15), 10.1029/2002GLO14891. Chu, P.C., L.M. Ivanov, L.H. Kantha, T.M. Margolina, and O.M. Melnichenko, and Y.A, Poberenzhny, 2004: Lagrangian predictabiltyof high-resolution regional ocean models. Nonlinear Processes in Geophysics, 11, 47-66.
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Physical Reality
Y
Physical Law: dY/dt = h(y, t)
Initial Condition: Y(t0) = Y0
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Atmospheric & Oceanic Models
X is the prediction of Y
d X/ dt = f(X, t) + q(t) X
Initial Condition: X(t0) = X0
Stochastic Forcing: <q(t)> = 0<q(t)q(t’)> = q2δ(t-t’)
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Model Error
Z = X – Y
Initial: Z0 = X0 - Y0
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Valid Prediction Period (VPP)
VPP is defined as the time period when the prediction error first exceeds a pre-determined criterion (i.e., the tolerance level ε).
VPP is the First-Passage Time
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VPP
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Conditional Probability Density Function
Initial Error: Z0
(t – t0) Random Variable
Conditional PDF of (t – t0) with given Z0
P[(t – t0) |Z0]
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Two Approaches to Obtain PDF of VPP
Analytical (Backward Fokker-Planck Equation)
Practical (Optimum Spectral Analysis)
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Analytical Approach
Backward Fokker-Planck Equation
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Backward Fokker-Planck Equation
Model Physics Stochastic Forcing
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Moments
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Example: One Dimensional Error (Nicolis 1992), Population (ecology), or General Production (economics) Models
1D Dynamical System (Maximum Growing Maniford of Lorenz System)
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Mean and Variance of VPP
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Analytical Solutions
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Dependence of tau1 & tau2 on Initial Condition Error ( )
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Practical Approach
Optimum Spectral Decomposition (OSD)
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Gulf of Mexico Forecast System
University of Colorado Version of POM1/12o ResolutionReal-Time SSH Data (TOPEX, ESA ERS-1/2) AssimilatedReal Time SST Data (MCSST, NOAA AVHRR) AssimilatedSix Months Four-Times Daily Data From July 9, 1998 for Verification
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Model Generated Velocity Vectors at 50 m on 00:00 July 9, 1998
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(Observational) Drifter Data at 50 m on 00:00 July 9, 1998
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Reconstructed Drift Data at 50 m on 00:00 July 9, 1998 Using the OSD Method (Chu et al. 2002 a, b, JTECH)
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Error Mean and Variance
Error Mean
Error Variance
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Exponential Error Growth
Classical Linear Theory
No Long-Term Predictability
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Power Law
Long-Term Predictability May Occur
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Statistical Characteristics of VPP
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Predictability Tube
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ConclusionsOSD is a useful tool for processing real-time velocity data with short duration and limited-area sampling.
The scheme can handle highly noisy data.
The scheme is model independent.
The scheme can be used for velocity data assimilation.
Phase Space Consideration