Laboratory in Oceanography: Data and Methods

1. Laboratory in Oceanography: Data and Methods Methods for Non-Stationary Means MAR550, Spring 2013 Miles A. Sundermeyer

2. Methods for Non-Stationary Means OA (cont’d) and Kriging • Recall, for OA • Assumed field is homogeneous and isotropic. • Assumed errors do not co-vary with themselves or with observations, and that errors have zero mean. • Estimated field based on observations and correlation matrix (assumes the observations are correlated with each other). • Computed expected error variances (Note, as long as stations don’t change w/ time, errors also don’t change with time. Can use this to explore possible station schemes to minimize error in maps.)

3. Methods for Non-Stationary Means OA (cont’d) and Kriging • Types of kriging • Simple kriging (OA, OI) – known constant mean, μ(x) = 0. • Ordinary kriging - unknown but constant mean, μ(x) = μ, and enough observations to estimate the variogram/correlation function • Universal kriging - assumes mean is unknown but linear combination of known functions, • Extensions • Lognormal kriging • Vector fields (incorporate non-divergence, or geostrophy) • Non-isotropic (challenge for coastal OA – see OAX from Bedford Institute of Oceanography) • Multivariate

4. Methods for Non-Stationary Means OA (cont’d) and Kriging • Extensions of simple kriging (OI,OA) • Consider problem of a localized tracer, such as dye-release experiment, river plume, or other localized field. • Suppose non-zero mean – can always subtract the mean • Suppose non-isotropic – can scale different directions (assuming correlation function is still the same) • Suppose spatially varying mean ... need universal kriging for this

5. Methods for Non-Stationary Means OA (cont’d) and Kriging Example: Dye mapping during Coastal Mixing & Optics Experiment (CMO)

6. Methods for Non-Stationary Means OA (cont’d) and Kriging • Example: CMO • Dye concentration varies spatially – approx. Gaussian in x and y at large scales. • Wish to map small-scale variability – capture variability within patch

7. Methods for Non-Stationary Means OA (cont’d) and Kriging Example: CMO

8. Methods for Non-Stationary Means OA (cont’d) and Kriging • Example: CMO • Start with large-scale interpolation

9. Methods for Non-Stationary Means OA (cont’d) and Kriging • Example: CMO • Start with large-scale interpolation (b=6 km, a=2) • “interpolate” smoothed map onto observation points as spatially varying mean.

10. Methods for Non-Stationary Means OA (cont’d) and Kriging • Example: CMO • compute covariance function of “residual” from first pass kriging (data minus spatially varying mean).

11. Methods for Non-Stationary Means OA (cont’d) and Kriging • Example: CMO • Do 2nd pass kriging on “residual” • Obtain kriging estimate and error map

12. Methods for Non-Stationary Means OA (cont’d) and Kriging • Example: CMO • Do 2nd pass kriging on “residual” • Obtain kriging estimate and error map

13. Methods for Non-Stationary Means OA (cont’d) and Kriging Nugget Effect Though correlation at zero lag is theoretically = 1, sampling error and small scale variability may cause observations separated by small distances to be dissimilar. This causes a discontinuity at the origin of the correlation function called the “nugget” effect.

14. Methods for Non-Stationary Means OA (cont’d) and Kriging • Anisotropy • Kriging/OA can handle different correlation length scales in different coordinate directions. • Can also handle time correlations for spatio-temporal data • Example: OAX (developed by Bedford Institute of Oceanography)

15. Methods for Non-Stationary Means OA (cont’d) and Kriging • Block Kriging • Use only data within certain range to estimate value at particular location. Minimizes size of inversion required for OA.

16. Methods for Non-Stationary Means OA (cont’d) and Kriging “Subjective” Objective analysis … Need to be mindful of decisions made during OA / kriging analysis http://people.seas.harvard.edu/~leslie/MBST98/ll_analysis.html

17. Methods for Non-Stationary Means OA (cont’d) and Kriging • References • A. G. Journel and CH. J. Huijbregts " Mining Geostatistics", Academic Press 1981

18. Laboratory in Oceanography: Data and Methods Methods for Non-Stationary Means (cont’d) MAR550, Spring 2013 Miles A. Sundermeyer

19. Methods for Non-Stationary Means Complex Demodulation • Basics idea of Complex Demodulation • Complex demodulation can be thought of as a type of band-pass filter that gives the time variation of amplitude and phase of a time series in a specified frequency band. • To implement: • Frequency-shift time series by multiplying by e-iwt, where w is the central frequency of interest. • Low-pass filter to remove frequencies greater than the central frequency. The low pass acts as a band-pass filter when the time series is reconstructed (un-shifted). • Express complex time series as a time-varying amplitude and phase of variability in band near the central frequency; that is, X’(t) = A(t) cos(wt -(ft)), where A(t) is the amplitude and f(t) the phase for a central frequency w, and X’(t) is the reconstructed band-passed time series. • (Note: the phase variation can also be thought of as a temporal compression or expansion of a nearly sinusoidal time series, which is equivalent to a time variation of frequency. )

20. Methods for Non-Stationary Means Complex Demodulation • Example: Idealized signal • 7 day record • Signal has period of ½ day (w=2 cpd) • A(t) has period of 3.5 days • f(t) has period of 7 days

21. Methods for Non-Stationary Means Complex Demodulation • The Math (simplified) ... • Time series is assumed to be a combination of nearly periodic signal with nominal frequency w, plus everything else, Z(t). • Amplitude, A(t), and phase f(t), of the periodic signal are assumed to vary slowly in time compared to base frequency, w. • Can write: • Step 1: Multiply by e-iwt => Y(t) = X(t)·e-iwt, which can be written as: • 1st term varies slowly, with no power at or above w • 2nd term varies at freq 2w • 3rd term varies at freq w (and none at zero freq)

22. Methods for Non-Stationary Means Complex Demodulation • Step 2: Low-pass filter to remove frequencies at or above frequency w. This smoothes the 1st term, and nearly removes 2nd and 3rd terms (i.e., the original signal who’s phase and amplitude we seek, as well as noise), giving: • where prime indicates smoothing. The choice of filter determines what frequency band remains. • Step 3: Isolate A’(t) and f’(t): see also: http://www.pmel.noaa.gov/maillists/tmap/ferret_users/fu_2007/msg00180.html

23. Methods for Non-Stationary Means Complex Demodulation • Example: Coastal Mixing and Optics Shipboard Velocity time (days)

24. Methods for Non-Stationary Means Complex Demodulation

25. Methods for Non-Stationary Means Complex Demodulation

26. Methods for Non-Stationary Means Complex Demodulation

27. Methods for Non-Stationary Means Complex Demodulation

28. Methods for Non-Stationary Means Complex Demodulation

29. Methods for Non-Stationary Means Complex Demodulation

30. Methods for Non-Stationary Means Complex Demodulation

31. Methods for Non-Stationary Means Complex Demodulation

32. Methods for Non-Stationary Means Complex Demodulation

33. Methods for Non-Stationary Means Complex Demodulation

34. Methods for Non-Stationary Means Complex Demodulation • Useful Tidbits: • Matlab has a “communications” toolbox with many implementations/functions • demod - frequency & phase modulation and demodulation • References • Bloomfield, P. 1976. Fourier decomposition of time series: An introduction, 258 pp., John Wiley, New York. • Chelton, D. B. and R. E. Davis, 1982. Monthly mean sea level variability along the west coast of North America, J. Phys. Oceanogr., 21, 757-784. • Bingham, C., M. D. Godfrey, and J. W. Tukey, "Modern Techniques of Power Spectrum Estimation,"  IEEE Transactions on Audio and Electro-acoustics, Volume AU-15, Number 2, June 1967, pp. 56-66.