1. Laboratory in Oceanography: Data and Methods
Methods for
Non-Stationary Means
MAR599, Spring 2009
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

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)
MAR599, Spring 2009
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 multiply 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 (unshifted).
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, giving:
where prime indicates smoothing. The choice of filter determines what frequency band remains.
Step 3: Isolate A’(t) and f’(t):
see also:

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:
Bloomfield, P Fourier decomposition of time series: An introduction, 258 pp., John Wiley, New York.
Matlab has a “communications” toolbox with many implementations/functions
fmmod, fmdemod - frequency modulation and demodulation
pmmod, pmdemod - phase modulation and demodulation
References
Chelton, D. B. and R. E. Davis, Monthly mean sea level variability along the west coast of North America, J. Phys. Oceanogr., 21,
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

35. Methods for Non-Stationary Means
Complex Demodulation
Example Applications
J. Hyatt and R. C. Beardsley - Observations of near-inertial motions in sea ice and the upper ocean mixed layer in Marguerite Bay, western Antarctic Peninsula shelf, Geophysical Research Abstracts, Vol. 7, 04162, 2005.