1. Sundermeyer MAR 550 Spring 2013 1 Laboratory in Oceanography: Data and Methods MAR550, Spring 2013 Miles A. Sundermeyer Methods for Non-Stationary Means

2. Sundermeyer MAR 550 Spring 2013 2 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.) Methods for Non-Stationary Means OA (cont’d) and Kriging

3. Sundermeyer MAR 550 Spring 2013 3 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 Methods for Non-Stationary Means OA (cont’d) and Kriging

4. Sundermeyer MAR 550 Spring 2013 4 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 Methods for Non-Stationary Means OA (cont’d) and Kriging

5. Sundermeyer MAR 550 Spring 2013 5 Example: Dye mapping during Coastal Mixing & Optics Experiment (CMO) Methods for Non-Stationary Means OA (cont’d) and Kriging

6. Sundermeyer MAR 550 Spring 2013 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. Sundermeyer MAR 550 Spring 2013 7 Methods for Non-Stationary Means OA (cont’d) and Kriging Example: CMO

8. Sundermeyer MAR 550 Spring 2013 8 Methods for Non-Stationary Means OA (cont’d) and Kriging Example: CMO Start with large- scale interpolation

9. Sundermeyer MAR 550 Spring 2013 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. Sundermeyer MAR 550 Spring 2013 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. Sundermeyer MAR 550 Spring 2013 11 Methods for Non-Stationary Means OA (cont’d) and Kriging Example: CMO Do 2 nd pass kriging on “residual” Obtain kriging estimate and error map

12. Sundermeyer MAR 550 Spring 2013 12 Methods for Non-Stationary Means OA (cont’d) and Kriging Example: CMO Do 2 nd pass kriging on “residual” Obtain kriging estimate and error map

13. Sundermeyer MAR 550 Spring 2013 13 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. Methods for Non-Stationary Means OA (cont’d) and Kriging

14. Sundermeyer MAR 550 Spring 2013 14 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) Methods for Non-Stationary Means OA (cont’d) and Kriging

15. Sundermeyer MAR 550 Spring 2013 15 Block Kriging Use only data within certain range to estimate value at particular location. Minimizes size of inversion required for OA. Methods for Non-Stationary Means OA (cont’d) and Kriging

16. Sundermeyer MAR 550 Spring 2013 16 http://people.seas.harvard.edu/~leslie/MBST98/ll_analysis.html “Subjective” Objective analysis … Need to be mindful of decisions made during OA / kriging analysis Methods for Non-Stationary Means OA (cont’d) and Kriging

17. Sundermeyer MAR 550 Spring 2013 17 References A. G. Journel and CH. J. Huijbregts " Mining Geostatistics", Academic Press 1981 Methods for Non-Stationary Means OA (cont’d) and Kriging

18. Sundermeyer MAR 550 Spring 2013 18 Laboratory in Oceanography: Data and Methods MAR550, Spring 2013 Miles A. Sundermeyer Methods for Non-Stationary Means (cont’d)

19. Sundermeyer MAR 550 Spring 2013 19 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 -i  t, where  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(  t -(  t)), where A(t) is the amplitude and  (t) the phase for a central frequency , 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. ) Methods for Non-Stationary Means Complex Demodulation

20. Sundermeyer MAR 550 Spring 2013 20 Methods for Non-Stationary Means Complex Demodulation Example: Idealized signal 7 day record Signal has period of ½ day (  =2 cpd) A(t) has period of 3.5 days  (t) has period of 7 days

21. Sundermeyer MAR 550 Spring 2013 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 , plus everything else, Z(t). Amplitude, A(t), and phase  (t), of the periodic signal are assumed to vary slowly in time compared to base frequency, . Can write: Step 1: Multiply by e -i  t => Y(t) = X(t)·e -i  t, which can be written as: 1 st term varies slowly, with no power at or above  2 nd term varies at freq 2  3 rd term varies at freq  (and none at zero freq)

22. Sundermeyer MAR 550 Spring 2013 22 Methods for Non-Stationary Means Complex Demodulation Step 2: Low-pass filter to remove frequencies at or above frequency  This smoothes the 1 st term, and nearly removes 2 nd and 3 rd 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  ’(t): see also: http://www.pmel.noaa.gov/maillists/tmap/ferret_users/fu_2007/msg00180.html

23. Sundermeyer MAR 550 Spring 2013 23 Methods for Non-Stationary Means Complex Demodulation Example: Coastal Mixing and Optics Shipboard Velocity time (days)

24. Sundermeyer MAR 550 Spring 2013 24 Methods for Non-Stationary Means Complex Demodulation

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34. Sundermeyer MAR 550 Spring 2013 34 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. Methods for Non-Stationary Means Complex Demodulation