Laboratory in Oceanography: Data and Methods

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Laboratory in Oceanography Data and Methods
Methods for Non-Stationary Means

• MAR599, Spring 2009
• Miles A. Sundermeyer

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Methods for Non-Stationary Means OA (contd) 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 dont change w/ time, errors also
  dont change with time. Can use this to explore
  possible station schemes to minimize error in
  maps.)

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Methods for Non-Stationary Means OA (contd) 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

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Methods for Non-Stationary Means OA (contd) 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

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Methods for Non-Stationary Means OA (contd) and
Kriging
Example Dye mapping during Coastal Mixing
Optics Experiment (CMO)
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Methods for Non-Stationary Means OA (contd) 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

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Methods for Non-Stationary Means OA (contd) and
Kriging
Example CMO
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Methods for Non-Stationary Means OA (contd) and
Kriging

• Example CMO
• Start with large-scale interpolation

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Methods for Non-Stationary Means OA (contd) and
Kriging

• Example CMO
• Start with large-scale interpolation (b6 km,
  a2)
• interpolate smoothed map onto observation
  points as spatially varying mean.

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Methods for Non-Stationary Means OA (contd) and
Kriging

• Example CMO
• compute covariance function of residual from
  first pass kriging (data minus spatially varying
  mean).

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Methods for Non-Stationary Means OA (contd) and
Kriging

• Example CMO
• Do 2nd pass kriging on residual
• Obtain kriging estimate and error map

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Methods for Non-Stationary Means OA (contd) and
Kriging

• Example CMO
• Do 2nd pass kriging on residual
• Obtain kriging estimate and error map

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Methods for Non-Stationary Means OA (contd) 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.
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Methods for Non-Stationary Means OA (contd) 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)

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Methods for Non-Stationary Means OA (contd) and
Kriging

• Block Kriging
• Use only data within certain range to estimate
  value at particular location. Minimizes size of
  inversion required for OA.

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Methods for Non-Stationary Means OA (contd) and
Kriging
Subjective Objective analysis Need to be
mindful of decisions made during OA / kriging
analysis
http//people.seas.harvard.edu/leslie/MBST98/ll_a
nalysis.html
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Methods for Non-Stationary Means OA (contd) and
Kriging

• References
• A. G. Journel and CH. J. Huijbregts " Mining
  Geostatistics", Academic Press 1981

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Laboratory in Oceanography Data and Methods
Methods for Non-Stationary Means (contd)

• MAR599, Spring 2009
• Miles A. Sundermeyer

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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. )

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Methods for Non-Stationary Means Complex
Demodulation

• Example Idealized signal
• 7 day record
• Signal has period of ½ day (w2 cpd)
• A(t) has period of 3.5 days
• f(t) has period of 7 days

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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 gt 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)

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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 http//www.pmel.noaa.gov/maillists/tmap/
ferret_users/fu_2007/msg00180.html
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Methods for Non-Stationary Means Complex
Demodulation

• Example Coastal Mixing and Optics Shipboard
  Velocity

time (days)
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation
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Methods for Non-Stationary Means Complex
Demodulation

• Useful Tidbits
• Bloomfield, P. 1976. 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, 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.

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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.