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NRCSE. Using wavelet tools to estimate and assess trends in atmospheric data Peter Guttorp University of Washington peter@stat.washington.edu. Collaborators. Don Percival Chris Bretherton Peter Craigmile Charlie Cornish. Outline. Basic wavelet theory Long term memory processes

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  1. NRCSE Using wavelet tools to estimate and assess trends in atmospheric dataPeter GuttorpUniversity of Washingtonpeter@stat.washington.edu

  2. Collaborators • Don Percival • Chris Bretherton • Peter Craigmile • Charlie Cornish

  3. Outline • Basic wavelet theory • Long term memory processes • Trend estimation using wavelets • Oxygen isotope values in coral cores • Turbulence in equatorial air

  4. Wavelets • Fourier analysis uses big waves • Wavelets are small waves

  5. Requirements for wavelets • Integrate to zero • Square integrate to one • Measure variation in local averages • Describe how time series evolve in time for different scales (hour, year,...) • or how images change from one place to the next on different scales (m2, continents,...)

  6. Continuous wavelets • Consider a time series x(t). For a scale l and time t, look at the average • How much do averages change over time?

  7. Haar wavelet • where

  8. Translation and scaling

  9. Continuous Wavelet Transform • Haar CWT: • Same for other wavelets • where

  10. Basic facts • CWT is equivalent to x: • CWT decomposes energy: energy

  11. Discrete time • Observe samples from x(t): x0,x1,...,xN-1Discrete wavelet transform (DWT) slices through CWT • restricted to dyadic scales tj = 2j-1, j = 1,...,J t restricted to integers 0,1,...,N-1 Yields wavelet coefficients Think of as the rough of the series, so is the smooth (also called the scaling filter). A multiscale analysis shows the wavelet coefficients for each scale, and the smooth.

  12. Properties • Let Wj = (Wj,0,...,Wj,N-1); S = (s0,...,sN-1). • Then (W1,...,WJ,S ) is the DWT of X = (x0,...,xN-1). • (1) We can recover X perfectly from its DWT. • (2) The energy in X is preserved in its DWT:

  13. The pyramid scheme • Recursive calculation of wavelet coefficients: {hl } wavelet filter of even length L; {gl = (-1)lhL-1-l} scaling filter • Let S0,t = xt for each t • For j=1,...,J calculate • t = 0,...,N 2-j-1

  14. Daubachie’s LA(8)-wavelet

  15. Oxygen isotope in coral cores at Seychelles • Charles et al. (Science, 1997): 150 yrs of monthly d18O-values in coral core. • Decreased oxygen corresponds to increased sea surface temperature • Decadal variability related to monsoon activity 1877

  16. Multiscale analysis of coral data

  17. Decorrelation properties of wavelet transform

  18. Long term memory

  19. Coral data spectrum

  20. What is a trend? • “The essential idea of trend is that it shall be smooth” (Kendall,1973) • Trend is due to non-stochastic mechanisms, typically modeled independently of the stochastic portion of the series: • Xt = Tt + Yt

  21. Wavelet analysis of trend

  22. Significance test for trend

  23. Seychelles trend

  24. Malindi coral series • Cole et al. (2000) • 194 years of d18O isotope from colony at 6m depth (low tide) in Malindi, Kenya 4.7 4.1 1800 1900 2000

  25. Confidence band calculation

  26. Malindi trend

  27. Air turbulence • EPIC: East Pacific Investigation of Climate Processes in the Coupled Ocean-Atmosphere System • Objectives: (1) To observe and understand the ocean-atmosphere processes involved in large-scale atmospheric heating gradients • (2) To observe and understand the dynamical, radiative, and microphysical properties of the extensive boundary layer cloud decks

  28. Flights • Measure temperature, pressure, humidity, air flow going with and across wind at 30m over sea surface.

  29. Wavelet and bulk zonal momentum flux • Wavelet measurements are “direct” • Bulk measurements are using empirical model based on air-sea temperature difference -1 0 1 2 3 4 5 6 7 8 9 10 11 12 Latitude

  30. Wavelet variability • Turbulence theory indicates variability moved from long to medium scales when moving from goingalong to going across the wind. • Some indication here; becomes very clear when looking over many flights.

  31. Further directions • Image decomposition using wavelets • Spatial wavelets for unequally spaced data (lifting schemes)

  32. References • Beran (1994) Statistics for Long Memory Processes. Chapman & Hall. • Craigmile, Percival and Guttorp (2003) Assessing nonlinear trends using the discrete wavelet transform. Environmetrics, to appear. • Percival and Walden (2000) Wavelet Methods for Time Series Analysis. Cambridge.

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