1 / 33

Wavelet Tools for Atmospheric Data Trends Assessment

Learn about using wavelet tools to estimate and assess trends in atmospheric data. Includes basic wavelet theory, trend estimation, oxygen isotope values in coral cores, turbulence in equatorial air, and more. Dive deep into continuous wavelets, multiscale analysis, trend significance tests, and more.

elawson
Download Presentation

Wavelet Tools for Atmospheric Data Trends Assessment

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related