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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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## NRCSE

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**NRCSE**Using wavelet tools to estimate and assess trends in atmospheric dataPeter GuttorpUniversity of Washingtonpeter@stat.washington.edu**Collaborators**• Don Percival • Chris Bretherton • Peter Craigmile • Charlie Cornish**Outline**• Basic wavelet theory • Long term memory processes • Trend estimation using wavelets • Oxygen isotope values in coral cores • Turbulence in equatorial air**Wavelets**• Fourier analysis uses big waves • Wavelets are small waves**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,...)**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?**Haar wavelet**• where**Continuous Wavelet Transform**• Haar CWT: • Same for other wavelets • where**Basic facts**• CWT is equivalent to x: • CWT decomposes energy: energy**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.**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:**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**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**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**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**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**Flights**• Measure temperature, pressure, humidity, air flow going with and across wind at 30m over sea surface.**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**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.**Further directions**• Image decomposition using wavelets • Spatial wavelets for unequally spaced data (lifting schemes)**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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