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NRCSE Using wavelet tools to estimate and assess trends in atmospheric data Peter Guttorp University of Washington [email protected] 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 t 2 1 A(,t) x(u)du t 2 How much do averages change over time? D(,t) A(,t ) A(,t ) 2 2 1 t 1 t x(u)du x(u)du t t Haar wavelet D(1,0) 2 (H) (u)x(u)du 1 2 , 1 u 0 1 (H) (u) , 0 u 1 2 0, otherwise where Translation and scaling (H) 1,t (u) (H) (u t) (H) ,t (u) 1 (H) u t ( ) Continuous Wavelet Transform Haar CWT: (H) W(,t) ,t (u)x(u)du D(,t) Same for other wavelets W(,t) ,t (u)x(u)du where 1 u t ,t (u) Basic facts CWT is equivalent to x: d 1 x(t) W(,u) ,t (u)du 2 C 0 CWT decomposes energy: x2 (t)dt energy 0 W2 (,t) 2 C dtd Discrete time Observe samples from x(t): x0,x1,...,xN-1 Discrete 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 Wj,t W(t j ,t) J Think of rt Wj,t as the rough of the j1 series, so s t x t rt 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: N1 J 2 2 2 2 X xi Wj S i0 j1 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 L1 Sj,t glSj1,2t1l mod(N2 j ) l0 L1 Wj,t hlSj1,2t1l mod(N2 j ) l0 t = 0,...,N 2-j-1 Daubachie’s LA(8)-wavelet 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 Multiscale analysis of coral data Decorrelation properties of wavelet transform Long term memory Coral data spectrum 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 Wavelet analysis of trend Significance test for trend Seychelles trend 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 Confidence band calculation Malindi trend 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 Latitude 6 7 8 9 10 11 12 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.