1. Using wavelet tools to estimate and assess trends in atmospheric data NRCSE

2. Wavelets Fourier analysis uses big waves Wavelets are small waves

3. 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 (m 2, continents,...)

4. 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?

5. Haar wavelet where

6. Translation and scaling

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

8. Basic facts CWT is equivalent to x: CWT decomposes energy: energy

9. Discrete time Observe samples from x(t): x 0,x 1,...,x N-1 Discrete wavelet transform (DWT) slices through CWT restricted to dyadic scales  j = 2 j-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.

10. Properties Let W j = (W j,0,...,W j,N-1 ); S = (s 0,...,s N-1 ). Then W = (W 1,...,W J,S ) is the DWT of X = (x 0,...,x N-1 ). (1) We can recover X perfectly from its DWT W, X = W -1 W. (2) The energy in X is preserved in its DWT:

11. The pyramid scheme Recursive calculation of wavelet coefficients: {h l } wavelet filter of even length L; {g l = (-1) l h L-1-l } scaling filter Let S 0,t = x t for each t For j=1,...,J calculate t = 0,...,N 2 -j -1

12. Daubachie’s LA(8)-wavelet

13. Oxygen isotope in coral cores at Malindi, Kenya Cole et al. (Science, 2000): 194 yrs of monthly  18 O-values in coral core. Decreased oxygen corresponds to increased sea surface temperature Decadal variability related to monsoon activity

14. Multiscale analysis of coral data

15. Long term memory A process has long term memory if the autocorrelation decays very slowly with lag May still look stationary Example: Fractionally differenced Gaussian process, has parameter d related to spectral decay If the process is stationary

16. Nile river annual minima

17. Annual northern hemisphere temperature anomalies

18. Coral data correlation

19. Decorrelation properties of wavelet transform Periodogram values are approximately uncorrelated at Fourier frequencies for stationary processes (but not for long memory processes) Wavelet coefficient at different scales are also approximately uncorrelated, even for long memory processes (approximation better for larger L)

20. Nile river 1 yr 2 yr 4 yr 8 yr ≥16 yr

21. Wavelet variance The wavelet coefficients pick up changes in the energy at different scales over time The variability of the coefficients at each scale is a variance decomposition (similar to Fourier analysis, although the frequency choices are different) The wavelet coefficients, even for a long-term memory process, behave (at each scale) like a sample from a mean zero white noise process (at least for large L)

22. Analysis of wavelet variance In the Nile data there is a visual indication that the variability is changing after the first 100 observations at scales 1 and 2 years. Let X t be a time series with mean 0 and variance.To test against we use the statistics

23. Testing for changepoint of variability Letand D=max(D +,D - ) measures the deviation of K k from the 45° line expected if H 0 is true. Asymptotically, D converges to a Brownian bridge, which can be used to calculate critical values. Alternatively Monte Carlo critical values, or simply Monte Carlo test.

24. Nile river

25. Test result ScaleDMC 5% point 10.15590.1051 20.17540.1469 30.10000.2068 40.23130.2864

26. When did it change?

27. Why did it change? Around 715 a nilometer was constructed and located at a mosque on Rawdah Island in the river. Replaced 850 after a huge flood.

28. 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: X t = T t + Y t

29. Wavelet analysis of trend where A is diagonal, picks out S and the boundary wavelet coefficients. Write where R= W T A W, so if X is Gaussian we have

30. Confidence band calculation Let v be the vector of sd’s of and. Then which we can make 1-  by choosing d by Monte Carlo (simulating the distribution of U). Note that this confidence band will be simultaneous, not pointwise.

31. Malindi trend

32. 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

33. Flights Measure temperature, pressure, humidity, air flow in East Pacific

34. Flight pattern The airplane flies some high legs (1500 m) and some low legs (30 m). The transition between these (somewhat stationary) legs is of main interest in studying boundary layer turbulence.

36. Wavelet variability The variability at each scale constitutes an analysis of variance. One can clearly distinguish turbulent and non-turbulent regions.