using wavelet tools to estimate and assess trends in atmospheric data peter guttorp university of...
TRANSCRIPT
Using wavelet tools to estimate and assess trends
in atmospheric data
Peter GuttorpUniversity of Washington
NRCSE
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?
€
A(λ,t) =1λ
x(u)dut−λ
2
t+λ2
∫
€
D(λ,t) = A(λ,t + λ2
) − A(λ,t − λ2
)
=1λ
x(u)du −t
t+λ∫
1λ
x(u)dut−λ
t∫
Haar wavelet
where
€
D(1,0) = 2 ψ(H) (u)x(u)du−∞
∞∫
€
ψ(H) (u) =
−12
, −1< u ≤ 0
12
, 0 < u ≤ 1
0, otherwise
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
Translation and scaling
€
ψ1,t(H) (u) = ψ(H) (u − t)
€
ψλ,t(H) (u) =
1λ
ψ(H) (u − t
λ)
Continuous Wavelet Transform
Haar CWT:
Same for other wavelets
where
€
) W (λ,t) = ψλ,t
(H) (u)x(u)du∝D(λ,t)−∞
∞∫
€
) W (λ,t) = ψλ,t (u)x(u)du
−∞
∞∫
€
ψλ,t (u) =1λ
ψu − t
λ
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Basic facts
CWT is equivalent to x:
CWT decomposes energy:
€
x(t) =1
Cψ 0
∞∫ W(λ,u)ψλ,t (u)du
−∞
∞∫ ⎡
⎣ ⎢
⎤
⎦ ⎥dλ
λ2
€
x2 (t)dt =W2 (λ,t)
Cψλ2dtdλ
−∞
∞∫
0
∞∫
−∞
∞∫
energy
Discrete time
Observe samples from x(t): x0,x1,...,xN-1
Discrete wavelet transform (DWT) slices through CWT
λ restricted to dyadic scales j = 2j-1, j = 1,...,Jt 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.
€
Wj,t ∝) W (τ j,t)
€
rt = Wj,tj=1
J∑
€
st = xt −rt
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:
€
X 2 = xi2
i=0
N−1∑ = Wj
2
j=1
J∑ + S 2
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 tFor j=1,...,J calculate
t = 0,...,N 2-j-1€
Sj,t = glSj−1,2t+1−lmod(N2−j )l=0
L−1∑
€
Wj,t = hlSj−1,2t+1−lmod(N2−j )l=0
L−1∑
Daubachie’s LA(8)-wavelet
Oxygen isotope in coral cores at Seychelles
Charles et al. (Science, 1997): 150 yrs of monthly 18O-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 18O isotope from colony at 6m depth (low tide) in Malindi, Kenya
1800 1900 2000
4.7
4.1
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
Latitude-1 0 1 2 3 4 5 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.