Using wavelet tools to estimate and assess trends

in atmospheric data

Peter GuttorpUniversity of Washington

peter@stat.washington.edu

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.