YODEN ShigeoYODEN ShigeoDept. of Geophysics, Kyoto Univ., JAPAN

March 3-4, 2005: SPARC Temperature Trend Meeting at University of Reading

1. Introduction2. Statistical considerations3. Internal variability in a numerical

model4. Spurious trend experiment5. Concluding remarks

Spurious TrendSpurious Trend

in Finite Length Datasetin Finite Length Dataset

with Natural Variabilitywith Natural Variability

Causes of interannual variations of the stratosphere-troposphere coupled system

Yoden et al. (2002; JMSJ )

1. Introduction

monotonic change response

(linear) trend

ramdom process (asumption)

Labitzke Diagram (Seasonal Variation of Histograms

of the Monthly Mean Temperature; at 30 hPa)

South Pole(NCEP)

North Pole(NCEP)

North Pole(Berlin)

Length of the observed dataset is at most 5050 years.

Separation of the trend fromnatural variations is a big problem.

Observed variations

Linear Trend of the Monthly Mean Temperature

( Berlin, NCEP )

A spurious trend may exist infinite length dataset with natural variability.

Nishizawa and Yoden (2005, JGR in press) Linear trend

We assume a linear trend in a finite-length dataset with random

variability

Spurious trend We estimate the linear trend by the least square method

We define a spurious trend as

( ) ( ), 1, ,i iX n an b n n N

ˆˆ ˆ( )i i iX k a k b

1

ˆ'

12 1' ( )

( 1)( 1) 2

i i

N

i in

a a a

Na n n

N N N

( )i n

2. Statistical considerations

N = 5 10 20

N = 50

Moments of the spurious trend Mean of the spurious trend is 0

Standard deviation of the spurious trend is

Skewness is also 0

Kurtosis is given by

3

2'

1212

( 1)( 1)a NN N N

standard deviation ofnatural variability

kurtosis ofnatural variability

+ Monte Carlo simulation with Weibull (1,1) distribution

Probability density function (PDF) of the spurious trend

When the natural variability is Gaussian distribution

When it is non-GaussianEdgeworth expansion of the PDF Cf. Edgeworth expansion of sample mean (e.g., Shao 2003)

2

'2

'

1

2' 0,

'

1( ) ( ) .

2a

a

x

a Na

f x f x e

Edgeworth expansion of the cumulative distribution function, of is written by

and and is the PDF and the distribution function of , respectively.

where is k - th Hermite polynomialand is k - th cumlant ( ).

Non-Gaussian distribution

(0,1)N

Errors of t -test, Bootstrap test, and Edgeworth test for a non-Gaussian distribution of for a finite data length N

But the length of observed datasets is at most 5050 years.

Only numerical experimentsnumerical experiments can supply much longer datasets

to obtain statistically significant results,although they are not real but virtual.

We need accurate values ofthe moments of natural internal variability

for accurate statistical text.

3D global Mechanistic Circulation Model: Taguchi, Yamaga and Yoden(2001)

simplified physical processes

Taguchi & Yoden(2002a,b) parameter sweep exp. long-time integrations

Nishizawa & Yoden(2005) monthly mean T(90N,2.6hPa) based on 15,200 year data reliable PDFs

3. Internal variability in a numerical model

stratosphere troposphere

Labitzke diagram for normalized temperature (15,200 years)

Different dynamical processes

produce these seasonally

dependent internal variabilities

↓“Annual mean” may introduce

extra uncertainty or danger

into the trend argument

Estimation error of sample moments depends on deta length N and PDF of internal variability

Normalized sample mean: (mN － μ)/σε

Standard deviation of sample meanThe distribution converges to a normal distribution as N becomes large (the central limit theorem)

sample variance [ skewness, kurtosis, ... ]

stratosphere troposphere

1

2Nm N es s

-=

Spatial and seasonal distribution of moments 10 ensembles of 1,520-year integrations without external trend

65

More informationmoments of variations → moments of spurious trends

Zonal mean temperature

How many years do we need to get statistically significant trend ?

- 0.5K/decade in the stratosphere 0.05K/decade in the troposphere

Max value of the needed length Month for the max value

Necessary length for 99% statistical significance [years]

87N 47N

50-year data 20-year data

[K/decade] [K/decade]

How small trend can we detect in finite length data with statistical

significance ?

Cooling trend run 96 ensembles of 50-year integration with external linear trend

-0.25K/year around 1hPa

Normal (present) Cooled (200 years) Difference

[K/50years]

4. Spurious trend experiment

JAN (large internal variation)

JUL (small internal variation)

Standard deviation of internal variability

25298.0)20/50( ,12 2

3

2

3

'

sNsaTheoretical result

Ensemble mean of estimated trend and standard deviation of spurious trend

Edgeworth test

Comparison of significance tests Edgeworth test: true The worst case in 96

runs but both test look good

t-test

Bootstrap test

Application to real data 20-year data of NCEP/NCAR reanalysis

t-test

Bootstrap test

5. Concluding remarks

Statistical considerations on spurious trend in general non-Gaussian cases:

Edgeworth expansion of the spurious trend PDF detectability of “true” trend for finite data length enough length of data, enough magnitude of trend evaluation of t-test and bootstrap test

Very long-time integrations (~15,000 years) give reliable PDFs (non-Gaussian, bimodal, …. ), which give nonlinear perspectives on climatic variations and trend.

Recent progress in computing facilities has enabled us to do parameter sweep experiments with 3D Mechanistic Circulation Models.

Ensemble transient exp.(e.g., Hare et al., 2004) vs. Time slice (perpetual) exp.(e.g., Langematz, 200x)

assumption:internal variability is independent of time

m - member ensembles of N - year transient runsestimated trend in a run:

mean of the estimated trends:

two L-year time slice runsestimated mean in each run:

estimated trend:

comparison under the same cost: mN = 2L

New Japan reanalysis data JRA-25 now internal evaluation is ongoing

Statistics of internal variations of the atmosphere could be well estimated by long time integrations of state-of-the-art GCMs. Those give some characteristics of the nature of trend.

Time series of monthly averaged zonal-mean temperature

January

Estimated trend [K/decade]

90N

Normalized estimated trend and significance

90N

Thank you !

Estimated trend [K/decade]

90N

50N

Normalized estimated trend and significance

50N

90N

1. Introduction

Difference of the time variations between the two hemispheres

annual cycle: periodic response to the solar forcing

intraseasonal variations: mostly internal processes

interannual variations: external and internal causes

Daily Temperature at 30 hPa[K] for 19 years (1979-1997)

North Pole

South Pole

Difference of Gaussian distribution and Edgeworth for a non-Gaussian distribution of for a finite data length N

3. Spurious trends due to finite-length datasets with internal variability Nishizawa, S. and S. Yoden, 2005:

Linear trend IPCC the 3rd report (2001) Ramaswamy et al. (2001)

Estimation of sprious trend Weatherhead et al. (1998)

Importance of variability with non-Gaussian PDF

SSWs extreme weather events

We do not know PDF of spurious trend significance of the estimated value

stratosphere troposphere

Normalized sample variance

The distribution is similar to χ2distribution in the troposphere, where internal variability has nearly a normal distribution

Standard deviation of sample variance Nss

/222

stratosphere troposphere

Sample skewness

stratosphere troposphere

Sample kurtosis

Years needed for statistically significant trend -0.5K/decade in the stratosphere 0.05K/decade in the troposphere

Significance test of the estimated trend

t-testIf the distribution of is Gaussian, then the test statistic

follows the t-distribution with the degrees of freedom n -2

22

1

' 1 ˆˆ, ( )212

( 1)( 1)

i

i

Ni

i i i in

at s X n a n b

Ns

N N N

i

2. Trend in the real atmosphere

Datasets ERA40

1958-2002 1000-1 hPa

NCEP/NCAR 1948-2003 1000-10 hPa

JRA25 1979-1985,1991-1997 1000-1 hPa

Berlin Stratospheric data 1963-2000 100-10 hPa

Time series of monthly averaged

zonal-mean temperatureJanuary

90N 50N

EQ

90N 50N

EQ

July

90N 50N

Same period (1981-2000)January

90N 50NJuly

90N 50N

Same vertical factorJanuary

90N 50NJuly

Mean90N

Mean difference from ERA40

50N

Mean difference from ERA40

standard deviation90N

stddev difference from ERA40

50N

stddev difference from ERA40