YODEN Shigeo Dept. of Geophysics, Kyoto Univ., JAPAN March 3-4, 2005: SPARC Temperature Trend Meeting at University of Reading 1.Introduction 2.Statistical considerations 3.Internal variability in a numerical model 4.Spurious trend experiment 5.Concluding remarks Spurious Trend in Finite Length Dataset with 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 50 at most 50 years. Separation of the trend from natural variations is a big problem.  Observed variations

Linear Trend of the Monthly Mean Temperature ( Berlin, NCEP ) A spurious trend may exist in finite 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 2. Statistical considerations N = 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 standard deviation of natural variability kurtosis of natural 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-Gaussian  Edgeworth expansion of the PDF  Cf. Edgeworth expansion of sample mean (e.g., Shao 2003)

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 polynomial and is k - th cumlant ( ).  Non-Gaussian distribution

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 50 at most 50 years. numerical experiments Only numerical experiments can supply much longer datasets to obtain statistically significant results, although they are not real but virtual. We need accurate values of the 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: (m N － μ)/σ ε  Standard deviation of sample mean  The distribution converges to a normal distribution as N becomes large (the central limit theorem) sample variance [ skewness, kurtosis,... ] stratosphere troposphere

 Spatial and seasonal distribution of moments 10 ensembles of 1,520-year integrations without external trend 65 More information  moments 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 Theoretical 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 runs estimated trend in a run: mean of the estimated trends: two L-year time slice runs estimated 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 ( ) 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 χ 2 distribution in the troposphere, where internal variability has nearly a normal distribution Standard deviation of sample variance

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-test If the distribution of is Gaussian, then the test statistic follows the t-distribution with the degrees of freedom n -2

2. Trend in the real atmosphere Datasets  ERA hPa  NCEP/NCAR hPa  JRA , hPa  Berlin Stratospheric data hPa

 Time series of monthly averaged zonal-mean temperature January 90N50N EQ

90N50N EQ July

90N50N  Same period ( ) January

90N50N July

90N50N  Same vertical factor January

90N50N July

 Mean 90N Mean difference from ERA40

50N Mean difference from ERA40

 standard deviation 90N stddev difference from ERA40

50N stddev difference from ERA40