Representing Extreme Behavior in Climate Models

1
Representing Extreme Behavior in Climate Models

• L. Marx and J. L. Kinter III
• Center for Ocean-Land-Atmosphere Studies
• NOAA/NCEP/EMC Seminar
• 24 May 2005

2
Outline

• Motivation How can non-normal climate variables,
  like precipitation, be represented in order to
  compare climate observations and climate models
  quantitatively?
• Introduction to L-moments
• An alternative to assuming a normal distribution
• Estimating the moments
• Fitting a standard distribution and testing the
  fit with Monte Carlo simulation
• Application to C20C and analyzed precipitation
  observations

3
Living in a Red-Noise World

• Extreme events unusual magnitude, size, duration
  or frequency
• The lower the frequency, the larger the variance
  - wait long enough, and distributions reach
  extremes with larger magnitude and longer
  duration
• Extreme events are, by definition, infrequent
• Big Thompson Flood, 31 Jul 1976 (a millennial
  flood)
• Hurricane Mitch, 26 Oct - 5 Nov 1988 (10,000
  deaths Honduras semi-destroyed)
• Upper Mississippi flood, May-Sep 1993 (50 deaths
  15B damage)
• Dust Bowl, 1930s (population migration)
• Sahel drought, 1960s-1990s
• Pluvial drought epochs (decades to centuries)
  in the paleoclimate record (millenia)
• Two kinds predictable and unpredictable
• Synoptic understanding
• Attribution and predictability
• Prediction
• Studying extremes is in vogue
• Political Science gains relevancy if it can
  improve understanding and eventually predict
  those events that have the largest impact ?
  prevention, mitigation
• Sociological Extremes are fascinating

4
Probability Distributions

• Normal probability distribution functions (PDF)

Normal cumulative distribution functions (CDF)
5
Probability Distributions

• Normal probability distribution functions (PDF)

90
Normal cumulative distribution functions (CDF)
10
6
Precipitation is Non-Normal
Daily data
Monthly mean data

• Most parametric statistical analysis is built on
  assumption of normal distribution
• Many climate variables, e.g. precipitation are
  non-normal

7
General Circulation Models

• Intended to encapsulate the relevant processes in
  the atmosphere (and ocean)
• Aid understanding of dynamics and processes
• Serve as a prediction tool
• How do we determine the fidelity of a GCM?
• Comparison of statistics of models and
  observations, including all moments and
  statistical characteristics (EOFs, SVDs, etc.)
• Correlation, RMSE,
• When is a given model good enough?
• What represents a successful simulation of
  extreme events?
• What can we learn about improving a given model
  based on its simulation of, say, the
  precipitation distribution?

8
December Means
COLA ensemble mean
CAMS (observed)

• Observed and 10-member COLA AGCM C20C ensemble
  mean December mean (1949-2003) Europe precip are
  comparable

9
December Means
GSFC ensemble mean
COLA ensemble mean

• Two 10-member AGCM C20C ensemble mean December
  mean Europe precip are comparable

10
December Top Decile
CAMS
COLA ensemble mean

• Top 10 averages Is this comparison meaningful,
  given the fact that the model result is an
  ensemble average while Nature has only one
  realization?

11
December Top Decile
GSFC ensemble mean
COLA ensemble mean

• Top 10 averages Are the two models giving the
  same result and how do the two compare to Nature?
  

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Brief Intro to L-Moments

• L-moments are summary statistics for probability
  distributions and data samples. They are
  analogous to ordinary moments - they provide
  measures of location, dispersion, skewness,
  kurtosis, and other aspects of the shape of
  probability distributions or data samples - but
  are computed from linear combinations of the
  ordered data values (hence the prefix L).

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Intro to L-Moments (cont.)

• The L-moments of a sample distribution of n
  elements, sorted in ascending order as X1n,
  X2n, Xnn are defined as
• ??????? E (X11)
• ??2????1/2 E (X22 - X12)
• ??3????1/3 E (X33 - 2 X23 X13)
• ??4????1/4 E (X44 - 3 X34 3 X24 - X14)
• L-moments defined in this way maintain certain
  mathematical and statistical properties analogous
  to the uniform distribution

Hosking and Wallis, 1997 Regional Frequency
Analysis An Approach Based on L-Moments
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Intro to L-Moments (cont.)

• The first L-moment, ??1??is obviously the mean
  of the distribution. Statistics analogous to
  other moments of the normal distribution may be
  defined as follows
• L-CV ??2??/ ??1?? (coefficient of
  L-variation)
• ?3 ??3??/ ??2?? (L-skewness)
• ?4 ??4??/ ??2?? (L-kurtosis)

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Estimating L-moments

• If we define a quantity br as
• br n-1 ?i (i-1)(i-2)(i-r) / (n-1)(n-2)
  (n-r) Xin
• then the L-moments can be estimated as follows
  
• ???????b0
• ??2????2b1 - b0
• ??3????6b2 - 6b1 b0
• ??4????20b3 - 30b2 12 b1 - b0
• L-CV, ?3 and ?4 can then be defined based on
  these estimates.

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Properties of L-moments

• Due to linearity, all L-moments higher than the
  first are more robust than their normally-defined
  counterparts. It is required only that the
  distribution have finite mean and variance.
• The L-CV is bounded between 0 and 1, and ?3 and
  ?4 are bounded between -1 and 1, unlike the
  normally-defined skewness and kurtosis which are
  unbounded
• Asymptotic approximations to sampling
  distributions are better for L-moments than for
  ordinary moments.
• For a purely normally distributed variable
• ????? ?
• ??2?? ? ? ?
• ?3 0
• ?4 0.1226

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Parametric Distributions

• Normal distribution parameters
• Gamma (Pearson type 3) distribution
• Generalized Extreme Value distribution

location, scale, shape
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Fitting Distributions

• Plotting distributions in ?3 - ?4 space makes it
  possible to see how they differ
• Two-parameter distributions (e.g. normal) are
  represented as points in this space more general
  (3-parameter) distributions are curves
• Fitting particular distributions to sample data
  and choosing the best fit is accomplished by a
  modification of the Hosking and Wallis index
  flood procedure that involves pooling data and
  Monte Carlo simulation to estimate goodness of fit

Hosking and Wallis, 1997
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A word about sample size
N 20
N 80
Hosking and Wallis, 1997
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One advantage of L-moments
Hosking and Wallis, 1997
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One advantage of L-moments better separation of
distributions
Hosking and Wallis, 1997
22
Pooling Data

• To get a more robust estimate of the L-moments of
  a climate variable at a given point in
  space-time, we can pool the data by including
  data from adjacent points in space (e.g., 8
  neighboring gridboxes), time (days before and
  after), or both, in the distribution
• A representativeness test must be applied to
  the pooled data, e.g. by Monte Carlo simulation,
  to ensure that the pooling has not altered the
  characteristics of the distribution artificially

23
C20C Experiment

• C20C Climate of the 20th Century
• To what extent can the observed climate anomalies
  of the past 130 years be simulated in current
  climate models?
• 10 ensemble members (analyzed initial states)
• 1871 (1948) - 2003
• Hadley Centre HadISST1 SST and sea ice
• Climatological initial snow depth, soil moisture
  and temperature predicted thereafter
• Climatological land vegetation values
• www.iges.org/c20c/

24
COLA C20C AGCM v2.2

• T63L18 resolution
• NCAR CCM3 spectral dynamics with semi-Lagrangian
  moisture transport
• Simple SiB land surface model
• Relaxed Arakawa-Schubert convection Tiedtke
  shallow convection
• Harshvardhan longwave radiation Lacis and Hansen
  shortwave radiation
• Mellor-Yamada 2.0 turbulence closure
• Predicted supersaturation and convective clouds
• Gravity wave drag
• Documentation in Schneider (2000) Kinter et al.
  (1997)
• Data available on C20C web site

25
GFSC NSIPP-1 AGCM

• 3 X 3.75 resolution (simulated climate similar
  to 2 X 2.5)
• Suarez and Takacs (1995) dynamical core
• Mosaic land surface model
• Relaxed Arakawa-Schubert convection
• Chou and Suarez shortwave and longwave radiation
• Louis (1982) PBL scheme
• Documentation in Bacmeister et al. (2000)
• Data available on C20C web site
• Thanks to Siegfried Schubert of GSFC for
  providing the data (Schubert et al., 2004)

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Obs/Analyses

• CAMS monthly mean precipitation analysis from
  land-based gauges
• Higgins North American daily analysis

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Monthly Means

• The monthly mean precipitation from the two C20C
  experiments (COLA and GSFC model integrations)
  was compared with CAMS analysis of rain gauges,
  using L-moments

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Monthly Mean PDF December
COLA
CAMS
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Monthly Mean PDF July
COLA
CAMS
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CAMS - Europe (land only)December means,
1949-2003
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• CAMS

• COLA

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• CAMS
• Nearly all points in Europe clustered
• between Normal Gumbel distributions

Normal
Gumbel
Normal
Gumbel

• COLA
• Much broader distribution

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• CAMS
• Nearly all points in Europe clustered
• between Normal Gumbel distributions
• Almost no points with L-skewness

Normal
Gumbel
Normal
Gumbel
Negative Skewness

• COLA
• Much broader distribution
• Many points in Europe have
• L-skewness

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CAMS - (2E,48N) - December
A word about fitting 2- and 3-parameter
distributions
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CAMS - (2E,48N) - December
Reasonably good fit to several 3-parameter
distributions
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CAMS - (2E,48N) - December
What about lower quintile?
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CAMS - (2E,48N) - December
Wakeby
Bottom Quintile
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CAMS - (2E,48N) - December
What about upper quintile?
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CAMS - (2E,48N) - December
Top Quintile
Generalized Logistic
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Normal Distribution
Gumbel Distribution
GEV Distribution
Relative Errors of Fit
Lowest 20
2nd quintile
3rd quintile
4th quintile
Highest 20
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EM 0.03
OBS 0.14
EM -0.09
OBS 0.18
M9 -0.13
M2 -0.05
M6 0.10
M7 0.07
(2E,48N)
(2E,60N)

• Selected points show
• Ensemble mean is poor representation of
  distribution
• Some members do a credible job of matching the
  obs
• ?3 end of distribution

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?(2)
CAMS
COLA-EM
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?(2) Ensemble mean suppresses variability
determined in this way
CAMS
COLA-10
COLA-EM
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L-CV
CAMS
COLA-EM
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L-CV Similar to ?(2) except that mean bias
reduces amplitude
CAMS
COLA-10
COLA-EM
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L-Skewness
CAMS
COLA-EM
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L-Skewness Ensemble mean exagerrates negative
skewness problems
CAMS
COLA-10
COLA-EM
48
L-Kurtosis Sample size probably too small, even
with 10 ensemble members
CAMS
COLA-10
COLA-EM
49
CONUS January Mean
CAMS
COLA
GSFC
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CONUS January L-CV
CAMS
COLA
GSFC
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CONUS January ?3
CAMS
COLA
GSFC
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S. Asia July ?3 - ?4
CAMS
COLA
GSFC
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S. Asia July ?3 - ?4
CAMS
COLA
GSFC
Negative skewness
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S. Asia July ?3 - ?4
CAMS
COLA
GSFC
Bulk of distribution
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S. Asia July Mean
CAMS
GSFC
COLA
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S. Asia July L-CV
CAMS
COLA
GSFC
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S. Asia July ?3
CAMS
COLA
GSFC
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Daily Values

• The daily precipitation from the COLA C20C
  experiment was compared with Higgins analysis of
  North American rain gauges, using L-moments

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Daily PDF July
COLA
Higgins
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CDFs

• CDFs at 82W,39N of Higgins and COLA members

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1- and 5-Day Pooling

• The effect of 5-day pooling is to greatly
  stabilize the statistical nature of the data
  sample.

62
Effect of Ensemble Average

• The upper panel is identical to the previous
  picture (just for 5-day pooling) except for a
  single member of the COLA ensemble
• The bottom panel shows the same thing for the
  ensemble average

63
Eastern CONUS 1 July Mean (5-day pooling)
COLA
Higgins
The GCM has a bias toward coastal and orographic
rainfall, dessicated interior in summer
64
Eastern CONUS 1 July Median (5-day pooling)
COLA
Higgins
The median is quite different from the mean
(non-normal) and accentuates the GCM coastal and
orographic bias
65
Eastern CONUS 1 July Top Decile (10) (5-day
pooling)
COLA
Higgins
The GCM high extremes are distributed more to the
east, compared to the observed, suggesting that
the extreme summer convection over the plains
is not well represented in the model
66
Show Animation
67
1 July ?3- ?4
COLA 5-day pooled data
Higgins 5-day pooled data
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1 July ?3- ?4
COLA 5-day pooled data
Higgins 5-day pooled data fits a ?-distribution
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1 July ?3- ?4
COLA 5-day pooled data displaced from
?-distribution
Higgins 5-day pooled data fits a ?-distribution
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Conclusions

• L-moments analysis is a promising technique for
  quantifying precipitation distributions
• Provides a method for distinguishing among
  distributions ? can help diagnose model errors
• In particular, ?3 extremes are missing
• Provides more robust estimates of characteristics
  of variability ? can be used in place of more
  traditional statistical measures
• Ensemble averaging masks considerable richness in
  the variability and distribution of precipitation