# Presentation: 904907

## 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

## 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

## 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

## Probability Distributions

- Normal probability distribution functions (PDF)

- 	Normal cumulative distribution functions (CDF)

## Probability Distributions

- 	Normal cumulative distribution functions (CDF)

**10%**

**90%**

- Normal probability distribution functions (PDF)

## Precipitation is Non-Normal

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

- Monthly mean data

- Daily data

## 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?

## December Means

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

- COLA ensemble mean

- CAMS (observed)

## December Means

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

- GSFC ensemble mean

- COLA ensemble mean

## December Top Decile

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

- CAMS

- COLA ensemble mean

## December Top Decile

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

- GSFC ensemble mean

- COLA ensemble mean

## 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*).

## Intro to L-Moments (cont.)

- 	The *L*-moments of a sample distribution of *n* elements, _sorted in ascending order_ as: **X**1:n, **X**2:n, ... **X**n:n are defined as
- 		     E  (X1:1)
- 		21/2 E  (X2:2 - X1:2)
- 		31/3 E  (X3:3 - 2 X2:3 + X1:3)
- 		41/4 E  (X4:4 - 3 X3:4 + 3 X2:4 - X1:4)
- 	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

## 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)

## 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)] Xi:n
- 	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.

## 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

## Parametric Distributions

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

**location, scale, shape**

## 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

## A word about sample size ...

**A word about sample size ...**

**N = 20:**

**N = 80:**

- Hosking and Wallis, 1997

## One advantage of L-moments:

**One advantage of L-moments:**

- Hosking and Wallis, 1997

## One advantage of L-moments: better separation of distributions ...

**One advantage of L-moments: better separation of distributions ...**

- Hosking and Wallis, 1997

## 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

## 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/

## 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

## 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)

## Obs/Analyses

- CAMS _monthly mean_ precipitation analysis from land-based gauges
- Higgins North American _daily_ analysis

## 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

## Monthly Mean PDF

**Monthly Mean PDF**
**December**

**CAMS**

**COLA**

## Monthly Mean PDF

**Monthly Mean PDF**
**July**

**CAMS**

**COLA**

## CAMS - Europe (land only)December means, 1949-2003

**CAMS - Europe (land only)****December means, 1949-2003**

## CAMS

**CAMS**

**                                 ****COLA**

## CAMS

**CAMS**
-  Nearly all points in Europe clustered
-   between Normal & Gumbel distributions

**Gumbel**

**Normal**

**Normal**

**Gumbel**

**                                 ****COLA**
-  Much broader distribution

## CAMS

**CAMS**
-  Nearly all points in Europe clustered
-   between Normal & Gumbel distributions
-  Almost no points with *L*-skewness < 0

**Gumbel**

**Normal**

**Negative**
**Skewness**

**Normal**

**Gumbel**

**                                 ****COLA**
-  Much broader distribution
-  Many points in Europe have
*L*-skewness < 0

## A word about fitting 2- and 3-parameter distributions ...

**A word about fitting 2- and 3-parameter distributions ...**

**CAMS** - (2°E,48°N) - December

## CAMS - (2°E,48°N) - December

**CAMS** - (2°E,48°N) - December

**Reasonably good fit to several 3-parameter distributions**

## CAMS - (2°E,48°N) - December

**CAMS** - (2°E,48°N) - December

**What about lower quintile? **

## CAMS - (2°E,48°N) - December

**CAMS** - (2°E,48°N) - December

**Bottom Quintile**

- Wakeby

## CAMS - (2°E,48°N) - December

**CAMS** - (2°E,48°N) - December

**What about upper quintile? **

## CAMS - (2°E,48°N) - December

**CAMS** - (2°E,48°N) - December

**Top Quintile**

- Generalized Logistic

## Relative

**Lowest 20%**

**2nd quintile**

**3rd quintile**

**4th quintile**

**Highest 20%**

**Normal Distribution**

**Gumbel Distribution**

**GEV Distribution**

**Relative**
**Errors of**
**Fit**

## (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 < 0 indicates lack of representation of upper end of distribution

- OBS
- 0.18

- M6
- 0.10

- M9
- -0.13

- EM
- -0.09

- M7
- 0.07

- OBS
- 0.14

- EM
- 0.03

- M2
- -0.05

**Notes:**
Left chart: (2E,48N); T3 = 0.18 (CAMS), 0.10 (#6), -0.12 (#9), -0.09 (EM)
Right chart: (2E, 60N); T3 = 0.14 (CAMS), -0.05 (#2), 0.07 (#7), 0.02 (EM)

## CAMS

**CAMS**

**COLA-EM**

******(2)**

## CAMS

**CAMS**

**COLA-EM**

******(2)**
**Ensemble mean suppresses**
**variability determined in this way**

**COLA-10**

## CAMS

**COLA-EM**

**L-CV**

## CAMS

**COLA-EM**

**COLA-10**

**L-CV**
**Similar to ********(2)**** except that mean **
**bias reduces amplitude**

## CAMS

**CAMS**

**COLA-EM**

**L-Skewness**

## CAMS

**CAMS**

**COLA-10**

**COLA-EM**

**L-Skewness**
**Ensemble mean exagerrates**
**negative skewness problems**

## CAMS

**CAMS**

**COLA-10**

**COLA-EM**

**L-Kurtosis**
**Sample size probably too small,**
**even with 10 ensemble members**

## CONUS

**January**
**Mean**

**CAMS**

**COLA**

**GSFC**

## CONUS

**CONUS**
**January**
**L-CV**

**CAMS**

**COLA**

**GSFC**

## CONUS

**CONUS**
**January**
******3**

**COLA**

**GSFC**

**CAMS**

## CAMS

**CAMS**

**COLA**

**GSFC**

**S. Asia**
**July**
******3 ****-**** ********4**

## CAMS

**CAMS**

**COLA**

**GSFC**

**S. Asia**
**July**
******3 ****-**** ********4**

**Negative**
**skewness**

## CAMS

**CAMS**

**COLA**

**GSFC**

**S. Asia**
**July**
******3 ****-**** ********4**

**Bulk of**
**distribution**

## S. Asia

**S. Asia**
**July**
**Mean**

**CAMS**

**GSFC**

**COLA**

## S. Asia

**S. Asia**
**July**
**L-CV**

**CAMS**

**COLA**

**GSFC**

## S. Asia

**S. Asia**
**July**
******3**

**CAMS**

**COLA**

**GSFC**

## Daily Values

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

## Daily PDF

**July**

**Higgins**

**COLA**

## CDFs

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

## 1- and 5-Day Pooling

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

## 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

## Eastern CONUS

**Eastern CONUS**
**1 July**
**Mean**
**(5-day pooling)**

**Higgins**

**COLA**

**The GCM has a bias toward coastal and **
**orographic rainfall, dessicated interior **
**in summer**

## Eastern CONUS

**Eastern CONUS**
**1 July**
**Median**
**(5-day pooling)**

**Higgins**

**COLA**

**The median is quite different from the**
**mean (non-normal) and accentuates **
**the GCM coastal and orographic bias**

## Eastern CONUS

**Eastern CONUS**
**1 July**
**Top Decile (10%)**
**(5-day pooling)**

**Higgins**

**COLA**

**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**

## Show Animation

## Higgins

**Higgins**
**5-day pooled data **

**1 July**
******3****- ********4**

**COLA**
**5-day pooled data**

## Higgins

**Higgins**
**5-day pooled data **
**fits a ********-distribution**

**1 July**
******3****- ********4**

**COLA**
**5-day pooled data**

## Higgins

**Higgins**
**5-day pooled data **
**fits a ********-distribution**

**1 July**
******3****- ********4**

**COLA**
**5-day pooled data**
**displaced from ********-distribution**** **

## 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 < 0 indicates that high value 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