t - Regional Climate Group

Weather Generator 

Downscaling Summer School in Lodz, 21 June 2007 

Deliang Chen 

www.gvc2.gu.se/rcg/dc 

Professor of Physical Meteorology and 

August Röhss Chair in Physical Geography (Geoinformatics) 

Department of Earth Sciences 

Göteborg University, Sweden 

Göteborg University http://www.gvc2.gu.se/rcg Regional Climate Group 

1

Topics covered 

• Statistical downscaling: Techniques 

and methodologies 

• What is a Weather Generator (WG) 

• Two types of WG 

• How WG is used in downscaling 

• Suggested further readings 


2

Acknowledgement 

• Some of the OHs appeared in my 

lectures are adopted from 

CCIS, Dr. Pierluigi Calanca and 

Dr. Rob Wilby. 


3

Temporal and spatial scales 

of climate 

(Global 

climate 

model) 

GCM 

RCM 

(Regional 

climate 

model) 


4

How to build the statistical model 

• Transfer functions: quantify relationship between large-scale 

climate (predictor) and local variables (predictand) by 

e.g. regression. Most often used for monthly temperature and 

precipitation. 

Related to Perfect Prognosis (PP) and Model Output Statistic 

(MOS) in numerical weather prediction. 

• Weather typing: relate a particular atmospheric state 

to a set of local climate variables 

(Lamb weather typing, Grosswetterlagen, others..) 

• Weather generators: generate realistic looking sequences 

of e.g. daily precipitation based on the large-scale 

atmospheric state 


5

Statistical downscaling: methods 

• linear – in large majority of studies 

– regression 

– canonical correlation / covariance (SVD) 

analysis 

• nonlinear 

– neural networks 

– analog 


6

Statistical downscaling: 

Reproduction of variance 

• underestimated variance in downscaled data 

• two ways of reproduction of variance 

– inflation (enhancement of anomalies by a 

constant factor) 

• physically questionable all forcing comes 

from large-scales 

– adding noise 

• white noise 

• regression residuals 

• noise with pre-specified statistical 

characteristics 


7


reproduction of variance 

observed 

downscaled: stddev = 58% obs 

downscaled inflated 


8



observed 

downscaled: stddev = 58% obs 

white noise 

downscaled with added white noise 


9

Statististical downscaling: 


obs down infl 

added 

white n. 

correlation 

with OBS 

1.000 0.826 0.826 0.598 

autocorrel. 0.587 0.619 0.619 0.305 


10

A successful statistical 

downscaling requires 

•A strong statistical relationship between local and 

large scale climate variables be established; 

•The statistical relationship should be physically 

interpretable and temporally stationary; 

•The large scale variables must be those that can be 

reliably produced by GCM; 

•Attempts should be made to understand the 

unexplained part by the statistical model and to 

assess/compensate its effect if possible. 


11

Added value of downscalings (SD,DD) 

120 

100 

80 

Hellstroem et al., 2001 

Vänersborg, One station in southern Sweden 

obs 

SDE 

DDE 

ECHA 

Precipitation (mm/month) 

60 

40 

20 

120 

100 

80 

1 2 3 4 5 6 7 8 9 10 11 12 

obs 

SDH 

DDH 

HadC 

60 

40 

20 

1 2 3 4 5 6 7 8 9 10 11 12 

Month 


b 

12

Differences between 

real world and GCM-world 


13

Two extremes to categories of downscaling: 

• Transfer functions relating atmospheric forcing to target 

variable 

• Stochastic functions and pure weather generators 

For both: variance explained as a function of the large scale flow, residual 

variance can only be stochastically generated. 

For future climate, only change due to the signal contained in the GCM 

scale forcing can be accounted for ….. 

Variance explained 

Synoptic Forcing 

Sub-grid scale forcings 

Synoptic scale 

Local scale 


14

Transfer Functions 

Area 

Grid Box 

Select 

predictor 

variables 

Calibrate 

and verify 

model 

Predictor variables 

e.g., MSLP, 500, 700 hPa geopotential heights, 

zonal/meridional components of flow, areal T&P 

Transfer function 

e.g., Multiple linear regression, principal 

components analysis, canonical correlation 

analysis, artificial neural networks 

Extract 

predictor 

variables 

from GCM 

output 

Drive 

model 

Observed station data for 

predictand 

Site variables for 

future, e.g., 2050 


15

Weather Typing 

Identify 

weather 

types 

Derive 

Observed 

weather 

variables 

Select 

classification 

scheme 

Relationships between 

weather type and local 

weather variables 

Pressure fields 

from GCM 

Calculate 

weather 

types 

Drive 

model 

Local weather 

variables for, 

say, 2050 

Statistically relate 

observed station or 

area-average 

meteorological data to 

a weather 

classification scheme. 

Weather classes may 

be defined objectively 

(e.g. by PCA, neural 

networks) or 

subjectively derived 

(e.g., Lamb weather 

types [UK], European 

Grosswetterlagen) 


16

Weather Typing 

Fundamental Assumption 

the relationships between weather type and local 

climate variables will continue to be valid under 

future radiative forcing 

ADVANTAGES 

• founded on sensible physical linkages between 

climate on the large scale and weather on the local 

scale 


17

Example: Investigate the link between circulation and 

winter temp. (1) 

(Chen, 2000: A monthly circulation climatotolgy for Sweden and its application to a winter 

temperature case study. Int. journal of Climatology 20, 1067-1076 

Approach: transfer function (multiple regression) 

Predictant=f(predictors) 

winter temp. 

circulation 

(strength of geostrophic flow 

and vorticity) 

T a =-3.48+0.079v+0.32V+0.17ξ 

circulation indices 


18

Example 1: cont…… 

How to quantify of atmospheric circulation 

Circulation indices and weather typing based on the system by Lamb 

ξ 

6 circulation indices: 

U 

V 

u: west-east geostr. wind 

v: north-south geostr. wind 

W: total geostr. wind 

ξ u 

: westerly shear vorticity 

ξ v 

: southerly shear vorticity 

ξ: total vorticity 

(after Chen and Li, 2003) 


19

Example 1: cont…… 

Temperature anomaly ( o C) 

7 

6 

5 

4 

3 

2 

1 

0 

-1 

-2 

-3 

-4 

-5 

-6 

-7 

-8 

January temperature in SW Sweden 

R=0.84 

N=122 

observation 

reconstruction 

-9 

1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 

Year 

Chen, 2000 


20

General categories of methodologies 

Transfer Weather Stochastic conditioned Stochastic 

Function Typing on weather type 

Stochastic / weather generators calibrated on observed 

data, conditioned on atmospheric state. 

• Very effective at capturing high frequency variance, 

peaks and extremes 

• Requires long term data sets to effectively define 

stochastic characteristics 

• Question of stationarity 


21

What is a WG 

WG is a and stochastic (based 

on observed probability) 

model of meteorological 

variables. It can generate 

unlimited length of data. 


22

The key problem->why WG 

Output of GCM 

~ 300 km 

~ 1 mo 

Ecosystem and other 

Impact models 

~ 1-10 km 

~ 1 h-1 day 

⇒ Disparity of spatial (and temporal) scales 


23

Stochastic weather generators: 

basic idea 

given slow set of statistics (monthly means and 

standard deviations, Y, from downscaling), 

generate the high frequency variability of the 

weather (y) assuming auto- and cross correlation: 

⇒ y(t) = O T [Y, y(t-1)] 

where O T is the time operator. 


24

Matching the scales: 

two common approaches 

(i) nesting Regional Climate Models (RCMs, 

horizontal resolution Δx < 10 km) into GCMs. 

Advantage: mathematical description of system is 

basically preserved! 

However: CPU ~ 1/(Δx) 2 or even 1/(Δx) 3 

⇒ Running a RCM at Δx ~ 1 km requires 

10 4 to 10 6 as much CPU as at Δx ~ 100 km! 


25

(ii) Statistical downscaling (space dimension) at 

low temporal resolution (say Δt ~ 1 mo), 

followed by stochastic generation of weather 

sequences (time dimension) at a fixed location. 

Advantage: computationally efficient 

⇒ large number of simulations feasible! 


26

Key publications in the development of daily WG 

Site(s) Observation Source 

Brussels Wet and dry days tend to cluster Quetelet (1852) 

Kew, Aberdeen, 

Greenwich, Valencia 

Probability of a rain day is greater if the 

previous day was wet 

Rothamstead, UK; Wet and dry spell lengths have a geometric 

five Canadian cities distribution 

Tel Aviv 

Use of Markov chain to reproduce geometric 

distribution of wet and dry spell lengths 

Combined Markov occurrence model with 

exponential distribution for rainfall amounts 

USA 

Generation of max/min temperature, and 

solar radiation conditional on rain occurrence 

USA 

Multi-site generalization of daily stochastic 

precipitation model 

Newnham (1916); 

Besson (1924); Gold 

(1929); Cochran 

(1938) 

Williams (1952); 

Longley (1953) 

Gabriel and 

Neumann (1962) 

Todorovic and 

Woolhiser (1975) 

Richardson (1981) 

Bras and Rodriguez- 

Iturbe (1976) 

Source: Wilby 


27

Precipitation occurrence process 

Most weather generators contain separate treatments of the 

precipitation occurrence and intensity processes. 

A first-order Markov chain for precipitation occurrence is fully 

defined by two conditional probabilities 

and 

p 01 = Pr{precipitation on day t | no precipitation on day t-1} 

p 11 = Pr{precipitation on day t | precipitation on day t-1} 

which are called transition probabilities. 


28

Precipitation occurrence processes (cont.) 

The transition probabilities for Cambridge, UK are as follows 

dry-to-wet (p 01 ) = 0.291 

wet-to-wet (p 11 ) = 0.654 

Therefore it follows (for a two state model) that 

dry-to-dry (p 00 ) = 1 - p 01 = 0.709 

wet-to-dry (p 10 ) = 1 - p 11 = 0.346 

This approach may be extended from a first-order to n th -order model 

by considering transitions that depend on states on days t-1, t-2…...t-n 

(as in Gregory et al., 1993). 


29

Precipitation amount processes 

Daily precipitation amounts are typically strongly skewed to the right. 

The simplest reasonable model is the exponential distribution, as it 

requires specification of only one parameter, μ, and whose probability 

density function is: 

f(x) 

= 

1 

μ 

⎡− x 

exp⎢ 

⎣ μ 

The two-parameter gamma distribution is another popular choice, 

defined by the shape α and scale parameter β: 

f(x) 

= 

⎤ 

⎥ 

⎦ 

α-1 

( ) [ ] 

x 

β 

exp - 

βΓ( α) 

x 

β 

Most weather generators make the assumption that precipitation amounts on 

successive wet days are independent. 


30

Precipitation amount processes (cont.) 

January precipitation at 

Ithaca, New York 1900- 

1998 represented by 

three pdfs: 

• exponential 

•gamma 

• mixed exponential 

Source: Wilks and Wilby (1999) 


31

Daily total (mm) 

140 

120 

100 

80 

60 

40 

Precipitation and its PDF 

[1] raw data 

20 

0 

0.4 

0.35 

0.3 

0.25 

0.2 

[2] empirical pdf 

0.15 

0.1 

0.05 

0 

0 10 20 30 40 50 

Daily to tal (mm) 


32

Other meteorological variables 

Condition the statistics of the daily variables (typically maximum/ 

minimum temperatures and solar radiation) on occurrence of 

precipitation (a proxy for other processes such as cloud cover). 

In the classic WGEN model, multiple variables are modelled 

simultaneously with auto-regression: 

z 

() [ ] ( ) [ ] () 

t 

= 

A z 

t 

- 1 

+ 

Where z(t) are normally distributed values for today’s nonprecipitation 

variables, z(t-1) are corresponding values for the previous day, and [A] 

and [B] are K × K matrices of parameters, and ε(t) is white-noise forcing. 

B 

ε 

t 


33

Other meteorological variables (cont.) 

The z(t) are transformed to weather variables dependent on rainfall 

occurrence: 

T 

k 

() t 

= { 

μ 

μ 

k,0 

k,1 

+ σ 

+ σ 

k,0 

k,1 

() t z () t 

k 

() t z () t 

k 

if day t is dry 

if day t is wet 

where each T k is any of the nonprecipitation variables, μ k,0 and σ k,0 

are its mean and standard deviation for dry days, and μ k,1 and σ k,1 

are its mean and standard deviation for wet days. 

Seasonal dependence of the means and standard deviations is 

usually achieved through Fourier harmonics (i.e., sine and cosines). 


34

WG type I: Markov chain (transition probabilities) 



35

WG type II: spell-lengths 



36

The Richardson type WG 

Precipitation Process 

Occurrence Amount 

Non-precipitation variables 

Maximum temperature 

Minimum temperature 

Solar radiation 

Model calibration 

Synthetic data generation 

Climate scenarios 



37

Weather Generators 

Precipitation Process 

Occurrence 

Amount 

Non-precipitation variables 

Maximum temperature 

Minimum temperature 

Solar radiation 

LARS-WG: 

wet and 

dry spell 

length 

Model calibration 

Synthetic data generation 



38

One application of WG: Create 

data for unsampled sites 

Assumption: compared to a 

meteorological variable it self, 

parameters of a WG for that 

variable can be more accurately 

interpolated from the sampled 

sites. 


39

Spatial Downscaling with WG 

Calibrate weather 

generator using 

area-average 

weather 

Calibrate weather 

generator for each 

individual station 

within area 

Calculate changes 

in parameters from 

grid box data 

Area 

Area parameter set 

Station parameter set 

Grid Box 

Apply changes in 

parameters derived from 

difference between area 

and grid box parameter 

sets to individual station 

parameter files; generate 

synthetic data for 

scenario 


40

Statistical spatial downscaling 

Source: Wilks (1999) 

Changes in station-series means 

and variances will be proportional to 

changes in the respective areaaverage 

(GCM grid) moments: 

μ 

down 

E S( T) 

E[ S( T) 

] 

station 

E S( T) 

= 

Tπ 

[ ] 

[ ] 

down 

GCMfuture 

GCMpresent 

where S(T) is the sum of T daily 

precipitation amounts,π is the 

unconditional probability of precipitation, 

and μ is the mean wet-day amount. 


41

Temporal Downscaling->high temporal 

resolution! – Use of monthly scenarios 

Observed station data 

WG 

Parameter file containing 

statistical characteristics 

of observed station data 

Monthly scenario 

information from 

GCM, RCM or SD 

Generate daily weather 

data corresponding to 

the monthly scenario 


42


Fundamental Assumption 

The statistical correlations between climatic variables 

derived from observed data are assumed to be valid under a 

changed climate. 

Advantages 

• the ability to generate time series of 

unlimited length 

• opportunity to obtain representative 

weather time series in regions of data 

sparsity, by interpolating observed data 

• ability to alter the WG’s parameters in 

accordance with scenarios of future climate 

change - changes in variability as well 

mean changes 


43


Disadvantages 

• seldom able to describe all aspects of 

climate accurately, especially persistent 

events, rare events and decadal- or centuryscale 

variations 

• designed for use, independently, at 

individual locations and few account for the 

spatial correlation of climate 


44

NCC/RCG-WG 

• Version 1: released in April 2004 with 

parameters determined for 672 Chinese and 

300 Swedish rainfall stations. Precipitation 

only. 

• Version 2.0: released with parameters 

determined for 671 Chinese (temperature and 

precipitation) and 300 Swedish rainfall 

stations. 

• bcc.cma.gov.cn 


45

Application of NCC/RCG-WG for China 

0.0~0.2 

Annual mean 

P(WD) 

0.3~0.5 

Annual mean P(WW) 

0.7~0.9 


46

Application of NCC/RCG-WG for China 

0.60 

转移概率 P(WD) 

转移概率 (PWW) 

0.50 

0.40 

0.30 

0.20 

0.10 

0.00 

1 2 3 4 5 6 7 8 9 10 11 12 

0.90齐齐哈尔乌鲁木齐北京常德海口 

0.80 

0.70 

0.60 

0.50 

0.40 

0.30 

月份 

月份 

1 2 3 4 5 6 7 8 9 10 11 12 

齐齐哈尔乌鲁木齐北京常德海口 

•Five stations are chosen 

to represent regional 

differences 

•For most months it is true 

that P(WW)> P(WD) 

•The seasonal cycle of 

P(WD) is relatively similar 

among different regions, 

while the seasonal cycle of 

P(WW)show greater 

difference. 


47

GAMMA distribution of daily precipitation 

The probability of daily precipitation may be 

described by a Gamma function: 

where 


48

Two examples 

Beijing 

Haike 


49

Two parameters of the GAMMA function 

ALPHA 

BETA 

1.40 

1.20 

1.00 

0.80 

0.60 

0.40 

0.20 

0.00 

50.0 

月份 

1 2 3 4 5 6 7 8 9 10 11 12 

40.0 


30.0 

20.0 

10.0 

0.0 

月份 

1 2 3 4 5 6 7 8 9 10 11 12 

The two parameters do 

not very much with 

the regions. This is 

especially true for 

ALPHA. -> great 

petential for 

accuarte spatial 

interpolations!!! 



50

Regional distribution of GAMMA parameters 

1~3 

>18 


51

Simulated and observed monthly and 

annual mean rain rates 

3000 

2500 

2000 

y = 1.0004x 

R 2 = 0.9968 

Annual 

precipitation 

模拟 

1500 

1000 

500 

700 

600 

500 

y = 0.9991x 

R 2 = 0.989 

0 

0 500 1000 1500 2000 2500 3000 

实测 

模拟值 

400 

300 

200 

Monthly 

precipitation 

100 

0 

0 100 200 300 400 500 600 700 

实测值 


52

Simulated and observed monthly mean rain days 

and mean annual number of days with precipitation 

rate greater than 20 mm/day 

30 

y = 1.0007x 

25 

R 2 = 0.9856 

模拟值 

20 

15 

monthly mean 

rain days 

10 

5 

30 

27 

24 

y = 0.9147x 

R 2 = 0.9826 

21 

0 

0 5 10 15 20 25 30 

实测值 

模拟 

18 

15 

12 

Day number 

with P>20 

mm/day 

9 

6 

3 

0 

0 3 6 9 12 15 18 21 24 27 30 

实测 


53

Simulated and observed daily mean and 

standard deviation of precipitation 

模拟湿日平均降水量 (mm) 

30 

25 

20 

15 

10 

5 

y = 0.9995x 

R 2 = 0.9869 

Mean daily 

Precipitation 

50 

40 

y = 0.8505x 

R 2 = 0.9471 

0 

0 5 10 15 20 25 30 

实测湿日平均降水量 (mm) 

Standard 

deviation 

模拟湿日降水量标准差 

30 

20 

10 

0 

0 10 20 30 40 50 

实测湿日降水量标准差 


54

Suggested Further Readings 

• IPCC TAR - Chapter 10 (www.ipcc.ch) 

• Richardson, C.W. (1981): Stochastic 

simulation of daily precipitation, temperature 

and solar radiation. Water Resources 

Research 17, 182-190. 

• Wilks, D.S. (1992): Adapting stochastic 

weather generation algorithms for climate 

change studies. Climatic Change 22, 67-84. 

• Wilks, D.S. and Wilby, R.L. (1999): The 

weather generation game: a review of 

stochastic weather models. Progress in 

Physical Geography 23, 329-357. 


55

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