Transcript wilby_weather_gen - global change SysTem for Analysis

Weather Generator
Methods
Dr Rob Wilby
King’s College London
A few wise words
“Probabilities direct the conduct of the wise man”
(Cicero, Roman orator, 106-43BC)
“The only certainty is uncertainty”
(Pliny the Elder, AD 23-79)
“As for me, all I know is I know nothing”
(Socrates, 470-399 BC)
Source: Katz (2002)
Presentation outline
• A brief history
• The “classic” weather generator approach
• Conditioning by atmospheric circulation patterns
• Weather generator applications
• Future directions
A brief history
Key publications in the development of daily
weather generators
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;
five Canadian cities
Tel Aviv
?
USA
USA
Newnham (1916);
Besson (1924); Gold
(1929); Cochran
(1938)
Wet and dry spell lengths have a geometric Williams (1952);
distribution
Longley (1953)
Use of Markov chain to reproduce geometric Gabriel and
distribution of wet and dry spell lengths
Neumann (1962)
Combined Markov occurrence model with
Todorovic and
exponential distribution for rainfall amounts
Woolhiser (1975)
Generation of max/min temperature, and
Richardson (1981)
solar radiation conditional on rain occurrence
Multi-site generalization of daily stochastic
Bras and Rodriguezprecipitation model
Iturbe (1976)
Distributions of daily wet (red) and dry (blue)
spell lengths at Cambridge, UK 1961-1990
approximated by geometric distributions
Precipitation (tenths mm)
600
500
400
300
200
100
0
1961
1966
1971
1976
1981
1986
1
Probability
0.1
0.01
0.001
0.0001
1
10
Spell length (days)
100
Distribution of daily wet day totals (tenths mm)
at Cambridge, UK 1961-1990 approximated
(poorly) by the exponential distribution
0.07
Probability
0.06
0.05
0.04
0.03
0.02
0.01
0
0
50
100
150
Precipitation (tenths mm)
200
250
The “classic” approach
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
p01 = Pr{precipitation on day t | no precipitation on day t-1}
and
p11 = Pr{precipitation on day t | precipitation on day t-1}
which are called transition probabilities.
Precipitation occurrence processes (cont.)
The transition probabilities for Cambridge, UK are as follows
dry-to-wet (p01) = 0.291
wet-to-wet (p11) = 0.654
Therefore it follows (for a two state model) that
dry-to-dry (p00) = 1 - p01 = 0.709
wet-to-dry (p10) = 1 - p11 = 0.346
This approach may be extended from a first-order to nth-order model
by considering transitions that depend on states on days t-1, t-2…...t-n
(as in Gregory et al., 1993).
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, , and whose probability
density function is:
1  x
f(x)  exp 
μ
 μ 
The two-parameter gamma distribution is a popular choice, defined by
the shape  and scale parameter :
f(x) =
(x β)α-1 exp[- x β]
βΓ (α )
Most weather generators make the assumption that precipitation
amounts on successive wet days are independent.
Precipitation amount processes (cont.)
January precipitation at
Ithaca, New York 19001998 represented by
three pdfs:
• exponential
• gamma
• mixed exponential
Source: Wilks and Wilby (1999)
Inverse normal transformation
140
Daily total (mm)
120
[1] raw data
[2] empirical pdf
[3] cumulative pdf
[4] normal pdf
100
80
60
40
20
0
[5] z-scores
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) + [B]ε(t )
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 (t) is white-noise forcing.
Other meteorological variables (cont.)
The z(t) are transformed to weather variables dependent on rainfall
occurrence:
Tk (t ) = {
μk,0 + σ k,0 (t )z k (t )
if day t is dry
μk,1 + σ k,1 (t )z k (t )
if day t is wet
where each Tk is any of the nonprecipitation variables, k,0 and k,0
are its mean and standard deviation for dry days, and k,1 and 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).
Daily weather generation (Markov chain)
Source: Wilks and Wilby (1999)
Daily weather generation (spell-lengths)
Source: Wilks and Wilby (1999)
Use of atmospheric patterns
Weather classification schemes may be used to
condition daily meteorological variables such as the
precipitation occurrence and intensity processes
Conditional probabilities of rainfall and mean
intensity at Kempsford, Cotswolds associated with
the main Lamb Weather Types (LWT), 1891-1910
LWT
Anticyclonic
Westerly
Cyclonic
Northerly
Northwesterly
Southerly
Easterly
Occurrence
(Pr)
Intensity
(mm/d)
0.125
0.599
0.792
0.426
0.384
0.621
0.561
1.58
3.04
5.15
2.16
1.58
4.43
4.61
Conditioning weather patterns may be derived from (a) observations;
(b) climate model output; (c) stochastic representations of (a) or (b).
Observed SD (mm)
Conditioning stochastic properties of daily
precipitation on indices of atmospheric circulation
65
60
55
50
45
40
35
30
25
Unconditional
25
35
45
Conditional
55
65
Model SD (mm)
Standard deviation of monthly precipitation at Valentia for
an unconditioned an induced SLP model (Kiely et al., 1998).
Conditioning variables:
day of the week (!),
month, season, year,
geography,
weather patterns,
moisture indices,
airflow/pressure indices,
hidden states,
NAOI and SOI, etc.
Multi-site daily weather
Repeat application of single-site methods (see example below)
Non-parametric (nearest neighbour, weather pattern) resampling
Spatially correlated random numbers
Fuzzy logic, neural networks
1
DET (w inter)
Eastern England
0.8
1
0.9
0.6
0.8
t(p)
0.7
SDSM
•
•
•
•
0.4
0.6
0.5
0.2
0.4
0.3
0
0.2
0.1
-0.2
0
0
0.1 0.2
0.3
0.4
0.5 0.6
0.7
0.8
0.9
1
Observed
Observed and downscaled
inter-site correlations for 12
stations in Eastern England
0
50
100
150
Distance (km)
200
250
300
Estimates of Kendall’s τ for the 90th percentile 20–day
winter maximum precipitation amounts across EE. Black
lines represent observations; blue/red are model estimates.
Applications
Generation of climate analogues
350
325
300
Ml/day
275
250
225
200
175
150
125
1971-90
historic
abstraction
1893
zero
abstraction
1893
historic
abstraction
1872
zero
abstraction
1872
historic
abstraction
Simulated 10-day annual minimum flow in the River Test under
extreme cyclonic (1872) and anticyclonic (1893) weather patterns.
Temporal disaggregation - Vegetation/Ecosystem
Modeling and Analysis Project (VEMAP)
• Daily Tmax/Tmin/PPT
using modified Richardson
(1981) approach;
• Parameterized using HCN/
Coop network and VEMAP
99-year monthly grid (0.5º);
• Separate parameters for
wet and dry periods (Wilks)
• Quality check of frequency
distributions/ extremes
• Not actual daily series
Source: http://www.cgd.ucar.edu/vemap/animations/index.html
Probability
Probability
.72
.7
.69
.71
.7
Detection of non-stationarity
.69
.68
.68
.67
.66
1900
.67
1910
1920
1930
1940
1950
Year
1960
1970
1980
1990
.66
1900
2000
.74
1910
1920
1930
1940
1950
Year
1960
1970
1980
1990
2000
1920
1930
1940
1950
Year
1960
1970
1980
1990
2000
1930
1940
1950
Year
1960
1970
1980
1990
2000
1930
1940
1950
Year
1960
1970
1980
1990
2000
.78
Edinburgh
.775
.72
Hastings
Probability
Probability
.77
.7
.68
.66
.765
.76
.755
.75
.64
.62
1900
.745
1910
1920
1930
1940
1950
Year
1960
1970
1980
1990
2000
.74
1900
2000
.76
.755
.75
.745
.74
.735
.73
.725
.72
.715
.71
.705
.7
.695
.69
1900
.78
.775
Kew
.77
.76
Probability
Probability
.765
.755
.75
.745
.74
.735
.73
.725
.72
1900
1910
1920
1930
1940
1950
Year
1960
1970
1980
1990
.78
.76
.76
.75
.75
.74
.74
.73
.72
1910
1920
Plymouth
.73
.72
.71
.71
.7
1900
Nottingham
.77
Oxford
Probability
Probability
.77
1910
.7
1910
1920
1930
1940
1950
Year
1960
1970
1980
1990
2000
.69
1900
1910
1920
Dry-spell persistence (p00) at selected sites in the UK
Source: Wilby (2001)
Statistical downscaling
Changes in station-series means
and variances will be proportional to
changes in the respective areaaverage (GCM grid) moments:
E[S(T )GCMfuture ]
E[S(T )station ]
E S(T )GCMpresent
μdown =
Tπ down
[
Source: Wilks (1999)
]
where S(T) is the sum of T daily
precipitation amounts, is the
unconditional probability of precipitation,
and  is the mean wet-day amount.
Future directions
Sub-daily models
Three steps in weather generator:
• Number of wet subperiods conditional on total daily amount;
• Relative distribution of rainfall amounts per wet period;
• Time series using Markov Chain Monte Carlo (MCMC) method.
Source: Bardossy (1997)
Seasonal forecasting
65
OF 1
2
Latitude (ÞN)
60
0.00
0.08
55
0.14
-0.02
0.02
0.06
-0.04 -0.08 -0.10
-0.18
-0.12
0.04
0.12
-0.16
-0.14
-0.06
50
0.18
0.16
45
0.08
40
-40
-0.10
0.10
EOF 2
-30
-20
-10
Longitude (ÞW)
65
-0.16
.95
60
8
OF 3
Using winter North Atlantic SST
anomalies to condition summer dry–
spell persistence (p00).
0.00
8
.9
0.04
Observed
Modelled
0.02
0.10
-0.00
0.08
-0.08
-0.02
50.75
0.02
-0.04 -0.02
-0.10 -0.08
-0.16 -0.14
.65
-0.22 -0.20
40
.6
.55
-40
1940
Hindcasts of summer dry–spell
persistence (p00) at Cambridge,
1946–1995, from preceding winter
SST anomalies.
-0.12
-0.06
-0.26
0.14
0.04
.7
45
-0.10
-0.04
0.06
.8
-0.16
-0.14
.85
55
Latitude (ÞN)
Probability
0.06
0
-0.00
0.12
-0.06
-0.12
-0.24
1950-30
EOF 4
1960
-20
-10
1970
1980
Longitude
Year (ÞW)
0
1990
2000
Source: Wilby (2001)
Summary of weather generator characteristics
Strengths
Weaknesses
 Computationally undemanding
thus enables generation of long
time-series and/or ensembles
 May be extended to multisite
generalizations
 Simultaneous generation of
several meteorological
variables conditional on
precipitation occurrence
 Applicable to climate analogues
 Requires classification (at the
very least wet/dry-day
definition)
 Precipitation amounts highly
sensitive to choice of probability
distribution function
 Adjustment of parameters can
have unexpected effects on
conditional variables
 Assumes stationarity of
conditional relationships
Further reading
Cameron, D., Beven, K. and Tawn, J. 2000. An evaluation of three stochastic rainfall models.
Journal of Hydrology, 228, 130-149.
Dessens,J., Fraile, R., Pont, V. and Sanchez, J.L. 2001. Day-of-the-week variability of hail in
southwestern France. Atmospheric Research, 59-60, 63-76.
Gregory, J.M., Wigley, T.M.L. and Jones, P.D. 1993. Application of Markov models to areaaverage daily precipitation series and interannual variability in seasonal totals. Climate
Dynamics, 8, 299-310.
Katz, R.W. 2002. Techniques for estimating uncertainty in climate change scenarios and impact
studies. Climate Research, 20, 167-185.
Kiely, G., Albertson, J.D., Parlange, M.B. and Katz, R.W. 1998. Conditioning stochastic
properties of daily precipitation on indices of atmospheric circulation. Meteorological
Applications, 5, 75-87.
Kilsby,C.G., Cowpertwait, P.S.P., O’Connell, P.E., and Jones, P.D. 1998. Predicting rainfall
statistics in England and Wales using atmospheric circulation variables. International Journal
of Climatology, 18, 523-539.
Richardson, C.W. 1981. Stochastic simulation of daily precipitation, temperature and solar
radiation. Water Resources Research 17,182-190.
Wilby, R.L. 2001. Downscaling summer rainfall in the UK from North Atlantic ocean
temperatures. Hydrology and Earth Systems Sciences, 5, 245–257.
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.