Weather Generator Methods

Dr Rob WilbyKing’s College London

“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)

A few wise words

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

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 theprevious day was wet

Newnham (1916);Besson (1924); Gold(1929); Cochran(1938)

Rothamstead, UK;five Canadian cities

Wet and dry spell lengths have a geometricdistribution

Williams (1952);Longley (1953)

Tel Aviv Use of Markov chain to reproduce geometricdistribution of wet and dry spell lengths

Gabriel andNeumann (1962)

? Combined Markov occurrence model withexponential distribution for rainfall amounts

Todorovic andWoolhiser (1975)

USA Generation of max/min temperature, andsolar radiation conditional on rain occurrence

Richardson (1981)

USA Multi-site generalization of daily stochasticprecipitation model

Bras and Rodriguez-Iturbe (1976)

Key publications in the development of daily weather generators

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Distributions of daily wet (red) and dry (blue) spell lengths at Cambridge, UK 1961-1990 approximated by geometric distributions

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Spell length (days)

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Distribution of daily wet day totals (tenths mm) at Cambridge, UK 1961-1990 approximated (poorly) by the exponential distribution

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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:

μx

expμ1

f(x)

( ) [ ]( )αβΓ

βx-expβx=f(x)

1α-

The two-parameter gamma distribution is a popular choice, defined by the shape and scale parameter :

Most weather generators make the assumption that precipitation amounts on successive wet days are independent.

Precipitation amount processes (cont.)

Source: Wilks and Wilby (1999)

January precipitation at Ithaca, New York 1900-1998 represented by three pdfs:• exponential• gamma• mixed exponential

Inverse normal transformation

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[1] raw data

[3] cumulative pdf

[2] empirical pdf

[4] normal pdf

[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:

( ) [ ] ( ) [ ] ( )tε+1-t=t BzAz

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:

( )( ) ( )

( ) ( )tztσ+μ

tztσ+μ{=tT

kk,1k,1

kk,0k,0

k

if day t is dry

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

Conditioning weather patterns may be derived from (a) observations; (b) climate model output; (c) stochastic representations of (a) or (b).

Conditioning stochastic properties of daily precipitation on indices of atmospheric circulation

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Model SD (mm)

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Unconditional Conditional

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

DET (winter)

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SDSM

Observed and downscaled inter-site correlations for 12 stations in Eastern England

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.

• Repeat application of single-site methods (see example below)• Non-parametric (nearest neighbour, weather pattern) resampling• Spatially correlated random numbers• Fuzzy logic, neural networks

Applications

Generation of climate analogues

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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

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Bradford Cambridge

Durham Edgbaston

Edinburgh Hastings

Kew Nottingham

Oxford Plymouth

Detection of non-stationarity

Dry-spell persistence (p00) at selected sites in the UKSource: Wilby (2001)

Statistical downscaling

Changes in station-series means and variances will be proportional to changes in the respective area-average (GCM grid) moments:

( )[ ] ( )[ ]( )[ ]

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GCMpresent

GCMfuturestation

down TπTSETSE

TSE=μ

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)

Hindcasts of summer dry–spell persistence (p00) at Cambridge, 1946–1995, from preceding winter SST anomalies.

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Seasonal forecasting

Source: Wilby (2001)

Using winter North Atlantic SST anomalies to condition summer dry–spell persistence (p00).

Summary of weather generator characteristics

Strengths Weaknesses

Computationally undemandingthus enables generation of longtime-series and/or ensembles

May be extended to multisitegeneralizations

Simultaneous generation ofseveral meteorologicalvariables conditional onprecipitation occurrence

Applicable to climate analogues

Requires classification (at thevery least wet/dry-daydefinition)

Precipitation amounts highlysensitive to choice of probabilitydistribution function

Adjustment of parameters canhave unexpected effects onconditional variables

Assumes stationarity ofconditional relationships

Further readingCameron, 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 area-

average 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.