1. Weather Generator Methods
Dr Rob Wilby
King’s College London

2. A few wise words “Probabilities direct the conduct of the wise man”
(Cicero, Roman orator, BC)
“The only certainty is uncertainty”
(Pliny the Elder, AD 23-79)
“As for me, all I know is I know nothing”
(Socrates, BC)
Source: Katz (2002)

3. Presentation outline A brief history
The “classic” weather generator approach
Conditioning by atmospheric circulation patterns
Weather generator applications
Future directions

4. A brief history

5. Key publications in the development of daily weather generators

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

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

8. The “classic” approach

9. 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.

10. 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).

11. 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:
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.

12. Precipitation amount processes (cont.)
January precipitation at Ithaca, New York represented by three pdfs:
exponential
gamma
mixed exponential
Source: Wilks and Wilby (1999)

13. Inverse normal transformation
[1] raw data
[2] empirical pdf
[3] cumulative pdf
[4] normal pdf
[5] z-scores

14. 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:
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.

15. Other meteorological variables (cont.)
The z(t) are transformed to weather variables dependent on rainfall occurrence:
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).

16. Daily weather generation (Markov chain)
Source: Wilks and Wilby (1999)

17. Daily weather generation (spell-lengths)
Source: Wilks and Wilby (1999)

18. Use of atmospheric patterns

19. Weather classification schemes may be used to condition daily meteorological variables such as the precipitation occurrence and intensity processes

20. Conditional probabilities of rainfall and mean intensity at Kempsford, Cotswolds associated with the main Lamb Weather Types (LWT),
Conditioning weather patterns may be derived from (a) observations; (b) climate model output; (c) stochastic representations of (a) or (b).

21. Conditioning stochastic properties of daily precipitation on indices of atmospheric circulation
Conditioning variables:
day of the week (!),
month, season, year,
geography,
weather patterns,
moisture indices,
airflow/pressure indices,
hidden states,
NAOI and SOI, etc.
Standard deviation of monthly precipitation at Valentia for an unconditioned an induced SLP model (Kiely et al., 1998).

22. 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
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.

23. Applications

24. Generation of climate analogues
125
150
175
200
225
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275
300
325
350
historic
abstraction
1893
zero
1872
Ml/day
Simulated 10-day annual minimum flow in the River Test under extreme cyclonic (1872) and anticyclonic (1893) weather patterns.

25. 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:

26. Detection of non-stationarity
Source: Wilby (2001)
Dry-spell persistence (p00) at selected sites in the UK

27. Statistical downscaling
Changes in station-series means and variances will be proportional to changes in the respective area-average (GCM grid) moments:
where S(T) is the sum of T daily precipitation amounts, is the unconditional probability of precipitation, and  is the mean wet-day amount.
Source: Wilks (1999)

28. Future directions

29. 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)

30. Seasonal forecasting
Using winter North Atlantic SST anomalies to condition summer dry–spell persistence (p00).
Hindcasts of summer dry–spell persistence (p00) at Cambridge, 1946–1995, from preceding winter SST anomalies.
Source: Wilby (2001)

31. Summary of weather generator characteristics

32. Further reading
Cameron, D., Beven, K. and Tawn, J An evaluation of three stochastic rainfall models. Journal of Hydrology, 228,
Dessens,J., Fraile, R., Pont, V. and Sanchez, J.L Day-of-the-week variability of hail in southwestern France. Atmospheric Research, 59-60,
Gregory, J.M., Wigley, T.M.L. and Jones, P.D Application of Markov models to area-average daily precipitation series and interannual variability in seasonal totals. Climate Dynamics, 8,
Katz, R.W Techniques for estimating uncertainty in climate change scenarios and impact studies. Climate Research, 20,
Kiely, G., Albertson, J.D., Parlange, M.B. and Katz, R.W Conditioning stochastic properties of daily precipitation on indices of atmospheric circulation. Meteorological Applications, 5,
Kilsby,C.G., Cowpertwait, P.S.P., O’Connell, P.E., and Jones, P.D Predicting rainfall statistics in England and Wales using atmospheric circulation variables. International Journal of Climatology, 18,
Richardson, C.W Stochastic simulation of daily precipitation, temperature and solar radiation. Water Resources Research 17,
Wilby, R.L 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 The weather generation game: a review of stochastic weather models. Progress in Physical Geography, 23,