1. Spatial-Temporal Modelling of Extreme Rainfall
Climate Adaptation Flagship
Mark Palmer and Carmen Chan
11th IMSC July 2010

2. Outline of talk What we wanted to do What data did we have
What data did we use
How we got the most out of it
Borrowing strength BHM’s
Combining data
Squeezing data
Colleagues:
Aloke Phatak, Eddy Campbell, Bryson Bates, Santosh Aryal, Neil Viney, Carmen Chan, Yun Li
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3. What did we want to do
Describe the characteristics of extreme rainfall over a relatively large area, which includes
both gauged and ungauged sites
Model the tails of the rainfall distribution
Be able to estimate return levels for various periods
Be able to calculate Intensity-Duration-Functions (IDF) curves
Be able to calculate Depth-Area (DA) and Areal-Reduction-Factors (ARF) curves
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4. Return Levels, Return Periods
The return period T, for a given duration(say 24 hours) and intensity i(d), is the average time interval between exceedance of the value i(d)
ie the Return Period is the reciprocal of the probability of exceedance of that event.
Corresponding to the return period T is the return level i(d)
“there is no universally agreed definition of this quantity, but one definition is that it is the level exceeded in any one year with probability 1/N. The less precise definition is the level which is exceeded ‘once in N years’ is problematical if there is a trend in the process (for example)’” Smith (2001)
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5. Intensity Duration Frequency (IDF) Curves
X-axis duration
Y-axis intensity
Each line corresponds to a fixed return period
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6. What data do we have Commonly Issues
Daily data (measured from >9am to 9am the following day)
Pluvio data
essentially continuous measurements), though generally discretised to small (say 5 minute) intervals
can then aggregate this data for any duration of interest, using a sliding window
Issues
Quality
Spatial sparsity
Temporal sparsity, gaps
Untagged accumulations
homogenization
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7. Apparent embarrassment of riches
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8. What data do we keep
The BoM high quality data set is too small (particularly spatially)
The minimal requirement is for seasonal data (eg summer winter) that we know the maximal value of!
Can have short time series
Can have temporal gaps
If there are problems with summer data, might still be able to use winter data for that year (and vice versa)
Any bit of ‘information’ in the data is potentially useful (better than completely omitting it)
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9. How to get the most out of it
Develop a BHM with both spatial and temporal components
Allows us to borrow strength (ie sites that are “close” together should have similar characteristics
Anticipate that small proportions of ‘unusual’ sites or rainfall measurements will be ‘smoothed’ over, rather than having an undue influence on the analysis
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10. Based on a remarkable distributional result:
Central to the statistical modelling of extreme values is the generalized extreme value (GEV) distribution.
If represents the maximum of a sequence of independent random variables (say daily rainfall over a year), then the distribution of (the yearly maximum rainfall), is given by
a very general result
Its relationship to modelling extremes is analogous to the use of the Normal distribution for modelling means or sums of random variables, regardless of their parent distribution (CLT).
the parameters the location, scale and shape parameters characterise the distribution of the extremes
we describe changes in extremes by changes in these parameters
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11. At-site model Generally Doesnt this waste data?
Use a Generalised Extreme Value (GEV) distribution to characterise extreme rainfall at a site
Assume that changes in climate are reflected by changes in the GEV parameters at a site
Assume these GEV’s vary smoothly spatially (Convolution kernel approach, Higdon, Sanso etc)
Take block maxima approach (ie just use the largest value from each season, year by year)
Doesnt this waste data?
What about Generalised Pareto (GPD) approach (why not)
So, what can we do to maximise information
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12. Getting most out of available data
Build a BHM to borrow strength
Use r-order statistics (smaller standard errors, better fit)
Reparameterise location to dispersion coefficient (Koutsiyannis, Buishand)
Combine over durations (Koutsiyannis et al., 1998)
Constant shape parameter (Nadarajah, Anderson, and J.A. Tawn (1998)
Constant dispersion coefficient (empirically observed)
Model scale parameter over durations (ie now have 5 parameters per site)
‘correct’ daily GEV to 24 hour basis.
The maxima from 24 hour aggregated data will be larger than the maxima of daily data [Robinson and Tawn, 2000], so estimate the extremal index, which is used to adjust the location and scale parameters of the GEV for the daily data to make it comparable with GEVs estimated from 24 hour data (the shape parameter remains unchanged
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13. Example of various durations:modelled scale parameter
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14. Example: BHM,fitted by MCMC
Spatial model
‘between’ covariates
eg elevation
‘within’ covariates
eg ocean heat
GEV parameters
‘within’ covariates
coefficients
Rainfall
Di-graph representation of the Spatial-Temporal model for extreme rainfall
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15. Results: Spatial plots of parameters
Model the scale parameter as a linear function of OHC
Fitted surfaces of GEV parameters from MCMC sampling; (a) dispersion coefficient, (b) scale (intercept) parameter, (c) linear ocean heat anomaly coefficient for modelling scale parameter, (d) log(shape) parameter, (e) theta parameter, (f) eta parameter
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16. Results: Return Level plots
Differenced return levels surfaces (2003 – 1953) for a fifty year return period (a), and associated standard error surface (b).
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17. Results: IDF Curves IDF curves
Sample of IDF curves for the pluviograph site , for a 50 year return period (Ocean heat anomaly = 0, i.e. effectively the historical long term average), drawn from the MCMC procedure, median value indicated by solid black line, broken lines indicate and quantiles.
IDF curves
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18. Conclusions:
Build a BHM to borrow strength, and combine many sources of data
Use r-order statistics
Combine over durations
Model scale parameter over durations
‘correct’ daily GEV to 24 hour basis
Reparameterise location to dispersion coefficient is useful
Would like to say something about the extremes of areal rainfall using this approach, but: Warning:
Assumption of conditional independence between sites after spatial modelling of GEV parameters is wrong,
try sampling from it
need to model this, eg Copulas, max-stable approach (composite likelihoods)
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19. Thank you Mark Palmer Phone: +61 8 9333 6293
Web:
Thank you
Contact Us Phone: or Web:

20. Koutsiyannis reparameterization
Reparameterise location to dispersion coefficient
properties of the GEV might give some help
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