Transcript Presentation Title, Arial Regular 29pt Sub title, Arial Regular 24pt

Spatial-Temporal Modelling of Extreme Rainfall
Climate Adaptation Flagship
Mark Palmer and Carmen Chan
11th IMSC July 2010
Outline of talk
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•
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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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What did we want to do
• Describe the characteristics of extreme rainfall over a relatively
large area, which includes
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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
63064 :-GE
80
0.0
40
60
Empirical
0.6
0.4
0.2
Model
0.8
Return Levels, Return Periods
100
1.0
120
Probability Plot
0.020
1 e-01
1 e+01
Return Period
“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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0.010
f(z)
120
0.000
• Corresponding to the return period T is the return level i(d)
20 40 60 80
1
1  F i  d  | , ,  
Return Level
T
1.0
160
• The return period T, for a given duration(say 24 hours) and intensity
i(d),
0.0
0.2
0.4
0.6
0.8
is the average time interval between exceedance of the value i(d)
Empirical
• ie the Return Period is the reciprocal of the probability of exceedance of
Return Level Plot
that event.
1 e+03
Intensity Duration Frequency (IDF) Curves
• Intensity Duration
Frequency
• X-axis duration
• Y-axis intensity
• Each line corresponds to a
fixed return period
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What data do we have
Commonly
• 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
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Quality
Spatial sparsity
Temporal sparsity, gaps
Untagged accumulations
homogenization
Apparent embarrassment of riches
-31
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-36
150
151
152
Longitude
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153
154
155
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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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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Based on a remarkable distributional result:
Central to the statistical modelling of extreme values is the generalized
extreme value (GEV) distribution.
• If Mn  max X1,  , Xn 
represents the maximum of a sequence of
independent random variables (say daily rainfall over a year), then the
distribution of M n (the yearly maximum rainfall), is given by
1/



x







G  M n  x   exp  1   
 
   
 


• 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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
At-site model
• Generally
• 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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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)
1/
1/ 
1/ 



 

 

 

 x    
 x   
x
 
G  M n  x   exp  1   

exp

1




exp

1




'








 











 
 
 



 

 

 

'


• 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)
d   , d 
d
d  

, d  
• ‘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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Example of various durations:modelled scale parameter
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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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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
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parameter
Results: Return Level plots
• 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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Results: IDF Curves
• IDF curves
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Sample of IDF curves for the
pluviograph site 566038, 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 0.025 and 0.975
quantiles.
Conclusions:
• Build a BHM to borrow strength, and combine many sources of data
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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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Mark Palmer
Phone: +61 8 9333 6293
Email: [email protected]
Web: www.csiro.au/cmis
Thank you
Contact Us
Phone: 1300 363 400 or +61 3 9545 2176
Email: [email protected] Web: www.csiro.au
Koutsiyannis reparameterization
Reparameterise location to dispersion coefficient
1/
1/


 

 

 x   
x
 
G  X  x   exp  1        exp  1      '   
    

 
 

 



• properties of the GEV might give some help
Mode  X    '
E  X    '
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


1


1     1    ' 1     1 







 
1 1 
 g1     '  g1 
 
  
