REVIEW OF SPATIAL STOCHASTIC MODELS FOR
RAINFALL
Andrew Metcalfe
School of Mathematical Sciences
University of Adelaide
Research Context
– Hydrology• ‘the natural
water cycle’
Rainfall is the driving input for water dynamics on a catchment
– Hydraulics• ‘man-made
water cycle’
Applications
• Drainage modelling
• Design of flood structures
• Ecological studies
• Other hydrologic risk assessment
www.apwf2.org
http://www.smh.com.au/ffximage
http://www.usq.edu.au/course/material/env4203/summary1-70861.htm
MurrayDarling
DroughtstrickenMurrayDarling River
PejarDam2006
AP/RickRycroft
DURATION
STOCHASTIC MODELS FOR SPATIAL RAINFALL
• Point Processes
• Multivariate distributions
• Random cascades
• Conceptual models for individual storms
Measuring Rainfall
FITTING MODELS
• Multi-site rain gauge
• Data from gauges can be interpolated to a grid. For example Australian BOM can provide gridded data for all of Australia
• Weather radar
• Weather radar can be discretized by sampling at a set of points
POINT PROCESS MODELS
LA Le Cam (1961)
I Rodriguez-Iturbe & Eagleson (1987)
I Rodriguez-Iturbe, DR Cox & V Isham (1987)
PSP Cowpertwait (1995)
Leonard et al
Rainfall is …• highly variable in time
Introduction Model Case Study Associate Research
Point rainfall models (a) event based (e.g DRIP Lambert & Kuczera)(b) clustered point process
with rectangular pulses (e.g. Cox & Isham, Cowpertwait)
Rainfall is …• highly variable in space
Introduction Model Case Study Associate Research
Spatial Neymann-Scott
• Clustered in time, uniform in space
• Cells have radial extent
Storm arrival
Cell start delay
Cell duration
Cell intensity
Aggregate depth
time
Cell radius
Simulation region
Aim• To produce synthetic rainfall records in
space and time for any region:
– High spatial resolution (~ 1 km2)
– High temporal resolution (~ 5 min)
– For long time periods (100+ yr)
– Up to large regions (~ 100 km2)
– Using rain-gauges only
Introduction Model Case Study Associate Research
Model PropertiesRainfall Mean
Auto-covariance
Cross-covariance
derive
Calibration ConceptMODEL
DATA
STATISTICSPROPERTIES
Objective function
calculate
Method of moments
PARAMETER VALUES
fn
optimise
Calibrated Parameters
PROPERTIES
Calibration ConceptMODEL
DATA
STATISTICS
Objective function
calculate
Method of moments
PARAMETER VALUES
fn
…
…
Calibrated Parameters
Efficient Model Simulation
M. Leonard, A.V. Metcalfe, M.F. Lambert, (2006), Efficient Simulation of Space-Time Neyman-Scott Rainfall Model, Water Resources Research
• Can determine any property of the model without deriving equations
Advantages
Disadvantages• Computationally exhaustive• The model property is estimated,
i.e. it is not exact
Efficient model simulation• Consider a target region with an
outer buffer region
• The boundary effect is significant
Efficient model simulation
• An exact alternative:
1. Number of cells
2. Cell centre
3. Cell radius
Efficient model simulation
Target
Buffer
• We showed that:
1. Is Poisson
2. Is Mixed Gamma/Exp
3. Is Exponential
Efficient model simulation
• Efficiency compared to buffer algorithmEfficient model simulation
Defined Storm Extent
M. Leonard, M.F. Lambert, A.V. Metcalfe, P.S. Cowpertwait, (2006), A space-time Neyman-Scott rainfall model with defined storm extent, In preparation
Defined Storm Extent
Defined Storm Extent• A limitation of the existing model
Defined Storm Extent• Produces spurious cross-correlations
• We propose a circular storm region:
Defined Storm Extent
• Probability of a storm overlapping a point introduced
• Equations re-derived
mean
auto-covariance
cross-covariance
Defined Storm Extent
Calibrated
parameters:
Defined Storm Extent
• Improved Cross-correlations
• But cannot match variability in obs.
• Other statistics give good agreement
Defined Storm Extent
January July
Defined Storm Extent• Spatial visualisation:
Sydney Case Study• 85 pluviograph gauges
•We have also included 52 daily gauges
Sydney Case Study
Introduction Model Case Study Associate Research
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Observed Data
Calibrated Model
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Results
Introduction Model Case Study Associate Research
1. 2.
3. 4.
mm/h
Potential Collaborative Research• Application of the model:
• Linking to groundwater / runoff models (water quality / quantity)
• Linking to models measuring long-term climatic impacts
• Use for ecological studies requiring long rainfall simulations
Introduction Model Case Study Associate Research
Introduction• Rainfall in space and time:
Why not use radar ?
IntroductionRadar pixel
(1000 x 1000 m)
Rain gauge (0.1 x 0.1 m) ~ 108 orders magnitude
Gauge data has good coverage in time and space:
Introduction
Aim• To produce synthetic rainfall records in
space and time:
– High spatial resolution (~ 1 km2)
– High temporal resolution (~ 5 min)
– For long time periods (100+ yr)
– Up to large regions (~ 100 km2)
– ABLE TO BE CALIBRATED
1. Scale the mean so that the observed data is stationary
Calibration
January
July
2. Calculate temporal statistics pooled across stationary region for multiple time-increments (1 hr, 12 hr, 24 hr)
- coeff. variation
- skewness
- autocorrelation
Calibration
3. Calculate spatial statistics
- cross-corellogram, lag 0, 1hr, 24 hr
Calibration
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Calibrated Model
January
4. Apply method of moments to obtain objective function
- least squares fit of analytic model properties and observed data
5. Optimise for each month, for cases of more than one storm type
Calibration
Results• Observed vs’ simulated:
– 1 site– 40 year record– 100 replicates
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m)
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Mean 1 Hour
Std. Dev. 1 Hour
Results• Annual Distribution at one site
Results• Annual Distribution at n sites
• Regionalised Annual DistributionResults
Results• Spatial Visulisation:
MULTI-VARIATE DISTRIBUTIONS
S Sanso & L Guenni (1999, 2000)
GGS Pegram & AN Clothier (2001)
M Thyer & G Kuczera (2003)
AJ Frost et al (2007)
G Wong et al (2009)
MULTIVARIATE DISTRIBUTIONS
• Gaussian has advantages
• Latent variables
• Power or logarithmic transforms
• Correlation over space and through time
• Multivariate-t
Copulas
• Multivariate uniform distributions• Many different forms for modelling correlation• In general, for p uniform U(0,1) random variables,
their relationship can be defined as:
C(u1,…, up) = Pr (U1 ≤ u1,…,Up ≤ up)
where C is the copula
RANDOM CASCADES
VK Gupta & E Waymire (1990)
TM Over & VK Gupta (1996)
AW Seed et al (1999)
S Lovejoy et al (2008)
CONCEPTUAL MODELS FOR INDIVIDUAL STORMS
D Mellor (1996)
P Northrop (1998)
FUTURE WORK
• Incorporating velocity
• Large scale models
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