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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Observed Data

Calibrated Model January

July

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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Me

an

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m)

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ain

fall

(mm

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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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