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

Murray Darling

Drought stricken Murray Darling River

Pejar Dam 2006 AP/ Rick Rycroft 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 km 2 ) –High temporal resolution (~ 5 min) –For long time periods (100+ yr) –Up to large regions (~ 100 km 2 ) –Using rain-gauges only Introduction Model Case Study Associate Research

Model Properties Rainfall Mean Auto-covariance Cross-covariance

derive Calibration Concept MODEL DATA STATISTICS PROPERTIES Objective function calculate Method of moments PARAMETER VALUES                                      fn                          optimise Calibrated Parameters

PROPERTIES                   Calibration Concept MODEL 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 algorithm Efficient 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 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 JanuaryJuly

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 January July

Results Introduction Model Case Study Associate Research 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 ? Introduction Radar pixel (1000 x 1000 m) Rain gauge (0.1 x 0.1 m) ~ 10 8 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 km 2 ) –High temporal resolution (~ 5 min) –For long time periods (100+ yr) –Up to large regions (~ 100 km 2 ) –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 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

Results Annual Distribution at one site

Results Annual Distribution at n sites

Regionalised Annual Distribution 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(u 1,…, u p ) = Pr (U 1 ≤ u 1,…,U p ≤ u p ) 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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