1. REVIEW OF SPATIAL STOCHASTIC MODELS FOR RAINFALL Andrew Metcalfe School of Mathematical Sciences University of Adelaide

2. Research Context –Hydrology ‘the natural water cycle’ Rainfall is the driving input for water dynamics on a catchment –Hydraulics ‘man-made water cycle’

3. Applications Drainage modelling Design of flood structures Ecological studies Other hydrologic risk assessment

4. www.apwf2.org http://www.smh.com.au/ffximage

5. http://www.usq.edu.au/course/material/env4203/summary1-70861.htm

6. Murray Darling

7. Drought stricken Murray Darling River

8. Pejar Dam 2006 AP/ Rick Rycroft DURATION

9. STOCHASTIC MODELS FOR SPATIAL RAINFALL Point Processes Multivariate distributions Random cascades Conceptual models for individual storms

10. Measuring Rainfall

11. 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

13. 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

14. Rainfall is … highly variable in time Introduction Model Case Study Associate Research

15. Point rainfall models (a) event based (e.g DRIP Lambert & Kuczera)(b) clustered point process with rectangular pulses (e.g. Cox & Isham, Cowpertwait)

16. Rainfall is … highly variable in space Introduction Model Case Study Associate Research

17. 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

18. 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

19. Model Properties Rainfall Mean Auto-covariance Cross-covariance

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

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

22. Efficient Model Simulation M. Leonard, A.V. Metcalfe, M.F. Lambert, (2006), Efficient Simulation of Space-Time Neyman-Scott Rainfall Model, Water Resources Research

23. 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

24. Efficient model simulation Consider a target region with an outer buffer region

25. The boundary effect is significant Efficient model simulation

26. An exact alternative: 1. Number of cells 2. Cell centre 3. Cell radius Efficient model simulation Target Buffer

27. We showed that: 1. Is Poisson 2. Is Mixed Gamma/Exp 3. Is Exponential Efficient model simulation

28. Efficiency compared to buffer algorithm Efficient model simulation

29. 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

30. Defined Storm Extent A limitation of the existing model

31. Defined Storm Extent Produces spurious cross-correlations

32. We propose a circular storm region: Defined Storm Extent

33. Probability of a storm overlapping a point introduced Equations re-derived mean auto-covariance cross-covariance Defined Storm Extent

34. Calibrated parameters: Defined Storm Extent

35. Improved Cross-correlations But cannot match variability in obs. Other statistics give good agreement Defined Storm Extent JanuaryJuly

36. Defined Storm Extent Spatial visualisation:

37. Sydney Case Study 85 pluviograph gauges We have also included 52 daily gauges

38. Sydney Case Study Introduction Model Case Study Associate Research January July

39. Results Introduction Model Case Study Associate Research 1. 2. 3.4. mm/h

40. 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

41. Introduction Rainfall in space and time:

42. Why not use radar ? Introduction Radar pixel (1000 x 1000 m) Rain gauge (0.1 x 0.1 m) ~ 10 8 orders magnitude

43. Gauge data has good coverage in time and space: Introduction

44. 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

45. 1. Scale the mean so that the observed data is stationary Calibration January July

46. 2. Calculate temporal statistics pooled across stationary region for multiple time-increments (1 hr, 12 hr, 24 hr) - coeff. variation - skewness - autocorrelation Calibration

47. 3. Calculate spatial statistics - cross-corellogram, lag 0, 1hr, 24 hr Calibration January

48. 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

49. Results Observed vs’ simulated: –1 site –40 year record –100 replicates

50. Results Annual Distribution at one site

51. Results Annual Distribution at n sites

52. Regionalised Annual Distribution Results

53. Spatial Visulisation:

54. 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)

55. MULTIVARIATE DISTRIBUTIONS Gaussian has advantages Latent variables Power or logarithmic transforms Correlation over space and through time Multivariate-t

60. 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

61. RANDOM CASCADES VK Gupta & E Waymire (1990) TM Over & VK Gupta (1996) AW Seed et al (1999) S Lovejoy et al (2008)

62. CONCEPTUAL MODELS FOR INDIVIDUAL STORMS D Mellor (1996) P Northrop (1998)

68. FUTURE WORK Incorporating velocity Large scale models

69. Danke schőn