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REVIEW OF SPATIAL STOCHASTIC MODELS FOR RAINFALL

REVIEW OF SPATIAL STOCHASTIC MODELS FOR RAINFALL. Andrew Metcalfe School of Mathematical Sciences University of Adelaide. Research Context. Hydrology ‘the natural water cycle’. Hydraulics ‘man-made water cycle’. Rainfall is the driving input for water dynamics on a catchment.

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REVIEW OF SPATIAL STOCHASTIC MODELS FOR RAINFALL

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  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’ • Hydraulics • ‘man-made water cycle’ • Rainfall is the driving input for water dynamics on a catchment

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

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

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

  6. Murray Darling

  7. Drought stricken Murray Darling River

  8. Pejar Dam 2006 DURATION AP/ Rick Rycroft

  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

  12. 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

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

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

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

  16. Cell radius Cell duration Cell intensity Storm arrival Cell start delay Aggregate depth Simulation region Spatial Neymann-Scott • Clustered in time, uniform in space • Cells have radial extent time

  17. Introduction Model Case Study Associate Research 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

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

  19. m1, m6, m24 s1, s6, s24 r1, r6, r24 fn = (m1 - m1)2 + (m6 - m6)2 + (m24 - m24)2 + (s1 - s1)2 + ... + ... Calibration Concept MODEL a=1, b=2, l = 3 PARAMETER VALUES DATA derive calculate STATISTICS PROPERTIES m1, m6, m24 s1, s6, s24 r1, r6, r24 Method of moments Objective function optimise Calibrated Parameters

  20. Calibration Concept MODEL a=1, b=2, l = 3 PARAMETER VALUES m1, m6, m24 s1, s6, s24 r1, r6, r24 DATA … … calculate STATISTICS PROPERTIES m1, m6, m24 s1, s6, s24 r1, r6, r24 Method of moments fn = (m1- m1)2 + (m6 - m6)2 + (m24 - m24)2 + (s1 - s1)2 + ... + ... Objective function Calibrated Parameters

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

  22. Advantages • Can determine any property of the model without deriving equations Disadvantages • Computationally exhaustive • The model property is estimated, i.e. it is not exact

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

  24. Efficient model simulation • The boundary effect is significant

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

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

  27. Efficient model simulation • Efficiency compared to buffer algorithm

  28. 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

  29. Defined Storm Extent Defined Storm Extent • A limitation of the existing model

  30. Defined Storm Extent • Produces spurious cross-correlations

  31. Defined Storm Extent • We propose a circular storm region:

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

  33. Defined Storm Extent Calibrated parameters:

  34. Defined Storm Extent • Improved Cross-correlations • But cannot match variability in obs. • Other statistics give good agreement January July

  35. Defined Storm Extent • Spatial visualisation:

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

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

  38. Introduction Model Case Study Associate Research Results 1. 2. mm/h 3. 4.

  39. Introduction Model Case Study Associate Research 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

  40. Introduction • Rainfall in space and time:

  41. Introduction Radar pixel (1000 x 1000 m) Why not use radar ? Rain gauge (0.1 x 0.1 m) ~ 108 orders magnitude

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

  43. 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

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

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

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

  47. Calibration 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

  48. Results • Observed vs’ simulated: • 1 site • 40 year record • 100 replicates

  49. Results • Annual Distribution at one site

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