1. Evaluating the risk of spatially extensive flooding Duncan Reed duncanreed@dwrconsult.demon.co.uk

2. 2 … a collective risk index for general use: An index of collective risk (4) But how is the effective number of independent sites, N e, defined?

3. 3 Gumbel reduced variate, y = - ln ( - lnF) lnN Distribution of maxima at a typical site Distribution of maxima over N sites FREQUENCY MAGNITUDE The clever bit … If sites independent Standardised extreme value x

4. 4 Gumbel reduced variate, y = - ln ( - lnF) lnN e Distribution of maxima at a typical site Distribution of maxima over N sites Standardised extreme value x FREQUENCY MAGNITUDE If sites partially dependent

5. 5 Gumbel reduced variate, y = - ln ( - lnF) lnN e FREQUENCY MAGNITUDE lnN CRINGE = 1 (fully dependent) CRINGE = 0 (fully independent)Network maximum curve – actual Standardised extreme value x

6. 6 N e = N CRINGE = 0 N e = 1 CRINGE = 1 A practical approach CRINGE = 1 – lnN/lnN e

7. 7 Spatial dependence model for rainfall extremes CRINGE = 0.919 - 0.085 lnAREA + 0.051 lnN + 0.027 lnD where: Nnumber of sites in network AREAarea spanned by network (km 2 ) Drainfall duration (days) [Re-expression of Dales & Reed, 1989]

8. 8 Definition of area spanned 2 1 3 d 12 d 23 d 13 d bar is mean inter-site distance Spanning area taken as AREA = 2.5 (d bar ) 2

9. 9 Example: 1-day maximum rainfall Number of insured units CRINGE AREA km 2

10. 10 Number of insured units CRINGE AREA km 2 Example: 90-day maximum rainfall

11. 11 Why effect greater for floods Extreme rainfalls that lead to flooding tend to be those occurring when soils are already primed. Such conditions tend to be strongly seasonal and – when they occur – more spatially widespread than typical rainfall extremes. Extreme rainfalls that lead to flooding tend to be those occurring when soils are already primed. Such conditions tend to be strongly seasonal and – when they occur – more spatially widespread than typical rainfall extremes. Highly permeable catchments struggle to flood other than spatially extensively. Highly permeable catchments struggle to flood other than spatially extensively.

12. 12 Koltun & Sherwood (1999) study pairwise dependence in flooding for streamflows in Ohio Koltun & Sherwood (1999) study pairwise dependence in flooding for streamflows in Ohio They index dependence by a ratio of counts: They index dependence by a ratio of counts: They have to make quite a lot of assumptions about what constitutes an “event” They have to make quite a lot of assumptions about what constitutes an “event” Spatial dependence in floods (1)

13. 13 Spatial dependence in floods (2) Koltun & Sherwood relate their index to: Koltun & Sherwood relate their index to: 1. Inter-catchment distance 2. Mean size of the catchments 3. Relative size of the catchments 4. Inter-catchment orientation They use logistic regression to force the dependence to lie between the independent and fully dependent cases. They use logistic regression to force the dependence to lie between the independent and fully dependent cases.

14. 14 Suggested developments (1) Use Dales & Reed model of spatial dependence in rainfall extremes to compare CRINGE factors for Portfolios A and B Use Dales & Reed model of spatial dependence in rainfall extremes to compare CRINGE factors for Portfolios A and B Adopt nominal values of rainfall duration, such as: Adopt nominal values of rainfall duration, such as:  D = 0.05 days for storm-sewers  D = 30 days for groundwater-fed rivers  D = 1.0 otherwise

15. 15 Suggested developments (2) Generalise a spatial dependence model for river flooding, taking the best from the Dales & Reed and Koltun & Sherwood approaches Generalise a spatial dependence model for river flooding, taking the best from the Dales & Reed and Koltun & Sherwood approaches Vary the method to allow CRINGE to reflect the much larger insured units at some sites Vary the method to allow CRINGE to reflect the much larger insured units at some sites Consider applications to extreme values of other variables, especially drought Consider applications to extreme values of other variables, especially drought