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Introduction to modelling extremes

Introduction to modelling extremes. Trevor Hoey Department of Geographical & Earth Sciences, University of Glasgow. Example: sediment entrainment. Entrainment occurs in zone of overlap between entrainment and resisting forces. Entrainment forces (turbulence). Resisting forces (friction).

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Introduction to modelling extremes

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  1. Introduction to modelling extremes Trevor Hoey Department of Geographical & Earth Sciences, University of Glasgow

  2. Example: sediment entrainment Entrainment occurs in zone of overlap between entrainment and resisting forces Entrainment forces (turbulence) Resisting forces (friction) Force

  3. Structure of turbulence Downstream Vertical Vertical Stress

  4. Stream flow www.nerc-wallingford.ac.uk/ih/nrfa/river_flow_data

  5. Flood Estimation • AIM: to estimate the probability of an extreme event occurring in a given time period • eg the probability of Glasgow being flooded

  6. Annual Floods • pq = the probability that discharge equals or exceeds q at least once in any given year; pq = annual excedence probability • (1 – pq) = probability that this flood does NOT occur in a given year • Assume: stationarity; no long-memory

  7. Recurrence Interval • Often refer to recurrence interval of floods (eg 1 in 200 year flood) • Recurrence interval: the average time between floods equaling or exceeding q • Recurrence interval (RIq) is the inverse of the excedence probability (1/pq)

  8. Flow frequency distributions River Dove

  9. Flow frequency distributions

  10. Estimating RIq • To estimate the q-year flood from N-years of data rank the data from highest (q1) to lowest (qN) • The excedence probability and recurrence interval can be estimated from the rank order • With N = 50, what is the rarest flood that can be estimated?

  11. Estimating Extremes: Graphical Method • Rank the data from highest (rank=1) to lowest (rank=N) • Estimate plotting positions from the ranks • Compute recurrence intervals • Plot of q(m) vs RIq(m) • Fit a line to the data • Extrapolate the best-fit line to the required RI

  12. Example: annual maximum data, Skykomish R, Gold Bar http://web.mst.edu/~rogersda/umrcourses/ge301/press&siever13.15.png

  13. Analytical Techniques • Fit an appropriate cumulative distribution function (CDF) to the data • Fitting requires use of estimation procedures (distribution shapes are not known in advance) • Use the CDF to estimate the discharge for a particular RI

  14. Analytical Techniques • Distributions used include: • Extreme value type 1 (EV1; Gumbel) • Log Pearson type III • Normal • Log Normal • Normal, log-normal require estimates of mean, standard deviation

  15. Extreme Value Distributions (EV) • Generalised Extreme Value (GEV) • Set k = 0 gives the EV1 distribution Q = discharge u,a = parameters k = shape parameter y,z = reduced variates

  16. Example Gulungul Ck example

  17. Summary • estimating extremes is inherently unreliable, even with large data sets • many environmental data sets are short, and require extrapolation beyond the period of record • various distributions may be used for estimation – which ones fit best in a particular situation is difficult to assess • data are assumed to be stationary – changing driving conditions, and long memory processes, may violate this assumption for many environmental data

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