QUANTIFYING THE RISK OF EXTREME EVENTS IN A CHANGING CLIMATE. Rick Katz. Joint Work with Holger Rootzén Chalmers and Gothenburg University, Sweden

Transcription

1 QUANTIFYING THE RISK OF EXTREME EVENTS IN A CHANGING CLIMATE Rick Katz Joint Work with Holger Rootzén Chalmers and Gothenburg University, Sweden rwk@ucar.edu Talk:

2 Quotes/Titles Milly et al. (Science, 2008) -- Stationarity is Dead: Whither Water Management? Lins and Cohn (JAWRA, 2011) -- Stationarity: Wanted Dead or Alive? Serinaldi and Kilsby (AWR, 2015) -- Stationarity is Undead: Uncertainty Dominates the Distribution of Extremes

3 Outline (1) Motivation (2) Example of Non-Stationarity in Extremes (3) Quantifying Risk under Stationarity (4) Quantifying Risk under Climate Change (5) Design Life Level (6) Discussion

4 (1) Motivation Trends in Precipitation Extremes -- Review paper for US National Assessment (Kunkel et al., 2013) Method -- Generalized extreme value (GEV) distribution for annual maxima (Linear trend in location parameter) -- Fit indirectly using peaks over threshold/point process approach -- Runs declustering (r = 1) -- Seasonality ignored

5 Change in 20-Year Return Level of Daily Precipitation Extremes

6 Issues -- Trend detection Difficult to detect trends in precipitation extremes (Much uncertainty in estimated return levels) Need to borrow strength across space (Regional analysis) -- Interpretation of change in extremes Changes in return levels (Meaning under non-stationarity?)

7 (2) Example of Non-Stationarity in Extremes Example (Mercer Creek, WA) -- Small watershed near Seattle -- Period of rapid urbanization (starting about 1970) -- Anticipated effects of land-use changes on runoff Increase in peak flow Not necessarily any effect on mean flow

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10 Model for Non-Stationarity in Annual Peak Flow -- GEV cumulative distribution function (CDF) G(x; μ, σ, ξ ) = exp{ [1 + ξ (x μ) / σ] 1/ξ } Location parameter μ Scale parameter σ > 0 Shape parameter ξ -- Assume μ & ln(σ) have piecewise linear trends (i) Constant (ii) Linear (iii) Constant

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13 (3) Quantifying Risk under Stationarity Stationarity -- Remains prevailing paradigm in engineering design

14 Return Period / Return Level -- e. g, so-called 100-yr flood Return level has 1% chance of being exceeded in given year Return period (or average recurrence interval ) corresponding to this probability of 0.01 is 1/0.01 or 100 yrs -- Suppose random variable X has CDF F Return level x(p) (with return period 1/p) is 1 p quantile of F That is, x(p) = F 1 (1 p)

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16 Interpretation of return level x(p) with return period T Pr{X > x(p)} = p, where p = 1/T (i) Length of time T for which expected number of events = 1 1 = Expected no. events = T p, so T = 1/p (ii) Expected waiting time Need to assume temporal independence Waiting time W has geometric distribution: Pr{W = k } = (1 p) k 1 p, k = 1, 2,..., E(W) = 1/p = T

17 (4) Quantifying Risk under Climate Change Non-Stationarity -- How to measure risk (over design life time period) Lack of appreciation of issue -- Avoidance of problem Treat climate change as shift to new stationary regime -- Use non-stationary quantiles Effective return period / Effective return level (As if move flood plain from one year to next)

18 Extension of Existing Concepts -- Could retain one of two interpretations under stationarity (i. e., frequency or waiting time) -- Notation Under non-stationarity, risk depends on year t For simplicity, assume design time period starts at year t = 1 Let pt (u) = Pr{Xt > u}, year t = 1, 2,...

19 (i) Expected number of events -- Let NT (u) denote number of events during t = 1, 2,..., T E[NT (u)] = p1(u) + p2(u) + + pt (u) -- Given specified return period T Set E[NT (u)] = 1 & solve for return level u Then u satisfies p1(u) + p2(u) + + pt (u) = 1 In other words, average of T probabilities is 1/T

20 Mercer Creek Example (Return to) -- Imagine engineer in 1955 Faced with flood design for next 50 years (i. e., 1956 to 2005) -- Suppose perfect knowledge available about effect of development in watershed on flood statistics Use observed shift in parameters of GEV distribution (Ignoring any uncertainty)

21 Frequency-Based Return Level (u 871 cfs for T = 50 yr)

22 (ii) Expected waiting time -- Assume temporal independence -- Let W(u) denote first time t that Xt > u Pr{W(u) = k } = { t=1,k 1 [1 pt (u)] } pk (u), k = 1, 2,... Given specified return period T: Set E[W(u)] = T Solve for return level u

23 Waiting Time-Based Return Level (u 871 cfs for T = 50 yr)

24 Return Levels for Mercer Creek Example -- Stationarity assumption Fit single GEV distribution: u 1036 cfs for 50-yr return level -- Frequency-based definition u 871 cfs for 50-yr return level -- Waiting time-based definition u 871 cfs for 50-yr return level Note: Two definitions produce approximately same return levels (but not exactly see Cooley, 2013 for an example)

25 (5) Design Life Level Abandon / Replace Existing Concepts -- Reasons for doing so (even under stationarity) 100-yr flood vs. probability of one or more floods within (e. g.) 30- yr time period -- Risk of failure (Court, 1952; Gumbel,1958) Under stationarity (and independence), risk of failure over T yrs.: 1 (1 p) T

26 Non-Stationarity (and Independence) -- Design life time period yr 1 to yr T Risk of failure = 1 {[1 p1(u)] [1 pt (u)]} -- Choose Design Life Level to achieve desired risk of failure (say α) -- Let Gt denote GEV CDF for year t Then CDF of maximum over t = 1, 2,..., T given by F1,T (x) = G1(x) G2(x) GT (x) So Design Life Level u (with risk α for time period yr 1 to yr T ) can be expressed as 1 α quantile of F1,T

27 Another Example (Manjimup, Australia) -- Manjimup, Australia winter maximum daily precipitation Region in state of Western Australia has experienced decline in precipitation in recent decades -- Fit linear trend in location parameter μ of GEV distribution (Note: More realistic model would allow for nonlinear trends in location and scale parameters of GEV)

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29 Future Design Problem -- Design life time period: 2011 to 2060 Extrapolate trend in location parameter μ of GEV dist. (i) Risk α = 0.05 Design Life Level: u = 121 mm (SE 36 mm) Standard error (SE) based on implicit delta method (ii) Risk α = 0.40 ( Risk over 50-yr period for 100-yr return level under stationarity) Design Life Level: u = 77.5 mm

30 (6) Discussion Uncertainty in Estimated Design Life Level -- Better method for quantification (e. g., based on profile likelihood or resampling) Uncertainty About Future Climate -- Peaks over threshold (point process) instead of block maxima -- Borrow strength across space (Regional analysis) -- Combine observed trends with future projections

31 Related Work -- Rootzén & Katz (2013) Paper in Water Resources Research -- Cooley (2013) Chapter in Extremes in a Changing Climate (AghaKouchak et al.) -- Salas & Obeysekera (2013) Paper in Journal of Hydrologic Engineering