# Assessing the Impact of Climate Change and Extreme Value Uncertainty to Extreme Flows across Great Britain

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Flood Risk in the UK and Climate Change

#### 1.2. Uncertainties Related to Climate Change

#### 1.3. Extreme Values Assessment and Automating Approaches

#### 1.4. Aims

^{3}/s) as this removes the issue of catchment area. The 1:100 year return period runoff is referred to as the “1:100 yrF” in this paper.

## 2. Materials and Methods

#### 2.1. The Future Flow Database

#### 2.2. Block Maxima and Peak over Threshold Methods

_{1}, X

_{2}, …, X

_{n}) a sequence of independent random variables and M

_{n}= max{X

_{1}, X

_{2}, …, X

_{n}} a sequence of block maxima [21]. If there are a

_{n}> 0 and b

_{n}, such that the probability $\mathrm{Pr}\left\{\frac{{M}_{n}-{b}_{n}}{{a}_{n}}\le z\right\}\to G(z)$, then:

- (i)
- G(z) belongs to the GEV family distribution:$$G(z)=\mathrm{exp}\left\{-{\left[1+\xi \left(\frac{z-\mu}{\sigma}\right)\right]}_{+}^{-1/\xi}\right\}$$
- (ii)
- For large enough u, the distribution function of y = X − u conditional on X > u, is approximately the GP distribution:$$H(y)=1-{\left(1+\xi \frac{y}{\sigma}\right)}_{+}^{-1/\xi}$$

#### 2.3. Automated Computing

- Identify a series of n suitable thresholds, u
_{1}< u_{2}< … < u_{n}, equally spaced, between the median and the 98% quantile of the data series. For j = 1, …, n, fit a GP model and estimate the likelihood estimator of the scale (σ_{uj}) and shape (ξ_{uj}) parameters for the declustered data above the threshold u_{j}. - Test the sequence (τ(u
_{j}) − τ(u_{j−1}))_{j}_{=2, …, n}for j = 1 against the Pearson’s Chi-square Test, with the test variable τ(u_{j}) = σ_{uj}− ξ_{uj}× u_{j}. For any u < u_{j−1}< u_{j}, the difference τ(u_{j}) − τ(u_{j − 1}) is approximately normally distributed with mean 0. If the null hypothesis of normality is rejected then u_{1}is not a suitable threshold. - Repeat Step 2 for j = 2, …, n if u
_{1}is not suitable, until finding and extracting the first threshold u_{j}for which the difference sequence fulfils the normality condition.

#### 2.4. Assessment of Uncertainties Associated with the Return Level Estimate

_{up}the upper 95% confidence limit, and CI

_{low}the lower 95% confidence limit. This coefficient is dimensionless and allows the comparison of CI ranges for locations with different 1:100 yrF estimates for both the GEV and GP models. Note that this is a relative measure of uncertainty since larger peak flows tend to have larger the confidence intervals associated which leads to higher absolute uncertainty. Thus the CI captures the uncertainty in model fitting, whilst combining the uncertainty from both methods allows the uncertainty in model structure to be captured.

## 3. Results

#### 3.1. Uncertainties on the Baseline

_{U}: Equation (3), the relative 95% CI range for 1 ensemble member); and the mean value across the 11 ensemble members (Step 2, Figure 2b) and the associated uncertainty (Step 2, RSD: Equation (4), the relative standard deviation across 11 ensemble members).

_{U}of 73% with GEV and 55% with GP), but for some sites can be higher, particularly on the east coast (e.g., river Waveney at Needham Mill: RC

_{U}of 79% with GEV and 86% with GP). Figure 3c shows that the mean 1:100 yrF tends to be slightly higher with the GEV (89% of stations, e.g., river Alness at Alness: 51 mm with GEV and 43 mm with GP) and Figure 3d shows that the CM uncertainty is lower with the GP (82% of stations), especially along the west coast (e.g., river Girvan at Robstone: 15% with GEV and 8% with GP).

#### 3.2. Change in Flood Peak by the 2080s

#### 3.3. Combined Uncertainties

_{GEV}) and max(E

_{GP}) the maximum estimate computed by the GEV and the GP, respectively, and min(E

_{GEV}) and min(E

_{GP}) the minimum estimate computed by the GEV and the GP, respectively.

_{up,GEV}) and max(CI

_{up,GP}) the maximum upper 95% confidence limit computed by the GEV and the GP, respectively, and min(CI

_{low,GEV}) and min(CI

_{low,GP}) the minimum lower 95% confidence limit computed by the GEV and the GP, respectively.

## 4. Discussion

#### 4.1. Main Results

#### 4.1.1. National Picture

#### 4.1.2. Cascading Uncertainties

_{U}(that quantifies the EV uncertainty calculated for ensemble member afgcx) increased from baseline to the 2080s for 63% and 73% of stations (GEV and the GP respectively). Similarly, the RSD (that quantifies the CMU across the 11 ensemble-members) increased from the baseline to the 2080s for 56% and 66% of stations (GEV and the GP respectively). When both models are merged, i.e., when the uncertainty related to the extreme value model structure is also taken into account, as illustrated in Figure 5, the total uncertainty range increases for each station, on both the baseline and the 2080s. It is clear from the results that the estimate and the uncertainty associated with it are strongly determined by individual constraints in each catchment. In some catchments CM uncertainties dominate (e.g., the Thames and the Don Rivers), while in others EV model uncertainty is critical (e.g., Skerne and Severn Rivers). Moreover, from the baseline to the 2080s, there is an increase in RC

_{U}for 64% of stations and in RSD for 59% of stations. This suggests that for the majority of catchments the total cascaded uncertainty increases from the baseline to the future, however in a significant number of catchments across the UK it remains stable or decreases.

#### 4.1.3. GEV vs. GP and Automated Procedure

#### 4.2. Limits

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Jonkman, S.; Vrijling, J. Loss of life due to floods. J. Flood Risk Manag.
**2008**, 1, 43–56. [Google Scholar] [CrossRef] - Guha-Sapir, D.; Below, R.; Hoyois, P. Natural Disasters Reported. EM-DAT: International Disaster Database; Université Catholique de Louvain: Brussels, Belgium, 2015. [Google Scholar]
- Kantamaneni, K.; Phillips, M. Transformation of climate: Will floods and coastal erosion crumble the UK economy? Int. J. Clim. Chang. Impacts Responses
**2016**, 8, 45–59. [Google Scholar] - Alfieri, L.; Feyen, L.; di Baldassarre, G. Increasing flood risk under climate change: A pan-European assessment of the benefits of four adaptation strategies. Clim. Chang.
**2016**, 136, 507–521. [Google Scholar] [CrossRef] [Green Version] - Wagenaar, D.J.; de Bruijn, K.M.; Bouwer, L.M.; de Moel, H. Uncertainty in flood damage estimates and its potential effect on investment decisions. Nat. Hazards Earth Syst. Sci.
**2016**, 16, 1–14. [Google Scholar] [CrossRef] - UK Parliament. Current Parliamentary Material Available on Flooding. Available online: http://www.parliament.uk/topics/Flooding.htm (accessed on 16 August 2013).
- Stewart, L.; Vesuviano, G.; Morris, D.; Prosdocimi, I. The new FEH rainfall depth-duration-frequency model: Results, comparisons and implications. In Proceedings of the 12th British Hydrological Society National Symposium, Birmingham, UK, 2–4 September 2014; Available online: http://nora.nerc.ac.uk/510563/ (accessed on 23 March 2016).
- Wilby, R.L.; Beven, K.J.; Reynard, N.S. Climate change and fluvial flood risk in the UK: More of the same? Hydrol. Process.
**2008**, 22, 2511–2523. [Google Scholar] [CrossRef] - Faulkner, D. Rainfall Frequency Estimation. In Flood Estimation Handbook; Institute of Hydrology: Wallingford, UK, 1999; Volume 2, 110p. [Google Scholar]
- Arnell, N.W.; Gosling, S.N. The impacts of climate change on river flood risk at the global scale. Clim. Chang.
**2016**, 134, 387–401. [Google Scholar] [CrossRef] - Di Baldassarre, G.; Schumann, G.; Bates, P.; Freer, J.; Beven, K. Flood-plain mapping: A critical discussion of deterministic and probabilistic approaches. Hydrol. Sci. J.
**2010**, 55, 364–376. [Google Scholar] [CrossRef] - Beevers, L.; Douven, W.; Lazuardi, H.; Verheij, H. Cumulative impacts of road developments in floodplains. Transp. Res. D
**2012**, 17, 398–404. [Google Scholar] [CrossRef] - Balica, S.; Beevers, L.; Popescu, I.; Wright, N. Parametric and physically based modelling techniques for flod risk and vulnerability assessment: A comparison. Environ. Model. Softw.
**2013**, 41, 81–92. [Google Scholar] [CrossRef] - Beven, K.; Hall, J. (Eds.) Applied Uncertainty Analysis for Flood Risk Management; Imperial College Press: London, UK, 2014.
- Von Christierson, B.; Wade, S.; Counsell, C.; Arnell, N.; Charlton, M.; Prudhomme, C.; Hannaford, J.; Lawson, R.; Tattersall, C.; Fenn, C.; et al. Climate Change Approaches in Water Resources Planning—Overview of New Methods; Report SC090017/R3; Environment Agency: London, UK, 2013.
- Prudhomme, C.; Jakoba, D.; Svensson, C. Uncertainty and climate change impact on the flood regime of small UK catchments. J. Hydrol.
**2003**, 277, 1–23. [Google Scholar] [CrossRef] - Wilby, R.L. Evaluating climate model outputs for hydrological applications. Hydrol. Sci. J.
**2010**, 55, 1090–1093. [Google Scholar] [CrossRef] - Augustin, N.; Beevers, L.; Sloan, W. Predicting river flows for future climates using an autoregressive multinomial logit model. Water Resour. Res.
**2008**, 44. [Google Scholar] [CrossRef] - Murphy, J.M.; Sexton, D.M.H.; Jenkins, G.J.; Boorman, P.M.; Booth, B.B.B.; Brown, C.C.; Clark, R.T.; Collins, M.; Harris, G.R.; Kendon, E.J.; et al. UK Climate Projections Science Report: Climate Change Projections; Met Office Hadley Centre: Exeter, UK, 2009. [Google Scholar]
- Prudhomme, C.; Haxton, T.; Crooks, S.; Jackson, C.; Barkwith, A.; Williamson, J.; Kelvin, J.; Mackay, J.; Wang, L.; Young, A.; et al. Future Flows Hydrology: An ensemble of daily river flow and monthly groundwater levels for use for climate change impact assessment across Great Britain. Earth Syst. Sci. Data
**2013**, 5, 101–107. [Google Scholar] [CrossRef] [Green Version] - Coles, S. An Introduction to Statistical Modelling of Extreme Values; Springer: London, UK, 2001. [Google Scholar]
- Centre for Ecology & Hydrology. Flood Estimation Handbook; Centre for Ecology & Hydrology: Wallingford, UK, 1999; Volumes 1–5. [Google Scholar]
- Bocchiola, D.; De Michele, C.; Rosso, R. Review of recent advances in index flood estimation. Hydrol. Earth Syst. Sci.
**2003**, 7, 283–296. [Google Scholar] [CrossRef] - Svensson, C.; Jones, D. Review of rainfall frequency estimation methods. J. Flood Risk Manag.
**2010**, 3, 296–313. [Google Scholar] [CrossRef] [Green Version] - Prosdocimi, I.; Kjeldsen, T.; Miller, J. Detection and attribution of urbanization effect on flood extremes using nonstationary flood-frequency models. Water Resour. Res.
**2015**, 51, 4244–4262. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Esteves, L.S. Consequences to flood management of using different probability distributions to estimate extreme rainfall. J. Environ. Manag.
**2013**, 115, 98–105. [Google Scholar] [CrossRef] [PubMed] - Madsen, H.; Rasmussen, P.; Rosbjerg, D. Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events, 1, At-site modeling. Water Resour. Res.
**1997**, 33, 747–757. [Google Scholar] [CrossRef] - Ferreira, A.; de Haan, L. On the block maxima method in extreme value theory: PWM estimator. Ann. Stat.
**2015**, 43, 276–298. [Google Scholar] [CrossRef] - Bayliss, A.C.; Jones, R.C. Peaks-over-Threshold Flood Database: Summary Statistics and Seasonality; IH Report No. 121; Institute of Hydrology: Wallingford, UK, 1993; 68p. [Google Scholar]
- Thompson, P.; Cai, Y.; Reeve, D.; Stander, J. Automated threshold selection methods for extreme wave analysis. Coast. Eng.
**2009**, 56, 1013–1021. [Google Scholar] [CrossRef] - Fukutome, S.; Liniger, M.A.; Süveges, M. Automatic threshold and run parameter selection: A climatology for extreme hourly precipitation in Switzerland. Theor. Appl. Climatol.
**2013**, 120, 403–416. [Google Scholar] [CrossRef] - Süveges, M.; Davison, A.C. Model misspecification in peaks over threshold analysis. Ann. Appl. Stat.
**2010**, 4, 203–221. [Google Scholar] [CrossRef] - Prudhomme, C.; Dadson, S.; Morris, D.; Williamson, J.; Goodsell, G.; Crooks, S.; Boelee, L.; Davies, H.; Buys, G.; Lafon, T.; et al. Future Flows Climate: An ensemble of 1-km climate change projections for hydrological application in Great Britain. Earth Syst. Sci. Data
**2012**, 4, 143–148. [Google Scholar] [CrossRef] [Green Version] - Murphy, J.M.; Booth, B.B.B.; Collins, M.; Harris, G.R.; Sexton, D.M.H.; Webb, M.J. A methodology for probabilistic predictions of regional climate change from perturbed physics ensembles. Philos. Trans. R Soc. A
**2007**, 365, 1993–2028. [Google Scholar] [CrossRef] [PubMed] - Brown, I.; Poggio, L.; Gimona, A.; Castellazzi, M. Climate change, drought risk and land capability for agriculture: Implications for land use in Scotland. Reg. Environ. Chang.
**2011**, 11, 503–518. [Google Scholar] [CrossRef] - Gilleland, E.; Katz, R.W. New software to analyze how extremes change over time. Eos
**2011**, 92, 13–14. [Google Scholar] [CrossRef] - Prescott, P.; Walden, A.T. Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Biometrika
**1980**, 67, 723–724. [Google Scholar] [CrossRef] - Katz, R.W.; Parlange, M.; Naveau, P. Statistics of extremes in hydrology. Adv. Water Resour.
**2002**, 25, 1287–1304. [Google Scholar] [CrossRef] - Scarrott, C.; MacDonald, A. A review of extreme value threshold estimation and uncertainty quantification. REVSTAT Stat. J.
**2012**, 10, 33–60. [Google Scholar] - Griffiths, J.; Young, A.R.; Keller, V. Continuous Estimation of River Flows (CERF)—Technical Report: Task 1.3: Model Scheme for Representing Rainfall Interception and Soil Moisture; CEH: Wallingford, UK, 2006; 45p. [Google Scholar]
- Moore, R.J. The PDM rainfall-runoff model. Hydrol. Earth Syst. Sci.
**2007**, 11, 483–499. [Google Scholar] [CrossRef] - Crooks, S.M.; Naden, P.S. CLASSIC: A semi-distributed rainfall-runoff modelling system. Hydrol. Earth Syst. Sci.
**2007**, 11, 516–531. [Google Scholar] [CrossRef] [Green Version] - Wilby, R.L.; Dessai, S. Robust adaptation to climate change. Weather
**2010**, 65, 180–185. [Google Scholar] [CrossRef] - Ludwig, F.; van Slobbe, E.; Cofino, W. Climate change adaptation and Integrated Water Resource Management in the water sector. J. Hydrol.
**2014**, 518, 235–242. [Google Scholar] [CrossRef] - Di Baldassarre, G.; Castellarin, A.; Montanari, A.; Brath, A. Probability-weighted hazard maps for comparing different flood risk management strategies: A case study. Nat. Hazards
**2009**, 50, 479–496. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) Uncertainty related to the EV model parameters: return level estimate (black dot) and its 95% confidence interval (CI, grey dashed line); (

**b**) Uncertainty related to the climate model parameters: smoothed empirical CDF of return level estimates for the 11 ensemble members (black line and dots); (

**c**) Combined uncertainty related to the EV and climate models: smoothed empirical CDF of return level estimates (black line and dots) and associated 95% CI (grey area) for the 11 ensemble members.

**Figure 3.**Results across Great Britain over the baseline for the GEV (on the left) and the GP (on the right) models. Step 1: (

**a**) 1:100 yrF (for ensemble member afgcx); and (

**b**) Extreme Value Distribution parameter uncertainty (relative coefficient of uncertainty). Step 2: (

**c**) Mean 1:100 yrF across the 11 ensemble membersl and (

**d**) Climate Model uncertainty (relative standard deviation).

**Figure 4.**Results across Great Britain of change from the baseline to the 2080s for the GEV (on the left) and the GP (on the right) models. Step 1: (

**a**) Change in 1:100 yrF (for ensemble member afgcx); and (

**b**) change in Extreme Value Distribution parameter uncertainty (relative coefficient of uncertainty). Step 2: (

**c**) Change in mean 1:100 yrF (across the 11 ensemble members); and (

**d**) change in Climate Model uncertainty (relative standard deviation).

**Figure 5.**Step 3: Combined uncertainties for the GEV (grey shades) and GP (blue shades) models on the baseline (left) and the 2080s (right) on: (

**a**) the Clyde River at Tulliford Mill; (

**b**) the Don River at Parkhill; (

**c**) the Ribble River at New Jumbles Rock; (

**d**) the Skerne River at Preston le Skerne; (

**e**) the Severn River at How Bridge; (

**f**) the Thames River at Eynsham; and (

**g**) the Avon River at Amesbury).

**Figure 6.**(

**a**) Definition of the total uncertainty range (TUR) of computed 1:100 yrF value and the CM uncertainty range (CMUR). (

**b**) Climate Model uncertainty range (CMUR) relative to the total uncertainty range (TUR) for the gauging stations across the UK over the baseline (black) and the 2080s (grey) computed with the GEV (left) and the GP (right).

**Table 1.**For each selected gauging station, on the baseline and the 2080s: combined GEV and GP model uncertainties related to the climate model (CCMU) and the total combined cascaded uncertainty (TCCU).

River | CCMU (mm) Baseline | CCMU (mm) 2080s | TCCU (mm) Baseline | TCCU (mm) 2080s |
---|---|---|---|---|

Clyde | 13.66 | 16.37 | 30.16 | 40.13 |

Don | 10.81 | 12.08 | 19.62 | 23.39 |

Ribble | 14.39 | 28.19 | 39.82 | 56.13 |

Skerne | 14.07 | 35.41 | 25.97 | 42.70 |

Severn | 3.84 | 10.08 | 8.63 | 18.11 |

Thames | 4.11 | 4.96 | 5.54 | 10.22 |

Avon | 3.39 | 13.64 | 7.41 | 14.15 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Collet, L.; Beevers, L.; Prudhomme, C.
Assessing the Impact of Climate Change and Extreme Value Uncertainty to Extreme Flows across Great Britain. *Water* **2017**, *9*, 103.
https://doi.org/10.3390/w9020103

**AMA Style**

Collet L, Beevers L, Prudhomme C.
Assessing the Impact of Climate Change and Extreme Value Uncertainty to Extreme Flows across Great Britain. *Water*. 2017; 9(2):103.
https://doi.org/10.3390/w9020103

**Chicago/Turabian Style**

Collet, Lila, Lindsay Beevers, and Christel Prudhomme.
2017. "Assessing the Impact of Climate Change and Extreme Value Uncertainty to Extreme Flows across Great Britain" *Water* 9, no. 2: 103.
https://doi.org/10.3390/w9020103