Uncertainty and sensitivity analysis of flood risk management decisions based on stationary and nonstationary model choices

Current practice in flood frequency analysis assumes that the stochastic properties of extreme floods follow that of stationary conditions. As human intervention and anthropogenic climate change influences in hydrometeorological variables are becoming evident in some places, there have been suggestions that nonstationary statistics would be better to represent the stochastic properties of the extreme floods. The probabilistic estimation of non-stationary models, however, is surrounded with uncertainty related to scarcity of observations and modelling complexities hence the difficulty to project the future condition. In the face of uncertain future and the subjectivity of model choices, this study attempts to demonstrate the practical implications of applying a nonstationary model and compares it with a stationary model in flood risk assessment. A fully integrated framework to simulate decision applied to hypothetical flood risk management decisions and the outcomes are compared with those of known underlying future conditions. Uncertainty of the economic performance of the risk-based decisions is assessed through Monte Carlo simulations. Sensitivity of the results is also tested by varying the possible magnitude of future changes. The application provides quantitative and qualitative comparative results that satisfy a preliminary analysis of whether the nonstationary model complexity should be applied to improve the economic performance of decisions. Results obtained from the case study shows that the relative differences of competing models for all considered possible future changes are small, suggesting that stationary assumptions are preferred to a shift to nonstationary statistics for practical application of flood risk management. Nevertheless, nonstationary assumption should also be considered during a planning stage in addition to stationary assumption especially for areas where future change in extreme flows is plausible. Such comparative evaluations would be of valuable in flood risk management decision-making processes.


Introduction
In hydrology and water resources planning, stochastic methods are routinely applied in the design process.Conventionally, it is assumed that the extreme hydrological events are stationary, which means that the probability distribution of the extremes remains time invariant over the design life/appraisal period of the planned structure.The assumption is, however, argued due to the fact that the extremes hydrological time series such as extreme precipitation and extreme floods and drought are driven by a complex interaction of different factors that inevitably changes over time.For example, the hydrological cycle of river basins is affected by the changes in land use from human intervention and urbanization [1].In some countries, changes in acceleration of runoff due to large-scale deforestation has causes unprecedented event of flooding.
There are also concerns over the effect of atmospheric circulation systems such as El Nino Southern Oscillation (ENSO), Pacific Decadal Oscillation (PDO) and North Atlantic Oscillation (NAO) to the pattern of observed extreme precipitation and extreme flood events [e.g.2,3].Increased greenhouse gases in the atmosphere that result in climate change is another reason for concerns over changes in extreme hydrological variables that may effectively result in changes in frequency and magnitude of extreme floods and drought [4].
The significant impact of the mentioned factors to streamflow remains inconclusive [5].Concerns over the possible nonstationarity and the impact of the conventional assumption of stationary in the case of designing flood protection are valid as the design structure are typically meant to be functional for decades.There has been a call to identify nonstationary probabilistic models instead of relying upon the stationary assumption in practical flood risk management problems [6].Yet abandoning stationary probabilistic model and identifying approaches of nonstationary for practical use in flood risk management raises challenges to flood risk analysts and the research community [7,8].
Concerns over nonstationarity have led many studies to conduct trend analysis on extreme precipitation and extreme streamflow observations [9] ( 2016) studies have also tried to address the changes in flood frequency by analysing changes over time slices of climate model-based future projections [e.g.10,11].The findings from the analysis of change in future flows projections have been taken as the basis of design and planning guidelines for flood risk management in England and Wales [12].Whereas many studies have relied on statistical trend tests such as Mann-Kendall trend test and regression analysis to observe change, there is a strong emphasis on the needs for developing more consistent non-stationary frequency analysis methods that can account transient nature of a changing climate [9].
The existence of nonstationarity in extreme streamflow is well recognized but research on the applicability of nonstationary model in flood frequency analysis is just recently being explored.A number of studies have conducted comparison analysis of design estimates from nonstationary probabilistic model to those of stationary [e.g.13,14,15], which lead to emergence of diverse opinions.Some indicate preferences on nonstationary model with some cautions [13,14,16], whilst others discard the notion that stationary is dead and suggesting that stationary should still remain the default assumption [15].
In the face of uncertain future and the subjectivity of model choice, this study attempts to incorporate nonstationary probabilistic model in a flood risk management decision analysis framework, in addition to the conventional stationary model.Different sources of uncertainty is also addressed and well embedded in the simulation study.In demonstrating the practical implications of applying nonstationary model additional to stationary model, GHFLVLRQ PDNHUV ¶ EHKDYLRXU LQ deciding upon an optimal protection level of flood protection is simulated.Decision uncertainty is represented by incorporating subjectivity of model choices between stationary and nonstationary probabilistic models in the decision making process.The work is implemented to a hypothetical case study where other components for cost-benefit analysis (i.e.cost and damage models) are derived accordingly to focus on the main aims of the study.The results are presented with explicit uncertainty range and the outcomes of the sensitivity analysis.
The paper is organised as follows.Section 2 presents the methodology for the study, which includes an integrated framework designed to represent decision PDNHUV ¶ UDWLRQDOH DSSURDFK LQ GHFLGLQJ XSRQ DQ RSWLPDO protection.Results are presented in Section 3 whilst Section 4 presents the discussion and conclusion.

Nonstationary underlying distribution and simulation of annual maxima flows
In this study, a nonstationary Generalized Extreme Value (GEV) model with time as covariate was chosen to simulate the annual maxima flow series.Assuming that the location parameter is linearly changing over time, the inverse cumulative distribution function (CDF) of the nonstationary GEV distribution was used to generate the flows.The CDF according to Jenkinson [20] can be written as follows: where, 11 different scenarios of future change were considered to test the sensitivity of outcomes.This is undertaken by specifying different changes of discharges per year (u 1 ) over the appraisal period for each scenario, ranging from 0 to 1.0 m 3 /s increase of discharge annually.The scenarios therefore is set to have a different rate of change by a factor of 0.1 m 3 /s, which means that the simulated future may have a weaker or greater representation of change over the future period as compared to the simulated historical underlying distribution (u 1 = 0.2 m 3 /s/year).Hence future underlying distribution with u 1 = 0.2 m 3 /s/year has the same rate of change as the simulated historical underlying distribution.Note that 1.0 m 3 /s increase per year will cause a 100 m 3 /s increase of discharge at the end of the 100 years appraisal period.This is extremely high and a higher value might not worth to be considered.
In order to simulate smooth transaction of values over the µKLVWRULFDO ¶ DQG µIXWXUH ¶ period, the location parameter, u, of the µfuture ¶ underlying distribution at the transition time (t = 51) should be the same as that of µKLVWRULFDO ¶ XQGHUO\LQJ GLVWULEXWLRQ +RZHYHU GXH WR WKH different rate of change between the two periods, u o of those periods are unique.Table 1 shows the computed u o associated with an assigned u 1 for the future period.To have an insight on the relative variability of the simulated annual maxima from different u 1 , tabulations of average annual maxima flows across 150 years when the historical u 1 = 0.2 m 3 /s/year and the future u 1 is 0.4 and 0.7 m 3 /s/year respectively are given in Figure 2.

Figure 1 :
Figure 1: Work flow showing the state of the underlying distribution, the model choice by decision makers and the final model used in the risk-based optimization methodology u t LV WKH ORFDWLRQ SDUDPHWHU ZLWK WLPH DV FRYDULDWH Į LV WKH VFDOH SDUDPHWHU DQG ț LV WKH VKDSH SDUDPHWHU 7KH location parameter as a linear trend has an initiation of u o and a rate of change u 1 over t.The generation of annual maxima flow series from the function should use realistic parameter values.For this study, available historical records of the Thames at Kingston gauging station ranging from 1883 to 2012 were fit into the model using L-moments [21].Initial check using the Akaike Information Criterion reveals that the time series is best represented by a GEV stationary model (i.e.without covariate) whilst the second best model is the GEV nonstationary model with a linear location parameter outperforming GEV nonstationary model with log-scale parameter.As the aim of the study is to evaluate the effects of nonstationary underlying distribution, estimated parameters of the nonstationary GEV distribution with linear location parameter were used to simulate the annual maxima flow series.The associated parameters estimated by maximum likelihood estimators are u o = 250.65,u 1 = 0.2, a = 95.7,ț = -0.046

Figure 2 :
Figure 2: Mean annual maxima flows associated with time over historical and future period for 300 simulated flow series.

2. 2 DOIFigure 3 :
Figure 3: histograms and boxplots of optimal protection design estimated based on nonstationary (NSE) and stationary (SE) model choices.The red line refers to optimal protection design based on perfect information (POP).

Figure 4 :Figure 5 :Figure 6 :
Figure 4: boxplots of investment costs of decisions based on nonstationary (NSE) and stationary (SE) model choices.The red crosses refer to investment costs based on perfect information (POP).

Figure 7 :Figure 8 : 1 4
Figure 7: Distributions of total benefit of with-project over the design life of protection, based on nonstationary and stationary estimates respectively for different possible future u 1 .
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Table 1 :
Summary of the nonstationary GEV location SDUDPHWHU FRPSRQHQWV IRU WKH XQGHUO\LQJ IXWXUH µGHVLJQ OLIH ¶ associated to the rate of change at t = 51.