Probabilistic mapping of flood hazards: Depicting uncertainty in streamflow, land use, and geomorphic adjustment

Journal Pre-proof Probabilistic mapping of flood hazards: depicting uncertainty in streamflow, land use, and geomorphic adjustment Timothy A. Stephens, Brian P. Bledsoe

PII:

S2213-3054(19)30042-6

DOI:

https://doi.org/10.1016/j.ancene.2019.100231

Reference:

ANCENE 100231

To appear in:

Anthropocene

Received Date:

11 April 2019

Revised Date:

28 November 2019

Accepted Date:

28 November 2019

Please cite this article as: Stephens TA, Bledsoe BP, Probabilistic mapping of flood hazards: depicting uncertainty in streamflow, land use, and geomorphic adjustment, Anthropocene (2019), doi: https://doi.org/10.1016/j.ancene.2019.100231

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Probabilistic mapping of flood hazards: depicting uncertainty in streamflow, land use, and geomorphic adjustment Timothy A. Stephens1,a and Brian P. Bledsoe1 1

Institute for Resilient Infrastructure Systems, University of Georgia, 200 D.W. Brooks Drive, Athens, GA 30602; [email protected] 2

Institute for Resilient Infrastructure Systems, University of Georgia, Athens, GA 30602; [email protected] Corresponding author

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Probabilistic floodplain maps were developed through Monte Carlo simulations of flood hydraulics Probabilistic floodplain maps reflect uncertainty in discharge, land use, and geomorphic adjustment. Flood hazard uncertainty is spatially variable and zones of hidden risk are identified. Nonstationary flood peaks increase the magnitude of flood hazard estimates. A simplified approach simulates uncertainty from ensemble results with high skill.

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Abstract: Spatial and temporal variability in precipitation, land use, and river channels introduce uncertainty in flood estimates and pose challenges for floodplain management and mitigation. Yet, standard deterministic methods for quantifying flood hazards and evaluating risk assume stationarity in a nonstationary world and fail to account for uncertainties as they translate to flood hazards. A need exists for improved methods to portray flood hazards that incorporate uncertainty and enable integrated water management. This paper presents novel techniques for evaluating flood hazards through probabilistic flood inundation maps that reflect uncertainty quantified through Monte-Carlo analyses of model inputs and parameters. These techniques represent a first in simultaneously varying discharge, friction parameters, and channel change in probabilistic floodplain mapping while maintaining the high level of detail implemented in regulatory hydraulic models. This study evaluated four methods for quantifying the 1% annual exceedance probability flood, including a nonstationary approach, at an urban watershed in Charlotte, North Carolina, USA. The range of variability in flood extents resulting from Monte-Carlo simulations was spatially variable, and the likelihood of inundation revealed areas of elevated or hidden risk that were not revealed by deterministic regulatory flood hazard boundaries. The nonstationary approach indicates a significant increase in flood hazards and suggests that the regulatory floodplain boundary underestimates and miscommunicates its intended risk status. A simplified approach for estimating uncertainty in flood hazards closely matched ensemble results, providing managers a practical method for conducting uncertainty analysis. These techniques can aid floodplain management by accounting for the inherent uncertainty in model estimates and the potential nonstationary behavior of flood hazards.

Keywords probabilistic floodplain map; uncertainty; hydraulic modeling; flood risk management

1. Introduction Floodplains provide numerous ecological, economic, and societal benefits including human occupancy. Yet, floodplains are also vulnerable to flood hazards that disrupt communities, impair

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productivity, and result in loss of life (White, 1945; Di Baldassarre et al., 2013). Flood hazards will likely increase in the future due to changing climates, land use, and population growth

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(Kundzewicz et al., 2014; Alfieri et al., 2017; USGCRP, 2017). In many countries across the world, programs such as the National Flood Insurance Program (USA) and the European Floods Directive

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have established goals of reducing flood risk through floodplain management standards. These

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standards are often based on regulatory boundaries that map areas inundated by floods with a specific annual exceedance probability. This procedure generally involves estimating the

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magnitude of a flood with a particular likelihood, simulating that flood in a hydraulic model, and delineating the resulting flood extents, which are depicted as a deterministic boundary.

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As such, deterministic flood hazard boundaries inform human development patterns, which also tend to closely follow them (Patterson and Doyle, 2009; Alfonso et al., 2016). Yet, researchers have identified many sources of uncertainty in estimating flood hazards with hydraulic models (Bales and Wagner, 2009; Teng, 2017). Further, human interactions with the landscape and

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atmosphere can alter the magnitude and frequency of floods and compound the uncertainty associated with model estimates. As significant portions of flood insurance claims have occurred outside regulatory flood hazard boundaries (Brody et al., 2012; Highfield et al., 2013), existing flood hazard maps have been widely criticized for inaccurately portraying flood hazards as a binary depiction of inundation probability and neglecting the inherent uncertainty in model estimates

(Galloway et al., 2006; Bell and Tobin, 2007). As humans continue to use deterministic estimates of floodplain boundaries that are imperfect realizations of reality, a need exists to quantify the uncertainty in these estimates. This study seeks to advance understanding of the distribution of uncertainty in regulatory flood hazard estimates, an endeavor with global significance considering the widespread presence of societies within floodplain corridors. We address the following questions using an urban study

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watershed in Charlotte, North Carolina, USA: 1) How does uncertainty in streamflow, river

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channel geometry, and land use affect the probability of flood inundation compared to deterministic, regulatory flood hazard boundaries?; 2) How does the resulting uncertainty in flood

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inundation affect the exposure likelihood of infrastructure where development patterns interact

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with deterministic inundation boundaries?; 3) How does an increase in flood peaks elevate the likelihood of exposure for areas adjacent to regulatory flood hazard boundaries?: and 4) How can

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an accessible and simplified approach accurately simulate the uncertainty distribution estimated from Monte-Carlo computations?

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To address these questions, we implemented a novel framework for probabilistic floodplain mapping that quantifies uncertainty through Monte-Carlo simulations of flood hydraulics. To our knowledge, this framework is the first to simultaneously account for uncertainty in flood magnitude, friction parameters, and channel change in Monte-Carlo simulations while

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maintaining the high level of detail and resolution implemented in regulatory hydraulic models. We compared different techniques for quantifying the magnitude of the flood with a 1% annual exceedance probability. One technique implemented nonstationary flood frequency analysis, advancing beyond the assumption of stationarity, and providing among the first probabilistic flood

inundation assessments to account for a systematic trend in flood magnitude. We compared MonteCarlo simulations with a simplified approach that is based on systematic sampling.

2. Background 2.1 Uncertainty in Flood Hazards Natural variability and limited knowledge generate uncertainty in hydraulic model results (Beven and Young, 2013). While random variability can typically be defined statistically (e.g., the

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probability of a flood event), a lack of knowledge may be difficult to quantify statistically due to

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changing characteristics in time and space (e.g., floodplain roughness). For practical purposes, both types of uncertainty are often defined statistically; and transparency is recommended in

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assumptions about uncertainties in hydraulic modeling and their implications for flood hazard estimates (Beven et al., 2011; 2015).

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Friction parameters characterize channel and floodplain roughness in hydraulic models

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according to land use, substrate, vegetation, and other factors causing uncertainty due to a lack of descriptive knowledge. Since, a theoretical calculation of friction parameters does not exist, the

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use of empirical formulas, visual inspection, and expert judgment make quantitative prescriptions of this parameter inherently subjective. Further, the influence of vegetation and land use on roughness often varies seasonally.

Discharge uncertainty represents the confidence in a flood magnitude associated with a

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specific probability of occurrence. Where stream gage information is available, uncertainty estimates will depend on the sample size, variability, and distribution (e.g. Log Pearson III). However, where stream gage information is lacking, gage extrapolation, hydrologic modeling, or regional regression equations can be used to estimate flood magnitude, and the resulting uncertainty will likely differ among methods. Trends in flood magnitude or frequency can further complicate quantifying the discharge with a particular probability of occurrence (Rosner et al.,

2014), and a large number of sites across the US have exhibited nonstationary behavior in the magnitude and frequency of flood response (Vogel et al., 2011; Barros et al., 2013; Mallakpour and Villarini, 2015; Slater et al., 2015). River channels adjust based on the rate of water and sediment supplied to them and the resulting magnitude and frequency of erosive forces relative to resistive forces (Knighton, 2014). Consequently, the capacity of river channels may oscillate about a mean value due to inter-annual

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variability in hydroclimatic controls, or the capacity of river channels may systematically increase

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or decrease based on changes in water and/or sediment supply. For instance, channels sometimes respond to increased runoff volume and energy by enlarging (Hawley and Bledsoe, 2013);

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however, many urban channels are laterally and vertically constrained with armoring. Where

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channel form is adjustable, incision and widening may either increase or decrease channel capacity depending on inputs of sediment from bank failures, instream wood, and other debris; and its

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subsequent interaction with infrastructure such as bridges and culverts. Alternatively, land disturbance from urban development and upstream channel erosion can increase aggradation and

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retention of sediment in the channel and diminish flood conveyance capacity (Bledsoe and Watson 2001) which can overshadow climate and land use effects of flood stages. However, the concomitant process of increasing flood frequency and reduced channel capacity has amplified flood inundation at stream gage locations across the U.S. (Slater et al., 2015), and geomorphic

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adjustment can influence interannual variability in flood inundation (Call et al., 2017).

2.2 Probabilistic Floodplain Maps Monte-Carlo (MC) simulations of flood hydraulics have proven useful for evaluating the

uncertainty in flood hazard estimates. Studies have investigated flood hazard uncertainty from friction parameters (Romanowicz and Beven, 2003; Bates et al., 2004; Pappenberger et al., 2005; Werner, 2005; Horrit, 2006; Papaioannou et al., 2017), discharge (Di Baldassarre, 2010; Aronica

et al., 2012; Neal et al., 2013), topography or geomorphic adjustment (Wong et al., 2015; Call et al., 2017), boundary conditions (Pappenberger et al., 2006; Domeneghetti et al., 2013), and model structure (Dimitriadis et al., 2016; Zarzar et al., 2018). However, discharge, friction parameters, and topography have been consistently documented as the most influential sources of uncertainty within a particular modeling scheme (Jung and Merwade, 2011; Teng, 2017). Jung and Merwade (2015) investigated the sensitivity of inundation extents to discharge, friction parameters, and

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topographic uncertainty at two separate reaches with unique valley shapes. Inundation extents were

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more sensitive to topography at one reach while discharge was more influential at the other.

uncertainty in discharge, roughness, and topography.

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Further, these relationships were nonlinear highlighting the need to simultaneously account for

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Probabilistic floodplain mapping frameworks have been developed that account for model input and parameter uncertainty through MC simulations (Merwade, 2008; Beven et al., 2011;

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Papaioannou et al., 2017). These frameworks implemented the generalized likelihood uncertainty estimation (GLUE) method (Beven and Binley, 1992) to determine the probability of inundation

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by weighting acceptable results and discarding unacceptable results based on a user defined likelihood function and an optimal solution. This application has proven useful for evaluating uncertainty induced by model parameters when flood observations are available. However, when considering flood hazard uncertainty from a hypothetical event, such as the flood with a 1% annual

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exceedance probability, an optimal solution is unknown. In these instances, result likelihoods should be based on adequate sampling and prior definition of model input and parameter distributions in MC simulations. Detailed hydraulic models, such as those used to estimate regulatory flood hazards, can represent spatially variable land use by prescribing friction parameters at a high resolution in an

attempt to accurately simulate flood conditions. However, MC simulations often implement model inputs and parameters at a coarse resolution compared to regulatory floodplain models (Bates et al., 2004; Papaioannou et al., 2017). Previous efforts have attempted to mitigate the effects of coarse parameter resolutions on model accuracy by defining acceptable and unacceptable model results using the GLUE method. However, quantifying result likelihood as a result of input and parameter distributions requires maintaining a resolution that sufficiently describes their spatial

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variability to avoid compromising the accuracy of the model. Despite previous efforts to evaluate

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uncertainty from friction parameters, a need exists to apply parameters in MC simulations commensurate with the resolution of regulatory floodplain models.

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Prior evaluations of uncertainty in flood hazard estimates have relied on the assumption of

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stationarity and have not investigated how the uncertainty in flood hazard estimates might change due to a nonstationary flood series. Yet multiple approaches for conducting nonstationary flood

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frequency analysis have emerged (i.e. Villarini et al., 2009; Obeysekera and Salas, 2013; Luke et al., 2017; Serago and Vogel 2018). When estimating the magnitude of a hypothetical event based

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on a probability distribution that changes with time, one must select a point in time to define the distribution. Recent evidence suggests that, where trends are present, an updated stationary distribution based on a nonstationary fitting period is more appropriate than projecting trends (Luke et al., 2017). Implementing an updated flood distribution based on a nonstationary fitting

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period might reveal how the distribution of inundation likelihood will shift due to trends in flood peaks.

The development of probabilistic flood inundation maps has primarily been limited to

application in a research setting where computer programming has enabled automation of the large number of hydraulic model simulations required by Monte-Carlo procedures (often >500). A

simplified approach that is less resource intensive yet adequately captures the uncertainty bounds in flood hazard estimates might make uncertainty analysis more accessible to a broader audience and user base.

3. Methods 3.1 Probabilistic Floodplain Mapping Framework The probabilistic floodplain mapping framework generally involved quantifying statistical

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distributions of model inputs and parameters, simultaneously sampling each distribution as input

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for each of a large number of model simulations, delineating the inundated area for each simulation, and probabilistically defining flood inundation based on the distribution of results

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(Figure 1). Discharge and channel change were treated as uncertain model inputs, and Manning’s n-values were the uncertain model parameters. The framework innovates on existing frameworks

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by simultaneously accounting for discharge, friction parameter, and channel change uncertainty

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while prescribing model inputs and parameters at the resolution of detailed regulatory models. Since observations of low frequency flood events (i.e. 1% annual exceedance probability)

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are rarely available, the uncertainty in model results was conditioned through statistical quantification of model inputs and parameters rather than an optimal solution and posterior likelihood function. This constrained model results to possible permutations of model inputs and parameters, and defining their distributions was a critical first step in the framework. For each

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uncertain input and parameter distribution, latin hypercube sampling (LHS) was employed to generate n number of parameterizations. Model simulations were then conducted n times where the inputs/parameters resulting from LHS were randomly sampled without replacement generating n water surface profiles. Model simulations were conducted 1000 times since this achieved statistical convergence.

The framework maintained the spatial resolution of friction parameters in detailed hydraulic models and implemented a unique distribution of uncertainty for each value. Uniform adjustment of the channel bed in the vertical plane simulated uncertainty in channel geometry. Significant changes in flow, such as a tributary confluence, dictated locations for discharge input throughout the model. Sections 3.4 – 3.6 describe each input and parameter distribution. Water surfaces resulting from MC simulations were overlain with the channel and

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floodplain topography to delineate n flood scenarios in raster format. The probability that a pixel

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is inundated given the occurrence of the flood with a specified annual exceedance probability was

∑𝑛 𝑗=1 𝑓𝑖,𝑗 𝑛

(1)

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𝑃𝑖,𝐴𝐸𝑃 =

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then calculated according to Eq (1):

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where i indicates the pixel, AEP is the annual exceedance probability, n is the number of simulations, and fi,j is the inundation status (1 = wet, 0 = dry) at a pixel for simulation j of n. While

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Eq (1) denotes the exceedance probability of a flood depth greater than zero at a location, modifying the inundation status (fi,j) in Eq (1) to indicate the exceedance of a particular water surface elevation (1 = exceeded 0 = not exceeded) can extend evaluations to additional water

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surface elevations or flood depths other than zero.

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Figure 1. Framework implemented for (a) conducting MC simulations of flood hydraulics to develop (b) probabilistic floodplain maps.

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3.2 Study Site

We implemented the probabilistic floodplain mapping framework at McMullen Creek,

Charlotte, North Carolina (NC), USA. An estimate of the 1% annual exceedance probability flood extents defines the deterministic regulatory floodplain at McMullen Creek. The drainage area at the downstream end of the study reach is 40 km2 with 23% imperviousness and an average slope of 6%. The channel is characterized by irregular geometry with multiple structure crossings (e.g.

culverts), and the legacy of anthropogenic impacts on cross-sectional shape and planform are evident as is typical of many urban stream corridors. According to the 2011 National Land Cover Database (Homer et al., 2015), land use in the watershed is 95% developed. A United States Geologic Survey (USGS) stream gage (02146700) is located at the approximate mid-point of the

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study reach (Figure 2a).

Figure 2. Study site at McMullen Creek, Charlotte, North Carolina, USA depicting (a) the modeling reach and (b) the mapping area with cross-sections labeled by river station in kilometers. Note, the Mapping Area is the same location in panels (a) and (b).

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The Mann-Kendall trend test indicates a statistically significant positive trend (tau = 0.35,

p-value < 0.001) (Figure 3) in annual peak discharge measurements at the stream gage from 1962 – 2017. However, data records from the USGS indicate the influence of urbanization on annual peak discharge since 1985.

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Figure 3. Time Series of annual peak discharge at USGS 02146700. The red line indicates a simple linear regression line, with a 95% confidence interval depicted by the dark gray shading.

3.3 Model Implementation

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We implemented the detailed hydraulic model used to determine the regulatory floodplain boundary at the study site. This model simulates one-dimensional steady flow hydraulics with the

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Hydrologic Engineering Center’s-River Analysis System 5.0.3 (HEC-RAS). HEC-RAS is capable of simulating steady and unsteady flows and 1-D and 2-D hydraulics. However, performing 1-D

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steady flow simulations reduced computational demand and removed uncertainty introduced from model structure on comparisons between MC simulations and regulatory flood hazard estimates. In a 1-D steady flow analysis, HEC-RAS satisfies the conservation of energy at cross-sections

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using the standard step method. A flood event producing the peak of record on August 27, 2008 enabled calibration of the model to water surface elevations at USGS 02146700 (Figure 2a). Model calibration matched observed water surface elevations at the gage within 0.2 m by adjusting Manning’s n-value for open space land use types from 0.11 to 0.055 – 0.075 (AECOM, 2011a). Realizations of the 1% annual exceedance probability flood event in MC simulations provided a comparison with the regulatory flood hazard boundary. The modeled reach in MC simulations is

approximately 17.5 km; however, we evaluated a sub-reach with probabilistic floodplain maps (Mapping Area in Figure 2b).

3.4 Discharge Distribution To account for differences in various modeling approaches that quantify flood magnitude, uncertainty distributions were estimated for four commonly applied methods: 1) stationary flood frequency analysis using stream gage data, 2) a regional flood regression equation, 3) a hydrologic

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model estimate, and 4) nonstationary flood frequency analysis using stream gage data.

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We used the statistical software package, Peak FQ (Version 7.2), to conduct stationary flood frequency analysis. The software fitted a log-Pearson type III statistical distribution to annual

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peak discharge measurements (stream gage in Figure 2a) with a weighted regional skew (Weaver et al., 2009; England et al., 2018). The fitted moments of the distribution were then used to

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exceedance probability (Cohn et al., 2001).

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calculate the mean and standard deviation of the 99th percentile, the flood with a 1% annual

A regional flood regression equation [Eq (2)] that models flood magnitude based on

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drainage area and percent imperviousness represented the uncertainty distribution at an ungaged site, (Feaster et al., 2014).

𝐷𝐴 0.5386

𝑄100 = 26.6 (2.59)

10(0.0028∗𝐼𝑀𝑃𝑁𝐿𝐶𝐷)

(2)

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where Q100 is the magnitude of the flood with a 1% annual exceedance probability (m3/s), DA is the drainage area (km2), and IMPNLCD is the percentage of impervious area from the 2011 National Land Cover Database (Homer et al., 2015). The regional regression equation was developed by fitting Q100 to drainage area and percent imperviousness at stream gages within the region. However, individual measurements deviate from the fitted relationship to varying degrees, and the standard error of prediction reported for the regression equation is -28% to +40%. For this

method, the value of Q100 from Eq (2) represented the mean of the uncertainty distribution, and the reported standard error of prediction provided a means to calculate the standard deviation. AECOM (2011b) simulated a 7.29-inch, 24-hour storm event in a hydrologic model (HECHMS) to estimate the 1% annual exceedance probability discharge that was used for delineating the regulatory floodplain boundary at McMullen Creek. However, the hydrologic model from AECOM (2011b) does not provide a readily attainable estimate of variance or prediction error.

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This method assumes that a storm event with a particular intensity, duration, and frequency will

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translate to the 1% annual exceedance probability discharge in the hydrologic model. Local flood response of small urban watersheds (< 100 km2) in Charlotte, NC was found to be highly sensitive

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to antecedent soil moisture and the spatio-temporal structure of rainfall rates that emanate from

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convective storms or tropical cyclones (Smith et al., 2002; Wright et al., 2013). Given the lack of quantifiable uncertainty from the hydrologic model output and the sensitivity of flood response to

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its inputs, the standard deviation of annual peak discharge at USGS 02146700 (Figure 2a) was used as a surrogate for the standard deviation of the uncertainty distribution from the model

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estimate. Despite this assumption, the hydrologic modeling method provides a valuable comparison.

To conduct nonstationary flood frequency analysis, a linear trend is fitted to annual peak discharge measurements (stream gage in Figure 2a) with time as the dependent variable. Thus, the

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mean of the annual flood peak distribution changes with time. Trends in the standard deviation and skew were not considered because available record lengths limit the ability to describe them (Yue et al., 2015). Bayesian Inference and Markov Chain Monte-Carlo simulations estimated the time variant mean of the Log Pearson III distribution based on the observed trend in annual peak discharge measurements and the method of Luke et al. (2017). Direct quantification of uncertainty

in the mean through Markov Chain Monte-Carlo simulations makes this method desirable over alternative methods that provide a single value. A nonstationary fitting period from 1962 – 2017, determined the updated distribution at 2017, which defined discharge uncertainty. A log-normal distribution with a skew of 0 approximated the distribution of discharge uncertainty, while the different methods determined the means and standard deviations. Table 1 and Figure 4 depict the means (µ) and standard deviations (σ) for the upstream end of the mapped

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area.

Figure 4. Distribution of uncertainty in the 1% annual exceedance probability discharge. Abbreviations represent methods for estimating discharge (FFA = stationary flood frequency analysis; HEC-HMS = hydrologic model; REG = regional regression equation; and NFFA = nonstationary flood frequency analysis).

3.5 Friction parameter distribution

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We assumed a lognormal distribution for the uncertainty in Manning’s n-value following empirical evidence (Burnham and Davis, 1986; Papaioannou et al., 2017). The existing Manning’s n-values in the calibrated regulatory model provided an estimate of the mean, and the standard deviation was determined after Burnham and Davis (1986) [Eq (3)]. 𝜎 = 𝑁𝑟 ∗ (0.582 + 0.1 ∗ 𝑙𝑛(𝑛))

(3)

where is Nr is a reliability constant ranging from 0-1 based on the confidence in the estimated Manning’s n-value (1 = perfect confidence, 0.5 = moderate confidence). We assumed a value of 0.5 to represent moderate confidence. Since HEC-RAS enables the prescription of variable Manning’s n-values at a resolution of the surveyed nodes within each cross-section, multiple distributions with a unique mean and standard deviation were prescribed within and among cross-

regarding the Manning’s n-values in the calibrated regulatory model.

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3.6 Channel change distribution

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sections depending on the spatial variance of roughness. AECOM (2011a) provides details

We utilized manual field measurement data at the stream gage (Figure 2a) to examine

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variability in bed elevation. Mean channel bed elevation for a measurement was estimated by

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subtracting the hydraulic depth from the water surface elevation (Jacobson, 1995; Stover and Montgomery, 2001). The water surface elevation was obtained by adding the recorded stage at the

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time of measurement to the gage datum, while the hydraulic depth can be calculated as the flow area divided by the top width. Site conditions during separate visits can cause cross-sectional field

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measurements to be taken at different longitudinal distances from the gage (i.e. 50 m or 100 m downstream). Eliminating measurements with an unknown location relative to the stream gage prevented mistaking changes in bed elevation due to cross-section location for geomorphic adjustment. Although measurements date back to 1962, this limitation greatly reduced the sample

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size to 61 measurements from 2008 - 2018. The stream gage was located downstream of a concrete box culvert, as most gage locations

are often placed in geomorphically stable locations. However, the bed material in the reach upstream and downstream of the gage location was mixed gravel and sand; while the bank cover was variable with mixed vegetation, exposed soil, and rock protection. Using the maximum standard deviation among unique measurement locations avoided underestimating potential

changes in bed elevation throughout the reach while constraining the distribution to empirical observations. A systematic trend in bed elevation was not evident at this site since 2008; however, there was random variance about a mean value. A normal distribution with a mean of 0 m and a standard deviation of 0.09 m approximated the uncertainty in bed elevation change.

3.7 Simplified approach The simplified approach for estimating the uncertainty in flood hazard estimates involved

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conducting deterministic model simulations where select quantiles of the model input and

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parameter distributions were systematically sampled. For instance, a pseudo 90% confidence interval on inundation extents from the simplified approach would be generated by running two

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deterministic simulations that implemented the 5th and 95th percentile of each model input and

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parameter distribution. Thus, one could evaluate a desired range of uncertainty by conducting two deterministic model simulations that bracket an upper and lower bound based on the uncertainty

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in model inputs and parameters. To evaluate the accuracy of the simplified approach to simulate the uncertainty in inundation extents derived from MC simulations, systematically sampled

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quantiles of model inputs and parameter distributions were implemented in deterministic model simulations. The Critical Success Index [CSI; Eqn (4)] was used to evaluate the deterministic simulation of a quantile with the associated non-exceedance probability of inundation extents (1 – Pi,AEP) from MC simulations. The CSI is a common index used to evaluate the performance of a

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simulation relative a validation area that is effective in floodplain mapping when the focus is on the spatial distribution of flood extents (Papaioannou, 2016). In this case, the validation area was defined by the area where 1 – Pi,AEP is equal to or less than the corresponding sampled quantile of the deterministic model run. 𝐶𝑆𝐼 =

𝐴 𝐴+𝐵+𝐶

(4)

where A = Hit – validation area correctly identified as inundated by the alternative approach, B = False alarm – incorrectly identified inundated area by the alternative approach outside the validation area, and C = Miss – validation area identified as dry by the alternative approach. In addition to the CSI, the ratio of total inundated area from the alternative approach to the total inundated area from the corresponding 1-Pi,AEP within the mapped domain was calculated.

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4. Results The variance in water surface elevations is spatially variable along the modeled reach

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(Figure 5). As expected, the standard deviation of water surface elevations increases with increasing standard deviation in the input discharge distribution. Vertical gray lines in Figure 5

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indicate the location of hydraulic structures, such as bridges and culverts. It is interesting that the

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peaks in the standard deviation largely coincide with hydraulic structures indicating greater uncertainty in flood hazard estimates at these locations. However, immediately downstream of the

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structure locations, the standard deviation of water surface elevations rapidly declines. This indicates variable sensitivity of water surface elevations to model inputs and parameters where

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structures impose hydraulic controls, such as backwater effects upstream of structures or critical flow downstream of structures. The shaded region in Figure 5, indicates the mapped area, and the

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standard deviation in this area is greatest at the downstream end.

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Figure 5. Standard deviation (σ) of the 1% annual exceedance probability water surface elevations (m) for the modeled reach. The shaded region indicates the mapped area, and the vertical grey lines indicate hydraulic structure locations. Abbreviations represent methods for estimating the distribution of the 1% annual exceedance probability discharge (FFA = stationary flood frequency analysis; HEC-HMS = hydrologic model; REG = regional regression equation; and NFFA = nonstationary flood frequency analysis).

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Probabilistic floodplain maps for each method of estimating discharge depict inundation extents that exceed the deterministic regulatory floodplain boundary (Figure 6). The regulatory floodplain boundary intersects varying probabilities of inundation within and among the different floodplain maps depending on location along the channel reach. Additionally, the gradient of

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inundation probabilities perpendicular to the channel vary along the channel reach indicating spatially variable confidence in flood extents. For example, a comparison of the west and the east side of the floodplain for the nonstationary flood frequency analysis method reveals greater uncertainty at the western portion of the floodplain through a more gradual transition in probabilities moving away from the channel. Contradictory to the standard deviation of water surface elevations, the locations with the greatest uncertainty are not necessarily observed at the

downstream end of the mapped area. The stationary flood frequency analysis method appears to contain the greatest contrast in inundation probabilities, with a majority of the cells having either a high probability (>75%) or low probability (<25%) of inundation indicating less uncertainty for this method compared to the others. The hydrologic model method contains the largest distribution of probabilities. These results are not surprising considering that the stationary flood frequency analysis method had the lowest standard deviation in discharge, while the hydrologic model

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method was assigned the highest standard deviation in discharge. The lower magnitude of mean

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discharge for the regional regression equation method likely accounts for the greater variance of inundation probabilities observed within the regulatory floodplain boundary. The nonstationary

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flood frequency analysis method appears to have the most linear distribution of uncertainty present

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which is visible along the western edge of the mapped area.

In order to view the spatial variance in inundation uncertainty, it his helpful to depict a

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zone of uncertainty as indicated by the area between a range of inundation probabilities, analogous to a confidence interval. Figure 7 shows the zone of uncertainty (red area) confined by a 90%

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interval (i.e. all inundation probabilities are greater than 5% or less than 95%). The variable width in the zone of uncertainty along the channel reach emphasizes the spatial variance in the uncertainty of flood hazard estimates for this channel reach. However, locations with a relatively narrow zone of uncertainty for one method do not necessarily have narrow zones of uncertainty in

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all other methods. The northern portion of the mapped area illustrates this by the relatively narrow zone of uncertainty for the stationary and nonstationary flood frequency analysis methods and a relatively wide zone of uncertainty for the regional regression equation and hydrologic model methods.

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Figure 6. Probabilistic flood inundation maps of the 1% annual exceedance probability flood relative to the regulatory floodplain boundary for different methods of quantifying flood magnitude.

Figure 7. Zone of uncertainty in the 1% annual exceedance probability flood extents as indicated by a 90% interval (red area) for different methods of quantifying flood magnitude.

Development within the mapped area extends to the outside boundary of the regulatory floodplain. In most areas, the probability of inundation immediately adjacent to the regulatory floodplain boundary is approximately 50% for the hydrologic model method indicating the areas of greatest uncertainty since they are equally likely to be inundated or not inundated. The nonstationary flood frequency analysis method reveals an increase in the probability of inundation in these areas to greater than 80%. Figure 8 shows the juxtaposition of development along the

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deterministic regulatory flood hazard boundary and an increase in inundation probability between

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the hydrologic model and nonstationary flood frequency analysis methods.

Figure 8. Juxtaposition of residential development along the regulatory floodplain boundary compared with the probabilistic inundation extents of the 1% annual exceedance probability flood for different methods of quantifying flood magnitude.

Figure 9 shows the distribution of inundated area according to exceedance probabilities (Pi,AEP) for each method. Shifts in the magnitude and range of inundation area with inundation probability among methods indicates the influence of the different means and standard deviations of discharge. The change in inundation probability as a function of inundated area is nonlinear with the exception of the nonstationary flood frequency analysis method. The nonstationary flood frequency analysis method inundates the largest area for probabilities greater than 15%. The

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regional regression equation method inundates the smallest area; however, the range of inundated

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area is greatest for the hydrologic model method due to the high standard deviation in discharge uncertainty. The gray vertical line indicates the area of the regulatory flood hazard map. The

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ordinate value associated with the intersection of the regulatory floodplain area and each method

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indicates the fraction of simulations that exceeded the area of the regulatory flood hazard area. For instance, this value is 50% for the hydrologic model method since the regulatory hydraulic model

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defines its mean condition. Approximately 62% of simulations exceeded the regulatory floodplain area for the stationary flood frequency analysis method, and its distribution of inundation area

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contains the least amount of uncertainty. For the regional regression equation and nonstationary flood frequency analysis methods, approximately 12 and 94% of the simulations exceed the

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regulatory floodplain area, respectively.

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Figure 9. Distribution of inundation area corresponding inundation probability (Pi,AEP) of the 1% annual exceedance probability flood for the stationary flood frequency analysis (FFA); hydrologic model (HEC-HMS); regional flood regression equation (REG); and nonstationary flood frequency analysis (NFFA) methods. The vertical gray line (NFIP) indicates the area of the regulatory floodplain boundary. Overall, the simplified approach appears to estimate the uncertainty distribution of

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inundated area from MC simulations with a high level of accuracy (Figure 10). Quantiles indicate the non-exceedance probabilities systematically sampled from model input and parameter

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distributions and are referenced against non-exceedance quantiles of inundation likelihood from Monte-Carlo simulations (i.e. low quantiles indicate a greater likelihood of inundation). For all methods, the simplified approach under predicts inundation area at low quantiles while over predicting inundation area at high quantiles. However, the median is close to unity. Lower CSI

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values are obtained at low quantiles (< 0.25) compared to high quantiles. Greater variance in the CSI exists among methods at low quantiles and appears to be associated with discharge magnitude since the regional regression equation method has the lowest CSI with increasing CSI values as the mean discharge increases among methods. However, the CSI for high quantiles (>0.75) is less distributed among the methods with average values of approximately 95%.

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Figure 10. The critical success index (left) and percent of inundated area (right) of quantiles from the simplified approach referenced to non-exceedance probabilities of MC simulations of the 1% annual exceedance probability flood for the stationary flood frequency analysis (FFA); hydrologic model (HEC-HMS); regional flood regression equation (REG); and nonstationary flood frequency analysis (NFFA) methods.

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5. Discussion

The results of this analysis emphasize the lack of information conveyed when portraying

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flood risk as a discrete boundary. A deterministic boundary might characterize two locations in close proximity by different classes of flood risk, one at risk and the other not at risk. In light of

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the information from probabilistic floodplain maps, these two locations may in fact have very similar if not equal likelihoods of being flooded. Development within the study area closely outlines the regulatory flood hazard boundary (Figure 8). Additional studies have identified similar patterns across a larger scale, where development immediately adjacent to this boundary has

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occurred at faster rates than in areas farther from regulatory floodplains (Ferguson and Ashley, 2017; Patterson and Doyle, 2009). Thus, increased development adjacent to static flood hazard boundaries, population growth, and the compounding impacts of changing land use, river channels, and climates have the potential to place an increased proportion of the population at risk of flooding.

The range of uncertainty in flood hazard estimates quantified in this analysis appears to be greater than other studies that conduct MC analysis with 1-D HEC-RAS simulations (Jung and Merwade, 2011; Papaioannou et al., 2017). This difference might reflect uncertainty distributions of model inputs and parameters, the use of the GLUE methodology, and valley shape. The knowledge and confidence in model inputs and parameters imposes a substantial control on the uncertainty in flood hazard estimates. Significant effort should be devoted to defining these

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distributions prior to uncertainty analysis. The framework developed in this study provides a useful

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method for extending uncertainty analyses of flood hazards with detailed hydraulic models beyond the observational record to hypothetical events.

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Variable widths in the zone of uncertainty among and within methods (Figure 7) suggest

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variable sensitivity in the nature of uncertainty to valley shape similar to the findings of Jung and Merwade (2015). At a cross-section, the shape of the valley controls the rate of change in

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inundation area per unit change in water surface elevation. Therefore, valley shape helps explain spatial discrepancies in the magnitude of uncertainty between water surface elevations and

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inundation area. Demonstrations of strong variation in within reach inundation dynamics as a result of valley morphology by Miller (1995) further support this. The uncertainty in flood extents for a wide flat floodplain might be much greater than a confined valley with greater uncertainty in flood depth. It is important to distinguish between the uncertainty in flood depth and area as they can

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represent a form of hazard intensity and extent, respectively. The distribution of water surface elevations, inundation probability, and inundated area

were highly sensitive to the method used in estimating discharge and highlights the full range of uncertainty that might exist in flood hazard estimates. An increase in flood peaks due, at least partially, to the impact of urbanization (Figure 3) provides strong evidence for use of the

nonstationary flood frequency analysis as the representative discharge distribution at this site. Similar findings of significant increases in flood peaks during periods of urbanization and population growth at an adjacent watershed further support the use of nonstationary flood frequency analysis (Smith et al., 2002; Villarini et al., 2009). If the observed trends in annual peak discharge (Figure 3) continue and the discharge distribution from this method remains valid, flood risk within and adjacent to the regulatory flood hazard boundary will greatly increase along with

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its effective occurrence probability.

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While differences in the sensitivity of flood hazard uncertainty to discharge quantification method are apparent, the individual influence of discharge, friction parameters, and channel

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change distributions within each method were not investigated in this analysis. Friction parameters

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or channel change could have a larger impact on model results in other scenarios and mask the sensitivity of discharge quantification method. This impact could result from model sensitivity to

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individual inputs and parameters or a greater uncertainty in model inputs and parameters. For instance, if the uncertainty distribution in channel change were to increase in its mean and standard

apparent.

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deviation, the difference among the four discharge quantification methods might not be as

The coincidence of peaks in water surface standard deviation and hydraulic structures indicates a high degree of uncertainty at these locations. This observation might be the result of

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backwater effects due to reduced conveyance capacity through structures. However, it is also likely that model structure adds a considerable level of uncertainty. HEC-RAS implements a rule-based decision tree to select from multiple numeric schemes for determining depths at hydraulic structures. Consequently, variations in the selected numeric scheme might exist at hydraulic structures within MC simulations due to variable discharges. Regardless, these findings have

significant implications for floodplain management and infrastructure design. Social and economic functions depend on bridges and other hydraulic structures as critical components of the transportation network. Exceedance of the structure’s design flood capacity could disrupt these functions or even result in failure. Yet, the high degree of uncertainty in inundation depth emphasizes the lack of confidence surrounding deterministic evaluations of flood risk at hydraulic structures.

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The implemented sampling procedure assumes independence between model input and

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parameter distributions. However, inter-dependencies and non-stationarities likely exist in reality. For instance, degradation of the channel bed can alter the characteristic grain size and thus the

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roughness. These inter-dependencies are highly complex, variable in space and time, and difficult

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to quantify. Although, the sampling procedure implemented makes this simplifying assumption, the large number of model simulations mediates its potentially deleterious effects on the inference

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of inundation likelihood. In the absence of a readily available scheme for probabilistically simulating the dependence structure between discharge, friction parameter, and channel change

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uncertainty, the sampling procedure implemented provides a suitable alternative. Applications of channel change uncertainty assumed negligible lateral channel adjustment and uniform response in the magnitude and direction of geomorphic adjustment along the entire channel reach. In reality, where erosion occurs, subsequent aggradation is likely to occur

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downstream. Additionally, empirical observations limit the transferability of the applied methods to characterize channel change uncertainty. Future research could focus on implementing parsimonious sediment transport models and evaluating stream response potential across a range of hydro-geomorphic settings (Bledsoe et al., 2017) to develop a more transferable characterization of channel change uncertainty. A systematic trend in bed elevation over the last decade was not

evident through manual field measurements at this site. Where trends are present, nonstationarities in channel geometry should be considered, and issues such as time variant parameter sets and trend projections will need to be addressed (Luke et al., 2017; Salas et al., 2018). The simplified approach provided a robust, yet practical alternative to conducting MC simulations. A reduced CSI at lower quantiles (< 20%), particularly for methods with lower discharge estimates, aligns with the results of previous studies that find reduced values of the CSI

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for smaller flood magnitudes (Stephens and Bates, 2014). This pattern indicates the influence of

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topographic breaks in the valley on the sensitivity of inundated area. Differences in permutations of sampled distributions between the simplified approach and MC simulations might influence

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differences in the CSI. For instance, the 5% inundation non-exceedance probability of MC

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simulations might contain some combination of parameters that do not necessarily represent the 5% quantile from each of their distributions. Whereas, the 5% quantile of each parameter

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distribution results in the 5% inundation quantile for the simplified approach. Although, the performance of the simplified approach is reduced below 90% at quantiles less than 20% in the

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regional regression equation and hydrologic model methods, much more concern is often placed on the conservative end of flood hazard estimates. Additionally, other methods performed well across the range of quantiles analyzed. Despite the ability of the simplified approach to quantify uncertainty in flood hazard estimates derived from MC simulations in this analysis, further

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evaluation of this approach is needed in contrasting hydro-geomorphic settings. Quantifying the uncertainty in regulatory flood hazard estimates can aid floodplain

management by identifying areas of elevated or hidden risk that were not made known through deterministic approaches. Further, the spatial variation in flood hazard uncertainty can be utilized in land use planning or prioritization of mitigation efforts. For instance, land use types that provide

multiple co-benefits and present low consequences if inundated, such as parks and greenways, can be planned in locations where the uncertainty interval is large. Areas where elevated or hidden risk are identified might serve as candidate locations for stream restoration projects that lower flood elevations while simultaneously enhancing biodiversity and water quality. The techniques presented in this analysis can be tailored to various appetites for risk (e.g. 70% vs. 90% zone of uncertainty) or extended to additional flood hazard metrics (e.g. depth) or flood magnitudes (e.g.

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enable spatially explicit descriptions of velocity and stream power.

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the 10 or 500-year flood event). Future research could incorporate two-dimensional models that

6. Conclusions

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We implemented a novel framework for developing probabilistic flood inundation maps

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through MC simulations by simultaneously accounting for uncertainty in discharge, friction parameters, and channel change. A detailed hydraulic model was applied where the resolution of

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model inputs and parameters were maintained in MC simulations. Quantifying the distribution of uncertainty in flood hazard estimates using four separate methods enabled a comparison of

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inundation probability to the deterministic regulatory flood hazard boundary in a 40 km2 urban watershed in Charlotte, NC, USA. One method included nonstationary flood frequency analysis providing the first probabilistic floodplain mapping analysis that accounts for trends in annual peak discharge measurements. This analysis addressed the research questions posed in the introduction

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as follows:

1. Uncertainty in streamflow, channel and floodplain roughness, and geomorphic adjustment resulted in flood inundation probabilities greater than zero that extended outside the regulatory flood hazard boundary for all methods. Further, the regulatory boundary intersected variable inundation probabilities within and among methods indicating spatially variable confidence in

predicted flood extents. The maximum uncertainty in water surface elevation did not coincide with the maximum uncertainty in inundation extent suggesting the influence of valley shape on the spatial distribution of flood hazards. 2. Inundation probabilities in probabilistic floodplain maps and the zone of uncertainty reveal areas with elevated or hidden risk that are not revealed by the deterministic regulatory flood hazard boundary. Despite a relatively entrenched valley, the 90% zone of uncertainty was

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substantially wide (> 150 m) in some locations, and 3 of the 4 methods indicate inundation

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likelihoods greater than 50% for infrastructure adjacent to the regulatory boundary.

3. While the regulatory flood hazard boundary indicates an 0% probability of infrastructure

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inundation during the 1% annual exceedance probability flood event, the nonstationary flood

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frequency analysis method reveals a probability of infrastructure inundation greater than 80% in some locations. Evidence of increased flood peaks within the last 40 years and the results of

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the nonstationary flood frequency analysis suggests the existing regulatory flood hazard boundary underestimates and miscommunicates its intended risk status.

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4. The simplified approach simulated uncertainty intervals of MC simulations with a high-level of accuracy across the entire distribution of uncertainty for two of the methods and at nonexceedance probabilities greater than 12% for the other two methods. This provides practitioners with an attractive approach for conducting uncertainty analysis if resources for

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conducting MC simulations of flood hydraulics are not readily available. However, additional evaluations of the simplified approach are needed to assess performance in contrasting hydrogeomorphic settings.

The extent to which humans occupy floodplain corridors worldwide and projected increases in population underscore the relevance of this approach for understanding the interaction between

uncertainty in flood hazard estimates, exposure likelihood, and human development patterns along deterministic floodplain boundaries. By facilitating more realistic depictions of flood hazards, the novel techniques described in this study can contribute to a shifting paradigm in flood management that acknowledges the inherent uncertainty in model predictions in both stationary and

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nonstationary environments.

Acknowledgement This work was funded by the NSF Sustainability Research Network Cooperative Agreement 1444758, Urban Water Innovation Network.

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A Hydraulic Multi Model Ensemble Framework for Visualizing Flood Inundation

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Uncertainty. JAWRA Journal of the American Water Resources Association.

Table 1. Characteristics of the uncertainty distribution in discharge pertaining to each method. µ log(Q) (cms) 2.17 1.97 2.14 2.28

σ log(Q) (cms) 0.07 0.14 0.21 0.08

Range (cms) 88 - 249 33 - 262 29 - 627 110 - 330

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Method Stationary Flood Frequency Analysis Regional Regression Equation Hydrologic Model Nonstationary Flood Frequency Analysis