Overview

The methods for assessing flood risk used by the U.S. Army Corps of Engineers are documented in Engineering Manual 1110-2-1619. A brief overview is provided here because assessing risk is what HEC-FDA is all about. Mario Andretti races cars and HEC-FDA assesses flood risk. Risk is broadly defined as a situation or event where something of value is at stake and its gain or loss is uncertain. Using HEC-FDA, we measure risk as expected annual damage (EAD), a metric that expresses risk as a combination of the likelihood and consequence of an event. Flood risk explicitly considers the probability and consequences of subjecting people and property to the entire range of likely flood events, with consideration of risk management possibilities provided by any structural or nonstructural measures. Flood risk can be conceptualized as a function of the hazard, performance, and consequences, where consequences are a function of exposure and vulnerability, as depicted in the figure below.

Flood risk can be conceptualized as a function of the hazard, performance, and consequences, where consequences are a function of exposure and vulnerability (USACE, ER 1105-2-101).

Each of the risk determinants has a specific implementation in HEC-FDA:

  • The hazard is the object that causes the harm –  in this case, a flood. In HEC-FDA, the flood hazard is described in terms of frequency, flow, stage, extent, and depth. The hazard for the primary compute of risk (measured as EAD) is modeled using an appropriate combination of the hydrologic and hydraulic input functions: discharge-frequency, regulated-unregulated flow transform, stage-discharge, interior-exterior, and stage-frequency. For example, the hazard of flooding from an undeveloped free-flowing stream would be modeled using an analytical discharge-frequency function and a stage-discharge function. For the purposes of aligning the hazard with the consequences (the stage-damage compute), the hazard is also modeled using geo-referenced hydraulic modeling (formerly known as water surface profiles), which describes stage and extent. 
  • Performance is the system’s reaction to the hazard. In the above figure, performance refers to the system features and the uncertain capability to contain/manage the flood hazard for the full range of possible events and for a specific event or load. We include performance in an HEC-FDA study as a top of levee elevation and a system response curve - a relationship between the hazard loading (stage) and the probability of system failure. 
  • Consequence is the harm that results from a single occurrence of the hazard. In HEC-FDA, consequences are typically measured in terms of economic damage for a given stage. Consequences are modeled using aggregated stage-damage functions (damage from a single occurrence of the hazard, for the entire range of potential hazard magnitudes). The aggregated stage-damage function is calculated by connecting the range of potential hazard magnitudes with exposure and vulnerability. We calculate depth above the first finished floor, find the percent damage for that depth, and multiply percent damage by structure value. 
    • Exposure describes who and what may be harmed by the flood hazard. Tools such as flood inundation maps provide information on the extent of flooding, which is used to define a boundary around who and what may be harmed. In HEC-FDA, we typically represent exposure with the use of a structure inventory which provides information on the population and property that may be affected by the flood hazard.
    • Vulnerability describes the susceptibility to harm of human beings, property, and the environment exposed to the hazard. In HEC-FDA, we represent vulnerability using depth-percent damage functions and structure characteristics like the elevation of the first finished floor. 

Computation of Flood Risk 

In the following section, we describe a basic approach for computing flood risk. Using HEC-FDA, we focus on economic flood risk which we measure as expected annual damage (EAD). EAD is computed in a few steps. First, we analyze available hydrologic and hydraulic data to derive a probabilistic model of the annual likelihood of flood inundation. Second, we use the probabilistic model of the annual likelihood of flood inundation to derive a related probabilistic model of the consequence of interest which takes the form of a damage-frequency function. Finally, we then compute EAD by integrating the damage-frequency function, a procedure that can be mathematically defined as:

E[X] = \underbrace{\int_{-\infty}^{\infty} xf_{x}(x)dx}_{1} = \underbrace{\int_{-\infty}^{\infty} \left(1-F_{x}(x)\right)dx}_{2}

where:

     X        =  the consequence which is inundation damage,

     E[X]   =  expected value of the consequence,

     x         =  a value of the consequence,

     fX(x)    =  the probability density function of X,

     FX(x)  =  the cumulative distribution function of X, FX(x) = P(X ≤ x) and (1- FX(x)) = P(X  ≥ x).

The above equation integrates over the complete probability domain and thus captures the complete range of consequence in a single value, EAD, the defining measure of flood risk in HEC-FDA. The first integration reflected in the above equation is similar to calculating the probability-weighted average consequence, and the second integration allows computation of the expected value as the area beneath the cumulative distribution. Thinking about the equation as calculating a probability-weighted average damage sheds light on the importance of higher-frequency consequences relative to lower-frequency consequences in their contribution to the average. For example, a very high-frequency event with very low consequences could matter as much if not more than a very low-frequency event with very high consequences (that's why we worry so much about ground-proofing high-frequency damages).

The figure below provides an illustration of the numerical approach used within HEC-FDA to apply the above equation. HEC-FDA carries out integration discretely based on the trapezoidal areas under the curve. For an interval of exceedance probability, the area under the curve is calculated as a trapezoid where the top of the trapezoid is a straight line between the two function values. This calculation is repeated on very small intervals of exceedance probability between 0 and 1. As shown in the right-hand side of the figure below (b), the accuracy of the EAD estimate can be improved by selecting exceedingly smaller frequency intervals (with the trade-off of computational expense). As the frequency intervals get infinitesimally small, the numerical approach to integration matches the parametric approach demonstrated in the above equation. 

Illustration of numerical approach used in HEC-FDA to apply the expected annual consequences (EAD) equation. (a) Provides the damages calculated for exceedance probabilities between 0 and 1. (b) Provides a snapshot of graph (a) for exceedance probabilities between 0.45 and 0.55.

Getting to Damage-Frequency 

Before EAD can be computed by integrating the damage-frequency function (which is the relationship between exceedance probability and damage used in the above equation), the function must first be developed. The function is typically developed by combining a series of interrelated functions, including a discharge-frequency function, a stage-discharge function, and a stage-damage function. To illustrate let's walk through an example of the development of a cumulative distribution function of damage for natural riverine conditions. The example process described below helps us think about how to use HEC-FDA to develop the damage-frequency function. 

  1. The process begins in subfigure (a) with a discharge versus frequency function. The word frequency is used interchangeably with exceedance probability here. The discharge-frequency function is often obtained from a frequency analysis of annual peak flows. In this example, the peak flow of the 0.05 annual exceedance probability (AEP) event is 4,000 cfs.
  2. Flood damage is correlated principally with flood stage, so the next step of the process is to translate the flow estimated from subfigure (a) to a stage, as shown in subfigure (b). In the example, the peak flow of the 0.05 AEP event with 4,000 cfs has a corresponding stage of 40 feet. Thus, the 0.05 AEP event has a stage of 40 feet.
  3. The next step of the process is to estimate the damage for a given stage using the stage versus damage function shown in subfigure (c). Calculating the stage-damage function is one of the most important functions of HEC-FDA, and will have taken prior to the development of the damage-frequency function. In the example, the peak flow of the 0.05 AEP event with 4,000 cfs and a stage of 40 feet causes damage of $400. Thus, the 0.05 AEP event has a damage of $400.
  4. The above steps are repeated over the full range of frequencies to develop the damage-frequency function, as in subfigure (d). The damage-frequency function is then integrated using the trapezoidal approach (approximation of Term 2 in the above equation) to calculate EAD. The trapezoidal rule is a required approximation of the above equation, because the damage-frequency curve is rarely an analytical probability distribution and is instead a graphical curve requiring a graphical or numerical solution.

Example of the derivation of the cumulative distribution function of damage using functions for natural riverine conditions.

Rather than carry out a frequency-by-frequency process, as indicated in Step 4, HEC-FDA uses the entire curve (which consists of more than 100 coordinates) at once. In other words, HEC-FDA composes the stage-discharge function with the discharge-frequency function to produce a stage-frequency function, and then composes the stage-damage function with the stage-frequency function. In HEC-FDA, this mathematically equivalent approach results in one realization of EAD.

Throw Uncertainty Into the Mix

Let's continue the discussion of the natural riverine example used above and add uncertainty. We are uncertain about the "true" measure of EAD because we are uncertain about the "true" relationships represented in the above figure. In other words, we are uncertain that a flow of 4,000 cfs has a probability of being exceeded of 0.05 - because the flow with that exceedance probability could be higher or lower! As a result, we model the series of interrelated functions with uncertainty. To include uncertainty, each point in the relationship is modeled with an x value and a distribution of y values. For example, a stage (x value) and a distribution of damage (y value). HEC-FDA applies a Monte Carlo simulation and randomly samples a y value for each x value for each summary relationship, combines each relationship to produce a unique damage-frequency function, and then integrates the function to calculate a unique realization of EAD. The process occurs many times to produce a stable distribution of EAD.