Similar to describing uncertainty about economic metrics such as EAD, explicit inclusion of uncertainty in a risk assessment enables an analyst to describe the uncertainty around system performance metrics as well. Distributions of uncertainty can be developed for all functions used in a risk analysis. Figure 4 illustrates an example of uncertainties over the range of a stage-discharge function. The uncertainty is described as a probability density function (PDF, or bell-shaped curve) for every discharge value, positioned on the most likely value.  The PDF exists on a Z-axis and so is viewed as coming out of the page.

Figure 4. Illustration of distribution of estimates of channel stage for specified discharge

Uncertainty in System Performance Metrics 

Annual exceedance probability (AEP) of a given threshold can be described using statistical summaries of a random sample of AEP estimates generated from Monte Carlo simulation (one estimate from each realization). The mean, median, and assurance (percent less than some value) of AEP from that sample are often reported and used as metrics. In Figure 7, an initial stage-frequency function is represented by the green line, while the thinner black lines represent randomly sampled realizations of stage-frequency that result from combining sampled discharge-frequency and stage-discharge functions. In the figure, an example stage of interest of 64 feet is shown. Each Monte Carlo realization has a different value of the AEP of 64 feet, shown as blue dots.  The mean AEP is the average of the sample of AEP of 64 feet across all stage-frequency realizations. 

Figure 7. Example of Mean Annual Exceedance Probability for a stage threshold

Figure 8 below shows the sample of these AEP values as a histogram.  The sample is compared to quantiles of interest to describe assurance of AEP of a given threshold. For example, the assurance that AEP is less than 0.01 is calculated as the fraction of realizations for which AEP is less than 0.01, as shown for 64 feet by the solid bars in the histogram in Figure 8, and equal to 44%.   

Figure 8. Example of distribution of AEP for a stage threshold  

Assurance

We use the term “assurance” to describe the uncertainty that the river-infrastructure system contains the hazard at a given likelihood or stage, either the likelihood that a quantile (e.g. 1% stage) is below some target stage threshold, or the likelihood that the AEP of a target stage threshold is below some probability (e.g. 1%). Assurance is calculable only when risk assessment is carried out with uncertainty.

Assurance of target stage

Assurance of a target stage reflects the chance that a stage quantile (the stage with a given exceedance probability, e.g. 1% stage) is less than the target stage or top of levee. In the example of a levee, assurance is the chance that a probabilistic channel stage such as the 0.01 AEP quantile does not exceed (overtop) the levee, given all uncertainties. Assurance is the term selected to replace “conditional non-exceedance probability” (CNP).  Figure 9 shows random realizations of the stage-frequency function that were generated by sampling and combining discharge-frequency and stage-discharge functions, and the sample of 0.01 AEP stage estimates is shown as the vertically-aligned histogram at the 0.01 exceedance probability.  Assume for example that the target threshold is a stage of 70 feet. Then assurance of the 0.01 AEP stage for 70 feet reflects the fraction of the 0.01 AEP stages from all realizations that are less than 70 feet, represented by the solid bars of the histogram of 0.01 stages.  This fraction, which is 86% in this example, reflects the likelihood that the 0.01 AEP stage is below the target stage of 70 feet, given the uncertainty in both discharge-frequency and discharge-stage.  In other words, given the uncertainties, there is 86% chance that the 0.01 AEP stage is below 70 feet.

Figure 9. Example of assurance for a stage threshold

Assurance of AEP 

Assurance of AEP is the likelihood that AEP falls below a given value, such as 0.01. Assurance of AEP provides the same result as assurance of a target threshold when no system response curve is included in a risk assessment. Figure 10 shows the many random realizations of stage-frequency curves, with a histogram depicting the values of AEP for stage = 70 feet.  In the figure, assurance of 0.01 AEP for the target stage of 70 feet is 86%, because 86% of the 0.01 event stages are less than 70 feet. In other words, given the uncertainties, there is 86% chance that the AEP of 70 feet is below 0.01.  Note, as shown in Figure 11 which includes both histograms, this assurance that the AEP of 70 feet is less than 0.01 is the same as the assurance that the 0.01 stage is less than 70 feet.  This result follows from the fact that any realization’s stage-frequency curve that has a 0.01 stage less than 70 feet must also have an AEP of 70 feet that is less than 0.01.  However, when a system response curve is included in the computation, assurance of AEP does not equal assurance of a target threshold.

Figure 10.  Histogram of AEP and Assurance of AEP<0.01 for stage = 70 feet.

Figure 11. Example of Relationship Between Assurance of AEP and Assurance of Target Stage

Rather than stating the likelihood that AEP is less than some specified quantile such as 0.01 for a given stage, a similar metric might take the 90% as given, and find the AEP that has 90% assurance at a given stage. Figure 12 shows that the AEP that has 90% assurance for a stage of 70 feet is AEP = 0.012.

Figure 12.  AEP with 90% assurance for a given stage


Assurance incorporating system response

When a system response curve is included in the assessment of a levee, then assurance includes the chance that water may move past the levee and inundate the floodplain, given the joint probability of hazard loading (discharge or stage magnitude) and structural failure (levee breach). When using the curve sampling approach to incorporating uncertainty, the two types of assurance (of a threshold, or of an AEP) incorporate a system response curve differently, and the results are no longer the same.

Assurance of a target threshold with system response

The assurance of a target threshold such as top of a levee incorporates the system response curve after all realizations are complete. Realizations of a channel stage quantile (e.g. 0.01) are discretized into ranges (the histogram shown for the 0.01 stages in Figure 13 panel (a)), and for each range below the top of levee, the likelihood of stage in that range is multiplied by the conditional likelihood that the levee does not fail in that stage range, shown in panel (b) of Figure 13.  (For ranges above the top of levee, the result is the same as levee failure, meaning likelihood = 0.)  Reducing each histogram bar in this way and summing the total results in the assurance of the target stage (top of levee) given the likelihood of levee failure that is captured by the system response curve (See equation #). This assurance must logically be less than the assurance without the system response curve as excluding the system response curve reflects an assumption of no chance of levee failure below the top.  In the example in Figure 13, the assurance of the stage 70 feet being less than the 0.01 stage is reduced from 86% to 51%, shown in (a) and (b), with the system response curve shown.


   


Figure 13.  Inclusion of System Response Curve in Assurance Metrics 

Equation #: Summation of Joint Probability for Assurance of Target Stage with System Response Curve

\sum_{j=1}^n\left(P(F)_{j+1}-P(F)_j\right)\cdot\overline{P}_{j+1,j}

Where:

P(F)= Probability of failure for a stage quantile or computation interval

\overline {P} = The average probability for a stage quantile or computation interval

n = The target stage or target computation point (top of the levee)

j =  The beginning quantile or computation interval


Assurance of AEP with system response

The assurance of AEP is also affected by including a system response curve.  In the Monte Carlo simulation approach to incorporating uncertainty, each AEP realization reflects the joint probability of loading and failure so that the system response curve is included at the realization level, rather than after all realizations are complete.  For assurance of AEP with system response, first find the average probability of failure for each stage quantile. Then calculate the incremental probability of stage exceedance for each stage quantile. After completing the incremental and average probability, sum the product for each stage quantile (see equation $ below).  In the following example, we demonstrate that assurance of AEP is different from assurance of a target stage when we include a system response curve in the risk assessment.  The distributions of annual exceedance probabilities of the stage 70 feet, with and without inclusion of a system response curve, are plotted from Figure 13 panels (c) and (d), respectively enlarged below, where solid the blue bars represent the relative frequency of AEPs that are less than or equal to 0.01. Note, the distribution of AEPs in panel (c) is the same distribution of AEPs represented by the blue horizontal histogram in Figure 10.  Also note that the relative frequency of AEPs less than 1% is lower when a system response curve is included in panel (d). The lower assurance of AEP reflects the higher chance that a leveed area could be inundated due to the imperfection of a lateral structure (levee) captured by the system response curve. Further observe in panel (b) and panel (d) (with system response) that the fraction of AEPs that are less than 0.01 at the target stage of 70 feet (49%) is no longer equal to the fraction of non-failure 1% stages (51%).   


Figure 14 contains a closer look at the histograms shown in panels (c) and (d), except with AEP on a linear axis.

Equation #: Summation of Joint Probability for Assurance of AEP with System Response

\sum_{j\mathop=1}^n\left(P_{j+1}-P_j\right)\cdot\overline{P\left(F\right)}_{j+1,j}

Where:

\overline{P(F)}= The average probability of failure for stage quantile or computation interval

P= The probability of a stage quantile or computation interval

n = The target stage or target computation point (top of the levee)

j = The beginning quantile or computation interval



Figure 14. Assurance of AEP. image reflects assurance after including a system response .