## What is meant by risk or uncertainty analysis?

It gives a framework that is financially critical for finding the exceptional designs of hydrological systems with a reduced cost in total inclusive of the project cost and the disaster mitigation cost. But there would be hardly any assessment of a decent quality damage function. Hence, errors in demonstrating the hydrologic events, incorrect model parameters, and so on may occur. Here, we will discuss a specific hydrological design that of a detention basin for studying and understanding the risk-based procedure, uncertainties, and probability analysis by following a systematic framework.

## Uncertainty analysis

The Uncertainty analysis can be classified into 2 Types:

- Aleatory uncertainty
- Epistemic uncertainty

The aleatory uncertainty is because of the built-in haphazardness of natural calamities. The epistemic uncertainty is attached to a lack of knowledge of the processes, utilized models, parameters specified, and so on. For instance, in gradual flow reduction systems like detention basins, aleatory uncertainty is the unpredictable occurrence of rainfall incidents generating runoff. The epistemic uncertainties may come by checking events related to land usage and soil characteristics along with runoff evaluations that affect the precision of the amount of surface runoff produced by rainwaters. This runoff quantity is not fixed that influences the performance reliability of the detention system as it contains a storage tank of limited capacity. Thus, having a detention system with a large storage tank shall have a lower failure probability even though the cost would rise.

## Uncertainty and risk theories

Uncertainty and risk theories are receiving quite a lot of attention and are extensively practiced in the hydrology spectrum. Combining uncertainty theory with reliability assessment and financial cost-benefit analysis has the ultimate life-cycle performance assessment for infrastructure development. The return period in the risk design procedure is a variable decision although being a primarily selected design parameter.

Many risk-based hydraulic design procedures assess the failure destruction expense by taking into account only aleatory uncertainty. For example, both aleatory uncertainty of floods and epistemic uncertainty of hydraulic model parameters were taken into consideration highway drainage structure design, storm sewer systems, and so on.

The predicted yearly damage cost can be received from the risk curve which represents the relationship between the probability of failure scenarios and failure consequences with respect to monetary damage. As failure-induced damage is a random variable, the expected damage utilization does not contain all the statistical features of damage into the optimal design. By only using aleatory uncertainty connected with floods, Su and Tung extended the conventional risk design by presenting a decision criterion using the Expected Opportunity Loss (EOL), which comprises the mean and variance of the damage cost of various designs.

## Aleatory uncertainty

### Risk-based design considering aleatory uncertainty

Risk-based Design with Monetary Damage Function: Annual expected failure-induced damage can be calculated by

$E\left(D|x\right)={\int}_{{q}_{\mathrm{dsgn}}\left(x\right)}^{\infty}D\left(q|{q}_{\mathrm{dsgn}}\left(x\right)\right)\mathrm{dF}\left(q\right)$

Where,

*q _{dsgn}*(x) = designated design capacity depending on the project capacity or design frequency

x and *F*(*q*) = cumulative distribution function portraying natural random occurrence of load *q*.

### Integrating epistemic uncertainties

The resulting mathematical expression for computing the system failure-related FoM (Figures of Merits) under crucial circumstances, be it the annual predicted damage or the overflow volume, will involve multiple integrations.

### Application to detention basin design

The epistemic uncertainties are integrated from model parameters resulting in determining inflow from abrupt natural rainfalls. In selecting the design capacity, the marginal cost criterion is utilized. For demonstrating the analysis procedure without preparing hydrologic rainfall-runoff simulation, a modified rational method was used for determining the size of storage in the detention basin.

## Sizing detention storage by the modified rational method

The rainfall intensity-duration-frequency (IDF) model mathematical expression is-

$i\left(T,{t}_{d}\right)=\frac{{a}_{T}}{{\left({t}_{d}+{b}_{T}\right)}^{{c}_{T}}}$

Where,

*i*(*T*, *t _{d}*) average min intensity of rainstorm event (in/h or mm/h)

*a _{T}*,

*b*,

_{T}*c*= model coefficients.

_{T}There exist many empirical or semi-empirical formulae for the computation of the time-of-concentration. One of the popular empirical formulae used is the Soil Conservation Service (SCS) is

${t}_{c}=\frac{{L}^{0.8}{\left(S+1\right)}^{0.7}}{1140{\gamma}^{0.5}}$

Where,

L = flow length (ft)

Y = average watershed land slope (%)

and S is the maximum potential retention (in) with CN = curve number representing flow retardance factor.

Land use variability, vegetation cover, soil characteristics (inclusive of soil moisture condition), and others give their contribution to variability in CN (SCS curve number) and runoff coefficient C. Both of them are the modal parameters of epistemic uncertainty.

## Epistemic uncertainty

### Presence of uncertainty in the design of detention storage

Due to the epistemic uncertainties, the design storage is uncertain. The presence of epistemic uncertainties induces uncertainty to design capacity. For a design frequency (*T _{dsgn}*), if S

_{T dsgn}(design capacity) is taken depending on the values of modal parameters. It says that when the rainfall calamity along the design frequency

*T*happens together, the detention basin will be unable to hold the new runoff by 50% probability. Thus, this high probability must be avoided and the forecast that predicted the chances of the basin resisting more water would stand false. If storage with zero failure probability is made then there would be an increase in the cost. So, the following expression needs to be satisfied in order to achieve minimum or almost no failure.

_{dsgn }${S}_{Tdsgn}{{S}^{\alpha}}_{Tdsgn}$

Where,

- S
_{T dsgn}= a deterministic constant for the design storage capacity of the detention basin *T*= design frequency_{dsgn}- α = performance reliability.

## Rosenbluth probabilistic point estimation method

When model parameters are random variables (RVs), in order to analyze the uncertainty may be imposed to quantify the uncertainty characteristics of model output or the design quantity. The Rosenbluth method is a probabilistic point estimation method that can be used for the estimation of statistical moments of the design quantity involving many random variables which are either correlated or uncorrelated. The sampling point (C and CN) can be plotted by deducing the data from the table format.

## Conclusion

The risk-cost derived from the risk curve interval portrays the probability-weighted damages under different threat scenarios of changing exposure likelihoods.

For instance, the estimation of the life-cycle performance of a green roof to evaluate the economic benefit on air quality improvement by transforming the estimated CO_{2} and nitrogen oxide uptake through emission credits.

## Context and Applications

- Bachelors in Technology (Civil Engineering)
- Masters in Technology (Hydraulic Engineering)
- Masters in Science (Fluid Mechanics)

## Practice Problems

1. What considerations must be taken for designing highway drainage structure design and storm sewer systems?

- Aleatory uncertainty of floods and epistemic uncertainty of hydraulic model parameter
- Probability distribution and floodplain estimation
- Environmental modeling and precipitation
- All of these

Correct option- a

Explanation: Aleatory uncertainty of floods and epistemic uncertainty of hydraulic model parameter considerations must be taken for designing highway drainage structure design and storm sewer systems.

2. What is the full form of EOL?

- Expected Optimum Loss
- Expected Opportunity Loss
- External Opportunity Loss
- External Optimum Loss

Correct option- b

Explanation: The full form of EOL is Expected Opportunity Loss.

3. Which method is used when the model parameters are represented as Random Variables (RVs)?

- Stochastic method
- Watershed deviation
- Non-exceedance probability
- Rosenbluth Probabilistic Point Estimation Method

Correct option- d

Explanation: Rosenbluth Probabilistic Point Estimation Method is used when the model parameters are represented as random variables (RVs).

4. Which abbreviation denotes design frequency?

- α
- S
_{T dsgn} *T*_{dsgn}- None of these

Correct option- c

Explanation: *T _{dsgn}* abbreviation denotes design frequency.

5. Which of the following is INCORRECT?

- CN is the SCS curve number and C is the runoff coefficient
- FoM stands for Figures of Materials
- Both a and b
- None of these

Correct option- b

Explanation: FoM stands for Figures of Merits.

## Related Concepts

- Urban drainage design and modeling
- Case study on regression analysis
- Detention storage frequency curve
- Detention tank storage design frequency curve

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