This Title All WIREs
How to cite this WIREs title:
WIREs Water
Impact Factor: 4.451

On return period and probability of failure in hydrology

Full article on Wiley Online Library:   HTML PDF

Can't access this content? Tell your librarian.

The return period measures the rareness of extreme events such as floods and droughts that might cause huge damages to the society and the environment; hence, it lies at the heart of hydraulic design and risk assessment problems. Indeed, return period is a commonly applied probabilistic concept in the hydrologic literature, which has attracted renewed interest stimulated by the need of efficiently dealing with complex processes in a changing environment. In this study, the concept of return period and the related risk of failure are presented by making use of a general mathematical framework. It helps for a better understanding of the return period and risk of failure formulations that are commonly adopted in practical engineering applications. The framework can be further applied under more general conditions. In particular, the extension of the return period concept to nonstationary and time‐dependent cases is discussed herein, by relaxing the hypotheses commonly (and sometimes implicitly) assumed that allows to derive simple analytic formulations. This article is categorized under: Engineering Water > Planning Water Engineering Water > Methods Science of Water > Methods
Time series drawn from a stationary and independent discrete‐time process Zjτ = 1 year), so that the lognormal marginal distribution PZ(z) = Pr {Z ≤ z} with mean and variance both equal to 1 fully characterizes the process. The probability distribution function PZ(z) is depicted for reference on the left of the upper panel; safe events B = {Z ≤ 4} are depicted in blue color, while dangerous events A = {Z > 4} are highlighted in red. The upper and mid panels depict the same realization of the process (i.e., the same time series); in the former case we discard what has happened in the past because te is unknown, while in the latter te is known, thus allowing for conditioning the probability of future events to the past ones. In the lower panel, a different time series drawn from the same lognormal and independent process is shown, where te is known and equal to 0. The gray boxes delineate the sequences of events that correspond to the definitions (1) and (2) used for the identification of the pmf of X, as in Equations (2) and (4)
[ Normal View | Magnified View ]
Cumulative probability distribution computed at present time (t = 0) of the time to the event A = {Z > 4}, based on the concepts of waiting and interarrival time: (a) nonstationary independent process, characterized by a lognormal distribution with mean and variance equal to 1 at time t = 0 and a mean value increasing with time at α = 0.05% rate per year, where the red line depicts the probability distribution of the waiting time; (b) stationary autoregressive AR(1) process, characterized by a lognormal marginal distribution with both mean and variance equal to 1 and lag‐1 correlation coefficient equal to ρ1 = 0.8, where both the waiting and interarrival time distributions are depicted (blue and magenta lines, respectively). Vertical dashed lines highlight the average values (i.e., the return period of the event A), while the horizontal solid lines depict the corresponding probability of failure. The cdf of the waiting time pertaining to the corresponding stationary and independent process is reported as a reference in both the panels. Note that all processes considered in the figure are discrete‐time processes sampled at the yearly scale (Δτ = 1 year)
[ Normal View | Magnified View ]
Left panel: contour plot of the probability of failure R or of the reliability Re due to the occurrence of the T‐year event as function of the design life l; the probability of occurrence of an event larger than the T‐year quantile in a time period equal to T tends to 1 − e−1 ≈ 0.63. Right panel: probability of failure R or reliability Re computed for T = 100 year event as function of l
[ Normal View | Magnified View ]
Geometric distribution of the time to the occurrence of the event A = {Z > 4} (X, waiting or interarrival time), for the same process considered in Figure ; E[X] ≃ 53 years (following Equation (7)) and . The average value of the distribution function, that is, the return period T(z), corresponds to a nonexceedance probability that corresponds the probability of failure R approximately equal to 1 − e−1 ≃ 0.63, as explained in detail in Section 2.3. For the sake of clarity, the distribution functions of the discrete random variables X are represented as continuous functions
[ Normal View | Magnified View ]

Related Articles

Understanding and Managing Floods: an Interdisciplinary Challenge

Browse by Topic

Science of Water > Methods
Engineering Water > Methods
Engineering Water > Planning Water

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

The latest WIREs articles in your inbox

Sign Up for Article Alerts