# Note 19-Bernoulli Model of Anual Floods

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Note 19 Bernoulli Model of Annual Floods The exceedance of a given annual flood discharge is commonly modeled using the Bernoulli Model (i.e., flipping a coin with unequal probabilities). Let Q = annual flood discharge q = a specific discharge of interest p q = P{Q>q} (Probability that Q exceeds q in any year) p q is referred to as an annual exceedance probability F(q) = P(Q<q) = 1-p q ( cumulative distribution function ; will be discussed in detail in Lecture 20.) As applied to annual floods, the Bernoulli Model assumes the following: o There are only two possible events- exceedance or non- exceedance of a specified discharge. o The magnitude of an annual flood discharge in a given year is independent of its magnitude in other years ( independence ). o The probability that an annual flood discharge exceeds any chosen threshold is constant over time ( stationarity ). Remember the concept of an annual flood event? We define a water year so that a single event cannot have peaks in consecutive years. For most locations in the U.S., major floods do not occur in September and October, so we commonly start each water year on October 1. o Hurricane floods are one exception. The water year can be changed to the calendar year (or some other year) for such exceptions.
Consider the flood series plotted below. If this flood had occurred so that the peaks were in different water years, only one could be used. So the water year should be changed in this case. Flood events are not always well modeled as stationary - For example, dam construction and changes in landuse and land management can change the probability of floods. - Climate change due to increasing greenhouse gases also changes in flood probabilities. - And even when climate and landuse/land management are apparently stable, flood probabilities can change in time. For example, the El Nino-Southern Oscillation (ENSO) affects flood probabilities in some locations. Most routine analyses of flood probabilities assume stationarity. But in specialized analyses it is possible to account for the presence of nonstationarity. However, given climate change, many experts recommend only using the last 30 years of flood data.
We apply the Bernoulli Model to annual floods, the probability model applied to coins with unequal probabilities of heads and tails. Let Q = annual flood discharge, q = a discharge of interest, and p q = P{Q>q} (probability that Q exceeds q) p q is called an annual exceedance probability Based on the Bernoulli model, the recurrence interval of q is given by RI q /(one year)= T = 1/p q = the average number of years between floods equaling or exceeding q Based on the Bernoulli assumptions, we can estimate various probabilities of interest: o P{q not exceeded in n years} = (1-p q ) n § If q is the design levee capacity, this is the reliability of the levee with respect to overtopping. o P{q exceeded at least once in n years} = 1- (1-p q ) n § If q is the design levee capacity, this is the risk of levee failure due to overtopping. Example probability calculations: o P{q 100 is not exceeded in 30 years} = (1-.01) 30 = .74 o P{q 100 is exceeded at least once in 30 years} = 1-(1.01) 30 = .26 o P{q 100 is exceeded at least once in 5 years} = 1-(1-.01) 5 = .05 o P{q 100 is not exceeded in 300 years} = (1-.01) 300 = .05