One Random Variable
Contents
One Random Variable#
Consider a river that is protected by dikes, earthen embankments with the purpose of keeping water in the river channel and preventing flooding of the hinterland. The dikes should be designed and build such that the height allows the river to safely pass the maximum discharge every year, measured in m
Fig. 1 Rating curve for hypothetical river. Relationship is
This is a difficult question because the actual discharge observed in a given year is random. In addition to other variables, it is largely dependent on rainfall in the drainage basin, which is impossible to predict far in advance—certainly not for the entire year. Furthermore, we don’t have a way to accurately extrapolate our past observations to identify a logical maximum discharge. For this reason, we will use a continuous probability distribution to quantify the probability associated with a specific maximum discharge,
Fig. 2 PDF and design value for river discharge with 0.01 probability of exceedance, or 100 year return period:
In particular, we note that failure occurs for high discharge, so we must use the complementary CDF to find the design value, in this case probability of exceeding
after which the water depth and minimum dike height can easily be determined from the rating curve:
Recalling the safety margin limit state from the previous section,
If the condition
Reflection on the Simple Example#
Is this an oversimplification of reality? Perhaps, but it gets pretty close. Many flood protection systems in the world use an exceedance proabability approach to probabilistic design: choosing a specific and making sure the component or system can handle the load. This works well when the primary source of uncertainty is the load: the river discharge in this case.
We also took for granted that a simple relationship between river discharge and water depth is available. However, this is never the case, as such a relationship depends on the cross-sectional shape and roughness of the floodplain, which not only varies along the river trajectory, but also changes due to natural phenomenon and human interventions. This introduces additional uncertainty into our assessment of whether the dikes are high enough.
Fortunately there is also extra conservatism built into this approach. For example, duration of high water plays a role: if discharge exceeds the capacity of the dike system, but only lasts for a short time (minutes or a couple hours), perhaps the dikes can withstand the overflow without eroding and causing flooding. This can be conceptualized by considering the joint probability of a high discharge and degradation of the dike leading to flooding:
where the conditional term represents the probability that the dike erodes (fails) given that the water depth exceeds the height. Assuming failure when the critical water depth is exceeded is conservative because it implies
Later chapters of this book will introduce methods for taking these realities into account in the design and decision-making process. For now, we will see how our probabilistic design changes when more than one random variable must be considered.
MUDE exam information
Given a probability requirement and distribution for a random variable of interest, you should be able to find the appropriate design value.
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Although the 20,000 m
/s value is tempting because it is the most safe option, it is probably prohibitively expensive to build!