Estimate a flooding probability

In this example, we estimate the probability that the ouput of a function exceeds a given threshold with the FORM method. We consider the flooding model.

Introduction

The following figure presents a dyke protecting industrial facilities. When the river level exceeds the dyke height, flooding occurs. The model is based on a crude simplification of the 1D hydrodynamical equations of Saint-Venant under the assumptions of uniform and constant flow rate and large rectangular sections.

Four independent random variables are considered:

• : flow rate [m^3 s^-1]

• : Strickler [m^1/3 s^-1]

• : downstream height [m]

• : upstream height [m]

When the Strickler coefficient increases, the riverbed generates less friction.

The model depends on four parameters:

• the height of the dyke: [m],

• the altitude of the river banks: [m],

• the river length: [m],

• the river width: [m].

The altitude of the dyke is:

The slope of the river is assumed to be close to zero, which implies:

if .

The water depth is:

for any .

The flood altitude is:

The altitude of the surface of the water is greater than the altitude of the top of the dyke (i.e. there is a flood) if:

is greater than zero.

The following figure presents the model with more details.

If we substitute the parameters into the equation, we get:

We assume that the four inputs have the following distributions:

• ~ Gumbel(mode=1013, scale=558), > 0

• ~ Normal(mu=30.0, sigma=7.5), > 0

• ~ Uniform(a=49, b=51)

• ~ Uniform(a=54, b=56)

Moreover, we assume that the input random variables are independent.

We want to estimate the flood probability:

References

• Iooss B, Lemaître P (2015) A review on global sensitivity analysis methods. In: Meloni C., Dellino G. (eds) Uncertainty management in Simulation-Optimization of Complex Systems: Algorithmsand Applications, Springer

• Baudin M., Dutfoy A., Iooss B., Popelin AL. (2015) OpenTURNS: An Industrial Software for Uncertainty Quantification in Simulation. In: Ghanem R., Higdon D., Owhadi H. (eds) Handbook of Uncertainty Quantification. Springer

Define the model

from __future__ import print_function
import openturns as ot

Create the marginal distributions of the parameters.

dist_Q = ot.TruncatedDistribution(ot.Gumbel(558., 1013.), 0, ot.TruncatedDistribution.LOWER)
dist_Ks = ot.TruncatedDistribution(ot.Normal(30.0, 7.5), 0, ot.TruncatedDistribution.LOWER)
dist_Zv = ot.Uniform(49.0, 51.0)
dist_Zm = ot.Uniform(54.0, 56.0)
marginals = [dist_Q, dist_Ks, dist_Zv, dist_Zm]

Create the joint probability distribution.

distribution = ot.ComposedDistribution(marginals)
distribution.setDescription(['Q', 'Ks', 'Zv', 'Zm'])

Create the model.

model = ot.SymbolicFunction(['Q', 'Ks', 'Zv', 'Zm'],
                            ['(Q/(Ks*300.*sqrt((Zm-Zv)/5000)))^(3.0/5.0)+Zv-58.5'])

Define the event

Then we create the event whose probability we want to estimate.

vect = ot.RandomVector(distribution)
G = ot.CompositeRandomVector(model, vect)
event = ot.ThresholdEvent(G, ot.Greater(), 0.0)
event.setName('overflow')

Estimate the probability with FORM

Define a solver.

optimAlgo = ot.Cobyla()
optimAlgo.setMaximumEvaluationNumber(1000)
optimAlgo.setMaximumAbsoluteError(1.0e-10)
optimAlgo.setMaximumRelativeError(1.0e-10)
optimAlgo.setMaximumResidualError(1.0e-10)
optimAlgo.setMaximumConstraintError(1.0e-10)

Run FORM.

startingPoint = distribution.getMean()
algo = ot.FORM(optimAlgo, event, startingPoint)
algo.run()
result = algo.getResult()
standardSpaceDesignPoint = result.getStandardSpaceDesignPoint()

Retrieve results.

result = algo.getResult()
probability = result.getEventProbability()
print('Pf=', probability)

Pf= 0.0005340887806479517

Importance factors.

result.drawImportanceFactors()