Thèse préparée par Charlie Sire

Title : Robust inver­sion under uncer­tain­ty for risk ana­ly­sis – appli­ca­tion to the fai­lure of defences against flooding

Beginning of the­sis : 2020
End of the­sis : 2023

Abstract : The risk of coas­tal or flu­vial floo­ding is aggra­va­ted by the fai­lure of defences (either natu­ral like dunes or arti­fi­cial like dykes). The stu­dy of flood hazard on the Dutch River System illus­trates this (Curran et al, 2019), as do seve­ral events that occur­red in the last decade such as the hur­ri­cane Katrina in 2005 in New Orleans (Sills et al, 2008). The fai­lure of defences is a fac­tor of floo­ding risk whose impor­tance will keep increa­sing in the future because of cli­mate change. Our ana­ly­sis of coas­tal and river floo­ding takes into account the fol­lo­wing variables :

  • Controlled variables, rela­ted to the geo­me­try and loca­tion of the flood pro­tec­tion embankments.
  • Uncontrolled variables cap­tu­ring the ran­dom­ness of natu­ral phe­no­me­na, such as hydro­graph para­me­ters for river floo­ding and off­shore hydro­dy­na­mic condi­tions (e.g wave cha­rac­te­ris­tics). These variables have known pro­ba­bi­lis­tic laws.
  • Drastically uncer­tain variables, that are uncon­trol­led and not well cha­rac­te­ri­zed nei­ther pro­ba­bi­lis­ti­cal­ly nor in regu­la­tions. They are rela­ted to the dyke breach parameters.

In this work, we inves­ti­gate a mathe­ma­ti­cal pro­ce­dure based on inver­sion to cha­rac­te­rize the pos­sible com­bi­na­tions of control­led para­me­ters (named excur­sion set) that lead to floo­ding with a pro­ba­bi­li­ty grea­ter than a given thre­shold α, a stan­dard safe­ty limit. If the floo­ding event is defi­ned as the water level at a spe­ci­fic loca­tion, model­led as the result of a func­tion F, excee­ding a thre­shold T, the pro­ba­bi­lis­tic excur­sion set to be inves­ti­ga­ted can be defi­ned as is a ran­dom variable giving the water level at a spe­ci­fic loca­tion for a com­bi­na­tion of control­led para­me­ters equal to xc.

Several ques­tions are addressed :

  1. First, how to represent excur­sion sets when there are more than two control­led variables and some uncer­tain variables do not have a known den­si­ty. Parallel coor­di­nates plots are consi­de­red and the rela­tion­ship bet­ween the pro­ba­bi­lis­tic excur­sion set Sα and the ran­dom excur­sion set  ,  (where Xu repre­sents the uncon­trol­led variables) is theo­re­ti­cal­ly investigated.
  2. Second, the nume­ri­cal simu­la­tions of the floo­ding are expen­sive to com­pute (typi­cal­ly seve­ral hours): meta­mo­de­ling tech­niques (main­ly kri­ging aka Gaussian Process) com­bi­ned with active lear­ning spe­ci­fi­cal­ly desi­gned to the esti­mate of the excur­sion set are used to reduce the com­pu­ta­tio­nal cost. The idea is to replace the nume­ri­cal simu­la­tions with an inex­pen­sive sur­ro­gate model, that inter­po­lates a few simu­la­tions points which are ite­ra­ti­ve­ly cho­sen to reduce uncer­tain­ty in the iden­ti­fi­ca­tion of the excur­sion set.
  3. Third, the inver­sion needs to be robust in the sense that it needs to consi­der the ran­dom nature of the uncon­trol­led variables. We gene­ra­lize pre­vious stu­dies that dealt with uncon­trol­led variables through a worst‐​case sce­na­rio (Richet, Bacchi, 2019) by consi­de­ring rare events, as the thre­shold α pre­vious­ly intro­du­ced must be very small.
  4. Fourth, the dras­ti­cal­ly uncer­tain variables need to be inte­gra­ted in the pro­ba­bi­lis­tic fra­me­work. Thus, the inver­sion will be com­bi­ned with an opti­mi­sa­tion to inves­ti­gate here the worst‐​case sce­na­rio, ie the sce­na­rio lea­ding to the highest pro­ba­bi­li­ty of flooding.

Keywords : Uncertainties, meta­mo­de­ling, active lear­ning, floo­ding risk

Thesis defense : Octobre 2023

Supervisor :

Partners or/​and fun­ders : IRSN, BRGM