A DECISION THEORETICAL APPROACH TO IDENTIFY OPTIMAL RISK MITIGATION STRATEGIES

Transcription

1 2 h Congess ITERPRAEVET 202 Genoble / ance Confeence Poceedngs A DECISIO THEORETICAL APPROACH TO IDETIY OPTIAL RISK ITIGATIO STRATEGIES Buno azzoana, Sven uchs 2 and ageh Kele 3 ABSTRACT Communes lvng n mounan aeas ae nceasngly affeced by consdeable damage o nfasucue and popey as a consequence of flood mpacs. The concepon of effecve flood sk mgaon saeges and he subsequen mplemenaon s heefoe essenal fo a connung susanable developmen n mounan aeas. Effecve flood sk mgaon saeges can be assessed by he expeced uly, whch allows he selecon of opmal managemen saeges fom a nomave pon of vew. The obecve of hs pape s o pesen he undelyng pocedue, and o deve fomal expessons o measue sk mgaon pefomance sang fom he basc heoem of aonal choce unde sk. oeove, an ovevew of exsng economc valuaon appoaches o aach moneay values o he elemens a sk s povded. Keywods: aual hazads, flood, flood sk managemen, expeced uly ITRODCTIO Takng no accoun ha he nenaonal communy as a whole s affeced by consdeable damage o nfasucue and popey as well as loss of lves, he ned aons Geneal Assembly desgnaed he 990s as he Inenaonal Decade fo aual Dsase Reducon (IDDR, ned naons Geneal Assembly, 989). Whn he assocaed nenaonal famewok of acon, he obecve of hs decade was o pomoe conceed acon n ode o educe loss of lfe, popey damage and economc dsupon caused by naual hazads, no only wh a pacula focus on developng counes, bu also wh espec o mos developed counes. Based on hs famewok, whch was connued by he Inenaonal Saegy fo Dsase Reducon (ISDR, ned aons Geneal Assembly, 2000) addonal emphass was pu on he necessy o consde he pocesses nvolvng he physcal and soco-economc dmensons of vulneably and sk. In coheence wh hese naves on a global level, he Euopean Commsson ssued he Decve on he Assessmen and anagemen of lood Rsk (Commsson of he Euopean Communes, 2007) as one of hee componens of he oveachng Euopean Acon Pogamme on lood Rsk anagemen. In hs Decve, he embe Counes of he Euopean non should povde flood sk maps on a local scale unl he end of 203, and subsequenly flood sk managemen plans focusng on he educon of poenal advese consequences of floodng fo human healh, he envonmen, culual heage and economc acvy. oeove, due o he oveall scaceness of publc funds, flood sk mgaon saeges should be anchoed o economc cea n ode o fulfl he equemens of cos-benef analyses (Weck-Hannemann, 2006; Peman e al., 20). As such, n seveal Alpne counes effos have been undeaken o lnk decsons on he mplemenaon of flood sk mgaon saeges o mehods of cos-benef analyses (Haeng e al., 2002; BLW, 2005). Howeve, sk managemen saeges, and n pacula he need o mpove cos-effcency f measues ae planned, ae subec o a bundle of unceanes; above all wh espec o () he sochasc naue of he modellng of hazadous pocesses, and (2) he sysems esponse n ems of D. Buno azzoana. Depamen of Hydaulc Engneeng, Auonomous Povnce of Bozen-Bolzano, Bolzano, Ialy (emal: buno.mazzoana@povnca.bz.) 2 Pv.-Doz. D. Sven uchs. Insue of ounan Rsk Engneeng, nvesy of aual Resouces and Lfe Scences, Venna, Ausa (e-mal: sven.fuchs@boku.ac.a) 3 D. ageh Kele. Insue of Geogaphy, nvesy of Ben, Swzeland (e-mal: mageh.kele@gub.unbe.ch)

2 vulneably and sk educon (uchs, 2009). Addonally, (3) he esmaon of he expeced uly n ems of economc benefs emans cucal, snce he decson makes sk aude s an essenal elemen of a compehensve plannng pocess ageed a makng explc he facos whch deemne he selecon of opmal sk mgaon saeges (Kuschwz, 2008). Based on hese equemens, n hs pape, we conbue o a fomalsed mehod o assess () he sk mgaon pefomance of nended mgaon saeges on a moneay bass (e.g. n ems of annual sk educon); (2) he cos-plan fo a vaey of possble sk mgaon saeges, whch s consdeed fom a lfe-cycle pespecve, and (3) he expeced uly assocaed wh each mgaon saegy. The lae s of pacula mpoance n ode o selec a mgaon concep whch s n lne wh he pefeence sucue of he publc decson make (who, n un, usually povdes he funds necessay fo he ageed mgaon alenave). Ths eles on he assumpon ha any publc decson make acs sk-avese bu wh an obecve evaluaon max n ode o spend publc money economcally effcen (uchs and calpn, 2005). Subsequenly, we oulne fom a mehodologcal pon of vew he pocedues necessay n ode o povde a oad map fo an achevemen of hese ams. Consdeng he peculaes of decsons on publc nvesmens ageed a he mgaon of naual hazad sk, he equed analyc effos need o be sucued and balanced (g. ). In pacula long-em capal commmens necessay fo mgaon expenses, and he nduced nedependences beween hese commmens and ohe mpoan economc acves n mounan egons (e.g., ousm and adng), eque such a conceed acon. lood hazad and sk Assessmen lood hazad and sk scenaos - pobably-conngen sk efeence pospec 2 lood hazad and sk assessmen Rsk mgaon saeges lood sk mgaon scenaos fo each sk mgaon saegy - pobably conngen sk mgaon pospecs Cos-plan ove he lfe-cycle fo each sk mgaon saegy Appasal of benefs (sk educon) and coss (cos-plan) 3 fo each sk mgaon saegy and deemnaon of he assocaed expeced uly (E) Pobably conngen ne pesen value pospecs o PV pospecs, pobably conngen uly pospecs, and assocaed expeced uly fo each sk mgaon saegy 4 Saegy selecon accodng o v-heoem (E max!) g. Pocedual oadmap fo a compehensve sk mgaon appoach based on he concep of expeced uly (E). Wh efeence o g., he fs sep (sub-pocedue ) of he poposed pocedual oadmap consss of deemnng conssen flood hazad and sk scenaos. The specfc ams nclude a spaally explc epesenaon of fequency and magnude (nensy) fo each of he undelyng flood hazad scenaos, and a quanfcaon of he assocaed consequences n ems of sk. The man esul s he so-called sk efeence pospec, whch seves as a bass fo pefomance compason of possble sk

3 mgaon opons. In hs pape, emphass s pu on he compuaonal aspecs of sk wheeas he peculaes elaed o he deemnaon of he undelyng flood hazad scenaos ae dscussed n moe deal elsewhee (azzoana e al., 20). The second sep (sub-pocedue 2) enals he geneaon of sk mgaon saeges ageed a maxmsng he educon of flood sk n accodance o he sysem. om a mehodologcal pespecve, subsanal effo has been undeaken o sysemase he desgn of sk mgaon saeges and o fomulae heuscs fo he pacones (compae fo an ovevew azzoana and uchs, 200). o each planned sk mgaon saegy, he sysem unde consdeaon has o be eanalysed fom a flood hazad and sk assessmen pespecve. Thus, he analyc seps of subpocedue have o be epeaed n ode o explcly modfy he sysems behavou ehe concenng flood hazad pocess unfoldng o concenng he deemnaon of expeced consequences on vulneably and elemens a sk exposed. An essenal equse fo he economc assessmen of sk mgaon saeges (sub-pocedue 3) s he assessmen of he benef and cos flows ove he lfe-cycle of he planned mgaon saegy. Concenng he benefs, n hs pape, we lm ou dscusson o he annual flood sk educon wh espec o he sk efeence pospec fo popey, nfasucue and human lves. A hs sage, a pobably conngen ne pesen value (PV) pospec can be deemned fo each saegy. The ange of PVs obanable by he ene se of mgaon saeges can be ansfomed no coespondng uly values eflecng he decson makes aude. Ths aude s nfluenced by a cean degee of sk aveson, bu also by ohe facos such as he dsbuon of popees and eal esaes (moe geneal, by so-called endowmen effecs). As such, he PVs ae ansfeed no coespondng uly values accodng o expeced uly heoy (compae Esenfüh e al., 200; Wakke, 200). A hs sage, pobably-conngen uly pospecs can be deved fo each ndvdual mgaon saegy. The opmal flood sk mgaon saegy s fnally seleced accodng o he von eumann- ogensen s Theoem (von eumann and ogensen, 953; Keps, 988) as he saegy whch maxmses he expeced uly (E). The heoecal undepnnngs of E Theoy ae noduced n he followng secon. Then, n a subsequen secon, he coe concep of sk pospec, used o deemne he sk mgaon pefomance of he avalable sk mgaon saeges, s explaned. oeove, an ovevew of exsng valuaon mehods o deemne he economc value of elemens a sk s povded. nally, he descpon of ecenly developed appoaches fo he quanfcaon of vulneably s povded, whch lnks he nensy of he pocess o he elave exen of he losses fo each elemen a sk. EXPECTED TILITY THEORY: A COCISE EXPOSITIO ollowng he pocedue oulned by Wakke (200), fsly, we oulne he essenal sucual assumpons fo decson unde unceany and sk, especvely. Subsequenly, we noduce expeced uly (E) and povde he basc heoem fo aonal choce unde sk. Sucual Assumpon [decson unde unceany]: S s a, fne o nfne, sae space, and R ( s he oucome se. Pospecs map saes no oucomes, akng only a fne se of values. The doman of pefeence s he se of all pospecs,.e., of all such maps; s a pefeence elaon n he se of pospecs; and non-degeneacy holds. Sucual Assumpon 2 [decson unde sk]: s a pefeence elaon ove all (pobably conngen) pospecs,.e., ove all fne pobably dsbuons ove he oucome se R (. Defnon : nde sucual assumpon 2; expeced uly (E) holds f hee exss a scly nceasng uly funcon () fom he oucome se R ( o he eal numbes R ), such ha he nex evaluaon epesens pefeences: ( p x...p m x m ) p( x ) p m ( x m ). Endosed wh hese sucual assumpons and he E defnon we sae he followng heoem: Theoem [E fo sk]. nde sucual assumpon 2, he followng wo saemens ae equvalen: () Expeced ly holds () sasfes: weak odeng, sandad gamble solvably, domnance and conssency. A fomal poof of hs heoem can be found n Glboa (2009). The condons lsed above o be sasfed by he pefeence elaon ae explaned n Wakke (200) and can be nuvely undesood

4 as pecse conssency equemens fo he decson make. We noduce now a geneal mehod fo he measuemen of uly, enablng he decson make o expess, fo example, PVs n ems of peceved uly. In geneal, assume wo fxed oucomes, ax > n, and assume ha we have nomalsed uly, ( n) = 0 and ( ax) =. o each oucome α, such ha ax α n, we can elc he sandad gamble (SG) pobably wh espec o n and ax, beng he pobably p such ha he equvalence n gue 2 holds. p ax α ~ -p n g. 2 The SG pobably p of α. Applyng E gves he SG equaon, ( α ) = p( ax) + ( p) ( n) = p. Expessed anohe way, (α) s he SG pobably p. The pobably-conngen sk pospec s a coe concep necessay fo he pocedual wokflow oulned n gue, and s, fomally noduced n he subsequen secon. RISK PROSPECTS Accodng o he concepualsaon of naual hazad sk (uchs, 2009; azzoana and uchs, 200), sk fo obecs exposed o he mpacs of flood hazad pocesses (e.g. debs flows) s quanfed on an annual bass as follows: R = h= H = p p E [ ED,,h ( I ( x,h ), s) ] = = h= h = In equaon () R s he annual sk n ems of expeced losses fo he elemens a sk ( =,..., ) nduced by he flood hazad scenaos ( =,..., ) unde he assumed exposue confguaons ( h =,..., H ). The pobably assocaed o a specfc flood scenao s denoed by p and he pobably of a gven exposue confguaon, namely a defned se of locaons x, h of he consdeed elemens a sk, s denoed by ED I x,s s he expeced damage (o loss) o an elemen a sk, gven a p. ( ) Eh,,h (, h ) pocess nensy I n x, h esulng fom he consdeed flood hazad scenao. In he adoped sac noon of sk (fo a heoecal eamen of dynamc sk, compae azzoana e al., 202), I coesponds o he maxmum nensy n ems of flow dephs o flow veloces. nally, s s a Boolean vaable amng a denfyng suaons whee moble obecs ae sheleed by envelopng obecs (e.g., people whn buldngs, cas n vehcle halls). Pacula aenon has o be gven o accuaely denfy he pobably p o be assocaed o each flood scenao. I s assumed ha a sascal analyss of exeme evens delveed a pobably dsbuon fed o he avalable annual dschage maxma a a elevan flow secon. In hs case fo each ecuence neval, RI, o each non-exceedance pobably s complemen o ), he coespondng peak dschage value p p ( p p ) + = 2, < = p + p p /. ( ) ( ) 2 2 () p (o exceedance pobably as Q s known. p <, and p ( p ) + ( p p )/ 2 = and s calculaed as

5 omally, fom an ex-pos pespecve, he hegh of loss fo a gven obec mpaced by a flood even can be assumed o be equal o he depecaed value of he pa of he obec o be ensaed, hence: D = C δ RV SE (2) whee: D s he moneay value of he losses abuable o flood mpacs; C ae he ensaemen coss, namely he coss o subsue he damaged pas of he obec by he coespondng new componens; δ s a depecaon coeffcen eflecng he depecaon of he obec (compae Gallean e al., 20); RV s he esdual value of he damaged poon; SE ae he pos-even expendues fo damage educon. To pefom an ex-ane esmaon of he expeced damage ED,, h we e-we he las equaon, noducng pope vulneably funcons, υ,,h = υ ( I ( x, h ),s ), eflecng fo a gven nensy I of he consdeed flood hazad scenao (a he locaon of he obec) he ao beween C and he new consucon coss of he ene obec V. Due o he ex-ane pespecve he expeced values of RV and SE wll be negleced and he expeced damage ED can be expessed as:,, h ED,,h = υ ( I ( x,h ), s) V δ (3) ow equaon () can be ewen as: R = h= H = p p E [ υ ( I ( x,h ),s ) V δ ] = = h= h = (4) Equaon (4) can convenenly be epesened as a pobably-conngen sk pospec RP (Table ). Tab. Pobably-conngen sk pospec RP. s p p s s p p p p p p p p p p p p p p p p p p p E Eh E H E Eh EH E Eh EH = = = = = = = = ED ED ED ED ED ED ED ED ED = =,, =,,h =,,H =,, =,,h =,,H =,, =,,h =,,H H ED h= = = R,,h s H ED h= =,,h = R H ED h= =,,h = R s R o convenence we denoe he pobably-conngen sk efeence pospec wh RP and he sk z pospecs coespondng o he, z =,..., Z, sk managemen saeges wh RP. Thoughou, we assume ha he decson make does no have any nfluence on he uh of he evens (achna, 987). Howeve, as oulned n Glboa (2009), he dsncon beween acs ove whch he decson make has conol and saes ove whch he decson make has no conol s one of he pllas of aonal choce. Hence, we assume ha he flood hazad scenaos (wh he assocaed pobables) ae he only elevan saes o be consdeed. azzoana e al. (n pess) noduce he dsncon beween a pocess loadng and pocess esponse sysem accodng o he pncple of aonal choce, hus, hazad pocesses unfoldng n he loadng sysem chaacese he sae space. R z The pobably-conngen sk mgaon pospec, RP = RP RP, s shown n Table 2. R z Tab. 2 Pobably-conngen sk mgaon pospec, RP = RP RP, of he saegy z. The R supescps R and z ae elaed o he sk efeence pospec, RP, and he sk pospec coespondng o he saegy z, especvely

6 s p p s s p R z RPs = RP s RP R z s RP s = RP s RP R z s RP s = RP s RP s In Table 2 he esuls achevable by saegy z wh espec o he sk efeence pospec fo all saes of he sae space, s S, ae oulned. Expessed n anohe way, Table 2 epesens a fne pobably dsbuon ove he oucome se R (, namely he annual sk educon, whch coesponds o he benefs assocaed wh he mplemenaon of saegy z. The mplemenaon of such a saegy z ncludes a sees of coss ove he planned lfe cycle. o analyc puposes s convenen o specfy he expendue flows C ( LC) z ove he lfe cycle duaon T n fom of a cos plan. Accodng o he wokflow pesened n gue we buld he pobably-conngen ne pesen value PV pospec (compae Table 3) fo all saeges, z =,..., Z, followng he geneal pncples of poec appasal (compae Peman e al. 20). Tab. 3 Pobably-conngen PV pospecs and he expeced values EVs fo all saeges, s p. p ( ) = T R R Rs C LC PVs = 0 ( + ) s. s p EV PV R s R ( ) ( ) = T R C LC s = 0 + C( LC) ( ) z =,..., Z. R = = T R R s EV PV = p PV s = = + z ( ) = T z R C LC ( ) s PV s = 0 ( + ) = T z R C LC ( ) PVs = T z Rs C LC PV EV = = p PV = = 0 ( + ) = 0 ( + ) Z ( ) = T Z R C LC ( ) s PV s = 0 ( + ) = T Z R C LC ( ) PV = T Z Rs C LC PV EV = s s = p = = 0 ( + ) = 0 ( + ) 0 R = R PV o a sk-neual decson make would be mpoan o selec he opmal flood sk mgaon saegy on he bass of Tab. 3. As such, he expeced values EVs of he pobably-conngen PV pospecs accuaely epesen he pefeences of he decson make, hence, he saegy wh he hghes EV should be seleced fom a nomave pon of vew. Howeve, n case ha a decson make does no ac accodng o hese pemses, and heefoe acs sk-aves, we wll explcly make use of he heoem [E fo sk], noduced above along wh he key conceps of expeced uly heoy. The ene oucome se, namely he se of all pobably-conngen PV,, z, s assumed o be known. We noduce he followng convenons: PV = 0 PV = =. ( ) 0 2. PV = ax ( PV ) = = We assume, moeove, fo a negave PV he coespondng dsuly s deemned by nveng he sgn of he uly of PV. Endosed wh hs seng we consuc he posve pa of he uly funcon, wheeas he negave pa s obaned by pon eflecon. Elcng he sandad gamble ( SG ) pobably p wh espec o PV = 0 and PV = ax of all posve pobably-conngen PV PV of Tab. 3, we oban he coespondng ules as = p. Endowed wh hese nodes he posve pa of he uly funcon s consuced by pecewse lnea PV

7 nepolaon beween he nodes. A hs sage Tab. 3 can be ewen by subsung he PV coespondng (compae Table 4). by he Tab. 4 Pobably-conngen uly pospecs and he expeced ules Es fo all saeges, z =,..., Z. s p s p s p E R R s R s z s s E Z s s E E R = = p = = = p = = = p = As a consequence of he heoem [E fo sk], fom a nomave pon of vew he aonal choce s he selecon of he sk mgaon saegy whch maxmses expeced uly E. ECOOIC VALATIO O ELEETS AT RISK In he adoped concepualsaon of flood hazad sk (compae equaons and 4) he expeced losses ae expessed monealy, whch enals an economc valuaon. lood damages can be classfed no dec and ndec losses. Whle he fome occu due o he physcal conac (mpac) of flood wae wh popees, people a sk, o any ohe obec (ez e al., 200), he lae alhough ggeed by he dec mpacs ae no spaally esced o he flooded aeas. uhemoe, ndec losses may exend well beyond he duaon of he flood even. Dependng on whehe o no flood losses can be assessed n moneay values, a fuhe dsncon no angble and nangble damages seems appopae (Pake e al., 987; Smh e al., 998). oe pecsely, angble damage s damage o capal socks o esouce flows whch can be specfed n moneay ems, wheeas nangble damage s damage o asses whch ae no aded n a make and ae heefoe dffcul o ansfe no moneay values (uchs e al. 2007a). Alhough he emnology of hs classfcaon s commonplace, nepeaons and delneaons dffe. Dffeen valuaon pncples ae employed o aach values o dsnc obec caegoes (Dees and Paul, 20, Peman e al., 20). In ode o povde he opmal supply of poecon agans flood hazads, he publc seco wll need, among ohe nfomaon, evaluaon of coss and benefs (uchs and calpn, 2005). Coss ae evaluaed n ems of he pesen value of he pevous nvesmen so ha he oppouny coss can be compaed o he uly ha would have esuled fom an alenave appopaon of he esouces. Hee, we pu he dsncon beween angbles and nangbles no a dynamc pespecve. Envonmenal valuaon s a apdly expandng feld. Refned valuaon echnques, based on fs pncples such as wllngness o pay (WTP) o wllngness o accep (WTA) ae n pepeual developmen. An eve wdenng specum of non-make commodes and sevces ae made accessble o economc valuaon (Pommeehne e al., 992; Peman e al., 20). Dffeen valuaon pncples ae employed o aach values o dsnc obec caegoes (uchs, 2009; Gallean e al., 20). Opeaonally we dsngush beween obec caegoes valued hough economc appoaches usng make values (e.g., ensaemen value fo sucues) and he caegoy of sascal lfe SL of people a sk, whee conngen valuaon (CV) mehods ae appled (Vscus, 2008). Wh efeence o he fome we noduce a geneal scheme o sucually dssec complex obecs and make hem accessble o economc valuaon n sk assessmen, whle he lae s eaed sepaaely. Hence, n dssecng a complex obec (e.g., a poducon plan) we dsngush beween:. vecally exendng fxed sucues (e.g., walls of he buldngs) mpaced decly by he flood pocess; R

8 2. pacula supesucues (e.g., oofs, decks) mpaced aely and heefoe ndecly by he flood pocess; and 3. nsallaons and/o moble obecs (e.g., machnes and cas) mpaced decly by he flood pocess. o compleeness wo supplemenay caegoes ae noduced ha ae affeced by floodng as well as sedmen and wood deposon pocesses: 4. sufaces (aeas) fo dffeen land use puposes (e.g., agculual land, bu also pakng los and oads); and 5. boc sysems (e.g. wood, bu also ochads). The dec economc efeence fo a valuaon of obec pas belongng o he caegoes () and (2) s he deemnaon of he ensaemen value. As suggesed by Gallean e al. (20), he ensaemen value can be calculaed as consucon value V by: m V q p (5) = n = = whee V s he ensaemen value of he consdeed obec; q s he equed quany of npu o pefom he consucon wokflow un ; and p s he unay pce of he consucon wokflow un. o he caegoy (3) he esmaon of he make value V of he componens of equpmen s calculaed as follows: D V = Ch + (6) 00 D Whee V s he mos pobable make value of he consdeed equpmen componen; Ch s he puchase pze; s he cos ncemen fom he yea of puchase o he yea of valuaon; D s he esdual economc lfe (n yeas); and D s he economc lfe span (n yeas); o caegoy (4) s elevan o deemne he coss of cleang-up opeaons and he necessay ensaemens o e-esablsh he ognal funconaly. In case of obec caegoy (5) he economc valuaon s caed ou by deemnng he capal value of he consdeed boc sysem hough suable capalsaon fomulas. The economc valuaon of he caegoy sascal lfe SL needs a sepaae consdeaon. In fac he esmaon of he value of a sascal lfe as a ypcal non-make value s defned as he ae a whch people ae wllng o exchange ncome fo he educon n moaly sk. I s calculaed by dvdng he annual mean o medan wllngness o pay WTP hough he coespondng sk vaaon. Seveal sudes amng a deemnng he VSL have been conduced n dffeen conexs (Vscus and Aldy, 2003), such as snow avalanches, whee he nfluence of mplc nfomaon assocaed wh he occuence of avalanches on WTP-values fo sk pevenon was quanfed wh a ange beween.8 and 5.2 mllon (Lene and Puckne, 2005). These esuls ae conssen wh ohe sudes (e.g. Alben e al., 2005)

9 VLERABILITY O ELEETS AT RISK The shf fom hazad o sk obvously eques a compleely dffeen appoach wh espec o necessay managemen ssues (uchs, 2009). Despe he compehensve expeences ha have been made by applyng he concep of sk o mounan hazad managemen, n pacula n Swzeland (Kenholz 994; Hollensen 995; Henmann 998; Kenholz e al. 2004), consdeable quesons wh espec o he mehods developed fo an opeaonal mplemenaon of he concep of vulneably sll eman open (uchs e al., 202). The assessmen of vulneably eques an ably o boh denfy and undesand he suscepbly of elemens a sk and n a boade sense of he socey o hese hazads (uchs, 2009). Sudes elaed o vulneably of human and naual sysems o mounan hazads, and of he ably of hese sysems o adap o changes n he funconal chan of hazads, ae a elavely ecen feld of eseach ha bngs ogehe expes fom a wde ange of dscplnes, ncludng naual scence, socal scence, dsase managemen, polcy developmen and economcs, o name only a few. Reseaches fom hese felds bng he own concepual models o sudy vulneably and adapaon, models whch ofen addess smla poblems and pocesses usng dffeen languages (Books, 2003). Howeve, apa fom he oveall dscusson on lngusc placemens and semanc dmensons of he em (Cue, 996; 2003), vulneably n he conex of flood hazads n Euopean mounan egons s, fom a pacone s sde such as he Ausan Toen and Avalanche Conol Sevce, usually defned as he physcal mpac of hazadous evens on elemens a sk. Accodngly, f quanavely assessed, vulneably s defned as he expeced degee of loss fo an elemen a sk due o he mpac of a defned hazadous even whn a defned peod of me and a defned locaon. These evens ae hemselves condoned by a cean nensy, fequency and duaon, all of whch affec vulneably. om hs echncal pon of vew, as a geneal ule, vulneably assessmen s based on he evaluaon of paamees and facos such as buldng caegoes o ypes, consucon maeals and echnques, sae of manenance, pesence of poecon sucues, and pesence of wanng sysems (ell e al., 2008). o hs eason, vulneably values descbe he suscepbly of elemens a sk o damage, facng dffeen pocess ypes wh dffeen spaal and empoal dsbuons of pocess nenses (e.g., flow dephs, accumulaon heghs, flow veloces and pessues). The evew of he concep of sk fo Alpne counes esuled n gaps concenng appopae ools fo he assessmen of vulneably of elemens a sk and of communes exposed. To ovecome hese shocomngs, sudes on vulneably have been undeaken amng a () he mehodologcal developmen of loss funcons wh espec o buldngs locaed n he un-ou aeas of oen pocesses (uchs e al., 2007b) and wh espec o fluval sedmen anspo (Toschng e al., 20); and (2) he concepualsaon of an oveachng vulneably model ncludng sucual, economc, socal and nsuonal vulneably (uchs, 2009) as well as he spaal chaacescs of vulneably (uchs e al., 202). In geneal, he damage ao s quanfed usng an economc appoach by esablshng a ao beween he loss and he econsucon value of evey ndvdual elemen a sk exposed (uchs e al., 2007b). In a second se of calculaons, hs ao obaned fo evey ndvdual elemen a sk s abued o he especve pocess nenses. The elaon beween damage ao and pocess nensy s defned as vulneably. Theefoe, nfomaon on he elemens a sk exposed on he ndvdual oen fans s necessay, as well as daa on he pocess nenses of he pacula hazadous evens. As a esul, scaeplos can be developed lnkng pocess nenses o obec vulneably values. These daa ae fuhe analysed usng egesson appoaches n ode o develop vulneably funcons whch seve as a poxy fo he sucual essance of buldngs wh espec o flood pocesses on oen fans (uchs e al., 2007b; Toschng and uchs, 202). Hence, he assessmen of physcal vulneably s an essenal equemen o quanfy he expeced damage n moneay ems. unconal vulneably appoaches have been poposed only fo a lmed numbe of obec caegoes (e.g. esdenal buldngs), howeve, n pacula vulneably funcons fo buldngs mpaced by debs flows (uchs e al., 2007b) and fluval sedmen anspo ae lmed (Toschng e al. 20). Accodng o he debs flow hazads, he elaonshp beween debs flow nensy n ems of depos heghs, ID ( x ) h d ( x ), a he locaon of he obec, x, and

10 υ, was found o f bes o he daa by a second-ode polynomal funcon, namely: vulneably, = υ ( h d ( x )) fo all nenses h ( x ) 2.5m υ = υ d 2 ( h ( x )) = 0. h ( x ) 0.0 h ( x ) d d d Accodng o he fluval sedmen anspo phenomena, he bes-fng funcon was expessed as υ = υ ( I ( x )) RST = e IR ST x an ( ( )) π whee I RST ( x ) s he elave nensy of he fluval sedmen anspo pocess a he locaon of he obec, x, expessed as ao beween he depos hegh h d ( x ) and he hegh of he consdeed buldng H. DISCSSIO AD OTLOOK In hs pape we pesened a compuaonal famewok fo he economc assessmen of flood sk mgaon saeges pllaed on he expeced uly heoy. In a fs sep mehods fo he quanfcaon of flood sk wee dscussed. Subsequenly, elaed conceps of pobably-conngen sk efeence pospecs and he sk mgaon pospec of poenal mgaon saeges wee noduced. In paallel, he necessy o specfy he cos plan fo each mgaon opon was hghlghed. By a moneay quanfcaon of he expeced benefs, expessed n ems of annual sk educon, and of he ncung assocaed coss, fomal epesenaons of he coespondng pobably-conngen ne pesen value PV pospecs wee deved. o a sk-neual decson make, he calculaon of he expeced value EV of hese pospecs s suffcen o selec an opmal mgaon saegy. To ovecome he consan of a decson make wh a possbly dffeen sk aude, and o esablsh a nomave bass fo aonal decson makng, elemens of he expeced uly heoy wee noduced. Successvely he pevously deemned fomal epesenaons of PV pospecs wee conveed no he coespondng uly pospecs. Recallng he essence of he v-theoem [E fo sk], he aonal choce s gven by he selecon of hs sk mgaon saegy whch maxmses expeced uly E. In hs pape we addessed wo supplemenay opcs, whch ae closely elaed o he oveachng umbella of sk assessmen, () he economc valuaon of he obecs a sk and (2) he assessmen of he vulneably. Conexually, we pesened especve quanfcaon appoaches and denfed eseach gaps o be closed n he nea fuue. The concepual sucue fo a sk-based poec assessmen pesened n hs pape allows o ncopoae fuhe aspecs fom he feld of sk analyses o he feld of aonal decson makng, such as dynamc sk analyses (.e. ackng he vulneably of he endangeed obecs houghou he duaon of he flood even). The heoecal seup povded n hs pape s vald fo he deemnaon of each avalable flood sk mgaon saegy, and of he coespondng expeced uly wh known pobables of he undelyng flood hazad evens. Ths s, scly speakng, he specal case of a moe geneal poblem of deemnng he expeced uly wh boh, unknown pobables and unknown ules. The eamen of hs ask becomes elevan n he conex of mpefec nfomaon, and eques he adopon of he Subecve Expeced ly Theoy (compae Savage, 954; Wakke, 200). REERECES Alben A., Coppe., Kupnck A., Smon.B. (2004). Does he value of a sascal lfe vay wh age and healh saus? Evdence fom he S and Canada. Jounal of Envonmenal Economcs and anagemen 48:

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